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Full text of "SPACE FLIGHT HANDBOOKS. VOLUME 1- ORBITAL FLIGHT HANDBOOK, PART 1 - BASIC TECHNIQUES AND DATA"

NASA SP 33 PART 1 



V&3-anoi 



SPACE FLIGHT HANDBOOKS 

Volume 1 



Orbital Flight Handbook 



NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 




SPACE FLIGHT HANDBOOKS 

Volume / 

Orbital Flight Handbook 



PART 1 - BASIC TECHNIQUES AND DATA 



Prepared for the 

GEORGE C. 

MARSHALL SPACE FLIGHT CENTER 

Huntsvllle, Alabama 

Under Contract NAS 8-5031 




Office of Scientific and Technical Information 

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION A QQQ 

Washington, D. C. IviDu 



FOREWORD 



This handbook has been produced by the Space Systems Division of the 
Martin Company under Contract NAS8-5031 with the George C. Marshall Space 
Flight Center of the National Aeronautics and Space Administration. The 
handbook expands and updates work previously done by the Martin Company 
and also incorporates, as indicated in the text, some of the work done 
by Space Technology Laboratories, Inc. and Norair Division of Northrop 
Corporation under previous contracts with the George C. Marshall Space 
Flight Center. The Orbital Flight Handbook is considered the first in 
a series of volumes by various contractors, sponsored by MSFC, treating 
the dynamics of space flight in a variety of aspects of interest to the 
mission designer and evaluator. The primary purpose of these books is to 
serve as a basic tool in preliminary mission planning. In condensed form, 
they provide background data and material collected through several years 
of intensive studies in each space mission area, such as earth orbital 
flight, lunar flight, and interplanetary flight. 

Volume I, the present volume, is concerned with earth orbital 
missions. The volume consists of three parts presented in three separate 
books. The parts are: 

Part 1 - Basic Techniques and Data 
Part 2 - Mission Sequencing Problems 
Part 3 - Requirements 

The Martin Company Program Manager for this project has been 
Jorgen Jensen; George Townsend has been Technical Director. George 
Townsend has also had the direct responsibility for the coordination 
and preparation of this volume. Donald Kraft is one of the principal 
contributors to this volume; information has also been supplied by 
Jyri Kork and Sidney Russak. Barclay E. Tucker and John Magnus have 
assisted in preparing the handbook for publication. 

The assistance given by the Future Projects Office at MSFC and by 
the MSFC Contract Management Panel, directed by Conrad D. Swanson, is 
gratefully acknowledged. 



CONTENTS 

Volume I, Part 1 - Basic Techniques and Data 

I Introduction 1-1 

II Physical Data H-l 

III Orbital Mechanics III-l 

IV Perturbations IV- 1 

V Satellite Lifetimes V-l 

The preceding contents are Part 1 of Volume I. The remaining two 
parts of Volume I contain the following: 

Volume I, Part 2 - Mission Sequencing Problems 

VI Maneuvers VI- 1 

VII Rendezvous VII-1 

VIII Orbital Departure VIII-1 

IX Satellite Re-Entry IX-1 

Volume I, Part 3 - Requirements 

X Waiting Orbit Criteria X-l 

XI Orbit Computations XI-1 

XII Guidance and Control Requirements XII-1 

XIII Mission Requirements XIII-1 

Appendix A A-l 

Appendix B B-l 

Index i 



CHAPTER II 
PHYSICAL DATA 

Prepared by: 

G. E. Townsend, Jr. 

S. L. Russak 

Martin Company (Baltimore) 

Aerospace Mechanics Department 

March 1963 

Page 

Symbols II- 1 

Introduction II- 2 

A. Astronautical Constants H-2 

B. Astrophysical Constants 11-15 

C. Conversion Data 11-50 

D. References 11-56 

E. Bibliographies 11-59 

Illustrations 11-63 



LIST OF ILLUSTRATIONS 



Figure Title Page 

1 Confidence Level for the Value of /u' as a Function 

of the Number of Data Points and Size of Interval. . . 11-65 

2 Present Standard and Model Atmospheres, and 

Proposed Revision of U.S. Standard Atmosphere. . . 11-66 

3 Temperature Versus Altitude, Defining Molecular 
Scale Temperature and Kinetic Temperature of the 
Proposed Revision to the United States Standard 
Atmosphere 11-67 

4 Molecular -Scale Temperature Versus Geometric 
Altitude Proposed United States Standard Atmo- 
sphere Compared with United States Detailed Data, 
Russian Average Data, and ARDC Model Atmo- 
sphere 1959 for Altitudes Above 80 km Only 11-68 

5 Density Versus Geometric Altitude for Proposed 
United States Standard Atmosphere Compared with 
United States Detailed Data, Russian Average Data, 

and ARDC Model Atmosphere 1959 11-69 

6 Pressure Versus Geometric Altitude for Proposed 
United States Standard Atmosphere Compared with 
United States Detailed Data, Russian Average Data, 

and ARDC Model Atmosphere 1959 11-70 

7 Molecular Weight Versus Altitude 11-71 

8 Average Daytime Atmospheric Densities at the 

Extremes of the Sunspot Cycle 11-72 

9 Density of the Upper Atmosphere Obtained from the 

Orbits of 21 Satellites 11-73 

10 Dependence of Atmospheric Density on A a = a - a 

in the Equatorial Zone (diurnal effect) 11-73 

11a Diurnal and Seasonal Variations in Atmospheric 

Density at 210 km Dervied from Observations of the 
Satellite 1958 6 2. (The lower x -scale gives true 
local time, the upper A a = a - a^. The parameter 

of the curves is A 6 = 5 - 6^-, where a is right ascen- 

sion, 6 is declination, 7r is perigee, O is sun.) 11-74 



Il-ii 



LIST OF ILLUSTRATIONS (continued) 



Figure Title Page 

lib Variations in Atmospheric Density at 562 km Above 
the Earth Ellipsoid Derived from the Observations 
of Satellite 1959 a 1 II -74 

lie Variations in Atmospheric Density at 660 km Derived 

from the Observations of Satellite 1958 j3 2 11-74 

12 Diurnal Variations of Atmospheric Density at Altitudes 
from 150 to 700 km Above the Earth Ellipsoid for 
|AS|<20° H-75 

13 Model of the Seasonal Variation of Mean Density to 

200 km II" 75 

14 Radiation Dose from Solar Flares Versus Skin 

Thickness 11-76 

15 Solar Proton Dose, May 10, 1959 Flare, 30-Hour 
Duration 11-77 

16 Solar Proton Dosages from February 23, 1956 

Flare 11-78 

17 Solid Angle Subtended by Earth as a Function of 

Altitude H-79 

18 Magnetic Dip Equator (1) from USN Hydrographic 
Office, 1955 and Geocentric Magnetic Equator (2) 

Inclined 13° to the Equator at Longitude 290° 11-79 

19 Inner Van Allen Belt 11-80 

20 Flux of Protons at One Longitude in the Van Allen 

Belt H-81 

21 Proton Differential Kinetic Energy Spectrum for the 

Inner Van Allen Belt 11-82 

2 2 Flux of Electrons in the Van Allen Belts 11-83 

23 Differential Kinetic Energy Spectrum Van Allen 

Belt Electrons 11-84 



II-iii 



LIST OF ILLUSTRATIONS (continued) 

Figure Title Page 

24 Electron Dose Rates 11-85 

25 X-Ray Dose Rates 11-85 

26 Cosmic Radiation Intensity as a Function of Geomag- 
netic Latitude for High Altitudes During a Period of 

Low Solar Activity 11-86 

27 Relative Biological Effectiveness for Cosmic Rays 
as a Function of Altitude and Geomagnetic Latitude 

During a Time of Low Solar Activity 11-86 

28 Cosmic -Radiation Dosage as a Function of Shield 

Mass 11-87 

29 Differential Energy Spectrum Measured During 

Rocket Flight NN 8. 75 CF 11-87 

30 Meteoric Mass Versus Apparent Visual Magnitude. . 11-88 

31 Meteoroid Frequency Versus Mass 11-88 

32 Average Meteoroid Distribution Curve from Micro- 
phone System Measurements 11-89 

33 Meteoroid Penetration Relations 11-89 



Il-iv 



I. INTRODUCTION 



The material within the manual is arranged in 
three major areas and these areas are further 
divided into related discussions. The classifi- 
cation of material is as follows: 

Basic Techniques and Data- -Chapters II 
through V. 



Mission Sequencing Problems- 
through IX. 



-Chapters VI 



Requirements--Chapters X through XIII. 

These areas encompass most of the material in 
the field of earth orbital mechanics. The intent 
in all of these discussions is to provide analytic 
relationships which define the problem, and to 
augment these discussions with an error analysis 
and graphical or tabular data. In some of the 
material, however, the number of variables is 
so large that it is not practical to present graphi- 
cal data; in others, the problem is so involved 
that it is not possible to obtain analytic solutions 
(such investigations were conducted numerically). 
In all cases, however, the prescribed purpose 
has been achieved without sacrificing the scope 
of the investigation. 

A brief resume of some of the more important 
features of these chapters is presented in the 
following paragraphs. 



IV. PERTURBATIONS 

Special and general perturbation techniques 
are discussed, and the results of several general 
perturbation theories are catalogued and compared. 
This presentation provides the reader with the in- 
formation necessary to evaluate the theories for 
each individual application and with an awareness 
of the subtle differences in the approaches and 
results. 



V. SATELLITE LIFETIMES 

The material of this chapter presents in suc- 
cession discussions pertaining to the aerodynamic 
forces in free molecular flow, to analytic approxi- 
mations for use in determining the lifetime of 
satellites in circular orbits in a nonrotating atmos- 
phere, and, finally, to decay rates in a rotating 
oblate atmosphere. Where possible, analytic ex- 
pressions have been obtained, but accuracy has 
not been sacrificed for form, and extensive use 
has been made of numerical computation facilities. 
Here again, however, attention to detail revealed 
several nondimensional decay parameters and made 
it possible to make these computations more effi- 
ciently. 



II. PHYSICAL DATA 

The material in this chapter reviews some of 
the work published by R. M. L. and by W. M. Kaula 
for the purpose of presenting a set of constants 
necessary in the computation of trajectories. 
Appendix B extending this data is an internally 
consistent set of constants developed by Dr. H. 
G. L. Krause. 

The chapter then discusses other geophysical 
factors which can affect the selection of an orbit. 
Included in these discussions is material on the 
radiation environment, the meteoroid environ- 
ment and the upper atmosphere and its variability. 

The chapter concludes with a discussion of the 
measurement of time, distance, mass, etc. This 
portion of the chapter contains tables constructed 
for the purposes of making the transformation of 
units as simple and accurate as possible. 



III. ORBITAL MECHANICS 

The discussions of this chapter present the 
basic central motion trajectory equations to be 
used in the balance of the text. Relations de- 
fining the 3-D motion are developed and a large 
number of identities and equations are presented 
for elliptic motion. These equations (numbering 
in excess of 400) are followed by approximately 
75 series expansions of the time variant orbital 
parameters with arguments of the mean anomaly, 
the true anomaly, and the eccentric anomaly. The 
chapter concludes with a discussion of the n-body 
problems. 



VI. MANEUVERS 

The general problem of orbital maneuvering 
is approached from several directions. First, 
the case of independent adjustment of each of the 
six constants of integration is presented both for 
the case of circular motion and elliptic motion. 
Then the general problem of transferring between 
two specified terminals in space is developed. 
These discussions, like those of the other chapters, 
are fully documented. 



The chapter concludes with a discussion of the 
effects of finite burning time, of the requirements 
for the propulsion system to accomplish the pre- 
viously described maneuvers, a discussion of the 
error sensitivities, and a discussion of the sta- 
tistical distribution of errors in the resultant 
orbital elements. 



VII. RENDEZVOUS 

Rendezvous is broken into two basic phases 
for the purpose of the discussion in this handbook. 
The first of these phases contains the launch and 
ascent timing problems, the problems of maneu- 
vers and of the relative merits of direct ascent 
versus the use of intermittent orbits or rendezvous 
compatible orbits. The second phase is the dis- 
cussion of the terminal maneuvers. Included in 
this final section are the equations of relative 
motion, a discussion of possible types of guidance 
laws, and information necessary to evaluate the 
energy and timing of the terminal maneuver whether 
it be of a short or long term nature. 



1-1 



VIE. ORBITAL DEPARTURE 



XI. ORBIT COMPUTATION 



The problem of recovering a satellite from 
orbit at a specific point on earth at a specific time 
is essentially the reverse of the rendezvous prob- 
lem, and the approach taken here is the same. 
First, an intermediate orbit is established which 
satisfies the timing constraints, then the maneuver 
is completed by deorbiting without requiring a 
lateral maneuver. For cases where this approach 
should prove impractical, data for a maneuverable 
re-entry is also presented. 

The presentation progresses from the timing 
problem to the analyses of the intervals between 
acceptable departures, the finite burning simu- 
lation of the deorbit maneuver, and the error 
sensitivities for deorbiting. 



The discussions of this chapter tie many of the 
previous chapters together since all trajectories 
to be of value must be known. The discussions 
progress from the basic definitions of the basic 
coordinate systems and transformations between 
them, to the determination of initial values of the 
six constants of integration, to the theory of ob- 
servational errors, and finally to the subject of 
orbit improvement. In this process, data is pre- 
sented for most of the current tracking facilities 
and for many basic techniques applicable to the 
various problem areas (e.g. , orbit improvement 
via least squares, weighted least squares, mini- 
mum variance, etc. ). The chapter concludes with 
a presentation of data useful in the preliminary 
analysis of orbits. 



IX. SATELLITE RE-ENTRY 

Once the satellite leaves orbit it must penetrate 
the more dense regions of the atmosphere prior 
to being landed. This chapter treats analytically 
and parametrically (i. e. , as function of the re- 
entry velocity vector) the various factors which 
are characteristic of this trajectory: Included 
are the time histories of altitude, velocity and 
flight path angle; also included are the range 
attained in descent, the maximum deceleration, 
the maximum dynamic pressure, and equilibrium 
radiative skin temperatures, as well as a dis- 
cussion of aerodynamic maneuverability. Thus, 
this chapter makes it possible to analyze the tra- 
jectory all the way from launch to impact in a 
reasonably accurate manner before progressing 
to a detailed numerical study of a particular vehi- 
cle flying a particular trajectory. 



Xn. GUIDANCE AND CONTROL REQUIREMENTS 

The discussions of this chapter relate the 
errors in the six constants of integration to errors 
in a set of six defining parameters. This 6x6 
matrix of error partials has been inverted to ro- 
tate the parameter errors to errors in the ele- 
ments. The result is that it is possible to pro- 
gress from a set of parameter errors at some 
time directly to the errors in the same parameters 
at any other time. This formulation has proved 
itself useful not only in the study of error propa- 
gation but in the analysis of differential corrections 
and the long time rendezvous maneuver. 

Also included in the chapter is information 
related to problems of guidance system design, 
the attitude disturbing torques and the attitude 
control system. 



X. WAITING ORBIT CRITERIA 

The balance of the book treats problems as- 
sociated with the flight mechanics aspects of 
specific missions. However, these are some 
problems which are not of this nature but which 
can influence the selection of orbits. (The radi- 
ation environment etc., of Chapter II is an example 
of this type material. ) Accordingly, Chapter X 
presents some information pertaining to the solar 
radiation heat level, and to the storage of cryo- 
genic fluids. This information is treated only 
qualitatively because it is outside the general 
field of orbital mechanics and is itself the subject 
for an extensive study. The material is included 
however, because of the requirement for fuel in 
many of the discussions of maneuver outlined in 
the rest of the text. 



Xm. MISSION REQUIREMENTS 

The purpose of this chapter is to present many 
problems which directly affect the selection of 
orbits for various missions and experiments. The 
data include satellite coverage (both area and 
point), satellite illumination and solar eclipses, 
solar elevation above the horizon, surface orienta- 
tion relative to the sun, sensor limitations (e.g., 
photographic resolution considerations, radar 
limitations), and ground tracks. Thus, giveh a 
particular mission, one can translate the accompa- 
nying requirements to limitations on the orbital 
elements and, in turn, pick a compromise set 
which best satisfies these requirements (when the 
radiation environment, meteoroid hazard and radi- 
ation heat loads have been factored into the selec- 
tion). 



1-2 



II. PHYSICAL DATA 



G 

i 

J n 

K s 

L 

L' 

m 

M„ 



P n<> 



SYMBOLS 

Semimajor axis of the instantaneous 
elliptical orbit 

Eccentricity of the instantaneous ellipti- 
cal orbit 

Flattening = (R equatoria i " R po lar ) " 

r> 

equatorial 

Universal gravitational constant 

Inclination of the instantaneous elliptical 
orbit 

Coefficients of the potential function 

Solar gravitational constant = G m 

Latitude 

Coefficient of the lunar equation 

Mass 

Mean anomaly of epoch 

Number 

Probability 

Legendre polynomial of order n 

Radius 

Radius of action (Tisserand' s criteria) 



U 






C 
o 



Coefficient obtained from t distribution 

Potential function 

Mean of a sample of size n 

Gravitational constant for a planet = Gm 

Mean of population from which sample is 
taken 

Parallax = ratio of two distances 

Variance of population from which sample 
is taken 

Estimate of the variance assuming the 
parent population is normal 



(•i-U *-■>*) 



Orbital period 

Longitude of the ascending node of the 
instantaneous elliptical orbit 

Argument of perigee of the instantaneous 
elliptical orbit 



Subscripts 



Lunar 
Solar 
Earth 
Planet 



II- 1 



INTRODUCTION 



In the study of trajectories about the earth, 
factors defining the trajectory must be accurately 
known. Since these factors fall into two areas: 

Astronautical constants 

Geophysical constants 

each of these general areas will be investigated. 
In addition, information which is not of a flight 
mechanics nature but which can effect the selection 
of orbits will also be presented. This type of in- 
formation includes: 

Radiation hazard data (all types) 

Micrometeoroid data 

Shielding data. 

Finally, information necessary to convert this 
data from one set of units to another will be pre- 
sented. This discussion goes beyond unit con- 
version, however, to include a review of time 
standards and measurement. This review is ap- 
plicable to the material presented in all of the 
chapters which follow. 

A. ASTRONAUTICAL CONSTANTS 

Three noteworthy articles dealing with the 
constants which define the trajectory of a mis- 
sile or space vehicle have been published within 
the past two years. These articles are: 



The discussion of these constants will be 
followed by a presentation of desirable data 
which is obtained from the constants and tables 
of conversions relating these quantities to the 
corresponding quantities in other sets of units. 
This latter set of tables is particularly important 
since there is much confusion as to the meaning 
of generally used units and the accuracy of the 
conversion factors. 

Dr. Krause' s paper, which is presented as 
Appendix B to this volume by consent of the 
author, presents a slightly different set of con- 
stants. This results from the fact that the 
approach taken was to produce an internally con- 
sistent set of constants based on the author' s 
adopted values of the independent quantities 
rather than to accept the slight inconsistencies 
resulting from the development of "best values" 
for each of the quantities. It is noted, however, 
that in nearly every instance Dr. Krause 1 s 
values differ from those quoted in this section 
by a quantity less than the uncertainties quoted 
in this chapter. Thus, the two approaches seem 
to complement each other. 

1. Analysis of Constants 

Although Baker' s exact analytical procedure 
is not known, his results indicate a process 
similar to the following: 

(1) Collect all available data pertinent to 
a particular quantity. 

(2) Obtain the mean and standard deviation 
of this sample 



"Analysis and Standardization of Astro- 
Dynamic Constants" by M. W. Makemson, 
R. M. L. Baker, Jr., and G. B. Westrom, 
Journal of the Astronautical Sciences, Vol. 
8, No. 1, Spring 1961, pages 1 through 13. 

"A Geoid and World Geodetic System 
Based on a Combination of Gravimetric, 
Astrogeodetic and Satellite Data" by W. 
M. Kaula, Journal of Geophysical Research, 
Vol. 66, No. 6, June 1961, pages 1799 
through 1811. 

"On a Consistent System of Astrodynamic 
Constants" by H. G. L. Krause, NASA 
Report MTP-P&VE-F-62-12, Marshall 
Space Flight Center, 12 December 1962. 

The first paper reviews measurements of 
heliocentric, planetocentric and selenocentric 
constants; the second treats the determination 
of the geocentric constants by statistical methods 
using the gravimetric, astrogeodetic and satellite 
data. The work reported in these papers is 
excellent and will not be reproduced since it is 
readily available. Rather the published data 
will be summarized and the best values selected 
for use in trajectory analysis. It is felt that 
this step is necessary because (1) there are 
small inconsistencies in the data, and (2) there 
is no mention in the first article of a method of 
analysis or an approximate confidence interval. 
"Confidence interval" will be used here to in- 
dicate that the sample interval brackets the true 
mean some prescribed percentage of the time. 



iy x. 

n /_, i 






(x, - xT 



2 n 
a = — 



(3) Throw out all points deviating from 
the mean by more than one standard 
deviation. 

(4) Recompute the mean and standard 
deviation. 

Assuming that the various pieces of data are 
of roughly the same accuracy (this assumption 
is necessary since the uncertainties quoted for 
the number are inconsistent) and that there is no 
uniform bias to the determinations, this procedure 
will result in a reasonable estimate for the 
quantity and its uncertainty, provided that the 
sample size is sufficiently large. However, 
there is no guarantee that the estimate will be 
reasonable for small samples. A general feel 
for the maximum number of random, unbiased 
determinations required for a specified accuracy 
of the resultant analysis can be obtained from 
Tchebycheff' s inequality. 



II-2 



[> 



(x 



<b] > 



nb 



h' (1 - p) 

= an estimate of the minimum 
sample size. 

Since the general accuracy of the determina- 
tions is quoted to about 1 to 5 parts in 

10 and since the standard deviations are of 
the same order, 



K 



(1 - P) 



K 



n a 10K 
« 100K 



P 
P 



90% 
99% 



where K is a constant of proportionality. 
Because the sample sizes are generally smaller 
than 10, it may appear that the confidence level 
for the quoted constants will be less than 90% 
but probably greater than 80% for most but not 
all of the constants. This, however, is not 
true as will be shown in the following para- 
graphs. 

Tchebycheff ' s inequality provides a general 
feel for the concept of assigning a probability 
of correctness to the quoted value of any of the 
discussed constants. However, the question 
arises as to the definition of the number K; 
moreover, even if K is defined, the estimates 
are in general too conservative. For this 
reason, the method described below will be 
utilized. 

Assuming once again, that the samples come 
from a normal distribution, the probability P 
that a given value will fall in a quoted region 
about the mean is 



yrr 



x + a — 
yrr 



However, care must be taken because the 
quantities )j.'and a used in this expression are 
the mean and variance of the true population, 

•■hi "•• 



not the estimates of |i'. 



and cr, 



*./x 



(x- 



)' 



While these 



estimates may be utilized there is no assurance 
for the correctness for any but the large sample. 
The solution to this problem is found in the "t" 
distribution 



- ,i< 



Jil (n-l) 1 ' 2 



VZ 



(x 
n <n l - 1) 



x) 



This distribution involves only ^' and the data 
x. and is of n - 1 degree of freedom. Since this 

distribution is also tabulated it is possible to 
write 

t, 



P (-t h< t <t b ) = C f (t; n- l)dt = P = l-b 



and convert the inequalities to obtain 



I 



(x. - xV 



n(n- 1) 



< M' 



I 



x + t, 



( Xi -xy 

n(n- 1) 



= 1 - b 



The coefficient t fa is called the b percent level 

of t and locates points which cut off b/2 percent 
of the area under f(t) on each tail (f (t) is sym- 
metric about t = 0). 




Thus, the problem of defining the probability of 
correctness which can be assigned to a quoted 
constant is one of defining t b . Since in all the 

work to be discussed la variation will be quoted, 

t, times the radical can be defined as a . This 
b 

assumption results in an estimate of the probable 
correctness of the quoted constant which is a 
function only of the number of data points. 



F 



At this point it is possible to refer to a table of a 
cumulative t distribution and obtain the estimate 
of the confidence level for a given value of t fa 

(i.e., a specified sample size). However, since 
this solution requires nonlinear interpolation, 
the confidence levels have been plotted as a func- 
tion of the sample size in Fig. 1. These data 
will be utilized for all estimates to be made in 
this section. 

In view of the facts that the original measure- 
ments do not agree to within the probable errors 
quoted for the experiments and that the confidence 
levels for the results are reasonable, this pro- 
cedure appears to be the most attractive means 
of resolving the confusion associated with these 



II-3 



constants until more and better data can be ob- 
tained. This is not meant to imply that Baker' s 
data should be used as presented because in 
several cases his constants deserve special 
attention. In any event, when superior data be- 
come available they should either be weighted 

r_ a (x.-x) 

heavily x obtained from = > l „ 

1=1 l 

or utilized in preference to any other value. 



] 



Kaula' s data will not be reviewed specifically 
because it is included in the analysis which fol- 
lows. However, in the discussion of the geo- 
centric constants, special note will be made of 
the agreement of Kaula' s data with Baker' s 
and that obtained by the criteria outlined above. 

2. Heliocentric Constants 



a. Solar parallax 



Planetary observations and theories of 
planetary motion permit precise computation 
of the angular position of the planets. Although 
angular measurements are quite accurate, no 
distance scale is readily available. Attempts 
to resolve this problem have led to the compari- 
son of large, unknown interplanetary distances 
to the largest of the known distances available 
to man, the equatorial radius of the earth. In 
the process, solar parallax was defined as the 
ratio of the earth 1 s equatorial radius to the 
mean distance to the sun from a fictitious un- 
perturbed planet whose mass and sidereal 
period are those utilized by Gauss in his com- 
putation of the solar gravitation constant (i.e., 
one astronomical unit). This definition renders 
unnecessary the revisions in planetary tables 
as more accurate fundamental constants are 
made available, since the length of the astro- 
nomical unit can be modified. 

In the broadest sense, the solar parallax is 
the ratio between two sets of units: (1) the 
astronomical set utilizing the solar mass, the 
astronomical unit and the mean solar day, and 
(2) the laboratory set (cgs, etc. ). 

Before reviewing solar parallax data obtained 
from the literature, it is worthwhile to consider 
the means of computing the values and their un- 
certainties. 

The first method, purely geometric, is 
triangulation based on the distance between two 
planets, between a planet and the sun, etc. One 
such computation was made by Rabe following a 
close approach of the minor planet Eros. The 
second method is an indirect approach based on 
Kepler' s third law (referred to in the literature 
as the dynamical method). The third method 
employs the spectral shift of radiation from 
stars produced by the motion of the earth. 
Perturbations on the moon produced by the sun 
constitute a fourth means of computing solar 
parallax to good precision provided that the 
ratio of the masses of the earth and moon is 
well known. A fifth approach utilizes direct 
measurements of distance between bodies in 
space obtained from radar equipment. 



Other approaches have also been advanced, 
but the five listed constitute the most frequently 
employed. 

Table 1 presents the adopted value of solar 
parallax (from Baker) along with the unweighted 
mean of the data and the mean of the adjusted 
sample. (Special note is made that the value 
adopted by Baker corresponds most closely to 
that of Rabe which has been widely utilized 
during recent years. ) The corresponding value 
of the astronomical unit is also presented. 





TABLE 


1 




Solar Parallax 






Adopted 

by 

Baker 


Jncorrected 
Mean and 
Standard 
Deviation 


Adjusted 
Mean and 
Standard 
Deviation 


Solar parallax 
(sec) 


8.798± 
0.002 


8.7995± 
0.0049 


8. 8002± 
0.0024 


Astronomical 
unit (10 6 km) 


149.53* 
0.03 


149. 507* 
0.083 


149.495* 
0.041 


Confidence 

level 


? 


99% 


92% 



The data in Table 1 show reasonably good 
agreement between the various estimates. 
However, it is interesting to note that the adjusted 
mean moved away from the value adopted by 
Baker. This behavior is undesirable but was not 
unforeseen because of the limitations of the 
method and the fact that more of the measure- 
ments were situated in this direction. However, 
most of the reported measurements were made 
before 1945 and the general trend during subse- 
quent years has been toward slightly lower values 
of the solar parallax. If it is assumed that this 
trend reflects increased accuracy in the measure- 
ments (resulting in part from the availability of 
radar data), and if the more recent measure- 
ments are weighted by the time of determination 
(since the uncertainty in the various measure- 
ments is much larger than the quoted error in the 
experiment), a value of solar parallax of 8. 7975 sec 
± 0.0005 is obtained. This value is almost ident- 
ical to Baker's which, as was noted, agrees with 
that of Rabe (generally accepted by those perform- 
ing astronomical computations). For this reason, 
and for consistency in calculations by various 
groups within industry and the government. Baker's 
value of the solar parallax should be used. How- 
ever, his assignment of probable error in this 
constant apparently is too large in view of the 
agreement of these data. A maximum uncertainty 
of* 0.001 is more realistic. 

b. Solar gravitational constant 

In 1938 it was internationally agreed (IAU 1938) 

that to maintain the Gaussian value of the solar 

2 
gravitational constant (K = Gm where G = 

Universal gravitational constant) in spite of 
changes in the definition of the sidereal year 
and the mass of the earth, the astronomical unit 
(AU) would be modified when necessary. Thus 
the solar gravitational constant has remained. 



II-4 



K 



T 



0.017, 202, 098, 95 



AU 



3/2 



solar 
day 



where 



m 



O = 



1 AU 

365. 256, 383, 5 mean solar days 

solar mass = 1 

ratio of earth mass to solar mass 
0. 000,002, 819 



This value of K is accurate to its ninth signifi- 
cant figure by definition. The precision in this 
determination is contrasted to the accuracy of a 
determination in laboratory units from the fol- 
lowing equation 



Gm 



where 



G = the universal gravitational constant 
in the cgs or English system of 
units (mass in same system). 

Utilizing even the most accurately known 
values of G and m (obtained from Westrom) the 
result is accurate only to its third place. 

2 



K 






6. 670 (1 ± 0.0007) 10 
1. 9866 (1 ± 0.007) 1 



'] 

33 ]| 



K = 1. 511 (1 ±0.0005) 10 13 cm 3 ' 2 /sec 

The evaluation of K in laboratory units using 

the solar parallax proves equally as inadequate 
since the uncertainty is large. When the adopted 
value indicated in Table 1 is used, K is found 
to be 



K. 



1.1509 (1 ± 0.00015)10 13 cm 3 ^ 2 /sec 



It is thus advantageous to compute in the 
astronomical system of units, converting only 
when necessary. This procedure assures that 
the results will become more accurate as better 
values for the astronomical unit are obtained 
and produces a much lower end figure error due 
to round -off. 

3. Planetocentric Constants 



a. Planetary masses 



Planetary masses are significant in comput- 
ing transfer trajectories to the planets and tra- 
jectories about these bodies. The two most 



common methods of determining planetary mass 
are by the perturbation actions on other bodies 
or by observations of the moons of the planet. 
While the accuracies of the two approaches differ, 
each involves such complex functions as near- 
ness of approach, mass of the planets, size and 
number of moons, etc. , that no general conclu- 
sion can be made as to the superiority of one to 
the other. 

Table 2 presents data reduced from deter- 
minations of the mass of each of the planets in 
terms of the solar mass, the related mass in 
kilograms, and the probable uncertainty in the 
measurement. In addition, since the number of 
points in the sample varies from planet to planet, 
this quantity is noted along with an estimate of 
the confidence level for the result. 

In each case shown in Table 2 the results ob- 
tained with the adjusted sample approach those 
of Baker to within the uncertainties quoted for 
the masses and are practically identical. How- 
ever, it should be noted that the uncertainties 
quoted for these masses are different at times. 
This discrepancy is believed to result from the 
somewhat arbitrary handling of the limits in the 
reviewed reference. On the basis of the data 
available, It seems more proper to use the 
standard deviation, as obtained from the adjusted 
sample, rather than Baker's value. 

b. Planetary dimensions 

While the physical dimensions of the planets 
have no effect on the trajectories of interplanetary 
vehicles and the dimensions are generally 
smaller than the uncertainty in the astronomical 
unit, the constants must be known for self-con- 
tained guidance techniques and for impact and 
launch studies. For these reasons the best shape 
of the various planets will be discussed. 

Table 3 presents equatorial and polar radii 
and a quantity referred to in the literature as 
the flattening which is defined to be 

R. — R 

- _ equatorial polar 

equatorial 

The table also presents comparisons of various 
data, the number of points in the sample and an 
estimate of the confidence level. 

The sample size for the planet Uranus is 
questioned because Baker references only one 
source for this planet and that is a weighted 
average of several determinations . In the tabu- 
lation on Mars, note should be made of the 
excellent agreement on the best value of the 
radius given by the statistical approach and by 
Baker, and of the slight discrepancies in the un- 
certainties of the radius and in the best value 
of the flattening. Therefore, it is once again 
proposed that Baker 1 s value of the radii and 
flattening (with one exception) be utilized but 
that the uncertainty obtained via statistics be 
associated with this number. The exception 
exists in the case of Mars for which it is pro- 
posed that l/f be 75 ±12, rather than Baker' s 
value (150 ± 50) since this estimate is consistent 
with the data. 



II-5 



TABLE 2 
Planetary Masses 



Planet 


Quantity of Interest 


Adopted by Baker 


Uncorrected Sample 


Adjusted Sample 


Mercury 


Solar mass /mass of Mercury 
Mass of Mercury in kg 

Sample size 
Confidence level 


6, 100,000 i 50,000 

?4 
0.32567 x 10 

4 


6.400,000 ± 630,000 

0.31041 x 10 24 
4 

81% 


6,030,000 ± 65,000 

?4 
0.32945 x 10 

3 

70% 


Venus 


Solar mass /mass of Venus 


407,000 ± 1,000 


406,200 ± 1,900 


407,000 ± 1,300 




Mass of Venus in kg 
Sample size 
Confidence level 


9d 

4.8811 x 10 
8 


94 

4.8907 x 10 
8 

97% 


4.8811 x 10 24 
6 

92% 


Earth-Moon 


Solar mass /earth-moon mass 


328,450 ± 50 


328, 500 ± 100 


328,430 ± 25 




Mass of earth-moon in kg 
Sample size 
Confidence level 


94 

6.04841 x 10 
6 


94 

6.04749 x 10 

6 

92% 


6.04878 x 10 24 
4 

81% 


Mars 


Solar mass /mass of Mars 


3,090,000 ± 10,000 


3,271,000 ± 795,000 


3,092,000 ± 12,000 




Mass of Mars in kg 
Sample size 
Confidence level 


6.04291 x 10 24 
6 


24 
0.60733 x 10 
6 

92% 


24 
0.64250 X 10 

4 

81% 


Jupiter 


Solar mass /mass of Jupiter 


1047.4 ± 0.1 


1047.89 i 1.87 


1047.41 i 0.08 




Mass of Jupiter in kg 
Sample size 
Confidence level 


1.89670 x 10 27 
8 — 


1.89581 x 10 27 
8 

97% 


1.89670 x 10 27 
4 — 

81% 


Saturn 


Solar mass /mass of Saturn 


3500.0 ± 3 


3497.3 ±4.5 


3499.8 ±1.7 




Mass of Saturn in kg 
Sample size 
Confidence level 


0.56760 x 10 27 
4 — 


0.56804 x 10 27 
4 — 
81% 


0.56763 x 10 27 
3 — 
70% 


Uranus 


Solar mass /mass of Uranus 


32,800 ± 100 


22,810 ± 60 







Mass of Uranus in kg 
Sample size 
Confidence level 


87.132 x 10 24 
2 


87.093 x 10 24 
2 

50% 


--- 


Neptune 


Solar mass /mass of Neptune 


19,500 i 200 


19,500 ± 200 







Mass of Neptune in kg 
Sample size 
Confidence level 


101.88 x 10 24 
3 


101.88 x 10 24 
3 

70% 





Pluto 


Solar mass /mass of Pluto 


350,000 ± 50,000 


333,000 ± 27,000 







Mass of Pluto in kg 
Sample size 
Confidence level 


5.6760 x 10 24 
3 


94 
5.9658 x 10 
3 

70% 






Underlined digits are questionable 



II-6 



TABLE 3 
Planetary Dimensions 



Planet 



Quantity of Interest 



Mercury Equatorial radius (km) 
1/f 

Polar radius (km) 
Sample size 
Confidence level 

Venus Equatorial radius* (km) 

1/f 

Polar radius (km) 
Sample size 
Confidence level 

Mars Equatorial radius (km) 

1/f 

Polar radius (km) 
Sample size 
Confidence level 

Jupiter Equatorial radius (km) 

1/f 

Polar radius (km) 
Sample size 
Confidence level 

Saturn Equatorial radius (km) 

1/f 

Polar radius (km) 
Sample size 
Confidence level 

Uranus Equatorial radius (km) 

1/f 

Polar radius (km) 

Sample size 



Neptune Equatorial radius (km) 

1/f 

Polar radius (km) 
Sample size 
Confidence level 

Pluto Equatorial radius (km) 

1/f 

Polar radius (km) 
Sample size 
Confidence level 



Adopted by Baker 



2, 330 ± 15 

? 

? 

4 
? 

6, 100 ± 10 

? 

? 

6 

? 

3,415 i 5 
150 ± 50 
3,392 ± 12 
9 
? 

71, 375 i 50 
15. 2 ± 0. 1 
66,679 ± 50 
2 
? 

60, 500 ± 50 
10.2 i ? 
54,569 ± 45 
2 
? 

24,850 ± 50 

? 

? 

? 

25,000 ± 250 
58.5 ± ? 
24,573 ± 250 
2 
? 

3,000 ± 500 

? 

? 

i 



Uncorrected Sample 



2, 355 ± 39 

? 

? 

4 
81% 

6, 154 ± 100 

? 

? 

6 

92% 

3, 377 ± 47 
108.4 ± 54 
3, 346 ± 55 
9 
98% 

71, 375 ± 20 
15.2 ± 0.1 
66,679 ± 50 
2 

50% 

60, 160 ± 480 
10.2 ± ? 
54, 262 i 450 
2 

50% 

24,847 ± 50 
14 ± ? ** 
23,072 i 50 
9 

24,400 ± 2100 
58.5 ± ? 
23,983 ± 2000 
2 

50% 

2,934 ± 500 
? 

? 

i 

20% 



Adjusted Sample 



2,333 i 11 

? 

? 

3 

70% 

6, 106 ± 12 

? 

? 

3 

70% 

3,414 ± 12 
75 ± 12 
3,403 ± 12 
5 
88% 



*Equatorial radius for Venus includes the distance from the 
of the dense atmosphere. 
**From K. A. Ehricke's book "Space Flight Trajectories. " 



surface to the outer boundary 



II-7 



(2) 



(3) 



As was the case with some of the planetary 
masses, there was insufficient data available 
to allow for refining dimensional computations 
for all planets. Even where such computations 
were possible the confidence level of the re- 
sultant quantity was low. 

c . Planetary orbits 

Because the motion of a planet about the sun 
approximates an ellipse for relatively long 
periods of time, it has become standard practice 
to express the paths in terms of an ellipse with 
time- varying or osculating elements. To assure 
that the terminology is familiar, the six ele- 
ments (or constants of integration) necessary 
to determine planetary motion are defined below. 

(1) Planar elements 

(1) Semimajor axis (a)- -This element 
is a constant, being one-half the sum 
of the minimum and maximum radii. 
Element (a) is also a function of 
radius and velocity at any point. 

Eccentricity (e)--This element is re- 
lated to the difference in maximum 
and minimum radii and is used to 
express a deviation in the path from 
circularity. 

Mean anomaly of epoch (M Q )--This 

element (referenced to any fixed 
known time) defines the position of 
the orbiting body in the plane of 
motion at any time. 

(2) Orientation elements 

(1) Argument of perigee ( u )--This is 
the angle measured in the orbital 
plane from the radius vector defining 
the ascending node to the minimum 
radius. 

(2) Orbital inclination (i)--This angle 
expresses rotation of the orbital 
plane about a line in the ecliptic 
(or fundamental) plane. 

(3) Longitude of the ascending node (£2)-- 
This is the angle measured in the 
fundamental plane from a fixed ref- 
erence direction to the radius at which 
the satellite crosses the fundamental 
plane from the south to the north. 

These osculating elements obviously are of 
primary importance in the computation of inter- 
planetary transfer trajectories. Thus, the 
procedure for obtaining these elements will be 
reviewed; then the values of the elements will 
be presented. It is assumed only that a table 
of the time variation of acceleration is available. 
One such table is presented in Planetary Coord- 
inates 1960 to 1980 available through Her Majesty's 
Stationery Office. 

This reference quotes position and accelera- 
tion components in ecliptic rectangular coordin- 
ates. The most direct transformation is thus 
via the vectorial elements P, Q and R (where F 



points toward perihelion, Q in the direction of 
the true anomaly equals 90° and R completes the 
right handed set). The computation proceeds as 
follows: First the velocity components at the 
instant are computed. This is accomplished by 
numerical integration of the acceleration com- 
ponents rather than by differentiation of the 
position data in order to obtain better accuracy. 



Argument 



Function 
(Acceleration) 



Thus ,_ at the argument t 



wK 



r *-'■• 







x - u » 6x+ jm fJ&3x ■■■] 



where 



w = the interval between points in mean solar 
days 



K = Gaussian constant 

s 

= 0.017, 202,098, 95 



AU 



3/2 



solar day 



( 5x -l/2 + 6x l/^) 

12 (« 3 x'_ 1/2 + 6 3 x 1/2 ) 



/u6x = 1/2 



* 3 " 1 

(10 X = 1 



and similarly for y and z. 
Now 

2 2 2 2 
r = x + y + z (evaluated at t„) 

2 "2 "2 -2 

v = x + y + z 



H = xx + yy + zz 

1 
a = T 

2/r -G* 

e sin E = H/ fa 



(1) 
(2) 



e cos E = rG - 1 (3) 

^pR = (yz - zy) x + (zx - xz) y + (xy - yx) z 



II-8 



1 - e 2 Q = r - sin E + 7a 1/2 



• (cos E - e) 



— -"1 -* 1 /2 

P = r — cosE + va sin E 



And finally 

sin i sin £2 = R 



sin i cos £2 = - R cos e - R sin £ 

y z 



cos i = R cos € - R sin « 

z y 



(4) 
(5) 
(6) 



And 



(1 ± cos i) sin (u ± R) = ± P cos « 

± P sin « - Q 
z x 



(1 ± cos i) cos (a) ± n) = ± Q cos « 

± Q sin c + P 
^z x 



(7) 



(8) 



where: « » obliquity of the ecliptic of date given 
below: 

t = 1960 « = 23°26'40. 15" sin <= 0. 39786035 cos f = 0. 91744599 

1962 23°26'39.21" 0.39785618 0.91744780 

1964 23°26'38.28" 0.39785201 0.91744960 

1966 23°26'37.34" 0.39784784 0.91745141 

1968 23°26' 36. 40" 0.39784368 0.91745322 

1970 23°26'35.93" 0.39783951 0.91745503 



Equations (1), '2) and (3) define a, e and E (analo- 
gous to M) at the selected epoch. Then Eqs (4) 
through (8) define the orbital planes and the quad- 
rants of the three orientation elements. 

Data for these six elements is presented in 
Tables 4 and 5. These tables present each of the 
six elements for a two-year period and the re- 
gression and precession rates of the nodal angle 
and the argument of perigee, respectively. These 
data are accurate to the last quoted digit for the 
quoted epochs and provide reasonably good ac- 
curacy when linearly interpolated. In order to 
maintain precision in such computations it is nec- 
essary to have the elements evaluated at much 
smaller time intervals. 

4. Geocentric Constants 

a. Potential function 

The potential function of the earth (i.e. , the 
relationship between potential energy and position 



relative to the earth) is not simply - 



Gm 



assumed in most Keplerian orbit studies because 
this approximation assumes that the mass is 
spherically symmetric. This assumption is suf- 



ficiently accurate for many preliminary studies 
but is not valid for precise orbital studies. For 
this reason it is general practice to expand the 
potential function in a series of Legendre polyno- 
mials. The coefficients of this series may then 
be evaluated from satellite observation. 

Since the perturbations in the motion (i.e. , 
deviations due to the presence of the terms in- 
volving mass asymmetry of the earth) are very 
sensitive to the uncertainties in the coefficients 
of the resulting potential function, one form of 
this function will be presented and discussed. 
The form selected, because of its simplicity and 
the fact that it was recently adopted by the LAU 
(1961), is that of J. Vinti of the National Bureau 
of Standards. The coefficients of other generally 
used expansions will be related to this set in later 
paragraphs. 



U = 



1 - y j (-) p < sin u 

Ij n ^r/ n 



n=2 

where 

H = gravitational constant = Gm„ 

J = coefficients 
n 

R = equatorial radius of the earth 
r = satellite radius 
P (sin L) = Legendre polynomials 
L = instantaneous latitude 
The first few terms of this series are: 



-J (3 sin L - 1) 



(5) (5 sin 3 L - 3 sin L) 

4 
(?) (35 sin 4 L - 30 sin 2 L + 3) 



3 
J 4 /R^ 4 



-^ (?) (63 sin 5 L - 70 sin 3 L + 15 sin L) 



T fi 

- 6 (B.) (231 sin 6 L - 315 sin 4 L 

51 ] 



+ 105 sin 2 L 



As is immediately obvious, this function contains 
the potential function for a mass spherically sym- 
metric earth and a series of correction terms re- 
ferred to as zonal harmonics. The odd ordered 
harmonics are antisymmetric about the equatorial 
plane (L = 0) and the even ordered harmonics, 
symmetric. This function was introduced merely 
to aid in the discussion of the factors affecting 
motion in geocentric orbits; therefore, the func- 
tion as a whole will not be discussed further but 
its coefficients will be treated. 



II-9 



TABLE 4 

Mean Elements of Inner Planets 
(from American Ephemeris, 1960, 1961, 1962; 
referred to mean equinox and ecliptic of date. ) 



Epochs: 1960 September 23.0 = J.D. 243 7200.5 

1961 October 28.0 = J.D. 243 7600.5 

1962 December 2.0 = J.D. 243 8000.5 



Planet 



Year 



i* 
(deg) 



f2* 
(deg) 



(deg) 



(AU) 



(deg) 



Mercury 



Venus 



Mars 



1960 
1961 
1962 

1960 
1961 
1962 

1960 
1961 
1962 



7.00400 + 1 
7.00402 + 1 
7.00404 + 1 

3.39424 + 

3.39425 + 

3.39426 + 

1.84993 + 
1.84992 + 
1.84991 + 



47.86575 + 325 
47.87873 + 325 
47.89171 + 325 

76.32625 + 247 
76.33611 + 247 
76.34597 + 247 

49.25464 + 211 
49.26308 + 211 
49.27153 + 211 



76.84441 + 426 
76.86145 + 426 
76.87849 + 426 

131.01853 + 385 
131.03394 + 385 
131.04934 + 385 

335.33609 + 504 
335.35625 + 504 
335.37641 + 504 



0. 387099 
0. 387099 
0. 387099 

0.723332 
0.723332 
0.723332 

1.523691 
1.523691 
1.523691 



0.205627 
0.205627 
0.205627 

0.006792 
0.006791 
0.006791 

0.093369 
0.093370 
0.093371 



152.303 
349.237 
186. 171 

108.652 

29.504 

310.356 

62.572 
272.180 
121.789 



*Plus variation per 100 days. 

**The large differences between the mean anomalies at epoch are due primarily to the shift in the epoch and 
not to perturbations. 



TABLE 5 



Osculating Elements of Outer Planets 
(from American Ephemeris, 1960, 1961, 1962; 
referred to mean equinox and ecliptic of date. ) 



Planet* 


Date 




i 
(deg) 


n 

(deg) 


(deg) 


(AU) 


e 


M 
(deg) 


Jupiter 


1960 Jan. 

1961 Jan. 

1962 Jan. 


27 
21 
16 


1.30641 
1.30626 
1.30616 


100.0560 
100.0651 
100.0725 


12.3279 
13.2393 
13.2614 


5.208041 
5.203825 
5.203520 


0.048, 335, 1 
0.048,589,9 
0.048,459,7 


249.7967 
278.7932 
308.6768 


Saturn 


1960 Jan. 

1961 Jan. 

1962 Jan. 


27 
21 
16 


2.48722 
2.48718 
2.48714 


113.3161 
113.3273 
113.3385 


92.1031 
90.7422 
89.3436 


9.582589 
9.580399 
9.581007 


0.050,548,4 
0.051, 145,6 
0.051,778,3 


188.9699 
202.4677 
216.0551 


Uranus 


1960 Jan. 

1961 Jan. 

1962 Jan. 


27 
21 

16 


0.77236 
0.77222 
0.77221 


73.7218 
73.6971 
73.6942 


172.5311 
172.8809 
172.3515 


19.16306 
19.13202 
19.11431 


0.046,906,5 
0.045,282,3 
0.044, 112,4 


329.2259 
333.0587 
337.7453 


Neptune 


1960 Jan. 

1961 Jan. 

1962 Jan. 


27 
21 
16 


1.77329 
1.77325 
1.77318 


131.3233 
131.3709 
131.4144 


25.9372 
22.4739 
26.5510 


30.23803 
30.17541 
30.09783 


0.003,139,4 
0.005,351,5 
0.007,911,7 


191.3613 
197.0665 
195.1770 


Pluto 


1960 Jan. 

1961 Mar 

1962 Jan. 


27 
2 
16 


17.16644 
17.17057 
17.16791 


109.8642 
109.8943 
109.8958 


223.8342 
224.3400 
224.5629 


39.52392 
39.38437 
39.29379 


0.251,35532 
0.249,400,9 
0.247,695,? 


316.9810 
317.9194 
318.8914 



*Osculating elements are given for every 40 days for Jupiter, Saturn, Uranus and Neptune, and for every 
80 days for Pluto. 



11-10 



Since the earth is almost spherically sym- 
metric, the J are all small compared to one (as 
n 

will be shown later); thus, the prime factor af- 
fecting motion is the gravitational constant, p., 
which is defined directly from Newtonian Mech- 
anics as Gm-, Data for this constant were not 

presented in the referenced paper (Baker) though 
a value was adopted. For this reason a review 
of some of the more recent determinations was 
made and a comparison constructed (Table 6). 

Baker's value corresponds to that of Herrick 
(1958) and no data were found which ascribe an un- 
certainty or confidence level to this value. The 
value corresponds very closely to mean of the ad- 
justed sample; for this reason an estimated un- 
certainty would be ±0.00004. 

While Herrick' s value appears valid, a better 
estimate in view of the work done by Kaula would 
seem to be Kaula' s value (or the mean of the ad- 
justed sample which is the same). It is proposed, 

therefore, that the value of m be 1.407648- 10 

16 3 . 2 _ „ .„„, 3, 

± 0.000035-10 



ft 3 /sec 2 or 398,601.5 



± 9. 9 km / 



sec . The selection of this constant, which is 
obviously related to the mass of the earth-moon 
system (previously adopted), does not produce 
large inconsistencies due to the fact that the con- 
version between solar mass and earth mass is ac- 
curate to only four places, and to this order the 
two answers agree. 

The remaining coefficients, J , are related 

to the earth's equatorial radius, the average ro- 
tational rate of the earth, the gravitational con- 
stant, and the flattening of the earth. For this 
reason, it is clear that the arbitrary selection of 
a set of constants will result in slight numerical 
inconsistencies. However, these uncertainties 
are small and of the same order as the uncertainty 
in the numerical values of the J . Data for the J fi 

are presented in Table 7. 



Baker's values of the J correspond almost 

identically to those of the adjusted sample while 
Kaula 1 s do not for J^, J g and J g . No satisfactory 



TABLE 6 
Gravitational Constant for the Earth 



Date 


ft /sec 


Author 




1957 


1 fi 
1.407754 x 10 


Elfers (Project Vanguard) 




1958 


1.407639 


Herrick 




1959 


1.40760 


Jeffreys 




1959 


1.40771 


O'Keefe 




1960 


1.407645 


Department of Defense (see Baker) 




1961 


1.40765 


Kaula 





Gravitational con- 

3 2 

stant (ft /sec ) 

(km /sec ) 

Uncertainty (1) 
(2) 

Sample size 

Confidence level 



Adopted by 
Baker 



(1.407639 x 10 
'398,599.9 



16 



± ? 

± 9 



Unadjusted 
Sample 



1.407666 x 10 
398,606.6 



16 



±0.000050 x 10 
±14.2 



16 



6 

92% 



Adjusted 
Sample 



1.407648 x 10 
398,601.5 



16 



±0.000035 x 10 
±9.9 

5 

88% 



16 



n-n 







TABLE 7 








Coefficients of the Potential Function 






Baker 


Kaula 


Uncorrected Sample 


Adjusted Sample 


J 2 


1082.28 x 10" 6 


1082.61 x 10" 6 


1082.396 x 10" 6 


1082.303 x 10" 6 


a(J 2 ) 


±0.2 x 10" 6 


±0.06 x 10" 6 


±0.241 x 10" 6 


±0.185 x 10" 6 


Confidence level 


? 


? 


98% 


95% 


J 3 


-2.30 x 10~ 6 


-2.05 x 10" 6 


-2.39 x 10" 6 


-2.39 x 10" 6 


a(J 3 ) 


±0.20 x 10" 6 


±0.10x 10" 6 


±0.23 x 10" 6 


±0.23 x 10" 6 


Confidence level 


? 


? 


98% 


90% 


J 4 


-2.12 x 10" 6 


-1.43 x 10" 6 


-1.82 x 10" 6 


-2.03 x 10~ 6 


ff(J 4 ) 


±0. 50 x 10" 6 


±0.06 x 10" 6 


±0.35 x 10" 6 


±0.24 x 10" 6 


Confidence level 


? 


? 


98% 


92% 


J 5 


-0.20 x 10" 6 


-0.08 x 10" 6 


-0.25 x 10" 6 


-0.19 x 10" 6 


a(J 5 ) 


±0.1 x 10" 6 


±0.11 x 10" 6 


±0.16 x 10" 6 


±0.08 x 10" 6 


Confidence level 


? 


? 


92% 


88% 


J 6 


1.0 x 10" 6 


0.20 x 10" 6 


0.68 x 10" 6 


0.83 x 10" 6 


a(J 6 ) 


±0.8 x 10" 6 


±0.05 x 10" 6 


±0.29 x 10" 6 


±0.10 x 10" 6 


Confidence level 


? 


? 


81% 


70% 



reason was obtained for this difference, though 
it is believed that the data utilized by Kaula in the 
determination of J., J and J fi may have been 

biased. This conclusion is strengthened slightly 
by the fact that the results of Kaula for these three 
constants are somewhat below the majority of the 
other independent determinations. Even if the un- 
certainty in these three values is increased an 
amount sufficient to include all values, no appre- 
ciable change will be noted in the computation of 
trajectories, since the numbers are very small 
compared to unity and are even small compared 
toJ 2 . 

It is proposed that the values adopted by Baker 
be accepted without change. This procedure seems 
justifiable on the basis of the data and has the ad- 
vantage that the set is presumably consistent. 
This advantage is not clear cut since, even though 
the J n 's are interrelated, the uncertainties in the 

values are relatively large. 

At this point Vinti's set of coefficients will be 
related to those utilized by other authors. Rather 
than discuss each potential, however, the poten- 
tials will be tabulated for comparison. Then, the 
coefficients of the various terms will be equated. 
This data is presented in Tables 8a and 8b. 



b. Equatorial radius and flattening 

The average figure of the earth is best repre- 
sented as an ellipsoid of revolution (about the 
polar axis) with the major axis the equatorial 
diameter. Obviously this model is not exact; 
however, the accuracy afforded is generally ade- 
quate when computing the ground track of a satel- 
lite, determining tracking azimuths, etc. For 
this reason the best values for the parameters of 
the ellipsoid are desired. These data are pre- 
sented in Table 9 in the form of values of the 
equatorial radius and flattening (previously de- 
fined) along with polar radii, also for each pair of 
values. 

Although the discrepancies in the sets of data 
shown in Table 9 are minor, they are sufficient 
to justify the selection of one particular set. 
Based on the data reviewed, it is felt that the 
data of Kaula is probably slightly superior to the 
remaining values. This conclusion is strength- 
ened by the good agreement between Kaula and 
some of the more recent standards. While this 
is by no means conclusive proof, the fact indi- 
cates a wide degree of acceptance. For this 
reason, an estimate of the confidence level would 
be greater than 90%. 



11-12 



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11-13 



TABLE 8b 

Comparisons of Constants Used in 
Potential Functions 



Vinti 


J 2 


J 3 


J Recommended 


Laplace 


-B 2 /R 2 


-B 3 /R 3 


-B 4 /R 4 




Jeffreys 


2 I 

1 J 


H 


■h" 




K«. 


2 A 2 


-A 3 /R :i 


8 A 4 




Brouwer 


2k 2 


■A /R 3 


8 k 4 




O'Keere, EckelB, Squires 


-A 2 , n /pR 2 


-A 3 . /*.R 3 


-A,. „/« K 4 




R. E. Roberson 


2*J 


None 


-»"2 




Garfinkel 


2k /B 2 


None 


k 1 /R 4 




St ruble 


^ 


None 


"i" 




Krause 


2k 2 /R 2 


None 


8 k 4 
' 3 H 1 




Sterne 


2B 

S 5 


None 


No. 




Her get and Mufien 


2k 2 /H 2 


None 


-8k 4 /R 4 




ProBkurin and Batrakov 


"l' 


None 


-3T D 




W. deSltter 


S' 


No„. 


-£« 





TABLE 9 



Equatorial Radius 


and Flattening 






Baker 


Kaula 


Uncorrected 
Sample 


Adjusted 
Sample 


Equatorial radius (km) 


6378. 150 
±0.050 


6378. 163 
±0.021 


6378. 215 
±0. 105 


6378. 210 
±0.045 


1/f 


298.30 
±0.05 


398. 24 
±0.01 


298.27 
±0.05 


298.27 
±0.03 


Polar radius (km) 


6356.768 
±0.050 


6356.777 
±0.021 


6356.831 
±0.105 


6356.826 
±0.045 


Sample size 


9 


? 


10 


7 


Confidence level 


9 


•7 


98% 


95% 



5. Selenocentric Constants 

The determination of the lunar mass has been 
made from the lunar equation (involved in the 
reduction of geocentric coordinates to those of 
the barycenter, i.e. , the center of mass of the 
earth-moon system), through the evaluation of 
the coefficient, L, defined to be 



L' = 



sin it. 



1 + . 



where 



w^ is the lunar parallax (i.e.. 



(& equatorial 

average lunar distance 



) 



Since there are no lunar satellites whose orbits 
can be used in determining lunar mass, the calcu- 
lations for the most part have been based on ob- 
servations of Eros at the time of closest approach. 

The method consists of finding the solar and 
lunar parallaxes, comparing the observed positions 



of Eros when nearest the earth with an accurate 
ephemeris, fitting the residuals to a smooth 
curve that has the periodicity and zero points of 
the lunar equation, and using the curve to im- 
prove the adopted value of L 1 . Once this is ac- 
m , 



complished 



© 



is evaluated from the previous 



equation. Thus, the first step in the evaluation 
of the lunar mass is the evaluation of the lunar 
parallax or equivalently the lunar distance. 

Baker presents data for the lunar distance 
evaluated by several different methods. These 
data have been used to produce Table 10. 

TABLE 10 



Lunar Distance 





Adopted 
by Baker 


Uncorrected 
Sample 


Adjusted 
Sample 


Lunar distance (km) 


384,402 


384, 402.6 


384,401.6 


Uncertainty (km) 


+ 1 


+ 2.6 


+ 1.1 


Lunar parallax (racl) 
(sec) 


0.016, 592,4 
3422.428 


0.O16. 592, 4 
3422.428 


0.016, 592, 4 
3422.428 


Uncertainty (rad) 
(sec) 


+0.000,000, 1 
+.021 


+ 0.000, 000, 1 
+ .021 


+0.000,000, 1 
+ .021 


Sample size 


6 


6 


5 


Confidence level 


? 


92% 


88% 



The data of Table 10 all agree very well and 
exhibit no inconsistencies of the type shown in 
other data. For this reason it is believed that 
Baker's value should be utilized as it is quoted 
in Table 10. It is interesting to note that the 
value of the lunar parallax and its uncertainty 
were the same for all of the evaluations. 

The next step in the evaluation of the lunar 
mass is the determination of the best value of 
the coefficient of the lunar equation. Once again 
several values are available, each determined by 
different individuals at different times. The re- 
sults of the analysis of these data are presented 
in Table 11. 

TABLE 11 



Coefficient of Lunar Equation 





Adopted 
hy Baker 


Uncorrected 
Sample 


Adjusted 
Sample 


Coefficient L'(sec) 


6.4385 


6.430 


6.4381 


Uncertainty (sec) 


10.0015 


±0.005 


±0.0016 


Sample size 


? 


8 


6 


Confidence level 


? 


97% 


92% 



Once again good general agreement is noted. It 
is proposed, therefore, that the value of L' be 
6.4385 ± 0.0015 with a confidence level of about 
90%. With this value of L' and that of lunar 
parallax adopted in Table 10, the best value of 
m 



the quantity 



\ 



sin ttj. L 

8.798 
0.016592 



is found as 



- 1 



8.7981 
6.4385 



1 = 81.357 



11-14 



The estimate of the uncertainty is obtained by 
differentiating this equation with respect to t 

and L' . It is not necessary to differentiate with 
respect to it*- since this constant is known to a 

much higher precision. 



/0.0015 o.ooiN 

^6. 4385 " STTPB) 



= 82.357 



0.0098 



Thus the best value of the quantity 



'« 



is 81.35 7 



± 0.010 with a confidence level of approximately 
90%. This value was obtained using Baker's data 
and is contrasted to his adopted value of 81. 35 ± 
0. 05. Since the uncertainty of Baker' s value 
seems inconsistent, it is proposed that the value 
and uncertainty developed here be utilized. 

The remaining information required pertains 
to the figure of the moon. The figure of the moon 
is best represented by a triaxial ellipsoid with 
the radii of lengths a, b and c where a is directed 
toward the earth, c is along the axis of rotation 
and b forms an orthogonal set. Very little data 
are available for these lengths. Some informa- 
tion, however, is presented in: 

Alexandrov, I, "The Lunar Gravitational 
Potential" in Advances in the Astronautical 
Sciences, Vol. 5, Plenum Press (N. Y.), 
1960, pages 320 through 324. 

This reference gives data for determinations of 
the dynamic dimensions and the methods of com- 
putation as: 





Forced 


Free 


Adopted by 




Libration 


Libration 


Baker 


Semiaxis a(km) 


1738.67 ± 0.07 


1738.57 ± 0.07 


1738.57 ± 0.07 


Semiaxis b(km) 


1738. 21 ± 0.07 


1738.31 ± 0.07 


1738. 31 ± 0.07 


Semiaxis cflcm) 


1737.58 ± 0.07 


1737.58 ± 0.07 


1737.58 ± 0.07 



There is no reason to assume a value other than 
that of Baker due to the general lack of data. 

6, Summary of Constants and Derivable Data 

Because several values have been discussed 
for each constant, there is need to combine in one 
table the best value, its uncertainty and approxi- 
mate confidence level. This is done in Table 12. 
Note is made of the source of each number given. 

In addition to a tabulation of constants, there 
generally exists a requirement for data which 
are easily derivable from this more basic data. 
Table 13 presents the mass, the gravitational 
constant (ju = Gm) and the radius of action* in 
metric, English and astronomical units. Table 14 

*Tisserand's criteria, r* = d (=^) where d 

is the average distance between the two bodies, 
m is the mass of the smaller body and M is the 
mass of the larger body. 



presents the geometry of the planets in metric 
and English units, and Table 15 presents surface 
values for the circular and escape velocities and 
for gravity. 

B. ASTROPHYSICAL CONSTANTS 

In the previous section certain of the astro- 
nautical constants were reviewed. The purpose 
of this section is to include other factors affecting 
the trajectory. Accordingly, atmospheric models 
and density variability will first be discussed. 
The discussions will then be oriented toward the 
definition of other factors which must be con- 
sidered in satellite orbit selection such as the 
radiation and meteorid environments. 

1. Development of Model Atmospheres for 
Extreme Altitudes 



In November 1953 an unofficial group of 
scientific and engineering organizations, each 
holding national responsibilities related to the 
requirement for accurate tables of the atmosphere 
to high altitudes formed the "Committee on the 
Extension of the Standard Atmosphere" (COESA). 
A Working Group, appointed at the first meeting, 
met frequently between 1953 and the end of 1956. 
This committee developed a model atmosphere 
to 300 km based on the data available at that time. 
This model was published in 1958 as the "U. S. 
Extension to the ICAO Standard Atmosphere, " 
(Ref. 1). 

At the time of the development of this standard 
only two methods of direct measurement of upper 
atmosphere densities were available: 

(1) High altitude sounding rockets. 

(2) Observations of meteor trails. 

Both methods have severe limitations in the 
interpretation of the measured data. First, the 
rocket made only short flights into the upper 
atmosphere and. the density measurements were 
made mostly inside the rocket's flow field, not 
in the undistrubed free stream. Second, meteors 
were visible only in a small range of altitude (85 
to 130 km) and their aerodynamic characteristics 
contained too many unknowns (unsymmetrical 
shapes, loss of momentum by evaporation of 
melting surface layers, etc.). 

The extent of the limitations of the rocket and 
meteor trail data became evident with the launch- 
ing of the first satellites. The orbital periods of 
the first Sputnik indicated that the densities of the 
upper atmosphere were off by approximately an 
order of magnitude. 

The Smithsonian 1957-2 atmosphere (Ref. 2) 
was developed based on the density estimates 
from the decay histories of the Sputnik satellites. 
This standard was soon superseded by the ARDC 
1959 Model Atmosphere (Ref. 3). Up to about 50 
km this atmosphere was the same as the U.S. 
extension to the ICAO Standard Atmosphere. 
Above that altitude some IGY rocket and early 
satellite data were used. Since all these data 
were obtained during the period of maximum 



11-15 



TABLE 12 
Adopted Constants 



Value obtained in this report. 
Gaussian value. 
Ehricke's value. 
Kaula's value. 





Best Value 


Uncertainty 


Approximate 

Confidence Level 

(%) 


Heliocentric Constants 


a 8. 798 sec 


b ±0.001 


90 


Solar parallax 


Astronomical unit 


a 149.53xl0 6 km 


a ±0.03 


90 


K 2 

s 


* 

°0. 2959122083 

3 2 
AU /solar day 


a ±o.oio- 10 


99+ 


Planetocentric Constants 








Mercury 


Solar mass/ mass Mercury 


a 6, 100, 000 


b ±65. 000 


70 


Equatorial radius 


a 2330 km 


b *U 


70 


1/f 


? 


? 


? 


Venus 








Solar mass/mass Venus 


a 407, 000 


b ±1300 


90 


Equatorial radius 


a 6100 km (incl 
atmos) 


b ±12 


70 


1/f 


? 


? 


? 


Earth-Moon 








Solar mass/earth-moon 
mass 


a 328. 450 


b ±25 


81 


Equatorial radius 


-- 


__ 




1/f 


— 


.. 




Mars 








Solar mass/mass Mars 


a 3, 090, 000 


b ±12, 000 


81 


Equatorial radius 


a 3415 km 


b ±12 


88 


1/f 


b 75 


b ±12 


80 


Jupiter 








Solar mass/mass Jupiter 


a 1047.4 


b ±0.1 


81 


Equatorial radius 


a 71, 875 km 


b ±20 


50 


1/f 


a 15.2 


b ±0.1 


50 


Saturn 








Solar mass/mass Saturn 


a 3500 


b ±2.0 


70 


Equatorial radius 


8 60, 500 km 


b ±480 


50 


1/f 


S 10. 2 


± ? 


? 
( continued) 


NOTE: 








Baker's value. 









11-16 



TABLE 12 (continued) 



Uranus 

Solar mass/mass Uranus 

Equatorial radius 

1/f 
Neptune 

Solar mass/mass Neptune 

Equatorial radius 

1/f 
Pluto 
'Solar mass/mass Pluto 

Equatorial radius 

1/f 

Geocentric Constants 

3 2 
y (km /sec ) 



Equatorial radius (km) 
1/f 
Selenocentric Constants 
Lunar distance (km) 

L' 

m e /m <r 

Semiaxis a (km) 
b (km) 
c (km) 



Best Value 



22, 800 
a 24, 850 km 
a 14. 

a 19, 500 
a 25, 000 km 
a 58.5 

a 350, 000 
a 3000 km 



e 398, 601.5 
a 1082.28xl0" 6 
a -2. 30 x 10" 6 
a -2. 12 x 10" 6 
a -0. 20 x 10" 6 
a -1.0xl0 -6 



Uncertainty 



"6378.163 
e 298. 24 



a 



384, 402 km 



6.4385 



81. 357 



1738.57 km 
1 1738. 31 km 



1737.58 km 



±60 
b ±50 

± ? 

b ±200 
b ±2100 

± ? 

b ±27, 000 
b ±500 
? 

e ±9.9 

a ±0.2 x 10" 6 
a ±0.2 x 10" 6 
a ±0. 5 x 10" 6 
a ±0.1 x 10 " 6 
a ±0. 8 x 10" 6 
e ±0.021 
e ±0.01 

a ±l km 
a ±0.0015 
b ±0.01 
a ±0. 17 km 
a ±0. 07 km 
a ±0. 07 km 



Approximate , 
Confidence Level 
(%) 



50 

9 
? 

70 
50 



70 
20 



95 
90 
92 
88 
70 
95 
95 

88 
92 
90 
50 
50 
50 



NOTE: 



Baker's value. 

Value obtained in this report. 
c Gaussian value. 

Ehricke's value. 
e Kaula's value. 



11-17 



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11-20 



solar activity, the resulting model was more 
representative of these conditions than average 
atmospheric properties. An example of the effect 
of solar conditions on upper atmosphere density 
is shown in the following sketches taken from 
Ref. 4. These sketches show the data calculated 
from the orbits of Explorer IX compared to 
earlier satellite data and the 1959 ARDC Model 
Atmosphere. Also shown are the portions of the 
solar sunspot cycle represented by the data. 










— t'ermd -A AHIK' runnel 


- 










/ 

/ 

— He] n>d uf I^jjIdm.t IX / 

1 
1 


\ 




t... 


1 1 




1 

1 
1 

1 

^ / 



a. U.S. Standard Atmosphere--1962 

The U.S. Standard Atmosphere--1962 was 
developed by four Task Groups of the Working 
Group of COESA. (Although U. S. Standard 
Atmosphere- - 1962 is the general terminology, 
the Working Group considers the region above 32 
km as "tentative" and above 90 km as "specu- 
lative. ") The recommendations of Task 
Group I for the region from 20 to 90 km were 
adopted. However, Task Group IV was appointed 
to resolve the discontinuity and inconsistency of 
the models prepared by Task Groups II (70 to 
200 km) and III (200 to 700 km). The reports 
of Task Groups I and IV (Refs. 6 and 7) have been 
used extensively in describing the new atmosphere. 

Suggestions agreed upon by the Working Group 
were that up to 79. 006 geopotential km (80. 000 
geometric km using the ICAO gravity relations) 
geopotential altitude would be the basic height 
measure. Geometric heights would be basic 
above this level. Above 20 km (the top of the 
ICAO Standard), temperature lapse rate is posi- 
tive at 1 deg/km to 32 km. This gives a value of 
228.66 which is in good agreement with measure- 
ments. From 32 to 90 km, the temperature lapse 
would be linear in geopotential height with changes 
(of whole or half degrees Celsius) to occur at 
whole kilometer levels. A 5-km isothermal layer 
(268.66 °K)at 50 km was suggested, and densities 

3 3 

close to 1 g/m and 0.02 g/m at 50 and 80 km 

(geometric), respectively were recommended. 

Re -examination of constants from those used 
previously resulted in new proposed values as 
follows : 





ICAO 


U.S. Kxt 


Proposed 


Units 


Universal gas constant 


8.3143G 


8.31439 


8.31470 


joules /g-deg 


Speed of sound 


331.43 


331.316 


331.317 


in ;' sec at 0° O 


Sutherland's constant 


120.0 


110.4 


110.4 


"K 



The new value of the gas constant decreases 
temperature values by 0.01° (0° C = 273. 15° K) 
and density and pressure values. The differences 
are summarized in Table 16 (from Ref. 6). The 
column labeled "n" is the adopted revision, while 
"H" and "d" refer to earlier revisions. The 
speed of sound at 0° C also changes slightly and 
the new relationship is 

C g = 20.046707 T 1/2 m/sec, T in °K 

The dynamic viscosity, /u, is slightly changed by 
the new value for Sutherland's constant, S, so 
that 



A new COESA Working Group was convened in 
January 1960. Using data and theories from more 
recent satellite and rocket flights, the Working 
Groups prepared a new standard atmosphere that 
was accepted by the entire committee on March 
15, 1962 (Ref. 5). This new U. S. Standard 
Atmosphere depicts a typical mid-latitude year- 
round condition averaged for daylight hours and 
for the range of solar activity that occurs between 
sunspot minimum and maximum. Supplemental 
presentations are being developed to represent 
variability of density above 200 km with solar 
position and a set of supplemental atmospheres 
that will represent mean summer and winter con- 
ditions by 15° latitude intervals to an altitude of 
90 km. 



= 1.458 x 10 



■6 T 3/2 / (T + s) 



In analyzing the temperature and density obser- 
vations an average temperature of 270. 65° K was 
indicated at 50 km, meeting the requirements of 
linear temperature lapse (above 32 km) that fit 
the observed data then placed the isothermal 
region at 47 km. The value of density at 50 km 
fell within the suggested value of the Working 
Group. From 30 to 50 km the new temperature 
■ profile is between the mean annual measured 
temperature for high and low latitudes as indi- 
cated in Fig. 2 (from Ref. 6). Above the iso- 
thermal layer, two temperature lapse regions 
define temperature to the next isothermal layer 



11-21 



TABLE 16 

Comparison of Properties of ICAO, U. S. Extension, ARDC 
1959 Model and U. S. Standard Atmospheres-- 1962 



Height 




Tempe 


rature 




Pi 


assure { 


mb's x K 


") 


Density (g/ 


3 , ft r 
m x 10 


) 


Geopot 
(km) 


U.S. Kxt 
56-58 


ARDC 
59 


"H" 


"N" 


U.S. Kxt 
56-58 


ARDC 

59 


"II" 


"N" n 


U.S. Kxt 
56-58 


ARDC 

59 


"H" 


"N" n 


88. 743 


196. 86 
0.0 


165. 66 
0. 


190.65 
0.0 


180. 65 
0.0 


2. 258 


1.353 


1.8980 


1. 6437 -3 


3. 995 


2. 846 


3. 4682 


3. 1698 -3 


79.006 


196. 86 
0.0 


165.66 
0.0 


190.65 
0.0 


180.65 
0.0 


1.224 


1.008 


1. 0868 


1.0364 -2 


2. 165 


2. 120 


1. 9859 


1.9986 -2 


79. 000 


196. 86 
0.0 


165.66* 
-4. 5 


190.65* 
-3. 2 


180.65* 
-4.0 


1.225 


1.009 


1.0879 


1.0376 -2 


2. 167 


2. 122 


1.9879 


2.0009 -2 


75. 000 


196.86* 
-3.9 


183. 66 
-4.5 


203. 45 
-3. 2 


196.65 
-4.0 


2. 452 


2. 1707 


2. 1771 


2. 1420 -2 


4. 3394 


4. 1176 


3. 7279 


3.7946 -2 


61.000 


251. 46 
-3.9 


246. 66 
-4. 5 


248.25 
-3. 2 


252.65 
-2.0 


2.0934 


2. 0372 


1. 8224 


1. 8209 -1 


2. 9002 


2. 8774 


2. 5574 


2. 5108 -1 


54.000 


278. 76 
-3.9 


278. 16 
-4.5 


270.65* 
0.0 


266.65 
-2.0 


5. 1637 


5. 1630 


4. 5834 


4. 5748 -1 


6. 4534 


6. 4664 


5. 8996 


5.9769 -1 


53.000 


282.66* 


= 282. 66* 
0.0 


270. 65 
0.0 


268.65 
-2.0 


5. 8320 = 


5.8320 


5.2001 


5. 1977 -1 


7. 1881 = 


7. 1881 


6. 6934 


6.7401 -1 


52.000 




282.66 
0.0 


270.65 = 
"D" 


270.65* 
0. 




6.5813 


5. 8997 = 
"D" 


5.8997 -1 




8. 1113 


7. 5939 = 
"D" 


7.5939 -1 


49.610 




282.66 
0.0 


268.66 
0.0 


270.65 
0.0 




8. 7858 


7.9969 


7.9772 -1 




1.0829 


1.0370 


1.0268 +0 


48.000 




282.66 
0.0 


268. 66* 
+ 2.5 


270.65 
0.0 




1.0673 


9. 5880 


9. 7748 -1 




1.3155 


1.2433 


1. 2582 +0 


47. 000 




282. 66* 
+ 3.0 


266. 16 

+ 2. 5 


270.65* 
+ 2. 8 




1. 2044 


1.0895 


1. 1090 +0 




1. 4845 


1. 4261 


1. 4275 +0 


32.000 




237.66 
+ 3.0 


228.66* 
+ 1.0 


228. 65* 
+ 1.0 




8.6776 


8. 6800 


8.6798 +0 




1. 2721 


1. 3225 


1.3225 +1 


25.000 


ICAO 


216. 66* 
0.0 


221.66 
+ 1.0 


221. 65 
+ 1.0 


ICAO 


2. 4886 




2. 5110 +1 


ICAO 


4.0016 




3.946ti +1 


20.000 


216. 66 = 
0.0 


216.66 = 
0.0 


216. 66* 


216. 65* 
0.0 


5. 4749 


5. 4748 


= 5. 4748 


5. 4747 +1 


8. 8035 


8. 8034 = 


8. 8034 


8. 8033 +1 


11.000 


216. 66* 
-6. 5 


216.66* 
-6. 5 




216. 65* 
-6. 5 


2. 2632 = 


2. 2632 




2. 2632 +2 


3. 6392 


3. 6391 




3. 6392 +2 


0.000 


288. 16 


288. 16 




288. 15 


1. 01325 




1. 01325+3 


1. 2250 = 


1. 2250 




1. 2250 +3 



'^Breakpoint in temperature gradient, given in deg/km. 



79 km (geopotential). The upper segment 61 to 79 
(km) is based upon observed densities which have 
been considered more reliable than measured 
temperatures. Adopted temperatures are seen to 
be at least 20° colder than reported temperatures 
near 80 km. The isothermal layer of 180.65° K 
above 79 km provides continuity for density in the 
region above the isothermal layer. The new density 
value at 80 km (geometric) agrees very closely 
with the target value. The properties of this por- 
tion of the new standard atmosphere are shown on 
Table 17 (from Ref. 6). 

The basic obstacle to a consistent, continuous 
standard atmosphere above 90 km was the de- 
velopment of a mean molecular weight (M) profile 
for the atmospheric gases together with a mole- 
cular scale temperature T M profile with linear 

lapse rates which would give the secondary atmos- 
pheric parameters in agreement with theoretical 
and empirical data. 



The boundary conditions applied to the model 
were: 

(1) The density, pressure and temperature 
at 90 km must coincide with those of 
Task Group I, namely: density 3. 1698 

_C O _ O 

x 10 kgm/m , pressure 1.6437 x 10 
millibars, molecular scale temperature 
180. 65° K. 



(2) The density at 200 km should lie within 

-10 3 

the range 3.3 i 0.3 x 10 kgm/m for 

mean solar conditions. 



(3) The model should agree as closely as 

possible with the densities in the altitude 
range 90 to 200 km recommended by 
Task Group II and based on rocket and 
satellite data. 



11-22 



Kilometers 
Geomet Geopot 



90.000 
89. 235 
87. 179 
85. 125 
83.072 
81.020 

79. 994 

78. 969 
76. 920 
74. 872 
72. 825 
70. 779 
68. 735 
66. 692 
64. 651 
62. 611 

61. 591 



52. 429 

50. 396 
48. 365 

47. 350 

46. 335 
44. 307 
42.279 
40. 253 
38. 229 
36. 205 
34. 183 

32. 162 

30. 142 
28. 124 
26. 107 
24.091 
22.076 

20. 063 

18. 051 
16.040 
14.031 
12.023 

11.019 

10.016 
8.010 
6.006 
4.003 
2. 001 
0.000 



88. 743 
88. 000 
86.000 
84.000 
82. 000 
80.000 

79.000 

78.000 
76.000 
74.000 
72.000 
70.000 
68.000 
66.000 
64.000 
62.000 

61. 000 



60 


572 


60 


000 


58 


534 


58 


000 


56 


498 


56 


000 


54 


463 


54 


000 



52.000 

50.000 
48.000 

47.000 

46.000 
44.000 
42.000 
40.000 
38.000 
36.000 
34. 000 

32.000 

30.000 
28.000 
26.000 
24.000 
22.000 

20.000 

18.000 
16.000 
14.000 
12.000 

11.000 

10.000 
8.000 
6.000 
4.000 
2.000 
0.000 



TABLE 17 

Properties, to 90 km, 
of the U. S. Standard Atmosphere- 



■1962 



Temperature 
Grad °K 



0.0 



-4.0 



-2.0 



0.0 
0.0 



-2. 8 



+ 1.0 



0.0 



-6. 5 



180. 65 
180. 65 
180. 65 
180. 65 
180. 65 
180. 6 5 

180. 65 

184. 65 
192. 65 
200. 65 
208. 65 
216. 65 
224. 65 
232. 65 
240. 65 
248. 65 

252. 65 

254. 65 
258. 65 
262. 65 
266. 65 

270. 65 

270. 65 
270. 65 

270. 65 

267. 85 
262. 25 
256. 65 
251.05 
245. 45 
239. 85 
234.25 

228. 65 

226. 65 
224. 65 
222. 65 
220. 65 
218.65 

216. 65 

216. 65 
216. 65 
216. 65 
216. 65 

216. 65 

223. 15 
236. 15 
249. 15 
262. 15 
275. 15 
288. 15 



Pressure 

(mb x 10°) n 



. 6437 
. 8917 
. 7613 
.0307 
. 8836 
. 5883 



1.0376 

1.2512 
1. 7975 
2. 5444 
3. 5530 
4. 8994 
6. 6776 
9.0034 
1.2017 
1. 5889 

1. 8209 

2. 0835 
2 . 7 1 90 
3. 5339 
4. 5749 

5. 8997 

7. 5940 
9.7748 

1. 10 90 

1.2591 



6294 
1203 
7752 
6544 
8430 
4610 



8. 6798 

1. 1718 
1. 5862 
2. 1530 
2. 9304 
3. 9997 

5. 4747 

7. 5045 
1.0287 
1.4101 
1. 9330 

2.2632 



6443 
5601 



4.7183 

6. 1642 

7. 9496 

10. 1325 



Density 
1 






0*\ 



-l-l 



+ 2 



1698 
6480 
3250 
7729 
1346 



1. 6562 



2.0009 



2 


3606 


3 


2504 


4 


4176 


5 


9322 


7 


8782 


1 


0355 


1 


3482 


1 


7396 


2 


2261 



2. 5108 

2. 8503 
3. 6622 
4. 6873 
5. 9769 

7. 5939 



8. 8033 

1.2067 
1. 6541 
2. 2674 
3. 1082 

3. 6392 

4. 1282 
5. 2519 
6. 5973 
8. 1916 
1. 0065 
1. 2250 



Sound Speed 

C-HL . io 2 ) 
V, sec l_ 



-1 



+ 1 



v 

+ 2 
+ 3 
+ 3 



6 944 
6944 
6944 
6944 
6944 
6944 



2. 6944 



2. 7241 
2. 7825 
8396 
8957 
9507 
0047 
0577 
1098 
1611 



3. 1864 



3. 2980 



9. 7747 


- 


1 


3. 2980 


1. 2582 


+ 


3.2980 


1. 4275 






3. 2980 


1. 6376 






3. 2 80 9 


2. 1645 






3.2464 


2. 8780 






3. 2115 


3. 8510 






3. 1763 


5. 1867 






3. 1407 


7.0342 






3. 1047 


9. 6086 


| 




3.0682 


1. 3225 


+ 1 


3.0313 


1 . 80 1 1 






3 . 1 80 


2. 4598 






3.0047 


3. 3687 






2. 9913 


4. 6266 






2. 9778 


6. 3726 






2. 9643 



2. 9507 



9507 
9507 
9507 
9507 



2. 9507 

2. 9946 
3.0806 
3. 1643 
3. 2458 
3. 3253 
3. 4029 



Dyn Vise 



gm 



10* 



3 


1990 


1 


6230 


3 


2240 


1 


6434 


3 


2489 


1 


6636 


3 


2735 


1 


6837 



1. 2163 
1. 2163 
1. 2163 
1. 2163 
1.2163 
1. 2163 

1. 2163 

1. 2399 
1. 2865 
1. 3323 
1. 3773 
1. 4216 
1. 4652 
1. 5082 
1. 5505 
1. 5922 

1. 6128 



1. 7037 

1. 7037 
1. 7037 

1. 7037 

1. 6897 
1. 6616 
1. 6332 
1. 6045 
1. 5756 
1. 5463 
1. 5167 

1.4868 

1.4760 
1.4652 
1. 4544 
1.4435 
1.4326 

1.4216 

1.4216 
1. 4216 
1. 4216 
1.4216 

1.4216 

1. 4571 
1. 5268 
1. 5947 
1. 6611 
1. 7260 
1. 7894 



"Altitude at which temperature gradient experiences discontinuity. 



11-23 



(4) At higher altitudes the density should 
match satellite density data under mean 
solar conditions and agree as closely 
as possible with the density values rec- 
ommended by Task Group III. 

(5) The molecular scale temperature gra- 
dients dT/dz should be linear and 

kept to a maximum of two significant 
figures and, where possible, to one 
significant figure. 

(6) The number of breakpoints or segments 
in the T M <z) function should be kept to 

a minimum, consistent with accurate 
representation of the properties of a 
mean atmosphere. 

(7) The value of T at 150 km should be as 
low as possible, consistent with the ob- 
served density values, to give some 
weight to Blamont's measurement of T 
at this altitude. (These temperature 
measurements are not consistent with 
temperatures deduced from density 
measurements. ) 

(8) The value of dT/dz should approach 
zero above 350 km. 

(9) The value of T above 350 km should lie 
in the range 1500 ± 200° K. 

b. Properties 

The model defined in terms of molecular-scale 
temperature as a function of geometric altitude is 
shown in Fig. 3 (from Ref. 7) together with the 
corresponding defining functions for the ARDC 
1959 model and the current U.S. standard atmos- 
phere (ARDC 1956). In Fig. 4 (from Ref. 1) the 
adopted profile (up to 300 km) is compared with 
profiles deduced from several types of observa- 
tions. 



presented in Table 18 (from Ref. 1). Table 19 
(from Ref. 1) shows the detailed properties of 
this upper part of the new atmosphere. A brief 
outline of the new standard from to 700 km in 
skeleton form is presented in Table 20 (from Ref. 
1). This table is included along with the data of 
Table 19 because of its compact form and be- 
cause of the fact that other data is also presented. 

TABLE 18 

Defining Properties of the Proposed 
Standard Atmosphere 





T» 








z 


, M 


L 






(km) 


C°K) 


(°K/km) 


M 


T 


90 


180.65 


+ 3 


28.966 


180.65 


100 


210.65 


+ 5 


28.88 


210.02 


no 


260.65 


+ 10 


28.56 


257.00 


120 


360.65 


+ 20 


28.07 


349.49 


150 


960.65 


+ 15 


26.92 


892.79 


160 


1110.65 


+ 10 


26.66 


1022.2 


170 


1210.65 


+ 7 


26.40 


1103.4 


190 


1350.65 


+ 5 


25.85 


1205.4 


230 


1550.65 


+ 4 


24.70 


1322.3 


300 


1830.65 


+ 3.3 


22.66 


1432.1 


400 


2160.65 


+ 2. 6 


19.94 


1487.4 


500 


2420.65 


+ 1.7 


17.94 


1499.2 


600 


2590.65 


+ 1.1 


16.84 


1506. 1 


700 


2700.65 




16.17 


1507.6 



The gradients dT M /dz increase steadily from 

0° K/km at 90 km to a maximum value of 20° K/km 
between 120 and 150 km, then steadily decrease to 
5° K/km at 200 km and finally to 1. 1° K/km at 600 
km. Because of the requirement that dT/dz tend 
to zero above 350 km, dT M /dz must be maintained 

at a small positive value determined by the rate of 
decrease of M in the same region. When dT/dz = 



dT M /dz 



T/M (dM/dz) 



where dM/dz is negative 

Figure 5 (from Ref. 1) presents density versus 
geometric altitude for the new standard compared 
with some U. S. and Russian data and the 1959 
ARDC Model Atmosphere. A comparison of the 
pressure versus altitude curves for the new U. S. 
standard atmosphere with other standards is pre- 
sented in Fig. 6 (from Ref. 1). Figure 7 (from 
Ref. 7) is a comparison of the molecular weight 
versus altitude for the different standards. A 
table of the defining properties of the 90- to 700- 
portion of the U.S. Standard Atmosphere 1962 is 



km 



z = geometric altitude 

T M = molecu l ar scale temperature = TM„/M 

T = kinetic temperature 

M = mean molecular weight 

M Q = sea-level value of M 

L = dT M /dz, gradient of molecular scale 
temperature 

2. Density Variability 

a. Measurements 

Variations in density of the upper atmosphere 
affect the orbital lifetime and re-entry of satel- 
lites. For these reasons considerable attention 
has been given recently to evaluation of these 
variations. 

Tidal variations in the atmosphere are at- 
tributed to gravitational variations caused by 
the sum and moon. This tidal energy is supplied 



11-24 















TABLE 19 














Defining Molecular Scale Temperature and Related Properties 
for the U. S. Standard Atmosphere-- 1962 




z 
(km) 


T 
(°K) 


L 

CK/km) 


H 

P 

(km) 


(mb x 10 n ) 


P 

n 


(mm Hg 
x 10 n ) n 


Log 10 p/p o ( 


5 P '°") 


, Log, A p/p 
n 6 10 K 'o 


90 


180.65 


f 


5. 438 


1.6437 - 
1.1448 1 


3 


1. 2329 -3 


-5.7899 


3. 1698 - 


6 -5.5871 


92 


186.65 


1 


5.623 




8. 5869 -4 


-5.9496 


2. 1368 


-5.7584 


94 


192.65 


3.0 


5.807 


8.0674 - 


4 


6.0511 




-6. 0990 


1. 4589 


-5.9241 


96 


198.65 


\ 


5.991 


5.7476 




4.3110 




-6.2462 


1.0080 ' 


-6.0847 . 


98 


204. 65 


6. 176 


4. 1372 




3. 1031 




-6. 3890 


7. 0428 - 


7 -6.2404 


100 


210. 65 


i 


6. 361 


3.0070 




2.2554 




-6. 5276 


4. 9731 


-6. 3915 


102 


220. 65 


1 


6. 667 


2. 2119 




1.6591 




-6. 6610 


3. 4924 


-6. 5450 


104 


230. 65 


5.0 


6.974 


1.6497 




1.2374 




-6.7883 


2. 4918 


-6. 6916 


106 


240. 65 






7. 280 


1.2460 ' 




9. 3456 




-6.9102 


1. 8038 


-6. 8320 


108 


250. 65 






7.588 


9. 5205 - 


5 


7.1410 ' 




-7. 0271 


1.3233 


-6. 9665 


no 


260. 65 


i 




7.895 


7.3527 - 


5 


5. 5150 -5 


-7. 1393 


9. 8277 - 


8 -7.0957 


112 


280. 65 


1 


8. 507 


5. 7609 




4. 3210 




-7. 2452 


7. 1512 


-7. 2338 


114 


300.65 


10. 


9. 117 


4. 5908 




3. 4434 




-7. 3438 


5.3196 


-7. 3623 


116 


320.65 






9.731 


3.7127 




2. 7848 




-7. 4360 


4.0338 


-7. 4824 


118 


340.65 






10.34 


3.0418 




2. 2816 




-7. 5226 


3. 1109 


-7. 5953 


120 


360. 65 


'f 


10.96 


2. 5209 




1. 8909 




-7. 6042 


2. 4352 


-7. 7016 


122 


400. 65 






12. 18 


2. 1204 




1. 5904 




-7. 6793 


1. 8435 


-7. 8224 


124 


440. 65 






13. 41 


1.8133 




1. 3601 




-7. 7472 


1. 4336 


-7.9317 ■ 


126 


480. 65 






14. 63 


1.5721 




1. 1792 




-7.8092 


1.1395 ' -8.0314 


128 


520. 65 






15. 86 


1.3787 




1.0341 




-7. 8663 


9. 2254 - 


9 -8.1232 


130 


560. 65 






17.09 


1.2210 - 


5 


9. 1584 -6 


-7. 9190 


7. 5873 - 


9 -8.2080 


132 


600. 65 






18. 32 


1.0905 ' 


I 


8. 1797 




-7.9681 


6.3252 


-8.2871 


134 


640. 65 






19. 55 


9.8118 - 


6 


7. 3595 




-8.0140 


5. 3357 


-8.3610 


136 


680. 65 


20. 


20. 78 


8.8852 




6. 6645 




-8.0571 


4. 5478 


-8.4303 


138 


720. 65 






22.02 


8.0923 




6.0697 




-8.0977 


3.9121 


-8. 4957 


140 


760. 65 






23. 25 


7. 4079 




5. 5563 




-8. 1360 


3.3929 


-8. 5576 


142 


800.65 






24. 49 


6.8124 




5. 1098 




-8. 1724 


2. 9643 


-8. 6162 


144 


840. 65 






25. 73 


6. 2908 




4.7185 




-8.2070 


2. 6071 


-8. 6720 


146 


880. 65 






26.98 


5. 8310 




4. 3736 




-8. 2400 


2. 3067 


-8.7251 


148 


920. 65 






28. 22 


5. 4233 


1 


4.0678 


i 


-8. 2715 


2.0522 


-8.7759 


150 


960. 65 




i 


29. 46 


5.0599 - 


6 


3.7952 -6 


-8. 3016 


1. 8350 - 


9 -8.8245 


152 


990.65 


i 


30. 39 


4. 7328 




3. 5499 




-8. 3306 


1.6644 


-8. 8669 


154 


1020. 65 


15. 


31.34 


4. 4359 




3.3272 




-8.3587 


1. 5141 


-8.9080 


156 


1050.65 






32. 28 


4.1655 




3. 1244 




-8.3861 


1.3812 


-8.9479 


158 


1080. 65 






33. 22 


3.9187 




2.9393 




-8. 4126 


1.2633 


-8.9866 


160 


1110. 65 




' 


34. 17 


3.6929 




2.7699 




-8. 4384 


1. 1584 


-9.0243 


162 


L130. 65 


10. 


34. 80 


3. 4848 




2.6138 




-8. 4635 


1.0738 


-9.0572 


164 


1150.65 


t 


35. 44 


3.2919 


' 


2. 4691 


' 


-8. 4883 


9. 9669 - 


10 -9.0896 


Z = g 


2ometric a 


Ltitude 














H = g 


sopotential 


altitude 














R 
R 


Z 

+z 












R = 


radi 


us of earth at 


45° 32' 4C 


" = 6356. 766 km 



11-25 



TABLE 19 (continued) 























z 
(km) 


T 
M 

(°K) 


L 

CK/km) 


H 

P 
(km) 


(mb x 10 n ) 


P 

(mm Hg 

n x 10 n ) n 


Log 1Q p/p 


\m / 


n L °SlO p/ Po 


166 


1170.65 


1 


36.08 


3.1128 - 


6 2. 3348 -6 


-8.5126 


9.2637 - 


10 -9.1214 


168 


1190.65 


10.0 


36.72 


2.9464 


f 2.2100 | 


-8.5364 


8.6211 - 


10 -9.1526 


170 


1210.65 






37.36 


2.7915 - 


6 2.0938 -6 


-8.5599 


8.0330 - 


-9.1833 


172 


1224. 65 




j 


37. 81 


2. 6468 


1. 9853 




-8.5830 


7. 5296 


-9.2114 


174 


1236.65 






38. 27 


2.5113 


1. 8836 




-8.6058 


7.0632 


-9.2391 


176 


1252.65 






38.73 


2.3841 


1.7882 




-8.6284 


6.6307 


-9.2666 


178 


1266.65 






39.18 


2.2648 


1.6987 




-8.6507 


6.2292 


-9. 2937 


180 


1280.65 


7.0 


39.64 


2.1527 


1.6147 




-8.6727 


5.8562 


-9. 3205 


182 


1294.65 






40.10 


2.0474 


1.5357 




-8.6945 


5.5094 


-9. 3470 


184 


1308.65 






40. 55 


1.9483 


1.4614 




-8.7161 


5. 1868 


-9. 3732 


186 


1322.65 






41.01 


1. 8551 


1.3914 




-8.7374 


4. 8863 


-9.3992 


188 


1336. 65 






41.47 


1.7673 


1.3256 




-8.7584 


4.6062 


-9.4248 


190 


1350. 65 






41.93 


1.6845 - 


6 1. 2635 -6 


-8. 7793 


4. 3450 -10 -9. 4502 


192 


1360.65 




i 


42. 27 


1.6064 


1. 2049 




-8.7999 


4.1130 


-9.4740 


194 


1370.65 






42.61 


1.5324 


1.1494 




-8.8204 


3.8950 


-9.4976 


196 


1380.65 






42.94 


1.4624 


1.0969 




-8.8407 


3.6901 


-9.5211 


198 


1390. 65 






43. 28 


1. 3961 


1.0472 




-8.8608 


3.4975 


-9.5444 


200 


1400. 65 






43.62 


1. 3333 


1.0001 




-8.8808 


3. 3163 


-9.5675 


202 


1410.65 






43. 96 


1. 2738 


9.5541 




-8. 9006 


3. 1458 


-9.5904 


204 


1420.65 






44.30 


1.2173 


9.1307 




-8.9203 


2. 9852 


-9.6132 


206 


1430.65 






44.63 


1.1638 


8.7291 




-8.9399 


2. 8340 


-9.6358 


208 


1440.65 






44.97 


1.1130 


8.3480 


' 


-8.9592 


2.6915 T -9.6582 


210 


1450.65 


5.0 


45.31 


1.0647 -( 


3 7.9862 -7 


-8.9785 


2.5571 -10 -9.6804 


212 


1460. 65 






45. 65 


1.0189 ' 


7.6427 




-8.9976 


2.4303 


-9.7025 


214 


1470.65 






45. 99 


9.7542 - 


7 7.3163 




-9.0165 


2. 3107 


-9.7244 


216 


1480.65 






46.33 


9. 3407 


7.0061 




-9.0353 


2. 1978 


-9.7462 


218 


1490. 65 






46.68 


8. 9475 


6.7112 




-9.0540 


2.0911 


-9.7678 


220 


1500.65 






47.02 


8.5735 


6.4307 




-9.0726 


1.9904 


-9.7892 


222 


1510.65 






47.36 


8.2177 


6. 1638 




-9.0910 


1.8952 


-9.8105 


224 


1520.65 






47.70 


7.8721 


5.9046 




-9. 1092 


1.8051 


-9.8316 


226 


1530.65 






48.04 


7. 5567 


5.6680 




-9.1274 


1.7200 


-9.8526 


228 


1540.65 






48.39 


7.2497 


5.4377 




-9. 1454 


1.6394 


-9.8735 


230 


1550.65 


i 




48.73 


6.9572 -" 


5.2183 -7 


-9.1633 


1.5631 -1 


3 -9.8942 


232 


1558.65 


; 




49.01 


6.6782 


5.0091 




-9.1811 


1.4927 


-9.9142 


234 


1566.65 






49.29 


6.4119 


4.8093 




-9. 1987 


1.4259 


-9.9341 


236 


1574.65 


4.0 


49.58 


6. 1577 


4.6187 




-9.2163 


1.3624 


-9.9538 


238 


1582.65 






49.86 


5. 9149 


4.4366 




-9.2338 


1. 3020 


-9. 9735 


240 


1590.65 






50.14 


5. 6830 


4.2626 




-9.2511 


1.2447 


-9.9931 


242 


1598. 65 






50.43 


5.4614 ' 


4.0964 




-9. 2684 


1.1902 


-10.0125 



11-26 



TABLE 19 (continued) 













P 




P 




z 


T 
M 


L 


H 
p 




(mm Hg 


Log 10 p/p 


(H . 10 n l 


, 


(km) 


(°K) 


(°K/km) 


(km) 


(mb x 10 n ) 


n x 10 n ) n 


L 3 ) 


n ^10^0 


244 


1606.65 


4.0 


50.71 


5.2496 - 


7 3. 9375 -7 


-9.2856 


1.1383 - 


10 -10.0319 


246 


1614.65 






50. 99 


5.0471 


3.7856 




-9. 3027 


1.0890 


1 -10.0511 


248 


1622.65 






51.27 


4.8535 


3.6404 




-9.3197 


1.0421 


T -10.0703 


250 


1630.65 






51.56 


4.6683 - 


7 3.5015 -7 


-9. 3366 


9.9738 - 


11 -10.0893 


252 


1638. 65 






51.84 


4.4912 


3. 3687 




-9. 3534 


9.5485 


-10.1082 


254 


1646.65 






52.13 


4. 3217 


3.2415 




-9.3701 


9.1434 


-10.1270 


256 


1654.65 






52.41 


4. 1594 


3. 1198 




-9. 3867 


8.7576 


-10. 1458 


258 


1662.65 






52.70 


4.0041 


3.0033 




-9.4032 


8. 3901 


-10.1644 


260 


1670.65 






52. 98 


3. 8554 


2.8918 




-9.4197 


8.0397 


-10.1829 


262 


1678.65 






53.27 


3.7130 


2.7849 




-9.4360 


7. 7058 


-10.2013 


264 


1686. 65 






53.55 


3.5765 


2.6826 




-9.4523 


7.3874 


-10.2197 


266 


1694.65 






53.84 


3.4457 


2. 5845 




-9.4684 


7.0837 


-10.2379 


268 


1702.65 






54.13 


3. 3204 


2.4905 


' 


-9.4845 


6.7940 


-10.2560 


270 


1710.65 






54.41 


3.2003 - 


1 2.4004 -7 


-9.5005 


6.5176 - 


11 -10.2741 


272 


1718. 65 






54. 70 


3.0851 


2.3140 




-9. 5165 


6.2537 


-10. 2920 


274 


1726.65 




l 


54.99 


2. 9746 


2.2311 




-9.5323 


6.0018 


-10. 3099 


276 


1734.65 


4.0 


55.28 


2. 8686 


2.1517 




-9.5480 


5.7613 


-10.3276 


278 


1742.65 






55.57 


2.7670 


2.0754 




-9.5637 


5.5316 


-10. 3453 


280 


1750.65 






55.86 


2.6694 


2.0022 




-9.5793 


5.3122 


-10. 3629 


282 


1758.65 






56.15 


2.5758 


1.9320 




-9.5948 


5.1025 


-10.3804 


284 


1766.65 






56.43 


2.4858 


1.8645 




-9.6103 


4.9021 


-10.3978 


286 


1774. 65 






56.73 


2.3995 


1.7998 




-9.6256 


4.7105 


-10.4151 


288 


1782. 65 






57.01 


2.3166 


1.7376 


i 


-9.6409 


4.5273 


-10.4323 


290 


1790.65 






57. 31 


2. 2369 - r 


1.6778 -7 


-9.6561 


4.3521 - 


1 -10.4494 


292 


1798.65 






57.60 


2.1604 


1.6204 




-9.6712 


4. 1845 


-10.4665 


294 


1806.65 






57.88 


2.0868 


1.5653 




-9.6862 


4.0241 


-10.4835 


296 


1814.65 






58.18 


2.0162 


1.5122 




-9.7012 


3.8707 


-10. 5004 


298 


1822.65 






58.47 


1.9482 


1.4613 




-9.7161 


3.7238 


-10.5172 


300 


1830. 65 


' 




58.76 


1.8828 


1.4122 




-9.7309 


3.5831 


-10.5339 


305 


1847.15 


i 




59.38 


1.7300 


1.2976 




-9.7677 


3.2629 


-10.5745 


310 


1863. 65 






60.00 


1.5910 


1. 1934 




-9.8041 


2.9742 


-10.6148 


315 


1880.15 






60.62 


1.4644 


1.0984 




-9.8401 


2.7135 


-10. 6546 


320 


1896.65 






61.25 


1.3491 ' 


1.0119 




-9.8757 


2.4780 ' 


-10.6940 


325 


1913.15 


3.3 


61.88 


1.2438 -7 


9.3293 -8 


-9.9110 


2.2650 -1 


1 -10.7331 


330 


1929.65 






62.50 


1. 1477 


8.6086 




-9.9459 


2.0721 


-10.7717 


335 


1946.15 






63. 13 


1.0599 1 


7.9499 




-9.9805 


1.8973 


-10.8100 


340 


1962.65 






63.76 


9.7957 -8 


7.3474 




-10.0147 


1.7388 


-10.8479 


345 


1979.15 


1 




64.40 


9.0604 ' 


6.7958 




-10.0486 


1.5949 ' 


-10.8854 



11-27 



TABLE 19 (continued) 













P 




P 




(km) 


T 
M 

<°K) 


L 

(°K/km) 


H 

P 
(km) 


(mm Hg 
(mb x 10 n ) n x 10 ri ) n 


Log 10 p/p l 


'3 ■ 10 1 


n L °gl0 p/ P0 


350 


1995.65 






65.02 


8.3866 - 


8 6.2905 -8 


-10.0821 


1.4641 - 


11 -10.9226 


355 


2012.15 






65.66 


7.7688 


5.8271 




-10.1154 


1.3451 


-10.9594 


360 


2028.65 






66. 30 


7.2018 


5.4018 




-10.1483 


1.2368 


-10.9958 


365 


2045.15 






66.94 


6.6810 


5.0112 




-10.1809 


1.1381 


-11.0320 


370 


2061.65 






67.58 


6.2024 


4.6522 




-10.2132 


1.0481 


-11.0677 


375 


2078.15 


3.3 


68.22 


5.7620 


4.3219 




-10.2400 


9.6595 - 


2 -11.1032 


380 


2094.65 






68.86 


5. 3567 


4.0178 




-10.2768 


8. 9092 


-11.1383 


385 


2111.15 






69.51 


4.9832 


3.7377 




-10.3082 


8.2233 


-11.1731 


390 


2127.65 






70.16 


4.6389 


3.4794 




-10.3393 


7.5957 


-11.2076 


395 


2144. 15 




' 


70.81 


4.3212 


3.2411 


' 


-10.3701 


7.0211 


-11.2417 


400 


2160.65 




» 


71.45 


4.0278 - 


8 3.0211 -8 


-10.4007 


6.4945 -] 


2 -11.2756 


410 


2186.65 






72.53 


3.5055 


2. 6293 




-10.4610 


5.5850 


-11.3411 


420 


2212.65 






73.61 


3.0571 


2.2930 




-10.5214 


4.8134 


-11.4057 


430 


2238.65 






74.69 


2.6714 


2.0037 




-10.5790 


4.1573 


-11.4693 


440 


2264.65 






75.78 


2.2339 


1.7543 




-10.6367 


3.5981 


-11.5321 


450 


2290.65 


2.6 


76.88 


2.0517 


1.5389 




-10.693'6 


3. 1204 


-11.5939 


460 


2316.65 






77.98 


1.8031 


1. 3525 




-10.7497 


2.7116 


-11.6549 


470 


2342.65 






79.09 


1.5875 


1.1908 




-10.8050 


2.3609 


-11.7151 


480 


2368.65 






80.20 


1.4002 


1.0502 




-10. 8595 


2.0595 


-11.7744 


490 


2394.65 




1 


81.32 


1.2371 


9.2792 -9 


-10.9133 


1.7998 


-11. 8329 


500 


2420.65 




, 


82.44 


1.0949 - 


8 8. 2124" 9 


-10.9664 


1.5758 -12 -11.8906 


510 


2437.65 






83.27 


9.7042 - 


9 7.2787 




-11.0188 


1.3869 


-11.9461 


520 


2454.65 






84.09 


8.6110 


6.4588 




-11.0707 


1.2222 


-12.0010 


530 


2471.65 






84.91 


7.6500 


5.7380 




-11.1221 


1.0783 


-12.0554 


540 


2488.65 






85.75 


6.8041 


5.1035 




-11.1730 


9.5250 -13 -12. 1093 


550 


2505.65 


1.7 


86.59 


6.0585 


4.5443 




-11.2234 


8.4238 


-12.1626 


560 


2522. 65 






87.43 


5.4007 


4.0509 




-11.2733 


7.4585 


-12.2155 


570 


2539.65 






88.28 


4.8197 


3.6150 




-11.3227 


6.6115 


-12.2678 


580 


2556.65 






89.12 


4.3058 


3. 2296 




-11.3717 


5.8673 


-12.3197 


590 


2573.65 






89.97 


3.8508 ' 


2.8883 




-11.4202 


5.2127 T -12.3711 


600 


2590.65 


i 




90.83 


3.4475 - 


9 2.5859 -9 


-11.4682 


4.6362 -13 -12.4220 


610 


2601. 65 






91.47 


3.0893 


2.3172 




-11.5159 


4.1369 


-12.4715 


620 


2612.65 






92. 13 


2.7705 


2.0780 




-11.5632 


3.6943 


-12.5206 


630 


2623.65 






92.78 


2.4865 


1.8650 




-11.6101 


3. 3017 


-12.5694 


640 


2634.65 






93.43 


2.2333 


1.6751 




-11.6568 


2.9531 


-12.6179 


650 


2645.65 


1.1 


94.09 


2.0074 


1.5056 




-11.7031 


2.6433 


-12.6660 


660 


2656.65 






94.75 


1.8057 


1.3544 




-11.7491 


2. 3679 


-12.7138 


670 


2667.65 






95.42 


1.6254 


1.2192 




-11.7948 


2. 1227 


-12.7613 


680 


2678.65 






96.09 


1.4642 


1.0983 ' 




-11.8401 


1. 9044 


-12.8084 


690 


2689.65 






96.76 


1.3200 


9. 9007 -10 


-11.8852 


1.7097 ' 


-12. 8552 


700 


2700.65 






97.42 


1.1908 - 


9 8.9317-10 


-11.9299 


1.5361 -1 


3 -12.9017 



11-28 



TABLE 20 
Skeleton of the U.S. Standard Atmosphere- -196 2 



Defining temperature and molecular weights of the proposed U.S. Standard Atmosphere and 
pressures and densities, where z = geometric altitude, h = geopotential altitude, T = kinetic te 
M = mean molecular weight, L = gradient of molecular scale temperature = dT M /dh (below 79 


computed 

mperature, 

geopotential 




km) = dT M 


/dz (above 79 geopotential km), T M = mol 


ecular scale 


temperature 


= (T/M) M Q ; 


anc 


M = sea 




level value of M. 


















z 
(km) 


h 
(km) 


T 
M 

(°K) 


L 

(°K/km) 


M 


T 
(°K) 


P 
(mb x 10 n ) 


n 


P 

K- ion ) 

\m / 


n 


0.000 


0.000 


288. 15 


-6.5 


28.966 


288.15 


10. 1325 


2* 


1. 2250 


3 


11.019 


11.000 


216.65 


0.0 


28. 966 


216.65 


2.2632 


2 


3. 6392 


2 


20.063 


20.000 


216. 65 


1.0 


28. 966 


216.65 


5.4747 


1 


8. 8033 


1 


32. 162 


32.000 


228.65 


2.8 


28. 966 


228.65 


8.6798 





1. 3225 


1 


47.350 


47.000 


270.65 


0.0 


28. 966 


270.65 


1.1090 





1.4275 





52.429 


52.000 


270.65 


-2.0 


28.966 


270.65 


5.8997 - 


1 


7.5939 - 


1 


61.591 


61.000 


252.65 


-4.0 


28.966 


252. 65 


1.8209 - 


1 


2. 5108 - 


1 


79. 994 


79.000 


180.65 


0.0 


28. 966 


180.65 


1.0376 - 


2 


2.0009 - 


2 


90.000 


88. 743 


180.65 


3.0 


28.966 


180.65 


1.6437 - 


3 


3. 1698 - 


3 


100.000 


98.451 


210.65 


5.0 


28. 88 


210.02 


3.0070 - 


4 


4.9731 - 


4 


110.000 


108.129 


260.65 


10.0 


28.56 


257.00 


7.3527 - 


5 


9.8277 - 


5 


120.000 


117.777 


360.65 


20.0 


28.07 


349.49 


2.5209 - 


5 


2.4352 - 


5 


150.000 


146.542 


960.65 


15.0 


26. 92 


892.79 


5.0599 - 


6 


1. 8350 - 


6 


160.000 


156.071 


1, 110.65 


10.0 


26.66 


1,022.20 


3.6929 - 


6 


1.1584 


6 


170.000 


165. 572 


1, 210.65 


7.0 


26.40 


1, 103.40 


2.7915 - 


6 


8.0330 - 


7 


190.000 


184.485 


1, 350.65 


5.0 


25. 85 


1, 205.40 


1.6845 - 


6 


4.3450 - 


7 


230.000 


221. 968 


1, 550.65 


4.0 


24. 70 


1, 322. 30 


6.9572 - 


7 


1.5631 - 


7 


300.000 


286.478 


1, 830.65 


3.3 


22. 66 


1, 432. 10 


1.8828 - 


7 


3.5831 - 


8 


400.000 


376. 315 


2, 160.65 


19. 94 


1, 487.40 


4.0278 - 


8 


6.4945 - 


9 


500.000 


463. 530 


2, 420.65 


2. 6 
1.7 
1. 1 


17. 94 


1, 499.20 


1.0949 - 


8 


1.5758 - 


9 


600.000 


548. 235 


2, 590.65 


16. 84 


1, 506. 10 


3.4475 - 


9 


4.6362 - 


10 


700.000 


630.536 


2, 700.65 


16.17 


1, 507.60 


1*. 1908 - 


9 


1.5361 - 


10 



to the atmosphere in the high density region and 
the diurnal tidal component propagates upward to 
about 105 to 305 km where it is damped. The 
semidiurnal components of the lunar and solar 
tidal variation, because of their shorter period, 
are usually detected between 50 and 80 km. The 
maximum density variation resulting from these 
tidal effects is of the order of 25%. At 96 km, 
Greenhow and Hall (Ref. 8) have found a diurnal 
density variation of about 13% and a semidiurnal 
variation of about 7%. Other causes of density 
variability are solar heating which may be ex- 
pected to vary with local time, latitude, season 
and altitude (as selective portions of the solar 
radiation are absorbed). In addition to gravita- 
tional and thermal causes of fairly regular den- 
sity variability there may be an irregular com- 
ponent analagous to storm systems in the lower 
atmosphere. 



