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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tTD
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
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Oi <D Si
m ^ ="
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M a a)
c m_ 2
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CO
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H
adi
min
om
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rn
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roto
ing
hed
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<
c
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C
o
c
o
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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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11-62
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
-
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-4
=
N
4
=
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^
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-
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<
1 *
1
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1 «
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-
1 4
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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
Q
10
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p t -i-- =EEE- =rpEE ===== ====,=.== :=p =]= - = =:
=;---- ■-■-■- i ■-!-.- ._• • i I.. 7 £|= =!47p= :]-(-:=;= -'.- : i ~
- : -p-.^r~U 5 ^-^^^-"-sfiWtirisrte-sk'
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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11-79
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a,
m
O
G
I
X
10
10
10
10
r A
A--Inner belt
B--Outer belt
4
T
f B
t
9
1
n
01
l
1.
10
Thickness of Aluminum (gm/cm )
Fig. 24. Electron Dose Rates
10
10
10
10
A--Inner belt
B- -Outer belt
1
r B
S k
1
1
i
10
Thickness of Aluminum (gm/cm )
Fig. 25. X-Ray Dose Rates
11-85
J3
o
u
10
10
Gee
7
3 mag
'45/
aetic
latiti
ade (J
:°)
1.6
-3
1/
if
j
/
/
I
/
J
1
1
-4
no
4
'0
10
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
o
6.
5.
4.
3.
2.
1.
> 30,000
6000
3500
< 400
1
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
A
O
\
\
y
\
□
A
V
^
f
C
'
c
□
20 i
4
5
6
1
o a
9
K
t
1
1.
4 i
i ii
Energy (kev)
Fig. 29. Differential Energy Spectrum Measured During Rocket
Flight NN 8. 75 CF
II -87
s
/
(
/- Whififil*. 13S2
v /
/ /• Wat sun. 1M1
/
^
Ap^'ireril Viiual Magnitude
Fig. 30. Meteoric Mass Versus Apparent Visual Magnitude
14
13
-
12
^v Whipple
and Milln
lan
11
10
2
o
J
9
8
Watson and \
McK
inley ^v
'
B
4
1
1
l
1
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
* ▼
Sputnik III
^ Space Rocket I
W Space Rocket II
• \
• \
^
H Interplanetary Station
A Pioneer I
?
$
B A®]
.1
<
£>
4
5
<•>
6
\
\
\
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]
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Fig. 7. (continued)
111-49
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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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IV-58
ILLUSTRATIONS
IV-59
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IV-62
AM = 3
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II
'+ ;
1-2.0
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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
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-6,200
-6,400
-6,600
6,800
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6.0-
7.0-i
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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.
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V-32
Schilling, G. F., and Sterne, T. E., "Densities
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Schilling, G. F., and Whitney, C. A., "Atmospheric
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Smithsonian Astrophysical Laboratory, Cam-
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Schilling, G. F., Whitney, C. A., and Folkart,
B. M., "Preliminary Note of the Mass-Area
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IGY Satellite Report Series No. 6, National
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Its Application to the Determination of Aero-
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Singer, S. F., "Studies of a Minimum Orbital
Unmanned Satellite of Earth (MOUSE). Part
II--Orbits and Lifetimes of Minimum Satel-
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Siry, J., "Satellite Orbits and Atmospheric Den-
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Spitzer, L., "On the Determination of Air Density
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Stafford, W. H., and Craft, R. M., "Artificial
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V-33
Yengst, W. C, "Comments on a Variable Atmos-
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Feb 9, 1961.
V-34
ILLUSTRATIONS
V-35
73
n<-rii
:
^ ,S,X
,u l; ,r
!,0„
I
14 Hi 1 1;
I
^L
^
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>
n)
Q
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
\ i
1 '
V
—
\
<5
«
U.b
0.4
0.3
0.2
0.1
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
"10
'in
-10
-10
■11
"10
-12
0.8
0.9
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
10 \- _ 10
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
O
6
.e
em
o
1U
9
—f-i
T i
— _
. — ,
- —
.. ._.
—
fl
V
—
:
'.
V 1
_-;__
fi
'
- ;
li
_____
- —
— -
=j
-■-' -
■ !_'■".
_...__.
K"AVi
4
!- : :
!■' : ;
P = P
"' : :
—
^eMt
. !:-:-- ! :
;
. 1 - ' :
■ V V
:.:: . :
" : r: -"-
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TSJT-
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_.
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i>:-: :: ' /
: !..L
1 4 I
:-;-!--]■
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■>
:::
Graphical differentiati
on
z. ' r i . : ■
r-:.-r:.
\^"LL
- : ~.
"' :j_ :
. . ...
: " . ■-
l n
fl
Diffei
ence •
able
-lid:
11
l+u :
11
ffit
: .j.r:
rrf**
rr :: :
. ' *T1"
- T '~L.:
. .... r _
0.
1
0.
2
0.
3
0.
4 0.
5
0.
6
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
0.01
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
10
10
P.
n!
rt
o
01
Q
01
0)
M
•rH
u
01
10
10
10
2
100
1 i i i i i
200
i i i i 1 i i
Altitude (km
300
iiiiii|Iiiiiiiii
)
400 500
1 1 ) ' ' ' J ' ' ' J ' ' I !
600
i 1
7#f'T
X' iM«MT
St"i-iX -^t!-- 1- ,---|- M "I ] i i -| ] !
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< . -
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i ; 2
ug = 0.6365 x 10 2 m 2 /k
,!!! = ■=: = Ebbk-d?i£i4- j!>*i ^^|-
: '^V ■ -
. :--:--!'" -.:: — = =
£ (1 ft Z /sl
g> -J !J_!. iEzl=F!;-±l^-l-:k =i=ii!>r- >Li -h
V - v : S
^=!i-.-i::r
: : :T -i=p T i r : '- !== ) = 'r"r;"dii"- -■-.EE_tv-l ::LI£
■ - - ; ■ J 1 I J ^^
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_!..._._.. .-1 :_
P^tf :t r M^ 7 ^
[-
■ ^*x
6
rrr~" .'! '1 i I
50
100
150 200 250 300
Perigee Altitude (stat mi)
350
400
Pif,. lie. Perigee Decay Rate Versus Perigee Altitude (Part II)
(see Fig. 12c for metric data)
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
100
150 200 250 300
Perigee Altitude (stat mi)
350
400
Fig. 12b. Perigee Decay Rate Versus Perigee Altitude (Part I)
(see Fig. lib for English data)
V-49
100
200
Altitude (km)
300 400
500
600
200 250 300
Perigee Altitude (stat mi)
400
Fig. 12c. Perigee Decay Rate Versus Perigee Altitude (Part II)
(see Fig. lie for English data)
V-50
Initial Perigee Altitude (km)
10
200 i-
100
50
10
20
10 -
10 J I
01
;>>
3
s
J
100
200
300
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1* s'= =
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j .- j-jF =■-- ;^
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V-51
Altitude (km)
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V-52
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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
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diction Caused by Sinusoidal Drag Variations
to the Amplitude of the Sinusoidal Variation
io»
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N-NLJMBER OF REVOLUTIONS AFTER THE EPOCH
,c?
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INTERVAL
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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
-
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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
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