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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 J c 3 N o» Jd 3 o <* ■p a o j UN t. U CM QJ —• .-I nS CO W m < cu c c cu — cu t-i o •O CO .5 gS * to 2; o n ^ 0! nl o : CLi CO J3 ht in cd = W 1-5 CU > CD •£3 oO W e d Kit. CO |W CO o 'STtJv in l<N CO i + c CO m pi C ca + CO ICO < r u o o b & i J c w « fc>a ca r u X\°u -rv ^ — ' „(ro ^ikro + i 1 « 1 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. 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CN ol CD col CO CM CN r- CO o o E o 1 ■* O) m ■>p| m CO 1 CO 1 co t- 1 CO CO CO m Tf CN o CO o rd oo t— o m o a CO t-" m CO ^P CO m r- DQ co" co" t- a> m CO CO CM o o c| *l col CO 1 o 1 O) 1 COl c>- ■* > U CNl TP ol 0) COl <3 o" o o o "** o CO t— 1 m CO m 1- o 1 ol CO 1 CO cd o OT OJ o CO m CO CM CO CO 3 »•-. o* CO m m ,_; o> CO CO o" ■* CO* TP k. —■ CN CN m •-• ^ CO m m CO ^ u —. COl c) ^ -*l cl r-f c CO H COl CN- ^1 CJ CO CO ol ■<p ^ ° CO co| CO 1 a> CO CO m 1 col col CO r-,1 m o o t- m m -P ■^P m E o CO OJ CO m CD* CN m m CO CO CO t- c- ^ CO "P CM •^ "^ ^ H rp C o o CU c rt Pu 3 O u 4> w 3 c arth arth-M [oon n u a C u p ti CO 3 C rt c 3 Q. 0) o 3 C 3 1 s > W W 2 5 ^ c/5 Z CU 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 «~ M Oi <D Si m ^ =" o » -° M a a) c m_ 2 ° c B CO ■s §> H adi min om -J rn K 3 u <; cZ^ H roto ing hed csg lien Orb Lau < c ClJ > C o c o "1 o 0. 001525 0. 001875 0, 001914 0. 001914 001922 002040 002062 0. 17588 0. 1008 0. 0774 0. 26078 0. 1494 1 1147 0,29726 , 0. 1703 0, 1308 0, 32565 | 0, 186G | 0. 1433 0, 34710 1 0. 1989 ' 1527 0. 42955 ' 0. 246 1 0. 1890 0. 5403 1 0. 4149 1, 0408 0. 7992 1,3578 ' 1.0411 1. 6982 1. 3040 2. 0337 1, 5616 1 64. 150 2. 9587 | 2. 2759 1. 3040 1. 0014 127.200 5,8720 | 4.5169 2.5882 ; 1,9874 176.000 8.1143 6.2418 , 3.5766 ' 2.7464 226.150 10.4359 8.0276 | 4.5998 [ 3.5321 275. 800 ' 12. G998 9, 7691 ; 5. 5977 4. 2984 S|SS gPS| 0, 001986 0. 002442 0. 002493 0, 002493 0. 002503 0. 0026 57 0, 0O2G8G ,„ 0. 003466 0. 004262 0, 0043 51 0. 004351 0. 004368 0. 004G37 0. 004687 0. 94296 1. 81645 2. 36609 2. 96365 3. 54915 *' 0, 004 51 0. 00555 0. 00561 0. 0056 1 0. 00566 0. 00601 0. 00608 0.2286 0. 3390 0. 3864 0. 4233 0, 4512 0, 5584 26,580 ! 1.2258 51. 300 2. 3614 66.700 ! 3,0759 33. 500 3. 8527 100. 000 ; 4. 6139 = 0.0980 0. 1200 0, 1228 0, 1228 0, 1230 0, 1305 0, 1320 4, 950 7. 349 8. 360 9. 160 9. 760 12. 110 c El O 1 ^ 0. 000743 0. 000981 0. 001003 0, 001003 0, 001006 0, 001059 0, 001064 0. 0G7G 0. 1071 0. 1228 0. 1280 0. 1467 0. 1753 0. 3189 0. G2G8 0, 86 12 1. 1161 1.2910 0. 9942 1. 9728 2. 7269 3.6443 4. 1415 1. 5559 2. 7413 4.2526 5, 7054 1. 5348 3. 0578 4.6479 5.7905 0. 0009G8 0. 