Nicolet (Ref. 9) indicates that atmospheric den- 
sity variations may also be produced by solar 
flares and sunspot activity. Sunspot variation ef- 
fects on density would be expected to vary from 
one year to the next with solar flare activity being 
associated with the sunspot activity. It is presumed 
that these effects would cause density variations 
of the order of 30 to 40% at altitudes of 200 km. 
The effect of the 11 -year sunspot cycle on density 
has been estimated by Johnson (Ref. 10) as shown 
in Fig. 8. The maximum decrease occurs at 
about 1000 km where density is lower by a factor 
of 100. The effect reverses at 1700 km. If these 
estimates are correct, then the solar cycle varia- 
tion may be the largest change in density. 

One of the most useful techniques in determining 
densities has been from changes measured in the 
orbits of satellites having fairly precisely defined 



11-29 



elements. King-Hele and Walker (Ref. 11) have 
determined density from 21 satellites. Figure 9 
shows the density ratio (to sea level density) from 
these determinations. These data confirm that at 
altitudes between 180 and 300 km "the density did 
not depart from the long term average of 1957 - 
1959 by a factor of more than 1. 5" as a result of 
latitudinal, seasonal or day -night effects, although 
it is possible that larger variations might have oc- 
curred over intervals of less than 1 day and not 
have been detected by this technique (which re- 
quires about 10 orbits for a determination). 

A grouping of the data from 180 to 250 km in 
Fig. 9 into those points up to January 1959 and 
after August 1959 would indicate density curves, 
respectively, 10% higher and 10% lower than the 
average shown on Fig. 9. This small decrease 
in density with time is attributed to the decrease 
in solar activity. 

At altitudes between 300 and 700 km, Fig. 9 
shows an increasingly pronounced day-night varia- 
tion. The authors note that this is a solar zenith 
angle effect and should not be attributed to latitude 
or season beyond the fact that solar zenith angle 
is related to latitude and season. 

In evaluating the large apparent day-night ef- 
fect shown, it should be noted that some of the 
variation is due to solar activity as the midday 
data all refer to early 1959 and the midnight values 
to late 195 9 and early 1960. 

Jacchia (Ref. 12) has found from observations 
of satellite motion that a large diurnal variation 
in atmospheric density primarily due to solar heat- 
ing effects occurs at altitudes greater than 325 km 
and decreases at the 200-km level. This bulge oc- 
curs in the general direction of the sun with a 25° 
to 30° lag produced by the earth's rotation. This 
atmospheric bulge represents the bulk of the den- 
sity variations at altitudes above 200 km with 
variations ranging from about 5% of the mean den- 
sity at 200 km to about 25% at 800 km. 

A separation of the day-night, seasonal, ter- 
restrial (latitude) and solar activity effects has 
been indicated by Martin and Priester (Ref. 13) 
using observations of Vanguard I. At 660 km, a 
factor of 10 day -to -night variation in density was 
determined. This is considerably larger than 
Jacchia's value at 800 km. The value of density 
shown in Fig. 10 is a function of the difference in 
right ascension A<* of the sun and satellite perigee 
(and therefore a function of true local time). The 
shift of maximum density at 660 km by 25° from 
local noon is well defined and in agreement with 
Jacchia. 

The seasonal and latitude effects are super- 
imposed and at 660 km and over latitudes and dec- 
linations 0° to 30° they are each about 1/10 of 
the day-night effect. The analysis of Discoverer 
satellite orbits (Ref. 14) has indicated that the 
latitude -seasonal effect was only about 20%. 
Kallmann-Bijl (Ref. 15) in a recent survey has 
indicated that the separation of yearly, latitudinal, 
seasonal and solar cycle effects still remains a 
problem and her belief is borne out by the lack of 
agreement among different estimates of these ef- 
fects. 



Data from Vanguard 2 and Sputnik in addition 
to Vanguard I data were further investigated (Ref. 
16) and yielded the diurnal (plus seasonal) density 
variations shown in Fig. 11. At 210 km the diurnal 
variation of density is about a factor of 2, at 562 
km it is between 5 and 6 and at 660 km it is al- 
most 10 as mentioned earlier. The difference in 
density between the solid and dashed lines is a 
measure of the seasonal effect at each altitude 
since 



A6 



O 



is the difference in declination between the satel- 
lite perigee it and the sun O. The seasonal den- 
sity decrease at an average As of about 40° is 
about 5% at each altitude. (Parkyn (Ref. 17) has 
determined the ratio of polar to equatorial density 
of 0.65 at about 250 km.) Figure 12 (taken from 
Ref. 17) is a model of the diurnal variations of 
atmospheric density. The "wiggle" at 200 km 
was first suggested by Kallmann (Ref. 18) and 
derived more exactly and with better definition 
by Priester and Martin (Ref. 19) using more data. 
The wiggle occurs in the Fl region of the iono- 
sphere and is considered as the beginning of the 
density "solar effect." It is caused by absorption 
of the relatively intense solar helium line at 304A. 
The diurnal variation of density at 200 km is small 
because of the poor heat conduction. The increas- 
ing diurnal effect "fan shape" with altitude results 
from the combination of absorbed solar electro- 
magnetic radiation and increasing heat conductivity 
of the atmosphere. Another density "wiggle" at 
300 to 500 km expected from the absorption of the 
584A solar helium line is apparently smoothed 
out by the large heat conductivity. 

The flux of solar radiations (short ultraviolet 
as well as perhaps X-rays and particles) which 
cause the diurnal density variation are themselves 
variables. Therefore a "solar activity effect" upon 
density (above 200 km) also occurs. The best in- 
dex of this effect is the intensity of radiation (in 
the 3 - to 30 -cm wavelength) from the sun which is 
emitted from the same solar regions (coronal 
condensations and flares) as the much more highly 
ionizing radiations which modulate atmosphere 
density. 

The relation between density and 20 -cm solar 
radio waves has been found to be approximately 
linear over the range of- values of solar flux be- 
tween 100 and 240 x 10~ 22 w/m 2 -cps. If 170 x 10 
is used as a standard flux, the density variation 
due to solar activity is about ±41%. This is over 
and above the diurnal variation. It is known that 
some of the ionizing solar radiations have their 
largest variations in intensity over relatively 
short intervals of minutes during solar flares. 
Short transients in density that result from the 
absorption of these radiations are not distinguish- 
able using the relatively long technique of varia- 
tions in satellite acceleration. On the other hand, 
some of the sources of increased ionizing radia- 
tion are relatively long-lived, as a 27 -day periodicity 
of density has been detected. This corresponds to 
the rotational period of the sun. 

An estimate of density at 1518 km has been 
made from the orbit of the Echo satellite (Ref. 20). 



11-30 



The variation in orbital period corresponded to a 

- 1 8 3 

mean density of 1. 1 x 10 gm/cm . However, 

at this altitude, density variations of 2 orders of 

magnitude are indicated, so the value of the mean 

is very limited. 

At lower altitudes, Quiroz (Ret'. 21) has con- 
structed a model of the seasonal variation of mean 
density as shown in Fig. 13. This author notes 
that the variations indicated on this figure join 
quite well with the factor of 1.5 at 220 km from 
Ref. 11. At altitudes up to 30 km there is con- 
siderably more data available. In Refs. 22 and 
23, summaries have been prepared and are avail- 
able for a number of specific stations and by lati- 
tude and season. 



behind the sun by approximately 25° in 
Jacchia's atmosphere.) 

3 
p = atmospheric density in slugs /ft 

(1 slug/ft 3 = 515. 2 kg/m 3 ) 

Priester's Vari ab le Model. Priester's model 
is similar to Jacchia's, since both are based on 
the correlation with the 20 -cm solar flux and the 
angle between perigee and the sun. In Priester's 
model, the atmospheric density is directly pro- 
portional to F 2Q , the 20-cm solar flux, and the 

peak of the diurnal bulge lags 1 hr (15°) behind 
the sun. 



b. Variable models from satellite orbits 
(Ref. 24) 

Jacchia (Ref. 12) and Priester (Ref. 25) both 
devised variable models of the upper atmosphere 
based on the observed correlation with the deci- 
meter solar flux and the angle between perigee 
and the sun. An annual variation in atmospheric 
density was then discovered by Paetzold (Ref. 
26) who constructed a variable atmospheric model 
based on all three effects. A C D of 2 should be 

used with these variable atmospheric models. 
(Paetzold has recently reported that he now uses 
C = 2.2.) In all the models .mentioned above the 

density is calculated as if all the drag were caused 
by neutral particles. At the higher altitudes charge 
drag may be important, but the gross effects of 
the interaction would be the same in any case for 
satellites with conducting skins. 

The model atmospheres based on satellite ob- 
servations are constructed mostly from accelera- 
tion data smoothed over 2 -day intervals. There- 
fore, these models can give no information about 
shorter term fluctuations. Little is known about 
short term fluctuations in the upper atmosphere. 

Jacchia's Variable Model . According to Jacchia, 
the density of the upper atmosphere is given by 
the following formula. 



P (h) F 20 



1.9 



/ 1 + 0. 19 e 



exp (0.01887h) 



cos 4//2 



PQ 



(h), which is the density when + = 180° and 



F 2Q = 1, is given by 



lot 



P (h) 



-15.733 - 0.006, 808, 3h 



+ 6.363 exp (-0. 008, 917h). 
The quantities appearing in these formulas are 

h = height in km (185-h<750) 

-22 
F„ n = 20-cm solar flux in units of 100 x 10 

w/ m - cps 

•\j = the angle between the satellite and the 
peak ot the diurnal bulge of the atmos- 
phere. (The bulge is assumed to lag 



P aetzold 1 s Variable Model . Paetzold' s at- 
mosphere is one of the more recent modes (July 
1961). It also covers the greatest range of al- 
titudes (150 to 1600 km), and uses the most depend- 
able and readily available solar flux data (the 10- 
cm measurements made by Arthur Covington at 
the National Research Council, Ottawa, Canada). 
Since Paetzold' s atmosphere includes more ef- 
fects, it is more complicated than Jacchia 1 s or 
Priester 1 s. 

In Paetzold' s model, the density of the upper 
atmosphere, p(h) is described by 



220 



log p(h) 



log p g (h) 



i 220 (h) 



10 



TIT 



a(h) g(a) - 0(h) f(6) 



where p (h) is the standard density function given 

in Table 21 . It represents the density in slugs/ 

ft (1 slug/ft = 515.2 kg/m )at the maximum of 

the diurnal bulge (local time, 9 = 14.00 hr), when 

the 10-cm solar flux, F 1fl is 220 (in units of 

?? 2 

10 w/m -cps), and when the annual variation 

is at its peak. The function i 22Q (h) represents 

the effect of solar ultraviolet emission, which is 
correlated with the 10-cm solar flux and with 
sunspots. The effect of the diurnal bulge is 
represented by 6(h), where 



0(h) 



e s (h) 



220 



.a x e(h) 



A 2 6(h) 



. i 220 (h) 



10 



/220 



T2TJ 
i 220 (h) + a(h > 



10 



+ a(h) g(a) 



2TT 



All three functions, 6„(h), 



given in Table 21. 

A 9(h) and A 9(h) are small 



A 9(h) and A 2 9(h) are 

Below 650 km, the corrections 
The function f(9) 



appears in Table 22. The annual variation in 
density is represented by the product g(a) a(h), in 
which g(a) is a function of the month of the year, 
and a(h) is a function of the height. 



11-31 





The Standard Functio 


TABLE 


21 


d Its Variations 






ns for the Air Density an 








( 1 naut mi = 


1. 852 km; 1 


slug /ft 3 = 


515.2 !E£) 






h 
(naut mi) 


P s lh) 
(slugs /ft 3 ) 


log P s (h) 


e s (h) 


a 220 (h) 


i 220 (h) 


A j 0(h) 


A 2 6(h) 


80 


7. 220 x 10" 12 


-11. 122 


-0.009 


0. 031 


0. 041 


0. 000 


0. 000 


85 


3. 845 




0.443 


-0.014 


0. 036 


0. 064 








90 


2. 098 




0. 694 


-0. 018 


0.041 


0. 091 








95 


1.347 | 




0. 879 


-0.023 


0. 047 


0. 121 








100 


9. 787 x 10" 13 


-12. 0133 


-0.017 


0. 053 


0. 156 








110 


7. 206 




0. 1438 


+0. 032 


0. 066 


0. 246 








120 


5. 135 




0. 2913 


0.070 


0. 079 


0. 325 








130 


3. 296 




0.4832 


0.049 


0.093 


0. 356 








140 


2. 060 




0. 6868 


0.054 


0. 108 


0. 373 








150 


1.423 




0. 8477 


0. 094 


0. 122 


0. 387 








160 


1.060 1 




0. 9756 


0. 133 


0. 137 


0. 398 








170 


8.046 x 10" 14 


-13. 0957 


0. 170 


0. 152 


0.409 








180 


6. 087 




0. 2167 


0. 207 


0. 168 


0.420 








190 


4. 612 




0. 3369 


0. 242 


0. 185 


0.431 


0. 001 





200 


3. 507 




0.4553 


0. 276 


0. 203 


0.442 


0. 001 





210 


2. 712 




0. 5671 


0. 314 


0. 221 


0.454 


0. 002 





220 


2. 151 




0. 6705 


0. 344 


0. 240 


0. 465 


0. 002 





230 


1. 714 




0. 7684 


0. 375 


0. 259 


0.476 


0. 003 





240 


1. 385 




0. 8604 


0.425 


0. 278 


0.487 


0. 004 





250 


1.130 1 




0. 9479 


0.462 


0.295 


0.498 


0. 005 





260 


9. 326 x 10" 15 


-14.0316 


0.499 


0. 312 


0. 509 


0. 007 





270 


7. 901 




0. 1107 


0. 536 


0. 327 


0. 520 


0. 009 





280 


6.474 




0. 1898 


0. 573 


0. 342 


0. 531 


0. 010 





290 


5.443 




0. 2650 


0. 605 


0. 356 


0. 542 


0. 012 





300 


4. 608 




0. 3376 


0. 642 


0. 370 


0. 554 


0. 014 





310 


3. 921 




0.4080 


0. 679 


0. 384 


0. 565 


0. 016 





320 


3. 352 




0. 4762 


0. 716 


0. 397 


0. 576 


0. 020 





330 


2. 873 




0. 54 30 


0. 753 


0. 410 


0. 587 


0. 023 





340 


2.473 




0. 6082 


0. 790 


0.422 


0. 598 


0. 028 





350 


2. 196 




0. 6717 


0.827 


0.433 


0. 609 


0. 033 





360 


1. 938 




0. 7340 


0.863 


0.444 


0. 620 


0. 038 





370 


1. 606 




0. 7953 


0. 895 


0.455 


0. 631 


0. 044 





380 


1. 397 




0. 8557 


0. 927 


0.467 


0. 643 


0. 049 





390 


1. 217 




0. 9153 


0. 960 


0.478 


0. 654 


0. 055 





400 


1. 063 




0. 9739 


0.992 


0. 991 


0. 665 


0. 061 





410 


9. 300 x 10" 16 


-15. 0316 


1. 025 


0.498 


0. 676 


0. 068 





420 


8. 161 




0. 0886 


1. 053 


0. 508 


0. 687 


0. 074 





430 


7. 174 




0. 1448 


1. 080 


0. 518 


0. 698 


0. 081 





440 


6. 316 




0. 2003 


1. 108 


0. 528 


0. 709 


0. 087 





450 


5. 564 




0. 2555 


1. 135 


0. 537 


0. 720 , 


0. 094 





460 


4. 905 




0. 3103 


1. 162 


0. 546 


0. 732 


0. 101 





470 


4. 333 




0. 3642 


1. 188 


0. 556 


0. 743 


0. 108 





480 


3. 834 


1 


0.4174 


1. 213 


0. 565 


0. 754 


0. 116 






11-32 



[ 1 nau 



TABLE 21 (continued) 

t mi = 1. 85 2 km ; 1 slug/ft 3 = 515. 2 M^ 

m / 



h 
(naut mi) 


P s (h) 
(slugs/ft 3 ) 


log p s (h) 


e s (h) 


a 220 (h> 


i 220 (h) 


AjOfh) 


A 2 e(h) 


490 


3. 395 




0.4701 


1. 239 


0. 574 


0. 765 


0. 123 





500 


3. 009 




0. 5223 


1. 264 


0. 583 


0. 776 


0. 131 





520 


2. 371 




0. 6256 


1. 310 


0. 602 


0. 798 


0. 145 


-0. 002 


540 


1. 875 




0. 7274 


1. 353 


0. 620 


0. 819 


0. 160 


-0. 007 


560 


1. 500 




0. 8278 


1. 396 


0. 637 


0. 836 


0. 175 


-0. 016 


580 


1. 195 \ 




0. 9276 


1.435 


0. 654 


0. 852 


0. 190 


-0. 024 


600 


9.477 x 10" 17 


-16. 0268 


1.471 


0. 671 


0. 868 


0. 206 


-0. 032 


620 


7. 499 




0. 1254 


1. 504 


0. 689 


0. 885 


0. 223 


-0. 038 


640 


6. 049 




0. 2225 


1. 536 


0. 706 


0.901 


0. 239 


-0. 038 


660 


4. 854 




0. 3186 


1. 565 


0. 726 


0. 917 


0. 255 


-0. 033 


680 


3. 882 




0. 4137 


1. 590 


0. 74 5 


0. 932 


0. 271 


-0. 024 


700 


3. 116 




0. 5075 


1. 611 


0. 754 


0.947 


0. 287 


-0. Oil 


720 


2. 538 




0. 5995 


1. 630 


0. 768 


0.961 


0. 302 


+0. 006 


740 


2. 059 




0. 6905 


1. 647 


0. 781 


0. 975 


0. 316 


0. 029 


760 


1. 666 




0. 7805 


1. 663 


0. 793 


0. 988 


0. 328 


0. 053 


780 


1. 356 




0. 8691 


1. 676 


0. 804 


1. 000 


0. 339 


0. 077 


800 


1. 115 




0. 9566 


1. 692 


0. 815 


1. 012 


0. 346 


0. 096 


825 


8. 692 x 10" 18 


-17. 0649 


1. 708 


0. 829 


1.028 


0. 354 


0. 114 


850 


6.786 \ 


0. 1721 


1. 720 


0. 843 


1. 043 


0. 360 


0. 126 



TABLE 22 
The Phase -Functions, f(9) and g(a) 



f< 


0) 


g(a) 


o h o 


0. 870 


12. Mon. 0. 120 


1. 


0. 94 5 


1. 0. 320 


2. 


0. 980 


2.0 0.265 


3. 


0. 995 


3. 0. 180 


4. 


1. 000 


4. 0. 170 


5. 


0. 975 


5.0 0. 300 


6. 


0. 850 


6.0 0.640 


7. 


0. 655 


7.0 0.980 


8. 


0.490 


8.0 0.900 


9. 


0. 295 


9.0 0.475 


10. 


0. 130 


10.0 485 


11.0 


0. 055 


11.0 0,025 


12. 


0. 030 




13. 


0. 010 


1.0 ... means the 
beginning of the 


14. 


0. 000 


first month, etc. 


15. 


0. 010 




16. 


0. 045 




17. 


0, 120 




18. 


0. 210 




19.0 


0. 300 




20. 


0. 400 




21.0 


0. 505 




22.0 


0.615 




23.0 


0.740 





The relative amplitude of the annual variation 
decreases toward a sunspot minimum. The prod- 
uct [g(a) a(h)] is represented by the equation 

g(a) a(h) = a 22Q (h){ g(a) + (220 - F) [0. 0043 

- g(a) 0. 0028]} + . . . 

The quantity g(a) appears in Table 22, while 
a„ 20 (h) is given in Table 21. 

Five special examples have been calculated 
in Tables 23 through 27 in order to demonstrate 
the effect of the different influences. The scale 
height H, mean molecular weight W, and temper- 
ature T, are given, in addition to the density p. 



11-33 



TABLE 23 







Standard Model 








log p (h) = log p s (h) 






This example contains th 


e greatest values of density and temperature which 


will occur in an 


average sunspot cycle. 










p(h) 








h 
(naut mi) 
(1 naut mi = 1. 852 km) 


(slugs /ft 3 ) 

/i s iHg =515.2 h^) 
\ ft 3 mV 


H(h) 
(naut mi) 
(1 naut mi = 1. 852 km) 


M(h) 


T(h) 
(°K) 


80 


7. 220 x 10" 12 


10. 1 


28.0 


589 


85 


3.845 




15.6 


27. 8 


899 


90 


2. 098 




21.0 


27. 7 


1192 


95 


1.347 1 




25. 7 


27.5 


1455 


100 


9. 787 x 10" 13 


28.5 


27. 3 


1603 


110 


7. 206 




27.9 


26. 9 


1541 


120 


5. 135 




27. 3 


26.4 


1469 


130 


3. 296 




29. 3 


25. 9 


1544 


140 


2. 060 




34. 2 


25. 3 


1734 


150 


1.423 




36. 7 


24.8 


1821 


160 


1.060 \ 




39.4 


24. 1 


1888 


180 


6.087 x 10" 14 


43. 7 


23. 


1987 


200 


3. 507 




49. 2 


21. 7 


206 7 


220 


2. 151 




54.2 


20.4 


2118 


240 


1.385 1 




57. 8 


19. 2 


2111 


260 


9. 326 x 10" 15 


61.4 


18. 2 


2110 


280 


6.474 




65. 1 


17.5 


2118 


300 


4. 608 




68.9 


16.8 


2130 


350 


2. 196 




73.4 


16. 1 


2125 


400 


1.063 < 




73. 1 


15. 8 


2116 


450 


5. 564 x 10" 16 


78.6 


15.7 


2107 


500 


3. 009 




81. 3 


15.6 


2105 


550 


1.650 


' 


84. 3 


15.5 


2118 


600 


9.477 x 10~ 17 


88.0 


15. 3 


2112 


650 


5.450 




93. 1 


14.9 


2130 


700 


3. 116 




99.6 


14. 2 


2130 


750 


1. 863 




108. 5 


13.4 


2112 


800 


1.115 1 




119. 3 


12. 5 


2118 


850 


6. 786 x 10" 18 


133.6 


11. 5 


2128 



11-34 



TABLE 24 



Solar Flux Effect 





log p(h) = log p 


a (h) - i 220 (h) 






This example represents 


the mean amplitude at a sunspot minimum, while the diurnal bulge and 


annual variation have their maximum values. 










p(h) 








h 

(naut mi) 

(1 naut mi = 1. 852 km) 


(slugs /ft 3 ) 

/IliM =5 15.2 *if) 
V ft 3 mV 


H (h) 

fnaut mi) 

(1 naut mi = 1. 852 km) 


M?h) 


T(h) 
CK) 


80 


6. 525 x 10" 12 


9. 7 


28.0 


569 


85 


3. 353 




14. 1 


27. 8 


784 


90 


1. 720 




18.9 


27. 7 


1066 


95 


1.028 t 


23. 3 


27. 5 


1344 


100 


6. 878 x 10" 13 


24. 5 


27. 3 


1468 


110 


4. 179 




25.0 


26. 9 


1383 


120 


2.449 




23. 8 


26.4 


1280 


130 


1.459 t 


25.8 


25. 9 


1357 


140 


8. 752 x 10" 14 


29. 


25.4 


1496 


150 


5. 905 




31. 5 


24. 8 


1554 


160 


4. 276 




33.4 


24. 


1593 


180 


2.498 




36.4 


22. 8 


1634 


200 


1. 372 


' 


40. 2 


21. 5 


1667 


220 


7. 542 x 10" 15 


44.4 


20. 1 


1693 


240 


4. 620 




47. 6 


18.9 


1708 


260 


3. 019 




50. 4 


17. 9 


1704 


280 


1. 972 




53. 2 


17. 1 


1700 


300 


1. 297 


1 


55. 9 


16.4 


1701 


350 


5. 685 x 10" 16 


59. 6 


16. 


1710 


400 


2. 513 




61.9 


15. 8 


1710 


450 


1.135 1 


' 


64.0 


15.6 


1707 


500 


5. 847 x 10" 17 


66.8 


15. 3 


1700 


550 


4. 185 




70.6 


14. 9 


1702 


600 


1.303 1 


' 


75. 8 


14.4 


1709 


650 


6. 764 x 10~ 18 


82. 5 


13.4 


1700 


700 


3. 544 




92. 


12. 2 


1700 


750 


1.963 




107. 3 


10.8 


1691 


800 


1.110 


' 


131.3 


9. 1 


1698 


850 


6. 343 x 10" 19 


169. 7 


7.3 


1708 



11-35 







TABLE 


25 








Day-Night Effect ("Diurnal Bulge") 








log p(h) = log p s (h) - e s (h) 






From this function the day-night variation can be seen. It represents the minimum of the diurnal 
variation, while the other influences retain their maximum values. 




p(h) 








h 

fnaut mi) 

(1 naut mi = 1. 852 km) 


(slugs /ft 3 ) 

(l*!M = 515.2 *%) 
\ ft 3 raV 


H(h) 

(naut mi) 
(1 naut mi = 1. 852 km) 


M(h) 


T(h) 
(°K) 


80 


7. 373 x 10" 12 


9. 7 


28.0 


562 


85 


3. 962 




14.4 


27.8 


838 


90 


2. 186 




18.4 


27. 7 


1054 


95 


1.419 




21. 2 


27.5 


1199 


100 


1.021 




23. 1 


27. 3 


1298 


110 


6. 788 x 10~ 13 


23.4 


26.9 


1280 


120 


4. 399 




22. 9 


26.4 


1241 


130 


2.945 




24.0 


25.9 


1250 


140 


1.822 




25. 1 


25.4 


1260 


150 


1.163 




26. 3 


24. 7 


1278 


160 


7.908 x 10" 14 


27. 6 


23. 9 


1288 


180 


4.485 1 


29. 6 


22. 7 


1303 


200 


2.279 \ 


31. 9 


21. 3 


1314 


220 


9.931 x 10" 15 


34.5 


19.9 


1318 


240 


5.413 




36. 7 


18. 7 


1311 


260 


3. 174 




38.9 


17.5 


1316 


280 


1.835 




41. 1 


16.8 


1316 


300 
350 


1.070 
3.854 x 1 


o- 16 


43. 1 
45. 5 


16.4 
15.9 


1312 
1330 


400 
450 


1.254 ♦ 
4.524 x 10" 17 


47.8 
50.0 


15.6 
15. 3 


1322 
1310 


500 
550 


1.773 t 
7.429 x 10" 18 


52.9 
58. 1 


14.9 
14. 


1310 
1312 


600 


3.274 1 


68.3 


12. 3 


1321 


650 
700 


1.523 I 

-19 
7.681 x 10 


83.5 
101.9 


10. 5 
9.0 


1332 
1369 


750 


4. 166 




131. 7 


7.2 


1370 


800 


2. 318 




179. 5 


5. 3 


1353 


850 


1. 333 


| 277.8 


3.6 


1327 



11-36 



TABLE 26 
Annual Effect 



log p(h) = log p g (h) - a(h) 

This example gives the density at the annual minimum, while the remaining influences are at 
their maximum 



h 

(haut mi) 

(1 naut mi = 1. 852 km) 



p[H) 

(slugs /ft 3 ) 



A slug 

\ ft 3 



515.2 



m 



r) 



80 
85 
90 
100 
100 
110 
120 
130 
140 
150 
160 
180 
200 
220 
240 
260 
280 
300 
350 
400 
450 
500 
550 
600 
650 
700 
750 
800 
850 



6. 702 x 10 

3. 548 
1.912 

1. 211 
8. 678 x 10 

6. 224 

4. 328 

2. 671 

1. 614 
1.085 
7.797 x 10 
4.482 

2. 397 
1.270 

7. 523 x 10 

4. 791 

3. 059 
1. 988 
8.818 x 10 
3. 777 

1. 725 

8. 257 x 10 
4.064 

2. 049 
1.045 

5. 524 x 10 

3. 073 
1. 747 
1. 004 



•12 



■13 



-14 



-15 



16 



17 



-18 



H(h) 

(naut mi) 

( 1 naut mi = 1. 852 km) 



7.9 
11. 6 
15.0 
18. 1 
20.4 
22.0 
22. 7 
25. 
29.4 
31. 8 
34.8 
37.9 
41. 3 
45.3 
48.9 
51.9 
55.0 
58.0 
60. 7 
62. 6 
65. 3 
68.4 
72.0 
76. 3 
82.4 
91.4 
106. 3 
128.4 
162. 8 



TVRh) 



T(h) 
(°K) 



28. 


469 


27.8 


668 


27. 7 


850 


27.5 


1002 


27. 3 


1119 


26. 9 


1208 


26. 4 


1212 


25. 9 


1312 


25.4 


1553 


24.8 


1623 


24.0 


1663 


22.8 


1697 


21.5 


1727 


20. 1 


1752 


18.9 


1759 


17. 9 


1754 


17. 1 


1754 


16.4 


1759 


16.0 


1755 


15. 8 


1760 


15. 6 


1757 


15.4 


1750 


15.0 


1748 


14. 5 


1741 


13. 8 


1750 


12. 6 


1740 


11.2 


1740 


9. 5 


1748 


7. 6 


1750 



11-37 



TABLE 2 7 
Total Variation 



log p(h) = log p s (h) - i 220 (h) - 6(h) - a(h) 

This is the lower limit which will be possible in an average sunspot cycle. 



h 

(naut m^ 

(1 naut mi = 1. 852 km) 



80 

85 

90 

95 

100 

110 

120 

130 

140 

150 

160 

180 

200 

220 

2 40 

260 

280 

300 

350 

400 

450 

500 

550 

600 

650 

700 

750 

800 

850 



\ ft 



p(h> 

(slugs /ft 3 ) 

1M= 515.2 



^r) 



6. 213 x 
3. 14G 
1. 616 
9. 738 x 
6. 365 
3. 396 
1. 7 ( 48 
1. 050 
6. 026 x 

3. 618 
2.318 

1. 141 

4. 851 x 

2. 000 
9. 621 x 

5. 048 
2.575 
1. 329 

4. 036 x 

1. 066 

3. 213 x 
1.035 
3. 768 x 
1.417 
7. 403 x 

2. 908 
1. 698 
9. 625 x 

5. 405 



10 



10 



-12 



-13 



H(h) 

(naut mi) 
(1 naut mi = 1. 852 km) 



10 



-14 



10 



-15 



10 



-16 



10 



10 



10 



10 



10 



-17 



■18 



-19 



-20 



-21 



7. 5 
10. 3 
12. 9 
14. 8 
16. 5 
18. 5 
18.8 
20.5 
21. 6 
22.0 

23. 3 

24. 5 
26. 6 
29. 4 
31. 5 

33. 
34.0 

34. 7 
37. 3 
39. 1 
41. 7 
46. 3 
54. 5 
72.8 

111. 
160. 4 
254. 1 
429. 4 
659. 1 



M(h) 



28. 
27.8 
27. 7 
27. 5 
27. 3 
26. 9 
26. 4 
25. 9 
25. 4 
24. 7 
23. 8 
22. 4 
20. 9 
19. 3 
17.8 
17. 1 
16. 6 
16. 2 
16. 
15. 8 
15. 3 
14. 4 
12. 7 
9. 8 
6. 6 
4. 5 
3. 96 
1.85 
1. 24 



T(h) 



42 9 

605 

739 

841 

928 

1026 

1017 

1071 

1099 

1091 

1098 

1087 

1088 

1098 

1091 

1084 

1080 

1080 

1085 

1094 

1107 

1117 

1108 

1102 

1118 

1071 

1079 

1080 

1115 



P s (h) 
"pThT 



1. 155 
1. 219 
1. 30 
1. 40 

1. 56 

2. 20 

2. 96 

3. 15 

3. 43 

4. 01 
4. 66 
6. 32 
8. 53 
11. 42 
15. 38 
20. 86 
27. 60 
35. 86 
54. 4 
99. 9 
173 
291 
489 
668 
736 
1071 
1096 
1162 
1252 



11-38 



4. Radiation 

a. Solar flare radiations 

One of the most extensive manifestations of 
solar activity is the chromospheric flare. Flares 
are ranked according to their area on the solar 
disk and their brightness (in the red line of Ha, 

6563 A) as indicated in Table 28 (from Ref. 27). 
The frequency of flares of different importance 
(or class) is shown in Table 29. 



TABLE 2£ 



Flare Characteristics 









Area 
Limits 


Ha Line 




Duration 


(min) 


10" 6 
Visible 


Width at 
Maximum 

o 


Class 


Average 


Range 


Disk 


(A) 


1- 


-- 


-- 


100 


1. 5 


1 


20 


4 to 43 


100 to 250 


3. 


2 


30 


10 to 
90 


250 to 600 


4. 5 


3 


60 


20 to 
155 


600 to 1200 


8 


3+ 


180 


50 to 
430 


1200 


18 



TABLE 29 





Flare Frequency 






Absolute 




Relative 


Frequency 


Class 


Frequency 


(R) 


1 


0. 72 


0. 044 


2 


0. 25 


0. 015 


3 


0. 03 


0. 002 



The number of flares per year varies with the 
cycle of sunspots and is defined by the Wolfe sun- 
spot number R, which is 

R = k (lOg + f) 

where f is the number of individual spots, g is the 
number of spot groups and k is an instrument and 
observer's correction factor. The mean sunspot 
period is 11. 07 yr with mean maximum and mini- 
mum Wolfe numbers of 103 and 5. 2, respectively 
(Ref. 28). The average time from sunspot maxi- 
mum to minimum is 6. 5 yr and the time from 
minimum to maximum is 4. 5 yr. The last sunspot 
maximum occurred in 1958 with a record number 
of 185. Thus, the next maximum will occur prob- 
ably in 1969. However, since there is a periodicity 
to sunspot cycle maximum which is not very well 
defined, it may be that the next maximum will be 
the end of the present period (with the 1969 peak 
exceeding the 1958 peak) or the beginning of the 



next period (with a sunspot number possibly as low 
as 50 during 1969). During 1958 more than 3100 
flares of Class lor greater occurred, while the 
number of flares during the last sunspot minimum 
in 1954 was only 16; none larger than Class 1 were 
reported (Ref. 29). Solar flares may have electron 

Q 

temperatures as high as 2 x 10 °K (Ref. 30) as 
compared to effective temperatures in the umbra 
and perumbra of sunspots of 4300° K and 5500° K, 
respectively. Prior to the IGY, high energy par- 
ticles from solar flares had been detected by 
ground-based measurements. Four such events 
were noted in the 15 yr preceding 1953. Three 
more of these events have occurred since that 
time, namely 23 February 1956, 4 May and 11 
November 1960. During the IGY and IGC-59 (July 
1957 to December 1959) 25 additional solar flare 
particle events were detected. These particles 
were detected by balloons and satellites but were 
not energetic enough to produce secondaries de- 
tectable at ground level. During this period 707 
Class 2 or larger solar flares occurred (of which 
71 were Class 3 or 3 + ). Therefore, although solar 
flares of Class 2 or greater have occurred on the 
average of once a day during solar maximum, 
only 25 times in 2 . 5 yr did these flares result in 
the arrival of flare particles in the vicinity of the 
earth. It should be noted here that during the last 
sunspot minimum (1954) no flares of Class 2 or 
larger occurred. 

The flare particles are mostly protons (alphas 
and some heavier nuclei have also been detected) 
with kinetic energies extending from a few million 
electron volts (Mev) to a few tens of billion elec- 
tron volts. These energies are considerably be- 
low the energies of cosmic ray particles although 
the particle flux is greater than the galactic cosmic 
ray flux. The first high energy solar particles 
were detected at ground-based cosmic ray (sec- 
ondary) monitors and one of the first names given 
them was solar cosmic rays. Other names are 
"solar proton event, " "solar flare radiation event, " 
and "solar bursts. " But solar high energy particles 
(SHEP) has been offered by a group of researchers 
in this field as a standard nomenclature. More 
confusing is the terminology "Giant" and "Large," 
sometimes used to describe the type of proton flux. 
Proton fluxes from the "Giant" flares of 23 February 
1956, 4 May 1960 and 11 May 1960 were not as large 
as from the "Large" flares of 10 May, 10, 14 and 
16 July 1959. Furthermore, the radiation doses 
from the "Giant" events were not as great as from 
the "Large" events. The only explanation for this 
ranking is that protons from the ' Giant" events 
produced secondaries in the atmosphere that were 
energetic enough to penetrate and be detected at 
the ground. A better way to describe these events 
is by their differential or integral kinetic energy- 
fluxes. Shown below are the differential spectra 
for two solar events, 2 3 February 1956 as derived 
from Foelsche's plot (Ref. 31) and 10 May 1959 as 
derived from Winckler's observations (Ref. 32). 



r 





■ (JN., 


= 7 . H'i 


i x io Ki':"" 




( ir: : 


inn ' iv r 


Sare 


'I ,,N 4 


= 2 . rt r> 


7 x 10 :1 KK" 
IxlH 11 KK" 


.. ()4i> 


! (IK; 
>1 !■'•■, 


f.SU - !■'■■ 1 
1CI1M • V.- 




1 i:N. 


- 2. VA 


1 x 10" KK' 


-.B-,C 


ill-:. 


r.o.m ■ v. 



K = i. \ (IN 



10' I!"'* ■' .IK 



11-39 



A reasonably simple yet unambigious ranking 
of the severity of these events can be seen directly 
from these equations to be the coefficient indicating 
the total flux of particles and the exponent indicating 
how these are distributed with energy. Figure 14 
shows the radiation dose inside different thicknesses 
of absorber for these events and clearly shows that 
the relative hazard from these events varies with 
the amount of shielding provided. 

Figure 14 also shows that the radiation doses 
to an unshielded astronaut exceed the lethal doses 
but are shielded rather efficiently by even small 
amounts of absorbers. The shielding afforded by 
the materials and equipment of two spacecraft is 
shown on Table 30. 



TABLE 30 



Solar Flare Event Radiation Dose Inside Mercury 

Capsule and Apollo Command Module 

(Including Secondaries) 



Vehicle 


10 May 1959 


23 February 1956 


Mercury 
Capsule 


3. 8 x 10 3 rem 


48. 33 rem 


Apollo 
Command 






Module 


60. 5 rem 


42. 5 rem 


Ambient 


~ 5 x 10 rem 

4 
(1. 8 x 10 assum- 
ing no protons be- 
low 20 Mev) 


2 
5. 4 x 10 rem 



The greater shielding inherent in the Apollo 
vehicle is apparent. However, it should be noted 
that the orbit of Mercury is such that the Earth's 
magnetic field would shield a large fraction of 
these solar particles. In Ref. 32 Obayashi and 
Hakura have developed a model of proton cutoff 
energies versus geomagnetic latitude during a 
solar plasma induced geomagnetic disturbance. 
At these times, the normal cutoff energies are 
reduced and the solar flare particles are "allowed" 
at normally "forbidden" regions near the earth. 
Using this model of cutoff energies to modify the 
incident solar flare proton spectra results in de- 
creasing values of dose from polar to equatorial 
latitudes. Satellites which spend little or no time 
at magnetic latitudes less than 50° will not en- 
counter solar flare protons. Correspondingly, 
polar orbital satellites will receive the highest 
dose. Figures 15 and 16 show dose versus orbital 
inclination for the two solar flare events at different 
values of shielding. The dose versus latitude cutoff 
for the May flare is seen to be much sharper than 
for the February flare. This is, of course, due to 
its relatively larger number of low energy particles 
which are excluded before the higher energy particles. 

Also shown in these figures are the free space 
proton doses given in Fig. 14 from Ref. 33. It is 
seen that even at a 90° orbit the satellite dose 

2 
under 1 gm/cm is reduced to about 40% of the 
free space dose. Actually, the doses within 
orbital vehicles will be even lower due to shadow 
shielding by the earth. This is a function of alti- 
tude as shown in Fig. 17. 



One further qualification in the use of Figs. 15 
and 16 is necessary because they are plotted in 
terms of magnetic inclination. Figure 18 shows 
the magnetic dip equator and a great circle approxi- 
mation. This latter curve may be used together 
with Fig. 17 to estimate the orbital dose. 

The following example is given for illustration. 
We will assume an orbital inclination of 60° , 500- 
km circular orbit extending to 60° N over 280° 
longitude. The assumed duration of the February 
flare event is about 1 hr as compared to about 
1 day for the May event. In 1 hr the magnetic in- 
clination of the orbit has changed little, so that 
the February flare dose may be read from Fig. 
16 at 60° + 13° (or 73°). This is about 35 rad 

under 1 gm/cm . However, during the day's dura- 
tion of the May event, the magnetic inclination has 
gone to 47° and back again to 73°. Averaging the 

dose at these two latitudes gives 1200 rad under 

o 
1 gm/cm . At 500 km the earth intercepts 0.314 
of the incident protons giving 35 (1-0. 314) o.r about 
24 rad from the February flare and 823 rad for 
the May flare as the final answers. In calculating 
dosages from the May 1959 event, the flux of pro- 
tons was assumed constant for 30 hr. This gives 

Q o 

a total flux of 3 x 10 /cm -ster above 20 Mev. 
In calculating dosages from the February event, 
the flux was assumed to decay immediately from 

_2 
the given value as t . This gives a total flux of 

OH ry 

1. 8 x 10 /cm -ster above 0. 60 Mev or 6. 33 x 10 / 

2 
cm -ster above 20 Mev. During maximum periods 
of solar activity, it is believed that the total yearly 
flux of protons with energies greater than 20 Mev 

is 10 -10 /cm -ster. Therefore, the maximum 
yearly dose would be equivalent to approximately 
,10 

■ ss 3. 3 times the May 1959 dose or 



10 



3 x 1C- 
10 



10 



-~ =s 158 times the February flare dose. 
6.33 x 10 

However, it is fairly certain that an event such as 
that of February 1956 occurs no more frequently 
than once every 4 to 5 years, so that the maximum 
total yearly dose (during the peak years of the sun- 
spot cycle) should be about 3. 3 times the May 10, 
1959 doses. This may be used to estimate the 
hazard relative to mission duration. 

b. Van Allen belts (geomagnetically trapped 
particles) 

In the vicinity of the earth, there are intense 
regions of charged particles trapped in the earth's 
magnetic field. In the four years since Dr. Van 
Allen confirmed the existence of these regions 
from measurements made on the early Explorer 
satellites, a considerable body of data has been 
gathered to "map" these regions. 

The trapped particles form a generally toroidal 
region beginning at approximately 500-km altitude. 
The earth's field is not geocentric and has a number 
of signficant anomalies from a dipole resulting in 
the radiation belt shape like that shown in Fig. 19 
(for part of the "inner" belt). Yoshida, Ludwig 
and Van Allen (Ref. 34) have shown that the loca- 
tion of the trapped particles is related to the dip 
latitude and scalar intensity of the real magnetic 
field. In effect, the belt varies over about 800 km 
in altitude and about 13° in latitude around the earth. 



11-40 



The belt position shown in Fig. 19 was deter- 
mined from the relationships found in the last 
reference and with the use of a spherical har- 
monic fit to the magnetic field obtained from 
D. Jensen of the Air Force Special Weapons 
Center. The adiabatic invariant integral has also 
been noted by a number of workers in this field 
as having a better physical basis for determining 
the structure of the belts. 

Most recently Mcllwain (Ref. 35) has shown 
that the magnetic intensity scalar B and the param- 
eter L define a practical and accurate coordinate 
system for the trapped particles. The parameter 
L is related to the adiabatic invariant integral and 
would be the equatorial radius of a magnetic shell 
in a dipole field. In the real field the physical 
interpretation of L is more complex. 

The energy spectrum and particle flux for in- 
ner belt protons were calculated using the experi- 
mental data of Freden and White (Ref. 36), Van 
Allen (Ref. 37), and Van Allen, Mcllwain and 
Ludwig (Ref. 38). Figure 20 shows the proton 
flux contours at one location over the earth, and 
Fig. 21 the differential kinetic energy spectrum 
of protons. The peak flux shown agrees with Van 
Allen's recent estimates. 

The model of electrons, by far the most abun- 
dant constituents of the trapped radiation belts, 
was determined using flux and spectral measure- 
ments of Holley (Ref. 39), and Walt, Chase, Cladis, 
Imhof and Knecht (Ref. 40), together with the 
Anton 302 geiger counter data from a number of 
satellites and space probes (Refs. 41 and 42). 
Figure 22 shows the electron flux contours at one 
location over the earth and Fig. 23 shows the dif- 
ferential kinetic energy spectrum. 

This spectrum agrees well in shape with the 

recent determination by Pizzella, Laughlin and 

O'Brien (Ref. 43) for the inner radiation belt at an 

altitude of 1000 km. The highest flux at this alti- 

6 2 

tude is 5 x 10 electrons/cm -sec-steradian as 

given by Frank, Dennison and Van Allen (Ref. 44). 

This agrees well with the flux at this altitude 

shown in Figs. 22 and 23. 

For the outer radiation belt. Van Allen has 
given the following peak electron distribution 

n _ q _ i 

10 cm sec above 40 Kev 

5 -9 -1 

10 cm sec above 2 Mev 

2 -2 -1 

10 cm sec above 5 Mev 

This is two orders of magnitude less in flux than 
Van Allen's earlier estimates of the outer zone 
electrons. Extending the new spectrum to 20 Kev 

no o 

gives 2 x 10' electrons/cm -sec or 1. 6 x 10 

o 
electron/cm -sec-steradian, which agrees closely 
with the peak outer belt flux shown in Fig. 22. 
Figures 24 and 25 show the electron and 
bremsstrahlung dose rates versus aluminum 
absorber from electrons at the peak of the inner 
and outer regions (Ref. 45). These may be com- 
pared with the Van Allen belt proton doses also 
shown in Fig. 14 as a function of absorber thick- 
ness for protons at the center of the inner belt. 



Proton doses for orbiting satellites may be ob- 
tained from Tables 31 and 32 as a function of 
orbital altitude, inclination and aluminum absorber 
thickness. Due to the belt asymmetry, the dose 
on each successive orbit differs. For example, 

at an orbital inclination of 40° (geographic) and an 

9 
altitude of 740 km under 6 gm/cm" of aluminum, 
the accumulated dose is 0. 0214 rem after the 
first orbit and 0. 0249 rem after two orbits. For 
integer orbits, the dose accumulation cycle should 
repeat itself every 24 hr. The doses in Tables 31 
and 32 are 12-hr totals, so that the orbital lifetime 
dose may be closely approximated by 2 (number 
of days in orbit) (12-hr cumulative dose). Table 
33 from Ref. 45 gives dose versus orbital incli- 
nation, altitude and absorber thickness for a 
satellite exposed to the electrons of the inner 
Van Allen belt. 

c. Primary cosmic radiation 

Steady-state cosmic radiation values (Ref. 46) 
that have been generally accepted for a number of 
years (Ref. 47) were based on the belief that the 
primary spectrum contained few particles in the 
energy region below a fraction of a Bev. This 
meant the ionization at geomagnetic latitudes 
greater than 60° was taken to be the same as that 
at 60° and this indeed appeared to be true during 
1950 to 1952. However, in 1954, a time of mini- 
mum solar activity, low energy protons caused 
an increase in the ionization levels at latitudes 
above 60° (Ref. 48). It should be remembered, 
though, that the most favorable periods for ex- 
tended space flight are these same times of low 
solar (but higher cosmic ray) activity, because 
of the great reduction in flare occurrences. For 
this reason, values of the ionization rate that in- 
clude the effect of the increase above 60° as 
would be expected during a typical time of solar 
quiescence are plotted in Fig. 2 6 as functions of 
altitude and geomagnetic latitude, both for near- 
earth and high altitude positions of measurement 
(Ref. 49). Not shown at the scale of Fig. 26 is 
that as the surface of the earth is approached, 
there is an ionization increase, followed by a 
decrease. The increase begins at 130,000 ft, 
continues down to heights of 80, 000 ft (at 90° 
latitude) or 50, 000 ft (at 0° latitude), and has its 
source in the shower, or cascade formation of 
mesons, nucleons, electrons and high energy 
photons, all of which are created by interaction 
of high energy cosmic particles with atmospheric 
constituents. The decrease in ionization with de- 
creasing altitude below 80, 000 to 50, 000 ft comes 
about through atmospheric radiation absorption, 
while the decrease with decreasing magnetic lati- 
tude results from the increased shielding offered 
by the earth's magnetic field against the lowered 
energy cosmic particles. Figure 26 shows that the 
increase in cosmic detector ionization at increas- 
ingty great distances from the earth arises from 
a combination of the decrease in the solid angle 
subtended by the earth and the decrease in geomag- 
netic field strength, with a corresponding decrease 
in the cosmic particle deflection. 

An estimate of the biological whole -body radia- 
tion intensity as a function of altitude and geomag- 
netic latitude can be obtained from Fig. 26 only 
if the conversion can be made from the ionization 
itself, in units of roentgen, to rem, the unit which 
gives an idea of the biological effectiveness of the 



11-41 



TABLE 31 

Inner Van Allen Belt Proton Radiation Dose (rems) 
Orbiting Aluminum Sphere 



Orbital 

Inclination 
(cleg) 


Orbital 
Altitude 


Aluminum Shield 
2 
Thickness (gm/cm ) 
No. Orbits 


Rems 
1.0 2.0 6.0 10.0 20.0 60.0 100.0 





555 km 
300 n mi 

740 km 
400 n mi 

1 110 km 
600 n mi 

1852 km 
1000 n mi 


1 
2 
3 
4 
5 
6 
7 

1 
2 
3 
4 
5 
6 
7 

1 
2 
3 
4 
5 
6 

1 
2 
3 
4 
5 


+0.00372 +0.00272 +0.00145 +0.00104 +0.00062 +0.00024 +0 00014 
+0.01852 +0.01354 +0.00720 +0.00517 +0.00312 +0.00120 +0 00070 
+0.02203 +0.01611 +0.00857 +0.00615 +0.00371 +0.00143 +0 00083 
+0.02744 +0.02006 +0.01067 +0.00766 +0.00462 +0.00178 +0.00103 
+0.03642 +0.02664 +0.01417 +0.01017 +0.00613 +0.00237 +0.00137 
+0.06091 +0.04455 +0.02370 +0.01701 +0.01026 +0.00396 +0.00230 
+0.07287 +0.05329 +0.02835 +0.02035 +0.01228 +0.00474 +0.00275 

+0.02093 +0.01530 +0.00814 +0.00584 +0.00352 +0.00136 +0.00079 
+0.08120 +0.05938 +0.03159 +0.02268 +0.01368 +0.00528 +0.00307 
+0.09957 +0.07282 +0.03874 +0.02781 +0.01678 +0.00647 +0.00376 
+0.15308 +0.11195 +0.05956 +0.04276 +0.02579 +0.00996 +0.00579 
+0.19437 +0.14215 +0.07563 +0,05429 +0.03275 +0.01264 +0.00735 
+0.24586 +0.17981 +0.09566 +0.06868 +0.04143 +0.01599 +0.00930 
+0.27285 +0.19955 +0.10616 +0.07622 +0.04598 +0.01775 +0.01032 

+0.63995 +0.46803 +0.24900 +0.17876 +0.10784 +0.04163 +0.02420 
+ 1.13415 +0.82947 +0.44130 +0.31682 +0.19113 +0.07379 +0.04290 
+1.62798 +1.19063 +0.63345 +0.45477 +0.27435 +0.10592 +0.06158 
+2.40827 +1.76130 +0.93707 +0.67274 +0.40584 +0.15669 +0.09110 
+3.02077 +2.20925 +1.17540 +0.84385 +0.50906 +0.19655 +0.11427 
+4.13293 +3.02264 +1.60814 +1.15453 +0.69649 +0.26891 +0.15634 

+8.14456 +5.95656 +3.16909 +2.27517 +1.37253 +0.52993 +0.30810 
+ 16.08871 +11.76655 +6.26020 +4.49436 +2.71130 +1.04682 +0.60862 
+24.51561 +17.92961 +9.53915 +6.84841 +4.13142 +1.59513 +0.92741 
+33.35166 +24.39190 +12.97731 +9.31674 +5.62049 +2.17006 +1.26167 
+41.75440 +30.53728 +16.24686 +11.66404 +7.03653 +2.71679 +1.57954 


20 


555 km 
300 n mi 

740 km 
400 n mi 

1110 km 
600 n mi 

1852 km 
1000 n mi 


1 
2 
3 
4 
5 
6 
7 

1 
2 
3 
4 
5 
6 
7 

1 

2 
3 
4 
5 
6 

1 
2 
3 
4 
5 


+0.07177 +0.05249 +0.02792 +0.02005 +0.01209 +0.00467 +0.00271 
+0.07767 +0.05680 +0.03022 +0.02169 +0.01309 +0.00505 +0.00293 
+0.07838 +0.05732 +0.03050 +0.02189 +0.01321 +0.00510 +0.00296 
+0.07838 +0.05732 +0.03050 +0.02189 +0.01321 +0.00510 +0.00296 
+0.07890 +0.05770 +0.03070 +0.02204 +0.01329 +0.00513 +0.00298 
+0.08052 +0.05889 +0.03133 +0.02249 +0.01356 +0.00523 +0.00304 
+0.08355 +0.06110 +0.03251 +0.02334 +0.01408 +0.00543 +0.00316 

+0.05174 +0.03784 +0.02013 +0.01445 +0.00871 +0.00336 +0.00195 
+0.07776 +0.05687 +0.03025 +0.02172 +0.01310 +0.00505 +0.00294 
+0.08903 +0.06511 +0.03464 +0.02487 +0.01500 +0.00579 +0.00336 
+0.08907 +0.06514 +0.03465 +0.02488 +0.01501 +0.00579 +0.00336 
+0.09400 +0.06875 +0.03657 +0.02626 +0.01584 +0.00611 +0.00355 
+0.12011 +0.08784 +0.04673 +0.03355 +0.02024 +0.00781 +0.00454 
+0.14274 +0.10439 +0.05554 +0.03987 +0.02405 +0.00928 +0.00539 

+0.60988 +0.44604 +0.23730 +0.17037 +0.10277 +0.03968 +0.02307 
+ 1.11837 +0.81792 +0.43516 +0.31241 +0.18847 +0.07276 +0.04230 
+ 1.36262 +0.99656 +0.53020 +0.38064 +0.22963 +0.08866 +0.05154 
+1.62606 +1.18922 +0.63270 +0.45423 +0.27402 +0.10580 +0.06151 
+ 1.86481 +1.36384 +0.72560 +0.52093 +0.31426 +0.12133 +0.07054 
+2.46111 +1.79994 +0.95763 +0.68750 +0.41475 +0.16013 +0.09310 

+7.25229 +5.30399 +2.82190 +2.02591 +1.22217 +0.47187 +0.27434 
+14.12855 +10.33298 +5.49749 +3.94679 +2.38097 +0.91928 +0.53447 
+19.89605 +14.55107 +7.74166 +5.55794 +3.35292 +1.29455 +0.75265 
+25.14740 +18.39168 +9.78499 +7.02490 +4.23789 +1.63624 +0.95131 
+30.67196 +22.43209 +11.93462 +8.56817 +5.16890 +1.99570 +1.16030 


■to 


555 km 
300 n mi 

740 km 
400 n mi 

1 1 10 km 
600 n mi 

1852 km 
1000 n mi 


1 
2 
3 
4 
5 
6 
7 

1 
2 
3 

4 
5 
6 
7 

1 
2 
3 
4 
5 
6 

1 
2 
3 
4 
5 


+0.03171 +0.02319 +0.01234 +0.00886 +0,00534 +0.00206 +0.00119 
+0.03866 +0.02828 +0.01504 +0.01080 +0.00651 +0.00251 +0.00146 
+0.03866 +0.02828 +0. 01504 +0.01080 +0.00651 +0.00251 +0.00146 
+0.03866 +0.02828 +0.01504 +0.01080 +0.00651 +0.00251 +0.00146 
+0.03866 +0.02828 +0.01504 +0.01080 +0.00651 +0.00251 +0.00146 
+0.03866 +0.02828 +0.01504 +0.01080 +0.00651 +0.00251 +0.00146 
+0.03866 +0.02828 +0.01504 +0.01080 +0.00651 +0.00251 +0.00146 

+0.05504 +0.04025 +0.02141 +0.01537 +0.00927 +0,00358 +0.00208 
+0.06403 +0.04683 +0.02491 +0.01788 +0.01079 +0.00416 +0.00242 
+0.06958 +0.05088 +0.02707 +0.01943 +0.01172 +0.00452 +0.00263 
+0.07104 +0.05195 +0.02764 +0.01984 +0.01197 +0.00462 +0.00268 
+0.07155 +0.05233 +0.02784 +0.01998 +0.01205 +0.00465 +0.00270 
+0.07749 +0.05667 +0.03015 +0.02164 +0.01305 +0.00504 +0.00293 
+0.08057 +0.05892 +0.03135 +0.02250 +0.01357 +0.00524 +0.00304 

+0.43148 +0.31556 +0.16789 +0.12053 +0.07271 +0.02807 +0 01632 
+0.81762 +0.59797 +0.31814 +0.22840 +0.13778 +0.05319 +0.03093 
+0.93977 +0.68731 +0.36567 +0.26252 +0.15837 +0.06114 +0.03555 
+ 1.02163 +0.74717 +0.39752 +0.28539 +0.17216 +0.06647 +0.03864 
+1.14910 +0.84040 +0.44712 +0.32100 +0.19364 +0.07476 +0.04346 
+1.52201 +1.11313 +0.5B222 +0.42517 +0.25649 +0.09903 +0.05757 

+4.77857 +3.49483 +1.85936 +1.33488 +0.80529 +0.31092 +0.18077 
+8.78610 +6.42576 +3.41872 +2.45438 +1.48065 +0.57167 +0.33237 
HI. 22799 +8.21165 +4.36887 +3.13652 +1.89216 +0.73056 +0.42474 
H3. 73962 +10.04854 +5.34616 +3.83814 +2.31543 +0.89398 +0.51976 
+ 17.46029 +12.76966 +6.79389 +4.87751 +2.94244 +1.13607 +0.66051 



11-42 





«~ M 




Oi <D Si 




m ^ =" 




o » -° 




M a a) 






c m_ 2 




° c B 


CO 


■s §> 


H 


adi 
min 
om 


-J 


rn 


K 3 u 


<; 


cZ^ 


H 


roto 

ing 

hed 




csg 




lien 
Orb 
Lau 




< 




c 




ClJ 




> 



C 

o 
c 

o 


"1 


o 


0. 001525 
0. 001875 
0, 001914 
0. 001914 
001922 
002040 
002062 


0. 17588 0. 1008 0. 0774 
0. 26078 0. 1494 1 1147 
0,29726 , 0. 1703 0, 1308 
0, 32565 | 0, 186G | 0. 1433 
0, 34710 1 0. 1989 ' 1527 
0. 42955 ' 0. 246 1 0. 1890 


0. 5403 1 0. 4149 
1, 0408 0. 7992 
1,3578 ' 1.0411 

1. 6982 1. 3040 

2. 0337 1, 5616 


1 

64. 150 2. 9587 | 2. 2759 1. 3040 1. 0014 
127.200 5,8720 | 4.5169 2.5882 ; 1,9874 
176.000 8.1143 6.2418 , 3.5766 ' 2.7464 
226.150 10.4359 8.0276 | 4.5998 [ 3.5321 
275. 800 ' 12. G998 9, 7691 ; 5. 5977 4. 2984 


S|SS gPS| 




0, 001986 
0. 002442 
0. 002493 
0, 002493 
0. 002503 
0. 0026 57 
0, 0O2G8G 




,„ 


0. 003466 
0. 004262 
0, 0043 51 
0. 004351 
0. 004368 
0. 004G37 
0. 004687 


0. 94296 

1. 81645 

2. 36609 
2. 96365 
3. 54915 




*' 


0, 004 51 
0. 00555 
0. 00561 
0. 0056 1 
0. 00566 
0. 00601 
0. 00608 


0.2286 
0. 3390 
0. 3864 
0. 4233 
0, 4512 
0, 5584 


26,580 ! 1.2258 
51. 300 2. 3614 
66.700 ! 3,0759 
33. 500 3. 8527 
100. 000 ; 4. 6139 




= 


0.0980 
0. 1200 
0, 1228 
0, 1228 
0, 1230 
0, 1305 
0, 1320 


4, 950 

7. 349 

8. 360 

9. 160 
9. 760 

12. 110 




c 


El 

O 


1 


^ 


0. 000743 
0. 000981 
0. 001003 
0, 001003 
0, 001006 
0, 001059 
0, 001064 


0. 0G7G 
0. 1071 
0. 1228 
0. 1280 
0. 1467 
0. 1753 


0. 3189 

0. G2G8 
0, 86 12 

1. 1161 
1.2910 


0. 9942 
1. 9728 
2. 7269 
3.6443 

4. 1415 


1. 5559 
2. 7413 
4.2526 
5, 7054 

1. 5348 
3. 0578 
4.6479 
5.7905 




0. 0009G8 
0. 001278 
0. 001306 
0. 001306 
0. 001310 
0. 001380 
0. 001386 


0. 0880 
0. 1394 
0. 1599 
0. 1667 
0.1911 
0. 22 83 


20.400 0.9422 0.72477 0.4153 
40.100 1,8519 i 1.42450 0.8162 
55. 150 2. 5444 \ 1. 95722 ! 1.1215 
71. 500 3. 2976 ■ 2. 53664 ! 1. 4535 
82,500 3.8144 ■ 2.93418 1.6313 


1. 2947 
2. 56 91 
3, 5512 
4. 7458 
5. 3934 


4. 5972 ; 3. 5363 ! 2. 02G3 
8.0995 6.2304 3.5700 
12.5644 | 9.6649 5.5380 
16.8568 12.9GG8 7.4300 
4. 5348 3. 4883 , 1. 9988 
9. 0345 ■ G. 9496 3. 9821 
13. 7327 : 10, 5636 G, 0529 
17, 1083 : 13. 1603 7, 5408 






0. 001689 
0. 0OB23 
0.0022 80 
0. 002280 
0. 002287 
0. 002408 
0. 002419 


0. 153 57 
0. 24335 
0. 27909 
0. 29091 
0. 33345 
0. 39837 


63. 550 2. 9373 2. 2595 
126.200 | 5. S287 4.4836 
174. 800 8. 0568 : 6. 1975 
233, 800 10, 7673 | 8. 2825 
265. 300 12. 2364 i 9, 4126 

1 




0. 00219 
0. 00290 
0. 00296 
0. 00296 
0.00298 
0. 00300 
0. 00314 


0. 1996 
0.3164 
0, 3628 
0.3 782 
0. 4335 
0. 5179 




99. 50 
176. 00 
272. 000 
3G5. 00 

98, 100 
195. 80 
298. 00 
3 70. 00 




° 


0.0471 
0. 0629 
0. 0643 
0.0643 
0. 0645 
0. 0680 
0. 0681 


4. 320 

6. 850 

7. 850 

8. 195 

9. 390 
11.210 


5 

1 | 




1 


- 


000453 
0. 00092 7 
0. 001036 
0. 001041 
0. 001041 
0. 001041 
(") 001103 


0. 0473 
I) 0865 
0, 1060 
0. 1158 
1229 
1393 


3833 
0. 7112 

1 0108 
1 3845 
1 5585 


1. 2047 0, 9251 
2.4758 1.9011 
3,6440 ' 2.7982 
4. 7650 3, 6590 
5.8333 4.4793 


Z«»". «*|* 




2 


0. 000592 
0. 001208 
0. 001350 
0. 001355 
0. 001355 
0. 001356 
0. 001436 


0. 10762 ' 0. 0600 
0. 19689 0. 1128 
0. 24168 ' 0, 1381 
0. 26274 0. 1505 
0, 27955 0. 1600 
0. 31G45 0, 1812 


0. 4991 
0, 9262 
1. 3164 
1. 8029 
2, 2096 


2. 2707 

4 3 700 
6 26 70 
8. 1980 

2 . 4404 

5 0211 
7, 3707 

j 10, 0219 


,„ 


0. 00103 
0, 002108 
0. 002356 
0. 002365 
0. 002365 
0, 00236 7 
0, 002 506 


0. 87110 
1.6164 
2,2973 
3, 146 5 
3.5421 


2. 1024 
4. 3207 
6. 3595 
8.3158 
10, 1803 


3. 9G29 
7. 6265 
10. 93 72 
14.3071 

4. 2590 
8. 7628 
12, 8633 
17. 4902 


* 


0. 00134 
0. 002G2 
0. 00306 
0. 00307 
0. 00307 
0. 00308 
0, 00326 


3. 025 j 0, 1398 
5. 545 '■ 0. 2560 
6.795 0.3138 
7. 400 0, 341G 
7. 875 0, 3625 
8.915 , 0,4110 


1, 1324 

2. 1013 
2. 9 865 
4, 0905 
4.6047 


2.7331 
5.6169 
8.26 74 
10. 8105 
13.2344 


5. 1318 
9, 9144 
14.2183 
18. 5992 

5, 536 7 
11. 3916 
16. 7222 
22. 7372 


= 


b c o o c o o 


24. 6 00 
45. 550 
64. 700 
88. 550 
99.900 


59.300 
121. 900 
179. 200 
244. 000 
286. 500 


11 1. 800 
214. 400 
308, 200 
404, 400 

120. 00 
24 7,40 
362. 10 
493 . 00 


1 
1 


1 


!5 



Z 


— cm cn -*■ ifi u> t- co 


- — •»•"»- = 






- -<•"-■* --«' 


< 


1 = 


j'i 

o c 

CO O 


CM O 


if 


3700 km 
2000 n mi 

407o km 
2200 n mi 



11-43 



TABLE 33 







Twelve -Hour Orbital Dos 


e (rad) Within Van Allen Belt 








Orbital 

Inclination 

(deg) 


2 

Aluminum Sphere Thickness (gm/cm ) 


0. 1 


1.0 


2. 


Altitude 


Electrons 


X-rays 


Electrons 


X-rays 


Electrons 


X-rays 


555 km 





4.598 x 10 3 


0.7569 


1. 137 x 10" 3 


0.2301 




0. 1575 


(200 naut mi) 


40 


1.444 x 10 3 


0.2377 


3.574 x 10" 4 


0. 0723 


<10" 5 


0. 0494 




90 


6.811 x 10 2 


0. 1121 


1.686 x 10" 4 


0. 0341 




0.0233 


740 km 





1. 1690 x 10 4 


1.9241 


2. 892 x 10" 3 


0.5849 




0.4003 


(400 naut mi) 


40 


5.046 x 10 3 


0.8306 


1.248 x 10" 3 


0. 2525 


<10" 5 


0. 1728 




90 


3.693 x 10 3 


0.6078 


9. 136 x 10" 4 


0. 1848 




0. 1264 


1110 km 





6.634 x 10 4 


10.9197 


1.641 x 10~ 2 


3.3196 




2.2716 


(600 naut mi) 


40 


4. 129 x 10 4 


6. 7964 


1. 021 x 10~ 2 


2. 0661 


-4 
< 10 


1.4138 




90 


2.359 x 10 4 


3.8825 


5. 835 x 10" 3 


1. 1803 




0. 8077 


1852 km 





2.625 x 10 5 


43.2147 


6.495 x 10" 2 


13. 1373 


1. 803 x 10" 4 


8.9898 


(1000 naut mi) 


40 


2.088 x 10 5 


34.3755 


5. 166 x 10" 2 


10.4502 


1.434 x 10" 4 


7. 1510 




90 


1.097 x 10 5 


18.0597 


2.714x 10" 2 


5.4901 


7.534 x 10" 5 


3.7569 



ionization. The factor of conversion, Relative 
Biological Effectiveness (RBE), yields a measure 
of the degree of localization, or nonuniformity, 
of tissue ionization. Ionization localization along 
the path of penetration is singularly noticeable 
for heavy (atomic number 6 or greater) particles. 
Although all atomic species through iron have 
regularly been observed, the biologically note- 
worthy heavy constituents of the primary radiation 
are carbon, nitrogen, oxygen, the magnesium 
and calcium groups, and iron. When these medium 
and high energy particles enter tissue, they first 
produce an ionization trail of great density. The 
high energy particles, in general, undergo nuclear 
disintegration during the penetration process, 
with a resulting large reduction in specific ioni- 
zation, since afterward the ionization is caused 
by several particles of reduced charge travelling 
in different directions. These primaries which 
have a reduced impinging energy have a signif- 
icant probability of being completely stopped 
through ionization only. This leads to extremely 
large specific ionizations near the ends of the 
paths, since the rates of energy loss increase 
as the particle energies decrease, down to very 
low energies. These thindown hits are capable 
of causing cell destruction. Their effects in 
nonreparable regions of the body, such as certain 
brain areas, have not yet been demonstrated. 
The RBE conversion from roentgen to rem ob- 
tained from a weighted analysis of particle type 
and tissue ionization characteristics between 30° 
and 55° latitude at the top of the atmosphere and 
extrapolation elsewhere, increases with increasing 
altitude and geomagnetic latitude, as seen in 
Fig. 27. This is explained by noting that at a 
position requiring decreased particle penetration 
of the magnetic field, there is a slight increase 



in the relative number of heavy constituents, 
compared with hydrogen and helium. At the 
same time, the heavy component energy range 
extends to lower values. It must be emphasized, 
however, that little actual biological experi- 
mentation has been performed to test the validity 
of the relation between ionization track density 
and the RBE for particles of large atomic 
number, which produce the greater fraction of 
the unshielded biological intensity. 

Shielding against cosmic radiation is not 
ordinarily advisable, since it requires thick- 

2 
nesses of aluminum greater than 25 gm/cm 

2 
for heavy particles, and at least 200 gm/cm 

2 
(400 lb /ft of shielded area) for hydrogen and 
helium, which have far higher penetrating power 
and constitute about 15 percent of the unshielded 
biological dose and 99 percent of the incident 
particle number. In fact, the biological dose 
increases for shielding thicknesses up to 15 

2 
gm/cm for the carbon, nitrogen, and oxygen 

2 
group, up to 10 gm/cm for magnesium, up to 

2 2 

6 gm/cm for calcium, and up to 5 gm/cm for 

iron. 

An estimate of the effectiveness of shielding 
against cosmic radiation is shown in Fig. 28 
taken from Wallner and Kaufman (Ref. 50). A 
comparison with the curves shown in Fig. 14 
shows the relatively slow decrease of dose with 
absorber thickness for cosmic rays as compared 
to other space radiations. The dose peak at 

2 
about 10 gm/cm is due to the increase of ionization 



11-44 



rate before significant numbers of particles are 
stopped in the absorbing material. 

d. Penetrating electromagnetic radiation 

Previous estimates of the high energy end of 
the solar system indicated intensities of the order 

-4 2 ° 

of 10 erg/cm -sec below 8A. Recent measure- 
ments indicated that during a solar flare (class 

2+) this intensity increased to about 10 erg/ 

2 ° 

cm -sec with 2 A as the lower limit of the radi- 
ation detected (Ref. 51). More recently, meas- 
urements have indicated that X-ray flashes during 
solar flares had energies as high as 80 kev (0. 15 

A (Ref. 52). 

During a class 2 solar flare on 20 March 1958 

an intense burst of electromagnetic energy was 

recorded which lasted 18 seconds (or less) (Ref. 

53). This was determined to have an intensity 

-4 2 

of 2 x 10 erg /cm -sec above 20 kev and 

peaking in the region of 200 to 500 kev (0. 06 to 

o 

0. 025 A). Measurements during a class 2+ flare 
on 31 August 1959 indicated a peak intensity of 

_ n 9 

4. 5 x 10 erg /cm -sec (-^20 kev) arriving at the 
top of the earth's atmosphere (Ref. 54). The 
spectrum decreases in photon count by a factor 
of 10 for an energy increase of about 20 kev. 
Although these photons are quite penetrating (the 
half -thickness value of aluminum for 500 kev 
photon is 3. cm) their intensity is so low as to 
produce an insignificant dose (of the order of 

10 roentgen from the March 1958 event). „ In- 
tensity enhancements in the region of 8-20 A were 
also observed during the August 1959 event. In 

2 
this region about 1 erg/cm -sec was measured. 

This would result in a much greater dose than 
the less intense higher energy photons; their 
penetration is very much less. The half -thickness 

values are less than 10 cm of aluminum. 

A solar X-ray spectrum from a class 2+ flare 
is shown in Fig. 29 taken from Ref. 30. X-rays 
with energies in excess of 20 kev appear to be 
emitted only for short periods (a few minutes) 
during large flares. The X-ray dose rate to an 
unprotected man from a flux as shown in Fig. 29 
would be about 3 rem/hr. However, since the 
emission lasts for much less than 1 hr we may 
conclude that high energy solar electromagnetic 
radiation will not be of concern to space flight. 
Saylor, et al. (Ref. 55) point out that ultraviolet 
light on bare skin can cause severe burns and 
even skin cancer. It will therefore be advisable 
to use windows or shutter arrangements to filter 
the otherwise unattenuated solar ultraviolet rays. 
In space there will be no warning glare of scattered 
light to alert the observer that his line of sight is 
approaching the sun. An inadvertent glance at the 
sun could cause temporary vision failure and ten 
seconds of exposure would cause permanent 
retinal burn. These authors conclude that pro- 
tection of the eyes against sunlight is a necessity. 



e. Radiation damage thresholds 

Of all the components of a space vehicle, 
man has the lowest threshold to damage by 
ionizing radiation as shown in Table 34. 

TABLE 34 



Radiation Damage Dose Limitations 


People 


Roentgen Equivalent 


2 ^ 
10 (sickness) 10 (lethal) 


Semiconductor 


10 (damage) 10 (failure) 


Electronics 


10 8 io 10 


Elastomers 


10 7 10 8 


Plastics 


10 8 10 9 


Metals 


10 15 


Ceramics 


io 17 



Ref. Nucleonics Sept 1956 

More detailed treatment of radiation damage 
mechanism are shown in Refs. 56 and 57 and 
in the very comprehensive Radiation Effects 
Information Center Series of Battelle Memorial 
Institute. 

Semiconductors are seen to be the second 
easiest damaged component. This is caused 
by the fact that their properties arise from their 
form of very nearly perfect single crystals. 
Most metals and ceramics used for structural, 
electrical or magnetic applications are already 
in a disordered polycrystalline form and their 
properties are only moderately changed by 
further disorder (ionization). 

It should be noted that certain types of sensing 
elements may give erroneous readings due to 
spurious signals from the Van Allen or other 
radiation environments. While this does not 
represent damage by radiation, it is neverthe- 
less undesirable and can be easily avoided by 
proper selection, design and calibration of these 
devices. 

As contrasted to actually "reading" unwanted 
signals from ionizing radiations in sensitive 
"front end" components it is known that electronic 
components and circuits may operate improperly 
while in the presence of large fluxes of ionizing 
radiation. Measurements made under conditions 
simulating a nuclear explosion in space have indi- 
cated that the threshold of susceptibility to these 

fi 7 

effects is at peak dose rates of 10 to 10 

roentgen per second. This again is greatly in 
excess of what will be encountered from the 
natural radiation environments. 

The radiation problem therefore reduces to 
protection of the crew. 



11-45 



Maximum allowable radiation doses for 
manned space flight have been revised upward 
from 25 rem considerably in the past year. 
Presently the Apollo maximum allowable emer- 
gency dosages are as shown in Column 4 of 



Table 35 from Ref. 58. The normal mission 
dosages are as shown in Column 3. These 
values are more meaningful than the single so- 
called "whole body" value used previously. 



TABLE 35 
Radiation Dosage 





5 Year Dose 
(rem) 


RBE 


Average Year 
Dose (rad) 


Maximum Single Acute 
Exposure (rad) 


Design Dose 
(rad) 


Skin body dose 
' 0. 07 mm depth 


1630 


1.3 


235 


500 


125 


Skin body dose 
extremities, 
hands, etc. 


3910 


1.4 


559 


700 


175 


Blood forming 


271 


1.0 


54 


200 


50 


organism 












Eyes 


271 


2.0 


27 


100 


25 



4. Meteoroids 

Empirical data on meteoroids has come 
either from optical and radar meteor obser- 
vations or from impact detectors on board 
rockets and satellites. In the first type of ob- 
servation, velocity and luminous intensity history 
are directly measurable. The mass and density 
of the meteoroid is then determined using the 
drag equation, the shape of the light curve and 
the vaporization equation. Due to the variety 
of assumptions and dependencies in this analysis, 
there is a large uncertainty in flux estimates 
from the same type of data. The relation between 
meteoroid mass and visual magnitude is shown 
in Fig. 30 from an early survey (Ref. 59). The 
relation between mass and flux is shown in 
Fig. 31 from a later survey article (Ref. 60). 
The flux uncertainty is dealt with in a number of 
other survey articles (Refs. 61, 62 and 63), and 
an examination of the assumptions employed in 
the analysis procedure will show why it is as 

3 
large as 10 . The best known model of the 
meteoroid environment was developed by 
Whipple in 1957 and summarized in Table 36. 
The following equation fits the distribution 
presented by Whipple in 1957. 



1 



-1 9 
3 x 10 m 



1 



where <j> is the flux/m -sec of particles with mass 
m grams and greater. This was revised by 
Whipple (Ref. 64) in 1960 to 



-12.6 



-1. 186 



10""'" m *' ' to include empirical 
data from rockets and satellites. A recent evalu- 
ation of rocket and satellite data (Ref. 65) (obtained 
from acoustic detectors) obtained 



10 



-17.0 



■1. 70 



applicable between 



10 to 10 gm. These distributions 



masses of 10 

are shown in Fig. 32 taken from the last cited 
reference. It should be noted that meteoroid 
masses of greatest interest to space vehicle de- 
signers lie between the mass regions measured 



by the meteor or satellite -borne microphone 
techniques. Observations of meteors simulated 
by shaped charge firings from an Aerobee Rocket 
(Ref. 66) have indicated that Whipple may have 
underestimated meteor luminous efficiencies. 
This may be accounted for by a downward revision 
by an order of magnitude in mass (Ref. 6 7) of the 
1957 flux estimate of Whipple so that 



1.3 x 



,„-13 - 
10 m 



Various investigators have put forth penetration 
models --some based on empirical equations derived 
from test data and some based on theoretical con- 
siderations and most all giving the penetration in a 
thick target. Since structural skins are usually 
made of aluminum alloy materials, a good basis 
of comparison is the penetration of meteorites into 
aluminum. Four penetration equations were in- 
vestigated to obtain a comparison of the meteorite 
penetrations given by the different equations. These 
equations were: 

a. Whipple ' s equation 

This equation is given in (Ref. 63) as 

P "K| <^> "' E " 3 

where 

P = penetration in a thick target 

K. = constant of proportionality 

E = meteorite energy 

p - target density 

c = heat to fusion of target material 

For a meteorite of diameter (d) moving at a 
velocity (V) cm /sec and with a meteoroid density 

p = 0.05 gm/cm and e = 248 cal/gm Whipple's 

equation is 



11-46 



TABLE 36 

Data Concerning Meteoroids and Their Penetrating Probabilities 

F. L. Whipple, Ref. 5 



Meteor 

Visual 

Magnitude 


Mass 
(g) 


Radius 
(u) 


Assumed 

Vel 
(km/ sec) 


KE 
(ergs) 


Pen. 

in Al t 
(cm) 


No. Strik- 
ing Earth 
(per day)** 


No. Striking 
3m (Radius) 

Sphere 
(per day)*** 







25.0 


49,200 


28 


14 
1.0 x 10 


21.3 


-- 


-- 




1 


9.95 


36,200 


28 


3. 98 x 10 13 


15.7 


-- 


-- 




2 


3. 96 


26,600 


28 


1.58 x 10 13 


11.5 


-- 


-- 




3 


1.58 


19,600 


28 


6.31 x 10 12 


8.48 


-- 


-- 




4 


0.628 


14,400 


28 


2.51 x 10 12 


6.24 


-- 


-- 




5 


0.250 


10,600 


28 


1.00 x 10 12 


4.59 


2 x 10 8 


2.22 x 10" 5 




6 


9.95 x 10" 2 


7,800 


28 


3.98 x 10 U 


3.38 


5.84 x 10 8 


6.48 x 10"° 




7 


3. 96 x 10" 2 


5,740 


28 


1.58 x 10 11 


2.48 


1.47 x 10 9 


1.63 x 10" 4 




8 


1.58 x 10~ 2 


4,220 


27 


5.87 x 10 10 


1.79 


3.69 x 10 9 


4.09 x 10" 4 




9 


6.28 x 10" 3 


3,110 


26 


2. 17 x 10 10 


1.28 


9.26 x 10 9 


1.03 x 10" 3 




10 


2.50 x 10" 3 


2,290 


25 


7.97 x 10 9 


0.917 


2. 33 x 10 10 


2.58 x 10" 3 




11 


9.95 x 10" 4 


1,680 


24 


2.93 x 10 9 


0.656 


5.84 x 10 10 


6.48 x 10" 3 




12 


3. 96 x 10" 4 


1,240 


23 


1.07 x 10 9 


0.469 


1.47 x 10 11 


1.63 x 10" 2 




13 


1.58 x 10" 4 


910 


22 


3.89 x 10 8 


0.335 


3.69 x 10 11 


4.09 x 10" 2 




14 


6.28 x 10" 5 


669 


21 


1.41 x 10 8 


0.238 


9.26 x 10 11 


1.03 x 10" 1 




15 


2.50 x 10" 5 


492 


20 


5. 10 x 10 7 


0. 170 


2.33 x 10 12 


2.58 x 10" 1 




16 


9. 95 x 10" 6 


362 


19 


1.83 x 10 7 


0. 121 


5.84 x 10 12 


6.48 x 10" 1 




17 


3. 96 x 10" 6 


266 


18 


6.55 x 10 6 


0.0859 


1.47 x 10 13 


1.63 




18 


1.58 x 10" 6 


196 


17 


2.33 x 10 6 


0.0608 


3.69 x 10 13 


4.09 




19 


6.28 x 10~ 7 


144 


16 


8.20 x 10 5 


0.0430 


9.26 x 10 13 


1.03 x 10 




20 


2.50 x 10" 7 


106 


15 


2.87 x 10° 


0.0303 


14 
2.33 x 10 


2.58 x 10 




21 


9. 95 x 10 


78.0 


15 


1. 14 x 10 5 


0.0223 


14 
5.84 x 10 


6.48 x 10 




22 


3.96 x 10" 8 


57.4 


15 


4.55 x 10 4 


0.0164 


1.47 x 10 15 


1.63 x 10 2 




23 


1.58 x 10" 8 


39.8* 


15 


1.81 x 10 4 


0.0121 


3.69 x 10 15 


4.09 x 10 2 




24 


6.28 x 10" 9 


25. 1* 


15 


7.21 x 10 3 


0.00884 


9.26 x 10 15 


1.03 x 10 3 




25 


2.50 x 10~ 9 


15.8* 


15 


2.87 x 10 3 


0.00653 


2.33 x 10 16 


2.58 x 10 3 




26 


9.95 x 10" 10 


10.0* 


15 


1.14 x 10 3 


0.00480 


5.84 x 10 16 


6.48 x 10 3 




27 


3.96 x 10" 10 


6.30* 


15 


4.55 x 10 2 


0.00353 


17 
1.47 x 10 


1.63 x 10 4 




28 


1.58 x 10" 10 


3.98* 


15 


1.81 x 10 2 


0.00260 


3.69 x 10 17 


4.09 x 10 4 




29 


6.28 x 10" U 


2.51* 


15 


7.21 x 10 


0.00191 


9.26 x 10 17 


1.03 x 10 5 




30 


2.50 x 10" U 


1.58* 


15 


2.87 x 10 


0.00141 


2.33 x 10 18 


2.58 x 10 5 




31 


9.95 x 10" 12 


1.00 


15 


1.14 x 10 


0.00103 


5.84 x 10 18 


6.48 x 10 5 





* Maximum radius permitted by solar light pressure. 
** These No. based on entrance to atmosphere at 100 km approx 
*** Includes earth's shading effect of 1/2 
V \U3 



t P 



c (sp*r) 



447 x 778.3 ft lb/lb for Al 
11-47 



where 



1.08x 10" 4 V 2/3 

P = penetration in thick target 

d = meteorite diameter 

V = meteorite velocity in cm /sec. 

Whipple's equation is theoretical and is 
believed to give penetration depths for hyper - 
velocity impacts that are too high. 

b. Kornhauser's equation 

This equation is given in (Ref. 68) as 



K 2<?) 1/3 <^") 



0.09 



where 



h = penetration (depth of crater) 

K„ = constant of proportionality 

T = kinetic energy of projectile 

E = modulus of elasticity of target 
material 



Eq = reference modulus 



This equation yields 



0.282 x 10" 4 V 2/3 



which is identical to Whipple' s except that the 
value of the constant is lower. 

c. Summer's equation 

This equation is an empirical equation based 
on experimental test data using many different 
projectile and target material combinations. As 
given in Ref. 6 9, the equation has the form of: 

p % 2/3 /v> 2/3 

P = 2.28 U>> (V) 



where 

P = penetration in a thick target 

d = diameter of projectile 

p = density of projectile 

p, = density of target 

V = projectile velocity 

C = speed of sound in target material 

For Whipple's meteorite density of p = 0.05 

3 P 

gm/cm , an aluminum target density of p. = 

3 5 

2. 8 gm/cm and C = 5. 1 x 10 cm/sec, the 

equation reduces to 



£ = 0.243 x 10" 4 V 2/3 

The agreement between this constant and that of 
Kornhauser is noted. 

d. Bjork' s equation 

This is a theoretical equation developed by 
Bjork (Ref. 70) using a hydrodynamic model to 
explain hypervelocity impact. He derived equations 
for the impact of aluminum projectiles on alumi- 
num targets and also iron projectiles on iron 
targets. In Ref. 71, Bjork gives the penetration 
of an aluminum projectile into an aluminum target 
as: 

P = 1.09 (m v) 1/3 
where 

P = penetration in cm 

m = projectile mass in gm 

v = impact velocity in km /sec 

Bjork in Ref. 72 states that the use of a correction 

factor of the form! — ] is subject to a great 

deal of conjecture as it rests on no theoretical 
basis. He also stated that he would favor the 
value of <j> = 1/ 3 and 9 = 1/3 in a general pene- 
tration equation such as: 



1/3 



equating the general and empirical relations. 

i no / vl/3 „ 1/3 -1/3 AA 
1.09 (mv) =K 3 m i° t \^) 

,09 . K 3 .,-"' (') "3 



For aluminum targets, p. = 2.8 gm/cm and 
C = 5. 1 km/sec, Kg = 2.63. 



Thus we may write 



P = 2.63m 



1/3 -1/3 

Pt 



ft) 



1/3 



Then, letting "d" equal the meteorite diameter 

3 
in cm and its density p p = 0. 05 gm/cm yields 

P - 2.63 (g. d p p ) p t ^j 



= 0.322 V 



1/3 



where 



P = penetration = cm 
d = meteorite dia = cm 
V = meteorite velocity = 



km 



11-48 



This probably stretches Bjork' s work more where 

than he would care to see done but it is necessary 
to obtain a comparison with the other formulas. 

e. Engineering model Then 



u = sin R /R. 
o 



For purposes of evaluating meteoroid effects 
upon propeilant storage vessel design, the follow- 
ing model has been recommended (Ref. 73). 



(1) 



The integral mass flux of particles 
is given by 



■13 



■10/9... , 2, , 

hits /m /sec, by 



$ =10 ' m 

particles of mass m gm and 
greater. Approximately 90% of 
the meteoroid flux is assumed to 

3 
have a density of 0. 05 gm/cm . 
The effective flux used in com- 
puting probability of hits is there- 
fore reduced by an order of magni- 
tude to compensate for the very 
low density meteoroids which will 
not follow the given penetration 
law. 

(2) The particle velocity (v) is 30 km/sec. 

(3) Penetration of impacting particles into 
a single thickness of steel is given by 

1/3 



S f = 1 - 1/2 (1 - cos u) 



1 - 



1 + cos (sin R /R) 



The integral mass flux thus becomes 

-14 -10/9 2 

$ = 10 m hits/m sec 

N (> m) = 8.64 x 10 _10 m " 10/9 hits /m 2 -day 

Eliminating the constant meteoroid velocity 
(30 km/sec), and expressing the penetration law 
in terms of mass gives 



m = 101.25 

as the mass in grams required to penetrate X cm 
of steel. With the flux and penetration expressed 
only by mass, it is convenient to combine the two 
relationships, obtaining 



P = 1. 



(mv) 



N (>m) = 8.64 x 10 



10 (P 3 /101.25)- 10/9 



(4) Aluminum is half as effective as steel 
in withstanding penetration. 

(5) The use of spaced sheets (Whipple 
bumpers) allows a reduction factor, 
B f = 5, in the total thickness required 

to withstand penetration. 

(6) Particle density, (p) is 3 gm/cu cm. 

(7) The area exposed to meteoroids is 
the total unshadowed surface area of 
the object. The shadowing can be ex- 
pressed in terms of an effective area 
by computing a factor to be multiplied 
by the actual area. This reduction 
factor will be in the ratio of a sphere 
with a conical segment removed to a 
sphere. The center of this sphere is 
the spacecraft and the conical segment 
is that volume intersected, as an ex- 
ample, by the Earth. Consider the 
following sketch 




= 1.46 x 10 

hits per square meter per day capable of pene- 
trating P cm of steel. The reciprocal of this 
relation is the average number of days between 
penetrations. To determine the thickness re- 
quired so that an area of A meters is not pene- 
trated on the average for at least T days, 



P = (AT • 1.46 x 10 



■7 



3/10 



8. 901 

7?~ 



(AT) 



3/10 



cm of steel 



This relationship is convenient to use for purposes 
of design after the effects of the time distribution 
of meteoroid encounters have been included. The 
Poisson distribution model has been used to elabo- 
rate on meteorite encounter probabilities. This 
distribution which is valid for uniform masses of 
low density is 



kt 



K' 



1 



where t is any selected interval, and -= 

average number of penetrations per day 

the probability of any number, K 

during time, t can be estimated. To determine 

the probability of no penetrations during T days 

(T = t) the relation reduces to 



is the 
Thus 
penetrations 



kt 



0. 368 



11-49 



so that the probability is 0. 368 that there will be 
no penetrations within the average number of 
days between penetrations. To find the time at 
the end of which the probability of no penetrations 
is 0. 99. 

0.99 = e" t/T 

t = -T In 0. 99 

t =0. 0101T 

For 0. 95 and 0. 90 probabilities, the correction 
factors are, respectively, 0.05 and 0. 10. For 
example, the average time between penetrations 

2 
for a 93 m steel surface 2. 5 cm thick is about 

1.6 x 10 days. There is a 0.368 probability 
that there will be no penetrations by the end of 
this time. For this structure, the limiting time 
for 0. 99 probability of no penetrations is 1. 6 x 

4 4 

10 days; for 0.95 probability, 8 x 10 days; and 

5 
for 0.90 probability, 1.6 x 10 days. 

Correspondingly, if the probability for no 
penetration of X thickness within T is 0. 36 8, then 
the thickness required for a 0. 99 probability of 
no penetrations in T days is 



international agreement in 1925, astronomical 
time is reckoned from midnight, so that the local 
time of day based on this origin is 

T = t + 12 h 

where t is the hour angle of the time reckoner. 
Because astronomers refer to two time reckoners, 
the sun and vernal equinox, there are two kinds of 
days; the solar day and the sidereal day. 



North celestial 
pole 



Observer's 
meridian 



Greenwich 
meridian 



(P kt at 0. 99) 



10/3 



pl0/3 
0.0101 




P, . at 0. 99 
kt 

for 0. 90 probability. 



= 3. 97P 



P, . at 0. 90 = 1. 96X 
kt 



More generally 
In (prob) = - 



t {1.46 x 10 ) A 
^TTT73 



The relationships between exposed area and 
time, aluminum thickness and oenetration prob- 
ability are illustrated in Fig. 33. 



C. CONVERSION DATA 

1. Definition of Time Standards and Conversions 
(Ref. 74) 

Time measurement may be based upon the 
period of motion of a stable oscillator, the decay 
of a radioactive isotope, or the period of any 
celestial body relative to the observer. The latter 
is the body chosen sometimes referred to as the 
time reckoner and a clock in most astronomical 
■research. The particular day is defined to be the 
time span between two successive upper or lower 
transits of the given time reckoner across the 
celestial meridian of the observer. Noon is the 
time of upper transit (the transit in the northern 
celestial hemisphere). Angles measured in the 
equatorial plane of the celestial sphere from the 
observer's meridian, O, westward are called 
local hour angles (see following sketch). Thus 
O is the local hour angle of vernal equinox. Then 

local time of day is the hour angle of the time 
reckoner for days beginning at noon. Since an 



The sidereal day is the interval between two 
successive upper transits of vernal equinox. 
Because this time reckoner is a point on the 
celestial sphere, an infinite distance from the 
earth, the sidereal day is the period of earth 
rotation relative to inertial space. Because side- 
real time is the hour angle of vernal equinox, it 
is given at any instant by the right ascension of 
a star that is crossing the observer's meridian 
at that instant. The best value for the sidereal 
day is 86164. 091 mean solar sec. 

The solar day, the interval between two suc- 
cessive upper transits of the sun, is 3 56 
longer than the sidereal day because the earth 
moves almost one degree each day in its orbit 
around the sun. Thus, the solar day is not ex- 
actly equal to the period of earth rotation. Also, 
the apparent sun (the sun we see) is not a pre- 
cisely uniform time reckoner because the orbit 
of the earth is slightly eccentric and the eliptic 
is inclined about 23° to the equatorial plane. Be- 
cause the apparent sun is a nonuniform time 
reckoner, the mean sun is used to measure civil 
time. The time unit is the average of the apparent 
solar days, the mean solar day and its length is 
defined to be 86400 mean solar sec. The differ- 
ence between apparent and mean solar time is 
called the "equation of time, " ET: 



ET = AT 



MT 



r M = A M " A A 



where 



AT = apparent time 
MT = mean solar time 



11-50 



t, = hour angle of apparent sun 

t = hour angle of mean sun 

Ivl 

A nT = right ascension of mean sun 
A . = right ascension of apparent sun 

Civil time, CT, is mean solar time measured 
from midnight, 

CT = x M + 12 h 

The local civil time at the Greenwich meridian 
is known as universal time, UT, or Greenwich 
mean time, GMT. 

The difference in local time at two places for 
the same physical instant is the difference in 
longitude, X: 

T l " T 2 = X 2 ' X l 

where \, in the astronomer's convention, is meas- 
ured positive westward from the Greenwich merid- 
ian. This equation applies for T measured in any 
system of local time, i.e., civil, apparent solar 
or sidereal times. For example. 



LMT = LCT = UT 



\ 



Fifteen degrees of longitude corresponds to an 
hour of time difference, so that for local mid- 
night at Greenwich, the corresponding local times 
at \ = 15° W and 30° W are 11:00 p.m. and 10:00 
p.m., respectively. The local time increases 
for eastward longitude changes. 

Since local civil times are the same only along 
a given meridian, some confusion is avoided by 
the use of time zones. The earth is divided into 
24 zones, each fifteen degrees of longitude wide. 
In the middle of each zone, at the "standard me- 
ridian, " local time differs from Greenwich time 
by an integral number of hours. The time read 
on a clock at any place, i. e. , standard time, is 
the local civil time of the standard meridian 
nearest the clock. Standard time differs in some 
places from zonal time where boundaries are 
twisted to suit geographical and political bounda- 
ries. 

Greenwich civil time is generally the system 
employed in astronomical almanacs. Therefore, 
conversions required most often are standard to 
GMT and GMT to standard. The conversion from 
a zone time to GMT is effected by dividing the 
longitude (in degrees) of the observation site by 
15 and obtaining the nearest whole number. This 
value is added to the zone time for sites west of 
Greenwich and subtracted for sites east of Green- 
wich. 



GMT = ZT ± 



T5" 



The same rule applies for conversion of standard 
times, except that the irregular boundaries for the 
time zones must be utilized. 

The preceding discussions provide the basis 
for an appreciation of the measurement of time 
intervals; however, in order to relate any two 



events in time it is necessary to refer them to the 
same time reference. For earth satellite prob- 
lems this requires only that an epoch be selected 
and that the universal time be recorded at the in- 
stant. A record of time by days and /or seconds 
from this epoch thus relates all of the events. In 
other problems where two or more bodies are in- 
volved such an arbitrary solution of the time origin 
for one body may lead to unnecessary complexity 
due to the fact that all of the various time scales 
must be correlated each time a computation is 
performed. To avoid such a situation the Julian 
day calendar was established by the astronomers. 
This calendar takes the origin to be mean moon 
4713 years before Christ and is a chronological 
and continuous time scale, i.e., days have been 
counted consecutively from this date to present. 
This practice avoids problems resulting from the 
nonintegral period of the earth (365. 2563835 mean 
solar days) and the difficulties of months of differ- 
ent length. On this calendar January (i. e. , 
mean noon January 1) 1900 is 2415020 mean solar 
days. The conversion of other dates in the later 
half of the 20th century is facilitated by Table 37 
obtained from The American Ephemeris and 
Nautical Almanac. 

2. Review of Standards of Length and Mass 

For many engineering purposes the conversions 
between units of measure need be known only to 
two or three significant figures. For this reason 
a general unawareness of the definition and use of 
these units has resulted and is evidenced by in- 
consistencies in the literature. The purpose of 
this section is to redefine a set of units and specify 
accepted conversions from this set to other com- 
monly used systems. 

a. Standard units 

The United States' system of mass and measures 
has been defined in terms of the metric system 
since approximately 1900; it was refined in metric 
terms in 1959. Therefore, care must be exercised 
to assure that proper standards are used for all 
precise computations. Before going further it is 
necessary to obtain an appreciation for the bases 
for measurement. 

7 
The meter was originally defined to be 1/10 
part of 1 /4 of a meridian of the earth. A bar of 
this length was constructed and kept under standard 
conditions in the Archives. Since subsequent meas- 
urements of the earth proved this definition to be in- 
correct, a new international standard, the Prototype 
Meter, was defined to be the distance between 
two marks on a platinum-iridium bar at standard 
conditions. This bar was selected by precise 
measurement to have the same length as the bar 
in the Archives. National standards were also 
produced and compared to the Prototype Meter. 
In October 1960, at the Eleventh General Con- 
ference on weights and measures, the meter was 
redefined to be 1,6 50, 763. 73 wavelengths of the 
orange -red radiation of Krypton 86. However, 
the bar standards are also maintained because of 
the ease of measurement. 

The kilogram was originally defined to be the 
mass of 1000 cubic centimeters of water at its 
maximum density (i.e. , 4° C). However, at the 
time the Prototype Meter was defined, the kilo- 



11-51 



TABLE 37 

Julian Day Numbers for the Years 1950-2000 
(based on Greenwich Noon) 



Year 


Jan. 0.5 


Feb. 0.5 


Mar. 0.5 


Apr. 0.5 


May 0.5 


June 0.5 


July 0.5 


Aug. 0.5 


Sept. 0.5 


Oct. 0.5 


Nov. 0.5 


Dec. 0.5 


1950 


243 


3282 


3313 


3341 


3372 


3402 


3433 


3463 


3494 


3525 


3555 


3586 


3616 


1951 




3647 


3678 


3706 


3737 


3767 


3798 


3828 


3859 


3890 


3920 


3951 


3981 


1952 




4012 


4043 


4072 


4103 


4133 


4164 


4194 


4225 


4256 


4286 


4317 


4347 


1953 




4378 


4409 


4437 


4468 


4498 


4529 


4559 


4590 


4621 


4651 


4682 


4712 


1954 




4743 


4774 


4802 


4833 


4863 


4894 


4924 


4955 


4986 


5016 


5047 


5077 


1955 


243 


5108 


5139 


5167 


5198 


5228 


5259 


5289 


5320 


5351 


5381 


5412 


5442 


1956 




5473 


5504 


5533 


5564 


5594 


5625 


5655 


5686 


5717 


5747 


5778 


5808 


1957 




5839 


5870 


5898 


5929 


5959 


5990 


6020 


6051 


6082 


6112 


6143 


6173 


1958 




6204 


6235 


6263 


6294 


6324 


6355 


6385 


6416 


6447 


6477 


6508 


6538 


1959 




6569 


6600 


6628 


6659 


6689 


6720 


6750 


6781 


6812 


6842 


6873 


6903 


1960 


243 


6934 


6965 


6994 


7025 


7055 


7086 


7116 


7147 


7178 


7208 


7239 


7269 


1961 




7300 


7331 


7359 


7390 


7420 


7451 


7481 


7512 


7543 


7573 


7604 


7634 


1962 




7665 


7696 


7724 


7750 


7785 


7816 


7846 


7877 


7908 


7938 


7969 


7999 


1963 




8030 


8061 


8089 


8120 


8150 


8181 


8211 


8242 


8273 


8303 


8334 


8364 


1964 




8395 


8426 


8455 


8486 


8516 


8547 


8577 


8608 


8639 


8669 


8700 


8730 


1965 


243 


8761 


8792 


8820 


8851 


8881 


8912 


8942 


8973 


9004 


9034 


9065 


9095 


1966 




9126 


9157 


9185 


9216 


9246 


9277 


9307 


9338 


9369 


9399 


9430 


9460 


1967 




9491 


9522 


9550 


9581 


9611 


9642 


9672 


9703 


9734 


9764 


9795 


9825 


1968 




9856 


9887 


9916 


9947 


9977 


*0008 


*0038 


*0069 


*0100 


*0130 


*0161 


*0191 


1969 


244 


0222 


0253 


0281 


0312 


0342 


0373 


0403 


0434 


0465 


0495 


0526 


0556 


1970 


244 


0587 


0618 


0646 


0677 


0707 


0738 


0768 


0799 


0830 


0860 


0891 


0921 


1971 




0952 


0983 


1011 


1042 


1072 


1103 


1133 


1164 


1195 


1225 


1256 


1286 


1972 




1317 


1348 


1377 


1408 


1438 


1469 


1499 


1530 


1561 


1591 


1622 


1652 


1973 




1683 


1714 


1742 


1773 


1803 


1834 


1864 


1895 


1926 


1956 


1987 


2017 


1974 




2048 


2079 


2107 


2138 


2168 


2199 


2229 


2260 


2291 


2321 


2352 


2382 


1975 


244 


2413 


2444 


2472 


2503 


2533 


2564 


2594 


2625 


2656 


2686 


2717 


2747 


1976 




2778 


2809 


2838 


2869 


2899 


2930 


2960 


2991 


3022 


3052 


3083 


3113 


1977 




3144 


3175 


3203 


3234 


3264 


3295 


3325 


3356 


3387 


3417 


3448 


3478 


1978 




3509 


3540 


3568 


3599 


3629 


3660 


3690 


3721 


3752 


3782 


3813 


3843 


1979 




3874 


3905 


3933 


3964 


3994 


4025 


4055 


4086 


4117 


4147 


4178 


4208 


1980 


244 4239 


4270 


4299 


4330 


4360 


4391 


4421 


4452 


4483 


4513 


4544 


45 74 


1981 




4605 


4636 


4664 


4695 


4725 


4756 


4786 


4817 


4848 


4878 


4909 


4939 


1982 




4970 


5001 


5029 


5060 


5090 


5121 


5151 


5182 


5213 


5243 


5274 


5304 


1983 




5335 


5366 


5394 


5425 


5455 


5486 


5516 


5547 


5578 


5608 


5639 


5669 


1984 




5700 


5731 


5760 


5791 


5821 


5852 


5882 


5913 


5944 


5974 


6005 


6035 


1985 


244 


6066 


6097 


6125 


6156 


6186 


6217 


6247 


6278 


6309 


6339 


6370 


6400 


1986 




6431 


6462 


6490 


6521 


6551 


6582 


6612 


6643 


6674 


6704 


6735 


6765 


1987 




6796 


6827 


6855 


6886 


6916 


6947 


6977 


7008 


7039 


7069 


7100 


7130 


1988 




7161 


7192 


7221 


7252 


7282 


7313 


7343 


7374 


7405 


7435 


7466 


7496 


1989 




7527 


7558 


7586 


7617 


7647 


7678 


7708 


7739 


7770 


7800 


7831 


7861 


1990 


244 


7892 


7923 


7951 


7982 


8012 


8043 


8073 


8104 


8135 


8165 


8196 


8226 


1991 




8257 


8288 


8316 


8347 


8377 


8408 


8438 


8469 


8500 


8530 


8561 


8591 


1992 




8622 


8653 


8682 


8713 


8743 


8774 


8804 


8835 


8866 


8896 


8927 


8957 


1993 




8988 


9019 


9047 


9078 


9108 


9139 


9169 


9200 


9231 


9261 


9292 


9322 


1994 




9353 


9384 


9412 


9443 


9473 


9504 


9534 


9565 


9596 


9626 


9657 


9687 


1995 


244 9718 


9749 


9777 


9808 


9838 


9869 


9899 


9930 


9961 


9991 


*0022 


*0052 


1996 


245 0083 


0114 


0143 


0174 


0204 


0235 


0265 


0296 


0327 


0357 


0388 


0418 


1997 




0449 


0480 


0508 


0539 


0569 


0600 


0630 


0661 


0692 


0722 


0753 


0783 


1998 




0814 


0845 


0873 


0904 


0934 


0965 


0995 


1026 


1057 


1087 


1118 


1148 


1999 


245 


1179 


1210 


1238 


1269 


1299 


1330 


1360 


1391 


1422 


1452 


1483 


1513 


2000 


245 


1544 


1575 


1604 


1635 


1665 


1696 


1726 


1757 


1788 


1818 


1849 


1879 



1900 Jan 0.5 ET = Julian Day 2,415,020.0 = Greenwich Noon, January 1, 1900, a common epoch 
1950 Jan 0. 5 ET = Julian Day 2, 433, 282.0 = Greenwich Noon, January 1, 1950, another common epoch and 

first entry in this table 



11-52 



gram was redefined to be the mass of the Proto- 
type Kilogram and, as was the case with the 
Prototype Meter, national standards were obtained 
by comparison to the Prototype Kilogram. This 
unit has not been changed to date though proposals 
have been made to base the measurement on some 
atomic standard. The conversion from mass to 
force is accomplished by the standardized con- 

2 
stant g„ = 9. 80665 m/sec . 

Effective July 1, 1959, the English speaking 
people defined their standards of length and mass 
in terms of the metric system of units. This was 
accomplished through the definition of an inter- 
national yard and an international pound. 

1 yard = 0.9144 meter 

1 pound (avdp) = 0.453,592,37 kilogram 

These two units constitute the basis for all measure 
with the exception of those accomplished by the 
U.S. Coast and Geodetic Survey which continues 
to use a foot defined by the old standard: 



The statute mile = 5280 international feet. 



1 foot = 



1200 
3~9~3T 



meter 



3600 

= 0. 91440182 meter 

Of course, other units of length, area, volume, 
etc. , can be related by their definition to these 
more basic units. These second generation units 
(for example: statute mile, nautical mile, etc. ) 
are in general peculiar to particular regions and 
thus only a few will be discussed in the following 
paragraphs. 

The astronomical unit (AU) is defined as the 
mean distance from the sun to a fictitious planet 
whose mass and sidereal period are the same as 
those used by Gauss for the earth in his determina- 
tion of the solar gravitation constant. This defi- 
nition enables the astronomer to improve his knowl- 
edge of the scale of the solar system as more ac- 
curate data become available but does not require 
recomputation of planetary tables since angular 
data can be computed with an accuracy of eight or 
nine significant figures. The best value of this 

c 

unit is presently 149. 53 x 10 km and the mean 
distance from the earth to the sun is presently con- 
sidered to be 1. 000,000,03 AU. 

The nautical mile was originally defined to be 
one minute of arc on the earth 1 s equator. On this 
basis the best value of this unit appears to be ap- 
proximately 6087 feet. Various attempts have been 
made to adopt a standard length, e.g., the British 
nautical mile was defined to be 6080 feet and the 
U.S. nautical mile was defined to be 6080.20 feet. 
In 1954, it was agreed to standardize the nautical 
mile by defining it in terms of the meter. As a 
result, the international nautical mile was defined 
to be 1852 meters, or, based on the conversion 
between feet and meters at the time, 6076. 10333 
feet. But with the redefinition of the foot (1 foot = 
0. 3048 meter) as of July 1959, the nautical mile 
changed once again to 6 076. 11549 international feet, 
approximately. This value has been accepted by 
the National Bureau of Standards and all respon- 
sible agencies. 



The meter was previously defined; however, 
many units of length have been defined based on 
the prime unit and related by powers of 10. Ac- 
cordingly the following prefixes have been intro- 
duced and are generally recognized: 

12 
tera, meaning 10 

9 
giga, meaning 10 

mega, meaning 10 

3 
kilo, meaning 10 

2 
hecto, meaning 10 



deka, meaning 10 
deci, meaning 10 



-1 



centi, meaning 10 

_3 
milli, meaning 10 

,.-6 
micro, meaning 10 

nano, meaning 10 

,„-12 
pico, meaning 10 

The yard = 0. 9144 meter 

= 3 international feet 

The foot = 0.3048 meter 

= 12 international inches 

The inch = 0. 0254 meter 
= 10 3 mils 

The micron = 10 meter 

The angstrom = 10 meter 

3. Mathematical Constants 

u =3. 141, 592,653,6 

2ir = 6.283, 185,307,2 

3ir = 9.424, 777,960, 8 

log 10 Tr = 0.497, 149,872,7 
log it = l. 144, 729,885, 8 
e =2. 718,281,828,5 

log 1Q e = 0.434,294,481,9 
e 2 = 7,389,056, 102 

log 10 = 2.302,585,091 
1/ir = 0.318,309,886,0 

l/2ir = 0. 159, 154,943,0 

l/3ir = 0. 106, 103,295,3 

360/2ir = 57,295,779,51 



11-53 



1/e = 0.367,879,441,0 

1/e 2 = 0. 135, 335,283, 1 



4. Time Standards 



1 second = 



10 



3.155,692,597,47 

times the Besselian (tropical, 
solar) year at 1900. and 12 hr 
ephemeris time 

-9 
1 mean solar sec >» (1 + 10 ) ephemeris 

seconds in 1960 

sidereal day = 86, 164. 091 mean solar 

seconds 



sidereal year 



365. 256,383, 5 mean 
solar days 



sidereal year = 3. 155, 814, 9x10 mean 
solar seconds 

5. Conversion Tables 

Ready conversions for the more generally 
used units of astronomical measurements will 
be found in the following tables: 

Table 38--Length Conversions 

Table 3 9- -Velocity Conversions 

Table 40- -Acceleration Conversions 

Table 41--Mass Conversions 

Table 42 --Angular Conversions 

Table 43 --Time Conversions 

Table 44- -Force Conversions 



TABLE 38 
Length Conversions 



1 Astronomical Unit ■ 

1 International Nautical Mile = 

1 Statute Mile - 

1 Meter - 

1 International Yard ■= 

1 International Foot « 

1 International Inch = 





International 








International 


International 




Astronomical Units 


Nautical Miles 
60.737, 90xlO 6 


Statute Miles 
92. 911. 52x 10 6 




Meters 
149. 5266 x 10 9 


Yards 


Feet 


Inches 


1 


163. 524. 3 x 10 9 


490. 5728x 10 9 


588.687, 4X10 11 


1. 238, 575x 10" 8 


1 


1. 150, 779, 447 




1852* 


2025. 371, 828 


6076, 115, 485 


72. 913. 385. 826 


1.076, 292x 10" 8 


0.868, 976, 242 


1 




1609. 344* 


1760' 


5280'' 


63, 360* 


0.66B, 777, 3xl0" U 


0.539. 956, 803x 10~ 3 


0. 621. 371, 192x 10 


-3 


1 


1.093. 613, 298 


3, 280, 839, 895 


39. 370. 078, 740 


0. 611, 529, BxlO -11 


0.493, 736, 501 x ID -3 


0. 568, 181, 818x 10' 


-3 


0. 9144* 


1 


3*" 


36^ 


0. 203, 843, 3xl0" U 


0.164, 5?8,833xl0" 3 


0. 189, 393, 939x 10' 


-3 


0. 3048*" 


0. 333, 333, 333 


1 


12* 


0. 169, 869. 4xl0~ 12 


0. 137, 149, 02Bx 10" 4 


0. 157, 828,282x10 


-4 


0.0254* 


0.027, 777, 777 


0.083, 333. 333 


1 



1 Astronomical Unit per 
Mean Solar Day * 

1 Astronomical Unit per 
Sidereal Day ■ 

1 International Nautitfal 
Mile per Hour ■ 

1 Statute Mile per Hour > 

1 Kilometer per Hour - 

1 Meter per Second ■ 

1 Foot per Second ■ 



TABLE 39 
Velocity Conversions 



Astronomical Units 
per Mean Solar Day 


Astronomical Units 
per Sidereal Day 

1.002, 737, 90 


International 

Nautical Miles 

per Hour 


Statute Miles 
per Hour 


Kilometers 
Hour 


per 


1, 


Meters per 
Second 

, 730,632 x 10 6 


Feet per Second 


1 


3.364,079 x 1G 6 


3.871, 313 x 10 6 


6. 230, 273 x 


io 6 


5.677,928 x IO 6 


0.997,269, 57 


1 




3. 354, 692 x IO 6 


3. 860, 743 x IO 6 


6.213, 260 x 


10 6 


1, 


. 725.907 x 10 6 


5.662,424 x 10 8 


0.297,258, 2 x 10" 6 


0.298,072, 1 x 


io- 6 


1 


1. 150,779,447 


1.852* 




0, 


. 514,444,444 


1.667,809,856 


0.258,310, 3 x 10" 6 


0.259, 017, 5 x 


to" 6 


0.868,976,242,6 


1 


1.609, 344* 




0, 


,447,040* 


1.466,666,666 


0. 160,506,6 x 10" 6 


0. 160,946, 1 x 


io" 6 


0. 539,956, 803, 4 


0.621, 371, 192 


1 




0, 


,277, 777, 777 


0.911, 344,415 


0.577,823, 6 x 10" 6 


0. 579,405,6 x 


io" 6 


1,943,844,491 


2. 236,936,288 


3.600* 




1 




3.260,839,895 


0. 176,210,6 x 10" 6 


0. 176, 602,8 x 


IO" 6 


0. 592,483,800 


0. 681, 818, 181 


1.097, 280* 




0, 


. 304B* 


1 



— Underlined digits are questionable. 
♦ Denotes exact conversion factor. 



11-54 



TABLE 40 
Acceleration Conversions 



1 Astronomical L'nit 

per Solar Day - 
1 Astronomical Unit 

per Sidereal Day 2 = 
1 International Kautic 

Mile per Hour 2 = 
1 Statute Mile per 

Hour 2 ■ 
1 Kilometer per 

Hour = 
I Meter per Second 
1 International Foot 



Astronomical Units Astronomical Units International Nautical 

Miles per Hour 



per Mean Solar Day per Sidereal Day 

1 1.005,483, 30 

0. 994, 546, 6f3 1 



0. 713, 419, 4 x 10 ' 



0.619,9 44,7 x 10"' 



0.385,2 09,6 x 10 ' 
0.049,923,97 



0.015,216,62 



0, 717,3 31, 1 x 10 
0.623,344, 2x 10" 5 



0. 387, 3 21,9 x 10 ' 
0.050, 197, 70 



0.015, 300, 26 



.401, 700 x 10 J 



1.394,056 x 10 J 



0.&68, 976,242,6 



0.539,956,803,4 



Statute Miles per 



604, 250 x 1(T 



1. 150. 773,447 



0, 621, 371, 192 



0.699. 784.017,6 x 10 0. 805. 297, 064, 9 x 10 

0.213.294. 168.6 x 10* 0. 245,245, 245, 2 x 10 4 



Kilometers per 
Hour 2 



Meters per Second 



International Feet per 

Second 2 



65,716,76 



2.595.989x10 20.03 0,46 

2. 581.832 x 10 5 19.92 ], 23 55.. 15 8, 38 

1. 852- 1.4 29, 012, 345 i 10 - " 1 4. 688, 360, 71 1 x 10*' 

1.609. 344* 1.241, 777,7 78 x 10 ~ 4 4. 074, 074, 074 x 1 0"' 



1 



0.771,604,938. 2 x 10 2. 53 1, 5 1 2, 264 x 10" 
I 3.280,839,895 



1.395.020,800 0.3048* 



TABLE 41 
Mass Conversions 



1 Solar Mass 

1 Earth Mass 

1 Moon Mass 

1 Slug = 

I Kilogram = 



Solar Mass 
1 

3.088, 062 x 10" 
3,697, 320 x 10" 
7.346, Jj^x 10 
5. 033, 73 x 10" 3 



1 Pound (avdp) - 2.283,26x10 
1 Ounce (avdp) = 1.427,04x10" 



1.229,14x 10 ■ 
0.244,25 x 10" 
0. 167, 36 x 10" 
0. 759, j_5 x 10" 
0.474,47 x 10" 



27, G 4E, SCO 

a 1 . 3 58 

1 

0. 198, 72 x 10" 

0. 136, _16 x 10" 

0. 6 17, fi3 x 10" 

0. 386, 01 x 10" 



23 



21 



Slugs 
1.361, 25 x 10' 
4.094. 2 x HV 
5,032, 3 
1 

6.852, 176, 612 x 1(T' 
3. 10B, 095, 016 x 10" 
1.942,559,385 x 10"' 



Kilo gram s 

l. sae^ x 10 30 

5. OTrj^J) x 10 24 

7. 34-^0 x 10 22 

14.51)3,902,876 

t 

0. 453, 592, 37* 

0. 283.495, 231 x !0~ 



Pounds 


Ounces 


[avdp. 


<avdp) 


4. 379, 70 x I0" f ° 


70.075, 3 x 10 3 


13. 172, fi x HI 24 


210. 76 x 10 24 


16. 191, x 10 22 


259.06 x 10 22 


32. 174, 048, 55ii 


514, 784, 777, 


2.204, f.22,621 


35. 273, 961,94 


1 


16.0=-' 


0. 062, 5--'' 


1 



-Underlined digits are questionable. 
* Denotes exact conversion factor. 



32. 174,048,556 ft/sec 



TABLE 42 
Angular Conversions 



1 Revolution = 

1 Radian = 

1 Decree = 

1 Minute of Arc = 

1 Second of Are = 

1 Aitirular- Mil = 

-Denotes exaet cor 



1 

0. 159, 154, 043 

2. 777,777,777 X 10" 
4. 629, 629, 629 X 10" 

7. 716,049,382 x 10" 

-4* 

1. 5625 X 10 



(i. 283, 185,307 

1 

1.745,329,252 x 10" 

2.!)08,882,086 X 10" 

4.848, 136,812 x 10" 

9.817,477,040 X 10" 









Minutes 


Seconi 


is 






Decrees 






of Arc 


of Ar 


c 




Angular Mills 


360.* 






21,600. 0* 


1,296,000. 


0" 




6400. * 


57. 295, 779, 511 






3, 437,746,771 


206, 264,806, 


236 


1018. 591,636 


1 






60.0* 


3,600.0* 








1. 666, 666,606 x 


10' 


-2 


1 


60. 0* 






0. 296,296, 296 


2. 777,777,777 x 


10' 


■4 


0. 016,666,666 


1 






4. 938,27 1. 605 X 10 


5.6250 x 10" 2 * 






3. 375* 


202. 5" 






1 



11-55 



TABLE 43 
Time Conversions 









Solar Year 








Julian 


Year 


Mean Solar 


Day 




Sidereal Day 




Mean Solar Sec 


Siderea 


1 Sec 


1 Solar or 
Besselian Year 


- 


1 








0. 


999, 978, 


641 


365.242, 198 








366.242. 198 




3. 155.692.59 X 10 ' 


3. 164,332, 


5 7 x 10 


1 Julian Year 


= 


1. 


000,021,358 






I 






365. 25 








366. 250,00 




3. 155,760* X 10 7 


3. 164,400, 


16x10 


1 Mean Solar Day 


• 


2. 


737,909,26 x 


10 


-3 


2. 


737,850, 


787 x 10~ 3 


1 








1.002.737.9O 




8640O* 


86636.555 




1 Sidereal Day 


• 


2. 


730,433,61 x 


10 


-3 


2. 


730,375, 


42 x 10~ 3 


0. 997, 269, 57 








1 




86164.091 


86401)" 




1 Mean Solar Sec 


« 


3. 


168,876,46 X 


10 


-8 


3. 


168,808, 


78 x 10" 8 


1. 157,407,40 


X 


10 


-5 


1. 160.576,27 x 


10 "^ 


1 


1. 002,737, 


90 


1 Sidereal Sec 


= 


3. 


160,224,08 x 


10' 


-8 


3. 


160, 156, 


58 x 10" 8 


1. 154.247. 18 


X 


10 


-5 


1. 157,407,40 x 


10" 5 


0. 997.269,57 


1 




Exact conversion 





































Kg (force) 



TABLE 44 
Force Conversions 

Pound (force) Newton 



1 Kg Force 
1 Pound 
1 Newton 
1 Poundal 
1 Dyne 



1 



0.453,592,370, 1 
0. 101,971,621,2 



2.204,622,621 

1 

0.224,808,943 



1.409,808, 183 x 10 3. 108,095,501 x 10 



-2 



1.019, 716,212 x 10 



0.224,808,943 x 10 



-5 



9. 806,65* 

4.448,221,62 

1 



Poundal 
70. 931,635,35 
32. 174,048,6 
7.233,013,85 



Dyne 



5* 



0.138,254,954 1 
-5 



10 



7.233,013,85 x 10 



9. 806,65 x 10 

4.448,221,62 x 10 5 

10 5 

0. 138,254,954 x 10 5 

1 



*Exact conversion 



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11-59 



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11-60 



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11-61 



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11-62 



ILLUSTRATIONS 



II -6 3 



100 



90 



80 



o 70 



> 

ID 



60 



c 

■v 

c 
o 



U 50 



40 



30 

























































































































\ 






































Confidence level for ±lcf interval 




















































\ 










±28 interval 






























/ 




































/ 




































/ 





















































































































































































5 6 7 8 

Number of Data Points 



10 



11 



Fig. 1. Confidence Level for the Value of y' as a Function of the Number of 
Data Points and Size of Interval 



f*****"* **Ci6 W.ANK NOT FILWPT 



11-65 



M 


O 


S-H 




o 




<D 


A 


B 




c 


a 



3 50 



o 
v 

O 




-300 



-250 



200 



T3 
C 
nj 

O 



- 150 



a 



- 100 



- 50 



180 



200 220 240 260 



280 



Fig. 2. Present Standard and Model Atmospheres, and Proposed Revision 
of U.S. Standard Atmosphere 



11-66 



Defining T M Task Group IV 

— "" ~~~~ Related kinetic temperature 
Defining T M of ARDC 1959 



U. S. Standard Atmosphere 196 2 



Defining T M of present U.S. Standard- -ARDC 1956 



700 



•a 

3 




1000 1500 2000 

Temperature (°K) 



2500 



3000 



Fig. 3. Temperature Versus Altitude, Defining Molecular Scale Temperature 
and Kinetic Temperature of the Proposed Revision to the United 
States Standard Atmosphere 



II-67 



3000 



2800 



2600 



2400 



2200 



2000 



t- 




3 




cd 




CI) 


1800 


a 




6 




CD 




H 


1600 


01 








a! 




a 




(/i 






1400 


t. 




at 




i — H 




3 




01 


1200 



1000 



800 



600 



400 



200 



^™ Proposed U.S. standard atmosphere 
— - (Task Group IV) 

_ «_ Proposed U.S. standard atmosphere 

(Task Group I) 

— — ARDC model atmosphere 

4 Ion and other gauges at WSPG (NRL) 

A Ion and other gauges at WSPG 

~~ (USAF- Michigan) P 

t> Ion and other gauges at / 

Churchill (NRL) / 

X Grenades / 

— "^ Russian containers / 

— Satellite drag model day 7 

_ — — Satellite drag model night / 

• Kallman. March 13 (computed from/ 

pressure scale height) / 

— Russian standard atmosphere, 1 




J 


^^ 


""^ 


*^i 




/ 


December 1960 

— O Sodium cloud resonance 

scattering 












*^^^ 




r*' 


^ 


- 






sJ* 


f ^^ 


rM • 


1 „ 


r^ 




— 


— ■ 
























- 














— 



















— 
















vj-. 






1 


PI 






Proposal ^ 


repared by T 


ask Group IV 


1 1 1 


1 


1 


I 


i 


I 


1 1 


1 


1 1 


1 


1 


1 



20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 

Altitude (km) 



Fig. 4. Molecular-Scale Temperature Versus Geometric Altitude 
Proposed United States Standard Atmosphere Compared 
with United States Detailed Data, Russian Average Data, 
and ARDC Model Atmosphere 1959 for Altitudes Above 80 km Only 



11-68 



Proposed U, S, standard atmosphere 

< Ion and other gauges at WSPG (NRL) 

t> Ion and diaphragm gauges at Churchill (NRL) 

A Radioactive ion gauges (USAF- Michigan) 

o Sphere drag (USAF-Michigan and SCEL-Michigan) 

d Bennett mass spectrometer at Churchill (NRL) 

"^ Russian average of containers and rocket data at central European 
Russia (data 110 km and below are for summer days) 

f Russian satellite-borne manometer for May 16, 1958 
(1300 to 1900 local time, 57° N to 65° N) 

V Manometer on Sputnik I 

X Grenade 

Satellite drag model, day active sun (3) 

Satellite drag model, night active sun (3) 

Satellite drag model, day quiet sun (1) 

Satellite drag model, night quiet sun (1) 

■ ARDC model 1959 



^\ 



) Jacchia 



to 

c 
<v 
Q 



u 

0) 

to 
o 



10 



10 = 



— 10 



10 



10 



-9 



< ID" 10 t 



10 



-11 




180 200 

Altitude (km) 



280 300 



Proposal prepared by Task Group IV 
October 15, 1961 



Fig. 5. Density Versus Geometric Altitude for Proposed United States 
Standard Atmosphere Compared with United States Detailed Data, 
Russian Average Data^and ARDC Model Atmosphere 1959 



11-69 



ARDC model atmosphere 1959 

■— - Proposed U.S. standard atmosphere 

-4 Ion and other gauges at WSPG (NRL) 

£> Ion and other gauges at Churchill (NRL) 

A Ion and other gauges at WSPG (USAF- Michigan) 

■\ Russian average of containers for summer days mid- European Russia 

G" Russian satellite-borne manometer for May 16, 1958 
(1300 to 1900 local time, 57° N to 65° N) 

x Grenade 



10 



10 



B0 

X 



B io s 



u 

3 io 

in 
u 

a. 



-J 


1 










































- 


-3 


Is- 










































- 


-4 


= 


N 




4 




































= 


-5 






^ 

-< 


» « 


[< 




^. 






























- 


-fi 








< 


1 * 


1 

« 


1 « 


> [ 
1 < 


u 

1 


> 
> 


> 1 

> 


> 


1 < 


> 

1 < 
















- 


-7 


- 


















1 4 


' < 


1 , 


-f* 


^ 


T^ 


T~! 




*7~7 




1 




= 










































E 



10 = 



10 = 



80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 

Altitude (km) 

Proposal prepared by Task Group IV 
October 15, 1961 

Fig. 6. Pressure Versus Geometric Altitude for Proposed United States 
Standard Atmosphere Compared with United States Detailed Data, 
Russian Average Data and ARDC Model Atmosphere 19S9 



11-70 



Proposed revision to U. S. standard 

ARDC model atmosphere 1959 

CIRA atmosphere 1961 

Present U.S. standard, ARDC 1956 



700 



600 



■a 

3 



< 300 




14 



16 



18 20 22 24 26 

Molecular Weight [kg/ (kg- mol)J 



28 



30 



Fig. 7. Molecular Weight Versus Altitude 



11-71 



500 



400 



~ 300 



0) 

•a 

3 



< 200 



100 




Density (gm/cm ) 
a. to 500 km 



2500 



2000 



~ 1500 



0) 
3 



1000 



500 



\ 


L*" 




Miiiimum of 
sunspot cycle 






















Maximum of 
sunspot cycle 



















































































10 



•20 



10 



■15 



10 



■10 



Density (gm/cm ) 
b. to 2500 km 

Fig. 8. Average Daytime Atmospheric Densities at the Extremes of the Sunspot Cycle 



11-72 




10 

Air Density 
Sea- Level Density 

Fig. 9. Density of the Upper Atmosphere Obtained from the Orbits of 21 Satellites 



10- 



a 

o 
i 

s 

60 

CD 

O 



o 
o 
to 

Q. 



S = 190 x 10 



2 / 

m c I i 




-60 



-40 



-20 



20 



40 

An 



60 



80 



Fig. 10. Dependence of Atmospheric Density on &a 
in the Equatorial Zone (diurnal effect) 



100 



120 



11-73 



180 240 300 60 120 180 



s 

o 

E 4 

an ^• u 



o 

<M 



3.0 





O 


O 


O 






o 

— -»- 

A 


-A 


^~A"" 


A 


O 


"0 £ 
A 















A 6 | < 20° 
20" <|A 6 | <60° 



12 

t 



16 



20 



24 



Fig. 11a. Diurnal and Seasonal Variations in Atmospheric Density at 210 km 
Derived from Observations of the Satellite 1958 6 2. (The lower 
x-scale gives true local time, the upper Ao ■ a - o . The 
parameter of the curves is A 6 ■ t„ - 6 where a is right ascen- 
sion, 6 is declination, tt is perigee, o is sun.) 



180 



24 



300 







60 



120 



180 



2.0 



1.0 



- 










- 


- 










- 


- 






Jr$ 




- 


- 






r *■ 


\? 


- 






77 








- 




7 / 




V 




I 




\ 
i 


> 




^B 




12 

t 



16 



20 



24 



Fig. lib. Variations in Atmospheric Density at 562 km Above the Earth 

Ellipsoid Derived from the Observations of Satellite 1959 a 1 



180 240 300 60 120 180 



B io 



- 










- 


- 




ft 
// 


A 




- 


Ep a __ 
"a, 


i 




1 


a\ 

1 


i 



|A fi | < 20° 

56" > IA5 I > 20° 



12 

t 



16 



20 



24 



Fig. lie. Variations in Atmospheric Density at 660 km Derived from the 
Observations of Satellite 1958 8 2 



11-74 



I 1 1 1 

All values are corrected to mean solar activity 
(solar flux of 20 -cm radiation S = 170 x 10~ 22 W/m 2 -cps) 
The indicator of the curve gives the true local time. 




100 



200 



300 



h (km) 



500 



600 



700 



Fig. 12. Diurnal Variations of Atmospheric Density at Altitudes from 150 to 700 km above the Earth Ellipsoid 
for I A6| <20° 



200 




+20 +30 +40 +50 



Percent Density Departure of Seasonal Mean 
from Annual Mean 

Fig. 13. Model of the Seasonal Variation of Mean Density to 200 km 



11-75 




o 

c 
M 
o 



a 



(Iran) -»soo 



11-76 



T3 

ni 



IB 

o 

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10 



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l-:i.T=ppi 7 3rEpp|== = = = ==j=bt~ = ;-:4: r- = = . 


?---. z=fr -fi - ■■ -- - ■ Mi 7|:i-7W==f=ia=?=i|=!.:! i- =i- -■; = 


p t -i-- =EEE- =rpEE ===== ====,=.== :=p =]= - = =: 


=;---- ■-■-■- i ■-!-.- ._• • i I.. 7 £|= =!47p= :]-(-:=;= -'.- : i ~ 


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J- ' ; 1 ! : ; 1 - P ' - j 1 1 i -" J- : - ; i - s 


4-rp-' -"Fi™ " 1 gm/sq cm of aluminur 


i pgg"4- : ,'Vi"r- i ■^Lhr" ±± f"~~'! ' :i ~ ""' : ^ T 


fe-b£;:^g±L^Mzi=^pj=|=^t=H^ 




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p -j-c^ ;— H- - j — = — : : p 


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Sp : ; r r-j E |:j : ^.^-j= t 7 " - j-7;-::= := |~Mp{lMd A: f : -;.jl 


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Ep r-E _ _ . .. =,t-E d- - -;: :. IE - : "'"^E: E - i E 


J ^pH_ .-^ -^-u. -— - ^~ ,__ = , i- 


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j^y-^^g-^^^4.-::i7^---"|Arfc-4^ Jj.i i- 


■.. -- i- . - f - = -yX--- — -J=— = : -:~. - - -:-: 


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|p j- i ^ |Mpi| 


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ff —- H- i ]. 1 ;;i:;t:;.|i "u -p -L *-ei=E i ifi : i - 44 1 p 


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S~ "-- - t " I -"] i--'! -:-.;--: t ; ---[- -j- j- pj- =-j^|— = - -pp -|- -1 ■ I i 


3-..=: 7--- f= r f4 "■ p = "fi ^? : - 7=3= =|=;^i=fi i Ja= = =.= 


4---.:-.',-'. -\-=i" ■■-:: -.;7!..-::!-: --EN.-7- -.EL -- -- : i - - p-^ -j.! [.! 


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££===:=!=-:-=:-:-•- ::-r4 -_ ippp = - ::: - z -4p =EE =E =)=: !"E 


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1 i-Lpi" J J j ■j.-i =i-pj.-4r-^-- ::--!^pp -4*t = nfc- 








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^fcj=^^^p|m^ai|i>3l^pppp ;■-;: 


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"ITT"" P 4 -J-----E = |- ■ -|-- 1 -•-- 






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7 7 - | ff ; : -- / f - -rj- 77 7 _.^j .;_ 


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^=^^^^ s _-«__^._^9* ^3^4-^^^ ^j:_ - ;[ - " U -* -rj- 


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^^ ^£=^^ =._ E" j"1:"-f~ -j -pi' -- - i=^fe^^ ~— -t"^" "J i -t^i : 


-.-^j[ j jjr =j " : ~~ - r-- _ " " " - _ - ^ — f - ~ ~ - ~ ~ - — - 


^!^^£;[^=-_- ". r. i ^■r = i=E = k=fiE= = ==^^SZpEjE. = T: f--t- 


„: jf t-i-1 £ -- J— - -:-- LI- U- — — j — = =. =j=t= — — — — = ~^F- == - 


— J — — — b: _ --- ! 1- - - :-; — 1 ^ F -1-1- j |- - - 






;-::#::-.#:= frt -. ^ : ^p f : e== =1=1^== ===i === 


E=^=ill = E|4p4pEEp |E=EEEEa=ppEEj=5E- l' 


jf ft f ~- -J..!r|C— :r.^-7-i--rTr-T^— - 


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60 



65 



70 75 80 

' Magnetic Inclination 



85 



90 



Fig. 15. Solar Proton Dose, May 10, 1959 Flare, 30-Hour Duration 



11-77 



100 



10 



-a 
a! 

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0.1 



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jpljjpiljM 




t- [I'I =fc= 1 -j e « f^j j+- - -p ,ii | i ■—*■■¥"'-""""+-■ 


Nir-M'^ - 'itij-|4J^ST| ffiiflrffflTl [4 '- w- - 444^^*44 '^ - 4444 j mrf /^\] 


[ NT] il^U^rfT 4j44iTM"TIT IT fTr444+""~ 10 -Rf I' "-'44" 6 nrJ rr^ 42 ^ 




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55 



60 



65 70 75 80 

Magnetic Latitude of Orbital Inclination 



85 



90 



Fig. 16. Solar Proton Dosages from February 23, 1956 Flare. 



11-78 



Xi 

O 



10 





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Fraction of Solid Angle Subtended by Earth 
Fig. 17. Solid Angle Subtended by Earth as a Function of Altitude 




200 



240° 



320° 0° 

Geodetic Longitude 

Fig. 18. Magnetic Dip Equator (1) from USN Hydrographic Office, 19S5 and 
Geocentric Magnetic Equator (2) Inclined 13° to the Equator at 
Longitude 290° 



160° E 



11-79 




Fig. 19. Inner Van Allen Belt 



11-80 




Protons / sq- cm-sec- ster 
with energy > 20 mev 



Fig. 20. Flux of Protons at One Longitude in the Van Allen Belt 



11-81 



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Asra-oas- mo/suojojd 



11-82 




Electronj/aq cm-sec-rter 
with enerfjr > 20, 000 er 



Fig. 22. Flux of Electrons in the Van Allen Belts 



11-83 





c 
id 

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u 

V 

a. 

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II -84 



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Fig. 24. Electron Dose Rates 



10 



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B- -Outer belt 








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Fig. 25. X-Ray Dose Rates 



11-85 



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u 



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15 



20 



25 



30 



35 



Altitude Above the Surface of the Earth (km x 10 ) 



Fig. 26. Cosmic Radiation Intensity as a Function of Geomagnetic Latitude 
for High Altitudes During a Period of Low Solar Activity 



c 

C 

o 



0> 

c 

> 

O 
0) 



M 
O 

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6000 






























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Altitude above the surface of the earth (km) 

































































































































































































10 



20 



70 



80 



Fig. 27. 



30 40 50 60 

Geomagnetic Latitude (deg +) 

Relative Biological Effectiveness for Cosmic Rays as a Function of Altitude 
and Geomagnetic Latitude During a Time of Low Solar Activity 



90 



11-86 




5 ' 40 ' 80 T20 '"" 160 

Shield Mass Density (gram/sq cm) 

Fig. 28. Cosmic-Radiation Dosage as a Function of Shield Mass 















































































































































& Viewing Bun only 
O Viewing space 
O Viewing earth 






























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t 


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Energy (kev) 



Fig. 29. Differential Energy Spectrum Measured During Rocket 
Flight NN 8. 75 CF 



II -87 











































































































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( 


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Ap^'ireril Viiual Magnitude 



Fig. 30. Meteoric Mass Versus Apparent Visual Magnitude 





14 
13 


- 














12 






^v Whipple 


and Milln 


lan 








11 


















10 
















2 


















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9 
8 
























































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1 



-8 -V -6 



-4 - :i -2 - 1 o 1 2 :i 4 

Log. n m (mass in grams) 



Fig. 31. Meteoroid Frequency Versus Mass 



11-88 



Q.-2L. 











| | 




• 


• 
• 




s*> Explorer VIII (preliminary 
® Vanguard HI 
^ Explorer I 
• OSU Rockets 




• 


\ 1 


-1 




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Sputnik III 

^ Space Rocket I 

W Space Rocket II 






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A Pioneer I 


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.1 




















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10 ' 10 10" 

Particle Mass (gm) 



Fig. 32. Average Meteoroid Distribution Curve from Microphone 
System Measurements 




Time x Area (days/meter ) 

Fig. 33. Meteoroid Penetration Relations 



11-89 



INTRODUCTION 



CHAPTER III 
ORBITAL MECHANICS 

Prepared by: 

J. Jensen, J. D. Kraft and G. E. Townsend, Jr. 

Martin Company (Baltimore) 

Aerospace Mechanics Department 

March 1963 

Page 

Symbols III-l 

A. Introduction Ill -2 

B. Motion in a Central Field III-2 

C. Lagrangian Equation Ill - 3 

D. Orbital Elements III-3 

E. Motion in Three Dimensions Ill -4 

F. Properties of Elliptic Motion Ill -4 

G. Lambert's Theorem Ill -7 

H. The N Body Problem Ill -9 

I. Series Expansions for Elliptic Orbits Ill- 12 

J. Nomograms Ill - 1 4 

K. Tables of Equations of Elliptic Motion Ill - 1 5 

L. Presentation of Graphical Data Ill -39 

M. References 111-39 

N. Bibliography 111-39 

Illustrations Ill- 41 



LIST OF ILLUSTRATIONS 

Figure Page 

la Semimajor Axis as a Function of the Radius and 

Velocity at Any Point 111-43 

* 

lb Velocity- -Escape Speed Ratio 111-44 

2 The Relationship Between Orbital Position and 
Eccentricity and Time from Perigee (Kepler's 

Equation) 111-45 

3 Three -Dimensional Geometry of the Orbit Ill -46 

4 Geometry of the Ellipse 111-46 

5 Geometry of the Parabola 111-47 

6 Geometry of the Hyperbola 111-47 

1 T 

7 The Parameter — = -?*— , as a Function of Semimaior 

n 2ir J 

Axis 111-48 

8 Velocity of a Satellite in a Circular Orbit as a 

Function of Altitude 111-57 

9 Parameters of Lambert's Theorem 111-60 

10a Lambert's Theorem (case 1) Ill -61 

10b Lambert's Theorem (case 2) 111-62 

11a Solution for Eccentricity Ill- 63 

lib Solution for Eccentricity 111-64 

12 Solution for Apogee and Perigee Radii 111-65 

13a True Anomaly as a Function of ** a /r n and 
, /r 
a r 

13b True Anomaly as a Function of r/a, e, and 7 .... IH-67 

13c True Anomaly as a Function of r/a, e, and 7 .... Ill -68 

14 Solution for the Eccentric Anomaly as a Function of 

9, and e or r /r 111-69 

a p 

15 Q- Parameter as a Function of Orbital Semimajor 
Axis and Radius 111-70 

16 Relationship Between Radius, Eccentricity, 
and Central Angle from Perigee- -in Elliptic 
Orbit Ill- 71 

17 Local Flight Path Angle Ill -72 

Ill-ii 



r_ /r ..." . f. . P . 111-66 



LIST OF ILLUSTRATIONS (continued) 



Figure Page 

18 Solution for the Semiparameter as a Function of r, 

V and y Ill -73 

19 Q- Parameter as a Function of Local Flight Path 

Angle and Eccentricity Ill -74 

20 The Solution for Local Flight Path Angle Ill -75 

21 Index for Figs 22a through 22i' Ill -76 

22a Mean Anomaly as a Function of Eccentricity and 

Central Angle from Perigee 111-77 

22a' Mean Anomaly as a Function of Eccentricity and 

Central Angle from Perigee Ill -78 

22b Mean Anomaly as a Function of Eccentricity and 

Central Angle from Perigee Ill -79 

22c Mean Anomaly as a Function of Eccentricity and 

Central Angle from Perigee Ill- 80 

22d Mean Anomaly as a Function of Eccentricity and 

Central Angle from Perigee Ill- 81 

22e Mean Anomaly as a Function of Eccentricity and 

Central Angle from Perigee Ill -82 

22f Mean Anomaly as a Function of Eccentricity and 

Central Angle from Perigee Ill- 83 

22g Mean Anomaly as a Function of Eccentricity and 

Central Angle from Perigee Ill -84 

22h Mean Anomaly as a Function of Eccentricity and 

Central Angle from Perigee 111-85 

22i Mean Anomaly as a Function of Eccentricity and 

Central Angle from Perigee Ill -86 

22i' Mean Anomaly as a Function of Eccentricity and 

Central Angle from Perigee Ill -87 



III-iii 



III. ORBITAL MECHANICS 



SYMBOLS 

a Semimajor axis 

A Right ascension 

b Semiminor axis 

e Eccentricity 

E Eccentric anomaly 

f Force per unit mass 

F Force or hyperbolic anomaly 

g Acceleration due to gravity 

h Angular momentum 

i Inclination angle of the orbit to the equatorial 
plane 

I Moment of inertia; integral 

K Kinetic energy per unit mass 

L Latitude 

m Mass 

M Mean anomaly 

n Mean motion (mean angular velocity) 

p Semiparameter or semilatus rectum 

P Potential energy per unit mass 

r Orbital radius 

r Apogee radius 

r Radius to semiminor axis 
m 

r Perigee radius 
P 6 

r Radial velocity 

r Radial acceleration 

t Time 



P 
T 

U 

v 



Time of perigee passage 
Kinetic energy per unit mass 
Potential energy per unit mass 
Velo city- 
Orbital velocity at apogee 
Orbital velocity at perigee 

Components of position 



a Angle of elevation above the horizontal plane 

fi Azimuth angle measured from North in the 

horizontal plane 

7 Flight path angle relative to local horizontal 

e Total energy per unit mass 

G Orbital central angle between perigee and 

satellite position 

6 Angular velocity 

6 Angular acceleration 

A Longitude (positive for East longitude) 

pi Earth's gravitational constant 1.4077 

x 10 16 ft 3 /sec 2 (398, 601. 5 km 3 /sec 2 ) 

v Angle between the ascending node and the 

projection of the satellite position on the 
equatorial plane 

t Orbital period over a spherical earth 

<(> Orbital central angle between the ascending 

node and the satellite (9 + u) 

u Argument of perigee 

Q Longitude of ascending node 

Q Rotation rate of the earth (27r rad each 

e 86164. 091 mean solar sec 



III- 1 



A. INTRODUCTION 

The purpose of this chapter is to present data 
pertaining to the more elementary laws and con- 
cepts of orbit mechanics. The bulk of the material 
consists of graphs and tabulations of formulas for 
motion in elliptical orbits. In addition, a brief in- 
troductory treatment is given of the theoretical 
background. Many excellent books are available 
in the areas of analytical dynamics and celestial 
mechanics (see the bibliography at the end of the 
chapter). Therefore this chapter will only treat 
the material in outline form with no particular 
attempt to present a generalized and rigorous 
treatise on classical mechanics. 




From Eq (4) it follows that: 



B. MOTION IN A CENTRAL FIELD 

To a first approximation the earth can, dy- 
namically, be considered as a point mass located 
at the geometrical center of the earth. This im- 
plies that the mass distribution of the earth exhibits 
spherical symmetry, an assumption that does not 
strictly hold true and will be discussed further in 
the next chapter. Additionally, the earth's mass 
will be considered infinite with respect to that of 
a satellite moving in its gravitational field. Finally, 
no additional forces will be assumed to act on the 
satellite. Under these assumptions the gravitational 
force F = ^- (u = the earth's gravitational con - 

r 
stant) acting on the satellite will be directed toward 
the stationary center of the earth. The ensuing 
motion will be planar. 

In a rectangular coordinate system (in the plane 
of motion) as shown in the sketch below (assuming 
m to be constant), we get 



x 

m 



= - -^7 cos e 



f cos 9 



(1) 



r = constant 



(5) 



This constant is the angular momentum defined from 
vector mechanics. Substituting Eq (5) in Eq (3) re- 
sults in 



f . 



Now letting r = — it follows that 
f=h 2 u 2 f u + dfu\ = MU 2 



where time has been eliminated by: 
1 



u 



1 du \ __ , du 

u 2 dG 8 " " h d0 

u 



and 



V, d f du \ - h 2 n 2 d U 

r = ~ h W (ae) - " h u T72- 



Lj 
de 2 



Equation (6) can be written 



(6) 



—2— = - JL sin 9 = - f sin 



y m 



f*=y 



(2) 




de 2 



+ u 



h 



the solution to which can be recognized as: 



+ Ccos (6 



or in terms of r the solution is 



The motion is, however, more easily found in a 
polar coordinate system (r, 6) as shown in the 
sketch below. 



In this system: 
F 



r 

m 



= 



- JL = r - r9 



+ 2r e =1 *- (r 2 e) 



(3) 



(4) 



1 +— C cos (8 - 6„) 



1 + e cos (6 - 9 Q ) 



(7) 



The last form of Eq (7) is the standard form of a 
conic with the origin at one of the foci. From 
Eq (7) it can be seen that the semiparameter p 

h 

(semilatus rectum) is p = — and the eccentricity 



e is — C = pC. If e < 1 the conic is an 



III-2 



ellipse; if e = it is a circle; if e = 1 it is a parab- 
ola, and if e > 1 it is an hyperbola. 



C. LAGRANGIAN EQUATION 

The preceding integration of the equations of 
motion is based on an elementary approach. At 
this point a brief digression will be made into the 
more general Lagrangian technique often used in 
orbit mechanics, and encountered in Chapter IV. 

The Lagrangian equation for a conservative sys- 
tem is: 



_d_ 
dt 



m 



9L 



= 



(8) 



where the Lagrangian is L = T - U, T is the 
kinetic energy of the system and U the potential 
energy. The q's are generalized coordinates. 



For a two-body central force case the Lagrangian 
is (in polar coordinates) L = T - U 
- U (r). With q 1 = and q = r we get: 



1 f 2 _!. 2 Q 2 \ 

2 m (r + r 9 ) 



_d /3L\ 



3L 
"30" 



_d_ 

dt 



2 ;, 

m r 



= 



(9) 



where p = m r is the angular momentum of the 
system 



and 



_d_ 

dt 



v 8r ' 



9L 



_d_ 

dt 



•2 3U (r) _ 

m rO s - 

3r 



or, since 
9U 



F(r) 



-rj- (mr) - niri) = - F(r) 



2 



(10) 



From Eq (9) it follows that r 6 = constant. (This 
is commonly referred to as the law of areas. ) 

The orbit can be found by eliminating t from 
Eq (10). From Eq (9) 

2 dG _ 
mr ^ - p e 

we can conclude that 



Substituting this in Eq (10) we get: 

9 

\) d f P dr\ P 



Pn 



dO I 

Vmi 



2 OT 



3 



F(r) 



(11) 



or using u = ■ 



£($♦») ■♦*<£-■"»«= 



which, since p = lira, is identical to Eq (fi). 



D. ORBITAL ELEMENTS 



Equation (7) for the conic which embodies 
ler's first law defines the planar orbit of the 



Et 
Kepler's 
satellite when the constants p, e and 0^ are prop- 



erly evaluated from a set of initial conditions, 
such as V, r and y , where }' is the flight path 
angle as shown in the sketch below. Note that 

. 9 
Or = V cos y and hence Or" 



= r V cos y 



h = 



constant = 




¥ x 



of a number 



The three constants p, e and 0„, or any 

of equivalent sets of constants, describe completely 
the geometrical properties of the ellipse in the plane 
of motion. From a kinematic standpoint one more 
quantity is needed to specify the position of the 
satellite in its orbit. Frequently this specification 
is given in the"form of the time of perigee passage, 
although a knowledge of the position at any time is 
sufficient. 

Finally the plane of the satellite orbit must be 
described with respect to some reference plane. 
This description requires that two additional quanti- 
ties be specified, for example, the inclination of the 
orbital plane with respect to the reference plane and 
the orientation in the reference plane of the line of 
intersection between the two planes. The complete 
specification of the orbit therefore requires knowl- 
edge of six quantities, commonly called six elements 
of the orbit. Under the simplifying assumptions 
made in this chapter with respect to the dynamics 
of the orbital motion, these elements will be con- 
stants, whereas in the actual physical situation they 
will generally be varying as functions of time. 

A set of orbital elements in common usage is: 



and 



I 2 = p 6 _d_ / P _d_\ 

Jl 2 dO ( 2 do) 

t mr \mr / 



Semilatus rectum = p 

Eccentricity = e 

Time of perigee passage = t 



III- 3 



Inclination of orbit plane (with respect to 
earth equatorial plane) = i 

Argument of perigee (with respect to ascend- 
ing node) = cj 

Longitude of ascending node (with respect to 
vernal equinox) = £2. 



E. MOTION IN THREE DIMENSIONS 
From the solution of the orbit as expressed in 



the orbital plane, i. e. 



1 + e cos 6 



» an expression 



can readily be obtained for the three-dimensional 
description of the motion in any coordinate system. 
For this purpose define a coordinate system (x, y, 
z) in the orbital plane with the x-axis pointing 
toward perigee, the y-axis pointing in the direc- 
tion of r at = 90° , and with the z-axis completing 
a right-handed Cartesian coordinate system. In 
this system the defining equations for the motion 
are x = r cos G, y = r sin 9 and z = 0. To trans- 
form these equations into the (x\y', z 1 ) system 
shown in the sketch, the following transformation 
applies: , 




>y' 



cos Q cos w 




- cos iisin d 


sin Q sin i 


X 


- sin £2 cos 


sin u 


- sin Q cos i cos u 






sin U cos d 




- sin S7 sin u 


- cos U 3 in i 


V 


- cos f2 cos 


l sin cj 


+ cos tJcos i cos u 






sin i sin u 




sin i cos w 


cos i 


Jl 



Hence, since x = r cos 9, y = r sin 9, z = 0, 
x' = A' r cos 9 + B' r sin 9, etc. , etc. 

where 

A' = cos £2 cos u - sin £2cos i sin u 

and 

B 1 = - cos £2 sin u - sin £2 cos i cos u 

Now, since the orbital elements £2, to and i are 
constant for this discussion the velocity com- 
ponents are: 

x ' = A 1 .(r cos 9 - r sin 9 9) + B' (r- sin 9 + 
r cos 9 9) 



where 
r9 
and 



■w 



(1 + e cos 9) 



ff si 



in 9 



Similar expressions are found for the other coor- 
dinates. To reduce this description in inertial 
space to one of position relative to the rotating 
earth the following transformation is required 



cos £2 t sin £2 t 

e e 



sin £2 t cos Q t 

e e 



where n is the rotational rate of the earth and 

e 
t is the time since the x -axis, being in the prime 

meridian, passed the x' -axis, the x 1 axis is ori- 
ented toward the vernal equinox. 

z (north) 
r 





The sketch also shows the right ascension A 
and the geocentric latitude L. 



and 



L = arc sin — = arc sin — 
r r 



The longitude relative to the prime meridian 

measured positive in the direction of rotation is 

thus A = A - n t. 
e 



F. PROPERTIES OF ELLIPTIC MOTION 

Before progressing to a detailed discussion of 
the motion, two general properties should be con- 
sidered. 



III-4 



liquation (5): r" = r (rb) = 2dA = h = constant 
expresses the conservation of angular momentum 
and is a consequence of the fact that the moment 
of force about the center of motion is 0. It is also 
the equivalent of the "Law of Equal Areas" known 
as Kepler's second law. It is a general law of 
central motion (i.e. , for any force directed toward 
a fixed center of attraction and hence having zero 
moment about this point) since it was obtained with- 
out recourse to any specific force law. Since 
1_ 
2 



(rO) is the differential area dA swept by the 



radius vector- 



one obtains A = -^ lit + constant, 



and hence, Kepler's second law: the radius vector 
of any given planet sweeps through equal areas in 
equal time. 

The time t to complete a revolution can easily 
be found since the area of the ellipse is 7rab and 
since b = I'Sp, one obtains 



277 



3/2 



F 



Hence, Kepler's third law: the squares of the 
periods of the planets are to each other as the 
cubes of their semimajor orbital axes, or 



(in our case, f (r) = — - ). Thus, V(r) = — and 

9 °u r 

v" = - — + constant, where the constant is found 
r 

to be equal to - /u/a for elliptical motion, zero for 
parabolic motion, and t y/a for hyperbolic motion. 
In terms of the initial conditions v and r, the mo- 
tion is elliptical, parabolic or hyperbolic depend- 



ing on whether v" 



2U 



is negative, zero or 



positive, respectively. This equation is inde- 
pendent of the initial flight patli angle v. For 
elliptical orbits the resulting semimajor axis is 
given by 



r/u 



I U - r v 



dig. i) 



V = 



IFW1 



For a circular orbit r= a and the circular orbit 
velocity is given by 



For a parabolic orbit a is infinite and the so- 
called escape speed or parabolic orbital velocity 
becomes 



It also follows from Eq (5) that = — ^— or 

r 
the angular velocity is inversely proportional to 
the square of the radius vector. 

An important integral of the equations can be 
obtained by multiplying Fq (1) by 2 x and Eq (2) 

by 2 y , and adding them . 



2 x x + 2 y y 



— (x x + y y) 



d 

dt 



(-0--^ («'♦>!) 



1¥ 



So far only the geometry of the orbit has been 
determined, and it has been obtained through the 
elimination of time from the equations. _ To com- 
plete tin solution for elliptic motion, time is 
reintroduced by substituting the area integral 



r 2 5 



] » a ( 1 - e 



[Eq (5)] , into the "vis viva" integral which in 
polar coordinates lor elliptic motion takes the 
form: 



2 -222 2 1, 

v = r +r -UK— - —) 

I d 



f d ,2. 
7 dt (r » 



2fr 



Thus 



If now f is a function of r only, the entire equa- 
tion can be integrated to yield: 



^(M 



2 2, .2 

a e - (a - r) 



9 

V = 



2 V(r) + c, 



f (r) dr + constant = 



dt = 



a/ju dr 



Ia 2 



(a - rr 



where V(r) in a physical problem is a single valued 
function of r. This equation is known as the "vis 
viva" integral. The velocity is, in other words, 
only a function of the distance from the center of 
attraction . V (r) is the potential of the force f (r) 



Now, introducing the mean angular motion 



2 77 



I i/H. 

a I a 



III- 5 



results in the equation 



This form can be further modified to yield the new 
estimate of E directly by substituting 



ndt = - 



dr 



2 2, *2 

I a e - (a - r) 



To clean up this equation a new variable E is 
inti'oduced defined by a - r = a e cos E from which 
r = a (1 - e cos E) and 

n dt = (1 - e cos E) dE. 

This equation is integrable and yields upon inte- 
gration 

n (t - t ) - E - sin E 

This equation is commonly referred to as Kepler's 
equation. 

Because of the importance of and general interest 
in circular velocity, period and the mean angular 
velocity (mean motion), these quantities have been 
computed and presented in various forms in Figs. 7 
and 8 and in Table 9 in both English and metric units. 

The quantity E is called the eccentric anomaly 
(anomaly = angle or deviation). Its geometrical 
significance is shown in Fig. 4. The angle is 
referred to as the true anomaly. The quantity 
n(t - t ) is the angle which would be described by 

the radius vector had it moved uniformly at the 
average angular motion. It is called the mean 
anomaly and designated by M = n (t - t ). 

Hence, M = E - e sin E. This transcendental 
equation in E is known as Kepler's equation. Time 
from perigee passage for elliptical orbits is now 
obtained from: 



t - t 






: , . = E + AE 
n + 1 n 



(E - e sin E). 



e (sin E - E cos E ) + M 
__ n n n 

1 - e cos E 



This series solution converges very rapidly and 
generally requires only two iterations for six or 
seven significant figures (given a two -place esti- 
mate). Since one means of obtaining such an 
initial estimate is a graph or nomogram, a nu- 
merical solution of Kepler's equation may be found 
in Fig. 2. 

A peculiar property of elliptic orbits is that 
the velocity vector- at any point can be broken into 

components, V. and V' d (V = V^ + Vj), such that 

V, is constant in magnitude and perpendicular to 

the radius from the point of attraction to the instan- 
taneous point in the orbit and V , is constant in 

magnitude and continuously directed normal to the 
major- axis of the ellipse. This behavior is illus- 
trated in the following sketch. 

Since V, is constant, only V, contributes to the 

acceleration, and solely by a change of direction, 
i.e. , the acceleration must be radial and such that 

a = a = - V, 6 
1- b 

where is the angular rate of the radius vector. 
Rut, the acceleration at any point can also be ob- 
tained from the gradient of the potential function 
(which, in the case of a spherical homogeneous 
earth, or one constructed in spherically concentric 

homogeneous layers is — ). 



The solution of Kepler' s equation for time 
as a function of position is direct since there 
exists a unique value of E for each value of r or 
9. However, the reverse determination (for 
position as a function of time) involves the solution 
of Kepler's equation for- E. This solution is trans- 
cendental and thus requires iteration for conver- 
gence to the proper value of E. The best form of 
this iteration (assuming that a reasonable estimate 
of E is available) is Newton's method which is ob- 
tained directly from the Taylor series expansion 
of M as a function of the estimate of E and the 
mean anomaly. All higher order terms are neg- 
lected. 




M 



M o + he-< m > AE + 



Therefore 



a = V, = 
r b 



6 r 



AE 



M - M, 



di < M > 




M - M Q 


(E Q - e sin E Q ) + M 


1 - e cos E 


1 - c cos E 



Now since the acceleration is directed toward 
the center of mass, the moment with respect to 
this center must be zero, or 







constant 



r V cos y 



III-6 



This equation is recognized as the equation for 
conservation of angular momentum, or the area 
law. 

Thus 



This representation of the orbit also offers a 
simple means of determining the direction of the 
line of apsides of the orbit . The line of apsides 
is determined from the preceding sketch by 



V, 



9 r 



h 



U 



cos y 



<F 



tan <j> = 



sin Y 



tan y 

V ~ r 

b - - 1 

-— - cos y p 



The second component of the velocity, V ,, 
can be evaluated from the law of cosines. 



V d 2 = V b 2 + V 2 - 2V b V cos y 



This equation reduces to the following upon 
substitution 



-d= PmTI 



= eV t 



The quantities V, and V. can also be evaluated 
from the sketch when it is noted that 



V = V,+ V , 
p b d 



V = V, - V , 
a b d 



Now assuming that the apogee and perigee radii 
are known 



>-ter("3 



v 



"2VTr 



b p 



(' - » 



eV, 



G. LAMBERT'S THEOREM 

In Chapter VI, the problem arises of determin- 
ing an ellipse from a given time interval between 
two points on an arc of the ellipse as described by 
the two radius vectors terminating on the arc. 
From Kepler's equation and the definition of the 
true anomaly, one obtains 

n At = E 2 - E 1 - e (sin E 2 - sin E ) 



A9 



'(^--feS 



From these equations the ellipse can be deter- 
mined. The simultaneous solution of these equa- 
tions for a and e is, however, very difficult since 
the numerical iterative solution is quite sensitive 
to the accuracy of the first estimates of a and e. 
This problem is circumvented by the use of Lam- 
bert's theorem which can be developed as follows: 
Let 

2G = E + E and 2g = E - E 



2 1 



r 1 = a(l - e cos E ) 



2 1 



a(l - e cos E ) 



Thus 



r^ + r = 2a(l - e cos G cos g) 



The total energy in the orbit can also be related 
to these fundamental quantities. This is accom- 
plished as follows: 



Let C be the chord joining the extremes of 
and r as shown in the following sketch. 



Potential ener 



unit mass 



Total energy 
unit mass 



EL 



r 



u 

2a 



= -KE 



U 
2a 



Kinetic energy 
unit mass 

+ Potential energy 



2a 



unit mass 
V 2 



V 




b sin E 



III-7 



2 2 

C = (a cos E - a cos E ) 



+ (b sin E 2 - b sin E^ 



But the quadratic forms in cos E , cos E and 
sin E , sin E can be reduced to functions of G 
and g to yield 



C =4a sin Gsin g 

+ 4a 2 (1 - e 2 ) cos 2 G sin 2 g 



This form of the time equation may seem to 
have no major advantages. Closer examination, 
however, shows that for the case where the At is 
specified for transfer from r. to r„ through a 

given AG, and it is desired to find the unique ellipse 
whose parameters are a + e, this form may prove 
preferable. This conclusion is based on the fact 
that for this case only one variable of interest a 
appears explicitly though it is necessary in the 
process to solve for the auxiliary parameters 
« + 6. One source of possible error is the selec- 
tion of the proper quadrants for the angles e and 6. 
This selection may be accomplished by referring 
to the following statements. 



Now introducing a new variable h defined as follows: 

cos h = e cos G 
leads to 



C = 4a sin g (1 - cos h) 



C = 2 a sin g sin h 



and 



r. + r„ = 2a (1 - cos g cos h) 



(1) 



sin t; is 



(a) the arc includes perigee 
and the chord intersects 
the perigee radius 

(b) the arc excludes perigee 
and the chord does not inter 
sect the perigee radius 



(That is, sin 6/2 is positive when the seg- 
ment of the ellipse formed by the arc and 
chord does not contain the center of mass.) 



Now introducing two new variables 

£ = h + g 

6 = h - g 
enables the following equations to be written 



(2) 



cos -~ IS + 



(a) the arc contains perigee 
and the chord intersects 
the apogee radius 

(b) the arc does not contain 
perigee and does not inter- 
sect the apogee radius 



cos -g- (e + 6) = e cos -g- (E + E ^) 



(That is, sin e / 2 is positive when the seg- 
ment of the ellipse formed by the arc and 
chord does not intersect the apogee radius.) 



r 1 + r + C = 2a {l - cos (h + g)} 



4a sin 



2 E 



(3) < 



< 7T 



'i + r 2 " c = 2a { *■ " cos (h " g) y 



• 2 6 
4 a sin •=■ 



These equations serve as the definition of the 
quantities e + 6. But 



n (At) = E - E x - e (sin E 2 - sin E^ 



= U - 6) - 2 sin -| U - 6) cos -i U +6) 



(4) -\<\ 6< 1 



More detailed discussions of the reasoning for 
selecting these quadrants are presented in Ref. 1. 



Graphical solutions to this form of the time 
equation are also possible. One such solution was 
prepared by Gedeon (Ref. 2). Let 



2s = 



r l + 



r + C 



e - 6 - (sin £ - sin 6) 



/hich is known as Lambert's theorem. 



and 



2 j. 2 

r + v 

r l 2 



2 r. r„ cos A 



III- 8 



Now define a function w 

IS 



± |1 - C/J 



where the + sign is utilized if AC < t and the 
- sign is for AG > tr . 



Expanding the previous solution nAt in a power 
series for the case that the empty focus falls out- 
side of the area enclosed by the arc and the chord 

yields 



nAt = V 2 



1 -(W) 



2n+ 3 



(3 



n = 



A = l 



1.3.5 .. . ( 2n - 1) . (2n - 1)1 
~2.4. 6.8 ... 2n Tnl 




Force center 



In a similar manner, a power series representa- 
tion can be obtained for the case in which the arc 
and chord enclose the empty focus 



nAt 



^t- jlz v -i a 

L<S/2a> 3/a L Q 



1 + (W) 
n 2n+ 3 



2n+3 



(A) J 




where the A are the same as those defined 
above. 

Graphical presentation of this material is 
found in Figs. 9 and 10. 



H. THE N-BODY PROBLEM 

The previous discussions have been directed 
toward the description of the motion of a particle 
in the gravitational field of a mass sufficiently 
large that the perturbation due to the particle is 
completely negligible. Indeed the attractions of 
all other masses on both the particle and the 
central mass were neglected. The discussions 
of this section are intended to provide the 
generalizations which are possible in order that 
the discussions of perturbation methods of 
Chapter IV will be appreciated. 



Consider the differential equations 
n 

- G m. > m. 



m. r. 

i l 



(r. 



r j> 



L l 
j = l 



'ij 



This set is of the order 6n due to the fact that 
there are 3n coordinates (x-^z.) expressed as 

second order differential equations. A rigorous 
solution thus involves the simultaneous solution 
of the n second order vector equations. 

Since these forces are all conservative, it is 
also possible to express the total force acting on 
the vehicle as the gradient of a work function. 
Let 



F. 



V. U 
i 



Then 



3U 

F . = m. x. = - -5 — 

xi l l 9x i 

9U 
F ■ = m. y . = - -n — 
yi l- 7 ! 9y. 



F ■ = m. z. 

Zl 11 



au 



iz. 

l 



i = 1, . . ., h 

iltiply F xi by x, F . by y, F zi by z and add 

n 

/ m. (x. x. + y. y . + z. z.) = 
l_i ill J \ J \ li 



i = 1 



n / 

I 

i= l \ 



9U • 30 

X. + -5 

X. 1 ay. 
1 J 1 



y. + -rt — z 

J i 9z. l 



Force center 



But if a potential exists, U is a function of the 3n 
variables x., y., z. alone. Thus, the right-hand 

side is the total derivative of U with respect to t. 
Thus, upon integration 



UI-9 



il 



where 



m. (x. + v. + z. ) = -U + constant 
1 x 1 J i 1 



T + U = constant (energy equation) 

Now, potential energy is the amount of work re- 
quired to change one configuration to another. 
Thus, since the bodies attract each other ac- 
cording to the law of inverse squares, the force 
between bodies is 



G. m. m. 

i i J 

2 

r . . 



r.. 



Thus, the work is moving along the radius r^. is 



w. . 



r 
i m j j 



dr. 



G m. 



Gm. m. 



r(°)« 



L r " r -lij 



Now if r (0) is °° , all possible system configura- 
tions are included. Thus 



w. . 



Gm. m. 

L_J_ 

r . . 



Now the total work is the double summation of 
the individual works 



j=l i=l 
*3 



Gm. m. 



The one-half arises from the fact that if i and j 
are both allowed to assume all values, each term 
in the series will appear twice in the equation. 
Now following an argument of Moulton (Ref. 3), 
it can be stated that since the potential function 
depends solely on the relative positions of the n 
particles and not on the choice of origin, the 
origin can be considered to be displaced to any 
new point, yielding: 



r! = r. + r n 
i i 



r Q =ax + fiy + oz 



Thus 



3U 
3T 



I 



au 



9x.' 
i 



9x. 

x! = x. + a ; 5 = 1 

i l o a 



But U does not involve a explicitly, since it is a 
function of relative position thus upon dropping 
the prime which is now of no value 



i = l l 

n 
Similarly for > = and ) 



i = l 



i = l 



9 U 

9z. 
i 



Thus 



) m. r. = 
= 1 
n 
I m i^i = C 



i=l 



and 



) m. r. 



Ct + B 



i = l 



But ) m. r. is by definition M R which is the 

product of the total mass of the system and the 
position vector for the center of mass. Thus 



M R = C t + B 

This equation states that the center of mass obeys 

Newton's law F = ma (where F = = the resultant 
force) and moves with a constant velocity in a 
straight line under the assumption that there are 
no net forces acting on the center of mass. This 
integral introduces six constants of integration 
to the system requiring 6 n such constants. Now 
consider: 



m. r. = v- U 
li v i 



r. x m. r. = r. x v- U 
l li li 



LX 

> r. x m. r. = ) r. x V- 

A 1 1 1 Li I l 



i = l 



i=l 



But the forces occur in equal magnitude and 
opposite directions for any given pair of masses. 
Thus, the right-hand side of the equation is zero 
when summed over all the masses and 



III- 10 



yr. x m. r. = 
i 11 

= 1 A^xm.r.) 
= 1 

n 
T 7. ^"W 



i= 1 



i=l 



m - 

— - Ri 
m 2 1 



Substitution of this equality eliminates R„ from 
the equations 



R, 



- Gm2 (l + _i) 



2 R 



12 



Thus by direct integration once again it is seen 
that the total angular momentum is conserved 



R. 



<r ; 



x m. r.) = 



i = l 



Since this is a vector equation, three additional 
constants have been introduced. 

One more relationship between the coordi- 
nates and velocities can be obtained from the 
energy integral, the general form of which was 
presented earlier. Thus, ten integrals exist. These 
ten are the only integrals known and are the only 
integrals available from existing algebraic func- 
tions. Thus, the general solution of the n body 
problem requiring 6 n integrals is at this time 
impossible even though several operations can be 
performed to eliminate two variables, the line of 
node and the time. (The latter simplification is 
obtained by expressing each of the coordinates as 
a function of a given coordinate.) The sole excep- 
tion to this rule is the 2 -body problem. 

Consider the equations of motion 



■Gfmj + m 2 ) 



R 2 = -G (m. + m 2 ) 



where 



R 12 " R l " R 2 



m _ 

(1 + — ) R 

m 



Thus 



R, 



-M - 
m. 2 



Gm r 



M 



■"12 

R 2 
^12 



, -— R 1 
1 m 2 1 



- r (r l ' ^ 
m. r. = - Gm. m 2 - — 

r 12 
■A £ 2 - T x ) 

m 2 r 2 = -Gm im2 — 

r 12 

Changing origin to the center of mass by sub- 
stituting 



R r 



G m. R 2 



m r; 



With this substitution, the differential equations 
become uncoupled in the coordinates. But these 
equations are immediately recognizable as the 
differential equation for a conic section with the 
center of mass at the focus. Thus, as before, 
the solution will be of the form 



Ri 



R 2 " r 2 R 



R, 



1 



1 + e 1 cos 9 1 



yields 



- n R l- R 2 

m l R l = -Gm im2 3 - 

K 12 



■-1 R 2 - R 1 

m 2 R 2 = -G mi m 2 ^ 3 



R 



12 



But the center of mass satisfies the equation 



m 1 R 1 + m 2 R 2 = 



1 + e„ cos 6 2 



But it is important to note that the elements of 
these conies are not the same though they must 
be related. Indeed, the effective masses as seen 
by the two bodies will be different. This latter 
requirement is the result of requiring that the 
line between the two bodies contains the fixed 
center of mass at any time. However, it is 
possible to obtain a set of six constants of in- 
tegration a 1 , e r i r Y a y t Q1 and a dependent 



set a 2> e 2> 



L 2' 



1 a 2 and t Q2 which will produce 



III- 11 



the desired motion. This is accomplished by 
considering various elliptic relations and the 
geometry of the plane of motion. To illustrate 
the relationships, consider the requirement that 
the mean motions be the same. 



r- n ,n-2 
\ e d 



^, n! dM n " 2 

n = 1 



(sin" M) 



(15) 



""3" 



\^2/ 2 m l 



The other elements are determined in an 
analogous fashion. 

I. SERIES EXPANSIONS FOR ELLIPTIC ORBITS 

Many of the solutions to trajectory problems 
can be greatly simplified by utilizing approximate 
forms for the parameters involved. The general 
forms of several useful series are developed in 
this section, and a list of expansions is given in 
Table 6 (see Section K). 

Kepler's equation can be rewritten as 

E = M + e sin E (12) 

By Lagrange's expansion theorem, this expres- 
sion can be developed (see Goursat and Hedrick, 
"Mathematical Analysis, " Vol. I, p 404) in powers 
of eccentricity, e. 



= M + 



TO 

I 



J n-1 



dM 



— r (sin 1 M) (13) 
n-1 



From Eq (12) it follows immediately that 

. _, E - M 
sin E = 



From. Eq (12) by integration, 

I = - \ (E - M) dM = - \ e sin E dM 

= - e \ sinE(l -e cos E)dE 

= - e \ (sin E - j sin 2e) dE 

and using an arbitrary integration constant c, 

2 

I = c + e cos E - ^- cos 2E (1G) 

but integrating Eq (15) with respect to dM, 

2tt 



2ir 2tt , „ 

r C / - 

1 IdM. = \ 



J 











I- iL_JdM + \ cosine terms] dM 



:V) 



dM 



(17) 



Similarly, from Eq (16), 



2lT 

IdM 







2lT 



c + e cos E - 



T~ 



") 



2E 1(1 - e cos K) dE 



(18) 



Equating Eqs (17) and (18), 



2it 2ir „ „ 

I (-V) dM = I (-V + V cosE ) 



dE 



Therefore, 

00 

sin E - £ e - 

n = 1 



n-1 jn-1 

2 r (sin 11 M) 

n-1 



dM 



(14) 



To obtain the expansion for cos E, the auxiliary 
integral function I is needed. 



I = 



-I 
•I 



(E - M) dM 



oo __ 1 

,-, n ,n-l 
\ e d 

L *■ d M 
n = 1 



n-1 



(sin M) dM 



L, n : J 



A n ' 2 

° , (sin 11 M) 

dM 



2ir 



L 



( e - ec+ r)' 



e 2 / e 3 \ 3e 2 

c - i- + (e - ec + ^-J cos E - ^4- cos 2E 



+ — cos 3E dE 



As for the complete integral, all the cosine terms 
are zero; it follows that, 



Finally, the auxiliary integral function becomes 
2 



I = e cos E + ^- (1 - cos 2E) 



(19) 



ITI-12 



Next, Kepler's equation is expressed in a 
functional form: 

F (E, e, M) ■ E - e sin E - M = (20) 



(r) 



1 + e + 2 






n ,n-2 

d (sin 11 M) 



n » 1 



dM 



n-2 



(27) 



The derivative of E with respect to e is found 
by the use of Jacobians as follows: 



sin E 



dE e = 

de VZ 1 - e cos E 



Differentiating, Eq (19) yields 

dl t-i , e e „ _ 

-»- = cos E +■» - *- cos 2E 



(21) 



From 


Eq (20), 










dE 


F M 
F E 


1 






a 


dM 


1 - e cos 


E 


r 


From 


Eqs (13) and (28), 








a 

T 


00 

n = 1 


d" 
dM n 


, . n 
(sin 


M) 



(28) 



(29) 



, „ dE . e , ,„dE 
■ e9ln % + T sln2E ¥ (22) 



Substituting Eq (21) into Eq (22) and collecting 
terms yields 



dl 

ar 



cos E 



(23) 



Finally, the expansion for cos F is found from 
Eqs (23) and (15) as 



It is known that 

x 

— = cos E - e 



\ 

VI -e sin E ) 



Combining Eqs (30), (24) and (14). 



^- n-1 j n-2 
\ e d 



n - 1 



dM 



n-2 



(30) 



(sin n M) (31) 



n = 1 



cos E = " Z fn - 1) ! —^2 <sin n M) 

(24) 



,-1 



,0 



Note: . (F) - \ FdM and !L_ (F) s F 

dM J dM 



From the basic equations of orbital mechanics, 

(25a) 



— • 1 - e cos E 
a 



From Eq (24), it follows that 

^ n .n-2 

- = 1 + ^ A 



n = 1 



Squaring Eq (25a), 



7 ttt it- ( sin M) 



(25b) 



Yl- 



Y 



n-1 .n-1 
e d 



L " 
n = 1 



dM 



n-1 



(sin M) (32) 



The relationships between the true anomaly and 
eccentric anomaly are expressed as follows: 



Vl-e sin E 



cos 6 



1 - 


e cos 


E 


COS 


E - e 


1 - 


e cos 


E 



Tl - e 

sr (r) 



dE 

de 



(33) 



The first equation follows from Eq (21) and the 
second by Eq (25a) 



d 
He" 



C-) 



-cos E + e sin E 



dE 

ar 



-cos E + e 
1 - e cos E 



Substituting Eqs (13) and (25b) into (33), 



t s— \— _n-l jn-1 

Vl - e ) 

dM 



e d . . n ... 

L ( n-1) ! J^=T (8m M) 



n = 1 



(34) 



© 



1 2 i 2 

1 + j e - 2e cos E + i-e cos 2E 



(26a) 



cos 6 



00 

- ) ne 



n = 1 



d n " 2 (sin n M) 

IT T7n^2 



dM 



(35) 



Comparing Eq (26a) with Eq (19), 
2 



(y 



1 + e - 21 



and immediately from Eq (15), 



(26b) 



The general form derivation of the time anomaly 
is somewhat more complicated and will not be 
attempted here. If a finite number of terms is 
carried, it follows from Eq (33) that 



d6 _ ti - e 

aw 



(1 - e cos E) 



i ■ *- 2 (f 



III - 1 3 



* '**■ * 



and after multiplying out 
follows by integration 



(—) , the true anomaly 



K- 



(!) 



dM 



Such an expression up to the sixth power of eccen- 
tricity has been derived by Moulton. 

This concludes the derivation of the series 
expansions in powers of increasing eccentricity. 
In general form these series are presented in 
Table 6 -la. The results are given in Section K 
in Table 6-lb for eccentricities up to sixth and 
seventh powers. 

Table 6 -2a gives the n-th power of sin M in 
order to simplify the use of the general equations 

13 
for expansions up to e . Table 6 -2b indicates 

the determination of numerical constants for the 
expansions. 

The general forms of the Fourier -Bessel ex- 
pansions are given in Table 6-3a with the cor- 
responding expansions of Bessel functions in 
"Table 6 -3b. Table 6-4 gives the Fourier-Bessel 
series expanded up to the seventh powers of ec- 
centricity. 

It has been shown by Laplace that for some 
values at M, the series expansions may diverge 
if the eccentricity e exceeds 0. 662743 . . . 
For small eccentricities, the convergence is 
rather rapid. Table 6-5 presents the series for 

small values of e (e 2 « 1) as a function of mean 
anomaly. Finally, Table 6-6 presents the 
variables as a function of the true anomaly rather 
than the mean anomaly. 



versus f 2 (15) and i l (Y) versus f 2 (v) on linear 

graph paper. It is important to note that the 
same scale must be utilized for each of the three 
curves. It is also important to note that the 
shape of the scales thus generated is defined en- 
tirely by the functional forms within the deter- 
minant. 

By utilizing this technique, the equations de- 
fining the two body problem have been analyzed. 
The type of presentation is considered to be, in 
many ways, superior to any other available be- 
cause of the fact that interpolation anywhere other 
than on a graduated scale is eliminated, and by the 
fact that more than a nominal number of variables 
may be handled without losing simplicity or accu- 
racy of presentation. The nomograph obtained 
for equations of three variables, generally results 
In three arbitrarily curved scales, U, V, and W, 
as shown In this sketch. 



U 



1 \.--~~ 



\ 



V 




J. NOMOGRAMS 

Many of the formulas of the previous sections 
are of sufficiently general interest to warrant 
numerical data being prepared for use in pre- 
liminary orbit aomputation. Accordingly, a set 
of figures will m presented relating the parameters 
which have bee« discussed. Use will be made in 
this presentation of the techniques of nomography 
(Refs. 3 and 4) and of more conventional forms 
of presentation. 

Before presenting the data however.it is de- 
sirable to discuss the basis for construction of 
a nomogram. If the equation can be expressed as 
a determinant with the three variables separated 
into different rows of the determinant and if by 
manipulation, the equation can be put in the fol- 
lowing form 



f x (a) f 2 (a) 1 
f x <I3) f 2 (R) 1 

f x (y) f 2 h) i 



= 



Then a nomographic presentation is obtained by 
plotting the values of i 1 (a) versus f 2 (a), ^ (B) 



For the simpler cases, the scales may be simply 
three parallel straight lines, or two straight 
scales plus one curved scale. In all cases, how- 
ever, the solution procedures remain the same. 



Given any two values of the two independent 
variables, say U = Uj, and V = V^ a straight 

line drawn between the two given points intersects 
the third scale at the desired value of the unknown 
function (W « W^. The straight line (U^ V^ 

W ) is called the index line or isopleth. It is 

immaterial which two variables are given and 
which is considered to be the unknown function. 

Four or more variables will generally result 
In a sequence of 3 -variable nomographs as shown 
in the following sketch. 



\ 



V, 



1 J— --"".'. 



scale 



\ 



W,.. 






w 



III- 14 



cmqinml pagc n 

OF POrm fUALTff 



Ungraduated auxiliary scales (e. g. , scale q in the 
given example) are employed, and the number of 
auxiliary scales is N-3, where N = number of all 
the variables (e. g. , N = 4 in the present example), 

A special case of the four-variable solution 
exists for equations of the form. 



f x (U) 



f 3 (W) 



These equations may be expressed in the form of 
a proportional chart illustrated below. 



V^-v 



u 




w 



Given any three values of three independent varia- 
bles U = U. . V « V. . W •= W. . the unknown X = X, 



v v V 



Vy W 



is found as follows: 



(1) Connect U. and V. with a straight line. 

(2) Draw a straight line through W and the 

intersection point T , reading X on 
the X scale. 



This table is so brief that no special 
index is required. 



Table 4 Elliptic Orbital Elements in Terms of 
r, v, y . 

This brief table enables one to deter- 
mine the orbital elements from given 
kinematic initial conditions. 

Table 5 Miscellaneous Relations for Elliptic 
Orbits. 

This table contains some of the special 
expressions not readily classified under 
the other tables such as energy relation- 
ship, time relationship and certain 
angular relationships. 

Table 6 General Forms of Series Expansions in 
Powers of Eccentricity. 

This table presents a variety of series 
expansions as follows: 

(la) General Terms of Series Expan- 
sions in Powers of Eccentricity 

7 
(lb) Power Series Expansions up to e 

(Eq 6-1 to 6-11) 

(2a) Expansion of Powers of Sin M 
(Eq 6-12 to 6-24) 

(2b) Pascal's Triangle and Its Modifi- 
cation 

(3a) General Forms of Fourier-Bessel 
Expansion (Eq 6-25 to 6-36) 



K. TABLES OF EQUATIONS OF 
ELLIPTIC MOTION 

Because of their applicability, the equations 
of elliptic motion have been collected and are pre- 
sented in the form of tables. The tabular content 
is as follows: 

Table 1 Elliptical Orbit Element Relations. 

This table presents a large number of 
formulas relating the various fixed 
parameters defining the ellipse. The 
index to Table 1 (next page) is a key for 
locating equations of a given parameter 
in terms of other parameters. For ex- 
ample, parameter b is expressed in 
terms of parameters a and e in Eq (20) 
of Table 1. 

Table 2 Time Dependent Variables of Elliptic 
Orbits. 

This table gives the relationship between 
the time varying parameters of the el- 
lipse. The index (next page) is a key to 
Table 2. 

Table 3 Elliptic Orbital Elements in Terms of 
Rectangular Position and Velocity Co- 
ordinates. 



(3b) Expansions of J n (ne) (Eq 6-37) 

7 

(4) Fourier-Bessel Expansion up to e 

(Eq 6-38 to 6-49) ,. 

(5) Expansions for Near-Oyrcular 
Orbits (Eq 6-50 to 6-61) 

(6) Expansions in True Anomaly and 
Eccentricity (Eq 6-62 to 6-76) 

Table 7 Hyperbolic Orbit Element Relations. 

This table gives the basic parameters 
for the hyperbola as follows: 

(1) Hyperbolic Orbit Element Relations 
Basic Constant Parameters (Eq 
7-1 to 7-56) 

(2) Time Variant Hyperbolic Relations 
(Eq 7-57 to 7-68) 

Table 8 Spherical Trigonometric Relations. 

This auxiliary table expresses the re- 
lationship between the various geometric 
elements of the three-dimensional orbit. 
An index to this table is found (next page). 

Indexes to some of the tables follow. 



111-15 



Index to Table 1 



Parameter 



t 42 

T 43 



31 
32 


50 
51 


33 


52 


34 


53 


35 


54 



r p 


v a 






t 99 


118 


100 
100a- 


119 


101 
100a 


120 
120a 




121 
120a 


102 




103 


122 


104 


123 


105 


124 


106 


125 




126 


107 


127 


108 


t 123 




129 


109 




t 110 


130 


111 
100a 


131 



147 

148 
1 146 

. 150 



I 154 



Index to Table 8 



t figure available 

NOTE: 

This Index to Table 1 Is a key for locating equations of a given 
parameter In terms of other parameters. For example, param 
eter b is expressed in terms of parameters a and e tn equation 
20 of Table 1. 



Index to Table 2 






X l 


* f(a, e 


, x 2 > 


Param- 














. 


eters 


f(E) 


«r) 


f(r) 


«v> 


f(v) 


«e) 


f(0) 


E 




1 
2* 


3 


4 


5 


+ 6 
7 
8 
2* 


9 


r 


10 

n* 




12 


+ 13 


+ 14 
15* 


t 16 
17 
U* 

15* 


18 


r 


19 


20 




24 


26 


27 


28 






21* 




21* 


22* 


23* 








22* 




25* 


25* 










23* 












r 


29 


30 
31* 


32 


33 


34 


35 
31* 


36 


V 


37 


f 38 
39 
40* 
41* 


42 




43 

40* 


44 
41* 


45 


Y 


46 


+ 47 
48 
49* 
50* 


51 


52 
49* 




t 53 
54 
55 
50* 


56 


e 


T57 


t 62 
63 
60* 


66 


67 


+ 68 




69 




58 






64* 








59 


61* 






65* 








60* 


64* 














61* 


65* 












9 


70 


71 
72* 


73 


74 


75 


76 
72* 




e 


77 


78 


79 


80 


81 


82 


83 



♦Function of more than one time-dependent variable 
^Figure available 
See Note with Table 1 



Para- 












meters 


i 


L 





V 


4> 


f(i, L) 






21 


31 


41 


(1. 0) 




11 




34 


44 


(1, y) 




14 


24 




46 


(1. <t>) 




16 


26 


36 




(L, 0) 


l 






37 


47 


(L. v) 


4 




27 




49 


(L. <t>) 


6 




29 


39 




(0, v) 


7 


17 






50 


(13, «>) 


9 


19 




40 




(U) 


10 


20 


30 






(1. L. 0) 








32 


42 


<i, L, ") 






22 




43 


(i, L, 4-) 






23 


33 




(i.0. v) 




12 






45 


Hi. &, 4>) 




13 




35 




(l. y, 4>) 




15 


25 






(l. fl. y) 


2 








48 


(L. fi, 4>) 


3 






38 




(l. y, 4>) 


5 




28 






O, y, ^) 


8 


18 









See Note with Table 1 



TABLE 1 

Elliptic Orbit Element Relations 
(see Fig. 4) 



* 



1 - e' 



fid - e ) 



2 J- I 2 

r + b 
a 

~~2r 



2 -Ll 2 

r + b 

P 

"~2r 



1 - e 



1 + e 



r 
P 



(Fig. 11) 

(Fig. 12) 
(Fig. 12) 



(1-1) 

(1-2) 

(1-3) 

(1-4) 

(1-5) 

(1-6) 
(1-7) 



III- 16 



a « 
v 



TABLE 1 (continued) p (1 26) 



(1-8) 



2 
e 



— 2" Vl + e) , 

v \ / / 

a /l - 

^l^) (1 - 9) /T 



r aYr+l (1-27) 



v 
P 



If (1-28) 

- e 



r 2 ..,i_^3/2 

a (1-10) 

2r a - p v/ (1 +e) 



- M 1 - e > (1-29) 

2TTTTT/2 



a 



2 

r 



P_ (1-11) = iu (l + e) 3/2 (1 _ 30 ) 

2r - p ? n/2 



'P 



a P (1-1 la) 



P~ = rt / P (1-31) 

aW2r - P 



a\4 

v (2*^~- v ) = r -*/, 

a Tp a' 'pVj 



Jf (1-12) 



a 



-,. P d-32) 



V 

p 



(2^~- v p ) "Vv a ( 2Ai -v a1 &) 



/ 



(1 - 13) = / (P^) 3/2 (1-33) 



+ r f , ,3/2 
a P (1-14) = / (PM) (1-34) 

VV2" " V a V^) 



M r 



2 ^ " r a v a 



2 ^ (1-15) = -/L _ (1-35) 



■«< 



'r r 
a p 



1 



(r v +Vr 2 v 2 + 8/ur J (1-16) 

v a p T a p a 



, 3 2 

'r v . 

a a (1-36) 



lv n a p T a p a T 2 K " r a v a 



» *-(r v +Vr 2 v 2 + 8,ur ) (1-17) - l/i- [f^ 2 v 2 + 8jur - r v 1 (1-37) 

4v„ p a T p a p 11T L" a p a a pj 

a p 

(1-18) -^ [fp 2 "a" + 8 ^p " r p V a] (1 " 38) 



P 
T 



2\x - r v„ 
P P 



Vf 3 2 
r P V P d-39) 

2m 



(■#) <» 8 " <•-> -V^T^ 



vv \ Zir / ? 2u - r v 

a p » ** p p 



b = aVTX (1-20) (V a + Vf^ 

" 1^P 



2^ (1-40) 



Vr a (2a- r J 



V- p < 2a " r p > 

2 ^* 3/2 v a 

2 

M +av a 

2fa 3 ' 2 v 



(1- 


■20) 


(1- 


-21) 


(1- 


-22) 


(1- 


-23) 


(1- 


-24) 











- V. - @ 2 


= 1' 


iua 

(Fig. 11) 


d-41) 


-V.-5 


(1-42) 


r 

« _* - 1 
a 




(Fig. 12) 


(1-43) 


r 




(Fig. 12) 


(1-44) 



'" + av p 



„ E. (1-25) 

2 III- 17 



TABLE 1 (continued) 



2 

" " aV a 








(1-45) 


° a. 2 


2 

a v p -m 








(1-46) 


2 . 
a v + u 
P 


°^W 


(1-47) 


2 u2 
r - b 
a 








(1-48) 


2 x i, 2 
r + b 
a 




k 2 2 
b - r 
P 








(1-49) 


b + r 
P 




= 1 -P- 








(1-50) 


a 










= -P - 1 








(1-51) 


P 










= i - v Je 

a ifi 








(1-52) 


= v JF. 1 
p T/U 








(1-53) 


r - r v - v 
= a p _ v p a 

r a + r r. v + V 
a p pa 








(1-54) 


2 

r v 
_ -i a a 




- 2 M - 


r v 2 ) 
a p J 


(1-55) 


M 




1 / J 2 2 , 
= Tt— 1 v wr v + 
2m \ p ' a p 


^ a 


(1-56) 


= * ( 2/J + r v 2 - 
2/T \ pa 


v Vr 
a » ] 


2 2 

V 

a a 


+ JfcrJ 


(1-57) 


2 

r v 
= P P _ i 








(1-58) 


h = iup = r 6 








(1-59) 


b 2 
P = IT 








(1-60) 


= a(l - e 2 ) 


(Fig. 


U) 




(1-61) 


r 

= — (2a - r ) 

a a 








(1-62) 


r 

= _P (2a - r ) 
a p' 








(1-63) 



r r 
a P 



4M 



\ a av a/ 

4m 

rrzr 
\ p av D / 



i« 



b VI - e' 



2b 2 r 
a 

b + r 
a 



2b 2 r 
P 

b + r 



r a (1 - e) 



r p (l +e) 






2r r 
a P 



r + r 
a p 

2 2 
r v 
a a 



(l-63a) 
(1-64) 

d-65) 

(1-66) 
(1-67) 

(1-68) 

(1-69) 
(1-70) 

(1-71) 
(1-72) 
(1-73) 
(1-74) 



a 



[4 M -V p y ra 2 v p 2 + 8Mr a + r a v p 2 ] (l- 75 ) 
5T £* + r P v a 2 - v aV r p 2 v a' + 8 " r p] (1 -™> 



2 2 
r v 
JP P_ 

4m 



(1-77) 
(1-78) 



♦tf 



r = a +>a 2 - b 2 
a ' 



a (1 + e) 



(Fig. 12) 



(i+fl) 



ap 

r 
P 



(1-79) 
(1-80) 

(1-81) 

(l-81a) 



III- 18 



TABLE 1 (continued) 



r = 2a - r 
a p 



2/na 



H + a v 
a 

.,2 2 

2a v 
P 

/u + a v 



■»t£ 




T^T" 



(^4) 



m (1 - e) 

2" 



P (1 + e) 
v p" (1 - e) 



r p 

"27 ^~p 
P * 






ypM~ 



2t£~- v 
Tp p 



■£ 



r 2ur 

p + P 



r v 
P P 



te — ; 

r v 
P P 



- 1 



2m 



v ( v 
a a 



r 

P 
2~ 



(1-82) 


r p - a - >4 2 - b 2 






(1-98) 


(1-83) 


= a (1 - e) 
P 


(Fig. 


12) 


(l-99) 
(1-100) 




^v^ 






(1-84) 


= ^p 

r 
a 

= 2a - r 
a 






U-lOOa) 
(1-101) 


(1-85) 


2a 








(1-86) 


1+ ^r 


(1 -lUz) 



(1-87) 

(1-88) 
(1-89) 
(1-90) 

(1-91) 
(1-92) 

(1-93) 
(1-94) 



(1-95) 



(l-95a) 



(1-96) 



(1-97) 



2a 



a v 
1 + — P_ 



>V£ 



1 + 
,2 



f-$ 





a 




_ 


P 






1 + e 




= 


1 
r a T- 


- e 

+■ e 




K<1 " 


\ 2 
e) 




2 M 

v a (1 


+ e) 




M (1 + 


e) 




2 

V 

P 






Pr a 






2r - 
a 


P 




Vpp 





>€- 



2\~ - v 
"p a 



J&L 



2 2 
r v 
a a 

2/j - r v 
a a 



1 

V- 



r 2ur r 

a ^ a a 

r~ + — t ~t 

V 



2m 

v Cv + V ) 
Pa p' 



(1-103) 

(1-104) 
(1-105) 

(1-106) 
(1-107) 

(1-108) 
(1-109) 

(1-110) 

(1-111) 

(1-112) 

(1-113) 
(1-114) 

(1-115) 
(1-116) 



111-19 



TABLE 1 (continued) 



- V 



-IMS) 

T ar 

a 



2a - r 



a v 



- / 2 »b 2 

f b (b + r ) 



-yi < i - e > 



(1 - e) 



-\r (1 + e) 

= V P \tts) 



YMP 

r 



Vi 



'I (»-!?- 



(1-117) 

(1-118) 
(1-H9) 

(1-120) 
(l-120a) 
(1-121) 
(1-122) 

(1-123) 
(1-124) 

(1-125) 

(1-126) 

(1-127) 
(1-128) 
(1-129) 
(1-130) 
(1-131) 
(1-132) 



v = 2- 
a 



»P P 



v, 

4 



2\xv 



r Tr + r ) 
a a p 



h- + ^ 'J 



2u - r v 
M P P 
r v 
P P 



"'♦*¥ 



■V* (^) 

4 



a 

a7 



4 (^ 






2 ^ r a 
fb 2 (r 2 +b^) 



2^ b' 

5 5" 

r (r + b') 
P P 



•# 



■vi 



(1 + e) 



(1-133) 



(1-134) 



(1-135) 



(1-136) 



Ml+eT 

r. (1 - e) 



(1-137) 



(1-138) 



(1-139) 



(1-140) 



(l-140a) 



(1-141) 



(1-142) 



(1-143) 



(1-144) 



(1-145) 



(1-146) 



(1-147) 



(1-148) 



111-20 



TABLE 1 (continued) 



*. 



(l+e) 



v a(r^) 



>£l 



iup 
r 
P 



v 
P a 



- 'if" 
"2^77 



^ 



r fr +' r ) 
Pa p' 



r v 
a a 



2u - r v 
a a 

r v 
a a 



-V^ + £ 



+ 2m _ V a 



(1-149) 

(1-150) 
(1-151) 

(1-152) 
(1-153) 
(l-153a) 

(1-154) 
(1-155) 



TABLE 2 

Time Dependent Variables of Elliptic Orbits 
(see Fig. 4) 



.os- 1 (JL££) (2-1) 

(Fig. 13) (2-2) 



1 / r sin i 



2 M 



iue ± e 



"22 ... 2. .21 

M e - ^a (1 - e ) r 



1/2 



2 2 ... 2. ■ 

/u e - jua (1 -e ) r 



T7? 






1 - (1 - e ) sec y) 



(2-3) 
(2-4) 

(2-5) 



,-iftf 



- "° 'Vrrfjirr-/ m,. u> cw» 



-1 



e + cos 8 
1 + e cos 9 . 



(Fig. 14) (2-7) 



E = 2 tan 



-1 






(Fig. 14) 
(2-8) 

1/4 



(2-9) 



r = a (1 - e cos E) 



iT 



2 sin E 



sin 9 



/na (1 - e ) 



iTT2 



(2-10) 
(2-11) 

(2-12) 



. T 2 2 ,, 2, -2l J 

ju ± |_/j e -^a(l-e)rj 

(Fig. 15) (2-13) 



2^a 

2— ~ 

a v + /u 



[l ± Vl - (1 - e 2 ) sec 2 Y 



J (Fig. 



a (1 - e ) tan y 
e sin e 



a(l -Q 
1 + e cos 6 



17) 
(2-14) 
(2-15) 



(Fig. Hi) (2-16) 



2r r 
^a p 

Tr + r ) + (r - r ) cos 



~6 



a p a p 

1/2 



a (1 - e) 



(Fig. 13) 

(2-17) 

(2-18) 



e sin E 



T a l _ e 


cos E 






. J^ 2 


2 2 

ar - r - a 


(1 - 


•'3 


T T 


2' 
ar 


r 




=yl 2 -Ma 


(1 -e 2 ) 

2 
r 




_ ia(l- 


e ) tan y 






r 




_lU(l- 


e 2 > /,... 


^ 



(2-19) 
(2-20) 
(2-21) 

(2-22) 
tan 6 (2-23) 



Va 2 . 2 ^ .2 ,. 2. 

4 M av - (ay + M ) (1 - e ) (2 _ 24) 
4fia 



v sin y 



(2-25) 



111-21 



TABLE 2 (continued) 



/, 2. . 2 
H (1 - e ) tan y 



M 



'1 - (1 - e 2 )sec yj 



1/2 



(2-26) 



M 



1/2 



a (1 - e ) 

, r 1/2 
2/u 6 



sin 6 



(2-27) 



X*«a (1 " e2 J 



T/4 "a"[' 



11/2 



11/2 



(ja(l-e') 9> 
(2-28) 



/u e (cos E - e) 
a (1 - e cos E) 

H a (1 - e ) - r I 
L 3 



= ^| cos e 
r 



(2-29) 



(2-30) 



(2-31) 



1/2 



<± m [fie - a (1 - e ) r J 

+ 2 / /2 [Me 2 -a(l-e 2 )i- 2 ] 
± p_ a(1 . e 2 ) -2] 3/2 J^l/2 a 2 (1 _ e 2 ) S 



(2-32) 



/ 2^.2 
f a v + n ) 

8^a 



(av 2 + M )(1 -e 2 ) 



2mJ 



(2-33) 



(1 - e 2 ) - (1 ±jl - (1 - e 2 ) sec 2 Y )J 



2 L Vl - (1 - e 2 ) sec 2 Y ] 



[«*f 



-j Me 2 g (1 + e cos 6) cos 6 

a (1 - e ) 



(2-34) 



(2-35) 



a (1 - e ) 6 



2> ^3/2 



[ia (1 - e 2 ) 

_t a 



[*ia (1 - e 2 )] 



TTT 



1/4 



(2-36) 



Vi 



(1 + e cos E) 
a (1 - e cos E) 



■vKM 



(2-37) 



(Figs. 1 (2-38) 

and 15) 



,£ 



(1 -e ) 



r cos y 



(2-39) 



(Fig. 18) (2-40) 



V H (1 + 2e cos 6 + 
r (1 + e cos 0) 



e ) 



(2-41) 



\pi (1 + e*) ± 2)n 



'2 „ 2. 2 

fie - a (1 - e ) r 



i 



l/2> 



1/2 



a (1 - e) 



H / lT Vl - (1 -e 2 ) 



H Z 1 * Vl - (1 - e^) sec y \ 
Vl±-*1 - (1 - e 2 ) sec 2 J 



1/2 



ju (1 + e + 2e cos e) 
a(l -e 2 ) . 



1/2 



, (2ae l/2 -La(l-e 2 )l 
" " ' } L ,1/4 ^ 



1/4. 



a La (1 - e )j 



(2-42) 
(2-43) 

(2-44) 
1/2 

(2-45) 



tan 



if 



sin E 



(2-46) 



l t/4 



■U/a" (1 - e ) 
cos V „ „>_ m) ( F ig. 17 ) (2 _ 47) 



a - r> 



cos - t / ^ J> (Fig. 17) (2-48) 

(r + r - r) 
a p 



^s-MMAiLL) (Fig. 18) (2-49) 



tan 



tan 6 



i tan 



-1 



\ a(l-e )/ 

r U (1 - e 2 )J 

j, 1/2 ± [ M e 2 -a(l-e 2 )r 2 ] l/2 ' 



(2-50) 



(2-51) 



= ± tan 



-1 



/ yi-e ) [^av 
\ (?.v 2 + m) (1 



Mav%) 2 (l-e 2 )J| (2 _ 52) 
e 2 ) 

(Fig. 19) 



) 



111-22 



y = tan 



TABLE 2 (continued) 
•lie sin t) 



1+e cos & J (Fig. 20) (2-53) 

, - 1 / e sin 9 



sin 



Vl+2e cos 6+e 2 



(Fig. 20) (2-54) 



cos ' W 1 + e C0S — 9 - \ (Fig. 20) (2-55) 



Vl+2e cos 6+e 



, ± tan "l ^a^^Ud-e^'-Ud-e 2 )] 

> , / a 2 (l-e 2 ) 6 

1/2 ^ 

"1 

; (2-56) 



1/2 



° S_1 (t^Ft) ^. H) (2-57) 



= 2 tan 



14) 

(2-58) 

(Fig. 14) (2-59) 
(2-60) 

(2-61) 

(Figs. 12 & 13) 

(2-62) 
. |"2r r -r (r +r ) "1 

= cos -1 [ r fr a -r) P J (Fi « s " 12 & 13) 

(2-63) 



sin" 1 ( 3in E ™^ 2 ) 
\1 - e cos E J 

:os I— [cos E - ejJ 
3in -l |ap sinEJ 

.os" 1 [-* (1 - e2 J- r ] 



= sin 



tan 



•1 | ~a(l-e 2 )tan Y 1 

•1 [a (1-e 2 ) tan y "1 
La (1-e 2 ) - r J 



3ln -l | f IV^] 

.os" 1 [ (av2+ j^' e2) ' 2M ] 

-i fi i 

,os ^_ |co 



(2-64) 
(2-65) 

(2-66) 
(2-67) 



= 2 1 j- J 2 ~ 27 

s y - 1 ± cos y ycos y-(l-e ) 



(Fig. 20) (2-68) 



.L \\^±] e -i[ 



(2-69) 



1/2 ., 2, 

(1 -e ) 



La(l -e 2 )J 



1/2 



(1-e cos E) 
1/2 



(2-70) 



(2-71) 



■ Jf* < 1 + 



e cos 9) 



(2-72) 



I 1/2 T 2 ., 2, -2l l/2 { 

- (V ± L^ e -a (1 - e ) r J J 

" r \ , h i/2 



1/2 



ja 3 (1-e 2 ) ] 



(av 2 +iu) 2 f"na (l-e 2 )1 



1/2 



^22 
4p a 



U (1-e 2 )] 



1/2 



a 2 U Vl- (1-e 2 ) sec 2 y ' 
-,1/2 



(2-73) 



(2-74) 



(2-75) 



3 /, 2. 
a (1-e ) . 



(1+e cos GT (2-76) 



9 1/2 
u 2e (1-e ) sin E 

a 



r 



(1-e cos E) 



(2ar - r 2 )(l-e 2 ) -a 2 (1-e 2 )' 



(2-77) 
1/2 



(2-78) 



-2i [a(l-. 2 ,]" 5/2 j„>« 



±Ue 2 -a (1-e 2 ) - 2 



(2-79) 



av + n 



. 2jja 



/ 2 4 ^ 
- (a v + 



(1-e 2 ) [2^av 2 (1+e 2 ) 

2 w, 4 1/2 

M ) (1-e W 



2p. (1-e ) tan y 



a 3 [l±yi - (1-e 2 ) sec 2 Y 



■r 



(2-80) 



(2-81) 



^ 3- (1 +e cos 6) 3 sin G (2-82) 



a 3 (1-e 2 ) 

n fa 2/3 



- (1-e ) 



^ |2a(l-e 2 )e 1 / 2 [,a(l-e 2 )] 
[t*a (l-e 2 )j 



1/4 



L/2 2 . 

-a Z (1-e ) 6 



1/2 



(2-83) 



III - 2 3 



TABLE 3 

Elliptic Orbital Elements in Terms of Rec- 
tangular Position and Velocity Coordinates 



L 
P 



(3-1) 
(3-2) 



= 2 (x + y + z ) - — (x +y + z ) 

■ I 1 ■ i h 

x . • •>2ir„ , 2 x 2 x 2 " 1/2 1 / 2 

+ (yz - zy) J [2 (x + y + z ) - — (x 

jr. . ., . . 

os <(xy - yx) [(xy - yx) + (xz - zx 

!, n l/2] 



j. ' 2 j. ' 2 s 
+ y + z ) 



. 2 ■ *2 

|(xy - yx) + (xz - zx) 



+ (yz - zy)' 
■1 



I 1 "} 



tan 



yx - xy 

J n 'a 



cot -1 [-^-cosn-i sinn] 



1/2- 



(3-3) 

\ 2 
zx) 

(3-4) 
(3-5) 

(3-6) 
(3-7) 



, -lr , 2 _,_ 2 , 2 " W 1 
sin I z (x + y + z ) J 

— [(xy - yx) + (xz -zx) + (yz - zy) J 

(3-8) 

r =Vx +y 2 + z 2 (3-9) 

v =Vx 2 + y 2 +i 2 (3-10) 

x = r [cos (u + 6) cos Q -cos i sin (to+ fa) sin Cl] 

(3-11) 

y = r [cos ( u + fa) sin U + cos i sin ( to+ fa) cos S7J 

(3-12) 

z = r sin ( w + fa) sin i (3-13) 

x = [cos fa (cos oj cos n - cos i sin U sin co ) 

+ sin fa (-sin to cos £1 

- cos i sin Q cos u ) •= — ; r- (3-14) 

-I 1 + e cos fa 

y = [cos fa (cos io sin Q + cos i cos fi sin u ) 

+ sin fa (-sin id sinQ 

+ cos i cos f2 cos u))| -= — : ? (3-15) 

-I 1 + e cos fa 

z = [cos 6 sin i sin u 

+ sin 6 sin i cos u)l -t— r — E- — c— (3-16) 

-1 1 + e cos 6 



"#[« 



(cos 6 + e) (-sin u cos n 

- cos i sin Q cos u ) (3-17) 

- sin fa (cos u cos £2 - cos i sin Q sin u) 
y = -IP- (cos 6 + e) (-sin u sin Q 

+ cos i cos Q cos u) ) 

- sin 6 (cos u sin q + cos i cos a sin W )J 

(3-18) 

z = V— (cos fe + e) sin i cos co - sinfa sin i sin <j I 
P L (3-19) J 

- 1 T ' ' ' 2 2 2-2 

■y = sin I (xx + yy + zz) (x + y + z ) (x 

•2 2 ' 1/2 1 
+ y +z z ) J (3-20) 



-i r 

= cos 



(xx p + yy p + zz p ) (x 
-1/2 „ -l/2-i 



2 , _2 % i " ,.. 2 , __ 2 , 2, '"l 



+ y+z) (x +y +z) 

17 p J p p 



(3-21) 



= cos (xx + yy + zz ) (x 

[_ n ■'■'n n 

-1/2 



2 + 2" 1/2 , 2 + 2 + 2,- 1/2 l 
r + z ) (x n + y n + z n ) J (3 

:os (xx +yy +zz)(x 

L n p ^ n- 7 p np n 



■22) 



+ 2 + 2 " 1/2 , 2 + 2 + 2." 1/2 1 
+ y + z ) (x +y +z) 



n n 

where: 

n = node 
p = perigee 



(3-23) 



n = tan 



■1 /yz -yz \ 

\xz - xz/ 



(3-24) 



TABLE 4 
Elliptic Orbital Elements in Terms of r, v, y 

(4-1) 



rv (Fig- 15) 



(Fig. 15) 



2 - Q 




2 2 

r cos y 


2m 
r~^ 


■ 1 


(r cos 


Y> 2 


2 
Q 


1 



(4-2) 



(4-3) 



(4-4) 



111-24 



V-(i-^F^-) 



(4-5) 



£ 



Q (2 - Q) cos% (Fig. 19) (4-6) 



— (r v cos Y ) 2 (Fig. 18) (4-7) 



Q 2 

— £ - COS V 



(4-8) 



Q 



/ v2 ry 2 

^-j = -^- (Figs. 15 and 19) {4 _ 9) 

-^7 [ 1+ V^"f (rvcos,) 2 4-^ 



(4-10) 



r 
2 - Q 



Q (2 - Q) cos ' v (4-11) 



] 



2 - 



2 - Q 



-j [l "V 1 ^ (rv C os Y ) 2 (|-^)] 



(4-12) 



Q (2 - Q) cos Y (4-13) 



a rv cos y 



1 "V 1 ■ ji (rvc ° 



] 



sy) 2 (- 2 - — ) 
' r n 



(4-14) 



1 - Q (2 - Q) cos y (4-15) 



] 



p rv c 



os Y I T 

i— l+Vl-- (rvcoSY) 2 (-f--^-> 
osy |_ * H r M 



Qcosy 



[.♦vr 



Q (2 - Q) cos y 



(4-16) 



(4-17) 



TABLE 5 
Miscellaneous Relations for Elliptic Orbits 



£ -- Ji 



2a 



(5-1) 



(see Eqs 1-1 through 1-19 for parametric 
variations of a) 



K + P 
2 



K 



(5-2) 
(5-3) 

(5-4) 



M = E - e sin E (Figs. 2 and 22a to i) (5-5) 

(see Eqs 2-1 through 2-9 for parametric 
variations of E) 



2 it 

T 



(Fig. 7) 



f 



3/2 



(5-6) 
(5-7) 



(see Eqs 1-1 through 1-19 for parametric 
variations of a) 



M 

"t-t 
P 

_ _ V- 

r 



(5-8) 
(5-9) 



r = a (see Eqs 1-1 through 1-19 for parametric 

(5-10) 



variations of a) 
M 



+ t 



3/2 



(E - e sin E) + t 

VT p 

(see Eqs 2-1 through 2-9 for parametric 
variations of E) 



(5-11) 
(5-12) 



■# 



(Fig. 8) 



(5-13) 



fsee Eqs 2-10 through 2-18 for parametric 
variations of r) 



V2v 



-^ 



(5-14) 



(5-15) 



(see Eqs 2-10 through 2-18 for parametric 
variations of r) 



v = sin (± e) 
'm 



(5-16) 



(see Eqs 1-41 through 1-59 for parametric 
variations of e) 



= tan 



m 



= cos (-e) 

■(■a 



• sin 



/a (Table 9 and 

;ira V-7T Fig. 1) 



(5-17) 

(5-18) 

(5-19) 
(5-20) 

(5-21) 



(see Eqs 1-1 through 1-19 for parametric 
variations of a) 



111-25 



TABLE 6-la 

General Forms of Series Expansions 
in Powers of Eccentricity 
(see Fig. 4) 



E 



sin E 



M+ / h~ n-1 (8ln M) ( 8_1 ) 



n-1 



n-1 d n-l 
dM 



i ^1- ~S=T (8inn M > < 6 - 2 > 



M) (6-4) 



n » 1 



cosE " " I (n4rr;^rr2-< 8inriM > (6 " 3 > 

n-1 aM 

(r) 



dM r 

n ,n-2 



1 +e 2 + 2 Y 

n^l" <™ 



^ d ^(sin n M) 



(6-5) 



1 + 



n ,n 

(sin 11 M) (6-6) 



n » 1 



dM'' 



x 

a 



z 

a 



Y « n " 1 d n - 2 

Z, (n -1): ^n^2 



n » 1 



(sin n M) 



(6-7) 



„ _ , dM ia a 



n-1 



sin 8 



cos 6 



_ oo 



(6-8) 



n-1 ,n-l 

e d . . n -», 

— (sin M) 



n-l < n -!>' dM"" 1 



(6-9) 



^- n-1 ,n-2 

• -I FTl: br <-"-> <6 - ,0) 



n = 1 



dM" 



e - J tt (&) 



dM 



(6-11) 



NOTE: Divergence for e > 0.662743.. . 



TABLE 6- lb 



Power Series Expansions up to e 



E = M + e sin M + |y- sin 2M 



+ - — k (3 2 sin 3M - 3 sin M) 



3!2 

+ - e - Tr (4 3 sin 4M - 4-2 3 sin 2M) + 
4!2 d 

vcontinued) 



TABLE 6- lb (continued) 



+ -2-j- (5 4 sin 5M - 5-3 4 sin 3M + 5-2 sin M) 
5!2* 

+ -2-c- (6 5 sin 6M -6-4 5 sin 4M + 5-3-2 5 sin 2M) 
6!2 S 



+ — %r (7 sin 7M - 7-5 6 sin 5M 
7!2 D 



+ 7-3-3 sin 3M -7-5 sin M) 



(Fig. 2) 



(6-12) 



sin E = sin M + S- sin 2M 



+ -2—, (3 2 sin 3M - 3 sin M) 



3!2 
4!2 



+ -^-T- (4 3 sin 4M -42 3 sin 2M) 



+-S-T- (5 sin 5M - 5- 3 4 sin 3M + 5-2 sin M) 
5! 2 

5 

+-2-cr (6 sin 6M - 6-4 5 sin 4M + 5-3-2 5 sin 2M) 
6!2° 



+ -^—tr (? 6 sin 7M - 7-5 6 sin 5M 
712^ 



+ 7-3-3 sin 3M - 7- 5 sin M) 

+ -£—*■ (8 7 sin 8M - 8-6 ? sin 6M 
8! 2 

+ 7-4-4 7 sin 4M - 8-7-2 7 sin 2M) 



(6-13) 



eda E - cos M + £. (cos 2M - 1) 



(3 cos 3M - 3 cos M) 



2! 2 

e 3 2 2 

+ -S— , (4 cos 4M-4-2 cos 2M) 

3! 2 S 

4 

+ - j (5 cos 5M- 5- 3 3 cos 3M + 5- 2 cos M) 

4! 2 

+ -£— r (6 cos 6M - 6-4 4 cos 4M + 5-3-2 4 cos 2M) 
5! 2° 



(continued) 



III -26 



TABLE 6-lb (continued) T ABLE 6- lb (continued) 

e 6 <i c: 

+ »- (1 cos 7M - 7-5° cos 5M cos 9 = cos M + e (cos 2M - 1) 

6! 2° 

3e 2 

+ 7-3-3 5 coa 3M - 7-5 cos M) + ^f~^ (3 cos 3M " 3 cos M > 

7 3 

+ -^—7 (8 6 cos 8M - 8-8 6 cos 6M + 4 -^— 7, (4 2 cos 4M - 42 2 cos 2M) 

7! 2' 3! 2 6 

+ 7-4-4 6 cos 4M-8-7-2 6 cos 2M) + 5 e 4 f5 3 pOR HM -R-a 3 cog 3M 



(6-14) 



+ A (5 d cos 5M - 5-3° cc 

4! 2* 



6 = M + 2esinM + H~ sin 2M 



2 +5-2 cos M) 



3 + 6 e g - (6 4 cos 6M - 6 -4 4 cos 4M 
+ ^- (13 sin 3M - 3 sin M) 5! 2 

4 + 5-3-2 4 cos 2M) 
+ l^. (103 sin 4M - 44 sin 2M) 

+ 4 (1097 sin 5M - 645 sin 3M + 50 sin M) + JTF ^ C ° S ™ " "'^ C ° S 5M 

+ |L (1223 sin 6M - 902 sin 4M + 85 sin 2M) + 7- 3 -3 5 cos 3M - 7 -5 cos M) 



9T0 
7 
jl-235 (47,273 sin 7M - 41,699 sin 5M f ~T (8 " cos 8M " 8 " 6 ° cos 6M 



» 7 t 8e 7 /0 6 Dnwr „ fi 



+ 5985 sin 3M + 749 cos M) + ? . 4 . 4 6 cQg 4M _ g.^6 ^ gM) 

+ (6-15) 



(6-17) 



sin 9 = Vl - e ^sin M + e sin 2M r P 2 



— * 1 - e cos M - £— (cos 2M - 1) 



+ — — ,- (3 2 sin 3M - 3 sin M) e , 

01 2 T < 3 cos 3M- 3 cos M) 

2! 2^ 
3 

+ C ■*■ (4 3 sin 4M - 4- 2 3 sin 2M) * 

3! 2 _ e ,„2 ___ „„ A „2 

4 4 4 3! 2 
+ -^— t (5 sin 5M - 5-3 sin 3M + 5-2 sin M) = 

4! 2 e_^ / K 3 _„„ c „ c .„3 

5 4! 2 

+ — — t (6 sin 6M - 6-4 5 sin 4M + 5-3-2 5 sin 2M) fl 

S 1 2 p » 4 

^— r-(6 cos 6M- 6-4* cos 4M 

6 5!^ 

+ e a (7 8 sin 7M - 7- 5 6 sin 5M 

6! 2 8 4 

+ 5-3-2 cos 2M) 

+ 7-3-3 6 sin 3M - 7-5 sin M) 7 



T (4 cos 4M - 4-2 cos 2M) 

1 9° 

3 3 

3- (5 cos 5M - 5-3 cos 3M + 5-2 cos M) 



e 7 ,„7 . „„ „ „7 , „ e!^ 

T 



e (7 5 cos 7M - 7-5 5 cos 5M 



7! 2 
f 7-4-4 7 sin 4M - 8-7 2 sin M) 



+ 7-3-3 5 cos 3M - 7-5 cos M) 



(6-18) 



(6-16) 



111-27 



TABLE 6-lb (continued) 



TABLE 6-lb (continued) 



£•] = 1 - 2 e cos M - |y (cos 2M - 3) 



- ^~y (3 cos 3M - 3 cos M) 



4 
e y <4 2 cos 4M ~ 4 "2 2 cos 2M > 



+ -j$- (103 cos 4M + 8 cos 2M + 9) 

5 
+ ^ (1097 cos 5M - 75 cos 3M + 130 cos M) 

6 
+ ^ (1223 cos 6M - 258 cos 4 M 



4! 2 



p 3 

3- (5 cos 5M 



5! 2 

5-3 3 cos 3M + 5-2 cos M) 

6 . 

■ e , (6 cos 6M 
6! 2 



4 4 

6-4 cos 4M + 5-3-2 cos 2M) 

e -%- (7 5 cos 7M - 7-5 5 cos 5M 



7! 2 



+ 7-3-3 cos 3M - 7-5 cos M) 



+ 105 cos 2M + 50) 

7 
+ 25 e 04Q (236,365 cos 7M 

- 83, 105 cos 5M + 17,685 cos 3M 

+ 13,375 cos M) 

+ (6-21) 



§• = - e + cos M + £. (cos 2M - 1) 



(o-19) 



+ — — k-(3 cos 3M - 3 cos M) 
91 9/ 



2! 2 



a 2 

— = 1 + e cos M + e cos 2M 
r 



+ — — *- (3 cos 3M - 3 cos M) 
3! 2^ 

+ ■ e , (4 4 cos 4M - 4-2 4 cos 2M) 
4! 2 J 



+ e j (5 5 cos 5M - 5-3 5 cos 3M 
5! 2 

+ 5-2 cos M) 

6 fi 

+ . (6 cos 6M - 6- 4° cos 4M 

6! 2° 



+ ■ e ., (4 2 cos 4M - 4-2 2 cos 2M) 
3! 2 6 



e 4 ,.3 



3- (5 cos 5M - 5- 3 cos 3M + 5- 2 cos M) 

1 ?/* 



4! 2 

+ — - — _- (6 4 cos 6M - 6-4 4 cos 4M 
5! 2° 

+ 5-3-2 4 cos 2M) 



+ e . (7 5 cos 7M - 7-5 5 cos 5M 
6! 2 6 



+ 7-3-3 cos 3M - 7-5 cos M) 



+ 5-3-2 cos 2M) 

e 7 7 7 

+ ■ fl (7 cos 7M - 7- 5 cos 5M 

7! 2 6 

+ 7-3-3 7 cos 3M - 7-5 cos M) 

+ (6-20) 

2 2 

y.1 = 1 + 2 e cos M + ^- (5 cos 2M + 1) 



e 7 ,„B 



+ — %-{8 cos 8M - 8-6 D cos 6M 
7! 2' 



+ 7-4-4 6 cos 4M - 8-7-2 6 cos 2M) 



(6-22) 



Z. « Vl - e 2 • sin M + 



• sin 2M 



+ Sy- (13 cos 3M + 3 cos M) 



?—~- (3 sin 3M - 3 sin M) + 

(continued) 



3! 2 



111-28 



TABLE 6-lb (continued) TABLE 6- lb (continued) 



3 



e 

4! 2" 
4 



+ , (4 sin 4M - 4-2 sin 2M) 

LI 9 J 



5! 2* 



(5 4 sin 5M - 5-3 4 sin 3M + 5-2 sin M) 



e 5 ,„5 , „„, „ ,5 



il 2 , e" e 4 


1-3 e 6 
" 2-4-6 


1-3-5 e 8 




" 2-4-6-S 





+ -H— -- (6 sin 6M - 6-4" sin 4M 

gi 2 2 4 6. 

D - ^ e e e 5 e 



1 ~"T "T - "1^ ~ 7W 



+ 5-3-2 5 sin 2M) 10 12 



e ,-,6 _,_ „„ ,,. K 6 



6 

+ e tf (7° sin 7M - 7-5 u sin 5M 
7! 2 6 



7 e 21 e J 

236" — " 1024 ■•■ < 6 ' 24) 



+ 7-3-3 6 sin 3M - 7-5 sin M) 

+ e - (8 7 sin 8M - 8-6 7 sin 6M 
. 8! 2' 

+ 7-4-4 7 sin 4M- 3-7-2 7 sin 2M) 



(6-23) 



TABLE 6-2a 
Expansions of Powers of Sin M 

sin 2 M = i(l - cos 2M) 

sin 3 M = *- (3 sin M - sin 3M) 

sin 4 M = i (3 - 4 cos 2M + cos 4M) 

sin 5 M = i— (10 sin M - 5 sin 3M + sin 5M) 

sin 6 M = K-„ (10 - 15 cos 2M + 6 cos 4M - cos 6M) 

sin 7 M = gij- (35 sin M - 21 sin 3M + 7 sin 5M - sin 7M) 

sin 8 M = yj^- (35 - 56 cos 2M + 28 cos 4M - 8 cos 6M + cos 8M) 

sin 9 M = 2^5- (126 sin M - 84 sin 3M + 36 sin 5M - 9 sin 7M + sin 9M) 

sin 10 M = h^to- (126 - 210 cos 2M + 120 cos 4M - 45 cos 6M + 10 cos 8M - cos 10M) 

sin 11 M * jo2j( 4 6 2 s *n M - 330 sin 3M + 165 sin 5M - 55 sin 7M + 11 sin 9M - sin 11M) 

sin 12 M = ^rTB- (462 - 792 cos 2M + 495 cos 4M - 220 cos 6M + 66 cos 8M - 12 cos 10M + cos 12M) 

sin 13 M = -rx^r (1716 sin M - 1287 sin 3M + 715 sin 5M - 286 sin 7M + 78 sin 9M - 13 sin 11M + sin 13M) 
4096 



NOTE: 

The numerical coefficients are easily obtained from the Pascal's triangle (cut in half), as shown in Table 6- 2b. 



Ill- 29 



TABLE 6- 2b 



TABLE 6 -3a 



Pascal's Triangle and its Modification 

1 

1 1 

1 2 1 

13 3 1 

14 6 4 1 

1 5 10 10 5 1 

1 6 15 20 15 6 1 

1 7 21 35 (35) (21) 7 1 

1 8 28 56 70 (56) 28 8 1 



Note: In the Pascal's triangle, each term is the 
sum of the two terms immediately above it (e.g. , 
35+21 =56). The coefficients for the expansions 

of sin wl in Table 6-2a result if the Pascal's 
triangle is cut in half as shown below. 



General Forms of Fourier-Bessel Expansion 
(see any reference on celestial mechanics, 
e.g., Smart) 



E-M + 2) ij (ne) sin n M (6-25) 

n = l 

CO 

sin E = - > i J (ne) sin n M (6-26) 

e £_j n n 



n=l 



cos E 



+ \. V 3e~ ^n*™ 5 ^ cos n M 



n = l 



(6-27) 



n 


The Coefficient 


s of 


Expansion of sin M 





1 














1 




1 












2 


1 




1 










3 




3 




1 








4 


3 




4 




1 






5 




10 




5 




1 




6 


10 




15 




6 




1 


7 




35 




21 




7 


1 


8 


35 




56 




28 




8 1 



9 » M +Y 2. sinnM S f '"' J n + k (ne) 

n = l k--„ (6-28) 

where 

1 i/i 2 3 5,7 

r 1-Yl-e e.e.e , 5e , 

f ' i 2" T + IB - + T2-B - + • • 

(6-29) 



sin 9 



cos 6 



" 2 "^^ I n-aT (j n (ne)jsinn 

"- 1 . (6-30) 

2 

= - e + — ) J (ne) cos n ivl 

e /_, n 



M 



n = l 
d 



(6-31) 



I ? ^ K (ne) i 



i. = l + 5j- -2e^ ^ ^ ^J n (ne)}cosnM 
n=1 n (6-32) 



(r) 



2 •* 2 V^ i 

" 1 + ^-" 4 I V J n 

LJ n (6-33) 



(ne) cos n M 



n = l 



- = 1 + 2 
r 



x 3e 



) J (ne) cos n M 

L n 

n«l 



(6-34) 



M 
35) 



! I T !e J n (ne) C08n 

n-1 n ( ' (6-3 

CO 

2. = 2. *4 - e 2 Y I J (ne) sin n M (6-36) 

a e » l_j n n 



Note: Divergence for e > 0. 662743 



111-30 



and 



TABLE 6-3b 

Expansions of J (ne) 
n 

ar { J n ( ne >} = J n <^> 



J n (x) = 


Y k x n + 2k 
k ^ Q 2 k! (n + k) ! 


J x (e) = 


3 5 7 
2" TF + Ttt ' 18,432 + • 


J 2 (2e)= 


2 4 6 8 
e e , e e , 

2" ""B- + 4T ~72TT + ' ' • 


J 3 (3e) = 


9 e 3 81 e 5 x 729 e 7 
T5~ " ' 255 10,240 


j (4e) = 


4 Q 6 8 
2 e 8 e _,_ 8 e 




3 15 45 " • ' • 


J 5 (5e) = 


625 e 5 15,625 e 7 , 
768 18,432 + • • • 


J 6 (6e) = 


81 e 6 729 e 8 , 
80 "560 + • • • 


J 7 (7e) = 


117,649 e 7 

92,160 " • " • 



J Q (8e) 



J 1 (e) 



J 2 (2e) 



J 3 (3e) 



J 4 (4e) 



J 5 (5e) 



J 6 (6e) 



J ? (7e) 



J (8e) 



512 e 



1 3 e 
7 



5 e 7 e 



~W "TOT fo7H2 

9 3 5 7 

e - 2 e + e - e + 



27 e 



3" 

2 4 R 

405 e , 5103 e 

T5 - 236— 10,240 



16 e" 

— 5~ 



64 



"T5~ 





18 


432 


729 e 7 
70 


+ . . . 



3125 e 109,375 * 

— 7BT 

243e 5 

~ 4iy~ 

823, 543 e 6 
92, 160 

4096 e 7 
315 " 



TABLE 6-4 



Fourier-Bessel Expansions up to e 



E 



/ 2 4 6 x 

+ (t"V + If -■■■) 



sin M 



3 e- 



27 e , 243 e 
TIB— + 3T21T 



sin 2M 
7 



■) 



sin 3M 



4 



e 4 e 
"3 IT 



••) 



sin 4M 



C 1 



25 e l 



o r 8T 
6 



3125 e' 
9216 



■•■) 



sin 5M 



("TIT- -•••) sin6M 



f 16, 807 e' 
V 46,080 



•••) 



sin 7M + 



(6-38) 



sin E 



sin M 



\ (6-37) 



/ e e '\ 

= V 1 T" + lW _ ^TeT + ■■•/ 

/e e 3 e 5 7 

+ (l " h + w "-rnr + • ■ •) sin 2M 

, /3 e 2 27 e 4 , 243 e 6 \ 

\-~S T21 STTO - " ■ ■ • ) 



sin 3M 



, / e" 4 e" , 4 e 
+ U T5~ + 



■). 



( 



125 e 4 3125 e 6 



♦( 



27 e 



5 



~9"2T6" 
243 



-53 . . ./ sin 4M 

+ . . .) sin 5M 



36" 



£>■■) 



sin 6M 



, / 16,807 e 
V 45,080 



'128 e' 
315 



••■) 



sin 7M 



sin 8M + 



(6-39) 



cos E 



e 
7 



( 



-a 2 c 4 „ 6 

3 e 5 e 7 e 

~5~ T9~2 9~2T6 



3 5 7 

+l e _ e + e _ e + 
+ V 2" J" + TT fit) 



♦ ...) 

. . . ) CO 



cos M 



s 2M 



(continued) 



III- 31 



TABLE 6-4 (continued) 



♦(¥--3f- ! &ir--)~ s " 



,/e 3 2e 5 + 8e 7 . \ 
4 .„„, 6 



cos 4M 



+ V ~3"81 3?T^ •■•/ °- u 

+ (81ei.81el + \ cog 



as 6M 



^ / 16,807 e 6 ^ „ na 7M 

+ V 46 , 080 "••■/ COS 7M 

+ (j^e ...) C os8M+ (6-40) 



( 



6 = M+ 2e - V + 4^ + 



e 3 , 5e 5 x 107 e 7 



T ~5E~ '?6~0T 



•-) 



+ . . . ) sin M 



^-HASf 8 ---) sln2M 



9 5 7 

'13 e 43 e , 95 e 



+ (^-^ + ^ 2 ----) Sin3M 

+ (ifcS^----) sln7M + (6 - 41) 



2 «„ 4 



, „ /, 7 e' 17 e* 317 e 6 x \ 
3 5 1Q 7 



F~ + T9T- " B51B- + • ' ' j Sin M 
( e " -T~ + ~T ""3TTT 



\ ± /9e 2 207 e 
+ . . . ) sin 2M + {-$ 128" 



3681 e 
5120 



--...) sin 



3M 



+ fl 7 li.34el + 121l. 7 -...)sin4M 



/625 e 4 29,363 e 6 
■\rm 9215 



+ I -^r= iT^TT! + ■ • • ) Sln 5M 



5 7 

+ I " e - 31 - 3 . e . + . . . ) sin 6M 



(81 



6 



-...) 



V 48,080 / 

(i°24_e_ 7 _ /) sln8 M + (6-42) 



cos 6 = -e + (l 



TABLE 6-4 (continued) 
2 



9 e 



^ 25 e 4 49 e 6 
+ _ 192 ""9TTB 

5 



. . .) cos M + ( e 



4 e 



3 e l 



2 e 



-•■) 



cos 2M 
6 



/9 e 2 225 e 4 , 3969 e 

+ \s~ — izs- 



-■) 



cos 3M 



+ /4_ef _ 12 e 5 4 .64 e 7 



V~3~ " " 

f625 e 4 
^ 384 



30,625 e 



/81 



"TO" 



486 e 
TUB" 

6 



7 



5T2TJ 

*5 ••• ) c ° 

cos 5M 



as 4M 



♦ ...) 

) CO 



s 6M 



/ 117,649 e" \ 

+ V 46,080 •••) 



cos 7M 



+ ( 10 3 2 i 4e7 -...) cos 8M+ (6-43) 



1 + 



( 



3e 
e --g- 



4 6 \ 

( 3e 



2 
e 

"7" 



as 2M 



45e J , 567e 
{-W ~ "T78~ "FTTTr 



- . . . ) CO 



is 3M 



■( 



4 
e 



2e c 



) 



cos 4M 



/125 4375e 7 , 
" VT8T " 9216 



•) 



cos 5M 



/81e b 



os 6M 



. . ) cos 

(»'-•■•) -™- (6 - 44) 



L) 2 . 1 + 3 e 2 



-(,e- 



T~ 



2 4 

e e 

,2" "TT 



+ |_ - .If.. + . . . ) cos M - U 
9lT 47jTJTT / v - 

6 \ 

. . . ) cos 2M + (continued) 



W 



[H-32 



TABLE 6-4 (continued) 



TABLE 6-4 (continued) 



+ fe 3 _ 9ef_ + 81e 7 _ ) cos 3M 
+ \"4~ TjT^ 235TT / 



+ f9e 6 _ Blei + ) co 



s 6M 



+ / 2401e _ ^ co 
+ V 23.640 •■■/ 



s 7M + 



(6-45) 



1 + (e - V- + 



3 5 

e e 



e 
57TB" 



4 6 

2 e , e 

e " T + 2T 



^ + .. . ) cos M + ( 
- . . . ) cos 2M 

J*°L - 8 j£ + ^- -•••) cos3M 

MT^ T21T 5120 / 

f4e 4 _16ef_ + ) co 



ds 4M 



^ /625e 5 15,625e 7 ,, ) co 



s 5M 



/81e _ "\ cos 6M 

- (lll^lL -...) cos 7M + (6-46) 

\ 46,08(1 / 



e 2 3e 4 , 15e 6 

[1 + S_ + 



/ 3e 3 ^ 65e 5 , 2675e 7 

+ . . . ) cos M 

+ (^_ + V + ^r- + •••) cos 2M 

/l3e 3 25e 5 . 393e ? 



) cos 3M 
(continued) 



/lOSe"* I l29e" ) 

5 .„ „„..7 



cos 4M 



/l097 



e J 16,621e 



. ) cos 5M + (- 



223e 
TBTT 



. I cos 6M 



+ f 47 '" 3e7 - . . ) cos 7M + (6-47) 

+ V 4608 ' 



3e 



/ 3e 2 + 5e 4 „ 7e 6 
[} ~-g- T9T S7T6" 



( 



e e 
7 ~'^ 



+ . . . J cos M + 

5 7 \ 

+ e _ e + _ ) cos 2M 

A 3e 2 45e 4 , 56 7e 6 „ ) co 
+ V-8- "T2S~ + ~5TZV ■■■' 

+ fe 3 _2e 5 + 8e 7 _...) co 

+ ( l25e4 - 43 J. 5e6 + . . . ) cos 5M 
+ V 384 9215 / 

_ + ) cos 6M 



s 3M 



s 4M 



81e 
"MO" 



81e 
T4TT 



+ (} \ f H t.- -..-) cos 7M 
V 46,080 / 

+ I 1 ™* ? - ... )cos 8M+ (6- 



48) 



a 



5e 2 He 4 457e 6 
r" " "IBS" T2-TT 



■T2~ 5T ~4T 



3e 



51e 4 , 5 43e 6 
TW 5T2TT- 



+ ,e 3 _l3el + l3el.. 
\T" 30 ^72^ 



I25e 



4625e 



27e° I35e 7 



+ l~8ir "2T4~ 



sin M 



sin 2M 



sin 3M 



sin 4M 



sin 5M 



sin 6M + 

(continued) 



III- 33 



TABLE 6-4 (continued) 



TABLE 6-6 (continued) 



/ l6 807e 6 _ \ 
V 46,090 • • • ) sl 



16,807e 
45,090 

I28e 7 
"TTT" " 



in 7M 



sin 8M + 



(6-49) 



cos E « coa 6 +•=■ (1 - cos 26) 

2 3 

- ^- (cos 6 - cos 36)+ ^- 



cos E » cos 6 + •=• (cos 6 - cos 36) 



2 cos 26 +~ cos 46 + 



i) 



(6-64) 



TABLE 6-5 

2 
Expansions for Near-Circular Orbit (e < < 1) 



E 



= M + e sin M + . . . 



sin E = sin M + 5- sin 2M + 



cos E 



|- + cos M + j cos 2M + . . . (6-52) 



9 = M + 2e sin M + . . . 

sin 6 = sin M + e sin 2M + . . . 
cos 8 = -e + cos M + e cos 2M + . 



(0 



(i 

(I) 



= 1 - e cos M - . . . 



= 1 - 2e cos M 



= 1 + e cos M + . 



= 1 + 2e cos M + . . . 



(6-59) 



y + cos M + 5- cos 2 M + .. .(6-60) 



sin M + 5- sin 2 M + . . . 



(6-61) 



TABLE 6-b 
Expansions in True Anomaly and Eccentricity 



E = 6 - e sin 6 + ^- sin 26 

- ~- (sin 6 + I sin 3e) + . . . (6-62) 



sin E * sin 8 - | sin 26 - ~ (sin 6 - sin 36) 



T 



sin 46 - . . . 



(6-63) 



M 



+ ^g- (3 cos 36 - cos 56) 



» 6 - 2e sin 6 +|- e 2 sin 2t 



+ 4e cos 6 + 



...] 



(6-65) 



(6-50) 
(6-51) 




- g- e 3 sin 36 + . . . 
2 


(6-66) 


(6-52) 


r 

a 


■ l-ecosO-y (1 - cos 26) 




(6-53) 




e 3 
- =-£ (cos 36 - cos 6) - . . . 


(6-67) 


(6-54) 
(6-55) 


a 

r 


2 3 

= 1 + e cos 6 + e + e cos 6 

-1^"e(l + ^- + ...)siiie 


(6-68) 


(6-56) 


r 


(6-69) 


(6-57) 


y 


= -^n e cos 6 1 + 2e cos 
2 




(6-58) 




+ ~ (cos 26 + 5) 





(7-70) 



"Va fl + e cos 6 + ^- (3 - cos 26) 

3 -, 

+ -^j- (4 cos 6 - cos 36 - 7) + . . .1 (6-71) 



= e sin 6 - ^- sin 26 + ^- sin 36 



T 



sin 46 + . . . 



(6-72) 



1 3 
sin y - e sin 6 - -|- sin 26 + ^ e (sin 36-3 sin 6) 

- -j^r e 4 (sin 46-2 sin 26) + . . . (6-73) 



coaV»l+t- (cos 26 - 1) + %- (cos 36 + 7) + . . . 



~T 



#[' 



1 + 2e cos 6 + i- (4 + cos 26) 



X 



+ 3e cos 6 + 



...] 



(6-74) 



(6-75) 



III- 34 



TABLE 6-6 (continued) TABLE 7-1 (continued) 



-^- e sin 6 ll + 3e cos 6 /e + 1 

a p f e - 1 



a 

3e 2 1 

+ ■^-(3 + cos 26) + ...J (6-76) 



v" (e - 1) 
TABLE 7-1 P 

Hyperbolic Orbit Element Relations 



b 2 



b - r 

r 
P 



e - 1 



p - 2r 



P 



* 



(7-1) / 2 



(7-2) 



p v p _ 2 yap 

P 



2 _ „ 2 „ r y E (7-19) 



^_E_ (7-3) P "V-'pV - 2 » 



"'£ 



e - 1 



#^ 



-yf + 1 (7-2D 



(7-5) 



« I 1 + e) (7-6) a 

^ i\ 2 

v^ (e - 1) av +m 

X P_ 

r 2 av„ -f 



V©^ 



b * a>e 2 - 1 (7-10) 

- fev~ (7-11) 



r v . 2 



= ^ p (r + 2a) (7-12) fa 2 

2JjTa 3/2 v » a(e 2 - 1) 

= — S E (7-13) 



r 

-P-(r + 2a) 
a P 

=- P =— (7-14) / 2a v v 2 



2-1 = „ P 



a v - a 
P 



III- 35 



(7-15) 



"fr, "l/s (7 " 16) 



c urDix jLiemem neiauuns i- 

(see Fig. 6) = r J — P (7-17) 

P TP " 2r 



<^V- ■ ± M*-= 



(7-20) 



r 
= -E + 1 (7-22) 



p x E (7-23) 

2 2 

P — (7-7) P 



(7-24) 



u F-, < 7 " 8 > .2 2 

v p( v p- 2 VI) »VV <™> 

b - r 
" r p (7-9) P 

r v Z - 2 M - £- - 1 (7-26) 

P P 



Ev p -1 (7-27) 



-P— P— -1 (7-28) 



(7- 


-29) 


(7- 


■30) 


(7- 


-31) 


(7- 


-32) 



TI 


tBLE 7- 


-1 (c 


;ontlmied) 




p - b Ve 2 - 1 


(7-33) 


2r b 2 
P 








(7-34) 


b - r 
P 




- r (e + 1) 

P 








(7-35) 


■ ' (^ 








(7-36) 


2 2 

r v 

« P P 








(7-37) 




a 








r = Va 2 + b 2 - 
P ' 


(7-38) 


= a (e - 1) 








(7-39) 


« a(lf7£~. 


■ 






(7-40) 



2n a 


2 
av p -n 


■-^Tl 


p 


lt y 1+ d) 2 


p 

1 + e 


_ » (1 + e) 



v 
P 



isr 



V 

p 



b/*- 



P ^..2 



(7-43) 



(7-45) 



TABLE 7-1 (continued ) 

p Vmp 

v p - / rl „ — \ (7-52) 



(7-53) 



(fb 2 + p 2 - b) 
. *£~(l+e) (7-54) 



= ^k(l+e) (7-55) 

r 
P 



'M£ (7-56) 

r 



TABLE 7-2 

Time Variant Hyperbolic Relations 
(see Fig. 6) 

Elements 

a = ^ (7-57) 

rv - Ip. 



V3 2 
r v cc 
— T- 
rv - 



(7-41) / 3 2 2 „ 

*' ,i ' /r v COS Y 

_ (7-58) 



rv - 2/j 






(7-42) / - o 9 9 

-'- •■-^. rv cosStrv - 2 M ) (7-59) 



2 2 2 
„ = r v cos > (7-60) 



(7-44) - r^ « — jj-i^ (vi + -^ rv ' cos v (rv " 2 <^ 



p rv - 2fi \» n 



1 (7-61) 



v = H fl + t/l+4 rv 2 cos 2 \(rv 2 - 2 M ) ) 

p rv cos y \ ¥ & / 



(7-46) (7-62) 



(7_47) Time variants 



'a" + b' - a F = iE 

' = cos 



(7-63) 



r~. — r ' = cosh- 1 [i(i + |)1 

V F/ e \V (7-48) i eK a/J _ 

▼a(e-l) -l fe + cos 9 (7-63a) 

" co&n I 1 + e cos ej 

£. . (7-49) -, 

a (f 7 ! - v = 2 ta ^ iVe-^4- tan 2-J 



(7-63b) 



(7-50) r - i ■ P na 6 <7-64) 



Tb(e-l) ' (con' 



(continued) 



III- 36 



TABLE 7-2 (continued) 



2 IP \ er 



TABLE 8 (continued) 



l/e - 1 






(- 1 ) /) l+t (7-65a) 



F + e sinh F 



^ e - 1 



1 . /e 

In 



e sm w 
1 + e cos 



/ 2 1 
/e - 1 



+ cos 6 + Ve 2 - 1 



sin 6 



1 + e cos 



+ t 






2 e - 1 

r p 



2e cos 6 



♦ .'] 

/ S v 

\r f2p + r (e 2 - 1)J 



1 + e cos 6 



6 » c 



Yl + 2e cos 9 + e^ 

-- 1 (^) 



(7-65b) 
(7-66a) 

(7-66b) 
(7-67) 

(7-68) 
(7-69) 



TABLE 8 
Spherical Trigonometric Relations 



cos (cos L sin 3) 
-1 /sin L sin j3 



= sin 



sin v 



tan 



tan 



-1 / tan L ] 
I sin <j> sin ji J 

1 / tan L \ 
^sin v/ 

-1 /cos L sin v\ 

= cos I p — t 1 

\ sm <p J 

\ sin <j) y 
. -1 /cos /3\ 

= sln \^rr) 

-tan" 1 ( C0 . S , g tan *) 
\ sin v / 



(8-1) 
(8-2) 

8-3) 

8-4) 

8-5) 

8-6) 
8-7) 
8-8) 



i » tan 



1 /cot j3 \ 
\cos §/ 

-1 /tan iA 

os (iam; 



T -1 /cos i\ 

L « cos I —5 — a ) 

VsTn/3/ 

, -1 /sin i sin v\ 

" sin \-surp — ; 

» tan" (tan i sin j$ sin <f>) 
= tan (tan 1 sin v) 

= tan (sin i cos vtan<(>) 

» sin (sin i sin 4>) 

■ sin 



sin 



•1 /tan v\ 
^a7T]3"J 

•1 /cos ft sin <j> | 
\ cos v ) 



-1 



tan (cos (3 tan <j>) 
-1 /cos <j> 1 



VJ 







j3 - sin' 



1 /cos i \ 

lcos"Ly 

. -1 lain i sin v\ 

sin r^Tn-x — / 

-1 (sin i cos 4>1 
cos V cost ) 

cos (sin i cos v) 

-1 /tan i sin v\ 
cos ( , tan,), ) 



= tan 



1 / cot i \ 

ycos 4>y 

. -1 /tan v \ 

tan ^iBrrj 

sin" 1 ( ^J 1 v \ 
ycos L tan if) 

-1 /tanL\ 

cos l^ah-Fy 

1 /sin v\ 



= sin 



v * sin 



sin 



=■ tan 



1 /tan l\ 

1 / s in L s in (3 

\~~sTn~T ; 

•1 / sin L \ 
I tan i cos ifl 



(8-9) 
(8-10) 



(8-11) 

(8-12) 

(8-13) 
(8-14) 

(8-15) 

(8-16) 

(8-17) 

(8-18) 
(8-19) 
(8-20) 



(8-21) 

(8-22) 

(8-23) 

(8-24) 
(8-25) 

(8-26) 

(8-27) 

(8-28) 

(8-29) 
(8-30) 



(8-31) 
(8-32) 
(8-33) 



III- 37 



TABLE 8 (continued) 



cos 



1 /cosj3 \ 
\sin i / 

- 1 / s in/3 cos 4 >| 
\ cos i / 



-I 



= tan (cos i tan<p) 

= tan (sin L tan/3) 
■ 1 /cos /3 sin cj)N 

V sirrc / 

/ cos 4> \ 
\cos L/ 



cos 



. -1 



= sin (sin/3 sin<J> ) 



8-34) 
8-35) 
8-36) 
8-37) 
8-38) 
8-39) 
8-40) 



. -1 /sinLA 
Sln Uin-T-,> 

-1 /cos L cos B\ 
COS { sini ) 



tan" 1 (-J™±—) 
V^sin l cos v) 



= cos (cot i cot 0) 

. -1 / tanv \ 
= Sin ( sin 1 tan p) 

•1 /tan v \ 
^cos 1 J 



tan 



tan 



= si 



-1 I sin L cos v\ 

{ cosfj j 

cos (cos L cos v) 

, -1 /sinv\ 

sin \sm) 



8-41) 
8-42) 
8-43) 
8-44) 
8-45) 
8-46) 
8-47) 
8-48) 
8-49) 
8-50) 



III- 38 



L. PRESENTATION OF GRAPHICAL DATA 

The figures presented at the end of this chapter 
will not he discussed here. A list of figures is 
given at the beginning of this chapter. 

M. REFERENCES 

1. Pluramer, II. C. , "introductory Treatise on 
Dynamical Astronomy, " Dover Press, New 
York, 1960. 

2. Gedeon, ''Orbital Segment .Mechanics," Norair 
Division of Northrop Corporation, Los Angeles, 
lie port ASG-TM-61-43, 1061. 

3. Moulton, IX R. , "Introduction to Celestial 
Mechanics," Second Revised Edition, MacMillan 
Company, New York, 1958. 

4. Epstein, E. I., "Nomography Interscience 
Publishers Incorporated, New York, 1958. 

5. Levens, A. S. , "Nomography, " John Wiley 
and Sons Incorporated, New York, 1948. 

N. BIBLIOGRAPHY 

Baker, R. M. L. , Jr. and Makemson, IVL W. 

'An Introduction to Astro-dynamics," New York, 
Academic Press, I960. 

Beard, R. I?, and Rotherham, A. C. , "Space Flight 
and Satellite Vehicles," New York, Putnam, 
19 57. 

Bellman, R. 

"Stability Theory of Differential Equations," 
New York, McGraw-Hill Book Company, Inc. , 
1953. 

"Dynamic Programming," Princeton University- 
Press, Princeton, New Jersey, 1957. 

Benedikt, E. T. , "Collision Trajectories in the 
Three -Body Problem, " Journal of the Astro- 
nautical Sciences, Summer 1959, Vol. 6, No. 2. 

Herman, A. I., "The Physical Principles of Astro- 
nautics; Fundamentals of Dynamical Astronomy 
and Space 1 light, " New York, John Wiley & 
Sons, Inc. , 1961. 

Bizony, M. T. , ed. , "The Space Encyclopedia, " 
New York, E. P. Button and Company, Inc. , 
1958. 

Bowden, G. E. , and Flis, J. , "Notes of the Sum- 
mer Institute in Dynamical Astronomy at Yale 
University, Yale University Press, New Haven, 
Connecticut, 1959. 

Brillouin, L. , "Poincare and the Shortcomings of 
the Hamilton-Jacobi Method for Classical or 
Quantized Mechanics," Archive for Rational 
Mechanics and Analysis, Vol. 5, No. 1, pp 76 
to 94, 1960. 

Brouwer, IX, andCTemence, G. M., "Methods of 
Celestial Mechanics," New York, Academic 
Press, 196 0. 



Corben, 11. C, andStehle, P., "Classical Me- 
chanics," New York, John Wiley k. Sons, Inc. , 
1950. 

Danby, J. M. A., "Fundamentals of Celestial 
Mechanics," MacMillan , New York, 1962. 

Darwin, G. H., "Periodic Orbits, " Acta Mathe - 
matica, Vol. 21, 1899. 

Dubyugo, A. I)., ''Determination of Orbits, " 
MacMillan, New York, 1961. 

Eckert, W. J., Brouwer, IX, and Clemence, 

G. M. , "Coordinates of the Five Outer Planets, " 
The American Ephemeris and Nautical Almanac, 
U. S. Government Printing Office, Washington, 
IX C. , Vol. 12, pp 1653 to 2060, 1951. 

Ehricke, K. A. 

"Space Flight," New York, D. Van Most rand 

Company, 1 tic. , 1 960. 
"Cislunar Orbits, " Convair Astronautics, 

AZP-004, March 1957. 
"Restricted 3 Body System Might Mechanics 

in Cislunar Space and the Effect of Solar 

Perturbations," A/.M-013, March 1957. 
'|The Solar System, " AZM-008, June 1957. 
"Space Flight Mechanics of Nonpowered Motion," 

AZM-010, November 1957. 
"Space Craft," AZM-020, February 1958. 
"Powered Space Flight Mechanics, 1 ' AZM-011 
"Celestial Mechanics ," AZM-009, August 1957. 

Felling", W. , "Summer Institute in Dynamical 
Astronomy at Yale University July I960," 
McDonnell Aircraft, St. Louis, 1961. 

Finlay-Freundlich, E., "Celestial Mechanics," 
New Yoj-k, Pergamon Press, Inc., 1958. 

Goldstein, 11., "Classical. Mechanics," Reading, 
Massachusetts, Addison- Wesley Publishing 
Company, 1950. 

Herget, P. , "The Computation of Orbits, " Uni- 
versity of Cincinnati, 1948 (published privately 
by author). 

Herrick, S. 

Astrodynamics and Rocket Navigation, " New 
York, IX Van Nostrand Company (to be pub- 
lished). 
"Tables for Rocket and Comet Orbits," U.S. 
Government Printing Office, Washington, 
IX C. , 100 pp, 1953. 

Herrick, S., Baker, li. , and Hilton, C, "Gravita- 
tional and Related Constants for Accurate Space 
Navigation," American Rocket Society Preprint 
497-57, 1957. 

Hohmann, W. , "The Attainability of Celestial 
Bodies," Munich, R. Oldenburg, 1926. 

Jastrow, li. , "Exploration of Space, " MacMillan, 
New York, 1960. 

Jensen, J., Townsencl, G., Kork, J., and Kraft, 
D. , "Design Guide to Orbital Flight," New York, 
McGraw-Hill Book Company, Inc., 1962. 



Ill- 3 9 



Kellogg, O. D. , "Foundations of Potential Theory, 
New York, Dover Publications, Inc. , 1953. 

Koelle, H. H. , ed. , "Handbook of Astronautical 
Engineering," New York, McGraw-Hill Book 
Company, Inc. , 1961. 

Kraft, J. D. , Kork, J. , and Townsend, G. E. , 
"Mean Anomaly for Elliptic, Parabolic and 
Hyperbolic Orbits as Functions of the Central 
Angle from Perigee, " The Martin Company 
(Baltimore), Engineering Report No. ER 12083, 
November 1961. 

Krogdahl, W. S. , "The Astronomical Universe, " 
MacMillan, New York, 1962 (2nd ed. ). 

Legalley, D. P. , ed. , "Guidance Navigation, 
Tracking and Space Physics, Symposium on 
Ballistic Missile and Space Technology, Los 
Angeles, August 1960,'' Ballistic Missile and 
Space Technology, New York, Academic Press, 
Vol. 3, 450 pp, 1960. 

MacMillian, W. D. 

"Dynamics of Rigid Bodies," New York, Mc- 
Graw-Hill Book Company, Inc. , 1936. 

"Statics and the Dynamics of a Particle, " 

Theoretical Mechanics, New York, McGraw- 
Hill Book Company, Inc. , Vol. I, 1927. 



Mehlin, T. G. , 1906, 
Wiley, 1959. 



'Astronomy, " New York, 



Moulton, F. R. 

"An Introduction to Celestial Mechanics, " New 

York, The MacMillan Company, 1914. 
"Periodic Orbits," The Carnegie Institute, 
Washington, Publication No. 161, 1920. 

Oertel, G. K. , and Singer, S. F., "Some Aspects 
of the Three Body Problem, " University of 
Maryland, Physics Department Report No. 
AFOSR TN 59-405, March 1959. 

Payne -Gaposchkin, C. , "Introduction to Astronomy, " 
New York, Prentice -Hall, Inc. , 1954. 

Plummer, H. C. , "Introductory Treatise on Dynam- 
ical Astronomy, " New York, Dover Publica- 
tions, Inc. , 1960. 

Proell, W. , and Bowman, N. J. , "A Handbook of 
Space Flight," 2nd edition, Chicago, Parastadion 
Press, 1958. 



Russel, H. N. , Dugan, R. S. 
"Astronomy, " 2nd edition, 
pany. Vol. 1, 1945. 



, and Stewart, J. Q. , 
Boston, Ginn & Corn- 



Scarborough, J. B. , "Numerical Mathematical 
Analysis, " Baltimore, The John Hopkins Uni- 
versity Press, 1955. 

Siefert, H. S. , "Space Technology," New York, 
John Wiley & Sons, Inc. , 1959. 

Siegle, C. L. , "Topics in Celestial Mechanics, " 
Baltimore, The Johns Hopkins University 
Press, 1954. 

Soule, P. W. , et al. , "Performance Manual for 
Orbital Operations, " Northrop Corporation, 
Report No. NOR 61-208, September 1961. 

Smart, W. , "Celestial Mechanics," New York, 
Longmans, Green and Company, 1953. 

Sternberg, W. J., and Smith, T. L. , "Theory of 
Potential and Spherical Harmonics, " Toronto, 
Canada, University of Toronto Press, 1944. 

Synge, J. L., and Griffith, B. A., "Principles 
of Mechanics, " New York, McGraw-Hill Book 
Company, Inc. , 1959. 

Timoshenko, S. P., "Advanced Dynamics, " New 
York, McGraw-Hill Book Company, Inc., 1948. 

Vertregt, M. , "Principles of Astronautics," New 
York, Elsevier Publishing Company, 1960. 

Watson, C. , "Theoretical Astronomy, " 2nd edition, 
Philadelphia, J. B. Lippincott Company, 1892. 

Whittaker, E. T. , "Analytical Dynamics, " New 
York, Dover Publications, Inc. , 1944. 

Wintner, A. , "The Analytical Foundations of 
Celestial Mechanics,' Princeton University 
Press, Princeton, New Jersey, 1947. 

Introduction to Ballistic Missiles. Prepared by 
STL for the Air Force Ballistic Missile Divi- 
sion, March 1960. 

Vol. I --Mathematical and Physical Foundations, 

ASTIA No. 240177. 
Vol. II --Trajectory and Performance Analysis, 

ASTIA No. 240178. 
Vol. III--Design and Engineering Subsystems, 

ASTIA No. 240179. 
Vol. IV- -Guidance Techniques, ASTIA No. 
240180. 

Notes on Space Technology. Langley Field, Vir- 
ginia, Langley Aeronautical Laboratory, Flight 
Research Division, May 1958. 



Ill- 40 



ILLUSTRATIONS 



III- 41 



a 

■«-» 

o 
o 

> 



3.5 — i 
3.4- 

3.3- 
3.2- 

3.1 — 
3.0- 
2.9— 
2.8- 
2.7 — 

2.6- 

2.5—1 

2.4- 

2.3- 

2.2- 

2.1- 

2.0- 

1.9- 

1.8- 

1.7- 

1.6- 

1.5- 



1.0- 



0.5— 



Note: 

See Fig. 15 for graphical trends 

and metric data 



•2.0 

-2.1 

-2.2 

-2.3 
-2.4 
• 2.5 
■ 2.6 
-2.7 
•2.8 
-2.9 
-3.0 



— 3.5 ° 



-4.0 

-4.5 

-5.0 

-5.5 
— 6.0 

-7.0 

- 8.0 
-9.0 
-10.0 



—15.0 

-20.0 
-25.0 



3 

■H 

3 



5.2 






5.4 




2.2 


5.6 






5.8 




2.3 


6.0 






6.2 




2.4 


6.4 






6.6 




2.5 


6.8 






7.0 




2.6 


7.2 










2.7 


7.6 






8.0 — '. 




2.8 


8.4 — '■ 




2.9 


8.8 

9.2—: 




3.0 


9.6 






10.0 







0. 



11.0- 

12.0- 
13.0- 

14.0- 
15.0- 



20- 



2 5- 

30- 

35- 
40- 

50- 
60- 

70- 

80- 

100- 



2.1 



3.5 



4.0 



4.5 



5.0 



6.0 



- 7.0 

- 8.0 

- 9.0 
-10.0 
-11.0 

-12.0 
-13.0 
-14.0 
-15.0 

-20.0 



u 
o 

!? 

e 
e 

en 



F ; i;;. la. Semi major Axis as a Function of the ilnJius and Velocity at any Point 
(hnglish Units - sec ! : i;',s. lb and lj for Other 'InitsJ 



^r,E WANK NOT FILMFT 



111-43 











1 I 




1 1 






























Hi ll Hi 


















1 












ill ill 






























V 






























III 






























11 






























ill 






























VI 






























iiii 






























jfiiiii 






























j 


























































































































































































i 






























iii 






1 
























u 






1 










\ 














A i 
















^ 


















1 














M 


a 
< 














\ 




1 


























M 
































'j 


, 


























u 
i 


+ 
a 

d 
u 

> 




w 


























\ 


</ 






















+ 

a 

u 

a 






V, 


' 
























w 
























n 






V 






















> 






\ 


\ 
































\ 





















,tt\u 



o 
o 



> > 



III -44 



180 



3.0- 



2w (t - t ) 

M » E- . E ■ e sin E 



2.0- 




M (rad) 




40 80 120 160 200 
E (deg) 



The Eccentric Anomaly, E (deg) 



1.0- 



rl.O 



■ 0.1 



-o.6 a 



0.4 W 



-0.2 



F i jj . 2. The Relationship between Irbital Position and hecent rici tv and Tine from Perigee 
(Kepler's hmiation) (also see J-'i;.;. JJ) 



ni-4. n 




Fig. 3. Three-Dimensional Geometry of the Orbit 




Fig. 4. Geometry of the Ellipse 



111-46 



Directrix 




Fig. 5. Geometry of the Parabola 



Focus 




Fig. 6. Geometry of the Hyperbola 



III - 47 



■15 



^|G 



at 
o 
VI 

T3 

a 



(1) 



o 
i— i 

X 



01 
C/3 



nt 
o 
w 

■a 

a 

I 

+-» 



1000 TB 



ht 0. 240 



•- 0.230 



900-i 



l - 0. 220 






.r0.2io 



2000 if; 



:?ooo 



■to. 35 



n 



u 

h 

\\ 
1500 -h 



-* 



+ 0.30 



800 
Fig. 7. The Parameter 



\{ 



tfO.25 

1000 Ji 

1 T 



::0.50 



+ 0.49 



0.48 



0.47 



2500 



4000 



rt°- 



46 



■P0.45 

: +0.44 

I 

ft 

i 

-1 

.tO. 43 
ttO.42 



0.41 



II 



C 1 



2000 Hi 



tO. 40 



TO. 39 



0.60 



0.59 



0.58 



-*0.57 



3500-: 



tO. 56 



3000. 



r0.55 



n 



0. 54 



*t 



0.53 



0.52 



r» 0.51 



as a Function nf Semimnjor Axis 
n 2T 

(l.n'jlisli Units - see Table 9 for 'Ictric Data] 



5000 Tr 

N 



Lf 0.70 



II 



ihO.69 



r*0.68 



1 

1*0. 



1 



67 



♦ 0.66 



4500+- 



0.65 



i : -0. 64 



h0. 63 



tl 



0.62 



?1 



4000- 



:• 0.61 



111-48 



*4 



II 


< 


-ilC 


u 






5 1 


0" 


8 









£ 


w 


5) 




co 


T5 




a 




as 


0) 


si 


"ni 


1 




CO 


a) 




J 








6000 T 



T* 



-0.77 



-0. 76 



5500 



5000 



0. 79 



iO. 78 






0. 75 



-0. 74 



0. 73 



-•0.72 



0. 71 



7000 tt 



.*0. 88 



+ 0.87 



+ 0.86 



-?0.85 



6500 -■- 



6000 01 



0.84 



r-0.83 



-0.82 



-■■0.81 



■1 
■i 
4 0.80 



8000 Tt 



7500 ~i 



7000 ii 



0.96 



■0.95 



0.94 



0.93 



0.92 



-0.91 



0.90 



.5-0.89 



1 



9000 tt 10,000 fwl. 12 



+ 1.04 



-1.03 



-1.02 



+ " 



8500 -- 



-1.01 



8000 



* 1. 11 



trl.09 



9, 500 



1.00 



t" 



:0.99 



r0.98 



0.97 



ill 9,000* 



1. 10 



1. 0E 



17 1.07 



1. 06 



fr 1. 05 



Fig. 7. (continued) 



111-49 





o 




H 




X 




+-» 


^«, 


<M 


o 




OJ 


rt 


in 










01 


. i= 


•■-i 


i-|n 


X 
< 


ii 


u 


i 


p 


-n|ci 


'c? 




S 


01 


-t-t 


1— 1 

cd 


s 


o 


1> 


OT 


w 


■a 


« 


s 


t— I 

o 


si 


2i 


OT 


tfH 




0) 


■a 


J 


§ 




x 




+-» 




-C 




M 



« 



11,000 15 12, OOOlg lS.OOOtF 14,000 



fi 



n 



1^1.18 



I? 



1.26 



:• 1. 25 



M 



ril. 17 



H 



■i 



t 1.24 



Ff 1. 16 

H 

10,500-- 11,500 



t? 1 - 15 

H 



-*1. 14 



1. 13 



H 



F1 



.* 1. 31 



1. 22 



; -1.29 



■1.21 



Et 



1.20 



lO.OOOlfi 11,000* 12,000ii 13.000.1 



D I 



'i 1.40 



1.33 



1.32 



1.23 12,500*- 13,500 

: ■* 1 or\ 



+ 1. 37 



1. 30 



I! 

ft 

t: 



±r l.2£ 



*"1.27 



1.39 



1.3E 



n 



t 1.36 



LJ 



;. 1. 35 



* 1.34 



Fig. 7. (continued) 



III -50 



TABLE 9 
Circular Velocity Period and Angular Rate 
(metric data; see Figs. 7 and b for English data) 



Velocity p |-| j5 1*7 6 )4« 6.141 8.137 B. 1 34 8.130 6.127 b ■ 1 7* 6. '20 8.117 6.114 fl. 1 in 8.107 8.104 8 . 1 00 8.097 8.094 8.090 8.0*7 

Period 1 =3 1.286 1.289 1 290 1.291 1.293 1.294 1.29b 1.296 1.299 1.301 1.303 1.304 1.306 1.307 1.309 1.311 1.313 1.3M (.313 

Anf. Vel. 4.390 4.384 4.878 4.872 4.866 4 . 8*0 4.634 4.846 4.842 4.636 *-830 4.824 *.8lft 4.812 4 . ftO* 4.800 4. 79« 4.788 4.7S2 *.?77 

Velocity i G84 6.030 6.077 6 074 6.070 8.067 8.064 6.061 8.057 8 . 084 8.031 8.047 6.044 8.041 8.038 8. 034 6.031 8.026 8.023 8.021 

Period l.*l7 1.319 1.320 1.322 1.324 1.323 '.327 1.326 1.330 1.372 1.333 1.335 1.337 1.338 1.340 1.341 1.343 1.343 1.346 1.348 

Al*. V«l. 4. 771 4.763 4.739 4.733 4.747 4.7,1 4-736 *■ 730 4.724 4.7l8 * . 7l 3 4.707 4.701 *.693 4.690 4.66. 4.678 4.673 4.6*7 4.661 

Velocity 8.018 6.0l3 6.012 8. 008 8.003 8.002 7.999 7.996 7.992 7.989 7.9£C. 7.9B3 7.96Q 7,970 7.973 7.970 7.967 7.964 7.9*1 7. 9(17 

Period 1.330 t.331 1.333 1.334 1.336 1.338 1.339 1.361 1.363 1 . 36* ».!Ct 1 . 36P 1.369 1.371 1.372 1.374 1.376 1.377 1.379 1.381 

Am V«l. 4.636 4.630 4.644 4.639 4.633 4.628 4.622 4.617 4.bl I 4 . t>03 4 . r; CO 4.594 4.569 4.383 4.578 4.572 4.56 7 4.362 4.33* 4.3B1 

Velocity 7. 934 7.931 7.946 7.943 7. 942 7.9*9 7.935 7.932 7.929 7.9;6 7.9;7 7.920 7.917 7.914 7.910 7.907 7.904 7.901 7.698 7.893 

Period 1.362 1-384 1.386 1.387 1.389 1.391 1.39; 1.394 1.396 1.397 1 . 7 C '' 1.400 1.402 1.404 1.403 1.407 1.409 1.410 1.412 1.414 

If V«l 4.343 4.540 4.334 4.529 4.524 4.318 4.513 4.306 4.502 4.«97 *.492 4.4GU 4.48; 4.476 4.471 4-4b3 4.460 4.433 4.430 4.444 

Velocity 7.892 7.889 7.B86 7.883 7.680 7.876 7.873 7.870 7.R fe 7 7.*b* 7 . S-. 1 7.35ft 7. 335 7.352 7.949 7.846 7.843 7.640 7.837 7.814 

Period 1.415 1.417 1.419 1.420 1.422 1 . *24 1.423 1.427 1.429 I.43O t . * 72 l.4i« 1.435 1.437 t.4j9 1.440 1.442 1.444 1.443 1.447 

Ang. Vel. *.*39 4.434 4.429 4.424 4.418 4.413 4.408 4.403 4.398 4 . 3v3 " . 3S£ 4.3&3 4.377 4.372 4.367 4.3&2 4.337 4.332 4. 34 7 4.342 

VelOCtty 7.831 7 826 7.823 7.622 7.619 7.816 7.813 7.810 7.607 7.504 7.t'01 7.79& 7.793 7.792 7. 769 7.786 7.783 7.780 7.777 7.774 

Period 1.449 i.43o 1.432 1.454 1.433 1.457 t.439 1 . 460 i.*i = 2 1.464 i.«l? imct 1.46" 1.470 1.472 1.474 1.476 r.*n \ . 479 i.*tti 

Mm- Vel. *-337 4.332 4.327 4.322 4 . J1 7 4.312 4.307 *.502 4.^7 4.292 *-iS 4.293 -.279 4. 273 4.268 4.263 4.236 4.25J 4.249 4.24* 

Velocity 7.771 7.768 7.763 7.763 7.760 7. 737 7.734 7.751 7.7*6 7. 743 7.742 7.739 7.73& 7.733 7.730 7. 72B 7.725 7.722 7. 7 19 7.7l6 

Period I .482 1.484 1.486 1.487 1 . 489 1 . 491 1.492 1 .494 1 . 496 1 . 497 ' • *99 1.501 1 .503 1 . 504 1 . 30« 1 . 308 l . 3U9 1 .31 1 ' .513 1.614 

ADf. Vel. *.239 4.234 4.229 4.225 4.220 4.213 4.210 4.205 4.101 4 . 1 96 •■ 191 4 . 1 £6 4. 182 4. 177 4. 172 4 . 1 68 4. 163 4.158 4. 154 4.149 

Velocity 7.713 7.710 7.707 7 705 7.702 7.699 7.696 7.693 T.,.90 7.667 7.685 7.632 7.679 7.676 7.673 7.670 7,666 7.6«»3 7.662 7.689 

Period 1.316 1.318 1.519 1.511 1.523 1.323 1.326 '.319 1.530 1.631 1.333 1.535 1.336 1.338 1.340 1.342 1.3*3 1.343 1.34' 1 . &46 

AW Vel. 4. 144 4. 140 4. 133 4.131 4. 126 4. 121 4. 1 1 7 4.112 4. 109 *. < 03 4.09S 4. 094 4. 089 4.083 4.080 4 . 076 4.071 4.047 4.062 4 . OS* 

Velocity 7.636 7.653 7.651 7.648 7..>«3 7.6*2 7.6 39 7.63 7 7.634 7.631 7.628 7.623 7.623 7.620 7.61 7 7.614 7.612 7.609 7.606 7.6A1 

Period I .350 I .352 1.554 1.533 1 .337 1.339 1.360 1 .56 2 1 .'564 1 .3(,6 1 .56 7 1.369 1.371 1.572 I .374 1.376 1.376 1.379 1.3R1 1 . o9J 

AdC. Vel. 4.033 4-049 4.044 4.040 4.033 4.031 4.027 4.022 4.,-il8 4 . d I 3 4.009 4.003 4. 000 3.996 3.992 3.967 3.983 3.978 3.974 3. 9?0 

7.601 7.398 7.395 7.392 7.590 7.587 7.58* 7.58! 7.579 7.576 7.373 7.370 7.566 7.^65 7.362 7.360 7.337 7.334 7 . ?13 1 7-849 

1 .334 1 .386 1 .388 1 .590 1 .391 1.393 1.595. 1 . 3 -:■ 7 1 .596 1 .600 1 -602 1 .603 1.605 1.6U' 1 . WJ9 l.oiu 1.612 1.614 1.616 1.617 

3.966 3.96' 3.957 3.933 3.948 3.944 3. 9*c< i.vJfc. 3.y31 3.927 3.923 3.919 3.914 3.910 3.906 3.902 3.898 3.»93 3.B69 3.883 

Velocity 7.546 7.343 7.341 7.338 7.533 7.53! 7.530 7.52? 7.5.25 7.522 7.*<9 7.517 7.314 7.311 7.309 7.306 7.303 7.301 7.498 7.493 

Period 1.619 1.621 1.622 1.624 1.626 1.628 1 . 629 1.631 1.633 1.633 ' ■ 6Ji> 1.638 1 . o*0 1,642 1.643 1 . t>43 1.t>47 1.649 '.630 1 . 6B2 

3.881 3.877 3,873 3.966 3.364 3.860 3.BS6 3.652 3.5*8 3.344 3.640 3.836 3.831 3.827 3-823 3.819 3.613 3.811 3.807 3.803 

Velocity 7.493 7.490 7.487 7.485 7.482 7.480 7.477 7.*7* 7.472 7.469 7.466 7.464 7.461 7.439 7.436 7.433 7.431 7.448 7.446 7.443 

2 Period 1.634 1.636 1.637 1.639 1.661 1.663 1,664 1.6-6 1 . toiiW 1 . fa 70 1 . 67 1 1.673 1.673 1.677 1.678 1.680 1.682 1.684 1.663 1.687 

*" AJ*. Vel. 3-799 3.793 3. 791 3.787 3.783 3. 779 3. 77? 3.77 1 3.7 b 7 3.763 3.739 3.755 3. 751 3.748 3.744 3.740 3.736 3.732 3.728 3.724 

§ Velocity 7.441 7.438 7.433 7.433 7.430 7.428 7.425 7.42 3 7.420 7.417 7.413 7.412 7.410 7.407 7.405 7.402 7.400 T.397 7.394 7.392 

Period 1.689 1.691 1.692 t 694 1.696 1.698 1.699 1.701 I . 703 1 . 705 1.707 1.708 1.710 1.712 1.714 1.713 1.717 1.719 1.721 1.722 

** AM Vel 3.720 3.716 3.713 3.709 3.703 3.701 3.697 3.693 J.6B9 3-686 3.682 3.678 3-674 3.670 3.6&7 3.663 3.659 3.633 3.632 3.648 

o Velocity 7.3u9 7.3B7 7.384 7.382 7.379 7.377 7.374 7.372 7.369 7.367 '.■»* 7.362 7.359 7.337 '.33* 7.332 7.349 7.347 7.3*4 7.342 

2 Period 1.724 1.726 1.728 t . 730 1.731 1.733 1.733 1.737 1 . 736 1.7*0 1-742 1.744 1.7*6 1.747 1.749 1.731 1.733 1.734 1.736 I - 738 

p Anf. Vel. 3-6*4 3.640 3.637 3.633 3.629 3.623 3.622 3.618 3.61* 3.6| 1 3.607 3.603 3.600 3.396 3.392 3.389 3.383 3.381 3.378 3.674 

§ Velocity 7.339 7.337 7.334 7.332 7.329 7.327 7.324 7.32 2 7.320 7.317 7.313 7.312 7.310 7.307 7.303 7. 302 7.300 7.297 7.295 7.293 

Period 1 -7f.O 1 .762 1.763 1.763 1 . 767 1.769 1.770 1 .772 1.77* 1.77* 1 . 778 1 . 779 1.781 1 . 783 1 .783 1.787 1.788 1.790 1.792 1.79* 

*" Am Vel 3.570 3.567 3.363 3.360 3.536 3.332 3.349 3.543 3.3*2 3. 338 J. 333 3.331 3.527 3.324 J. 320 3.317 3.513 3.310 3.306 3.303 



§Veloc 
Perlo 
Am. 

fVelc 
Perl 
Am. 



Velocity 

Period 
Ar«. Vel. 


7.2 90 
3-499 


7.2B6 
1 .797 
3.496 


7.285 
1 . 799 
3.492 


1,263 
1 .601 
3.489 


7.280 
1.8Q3 
3.465 


7. 
3. 


2T8 
903 

4£2 


1.806 
3.47* 


3.475 


7 . : 7 1 


3.4L-6 


7.-1. 1: 
3.*t5 


?!4.':.i 


\'. + '*i 


3! 434 


':\l?\ 


1.*8 23 
7. «*7 


1 .624 
3.444 


7.249 
1.826 
3.441 


7.247 
1 .828 
3.*37 


7.24* 
I.83O 
3.434 


Velocity 
Period 
Am- Vel. 


1 !s32 
3.430 


7.240 
1.633 

3.427 


7.237 
1.633 

3.424 


7.233 
1 .837 

3.420 


7.233 

1 . 8 J9 
3.417 


7. 
3. 


2 30 
84I 


7. 4 Id 


?.4t* 


'.2-1 


7.221 

1 .*4S 

3. 4 CO 


t . e-50 
3. 3-"' 7 


3. 39-t 


"■'■ ~-'A 


3. 3W ' 


7 . ^' 
1 .637 
7. ,"8* 


7.207 
1 .859 
3. 380 


7.204 
1.861 
3.377 


7.202 
1.862 
3.374 


7.200 
1 .864 
3.370 


7. 197 
1 .866 
3.3*7 


Velocity 
Period 
Am- Vel. 


7. 195 
1 .$68 
3. 36* 


7. 193 
1 .870 
3.36! 


7. 190 
1.871 
3. 357 


7. 188 
1.873 

3. 33* 


7. 186 
1.675 
3.351 


7, 
3. 


877 

34* 


K 8 7 9 
3. 344 


1 .601 
3. 341 


\ . 662 
3. 338 


7. 174 
1 .884 
1. 335 


7. 172 

1 . 6fc6 

3.331 


1 .866 
3. 323 


1 . e*'o 

l. 323 


7.165 
1.892 
3.322 


7.162 
1-893 
3.318 


7. 160 
1 .895 
3.313 


7. 158 
1.897 
3.312 


7.136 
1 .899 
3.309 


7. 133 
1 .901 
3.30* 


7.131 
1 .903 
3.303 


Velocity 
Period 
Am- Vel. 


7. 149 

1 .904 
3.299 


7. 1*6 
1 .906 
3. 29b 


l!906 
3.2 93 


7. 142 
1.910 
3.2 90 


7. 139 
1.912 

3.287 


7. 
1 . 


1 '7 

.914 
. 2fe* 


7. 135 
1.915 
3.250 


7 . 1 3 J 
1 .917 


7. 1J0 

1.9(9 
3.27* 


7. 1 28 

1 .921 
3.271 


7. 126 
1 .92 3 

3.262 


7. 124 

3. -65 


7. 121 
I. 262 


7. 1 19 
1.926 
3.259 


7.117 
1 .930 
3.233 


7.1i5 

1.9J2 


7. 1 12 
t.93* 
3.2*9 


7.1 10 
1.936 
J. 246 


7. 108 
1.937 
3.243 


7.103 
1.939 
3.240 


Velocity 
Period 
Am- Vel. 


7. 103 
3.237 


7. 101 
1 .943 
3.23« 


7.099 
1.943 
3. .31 


K947 

3.226 


7. 09* 
1 . 94 6 
3. 225 


7. 


092 
.9 50 
■222 


7.090 
1 . 95 2 

3.219 


7.r.:?E 
1 .954 

3. i 1 6 


7.f*5 
I . 956 


7,063 
1.956 


7.081 
1 .960 
3.206 


7.079 
1 . 961 
3.203 


7.076 
1 . 963 
3.20C- 


7.074 

1 .965 
3. 197 


7.072 
3. l'9* 


7.070 
1.969 
3. 191 


7.068 
1.971 
3. 188 


7.063 
1.973 
3.183 


7.063 
1 .974 
3. 182 


'.061 
1 .976 
3.179 


Velocity 
Period 
Am. Vel. 


7.059 
1 .978 
3. 176 


7.036 
1 .980 
3.173 


7.054 
1.982 
3. 170 


7.052 
1.984 
3.168 


7.050 
1 . 965 
3. 163 


7. 


.040 
.987 
. 162 


7.0*6 
1.989 
.-.139 


7.04 3 
1.9C1 
3. 156 


1 . 993 
i. 1-53 


7 . 39 
1 . 993 
3. 150 


7.037 
1 . 997 

i. 147 


7.031 
1 . 999 
3. 1*4 


7.032 
2.000 
3. 1 41 


7.030 
2.002 
3. 1 38 


7. 028 
2.004 
3. '33 


7.026 
2.006 
3. 132 


7.02* 
2.008 
1. 129 


7.021 
2.010 
J. 126 


2^012 
3.12* 


7.017 
2.013 
3. 121 


Velocity 

Period 
Am- Vel. 


7.013 
2.015 
3. 118 


7.013 
2.017 
3. 1 13 


7.01 1 
2.019 
3. 1 12 


7.009 

2.021 
3.109 


7.006 
2.023 
3. 106 


7. 
3. 


.0(14 

.025 
. 103 


7.002 
2-02u 
3. 101 


7.000 
2.0*8 
3.098 


£.996 
2.Q30 
3.095 


6.996 
2.032 

3.092 


6.993 
2.03* 

3.089 


6.991 
2-036 
3.086 


6.989 

2.038 
3-083 


6.997 
2.O40 
3.081 


6.985 
2.0*1 

3.G78 


6.983 

2-0*3 
3.075 


6.981 
2.0*5 
3. 075 


6.978 
2.047 
3,069 


6.976 
2.049 
3.06' 


6.9 74 
2.001 
3.U6* 


Velocity 
Period 
Am- Vel. 


6.972 
2. Q33 
3.0*1 


6.970 
2.035 
3.03B 


6.966 
2.056 
3.053 


6.966 

2.038 
3.053 


6.96* 

2.06 
3.030 


6. 


. 9b 1 
. o,:2 
.0*7 


fa. 939 
2.064 

3. 0*4 


2.0fa6 
3.0*1 


b.955 

2.066 

3.07? 


6.933 
2. 070 
S.02C 


6.931 
2.072 

3-033 


6.949 

2.073 
3.030 


6.947 

2.075 
3-023 


t..9*3 

2.077 
3.025 


h . 9* 3 

2.079 
3.^22 


6.9*0 
2.081 

3.0l9 


6.938 
2.063 
3.017 


6.936 
2.083 

3.014 


6.934 
2.0B7 
3.011 


6. VJ2 
2.088 
3.U0S 


Velocity 
Period 
Am. Vel. 


6.930 
2.090 
J. 006 


6.926 

2.092 
3.003 


6.926 
2.09* 

3.DO0 


6.924 

2.096 
:.99S 


6.922 
2.099 

: . 993 


t 


.920 
. 100 


2'. 102 

:.9T'0 


2.9E7 


. 9 1 3 

2. 106 

r.^64 


6.911 

2. 107 

:.9ei 


6 . f09 
2. 109 

2.979 


6. 907 
2.111 


6.905 

:!-"-7j 


6.?03 

2.H5 
2.971 


G.90I 
2.117 
2.968 


6.899 
2. 1 19 
2.963 


6-897 
2. 121 
2.963 


6.893 

2.123 
2.960 


6.893 
2. 124 
2.938 


6.891 

2.126 
2-9B3 


Velocity 
Period 
Am. Vel. 


6.869 
2. 128 
2.932 


6.887 
2.130 
2.930 


6.884 

2.132 
2.9*7 


6.832 
2.134 
2.9** 


c.680 
:. 136 


*• 


. 1 38 

. 9 %■- 


'. . &76 
2. 1*0 

2.937 


6.674 
2.934 


6.872 
2. 1*3 
2.931 


6.870 
2. 1*5 
2.929 


2.147 


6.866 


2.151 


£,.*V>2 
2. 153 
2.918 


6.860 

i.133 
2.916 


6.B38 
2. 157 
2.913 


6.856 
2. 159 
2.9H 


6.834 
2.161 
2.908 


6.832 

2. 163 
2-905 


6.880 
2. 1t>« 
2.903 


Velocity 
Period 
Am- Vel. 


6.848 

2.166 
2.900 


6.846 
2.168 
2.898 


6.8*4 
2.170 
2.893 


6.8*2 

3.172 
2.893 


6.6*0 
2. 174 

2.390 


t 


. ess 


6.8 36 


2! liO 

2 . i s ;■ 


u.*j2 

2 ■ *S0 


5.830 
2.16* 

2.677 


6.826= 

2. 1&6 

2.375 


6.326 
2. 187 


6. £2* 

2 . 1 6 9 

2. ::7Q 


6.822 
2.191 

2.86 7 


6.820 
2.193 

2.663 


6.816 
2.193 
2.862 


6.616 

2-197 
2.860 


6.814 
2. 199 
2.837 


6.812 
2.201 
2.833 


6.810 
2.2o3 
2.882 


Velocity 
Period 
Am. Vel. 


6.808 
2.203 
2.830 


6.806 
2.207 
2. 847 


6.80* 
2.209 
2.6*3 


6.802 
2.21 1 
2.8*2 


6.800 
2.2'2 
2.640 


I. 


. 377 


2-216 


is . 7 9 4 
2.-18 


6.79 2 
2. 220 
2.r-J0 


6.790 
2-828 


t.788 
2.22* 

2.625 


6.786 
2.6;3 


-.' 7: ;- 


2. 2 30 
2 . 8 1 ■» 


6.780 
2.232 


6.779 
2.234 
2.813 


6.777 
2.236 
2.9H 


6.773 
2.237 

2.808 


6-773 
.t.239 
2.806 


6.771 
2.2*1 

2.803 


Velocity 
Period 
Am. Vel. 


6.769 
2.243 
2.801 


6.7<7 
2.243 
2.798 


6.763 
2.2*7 
2.796 


6.763 
2.249 
2.794 


6.761 
2.231 

2.791 


6. 
2. 


. 759 
. 25 3 


t.. 737 
2.255 

2- 76>> 


6.735 


6 . 75 3 
..7&2 


6. 751 
2. 261 
2. 779 


6.749 
2.26 3 
2.777 


6. 747 
2.265 
2.775 


6. 7*C 
T.172 


2.770 


6.742 
2. 270 
2.767 


6.7*0 
2.272 
2.765 


6.738 

2.274 
2,763 


6.736 
2.276 

2.760 


6.73* 
2.278 
2.738 


6.732 
2.280 
2.736 


Velocity 
Period 
Am- Vtl. 


6.73D 
2.282 
2.733 


6.728 
2.284 
2.731 


6.726 
2.286 
2.7*9 


6.724 
2.288 
2.746 


6.723 

2.290 
2.744 


2. 


;i:i 


6.71v 
2. 73';. 


6 . 7 1 7 


ii.715 
i.2y8 

2.753 


u.713 

2 . 300 

2. 7J2 


6.71 1 
2. 702 

2-730 


6.70^ 
2. 304 


b. 707 
2. J05 


(,.703 
2.307 
-.723 


6.704 
2. 309 

2.721 


6.702 
2.311 

2.716 


6.700 

2.313 
2.716 


6.698 
2.313 
2.714 


6.696 
2.31? 
2.712 


6.694 
2.319 
2.709 


Velocity 
Period 
Am- Vel. 


6.692 

2.321 
2.7Q7 


6.690 
2.323 

2.705 


6.6B9 
2.323 

2.702 


6.687 
2.327 
2.700 


6.663 
2.329 
2.698 


6. 


.663 
. 331 

. '=96 


6.ESI 
2. 333 


u . f 7 
2.335 


t>,, 3 77 
2. 337 
2. =69 


6.675 
2.339 

2.687 


6.C7* 


6.672 

2. 3*3 

2.682 


L-.6 70 

: . J4-. 

2. '.-.60 


0.668 
-. 347 

2.(s78 


6.666 

2. 349 
2.675 


6.66* 
2.350 
2.673 


6.662 

:-33: 

2.671 


6.661 
2.334 
2.669 


6.639 
2. 136 
2.666 


6.687 
2.336 
2.66* 



Velocity — Velocity in Kilometers per Second 

Period — Period in Hours 

Ang. Vel. Angular Velocity in Radians per Hour 



IH-51 



TABLE 9 (continued) 



Anc. Vel. 



■J. f,"i^ 
2. J60 


I'.iil 


2.3^4 
2. '^59 


6.6*5 
2.366 
2.655 


6.^48 
2. 369 
2.633 


2*37.:. 
i.toSt 


S..64* 

2. 372 
2.649 


6.t,42 
2.374 
2.b47 


u.,,40 
2 . _'"6 
2.^44 


6.638 
2-376 
2.642 


6.637 
2-3P0 
2.6*0 


6.635 
2.382 
2-638 


6.6J3 
2- 3S4 
2.636 


6.631 

2. 38b 
2.633 


6.629 
2. 388 
2.631 


2* "?90 
2.629 


6.626 
2-392 
2.627 


6-624 
2.394 
2.625 


6.622 
2.396 
2.623 


6-620 
2.39* 
2.620 


6.616 

2.400 
2 . t. 1 £ 


:!»o2 


6 . 1 5 
2.40, 


6.613 
2.406 
2.612 


6.61 ' 
2. *G8 
2.610 


2.4|0 

2-607 


6.607 
2-412 
2.603 


6.606 

2.414 
2.603 


(j.604 
2. 4I6 
2.601 


6.602 
2.4,8 
2 - 399 


6.600 
2.420 
2.597 


6.598 
2.422 

2.595 


6.397 
2.424 
2-39.1 


6.395 

2.426 
2.590 


2.428 
2.383 


6.391 

2.-29 
2.5?6 


6.369 
2.431 

2.364 


6.586 
2.433 
2.382 


6-586 
2.433 
2.580 


6.0*4 
2.437 

2.676 


6.5E2 
2.57c 


*.3?0 
2. 441 
2.574 


1-.579 
2.443 

2.37| 


c.377 
2.445 

2.569 


6-575 
2.44/ 

2.567 


6.373 
2.449 

2.5(j3 


6.372 
2.451 
J. 363 


6.370 
2.433 
2.361 


6-366 
2.453 

2.339 


6. 566 
2.437 

2.337 


6-364 
2.459 
2.333 


6.563 
2.461 

2.553 


6.361 
2.4*3 
2.331 


6.55) 
2.463 

2.345 


■i.357 
2.467 
2.347 


6.53t 
2 . < 69 
2.344 


6.534 

2.471 
2.542 


6.332 

2.473 
2.540 


6.530 
2.473 
2.338 


6.649. 
2. 477 
2.536 


6.3*7 

2.5 3" 


6.545 
2.*ei 
2.532 


6.543 
i. «9 3 
2.530 


6.542 
2.495 
2.329 


6.540 
2.487 

2.526 


6.538 
2.489 

2.524 


6.336 
2. 491 
2.322 


6.335 
2.493 
2.320 


6.333 

2,493 
2.316 


6.531 

2.497 
2.516 


6.529 
2.499 
2.314 


6.S28 

2-301 
2-312 


6. 526 
2.503 
2.310 


6,524 

2.303 
2.306 


£.322 

2.307 


6.32" 
2-309 

2.304 


5-319 
2.311 
2.502 


6.517 
2.313 

2.500 


6.^15 
2.513 
2.498 


6.B14 
2. 317 
2.496 


6.51- 


u.MO 
2.521 
2.49; 


».508 
2.523 

;.4*<o 


b.507 
2.325 
2.488 


6.503 

2.527 
2.496 


6.503 
2.323 
2.4 64 


6.302 
2.331 
2.462 


6.300 
2.333 
2.480 


6.498 

2.336 
2.476 


6.4 96 
2.536 

2-4 76 


6.495 
2.340 
2.474 


6.493 

2.342 
2-472 


6.4-J1 
2-344 
2.470 


6.469 
2.346 

2.468 


6.469 
2.546 
2.466 


6.486 
2.33O 

2.«64 


6.484 
2-332 

2.462 


6.463 

2.534 
2.460 


6.461 
2.336 

2.459 


6.47* 
2.638 
2-407 


6.479 
2 . 560 
2.4?3 


6.47t> 

2.562 
2.453 


6.474 

2.451 


6.472 
2.366 
2.449 


6.471 
2.366 
2.447 


6.469 
2.370 
2.445 


6.467 
2.372 
2.443 


6.466 
2.374 
2.441 


6.46 4 
2.376 
2.439 


6.462 
2.B78 
2.437 


6.461 
2.560 
2.433 


6. 459 
2.582 
1.433 


fc.457 
2.394 


6.453 
2.58t 
1.430 


6.434 

2.366 

2.429 


6.432 
2.390 
2.426 


6.4 5c 
2.392 
2-424 


2*594 

2.422 


6.447 
2.396 

2.*20 


6.445 

2.69* 

2.41* 


2.bOG 


6.442 
2.602 
2.4(4 


6.44Q 
2.604 
2.413 


6.439 
2.606 
2.41 1 


6.437 
2.608 
2.409 


6.435 
2.610 

2.407 


6.434 
2.612 
2.403 


6.432 
2.614 
2.403 


6.430 
2.617 
2. 401 


6.429 
2.M9 
2.399 


6.427 
2-621 
2.398 


6-423 
2.U23 
2. 396 


2.623 
2. 394 


6.422 

•J.62' 
2.392 


6.420 
2-629 

2.390 


f .419 
2.631 

2.388 


6.417 
2.63J 

2. 396 


6.415 

2.635 
2.385 


6.414 
2.637 
2.383 


6-412 
2.639 
2.381 


O.410 
2.641 

2.379 


6-409 
2.643 
2.377 


6.407 
2.6*3 

2. 375 


6.403 
2.647 
2.374 


6.404 
2.649 
2.372 


6.402 
2.631 
2- 370 


6.400 
2.653 
2.368 


6.399 
2.635 
2.366 


6.397 
2.63? 
2.364 


6.396 
2.639 
2.363 


6.394 
2 . 66 1 
2.361 


6. 392 
2.663 
2.339 


6. 391 
2.66b 
2. 337 


6.3S9 
2.666 
2.353 


6.387 
2.670 
2.334 


C. 396 
2-l>72 

2.332 


6. 384 
2.674 
2. 330 


6.382 
2.676 
2.348 


6.381 
2.678 
2.346 


6. 379 
2.680 
2.343 


6.376 
2.682 
2. J43 


6.376 
2-664 
2.341 


6.374 
2.686 
2.339 


6.373 
2.688 
2.337 


6.371 
2.690 
2.336 


6.369 
2-692 
2.334 


6.366 

2.694 
2.332 


6. 366 
2.696 

2. 330 


6.365 
2.698 
2.329 


6.363 
2. TOO 
2.327 


6 . 3C- 1 
2*. 325 


6.360 
2.703 
2-323 


6. 336 
2.707 
2. 321 


6. 337 
2. 709 
2.320 


6.333 
2. 71 1 
2.318 


6.353 
2.713 
2.316 


6.352 

2.713 
2.314 


6.330 

2.717 
2.313 


6. 149 
2.719 
2.31 1 


6.347 
2.721 
2.309 


u.345 
2.7:3 
2.30? 


6.344 

2.7;5 
2.30 6 


6.342 

2.727 
2. 304 


6. 340 

2.729 
2.302 


6.339 
2.731 

2. 300 


6.337 
2.733 
2-299 


6.33c 

1-733 
2.297 


6. 334 
2. 738 

2.293 


6.333 

2. 740 
2.293 


6.331 
2. 7*i 
2-292 


6.329 

2.744 
2-290 


6.323 

2.238 


b.326 
2-748 
2.287 


i=.325 

2. 750 

2.265 


2*263 


6.321 
2.754 

2.291 


6.310 
2.736 
2.260 


6 . 7 1 e 

2.7^.6 
2.276 


6.317 

2.760 
1.276 


6-313 
2.762 
2.273 


fc.3l3 
2.764 
2. 27 J 


6.312 

2. 271 


6.310 
2.769 
2.269 


6.309 
2.771 

2.268 


6-307 
2.773 
2.266 


6.306 
2.773 
2.264 


6.30* 
2.777 
2.263 


6.302 

2.779 
2.261 


6. 301 
2.761 

2.239 


6. 299 
2.783 
2.236 


6.29B 
2.785 

2.256 


6.296 
2.7g7 
2. 234 


6. 293 
2.785 
2.233 


6.293 

2.791 

2.251 


6. ;92 
2.79* 
2. ..49 


6.290 
2.796 
2. 248 


6.268 
2.79 8 
2.24^, 


6. 267 
2.600 
2.244 


6.265 
2-802 
2.243 


6.284 
2.004 
2.241 


6. 282 
2.806 
2.239 


6.281 
2.808 
2.238 


6.279 

2.6*0 
2.236 


6.278 
2.812 
2.234 


6.276 
2.814 
2.233 


6.274 
2.816 
2.231 


6.273 

2-819 
2.229 


6.271 
2.621 

2.228 


6.270 
2.923 

2.226 


6.268 

2.02S 
2.224 


6. 267 
2.827 
2.223 


6.265 
2.629 
2.221 


L-.264 

2.631 

2.219 


6.262 
2.633 

2.216 


6.261 
2.9J5 


6. 259 
2.8 37 
2.2 ( 4 


6. 237 
2.639 
2. 21 3 


6.-36 
2.642 

2.2' 1 


6.254 
2. 84* 

2-210 


6. 25 J 
2.844 
2.20* 


6. 251 
2.848 
2.206 


6.230 
2.630 
2.205 


6.248 
2.B32 
2.203 


6.24 7 
2.834 
2.201 


6.243 
2.936 

2.200 


6. 244 
2.836 
2. 196 


6.242 

2.860 
2.197 


6.241 
2.662 
2. 193 


6.239 

2.663 
2. 193 


6.238 
2.667 

2.192 


6. 236 

2.669 
2. 190 


6. 235 

2.671 
2. 169 


b.233 
2.873 
2. 187 


u.23l 
2.675 
2. 185 


6.230 
2. 184 


6. 228 
2.879 
2 . 1 62 


b.227 
2.861 
2. 181 


6.223 
2.663 
2.179 


6.224 
2.886 
2. 177 


6.222 

2. est 

2. 176 


6.221 
2.890 
2.174 


6.219 
2. 892 
2.173 


6.218 
2.894 
2.171 


6.216 
2.896 
2. 170 


6.213 
2.898 
2. 168 


6.213 
2.900 
2. 166 


6.212 
2.902 
2. 165 


6.210 
2.905 
2. 163 


6.209 
2.907 
2. 162 


6.207 
2.909 
2. 160 


6.206 
2.91 1 
2 . 1 59 


2.913 
2. 157 


6.20 3 
2-915 

2. 155 


6.201 
2.154 


fc.2 00 

2. M 5 
2. '32 


6. i?e 

2. 131 


b. '97 


6.193 

2.i'26 


6..D4 

2.928 


6. 192 
2.930 
2. 143 


6.191 
2.532 
2.143 


6. 189 

2.934 
2. 141 


6.188 
2.936 
2. 140 


6. 186 
2.938 
2. 138 


6.183 
2.940 
2. 137 


6.183 
2.943 
2.133 


6. 162 
2.943 
2. 134 


6. 180 
2.947 
2. 132 


6. 179 
2.94 9 
2. 131 


6. '76 
2-951 
2. 129 


6. 176 
2.953 

2. i:s 


6.17? 
2.935 

2. 126 


6. 173 
2.957 

2.125 


6. 1 72 
2.939 
2. 1 23 


6. 1 70 
2.962 
2. 122 


2-9 & 4 
2.120 


6. 167 
2. -'66 
2-119 


6. 166 
2.966 
2.117 


2.970 
2. 113 


6. 163 
2-972 

2. 114 



I 


Velocity 
Period 
Aaf. Vel. 


6. 161 
2.974 
2.112 


6. 160 
2.976 
2. Ill 


6. 138 
2.979 
2- 109 


6.137 
2,981 
2.108 


6. 133 
2.983 
2. 106 


6. 134 
2-983 
2. 105 


6. 133 
2..S87 

2-103 


6.131 
i.9ti> 
2.102 


6. '50 
i.991 
2.100 


6. I4g 
i.V-jt 

2.099 


2*097 


6. 145 

2.956 

2.O96 


1 


Velocity 
Period 
Anc. Vel. 


*. 132 
1.017 
2.0*3 


«. 131 

3.019 
2.081 


6. 129 
3.021 
2.080 


6.128 
3.023 
2.078 


6. 126 
3.023 
2. 077 


6. 125 
3.028 
2.073 


3! 030 

2-Q74 


6. 122 
3.032 
2.072 


6. 121 
3.034 
2.071 


6. 1 19 
3.036 
2.069 


6.1 16 
3.038 
2.066 


6. 116 
3.040 
2.067 


g 


Velocity 
Period 

An*. V«l. 


6.103 

3. o«c 

2.034 


6- 102 

3.062 
2.032 


6. 101 

3.064 
2.051 


6.099 
3.066 
2.049 


6.098 
3.068 

2.048 


6.096 
3.070 
2.046 


6. 095 
3.073 

2.043 


6.094 
3.075 
2.043 


6.092 
3.077 

2.042 


6.091 
3.079 
2.041 


6.069 
3.061 
2-039 


6.0S6 
3.083 

2.036 


I 


Velocity 

Period 
Aug. Vel. 


6.073 
3.103 
2.025 


6.074 
3. 103 
2.024 


6.072 
3. 107 
2.022 


6.071 
3.109 
2.021 


6.070 
3. 11 1 

2.01V 


6.068 
3. 1 14 
2.018 


6.067 
3. 1 16 
2.017 


6.065 
i. 1 18 
2.015 


6.064 
3.120 

2.01* 


6.063 
3. 122 
2.012 


6.061 
3. 124 

2-011 


6.06D 
3.126 
2.010 


i 


Velocity 
Period 
Aug. Vel. 


6.047 
3. 146 
1 . 997 


6.046 
3.148 
1 .996 


6.044 
3. 130 
1.993 


6.043 
3.152 
1.993 


6.042 
3.133 
1 .992 


6,040 
3. 157 
1 - 990 


6.039 
3. 139 
1.989 


6.036 
3. 161 
1.968 


6.036 
3.163 
1 .986 


6.033 
3. 163 
1 .983 


6.033 
3. 168 
1 .984 


6.032 
3. 170 
1.982 


1 


Velocity 
Period 
Anc. Vel. 


6.020 
3. 189 

1.570 


6.018 
3. 191 
1.969 


6.017 
3. 194 
1 .96 7 


6.016 
3. 196 

t.ofie 


6.014 
3.198 
1 .965 


t>.U13 
3.200 
1 . 963 


6.01 1 

3.202 
1.962 


b.010 
3.203 

1 .9b1 


6.009 
3.207 
1.939 


6.007 
3.209 
1 .936 


6.006 
3.21 1 
1.937 


6-003 
3.213 

1.933 


§ 


Velocity 
Period 
Anc. Vel. 


3.993 
3.233 
1.944 


5.991 
3.235 

1 .942 


3.990 
3.237 

1.94 1 


3.96 8 
3.23? 

1.940 


3.967 
3.242 
1 .938 


5.986 
3.244 
1 . 937 


5.964 
3.246 


3.96* 
3.248 
1 . 934 


3.982 

3-230 
1 .933 


6.980 
3.233 

1.932 


3.979 
3.255 

1.930 


3.97B 
3.257 
l.»29 


§ 


Velocity 
Period 
Anc. Vel. 


3.966 
3.277 
1 .918 


3.^64 
3- 279 

1.916 


3.963 

3.261 
1.915 


.*! 263 

1.914 


3. 360 
3.2S3 


5.939 
3-286 
1.9lt 


5. V*6 
3. 290 
1.910 


3.?36 


3.95S 
3-29* 
1 . 90 7 


5.934 
3-296 
t .906 


3.952 
3.299 
1.905 


3.931 
3.3 1 

1 .»04 


i 


Velocity 
Period 
Anc. Vel. 


3.939 
3.321 
' .892 


3.938 
3.323 

1.891 


3.937 
3-323 
1.890 


3.933 
3.327 

1 . 8S3 


3.934 
3.329 
1 .86' 


3.933 
3.332 

1.896 


5. 931 
3-334 
1.683 


3.93C 
3.336 
1.88J 


3. 929 
3. 338 
1 .882 


8.927 
3.341 
1 .661 


5.926 
3.343 
T. 880 


S.«23 
3.3*3 

1.878 


i 


Velocity 
Period 
Anc. Vel. 


3.^1 J 
3. 3t3 
1 .96 7 


3.9i; 

3. 367 

1.S66 


5. 91 1 
3.369 

1.863 


5.909 
3.371 
1 .864 


3. 90S 
3.374 
1 .862 


3.907 
3.376 
1.861 


3.903 
3.376 
1.860 


3.904 
3.360 
1 .839 


3.903 
3.383 
1.838 


0.901 
3.383 
1.856 


3.900 
3.387 

1 .833 


3.899 
J. 309 

1 -854 


1 


Velocity 
Period 
Anf . Vel. 


5, 987 
3.409 
1 .643 


s.eafa 

3.41 1 

1.84? 


3.695 
3.4U 
1 .84 1 


5.684 
3.416 
1.639 


5.882 
3.418 
1 .838 


3.681 
3-420 
1.837 


3.880 
3.423 
1.836 


5.878 
3.423 
1.835 


3.877 
3.427 
1 .633 


0.676 
3.429 

1.832 


3.873 
3.431 
1.83J 


3.873 
3.434 

1.830 


1 


Velocity 
Period 
Ang. Vel. 


3-3*2 
3.434 
1 .819 


5.861 
3.436 
1.818 


5-839 
3.438 
1.817 


3.838 

3.«60 
1 .816 


5.857 
3.4«3 

1-613 


5.856 
3-463 
I.813 


5-834 
3.467 
1.812 


5.853 
3. 4 69 
1.81 1 


3.632 
3.472 
1 -810 


0.851 
3.474 
1 .809 


3.849 
3.476 

1.808 


3.848 

3.478 
1.806 


i 


Velocity 
Period 

Ang. Vel. 


5.837 
3 . 499 
1 .794 


3.836 
3.301 
1.793 


5.834 
3.303 
1.794 


5.333 

3.503 
1.792 


3.B32 
3.308 
1 .791 


3.831 
3.310 
1.790 


3.829 
i.312 
1.769 


3.826 
3.314 
1.788 


3.827 
3.516 
1.T87 


8.826 
3.0t9 
1.786 


5.824 

3.321 
1 .784 


3.823 

3.323 
1 .783 


1 


Velocity 
Period 
Anc. Vel. 


3.812 
3.343 

t.773 


3.811 
3-346 
1.772 


3.810 
3.346 
1.771 


3.806 
3.330 
1 .770 


3.607 
3.332 
1 .769 


5.806 
3.533 
1.768 


3.803 
3.057 
1.766 


3.803 
3.339 
1.765 


3-802 
3.362 
1.764 


8.8OI 
3.564 
1.763 


3.800 
3.566 
1.762 


3.799 
3-568 

1.761 


i 


Velocity 
Period 
Anc. Vel. 


3.788 
3.389 
1.731 


3.786 
3-591 
1.730 


5.783 
3.393 
1.749 


3.784 
3.393 
1 .748 


3.763 
3-398 
1 .746 


3.781 
3.600 
1.743 


5.780 
3.602 
1-744 


5.779 
3.604 
1.743 


5.776 
3-607 
1.742 


0.777 
3.609 
l.74» 


3.775 
3.61 1 

1.740 


3.774 
3.614 
1.739 



3.Q0Z 
2.0^3 


3.004 
2.0?2 


3.006 


-'.009 
2.089 


3.011 
2.087 


3.013 
2.086 


3.015 
2.084 


b. 1 13 
3.045 


J. 04V 
2.C62 


t . t 1 1 
3.049 

2.U61 


6-109 
3.031 
2.039 


6. 106 
3.033 
2.036 


6. 1O6 
3.033 
2.036 


6. 103 
3.006 
2.005 


6.035 

3.0B6 
2.035 


6.064 
3.090 
2.034 


6.002 
3.092 
2.032 


6.O81 
3.094 
2.031 


6.079 
3.096 
2.029 


6.078 
3.098 
2.028 


6.077 
3. 101 
2.02* 


6.057 
3.131 

2.00 7 


6.056 
3.133 
2.006 


6.034 
3.133 
2.004 


6.033 
3. 137 
2.003 


6-031 
3.139 
2.001 


6.050 
3. |*2 
2.0OU 


6.049 
J. 1*4 
l.**9 


6.029 
3.174 
1.980 


6-028 
3.176 
1.978 


6.027 

3.178 
1.977 


6.023 
3. 161 
1.973 


6.024 
3.183 
1.974 


6-022 
3.183 
1.973 


4.021 
3. 187 

1 . tfi 



6.003 6.002 6.001 3.999 5.998 3.997 3.993 8.*+4 
3.213 3.218 3.220 3.222 3.224 3.22* 3.229 X.2>1 
1 . 954 1.953 I . 951 1 . 930 I . 949 I . 947 1 . 9*6 1 . 949 



3.976 3.973 3.974 
3.259 3.261 1.264 

1.928 1.927 1.923 



3.930 3.948 
3-303 3.303 
1 . V02 I . 90 I 



3.947 
3.J07 
1 -900 



3.972 
3.266 



3.946 
3.3i0 
I .898 



3.924 5.922 3.921 3.»20 
3.3*7 3.349 3.332 3.33* 
1.877 1.8/6 1.8/3 1.873 



3.698 S.B96 3.893 
3.391 3.394 3-396 
1.833 1.631 1.830 



3.822 
3.323 
1.782 



3-89* 
3.398 
1.849 



S.9T0 3.9*8 B.*«T 

3.270 J.2T2 1.274 

1 .921 1 .920 I .919 

5.943 3.942 0.941 

3.314 3.JI4 3.3(6 

1.896 1.893 1.893 

3.917 5.91* B.»I4 

».J3» 3.3*0 3. Ml 

1.871 1.670 1.8*9 



3.892 3.691 3.890 B.**9 
3.400 3.403 3.405 J.407 
1.846 1.647 1.845 1.644 



3-97| 
3.2*6 

1.923 

5.944 

3-312 
1.897 



1.H7J 



3.872 3.871 3.870 3.868 
3.43* 3-438 3.440 3.4*3 
1.829 1.627 1.826 1.823 



5.867 3.86* 3.6*4 
3.445 3.447 3.4*9 
1.624 1.623 1.822 



B.6»3 

3-462 
I -820 



3.846 3,»44 3.843 3-842 
3.483 3.483 3.487 3-490 
1.804 1.803 I.B02 1.801 



3.841 
3.492 
1.799 



3.939 
3.494 
1.796 



3.821 3.819 t.Bie 3.B17 3.81* 5.813 S.813 

3.32b 3.330 3.332 3.534 3.33V 3.339 3.041 

1.761 t.760 1.779 1.778 1.777 1.773 1.774 

3.796 3.793 3.794 3.792 3.791 3.790 5.789 

3.37J 3.373 3.377 3.380 J. 302 3.384 3.88* 

1.739 1.738 1.736 1.733 1.734 1.733 1.752 



3.772 
3.618 
1.737 



3.767 3.766 0.765 
3.627 3.629 J. 612 
1.732 1.731 1.7 JO 



Velocity — Velocity In Kilometers per Second 

Period — Period in Hours 

Ang. Vel. — Angular Velocity in Radians per Hour 



III- 52 



TABLE 9 (continued) 



£ ^ l Tl tf ^- 7 "- ^ 7M '- 759 5 - 7Hf - " : -"* —■' ''■ 7 " : 3.7-«7 f.*44 ;■■'■*: 5.--0 T..T-7 ■ . 7 J- -. ,- ^ 5.7'0 -.7:, 5.7> 3 7; 3 5 7". 5 7,3 

8 Period 3.63* j. u3B 3 . b4 j V) *fl 3., .5. j..=.-: j. b M *.„.,<, j. t7 o 7....-f 5..--:? 3.6*4 ,-. t ^. 7., -.* 3.6^6 >. ?n- V 707 3" -17 3 7}b 3' 72? 

- Ang. Vel. -.1.497 41.445 41.393 41.341 41.190 41.23: 41.167 41. Ufa 41.035 4 1 . U 74 4 . ■ t. * 40. 93. -U.&r. 40.;^ -O.-l -,,.7/, 4,j. £& 4G..30 ,0.580 4U.S3C 

g Velocity ..^ u . 5 . 7I , «, <TI) 5-T09 =. 70T 5>7o4 5-T02 . 7u(l i|ii _,, T ^.,, 5^,,, 5ibl0 ^ . C| r| 

5. Period 3.7:5 3.73o 3.734 3. 739 7.744 3.748 3. 753 3.757 3.~,2 :.7 L , 3.77. 7.77, 3. 730 3.7*5 7.7V9 . . 7 > j.T^ 3., 03 3.608 3.S.3 

- Ang. Vel. 4U.4C0 40. *31 40. j- i 4U.331 40.18.' 40.235 »0.ltj 40.134 -4,1.-1 . ob^ 40. --'36 3^'. "if:- 39. 'J ^ •'.>--'i_i J -'.a 4. _-9.,"-ii 3'.i.'-.5 j -\ u. ' ■ 7 3V.649 39.601 35.Sfj 



Velocity 
Period 



5.b:'l 5.62V 
3. t9i 



a 



« Ang. Vel. 3V.505 39. "57 J ■:■.■* t 39. 362 39.3.5 39.2i-7 39.220 39. | 7J .,v.l';,_, 39.07;, 7' 7 ' . o '2 3 - . .' ^, 5 -' ! ■" 3 ' := ! S, 9 2 J-!;-i^ Jt:.'?^' Jt! 75: JB.70t 38.'b60 5W " 6 1 4 

g Velocity •.<,:■! 5.6.2 5.62o 5 (-■ i 8 5..16 3 n j •■ ,-. i 5 L r,"' 5 ,,j7 «, ,,-,5 5 , n ■■ ^ ,.,,-, «• = ,._ = ..-,, ■= c 

<e Period 7 . ■;, 1 o 7 . 9 1 5 3.919 3.924 3 , 92 Q 3 . _i ; ~ 3 . '" 3t. -■ . "■* i 3.947 j . '^i. 1 ? . ■ .' 7 3- . -, t . ^ ?' ■!■ „ ". 3 ' .-, - 1 V ■-. -* ?" ■", e. , -'i,,^. >'■-;, .-, t " r'n ?;,., 

- Ang. Vel. 36 . 5.,8 ?c,.5.2 3f.47 u 39.431 3£.3S5 33.340 3.T-.194 38.149 36.204 36 . 1 5& 3J.lt.? 't . 0,,*. ?■:. . o.'4 3?'. '-■". ?.-.■', 34 --7.3:".<i '7. ..45 -7. £00 • 7 . 75 " 37.7.2 
g Yf 1 ^ '-^ S.57& 5.576 5.574 5.572 ,.570 5.5.7 5. ',5 5 . ,,,3 5.=,-. -.V- 5.-5 7 5. 5 ^ 5.,^ , 5 ,o -■ ; ^ ^54C 5 544 5^ u ^,3. 

„ Period 4.003 -.0q& 4.013 4.01? 4.022 4.027 4.^; 4.oJt * . r - 1 ,."», 4. .■.-. ,. . t.-. -.,-,, u ».ci, ; uri 4" < : ' "4 i'07- 4"os- ;"n&- i'c^ 

" Ang. Vol. 3'.i.(.-3 '7.6;* 37.5so 37.536 37. 4 y2 37. 446 37.4L4 37.361 77.717 3 r . ; Tt ;'..."■ 77 . 1:7 77.1«4 37. ,1.11 ;;,,■-,, 77.m5 3 L ,!'J7; "6 9^0 3i.!f87 - u ' b4< 

§ Velocity 5.537 5. -535 5.533 5.531 5.529 5.5;7 5.515 5.522 5. 'In 5 . 5 1 3 c .ml ' . =; 1 - =. ' 1 "■ «; -. t n - ',;? - 'n, - ■= 1 4 5 501 - 4--. 5 4 -.? 

Period ».o:»£ 4.13: <♦. 1 7 4.1 1 i 4.116 4.121 4. lib 4. 1 3t -. 1 35 4. 140 4. I--, ... 150 -.154 4 1 53 4 Il4 , Kv 4" 1 7 1 4 ' 1 76 4 " 1 & f 4 ' T-.B 

- Ang. Vel. 3l-.E:o: ru.759 3 U .71 7 3G.C75 36.633 J.i.7^0 3.^.546 36.506 36. ,65 36.423 .'o.TI 3l . :tu 3i : . 2='6 3b. .57 .^..,5 3(. . 1 '4 3^133 3g!u9I 3C ! Cj;.£i 3<,'.(Z? 

g Velocity --..,"5 5.493 5.4?t 5.489 5.487 5.4^5 5. 4&-. 5.481 5. 4 7^ ^.477 '..,."-4 5.47" 5 4 7n ■■,.!* 5 j,f. 5 4.4 - 4)i - - 46 n . ..^ ^ 4 . ■ 

« Period *. is>; «.iv ». 2u2 4.2u/ 4.212 4.itt> 4.^21 4.:^b 4.131 4.^.5 «..*u 4.145 4. /'So 4.255 4.25' 4.:„4 4' «. ( n 4 '>;, :*—'." '" - 

- Ang. Vel. 35.LV? 75.918 35.9*7 35.946 35.30b. 35.7i,5 35.725 35.1-94 35.b44 ^.L.114 .-' . 5 L .4 -5.5j4 35. 4f-,^ 35 444 35 . «04 = c . ! b 4 -^'3-4 '-."e*-, 



Velocity 5.454 5.452 5.450 5.446 5.446 5.444 5.44; 

-J •* Period 4.28^ 4. 29" 4. 2^8 4. 303 4. 307 4.312 4. 317 

2 2 Ang. Vel. 35.li.-o 35.127 35.086 35.048 35.009 34.970 34. ?3] 

B g Velocity V414 5.412 3.410 5.403 5.40.; 5.404 5.402 

9 w Period 4.;t:» 4. jay 4.394 4.3-19 4.4O4 4.4o ; J <.4i, 

5 " Ang. Vel. 34.?.-, 3 34.355 34.31& 34.260 34.242 34.205 34.lb7 



15.. .E.5 35. 2«5 35. 2jC 

5. 420 =. 41 ti 5. 41f. 

4.370 4. 3,-;, 4. 2&0 

'4, 5de 74.46--' 34. 431 

5. 3F0 5. 37fa 5. 376 

3.-5 6 33^ 72; _j.3io.ft5 



g Velocity 5.^4 5. 372 5. ?7t 5. 369 5. 367 5. 365 5.363 5.36 1 5. 35 > 5. 357 5 . 75 5 5. 35? 5. 351 5. 74' 5 . 34 7 -. 345 5. 344 -.34. 5. 340 

" Ang. Ve). 73.6-S- 33.612 33-575 33.539 33.503 33. 4 b C 31.43Q 33.394 J3 . 3Z6 31.3U 33.2tC 33.'250 Js'.H* ■■ 7 '. f ,' :- i:'.i A i 3 1 ! 1 ;i ; J *. r 71 ? 3 ! n n* j^ior.i' 



Velocity 5.77k- 5.33* 5.332 5.330 5.328 5.3^6 5.324 5.J23 5.721 5.319 

Period 4. 57> 4. 5 34 4. 56S' 4.5 94 4.59 9 4. 604 4. 6 09 4.614 4.619 4. 6 24 

Ang. Vel. -■l.-.' T '2.695 32.360 3^.324 32.789 32.754 32.719 32.685 '2. 650 32.615 






15 3 70.32 7 
5b 5.15? 



g Velocity «,.-r. 8 ,. :3l ; S-: c, 4 ? _ : .^3 - i:?1 ,.2^9 5.297 5.235 5. .'63 5.2 8 i 5.^n 5. 27fc -.17, 5. T-. - 7. 5 ^0 5 -,' 

« Period 4. L .7£ 4. '^93 4.^3^. 4.^3 4.693 4 . 703 4. 70 7 4.712 *. 71 - 4. 7 27 ■.. 717 4 . 7 3 2 , . 7 _-. 7 4. T». 4.-47 4. 752 4! 757 

— Ang. Vel. 32. .37 ^2.203 32.1b-' 32.135 32.101 32.067 3;. 033 32.000 3 1 . ''t >■ 31.^73 7 1 . e'^ 71-?6b 3l.Si; 31.7''9 -1.7,6 3^.737 3 1 . C 9' 

f Velocity -,. :6 , 5.25^ 5.258 5.25'i 5.254 5.252 5.250 5.24-9 5.24? 5.245 '..24= 5 14] 5 "■ 1 ■:, ' 7. «■, -. , .- ,-, 4 ^ ■, ,-. 

Period 4.777 4.7^2 4.737 4.792 4.797 4.602 4.807 4.912 4 . s 1 7 4.t21 4 . I. - 4.672 4.837 4.*4. ..^47 ,*iv 4''-57 

— Ang. Vel. - 31 ■ -"■ 7 ?' ■ '35 31 . 502 3' .4b 9 7' . 436 31 .404 3. . 37. 31 . 339 31 . 30 6 31 . 274 -.1.2-1 7 1 . 1 ;, ■ 3 , . ' 77 71.1,1, : : . , 1 3 71 . ,,;,* - , . o,^ 

g Velocity 5,-s 5.223 5.2-2 5.120 5 . 2 1 3 5.21b 5.114 '.;I7 5.211 5.2.1.-' 5.2,7 5.-1^ 5. ."'4 5.202 5.200 5.v,-L- 5.197 

2 Ang. Vel. 30. -1.1 30.469 30.85'- 30.826 30.7S'« 30.7^3 3(7.73i 3O.7O0 .'H'. ,_,<_■-■ 7'9.L-=7 -li.i..0l. 30.575 ;ri>,44 l\-'.',:'i ?\/.*zl 'C^'l 5.0! 4.;.i 

o Velocity 5-ISk.i 5.t8d 5.16,- 5.184 5-1S3 5 . 1 s= I 5.17s 5.177 5.1.-6- ;..I7. *-■ . 1 7 1 5.170 5. ■■>--■ 5 1,7 ' , t = «, 1-4 », ,. -. 

<0 Period -.-'77 4.9il 4.967 4. -.93 4 . i-k ^.0,i- 5.00a 5.ol3 5..H& 5.02" "."'.-- 5.633 5.>''J4 ..',,,1 '.! u s s!n'^ 5*&5f 

— Ang. Vel. 3U.29U. 30.26t. 30.235 30.205 70.174 30.144 30.113 30.033 30.:-l. 70.0.2 l"'."'''l 29. 9 U ; 19.^3. 1'"'. ,0. 2 S . ■= ?2 .i.t\; . .. :. 1 , 

» '* » » *0 M. « 70. H. W. 140. IN. .» 1» 140 ISO 1« 

o Velocity 5.155 5.15; *..t5- 5 150 5 . ■+;■ 5 1 4 ,. 5 >4= = i»f « 1 - 1 ■■ 1 ' 1 -^ e ■? «. 

o Period 5! 07"' 5! 084 5.. 0i9 5^o94 5! nni 5! 1 n, 5" t rr, ; .| , "- - . n -1 5! 1 ;i, ' ' 1 ■ '■ 5" i ?5 «" ■^ •' 14- - * ' «m • " ! ^ "'■ '-^ 

2 Ang. Vel. 2'.b9. .9.663 I' ;■ . ,3 7 7 2 9.1,., 14 ."'.574 /'.'■^, I ■. - 1 5 2 ? . ., £ 6 .^457 ,^4*7 .■:■!."- ,"i!jo9 ;-''-40 . ■] \ 7 ■ t ■■!''i'' : ' vl'-^f -■V'- 4 

Velocity 5 ,-■■ - 1 1 ■, •, t . . «■ n . . ,,„ «; 1 ■ = -11, -,,.., - ,,.. . .,-, . , c .-. c , - 

S Period -5- : 1 = 1 5.,;,. .;:,>.■; •;,-,; -: :J1 5_,-„; s ; : , t -j;^, <_, ^.;.^ 7 r:'_r ^;^ 7 ' ^;^! ,°-^ r^; =-'-._'^ ? - ' 1 '* 

-^ Ang. Vel. ."■■103 ;-".>-. 1 j .■-,,0 r i 19.022 1S.-.94 l--.. r ',5 1.-.977 ;■ ^. . --i £ .f;.. : ; .0 26.652 ..],,..j .■■_ . 795 2&!7(.-7 2 ^ ' 7 3 ") 1S.7M ■. 

o Velocity 5 rno ^ np, 5 ns» 5 ,-,97 5 ,-.01 ^ ,-,-;.-, = .-,-- c ,-,,■. ■= n-^ t r- T E r , , , 

2 p.r.od ;:.;1 -:.88 ?:°;i ,:,fe ,:„; ^:,;; ,:,;. .:'.;," ,•,;: s:':, ;-';;; :- c ;;° ;•■ ; :■.;;- ;-^ '-^ '■," '-°'° =■«» ^.^ 



2P.,;, 2e.U'4 .£,.077 28.05.J ,6.023 



a 



- Ang. Vel. 

g Velocity ,.,,, .. .„ _...._._ ,.„,.. , m _,,,_,_, , , _ 

» Period , , B; 5.3,; ,.3»- 5. , , ,.„, ,..,, ,..,, ..... ,.,,„ .„.,. ,.,,. . . ., , ..„„ ^ * ; ^ ^-J" ^^^ -.J.| - 0.J t.0|. 

- Ans.V.1. ■■'.« .7.5.5 27.».. .7.5,, ,7. t6 , ,7.3.2 , ,' . 8 .V, 27. M , 27.7 8 , .7.7,, ,,. 7 ;. 27.7„ : 2,.„7 b ,,.,«, : ..^ 3 ,..,,: ,?;,„ ^^^U ^^iSTs 27!^ 

Q Velocity •; ,-,■* ■, m r , - . «,-,', b , , , a * r . r- .,!.■; c.,- c-l.- c-.:t =-,-^ 

2 A«.V.l 27.. M 27. -« „..,. 27.383 27 . 3.2 ,7.33. 27.3,5 27. , B - 27.2,5 27.233 Ua ^. ^ ^1U ,V\11 J\^ ^ zV.ll', iV.lll ^ ^1^ 

§ Velocity 4.991 4.990 -.93- 4. j- 1 - - ■"— = - -- ' "" " - - '- - - "■ - — 

Period 5.595 5. GOO 5.O.05 5. u ..„. ..... , „_, . „_.,. . u , . ^.^^ 3 ,^ u ,,,„, - , b „ . ,--. . , 7tl _ 

- Ang. Vel. 26.953 2b.?29 26. 902 26. 8 77 it.^l 2„.^27 2b. fe 02 26. 7 77 2 6.7 5 2 2&.721 26.70^ 26.677 26.652 26.626 26^03 26 ' 578 2e'«3 «' 

I i» 5:SS ;:'-2? 5:^ 5:^ ;:?3? ;:?H ;:?!; ; : ?1! 5:?:; ;:K ?:?;! ; : ; S :••» ;;- S: »; ; : - -» j-» "; : «j -- 

26.1*0 26.1,6 26.052 26.066 26.04. 2d.02u 25.557 



An«.vei. ,-i:;M j;:.;, =;:4i: 3 r.,«; 3 ;:.;; ,;:.ii :;•,«: ,;•;:; ^r«; ,>■;" ,..•:?; .?■?;;.'■!" ,!-? : : .^- 77 : _' : "" '•!" .5-«" '•■"' =:«^ 



3 | SIST S:S ;:5» ;.;?; ; : ;is ;-», ; : ^ ; : su -?» ; : -| ; : ;y ; :S = ; : - -- ..... : . s; ..... ...=« ...» ..«,, ..502 

E 2 Anf. Vel. 23.573 25.545 25.525 25.502 25.676 25.85. 25.831 25.607 25.764 25.760 ,5.737 25.7,. n.hu li.lii i!!,.;; ii'.t% 2?.j' 7 2?'SyJ '',?, 4's" 

□ 8 ™7?', ty *-" M •'' CS ' •• 897 .-»" .-8 9 . .- 8 » 3 .■»" .- s '0 .-See 4.887 ..88.- 4.864 ..883 ..S8, 4.886 4.878 4 877 . »7. . „. . „,. 

y g Period 5. .1. n.5,B 5.923 5.325 5.534 5.535 5.945 5.950 5.955 5.961 5.-I.6 5.571 5 577 5 5a . 5 

o. -• Ang. Vel 2,-50' 25-.8i .5. .79 .7.4m ,5. 41.- 25-J90 ."5. -*6 ^ .r..j44 25.3,1 25. .99 25.276 25.25? 25!2il 25.208 ;','. 

g Velocity 4.871 4.87U 4.866 4.867 4.865 4.864 4.662 4.661 4.858 4.858 ..857 4.8,5 ,.8,4 4 85, 4 83, 4 8.9 . *.o . c.7 . ~.« . ^ 

g Period 6.0,0 „.0„ 6.030 6.036 6.04, 6.0.7 6.052 6.057 6.063 6 . 0„3 6.073 6.07- 1 ^"o t.Ut t.Vol '2 t"] t *,$ 1 ' *" 

2 Ang. Vel. 25.05, .5.028 25.006 24.984 ,4.96. 2.. 939 24.917 24.89, 24.873 2,. 851 ;..e,<. 24.807 24.785 24:" 5 3 2. . Vt 1 2.. 7?S :,'S A'.VA £'.Hl A'.lll 

§ Velocity ,. 8<2 .. 8 ,, 4 „,, ,. aJ8 37 . .„_„ tJ 32 

Period 6.127 6.133 6. ,38 6.,,, 6.149 6.155 6.160 6.165 o.,71 6 . , 76 6. ,82 . . Tj*7 „ 19, * ,TO o''o3 ^''n' °t. ?'? ! S ?' \l *■?" 

E Ang. Vel. 24.6,0 24.588 24.367 24.5,5 ;,.5,3 24.502 .4.480 ,4.459 24.437 ,4.4,6 ,..3-., 24.37! 2.. 3,2 , . ! 3 3 i!.:,." ^ijse 24.'2..7 .VM, £.l£ 2^203 

§ of'^ 17 "■ S ". 4 "■?'? *■?" "■ 8 '° "■ S ° 8 '- 807 *- 806 •' 804 *- 303 .- 8UI "■ B0LI .- 7 "' 9 '•'" .-'96 ..79, 4.753 . 752 4 790 . 789 4 788 

f. Period 6. ,36 ...., t .,47 6.2,2 6. ,56 6.2 t 3 6- 26 5 6.274 ».,79 6.2 8 5 ,.„ 6 . 2 9, 6.3o, 6.307 6 7,2 6 3,6 6 323 ' , ' , ,! ?'-!„ 

- Ar*.V.l. 24. ,82 24. ,6, 2.. ,40 24. ,,5 ,4.098 2.. 07, 2..056 24.035 „.,„. .3.9.., 23.573 .3.2-2 22.97, 23.9,0 J.lUo 2 5:»65 23.I" iVAil 2^* 23\'l° 

S S'^' 7 '• 7Bt 4 - 78 ' *' 78J *- 782 .■ r8 ' 4 - 77 ' .- 778 4 - 777 «- 77 5 4.774 4.773 ..77, ,.770 4 76b 4 767 . 7 66 . 76. . ,6 3 . „ ■. . ,-„ 

J. Period 6.3.5 6.350 6.356 6.36, 6.367 6.37, 6.376 6.383 6.389 6.394 „ . . 6.4D5 6.4, l.<,iZ „ I,, 1 ,"j V,%i t'' , T.l Vi"i 

E AD,. V.l. 23.766 23.7.6 23.72, 23.705 23. ,6. ,3.66< 23.644 23.6,3 2 3 . t 3 „.«. 23.56? 23.54, 23.522 23.502 A.Ui iV.ltl iV.." 23.' "2 23.'402 2V III 

1 S!^ ::s; : : :s : : :a : : : 7 ? : : n; ::i^ :::.; .::;? ::is ::;s ::':s j : ;:; s.-ss? ::;s. s:s; ::?s j-;j 7 tin :-s; :-:g 

E An,. V.l. 23.362 23.3.2 23.322 23.303 23.283 23.2.3 23.2,3 .3.22. „.,_,. 23., ». 23. ,65 23. , .5 23., 2, 23.106 2 3 . ot 2 Y.£l 2^.11] 23: 28 23.'„08 22 . «1 

1 ^Lf j:S ;:jj! ;:JS : : S ::?S? ::J|S ::';: -;: - ; -s ::^ US £:.'!! i:a; j : ;ii <: 7 :i : : jy ;-- ;■- ;•- 

-4 Ang. Vel. --.^0 .2.950 2.. 931 22.9.. 22. =92 22. 67j 22.854 .2.,:35 .2.B16 22.796 22. 7 77 2 2. 758 2;. 739 22.720 2 2.701 ;2.6«2 2^66* 22^44 2:^26 22 . 607 



Velocity Velocity in Kilometers per Second 

Period Period in Hours 

Ang. Vel. — .Angular Velocity in Radians per Day 



III -53 



TABLE 9 (continued) 



a »..««, ..,oo ,.?o. ..?« ..70, ..,o, -,, ;: ..^ .,;; -.,5 -. r; ..»;; -..o, ..-o ^ ..<-« ...^ ..»« ..,*. ;.«| ••!£ 

8 Period t.L-To o.ob. 6.667 '•«'- .■;■;" .''" "'. . ',,: --'.W -; .7> -2..F, 2.-,- - : .-'- -2.7.3 :..:-? 22.308 22.290 22. .72 22.23! 22.233 

S Ar«. V«l. 22.533 22.36? ...350. -.33. ...,3 ....-4 ...4. ...45. _...-* ... 

g viocirr ..6bo ..i7> ..»" •.»:* •■■.■>■ «■»•-: *-!; *-^; ;•»,■? *•■" ;•-,; *- s r ;•;•' ":?'■:- l'.ti r.:5il ;:S" «:«; ::>« *:.£ 

8 Period 6.788 6.793 l. 7» '-BO. ^..tl °-f.o ;•;■;' v:',, - '£,7 a - \" : 0,7 ', >,. 21 541 LMl ; , . 9.5 2, .927 2 1 . -'05 2 , . 89 , 2I.87! 



..632 4.631 
7. Dili 7.007 



™*. ,?-:ss 4-l°7 3 t-2;;, 2^: :':;'.. .-k':? .;:?.: 2?:?n ^:,;: 2?:K 2';::''; :,:^ .'^'.l ; \:^: 2kc , c S 2;:,,, 2 I ;:,7 J 2,:,,6 *:.». .,:«■ 



" Ang. Vel 

f Velocity ,.654 4. £.53 4.b52 4.65. 4.649 4.^48 *.b*7 

P«rlO(l b.^OO ii. 305 6.911 6.917 ■ ' - - - ■ 

2 Ang.Vel. 21.855 21.837 21. Bio 21. SO* I 
g VelOCit, 4.629 4.628 ..6,7 4.,26 ..,24 4.623 4.«Z 4.«i. 4.,,, 4.6,8 ;..; ;-.,. ..M* *.««3 4.bJ| - *' J 4.-»g* 4.60. « 607 4 606 

§ Period 7.o«3 7.0,8 7. os, 7.030 7.w 7.0,. 7.0,. 7..,5 ; 7.,** _,.o., ; .c _7.u ,.ue. .. .. , ■« j; J; ^ 21 ;* 2 ,,^ 

3 Ang. Vel. 21.50* 21.466 21.465 .1.452 21.435 21.-17 .1.400 .I.i6. ^I.'ob .M« *,.-.i -i.-i« -'■- - ■ - - ''-O- 

_ v.lrwltv . ~. 2, *„■> .-.-.- * -i-ji i mn J tcia - 5<;-t 4 5-.,, 4 5'-'5 4.t94 4.-92 *.5 l J' -.59.J 4.5S9 4. -38 4.386 4.565 4.584 4.593 4.661 

8 2JSP Vfi ,'t?' • VT.i *1.° 'il!. "'iii 7:,66 7 W2 7.177 7. ,57 7., B <, 7.19. 7.200 7. 2 06 7.2,, 7.2,7 7.223 7. 2 29 7.2,4 

S Anj!^.l. 2':!« 2,'Mi :<'Ml :,:" :,:.''• 21.07, 2, .06, 2,.n.. 2, .027 21.0,0 20.™. 20.9?T 20.-76, 20.9.. 20.927 20 . 9 , , 20.89. 20. 8 76 2 0.86, 20.9.3 

§V.lnrlrv . -co . 379 . 378 4 37" 4 375 4.37* 4. 573 4.37. 4.57, 4.565 4.5..L- 4. So? 4.56(. 4.365 4.504 4 . 3o2 4.36, 4.560 4.33? 4.539 

SSr 7 V'!S I' "J i'-3, I,-' '"i' ' '.; ,27. 7-6O 7.2S6 7.29, 7.297 7.30, 7.309 7.31. 7. 320 7.326 7.332 7. 337 7.343 7.J49 

3 2?™. «:!« 2^:1" d:iV, ,1:^ .oMi z».r4 :l:."l ,1:^. 20:;,, 20.^, .o. B o5 ; o.6, s . . U . (J . 20.0,., 20.6.00 20.3,. 2 .56 t 20. ,52 20.330 20..20 

§ V.lOCltf 4 336 4 335 . 334 < 55- ..552 ..350 ,.549 4.3." ..54? ..5.6 4.5.3 ..347 ..5.. 4.5-, 4.540 ..339 4.538 4.336 4.335 4.334 

D..4nd 7 - 333 7*360 7''66 7 372 7.376 7.383 7.3S9 7.7-5 7..0, 7.406 7..I2 7.4, & 7.4^. 7.429 T.4J3 7.44, 7.447 7. ,53 7.458 7.464 

2 ^ TO. 2^304 20 1 J" 23:.'° 20:;5* :0...O 20..;. 20..0 E 20. -,2 20.:7 O LO.S^O 20.3.5 20.325 20.31. 20.297 20.28, 20.266 20.250 20.23. 20.2,9 20.203 

3 S PerST 7'.7o W?7 W,'l '^' "".^ V Hi VM- V.IX V.l\Z VM -\Vil 'M -.V,l ':VA ?:U7 ;:,;? V.lll *:l'd ;:?;. V.Hi, 

i I JS!«i. 2J:is? 2 :?72 2J:t?6 .0:;., .;.:tJ 2,:,,o 1:^ 20.07- . u .o„3 .o.o. E .- L ,. c ,, ; ,0.0,7 20.002 ,9.. JB6 ,9.97, ,,.956 ,9.,.o ,9.923 ,9.9,0 ,,..9. 



3 



Ang. Vel. 

g g Velocity 4.510 4 . 5o& •»- 5 07 

9 « Period T.s&tr, 7.5?i 7.5 C ^ 

S 2 Ang. Vel. i n .*7? 19.8^, iy.&"v 

o Velocity 4. ,87 4.48b 1.49? 

S Period 7. 702 7.70& 7. 71 4 

S Ang. Vel. i957? i?-5i>4 lv.54S 

o Velocity 4.464 4.4(,.t h.*62 

5 Period 7.61? 7.625 7.831 

S Ang. Vel. 19.286 19.1-71 19.257 

o Velocity 



..■■95 
■ . & '2. 1 

f .(bS3 


7^667 


7. £73 


7is7v> 

13.&3& 


?! «..£5 
I9.^2'3 


7'. 1,90 
19.608 


7.696 

1 -E- . 594 


>.472 

'.779 


4.471 
7.7S4 
1 i',37 3 


4.470 

7 . 790 
l^. 358 


4.469 

7. 79fc 
19.344 


4.4t6 

7,«01 
19. 329 


*.,6 7 

7.8U7 
19.315 


4.46? 

7.81 i 

1 9 . 300 


>,*50 
: . 3 95 


4.4*.9 

7.501 

I9.f.£5 


4.44S 

7 . 907 
1SJ.071 


,.♦47 
7.513 

19.057 


4. 445 
7.^1? 


7.V25 

1-J.Oit 


4.4,3 

7.931 

19.014 



-,..» -.-... -.^.- 4.422 4.421 
025 8.031 &.0J7 B.043 6.U49 



§ ^!°?.i. iJ^oo ,l:ik , s :9,: ,»:-.3 3 .b:^:. „:9>f, ,*:9,; i.:™: , 6 . 8 o t , E . 8 74 ,:..,o ,,.*.< u. 8! . ,=.»,* , s .9 . .3.7,0 ,8.777 ,..,.3 ,..7.9 ,..7„ 

O V.lOClly , ,; 4.4,9 ...U 4.4,7 ..4,6 4.4,3 4.4,4 4.4,3 4.4,. ..4,1 ...,0 4.409 ...07 ...06 4.405 4.404 4.403 4.402 4.40, 4.400 

S P.«Od" »:U° 8.0„i 6.0.7 S.073 3.076 3 . 094 8. '0 6 . o9, . . , 02 . . ,06 , . , ,; 8 ; 20 6. , 2, 6 . , 3 ; | . , « 8 . ,4. 6. ,50 6 . , 36 62 . 



* Ang. Vel. 18.721 16-706 1 8- 

§ Velocity 
Period 
<n Ang. Vel. 



. Ii» B.14, S.13U C.13U O.ltii O.I ISO 

.530 18.517 18.503 16. ,90 18.476 18.46J 

4.284 4.383 4.382 4 . J6 1 4 . JeO 4.379 

6. 257 8. 263 8. 269 8.273 6. 281 8. 287 

!$.263 18.250 16-237 16.223 16. 210 18.197 



a Velocity < 176 *377 4 37,. 4 17, 4.'T7 4,17: *.371 -.770 4 . J-. f *.3*Z A. XI 4.7b- 4.Jb5 4.?64 *.36I 4.362 4.361 4.JS0 4.359 4.356 

I 21Sl l'*Z\ t-ll t'%-, 8' U I : MI ^ 323 ^.-'2- 8.?:5 :>.J-> S.3,7 8.35' c.3?9 e.^5 *.371 ^.377 e.i^J 8.JB5 8.395 8.4 1 6-4Q7 

S S^el. .-i*- il^Ti .S:;^ .I:;,3 .*..3: -:.--, . E ..0 L .S.,:i ,= .060 li.O^ 1 & .05, ,S. 4l .6.0.8 1».C5 18.002 ,7.989 -7.976 -7.963 w.WO ,7.938 

10 . M. 30 40 » tO 70 » M 100 HO 1M 130. 1*0. 150 1*0. 170 l»- '», 

§ Velocity 4, 357 4, 3?<_- 4, 355 ..•% -. 3-.J ■.."-: 2 -. 7* I ■*. - -. :~i -.:■*' ■• - -"-'■.■. *. J, 5 •> ■ i44 ,. 3< 

Period ?. ,1 7 5. *1S' 8. "-5 -:.-. 31 :- .-,.'" ::.-•-.' i.. +*' ' . ■< " " ■ -r ' - - "l"' t . ,7 " - . «7:- ^-4^5 *-"^ 

« Ang. Vel. iT.,-25 t7. 9i: 17. a;--. 17.fi... 1~..74 1, ".-..I I'.t*- '.'. : ; .. W...-' t 7 . .- 1 1.. 7 -.■■:■ 17.. '£5 . - . t 

g Velocity „..■-.,.; 4.33^ 4.3.'- *.'■;: -.??. 4 . .' : I -.-K -■■- *-?Z* -..3.7 *.J2t. 4.325 -.3i* 4.3. 

S Period &.53J 8.539 6.5,5 :.55l ,.*■■= 7 ::.'..? :.5'. r ' :.T* o . '^ > >."■'> ■"■V:'- & . b 00 S.^'--: 8.^ 

« Ang. Vel. 17.672 ,7.6-59 17.6-7 1 7 . ,.. 3 . I 7 - ,_ 2 2 ,;. -u? IT./ 7 1 . ° ■■ '■-' - l- -lO 1 7 . ,4 , I , . 5 . .. 1 .■ . 5-J 1 ...1 



542 
,91 

'47 


e. 


503 


4. 340 

e.5o<> 

1 7. 7 22 


4.3 39 

S.515 
1 7 . 70*j 


,.338 4.337 

8.521 6.327 
17.697 ,7.684 


122 


u. 


6 24 


4.320 

8.630 


V.Vil 


,.3.6 4.3.7 
8.642 8.6*8 



.202 -...'CI 4.300 4.2-9:' 4.296 4.297 
. 7 7 c - -■>■. 7,-. 8. 751 6. ^56 8.764 6. 770 
.:55 W.243 17. 2?> .7.2,9 17.207 17.195 



* Period e.n.54 s.6t.o f.t«- \-'>j- ;-'.^' 7r ■;''•"'" ~^\V %''S'Z 'i'\'-. J - ,-'\% ,-'»n. ,"-''■'--] ,^' ' 7 I 17 

S Ang. Vel. ' 7. 4 25 17.412 1 7. -t.iri I ,■ . J,-. ■ 7. ■ 7... '7,'n 1 ''. ■ ■- !■..-..■' 1 . V ■ . 1 ■ - -' -■ ■ - 

g Velocity ,.;^t. 4.295 ,..'.'. *.2">3 -..:'- 4.2-.1 h.-9i_ h_-.'J 4.;eJ=. •.:.' J . : .. < ^.-S'. «.-!.-4 4-tj'J 4.;d2 

S Ang.Vel. i?!ipj i^m i^il'. i^m: ttI'i^ i^^', i^ii: wiiut- iv!^, ,7.l-t, ,7.o. : 5 .7.05? it.l'*i it. 029 i7. i7 

O Velocity 4 --v, 4 -7- 4 ;7- ■* 21' - 172 4.271 4.270 *.:'-■: *..*V 4.2b7 4.JLL 4.2'.5 «.._«.■♦ *.:t. ,.262 «.i-.l 4.ib0 4.259 4.256 4.258 

S iu.i~i .,' rig fi'?,-m e'.«.ri i ?1» i,^:' ?.v-9 e.97; i.V<*\ 8.-:-7 t . "'5 3 i. .<*,.■ a.':\\ >.S'7_ c.?78 8.9fc4 a-990 8.99b 9. 002 9.008 9.015 



_ Period 
n Ang. Vel 



6.762 16.751 16.739 ,6.7;e 



Velocity 4 ~- 4 25r 4 25^ 4 -5- 4.153 4 . 2-2 •.15.1 *.25t' 4..-V «. . 2-f 4.^,7 4.;*u ^.2,-. 4.24- 4. 243 4.2,2 ,.24, 4.240 4.239 4. 2 3 6 

X P.Hnd =.7"-'., *j'q^ 9'033 <■ ' "' ^O," 9.052 9054 3.064 9. 070 9.07,: '.0^2 I.Of:* ' . 0V* 9.101 9-107 9.1,3 9.119 9.125 9. ,32 9.138 

3 Ang!v.l. l,:^? .6I703 U^L U.t8J K.^t ,6.660 1,.64, i,. S H tt .^G ,-=.6.= I6.6O3 '-^2 ..>.5^l 16.570 16.558 16.3.7 l6 .336 .b,323 16.5.* 16.B02 

o Velocity . -.7-- , -ic * 2'5 4 M, * -.3* 4 13 4 2'"' 4 23. 4.230 ,.221 4.-.2S «-'27 h.226 4.225 4. 2 24 4.223 4.222 4.22. 4.220 4.2.9 

§ pT,S mm 9'tso *' v "/T ' *:;'* * 7^ >iT .'^ 9.^3 ..^ ?.:c %:i: ',*.* 5. 2 24 9.231 9.237 ^..243 9.249 9.233 9.2*2 

« PerlOa ,'•]** ,;*!'" l: 1",, ,,;■,., , ;, II7 ,r; ,7r- i.. 4^ 112.4 14 li.4Li3 lb.3','1 16.3fQ It. 2b* U . 338 1S.347 i b . 336 1 r, . 326 .6.31! .6.304 16.293 16.262 



_ Ang. Vel. 

j o Velocity 

3 * Period 

E n Am. Vel. 



20! 4. 202 4.202 
367 9.374 9. jgO 
.098 U. OS? lb. 077 



8 Velocity 4 -no 4 .99 4 1*& 4. I ■/? -..,9c 4. , -15 ,. 14 4. IV I 4. 1 ■•: 4. 191 4. 1.0 «. t&v 4. 169 4.18= ,.187 ,. 1st, «, 1 85 4.164 4.183 4 . 1 82 

« Period ^ * U,ft ,,4 05 4.11 " 41.- * - ■■■> 4 4'n - 4 ■ . . ,*2 ''.4,3 ?.4 e 5 9.4iil 9.4-7 v. 473 9.480 9.486 9.492 9.49b 9.305 9. 51 I 

§ Ang.vel. '6.^5 ,c'.:5,5 iJj! icio;: , ■: ! o 1 7 .^'-^ i5>?: i-ijft ,-..-.?i i-.: : t,(t i^.-- .5.?ji t-.o: e 15.91B .5.-07 ,5.9.? i5.ee, T5.5TL ,5. 8U 5 ,3.855 

a Velocity' ...i*. 4 IE0 4 1 7 4 4.173 -.17-; 4.I77 ,.176 4. .75 4.174 ,.f? 4. it; 4.171 -..170 4., u 9 -.168 4-U.6 4.167 4.166 4.165 4.164 

S Period '• 5T7 <■■ 52^ « *3'< 9.5 '1- ^.5.2 9.-4? 9.535 9.5-,. ? . 5o 7 9.57, ^.^0 -..336 9.592 D.599 9.^03 9.411 9.618 9.624 9.630 9- 636 

S Ang.Vel. 15.843 ,5.834 .5.^24 < 5 . 8 « 3 15.SCJ t 5.-v3 .5.782 15.77; ,5.7^2 ,5.75, 15.7-1 ,5.771 ,5.720 15.-, Q ,5.700 .5.69r : ,5.679 13.669 13.639 IB. 649 

§ Velocity 4 „;, 4. 162 . 161 * lb 'J 4. I«9 4.15? 4. 158 4.137 4. 15.2. 4. ,55 4. 154 *. 15J ,.1\ 4,151 -. ]5J 4. 1,9 ..!«? 4 . 1 48 4. ,47 4 . I 46 

Period 9.643 9.649 9.655 9.,_62 9. & bS ,'.C7, <.. U t 9.6&7 9.6-->? *.«9^ "'-70: 9 . 7 1 2 i.7l£ 9.7.5 .. . 7 3 1 9.737 9.744 9.730 9.736 9.762 



Ang. Vel. 
Velocity 



15.477 ,3.467 ,5.457 .0. 



Velocity , ,4? 4.144 4.14-t ,'42 4.141 4.141 4.140 4.U9 4..3t *.1j7 4.1?.. 4.135- 4.13- 4. .33 -.133 4.132 4.131 4.130 4.129 4.12B 

Period 9 7 6 1 9.775 ft 79J n 788 9. 794 J.eoo 9.e07 9.81 3 9. J,9 --^ 8:r. 9.t.'2 9.8 38 9.^45 9.63. 9.637 9.efa4 9. 67Q 9.876 9. 883 9.88* 

Ang.Vel. .5-4 37 ,5.427 15., 17 .5., 7 ,5.39? .5.387 15.377 15.3„7 .5.357 ,5.:,' 1 5 . 32 7 .5.327 15.318 13.308 !5.296 15.288 .5.278 .3.268 |5.259 15-249 

Velocity 4, 27 4. ,26 4. 125 4. 123 4. .24 4.123 4.122 4. ,2, 4. ,20 ,.119 -.1,8 4. 118 4.H7 4. ,16 ».,|5 *-114 4. ,13 4, I, 2 4.111 4. 111 

Period Mt, * %S! * 4& v »1, 9; 9 2. 9.92" 9.933 9.9,0 -.946 9. 9 5 3 &.?59 9.9b5 9.972 9.978 9.984 9.991 9.99? 10.003 .0.0.0 .0.016 

Anc Vel l3.1'9 ,3.:29 15.22u 15.210 .5.200 1 5 . , 90 15.16. 15. ,71 ,5.161 ,5.152 ,5.14. 13.13. 15.123 I5.11j 15.107 ,3.094 15.06, 15.075 13.065 15.085 



Velocity 4 ,10 4 10'. 4 ,03 , ,07 ,.106 4.t0* 4.1OS *.,0- ,.10I 4. '02 4.101 4.10U 4.099 4.098 4.698 4.097 4.096 4.093 4.094 4.093 

1 Period lu 022 10.0^' '0'.O35 '0.042 10.048 10.034 10-061 .0.067 .0.07,- , C . 080 1 0-086 -u.0^3 1 D. 0«>° I'i.lOf lO.Hi 1O.U8 10.12? 10.131 .0..37 10.U4 
1 Ang.Vel. 15. 04b 13.036 15.02-' .5.017 I5.00e ,4.998 ,4.969 14.979 1,-970 14.9,2.0 1 4 . 95 1 14.9,1 .4.932 14. --.22 ,4.913 14.903 l,.894 14.863 14.875 14.666 

Velocity 4 n-:.^ 4 09 '■ 4 091 4 090 4.089 4.088 4-087 4.066 4.0*6 4.065 4.084 4.083 4.082 4.081 4.080 4.080 4.079 4.078 4.077 4.076 

Period I0'l5fi ,n 157 to' 161 io'k-9 ,f..i7< ; - in. ,82 10.169 iQ.1^-5 ir..;n. ,n.206 1O.21* 10.22, '0.227 m.,.33 10...0 .0-246 10.253 10.259 .0.266 10.272 



Ang. Vel. 14.357 14.64? 



Velocity — Velocity in Kilometers per Second 

Period — Period in Hours 

Ang. Vel. — Angular Velocity in Radians per Day 



*.;-£..- 14^73 |4.7 U 3 '4.75* 14.745 1».73i> 14.726 14.717 I4.70& 14.699 ,4.690 1,.68r. 



111-54 



TABLE 9 (continued) 



Velocity 
Period 
Ang. Vel. 
Velocity 
Period 
Ang. Vel. 

Velocity 
Period 
Ang. Vel. 
Velocity 
Period 
Ang. Vel. 
Velocity 
Period 
Ang. Vel. 

Velocity 
Period 
Ang. Vel. 
Velocity 
Period 
Ang. Vel. 
Velocity 
Period 
Ang. Vel. 

Velocity 
Period 
Ang. Vel. 

Velocity 
Period 
Ang. Vel. 
Velocity 
Period 
Ang. Vel. 
Velocity 
Period 
Ang. Vel. 

Velocity 
Period 
Ang. Vel. 

Velocity 
Period 
Ang. Vel. 
Velocity 
Period 
Ang. Vel. 



>72 4.071 
to* 10.311 
>?5 14.626 



4.058 4 
10.4 14 10 
14.481 '4 



4.05-i 
10.426 ' 
14.463 ' 

4.039 
10.556 ■ 
14.286 ' 



4.009 

10.797 
t J. 967 

3.993 

10.927 

tJ.eoo 

3.977 
1 1 . 039 
13.636 



T-.946 
11.323 
13-318 

3.931 
I 1 .456 
13. 163 

3.913 

I 1.390 
13.01 t 



4.008 4. 007 

ID. 803 10. 810 ' 

13.959 1 3. v50 ■ 

3.992 3. 591 

10.934 1Q.-:'4 1 ' 

13-792 13, 7=>3 

3. 976 3. 9 7u 

1 1 . 065 1 1 .072 ' 

13.628 13.620 ■ 

3.961 3.960 

1 1 . 197 11 .204 

13.467 13.435 

3,945 3.^44 3.' 

I .330 ti. 33& » i .. 

3.310 13.3Q2 13.: 

3. 930 3. 925 3.^ 



i.03' 4.038 

i.5(>2 l0.5fc5 

..277 14.269 

1.022 4.021 

t.692 10,699 

1.103 14.095 

. .ooe. 4.003 

1.623 10.829 

1.933 13.923 



4.070 
10.317 1 
1*.61b I 

4.U33 
10.446 I 
14.43C I 

4. on 

10-573 ' 
14.259 ' 

4.020 
10. 703 
14.086 



4.063 
0.32! 
4.t 07 



3.991 

0.9-7 
T.lT. ■ 

3.973 
1 .079 
3.61 2 

3.959 
1.211 
J.*31 



;.-?90 J. 939 

■.954 10.960 

J.76"7 IJ.739 

3.974 3.973 

1 .093 11 .091 

1.603 13.393 



3.988 

10.967 

13.730 

3. 971 
1 1.099 
13.397 



3.9i3 1.914 

1.396 11.603 

3.004 12.996 

3.900 3.699 

1 I .730 11.737 

12. B55 12.648 

3.8S3 3.684 

l 1.863 ' 1 .872 ■ 

12.710 12.702 



3.913 
1.610 
2 .969 



3.B71 
1 I .993 
12.374 

3. 637 
12. 129 

12.43 3 



3. 670 


3.870 


3.869 


2. 000 


12.007 


12.013 


2. 567 


12.539 


12.332 



3.836 3.833 
2. 133 12. M2 

2.4^6 12.419 



3.834 
12.149 
12.412 



1,3*'. 3.942 

. *30 11.336 

J. 266 13.279 

1.928 3.927 

.463 11.489 

!. 132 13. 123 

! . a 1 2 3.912 

.616 11.623 

I. 961 12.974 

1.898 3.897 

.750 11.737 

■.S3' 12.826 

S.S83 3.982 

.365 11 .892 

i.eee 12.681 

1.866 3.867 

i.G20 12.027 

1.5*3' 12.338 

1.834 3.833 

!. 136 12. '63 

i.403 12.396 



4.020 
0.712 
4.078 

4.003 
10.842 
I3.9UB 

3.967 
10.973 
13.742 

3.972 
1 1 .103 
13.379 

J.93fa 
11.237 

13.420 

3.94! 
1 1.370 
13.263 



3.926 

13.1 17 

3.911 
1 1.630 
12.966 

3.896 

1 1 . 76* 
1 2.019 

3.861 
It. 891 ' 
12.674 

3.8*7 
12.034 
12-331 

3-832 
12.169 
12.392 



3.910 
1 1.636 
12.939 



3.866 

12.040 
12.524 

3.632 
12. 176 
12.363 



4.0b9 4.068 

10. J30 10.336 

14.598 14.589 

4.032 4.0B1 

10.439 to. 463 

14.418 14.409 

4.035 4.034 

IO.38& 1O.B93 

14-242 14. 233 

4.019 4.0i8 

10.718 1U.723 

14.069 14.061 

4.003 4.002 

10.649 10.633 

1 3.900 13.891 

3.987 3.986 

10.980 10.986 

I 3.734 13.726 

3.971 3.970 

11.112 11.118 

13.371 13.563 

3.953 3.934 

11.244 1 1.250 

13.412 13.40* 

3.940 3.939 

II .376 I 1.383 
13.233 13. 24B 

3.923 3.92* 

11.309 I 1.816 

13. 102 13.094 

3.909 3.909 

1 .64 3 11 .600 

2.932 12.944 

3-895 j.o-., 

I 1.777 1 I .784 

12.8U4 12.797 

3. 660 3.879 

1 1 .<»12 11 .9i9 

12.639 12.652 

3-863 3.865 

12.047 12.054 

12.317 12.510 

3.831 3.850 

12.183 12. 190 

12. 378 12.371 



4.0o7 
10.343 

14.380 



4.03* 

10.601 
14.224 

4.017 

10.731 

14,03 2 

4.001 
10.862 
13.S83 

3.983 
10.993 
13.717 



4.55 J 

4.04a 



10.7JS 10.7*. 

1 4.04' 14. U J' 

4.000 3.9 C - 

10.868 '0.87; 

13.873 U.Sb, 



4.014 

10.737 

t4.o' e 



3.999 

10.882 I 

t3.858 1 



3.934 
1 I . 237 
13.396 

3.938 
I 1 . 390 
13.240 



3.984 

1 1 . ooo 

13.709 
3. 966 

13.547 
3-933 

13.36B 

3.9'7 
' I . 396 
1 7. 23: 

3.923 3. 9.2 

11.327 I'.!P 
13.087 13.079 

3.908 3.9o7 
I 1.637 1 1 .663 
12.937 12.T-29 

3.B»3 7.8=': 



3.878 
11 .923 
12.6*3 

3.8(4 
12.061 
12.303 



3. -'63 

1 . 00b ■ 
3.701 ■ 



3. "32 
1.27 D 
i. 380 



?.967 
I 1 . 144 
I 3.331 

3.951 

I 1.277 
1 3. 372 

3.9 30 
t 1.410 
13.217 

3.921 
r 1.5*3 

13.064 

3.906 
I 1 .677 
12,914 



7.982 J. 981 

11.019 1 1 .0:6 

13. 683 1 J.b77 

3.96k. 3.965 

11.131 ' 1 . 1-8 

13.32 3 1 3.3)3 

'.93' 3.930 

1-263 11.290 

3.3b* ) 3. 357 

3. "35 3.9?4 

11.416 1i ,*;r 

>S.2l<> 1 ?.2Q> 

3.920 3.919 

11.54^ 1-.536 

I 3. 057 1 3- 049 

3.-;03 3.904 

1 1 .683 1 1 .690 

i:,o(17 I 2.900 



10. 51 1 10.51 
'4. 3*7 1*. '3i 

*.029 4.021 



3.996 3.9^5 

10.90) 10.908 

I3.*33 1 3.813 

3. "60 3.979 

1.032 11.03"? 

i.666 13.660 

3.963 3. 964 

1.16* I I . 1 7l 

3-507 I3.»99 



«.04< 4.0*3 

10.523 10.530 

1*. 37.0 1*. 321 

4.027 4.026 

10. '.-53 IO.06O 

1*. 155 14. 1»C- 



3.9^5 3.994 

10,91 4 10. 921 

17. 61' 13-808 

3.979 3.978 

1 1 .046 l1 .052 

1 3.b52 13.6*4 

3-96 3 3-962 

1 1 . )78 1 1 . \9i 

13.«?i ,3.*e3 

3.9*7 



1 1.310 
1 3. 3 



.3)7 

13. 323 



3. 933 3. c '32 



3.0*1 I 3.034 

3.903 3.9U3 

1.697 11.70 3 

2.69; I 2. 985 



760 12.733 "2.7*. 



2.0*8 IT. 074 



3.S73 3.075 
1 1 .93? 1 1 .959 
I2. u '6 i2.bfl9 



7.S* 7 3.846 
1.-.22* 12.231 
1 2. 336 )2. 729 



3.87« 3.873 
1.96U 11,973 

2.602 12.593 



3. "*5 3.84, 
12.137 12.2* 
'2.3:3 1:. 31' 



3.917 3.916 

1 1 .57C 11 .587 

1 3.026 13. 019 

3.902 3.90' 

1 1 .710 11.717 

1 2.877 12. 670 

3.697 3.96i? 

11.8*3 H.85' 

12.731 12-72* 

3.873 3.87: 

1 1 .960 1 1 .986 

12.368 )2.5e' 



.8*4 3.6*3 

. 25 ' 1 2 . 236 
. 3C9 12. 702 



© Velocity 

8 Period 

S Ang. Vel. 

g Velocity 

g Period 

K Ang. Vel. 

g Velocity 

9 period 

** Ang. Vel. 

velocity 

1 Period 

E Ang. Vel. 

Velocity 

3 Period 

fi Ang. Vel. 

9 Velocity 

8 Period 

S Ang. Vel. 

8 Velocity 
Period 

A Ang. Vel. 

o Velocity 

9 Period 
« Ang. Vel. 
g Velocity 
8 Period 

« Ang. Vel. 

e Velocity 

S Period 

«S Ang. Vel. 

Velocity 

8 Period 

S Ang. Vel. 

c Velocity 

« Period 

8 Ang. Vel. 

o Velocity 

3 Period 

8 Ang. Vel. 

g Velocity 

* Period 

» Ann. Vel. 

g Velocity 

8 Period 

« Ang. Vel. 



3.828 

2.401 12.408 

2. 1*0 12. 133 

3.81* 3.61J 

2.338 12.343 

2.027 12.020 

3.800 3. EDO 

2.676 12.683 

I .897 1 1 .B90 

3.787 3.786 

2.A1* 12. en 

1.76P 11.762 

3.773 3.772 

2.932 12.939 

l.<;*3 11.636 



3.627 3. *27 



J. 760 
3.091 

1.319 

3.746 

3.23' 
1.397 



3.739 

3.096 
1.313 



3.733 
1 J.37G 
1 1.272 



3.7V9 
1 2.66V 
1 1 . 884 

I. 76-. 
12.629 
11.736 

3.772 

12.946 
11.630 

3.739 

t J . 1 ri?/ 
1 1.307 

3.745 
13.2*3 
1 i.3a5 

3.732 
1 3. 395 



13. 639 1 *. 666 



3.312 
12.531 ! 

12.. "-0 7 1 

3.796 
2.696 1 
1 .877 1 

3.7*3 
2.83* 1 

1 .74" 1 

3.771 

2.-77 3 1 
1.624 1 

3.736 



.501 



3.682 3.661 

3.936 13.943 

0.62' 10.813 

3.670 3.669 

••078 14.0B3 

0.71 I 10.706 

3.657 3.637 

*.22l 14.228 

0.60* 10.398 ' 



3.636 
*.233 
0.593 



3.6'>8 

14, 100 1 

10.693 1 

3.635 

14.243 1 

10.348 1 



3.839 
'2.?99 
'2.261 

3.823 
t 2 . 4 Jl 

1 2 - ' 26 

3.611 

12.572 
1 I . 99* 



3.836 3.S37 
12.'30b \?.1H 
12. 23« l2.24o 



3.783 
12. 8*8 
It. 737 



'.810 3.S09 

12.579 12.580. 

t 1 .968 I I . c, 8l 

3.796 3.79^ 

12.717 12.72* ' 

It. 656 It. 851 1 

3.763 3.7a2 

17.83- 12. 6«2 ' 

1 1. 77o ' 1.7.'* ' 



3-736 
13.126 

1 1.408 



3. 7 JO 

13.406 
1 1 - 2*9 



1 .973 H .968 

i.795 

2.731 1 

1.84 5 1 



3.7*2 3.74- 
■ 3.27J I 3.; 80 
1 I. 361 1 1. 355 

3.729 3.729 
13.413 1 3.4iii 1 
1l.?*3 ".277 , 



3.692 
13.029 

1 10.904 

1 3.679 

13.971 

1 1O.T9J 



3.654 

I * . 237 

to. 577 



3.-.-9t 3.6*0 

11.83*. 13.843 

ir..89« 10,893 

3.678 3.676 

i].97i 13.985 

10-788 '0.7R2 

3-*66 3.663 

t4.l2i 14.t28 

10.679 10.674 

3.634 3.633 

4.264 '4.271 

0.372 10.3C6 



3.793 3.793 

12.743 12. ?32 

1 1.832 1 t-S:b 

3.780 3.77V 

12.88 3 12. "90 

1 1.703 11 .699 

3.7*6 '.76,- 

13.022 1 3.029 

I 1.500 1 ' .57* 



'.73* 3.734 
3.147 1 J. 164 
I . «70 11.464 

3.7*1 3.740 
3.2?7 13.294 
1 .349 11 .343 

3.728 3.727 
3-»;7 13.*3* 
1-23' 11.223 

3.713 3.714 
13.568 13.573 
11.114 1 t . 1 09 

3.7"2 3.702 
13.709 13.716 
1 t . 000 10. 994 

3.U90 3.689 
13.B30 13.030 
10.68/ 10.8B2 

3.677 3.676 
1 3.993 U.OOO 
1 ft. 777 10.7TI 

3. u »5 3.66* 
14. 133 14. U2 
■0.66B 10.663 

3-652 3.682 
4.276 14.786 
0.361 '0.356 



. 819 

.778 
. 8-J7 
. 693 



!.*}.3 3.6*2 

1 i. 333 12. 3^-0 

1 r.207 1 ;. 200 

3.,31'j 3.618 

1 2.*9J 12.497 

12.073 12.0'.-7 

'.SO* 3.804 

1 :. u r7 12-63* ■ 

' 1.942 I t .93" ' 

3.791 3.791 

1.1.763 12.772 ' 

.813 11 .gi)7 



j . & 1 a 3. 8i7 
:.504 12.311 

r.ObO 12.033 



11,929 11. '•■22 

3.790 3.78! 

12. 779 12. 78( 

1 1 . 800 ' 1 . 79- 



.816 3.816 3.81S 

.318 12.124 12.331 

'.''47 12.0*0 12.03* 

.802 3.802 3.901 

.653 12-662 12.669 

.9'6 11 .909 1 1 .903 

3.789 J.78B 3.78? 

12.793 12.800 12.607 

1 1.767 I 1.781 1 I .773 



3.776 r.TT 

1 2.924 i:.9Ji 
1 1.667 1 I .661 



3.774 
12.938 

1 1 . 633 ' 



'. 76 T 3. 762 3.762 J . 76 1 3.760 
1.056 13.063 13.07C 13.077 13.084 
.530 1i.5*3 H.5J7 11.331 it.SVJ 



■.73^ 
!. '73 
.4*6 



3.740 3.7;9 

13.301 13.308 
1 1.337 1 1 .332 

3.727 3.7:t> 

13.441 13. 44S ■ 

1 1.219 11.213 ' 

3.71* 3- 713 

13.3*2 ' J -58 n 

11. 103 n .097 ' 

3.701 3.700 

13. 723 13. 730 ' 

10.989 TO. 983 ) 

3.(99 3.6?8 

13.863 '3.872 1 

10.B76 10.871 I 

3.676 3.673 

t*.007 14.01* 1 

10.766 t0.7,:o I 

3.663 3.663 

14. 130 )4. 137 I 

10.637 10.632 1 

J. 631 3.631 

14.293 14.300 1 

10.331 10.543 1 



3.731 3.730 

13, 182 1 3. I=r9 

I 1.44C 1 1 .434 

■ 3.738 3.737 
i 13. 322 13.329 

1 1.320 I I .314 

3.723 3.724 

i 13.4K.2 I 3.4fc.9 

' M . 202 I I . 196 

3.712 J,7H 

. 1 3.603 13.&I0 

1 1 .086 1 1 .090 

i 3.ij99 3.696 

■ 1 3.74* 13.731 

' 10.972 10.966 

3.686 3. C86 

1 1 3.88b 13.893 

10.860 10.83* 

3.674 3.673 

14.028 14.033 

10.749 10.7*4 

3.662 3.661 

14.171 14.17B 

10.641 10.636 

3.649 7.649 

14.314 1*. J2' 

10.335 1C.329 



3.750 3.7*9 3.7*8 

T. 19C '3. 20 J 13.210 

1.426 11. 4 22 11.415 

3.737 3.736 3.733 

3.336 13. 3*3 13.330 

1 . 308 1 I. 302 1 1 .296 



J. 746 3.747 

13.217 13.224 

11.404 1 I.4U3 

3.733 3.734 

13.337 13.36* 

1 1 . 290 1 1 . 2B4 



T.72J 3.722 3.722 3.721 

13.483 13.490 13.497 |3.80* 

11. 190 1 t. 184 1 1.178 1 1 . 172 11 . 167 

3-7)1 3.710 3.709 3.709 3-706 

13.617 13.624 13.631 13.630 13.6*3 

11.07* 11.068 11.063 11.037 11.051 

3.699 3.697 3.697 3.696 3.693 

13.738 13.763 13.772 13.780 13.787 

10.960 10-933 10.9*9 10.9*3 10.930 

3.6?3 3.665 3.68* 3.683 3.6B3 

3.900 13.907 13.91* 13.921 13.929 

0-8*9 10.643 to. 837 10.832 10-826 

3.673 3.672 3.672 3.671 3.670 

4.0*2 14.030 14.037 14.064 14.071 

0.739 10.733 10.728 10.722 10.717 



3. 660 3. 660 3 . 659 3. 639 3 . 658 

14.183 14.192 14.2D0 I*. 207 14-214 

10.630 10.623 10.620 10.614 10.609 

3.6*8 3.6*7 3.646 3.6*6 

14.336 14.343 14.730 t*.3?7 

10-319 10.314 10.308 10.B03 



Velocity Velocity in Kilometers per Second 

Period — Period in Hours 

Ang. Vel. — Angular Velocity in Radians per Day 



III- 55 



TABLE 9 (continued) 



g vew 



I 
I 
1 
I 

ii 
11 
f I 



! 



Velocity 
period 
Ang. VeL 
Velocity 
Period 
Aag. VeL 

velocity 

Period 

Aag. Val, 

Velocity 

Period 

Abb. Vel. 

Velocity 

Period 

Ang. Vel, 

Velocity 

Period 

Aag. Vel. 

Velocity 

Ptrlod 

Am. vei. 

Velocity 
Period 
Aug. Vel. 

Velocity 
Partot* 

. lei. 
Velocity 
Period 
Ang. Vel. 

Velocity 
Period 
Aag. Vel. 
Velocity 
Period 

:. vei. 

Velocity 
Period 

Aag- Vel. 
Velocity 
Period 
Ang. Vel. 



3. u 45 J. 042 3.63y 

14.34.4 (4.400 14. *H i 

10.498 10.472 10.4*6 i 

J. 386 3.383 3.3B0 

|3.G89 i3.*23 13. 162 ' 
9.49* 9.970 9.9** 



3.633 J.6 31' 

14T308 |4.*l44 

10.394 10. 368 

3.374 '.371 

13-233 is. 272 

9.696 9.8T4 



3.613 3.612 3.609 
(•.723 14.761 1*. 798 
10.241 10. 21ft (0. 191 



3.329 J. 327 J. 324 3.321 3.316 
15.823 13.662 13.999 13.936 13.973 
9.529 9.307 9.483 9.463 9.441 



3.3*3 
9.397 



3.5(0 3.307 3. 603 
6.093 16. 1?2 \r. 160 
9.373 9.333 9.3J2 



3.473 3.473 3.470 3.468 
(6.372 16.610 *6.6*8 16.683 
9.099 9.079 9.058 3.033 



3.424 3.421 3.419 

17.331 17. 369 17.406 

8.701 8.682 6.663 

3.373 3.372 3.370 

16. |01 18. WO 18. 1 79 

8.331 6. 313 8. 293 

3.32B 3.323 3.323 

(9.483 16.922 19.961 ' 

7.986 7.969 7.933 

3.282 3.280 3.279 

(9.673 (9.7)3 19.733 ' 

7.664 7.649 7.633 



3.239 3.237 3.235 3.232 
20.478 20.518 20.331 20.399 
7-364 7.349 7. 3J3 7.321 



3.416 J. 41* 
17.446 17.4 ? 4 
8.644 a.t.23 



3.4U6 J. 40* 3.402 
17.599 l7.u38 17.676 
8.368 6.330 8.331 



3.273 
19.833 
7.603 



3.263 3-262 
19-993 20.133 
7.3*2 7.327 



e VtfOCltJ 3.(97 3.(93 3.193 3.191 3.189 3.(87 3.183 3.(83 3.101 3 

8 P»rt«* 21.291 21.3J2 7i.3TJ 7T.<T* 21-453 2 l.4\j6 2(.53 S Tt 1.579 27 . 6 7D ;T 

II *"g Xml 7.082 7.069 7.033 7.042 7.028 7.015 7.002 u.98£ 6 . c. 7 5 6 



bfcl 



e Vale 

3 peri 
3 Aag. 



3.041 3.039 3.038 3.0?6 
24.736 24.779 24.622 24 . ' a 5 
6.096 6.086 6.073 i..0b5 



3.010 3.008 3.006 3.003 3.003 
23.514 23.336 23.602 23. (,43 25.68-1 
3. 910 3.«00 3.890 5.880 5.870 



3-001 3.000 
25.732 23.77fc 
5.960 3.650 



3.032 3. 031 3.029 

i*.9bl 2*. 99* 2». Tij6 

0.0** 6.0*3 ft. 02? 

2.998 2. 9"6 2. 9^5 

'5.82U 23.663 23. V07 

3.8*0 3.831 t.?2l 



3.606 3.603 J. 600 3.397 3.393 

4.834 14.870 14.906 I4.94J 14.979' Ij.u» 

0.166 (0.141 10.116 10.092 10.067 10.043 (0.018 



t 13. Oil IB. 002 



3.337 3.534 
13.*33 13-492 



3.332 3.349 3.346 3.543 3.340 3.33B 3. SJS 3.632 

._ 3.329 13.366 13.603 13.6*0 13.676 (5.7(3 I S. 750 1B.?»B 

734 9.7(1 9 688 9.663 9.642 9.6(9 9.397 9.574 9.B«2 



3.502 3.499 



4 97 3.4 9* 3.491 3. *69 3.466 
234 16.272 lb. 309 (6.347 16.38* 16.42; 
289 9.267 9.2*6 9.223 9.20* 9.183 



3.483 3.481 3.478 
16.459 t6.497 16.B34 
9.162 9.141 9.120 



3.463 3.462 3-*b0 3,*57 3. ,55 3.4B2 
16.723 16.761 lb. 799 16.836 16.87* 16.91? 
9.017 6. '97 B.977 S.937 6.936 8.°16 



3.439 3. 
(7. 102 17. 
9.817 8. 



*J7 3.*34 3.432 3.429 3.424 
140 17.178 17.217 l7.23f 17.283 
798 8.77B 8.739 8.739 t. 720 



3.367 3.363 3.363 3.360 3. 35B 3.336 3.353 

18 218 18.;-37 18.296 '8-335 '8.373 (8.4'i 18-452 

8.2?7 8.260 8-2*2 8.223 8.2^7 & . 1 90 8-173 

3.3H 3.309 3.307 



3. 3*-9 3. 197 3.3'.'* 3.392 3.389 3-387 3.384 3.382 3.380 3- 3^7 

17.713 17.753 17.??2 17.630 I7. e e9 i7.9q8 17.94* 17.983 18. o2* l6-0*J 

8.512 6.494 8.476 8.457 6.439 8.421 8.403 8.J83 8.3*7 8-349 

3.351 3.348 3. J*6 3.34* 1 . 34 1 3.339 *. 337 3.334 3.332 3.330 

16.491 (8.330 18.369 16.608 16.647 18.686 18.723 18.763 18.8O* (8-8*3 

6.133 8.136 8.121 8.104 B.Q87 8.070 8-033 8.036 8.0(9 8.0Q3 



.302 3.300 3.298 3.296 3.293 3.291 
.317 19.33' 19.^96 19.436 19.476 19.313 
.606 7.790 7.77* 7.739 7.743 7.727 



3.289 3.287 3- 2*4 
9.333 19.593 1§.6J3 
7.711 7.696 7.680 



3.230 3.228 3-226 3.224 3.222 3.220 
20-440 20.680 20.721 20.761 20.8u2 20-843 
7. 306 7. -92 7. 2 7& "> . 26 1 >'. 2«V 7. 233 



3.260 3.238 3.236 3.234 3. 252 3.249 3.24 7 3.243 3.243 3.2*1 

20.075 20-K3 20. 133 20. (96 20-236 20.276 20.316 70.357 70. JTT 20-437 

7.512 7.497 7.*B2 7.407 7.432 7.437 7.422 7.408 7.393 7.378 

3.218 3.216 3.213 3.211 3.209 3.207 3.205 3-203 3.201 3.199 

20.683 20.9?* 20-963 21.003 21.046 21-08* 2l.'2e !(.'*& 21-210 21-2B0 

7. i21 7. 207 7. 193 7. 1 79 7. 163 7. 131 7. 1 37 7. 1 24 7. HO 7.09* 



.173 3. (73 3. 171 
.743 21.763 21.826 i 
.933 6.922 6.909 



3. 165 
21.930 ; 
6.870 



3-137 3. 133 3. 133 3.131 3.149 3. '47 3. 1 *5 3. 143 3. 14! 3. 1 39 

22.1** 22.157 22.199 22.240 22.282 22.121 22.363 22.40C- 22.4*6 22.490 

6.819 6. 806 6-T93 6.780 6.768 b. 7»3 <,. 74J 6.730 6.7'8 6. 7o3 

3.118 3-1(6 3.1(4 r.lU 3.110 3.10O 3.1 r, 7 *.|Q^ 3.103 3-'0( 

22.930 22.992 23.03* 23.07b 23. lie 23.160 23.202 23.2*3 23-267 23.329 

6.371 6.339 U.347 6.333 6.323 6.311 6.499 6.487 6.476 6.464 

3.08( 3.079 3.077 3.073 3.073 3.0 72 3.07Q 3.068 3.066 3.064 

23.793 23.837 23.880 23.922 23.9b3 2*. 008 24.050 24.u93 24.136 24.176 

6-337 6.326 6.313 6.304 6.292 6.2<ji 6.27Q 6.259 6-2*8 6.137 



3. 1 37 3. 133 3. 133 3.13! 3. 129 3. 1 28 3. 1 26 3.12* 3-122 3. '20 

22.332 22. 5?3 22.613 22. 6^7 ;;.699 22. 7*0 22.7*2 22. B2* 22.866 22-*08 

6.693 6.660 6.668 6.656 6.643 6.631 6.6(9 6.607 6.593 6.383 

3.099 3-097 3.095 3.09* 3.092 3.090 3.086 3.086 3. 084 3.083 

23.37' 23.*'3 2j.*3b 23.496 21-3*0 23-383 23.623 23.6*7 2J.T1O 23-732 

6.432 6.441 6.*2'i 6. 41 7 b.406 6. 394 6. 383 6.37( 6. 360 6. 349 

3.062 3.061 3.039 3.037 3.053 3.03* 3.032 3.030 3.048 3.046 

24.221 2*-2b4 2*. 306 24.3*9 24.392 2* . * 33 2*.47e 24.321 24.564 24.607 

6.226 6-213 6.204 b.l93 6.182 6- (71 6.161 4.130 6.139 6.128 



3.027 3-023 3.024 J. 022 3.020 3-Ote 3.017 3.015 
23.081 25.12* 23.167 25.211 23.234 25.297 23.J41 23.3*4 
6.012 6.002 5-992 3.981 3.971 3.9&1 3-93i 3.941 



3. Oil 3.012 
13.428 20.471 
5.930 3.920 



2.993 2.99i 2.990 2.988 2.986 2.985 2.983 2.981 2.980 2.97fl 
23.951 23.99* 2o.Q3e 26.082 26.126 26.170 26.213 26.237 26.30' 26.3*3 
5.6H 5.801 5.791 5.7*2 S".T72 5.762 3.TJ3 3. 743 5.73J 3.724 



Valoctty 
Ptrlod 
Ang. Val. 

Valoclty 
Parlod 
Val. 

Velocity 

Parlod 
. Val. 

Valoctty 
Parlod 



S Vsl ° 
S Pari 
* Ang. 



2.976 2.975 2.973 2.971 2.970 2.965 2.9ml 

26.389 26.4J3 26.477 26,321 26.563 2l-.'>0$ 2, : .^« 

3. 7(4 3-703 3.695 5. 68iS 3.676 5.6<-7 5.6"' 

2.944 2.942 2.V4Q 2.939 i.937 2.936 2.9,'* 

27.274 27.3(8 27.363 27.407 27.432 27 . 49t, 27.3*1 

3.329 3-320 5.3d 3.302 5.493 3.*84 5.*7- 

2.912 2.9(1 2.909 2.908 2.906 2.90* 2.903 

X?.-?l 3 2J._23ft 28.303 28.3*8 2*".39? 2e.438 26. 



B Period 28.(68 28.2(3 28, 

5 Ang. Val. r»J V »3 i 



v^*? 


_<>. ?• ;■ 

5.tJS' 


3 . ■": 30 


2 . ytO 
2i..e30 

3 . ,;20 


2.^38 
26. 874 
3-61 1 


i . 9 S 3 
.■''.3*1' 
5.466 


3.438 


2.929 

27.-5 75 
3-*49 


2.92? 
27.720 
5.440 


2.926 
27. 764 
3.4!' 


2.901 
r6.*63 


2.900 
28.328 


2.B98 
29. B7? 


2.6V7 
26.';I9 


2.B93 



' 2.935 2 . 55 3 2 . 932 I - 93c 2 . 9* B 2 . 9* 7 2 . 943 

| 2i.9bJ 27. CO? 27.032 27.096 27.140 27.ifl3 27.229 

5.393 3,38* 5.37* 3.363 3.336 3.34? 8.8JI 



3.*23 3.* 



9*3 2.922 2.920 2.918 2. '.'< 7 2.9(5 2.914 
83* 27.999 27.943 27.988 28.033 28.078 2t.'23 
03 5.396 3. 388 3.3?» 3. J7I «. 36! 



5. *1( 5.703 



S.->7e 5. i£9 3. . 



§Velc 
Pari 
Ang. 

9 Valo 
9 Pari 
» Aug. 

e Valo 
g Part 
8 Aag. 

| 
§ 



Valoctty 
Period 
!. Val. 

Valoctty 

Period 

t. Val. 
Velocity 
Parlod 
Ang. Val. 

Velocity 
Parlod 
Ang. Val. 
Valoclty 
Parlod 
Aag. Val. 

Valoctty 
Bd 
Val. 

Valoctty 

Parlod 

Val. 
Valoclty 
Period 
Ang. Vel. 



8 *•** 

8 perl 
X Ang. 

§V.lc 
Pari 
•» Aag. 



Valoclty 
Parlod 



Valoctty 
Parlod 
Ang. Val. 



2.882 2.680 I.8?9 2.877 2.87c. 2.874 2.87J 2.871 2.970 2 . Bf.8 

29.072 29.117 29.163 29.208 29.254 29.2^9 2". 3*T 29.3^0 29.«36 :y.4P1 

3.187 3.179 3.l?1 3.163 3.153 3.147 5. "39 3.131 5.123 3.(13 

2.852 2.831 2.8*9 2.846 2.846 2.843 2.643 2.6*2 2-6*1 2.839 

29.983 30.031 JO-077 30.(23 30.(69 30.2lS 3Q.26i 30.307 30.333 30.399 

3.02* 3.021 3.014 3.006 4.T.98 4.9">1 «.9b? *.97b *."68 *.9^1 

2.823 2.622 2.821 2.8 l 9 2.818 2.816 2.813 2.814 2.812 2.811 

30.907 30.934 31.000 31. 0*7 31.093 3|.(*0 31.166 31-233 jl.279 3|.326 

4.879 4.872 4.6T4 *.83"7 4.630 4. »4 J 4.635 4.8.6 4.921 *.rf* 

2.79* 2.?94 2.7J3 2.792 2.790 2.7e9 2.7B7 2.786 2.783 2.763 

jl-839 31.886 31.933 31.980 32.027 32.07* 32. '21 J2.lf8 32.21? 32.262 

4.736 4. 779 4.772 4.713 4.706 4.70? 4.695 4.089 4.681 4.674 



^.769 2.7*7 i.T«6 2.76T 2.7&T ^.7^2 2.76 
J2.780 32-828 32.875 32.922 32-970 33-017 33. Ob. 
4.600 4.394 4.387 4.380 4.374 4.5b7 4.*6 



2. iZ r > 2.758 2.757 
33. I (2 33. 159 33-207 
4.554 4. 346 4.541 



3 2.732 2.731 
3 3*. 113 3*- '61 



2.742 2.74( 2.7*0 2.739 2.737 2.736 

33.730 33.778 33.626 33.874 33.922 33-9 ( ,9 

4.47f 4.4*4 4.4~58 4.432 4.4*5 4 . * 39 

2.7|7 2.716 2.714 2-713 2-712 2.711 

J4.690 34.738 34.786 34.934 34.882 34.931 

4.J47 4.341 4.335 4.329 4.323 4.3)7 

2.692 2.691 2.690 2.688 2.687 2.686 2.685 2.684 2.(82 2.681 

33.638 33.706 33.755 35.804 35.832 33.901 33.930 33.998 36.1M7 36.096 

4.229 4.223 4.2(8 4.212 4.206 4.200 4.(93 4.189 4.183 4.178 



2.735 2.7J 
I4-OI7 3*.0t 
4.433 4.*: 



2.709 2. 70S 2. 707 7. 706 
34.979 33.<i27 33.076 33.124 
4.311 4. 305 4.2 a 9 4.293 



2.668 2.6*7 2.666 2.664 2.66J 
36.*34 36.684 36.733 36.7B2 J6.831 
4. 1 16 4.171 4. (05 4. 100 4.094 



2.309 
-'1.372 , 
4.807 

2.782 
32. 309 ; 

*.6£7 

2.753 
33-234 : 
4.535 

2.7 30 
34.20? . 

4 . 4fiB 



2.8*5 
29.573 
5.099 



2.63b 
0.491 
4.946 

2-808 



2.894 2.892 2.891 2-839 2.666 2.886 2-885 2. 66 J 

2&.709 2e.73* 28.800 28-8*3 28.890 26.935 28.961 29.026 

5.253 3.24* 3.236 3.228 5.220 5.2IT 5.203 g.'95' 

I.864 2.66< 2.86 1 2. 859 2.856 2.837 2.833 2-8B4 

-n .-.*» 5n .-,-.. ,r, ,.!-, -«.7 Sfe 29.601 29,6*7 29.093 29-93* 

.066 5.060 3.032 3.043 B-0T7 



3.091 5.083 3.076 5 



30 



'.8 35 
1.337 
..938 



1.633 2.832 2.031 
.383 30.630 3o-*-"?6 
1.931 4.923 4.9i6 



2.829 2.82*8 2.626 2.B29 
30.722 30.768 30.B13 30-86i 
4.906 4.901 4.g94 4 . 86C 



2.807 2.805 



.800 



2. 7^8 
U.257 
4.40: 

2.?03 
13. 221 
4.261 



2.30 1 
M .559 3' 

4.778 . 



2.803 2.801 



2.800 2.796 2.797 
11.699 31.7*6 31 .7« 
4.757 4.T50 4.7*1 



.778 2.777 2.773 2.774 2.773 2.771 2.770 

■.*30 32.497 32.344 32.391 32.639 32.686 32-733 

.6*7 4.640 4.634 4.627 4.620 4.614 4.607 

.75? 2.730 2.7*9 2.748 2.746 2.745 2-"»44 

.397 33.44* 33-«92 33. 34o 33. 3B7 33.*33 33-6iJ 

.515 4.509 4.502 4.496 4.4*0 4.4WT 4.4T7 

.726 2.72* 2.723 2.722 2.771 2. 719 2.M6 

.333 34.4QI 34.449 34.497 34.545 34.393 34.641 

.J^O 4.3F4 4.T77 4.3Ti 4.3F3 4.J59 «. J5j 



2. 697 2.b9o 2.693 2.693 
3 33.51^ 33.360 30.609 
2 4.24b 4.241 4.233 



■.3tb 35.415 33. 



36. U3 36. 19* 36. 2*3 3".2^ 



2.662 2. 661 2.660 2.636 2. 637 
!b-BB0 3b. 929 36.V7B 3?.0«B 37.077 
4.0»9 4. OB 3 4.07B 4.073 4.067 



2.6*4 2.643 2.642 2.641 2.640 2.639 2.637 2.636 2.633 2.634 

n. te 20 3?.*T0 J7.?I9 37.769 37.616 37.B*8 37.917 37.96/ 3B. ol 7 36. 0*6 

4.008 4.003 3.998 3.993 3.987 3-982 3.977 3.g72 3.9&7 3.9*1 

2-622 2.620 2.619 2.618 2-617 2.616 2.6'3 2.61* 2-613 2. oil 

38.614 36.664 18.714 36.764 36.814 38.864 38.914 38.964 39.0(5 39.063 

3.905 3.900 3.893 3.890 3.883 3.880 j.Kl 3. 670 3. 865 3.860 

2.399 2.398 2.397 2.596 2-393 2-394 2.593 2.592 2.590 2.889 

39.*|7 39.6*8 39.718 39.769 3^.819 39.969 39.9*0 39.t)70 40-021 40-071 

3.80* 3.801 3.797 3.792 3.787 3.782 3.777 3.773 3. 7btf 3.76? 



2.6J3 

39. lit 
3 . -.3C 



2.632 
J.931 



2.^73 2.I-T4 r.b7j 2.672 2.„70 2.669 

3-- . 340 3fe.?-39 ?C-.*J8 36.487 36.336 3*. 385 

4. 150 *. 1*« *. 1 36 *. 133 *. 127 * . 1 22 

2.631 2.630 2.b49 2.b4fi 2.b«7 2. 6*6 

J7.i23 3^-373 3V«22 37.*72 37-521 J?.37' 

*. 040 4.033 4.0 30 4.024 4.019 4. On 

2.ui8 2.6i7 ;.62b 2,623 2 . o2« 2.623 

3^.'i5 36--'o3 38.415 J6.4t5 3B.5i5 38.365 

J. ■-> 3b r. 9]1 J. "25 J.".0 J.<*13 3. 9i0 



.;0' 2. uOC 2-605 2. L-04 2.bPJ 2 . bO ' 2-600 
'.2i*3 V3.;i6 3-}. 3 fa( , 3y.4l to 39,466 1<-.517 39.3*7 
.840 J.tJ6 (.331 3.626 3.B21 J. Bib 3. 611 



2. 5&6 
3.?58 



2.587 
•0. '73 • 
3.754 



.385 i.*.i34 2. 3^! 2.58; 

1.27* 40. 3.3 40- '7y *o. 42b ' 
.744 J. 7,0 3.735 3.730 



2.^60 2.B79 

40.32? 40. 378 
3.721 3.71* 



Velocity — Velocity in Kilometers per Second 

Period — Period in Hours 

Ang. Vel. — Angular Velocity In Radians per Day 



IH-56 



a) 
o 
W 

■a 

c 

ai 

JS 

i 

-*-■ 
X! 

3? 

s 



420 



246 50 



-:- 24700 



400 ■-.- 



380 - 



■-- 24750 



:r 24800 



360 - 



*T 24850 



340 if 24900 



320 -r i 



300 



280 .;■ 



r- 24950 



260 



240 



220 ■ 



200 - 



180 " 



160 :i 



140 -!! 



120 



100 -: 



25000 



25050 



■■- 25100 



25150 



25200 



■-• 25250 



25300 



25350 



■■•25550 



25450 
25500 



■ 25600 



rr 23650 



23700 




760 



740 



f- 23850 



720 z 



700 *- 



680 



660 t 



640 



620 



600 



580+: 



560 



540-::. 



:i "24100 



25400 520 f : . 24400 



500 



480 ■!! 



460- 



440 ■:! 



"24550 



^25650 



1180 



1160 



23900 

23950 
24000 

24050 



•H 
.If 

it 



22900 



■--' 22950 



23000 



-•24450 



24150 
24200 

24250 
24300 
24350 



980 



880 



r-24500 



24600 
24650 



820 



■22750 



t- 23100 



23150 



r- 23200 



23250 



i- 23300 



23350 



23400 



tt23450 



'-•23500 



rr 21650 



it 21700 



21750 



ft 21800 



21950 



r + 22000 



23050 U 22050 



■tl 



T- 22100 



i- 



23650 



22150 



■■* 22200 



-t 22250 



;-22300 



22350 



22400 



■■- 22550 



22600 



22650 



Fig. 8. Velocity of a Satellite in a Circular Orbit as a Function of Altitude 
(Lim-lish Unit - see Table 9 for Metric Data) 



III- 57 



3 



o 






o 
o 

r— ( 

> 



o 

rH 

o 

O 



ni 
o 
en 

T3 
C 
cS 



s: 

bjD 

•r-l 



«r 20650 



2250 



::t 20700 



t- 20750 



2200 ft! 20800 
20850 
2150 34,20900 



2100 



2050 



2000 :■- 



1950 



1900 -: 



■:-21400 2450 '> 



1850 '!■ 



1800 ! a 



— 19650 



20950 
21000 
21050 
21100 
4J-21150 
1-21200 
21250 



! ; -21300 



21350 



2900-:! 



2850-;; 



2800t 



2750- 



19700 



19750 



3700 4- 



3650. 



3600* 



St 19800 

19850 3550& 18850 




19900 



2700.. 



2650 f : 



2600 |: 



19950 
■U 20000 



H20200 



2550 



2500 ■'■.'. 



T20350 



:-■ 21450 



2400 -13 



1*21500 



21550 



2350 - 



1700IJ21600 
-^•21650 



2300 - :: 



20050 



20150 



■20400 



;t20550 
f~20600 



{:.i 



3500! 

t- 



3450 t 



0? 



20100 335 ° 



3300 .: 



3250 



20250 

20300 3200*19300 



8900 



3*18950 
19000 



*4 19050 



19100 



-7 19150 



I 19200 



H. 



3150 



3100 I 



.;?20450 

"H'20500 3050llf 19500 



3000 



19350 



19400 



19450 



19550 



ffi-20650 2950tti 19650 



Fig. 8. (continued) 



4550J 

4500f 
4450! 

4400.: 

4350 
4300 



17650 



17700 

17750 
17800 
17850 

17900 

17950 

*18000 



4250 



42 00 T-1 



4150 



4100 T 

T 
f 



11 
-ft 19250 4050 t- 



3950r : 



3900 + 



38504; 



18050 



M18100 

ii 

P18150 

Ii 



18200 



18250 
4000* 18300 



r*18350 



|t 18400 
"4-18450 



it 18500 



.il 

I! 



18550 



" : '19600 3750 



if 186 00 
^18650 



III-5{ 



I 






o 
w 

■a 

c 

n! 



u 

o 

> 



U 



o 

a 

i 



5600 



■16700 
16750 

5500ffl. 16800 



5400 :;■; 



:g- 16850 
16900 



5300:: 



5200 ■'■'-. 



5100 



16650 



16950 
t 17000 



17050 

£ 17100 

17150 



:|17200 6200J 
5000 i-17250 



4900 



4800- 



4700 



4600 



■£■17300 

17350 

;£I7400 

17450 



B17500 
17550 
17600 

^17650 



6900 ; 



6800::: 



6700:i: 



6600 :;; 



6500: 



64001 :. 



6300 : : : 



6100:;: 



6000' 



5900 



5800 '■:- 



5700:: 



15650 
15700 
15750 
15800 
15850 
15900 

15950 
16000 
16050 
16100 
16150 
16200 
16250 
16300 



16350 
16400 

16450 
ft 16500 
16550 
16660 



14650 



8400ii 1470 o 



8300 ft 14750 



8200-1 



J14800 



8100 



8000 



7900 



7800 



7700 



76001^ 



m 14850 
14.900 
14950 
15000 
15050 
15100 

■#15150 
15200 



7400 ■•■■ 



7500 ■!* 15250 



7300 ■;; 



15350 
.ft 15400 



7200 .;"15450 8800 ■ ■■ 



^^so 



1530C 



■""15500 
7100 # 

■15550 

7000 -ffi-15600 
15650 



10000 flj 



9900 



9800 



9700 



9600 



9500 -ed 



9400 



9300 



9200 



9100 



9000 



8900 



14300 



14350 



8700 



8600 



8500 



14500 



14550 



14600 



14650 



Fig. 8. (continued) 



III- 59 



Semiperimeter, S/r 



CD 
<1 



C 
< 

a! 
U 

c 
u 




0.1 0.2 0.3 0.4 0.5 0.6 0.7 

Shape Factor, W 
Fig. 9. Parameters of Lambert's Theorem 



0. 8 0. 8 1.0 



III -60 



1. 







-0.6 



0. 1 



0.2 0.3 0.4 



Mean Anomaly, n At 
J s 

Fig. 10a. Lambert's Theorem (case 1) 
111-61 



0. 6 



0. 8 1.0 



U HHHmrumffM 




0.1 0.2 



0. 3 



0. 4 0. 5 0. 6 

Mean Anomaly, n„ At 

Fig. 10b. Lambert's Theorem (case 2) 
111-62 



1.0 




f 0. 7 



a 

a 






0. 8 



0.9 



1.0 



Fig. 11a. Solution for Eccentricity 



111-63 




0.2 



0. 1 



0.2 0.3 



0.4 0.5 O.fi 

Eccentricity e 



0. 7 0.8 0. 9 1.0 



Fig. lib. Solution for Eccentricity 



111-6 4 



20.0— j 



19.0- 



1B.0— 



17.0— 



r ' a (1 + e) 



r * 


a U 


_ 


e 


P 
















a 


1 + 


p 












r 


1 - 


p 




P 









■2. 



-3. 



16.0— 



- 4.0 



15.0 — 



-5.0 



14.0 — 



13.0- 



E 12.0- 



11.0- 



X 10.0- 



a. o.o — 



8.0- 



6.0- 



5.0- 



4.0— 



3.0— 




1.0 



0. 2 0.4 0. 6 0.8 1.0 



— a.o 



Fig. 12. Solution for Apogee and Perigee Radii 
111-65 



■} cm o t~ in w o 



I I I 

I I I I I 

I I I I I 

I I I I I 

l l 



I I I I I I 



I I 1 I 



I I I I I I I I I 



III! 
i i i i i 



I I I I 



I I I 























UTUUU 




















II 


1 


tuttut 






















1 


mum 


















/ 


1 


// 


Umttt 


:t::: : 


















I 


7, 


mutt 


t 
















J 


1 


// 


mutt 


t" 


















J 


1 / 


uttttt 


r 






















tuttt 


1 






















tutu 


P 






















uttuL 


u 




















tutu 


r ^~ ' 




















tuttt 


44 " 




















Utttt 


44 " 


T 




















ttttt 


44 " 


*\J*- 




















ttutt 


ttt 




















//m i 


ttt 
























tut 
























tut 
























' tt 
























/ n: 




























































/ / 






















/ / 























































































































(Sap) e 'X^BUiouy 3nj X 



III -66 



180 



:;:ni 



. o_. o . 



o o o o p P P P 




0. 1 



0.2 
Eccentricity, e 



0. 4 



Fig. 13b. True Anomaly as a Function of r/a, e and y 

III -67 



180 



>I2 - r/a = 
„ „ ooooooooopopo op pp 

• - -J -J °"» ? a 5 jj, ^Mooaa ii. o 1 »J.g- 




= V2 - r/a 



Fig. 13c. True Anomaly as a Function of r/a, e, and y 



111-68 



(Sap) e "as^Tjad uioa; a^uy ibjju33 



J I I I I I I I I I I I L 




(JJap) a '/-[Buiouy OTJIuaaaa 



111-69 



Radius (ft or m x 10 ) 
14 12 10 8 6 4 2.5 2.5 4 6 8 10 12 14 
2.1 2.1 




Q-(7) 
c 

Fig. 15. Q-Parameter as a Function of Orbital Semimajor Axis and Radius 



III -70 



2.0 



1.8 



l.G 



1. 4 



1.2 



1.0 



0. 8 



0. 4 



0.2 




1. 2 



1. 1 



1.0 



0, 



10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 

True Anomaly, (deg) 



Fig. 16. Relationship Between Radius, Eccentricity and Central Angle 
from Perigee in an Elliptic Orbit 



111-71 



1 


IV 


\\ 


v 


\ ' 


A 


L 


A 


\ 




































) 


\\ 


\ 


\v 




\ 


\ 


\ i 


K 






























e^- 9 






w 


A 


A 


Y 


\ 


A 


s 


\\ 






































\\\ 


V 


A 




\ 


\ 
































<j 


Z'9 






\ 


A 


A 


v 


\ 


, \ 


\ 


\* 
































o 






A 


v 


A 


\ 














M x 






















CD 






\ 


A 


v 


V 


V' 










































— i 


v 


A 


u 


\ 




































s" 7 ? 










^ 


\ 


\\ 


k ' 


































gTTr 










N 


\ 





\ 


\ 


\, 
































u s 


o 












\ 






\ 
































o*g 


m 












r^ 






> 






























sp? 






Note: The values for the lines of constant r It are equal 

to the values of the abscissa r It at the points where 

a p 

the two intercept. 








































s-* 














































2> 














































o 










































O'fr 


t 








































SL-E 














































ire" 












































c, 








u 

a 
u 


(X 

u 

n) 
U 




































o 






































Hre 


<n 






+ 

ni| a. 
































sp? 
















































ii 

w 



o 








/ 


























Z'Z 












/ 


/ 


/ 
























p 








< 


< 


( 


























O'Z 


r 1 














U A cb 

O O iH 

-tA "-H ^ 

II 

- Sn 
ni 




















































ff^ 








y 


OS'I 




































































































o 








< 


i. 








( 


D 










O 










o 










D 



4) 



4) 

s- 



(Sap) X. 'aiSuy H^a mSHJ IBOoq 



III- 72 



(gap) a^Suv m^d imHti j 



I i i I i I l l I ll i l i iIiimIiiiiIi.iiIiiJ I 



m 


c 




<D 




In 


■O 


-M 




-3 








••-1 




;d 


M 


a 


En 


SO 


4) 


tl 


0) 


o 

<t-i 


W 




S 




£ 






(, E x ^<~luj) a;i.j(j[3j\ 



f ■ ■ < ' '/' T ' V ' f ■ ' ■ ' ■ ''i ■ ' ■'■ i'i 'i ^i 

m to r- co e» O m o ir 



(01 * sri J) K1100I3A 



III -73 



>l> 




(dog) 



Fig. 19. Q-Pa 
and 



rameter as a 
licccntricity 



Function of Local Flight Path Angle 



III -74 



I- 45 



30 
20 
10 

-20 
-30 



# 


e = 0.5 


Pi 


fffl 


"I 1 


:i li 


!!!! 


1 


[jit 


n 


i 


:; it 




•:!• 


if 1 


ijt 


i$ 








)m 


IF 


0.25 


liii 




jlP 




•r 1 


l!:i 




I 


it 


II 


!=ii 






r b 


ti- 
ll 





liii 




III 


M 




ill 


;;j| 


til 


tilt 


illl 


'III 


h 


1^1 




jji 


ill 


■ : !l 




H:: 


li- 


lit? 




& 


a 


ill; 


l' ! 


ill! 


1 


i=ii 


fe 


Ill; 




ill! 


ill 




'■■■ 



90 180 270 360 
8 (deg) 



Fig. No. -i~ii 

ERNo. — — 

Job No. I- 1*4 I'* 

Drawn 

P.U. 




— -30 



--35 



I- -40 



Fig. 20. The Solution for Local Flight Path Angle 



III -75 



180 




1.0 2.0 

Mean Anomaly, M (rad) 



3.0 



Time from perigee: (t - t ) = — M 

P n 

where - is tabulated as a function 

of a in Fig. 7 and Table 9. 



Fig. 21. Index for Figs. 22a Through 22i' 

(circled numbers in field designate areas covered by corresponding figure numbers) 



111-76 



Eccentricity 



O '-C CM CC Tf* O00 Tl* O tD EN033tC-fMO<C(Oi"^ O CO UJ't MO CO «3 -^ CN O 00 CO -r CM 

cjj cc cc t~- r- c-co co to in mcif ^tt^nnnnn NNNNNHHrtHnoooo 



oooooooo o o oooooooooooooooooooocooooo o 




0.1 0.2 0.3 0.4 0.5 0.6 

Mean Anomaly (rad) 



0.7 



0.8 0.9 



Fig. 22a. Mean Anomaly as a Function of liccentricity and Central Angle from Perigee 



III - 7 7 



Eccentricity 



O CO fg 03 rt" C CO C-J CO 

cc t- t— cc co co 10 in «y 



-r o 



cm c 
oooooooooo o © © 



CO -Cf C] C cc to 1> CN O cc c 
co co co crj c-j cm cm cn c\) •— 



0.08 




0.05 0.10 

Mean Anomaly (rad) 



Fig. 22a' Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee 



III -78 



100 



80 I 



60 



o K 'r -r r-i c :c ■- t mo :c trs 
.-. ro cc r; ;c 33 r- l- t- t- t~ er. ir> 



1'a t'l.'Il! t'k'ity 

!■■] O 



m uo i7 ic -r 



o o 



.' ' - f : / ■ / 



o o o 



/ / ■:/■ /■ '/: / 7 7 /:■ / ;/" / 

• / / 1 •' '' -' / / / / ■ J 



/ ■;■'■/ •/-■/, : ■ . , , , 

/ 7 



/ • / ■■/ / / / 

r....; I if... -J.:. J. ... J.. . 



7- /■!■/-:/-. o.4o 



? 7! / .'! / 7 / ■ f / : I !/■ / /: / 7 / 7,/ 7 7 /• 
: 7 /■//:// / / / .•' ./ 7 / ■/■ / 7: / / 7 /• / 7 A 38 

: < / / / ■/ 7 ; / / / ; / . i \l / ' / / 7 / 7 / / / ; 

///// //-/■/■://■■/■/ //!// 7 ■/■///■■/ //- 36 

1 '■ r ■ -i ' i ''■•'■•■/ /•■ 4-../..;./-..-^- /--/■■■/••■/ / '.•' '~7 ■■/ - /■ 7 ■,, 

:!■!!; ! ! : ' : / I / : / / /' ' / / 7 / : / / /: 
' I t ■■' ' ' / .' / ' / / t ! ' f f ■' '' '■ 



/ A 



* i 



' i 



777 



W ;-t f 



;>U : , i,: ;iul 



/'/' /:/ /V /77// / ////// / /=/ / /"" 
/■/7 7 //■/•/ / ' / ■/ 7 7 A/7 //■/■ / 7/' / /-7 i 

A//// /■/ /■//////// /:///:/ / / / / / ,o.2o t 

^ ; / < V •» / / -/ /■■•■/■-,• - h ■+-)-■+ 7--r ■■-/■■* ■•/- -/ -7--/-/ /■-.-'■■ <0.1« 

' .' ' ' / / ; ///// / ///;// / / / ■ / / / ; ' ;/ / / / " 

M7 i ////7//7///!///7/7///7^7/ / : 
7; :; 777/ ; J if f / f? //■/■/// / /;// // 7"-'" 
/ \ \ I m 7/ / ./ / /■ / / / / ■■'!]//■ I I I / / > ■/] / / / / 7 J - l - 
;;■;;■// ■' '' ////// ; 1 1 / / // /V 7 / //: / 7uo 

illl 11: ! If /. /:/ ; 7 / / ,'./:/. 7}/./;./ A/ / ./ /./.1/..7- / / -7 
777;/'////// / / / /;/ // / f// / 7 / / / 7 / / 7 7/ Jo.om 

777/7///F II I//./ 7/V//// ///■■/ A// '//A - 04 
■ 77 77//7//;//7////L//7 /;//■/ ////// ,• ^ J 

o.i 0.2 o.:? 0.4 o.r> 



/ 7 77 7/ /■/ / / / / 7 7 

o.c 0.7 0.8 0, 



o.:? 0.4 o.r> o.( 

Mean Anomaly (rad) 

Fig. 22b. Mean Anomaly as a Function of Kccentricity and Central Ancle from Perigee 

III -79 



Eccentricity 



150 



140 .. 



130 



"3b 



U 



120 



100 




no ..— .- 



0.3 0.4 0.5 0.6 

Mean Anomaly (rad) 



Fig. 22c. Mean Anomaly as a Function of Eccentricity and Central Angle from I'erigee 



III -80 



T-i r-i ") ~i --J 



rry-7 / /// 



/ / /■////.'// 



7'/7777r / 77777V7T / 






///'/// 



' ■ V///V ///////// A 

'////////7////A ■ / /// / / 



777Y/77777'77777 



//' -■' // /// A '/ S////7 // ///////////////// ///// . 
■ ■' / / A A / - / //A / AA //A //// / / '■'/ // ■' - ''/A / 

'/////// ''/////// / ///y/////////y / ^/////// 

■ : / / / • / / ,' f / / i / / / / / -' / .' , / * ' ■ r / y .- ' / / /,-.';• 



■■////>■., 



A/ A/// 



// '// 



A 



I 



■ / ' / / -V 



- ,'' / 



//>'///.. 



' ' /' - / ,■ / 



7. 



'■-'/v •■■■;'/: 

///A A 



1 / / / 
/ ■" / , 

7 A A 
/•■ / /. / 



/ , ,• 



/ 1 

' / ■ ,- 

■' r 



Fig. 22d. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee 



111-81 



.^CC'lUrUM! y 



"# :m 



140,. 9 






OS JO T M 

1.0 in ir; m 



o o 

y- -7- ;'-' y ,r s 

/ ■*■/. ,//// „ 

. y / , s / / 
ye ■ ■ 

/ 






'//"', '-■■/.- v///////^y^//////AZo // / / ^ 
- ' ' ■///sy//////////Y'' / /A / -4 / //* 



. .'. / / / 

/ /■ / / s 

/ 



■ / . 

/ ,.' /' 




/////y///^///^/////////V/////^/////^ 

/ //////////y////////////Ay/y/A^//^- n 

■: // / / / / / •' / // / / / / // / / / '/ ////////'//// /// •- 

> v, '/'A ■-/. ■'/////<////// ■y///////////////o :> 

' / / /// / // V • ■'///// ////////// //.//. 



/ ', / , /',•/.■/ 

.///// //y/////^'///// //a <//////////7//////. l 

// ////////////// ////// /y / /////////////// • 



iwit / / / / / / 

W///// / // , ' ,// ' ,/ ' 

/ / / t / / ,■ 



/ '' ,■■■ // / / // 7 / ,■■/// / ' //////////// / 






///A'// ■" 






- f ' ■' ' '///// ///// 






////. 'J- 10 

,■' / 

08 

0.06 



/ / / ■ . ' 



/ / / ,■ ' / 



/'/ .'/'/ 



■'////>//#//Ay^ 

7'A///u//-/////////////yfy//////7" "' ■ 



///////, 



'' / ' ' ■ ■ : / ■ .' / ////// / / / / , /' / / / / / / / / / // / '-' / / 



90 
0.8 



0. 1! 



1.0 



1.1 1.2 1.3 1.4 1.3 

Mean Anomaly (rad) 



Fig. 22e. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee 



III -82 



.80 



170l 



160 i 



150 



140 



130 




0.5 0.6 



o. a o. 9 l.o i.i 

Mean Anomaly (rad) 



i. :•! 



. 4 



Fig. 22f. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee 



III -83 



,ceun; rielty 




!)0 



1.6 



] . a 



1. 9 



2.0 



2. 1 



2. 2 



Mean Anomaly (rad) 



2. 3 



2. 4 



Fig. 22g. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee 



HI- 84 



10. 00 

0. 8(i 




1.7 1.8 1.9 2.0 

Mean Anomaly (rad) 



Fig. 22h. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee 



III -85 



2.30 



2.40 



2.50 



Mean Anomaly (rad) 

2.60 2.70 2.80 



2.90 



00 



3. 10 




Fig. 22i. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee 



111-86 



Mean Anomaly (rad) 

3.0 



3. 1 




-- .1 80 



■|17!/ 



im 



I7G 



175 ■- 



17-1 



173 



172 



171 



. 1 70 



160 



Kit! 



0. 10 



Fig. 22i'. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee 



III -87 



CHAPTER IV 
PERTURBATIONS 

Prepared by: 

G. E. Townsend, Jr. 

Martin Company (Baltimore) 

Aerospace Mechanics Department 

March 1963 

Page 

Symbols IV- 1 

A. Introduction IV-2 

B. Special Perturbations IV-2 

C. Methods for Numerical Integration IV-7 

D. General Perturbations IV- 14 

E. References IV-50 

F. Bibliography IV-5 2 

Illustrations IV-59 



LIST OF ILLUSTRATIONS 



Figure Title Page 

1 Comparison of Perturbation Magnitudes (for 
equinoctial lunar conjunction) IV-61 

2 Solution for the Secular Precession Rate as a 
Function of Orbital Inclination and Semi- 

parameter IV-62 

3 Change in the Mean Anomaly Due to Earth's 
Oblateness IV-63 

4 Solution for the Secular Regression Rate as a 
Function of Orbital Inclination and the Semi- 
parameter IV-64 

5 Change in the Anomalistic Period Due to the 

Earth's Oblateness IV-65 

6 Change in the Nodal Period Due to the Earth's 
Oblateness (for small eccentricities) IV-66 

7 The Variations of the Radial Distance as 

Functions of the True Anomaly e and IV-6 7 

8 Maximum Radial Perturbation Due to 

Attraction of the Sun and Moon IV-68 

9 Satellite Orbit Geometry IV-69 

10 Effects of Solar Activity on Echo I 

11 Apogee and Perigee Heights on Echo I 

(40-day interval) IV-70 

12 Minimum Perigee Height as a Function of 
Days from Launch, Showing Effect of 
Oblateness, Drag, and Lunisolar Perturba- 
tions IV-71 

13 Minimum Perigee Height of Satellite as a 
Function of Days from Launch (8 to 14 hr, 

expanded scale) IV-72 



IV-ii 



LIST OF ILLUSTRATIONS (continued) 



Figure Title 



Page 



14 Comparison of Approximate and Exact 

Solutions of Satellite Motions IV- 73 

15 Minimum Perigee Height of Satellite as a 
Function of Days from Launch, Showing Effect 

of Neglecting Oblateness IV- 73 

16 Minimum Perigee Height of Satellite as a 
Function of Days from Launch, Showing Effect 

of Neglecting Moon, Sun IV- 74 

17 Minimum Perigee Height of Satellite as a 
Function of Days from Launch, Showing Effect 

of Sun and Moon 90° Out of Phase IV- 74 

18 Minimum Perigee Height of Satellite as a 
Function of Days from Launch, Showing 

Effect of Changing Orbit Size IV-75 

19 Minimum Perigee Height of Satellite as a 
Function of Days from Launch, Showing Effect 

of Change in Inclination IV-75 

20 Minimum Perigee Height of Satellite as a 
Function of Days from Launch, Showing Effect 

of Change in Argument of Perigee IV- 76 

21 Minimum Perigee Height of Satellite as a 
Function of Days of Launch for About a 

10-Year Period # IV-76 



IV-iii 



IV. PERTURBATIONS 



A 
a 
B 
C, 



1 

J 

n 

L 
M 



SYMBOLS 
Right ascension, area 
Semimajor axis 
Ballistic coefficient C,.A/2m 
Drag coefficient 
Eccentric anomaly 
Eccentricity 



[6, 



Universal gravitation constant |_6.670 
(1 ± 0.0007) x 10" 8 cm 3 /kg-sec 2 ] 

Magnitude of the angular momentum 
per unit mass; step size in numerical 
integration 

Orbital inclination 

Coefficients of the zonal harmonics in 
the Vinti potential 

Latitude 



Mean anomaly n(t - I.) - E - e sin E 



Mass 

Mean motion = 2tiIt 



Vw^ 



Semilatus rectum = a( 1 - e ) 

Perigee radius = a(l - e) 

also quantity in Encke's equation 



- 1 



R, S 



W Vehicle coordinates R - r, S normal 
to R in the plane of instantaneous mo- 
tion (S • V - positive number), W com- 
pletes the set 

Radius 
dr 

ar 



t 
u 

V 

x, y, z 

r, r 
p 

r 

c 

1 





:<%< 



2 "> 

d r/dt" 



Time 

Potential function 

Velocity vector 

Equatorial Cartesian coordinates 

Angular coordinates of perturbing mass 

cos _1 (r • V) - 90° 

-/u/2a = energy per unit mass 

Dimensionless parameter - 1 

True anomaly 

= GM - masses' gravitational constant 

Disturbing potential 

Perturbed 4> in Anthony and Eosdick 
theory 

(x - x i ), (y - y J, (z - y, ) in Encke' s 

solution--also vehicle-centered coordi- 
nates 

Orbital period 
Time of perigee passage- 
Central angle measured from the as- 
cending node 

Right ascension of the ascending node 

Rotational rate of the earth, 

1 revolution every 8(i, lf>4.001 mean 

solar seconds 

Argument of perigee 



IV- 1 



A. INTRODUCTION 



The Keplerian relations, as discussed in 
Chapter III, give convenient approximations for 
use in preliminary orbit computations. However, 
in order to obtain precise earth satellite orbits, 
the various perturbing factors which give rise to 
accelerations (in addition to that of the central 
force field) and cause the motion to deviate from 
pure conic form must be considered. These per- 
turbative accelerations may be due to the mass 
asymmetry of the earth, the gravitational attrac- 
tion of other bodies, atmospheric drag, electro- 
magnetic drag, radiation pressure, thrust, or 
may be required to account for relativity effects. 
These factors affect the motion of the satellite to 
a varying degree depending on the shape and 
mass of the satellite and the type of trajectory. 

Special perturbation methods involve the 
formulation of the differential equations of mo- 
tion in such a manner that the computation of an 
orbit is achieved by numerical integration. The 
perturbation method to be used is determined by 
the type of problem that is under consideration. 
Similarly, all combinations of integration tech- 
niques and perturbation methods are not equally 
suited to the solution of a particular problem, 
even though the use of such combinations is pos- 
sible. Because numerical integration is subject 
to the inevitable accumulation of errors which 
eventually destroy the validity of the results, 
special perturbation methods are restricted to 
the prediction of earth satellite orbits for times 
dependent upon the desired accuracy, the for- 
mulation of the problem and the number of digits 
carried in the computations. 

One source of error in the numerical integra- 
tion process is roundoff error, resulting from the 
limited number of digits which can be carried in 
computation. The roundoff error is not reduced 
by double-precision computation where tabulated 
values to be interpolated at each integration step 
are known to less than single-precision accuracy. 
This error obviously increases with the number 
of computations, which in turn increases with 
decreased integration step size. Roundoff propa- 
gates through the numerical integration so that, 
assuming a normal error distribution, the absolute 
error incurred in double integration is 

(the product of the number of steps and 

3/ 2 
the original roundoff) 



r 

~3" 
r 



+ F 



(1) 



where F is the sum of the accelerations due to 
the various perturbing forces. If F = 0, there 
are no perturbations and the motion is Keplerian. 

If the position coordinates of the vehicle and 
the perturbation accelerations are given in rec- 
tangular equatorial coordinates, Eq (1) can be 
written: 



x y°° 



(2) 



Z x i 



here ) x. is the sum of the perturbation ac- 



celerations. These terms are discussed in the 
following paragraphs. 

a. Vinti potential 

If the earth were homogeneous in concentric 
spherical shells, its potential would be that of a 
point mass. The effects of the flattening of the 
poles and lack of symmetry about the equator, 
however, manifest themselves as perturbative 
forces on satellites in the vicinity of the earth. 
The acceleration due to the oblateness of the 
earth can be written in a simple form attributable 
to J. Vinti of the National Bureau of Standards: 



yx 
3 



(?)' 



r 



',M£)'i(>-'£ 



+ J, 



R\ 4 5 



3+42 



~2~ 

r 



63 



r 



+ J 5(f) 7 4 1-693^ + 630^-105 



r 



~1 

r 



(3) 



A second source of error is truncation. This 
error arises because of the finite polynomial 
approximations in the integration formulas. Since 
the terms in the polynomials involve powers or 
differences of the integration interval, the trun- 
cation error can be reduced by choosing a smaller 
integration step. Therefore, increasing the num- 
ber of integration steps decreases the truncation 
error, but increases the roundoff error. 



B. SPECIAL PERTURBATIONS 

1. Perturbative Forces 

The equation of motion of a perturbed orbit is 
of the form: 



1X2. 

-^ 
r 



+ J , 



^\V\l 



+ j,m -= 



693 



6 
z 

~6" 

r 



r 3 



5* 
r 



1 + 10 



z 35 z 

r r 



f) 4 | (- 15 + 70 V 63 7, 



£) 5 a("-3»4 + °«4 



IV-2 



where J. are the harmonic coefficients. Since 

the earth is almost spherically symmetric, the 
J i are all small compared to 1 (see Chapter II). 

b. Perturbative terms due to remote bodies 

The perturbative terms due to remote bodies 
which can be considered as point masses can be 
written directly from the integrals for the n-body 
problem as developed in Moulton (Ref. 1) and in 
other texts on celestial mechanics. 



7 ■%<%■?) 

" \ / y Ai * 
y =2 V 




(4) 



i=l \ r Ai 



z. 
l 

~7 



where r A - is the distance from the satellite to the 

ith body and r. is the radius from the center of 

the earth to the ith perturbing body. For the case 
of an earth satellite, lunar and solar attractions 
are the major sources of perturbations for short 
term orbits. The order of magnitude of these 
perturbing forces may be observed in Fig. 1. 
(Subsequent discussions appear in Section C of 
this Chapter. ) 

c. Thrust 

If thrust is applied, it may also be handled 
as a perturbation. The general procedure, how- 
ever, for large thrust-to-mass ratios is to treat 
the thrust periods in a different fashion by con- 
sidering the vector sum of the thrust and central 
force terms as defining the reference trajectory 
rather than the central force term alone. Since 
the thrust vector is determined by the maneuver 
requirements and the guidance law to be utilized, 
no analytic solutions are available for this ref- 
erence trajectory; thus, numerical integration is 
necessary. Indeed, no single form of the per- 
turbing acceleration can be written other than its 
resolution in terms of generalized vectorial com- 

T T T 

ponents; for example: — — , — £- and z 
mm m ' 

d. Atmospheric lift and drag (Ref. 2) 



x -- D 2 s 2 J|*t'Y <v) a(H) Y (or) vl| 



- H< A + 




- A< Wv) o- (H) Y M -^ j^- \(r x l) sin g 



* <i* 



ML H} 



cos £ > x -» y, z 



(5) 



where the vehicle velocity relative to a 
rotating atmosphere with cross winds is 
given by 

v x = x +y n e + q (cos a sin 4>' cos j3 

+ sin a sin /3) 

v y = y - x" e + q (sin a sin <t> • cos £ 

- cos a sin £) 

v z = z - q cos <t»' cos /3 

where 

A «= constant fitted to the Mach 

number variation of the drag 
coefficient with a mean sonic 
speed = 1 

A * Initial projected frontal area of 
o 
the vehicle, m 



B 



-D„ 



v- 

f(r) = 
H 



m ■ 

Q - 

q 

r ■ 

s >= 



constant fitted to Mach number 
variation of the drag coefficient 
with a mean sonic speed 

/C T 




reference (hypersonic continuum) 
value of the drag coefficient (0. 92 
for a sphere, 1. 5 for a typical 
entry capsule) 

lift coefficient 

local sonic speed in terms of sur- 
face circular satellite speed 

C D A 0<>0 V C0 2/2 So m O 

2 
" D ctv(ct) 

acceleration of gravity at unit dis- 
tance (surface of earth) 

altitude above an oblate earth « r - 



' + 



+ fsinV +^- (I - Jj sin 2 2<t» 

where the flattening f ■ 2^-3 (units 
of earth radii) 

mass of space vehicle (kg) 

unit vector in the orbit plane perpen- 
dicular to the line of apsides 

speed of the cross wind measured in 
a system rotating with earth's angular 
rate (units of surface circular satellite 
speed V CQ ) 

radius from the geocenter to the vehicle 

speed of the vehicle with respect to an 
inertia! frame, directed along Q 



IV-3 



d-3. 



v co 

x,y, z = 

a - 
- 
V(v) = 

V(a) = 

Q e = 
M' = 

£ = 

p 

<T = 

<f,. = 



Surface speed for circular orbit -- 
7905. 258 m/sec 

equatorial coordinates in units of 
equatorial earth radii 

right ascension of the vehicle (radians) 

azimuth of the direction from which 
the wind is coming 

C (v/C )/C„ , the drag coefficient 

D s u o 

variation with Mach number 

C„ (<r)/C n , the drag coefficient varia- 
D L> 

tion in the transitional regime 

constant relating to the rotational rate 
of the earth, 0.058834470 

m Q /m 

bank angle 

3 

atmospheric density, kg/m 

"sea level" atmospheric density, 
1.225 kg/m 3 

P 
p 

geocentric latitude, radians 



e. Radiation pressure 

A body in the region of the earth is subjected 
to solar radiation pressure amounting to about 

4. 5 x 10~ 5 dyne/cm 2 , the order of the force being 
the same for complete absorption and specular 
reflection of the radiation. Radiation pressure 
is an important source of perturbations for satel- 
lites with area -to -mass ratios greater than about 
25 cm 2 /gm. The effects of radiation pressure 
on lifetime are discussed in Chapter V and also 
in Section C-7 of this chapter. 

The rectangular coordinates (X-axis toward 
vernal equinox) of the accelerations are: 



x = f cos A 



v = f cos i sin A 

•> 

z = f sin i sin A @ 



(6) 



where: 



i = inclination of the ecliptic to the equator, 
° 23.4349° 

A = mean right ascension of the sun during 
the computation 



4. 5 x 10" 



(=)■ 



f. Electromagnetic forces 

As a satellite moves through a partly ionized 
medium, the incident flux of electrons on the 
satellite surface is larger than the ion flux, so 
that the satellite acquires a negative potential. 
On the day side of the earth, this effect is op- 
posed by the photoejection of electrons. Jastrow 
(Ref. 3) estimates that the satellite potential may 
approach -60 volts on the day side and will not be 
greater than -10 volts on the night side. 

In addition to the potential acquired by ionic 
collision, the motion of a conducting satellite 
through the magnetic field of the earth causes 
the satellite to acquire a potential gradient which 
is proportional to the strength of the magnetic 
field and the velocity of the satellite. The inter- 
action of the electric currents thus induced in the 
satellite skin with the magnetic field causes a 
magnetic drag to act upon the satellite; this drag 
is proportional to the cube of the satellite dimen- 
sions. 

If these forces are found not to be negligible, 
they can be included directly by the use of Max- 
well 1 s equations or indirectly by use of an at- 
mospheric model which takes the effects into ac- 
count. 

g. The effects of relativity 

Perturbations caused by relativity are of the 

V 2 
order a = — E- = -Ay , where c is the speed of 



c re 

light. Since a is a very small quantity and any 
measurable deviations occur only after a long 
period of time, relativistic effects can usually 
be ignored in the case of earth satellites. A mod- 
ification of Newton's law as a consequence of the 
theory of relativity can be found in Danby (Ref. 4). 

Substitution of these perturbative accelera- 
tions (a through g) in Eq (2) yields the complete 
equation of motion. 

2. Special Perturbation Methods 

Three special perturbation methods currently 
used for computing earth satellite orbits will now 
be discussed with an evaluation of the main ad- 
vantages and disadvantages of each. 

a. Cowell's method 

In Cowell's method, the total acceleration, 
central as well as perturbative, acting on a 
satellite is integrated directly by one of the 
numerical integration techniques (Section B of 
this chapter). The equations of motion which 
must be integrated twice to obtain position co- 
ordinates are: 



MX 

r 



£ ° x °i' x -y. z - 



T- 



These equations are symmetrical in the rec- 
tangular coordinates and are simple in form; 
they apply to elliptic parabolic and hyperbolic 
orbits, and require no conversion from one co- 
ordinate system to another. 



IV-4 



A disadvantage of the method is the large 
number of places which must be carried because 
of the large central force term to prevent loss 
of significance for the small perturbations. Also, 
since the total acceleration, which is subject to 
fairly rapid changes, is being integrated, it is 
necessary to use a smaller integration step to 
maintain a given accuracy. This requires an 
increase in the number of integration steps and 
the inherent roundoff error accumulation. De- 
tection of small perturbation effects such as 
those caused by radiation pressure may be im- 
possible due to roundoff and truncation errors. 
Cowell's method is especially useful when the 
perturbation forces, such as thrust, are of the 
same order as the central force. 

b. Encke's method 

In the Encke method, only the deviations of 
the actual motion from a reference orbit, which 
is assumed to be reasonably close to the actual 
orbit, are integrated. Usually a two-body ref- 
erence orbit is used since the position at any time 
on this orbit can be determined analytically. How- 
ever, more complicated reference orbits such as 
Garfinkel's solution (Ref. 5), which is known 
analytically and which incorporates some of the 
oblateness effects in the earth's gravitational 
potential, might be used on an earth satellite 
nrbit. 

Let x, y, z denote the actual position of the 

satellite and x , y , z the position on a Keplerian 
e J e e 

reference orbit. 



Because of the possible loss of significance in 
subtracting nearly equal quantities in Eq (10), it 
is necessary to rewrite Eq (10) in better compu- 
tational form. 

Substitute Eq (9) into the defining equation for 



2 4- 2 + 2 

x + y + z 



= (x e + ?) 2 + (y e + r,) 2 + (z e + ?) 2 
= r e 2 + 2 [C(x e + i?)+ n(y e + in) 
+ ?(z e+ Jf)] 
Define q to be: 

q = -V [ ?(x e + 7 5) + T ><y e + £' l) 



+ ?(z e + i e>] 



So that Eq (13) becomes 
2 



(11) 
(12) 

(13) 



fc) 



= 1 + 2q or 



m 



(1 + 2q) 



3/2 



(14) 



(15) 



Encke's series, using a binomial expansion, is 
defined by: 



The equations of motion in an inertial frame 
of reference are then: 



^ + I * i 



e 
r 



x -y, z 



x -» y , z 
e J e e 



(7) 



(8) 



Let the deviations from the reference orbit 
be ?, n, ? so that: 



1 



y - y e 



(9) 



Differentiation of Eq (9) and substitution of Eqs 
(7) and (8) into the result yield: 



x -*■ y, z for f -* r\, ? 

I *i < 10) 



x 

~3" 
r 



i 
3-| 



- O + 



L x i 



-m 



1 - (1 + 2q) 



-3/2 



CO 

I 



( _ 1)k -l(2l lii )) q k 



k=l 

-1/2 < q < 1/2 



2 k (kl)' 



fq 



(16) 



Substitution of Eq (16) into Eq (10) yields Encke's 
formula: 



f = J±^ (fqx - €> + £ *i 



(17) 



This equation, which employs series expansion, 
yields more accurate deviations when the terms 
are small. When the terms exceed a certain 
limit, a process of rectification is initiated, 
that is, a new reference orbit is computed. The 
limits on q needed for rectification are estab- 
lished as: 

n+1 



AC 

Vi 



(18) 



where Af is the allowable error in | and a . is 

the coefficient of the first neglected term of the 
Encke series. 



IV-5 



In contrast to Cowell's method, only the dif- 
ferential accelerations due to perturbations are 
integrated to obtain deviations from a two-body 
orbit. These deviations are then added onto the 
coordinates of the satellite as found from the 
two-body orbit to obtain the actual position of the 
satellite. Since the deviations are much smaller 
and, therefore, need not be determined as ac- 
curately, it is possible to maintain a given ac- 
curacy with larger integrating steps. As a con- 
sequence of the larger integrating steps, there 
is less danger of serious roundoff accumulation. 
Moreover, the integration errors affect only the 
least significant figures in the deviations and, 
when added to the much larger positions deter- 
mined from the reference orbit, should have a 
less serious effect on the overall accuracy. Al- 
though the roundoff error is less, Encke 1 s 
method involves expressions that are much more 
complicated and often less symmetric than 
Cowell's simple formulas. In addition, both the 
necessity of solving the two-body formulas at 
every step and the possible need for rectification 
introduce additional sources of error. In the 
former case, the frequency of rectification af- 
fects the attainable accuracy and also introduces 
small errors in the determination of the mean 
anomaly M. For the case of nearly parabolic 
orbits, errors in the use of the two-body formu- 
las in an unaltered form are especially critical. 
This is due to the fact that when the eccentricity 
e ~ 1, and the eccentric anomaly E is small, can- 
cellation errors arise in forming the radial dis- 
tance r = a (1 - e cos E) and the mean anomaly 
M = E - e sin E. In addition, small division er- 

2 
rors will be introduced in forming p/a = (1 - e ). 

The Encke method is especially suited to 
problems in which the perturbative accelerations 
are not large and have their major effect over a 
limited portion of the orbit, e. g. , lunar and in- 
terplanetary orbits except microthrust or long- 
thrust trajectories. 

c. Variation-of-parameters method 

The variation-of-parameters or variation-of- 
elements method differs from the Encke method 
in that there is a continuous set of elements for 
the reference orbit. The reference motion of the 
satellite can be represented by a set of param- 
eters that, in the absence of perturbative forces, 
would remain constant with time. The perturbed 
motion of a satellite may thus be described by a 
conic section, the elements of which change con- 
tinuously. The variable Keplerian orbit is tan- 
gent to the actual orbit at all times, and the ve- 
locity at any time is the same in both orbits. 
This reference orbit thus osculates with the ac- 
tual orbit. The variations in the elements used 
to describe the osculating conic can be integrated 
numerically to solve for the motion. 

Any set of six independent constants can be 
utilized for this purpose though it is conventional 
to use the geometrical set a, e, T , u, f2 and i. 

Lagrange's planetary equations, which specify 
the variations for this set of parameters, are 
derived in Section C of this chapter. 

It is also possible to choose a different form 
for the reference motion. As in Encke 's method, 



Garfinkel's solution which includes part of the 
perturbative forces caused by the nonspherical 
shape of the earth might be employed. If the 
drag force predominates, as in the case of entry, 
a rectilinear gravity-free drag orbit as applied 
by Baker (Ref. 6) can be used instead. 

Many variation-of-parameters methods have 
been proposed including those of Hansen, 
Strb'mgren, Oppolzer, Merton and Herrick. 
These methods differ in the choice of elements 
or parameters and of the independent variable. 
Of these, the parameters suggested by Herrick 
(Ref. 7) will be briefly described here. 

Let x , y be rectangular coordinate axes 
in the instantaneous orbit plane with x the 

axis along the perigee radius as shown. Let P 
be the unit vector in the orbit plane in the di- 
rection of perigee, Q be the unit vector perpen- 
dicular to P in the direction of motion along the 
y -axis and W be the unit vector normal to the 
orbit plane in a right-hand system. 

The parameters selected by Herrick for or- 
bits of moderate eccentricity are vectors A(t) 
and B(t), the mean anomaly M and the mean 
motion n. The vectors A and B are defined by: 

A = eP 



B = e y p Q 

M = n(t - t Q ) 




where 

a = semimajor axis 

e = eccentricity 

p = semilatus rectum 

k = VgM 



IV-6 



The differential equations in the parameters 
have the form: 



A = A Q + k g \ A' dt 



s 



B = B Q + k e \ B 1 dt 



s 



n(t) 



n +k eJ n ' 



dt 



t 



M(t) = M Q + n Q (t - t Q ) + k g CC n' dt dt 







t 



k 4 



+ k \ M dt 
X 



and the perturbative variations A', B\ n 1 , M' 
are defined as: 



D = e ya sin E 



-* B 



H = e x = -r • A 

CO 

Y^Td' = r • F = xF + yF + zF 
* x J y z 



»T ■»£•*■»(£ '.♦£',♦&'.) 



H' = 2 DD 1 - 



r 2 dD 1 
~3T 



^ 



r—W 



dH' _ r-D' 

r ^L - E£ H' - F H 



^' - 'T - S 



V7A.=r- 



dD' dr 

"ar 



D' - F D 



e J~pV = A • B' = A B' + A B' + A B ' 
T x x y y z z 

ja~M> = yp~v' -2D 1 
. _ 3 n a dD 1 

The Herrick elements must be related to the 
rectangular coordinates and to the usual elliptic 

elements because the perturbative forces F are 
given in rectangular coordinates. It is thus 
necessary to go through the two -body formulas at 
every step, as in the Encke method, and through 
some complicated conversions as well. 



The essential characteristic of this method is 
that the integration is carried out on parameters 
which are much more slowly changing functions 
of time than rectangular coordinates. Since they 
vary slowly, the error accumulation from the 
calculation of the derivative is, for a long time, 
far beyond the eighth significant digit of the 
initial calculation. Thus, it is expected that 
truncation error would appear only for very large 
intervals and much larger integrating steps can 
be taken for a given accuracy. Since in this 
method a system of first order equations is 
being integrated, there is less danger of round- 
off error accumulation. A disadvantage is that 
the programming and numerical analysis in- 
volved in this method are the most complicated 
of the three methods discussed. Because of 
this, the computing time per integration step is 
at least twice as long as for a Cowell method. 
The Herrick formulas given here lead to special 
difficulties on low eccentricity orbits because of 
small division problems. Similar difficulties 
arise with other variation-of -parameter methods 
for low inclination orbits, as well as for hyper- 
bolic and parabolic orbits. Such cases all re- 
quire special consideration, thus detracting from 
the usefulness of parameter methods as basic 
integration tools. A new method due to Pines 
(Ref. 8) is apparently suitable for all earth 
satellite orbits. The variation of parameters 
method is primarily applicable to missions in 
which small perturbations act throughout the 
orbit, e.g., microthrust transfer. 



C. METHODS FOR NUMERICAL 
INTEGRATION (REF. 9) 

Of the factors affecting the choice of an in- 
tegration method for space trajectory calcula- 
tions, the two most important are speed and ac- 
curacy. Other factors, such as storage require- 
ments, complexity, and flexibility, are of sec- 
ondary importance with most modern computers 
such as the IBM 7090. A good integration sub- 
routine should have the following features: 

(1) It should permit as large a step -size as 
possible. Thus, higher order methods 
should generally be given preference 
over lower order methods. 

(2) It should allow for the automatic selection 
of the largest possible integrating step 
for a required accuracy. The procedure 
for increasing or decreasing the step- 
size should be reasonably simple and 
reasonably fast. 

(3) It should be reasonably economical in 
computing time. 

(4) It should be stable; that is, errors in- 
troduced in the computation from any 
source should not grow exponentially. 

(5) It should not be overly sensitive to the 
growth of roundoff errors, and every 
effort should be made to reduce roundoff 
error accumulation. 

Some of the more commonly used integration 
methods are compared in detail on the basis of 
these criteria. 



IV-7 



1. Single Step Methods 

Of the various Runge-Kutta methods the Gill 
variation is most popular. It was devised to re- 
duce the storage requirements and to inhibit 
roundoff error growth. There seems to be little 
reason to choose the Gill variation over the 
standard fourth order method when modern com- 
puters are available, because the storage savings 
are insignificant and the roundoff error control 
can be achieved more simply and more effectively 
by double precision accumulation of the dependent 
variables. 

The process of double precision accumulation 
can be used with any integration method. It is 
extremely effective in inhibiting roundoff error 
growth and very inexpensive in machine time. 
The process consists simply of carrying all de- 
pendent variables in double precision, computing 
the derivatives and the increment in single pre- 
cision, and adding this precision increment to 
the double precision dependent variables. For 
integrating a single equation of the form Y' = 
dy/dt = f(t, y), the formulas for the standard 
Runge-Kutta fourth order method are 



= hf, 



(v y n ) 



hf t_ + 



n 



hf 



= hf 



(v 



h 

2"* 

+ h, 



y + 



y + 
•'n 



2 
T 



y + k c 

■'n 3 



) 



(19) 
(continued) 



y n+ l=W( k l + 2k 2 +2k 3 + k 4) 

where h denotes the integration step -size and n 
denotes the integration step. 

Runge-Kutta methods are stable, follow the 
solution curves well, have a relatively small 
truncation error among fourth order methods, 
and do not require any special starting proce- 
dure. However, 

(1) They tend to require more computing 
time, since four derivative evaluations 
per step must be made compared to one 
or two for other multistep methods. 

(2) The usual fourth order methods restrict 
the step -size for a required accuracy. 

(3) There is no simple way to determine the 
local truncation error and, as a conse- 
quence, it is difficult to decide on the 
optimum step-size for a required accu- 
racy. 

Various suggestions have been made for over- 
coming this deficiency. The same trajectory 
could be integrated twice: first with step -size 
h and then with step -size h/2. The difference 
between the two values at a time t can then be 
used to decide whether the step -size should be 
increased or decreased. This process involves 
three times as much computing and, therefore, 
cannot be seriously considered. The simplest 



method, proposed by Aeronutronic, is to integrate 
over two intervals of length h and then to re- 
compute the dependent variable using Simpson's 
rule. 



(s) 

y n + 1 y n 



h 



— (v' , + 4y' + y' 
3 \ y n+ 1 y n ■'n 



The difference between this value and that 
obtained by the Runge-Kutta method at time 

t , , is then used as a criterion. This pro- 
n + 1 

cedure is relatively simple and inexpensive, but 
there is no mathematical justification for it. 
Any decision to change the step -size based on it 
might be erroneous. 

Other single step methods include several 
attributable to Heun, the improved polygon or 
Euler-Cauchy method, and a method employed 
by C. Bowie and incorporated in many Martin 
programs. Bowie's method is outlined below. 



A 
*0 = 



x h/2 x + x 5" 



"h/2 



y + y ( 



h 
5" 



'h/2 



, • h •• h 
x + X 2" + X T 



'h/2 



= y + yo2 + y o T 



= x Q + x Q h 



y h =y + y h 



X Q h 



y h = y o + y o h 

Step A 

= f 



h/2 x h/2* J h/2 e h/2' 



A ^T 

.- h 2 
^0-2- 



S h 



f h' y h = g h 



i h/2 



x + 2-4 { 



5x + 



yh/2 = h + -k { 5y o + 



x h/2 " X hJ 

y n /2 -'4} 



x h/2 = x + x o!? + 9Tr ( 7x + 6x h/2 " x h) 
h h 2 

y h/2 = y o + y o ? + m ( 7 y<> + 6 y h/2 - yh) 

h { X + 4x h/2 +x h} 



*h 



X + !T 



IV-8 



^ = yo + 



^{^o + 4 y h /2 + > ; h} 



and the Adams -Moulton formulas are 



x h x + x 



y h =y + .y 



h + ^|x 0+ 2x h/2 J 
,2 

^o h+ T-{yo +2 y'h/2} 



Step B 

x h/2 = f h/2* y h/2 = g h/2' x h = f h' y h = g h 

x h = x O + !r{ x + 4x h/2 + x h} 

^h = y o + !r {yo + 4 y h /2 + y'h} 

x h =x + x h+^{x + 2x h/2 | 



hi \ 

y h = yo + yo h + ^\y + 2 y h i2J 

If the functions f, g do not actually involve x, 
y it is clear that x, ,„, y . need never be com- 
puted and that x, , y, need only be computed at 

the point they occur for the last time in the above 
list. 

It will be noted that the process as described 
above involves two iterations and requires that 
the functions f, g be evaluated five times. If 
further iterations are desired, one simply goes 
back to the point marked "A" when he completes 
all the steps of the preceding page. Note that 
Steps "A" and "B" are identical, though the 
formulas immediately following them are not. 

If the number of iterations are continued un- 
til there is no (sensible) change, the solution is 
exact on the assumption that '£ and y vary quad- 
ratically over each interval. Since this assump- 
tion is exactly realized only in trivial cases (for 
which it would be unreasonable to use any step- 
wise method), the optimum procedure seems to 
be to do only the two iterations as the list of 
steps implies. Put another way: when the over- 
all accuracy is not sufficient, it is better to 
shorten the time interval than to increase the 
number of iterations beyond two per interval. 



2. 



Fourth Order Multistep Prediction -Correct 
Method ' ~~~~ 



Of this type, for a first order system y' = 
f (t, y) are the Milne and Adams -Moulton methods. 
The Milne formulas are: 



^ 



y (p > = y 

^n+l y 



n - 

14 



+4J1 
3 3 

5 v 



(K 



2y n - 2 ) 



h-V(i) 



y n+i = y n -i + t (y n +i + 4 y; + yn -1) 

h v 



>(20) 



- m y (i) 



Vi 



y (c } 



'n + ^( 55y „ -^a-l^n 



-9y' 
•'n 



3) 



, 251 , 5 v , , 

+ 720- h y ^ 



= y + 

J n 



+ y' 

•'n 



^(9y n+1 +i9y n+1 -5y n 



\(21) 



i) ~TZd 



19 , 5 v . . 
h y (r|) 



For these methods, as well as for all multi- 
step methods, special formulas must be used to 
obtain starting values at the beginning of the in- 
tegration and wherever it is desired to double or 
halve. A Runge-Kutta method is the most con- 
venient for obtaining these starting values. The 
difference between the predicted and corrected 
values provides a good estimate of the local 
truncation error and this estimate can then be 
used to decide on whether to increase or reduce 
the step-size. 

The Milne method has a somewhat smaller 
local truncation error, but for some equations it 
may be unstable (i. e. , errors introduced into 
the computation will grow exponentially) and, 
while some techniques have been suggested to 
eliminate this instability, it is probably advisable 
to avoid the use of the Milne method. 

The Adams -Moulton formulas are uncondi- 
tionally stable and lead to a fast and reasonably 
accurate method. Its principal disadvantage is 
its low order of accuracy which restricts the 
integration step -size. 

3. Higher Order Multistep Methods 

Variation-of -parameter methods lead to 
systems of equations which are essentially first - 
order in form as contrasted to Cowell and Encke 
methods which lead to systems of second order 
equations. For second order systems, special 
integration methods are available. 

Before considering these, the Adams back- 
ward difference method applicable to first 
order systems must be mentioned. If the sys- 
tem has the form y' = f(t, y), the Adams 
formulas are 



'n+ 



N 

1 = y n + h I °kv k f 



(22) 



k=0 



.k . 



where V is the backward difference operator 
defined by 

V k f =V k_1 f - V k_1 f • V°f =f 
n n n-1 n n 

The first few values of a\ are (1, 1/2, 5/12, 

3/8, 251/720, 95/288) for k = 0, 1, 2, 3, 4, 5. 
If Nth differences are retained, the principal 

part of the local truncation error is 0(h ). 
If Nth differences are retained, then N + 1 
consecutive values of y. must be available, and 



IV-9 



these must be supplied by some independent 
method. This Adams formula is of the open 
type and, therefore, not as accurate as a closed 
type formula of the same order would be. How- 
ever, it involves only one derivative evaluation 
per step and this, combined with the smaller 
truncation error, leads to a very fast, stable 
integration method for first order systems. 

The Adams method can be modified for 
second order systems. Thus, if the system to 

d 2 
be solved has the form y" = — £ = f(t, y, y'L 

dr 

the method consists of applying the formulas 



IN 

y n+ i = y' n + h I °k v k f 

k=0 n 

N 



) (23) 



k=0 



The first six values of a , are the same as those 
given above, while the first six values of j3^ are 
(1/2, 1/6, 1/8, 19/180, 3/32, 863/10080). 

In contrast to the straight use of differences 
as exemplified by the Adams method the Gauss - 
Jackson method makes use of a summation 
process. The formulas may be expressed in 
terms of differences or in terms of ordinates. 
In ordinate form, predicted values for y at time 
t = t are given by the equations 

n-1 



< = h2 (" f n + I C k f k) 

x k=0 ' 

H="(v.,4! dA ) 



(24) 



where the first sums 'f , ,„ and the second 
n-1 1 1 

sums "f are defined by the recurrence relations 



'f = f + 'f 

n-1/2 n-1 n-3/2 

"f = 'f + "f 

*n n-1/2 n-r 



(25) 



Using these predicted values, y , d/dt(y n >, and 

the attractions f may be computed from the 

equations. The following corrector formulas 
can then be used to obtain improved values for 

v d / dt <y n > 



^ 



y c n = h2 k + I c k f k 

\ k=l ' 



(26) 



The coefficients c, , d, , c,, d , depend upon 

the number of differences retained. For n = 11, 
the coefficients are given in Ref. 10. With a 
single precision machine, it is recommended 
that eight differences be retained in these for- 
mulas. The starting values as well as the first 
and second sums must be supplied by an in- 
dependent method. The difference between the 
predicted and corrected values can be used to 
decide whether to double or halve the step-size. 
A convenient method for starting or changing the 
step-size is the Runge -Kutta method, but, since 
this is a lower order method, several Runge - 
Kutta steps will have to be taken for each Gauss - 
Jackson step. 

The Gauss -Jackson second -sum method is 
strongly recommended for use in either Encke 
or Cowell programs. For comparable accuracy, 
it will allow step-sizes larger by factors of four 
or more than any of the fourth order methods. 
The overall savings in computing time will not 
be nearly so large, however, because per step 
computing time is somewhat greater and because 
the procedure for starting and changing the in- 
terval is quite expensive. As compared with 
unsummed methods of comparable accuracy, the 
Gauss -Jackson method has the very important 
advantage that roundoff error growth is inhibited. 
It can be shown that, in unsummed methods 

3 /2 
roundoff error growth is proportional to N , 

where N is the number of integration steps com- 

1 /2 
pared with N for summed methods. The 

Gauss -Jackson method is particularly suitable 
on orbits where infrequent changes in the step- 
size are necessary. Frequent changes in the 
step-size will result not only in increased com- 
puting time but in decreased accuracy as well. 

Finally mentioned is a higher order method, 
associated with the name of Obrechkoff, which 
makes use of higher derivatives. A two-point 
predictor -corrector version as applied to a first 
order system y 1 = f(t, y) makes use of the for- 
mulas 



y (p) =y 
y n+l y n-l 



2h 



K - 3y n-l) 4" K 



+ 7y n -i) ^K-K-x) 



. 13h vii , t , 
+ 63W y (l) 



>(27) 



(c) , 

y n + l=y n+ £ 



2 

(y n +i + y n ) -Jo- (y n +i - y'n) 



T2D 



(' 



n+1 



+ y 



s; 



h vn 

100, 800 y 



<£) 



where the higher order primes mean the higher 
order derivative of y with respect to t. The dis- 
advantage of this method is that the higher deriv- 
atives of the dependent variable must be available. 
Thus, to use these formulas, the first order sys- 
tem would have to be differentiated two times. 



IV- 10 



Moreover, as the force terms in the equations of 
motion change, these higher derivatives will also 
have to be changed. Thus, in spite of the favorable 
truncation error, this method cannot be recom- 
mended as a general purpose subroutine for space 
trajectory computations. However, the method 
appears clearly tailored to the; lunar trajectory 
problem (Kef. 11). 

4. Special Second Order Equations of the Form 
y^fdUyr 

The free -flight equations in the absence of 
thrust or drag forces can be written in the form 
y" = f(t, y) with missing first derivative terms. 
Some formulas which take advantage of this form 
have been proposed. The following special 
Runge-Kutta method, for example, requires only 
three derivative evaluations per step and, thus, 
results in a saving of about 25 percent over the 
standard Runge -Kutta formulas: 

k. = hf(t , v ) ^ 

1 n • n I 



k 2 =hf 



k„ = hfYt + h, y + hy' + } ± k„\ 
.5 \ n J n J n 2 2 / 



y n +i =y n +h 



[y n+ l/6(k 1 + 2k 2 )] 



(28) 



y n+1 =y n+ l/6(k 1 + 4k 2+ k 3 ). J 

A predictor -corrector method (due to Milne 
and Stormer) adapted to this form makes use of 
the formulas 

h 2 ^ 

y n+l = ^ + >n-2 " y n-A + X (5f n + 2f n-l 



, - , . , 1 I h 
+ ;,t n-2 ) + W 



y vl (l) 



- n-1 12 n+1 n 



n-1 



n vi , , 

MU y (ti) - 



(29) 



J 
These formulas appear to achieve a local trun- 
cation error of 0(h ) while retaining only four 

ordinatcs, compared with an 0(h' ) error for 
other fourth order methods. However, this 
advantage is illusory since the overall error is 

4 
still 0(h ) as in fourth order methods. In ad- 
dition ttiese formulas are somewhat unstable rel- 
ative to roundoff error propagation. In practice 
there appears to be little to recommend the Milne - 
Stormer method. 

The characteristics of these various integra- 
tion routines are summarized in Table 1. 

5. Evalu ation of Integration Methods 

The more important integration methods in 
general tisage will be evaluated below as they 
arc utilized with the various special perturbation 
formulations. 

a. Cowell method 

For the Cowell method, the choice of an in- 
tegrating routine is very important because of 
the greater danger of loss of significance due to 
roundoff error accumulation. The Gauss- 



TARLE 1 



Comparison Criteria 



Method of Numerical 
Integration 


Truncation 
Error 


Ease of 
Changing 
Step -Size 


Speed 


Stability 


Roundoff Error 

Accumulation 


Single Step Methods 












Runge -Kutta 


h 5 


* 


Slow 


Stable 


Satisfactory 


Runge-Kutta Gill 


h 5 


* 


Slow 


Stable 


Satisfactory 


Bowie 


h 3 


Trivial 
{step -size 
varied by 
error con- 
trol) 


Fast 


Stable 


Satisfactory 


Fourth Order Multistep 
Predictor-corrector 












M ilne 


h 5 


Excellent 


Very fast 


Unstable 


Poor 


Adam s-Moul ton 


h 5 


Excellent 


Very fast 


Unconditionally 
stable 


Satisfactory 


Higher Order Multistep 












Adams Backward 
Difference 


Arbitrary 


Good 


Very fast 


Moderately 

stable 


Satisfactory 


Gauss -Jackson** 


Arbitrary 


Awkward and 

expensive 


Fast 


Stable 


Excellent 


Obrechkoff 


h 7 


Excellent 


*** 


Stable 


Satisfactory 


Special Second Order 
Equations [y" = f(t, v)] 












Special. Runge -Kutta 


i, 5 


* 


Slow 


Stable 


Satisfactory 


Milne -Stormer 


h 6 


Excellenl 


Very fast 


Moderately 
stable 


Poor 



*R-K (single step} triv: 
*G;iuss -Jackson is for 
*Speed of Obrechkoff <j, 



Lai to change stepb, very difficult to determine proper f 

second order equations. 

epends on complexity of the higher order derivatives re 



IV-ll 



Jackson method of integration is recommended 
for Cowell programs because it allows larger 
step-sizes and because it inhibits roundoff error 
growth. 

b. Encke method 

For the Encke method, the choice of an in- 
tegration method is less important relative to 
accuracy. There is some advantage in computing 
time, however, in choosing a single step method 
which will allow frequent changes in step-size 
without the necessity of going through an expen- 
sive restart procedure. For lunar flights, it 
has been found that the Obrechkoff method is es- 
pecially useful in reducing computing time, but 
this method does not appear to be easily adaptable 
to other types of orbits or to other formulations. 
Although the Gauss -Jackson method is recom- 
mended in Encke programs, its advantages over 
other methods are not as great as in Cowell pro- 
grams. 

c. Variation-of -parameters method 

For variation-of -parameters methods, the 
Adams backward difference formulas are re' 
commended among higher order methods and the 
Adams -Moulton formulas among lower order 
methods. 

In general, multistep integration methods 
which allow for automatic adjustment of the size 
based on an error criterion are preferred. 

With any integration method, the process of 
double precision accumulation of the dependent 
variables should be used to prevent excessive 
roundoff error growth. 



Summary of Studies on Special Perturbation 
Methods 



In order to provide the mission analyst with a 
set of guide lines in determining the best integra- 
tion methods for various special perturbation 
methods used in computing precise satellite tra- 
jectories, it is useful to examine the results ob- 
tained by others in the industry. This section is 
intended to show the interrelation of the mission, 
formulation of the problem, and method of inte- 
gration so that the most efficient, accurate, and 
economical balance is achieved. Several serious 
questions, which must be carefully considered 
by the mission analyst, are raised in connection 
with the balance between the type of orbit and the 
scheme of integration. 

a. Aeronutronic report (Refs. 12 and 13) 

The Cowell, Encke and Herrick methods are 
compared for the following problems: a selenoidal 
satellite which is physically unstable, but for 
which an analytic solution is known; a low thrust 
trajectory; a high thrust trajectory and a ballistic 
lunar trajectory. In all cases the integration is 
carried out with a Runge-Kutta method with 
variable step-size adjustment. Their conclusions 
are: 



(1) For the Cowell method, the effect of 
roundoff error is felt very quickly -- 
within a few hundred steps. 

(2) Overall, the Encke and Herrick 
methods are computationally more 
efficient than the Cowell method. 

(3) On ballistic lunar trajectories, the 
Encke method is best. The Cowell 
method requires almost ten times as 
many integrating steps as the Encke 
method and three times as many as 
the Herrick method. 

(4) On continuous low thrust trajectories, 
the Herrick method is superior. 

(5) On trajectories where high thrust 
corrective maneuvers are introduced, 
the Cowell method is superior. 

Although the trend of the conclusions in this 
study is probably correct, there are serious 
questions as to the validity of the conclusions on 
the degree of superiority of the perturbation 
methods. For one thing the method of integra- 
tion (Runge-Kutta) favors the perturbation meth- 
od. For the Cowell method, the choice of in- 
tegration method is much more important, as 
indicated earlier. Experience has shown that 
roundoff error effects are not nearly so critical 
as concluded here. Both the use of the Gauss - 
Jackson integration method and double precision 
accumulation make roundoff error much less 
serious for the Cowell method than indicated 
here. The evidence presented, moreover, is 
not conclusive relative to accuracy. The nu- 
merical results, for example, are not given at 
corresponding times, and no accurate standard 
for comparison is available except for the un- 
stable selenoidal satellite. The selenoidal satel- 
lite is by no means typical of the earth satellite 
problems and any generalizations of results 
based on a study of this orbit must certainly be 
viewed with skepticism. 

b. Republic Aviation report (Ref. 14) 

The orbit selected is that of a vehicle moving 
in the gravitational field of two fixed centers. 
An analytic solution in terms of elliptic functions 
is available for this orbit so that an accurate 
standard is thus available. This study compares 
the Encke, Cowell and Herrick methods with two 
different integration routines: a fourth order 
Runge-Kutta method and a sixth order Adams 
method. The conclusions of this study are: 

(1) The Encke method was superior to the 
others in both accuracy and machine 
time. For an integration over a 100- 
hr period the Encke method required 
0. 5 min, the Herrick method 2. 5 min 
and the Cowell method 3.5 min. All 
of those programs used the same in- 
tegration method and the results were 
comparable as to accuracy. 

(2) The Herrick method is superior to the 
Cowell method relative to attainable 



IV-12 



accuracy and slightly better relative 
to computing time. 

(3) An integral of the motion, such as 
the energy integral or a component 
of the angular momentum, is a poor 
positive test of accuracy. 

(4) The Adams method is considerably 
faster than the Runge-Kutta method 
by a factor of almost three. 

(5) Double precision accumulation is 
very effective in reducing errors 
due to roundoff. 

(6) The largest error in the Encke and 
Herrick methods arises from errors 
in solving the two -body formulas, 
particularly as such errors affect the 
mean anomaly calculation. 

The conclusions of this study appear to be 
well grounded. The only serious consideration 
is that the orbit selected is quite specialized 
and that no strong perturbations such as those 
due to oblateness or thrust are considered. Thus 
the extent to which these results can be assumed 
typical for satellite orbits is in some doubt. 

c. Experiments at STL 

The relative efficiency of the special per- 
turbation methods is a function of (1) the type of 
orbit and (2) the method of integration. A given 
integration subroutine may favor one of the 
methods over another, so that the use of the 
same subroutine for all methods does not con- 
stitute a fair test. 

In general there appears to be no doubt that 
the Encke method is computationally the most 
efficient on ballistic lunar trajectories. For 
comparable accuracy, however, the advantage 
in computing time is probably on the order of 
two or three, rather than ten as is sometimes 
quoted, when any of the standard integration 
subroutines are used. 

There is no doubt that the Cowell method 
requires much greater care to ensure that 
roundoff errors do not become a serious factor 
in the accuracy. However, effective methods 
are available to curb roundoff error growth. 
When these are used, the Cowell method is still 
a very useful tool for many space computations. 

None of the orbits considered in the reports 
by Aeronutronic and Republic Aviation appear 
to be applicable to the earth satellite problem in 
which a small but significant force, such as that 
of oblateness, is continuously applied. 

To obtain information about the comparative 
performance of these special perturbation meth- 
ods on earth satellite orbits, a numerical study 
was recently completed at STL. An idealized 
orbit was selected for the study with initial ele- 
ments: 

a = 1.5 earth radii 



0.2 




45° 




co = M Q = 





155 m in 




800 mi 




3200 mi 





e 

i 

Q 

period of the un- 
perturbed orbit 

perigee distance 

apogee distance 



The only perturbation force considered was that 
due to the second harmonic in the earth's gra- 
vitational potential (J,)- An accurate standard 

against which to check the programs was pro- 
vided by a double precision Cowell program. 
The double precision program yielded results 
on the unperturbed orbit (J„ = 0) which agreed 

with the known analytic solution to a few digits 
in the eighth significant figure. For the per- 
turbed orbit, the results provided by the standard 
are correct to at least seven significant figures. 

Single precision floating point programs for 
the Cowell, Encke and Herrick methods were run 
on an IBM 7090 and compared with the double 
precision standard. Great care was used to en- 
sure that all physical constants and initial con- 
ditions were identical in all programs. The in- 
tegration was performed over 64 revolutions 
with output at 20-min intervals. Table 2 gives 
the method of integration used, the local trunca- 
tion error criterion, the number of integration 
steps required, the computing time for 64 revo- 
lutions, and the maximum error in the distance 
Ar over the 64 revolutions. For each method 
several runs were made with successively 
tighter error criteria, and the most accurate of 
these was selected for the comparison. While 
the Cowell method required almost twice as 
many integrating steps, overall computing time 
was only slightly greater than the Encke method 
and, moreover, the accuracy was somewhat bet- 
ter. The Herrick method gave the best accuracy. 
The relatively large computing time required 
by the Herrick method is partially accounted for 
by the fact that the Adams -Moulton formulas 
(fourth order) are of lower order than the Gauss - 
Jackson formulas (sixth order). Since the latter 
will allow integrating steps perhaps twice as 
large for the same accuracy, the adjusted com- 
puted time would be comparable to that for the 
Cowell method. 

A more detailed comparison of achievable 
accuracy is contained in Table 3 where the maxi- 
mum errors in the distance r, the mean anomaly 
M, the semimajor axis a, and energy integral E 
are given on the 20th, 40th and 64th revolutions. 
It is clear that the Herrick method consistently 
yields the most accurate results and the Encke 
method yields the worst results. For all meth- 
ods, there is a strong correlation between mean 
anomaly errors and position errors, indicating 
that the error is largely along the path of the 
motion. This conclusion also follows from the 
energy integral errors which are seen to be rela- 
tively constant and much smaller than the position 
errors. It may also be concluded that the con- 
stancy of the energy integral is a poor positive 
test of accuracy in the position coordinates. The 



IV-13 



TABLE 2 
Numerical Results --Special Perturbation Methods 



Formulation 


Method of 
Integration 


Error 
Criterion 


Number of 
Steps 


Computing 
Time 
(min) 


Maximum 
(ft) 


Ar 


Cowell 


Gauss -Jackson 


IxlO" 10 


10, 200 


5.75 


800 




En eke 


Gauss -Jackson 


7X10" 10 


6395 


5.31 


1700 




Herrick 


Adams -Moulton 


5X10" 10 


7000 


11.45 


400 





TABLE 3 
Maximum Error --Special Perturbation Methods 



Method 


Cowell 


Encke 


Herrick 


Revolution 


20 


40 


64 


20 


40 


64 


20 


40 


64 


Ar x 10 6 
(er) 


1.2 


2. 2 


4.0 


2. 2 


6 


8.4 


0.2 


0.8 


2 


AM x 10 3 
(cleg) 


0.3 


0.6 


1 


1 


2 


2.7 


0.1 


0.2 


0.6 


Aa x 10 7 
(er) 


1.6 


1.4 


1 


3 


3.5 


3 


2. 2 


2.2 


2. 2 


AE x 10 9 


1 


1 


1 


4 


6 


9 


2 


2 


2 


I min / 





















error in the semimajor axis is also seen to be 
smaller than the position errors, indicating that 
the geometry of the orbit is much more accurately 
determined than position in the orbit. 

Although these results show that the Herrick 
method yields the most accurate results and the 
Encke method takes the least computing time, the 
order of magnitude of the difference is not suffi- 
cient to lead to a clear preference for any one 
method. Some improvement in the Encke and 
Herrick results could probably be obtained by 
even more careful analysis of the two-body 
formula computations. The Encke method, for 
example, is quite sensitive to the frequency of 
rectification and some improvement might be 
obtained by experimenting with rectification. 
There appears to be little reason to prefer 
either the Encke or the Herrick methods on 
earth satellite orbits of moderate eccentricity 
particularly, since they are considerably more 
complicated and require much more careful 
numerical analysis. In addition, special difficul- 
ties will arise in limiting type orbits (low eccen- 
tricity, high eccentricity, critical inclination) 
which do not arise when the Cowell method is 
used. 



D. GENERAL PERTURBATIONS 

Chapter III presented the discussion of motion 
about point mass (or a spherically symmetric 
mass). Although that discussion is revealing, it 
does not in general constitute a solution to the 
problem because the assumptions utilized prevent 
the solution from behaving as it should for the 
true gravitational field. In the preceding sections 
of this chapter, discussions have been presented 
which circumvent these limitations; however, 
in the process much generality has been lost since 
nothing can be said for trajectories beyond the 
neighborhood of the numerically obtained trajec- 
tory and nothing can be said about the long-term 
behavior of the orbit. (Before proceeding, it 
must be added in defense of numerical integra- 
tion that the solutions thus obtained are valid to a 
very high order of approximation. ) For these 
reasons it is desired that analytic expressions be 
presented which can be utilized to describe the 
motion of a satellite to varying orders of approxi- 
mation. The approach taken here will be first to 
discuss the variation of the orbital elements and 
secondly, the first order secular or cumulative 
perturbations which can be added as linear func- 
tions of time or as discrete corrections to the two- 
body solution to improve the fit of the resulting 
motion. Then as a third step, the various general 
perturbation theories (i.e., approximate analytic 



IV-14 



solutions for the perturbed motion obtained by 
series expansion) which present second order 
secular and periodic effects will be discussed. 
The advantages and disadvantages of this ap- 
proach are summarized at this point. 

Advantages of general perturbation methods 



(1) They are very fast both because no 
step-by-step integration is necessary 
to obtain the elements at a given time 
and because the computing time per 
point is very small (on the order of 

1 sec per point on an IBM 704). 

(2) The accuracy of the computation is 
limited only by the order to which the 
expansion is carried out, and not by 
the accumulation of roundoff and trun- 
cation errors. 

(3) They can maintain reasonable accuracy 
over many hundreds of revolutions. 

(4) They allow for a clearer interpreta- 
tion of the sources of the perturba- 
tion forces and the qualitative nature 
of an orbit. 



Disadvantages of general perturbation meth- 
ods are: 

(1) Nonoonservative forces, such as drag, 
are not easily included in the theory. 
No simple and adequate theory has yet 
been prepared which includes such 
forces in a form suitable for numerical 
computation. 

(2) The effect of other forces, such as 
luni -solar perturbations and radiation 
pressure, are difficult to incorporate 
since they involve substantial amounts 
of new analysis and checkout. 

(3) '['he series expansions are very com- 
plicated, and programs based upon 
them are complicated to write and 
difficult to check out even for a first 
order theory. 

(4) There is a serious degradation in ac- 
curacy for special types of orbits in- 
cluding the important case of nearly 
circular orbits (e - 0), highly ellipti- 
cal orbits (e - 1) and orbits near the 
critical inclination (i - 63.4°). 

(5) Although agreement with observations 
does confirm practical convergence, 
no mathematical proof of convergence 
has yet been given for any of the 
general perturbation methods, nor are 
any estimates of the error in the trun- 
cated series available. 



Finally, these discussions will be followed by 
those of atmospheric effects and extra-ter- 
restrial effects. 



1. Rat es of Chan ge of Satellite Orbital Elements 
Cau sed by a Pertur bing Force (HefT 15) 

The instantaneous rates of change of satellite 
orbital elements caused by a perturbing force, 
as given, for example, by Moulton (Ref. 1, pp 
404 and 405) are derived from astronomical 
perturbation theory involving tedious mathemati- 
cal transformations. The purpose of this de- 
velopment is to give a simplified derivation of 
the same equations by using only elementary 
principles of mechanics. It is hoped that this 
approach will make the equations more meaning- 
ful and the discussions which follow later in the 
chapter more readily appreciated. 

Consider a satellite of mass m moving in the 
inverse square force field of the earth. Its or- 
bit is a Kepler ellipse (Ref. 1, Chapter V) 
specified by the following orbital elements a, e, 
h, w, i and M n (see following sketch). The 

location of the satellite in its orbit is given by 
the angular position 4> which is measured in the 
orbital plane from the node. The angular dis- 
tance of the satellite from perigee is called Un- 
true anomaly, 9. Therefore, 



+ 



(30) 



The radial ciistance, r, from the center of the 
earth to the satellite is given by 



P 



1 + e cos 



(31) 



The satellite's energy per unit mass, c, and its 
angular momentum per unit mass, h, are related 
to the orbital elements by the equations 



and 



2a 



r 2 



u p = na 



f[ 



(32) 



(3 3) 



where: u = CM (the product of the gravitational 
constant and the earth's mass) and a dot over a 
quantity indicates a time rate and 



(34) 



Now suppose that a perturbing force K acts on 
the satellite. The orbit will no longer be a Kepler 
ellipse, but at every instant we can associate an 
"instantaneous osculating ellipse" with the new- 
orbit by choosing the Kepler orbit corresponding 
to the instantaneous radius and velocity vectors 

of the satellite and to the potential energy, - ti , 

of the satellite in the gravitational field of the 
spherical earth. This is the orbit the satellite 
would follow if the perturbing force were re- 
moved at that instant. The true orbit can thus 
be specified completely by a series of elements 
of the instantaneous osculating ellipse. There- 
fore, the set of differential equations which shows 
how these elements change with time is equivalent 



IV-15 



Z-axis 




\ Y-axis 



to the Newton or LaGrange set involving the co- 
ordinates and their rate of change with time. 
With this discussion as background, the rates of 
change of the orbital elements a, e, n, u and i 
will now be derived. 

Following Moulton (Ref. 1, p 402), the per- 
turbing acceleration, F'/m, may be resolved into 
a component R along the radius vector (meas- 
ured positive away from the center of the earth), 
a transverse component S in the instantaneous 
plane of the orbit (measured positive when 
making an angle less than 90 deg with the velocity 
vector V), and a component W normal to the in- 
stantaneous plane (measured positive when 
making an angle less than 90 deg with the north 
pole or z -axis). 

Let the unit vectors along the three direc- 
tions be denoted by n ,, n and n w . That is, 



dt m 



(37) 



/here V is the instantaneous velocity vector, 



= fn r + r6n v = e ^n>rn g ). 



(38) 



Now from the definition of the instantaneous os- 
culating ellipse, it is clear that its velocity 
vector is the same as the instantaneous velocity 

vector of the actual orbit. Therefore 6 and ^ 

in Eq (38) may be evaluated from Kqs (31) and 
(33) to obtain 



V = 



{l^J 



/re sin - + ^ \ (3 
V 1 + e cos r s/ 



9) 



m (Rn , + Sn + Wn 



(35) 



To find the rate of change of the semimajor 
axis, a, refer to Eq (32) for the relationship to 
the energy 



da 
dT 



2a 2 dc_ 

u at 



(36) 



Forming the dot product with F/m and substi- 
tuting the resulting expression for ^- in Eq (36) 

yields 



da 

dt 



2e sin 



nVl 



2 



la Vl -e 2 



(40) 



The energy change (per unit mass) may be found 
from the definition of the work done on the satel- 
lite by the perturbing force. 



da 



which is the expression given for- -yp by Moulton 
(lief. 16). 



IV-16 



To derive the changes in the other orbital 
elements, it is necessary to know thc^rate at 
which the angular momentum vector h (per unit 
mass) changes. This rate of change of h is then 
known to be equal to the summation of the ex- 
ternal moments acting on the satellite. 



dh 
dt 



1 



(r x F) 



rn x (Rn + Sn + Wn ) 

r r s w 



rSn - rWn 

w s 



(41) 



The rate of change of h can also be written as 

(42) 



dh dh -» , . da -» 
dt dt w dt s 



where da is the angle through which the angular 
momentum vector is rotated in time dt. There- 
fore, 



dh 

-rr- = rS 

dt 



(43) 



Now, the eccentricity of the orbit may be ex- 
pressed in terms of a and h through Eqs (33) and 
(34) which yield 



hi 

Ida 



1/2 



(1 - p/a) 



1/2 



By differentiating, the following is obtained 

h L dh __ h da\ 
2uae l dT a 3T J 

Vl -e 2 /„ dh J] 2 da\ 



de 



2na e 



(45) 



dh 



Upon substituting Eqs (40) and (43) for -rr- and 

da 

-rr- , Eq (45) takes the final form, 



de 

ar 



'1 - e sin 9 



R + 



fl 



2 
na e 



a 2 (l 



e ) 



S. 
(46) 



and 



da 
dT 



rW 
h 



(44) 



The motion of the node is the same as the 
motion of the projection of FT' on the equatorial 
plane (see the following sketch). Let the sub- 
script p denote the projection of any vector on 
the equatorial plane. Then it can be seen that 




Y-axis 



(*)'(£-) 



Node 



X-axis 



IV-17 



h = projection of h on the equatorial 
p plane. 

-> -» 

2£- ) = projection of -rr- on the equatorial 
'p plane. 



( h ar%) = " rW [><- c ° s <> 



cos 1 sin P 



- sin <j) cos p) +j (-sin 4> sin P 
+ cos <)> coo i cos P)J- 



ft) * ( 



g|- y = the component 
P 



-( 



dh\ 
3*7 p 



which is normal to h . 



Thus, upon performing the cross product, Eq 
(47) becomes 



dP 
dt 



rW sin $ 



4V^J 



(48) 



dP 



YT 



(&) 



v (» £ »«,) 



(47) 



The change in the orbital inclination is re- 
lated to the change in the node. This can be 
seen by referring to the following sketch in which 
two positions of the node, P,, and P., are shown 
with 



Afi = P 1 - P Q 



and 



Ai 



h ~ V 



But 



h = h sin i (i sin P - j cos P) 



where i and j are unit vectors along the X- and 



Y-axes, respectively, and 



By spherical trigonometry, it can be shown that 
sin Ai = sin i. cos i„ - sin i„ cos i. 



1,, 

: — [cos i.-. sin 4>_ (1 - cos AP) 



:os <t>„ sin Ap] . 




Y-axis 



X-axis 



IV-18 



Differentiating and taking the limit as AS2 
>0, the following is obtained 



di sin i , d£l 

tt - —■ — r cos A 
at sin 6 



dt - 



Therefore, 
di 



rW cos 



, 2 VT 



(49) 



(50) 



The change in the argument of perigee, to, 
arises from two sources. One is the motion of 
perigee caused by the forces in the orbital plane 
tending to rotate the ellipse in its plane. The 
other change occurs because cj is measured 
from the moving node (see preceding sketch). 
To evaluate the latter changes, assume that the 
in-plane perturbing forces are zero. Then the 
change in u equals the change in $. According 
to the relations in a spherical triangle, 



cos An cos 



>„ + sin Afi sin 4>„ cos i„. 



Differentiating and taking the limit as An -> 0, 
yields 



d<j) _ /du\ 
dT " \-3T) 



W 



. dn 

cosl ar 



-r sin 4> cot i 



VT 



w, 



(51) 



where the subscript W means that this is the 
change in oo contributed by the nodal motion 
which is caused by the component of the per- 
turbing acceleration, W, normal to the orbital 
plane. The change caused by the in -plane com- 



The 



R, S 



ponents, R and S, is denoted by I pre- ) 

effect of these in -plane forces is to change the 
instantaneous velocity vector which must, at 
every instant, remain tangent to the instantaneous 
osculating ellipse. This ellipse will therefore 
have a changing perigee position. The resulting 
rate of change of the argument of perigee will 
clearly be 



(5?) 



R, S 



dO 

ar- 



(52) 



d9 



Here -rr- , the rate of change of the true anomaly 

caused by the perturbing force, must not be con- 
fused with 5 which is the rate of change of 9 in 

H 9 
an unperturbed Kepler orbit. To evaluate -tt- , 

refer to the following sketch. 

After the force m (Rn +- Sn ) has been ap- 
plied for the time dt, the velocity vector is 
changed from V* to V*+ dv, the true anomaly 
from 8 to 9 + d9 and the angle v, between n 

-> s 

and V, is changed from y to y + dv. The ex- 
pression for y is obtained from the angular 
momentum, 



h = rV cos y. 



Since h 
that 



2 .22 

r 8 and V = (r + r 



cos y 



1 + 



3" W 



1/2 



-1/2 



it follows 



dr 



Computing -rg from Eq (31) yields 



(Rn + Sn ) 
r s 




Old perigee 
New perigee 



IV-19 



and 



cos V 



sin "V 



1 + e cos 6 



Vl + e 2 + 2 



e cos 6 



e sin 9 



Vl + e + 2e cos 9 



(53) 



(54) 



Differentiating Eq (54) with respect to time and 
using Eq (52), it is found that 



(£) 



R, s L 



1+e + 2ecos9 
e (e+ cos 9) 



sin 9 



de 
dt 



d\ 

"at 



(55) 



V 1 + e +2ecos9 
If N is the component of the force normal to V, 
Ndt 



dv = 



~v — 



But 



and 



N = R cos y - S sin y, 





V = 


h Vl + e 2 


+ 2e 


cos 


e 




r 1 + 


e cos 9 




Th 


erefore 












dt 


| r(l + e 


cos 


6) 






)h(l + e 2 + 


2e cos 


0) 



• [R(l + e cos 6) - (e sinG)s]j" (56) 

de 
Equation (56), along with Eq (46) for -ry-, yields 



lar) r, 



VTT 



[- (cos e) r 



S nae 

(57) 

+ sin 9(1+ ,— - — - B ) si 

1+e cos 9 J 



The total rate of change of the argument of 
perigee is 



dw _ /duA ,/du\ 

dt \dt/W \dWR,: 



(58) 



The final element, mean anomaly at epoch, 
which provides the position of the satellite at 
any time also has a time rate. This relation- 
ship is obtained directly from Kepler's equation 

a = M„ = E - e sin E - nt 



IV-20 



and can be found by using the equations already 

obtained for -nr and -n- , with the relationship 
dt dt ^ 

between E and given by 
„ cos E - e 



1 - e c 


os E 

sin E 
:os E 




Vi 2 
sin 9 = y ] - e 
1 - e < 




The result is 




da __ 1 /2r _ 1 - e 2 
dT na \ a e 


cos ej r 


. d - e 2 ) 
nae 


1 + 
L a( 


rr (sin 9) S 

1 - e^)J 



- t 



dn 

ar 



(59) 



Note is made at this point that the last term has 
been omitted in Moulton, Ref. 1, p 405. 

This completes the set of equations for the 
orbital elements. The remaining 5 are sum- 
marized below for reference: 



da 

ar 



2e sin 9 n x 2a 11 - e 
rt + 



nVl - 



de 
dt 



_V^ 



e sin 9 



na 



,V7T7" 



a 2 (l 



e ) 



dn 

"ar 



di_ 
dt 



dqj 

dt 



r sin <fy 



W 



VT 



r cos <j) 



VT 



w 



r sin <b cot i 



2 J7 2~ 

1 1 1 - e 



- W - 



V7 



2 

e cos 



'L_L±_ fl+T-r ^sinOS 

nae \ 1 + e cos / _, 



(60) 

If at this point we introduce a disturbing 
function rather than the four components, we can 
put these equations in the Lagrangian form 



R 



3F 



o _ 1 9" 
b = F 7f 



(61) 



W = 



r sin <(> 7T 



J 



da = _2_ dj~._ 
dT na dcr~ 



\ 



de 

dt 



dt£ 
dt 



dM 
dt 



di_ 
dt 



dS2 



tl - e' 
na e 



M - e — 

dcr 



d" 

duj J 



V," 2~ 

Tl - e d~ 



cot 1 



~ 2~~ 

na e 



de 



2 »n 2~ 

na Tl - e 



d~ 
Ji 



2 d~ _ 1 
na da 



a- 



>(62) 



+ n 



: v7 



e sin i 



2 VT 

na Tl 



3" 
"BT 



e sin l 



J 



2. First Order Secular Perturbations 



For an oblate body having axial symmetry, 
the gravitational potential at any extension point 
may be represented by Vinti's potential 
(Chapter II). If for the present analysis we 
neglect terms with coefficients the order of 

J J (i.e., J, 



3' 



J .... ) we can write the work 

4 



function (minus the potential) as: 

2 



U 



but since 
1 1 



r 



1 + 



1 + 



''4 (?) 



(3 sin L 



/R\ 2 ,, . 2. . 2 
I — J (3 sin i sin 



1) 
(63) 

<f -1) 



+ u is a periodic quantity, sin if 



1 



2 2 cos 2<J> has a nonperiodic part •=•. 

Thus, the potential ,1 will produce secular changes 
in the orbital elements as well as periodic 
changes. Before the magnitude of this change 
can be evaluated, however, the constant part of 

the function (a/r)' must be evaluated. Following 
the method of Dr. Krause (Ref. 16) we have: 



(f)' 



'0 

~2~ + C ; . cos M + C„ cos 2M + 



. + c cos n M 
n 



where 



1 f 2rr a n 
- n = T ] Q < F > cos n M d M 



The C are simple functions of the eccentricity 
as may be seen in the expansions of Chapter III. 



Thus, 



^0 
T 



1 



,2tt 



(p ) dM 



2tt 



~ (1 - e 2 )" 3/2 K (1 + e cosO) dO 



2tt 



= (1 -e 2 ) 3 ' 2 



and 



secular 



secular 

r r2 

'2 ~TT 
a 



(1 - o-) 



-3/2 



(1 - 3/2 sin - i) 



(6 4) 



(65) 



At this point we refer to the Lagrangian 
equations of Section I)-l of this chapter and con- 
clude that the secular variations in the elements 
are expressible to the first order in J as: 



A a = 
A e = 



(66) 
(67) 



A co = 3tt J 2 n±\ (2-5/2 sin 2 i) (rad/rev) 

(6 8) 

3 ° ' 



A M 
nt 



2" J 2 



(rad/rev) 



e 2 (1 - 3/2 sin 2 i) 



A i = 

A n = -3t J, 



(6 9) 



(70) 



(«)' 



cos i (rad/rev) (71) 



The physical significance for the fact that the 
secular variations in a, e and i are zero may be 
seen by looking at the potential function itself. 
The fact that J„, J, and J. are small implies that 

to a first approximation the orbit will be nearly 
elliptical. Although one cannot assign an un- 
ambiguous major axis or eccentricity to the per- 
turbed satellite orbit, the experience of astrono- 
mers has shown that it is convenient to refer the 
motion to an osculating ellipse. This is the 
orbit in which the satellite would move if at some 
instant the perturbing terms were to vanisli (J„ = 

J, = J, = 0) leaving the satellite under the at- 
traction of the "spherical" earth. Hence the 



IV-21 



actual position and velocity vector at each point 
define the osculating ellipse in terms of a set 
of elements a, e, and i, where a and e are the 
semimajor axis and eccentricity and i is the 
inclination of the plane of the ellipse to the 
equator. 

The major axis a may be specified in terms 
of the energy E, associated with the osculating 
ellipse. When J„, J„ and J. are set equal to 

zero to calculate E, only the potential energy is 
altered, and it can be seen that unless r exhibits 
a secular (nonperiodic) variation, which is not 
possible here since we are dealing with bound 
orbits, only periodic variations in E can occur. 
Hence there can be only periodic variations in a. 

2 
Although p, i. e. , a (1 - c ), is a constant of 

the motion, the total angular momentum h is not 

constant, because the equatorial bulge produces 

a nonradial component of force. But by the same 

arguments as above, the torque, and hence h, 

can exhibit only periodic variations. Further, 

since at each equatorial crossing the momenta 

are related by 



(h cos i) 



The perturbed anomalistic period can be 
evaluated from the average angular rate using 
the method of Kozai (Ref. 18) and a relation 

2 3 

analogous to n a = u. 



-2-3 - L 3 T /R\ ,. 

n a = »=» {1 -^ J^-J (1 



3 . 2 .. J~. 2 

2 sin l) T 1 - e 



where 



n = perturbed mean angular rate 
a = mean value of the semimajor axis 

2 



a jl - 3/2 J 



.(?) 



(1 - 3/2 sin i) 



VT 



N 



constant. 



iu = effective gravitational constant as 
sensed by the satellite in its orbit. 

This process yields 



where N means node, it follows that the orbit 
inclination i behaves similarly. The same may 
be said for the orbit eccentricity, since the 
equation for eccentricity depends explicitly only 
on I hi and a. 

It is noted at this point that since 3 of the 6 
elements vary, the satellite periods will vary. 
The plural of period was intentionally utilized 
at this point because of the manner in which three 
distinct periods are defined (Ref. 17). 

Anomalistic period is defined as the time 
from one perigee to the next. In that time the 
elliptic angles (true, mean, and eccentric 
anomaly) increase by 360°, while the central 
angle ji increases by more or less than 360°, 
depending on whether the apsidal notation is 
against or in the direction of satellite motion. 

Nodal period, also called synodic or draconic 
period, is defined as the time from one ascending 
node to the next. In that time the central angle ji 
increases by 360°, since $ is measured from the 
instantaneous position of the ascending node. 
The satellite does not, except at an orbit in- 
clination of 90°, return to the same relative 
position in inertial space after one nodal period 
due to the regression of the nodes. 

Sidereal period is defined as the time for the 
satellite to return to the same relative position 
in inertial space. In that time the satellite 
central angle as measured from a fixed reference, 
which is not to be confused with the central angle 
as measured from the ascending node, increases 
by 360°. In artificial satellite theory, the sidereal 
period is less important than the other two periods, 
it is rarely used, and it will not be discussed any 
further. 



2?r 



2tt ,-*3/2 
— (a) 



3 J 2 R e 
a 2 (l - e") 



2.3/1" 



3 cos i_ -1 



'(72) 



For a near -polar orbit the anomalistic period 
is longer than the unperturbed period, while for 
a near -equatorial orbit the anomalistic period 
is shorter. At inclination angles of i„ - 54. 7° 

and in - 125.3°, 3 cos \q = 1, and hence the 

anomalistic period equals the unperturbed period. 
Physically this is due to a combination of the 
mass distribution of the earth and the apsidal 
rotation at these inclination angles. 

The perturbed nodal period, however, has 
been subject to much more confusion since the 
results of many of the authors are in conflict. 
Upon review of this work, however, it is felt 
that to the order J„ the results of King Hele 

(Ref. 19) and Struble (Ref. 20) are the most pre- 
ferable for small eccentricities. (Additional 
discussions and proofs appear in Ref. 17.) This 
result is: 



T = 27T 

n 



#~!> - ».(*.) 



7 cos i - 1 



(73) 



These two period expressions (Eqs 72 and 73) may 
be seen to differ in both magnitude and in the 
algebraic sign of the corrective term. This 



IV-22 



apparent discrepancy is due to the fact that the 
perigee is moving. Thus at the time the perigee 
has rotated through 360° the number of nodal and 
anomalistic periods should differ by 1. 

Equations (68), (69), (71), (72) and (73) are 
presented in graphical form as Figs. 2, 3, 4, 
5 and 6, respectively. 

3. Higher Order Oblateness Perturbation 

The errors inherent in numerical integration 
are not conducive to accurate computation of 
orbits over long time intervals. For this rea- 
son, general perturbations (analytic approxi- 
mate solutions for the perturbed motion obtained 
by series expansions) are more useful in mis- 
sions of long duration. 

a. Oblateness of the earth 

The potential function of the earth can be 
accurately expressed as an infinite series of 
zonal harmonics. 



U = -=- 



L 

k = 2 



©' 



P R (sin L) 



where P, (sin L) is the Legendre polynomial 
of order k, given by 

p k (x) - "IT- ^k (x2 " 1)k 
K 2 k k : dx k 

Phis is the form of the potential function given 
by Vinti. The recommended values of the co- 
efficients J. and several expansions are given 

in Chapter II. The potential function determines 
the motion of a small body in the earth's field by 



au 

Jx 



>y, z. 



The classic approach of the general perturbations 
method is the analytic integration of one of the 
sets of equations for variation of parameters, 
i.e., a set similar to that of Section C-l (this 
chapter) with the perturbing function H defined by 



(1) Libration, min. q. < q. < max q. 

i — n — n 

(i = 1, 2, 3). 

(2) Circulation, -„ < q. < „ . 

These two possible regions are shown in the 
following sketch. 



Libration 
region 



Circulation 
region 




Element 
value 



In the neighborhood of the so-called critical in- 
clination, the elements which become in- 
determinant merely leave the circulation region 
and enter the libration region. Since the theory 
isn't prepared to handle points of this type along 
with the more regular points, it ceases to apply 
in this region. This behavior is no reflection 
on the theory in general, since other approaches 
can be utilized in these neighborhoods. 

In the latter cases (i. e. , e = or i = 0) the 
problem is one of indeterminacy in one or more 
of the elements being utilized to describe the 
motion. More specifically, the angle u cannot 
be utilized for e = because of the fact that the 
line of apsides cannot be located. Similarly, the 
nodal angle Q becomes meaningless if the plane 
of motion is the primary plane of reference. 
Special sets of elements have been developed 
however, which may be utilized effectively for 
very low eccentricity orbit. These sets will not 
be discussed. 

One set of solutions obtained using this 
method including J„ and J , terms in secular 

perturbations, J„ to J r terms in long period 

perturbations and J„ terms in short period 

perturbations, is presented below. This form 
is exactly analogous to those referenced pre- 
viously; however, there are differences in the 
notation and in the coefficients 



Secular terms 



This approach has been taken by several 
authors [Brouwer (Ref. 21), Kozai (Ref. 18), 
Garfinkel (Ref. 22), Izsak (Ref. 23) and Krause 
(Ref. 16) to name a few] . The method results 
in easily visualized perturbations since the 
variables are geometric quantities. However, 
because of a failing peculiar to the method of 
analysis, the equations exhibit singularities in 
certain elements in the vicinity of the "critical 
inclination," i.e., i = 6 3.4° and for i = or 
e = 0. In the first case a physical explanation 
exists in that since the momenta of the canonical 
equations are bounded, the system is conditionally 
periodic. This situation admits 2 possibilities: 



M 



1 + | jj 






a o * a o 

■ <-i + 3cos 2 i > ^^^yr^f 

.[l0 + 16 Vl - e 2 - 25 e 2 



2 + (-60 - 96 yi - el: 



+ 90 e 2 ) cos 2 i + (130 + 144 \l - 



continued 



IV-23 



25eJ)cos 4 i ]- 1 f J^/f 7 ^ 



M t = 



ii J 2© 2(1 - e o 2 > 3/2 ^- llcOB21 '- 



30 cos 2 i Q + 35 cos 4 i Q )i + M Q (74) 



V?'tf '.(£)-'* 



s a Q Ta Q j4 2 



2 . , 

COS 1„ ) 



40 cos i Q Y 5 J 4 /R_Y - - 2 

, , 2 . r 16 J_ IpJ 
1 - 5 cos i/ 2 



fe) (1 ' e o) 



3/2 



x 



4 
8 cos i 

1-3 cos i n — 

u 1-5 cos" 1 i r 



sin 2 a, 



1 - ep + 126 e Q ) cos i Q 

+ (430 + 360 Vl - e 2 - 45 e 2 ) cos 4 i Q j 

(i) 4 [l2 + 9e 2 + (-144 

os 4 ljj 



128 J 4 



- 126 e 2 ) cos 2 i Q + (196 + 189 e 2 ) c 



J (1 - e ) 3/2 

+ -i ^4 — — sin i„ cos u 

2 J„ p„ e„ 0s 



*l = Y T2 J 2© 



2 ^0 

2 

] 





(79) 



2 + e 2 - 11 (2 + 3 e 2 ) cos 2 i Q 



4 . 
cos 1 

40 (2 + 5 e^) 



, ,,„ 2 6 . 

400 e„ cos i n 



1 c 2 . / 9 \2 

1 - 5 cos i Q ^ _ 5 cQS 2 



r, , 2 ■ V 

11-5 cos i 0/ / 



= J* '^\2 
5 4 

32 J r 






2 2 2 

2 + e Q - 3 (2 + 3 e Q ) cos i Q 



(75) 






9 e^) cos i Q + (-40 - 36 



vr 



4 . 

2 COS X 
(2+5 e^) -^ 



1-5 cos i r 



on 2 6 . 

80 e n cos i 



, sin 2^ 



;i - 5 cos i„) 

, . . 2 . \ (80) 

1 J 3 R / Slnl e C ° S l o ) co= .. 
sin i„ / s 



2 J 2 P \ e 



"0 



5 e 2 )cos 3 i ]-i|j 4 (^) 4 (2 + 3e 2 ) 



(3 



U i = ) 1 



6 J 2 V 



R\ 2 

— ) e„ cos i n 

Pn/ ° ° 



80 cos i„ 

11 + g— 

L 1-5 cos i. 



- 7 cos i n ) cos i Q > + fi Q 



b. Long period terms 



(76) 



1-11 cos i r 



40 cos 



1-5 cos 



4 • -l T 

2 . J 16 J, 
:OS l„ J 2 



4 © •» " 



.;>[ 



4 . 

COS 1, 



1 - 3 cos i 



. 2 . 

l-o COS l r 



1 J 3 R ,. 2. . . 

tt -r— — (1 _ e„ ) sin 1- sin to 

2 J 2 p n s 



e e f 



I 2 

(1 - e Q ) tan i Q 



cos 2w 

s 

(77) 
(78) 



200 cos i 



(l - 5 cos i„j 



5 J 4/R\ 2 2 
2 ,21 16 J^fe) 6 0— 







2 4 

16 cos ig 40 cos ig 



1 s 2 ■ 9 2 

1-5 COS lr, /, - „„' ; \ J 

(1 - D COS 1„J J 



A ^_R e COSi cos , 

2 J 2 P g Sin Ig 



sin 2" 



(81) 



b. Short period terms 
3 



J 2T U!- Sin2i 0) {t) 



P 2 z a n j 

.3 



-3/2" 



(l-e<) 



+ 



MJ sin 2 i cos 2 (6 + o> s + u) e )J. (82) 



IV-24 



(1 - e Q 2 ) 



"P 2 e r 



I'.lfh'-'li 



■3/2 



U-e 2 ) "'" [ + 3sin 2 i (^) 



(1 - e Q ) |cos2(6+(. s +u ( ) 



"l J 2© Sin2i [3 e Q cos O + 2c s + 2 
+ e cos (38 + 2w + 2co f ) > 



(83) 



V^ ^ 



(0 cos 4 o sin [ o [ 3 co 



s 2 (6 + co + u, ) 
s * 



+ 3 e cos (9 + 2^ + 2" { ) + e n cos (39 
+ 2co s+ 2^)] 



" = " -7 Jo f— ) cos i n [6 (9 - M - M, 
p 4 2 Vp„7 [ s * 



+ e sin 8) - 3 sin 2 (9 + u + co { ) 



- 3 e Q sin (9+2" + 2" ( ) 

- e,, sin (38 +2" +2%) 

s 1 J 



9 3/2 
M p = " — 8T J 



(84) 



(85) 



"0 



2 
+ 3 cos i„) 



+ 3 sin i r 



'©'I"-' 






sin (9 + 2" + 2co { ) 



-0' r 



sin 6 



-(-^) (1 - e„ 2 ) - ^ + If + sin (39 + 2c 



+ 2" 



(1 - e o> T 

"p = * - J 



!■ 



.. -(0^" 



'0 



+ 3 cos 2 i fl ) Q (1 -e 2 )+^+l 



'0' ' r 



+ 3 sin 2 i r 



sin (8 + 2 1" + 2", 



sin 
2 



<<>-(?) 



(1 - e 2 )- Ji+ 1 ( + sin (39 + 2^, + 2^, 



(^) 



«-«M + i 



^(^j j 6 '" 1 + 5 cos 2 i Q ) (9 - M s - M f 

+ e Q sin 9) + (3 _ 5 cos 2 y [3 sin 2 (8 + " g + " f ) 

+ 3 e n sin (0+2" + 2co ) + e n sin (39 
s l 



+ 2u> + 2co f ) 



J} 



(87) 



where 



E - e„ sin E = M + M, , 
s l 



tan 



■£ 



, E 
- e - tan ^ 

Z 



The solutions for the perturbed elements are then 



X = X + X . + X 

sip 



where 

x = a, e, i, ", £2, M. 



These expressions provide all of the in- 
formation necessary to describe the motion of a 

2 
satellite to the order J„ . However, there exist 

requirements in many studies for the perturbed 
expressions for r and 4> , (§ = + uj). This in- 
formation can be obtained from the equations 
presented above; however, the procedure is 
lengthy and unnecessary in view of some of the 
work quoted in (lief. 18) by Kozai. This ref- 
erence gives r and $ to the order J„ 



r =r + 2 J 2 R p (1 " 2 Sln ° 



-1-^(1 



- Vl - e 2 ) cos 9 + r 1 



1 



1 - e* 



+ 4 Jn R 2 - sin 2 i cos 2 (9 + co) 
4 2 p 



►o*! J 2 



(?) {H« 

(>-W')[£('-^-V : 



(88) 



in i 1(6 - M + e sin 9) 



1 - e /sin 



*M 



1 - e sin 29 



(i 



I sin 2 i) e sin (9 + 2u) - U 



7 2 1 e 2 

- -r^sin il sin 2 ( 9 + co) - -^ cos i sin (39 
12 / 



+ 2co) 



(89) 



where r n and 4 > n are values computed from mean 
orbital elements. 



IV-25 



Oblateness of the central body tends to make 
a twisted space curve out of the satellite orbit. 
It is customary to map this orbit as a plane curve 
on the orbital plane which contains at any instant 
the satellite radius and velocity vectors. In this 
plane one may either approximate the trajectory 
by an osculating ellipse (the astronomical ap- 
proach) or try to assume the actual equation of 
the plane curve to the desired accuracy. This 
latter approach is the one taken by R. Struble 
(Refs. 20 and 24). Another significant difference 
is that in this work some of the conventional 
orbital elements become variables to the order 
J . Struble in this reference derives per- 
turbations based on the following model 

i=u=— |l+e cos (? - ") - J 2 c + J 2 dj 

fr , e, w, c, d variablej . 

In the solution obtained, the short period pertur- 
bations are isolated in the c and d variables, while 
r , e and <» have only long period oscillations 

(with a secular variation in «). The independent 
variable ? is related to the central angle from the 
node, <t>, but provides simpler solutions than <t>. 
In particular. $ = <t> when J 2 = 0. The solutions 

for some of the elements, accurate to the second 
order, are included below. Note is made of a 
shorthand notation employing a set of inter- 
mediate variables n 2 • • • ^6 and w l and v 2 - 
These terms are presented following the 
equations for the terms c and d defined in 
Eq (90). 



(90) 



1 + e cos (3> 



CO) 



J 2 c 



J 2 2 d 



3 



. & 



3 sin 






(91) 



r sin e* 



dA 



where A is the right ascension and 6* = 90 - L. 



= e ' 2 J 2 e 



9 - 1 

(fJ (5 cos 2 i Q -lXl^ cos 2o 



+ 4-1.. sin 4") 
4 4 



(92) 






cos i„ - 1) 



9 T 2 

+ n 5 4 J 2 



? + 



+ ! J 2 



(-^\ (5 cos 2 i Q - D" 1 (n 6 sin 2c 



-^ t| 4 sin 4u) 



(93) 



i=i + 



(94) 



I J Q?\ sin 2 i Q [e cos (<t> + u) 
+ cos 2<i> +| cos (3? - to)J 

+ l! J 2 *1 (^) 4sin21 

J 2 (t) Sin 2 l ^ C ° s2 '<> " J ) 
-14 + 15 sin 2 i Q - 5 -j(G - 7 sin 2 i )J 



x l 00 + 32 



14 + 15 sin 2 i Q - 5 -j(6 - 7 sin 2 i Q )| cos 2<- 
1 (95) 



♦ = * + 1 J 2 (t) [ 4 e C ° s2 L Sin ^ " W) 



+ 2e (1 - 2 cos i„) sin (4> + u ) 



+ (1 - 3 cos i n ) sin 2? + „ e (1 



4 cos 2 i () ) sin (3$ -<•>)] +| J 



2 V 2 (96) 



Now adopting the shorthand notation 

d - - 35 ii. 

D l TS T7T 
J 2 

The short period terms c, d can be written 

2> 



i(^) 



sin l 







(2 +^-) cos 2$ 



+ e cos (3^ - u ) + 4- cos (4 ^ - 2 U ) 



3 2 



cos 2 1 

2 



© 



^2 . 

4 



+ UP-) e (2-3 sin 2 i ) cos (2? - 2 J (97) 

[m D i sin2l o 



-(* + h D 1 )sin 4 l ]co S (2?-4 (1) ) 



continued 



IV-26 



/45 _. , 281\ . 4. ( , 4 l/3 n 1\ 
\T D l + -W sln 'of +e j(7 D l-¥) 

/3 , 15 „ \ .2. 
U + TT D lJ sln l 

(fff + Tff D l/ sin *() j J cos ^ " 2u ' 

1 far, 8 "\ < 2 - a-/37 3 _ \ . 4. 
7 [W D 1 -7J sln l + (T7-7 D lJ sln l 

J(n D i " w) sin2i o 
"(if D i + T-) sin4i o[J cos 2? 

1 f 2 (3 .2. , /23 _. 41\ . 4. 
"7 L C i^ Sm 1 + (TB- D l-5?j Sin l 

D x ^g. sin 4 i J cos (2? + 2 J 

3 |S°i-i)-(J^°0- taai o 

D i' ain4l ojj 



+ e 



+ e 



+ e 

1 
~ff 



,191 15 



cos (3?- 3 J 



1 
"8" 



e (3D 1 - |) sin 2 i 



- (§ + J D x ) sln\ j + e 3 \<1™ D, - ^-) sin 2 l 
cos (3 $~ - u ) 



.33 , 41 „ > , 4. 



1 
ff 



e ^H D 1 "i- )sin4i n^ +e 



3 1 



sin i r 



{ TU + ix D i )sin l 



cos (3"$ + io ) 



Jp J3 sinVos2io |] 

-^[ e4 !! D i-4 + ^ D i )sin2i o 

D i> sln4i o|] 



+ f 9 + 15 



1 
T5" 



cos (4 <(> - 4 U ) 



(^D.-Dsln 2 ^ 



-(§ + ^ D l )sin \ 

+ e4 t ( W D l-W )Sin % 



^2- + TO D l )sl A) 



cos (4<f - 2 U ) 



1 
T3" 



|sin 2 i + (|D 1+ |) S ln 4 i 



, 2 \ 17 .2. . .19 _ , 41. . 4, 

+ e •(- -m- sin i„ + (,--- D. + Txr )sin i- 



7Z' 



, 4 169 n .4, 

+ e W D i sin l 



T ^TB" "l ^T4T'' 
cos 4^ 



"T5" 

1 
"2T 



2 « . 2. 2 S 

e ^ sin i Q cos i Q 



cos(4^ + 2 u ) 



3 1,243 „ 



'7577^1 -T4T )ain *c 



,35 , 81 _. . . 4. 
W + M D l )sln *<) 



cos (5^ - 3 J 



1 
"2T 



T7 



,33 



35, 



sln^l. + (^D.- ")sln*l 



l ^TU U 1 "14T 



j- 3 ) ,1 n ± 73, ,4, 

+ e i- ( mr D i + MT )sln i 



cos (5^ - u ) 



- 1 I" 1 



3,2, 2. 

^ e sin i Q cos i Q 



1 
"2T 

1 
T5- 



3*3 .4, 
e TC sin i 



63 



cos (5^ + u) 
cos (5? + 3 U ) 



\ZSU D l sin *o 



( TOr D i + T6- )sln4i o 



cos (,6~§ - 4 U ) 



1 f" 2 (1 .2. , ,53^ 23, . 4. / 
W L C |^4- Sln l Q +{ m D l-T& )sin *()[ 



"3T 



, 4 \ 123 . 4. 

+ e <D, , , „„ sin 1. 



'\ TTSTT 



j. ! [ 2 ( 3 ,, 
+ T5" [_ e |3T (4 

1 T 3 (287 

■if [ e I 147 



sin i - 5 sln 



cos (6^ - 2J 

cos(6?+2u,) 



'Ml 



, 2, 
sin i. 



_,_ ,13- ^ 845, . 4, 
+ %ff D l + -OTl> Sln ^ 



3 . 4, 
1l4T sIn ! 



cos (7? - 3cj ) 
cos (7^ - a,) 



"iff P j 

"ffff [ e4 j D l isin 4 l [J cos(8?-4J 

+ [e 2 |(3D 1+ i^ 



. 4 ),1 , 33 „ , , 2. 
+ e {(rTr +T-P- D^sin i 



33 



)sin 2 i Q -(*| +^.D 1 )sin 4 - 



L 



- < 79 + 3 



Dj^Js^IqI cos 2o 



e4 1D 1 T2ff aln '* 1 



COS 4(; 



(98) 



IV-27 



Finally the pseudo variables r\ „ . . . r\ 
and v, and v„ can be defined in terms of the 
true variables. 

n 2 - j(3 Dl -i) + ( ^-l5 Di)sln 2 lo 

+ ^ Dl -^i)sin\ + e^ [(» Dl -l, 
^ ,53 45 „ . . 2, , ,45 _. 95* , 4. 



+ fe* 



^ D i-i ) - { ^ D i )sln \ 



+ ( 9 + 45 



D l> **\ ]j 



(99) 



"3= !<7 + ^ D l )sln2i 0- ( ^4 D l )sIn4l (100) 



S D i> sln \J 



-e 2 4D 1+ ^) S ln 2 l + ( 1 J- B . 

r, 4 - je 2 [^sin 2 i -||sin 4 l ]j 

_ 5 ,24 _ .. , ,151 93 _ > . 2. 
n 5 |(^D 1 -4) + (- TT -^D 1 )sin i Q 

_, ,21_ 229, .4. , 2 [",11 9 „ . 

+ ( -2- D i--2T )aln i o + e [^nr-Tr !* 



(101) 



-4 D i + § )sln2l o + ( ¥ D i-^ )sin41 



„I 



■ 7 "M 9 1 7 Q 4(102) 

*!«" i4 + " D l ,sini 0- ( T + 7 D l )slnl 



+ e 



[< 14 "7 D 1 

D^sln^jj 



,,,1119,, 158 . . 2. 
)+( ^B- D l"T- )sin l 



. ,289 29 
|g. D x (6 - 7 sin i Q ) cos (? - 3 J 



(103) 



+ ^r (36-89 sin 2 i ) cos (? - J 



■3T 

e 



[ 3D 1 



(4 + e 2 ) - 28(6-7sin 2 i ) 



- 7e 2 (2 - 3sin 2 i Q ) 



cos ($ + to) 



e 3 2 

+ ^-sin i Q D x cos (^ + 3 J 



7sin 2 l ) - 7(4-5sln 2 i ) 



+ ^-(9 -25 sin 2 i Q ) cos (2?- 2 J 

+ T¥ [ 2D 1< 6 

+ e 2 3Dj (6 - 7sin 2 i )-^.(2-3sin 2 i ) j cos 2? 

+ TC [6D lS in 2 i -(2-sln 2 i )] 



cos (2? +2 J 



■23? 



|28 (2 - sin 2 i ) + 9D X (4 +e 2 ) (6 - 7 sin 2 i Q ) 



- 21e 2 (2-3sln 2 i ) 



cos (3^ - u ) 



"2T 



D 1 (4 + e 2 ) sin 2 i Q 



- 2 (3 - 2sin i )J cos (3?+ J 



2 

+ e 



7 (10 -9 sin i Q ) 



+ 18D 1 (6 - 7sin i Q ) 



cos (4'$' - 2 U ) 



T34" 



[ 18D 1 



(2 + 3e 2 )sin 2 i n -6(3+sin 2 i n ) 



- e"(12-7s 



in 2 i Q )] 



cos 4<iT 



+ ^ r D 1 (6-7sin i Q ) cos(5?-3 u ) 



- 20 
2 



J [ 27D 1 
(3 + sin 2 i )J 



(4 + e 2 )sin 2 i r 



cos (5^ - u) 



18D 1 sin 2 i - (2 + sin 2 i Q ) cos (6?- 2J 



+ |^. D x sin 2 i Q cos (7?- 3 J 



']l 



(104) 



and 



j[-D ie 3 ^(6-14sin 2 i 

sinff - 3 u )+[j. j(^ D !-i 



+ 7 sin i Q ) 



+ (4 #-? D i )sin2l o + ( 9D i-^ sin \ 



+ e cos i Q U-J2 D 1 - j) 



+ (|-|D l)s ln 2 i j 



sin Of - oj) + 



[. j(3-J Dl ) 



53, 



+ (|d i -^.)sin"l + (ll-3D 1 )sin'*l c 



+ e ' |4-^ D l) + 4 D l"F )sln2l 
+ (J-|D 1 )sin 4 i j] sin(?+ u ) 

- [e 3 D 1 ^ sln 2 i (l-2sin 2 i )n 8 in(?+3 u ) 

+ 7 L e | ( T D l--6- )+( H¥-2¥ D l )sin ^ 
+ (|D 1 -|^.)sin 4 i |] sln(2?-2j 



IV-28 



+ (g-|D l)si n% + e^ |(l-9 Di> 



, ,69 _ 91.2. 
+ ( T¥ D 1 -3TT sin l 



, ,155 15 _ . .4. 

+ Hir--4- D i )sin x 



, ,163 3 „ , .2. 
+ ( T4T"F D l )sln r 



sin 2^ + ^. e 2 J- 



1 



+ ( tV D l _ Ul ) sin \ J sin (2? + 2u) 



e 2 iD 1 ^(4-Hsin 2 i 



+ 7sin 4 i )|J sin (3«7 - 3,J 



,25 7 

lr 2TG" 7 



D l)s in 4 i j] + e 3 j^-^D^ 



+ ( H D i "lr )sin2i o 



,17 4 

+ ( 7 - F Djjsin i ( 



sin(3^ -<J + 



[• 



(l + |D 1 )sin 2 i + 4 Di + n )s . n 4. o J 



2. , 3 5 ,2 
'1 oin 1 ( TC"2T sln : 0' 



sin(3^+ M ) 



2 5 / 5 , 9 _. , , ,129 _ , 91 > .2. 
6 <"<T7 + T4- D l ) + ( 3^ D l + T^4- )sln l 



,27 59 . . 4. 

( T7J D 1 + W sln 'o 



+ 1 l" 1 - f 1 D + 5 



sin(4^ - 2 U ) 



, . 2. , ,5 „ 1 . . 4. 
)sin i +(jD fH ) S m i Q 



+ e 



2)1 , 65 , 3 



T " ( T^ + T D 1> 



, ,15 _. ^ 41 . .4. 
+ ( TTT D 1 + W Sln >0 



. 2. 



sin 4<jT 



e " 1D i ( 'l¥ + W sln2i o 



11 . 4. , 
4TP sm 1 ) 



J sin (5?- 3,,,) +3. e J 
( ! D l + ^ sin2i + (TO D l-A> s AS 



5 

T7 



-e 3 D^in^^-^sin 2 ^) 



2 1 , 1 3 _. . . 2 

e Jttt - Onr + T D,)sin 1 



sin(5^ - c) 







( T6- D i--m )sin4i 



T7 " V TB" F^l ; 

sin(6? - 2 U ) 



1 T 3 L 

7 [_ e Pr 



2 1 2 

"7" | c ]"l sin ^m^ 7 " 8sin i ^ 



sin(7^-3 M ) 
(105) 



In these equations ^ Q , i„„ and e„ are inte- 
gration constants and as before the singularity 
at i = 63.4° occurs. However, Struble notes 
that for this inclination the motion is given by 
the simple pendulum equation and concludes, as 
was done earlier, that an oscillation occurs in 
the element ^ . 

Still a third approach, though somewhat more 
similar to the second than the first, to predicting 
the motions of a satellite has been developed by 
Anthony and Fosdick (Ref. 25). This work, 
based upon the method of Lindstedt, is the re- 
sult of series expansions for all variables in 
power series of the small parameter J„. Since 

the higher order coefficients (J , etc.) are 

neglected, these series are truncated following 
terms of the order J . This being the case, 

each of the variables may be represented as 



u (£)+ 3/2 J 2 Ul (I) 



4> = P (£)+ 3/2 J 2 P. (i) \ (106) 



)' = (90 - L) = 7T/2+ 3/2 3 2 6 l 'd) 



J 

where the new variable i is defined by 

4> =4(1+ 3/2 J 2 4,^ 

i> . = constant to eliminate secular 
variations in u 

and u = 1/r (for Kepler ian orbit) 



IV -29 



Now starting the solution _for the motion at an 
apse (i.e., at a point where r = 0), the equations 
of motion were found to be as follows: 



General First -Order Results (Arbitrary 4 Q ) 



* = 4 



, 3J 2 / R\' 



(2-3 sin 2 i) 



(given 



4> n , use this equation to find 4 Q (107) 
in 2i R . 2 i 

T~(h) r sinl ° 



J sin 2i 



4c' 



- 2ti sin 4 cos (4 - 4 Q ) 

+ (3 + 2ti) cos i„ sin (4 - 4 Q ) 

- t] sin 4 cos 2(4 - i Q ) 

- n cos 4 sin 2 (£ " ^o* 

- 3(4 - 4 ) [cos 4 (4 - 4 > 

- sin 4 Q sin (4 - 4q)J \ 



(108) 



P = r> = r Q V Q I 1 



°2 

4c 



(Ji) 2 ^ , [«, 



+ 4r]) cos 24 Q - 3ti cos 24 q cos (4 - 4 Q ) 
+ 3n sin 24 Q sin (4 - 4 Q ) 

- 3 cos 24 Q cos 2(4 - 4 Q ) 
+ 3 sin 24 Q sin 2(4 - 4 Q ) 

- n cos 24q cos 3(4 - 4g) 
+ n sin 24 Q sin 3(4 - 4 >]| 



(109) 



l + n cos (4 - 4 Q ) 



T 2 

2 (2X 

16^ W 



(HO) 



1 + n 
r \ T~+ r, cos (4 - 4^1 



J 2 /R.X 2 _K ) 

^ V r 0/ (1+ ri) [l+n cos (4 -4 )] T ) 



,r 2 



+ J 2 /RV M ll 



1 + T| + 2T| cos (4 - 4q) 



(112) 



V 



/ y i + ^ 2 + 2 



2tj cos (4 - 4 Q ) 



|(iL) 2 Ml {<i*„» 2 [I*, 2 



♦ ^»- 2 



+ 2) 7 cos (? - f )] 



r 1 



where 



i 
i 



(113) 



-36 - 18r, 



L. = {24 + 12r, 2 + (sin i) [-3 
+ (24 + 32r| + 3r, 2 ) cos 2 4q]| 
+ i- 24 - 8r, 2 + (sin 2 i) [(-20 - 27r, 
+ 4n 2 ) cos 24 Q + 36 + 12ri 2 ]} cos (4 - 4 Q ) 
+ |- [8 + 15n 

+ 16r, 2 ] (sin 2 i) sin 24J sin (4 - 4 Q ) 
+ j- 4n 2 + [6r, 2 + (-4 

- 6n 2 ) cos 24 Q ] sin i> cos 2( 4 - 4 Q ) 
+ |(4 + 6ti 2 ) (sin 2 i) sin 24 Q | sin 2 (4 

- 4 Q ) - J5n (sin 2 i) cos 24 Q 1 cos 3(4 

- 4 Q ) + |5n (sin 2 i) sin 24 Q } sin 3( 4 - 4 Q ) 

- L 2 (sin 2 i) cos 24 Q | cos 4(4 - 4 Q ) 
+ It, 2 (sin 2 i) sin 24 Q | sin 4(4 - 4 Q ) 

(114) 

M 1 = !l6(3 - 3ti - r, 3 ) + (sin 2 i) [24(- 3 

+ 3n + n 3 ) + 8(3 - r, - 6T, 2 - 3^) cos 2£^ty 

+ J4(-12 + 12t! - 4n 2 + 3ii 3 ) 

+ (sin 2 i) [6(12 - 12t, + 4iq 2 - 3r, 3 ) 

+ (-40 - 18r| + 8n 2 
3 



+ 12^°) cos 24f,]^ cos (4 -4 ) 

5 2 

2n 

3, , . 2 



^0]} 
+ i- (16 + 66t! + 32t 

+ 6r, 3 ) (sin 2 i) sin 24 A sin (4 - 4 Q ) 

+ |l6r, 2 + (sin 2 i) [-24T, 2 + (16 

+ 24t! 2 ) cos 24 Q ]| cos 2(4 - i Q ) 

- /(16 + 24r, 2 ) (sin 2 i) sin 24 1 sin 2(4 

- 4 ) + {4n 3 + (sin 2 i) [- 6 n 3 

+ (26n + 9t| 3 ) cos 24 ]| cos 3(4 - 4 Q ) 

- |(26t, + 9r, 3 ) (sin 2 i) sin 24 Q l sin 3(4 

- 4 Q ) + |l6Ti 2 (sin 2 i) cos 24 | cos 4(? - 5 Q ) 

continued 



IV-30 



- il6-n 2 (sin 2 i) sin 24 Q } sin 4(4 - 4 ) 
3ti (sin i) cos 2| Q J. cos 5(5 - 4 Q ) 
3T) 3 (sin 2 i) sin 2| | sin 5(4 - 4 Q )- 

(115) 



+ 1 



vS 



(116) 



(117) 



Under the assumption that the trajectory is 
nearly circular these equations can be simplified 
to yield 

Nearly Circular Orbits (Arbitrary 4 Q ) 



3J r 



1 + 



®' 



(2-3 sin i) 



4 (given 



<$>., use this equation to find 4q) (118) 



3J„ 

4 



( JL] sin 2 i I cos 4 n si" (£ " 4 n ) 
2 4 ^r 0/ / ) 

(4 - 4 Q ) [cos 4 cos (4 - 4 ) 

sin 4 sin (4- 4 )]j (119) 



[cos 24 Q 



I 3J„ / R \ 2 , 
P = r„V h - —A l — \ sin i 
00 I 4 VO/ 

- cos 24 Q cos 2(4 - 4 ) 
+ sin 24 Q sin 2(4 - 4q)]| 

u = _L r 1 - T! 1 - cos (4 " 4 > 

+ ^(^) 2 j 6 [1 - cos (4 - 4 )] 

+ (sin 2 i) [- (9 - 6 cos 24 Q ) 
+ (9 - 5 cos 24 Q ) cos (4 " 4 ) 

- 2 (sin 24 Q ) sin (4 - 4 Q ) 

- (cos 24 Q ) cos 2(4 - 4 ) 
+ (sin 24 Q ) sin 2(4 - 4 Q )] 

1 + n ll - cos (4 - 4 >} 

g) 2 {e [1 - cos (4 - 4 )] 

+ (sin 2 i) [-(9-6 cos 24 Q ) 
+ (9 - 5 cos 24 Q ) cos (4 - 4 ) 
- 2(sin 24 Q ) sin (4 - 4 Q ) + 



(120) 



(121) 



u 2 
T 



(cos 24 Q ) cos 2(4 - 4 ) 



+ (sin 24 Q ) sin 2(4 - 4 )]j 

: V Q 2 ll - 2 n |l - cos (4 - 4 )} 

+ J 2 {lS i 3 [l " C ° S ^ _ ^ 
+ (sin 2 i) |(~ 2" + 2" cos 2 ^0/ 

+ (I " I cos 24 o) cos (e " l ) 



(122) 



- (sin 24 Q ) sin (4 - 4q) 
+ (cos 24 Q ) cos 2(4 - 4 ) 



(sin 24 Q ) sin 2(4 - 4 



V = V, 



~2" 



+ (sin i) 



1 - r, |l - COS (4 " 4q)| 

Q 2 | 3 [l -cos (4 -4 )] 
|-| + |cos24 ) 

+ (I " I cos 2 ^o) cos (l " e o ) 

- (sin 24„) sin (4 - 4J 



+ (cos 24 Q ) cos 2(4 - 4 Q ) 



(123) 



(sin 24 Q ) sin 2(4 - 4 



4 



(124) 



The solution obtained using these equations 
exhibits no singularity at the "critical inclination" 
and indeed is well behaved at every point. For 
this reason this set of equations, though not pre- 
cise, seems well suited to analytic studies involv- 
ing computer programs. 

4. A nalytic Comparison of General Perturba- 
tTbns Formulations' 

Recently several analytical methods of deter- 
mining the oblateness perturbations have been 
published (Refs. 18 and 23 to 28) in which basically 
different mathematical approaches are employed. 
These approaches include: 

(1) The classical approach of general 
perturbation theory in celestial me- 
chanics, using the concept of an oscu- 
lating ellipse and solving for the varia- 
tions in orbital elements. 

(2) Integrating the equations of satellite 
motion by seeking a solution in the 
form 

_ = — [l + e cos (^ - u) - J 2 c + J 2 dJ 



continued 



IV-31 



where c and d are unknown functions 
in terms of short -period perturba- 
tions (to be determined by the integra- 
tion process), while r„, e and to ex- 
hibit only long -period perturbations. 

(3) Direct approximate integration of the 
equations of motion with oblateness 
perturbations, solving directly for the 
instantaneous coordinates of the body 
in orbital motion. 

Depending on the variables and mathematical 
tools used, the final solutions of various authors 
are seemingly different and physical interpreta- 
tions of certain important variables are some- 
times hard to visualize. The transformations 
between the different sets of variables employed 
in the literature have not been obtained previ- 
ously. 

Due to these facts a somewhat bitter contro- 
versy has arisen about the merits of classical 
celestial mechanics (Refs. 20, 23 and 29) for the 
solutions of near -circular orbits. The present 
analysis, which was made by J. Kork (Ref. 30) 
compares the solutions obtained by all the above 
mentioned authors for nearly circular orbits 
within the first order accuracy in the oblateness 

parameter J (i. e., neglecting J , J J terms). 

a. Kozai's formulation (Refs. 18 and 26) 

Upon a change in the notation utilized by Kozai 
to that utilized by Vinti and upon changing the 
symbols to be consistent with those presented in 
Chapter III, the first order perturbation in posi- 
tion may be written 



<r,r = a 



- 1 



\l J 2 (a?) i 1 -| sin2i ) 



[d-r 51 — r ' 



e ) cos 8 + - 
a 



5 * = | J 2 



\ J 2 (aV) 



fi 



1 - e J 



sin i cos 2 (8 + gj )! 



'(125a) 

|(2 - g- sin i) (9 - M + e sin 9) 



. ,, 3 . 2 .. T 2 „ 2 

+ (1 - 2 sin i) [^ U - e 



Vl - e 2 ) sin 6 + I (1 - Vl - e 2 ) sin 29J 

/I 5 . 2 A . . „ , 

lj-jr sm ll e sin (0 + 2o> ) 



G- 



7 2 \ 

T-Tj- sin ij sin 2(0 + tu) 



e 2 

p- cos i sin (30 + 2oj)! 



(125b) 



and the secular perturbations in the orbital ele- 
ments are 

2 



'0 2~ J 2 



® ' (= 



o 5 • 2 .I, 

2 - ;y sin i) t 



(126a) 



r, '"0-7 J 2 (f) 



M = M Q + nt 



n = n + 2 J 2 



nt cos i (126b) 

(126c) 

(B)\ (i.j.,.',)^: 



. 3 . 2 A J, 2~ 
1 - ir sin ij Tl - e 

(126d) 

where w Q , n and M„ are the mean values at the 

epoch, i.e., the initial values of the osculating 
elements from which the periodic perturbations 
have been subtracted. 

There are no first order secular perturba- 
tions of the semimajor axis, a, of the eccentricity, 
e, and of the inclination, i. 

The mean value of a (i.e., a) is given by Kozai 
in terms of the unperturbed semimajor axis a n , 
as 

* = a \-l*2 (^ (l-|sin 2 i)^ 

(127) 

Notice that the classical relationship n„ 2 a 3 = p, 
becomes in these variables 



-2-3 



3 



n— =„ 11 -JJ 2 (5) 



3.2 

ly sm 1 



,) i 



'1 - e' 



(128) 



The value of the mean semimajor axis, a, has 
been already used in the derivations of Eq (5). 

If the eccentricity, e, of the orbit is a small 
quantity of the first order or less, Eqs (125) can 
be reduced to the simple form given below (Ref. 
26). 



6r 



1 — /r\^ 2 

= ¥ a J 2 \a ) sin i cos 2>t 



1 - ■ 2 . 
= * a ( sin i cos 2\ 



(129a) 



■3 /-p \ /l 7 9 \ 

H = ' 2" J 2 U ) \2" " IT sin V sin 

(l 7 . 2 .\ . 
-- t I j - Yz Sln l / sin 2x - ( 

where (within a first order accuracy) 
\ = M + u 

.2 



2\ 
129b) 



_ 3 
c " 2 J 2 



(?)'•*', ©' 



Since E is a small quantity, and since the relation- 
ship between M and is (Ref. 31 ) 

M = - 2e sin + . . . 

it can be shown that for small eccentricities, 
i.e., e = 0( f ) 



IV-32 



1 + € cos 2\ ~ 1 + e cos 2 (6 + u ) 
+ 4t sin sin 2(9 + u ) 
ss 1 + « cos 2<J> 

and similarity 

1 + e sin 2\ = 1 + e sin 2<\> 
Thus Eqs 129a and b can be written also as 



(130a) 
(130b) 



1 - • 2 . 
or = £ a e sin i cos 24> 

(17 2 \ 

2 ' Y2 sin sin 2< t> (131) 

Finally, the expression for the instantaneous ra- 
dius vector in near -circular orbits can be written 
as 



e Q cos (<(> - go) 



i', 



®' 



sin i cos 



2*] 



(132) 



From Eqs (126) and (130a) it can be seen that 
for small eccentricities the average angle from 
node to perigee co can be approximated for one 
revolution by its initial value, oj-. 

Kozai's solution for near-circular orbits con- 
sists basically of two independent components 
varying about a mean radius, a. These com- 
ponents are: 

1 2 

(1) An oblateness term, j e sin i cos 2$ 

which has a period of ir (double periodic 
within one full revolution) and depends 
mainly on the shape of earth seen by 
the satellite vehicle (i. e. oblateness 
parameter J„ and inclination of the or- 
bit, i) but is independent of the orbital 
eccentricity, e, and nodal angle to 
perigee, u . The oblateness term de- 
pends also on the semimajor axis 

2 



through the term s = =- J„ 



©' 



(2) An elliptical term, e„ cos (t 



'0 



) de- 



pending only on the geometrical prop- 
erties of the orbit, e_ and oj _ but being 
s 

completely independent of the oblateness 
of the planet or the orbital inclination. 

It is obvious from the mathematical form of 
Eq (132) that depending on the relative size of the 
oblateness and ellipticity terms, in connection 
with proper phase shifts between the two, two, 
three or four "apses" can be obtained during a 
single revolution (i.e. points where r = 0). 

This fact will be graphically illustrated in the 
discussion of Izsak's work. 

b. Struble's formulation 

If only terms to the first order in J are re- 
tained, Struble's main results, periodic in ra- 



dius, can be presented in the following form (Ref. 
24, p 93). 

- = — [l + e cos (JjT - w ) - J 2 c] (133a) 





I = J£ 
P„ 



r 



T cos2i + ¥r 



+ | J 2 



) ( 2 " 3sin2i o) 



(133b) 



4e cos i sin (<j> - co ) 



2 _ 

+ 2e (1 - 2 cos i Q ) sin ($ + u ) 

2 — 

+ (1 - 3 cos i ) sin 2 <(> 

2 2 — 1 

+ ■§■ e (1 - 4 cos i ) sin (3<j> -to) 



(133c) 



where 



4 {^f sin2i l + cos2 < 



e cos (3<t> - cj)+ -r cos (44> - 2oj ) 



i 3e „ 

+ — *- cos 2o 



i© ! 



e 2 (2 



3 sin i Q ) cos (2<j> - 2cj ) 



(133d) 



dA 



P — 

P m = r sin 6'-gy- = angular momentum 
about the polar axis 



G 1 = 90° - L 



(133e) 



In Ref. 32 it is shown that the angular mo- 
mentum orbital plane is given by 

h = r (0 + oj + cos i 6) = yfp (134) 

From Eqs (133) and (134) it can be shown that 

2 

P. 



. 2 . 

p = JJTp cos i or - = ti cos rl — _- 
m » ^ p 2 



(135) 



For small eccentricities of the order J,. 



1 + e cos (<)> - oj)« 1 + e cos ( $ - u ) (136) 

at least for one revolution. Similarly all terms 

2 
containing e , J e, etc can be neglected. Using 

Eqs (135) and (136) the results given in Eqs. (133) 
can be simplified to read 



1 t /rV ■ 2 . „ J 

tJ, ( — I sin icos 2* 

¥ 2 \ r 0/ J 



r = r n |1 - e cos (9 - u ) 

I «■ 1 



(137a) 



IV-33 



r Q = P II 



i-nm 



(2 



3 sin 
(137b) 



■>] 



Furthermore it should be noted that for small 
eccentricities 



fe) 



p = a(l -e ) = a 



(138) 



Remembering this approximation and comparing 
Eq (137b) with Eq (127) similarily Eq (137a) with 
Eq (132) it becomes obvious that for e = O (J 2 ) 

the first order results of Struble are identical with 
Kozai's formulation and the constant r Q is given 

simply by the mean semimajor axis: 



r Q = a 



(139) 



Izsak's formulation (Ref. 23) 



The instantaneous radius is given by Izsak as 
follows 

T 12 

r = a* 1 - e cos (<t> - w ) + j e cos 2(ob - &> ) 



I'. (?)' 



sin i cos 2w + 



•■] 



where 



H- 2 4' 2 (i)V^" 2 '3 



w = (1 + E ') 9 + o, 



(140) 



(| =a constant for the motion of the perigee 
of the order J„ 



For e =0(J 2 ) tne solution for one revolution is 
simply 



[' 



r = a* 1 - e cos (4> - u) 

+ \ J 2 (!) Sin2 ic ° S 2 *1 (141) 

Comparing Eq (141) with Eq (132) it is seen that 
Izsak's solution can be also reduced to the form 
given by Kozai and the parameter a* is simply 
a* = a. 

An interesting feature of Ref. 23 is a set 
which represents parametric families of curves 
obtained by solving Eq (141) of this study nu- 
merically for various values of e Q (0. , 0. 00012, 

0.00030, 0.00049) and for three particular cases 
of o n (0°, 45°, 90°). The curves show clearly the 

possibilities of 2, 3 and 4 "apses" (i.e. points 
where r = 0) during one revolution, depending on 
the relative sizes of ellipticity terms with respect 
to the oblateness terms and also on certain phase 
shifts between them. These figures have been 
reproduced and are presented for convenience 
as Fig. 7. 



d. Equations derived by Anthony and Fosdick 

The form of the resulting equations in Ref. 25 
is completely different from the results obtained 
by the authors considered previously. In Ref. 28 
the equations of motion in spherical coordinates 
are integrated directly and certain new variables 
are introduced, which do not have a simple phys- 
ically intuitive connection with the variables used 
previously. There may exist some doubt, how 
the initial value, 4 Q , of the "independent variable 

for which the first -order analytical results for r 
and V are periodic" compares with the classical 

V„ 2 



j , and how the analog of eccentricity t| = 



V 



may depend on the classical eccentricity, e. 
These transformations are far from obvious, 
thus, they are derived in this section by reducing 
Anthony's solution to an analytical form similar 
to Kozai's results and then comparing the coef- 
ficients term -by-term. 

The equations for arbitrary near -circular or- 
bits are given as Eqs (118) through (124) assuming 
n = 6(JJ. Certain terms in these equations can 

be simplified by using the equality 

cos 2| Q cos 2(| - l ) - sin 2i Q sin 2<£, - Z Q ) 
= cos 2£ (143) 

/ R \2 _ 
Next, using the previous notation £ = J I — j = 

| JL] the expressions for r and V can be 
\ r 0/ 
written as follows 



3 J 
2" J 2 



jl +T1 [j 



i - cos (i - e )J 



+ c cos a - £ ) 



- ^-t sin 2 i [(-9+6 cos 2£ Q ) 

4- (9 - 5 cos 2i Q ) cos (| - £ Q ) 

- 2 sin 2| Q sin (i - i Q ) - cos 2£j? 

V = V Q < 1 -ti + r| cos (i - i Q ) 

' i 2 

+ £ - « cos (i - i ) + j £ sin i 

• _(-! + ! cos2 ^o) 

+ (I - | cos 2£ J cos (| - £ ) 

- sin 2| Q sin (% - | Q ) + cos 2£ 



(144a) 



where 



£ (2 - 3 sin i) 



(144b) 



(144c) 



Notice, that in Eqs (144a) and (144b) the sine and 
cosine terms appear combined with a small con- 



IV-34 



stant of the form a. cos |, where | = (1 - a„) 4>. 

Since for the nearly circular orbit considered here 
both a. and a„ are of the order s , it follows by a 

reasoning similar to Eqs (130a) and (130b) that 



1 + a. cos 4 = 1 + a i cos 4>, etc. 



(145) 



Equation (145) indicates that for the purposes 
of this analysis it does not make a noticeable dif- 
ference, if during any single revolution 4 is simply 
visualized as the central angle from the ascending 
node, c)>. 

Next, collecting the cosine and sine terms in 
Eq (144a) 



r = r Q (1 + A ) 



i - a x cos (4 - e > 



1 2 

+ A„ sin (4 - 4 Q ) + ,- e sin i cos 24 



(146) 



where 



3 2 2 

A„ = t| - c + tj c sin i - e sin i cos 24 r 



3 2 5 2 

+ =- € sin i - *- e sin i cos 24 n 



1 . 2 . . ot 
o- € sin i sin 2|„ 



By trigonometry 



A 1 cos x + A sin x 



V^l 2 + k 2 C0S 



x + tan 



Thus Eq (146) becomes 



r = r Q (1 + A Q ) 



Va x 2 + a 2 2 cos (^ -e c 



1 2 

+ ct„) + £- « sin i cos 24 



(147) 



where 



*o = tan_1 Wj 



Kozai's form of radius, given by Eq (132) can 
be written as follows 



r = a 1 - e cos ( $ - u n ) 



. 1 .2 
+ 7t ( sin l c 

b 



os 2$ 



(148) 



By comparing Eq (147) with Eq (148), while re- 
membering that within the first order accuracy 
4 ~ <J>, the following important transformation 
equations can be derived by equating the corre- 
sponding coefficients of two Fourier series ex- 
pansions of the same function cj>. Thus, Anthony's 
variables are related to Kozai's formulation by 
the following equations: 







1 + ,, 



3 2 

i + p- e sin i + 



continued 



sin i cos 24 n 



(149a) 



,3 . 2 . 

* + ^ i sm i 



5 . 2 . n£ 

7t s sin i cos 24 r 



+ (tv « sin is 



in 2| ) 2 



1/2 



(149b) 



tan 



(- 



"! 



4 e sin i sin 24 



o) 



2 . 



e sin i cos 2t 



»)■'] 



(149c) 



The inverse transformation equations for t) and 
r„ can also be obtained from Eqs (149a) and 

(149b) to be: 



(I 



■ 2 • • «\' 
i sin i sin 2| ) 



1/2 



3 2 

+ € • j! sin i 



5 . 2 . „, 

p- e sin i cos 2| 



(150a) 



3 . 2 . 
Tj- £ sin i 



. 2 . 

sin i 

1 ( 2 A ■ 2 • ■ nt V) 1 

1 - le - f =■ e sin l sin 24 0/ / y 



+ e sin i cos 24 n 



(150b) 

/2 



,1 • 2 . „, 

+ =■ e sin i cos 24 n 



4 Q = 4 Q <u . i. e. «) 



(150c) 



Unfortunately, Eq (149c) is transcendental and 
the third transformation must be found by nu- 
merical successive approximations. Character- 
istic solution curves for Eq (150c) can be obtained 
by the following procedure: 

(1) For a given e, i, e solve for various 
values of u. by assuming values for 

4 in steps of 10°, for example. 

(2) Plot the data and obtain a value of 4 n 
corresponding to the given oj„. 

For step (1) it is advantageous to write Eq 
(149c) in the following form 



4 Q - tan 



1 . 2 . . „, 
=- e sin i sin 2£,(-| 



If " (3* sin i sin 2 ^o) J 



72" 



(151) 

Note: 

If in Eq (151) the eccentricity becomes smaller 
than a critical value e* = ^ sin i, the values of 4 n 



IV-35 



can no longer be picked arbitrarily. This fact 
is illustrated by assuming e = in Eq (149b) and 
observing that the required value of £, Q = 0°, 90°, 

180°, 270°. Physically this means that for e = 
the "apoapsis" always occurs at the equatorial 
crossings ( ^ = 0°, 180°) and "periapsis" always 

occurs at the maximum latitude (£ Q = 90°, 270°), 

there being four "apsidal" points during one 
revolution. 

It is noted once again that cj Q gives the loca- 
tion of the minimum point of the eccentrical com- 
ponents of orbital radius, while 4 Q , gives the ex- 
treme of the radius. 

Finally, it should be remarked that the state- 
ment made in Ref. 28 

"e = \r\\ for an elliptical orbit" 

is misleading since it is true only for the non- 
oblate case, while in general e = e (ti, e, i, i Q ) 

and must be computed by Eq (149b). Only for 
large eccentricities is the approximation e = |r|| 
valid for rough engineering estimates. 

e. General comparisons 

It was shown above that to the order J 2 in 

oblateness all the methods considered are identi- 
cal at least in the case of nearly circular orbits. 
Mathematically, Kozai's formulations for the 
instantaneous radius, Eq (132), and secular per- 
turbations, Eqs (126) are generally the simplest 
to use. However, if for any fixed orbit the or- 
bital injection conditions are desired, the results 
of Anthony and Fosdick merit investigation. It 
was thus shown that the classical method of oscula- 
ting ellipses is still valid for nearly circular or- 
bits and that it provides a somewhat clearer ge- 
ometrical interpretation of end results. 



node and the disturbing body, and r be the 

central angle between perigee and the disturbing 
body. Also, let i be the angle between the ve- 
hicle orbit plane and the plane containing the 
origin, perigee and the disturbing body. 



Perigee 



— Y 




Disturbing 
body 



The deviations in the elements are derived in a 
system based on this latter plane. In this system, 
n = 0. u> =0 and i is the inclination. The 

P P P 

solutions obtained for the perigee system are then 
transformed into the solutions in the original X, 
Y, Z system. The solutions are: 



M, 



<M 



r sin T cos T sin i sin 9 
p P P P 

r d (1-e) 3 (1+ e cos 9)* 



+ 2e 2 ) e - 3 (l-9e 2 - 2e 4 ) cos 9 



(13 



4 2 

e (l-6e ) cos 



r (1 + e) sin V sin i cos i 



M r d e 2 (1+ e cos d) 6 



(1 



5. Sola r and Lunar Perturbations 

The problems of defining the changes in the 
motion of an earth satellite due to the presence 
of distant gravitating masses and the discussion 
of the stability of an orbit are of necessity closely 
related. This relationship exists because the two 
analyses differ only in the time intervals consid- 
ered and the fact that forces other than those pro- 
duced by external masses (for example atmospheric 
drag) must be included in the discussion of sta- 
bility. For this reason much of the material 
presented in the following paragraphs is applicable 
to subsequent discussions. 

Analytic expressions for the perturbations due 
to the gravitational attraction of a third body may 
be derived by techniques similar to those used in 
the oblateness derivations. This approach has 
been taken by Penzo (Ref. 33) with the result that 
one set of equations for the variations in the or- 
bital elements may be obtained. This solution is 
outlined below: 

Choose geocentric coordinates with the X-Y 
plane being the orbit plane of the disturbing body. 
Let r be the central angle between the ascending 



+ 3e cos 9) 



i d 3r 3 (1+ 4e 2 ) sin r cos T sin l p E 



+ C; 



^n„ 



r d (l-er 



v>~ 



(152) 



9 9 
r (1+ e) sin r cos i sin 9 " 
p P P 



r d (1 - e) (1 +e cos 6) 



3e 



+ 3(l + e 2 ) cos 9 + e (1+ 2e 2 ) cos 9 



u , r ( 1 + e) *in T cos T (1 + 3e cos 0) 
M d p v P_ FL 



r ,e 2 (1 + e cos 9) 3 



u , 3r 3 (1+ e) sin r cos i 
d P P P E + C 



r d (l-e) 



^ 



Q 



(153) 



IV-36 



^p = -° os i p AQ p 



r 3 (1+e) 3 sin 2 r cos i 
p P P_ 

r d 2e 4 (l + ecose) 3 



+ 3e (4 - e 2 ) cos 6 + 12e 2 cos 2 9 



5e 



r (1+e) sin 6 

p 

r, e 3 (1-e) 2 (1+e cos 8 ) A 



44 e 2 + 13 e 4 - 2 e 6 ) + 3 e (4 - 25 e 2 



(6 



+ 3e 4 ) cos 6 + e 2 (8-37e 2 



4 2 

+ 2 e ) cos 



2 2 2 

(cos T - sin r cos 1 ) 
p P P 



+ e 



f(2 + e 2 ) + 3e( 1 + e 2 ) cos 6 



+ e 2 (1+ 2e 2 ) cos 2 



ej (1-3 sin 2 r cos 2 1 )| 



M d 15 ^pr p e sin2 T 



cos 1 

p p 



2r d (l - e) 3 yT 



E + C 
e 

(156) 



where n , and r . are the gravitational constant and 

orbital radius (assumed constant) of the disturbing 
body, respectively, and the C^C^, etc., are con- 
stants of integration, i.e. , they are functions of 
the initial conditions. 

The transformations to the elements in the X, Y, Z 
system are 



Al = — , — r (cos a sin i 
sin l L p 



sin a cos i cos V) A 1 
p P P 



sin a sin i sin r 



P J 



(157) 



An = 



3r (1 + e) (4cos z r . s in z T cos i -P 



r d (1-e) 



f- 



+ C, 



(154) 



cos v sin 



I 



sin 1 sin r cos i (cos a sin i 
P P P 



- sin acos i cos r ' ) - sin 2 i cos i p sin r p J A i p 

+ (sin 2 isini cos T - sin 2 i sin 2 T p cos i sina)AJ2 p 

(158) 



sin a 



cos u sin i 



r 2 

(sin a sin r cos isini 

3 . L P F 



sin i cos V ) Afi 



o 2 2 

2a p 

r.e 2 (1+ ecos 6) 2 



+ sin V cos i (cos a slni 
P P 



sin acos i p cos r) M ] + A < 



3e sin 2 r cos i sin 6 cos 6 
L p p 

-6e (cos 2 T - sin 2 T cos 2 1 ) cos 6 

. 3 cos 2 T + 3(l+e 2 ) sin 2 T cos 2 i p - e 2 ]+ C a 



Ae = 



P 

Aa 



(155) 



2ea 



ea T p. j p 
8) 



^OT> 



r sin 2 F cos i sin 6 r , 

P £ P,. [e(2e 4 



2r (1 -e) 3 (l + ecos9) 3 



9e 



.2 o _4^ Q j.„/o _ Q „ 2 _ a ^ 4, » ™= 2 e 



+ 3(2 - 9e -3e ) cos 9 +e(2 -9e -8e ) cos ej 



M d ykP r (1 + e) 



T —7 



P '" ' w (cos 2 r -sin 2 T cos 2 i) (1 



r, e (1+e cos 6) 
d 



+ 3 ecos 8) 



where 



sincr 



sin to sin i 
sin r 



The assumptions in the derivation of these solu- 
tions are that r d >> r and that the disturbing 

body does not move during the interval of varia- 
tion. 

Thus, in order to solve for the perturbed mo- 
tion of a satellite it would be necessary to compute 
the perturbations (for some small time, say 1 
period) due to each body being considered, resolve 
these perturbations into a common coordinate sys- 
tem, add the resultant motions, adjust the orbital 
elements and then continue the computation. This 
is obviously a lengthy procedure and is not intended 
to be performed by hand. 

Another approach to perturbations has been 
reported by Geyling (Ref. 34), who presents the 
effects of these remote bodies in terms of varia- 
tions in the position of the satellite in cartesian 
coordinates. Only circular satellite orbits, how- 
ever, are considered. 



IV-37 



Choose X, Y, Z axes such that the orbit 
plane of the disturbing body is the X-Y plane, 
the X axis being in the direction of the satellite ' s 
ascending node. The deviations from the nominal 
trajectory will be given in the £,, r|, ? system, 
which moves with the position in the nominal or- 
bit, i is radial, and r\ is in the direction of mo- 
tion. 




Kft 



3 "d r c 

~s 73 T 



2 sin 2 i sin 4, - f sin 2 i cos 4, 



_ (1 - cost) s in i . . - 

2A(X+TJ sin ^ + 4>) 



- (! + co s i) sin i , , - 

y x. ( \ - n — sin &* 

+ kg cos <|> + k sin 4, 



-4>>J 



(162) 



where r = radius of the circular nominal orbit, 
c 

and the k' s are constants to be evaluated from 
initial conditions. These solutions are indetermi- 
nate for X. = 0, ±1/2, ±3/2, ±1. However, for 
X = 0, i.e., for a stationary disturbing body, the 
particular solutions are 



The position of the disturbing body in the X-Y 
plane is given by the central angle $" = $7 +• \ f 

where <jT is an initial value at t = f = and \ is 

the ratio of the angular velocity of the disturbing 
body to that of the vehicle. Geyling's solutions 
are 



t 3 "d r c 

r d 



■n- (2 cos i - sin i) 



+ ^. sin i cos 2 <j> - 2 sin icos2<j> 



t ' 



3 " d 



2 ,„ 2 . . 2 .. 

g- (2 cos 1 - sin 1) 



4 2 

+ 3- sin i cos 2 4> + 



4\ - 1 



sin i cos 2<J> 



(X + 2) (1 - cos if 
<\ + l)(2\+l) (2\ + 3) 

*2 



cos 2 C$ + <(>) 



(X- 2) (1 +cosl)' _„„,. ., 

+ {k- 1) (2V- 1) (3X.- 3) cos2 <♦ " * ) 



9 2 

+ -j (1-cosi) cos 2(<)> + 4> ) 

2 9 

+ y (l + cosi) cos 2(c)) - J) 



3 "d r c 

11 = " » 7T- 



4 /o 2. ,2, 
•it (2 cos i - sin i 



+ 3 sin 2 i cos 2 $ Q )f - ^1 sin 2 i sin 2<t> 



(163) 



+ k + k sin 4 + k cos . 



(160) 



3 



4 (2 cos 2 i -sin 2 i)f 



11 sin 2 i sin 2 4. - 2 3l " t sln2 4T 



X(4x 2 -1) 
2 



_ (4x + 12\+ 11) (1 - cos t) 
4 (\+l) 2 (2\+ 1) (2x-t-3) 

+ (4X 2 - 12X.+ 11) (1 + cosi) 2 
4(X- 1)^ (2X - 1) (2X- 3) 

+ k 4 + k 5 4 + k g sin 4. -t k ? cos 4, 



sin 2 (4> +<(>)' 



sin 2 {<)> - <(>) 
(161) 



11 2 ~~ 

Yj- (1 - cos i) sin 2(<)> + <t> ) 



j^(l +cosi) 2 sin2((t> - 4> ) 






(164) 



1 + cos i) sin i cos (<j> - 2 4> n ) 



- (1 - cos i) sin I cos (4> + 24> n ) - sin 2 i cos 4> : 
+ •= sin 2 i sin 4> + -p (1 _ cos i) sin i sin (4> 
+ 2 $ Q ) - I (1 + cos i) sin i sin (<j> - 2^ Q )> (165) 



IV-38 



Again, if more than one disturbing body is 
considered, it is necessary to consider them in- 
dependently, compute the resultant displacements 
r|, |, £ in the respective coordinate systems, re- 
solve the displacement vectors and add. Despite 
the limitation imposed by the assumption of cir- 
cular orbits, this approach affords a simple means 
of computing realistic coordinate variations for 
many satellite orbits. 

The magnitude of these radial perturbations 
for near earth circular orbits can be seen in 
Fig. 8. This data is based on the work of Blitzer 
(Ref. 35). 

Another approximate method for computing 
the effects of external masses on the orbit of an 
earth satellite has been reported by M. Moe (Ref. 
36). This work is outlined below: 

First consider the perturbations of a satellite 
orbit due to a disturbing body assumed to be in 
the X-Y plane. The geometry is shown in Fig. 9. 
The orbit will be described in terms of the oscu- 



m , n. 



lating ellipse whose elements are a, e, 

cj, and i, and expressions will be derived to com- 
pute the approximate changes in the elements 
during one revolution of the satellite. The param- 
eters i, to, n, and Tare taken relative to the dis- 
turbing body plane. For an earth satellite, this 
is either the ecliptic or the earth-moon plane. 

Now, if the equations for the variation of ele- 
ments of Section C-l of this chapter are utilized 
together with the components of R, S and W, the 
approximate changes in the elements can be evalu- 
ated. Moulton (Ref. 1, p 340) gives the form of 
these forces. Under the assumption that the ratio 
of orbital radius to the distance to the disturbing 
body is small these components may be expanded 
in powers of r /a, and all but first order terms 

can be neglected. This procedure yields: 
R = K d r (1 + 3 cos 2 F ) 

S = 6 K.r [cos T sin (cj + 6) - sin T cos (oj 
+ B) cos i] cos r 



W = -6K.r cos F sin i sin T 



where 



K d = M d /2a d = MH 



a , = assumed constant, 
d 



Letting c stand for any orbital element and A e 
for the change in that element after one revolution 
of the satellite (from perigee to perigee), we have 



t = 2ir/n 

t = o 



2m 



a. = ^ dT dt = I -- de (i66) 



de dt 

ar as 



= o 



where t is time measured from perigee passage 
of the satellite. Since Ae is supposed to be 



small compared to i , it is permissible to approxi- 
mate all variables in the equations for element 
variations for dc /dt by the values they would have 
in the unperturbed orbit, and to approximate dt/ 
d6 by its relationship to the conservation of angu- 
lar momentum, h 

dt _ r 2 
ao ' IT 



2 J 2 

where h = na Tl - e is assumed constant. 

Since the angular velocity of the satellite is usu- 
ally large compared to the angular velocity of the 
disturbing body, we may further assume that T 
is constant during the time the satellite takes to 
complete one revolution. Then integrals of the 
type in Eq. (166) can be evaluated easily. The 
results are 



Aa = 



(167) 



Aq = 15 Hira 4 e Vl - e <sin 2 T cos 2uCOS 



o 9 2 ) 

sin 2 u (cos T - sin T cos i)| 



where q = r = a (1 - e) 



£ e = -- Aq 

a n 



(168) 



(169) 



Ai = - 3 Hra (2 sin 2 r sin i [l - e 2 (1 

2 yT^ l 

- 5 cos 2 oj)J + 5 e 2 sin T sin 2 u sin 2 ij 

(170) 
£ fi = 1 3 HlTa {5 e 2 sin 2 T sin 2 u 

2 £~7 x 

9 r 2 2 

+ 4 sin T cos i 1(1 - e ) cos to 

+ 4 (1 +4 e 2 ) sin 2 tuj} (171) 

2 I 2 

Au=- cos i Afi + 6 Hira Vl - e 

< 5 sin 2 r sin cj cos <<j cos i 

2 2 2 

- 1 +3 sin T cos i - (5 sin to 

9 2 2 \ 

- 4) (cos T - sin r cos i)| 



(172) 



where 



M 



H 



D 



GM 



D 



^E " 2 ^I^ 



Here, M„ and M„ are the masses of the earth and 
the disturbing body, a D is the average distance to 



the disturbing body. 



IV-39 



G is the universal gravitational constant and n 
is the satellite's mean angular motion. 

For the moon as the disturbing body 

H = H m = 0.68736 x 10" 18 (naut mi)" 3 
= 10.8207 x 10" 20 km" 3 



= 2. 80763 x 10"° (earth radii)" 
If the disturbing body is the sun, then 



Recall from Eq. (157) that 

Aq = A ]sin 2 T cos 2 c cos i 

' / 2 2 

- sin 2 u> (cos r - sin r c 



where 



os 2 i)j 



A =15 HTra 4 e Vl - e 2 



Using trigonometric identities, the expression 
for Aq can be written in the following form: 



H = H g = 0.31584 x 10" 18 (naut mi)" 3 



4.97207 x 10" 20 km" 3 



1.29010 x 10"° (earth radii) 



-3 



Note that H m = 2. 17631 H g , but remember that 

the fundamental planes are different for the two 
perturbations. Assuming that the other variables 
(a, e, i, and u.) remain constant during one period, 
Aq can be integrated from to it (the period of T) 
to give the approximate total change. Dividing by 
tr gives the average change in q for one revolution 
of the satellite. Similarly, formulas for the 
average change in the other parameters can be 
determined to be: 



Aq 



Ae 



7. 5 Hrra e 



- Aq 
a ^sec 



if 



A " „„ = 6 Hira 3 Vl - e< 



■ 2 .. 

sin i) 



2 2 

e sin 2 u> sin i 



(173) 

(174) 

2 
. , 5 sin (i) , 2 
1 + 7 (e 

- 2(l-e' :: ) 

(175) 



Ai 



An 



-3.75 Hrra" 5 , 2 . _ . „ ., 
(e sin 2 uj sin 2 1) 



i« 



'1 - e' 



-3 Hita cos 



(176) 



vT 



i [d-e 2 ) 



2 

COS CJ 



(1 



a 2 \ ■ 2 
4 e ) sin 



»] 



(177) 



where the subscript sec means secular. To com- 
pute the changes per unit time, divide by the 
period of the satellite in the specified time units. 
Note also that H and a must be in units consistent 
with those used for q. 

The above expressions indicate the secular 
trend in the various parameters due to a disturb- 
ing body, for example, the moon. To illustrate 
the meaning and importance of these formulas, it 
is helpful to return to the complete formula for 
the perturbation of perigee distance q. 



Aq = Aq + Aq 

x ^per ^sec 



where subscript per means periodic 

Aq per = A L sin 2 r cos 2 w cos i 

1 2 1 

- ^ cos 2 T sin 2 u (1 + cos i) 



and 



Aq - - >r 

H sec 2 



1 2 

A sin 2 (jj sin i. 



Thus Aq can be expressed as the sum of two 
terms; the first of which is a periodic function 
of r , and the second is independent of r . This 
nonperiodic or secular term is precisely A a 

J M sec 
which was previously derived. 

The effect indicated by the periodic term 
(Aq ) can be better understood if its form is 

changed as follows 

Aq per = AB * sin 2 r cos « - cos 2 r sin a) 

= AB sin (2 V - a) 
where 



Tcos i + 



1 . 2 „ . 4 . 

. sin 2 oi. sin i 



and a = ± cos 



- 1 cos 2 u cos i 
~B" 



with the minus sign 



holding if sin 2 w is negative. 



The formulas for A u, A i, and At. can each be 
expressed in a similar form, and in each case the 
secular terms have already been derived. Since 
the forms of the periodic terms are not important 
for most purposes, they will not be given. 

From this point the method of computation 
parallels Penzo's. 

6. Drag Perturbation of a Satellite Orbit 

The effect of air drag on the osculating orbital 
elements of a satellite can be determined using 
the approach outlined by Moe and discussed under 
solar lunar perturbation. The effect on each ele- 
ment is expressed as the change in that element in 
one orbital revolution. That is, if the elements 
at a certain perigee are a, e, i, co, and fi, then 



IV-40 



the elements at the following perigee will be 
changed by the amounts A a, Ae, Ai, Aw, and 
Aft (Refs. 37 and 38). 

a. Perturbation equations and the drag force 

To obtain expressions for these changes, 
start with Eqs. (178) through (181), relating the 
time derivatives of the orbital elements to the 
components of a general perturbing force. A 
particular form of these equations, given by 
Moulton (Ref. 1, pp. 404 to 405) and Moe (Kef. 
39), is 



da 
dT 



de 

at 



df 



di 

ar 



2e sinO 



R + 



2a Yl 



vT 



(178a) 



sin 



na 



R + 



r S 
dft _ r sin (0 + u) 



Vl - e 2 

2 
na e 


2 i, 2 \ 
a (1 - e ) 

r 




(178b) 


V 


(178c) 



na 



iT 



r cos (0 + o.) 



W 



(178d) 



VI ^ 



and 



where 



w 


= -B p (r) V V sin 3 


B 


C D A 
2m 


m 


= mass of the satellite 



(179c) 



C = drag coefficient 

A = effective area of the satellite 



r = radius vector from the center of the 
_ earth to the satellite 

p(r) = density of the atmosphere at r 

V = velocity of satellite relative to the 

atmosphere 

V„ = velocity of satellite relative to inertial 
space 

V = velocity of the atmosphere relative 

to inertial space 

3 = the angle between V and the plane of 
the orbit 

b. Assumptions and approximations 



ar 



r sin (0 + tu) cot i 



yT 



w 



t 



1 - e 



1 + e cos 



e cos R 
lae 

r ) sin 9 S 
()78e) 



R is the component along the radius vector 
(measured positive away from the center of the 
earth), S is the transverse component in the in- 
stantaneous plane of the orbit (measured positive 
when making an angle less than 90° with the 
satellite's velocity vector), and W is the com- 
ponent normal to the instantaneous plane (meas- 
ured positive when making an angle less than 
90° with the north pole). 

When the disturbing force is caused by air 
drag, the perturbing acceleration is 



1 



(r) V' 



C D A 



B p (r) V 



which has the components, 



R = -B p (r) V V 



e sin 



Yl + e 2 + 2e cos 6 



S = -B p (r) V 



-V cos /3 
a 



V Q (1 + e cos 9) 
Vl + e 2 + 2e cos G 



(179a) 



(179b) 



Equations (168a), (168b) and (168c) can also 
be expressed in terms of the eccentric anomaly 
E, instead of the true anomaly Q. This step is 
desirable since the integration of Eqs. (167a) 
through (16 7e) over an orbital revolution can be 
most easily carried out by using E as the variable 
of integration (limits to 2tt). To facilitate the 
integration, the following assumptions and ap- 
proximations are made: 

(1) The density, p (r), is spherically sym- 
metric. It is assumed to change ex- 
ponentially above perigee height, i.e. , 

-(h - h n )/H 
P (r) = Pp e p (180) 

where p is the density at perigee. It 
is a function of the height, h , of peri- 
gee above the surface of the earth. II 
is the scale height at perigee altitude 
and h is the height of the satellite above 
the surface of the earth. 

(2) In integrating the effect of the perturbing 
force over one revolution, the satellite 

is assumed to move along the unperturbec 
Kepler orbit. This is a good approxima- 
tion because the perturbation has little 
effect on the orbit over one revolution. 
This is not true during the last few 
revolutions of the lifetime. Other 
methods must be used to determine the 
effect of air drag during that short 
time. 

(3) The integrand is expanded in the quanti- 
ty e (1 - cos E) (which is always small 



IV-41 



wherever the perturbing force is im- 
portant). Only the most important 
terms of the series are integrated. 

(4) The entire atmosphere rotates at a 

uniform angular rate equal to the rate 
of rotation of the earth about its axis. 

Several investigators (Refs. 40 and 41) have 
carried out integrations using variants of the 
above approximations. Sterne (Ref. 41), for 
example, in addition to treating the problem with 
a spherically symmetric atmosphere, also made 
a more refined analysis taking account of the 
atmosphere ' s flattening. However, for altitudes 
above 200 naut mi or 3 70 km, the neglect of the 
diurnal bulge causes errors, which overshadow 
the improvement obtained by considering atmos- 
pheric flattening. This was shown by Wyatt (Ref. 
42). Moreover, fluctuation in the density of the 
atmosphere causes uncertainties large enough 
that highly refined expressions for the changes in 
orbital elements are not warranted for most pur- 
poses. 

c. Approximate changes in osculating orbital 
elements 

Given below are methods useful in simplified 
programs, based on approximations (1), (2), (3) 
and (4). Most of the results were obtained in 
series form, but only the dominant terms are 
given here. For higher order terms see Sterne's 
paper (Ref. 41). 

The case of ae/H > 2. When the parameter 
ae/H x 2, the changes in the orbital elements per 
revolution are 



ae/H 



Aa = -Q 



Ae = -Q 



1 + 



1 - 8e + 3e 



8c (1 - e ) 



(181a) 



(v) 



l _ (3 + 4e - 3e ) 



Be (1 - e ) J 



(181b) 



Ai = -D(l - e) 2 |cos 2 u, + 



p-sc- [«(j-H) 



_l i/if* , 9e + 6e - 151 2 
+ |4f + n 1 cos OJ 

(1 - er 



(181c) 



AQ = -D(l - e) 2 Jl +^ 



9e + 6e - 15 



4f* 



(1 - ef 



A oj = - A£2 cos i 



> sin 


to 


COS 


CO 


(181d) 
(181e) 



where 



„ OT , 2 , (1 + e) 

Q = 2B p a f — - 

P 2 

(1 - e) 



m (!-) 



1/2 



2U 

f = 1 - -— (1 - e) 
n 



/l-eN l/2 



f* 



1 - e 



f-fc 



D = 2ttB — ap f l/2 (2irc)" I/ ' 2 
n ^p 

n = angular rate of rotation of the earth's 
e 

atmosphere in inertial space (2it in 

approximately 24 hr) 

It might also be useful to know how the radius of 
perigee, q, changes in a revolution; q is simply 
related to a and e through the equation 

q = a (1 - e) 

Thus, the change in q, when ae/H > 2, is 

A « - -Q (l^l) 2c < 181f) 

and the change in the period can be found from the 
change in a through the relation 



At/t 



(I) 



Aa/c 



The case of ae/il < 2. When the parameter 
ae/H_2, the appropriate changes are 



,3/2 



(1 - 2e)I Q (c) 



+ 2e I, 
G 



(182a) 



(l-e) 1/2 L 
l (c >] 

+ | [l Q (c) + I 2 (c)l | (182b) 

Ai =-K j ["l o (c) - I 2 (c)] + (cos 2 oj) [l 2 (c) 

- 2e I x (c)j( sin i (182c) 

A Q = -K [l 2 (c) - 2e Ij (c)] sin a cos oj 



A< 



-An cos i 



(182d) 
(182e) 



and 



(c)] (182f) 



3e) Ij (c) - | I 2 



where 



C D A 2 -c 

G = 2tt a p f e 

m *p 



IV-42 



K 



C D A Q , 



— a p 
n 'p 



VF, 



and I is the Bessel function of imaginary argu- 
ment and nth order. The secular time rate of 
change of the elements may be obtained by 
dividing Eqs. (181a) through (18 If) and Eqs. (182a) 
through (1821) by the Kepler- period, 
.3/2, r - 



2-na'' 



From Eqs. (181) and (182) it can be seen that 
the rotation of the earth's atmosphere relative to 
the satellite affects the inclination, node, and 
argument of perigee of the orbit. If there were 
no atmospheric rotation lo ^ =0), only the semi- 
major axis and eccentricity (hence the height of 
perigee) would be affected. 

The orbital parameters most sensitive to 
drag are the heights of apogee and perigee, the 
period, and the eccentricity. The reason for this 
sensitivity is primarily the fact that V relative to 
the atmosphere is not vastly different than V rela- 
tive to space. Thus, the perturbing force is 
nearly planar and therefore affects semimajor 
axes and eccentricity. 

The procedure for evaluating the effects due 
to drag is now clear: First the element variations 
are computed, then the elements are adjusted and 
the process continued. If a sufficiently small in- 
terval of time is utilized for the stepping proce- 
dure, say 1 revolution for satellites above ap- 
proximately 180 km, then the element changes 
will be sufficiently small so that they may be 
added to those produced by the sun, moon, ablate - 
ness, etc., to produce a first order approximation 
to the total solution. Numerical data and discus- 
sions of the planar effects are presented in Chap- 
ter V (Satellite Lifetime). Thus, graphical data 
will not be included at this point. Data for the non- 
planar parameters will not be prepared because 
of the fact that too many parameters are involved 
to make such a presentation meaningful. Rather 
it is suggested that these effects be evaluated for 
each orbit. 

d. Contribution of random drag fluctuations 
to error in predicted time of nodal cross- 
ing of a satellite, assuming perfect initial 
elements* 

If the period is known to be exactly P(0) during 
the zeroth revolution, then the period will be pre- 
dicted to be P'(n) during the nth revolution. This 
prediction will be based on the average rate of 
change of period during the preceding revolutions. 
But suppose there are random fluctuations about 
the average change in period. Let these random 
fluctuations be p,, p„, . . ., p., . . ., p N . 

Then after N revolutions the period will actually 
be 



N 



P(N) = P'(N) + 



I 



y- 



i 



*This subsection was included as Appendix E, 
Special Derivations" in Flight Performance 
Handbook for Orbital Operation, STL report 
prepared under Contract NAS 8-863. 



The time of nodal crossing will be predicted 
to be 

.N 

t'(N) = t(0) + ") P'(n) 
n = l 

while the actual time of nodal crossing will be 



N 



N 



(o) + y 



t(N) = t(0) + x P'(n) + ) r(n) 
n=l n=l 



where 

n 

r(n)s I p y 

The error, E(N), in the prediction is 
N N n 

E(N) = -£ r(n)--l £ P f 
n=l n=l j=l 

This double sum can be written out explicitly as 

E(N) = - T( Pl ) + ( Pl + p 2 ) + . . . 

+ (p 1 + p 2 +. . . +p N )]. 

Rearranging terms, we obtain 

E(N) = - Tn Pi + (N - 1) p 2 + . . . + p N ~J . 

(183) 

Case a: Fluctuations Indepen dent from Revo- 
lution to Revolution? If each p. is independent 

and has the standard deviation F, then the 
standard deviation of E(N) is , , „ 

/ N \^ 2 

G„„„(N) = E(N) „ =lF ^ ~ 2 



n=l 



= f[n(N + 1) (2N + l)/6] 1/2 . 

(184) 

Case b: Fluctuation s Correlated over 25 
Revolutions? On the~bther hand, suppose th a t 
the random drag fluctuations are perfectly cor- 
related over intervals of 25 revolutions, but in- 
dependent from one interval to the next. A 25- 
revolution interval is chosen because it is the 
usual smoothing interval in published orbits. 
We begin with Eq (183). 

Since the accelerations are assumed to be 
correlated over intervals of 25 revolutions, 



p q+l = P q+2 



P q = P A 



= P q+25 = P B 



p q+26 P q+27 ' 



= P q+50 = P C 



IV-43 



The fluctuations in acceleration about the 
smoothed value are illustrated in the following 
sketch. 




'////////ti n 



• i i 



q + 25 
n = Revolution number 



50 



The possible values of q range from 1 to 25. 
In the absence of particular information, all 
values of q will be assigned equal weights. When 
n = 1, p = p.. When n = 2, p will equal p. if 

2 < q < 25, and p = p if q = 1. When n = 3, 

p = p . if 3 < q < 25, and p = p„ if q = 1 or 2, etc. 

The equal weighting of the 25 values of q can be 
expressed by averaging over the ensemble of 
possible values, that is 



J l 



J A 



(1/25) (24 p A + p B ) 
(1/25) (23 p A + 2 p B ) 



> 25 =(l/25)(p A+ 24p B ) 



J 26 _ H B 

3 27 = (1/25) (24 p B + p c ) 



p 50 = (1/25) (p B + 24 p c ), etc. 

The timing error, averaged over the ensemble 
of possible values of q, is found by substituting 
these p. 's into Eq (184). 

ETnT = - [Np A + (N - 1) (24 p A + p B )/25 

+ (N - 2) (23 p A + 2 p B )/25 

+ ... + (N - 24) (p A + 24 p B )/25 

+ (N - 25) p B + (N - 26) (24 p R 



+ p c )/25 + ... + (N - 49) (p B 
+ 24 p c )/25 + (N - 50) p c 
+ (N - 51) (24 p c + p D )/25 + ...J , 
for all (N - k)>0 . . . (185) 



Collecting coefficients of p., p R , and p„ 



Let 



ETNT = - (p /25) [25 N + 24 (N - 1) + ... 
+ (N - 24)] - (p R /25) [(N - 1) 
+ 2(N - 2) + . . . + 24(N - 24) 
+ 25(N - 25) + 24(N - 26) + . . . 
+ (N - 49)] - ( P( .,/25) [(N - 26) 
+ 2(N - 27) + . . . + 24(N - 49) 

for all (N - k) -. . . . 

a(N) = [25 N + 24(N - 1) + . . . + (N - 24)] . 
b(N) = [ (N - 1) + 2(N - 2) + . . . + 24(N - 24) 

+ 25(N - 25) + 24(N - 26) + . . . 

+ (N - 49)] 
c(N) = [(N - 26) + 2(N - 27) + . . . 24(N - 49) 

+ 25 (N - 50) + 24(N - 51) + . . . 

+ (N - 74)] 
d(N) = [ (N - 51) + 2(N - 52) + . . . 

+ 25(N - 75) + . . .] 



e(N) 



etc. , 
for all (N - k) -> 0. 



If the standard deviation of p. is <t, and each p. 

is ind ependent, then the standard deviation of 
e7nT is 

K (N)h |eTMt1 =(a/25) Ta 2 (N) 

rms L Jrms L 

1/2 



+ b 2 (N) + c 2 (N) + ...1 



(186) 



In case N <_ 25, a(n), b(n), and c(N) are calcu- 
lated as 

b(N) = (N - 1) + 2(N - 2) + . . . + 24(N - 24), 



N-l 



for all (N - k) -> 
and for N s_ 25 

N-l N-l 



I'-I 



= ^ q(N - q) = N 

q = l 1 1 

= N 2 (N - l)/2 - N(N - 1) (2N - l)/6 
b(N) = [N(N - l)/2] [N - (2N - l)/3] . 

for N < 25 

a(N) = 25(N + N - 1 + ... + 1) - b(N) 



IV-44 



a(N) = 25 N (N + l)/2 - b(N). 

for N < 25 

c(N) = 0. for N < 25. 

In case N is greater than 25, the contribution 
of the first 25 terms in Eq (185) to b(N) is 



24 



24 



24 



b l (N) = A q(N - q) = N^ q -^ q^ 
q=l 1 1 



b (N) = 100 (3 N - 49), 
for 



N > 25. 



a(N) is then given by 

a(N) = 25(N + N - 1 + . . . + N - 24) - b^N) 



a(N) = 025 (N - 12) - b^N), 



for 



N -, 25. 



We define b„(N) to be the contribution to b(N) 

of all those terms of the second 25 terms in Eq 
(185) for which the quantity N - k is positive. 
For N ^ 25, b 2 (N) = 0, and for N ^ 26, b 2 (N) is 
given by 



b 2 (N) = a(N - 25), 



for 



N -, 26. 



b(N) is given by 



b(N) = b^N) + b 2 (N). 
The quantities c(N), d(N), etc., are given by 
c(N) = 0, for N - 26 

c(N) = b(N - 25), 

for N ^ 27 

d(N) = 0, for N <_ 51 

d(N) = b(N - 50), 



for 



Nn 52 



etc. 



Compari son of Case a and Case b . The limits 
of the equations for correlated and uncorrected 
errors will now be calculated, to show how the 
two cases are related. For uncorrected errors 
(Case a), take the limit of Eq (184). 

lim F [N(N+ 1) (2 N + 1)/6] 1/2 = F(N /3) 

(187) 

For correlated errors (Case b), take the limit of 
Eq (186) 



im (a/ 25) |[625 (N - 12) - 100 (3N -49)] 



+ [lOO (3 N - 49) + 625(N - 37) 

- 100 (3 K - 124)1 2 + 

J continued 



+ [lOO (3 N - 124) + 625 (N - 62) 
- 100 (3 N - 199)J + ... \ '" 

= lim a { [13 (N - 8)1 2 + [25 (N - 25)] 2 

+ [25(N-50)] 2 + ...} ^ 2 . 

Let N = 25 M, where M is an integer. Then the 
above limit becomes 

lim (25) 2 ct { M 2 + (M - l) 2 + (M - 2) 2 . . . 
+ l 2 - M 2 + [(13/25) (M - 8/25)] 2 } 



= lim (25)' 

M -, „ 



■{ 



M (M + 1) (2 M + l)/6 



1/2 



- M 2 + [(13/25) (M - 8/25)] 2 | 

= lim (25) 2 a { M (M + 1) (2 M + I) I s\ l 
M ->,*, ■ <■ ' 

? , 1/2 3 1/2 

= (25T a (M 13) -= 5(j (N /3) 



/2 



(188) 



Thus, the limits (5) and (6) for correlated and 
uncor related errors approach the same asymp- 
totic form for large N. This makes it possible 
to evaluate the constant F, which must equal 5a. 
The relationship F = 5a corresponds exactly to 
the situation in the theory of errors, in which 
the standard deviation of the mean of k indepen- 
dent observations equals the standard deviation 
of one observation divided by the square root of k. 
The asymptotic form Eq (188) is a convenient 
approximation to represent the error contributed 
by random fluctuations, when the initial elements 
are perfect. The satellite accelerations, i.e., 
the rate of change of the period published to 
July 1961, furnish no evidence for choosing be- 
tween Case a and Case b, because they are 
smoothed over intervals of 25 revolutions. 

7. Radiation Pressure 



Above a height of 500 naut mi or 926 km, 
radiation pressure usually has a greater effect 
on the orbit of an artificial satellite than air drag 
(though for ordinary satellites, the effects of 
radiation and drag both are very small). How- 
ever, both effects are significant for balloon 
satellites since the area-to-mass ratio is large. 
(The area-to-mass ratio of the Echo I balloon 
satellite was 600 times that of Vanguard I.) At 
first glance it may appear that it is possible to 
handle this force as was done in the previous 
sections. However, this is not the case because 
of the fact that the earth affords a shield from 
the sun 1 s rays during a portion of the orbit. This 
shadow effect is investigated in detail in Chapter 
XIII. 

Kozai (Ref. 43) has integrated the pertur- 
bations of first order over one revolution, in 
terms of the eccentric anomaly, E. The satellite 
leaves the shadow when E equals Ej, and enters 

the shadow when E equals Eg. (Reradiation 

from the earth is ignored. ) 



IV-45 



The perturbations over one revolution are 
given by 



6 a = 2a 3 F (S cos E + T Vl - e 2 sin E) 



6 e 



= a 2 F Vl - e 2 I S Vl - e 2 cos 2E 



+ T (-2e sin E + ^ sin 2E) 



*! 



C TdE 



p (189) 

^2 



(190) 



5i = a'F 



W 



vT 



■[{«'* 



e ) sin E 



e 
4~ 



+ VT 



unless other values are written; S and T are the 
expressions of S (6) and T(6), in which <J> is re- 
placed by io ; that is, 



S = S(0), 
T = T(0). 



(195) 



If the satellite does not enter the shadow dur- 
ing one revolution, the terms depending explicitly 
on E vanish, and, in particular, 8a vanishes. 

In the expressions of 6oo and 5fi, indirect 
effects. of the solar radiation pressure through 
(1) and U must be considered as 



d5w do. . , doj c . ( dto t 

-n- = -j- iet-jr 6i+-j- 5a, 
dt de di da 



d5ft dS2 5 , dn . . ^ dft , 

-37- = j— 5e + 3-- 5l + j— ° a - 
at de di da 



(196) 



- j co s 2E) sin u 



3 
2 e 



f W COS o; dE 



(191) 



sin i SQ = a 



^H 



+ e ) sin E 



T sin 2E > sin w - Vl - e (cos E 



(192) 



t cos 2E) cos oj 


E 


E 


tj- e \ W sin u dE 


' J 


J 



5(i> = - cos i 512 + a F 



vTT^ 



S(esinE 



+ ^ sin 2E) 



cos 2E. 



+ T Vl - e 2 (e cos E 



if 



SdE 



(193) 



ATT 

6M = - | C ^- dM - Vl - e 2 5u 



The disturbing functions S(0), T(6), and W 



S(6) = - cos k- cos y cos (X n - 4> - SI) 



■ 2 i . 2 ( . . 

sm y sm 2 cos * n n " * ' 



k- sin i sin e jcos ( \„ - <t> ) 



W 



cos (-X Q - ^ )| 



• 2 i 2 « . . 

sin *■ cos n- cos (ft - x - <t>) 



5s -5 sin g- cos (-Xq - <j> -ft), 

(19V) 



sin i cos ,,- sin ( \., - £2) 



sin i sin „- sin (x„ + ft) 



cos 1 sin « sin X 







(198) 



where \„ is the longitude of the sun, and e is the 

obliquity. The expression of T(6) is obtained if 
cos in S(6) is replaced by sin except for the trig- 
onometrical terms with an argument i, e, i/2, 
or e /2. 



- Si - e cos i 6ft 



2a 2 F 



is (1 + e 2 ) sin E - I sin 2e| 



T Vl - e 2 (cos E - I cos2E) 



SdE 



(194) 



where the limits of integration are E. and E„ 



The conventional symbols are used for the 
orbital elements: a is the major axis, e the ec- 
centricity, i the inclination, ft the node, 10 the 
argument of perigee, M the mean anomaly, and 
9 the true anomaly. In addition, 



and 



+ 00 



a (1 - e"); 



n 2 a 3 F S(9), n 2 a 3 F T(6), and n 2 a 3 F W are three 
components of the disturbing force due to the 
solar radiation pressure in the direction of the 



IV-46 



radius vector of the satellite, in the direction 
perpendicular to it in the orbital plane, and in the 
normal to the orbital plane; and F is a product of 
the mass area ratio, solar radiation pressure, 
and a reciprocal of GM. 

The smallness of the effect of radiation pres- 
sure on an ordinary satellite is illustrated by the 
orbit of Vanguard I (Rets. 44, 45 and 46). 
Radiation pressure periodically changes its height 
of perigee by about one mile. The effect of rad- 
iation pressure on the period is obscured by the 
fluctuations in air drag. Both radiation pressure 
anil air drag would have had very small effects on 
a conventional satellite at the original perigee 
height of Echo I, but both effects were magnified 
by the area-to-mass ratio, which was, 600 times 
that of Vanguard I. The consequent large effects 
on the rate of change of period are shown in 
Fig. 10, which originally appeared in Ref. 45. 
The correlation of air drag with the decimeter 
solar flux is also shown to persist to this great 
height (see Chapter II). Note also in Fig. 10 that 
radiation pressure sometimes lias no effect on 
the period. This occurs when the whole orbit is 



[E„ = E-, + 2 i in the expression for 
-Sa of Eq (194).] 



in sunlight. L u„ - u. 



The radiation pressure sometimes acts to in- 
crease the period. Echo I was the first satellite 
for which this was observed (Ref. 45). It was 
also the first satellite for which the eccentricity 
was observed to increase. This can be clearly 
seen from the increasing distance between peri- 
gee and apogee in Fig. 11, which is modified 
from the NASA Satellite Situation Report of 
July 18, 1961, though for most satellites the 
eccentricity has decreased during the lifetime. 
Detailed behavior of a satellite due to this per- 
turbation cannot be tabulated in a parametric 
form due to the large number of factors affecting 
the solution. These factors include longitude of 
the nodes, orbital inclination, position of the 
earth in its orbit and semimajor axis and eccen- 
tricity of the orbit. Thus, it is necessary to ob- 
tain a particular solution for the perturbed rates 
of the elements given a set of desired elements, 
then incorporate them in a numerical manner with 
the rates produced by other forces. 

The analyst is urged to consult a growing body 
of literature for this perturbative influence. 
Some of these references have been collected and 
presented as Refs. 1, 34, and 43 through 57. 

8. Satellite Sta bility 

The study of satellite stability concerns the 
long term orbital behavior of artificial satellites. 
It attempts to provide the mission analyst with 
answers to such questions as: How will the various 
orbital elements change? What will be the magni- 
tude of these changes'? Will their pattern be highly 
erratic or regular? Will there be a change in the 
pattern from erratic to regular or vice versa 1 ? 
In order to answer these and other questions it is 
necessary to combine the perturbing forces acting 
upon the satellite orbit and their effect upon the 
various orbital elements of interest for a particu- 
lar mission. 



This section discusses some approximate 
methods for dealing with satellite stability 
problems. The formulas and methods given 
here can be used to: (1) construct approximate 
computer programs, which arc much faster and 
cheaper than ''exact" programs; (2) solve some 
satellite stability problems without the need for 
a high speed computer; (3) help in gaining more 
insight into the behavior of satellites. 

Section C2 of this chapter discussed the ap- 
proximate method of IY1. Moe and presented most 
of the formulas which will be used in this sec- 
tion. The following discussions present some of 
the results obtained using this method. Although 
only earth satellite results are given here, these 
methods have also been used extensively for 
lunar satellites and can be applied to orbits 
about other planets. Part 2 illustrates a method 
for computing satellite trajectories by hand. 

Care must be taken not to use the methods of 
this section on orbits which are physically too 
large, in which case the approximations for 
luni -solar perturbations break down. While 
definite rules cannot be laid down, Table 4 
should prove helpful. The table lists ttie various 
bodies and the approximate upper limits where 
"very good, " "good, " and "fair" results can be 
obtained. The parameter used is the period of 
the satellite in days. 



TABLE 4 



Validity of the "Approximate" Method as 
a Function of Orbital Period (days) 



Ear tli 



Moon 



Mars 



Venus 



Mercury 



Very 
Good 

2. 

0. 5 

45. 

15. 



Good 



60. 



Fair 



90. 



5. 



25. 



8. 



35. 



10. 



A special case arises for very remote earth 
satellites which do not pass near the moon. 
These may also be treated by approximate meth- 
ods and in these cases some orbits with periods 
as long as 45 days can be studied. For this class 
of orbits the effects of the moon are ignored and 
the sun is treated as the only disturbing body. 
Another class of orbits for which the methods of 
this section are not very helpful is the very 
near eartli orbit where drag and oblateness 
perturbations are predominant. 

Accurately predicting the future history of an 
artificial satellite is difficult and expensive. 
Fortunately approximate methods often give good 
results. This section discusses approximate 
methods which have been extensively used for 
terrestrial and lunar satellite orbits. 

It is convenient to consider the stability of the 
orbit of an earth satellite as a two -body problem 
with perturbations introduced by the sun, moon, 



IV-47 



earth shape, drag and radiation pressure. These 
effects must be analyzed separately and then 
combined. This procedure is accomplished only 
after allowing for the fact that the various equa- 
tions refer to different planes; the results can 
then be summed to yield the new orbit. The 
process can then be repeated. 

Performing this operation by slide rule or 
desk calculator is very slow and requires about 
8 hr to compute the change for one revolution, or 
1 man -year for 1 month of the satellite's orbit. 
However, the combined equations can be eval- 
uated on a high speed computer such as the IBM 
7090 at the rate of about 5 rev/ sec. Subsequent 
paragraphs of this section discuss results ob- 
tained in the latter manner. 

When high speed computers are not available, 
good results can be obtained by using the secular 
terms to estimate the results over many revolu- 
tions. This method is illustrated in Part 2. 



but the period is one -half year and the amplitude 
is about 200 naut mi or 370 km. Figure 12 is a 
graph of perigee height versus time. Note that 
the moon waves are shown only for the first 100 
days. The rest of the curve shows the envelope 
of minimum perigee height. This simplification 
is adopted for all similar graphs in this section. 
Note also that the moon waves should be just a 
sequence of separate points plotted at 1.73-day 
intervals since perigee is reached only once 
each revolution of the satellite whose period was 
1 . 73 days. 

Now consider the combined secular effect 
caused by the sun and moon. This is given by 
the following formula which is derived in Section 
C5 of this chapter. 



A q 



K* 



sin 2o.- sin i 



+ A sin 2t, 



2 . 



(199) 



Part 1: Sample Results by "Approximate" 
Method. Early in 1961, a~study (Ref. 58) was 
made at STL to determine the lifetimes of earth 
satellites in highly eccentric orbits. The project 
was the Eccentric Geophysical Observatory 
(EGO). Some of the results of this study will be 
used to illustrate the approximate method and 
the general problem of orbital stability. 

The experimental objectives of Project EGO 
made it desirable to keep perigee height as low 
as possible consistent with lifetime require- 
ments. A graph of the suggested nominal an- 
swering these requirements is shown in Fig. 12. 
This graph will be discussed in detail since it 
illustrates most of the important features of this 
type of orbit. The initial conditions in terms of 
equatorial spherical coordinates are given in the 
figure. These were the suggested burnout con- 
ditions of the missile which were to inject the 
satellite into orbit. The resulting orbital param- 
eters in terms of equatorial coordinates are as 
follows: 



where 



a = 32, 879 naut mi 
= 60,892 km 

e = 0. 891057 

i = 31. 289° 
a 



U = 41. 796° 
a 



= 135.617 



Launch time = 
3 hr 30 min GMT 

Launch date = 
1 April 1963 



The most important parameter in the EGO 
study is perigee height or equivalently perigee 
distance q and to the first order, the only per- 
turbations affecting q are caused by the sun and 
the moon. The periodic term for the lunar per- 
turbations of q may be written as 



q 



per 



A B sin (2T + 
m m m 



*m> 



and ( v are as given in Section 
m fo 



where A , B , 
in m 

05 of this chapter. Therefore the moon causes 

the satellite's perigee to alternately rise and 

fall. The period is one-half the moon's sidereal 

period or a little less than fourteen days. The 

amplitude for EGO-type satellites is about 40 

naut mi or 74 km. The sun has a similar effect 



A = 15 II tt a 4 e VI - e 2 
m m 



f> 



and 



15 H ira e 
s 



V7 



Recall that H and H are positive constants, 
m s ' 

Note that the subscripts m and c indicate moon 

plane and ecliptic plane parameters. Equatorial 

parameters will be indicated by the subscript a 

in the following discussions. 

Initially, the nominal orbit had equatorial 

parameters i =31.29°, V = 41. 80° and 
' a a 

co =135.62°, and gj =94.68°, i =20.30°, 
a m t 

Si =87.47°, and oj = 85. 69°, respectively. At 

the end of 402 days, the orbit parameters take 
on the values: a = 32, 793 naut mi or 60, 733 km, 
e= 0.8893, i =37.58°, Si = 8. 55°, c, =181.38°, 



= 16. 11°, 



= 187.07°, i = 14. 75", and 



o. = 167. 96°. Note that the secular trend is now 

nearly 0, which is again shown in Fig. 12. At 
the end of 554 days, the orbit parameters are: 
a = 32, 779 naut mi or 60, 707 km, e = 0. 8902, 

„ = 195.01°, 



i a = 36.87°, U a = -1.65 

i = 16.77°, w 
m 



214. 50°, i = 13. 45° 



and 



oj = 198.43°. The secular trend is now negative. 



Now a brief discussion will be given of the 
other figures in this section. In the initial EGO 
study (Ref. 58), the burnout conditions of the 
missile were given. The only variation per- 
mitted was in time of launch. A series of satel- 
lite lifetime runs (Ref. 59) were made on the 
IBM 7090 with 1 April 1963 as launch day. The 
first run was at hr GMT, the next at 2 hr and 
so forth to 24 hr. The results are illustrated in 
Fig. 13. 

At first glance, it is surprising that merely 
changing the launch time would have such a large 
effect on the satellite's future history. This 



IV-48 



behavior results since changing the launch time 
of day changes the satellite's nodal longitude 



(U). At h, V 



-10.849. From then on Q 



i , and 
m 



increases by 30.083° for each 2 hr added to the 
launch time. This, of course, is due to the 
earth rotating 360. 996° in 24 mean solar hours. 

Changing ii does two important things. First, 

it changes the phase of the sun and moon desig- 
nated by T and Y . For EGO -type satellites, 

the moon's periodic effect is only about 40 naut 
mi or 74 km in amplitude and hence is not too 
critical. The sun's periodic effect, however, is 
very important. Secondly, changing il changes 

the ecliptic and moon plane parameters of the 
orbit and hence changes the secular trend of the 
satellite. The secular trend is large and posi- 
tive for the 8-, 10-, 12-, and 14-hr orbits. 

In Fig. 14 comparison is made between ap- 
proximate results as obtained from the Satellite 
Lifetime Program (Ref. 59) and results obtained 
by integrating the equations of motion in a way 
that is essentially exact. Note that the agree - 
ment is good. 

Figure 15 illustrates how oblateness indirectly 
affects perigee height even though its direct 
effect is zero to first order. It does this by 
changing the equatorial inclination i and the nodal 

longitude ii a . This in turn changes the ecliptic 

and moon-plane parameters i , 

This then changes the secular effect as is shown. 

In Fig. 16 the effect of leaving out the effects 
of sun or moon is demonstrated. Here the nomi- 
nal graph is shown in comparison with the same 
orbit computed with the sun only and with the 
moon only. Note especially the difference in 
secular trend. 

The effect of making various changes in the 
initial parameters of the nominal orbit is shown 
in Figs. 17, 18, 19 and 20. 

The graph of the 6-hr orbit for a period of 

10 yr is shown in Fig. 21. This orbit illustrates 

an important phenomenon. From the secular 

trend in perigee distance given by Eq (185) it 

follows that A q depends mainly on the incli- 

sec J 

nation and argument of perigee. The inclination 
does not change very rapidly; however, the argu- 
ment of perigee is perturbed very much by oblate- 
ness and to a lesser extent by luni-solar effects. 
As i increases, oblateness perturbations get 

smaller (0 < i < 03. 7') and as a result cj and 

o. change slowly. Thus the secular term can be 

nearly constant over a long period of time. If 
this happened when the secular trend was down, 
the satellite would probably expire. This effect 
also explains the short life of most lunar satel- 
lites (Ref. 58). 

Part 2: Hand Calculation of an Earth Satellite 
Orbit! The detailed revolutum by revolution ap - 
proximate calculation of a satellite orbit is too 
slow and tedious to be practical by hand. However, 



the process can be accelerated by treating the 
periodic and secular terms separately. 

To illustrate this method, part of the tra- 
jectory of the EGO Nominal will be calculated 
(see Fig. 12). 

Consider first the periodic term for the 
lunar perturbations (given in Section C2 of this 
chapter). 



Aq , .. = A B sin (2T , - a ) 
per(mt) m m mt m 



where 



il = 0.68736 x 10 ~ 18 (naut mi)" :i was 
evaluated in Part 2. 



A = 15. 3 naut mi = 28. 3 km 
m 

H = 0. 961 
m 

a = -170. 64° 
m 

(Note that the minus sign is taken when 

sin 2gj is negative. ) 
m h 

The parameter Y , denotes the angular 
mt h 

position of the moon measured from the satel- 
lite's ascending node at time t (see Fig. 9). 
This parameter is given by the following formula. 



r = (t - t ) n - !.■ 
mt m m mt 



where 



t = time the moon was at its ascending 
equatorial node 

n = moon's angular- rate = 

m t 

m 

i. ! , = satellite's moon-plane ascending 
node measured from the moon's 
equatorial node 

t = time . 

If time is measured in days, and angles in degrees 
and if the initial time t., = 

then 

t = -6.9658 davs (ephemeris) 
m ^ 

n = 13. 176°/day 
m J 

V = 67.58° 
m 

t = (initially) 

T = 24. 14° 
mo 

T = 24. 14 + 13. 176° 
mt 

where t is measured in days. 

Substituting the computed values of A , B , 

and „ gives m m 

m b 

Aq , . = 14. 7 sin (2 T +170.60) 

per(mt) mt 

= 14.7 sin (218. 92 + 26. 352 1). 



IV-49 



The period of the satellite once again is 1.73 
days. Hence the periodic term alone indicates 
that the moon's gravitational field will push the 
satellite down for four revolutions. The satellite 
will then be at a minimum height as far as the 
periodic effect of the moon is concerned. From 
then on this periodic motion can be ignored (see 
Fig. 12). 

Evaluating Aq , ,■. for time t = 0, t = 1. 73, 
b T>er(mt) 

t = 3.46, and t = 5.19 days, and then summing 

gives the initial downward push by the moon to 

be 36. 2 naut mi or 67. km. 

Consider now the periodic term of the sun's 
perturbation in perigee distance as measured 
from the center of the earth (q) 

Aq , ,, = A B sin (2 T . - a ) 
^per(et) t t etc 



1 / 2 

Aq = - «- [ A sin 2uj sin i + 

M sec 2 V m m m 



+ A sin 2 .. 



Aq = +0.0319 naut mi /rev. = +0. 0591 km/ rev 

sec 

Assuming the various parameters are relatively 
invariant during the first 164.35 days, the secular 
rise in perigee height for this period can be com- 
puted as 

v Aq = l^Lj^L (0.0319) = 3.0 naut mi or 

'sec 1. 73 

5 . 6 km . 

The combined periodic and secular results indi- 
cate that perigee height should have decreased by 

3 6.2 + 21.0 -3.0 =54. 2 naut mi or 100. 4 km. 



where 

A = 7. 03 naut mi = 13 km 

£ 

B = 0. 961 

£ 

a =171. 38°. 

€ 

The parameter r is given by 

r = (t - t ) n -si • 

6 t £6 6 t 



This checks reasonably well with the results 
shown in Fig. 12. 

Better results could be obtained by summing 
the secular perturbations over perhaps 20- or 50- 
day intervals and taking into account changes in 

the parameters e, i , u. , i and <~ (in such com - 
1 m m £ e 

putations the periodic terms in these parameters 
are not important). The main difficulty here 
would be in converting solar and lunar perturba- 
tions into changes in the equatorial parameters. 



t = - 11.4258 days 

£ 

n =0. 9856° /day 



S. , = 87. 47° when t = 

€ t 



r e0 = -76.21". 



Thus 



r = - 76.21 + 0. 9856 t° 

£ t 

whore t is measured in days. 



Combining the above equations gives 

Aq , . = 6. 59 sin (2 r , - 171. 38) 

per (ft) et 

= 6. 59 sm (36. 20 + 1. 9712 t). 

Note that the sun's periodic effect is initially 
upward. But after about 146 days, this upward 
move is cancelled. The satellite than has about 
18. 4 days or eleven revolutions to reach a min- 
imum, i: valuation Aq . at time t = 147.05, 
n per (ft) 

t = 148.78, t = 150.05 1 , • ■ • , t = 164.35- that 
is, once each revolution from time t = 147.05 to 
t = 164.35--and summing yields the net downward 
push of the sun as 21 naut mi or 39 km. The 
satellite will then be at a minimum height as far 
as the periodic effect of the sun is concerned. 
From then on this periodic motion can be ignored 
(see Fig. 12). 

Now consider the combined secular effects of 
the sun and moon on perigee distance q: 



Using this method with, say, 50-day steps 
should yield results of fair accuracy for many 
satellite orbits. For example, the hr, 2 hr, 
8 hr, 10 hr, 12 hr and 14 hr would be quite easy 
to compute by hand (see Fig. 13). Hand com- 
putation of the orbit of a lunar satellite is also 
easy because the moon's equator is very close 
to the ecliptic, and because the sun's effect is 
very small compared with the effect of earth. 



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Blitzer, L., and Wheelon, A. D. , "Oblateness 
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IV-53 



Dobson, W. F. , Huff, V. N., and Zimmerman, 
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IV-54 



King-Hele, D. , 

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Krauss, L., and Yoshihara, 



H. 



'Electrogas - 



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Lass, H. , and Solloway, C. B. , "Motion of a Satel- 
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Levin, E. , "Satellite Perturbations Resulting 
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IV-55 



Observations of Asteroids, " Bull. Inst. Theo- 
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Mersman, W. A. , "Theory of the Secular Varia- 
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Miller, G. K. , Jr. , "Determination of Ballistic 
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Moe, M. M. , and Karp, E. E. , "Effect of Earth's 
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Moran, J. P. , "The Effects of Plane Librations 
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Mullikin, T. W. , "Oblateness Perturbations of 
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Musen, P. , 

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Musen, P. , Bailie, A. , and Upton, E. , "Develop- 
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Nonweiler, T. R. F. , "Perturbations of Elliptic 
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IV-56 



Ballistic Missiles and Space Technology, 
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Sarychev, V. A. , "Influence of the Earth's Oblate- 
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Sedwick, J. L. , Jr. , "Interpretations of Observed 
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Socilina, A. S. , "On Accumulation of Errors in 
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Struble, R. A. , and Campbell, W. F. , "Theory of 
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IV-57 



"Motion of an Artificial Earth Satellite in the 
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Upton, E. , Bailie, A. , and Musen, P. , "Lunar 
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Vinti, J. P. , 

"Theory of the Effect of Drag on the Orbital 
Inclination of an Earth Satellite, " Journal of 
Research, National Bureau of Standards, 
Vol. 62, No. 2, February 1959, pp 79 to 88. 

"New Method of Solution for Unretarded Satel- 
lite Orbits," Journal of Research, National 
Bureau of Standards, Vol. 62B, No. 2, 
October to December 1959, pp 105 to 116. 

"Theory of an Accurate Intermediate Orbit for 
Satellite Astronomy, " Journal of Research, 
National Bureau of Standards, Vol. 65B, 
No. 3, September 1961. 

"Intermediary Equatorial Orbits of an Artificial 
Satellite, " National Bureau of Standards, 
Report No. 7345, October 1961. 

"Formulae for an Accurate Intermediary Orbit 
of an Artificial Satellite, " Astron. J. 66, 
No. 9, November 1961, pp 514 to 516. 

Ward, G. N. , "On the Secular Variations of the 
Elements of Satellite Orbits," Proc. Roy. Soc. 
(London) 266A, 27 February 1962, pp 130 to 137. 

Westerman, H. R. , "Perturbation Approach to the 
Effect of the Geomagnetic Field on a Charged 
Satellite, " ARS Journal, Vol. 30, No. 2, Feb- 
urary 1960, pp 204 and 205. 

Whittaker, E. , and Robinson, G. , "The Calculus 
of Observations, " Blackie and Son, London, 
1924. 

Wong, P. , "Nonsingular Variation of Parameter 
Equations for Computation of Space Trajec- 
tories, " ARS Journal, Vol. 32, 1962, pp 264 
and 265. 

Yatsienskii, I. M. , "Effect of Geophysical Factors 
upon the Motion of an Artificial Satellite, " 
Uspekhi Fiz. Nauk, No. la, 1957, p 59. 



IV-58 



ILLUSTRATIONS 



IV-59 




o 
c 

'3 

cr 

S-, 

o 



CO 

■a 



c 
he 

a 

c 
o 



JO 

a, 



in .° 



o o 
U o 






(J = jCjJABjg -[8A9t B3S) SSB]/M 1T U H • ia d SOJOjI 



"«>Cli>tf4€ «**GE BLANK NOT FILWFt 



IV-61 



90- 
80- 



70- 



60- 



i> 




T) 






M) 


•^ 




^ 




c 




o 












rt 




a 








O 




S3 




t— i 




, i 




at 












u 


40 


O 





30- 



20- 



10- 
0^ 



3ttJ 2 (— ) (2 - | sin 2 i)rad/rev 



J 2 = 2/3 J 




20 40 60 80 100 
Inclination (deg) 



,6,000 



2.0 



,8, 000 



X 



,9,000 



3.0/X/10, 000 Of 




12,000 



14,000 
+ ">^ .16,000 



> 4.0/ 

'$ 5.0^/18,000 

6.0^ 
7.0' 



r-24 



-22 



-20 



-If 



-16 



-14 — 



-12 % 



T3 



■10 .3 

a)" 

PS 
■8 § 



41 
O 
0) 
Li 
P. 
Li 

3 
O 




-6 



-4 



-2 



--2 



--4 



C--6 



Fig. 2. Solution for the Secular Precession Rate as a Function of Orbital Inclination and Semiparameter 



IV-62 



AM = 3 
M 2~ J 2 



Tf 



1 -e' (1 - 3/2 sin i) 



~ 2.0 



90- 
80- 



70- 



60- 



<D 




TJ 




C 


rso 


O 












m 




C 








r— 1 




o 




c 




•—4 




r— 1 




til 




■■-J 


40 


JJ 




In 










30- 



20- 



10- 
0- 



T3 
a! 
S-i 



1.0 



-1.0 



-2.0 



; t&itl 11 Willi 1 




r: ::ip::±I 


-+l : 


ft 


i + " T 1 1 1 f Mri>lT" 3 


:±fff; 


g R/p = i; 


-If 


"+^"iiffinffli 






ffij:: 








ft Trfflii" 1 


trt 


: tg'i(|m|fflj 




lis- 


if .';•■ 


-f 


-f$$ffi^ffi f 








± 








jll: 








_i: 


TTrrmi / 






- tt "H " 


■T- 


: f fffj|[j R/P - -l/i 


TrHrrr 


-ffiff : 


::t: 








II 


'+ ; 



1-2.0 



10 20 30 40 50 60 70 80 90 
Inclination, deg 



1.0 



<* % 2.0 



-1.0 




^-1.0 



Fig. 3. Change in the Mean Anomaly Due to the Earth's Oblateness 



IV-63 



1. 5 




1—6,000 



2.0- 



o 






<a 

CO 



-6,200 

-6,400 

-6,600 
6,800 
7,000 



B 
-7,500 "^ 



2.5- 



-8,000 g 



.-8,500 



3.0- 



-9, 000 



3.5- 



4.0- 



-9,500 
-10,000 



5.0-" 

6.0- 

7.0-i 

10.0- 



--11,000 

-12,000 
13,000 
14,000 
15,000 



10 20 30 40 50 60 70 80 90 

Inclination (deg) 



d 
U 

a 



a; 
co 



= 3 Mf) 



J 2 = 2/3 J 




cos i rad/ rev 



Fig. 4. Solution for the Secular Regression Rate as a Function of Orbital 
Inclination and the Semiparameter 



IV -64 



— ^=-= (^- - l\ 1 = 3 J ( n ) 2 ( 3cos 2 i- l \ 

;VT~7 U 'VT7" Mpj ^ — 5 — > 



90- 
80- 



70- 



60- 



3 

c 50- 
o 

c 



O 



40- 



30- 



20- 



10- 
0- 



o 



t- 

<l 



|t- 




-0.5 



20 40 60 80 100 120 

Inclination (deg) 



1.0 



1.5 



r 9t 



"to*/*' 



3.0 



4.0 



p-1 





~-0 


9 


ro 


8 


~-0 


7 


~—o. 


6 


r°- 


5 


_ 




r°- 


4 


E-o. 
r-o. 


3 7 

o 
2 * 


^-0. 


n! 
1 < |"- 


r-o 




r--° 


. 1 


r--° 


.2 


r-0 


3 


r—0 


4 


~--0 


5 


t--0 


6 



Fig. 5. Change in the Anomalistic Period Due to the Earth's Oblater 



IV-65 



90-. 
80- 

70- 



60- 



bfl 

I 5 °" 



40- 



30- 



20- 



10- 
0- 



^=(f-.v- 2 (D 2 K^) 

n \ n / 



o 
x 



< 



It- 




20 40 60 80 
Inclination (deg) 



1.0 



r-2.5 



-2. 



— 1.5 



--1.0 




li- 



— 0. 5 



•— +0. 5 



Fig. 6. Change in the 



Nodal Period Due to the Earth's Oblateness (for small eccentricities) 



IV-66 



e = 0.00049 
e = 0.00030 
e = 0.00012 
e = 0.0 




Fig. 7. The Variation 



135 180 225 270 
True Anomaly (deg) 



315' 



360 



True Ano m SeaX Radial ^^ " FUnCU ° nS ° f th ° 



IV-67 



Circular Orbit Radius in 10 km (1 ft 
50 60 70 b|o 




B 
.^ 

c 
o 

XI 
t< 

3 

£ 
a* 



a 



S £ 

3 _ 

s ~ 

■l-« 
nl 

s 



:--! 



i— 



Circular Orbit Radius (ft x 10 ) 
Fig. 8. Maximum Radial Perturbation Due to Attraction of the Sun and Moon 



IV-68 




Ascending node 
of satellite orbit 



To disturbing body 



Fig. 9. Satellite Orbit Geometry 



IV-69 



a> ~ 




M 73 




c ° 


o 


m ■£ 


^ 


Oft, 


>> 

nt 




T) 


o « 




-C 




ID ■"-■ 


73 



o 



0.5 
0.4 
0. 3 
0. 2 
0.1 

240 
2 20 
200 
180 
160 
140 
120 
100 



Attributed to: 
. . . Air drag 
ooo Solar radiation pressure 



„o°°o 



' 00 oOooooooooo**.° . 



..'»»' 



Ad°° 



o°°°o°Oo • 



— oooooooooo 



-L- 1_ 1 1 I I I L 



37, 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 306" 310 



Aug 12, 1960 Sep 



Oct 



Nov = T 







Dec 



Jan 1961 



Fig. 10. Effects of Solar Activity on Echo I 



p 
a_ 
<u 

0) 
OD 

U 
D 

ft, 

T3 
C 
a) 

0) 
<D 

O 

a 
< 



01 

X 



8U0- 




















700- 




















600- 






















iApogee 












500- 


1 Perigee |||||||l|||||l |Wffl 










400- 


















300- 




















200- 























Aug 
12 

1960 



Sep 
21 



Oct Dec Jan Feb 
31 10 19 28 

1961 



Apr 
9 



May 
19 



Jun 
28 



1400 

1300 

1200 

1100 

1000 

900 

800 

700 

600 

500 

400 



Aug 
7 



a 




<u 


.— ^ 


b£ 


b 


u 


X 


ft, 


in 


T) 


oo 




-" 


a> 


ttl 


Hi 


•r-> 


to 

O 


a 


(1, 




-"I* 


3 



<~ c 



Fig. 11. Apogee and Perigee Heights on Echo I 
(40 -day interval) 



IV-70 



(un\ gcj8 -\ = mi jncu i) 




X3 





C 




o 










r 


al 


o 


XI 


c 


tj 


t 


3 


a! 


^ 


>— * 


fl) 


s 


k 


o 


!h 


s- 


a) 








o 


U) 




>-, 




nl 


C 


n 


3 


C4— 


u 





TI 


a 
o 


c 

a! 










r> 


Wi 


c 


ai 



fa Q 



£ 


CD 


hi 






ai 






£ 


X) 

o 


<D 






n 


hr 








t~ 


o 


<ii 


a) 


Oh 


^ 


F 


fa 




M 


F 


C 




•< 


c 


o 


_ 


.c 


<5 


(/J 



CO 

fa 



(itu }neu) }q3f3H asSi-tad uitilutuii/\[ 



IV-71 




000 



-■6000 



5000 





6 






^ 


^ 










^ 


h 




r, 


y: 


4000 


M 


(M 




<u 


ic 




X 


CO 




0) 


-* 




<u 






Ul 












1- 

0) 


s 




CU 


3 


3000 


E 

s 

'c 


cd 
C 



2000 



1000 



100 200 300 400 500 600 700 800 900 1000 
Time from Launch (days) 



Fig. 13. Minimum Perigee Height of Satellite as a Function of Days 
from Launch (8 to 14 hr, expanded scale) 



IV-72 



J3 

■an 
% 

X 
a 

01 
60 



380 



340 



300 



260 



220 



180 



140 



100 




20 30 40 50 60 70 

Time from Launch (mean solar days) 

Fig. 14. Comparison of Approximate and Exact Solutions of Satellite 
Motions 



1200 



J5 
M 

'a 

X 

0) 
0) 




Fig. 15. 



300 400 500 600 

Time from Launch (days) 

Minimum Perigee Height of Satellite as a Function of Days 
from Launch, Showing Effect of Neglecting Oblateness 



IV-73 



.a 

n 

X 
<u 

V 

u 

01 

£ 

3 



1600 




150 200 250 300 350 400 450 500 550 600 650 700 750 
Time from Launch (days) 

Fig. 16. Minimum Perigee Height of Satellite as a Function of Days from 
Launch, Showing Effect of Neglecting Moon, Sun 




50 100 150 200 250 300 350 400 450 500 550 600 

Time from Launch (days) 



650 700 750 800 850 900 



Fig. 17. Minimum Perigee Height of Satellite as a Function of Days from Launch, 
Showing Effect of Sun and Moon 90° Out of Phase 



IV-74 



ii;itl600 
1500 



3 

a 



J3 

K 

0) 
01 
Oj0 



£ 

3 




50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 

Time from Launch (days) 

Fig. 18. Minimum Perigee Height of Satellite as a Function of Days from 
Launch, Showing Effect of Changing Orbit Size (A r = 100 

mm ' 



<D g 
X M 
CD OJ 

coco 



3 ^ 



.S c 



max 



6000 naut mi) 



3 



CP 
M 




50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 

Time from Launch (days) 



E 



hi) 


^ 


CI) 


E 


X 


^ 




CM 




in 


hn 


00 






i< 


^H 


CU 




eu 


" 






E E 


-i 








fa 


3 



a c 
2^ 



Fig. 19. Minimum Perigee Height of Satellite as a Function of Days 
from Launch, Showing Effect of Change in Inclination 



IV- 75 




T5 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 

Time from Launch (days) 



1400 

1300 

1200 

1100 

1000 

900 

800 

700 

600 

500 

400 

300 

200 

100 








a 11 

•rH 

t-l H 

(£'' 
■r-t 

a s 

S 2 



§C 



Fig. 20. Minimum Perigee Height of Satellite as a Function of Days from Launch, 
Showing Effect of Change in Argument of Perigee 




1200 1600 2000 2400 2800 

Time from Launch (days) 



3200 



6 



J3 



m 



U " 

s s 

U 
■as 



Fig. 21. Minimum Perigee Height of Satellite as a Function of Days from Launch 
for About a 10- Year Period 



IV-76 



CHAPTER V 
SATELLITE LIFETIMES 

Prepared by: 

G. E. Townsend, Jr. 
Martin Company (Baltimore) 
Aerospace Mechanics Department 
March 1963 

Page 
Symbols y_4 

A. Introduction y_2 

B. The Drag Force y-2 

C. Two-Dimensional Atmospheric Perturbations . . . v-8 

D. Three -Dimensional Atmospheric Perturbations . . V-21 

E. The Effects of Density Variability v-25 

F. References V-30 

G. Bibliography V -30 

Illustrations V-35 



LIST OF ILLUSTRATIONS 



Figure Title Page 

1. Drag Coefficient for a Sphere at 120 km 

Versus M V-37 

2. Cone Drag Coefficient, Diffuse Reflection V-37 

3. Drag Coefficient for a Rich Circular Cylinder 
with Axis Normal to the Stream at 120 km 

Versus M V-37 

00 

4. Comparison of Drag Coefficient of a Trans- 
verse Cylinder for Specular and Diffuse 

Reflection V-38 

5. Cone Drag Coefficient, Comparison of Free 
Molecular and Continuum Flow Theory; a- = V-38 

6a. ARDC 1959 Model Atmosphere V-39 

6b. ARDC 195 9 Model Atmosphere V-40 

7a. Logarithmic Slope of Air Density Curve V-41 

7b. Logarithmic Slope of 1959 ARDC Atmosphere V-41 

8. Values of True Anomaly as a Function of 
Eccentricity for Which pip (h ) = Constant 

(exponential fit to ARDC 1959 atmosphere) V-42 

9. Nondimensional Drag Decay Parameters for 

Elliptic Satellite Orbits V-43 

10. Decay Parameters P and P for Elliptic Orbits V-44 

11a. Apogee Decay Rate Versus Perigee Altitude V-45 

lib. Perigee Decay Rate Versus Perigee Altitude 

(Part I) V-46 

lie. Perigee Decay Rate Versus Perigee Altitude 

(Part II) V-47 



V-ii 



LIST OF ILLUSTRATIONS (continued) 

Figure Title Page 

12a. Apogee Decay Rate Versus Perigee Altitude V-48 

12b. Perigee Decay Rate Versus Perigee Altitude 

(Part I) V-49 

12c. Perigee Decay Rate Versus Perigee Altitude 

(Part II) v " 50 



13. Satellite Lifetimes in Elliptic Orbits V-51 

V-52 



14. Generalized Orbital Decay Curves for Air 

Drag 



15. Comparison of Errors in Orbital Prediction 

for Correlated and Uncorrelated Atmospheric 
Density Fluctuation V-5 3 

16a. The Ratio of the rms Error in Orbital Pre- 

diction Caused by Sinusoidal Drag Variations 
to the Amplitude of the Sinusoidal Variation V-54 

16b. The Ratio of the rms Error in Orbital Pre- 

diction Caused by Sinusoidal Drag Variations 
to the Amplitude of the Sinusoidal Variation V-54 

16c. The Ratio of the rms Error in Orbital Pre- 

diction Caused by Sinusoidal Drag Variations 
to the Amplitude of the Sinusoidal Variation V-5 4 

16d. The Ratio of the rms Error in Orbital Pre- 

diction Caused by Sinusoidal Drag Variations 
to the Amplitude of the Sinusoidal Variation V-54 

17. The Ratio of the rms Error in Orbital Pre- 
diction Caused by Random Drag Fluctuation 

from Period to Period V-55 

18. The Ratio of the Error in Orbital Prediction 
Caused by Smoothed Observational Errors to 

the rms Error of a Single Observation V-55 



V-iii 



CHAPTER V. SATELLITE LIFETIMES 



A 
a 
B 

C D 

C L 

c /c 
P v 

D 
E 

E T 
erf (x) 



Fj (z, e ), 

F 2 (z. € ) 

G 1 (z, e ). 

G 2 (z, E ) 

H 

h 



I n (z) 



K 



K N 

f , m, n 

M 

M 



P + , P" 



P 
R 

R r 

r 

S 



SYMBOLS 
Area 

Semimajor axis 
Ballistic coefficient C A/ 2m 
Drag coefficient 
Lift coefficient 
Ratio of specific heats 
Drag force 
Eccentric anomaly 
Total energy 
Error function of argument x 

Nondimensional decay parameters 

Nondimensional decay parameters 

Angular momentum per unit mass 

Altitude 

Modified Bessel function of n tn order 

Orbital inclination; smoothing interval 

Inverse of square of most probable 
velocity, negative log slope of atmos- 
pheric density 

Knudsen number 

Direction cosines 

Mach number 

Molecular speed ratio 

Mass 

Disturbing force normal to velocity 
in the plane of motion; number of 
revolutions since epoch; number of 
molecules hitting satellite surface. 

Drag parameters for low eccentricity 

Semilatus rectum 

Universal gas constant; radius of the 
earth; radial component of disturbing 
force 

Reynolds number 

Radius 

Circumferential component of dis- 
turbing force 



T 

T L 

t 

V 
W 

x, y, z 

Z 
a 



r(n) 

7 



\ 



p 

(J 



n 

a e 

to 

Subscripts 

c 

i 

o 

P 
r 



Tangential component of disturbing 
force; temperature 

Lifetime 

Time 

Velocity 

Component of the disturbing force 
normal to plane of motion 

Position coordinates in Cartesian 
coordinates 

Lifetime parameter Kae 

Angle of attack 

Emissivity of surface 

Gamma function 

Flight path angle 

Eccentricity (to differentiate from 
base of natural logs) 

True anomaly; 1/2 angle of a cone 

Mean free path 

Gravitational constant for the earth = 
GM 

Yaw angle 

Atmospheric density 

Stefan -Boltzmann constant; statistical 
variance; ratio p/p n 

Orbital period 

Right ascension of the ascending node 

Rotational rate of the earth's atmos- 
phere 

Argument of perigee 

Circular 
Initial incident 
Original 
Perigee 
Relative 
Wall; surface 



V-l 



A. INTRODUCTION 

For most of the low altitude orbits for satel- 
lite payloads it is either interesting or necessary 
to study the effects of the atmospheric perturba- 
tions on the orbital elements of the satellite and 
on the lifetime. (Some material of this sort is 
in Chapter IV; however, the scope of the previ- 
ous discussion of this subject is not adequate for 
the present task. ) Many analytic approximations 
to these effects are presented in the literature; 
however, in obtaining these solutions approxima- 
tions have been made which at times drastically 
restrict the validity of the results. For this 
reason, it is the purpose of this chapter to present 
not only the information but also higher order 
solutions to the nonlinear equations of motion for 
the effects of atmospheric drag. The combina- 
tion of these effects with those due to gravitational 
accelerations, etc. , will not be discussed beyond 
the statement that such a process requires the 
simultaneous utilization of special perturbations 
and general perturbation techniques as discussed 
in Chapter IV. (The present analysis, of course, 
falls into the latter category. ) As a matter of 
fact, special perturbations will be utilized even 
in this study in the integration of the analytically 
determined decay rates. 

It is believed that this approach is inherently 
more accurate than those utilizing either general 
or special perturbation techniques alone. It 
should be noted in support of this statement, that 
even though numerical integration of the equations 
of motion has become increasingly popular with 
the advent of faster digital computers, special 
perturbations have three definite limitations: 

(1) Loss of numerical accuracy, if long 
integration times are involved (hun- 
dreds or thousands of revolutions). 

(2) Long running times even with IBM 7090. 
or 7094. 



B. THE DRAG FORCE 

As a preface to the discussion of atmospheric 
perturbations, certain phenomena and techniques 
must be presented. These discussions will be 
divided into three general areas: 

(1) Gaseous flow regimes. 

(2) The force exerted by the atmosphere 
on the vehicle. 

(3) Tumbling satellites. 

Each of these areas will be divided in turn into 
discussions of the factors necessary in subse- 
quent discussions. In particular they are slanted 

C D A 
toward the evaluation of the quantity ->>— - which 

will be designated the ballistic coefficient. 
1 . Gaseous Flow Regimes 

The work in the field of aerodynamics has 
been divided into investigations in four general 
regions or flight regimes: 

(1) Continuum flow. 

(2) Slip flow. 

(3) Transition flow. 

(4) Free molecule flow. 

These regimes are defined in terms of the Knudsen 
number: 



K 



X 



mean free path 



N " J characteristic length of body 



IT C 



P _M for small R N (Ref. 1) 
C v N 



(3) Lack of general trends, since only iso- 
lated particular cases are solved. 

As an additional step to enhance the value of the 
results, the analysis will be conducted, where 
possible, carrying the density as a parameter. 
Thus, the final result of the study will be of value 
for all atmospheres. This advantage is quite 
significant due to the fact that the atmospheric 
models are constantly changing and the fact that 
there are seasonal and other variations (discussed 
in Chapter II). 

In order to develop an appreciation of the ma- 
terial and methods of analysis, this chapter will 
be presented in three basic parts: 

(1) The drag force. 

(2) Two-dimensional atmospheric perturba- 
tions. 

(3) Three-dimensional atmospheric perturba- 
tions. 



M 



for large R 



•N 



where 



c /c = ratio of specific heats 
P v 

M = Mach number 
V 



JIgRT 
v 



R = Reynolds number 

Though there is overlap of the regions, and though 
no truly definitive numerical values of K^ for 

these regions exist, generally accepted values 
for the four flight regimes are: 



V-2 



Continuum flow--K < 0.01. 
Slip flow--0. 01 < K < 0. 1. 
Transition flow--0. 1 < K < 10. 

Free molecule flow--10 < K . 

N 

These flow regimes are illustrated in the fol- 
lowing sketch (Ref. 1): 





l ' 


if 


! Ill [ 
i 1 




i 


/ 1 1 ' 






■ 






II 






4rf 










i 








1 
/ 
























1 
// 








/ 






n 








1 


/ 
































; 


















ir 
1 






/ 








/ 








/ 


L± 




^ 


1 






/ 
1 






1 I 




i 




1 


i 


I 








1 








As / 






/i 










i 

n 








1 






. 






-t 












! 








1 














/^ ,„ 2 


i — 










^ 


/ 






/ 










yf 


V Tr N / 




--a, 


^ 




_^ 


/, 




-V 


S 




<i- 


? 


-W 







Reynolds Number 



It is noted that in addition to the defining lines men- 
tioned above, a second set of lines denoting alti- 
tude is also included on this figure. It is also 
noted that for any satellite above the altitude of 
100 stat mi (161 km), the flow is always free 
molecule and that free molecule flow could be 
considered to extend down to as low as 75 stat mi 
(121 km) without introducing significant errors 
in the analysis. Since this region (121 to 161 km) 
is the lowest possible altitude for even moderate 
durations in orbit, the entire lifetime analysis 
can be conducted, based on the assumption of 
free molecule flow. This assumption, however, 
makes it necessary in subsequent calculations to 
stop the decay analysis or integration at the afore- 
mentioned altitude of 120 km (-=400, 000 ft). At 
this altitude the mean free path is 20.49 ft (6.25 
meters); thus the Knudsen number for all but ex- 
tremely large vehicles is such that the analyses 
will be valid. 

2 . The Forc e Ex erted by the Atmosphere on 
the Vehicle ~ 

In order to determine the drag coefficients 
analytically it is necessary to study the mech- 
anism by which the force is exerted on the satel- 
lite. This step will be accomplished in the fol- 
lowing analyses utilizing the work reported in 
Ref. 2 as the basis for the discussions. 

Let x 1 , y' and z' be the velocity components 
of a molecule of gas relative to the mean velocity 
of the gas. In addition, assume that the distri- 
bution of these velocities is normal--i.e. , that 
the number of molecules with velocities in the 
region x to x + dx, etc. , is 



dN = N 



3/2 



+ z' 2 )] dx' dy 1 dz' 



exp \-K (x' 2 +y' 2 



where 

N 

K 







= the number of molecules per unit 
volume 

= the reciprocal of the square of the 
most probable velocity - 



2RT 



R 

T 



= universal gas constant 
= absolute temperature 



These molecules impact on a surface whose 
velocity components in the same coordinate sys- 
tem are IV, mV, nV (I, m and n being the direc- 
tion cosines for V). Thus, the velocity relative 
to the surface is 

x = x' - IV 

y = y' - mV 

z = z' - nV 

and the distribution of the impacting molecules 
with veloc ities x + I V to x + I V + dx , etc . , is : 

dN = N (f) expf-K [(x+*V> 2 

+ (y + mV) 2 + (z + nV) 2 ]| dx dy dz 

It is noted at this point that while either positive 

or negative values of y and z are permissible, 

only negative values of x will yield impacts; thus 
the total number of particles of all velocities 
hitting the surface is 



3/2 



N 



- 00 _ 00 _ eo 

-K[(x +iV) 2 + (y +mV) 2 
+ (z +nV) 2 ]}xdz 



N r 



-I 2 v 2 K 



^ttK 



N S.V 
+ -^— [l+erf (IV fio] 



where 



erf (I V (K) = — I 



*v(k" 



ds 



At this point it is possible to relate the number 
of particles hitting the plate to the mass and hence 
to the momentum transferred. The force acting 
on the surface is the integral of the momenta 



V-3 



imparted by the molecules for all possible veloc- 
ities. Assuming for the moment that complete 
energy transfer is made and that the direction 
cosines of the stream are t ' , m'and n 1 , this 
pressure on the surface is: 



Sphere (A = tt r ) 



Specular C_ = erf (M ) 
D °° 



2 + ■ 



M 



-^1 

2 M J 



- P 



o/ „ 00 oo 



+ m' y + n' z) exp {- K [ (x + IV)' 
+ (y + mV) 2 + (z + nV) 2 ] } d z 




+ I (i I ' + mm' + 



L2K V 



,,] 



•[l + erf (I VVK)] > 



This estimate is not correct, however, because 
of the molecules impacting the surface. Some 
are reflected specularly (i.e. , according to 
Snell's law), while the others are temporarily 
absorbed and reflected diffusely (i.e. , in random 
directions) at a later time. For specular reflec- 
tion, the effective pressure is thus. 



eff 



2p 



while for diffuse reflection, the equation remains 
unaltered. Thus, the two types of reflection 
bracket the actual process and the true force can 
be written 



(2 



' p incident P reflected 



where 



f is the fraction of the total molecules which 
is diffusely reflected. (Experiment indicates 
the value lies in the range . 9 < f < 1 . . ) 

At this point attention is turned to the computation 
of the drag and lift coefficients, defined as follows: 



C D A = r 



d . 5Jp D dA 



P V 



P V 



C L A = T 77T 

■o- P V 



k 



dA 



P V 



Since dA is a function of geometry and orientation, 
these coefficients can be defined for various shapes. 
The succeeding paragraphs present data for C D 

both for specular and diffuse reflection (see Ref. 2). 
Note is made that the surface temperature, which 
is calculable as a function of the same set of 
variables, has been included in the diffuse re- 
sults. The derivations are in themselves not 
unique or necessary for this discussion; thus, 
only the final forms will be presented. Additional 
material may be found in the reference and in the 
literature. 



-M 



TT 



1 + 2 



M 



(la) 



Diffuse C D = C D 



specular 



K 3 M . )' T i 



(lb) 



where T is the surface temperature obtained 

w 
by iterating the following equation: 

8/k P u T 3 + 1 



" w \ 3 p R $ 
M = speed ratio 



)---A 



w».': 



v 



J2RT 



T. = temperature of incident stream 



P 



= surface emissivity 

= Stefan-Boltzmann constant 



pN 



surface 
-M 2 



+ erf(Mj (M^^-) 



for a monatomic atmosphere of 
oxygen and nitrogen in the shadow. 

Since the properties of the atmosphere are 
integrally associated with this evaluation of these 
coefficients only specific data can be generated 
for C . An example of the application is pre- 
sented in Fig. 1. This figure, obtained from 

Ref. 2, presents C^ as a function of M and for 

r L) °° 

an altitude of 120 km. Though computations for 
this figure were made with atmospheric data 
available in 1949, the variations which are shown 
are representative and the limiting values, which 
are rapidly approached, valid for this reference 
altitude. Data for other altitudes must be gen- 
erated as needed. 

Flat plate at angle of attack a to the flow (A = ab) 




For this body configuration the drag coefficients 
vary according to the following equations: 



V-4 



2 2 

. • 2 -M sin a 

„ , „ 4 sin a «. 

Specular C = e 



M [^ 

00 ' 

' 2 s in a . . ■ 3 \ 
„ — + 4 sin o\ 

M ' I 



erf (M sin a) 



(2a) 



tvt 2 ■ 2 

-M sin » 



Diffuse C 



D 



M 



+ 2 sin a (1 + ^-^A erf (M sin cr) 

1 2 M Z > 



I it sin 



TvT 



T. 



(2b) 



where T is obtained from 

w 



1 ? S 1 

4 V^ + | R T. - ^ R T 

2 2 i 2 w 



Cone with axis parallel to flow (A = tt r ) 



Specular C T 



2 sin 6 



2 2 

-M sin 9 



M 



3 1 



+(' 1 ^— + 2 sin 2 9j [1 + erf (M^sinG)] 



(3a) 



Diffuse C T 



1 
M sin o nr 



J^ 



2 M 



w 
i 



2 2 

-M sin 



1 + 



2 M 



n9 [T 




2 M 

where T... is obtained from 
3 



w 



1 + erf (M sin 9) 
(3b) " 



T 



4 Jtr K (3 a T 
3 p R 



w -)--i K M » 2 



and where 9 is the half angle of the cone. These 
results can be extended to nonzero incidence angles 
by utilizing the flat-plate results mentioned earlier. 
Such calculations are presented graphically in 
Fig. 2 (Ref. 3). 

Right circular cylinder with axis perpendicu- 
Tar to flow (A = 2 r"T) 



Specular C 



D M 



I 

M = 



M 2n r |2n + 3 



(-D 1 



n ! T(n + 2) 



M L 



m 2n r 



(-D 1 



2n + 1 



n=0 



n ! 1' (n + 2) 



n =0 



M 



2n 



(-1)' 



2n + 3 



T (n + 3) 



2 r 



2n + 1 



T(n + 2) 



(4a) 



Diffuse C 



1 



D M 



n =0 



>(^S 



M 

n~! V(n + 1) 



3/2 



4 M 



T. 



M + 



M 



M 



2n 



r 



(-1) 



*fL) 



n=0 



T(n + 2) 
(4b) 



where T is computed from 
w ^ 



tt (3 a [k ' 

p~tt 



3 5 



2 U 



+ M 




m 2n r' 2 



(-1)' 



n + l\ 



T(n + 1) 



r-^ IV! 

> (-1)" - 
/_, n 



n r ^ 
T(n + 2) 



Figure 3 presents data comparable to that 
discussed in conjunction with the sphere. Of 
particular interest is the fact that this coefficient 
approaches a limit which is not unlike that of the 
sphere. 

2 
Circular -arc ogive (A = tt r ) 

This figure is constructed by rotating an arc of 
a circle about its chord then cutting the body of 
revolution perpendicular to the axis at its mid- 
point. The angle of the nose (29) analogous to the 
half angle of the cone is utilized to describe the 
shape. 



Specular C 



D 



1 ([4 

1 - cos 9 S [J 

2 2 

-M Z 9 Z 



1 



M 



(1 - cos 9) 



29' 



cos 9 + 



+ erf (M 9) 



4 M 



"3" + 2TT 



"3" 



2 M 



M 



(5a) 



V-5 



Diffuse C 



+ erf (M 6) 




(1 -cos 6) 



-M 



2 „2 



2 M 



, V q 



4 M 



1 + Q -1^- (l + erf (M el) 

12 M 4 6M » 



-M 



2 n 2 



+ e 



1 



6 M 



12 M 



(5b) 



where T is obtained from 
w 

4 pN 



PjT 



N 



L 



i V 2 +|rT. 4rT 
2 2 i 2 w 



To provide a feel for the validity of these re- 
sults, tests have been performed (Refs. 3 and 4) 
and data prepared for the transverse right circular 
cylinder. The results of these tests are shown 
in Figs. 4 and 5. These figures depict the varia- 
tion in the critical region for molecular speed 
ratios in the vicinity of 0. 7 to 2. 5. The agree- 
ment between these data and the theoretical values 
is observed to be very good. Also noted is the 
tendency for the results to agree better at higher 
values of the speed ratio with the specular reflec- 
tion theory than with the diffuse theory and vice- 
versa at the lower speeds. 

3. Tumbling Satellites 




Side 



Now approximating the effective drag coefficient 
based on one of the surfaces (say A ) 

* A 2 

Cr~, = C n cos a cos y + C„ -,r— cos ~ sin a 
D D : D 2 A l 

A 3 A 4 

+ C„ tt — cos a sin ~ + C„ -* — sin a sin 

U 3 A 1 U 4 A l 



where a and H are uniformly randomly selected 
variates always lying in the range to it/ 2 



C ' is the affective drag coefficient for 
the body 

A is the reference area for the nth geo- 

metrical shape 

Since the distributions of a and ~ are known 

2 



'D 



(the joint density function is (-1 ), it is desired 
to determine the distribution of the function C T 
This is accomplished as follows: 

g (C D *. a) = f [a, y (C D *. a)\ 



9 C 



D 



The preceding discussions have presented 
data for bodies fixed relative to the flow field. 
However, in most satellite applications this is 
not the case. The first class of such exceptions 
consists of those satellites which by design orient 
themselves relative to the earth or space in order 
to perform some mission. The time history of 
attitude for this vehicle is thus known, and a time 
history of the drag coefficient can be constructed. 
The second class of vehicles consists of those 
which tumble in both time and space, thus com- 
plicating their aerodynamic description. One 
path around this impasse is to describe the param- 
eters statistically and assume that they are inde- 
pendently distributed. This approach, while not 
rigorous for either class of exception, provides 
a convenient means of computation for the latter 
case and an approximate method for long time 
intervals in the former case. Consider the fol- 
lowing sketches. 




but H (C n , a) must be obtained from 



?_ = a. cos H + a„ sin tr 
= a cos (k - w) 



where 



a, = C„ cos a + C„ ■* — sin a 
1 D 1 u 2 1 

A 3 A 4 

a 2 = C D 3 Ay COSff + C D 4 Ay sina 



a„ cos w = a 1 



a„ sin w = a„ 



tan (ag/aj) 



2 j. 2 
a l +a 2 



thus 



also 



.-1 /' C D 



a c 



Top 



D 



+ w 



- a 1 sin - + a_ cos H 



-1 



V-6 



or, 



3 C 



D 



1 / C D , 
cos I ] -f w 

a 3 



+ a„ cos 



■1/' C D 



+ w 




2 2 2 

a l + a 2 = ^5 cos Q + C fi cos a sin " 



+ C„ sin a 



2 2 ? 

a 2 " a l = ^8 cos " ° + Cq cos » sin « 



+ C 1Q sin a 



At this point it is noted that the area A can be 

2 9 1 

selected so that a > a "; thus, since a and -- 

are always between and tt/ 2 the function defined 
is everywhere positive in every term. Thus, 
the absolute value signs can be dropped 

and 



thus 



a 3 " C D / a n + ~D 



la„/ a„ [ a~37 



+ a„ 



a,/ 



2 „ * 
a 3 -C„ ,a r 



MS) 



7~2 



a 3 " C D 



( a 2 2 " a l 2 ) 



8 (C D :! < °> = (I) 



a 3 " C D 



The distribution of C 'is obtained at this point 
D 

by integrating g (C ' , a) with respect to a over 

the range to tt/2. First, however, it is nec- 
essary to replace a in the joint density function. 



g(c D ) 



> d. cos 



i=0 



. do 



Z"V 



-l .1 

» sin 



i=0 



(6) 



This function may be approximated analytically 
upon studying the behavior or integrated numer- 
ically. Analytic integration, however, does not 
appear attractive. It is noted that for the special 
case of 2-D analysis this problem is circumvented, 
since ^integration is not required. For this case 
g (C ''") is obtained directly to be: 



« «v ■ 



2 , 2 
a l +a 2 



[*1 



2 +a 2 -C * 2 (a 2 -a 2 ' 
+ a 2 C D \ a 2 a l 



where 



a 2 = C 2 
a l C D1 



« < C D*' *»■(!) 



2 , 2 
a l +a 2 



2 lx 9 2 

a l = * C D1 oos <* + C D2 A - " sin a ' 



a l 2+a 2 2 - C D* h"- ai 2 



A, 



= C 



D3 



m 



A 2' A 4* C D2 and ^n4 do not a PP ear m tnis form 
for the reason that only a 2-D analysis is made. 
Thus, if the vehicle is tumbling in a known plane 
this much simpler solution can be utilized. 



- (C, cos a + C 9 sin a)' 



2 2 9 9 

Cj cos a + 2 Cj C 2 cos a sin a + C 2 sin a 



2, 3 4 r 

*2 = (C D3 A^~ C0S Q, + C D4 Ay sin a) ' 



= (C„ cos a + C. sin a-)^ 



The density function is known or at least de- 
finable for the 3-D case and known analytically 
for the 2-D case, the problem turns to one of 
evaluating the moments of the distribution. These 
moments may be obtained directly from the mo- 
ment generating function in the following manner: 



m(t) 



fc 



tu(x. 



X ) 

n 



f(x, 



x n> 



C 3 cos a+ 2 C 3 C 4 cos a sin a + C. sin 2 a 



TT dx i 



V-7 



4m(t) 



dt 
where 



t=0 



fi' = the mean 

9 2 

a = /j' - /n' = the variance 

Substitution for this problem into the previous 
formula yields: 

ir 77 

2 * * r 





m(t) 



h. cos a cos ~. 



where 
h. 



+ h„ cos a sin y. + h, sin a cos 
2 ^ 

+ h. sin » sin ~; Id »d^ 






1. 2, 3, 4 



But this problem, like the first, is not easily 
integrable. Thus, a numerical evaluation is sug- 
gested for each case of interest. In fact, even 
for the 2-D case, in which 



m(t) 



-) e 



't 



'1 



dC, 






where 



C 2 +C 2 
C D1 D3 



*! 



C 2 -C 2 
^Dl D3 



C 2 + C 2 
U D1 D3 



A 3 
*~1 



A 3 



an analytic form is not readily available. 

Since the mean is not available in analytic form, 
little can be said relative to the best value of C 



'D 



A 1 in the general problem. Many investigators 

avoid this problem by using the approximation 
derived from consideration of a spherical satel- 
lite. 



C D* A = °D h (A surface> 

sphere 



surface of sphere 



projected area of sphere 



-1 



sphere 



A 



surface 



Though this may seem to be a crude approximation, 
there are many cases in which it is reasonable. 
In fact, Ref. 5 reports an investigation in which 
a body randomly tumbling (about three principal 
axes) is analyzed and in which the author concludes 
that for convex surfaces the average drag on a 
surface element in random orientation is the same 
as that on a sphere of equal area. This work thus 
lends credibility to the previous assumption and 
provides a numerical value which can be utilized 
as an initial estimate in the numerical calculations 
outlined previously. 

C. TWO-DIMENSIONAL ATMOSPHERIC 
PERTURBATIONS (REF. 6) 

The motion of a point mass in a nonrotating 
atmosphere surrounding a central force is given 
by the following set of simultaneous differential 
equations 



-^ - B p r V 



(7) 



Ji(r 2 9) = - BpVrl 



where 
V 



' ? .2 
(rer + r 



Id = earth's gravitational constant 
9 = ^- = angular velocity (rad/sec) 



C D A 
2m 



ballistic coefficient 



(8) 

It is noted that this set of equations is nonlinear 
and that a solution can be obtained only by nu- 
merical integration. This fact is somewhat dis- 
concerting, since these equations neglect atmos- 
pheric rotation, which introduces considerations 
of a third dimension and complicates the analysis 
further by entering the equations explicitly in the 
drag term. This latter factor results in the re- 
placement of V as defined previously with 

V = velocity relative to the atmosphere 
r 

V + V . 

atm 

Thus, if analytic approximations are desired, it 
becomes necessary to divide the problem into two 
phases--a perturbed orbit phase and an aerody- 
namic entry phase. In the first phase, a region 
is considered where the orbit is determined by the 
inverse square gravity field and only small per- 
turbations are caused by the relatively small drag 
forces. In the entry phase, the aerodynamic forces 
(lift, drag, etc.) become the important factors 
influencing the trajectory of the satellite and grav- 
ity forces become less important. This last phase 
is by far the more complicated, and fortunately 
for a lifetime study it can be neglected, since rel- 
atively short periods of time are spent at the alti- 
tudes where drag forces become dominant. Thus, 
the present problem is the analysis of only the 
first phase. References 7 through 20 present a 
portion of the pertinent literature and will be 
discussed as the presentation progresses. 



1. Near-Circular Orbits (approximate solution) 

To initiate these discussions, consider the 
decay of a circular orbit. The energy loss due to 
drag during one revolution, Ae , is given by the 

loss in total energy 



AE 



D 



E - !•' 

Tl T2 



v2 , 
c 1 

~7" 



j" 



V 



c2 



r 



(9) 



Using the equation for circular velocity and letting 
Ar = r 2 -r-j. 
,uA r 



AE 



D 



2r l r 2 



(10) 



The energy loss per unit mass due to drag is also 
equal to the drag force per unit mass integrated 
over a full revolution 



AE 



D 



D 



ds 



(11) 



Assuming small altitude losses during each single 
revolution 



Equation (16) shows that the decay rate for this 
special case is a linear function of the ballistic 
coefficient. This fact will be utilized in much of 
the future work in order to restrict the number of 
variables in the analysis. Equation (16) is not 
directly integrable because of the odd fashion in 
which the true density varies. However, if the 
density is assumed to vary exponentially with 
altitude, approximate lifetimes for circular orbits 
can be obtained: 



\ «" "J 



f 



dr 



2Bp Q e 



IsTW 



"^ 



(17) 



where 



r.. = the final radius = R + 120 km 

r + r f 
p Q = the density at the -5-^ (see Figs. 6a 

and 6b) 

K = the negative of the logarithmic density 
slope (see Figs. 7a and 7b). 

(Note: This data is for the 1959 ARDC Atmos- 
phere. Data for the U.S. Standard 1962 Atmos- 
phere is presented in Chapter II. Either can 
be utilized if the lifetimes are adjusted, as will 
be discussed on p V-20.) 

Thus 



* E D~8^-(h) 2. (-i 



r, + r. 



(12) 



r l + r 2 
where g = an average radius for the 

revolution. 

Now using the approximation that the circular 
velocity is averaged approximately as 



V 2 = .__ 
c r . + r„ 



2/u 



(13) 



Eqs (12) and (13) and the relation — = flpV 2 yield 

m Mr J 



AE^ = 2tt/uB p 



(14) 



A r 2 

If — — << 1, then r r„ - r and Eq (10) with 

*- i J- £ cLV 

Eq (14) results in the decay rate of the orbital 
altitude per revolution 



A r 
rev 



4TrBp r 

av av 



(15) 



A' 



This decay rate can be converted to —5- by 

sec J 
considering that the orbital period for this per- 
turbed circle is 



t = 2tt 



av 



Thus 



AT 



- 2B p J/nr 

av f av 



(16) 



f -K r , 
e dr 



jTBp c 



F 



let 



x = Kr 
2xdx = 



2 x 
K dr or dr = ^- dx 



Thus 



1 



|T~r dx 



r e 

r 



-Kr , 
e dr 



2 C -x" 
\ e 



dx 



fl[ erf (f^f) " erf (po)" 



and 



-Kr 



2 {HB Pq 
- erf ( ^7 f j 



erf 



if^o) 



(18) 



The disadvantage of utilizing this form for the com- 
plete lifetime is that the density does not vary 
exponentially, and thus the approximation becomes 
poorer as the difference in r and r. becomes large. 

This deficiency can be circumvented through the 
simple expedient of breaking the true radial incre- 
ment into several subdivisions and evaluating the 
times required to descend through each interval. 
These times can then be summed to yield the life- 
time. Computations utilizing this philosophy will 
yield accurate estimates provided that the intervals 
are no larger than 50 stat mi or 80 km. 



V-9 



The case of even slightly elliptic orbits must 
be treated in a different fashion since the assump- 
tions made in generating circular orbit lifetimes 
are not valid for other orbits. Thus, it is neces- 
sary to consider the equations of variation of ele- 
ments derived in Chapter IV or to approximate the 
motion in some other fashion. If the latter approach 
is taken, one possible avenue of investigation is to 
linearize the equations of motion by expanding the 
variables in Taylor series and retaining only first- 
order terms. This approach is valid only for 
small variations in the parameters. One such in- 
vestigation is reported in Ref. 12. The author 
utilizes a small parameter /3 1 defined as 



H' =Bp Q r 



(19) 



All orbital parameters are expressed as power 
series of /3, considering only the first order terms 



AV 



Ar 
rev 



(23) 



Now, from the first two relationships in Eq (22), 
exactly the same relationship follows: 



Ay 
rev 



V 
c 



A r 
rev 



This implies that for a first order approximation 
in B p„ r„ the speed at any given altitude remains 

exactly equal to the circular speed during the drag 
decay of a circular orbit. 

And, from Eq (21) for n = 2it the corresponding 
angle 9 is obtained as 







= 2tt + 6tt B p„ r„ 



(24) 



r +P ' r l ^ 



V = V Q + 0" Vj 
H = H + P H l J 



(20) 



where 

2 ' 
H = r is the angular momentum per unit 

mass (to differentiate from h = altitude). 

Substituting Eq (20) into the differential equations, 
Eq (7), the following relationships are obtained 



1 + 



Bp Q r 



M 



4cose 0+ |8^ 



r = r r 



V = V 



1 +2Bp Q r (sine Q 



V 



H 

where 

e o 



H r 



l + Bp r (-2sine 0+ e ) 



1 - B ' 3 o r o 9 o 



(21) 



V t 
c 



Expressions for these quantities on a per revo- 
lution basis are next obtained from the differences 
in Eq (21) evaluated at the limits 6 = and 2tt: 



Ar 
rev 



47TB p Q r 



— = 2T7Bp n r„V 
rev c 

^ = -2Bp„r n 
rev 



But, for circular orbits V - J — 

c I r 



(22) 



and 



dV 
c 



o— J— » giving the following condition: 



Equation (24) indicates that the line of apsides is 
advancing by the amount 



6tt B P r (rad) 



(25) 



Since the equation for the change in the radius per 
revolution is the same as that for the circular 
orbit. The lifetime of this slightly elliptic orbit 
will be the same as that presented earlier. Ac- 
tually, as will be shown later, the lifetime is 
slightly longer, but a quantitative analysis is left 
until subsequent paragraphs. These subsequent 
discussions will concern the behavior of these and 
other more elliptic orbits. 

2. Elliptic Orbits (approximate solution) 

The type of expansion outlined for near-cir- 
cular orbits can also be utilized for elliptic orbits 
as was shown in Ref. 12. This reference pre- 
sented power series expansions for decay rates in 
elliptic orbits utilizing the small parameter 



=B P (h P o )r po 



(26) 



where 



p(h ) » air density at perigee radius 



l P o 



Initial perigee radius. 



Next, a density ratio is defined 

^0 "P/P (h p0 ) ' 
For these orbits Eq (7) becomes 



rfc) « -- 



£<r 2 e) 



/3a. 



rV 



r 



0*. 



pO 



r6V 



(27) 



r 



pO 



, and 



Using a change of variables u 

neglecting higher order terms in 3, the 
power series expansions assume the fol- 
lowing form: 

u »u Q + |8u 1 

V - v + p v 1 

H - H n + H, 



(28) 



V-10 



Now the ratio of the Initial speed at the perigee 
radius to the circular speed at r is defined as 



V 



pO 



(29) 



and the corresponding eccentricity is expressed as 

2 



c 2 -i 



m 



i 



(30) 



An exponential atmosphere is assumed in the form 
-K(r - r J 



a °*^p7 



pO' 



(31) 



The differential equations given by Eq (27) are then 
solved for the two cases below: 

Case I : near- circular orbits 

Case II: eccentric orbits 

Case I -- near -circular orbits. The solutions 
derived by Kef. 12 are summarized below. First, 
the orbit parameters: 



H - r „ V n 
pO pO 



1 " B e<\> r po 



+ ! (Kr P o<> 2 - T^po* 



> [l - K : 

> 3 ] 



PO" 



+ sin 6 Tkt Q 6 (l - K 



pO 



L" pO V pO" 2 



7 Kr pO« 



■)1 



. Ko'f n 



+ Bl n 30V_i-xjJ 



(32a) 

— -*+' H |l-2Bp(h)>f + '\\\l 
r„ n 1+e cos y )| ^ v p 1 + « cos t) j 



l p0 



-Kr p0 « + |(Kr p0 ,) 2 -fV(Kr p0 O 3 



T4T (Kr p0 € ' e cose 



l 1 -* 15 ^' 



+ I< Kr pO° 2 + TS< Kr pO«> 3 



J ]sin 



(Kr„«) 



pO 
23" 



(1 - Kr Q «) sin 2£ 



(Kr Qt) * 

4r™ sin 39 



(32b) 



Second, the decay rates obtained from the above 
equations : 



M.- 2 ,Bp(h p0 )V p0 r p0 2 



[« 



pO 



+ l (Kr po £ ) 2 -TV( Kr P o e n 



(33a) 



Ar 2 

rif-'le = 2nf r (e - 0)* - 4TrB ^ h p0 )r p0 

- Kr P o' + ^ Kr P o e)2 -Tli< Kr po< 

(33b) 



Ar 
a 

"rev" * r (e - 3tt)" r (e » ir) 



4lTB ^ h pO )r pO 



(-) 



1 - Kr _ t 
pO 



+ !< Kr P o<> 2 



55 tv \ 3 

Ttt (Kr pO () 



(33c) 



Note that for « « both Eqs (33b) and (33c) reduce 
to the circular decay rate given previously by 
Eq (22). 

The given series expansions are adequate only 
for small values of Kr . s, the upper limit being 

suggested asKr n e <0.5. Reference 12 gives the 

following table, indicating the upper limits of 
eccentricity for various altitudes from sea level 
satisfying this condition: 



(km) (stat mi) 



K 



(ft" 1 ) 



(m" 1 ) 



161 
322 
483 



100 
200 
300 



9.3 x 10" 6 30.5 x 10" 6 
5.1 x 10" 6 16.7 x 10" 6 
3. 65 x 10~ 6 12. x 10~ 6 



0.0025 
0.0045 
0.0061 



(1 stat mi = 1. 609 km; 1 ft = 0. 3048 meter) 



Case II-- elliptic orbits. For values of 
Kr _ « >1, terms up to the seventh power were 

carried. The resulting series expansions are 
shown below. 

1 - e p° ( Cl e 



H * r n V n 
pO pO 



+ Y C n + l slnne J 



(34a) 



n=l 



1 + i 



r _ 1 + c cos 
pO 



1 - 



- Kr .€ 
Bp(h n )r n e P° 



"pO 7 pO 
1 + « cos t) 



2C 1 6 - C 2 6 coe e + C* sin I 



|-C 3 sin 26 - ^C 4 sin36-^.C 5 sin4e 



■^ Cg sin 56 + ^ C ? sin 66 + ® sin 76 



(34b) 



V-ll 



where 



c i- 1+ i< Kr po e)2+ ir< Kr po <)4 



C 2" Kr pO' + i (Kr pO° 3+ TTO (Kr pO €)5 

+ W32 < Kr po £)7 + -"- 

C 3"F (Kr p0 4)2+ ^ (Kr pO° 4 
+ WT? (Kr pO £)6 + --- 

C 4-^ (Kr pO €)3+ T^ (Kr pO° 5 

C 5 " TTO< Kr p0 6)4 + T53F (Kr P O t)6 + ••• 

c 6 ' mm (Kr po* )5 + W?w (Kr po° 7 

C 7"T3CTIT (Kr pO e)6 + -" 

C 8 " 2,358, 720 ^pO** + '" 

C*- -2C 1 + C 2+ |C 3+ JC 4+ ^C 5 

T2" L 6 "37 7 T7^8 •" 

The accuracy of the series solution 1b limited 
to a region near the perigee, due to expansion of 
a n aroung the perigee point. Therefore a limiting 

central angle, e llm » was designated, such that 

„ p . < 0. 01 for 6 < e., . The limiting angle is 
p(h ) — — lim 

given as 

Kr 
P 

For ,, p ■ < 0. 1 the constant 4. 60 is replaced by 

«v ~ 

2. 30. Figure 8 presents 9 plotted versus the 

orbital eccentricity for two values of density 
ratios and two initial perigee altitudes. Since 
the air density has decreased to 1% of the perigee 
value at a central angle of ^ lim > the following 

assumptions can be made: 

(1) The drag effects are negligible for the 
arc BCD. 

(2) All the drag takes place in the region 
DAB. 

(3) A symmetry exists about the line AOC 
(i.e., Drag DA - Drag^). 




lim 



B \ l im 
lim 



+ 2it 



Therefore, the change of orbital radius at a cen- 
tral angle ltm is expressed as 



7§i ' r B' " r B " r(e iim + 2W) " r(e ilm> 



From Eq (34b) 



(35a) 



Ar , \_ ^ (h P )r p0 e 



2 - Kr p0 6 



I 



1 + <= cos u 



c„ t; cos e + . . 



2C 1 e 



lim 

- e 



lim 



But 



-M 



Aa 



From the chain rule 



*'-£)*■♦©" 



(35b) 



(36a) 



(36b) 



and from Eqs (36a) and (36b) it can be shown that 
the following orbital parameters can be obtained 
from Eq (35b): 



Aa* (1 V C0Be) Ar 

(1 - O (1 - cost) 

Ah - 2(1 + ^ cose) 2 A „ 
a (1 - e) (1 - cosG) 



(37a) 



(37b) 



Equations (37a) and (37b) are based on the assumption 
that Ah » Ah . Thus the apogee decay rates can 

a p 

be obtained by the expansion of a small parameter 
method by Eqs (35b) and (37b). For perigee decay 
rates no information is given by this solution. 

3. Variation of Elements 

As was noted in the previous paragraphs, a 
second method of solution for the effects of drag 
is available in the form of the equations for varia- 
tion of elements. These equations will be utilized 
in the investigations of elliptic orbits which follow. 



V-12 



Since the interest in this discussion is in the 
solution for the lifetime of a satellite in a nonro- 
tating atmosphere, the disturbing acceleration 
will be due to drag and will act along the velocity 
vector that is tangent to the path. Thus, since 



R = 



(1 + € cos 6) T 
| 1 + e + 2« cos 

(e sin 9) T 
J 1 + e + 2e cos 



(s sin 9) N 



1 + e + 2« cos 9 

(1 + c cos 9) N 

2 " 

1 + « + 2e cos 9 



where 



S = circumferential disturbance 

Ft = radial disturbances 

T = the tangential acceleration 

N = the normal acceleration s 

« = the eccentricity to differentiate from the 
base of natural logarithms 

The equations of variations of constants can be 

written as 



da 

ar 



de 

ar 



ar 



V777T 



2e cos 9 



n ^1 



AH 



(cos 6 + e) 



la -y 



1 + e +2* cos 
2 sin 6 



^1 + *' 



+ 2( cos 9 



£ = - [2(1 -e 2 )(l + e 2 

+ e cos 9) sin el [na « (1 

2 1 

+ 6 cos 9) (1 + e + 2e cos 9) 



dfi 

ar 



o , 



di 

ar 



/2]" 1 
(38) 




Consider a slowly decaying elliptical orbit as 
shown on the sketch. Take points 1 and 2 as 
shown in the sketch in such a manner that the 
angle from perigee is constant. 

Then Q^ = 9 g , r J > r 2 and p < p From the 
basic equations of elliptic orbits 



V 



a 



1 + 2c cos 6 + 



(39) 



From Eq (38) 



2 B p sin 9 <y£. f 1 + {2 + 2 2 C COS 6 ) 



The ratio u>/a>„ becomes 



(40) 



1 P l 6 2 / a 2\ l/2 / 1 + t j + 2 *l cose A l/2 

2 " "2 € 1 V a J \ 1 + .* + 2._ cos 6 J 




Then for the first order of eccentricity 

1/2 



f\ '2 



P 2 £ 



2_( a J,\ / 1 + { l cos6 l \ 
lUi/ V + '2 cos °l/ 



(41) 



1 + « . cos 8 

But, t-t — i -i- „ 1 

1 + e 2 cos B 2 



where 



2tt 

T 



D 



y~ mean angular velocity 



drag deceleration. 



From Eq (38) it follows that for a nonrotating 
atmosphere, drag does not cause any variations 
in the inclination or the nodal position of the orbit. 
Aerodynamic drag will, however, cause a forward 
rotation of the perigee in the orbital plane, as was 
shown quantitatively in Eq (25). An appreciation 
of the reason for this advance can be obtained 
from the following qualitative analysis. 



2 P-[ 

< 1, — < 1 and — < 1 

p 2 



1 



Therefore 



< 1 and the perigee advances 



due to air drag as was stated. This advance does 
not affect the lifetime of the satellite to the order 
of approximation of this analysis; however, since 
the atmosphere is not considered to rotate, den- 
sity need not be considered to vary with posi- 
tion around the earth. Thus, the orientation of 
the orbit while it changes does not change the de- 
cay history (again, to this order of approximation). 
For this reason, attention can be focused on the 
change of the three elements in the plane of the 



V-13 



orbit (a, e and a). Further, since a relates posi- 
tion in the orbit as a function of time and not a 
change in the size or shape of the orbit, the ele- 
ments of primary concern are a and t. Variations 
in both of these elements are discussed in the fol- 
lowing paragraphs. However, before these dis- 
cussions it is desirable to relate the change in 
altitude of apogee and perigee to the changes in 
the elements a and e. 

The altitude variations during one revolution 
are quite large for elliptic orbits with high eccen- 
tricity, and therefore it is necessary to pick certain 
reference points during one revolution, for which 
the altitude, air density and decay rate can be 
found more easily. Since this geometry of a two- 
dimensional ellipse is completely determined by 
the perigee and apogee altitudes, and since air 
drag occurs primarily in the vicinity of perigee, 
apogee and perigee radii will be utilized as the 
reference points. These radii are expressed in 
terms of the semimajor axis and eccentricity as 



(42) 




Assuming an orbit with a very high eccentricity, 
the significant part of air drag takes place near 
the perigee and the maximum variations of orbital 
parameters can be found approximately by setting 
cos 9 a 1.0. Equations (38) become 



da 



de 

dT 



• 2 (1 + «) T ->j 

n^TTT^ 

. 2^TJ T 



(46) 



and the ratio of a to c is found as 



da 

ar 



7 t^- 



r 



d< 

dT 



(47) 



Substituting Eq (47) into Eq (45) yields 

da 



dh 
a 

ar-° 


/ 2a 


1 


dc 

ar 


dh 

-ar- 


dc 
a oT 


- a 


dc 

ar 



ar 



) 



(47a) 



Now, orbital altitude is given by h. = r. - R , 

& J i i e 

where R is the radius of the equivalent spheri- 
cal earth. Therefore the partial derivatives be- 

dh. dr. 

i l 

come, since -* — = -s- — 

8x 9x 



1 + c 



8h 
a 

8h 
a 

"5i~ 



And from the chain rule for derivatives 



9h 



8h 



) (4 3) 



dh 8h 

a . a 

"dT" "BIT 



dh 8h 

P - P 
"dT" 8"a~ 



da 

ar 



8h 



dc 



"ST- ar 



A 8h 

da p 

ar "si - ar 



dc 



(4 4) 



Substituting Eqs (43) into Eqs (44) yields 



Equations (47a) indicate that orbits with large ec- 
centricities tend to become more circular during 
the drag decay process. For highly elliptic orbits 
the perigee decay rate is zero for a first approx- 
imation and in all cases it is considerably smaller 
than the apogee decay rate, as proven by numeri- 
cal integrations (Ref. 10). 

Now continuing, using the expression for 
drag deceleration 



T -R - 

m 



BpV 



(48) 



Equations (38) become 

2 
da 2a n ,.3 

-rr = BpV 

dt p ^ 

^ =-2pV (cos G + € ) 

Substituting for V and 6 from 



(49) 



V 



U~ 



2 e cos e + e ' 



1 



(50a) 



dh , , •> 

a ,, , , da , de 

■ar " (1 + <) ar + a ar 



dh 
P 

aT~ 



,, , da dc 

(1 - €) ar " a ar 



(45) 



Thus, after the time derivatives of semimajor 
axis and eccentricity are determined from the 
Lagrange planetary equations, the time rates 
of the perigee and apogee altitudes can be found 
by substitution. The instantaneous orbital alti- 
tudes can be determined by integrations of Eq 
(45) either by numerical or analytical expres- 
sions. 



dt 

ae 



(50b) 



1 



the equations for the variation of elements can be 
expressed as derivatives with respect to the cen- 
tral angle 9. At this point it should be noted that 
Eq (50b) applies rigorously only if angular mo- 



[vp = na I 1 ■ 



mentum is conserved, i.e., r 9 = i up = na" 

In Ref. 17 the correct expression is given in terms 

of the osculating elements as 



+ u + COS 1 



dt 



[mp 



(51) 



V-14 



However, as seen from Eq (25) 



Acq 
AW 



37rBp„ r_ (rad/rad). 



But since 1 » - , Eq (50b) is justified for the 

e 

present analysis. Thus, Eqs (49) become 



da - o 2 p„ (1 + 2t cose + Q 
■fig ^ a a P rj — 



3/2 



(52a) 



(1 + c cos 6) 



^-- 2aBp(l - « 2 ) 



(cos e + t) 



(1 + 2e cos + O 



1/2 



>- (1 + c cos 0) 



T 



(52b) 



Next, the functions of the central angle are 
expressed as functions of the eccentric anomaly 
by the following relationships : 



r = a(l -■ e cos E) 

' 2 



sin 9 * 



cos 8 



V 


- t 


sin E 


1-6 COS 

cos E - € 


E 


1 


- < cos 


E 




v 


!-.> 


dE 


T^ 


e cos 


E 



(53) 



d6 



Substituting Eq (53) into Eq (52) and using the approx- 
imate symmetry relationship of drag decay functions 

2* 



I fd6=2 I 



f de 



o o 

The decays per revolution are found by the follow- 
ing integrals: 

3/2 
Aa _ ,.2, 
rev 



-4a 



! b Po r £. n±±^^\ dE 

U J P (1-c cos E) T/2 



(54a) 

^^aBp^l-E 2 )]^ illljW'* EdE 

p (l-ecosE) 1 '^ 

(54b) 
Note that Eqs (54) basically involve the application 
of the Krylov and Bogoliuboff averaging method (Refs. 
13 and 14), by which approximate differential equa- 
tions are obtained for the variation of orbital 
elements by averaging the original equations over 
one full revolution (i. e. , E = to E » 2n). This 
removes all trigonometric terms from Eqs (54) 
and is actually equivalent to a conservation of 
energy approach (Ref. 14, p. 238). 

The fraction in Eqs (54) can be expressed in a 
simplified form by employing power series ex- 
pansions as: 



Aa A 2 a C 

— » -4a Bp„ \ 
rev ^0 J 





'0 L 



2e cos E 



+ (continued) 



j. 32 2_ , 3 3^74 4 
+ tj- € cos E+ e cos E+j( cos E 



dE 



u 



J 



1 2 



1 3 



(55a) 



cos E + e cos E 



3 4 



+ 7 € cos E+ T € cos E + - € cos E 



5" 



dE 



(55b) 



In general, the density function -2- is empiri- 

p 
cally found (see atmospheric models) and cannot be 
expressed in a simple exact analytical form. Thus, 
the analytic integration of Eqs (55) is not possi- 
ble. Numerical integrations of Eqs (54) or (55) 
can be performed on a high speed digital com- 
puter, however. If this step is to be taken, the 
density is related to eccentric anomaly in two 
steps: 



(1) 



(2) 



Altitude: 

- R 
e 

Density: 
tables. 



Defining S = 1 + 2 e cos E + £ e 2 



h=r-R g = a(l-€ cos E) 

p(h) from atmospheric density 

(56) 



2 ~ 
cos E 



and dropping terms higher than the second power 
of eccentricity (Ref. 12) has numerically com- 
puted the function of the integrand in Eq (55a) for 
Explorer IV, considering both Smithsonian 1957-2 
and ARDC 1959 model atmospheres. 

The most important conclusion from this study 
and related studies performed elsewhere is that 
even for orbits of relatively small eccentricities 
(Explorer IV had e = 0. 14). The most significant 
portion of the drag perturbation takes place in the 
vicinity of perigee in a region where |E| < 40° . 
Utilizing this conclusion (not the limit on IE I) 
and approximating the density in this region by 
an exponential, Eqs (55) can be put in an integra- 
ble form. Let 



-K(h-h p ) 



J 



(57a) 



where K is the negative logarithmic slope given 
in Figs. 7a and 7b. Equation (57a) implies a 
straight line variation of p versus h on a semilog 
paper, which does not exist for any altitude range. 
Nevertheless, for a relatively small region, say 
50,000 ft (15 km) around the perigee point, this 
approximation is valid to a very high order if an 
instantaneous value of K is selected. 

Using relationships r = a(l - e cos E) and 
r = a(l - e), Eq (57b) can be written as 



-Kat Kae cos E 
e 



(57b) 



Now substituting Eq (57b) into (55a, b) yields 



V-15 



Aa . 2 D -Ka< 

= - 4a Bp n e 

rev K 

( 

+ . . . ) dE 

^S- « -4aBp n (l-t 2 )e 
rev r 



e Ka e cosE (1 + 2( cQs E 

(58a) 
-Ka* f e Ka.cosE (cog E 



3 2 3 74 3 5 jll 6 , 



F' 



,3 2J 3,7 4, 



TF' 



+ <r cos E + . . . ) dE 



(58b) 



1 3 3 7 4 

a 2 = T « " F « " F « 



The integrals above could be evaluated in the 
form of modified Bessel functions of imaginary 
argument, if the brackets contained a series of 
sine terms. Therefore, at this point a further 
crucial approximation is introduced. It is as- 
sumed that significant drag exists only near the 

perigee. This assumption breaks down for very 
small eccentricities (i. e. , as e -*0), but the va- 
lidity of it is good for moderately elliptic orbits. 

2 n 

Assuming that sin E << 1 then cos E can be 

written as an infinite series of sines for odd n or 

as a finite polynomial in sines for n even. The 

first five sine expansions are aa follows: 



isin 4 E --^-sin 6 E 



cos E 


1 ? 

- 1 - -j- sin E 




5 ■ 8 „ 
- T5¥ sin E 


2 ^ 
cos E 


= 1 - sin 2 E 


3 ^ 
cos E 


-1-2- sin 2 E 



3 . 4_ , 1 , 6„ 
Tj- sin E +-t£ sin E 



> (59) 



+ TM 



sin E + 



4 
cos E 



2 sin 2 E + sin 4 E 



5„ , 5 . 2„ , 15 ■ 4 ,-, 
cos E = 1 - w- sin E + — tt- sin E 



TB 



sin E 



T2F 



sin E + 



Substituting Eq (59) into Eqs (58a, b) the fol- 
lowing expressions are obtained: 



Aa 
rev 



-4a 2 Bp 



*-s 



z cos E . . 2 _ 

e (o n - a- sin E 



4 fi fi 

a„ sin E - a, sin E - a . sin E-. . . )dE 



— * -4aBp 
rev ^ 



0'- S 



z cos E lo 



(60a) 



sin 2 E 



2 sin 4 E - /3 3 sin 6 E - |3 4 sin 8 E-. . JdE 

(60b) 



where 

z h Kac 



and the constants a., P. are power series in terms 

4 
of eccentricity, up to « , as follows: 



1 13 

a 3 = F * " TF « 

56 3 3 



(61a) 



a , , 12131415 16 

^0 cl + £ "T e 'I' ~ F e "F 6 " TG < " 

« .i +£+ ic 2 + ^c 4 + ... 



'l 'J 



IF 



1 5 2 13 33 4 



3 2 " F " TB ' " 2" ( " 



"BT* 



a ! 3 2 19 4, 

?3 * TB ~ "F2 { + TZB e + • • " 

5 13 2 A 27 4 ^ 

P4 " T78 ~ IW € + TTj^ e + --- .„... 

(61b) j 

It is noted that Eqs (60a, b) conform to the 
modified Bessel functions of imaginary argument, 
which can be written as 

( l 2\* n« 

_ . . \Z I ( z cos E , 2p 

1 (z) = -i '- \ e sin v 

P , , U ,U J „ 



r(p + i) r (i) 



EdE 
(62) 



where: 

p = (1, 2, 3 ---) 



r (n + 1) = h r (n) 



and 



<h"F 



The integrals in Eqs (60a, b) can now be expressed 
in terms of Bessel functions as 



zcos E ,_ t / \ 

e dE = « I„ (z) 





C Z CO 

J e 



s 



C z CO 

J e 



sE , 2_, ._. « l l lz) 
sin E dE = — — 



zcosE . 4^ .„ 3 « I 2 (z) 
e sin E dE = 9 — 



\ (63a) 



"E • ^h*. 3 " 5,tI 3 (z ) 
sin E dE = — — 75 



f z 

J eZc ° sE sin 8 



EdE = 



3-5-7n I (z) 



V-16 



NOTE: For modified Bessel functions I Q (0) = 1 



and I 2 (0) = I 3 (0) = . . . =1 (0) = 0, so that for 

z = 0, Eqs (63a) are seemingly indeterminate 
for p > 2. The limiting values, however, can 
actually be found to be finite: 



lim 
z_^0 



I p (z) 



1 



2 P (p!) 



(63b) 



Now in terms of modified Bessel functions the 
integrals of the orbital decay rates can be ex- 
pressed as: 



I 



n ffI i (z) 



3*I 2 (z) 



3- 5 it I 3 (z) 



3-5-7«I 4 (z) 
"4 T ■••(64) 



(and a similar equation involving j3. ). 

Thus, both Aa and Ae can be expressed as scries 
of the same form but differing coefficients. How- 
ever, the computation of these changes is unnec- 
essarily complex due to the fact that higher order 
modified Bessel functions can be reduced to a 
linear combination of orders zero and one (I n (z) 

and IjCi)) by the use of the reduction formula 



2p 



W z)=I P-i (z) --ir r p (z) 



(6 5) 



840a 4 (z +6) 



I x (z) 



(6 8a) 



F 2 (z, O = e" 



3/3, 



60/3. 



105/3 4 (z + 24) 



z 

6/3 



...]l (z) 



2 15^ 3 (z i +8) 



(68b) 



840^ 4 (z^ +6) 




Note is made that Ref. 16 tabulates e Z I„(z), 
e l^z). Note also that the following asymptotic 
series are given in Ref. 16, p. 271 for large z: 



e- Z I (z) 



(2ttz) 



1/2 



1 + 



l 2 -3 2 



1! 8z 2 ! (8z) 



,l 2 -3 2 .5 2 + 1 2 .3 2 .5 2 .7 2 
Z~ + , , ,„_ v 4 + ' 



3 ! (8z)" 



4! (8z) 



(69a) 



J Uz) 



(2ttz) 



12 ) 



!.- 



1-3 



9 

1-3-5 



l!8z 2! (8zT 



The reduction formulas up to the order four 



I 2 (z) =I Q (z) -f Ij(z) 



r 3 (z) 



I 4 (z) 



1 + 



2 I l l 
z / 



(z) 



-f~ V z > 



}(66) 



'^)V-' 



'h?) 



Ij(z) 



Now using Eqs (66) the decay rates of elements 
can be written in the final form for elliptic orbits 



l 2 - 3 2 - 5- 7 l 2 -3 2 -5 2 -7-9 



3 ! (8z) J 



4! (8zf 



(69b) 

Note is made at this point that decay rates as 
predicted by these formulas have been checked 
against the numerically determined rates and ' 
agreement shown to be good for the cases of mod- 
erate eccentricity. In no case, however, should 
the method be employed for eccentricities less 
than approximately 0.03 since the assumptions 
made previously restrict the range of applicability 
of the method. The value 0. 03 was determined 
numerically. 



Aa 

rev 

Ae 
rev 



4* a ^Bp F x (z, O 



4naB P() F 2 (z, O 



(67a) 



(67b) 



where the following nondimensional functions are 
used- 



F : (z,«)= e" 



60c 



105c? 4 (z + 24) 



z 
6 a, 



I (z) 



2 15c 3 (z +8) 



z z 

+ (continued) 



Now, noting that a = _ p , Eqs (67a, b) can be 
written in the following form: 



da 

ar 



de 

ar 



2Bp 



pr r 



OWT 






(70) 



But, since (-2Bp \^ r p ) is simply the decay rate 

for a circular orbit at initial perigee altitude, 
'dr \ 
3F/ = 0' * ne equations can be rewritten as 



dS "f ^P] (1 «)" 1/2 F 



(Via) 



V-17 



de 1 f dr p 

ar a \ "ar, 



(i - •) 



-1/2 



£ = 



(71b) 



From Eqs (45) and (71 ) the final decay rates are 
obtained 



dh a ( dr p 

ar - \~w> 



dh /dr 

ir Tar 



(i-o" 1/2 Gl 



(1 - «) G, 



(72) 
0. 03<€<0. 4 



where 

'dr 
P 

-ar 



-2Bp ^ 



^r 



}j « (1 + f) F x + 



(1 



OF 1 -F 2 



(nondimens ional ) 
(nondimension&l) 



At this point it should be noted that the functions 
G. and G„, although they are relatively complicated, 

are nondimensional and need be computed only once. 
In the present study these nondimensional drag de- 
cay parameters for elliptic satellite orbits were 

4 
hand computed, carrying terms up to € . The re- 
sulting parametric curves are presented in Fig. 9. 
Thus, the upper limit on c, e < 0. 4. 



This figure shows G 2> the perigee parameter, 

to be independent of e to a high order of approxi- 
mation though there is a variation of G 2 with the 

parameter Z. This behavior is not the case with 
G. , the apogee parameter, the reason for this 

behavior being that apogee decays much more ra- 
pidly than perigee for an elliptic orbit. Special 
attention is also drawn to the curves denoting low 
eccentricities. These curves will be discussed in 
subsequent paragraphs. 



dr 
a 

ST 



dr 
P 

-ar 



K 



P \ -C , ., b . 
?0 



K|/-°)e-C(a->> 

Po 



(75) 



But since V r « ^r (1 + t), Eq (92) can be 

(3L / 



written as 

dr 
a 

-ar 



1 + e 



'(-0 



dr /dr \ r~ 



+ 1 



(76a) 



(76b) 



where 



(50 n --"poiFT 



i 



P + =e 



€=0 

C 



(a + 5-) 



e- C (a-|) 



L n (nlf (1M (2!T 



n»0 



C> 



^2- 



£0 wr<n + i) 



1 + 



2(1!) 



V 



(77) 



3(2.' ) 2 



(c/2r 



kr t 
P 



J 



4. The Case of Small Eccentricities 

Since the Bessel function expansions of the 

previous section are not valid for eccentricities 

below 0.03, an alternate approach will be applied 

in this region. This approach was developed by 

Perkins (Ref. 8) and again assumes an exponential 

-kAr 
atmospheric model p = p Q e .In this analysis 

a nondimensional parameter C and a drag constant 
K are defined to be 



T /V„\2 "I kr t 

. c -*pHvl]-rfr 



(l + €) 



K^g, 



C D A 



w 



"o r o 



2Bp n r 
p. 



(73) 



(74) 



Using Laplace transformations, the decay rates 
are found as 



and 

The nondimensional parameters P and P of 
Eq (76) are plotted in Fig. 10. The trends of 
the curves are noted to be the same as those ob- 
tained by numerical integrations. 

Figure 10 is, of course, limited to small eccen- 
tricities, as can be seen from the following ex- 
ample: 

Assume: 

h . = 85 stat mi = 448, 800 ft = 136, 794 meters 
Pi 



Pi 



2. 135, 170 x 10 7 ft = 6. 507998 x 10 6 



meters 



0.02 



V-lf 



Solution 
From Fig. 7a: 
k 





p 



= 1.98 x 10" 5 /ft = 6.50 x 10" 5 /meter 
= 7. 15 x 1(T 12 slug/ft 3 

- 3. 684 x 10 kg/meter° (from Chapter II) 



. dt / 



= 



= " 2B Po^ r P 



-7. 84 fps = -2. 39 
mps 



From Eq (73): 
kr 



C = 



IT 



P 



= 8. 24 



From Fig. 10: 

P + = 2.73, P" = 0.0088 



From Eq (76a): r & = (_jJPJ j i + £ p+ 



6 = 

= 2.16 fps = 0.658 mps 
dr 



/Or \ 
From Eq (76b): r = -P J 



1 + 6 P" 



= 0.070 fps = 0.021 mps 

Consider the same example for a slightly 
larger 6. If £ = 0. 04, then C = 16. 1 and x = 64 
Proper convergence of Eq (77) now requires an 
extremely large number of terms (at least 25) 
thus making the solution impractical. 

Thus, since Perkins' methods and the Bessel 
method are applicable in different regions and 
since the solutions have the same form, i. e. , 

^dr 



f^) .„ f 



1 + e P € < 0. 03 



dr 

I 

dt 



if 1 + e G 1 (z) t > 0.03 



and similarly for r p . Perkins' parameters P + and 

P , can thus be considered to be analytic extensions 

of the parameters G y and G.-,. This fact was noted 

to be responsible for the low eccentricity curves 
of Fig. 9. J 

5. Apogee and Perigee Decay Rates and Satellite 
Lifetimes 

The previous Subsections C-3 and -4 have pre- 
sented in nondimensional form equations and graphi- 
cal data for r & and r However, before determin- 
ing an estimate of the lifetime of a satellite it is 
necessary to dimensionalize the various param- 
eters. This has been done in Figs. 11a, b, c 
and 12a, b, c, which present apogee and perigee 
decay rates both in English and metric units for 



altitudes in the range 75 to 400 stat mi (120 to 
640 km) and eccentricities from to 0. 4. It is 
noted that there are bumps on these curves. 
These irregularities are the direct result of 
similar behavior for the density slope of the 
ARDC 1959 atmosphere. Correction of this data 
for atmospheric variation will be discussed in 
Subsection C-6. Changes resulting from changes 
in the model atmosphere (e.g. , 59 ARDC to 62 
U.S. Standard) require recomputation of Fiss 11 
12, 13 and 14. S ' ' 

These decay rates must be integrated to yield 
the lifetime. As was mentioned earlier, this 
portion of the analysis will be conducted numeri- 
cally. The reason for this step is simple --it is 
not desired to introduce further approximation, 
which could materially affect the accuracy of 
study. To be sure, approximations have been 
made to this point; however, the validity of each 
has been well founded. If a further assumption 
were made to obtain an integrable form, the 
accuracy would suffer materially and the attention 
to detail exhibited earlier would be for naught. 
Some have argued that since the atmosphere is 
not known and since the other approximations have 
been made, such core is unnecessary. While this 
is true to a degree, a philosophy such as this will 
never yield good estimates even as the various 
density variability factors become known, while 
the philosophy of this section will reflect such 
improvements. 



is 



The integration procedure for this computation 



At. 



< Ah a> 
.1 



where 



(Ah ) is the j-th apogee altitude increment 



ffl.' 



is the apogee decay rate at this altitude 



thus 



reentry 



At. 



j = ° 



This integration is very simple and can be rapidly 



performed even for small values of (A h ) 



type of integration also admits several refinements 
involving the use of iteration and average decay 
rates rather than instantaneous rates. However, 
if the step size is sufficiently small this is not 
necessary. The correct value of (A h ) is deter- 
mined by the repetition of the same integration 
until the values of T^ for successive values agree 
to within a prescribed error. This step size need 
not be the same for all orbits, but for orbits of 
similar a and e , the step sizes generally are the 
same (a value of 500 ft or 150 meters was utilized). 
The results of this integration are presented in 
Figs. 13 and 14 in both English and metric units 



V-19 



for a value of B = 1 



ft' 
slug 



or 0.6365 x 10 



-2 meters 



kg 



Decay histories for typical satellites were added 
in dotted lines in order to indicate the changes in 
eccentricity and perigee altitude as functions of 
time. 

Lifetimes for all other values of B are obtained 
via the approximation 



L, 



B 



T L 2 * 2 



observed 



2 P p~a (G L + G 2 



This approach compensates for a variety of sins 
since the nature of the body in question, the mass, 
the nature of the tumble, and even variations in 
the density of the atmosphere are factors included 
in the correction. 

TABLE 1 
Comparison of Satellite Lifetime Estimates 



Effective B<- 



L r 



The basis for this approximation is that the decay 
rates were all noted to be linear functions of B. 
Thus, since B is a constant, it does not affect the 
integration, and as a result lifetime is inversely 
proportional to B. This behavior is true in free 
molecular flow; however, as B is made signifi- 
cantly larger or as the altitude is decreased, the 
vehicle leaves the free molecule region, and the 
assumptions of this chapter deteriorate. Thus, the 
simpler conversion must not be used indiscrim- 
inately. If there is a question as to the regime 
of flight, specific data should be prepared. Other- 
wise the conversion is justifiable. 

Though much has been written on the variation 
of lifetime with eccentricity, it is noted that these 
figures show the extreme sensitivity of this param- 
eter even for small eccentricities. This sensitivity 
explains why satellites with the same total energy 
per unit mass (i.e., same a) do not necessarily 
have the same lifetime. 

6. Comparison with Satellite Data 

In the final analysis, the value of a computational 
technique such as this must be assessed in terms 
of its ability to predict phenomena correctly. Thus, 
the actual lifetimes of several satellites will be 
checked in order to provide this information. 
First the value of B to be utilized must be com- 
puted for initial determinations of lifetime or for 
preliminary estimates. The value of B must be 
computed based on estimates made earlier in the 
discussion of free molecular flow. However, 
once the initial tracking data from the satellite is 
available, a more accurate method is available. 
This method is based on the formulas developed 
for the change in the element a. 



Name 

Sputnik 1 
Sputnik 11 
Sputnik III 

Explorer 111 

Explorer IV 
Score 

Discoverer 1 
Discoverer II 
Discoverer V 
Discoverer VI 
Discoverer VII 
Discoverer VU1 
Discoverer XI 
Discoverer XIII 
Discoverer XIV 
Discoverer XV 
Discoverer XV11 



(ftVslug) 

0. 69 
1.00 

1. 13 



3. 69 
1. 55 
2.98 
-1 . 5 
1. 50 
1. -16 
1. 13 
1.53 
1. 38 
1.65 
1.04 
1. 30 
1. 50 
0.95 



(in /kg) 

0. 44 X 10' 
0.64 

0. 72 



0. 38 
1.91 
0. 95 
0. 95 
0. 93 
0.72 
0. 97 
0. 88 
1.05 
0. 66 
0. 83 
0. 95 
0.61 



Estimated 

Lifetimes 

(days) 

145 



84 
46 9 



12. 6 
11.0 



14 
100 



87 
24 



51 



Actual 
Lifetimes 
(Rel. 15) 

( days) 

92 
162 
202 
693 

93 
455 

34 



19 
109 
11 
97 
29 



(.Computed from the satellite data of the initial decay rates of 
semimajor axis. 
(1 ft 2 /slug = 0.6365 x 10" 2 m y'ke) 

Since effective ballistic coefficient is considered 
the more accurate, it was used in the construction 
of the following table. 

Two things in Table 1 are important and should 
be noted. First, the values of B gff as computed 

from the orbital decay during the first few orbital 
revolutions are not in all cases in good agreement 
with the values predicted theoretically. Consider 
the following examples: 



B 



eff B t , 
„ theo 

Satellite (ft /slug) Agreement Remarks 

0.69 0.603 Good 



Sputnik I 



Neglecting 
antennas 



r + r 
a p 



Explorer III 3.69 3.71 



Good 



Random 
tumbling 



r + r 
a P 



h + h 
a p 



a = T~ *~ = ~~ " T 

Thus, if a is known, an effective ballistic coeffi- 
cient B ff can be found by utilizing the computed 

h and n for B = 1 (rather than the observed 

a p 

values). Thus 



2 a 



'observed 



eff 



(fl a + ^ 

F theoretical 



Explorer IV 1.55 3.21 



Poor 



Random 
tumbling 



This being the case, it is necessary to update 
the knowledge of B as data becomes available 
in order to obtain reasonable lifetime estimates. 
The second point is that the agreement between the 
computed data and the true data is good. To pro- 
vide an appreciation of the level of improvement, 
several previous works in the field were reviewed 
(Refs. 7, 9, 10, 11, 12 and 15). Data for these 
references are not included here because of the 



V-20 



fact that different atmospheric models and differ- 
ent data for the satellites have been assumed and 
different corrective procedures (i. e. , B „„) 

utilized in the correction of the results. As a 
general rule the estimates obtained here are 
superior to these works, though there were cases 
for which other curves were more accurate. 
Since this was expected, the relative value of the 
approach was determined by a root mean square 
estimate of the errors in the predicted lifetimes. 
(The results included here produced approximately 
13% error, while those of the literature varied 
from approximately 15% to 3 5%. ) 



atm 



sini sin(B + u) 



A 

s 


A 

, w 


sini cos (8+ u) 


! cos i 





; ° 



r A A-, 

r U I cos i S - sin i cos (0 + u) WJ 



Secondly, the vehicle velocity 



V=rR+r9S 



thus 



This improvement in the agreement seems 
very significant. However, the magnitude of the 
final error is still large. The reason for this 
large error lies in the fact that the method does 
not provide for atmospheric rotation, for density 
variability for variations in B, or for the oblate 
nature of the atmosphere. This being the case, 
subsequent paragraphs will be devoted to refining 
the previous work. 



D. 



THREE-DIMENSIONAL ATMOSPHERIC 
PERTURBATIONS 



Due to the fact that the atmosphere rotates, 
the velocity of the vehicle relative to the atmo- 
sphere will not be the velocity of the vehicle rela- 
tive to space. Thus, the drag force will not lie in 
the plane of unperturbed motion and each of the 
six elements or constants of integration will be 
affected rather than just the three considered 
previously. Since the equations for variation in 
the elliptic constants have previously been de- 
veloped, it thus remains to describe the perturb- 
ing force and discuss the resulting motion. 

1. The Perturbing Force 

The drag acceleration which acts on the vehicle 



n 2 A 

— = -B p V V 
m r r 



where 



V r =< V - V atm> 



V . =fi xr 
atm e 



This acceleration must now be resolved into com- 
ponents in order to permit evaluation of the re- 
sultant motion. The specific set of components 

to be utilized is the set R, S, W discussed in 
Chapter IV. 

/\ 

R is measured along the radius 

S is measured in the general direction of 
motion perpendicular to R 

/\ 

W completes the right handed set. 

First, the atmospheric velocity 



V = r R + (r i 
r 



r Q cos i) S 
e 



and 



+ r f2 sin i cos (9 + u) W 
e 



|v I 2 = r 2 +(r9) 2 - 2r 2 G fi cosi+(rQ cosi) 2 
r I g e 



tT"r n si 
L e 



sin i cos 



(6 + u>l] 



V 2 - 2HVi cos i + r 2 Q. Tcos i 
e e u 

2 2 "I 

+ sin i cos (9 + u)J 

V 2 - 2Hf2 cos i+ r 2 Q 2 fl 



sin i sin (8 + u)J 



where 



H= the angular momentum per unit mass 

This result was also obtained by Sterne (Ref. 
18) and Kalil (Refs. 19 and 20). Now at this point 

the function |v | must be expressed in terms of 

the eccentric anomaly in order to facilitate inte- 
gration with respect to time. 



V 



M 


1 + € cos E 


a 


1 - t cos E 



a 2 (1 - 2c cos E + i 2 cos E) 



thus 

2 _ n 1 + e cos E 



V 



a 1-6 cos E 
2 



"eP' € ■ !-« cosE 

"5 cosl l+ £ cosE 



+ %_ U ; « c° s gf . gin 2 . sin 2 + u)) 
2 (1 + e cos E) 
n 

n = (ii/a 

But, as was noted by Sterne, U : /n can be no 

larger than approximately 1/15 for earth satellites; 

thus V can be obtained in an approximate sense 
r 

by the binomial expansion of the quantity within 

the braces by neglecting terms of the order 



V-21 



(n l /n) . This step appears justifiable in view 

of the fact that there is such a large uncertainty 
in the atmospheric density at any time and in the 
aerodynamic characteristics of the vehicle. Under 
this assumption, V can be expressed as 



1 + £ cos E 
1 - € cos E 



n Ji-t 



1 - £ cos E 
1 + e cosE 



This equation shows that to the order of corrective 

^ 2 1 



or 



45fT 



i /i r 

terms smaller than approximately 7 f y-rl 

the effect of the earth's rotation is a simple func- 
tion of the inclination and of time. The form of 
this corrective term being sufficiently simple, the 
subsequent integration of the equations of motion 
appears attractive. Now, the drag acceleration is: 



d€ 



^- = jj cos <j> [R sin 9 + S (cos 6 + cos E)] 
di _ r cos (6 + w) 



dt 



2 . 



W 



n a sin i cos <j> 
df2 _ r sin (9 + u) 



dT 



-W 



n a sin i cos <b 



2 2 

du _ a cos 4> cos 8 R - r sin 6 (2 + € cos 9) , 



dT 



2 • X 

n a sin d> cos 



2 
n a cos if tan i 



W 



d€'_ 2r R , „ . 2 4> d (u + «) 
dT T + ' i sin -2 at 



aJ (l-£CosE) 3 



1 -C 



1-6 cos E 
1 + e cosE 



e sin E R 



{(' 






+ [ il - € c 2 - « o cos i J — (1 - £ cos E) 2 j S 



+ Q e sin i cos (9 + u) ( ^- (1 - e cos E) 2 W 



where 



C = 



U || 1 - £ cos i 



where 



, „ , . 2 idfi 

+ 2 cos <)> sin -p -TT- 



1/2 



sin 4> = (1 - £ ) as is customary in some 
of the astronomical texts 

£' = mean longitude at the epoch 

R, S , W = the components of the disturbing 
acceleration 

At this point it is noted that since 



But 



cos (6 + u) = cos 9 cos w - sin sin u 



n (t - t Q ) = E - £ sin E, 
E = 1 ~ W 

1 - £ cos E 



cos E 



sin E I 1 



1 - £ cos E 1 - £ COS E 

Thus the final form of the drag acceleration is 



R.= _ Bp A< l + *cosE 

a (1 -ecosE) 3 



[-c^ 



£ cosE 



£ cosE| I . „ g 
1 ' £ sinE R 



.{{: 



\-(u e sin i I — (1 - £ cos E)J (( 



a " (1 - £ cos E) 2 ] S 



£ cos E)l ((cos E - e) cos u 



/ 



(sin E » 1 - £ ) sin 



u) W 



2. The Change in the Orbit 

At this point it is necessary to refer to equa- 
tions for the time variations of the orbital elements 
(Eqs (60), Chapter IV) or to the form utilized by 
Sterne and presented in Plummer (Ref. 21): 

gr- = - [R tan <)> sin 9 + S sec $ (1 + £ cos 9)] 



Also from Chapter III, 
cos E - £ 



cos 9 = 



r 



sin 9 



.rr. 



£ sin E 
1 - £ cos E 



Thus the expressions for the changes in the 
orbital elements obtained by substituting for 
R, S and W can be transformed into functions of 
the independent variable E and its time rate 

E. Integration for the secular change in each 
element would then be possible (utilizing the 
limits for E of to 2tt) if the density could also 
be expressed as a function of the variable E. 

As was noted in previous sections of this 
chapter, the density of the true atmosphere does 
not vary exponentially with altitude. However, 
as was also noted for small variations in the 
altitude the approximation is valid. Selecting 
once again the perigee altitude as the reference 
for the approximation (since the largest portion 
of the drag force occurs near perigee), the den- 
sity can be written as 



P = P, 







-K (h - hp) 



V-22 



a 4 



where 

p„ = density at perigee 

h = a (1 - e cos E) - R [l 

, . 2 . . 2 . ,i 

- f sin l sin (9 + a>)J 



(1 - - R [l - f sin 2 i sin 2 u] 



h = a 

P 

h - h = at (1 - cos E) + R e f sin 2 i[sin 2 (e+u>) 



. 2 

sin 



«3 



R„ 



v e = earth's equatorial radius 
Thus the approximate density is 

P = Pq expQ-Z (1 - cos E) + q (sin (9 + u) 



. 2 



u)] 



where Z was previously defined to be Kae, and 
where 



q = K R f sin i 
n e 

At this point Sterne presents a Taylor expansion 
of p in the form 



Since the angle w is approximately constant during 
any single revolution, the q. can be treated as 

approximate constants when integrating over one 
revolution, without the introduction of appreciable 
error. 

It is noted that according to the remainder 
theorem for alternating series, a series whose 
terms are alternately positive and negative, and 
such that their absolute values form a monotone 
null sequence, is convergent (this is the case 
here for the series expansion of the atmospheric 
density). This being the case, the absolute value 
of the remainder after n terms of such a series 
does not exceed the absolute value of the (n + 1) st 
term. Hence, the relative error introduced in 
the series expansion of the atmospheric density 

by retaining only terms through q D is 

A p K krrji ex p w 

Thus, by retaining terms through q , the relative 
error in p is 3. 4% at altitudes of 100 naut mi (185 km) 
where q ■* 0. 5, and only 0. 16% at altitudes of 
200 naut mi (370 km) where q -v. 0. 2. 

Upon substitution of this density model into 
the equations of variation of constants and perform- 
ing the integration, Sterne reported the following 
secular changes in the elements: 



P = P Q e e 



-Z Z cos E 



7 ±$— (sin 2 ( 9 + U ) - sin 2 u) -e 



i = 



-Z Z cos E 
; e 



q' 



■ 2m _ 
sin E 



m = 



2m ,. „,2m 

(1 - ( cos E) 



In the series, the terms which are odd functions 
of 9 are also odd functions of E and may be ig- 
nored since they will not contribute to the com- 
plete integral for the secular changes in the 
elements. Using the even part of the series 

through terms in q , which gives the series ac- 
curately to about 1 part in 1000 for the altitudes 
in which this study is concerned, Kalil obtained 



q = 1 



2 n 9 

q 1 = (1 - c ) (-q cos 2u> + \ gin^ 2u) 



^3 



(I-, 2 ) 2 


2 

q 

~2~ 


4 ~| 


+ Trx sin 2(j 


(I-, 2 ) 3 


c 



4u - SL cos 2u sin 2u 



3 

J_ 

2 



cos 2u + %- cos 2u sin 2u 



4 
1 
2"4" 



2u 



+ ^j- cos 2u sin 2u - ^ sin 4 

.4 4 



4 
i 
YZ 



2.4 



(1 - c ) 



5L cos 2u - % sin 2 4u + %? sin 4 2u 



24 



16 



24 



(Aa) sec = - 2B ^ a 



(1 +0 



3/2 



(1 - c) 

9f n 



T72 



1 - 
1 + € 



'ote 



l+ A + 



128Z 



(A«> sec = -2B (1 - «<) rl . ( 



^ 



1 - C 



1 - e 



1 + e 
4e C 



p oQk 



(Ai) 



8Z 
B 



1 (3 + 4. N + ^ - + t 4tC \ + 1 

'^ ini(1 - (2)(1 - c fe) -Po/^k 

+ cos 2 u |"l - gL. A 5 + 4, N + 4t 5 + 6 A + ... 
(AC!) sec = "I "e sin 2 " (1 " e2 > (1 - C f^|) apfT 



•^l-i z (15-4e 1^^+46 N) + 



(Au) sec = - cosi < An > S ec 
(Ae,) sec = (1 - cos i)(^) sec 



V-23 



or 




(AM) 

sec 





where 




t l = 1 - 8c 


N 



4« 



r+ 8q 



Ni^T? 



B„ 



(1 -C)' 



4 1 33 2 



iKa 



3q r 



(6 - 5C) 



(1 - C ) 2 + ^^ (10+ 17C> 



t , , 8 6 2 (1 + 5€ 2 ) ,16 eN (5e 2 - 1) ^ 32 ,2 M 2 

2 d (1 - £ V 1-6 



T' 



+ -| J*! (7 - lOd + 6C 2 ) 



N = 



-i|.q (1+lOe + 8€ N) + i^-q 2 (1 + 4€) 

1 + C 

1 -C+(+«C 



These results are believed valid for all of the cases 

for which Z > 2 to the order of q 2 and represent 

the solution well for such cases. However, if 
Z < 2 a more general solution is necessary. This 
solution suggested in Sterne's paper (carried out 
for the element a) is reported for the elements 
a and e by Kalil. The results are shown below. 

5 

(Ar) sec = -GTrrBad - C) 2 p Q e" Z £ A^ (Z) 



(Aa) = -4tt Ba 2 (1 -C) 2 p n e 

sec 






A I (Z) 
n n 



(Ae) 



■kBa (1 - c 2 ) P Q e" Z 



B I (Z) 

n n 



n=0 



irhere the constants evaluated for small eccentric- 



ities (i. e. , e << 1) are presented below: 
^0 



A n = 1 + , 2 (j 2 + \) 



A 1 = 2j,-i z ( j 2 4) + | 



l + e 2 (j 2 + 4j + I> 



A 2 =2q 1 |(j + l)-3^q 1 (r + 4 j + ^) + 3- 2 



A, = 6| q 9 (j+ 2)+ 15-4 



"3 "Z 4 2 
15q 



^"l 3 " 



105 q. 



* 9 4S 

2e <j+ 3) + ^- (j + 12j + ^) 



A 5 = 210 q 4 -^ (j + 4) 



B Q = e (2C+ 1) 

B l = ^- C ) 2 -^f + ^a< 3 - 2C) 



3q, 



97 
~2~Ka 



+ € (5 - 4C) 



15q, 



(1 -C) 2 --[^(7 - ■ 2 ^C+6C 2 ) + 6 2 (^- 



~^L 



30C+21C ) 



Ka 



M 105, 



_, + 73 ^(9-14C + 8C ) 



q 3 2 

B c = -4(105^) 



-^ - 33C+ 21C 2 



+ — i (105) 



-I Z' 



(1 



l + c 



"Ka 



(9 



29 



2,89 



C+8CT) + e {=£-- 56C) 



K = negative log density slope 

The symbols C, Z, e and q. are the same in this 

set of equations as previously defined. The re- 
duction formulas discussed earlier can also be 
utilized, to relate all of the higher order Bessel 
functions to the fundamental functions I„ (Z) and 

I 1 (Z). This step simplifies the numerical evalua- 
tion of the time history of the decay; however, it 
only serves to make the functional form of the 
resultant equations more complex. For this 
reason the equations are left in their present 
form. 

This set of equations is believed valid for 
satellite orbits extending down to approximately 
180 km with errors less than several percent. 
Thus, if the inclination of the orbit were to be 
specified, the equations could be integrated 
numerically to yield realistic lifetime and decay 
histories for the vehicle as was done in the 
discussion of the nonrotating atmosphere. The 
possibility of being able to construct a family of 
lifetime figures for various inclinations is also 
noted, though to date this has not been accom- 
plished. Indeed, this step does not appear at- 
tractive for general computations because the 
procedure would result in an error source when 
data is applied for values of B other than that 
utilized in the construction of the figures. Thus, 
the most attractive procedure involves the numeri- 
cal integration of the decay rates for each satellite 
of interest. This approach, though more cumber- 
some, will be more numerically exact and should 
result in errors approaching an order of magnitude 
less than those obtained with the nonrotating at- 
mospheric analysis. 



V-24 



Though numerical data is not presented, 
several general observations will be made. First, 
the equations show that the effect of the atmospheric 
rotation is to decrease inclination for all orbits 
(inclination defined 0" < i < 180°). Secondly, the 
effect is to decrease the rate at which a and t 
vary for i < 90° and increase the rate i > 90° . 
Thirdly, rotation produces secular regression 
and precession of the osculating ellipse. 

Numerical computations reported by Sterne 
substantiate not only these general trends but 
also to a good degree, the numerical values of 
the perturbed elements. This being the case, the 
theory as evinced by the equations of this section 
is believed to represent the best theoretical esti- 
mate of the behavior of the vehicle. 



E. THE EFFECTS OF DENSITY VARIABILITY 

(Ref. 22) 

To this point the approximations made in the 
discussion of atmospheric effects have been re- 
fined to include oblateness and rotation. Still 
no mention has been made of the effects of density 
variability. If the time intervals are large and the 
altitudes sufficiently high that the forces are not 
extremely large, the density variability effects 
will tend to null out due to the fact that the model 
atmosphere approximates average conditions. 
These cases are treated in previous discussions 
to varying degrees of approximation. However, 
if the time intervals are short or the densities 
more significant, the effect of variability will be 
more pronounced, and the equation should be 
integrated with the estimated density rather then 
with the model density. One approach to the 
problem of analysis of this latter case was shown 
in Chapter IV-C-6-d, which discusses random 
drag fluctuations. The following paragraphs 
(Ref. 22) extend this approach and provide some 
numerical data which is of general interest. The 
parameter of these discussions is the time of 
nodal crossing, a readily observable and easily 
computed quantity; the other parameters, be they 
orbital elements or position and velocity, should be 
checked as time permits. One such investiga- 
tion is reported in Ref. 23. 



1. 



Errors in the Time of Nodal Crossing due 
to Drag Fluctuations Alone 

The contribution of random drag fluctuations 
to the rms error in predicted time of nodal 
crossing depends on the correlation function of 
the random fluctuations, which is unknown. Upper 
and lower bounds, however, can be constructed. 
These bounds on the random error are given in 
Fig. 15. In the upper bound, the random drag 
fluctuations are assumed independent from one 
revolution to the next. In the lower bound, the 
random fluctuations are assumed perfectly cor- 
related over intervals of 25 revolutions, but un- 
corrected from interval to interval. The curves 
actually show the ratio of the standard deviation 
of the prediction to the standard deviation of the 
random fluctuation, a, which is calculated from 
observations smoothed over intervals of 25 
revolutions. 



The estimation of a is thus necessary to trans- 
late the data of this figure to errors in the pre- 
dicted time. No completely satisfactory method 
is available to perform this function; however, 
observations of satellites with perigees in the 
range 220 to 650 km indicate that a (in minutes/ 
revolution) is given by the empirical equation 



= 2. 2 x 10 



h 



(78) 



where h is the height of perigee in km, and t is 

the smoothed rate of change of period (unperturbed 
by sinusoidal and random fluctuations) in minutes 
per revolution. 

For orbiting satellites the smoothed rate of change 
of period, t, can be determined from observations. 
For satellites not yet launched, the values obtained 
from the previous discussions can be used as an 
estimate for the smoothed rate of change of period. 

A simple approximation for the prediction 
error caused by both of the assumed random drag 
fluctuations is dashed in between the two bounds 
in Fig. 15. It is 



1/2 



G rms < N >/ CT= 5 < N ' 3 > 



(79) 



where G (N) is the rms error in the predicted 

rms 

time of nodal crossing (in minutes), N revolutions 

after the orbit was perfectly known. Equation 

(79) is asymptotic to both bounds and all three 

curves derived in Chapter IV. 

The contribution of a different assumption 
(i. e. , of a sinusoidal drag variation) to the error 
in the time of nodal crossing is given by 

(80) 



H (N) = (2)" 1/2 A (k) 2 

rms 



- (kNr/2 



1 - cos (kN) 
kN - sin(kN) 2 



1/2 



where: 



A 



the rms sinusoidal prediction error 
(in minutes) for arbitrary initial 
phase of the sinusoidal drag 

I I -3 (81) 

1.8 h D x 10 (empirically 

P ' ' 
determined for same conditions as 

cr, Eq (78)). 
perigee altitude(km) 

(1.61 t) 10~ 4 

the period in minutes 



Thus the sinusoidal and random errors can be 
combined to give the rms error in timing of an 
orbital prediction when the initial elements are 
perfect: 



V-25 



(N) = 



G 2 (N) + H 2 (N) 

rms rms 



1/2 



(82) 



Now, if the local speed of nadir point is V n , and 

changes only slightly during the N periods over 
which the prediction is made, then the correspond- 
ing positional error tangential to the projection of 
the orbit on the earth is 



X (N) 



v o A 



r n (N) 



(83) 



2. Errors in Orbital Predictions When the Elements 
and Rate of Change of Period are Obtained by 
Smoothing Observations - 

In the preceding simplified formulas, a perfect 
knowledge of the orbit at the initial time, or epoch, 
has been assumed. In actual orbital predictions, 
the elements at the epoch and the rate of change of 
period are usually found by some smoothing pro- 
cedure, using data containing observational errors. 
(Discussions of the errors made by various satellite 
tracking devices appear in Chapter XI.) Thus, to 
be rigorous these error sources must also be in- 
cluded in the analysis. 

Suppose that the rate of change of period is cal- 
culated from M(< i) "measured" times of nodal 
crossing, which are uniformly distributed through- 
out an interval of i revolutions. Assume that there 
are three independent causes of fluctuations in the 
"measured" time of nodal crossing: 

(1) A 27-day sinusoidal variation in the rate 
of change of period 

(2) A random fluctuation in the rate of change 
of period, which is independent from 
revolution to revolution 

(3) A measurement error introduced by the 
tracking device. 

Of course, only (3) can be regarded as an error of 
measurement, but (1) and (2) will contribute an 
error to the smoothed values of the period and the 
rate of change of period. The errors will be 
given as a function of the number of revolutions 
N, after the epoch. The epoch is taken to be at 
the center of the smoothing intervals. 

(1) The contribution of the smoothed sinu- 
soidal drag variation to the rms error 
in an orbital prediction which runs for 
N revolutions from the epoch is 



S(N) 



A 



where 



cos kN 



F 



2i 

TT 



f 2 + f) 



(*) 



1/2 



(84) 



64 . /ki\ 

737 sin It) 

i k 



cos (£) 



N 



.2 (i + 2)\ 



T2~ 



= sin kN - kN + 8N £ i ( 


i + 2) k] 


T i • .2,2] 
.[cos^-l+i-^ 





and A is given by Eq (81), i is the smoothing in- 

_4 
terval in revolutions, and k = 1.61 x 10 t, 
where t is the period in minutes. 

As the smoothing interval, i, approaches zero, 
Eq (84) approaches Eq (80), which represents the 
sinusoidal error when there is no smoothing. The 
quantity S(N)/A is graphed in Figs. 16a through 
16d. 

(2) The contribution of the smoothed random 
fluctuation to the rms error in orbital 
prediction is 



R(N) 



*iv + 2 & 

- *6(¥) 3 - (?) 2 



64 /N 



(f) 

2U J 



1/2 



for i >> 1 
where a is given by Eq (78). 



(85) 



Equation (85) should be compared with its 
unsmoothed counterpart, Eq(79). The quantity 
R(N)/(5(j) is graphed in Fig. 17. 

The contribution of smoothed measurement 
errors to the rms error in the predicted time of 
the Nth nodal crossing is 



O(N) 



C7 Q (M) - 1/2 (if 2 



( 



(ir 



M (M +2)" 1 + (16/9) (M +2) 2 /M 2 



+ 256 N 4 + 16 (Ni) 2 I M (M + 2)" 1 



•■0 

1 

+ 2)" 2 J + 32 Ni 
(i) 2 /(3M) - 4N 2 (M + 2)" 1 j 



- (8/3) (M + 2)/M 

- 2M (M 



(86) 

where all the observations are assumed to have 
the same standard deviation, cr n , and M is the 

number of observations in a smoothing interval 
of i revolutions. The quantity 0(N)/a n is graphed 

in Fig. 18. The observational errors, a n , made 

by various tracking devices are given in Chapter XI. 
In order to have the error given by Eq (86) in 
minutes of time, it is necessary to use <r n , the 

error of a single observation in minutes of time. 
Angular errors, A 9 (in radians), can be approxi- 
mately converted to timing errors, <j Q (in minutes) 



V-26 



by 



(' * sM 



AO 



(87) 



where h is the height of the satellite, and R is 
the radius of the earth, and V Q is the local speed 
of the nadir point in units of length per minute. 

Doppler errors are more difficult to convert 
to errors in timing. They are subject to refraction 
and azimuth uncertainties, and it is difficult to 
tell how many independent observations are made 
in one pass. In addition, refraction and oscillator 
instability can create biases as large as the random 
errors of observations, and these biases cannot be 
reduced by smoothing observations from a pass 
over a single station. The observational error 
in minutes for one independent doppler observation 
is approximately 



,(t.-t.) . Ar 
f l (r. - r.) 



(88) 



where the range rate changes from an initial value 
of r. to a final value r f during the time (t - t.), in 

minutes, that a doppler signal is being measured 
by the station. The range-rate error in a doppler 
observation is A f. For a typical case, (t - t.) is 
10 minutes, and (r. - f f ) is 20, 000 feet per second 
(or 6100 mps). 

/o^ There ls an im P° rta nt difference between Eq 
(87) on the one hand, and Eq (88) on the other 
Equation (87) is applicable to each individual ' 
observation, hence to the average of a group of 
observations. Equation (88) only represent average 
conditions, so they only apply to the average of a 
group of observations, such as would be used with 
Eq (86). 



The errors are given as a function of the number 
of revolutions after the epoch assumed to be at 
the center of the smoothing interval. Now assum- 
ing that the observational, sinusoidal, and random 
errors are independent, they can be combined to 
give 



(N) =|[0(N)] 2 + [S(N)] : 
-oV/2 



[K(N)] : 



(89) 



where E rms (N) is the standard deviation of the 
predicted time of the Nth nodal crossing after the 
epoch, when the elements and rate of change of 
period are obtained by smoothing observations. 
E rms (N ^ re P re sents the error tangential to the 
orbit of the satellite projected on the celestial 
sphere. Errors at right angles to the orbit are 
usually an order of magnitude smaller. 

Errors in actual predictions issued by the 
Vanguard Computing Center, NASA Computing 
Center, Smithsonian Astrophysical Observatory, 
and Naval Weapons Laboratory are compared 
with the theoretical model in Tables 2 and 3. 
Table 2 contains the errors in one to two-week 
predictions made near the peak of the sunspot 
cycle. Table 3 shows the errors in predictions 
half-way between sunspot maximum and sunspot 
minimum. In the tables, N is the number of 
revolutions predicted, beginning at the center of 
the smoothing interval. The smoothed rate of 
change of period is f (minutes per revolution). 
The root-mean-square prediction error F 
i^n /• ■ ^ > • ' " r ms 

(N) (in minutes), includes the contributions of 
observational errors and drag fluctuations The 
theoretical prediction error caused by observational 
errors alone is designated by O(N). 









TABLE k 












Prediction Errors 


Near Peak of Sunspot Cycle 






Satellite 


Dates 


No. of 
Predictions 


-T 

(Min/ Rev) 


N 
(Rev) 


O(N) 

(Min) 


E rms (N > 


Actual 
(Min) 


Theoretical 
(Min) 


Explorer IV 


1958 


8 


2. 15 x 10" 3 


165 


0.024 


3.2 


3.7 


Sputnik III 


1958 


7 


1. 32 x 10" 3 


220 


0. 01 


3. 3 


1.9 


Vanguard I 


Fall, 1958 


20 


5.5 x 10" 5 


154 


0. 056 


0.25 


0.22 


Vanguard I 


Summer, 
1959 


11 


2. 1 x 10" 5 


154 


0. 056 


0. 13 


0.097 


Vanguard I 


Winter, 
1959 to 1960 


7 


6.5 x 10~ 6 


154 


0.056 


0.062 


0.061 


Atlas -Score 


Dec. 1958 to 
Jan. 1959 


1=:= 


2. 2 x 10" 2 


271 


0.3 


67.0 


74.0 



**™ g ^ ol ? set T ati °? has ™> statistical significance. This case is included merely to show how large the 
error can be when the rate of change of period is large. g 



V-27 



TABLE 3 

Prediction Errors Half-Way Between Sunspot 
Maximum and Minimum 



Satellite 



Tiros II 
Vanguard I 
Transit III-B 
Echo I 



Dates 



Dec. 1960 to 
May 1961 

Oct. 1960 to 
May 1961 

Feb. to Mar. 
1961 

Oct. to Dec. 
1960 



No. of 
Predictions 



12 



12 



10 



(Min/Rev) 



3. 7 x 10 



7.4 x 10" 



1.05 x 10 



6. 8 x 10 



N 
(Rev) 



TABLE 4 

Errors in Individual Orbital 
Predictions for Vanguard I 



Number 
of Pass 



2309 
2986 
2836 
2234 
2459 
2535 
3173 
1934 
2911 
2610 



Errors 

(seconds 

of time) 



+37 
-25 
+21 
-21 
+ 17 
-16 
+ 14 
-14 
+ 12 
-12 



Number 
of Pass 



2159 
1708 
2685 
2009 
1633 
2384 
2760 
2084 
1858 
1783 



Errors 
(seconds 
of time) 



-12 
-12 
-11 

- 9 

- 7 
+ 6 

- 3 
+ 2 
+ 2 
+ 1 



rms 



15 seconds = 0.25 minutes 



It is interesting to note that observational 
errors were the principal cause of errors in 
orbital predictions for only one of the cases shown, 
that of Vanguard I with its perigee in darkness 
(Winter 1959-1960). In all the cases, the pre- 
diction errors attributable to observational errors 
were smaller than the total error for Vanguard I 
in darkness. If the errors in predictions had 
been caused mainly by observational errors, then 
the prediction errors would have been independent 
of the smoothed rate of change of period. A de- 
tailed discussion of the theory and the method of 
calculation is given in Ref. 21. 

Theoretical calculations of the errors in 
orbital predictions by the methods described above 
are subject to uncertainties because of variations 
in methods of fitting, spin of nonspherical satel- 
lites, and sampling errors as well as uncertain- 
ties in the estimates of the smoothing intervals. 
The uncertainty in the theoretical rms error is 
approximately +100 to -50 percent. All of the 
examples in Tables 2 and 3 were within these 



250 



150 



22 



145 



O(N) 

(Min) 



Actual Theoretical 
(Min) (Min) 



0.08 
0.06 
0. 04 
0.04 



E (N) 

rms 



0. 12 



0. 12 



0. 74 



4.4 



0.0E 



0.06 



0. 50 



3.3 



bounds. Deviations from the theoretical model 
have tended to be on the high side so far (1958 to 
1961). During the two years near sunspot mini- 
mum, the percentage variations of the decimeter 
solar flux (which is correlated with atmospheric 
density) are only one-third as large as during the 
rest of the sunspot cycle, so the deviations from 
the theoretical model can be expected to be on the 
low side during 196 3 and 1964. 



E (N) in Tables 2 and 3 is, of course, 

rms 
a root-mean-square error. The error in an 
individual prediction can be larger or smaller 
than the root-mean-square value, and can be 
positive or negative. The distribution function 
appears to be normal. Table 4 shows the individual 
errors in twenty predictions made for Vanguard I 
when its perigee was in sunlight (Fall, 1958). 

3 Err ors in Orbital Predictions When the Rate 
of Change Period is Calculated from a~ 
S tandard Atmosphere ~~ 

The usual way of making satellite orbital 
predictions is to compute the elements and rate 
of change of period at the epoch by smoothing all 
the observations made during a certain time in- 
terval (usually a few days). This orbit is then 
projected forward in time. All of the predictions 
listed in Tables 2 and 3, with the possible ex- 
ception of the predictions for Transit III-B, were 
made by this method. The theory appropriate to 
this method of making predictions has been de- 
scribed above. The theory for the case in which 
the rate of change of period is derived from a 
standard atmosphere will now be described. Such 
a method might be used when there are not enough 
observations to determine the rate of change of 
period. In this case, the error can be separated 
into three parts, described under the following 
headings: 

(1) The error in the period and the time 
of nodal crossing. 

(2) The error caused by computing the 
rate of change of period from a standard 
atmosphere. 



V-28 



(3) The error caused by the sinusoidal and 
random drag fluctuations. 

(1) If the period and the time of nodal crossing 
at the epoch are obtained by a single orbital fit 
over N revolutions containing M independent ob- 
servations, then the errors in the period, X~t (in 
minutes), and time, At (in minutes), caused by 
observational errors, are 



At 



M 



■1/2 



(90) 



where the errors in predictions contributed by 
the time of nodal crossing, the period, and the 
rate of change of period are At, NAt, and 

2 
(N /2) t, respectively. 

If the coupling among the period and the time 
of nodal crossing (which should not cause much 
error) is ignored, then the root mean square 
error in a prediction made with a standard 
atmosphere, N revolutions after the epoch, is 
approximately 



and 



At 



4a i 



M 



•1/2 



(91) 



where a is the error of a single independent ob- 
servation (in minutes of time) and may be obtained 
from the observational errors in angular and dop- 
pler units by Eqs (87) and (88), respectively. 

In the case of precision doppler observations, 
an alternative method of calculating the period is 
feasible but is not recommended, because it pro- 
duces large errors in the period. This method 
is to compute independent values of the elements 
from each pass of doppler data recorded by a 
station, and average all the sets of elements de- 
rived during i revolutions. The errors in period 
and timing (caused by observational errors) pro- 
duced by this method are roughly 



At 



ff (M) 



•1/2 



(90a) 



E * (N) 

rms 



(At) 2 + (NA?) 2 + 



G (N) 

rms 



ft-"* 



(93) 



H (N) 

rms 



1/2 



where the epoch is taken to be the center of the 
smoothing interval employed in calculating the 
period and time of nodal crossing. Equation (93) 
applies in cases in which a standard atmosphere 
is used for calculating the rate of change of 
period. The error E * (N) is tangential to the 

orbit of the satellite projected on the celestial 
sphere. The error at right angles to the orbit 
is usually smaller. 



and 



4. Example 



At 



CT (w) 



1/2 



t„ - t. 
f l 



(91a) 



Problem: 



where (t f - t.) is the time interval during which 

a single station is recording doppler data during 
a pass. 

(2) The rate of change of period f can be ap - 
proximately calculated by using the theory of drag 
perturbations in Chapter IV and one of the stand 
ard atmospheres described in Chapter II. This 
method is not precise and a certain amount of error 
is thus inserted. However, the magnitude of this 
error can not be described analytically and must 
thus be accepted. 

(3) The errors caused by sinusoidal and ran- 
dom drag fluctuations are given by Eqs (80) and 
(79), respectively. The reason for using models 
which do not include smoothing is that f is ob- 
tained from a standard atmosphere. 

Now that the three factors have been discussed, 
the predicted time of nodal crossing can be 
written in the following form: 



t (N) = t + Nt + 



(i-y 



(92) 



Calculate the root-mean-square error in an 

orbital prediction for Explorer IV, 165 

revolutions from the center of the smoothing 

interval. The period at the time of interest 

was 109 minutes, and the heights of perigee 

and apogee were 142 and 1190 naut mi or 

263 and 2200 km, respectively. The smoothing 

interval is estimated to be i = 100 revolutions, 

the number of observations, M = 25, and the 

prediction interval, N = 165. The smoothed 

-3 
rate of change of period, r = -2. 15 x 10 

min/rev, and the observational error is es- 
timated to have been 0. 7 milliradian. The 
elements and rate of change of period were 
derived by smoothing observations. 



Solution: 



The errors given by Eqs (84) through (89) are 
appropriate. The average height of the 
satellite h, was 666 naut mi or 1232 km and 
the approximate speed of the nadir point was 
V n 2 2tt R /P = 198 naut mi per minute or 

367 km/min, so Eq (87) gives for the average 
error of an observation, a„ = 2 x 10 minutes. 
From Fig. 18, 0(N)/a Q = 12, so the con- 
tribution of observational errors to the error 



V-29 



.0. 



in an orbital prediction is 2.4 x 10 -2 minutes. 
The normalized random error, R(N)/(5a) is 

3 
1.6 x 10 , from Fig. 17. According to Eq 

(78), a is 3. 7 x 10 minutes per revolution. 
Therefore, the prediction error caused by 
random fluctuations is 2.95 minutes. The 
normalized sinusoidal error is S(N)/A = 7. 5 x 

3 
10 , interpolating between Figs. 16b and 16c. 

According to Eq (81), A is 3. 06 x lO -4 minutes 
per revolution. Therefore, the prediction 
error caused by the sinusoidal variation is 
2. 3 minutes. Combining the three errors 
by Eq (89), the theoretical error of prediction 
is 3.7 minutes. For comparison, the root- 
mean-square error of eight predictions issued 
by the Vanguard Computing Center was 3. 2 
minutes. 



F. REFERENCES 



12. 



13. 



14. 



15. 



16. 



17. 



1. Eckert, E. and Drake, R., "Heat and Mass 
Transfer, " McGraw-Hill Publishing Company 
(New York ), 1959. 

2. Ashley, H. , "Applications of the Theory of 
Free Molecule Flow to Aeronautics, " IAS 
Journal, Vol. 16, 1949, pp 95 to 105. 

3. Stalder, J. R. and Zurick, V. J., "Theoretical 
Aerodynamic Characteristics of Bodies in a 
Free Molecule Flow Field, " NASA TN2423 
1951. 

4. Stalder, J. R., et al. , "A Comparison of 
Theory and Experiment for High Speed Free 
Molecule Flow, " NASA TN 2244, 1950. 



5. Schramberg, R. , "A New Analytical Re- 
presentation of Surface Interaction for 
Hyperthermal Free Molecule Flow with 
Application to Satellite Drag, " Heat Transfer 
and Fluid Mechanics Institute, June 1959. 

6. Kork, J., "Satellite Lifetimes in Elliptic 
Orbits, " Chapter V, "Design Guide to 
Orbital Flight, " McGraw-Hill, 1962. 

7. Roberson, R. E. , "Effect of Air Drag on 
Elliptic Satellite Orbits, " Jet Propulsion, 
Vol.28, No. 2, February 1958, p 90. 

8. Perkins, F. M. , "An Analytical Solution 
for Flight Time of Satellites in Eccentric 
and Circular Orbits, " Astronautica Acta, 
Vol. IV, Fasc. 2, 1958, p 113. 

9. Breakwell, J. V. and Koehler, L. F. 
"Elliptical Orbit Lifetimes, " American 
Astronomical Society, Preprint No. 58-34, 
August 1958. 



Dynamic Analysis and Design Performance 
Requirements for Satellite Vehicle Guidance 
Systems, " Martin-Baltimore Engineering 
Report ER 10470-6, 31 January 1959. 

Billik, B. , "The Lifetime of an Earth 
Satellite, " Aerospace Corporation, TN- 
594-1105-1, December 1960. 



Fosdick, G. E. , "Orbital Lifetime and 
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Sterne, T. E. , "Effect of the Rotation of a 
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on Dynamical Astronomy, " Dover Publications 

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G. BIBLIOGRAPHY 



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18. 



19. 



20. 



21. 



22. 



23. 



V-30 



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12, 



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V-31 



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V-32 



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V-33 



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V-34 



ILLUSTRATIONS 



V-35 













































































































































73 




n<-rii 










































: 


^ ,S,X 


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I 


























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14 It. Us 



F-ig. 1. Drag Coefficient for a Sphere 
at 120 km Versus M 



Fig. 3. Drag Coefficient for a Kich 
Circular Cylinder with Axis 
Normal to the Stream at 120 km 
Versus M 



,^> |Semivertex angle 
32 |C- j of cone, 




V<„ Angle of attack, a' 



Molecular speed 
ratio, M 



Fig. 2. Cone Drag Coefficient, Diffuse Reflection 



*«*tt«0*N« Mtit BLANK NOT FH.WKI 



V-37 



12 



10 



Q 
U 



o 
U 

W> 
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i 












_ _ _ 


Diffuse reflection 
Diffuse reflection 


O Helium 
Q Nitrogen 


\ 


o 


































V 

^ ^^^ ^ 
































^ m 











0.5 1.0 1.5 2.0 2.5 

Molecular Speed Ratio, M 



3.0 



3.5 



4.0 



Fig. 4. Comparison of Drag Coefficient of a Transverse Cylinder for Specular and 
Diffuse Reflection 



CD 
O 

o 

P 



1.1 

1.0 
0.9 


-- 












1 






_ i 






O 30° semivertex angle cone 
cylinder- -experimental 

Free molecular flow theory 

Continuum flow theory 

Inelastic Newtonian flow theory 













u 






0.8 
0.7 
0.6 




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V 






— 


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« 














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0.4 
0.3 
0.2 
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M = 1.0 



















































































4 6 8 10 12 14 

Molecular Speed Ratio, M 



16 



20 



Fig. 5. Cone Drag Coefficient, Comparison of Free Molecular and Continuum 
Flow Theory; a = 0° 



V-38 



Altitude (km) 
150 200 



250 



10 



10 



-0- 



10 



10 



■nL 




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'in 



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■11 



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10 



13 



Altitude (ft x 10 u ) 



Fig. 6a. ARDC 1959 Model Atmosphere (1 slug/ft 3 = 512 kg/m 3 ) 



V-39 



300 



10 



■12„ 



Altitude (km) 
350 400 



450 
i 



10 



■13 



'- 10 



-14 
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3 



a 

v 

Q 



10 



-15. 



10 



-16. 



10 



•17L 




0. 9 



1.0 



Altitude (ft x 10 ) 
Fig. 6b. ARDC 1959 Model Atmosphere 



V-40 




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em 
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Altitude (meters x 10 ) 
Fig. 7b. Logarithmic Slope of 1959 ARDC Atmosphere 
V-41 



120 r— 



100 



be 
a; 
-a 



CD 




0.3 0.4 0.5 

Eccentricity 



0.8 



Fig. 8. Values of True Anomaly as a Function of Eccentricity for Which p/p(h p ) = Constant 
(exponential fit to ARDC 1959 atmosphere) 



V-42 




Orbit Parameter, z 



Fif>. 9. Nondimensional Drag Decay Parameters for Elliptic Satellite Orbits 



V-43 



1.0 



0. 10 



On 
C 



w 

03 



>5 

a) 
O 

Q 



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0.001 




6 8 10 

Orbit Parameter, C 



Fig. 10. Decay Parameters P + and P" for Elliptic Orbits 



V-44 



100 



200 



Altitude (km) 
300 400 




100 



150 



200 250 300 

Perigee Altitude (stat mi) 



350 



400 



Fig. 11a. Apogee Decay Rate Versus Perigee Altitude 
(see Fig. 12a for metric data) 



V-45 



Altitude (km) 
300 400 




200 250 300 

Perigee Altitude (stat mi) 



400 



Fig. lib. Perigee Decay Rate Versus Perigee Altitude (Part I) 
(see Fig. 12b for metric data) 



V-46 



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10 



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o 

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Q 

01 
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M 

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u 

01 



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10 



10 



2 


100 
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200 
i i i i 1 i i 


Altitude (km 

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) 

400 500 

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ill 




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i i • ■ ■ V " 




|- - ^ — S : i- f --; T Si^ ^ " 


\^ =-= i S ; I 1 ^ : ^L' : / \ 1 1 


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Pif,. lie. Perigee Decay Rate Versus Perigee Altitude (Part II) 
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V -47 



200 



Perigee Altitude (km) 
300 400 



500 



600 




10 



50 



100 



150 200 250 300 

Perigee Altitude (stat mi) 

Fig. 12a. Apogee Decay Rate Versus Perigee Altitude 



350 



400 



V-48 




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150 200 250 300 

Perigee Altitude (stat mi) 



350 



400 



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V-49 



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200 



Altitude (km) 
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500 



600 




200 250 300 

Perigee Altitude (stat mi) 



400 



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Initial Perigee Altitude (stat mi) 



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V-51 



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Fig. 14. Generalized Orbital Decay Curves for Air Drag 



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«5 


*r~- 




.7; 




i::! 1 !: \*$- 


'HHil 


?jl; 


.;;• :- : : 




■ = :-=: 




. ; J.;:-.: 


: ;:;'. :il' 










: ' t : 


'f 


: " t 


i;;i|p-Ez; 




iii! 


'ri^:: i; 


i: : -:;:; :; - ; " 






f': 


f. ~ 


■- ■ ' 


1 / 


t 


ill' 

iii; 


-r 


-.; p. 


•;-}Ht ■■!:■.•■■ 


■-J1 


':'::: 


f:f Br. 


:.:;:: ; ;.::i: : 


■ ;,, ;;;-i 




f: 


.,::::::: 





10 100 1000 

N = NUMBER OF REVOLUTIONS SINCE ORBIT WAS PERFECTLY KNOWN 



Fig. 15. Comparison of Errors in Orbital Prediction for Correlated and Uncorrelated 
Atmospheric Density Fluctuation 



V-5 3 




100 200 300 400 5O0 600 

N; NUMBER OF REVOLUTIONS AFTER THE EPOCH 




i (1^115 MINUTES1I 



100 ?00 300 400 500 600 

N=NUMBER OF REVOLUTIONS AFTER THE EPOCH 



hig. 16a. The Ratio of the mis Error in Orbital Pre- 
diction Caused by Sinusoidal Drag Variations 
to the Amplitude of the Sinusoidal Variation 



Fig. 16c. The Ratio of the rms Error in Orbital Pre- 
diction Caused by Sinusoidal Drag Variations 
to the Amplitude of the Sinusoidal Variation 



io» 





i 










i 



















SMOOTHING 
NTERVAL 






<C' 








n 


// 














































|(T= 100 MINUTES)! 






















/ A 


L 


t _ - ._ 







- — 





















100 200 300 400 500 

N-NLJMBER OF REVOLUTIONS AFTER THE EPOCH 



,c? 



















-- h-~ 










SMOOTHING 

INTERVAL 














_■■ 


, 




// 1 








\ 






it 


A / 
ft / 

/ 








ii i 










i / 













ft f 
1 1 










in 
1 1 












| (T=I40 MINUTES) 1 


1 

1 
ll 




















II 


— 










J; 

/ A 
/ i\ 




















/ ' 

/ i 
i 













100 200 300 400 500 

N;NUMBER OF REVOLUTIONS AFTER THE EPOCH 



Fig. 16b. The Ratio of the rms Error in Orbital Pre- 
diction Caused by Sinusoidal Drag Variations 
to the Amplitude of the Sinusoidal Variation 



Fig. 16d. The Ratio of the rms Error in Orbital Pre- 
diction Caused by Sinusoidal Drag Variations 
to the Amplitude of the Sinusoidal Variation 



V-54 





- 


— 


--■ 





. 


SMOOTHING 
NTERV4L 










i= 12 








-- 






i= 48 











^^, 


i= 80 
i= 160 


"^»" 


i=240 










s^^ 










































1 






— in 














































7/ 


I 





































— ■ 


. 


1 






- 






I 








r 














j 







00 200 3O0 400 WO 

N = NUMBER OF REVOLUTIONS AFTER THE EPOCH 



Fig. 17. 



The Ratio of the rms Error in Orbital Prediction Caused by Random Drag 
Fluctuation from Period to Period 




100 200 300 400 500 

N= NUMBER OF REVOLUTIONS AFTER THE EPOCH 



Fig. 18. The Ratio of the Error in Orbital Prediction Caused by Smoothed 
Observational Errors to the rms Error of a Single Observation 



V-55