001278 0. 001306 0. 001306 0. 001310 0. 001380 0. 001386 0. 0880 0. 1394 0. 1599 0. 1667 0.1911 0. 22 83 20.400 0.9422 0.72477 0.4153 40.100 1,8519 i 1.42450 0.8162 55. 150 2. 5444 \ 1. 95722 ! 1.1215 71. 500 3. 2976 ■ 2. 53664 ! 1. 4535 82,500 3.8144 ■ 2.93418 1.6313 1. 2947 2. 56 91 3, 5512 4. 7458 5. 3934 4. 5972 ; 3. 5363 ! 2. 02G3 8.0995 6.2304 3.5700 12.5644 | 9.6649 5.5380 16.8568 12.9GG8 7.4300 4. 5348 3. 4883 , 1. 9988 9. 0345 ■ G. 9496 3. 9821 13. 7327 : 10, 5636 G, 0529 17, 1083 : 13. 1603 7, 5408 0. 001689 0. 0OB23 0.0022 80 0. 002280 0. 002287 0. 002408 0. 002419 0. 153 57 0. 24335 0. 27909 0. 29091 0. 33345 0. 39837 63. 550 2. 9373 2. 2595 126.200 | 5. S287 4.4836 174. 800 8. 0568 : 6. 1975 233, 800 10, 7673 | 8. 2825 265. 300 12. 2364 i 9, 4126 1 0. 00219 0. 00290 0. 00296 0. 00296 0.00298 0. 00300 0. 00314 0. 1996 0.3164 0, 3628 0.3 782 0. 4335 0. 5179 99. 50 176. 00 272. 000 3G5. 00 98, 100 195. 80 298. 00 3 70. 00 ° 0.0471 0. 0629 0. 0643 0.0643 0. 0645 0. 0680 0. 0681 4. 320 6. 850 7. 850 8. 195 9. 390 11.210 5 1 | 1 - 000453 0. 00092 7 0. 001036 0. 001041 0. 001041 0. 001041 (") 001103 0. 0473 I) 0865 0, 1060 0. 1158 1229 1393 3833 0. 7112 1 0108 1 3845 1 5585 1. 2047 0, 9251 2.4758 1.9011 3,6440 ' 2.7982 4. 7650 3, 6590 5.8333 4.4793 Z«»". «*|* 2 0. 000592 0. 001208 0. 001350 0. 001355 0. 001355 0. 001356 0. 001436 0. 10762 ' 0. 0600 0. 19689 0. 1128 0. 24168 ' 0, 1381 0. 26274 0. 1505 0, 27955 0. 1600 0. 31G45 0, 1812 0. 4991 0, 9262 1. 3164 1. 8029 2, 2096 2. 2707 4 3 700 6 26 70 8. 1980 2 . 4404 5 0211 7, 3707 j 10, 0219 ,„ 0. 00103 0, 002108 0. 002356 0. 002365 0. 002365 0, 00236 7 0, 002 506 0. 87110 1.6164 2,2973 3, 146 5 3.5421 2. 1024 4. 3207 6. 3595 8.3158 10, 1803 3. 9G29 7. 6265 10. 93 72 14.3071 4. 2590 8. 7628 12, 8633 17. 4902 * 0. 00134 0. 002G2 0. 00306 0. 00307 0. 00307 0. 00308 0, 00326 3. 025 j 0, 1398 5. 545 '■ 0. 2560 6.795 0.3138 7. 400 0, 341G 7. 875 0, 3625 8.915 , 0,4110 1, 1324 2. 1013 2. 9 865 4, 0905 4.6047 2.7331 5.6169 8.26 74 10. 8105 13.2344 5. 1318 9, 9144 14.2183 18. 5992 5, 536 7 11. 3916 16. 7222 22. 7372 = b c o o c o o 24. 6 00 45. 550 64. 700 88. 550 99.900 59.300 121. 900 179. 200 244. 000 286. 500 11 1. 800 214. 400 308, 200 404, 400 120. 00 24 7,40 362. 10 493 . 00 1 1 1 !5 Z — cm cn -*■ ifi u> t- co - — •»•"»- = - -<•"-■* --«' < 1 = j'i o c CO O CM O if 3700 km 2000 n mi 407o km 2200 n mi 11-43 TABLE 33 Twelve -Hour Orbital Dos e (rad) Within Van Allen Belt Orbital Inclination (deg) 2 Aluminum Sphere Thickness (gm/cm ) 0. 1 1.0 2. Altitude Electrons X-rays Electrons X-rays Electrons X-rays 555 km 4.598 x 10 3 0.7569 1. 137 x 10" 3 0.2301 0. 1575 (200 naut mi) 40 1.444 x 10 3 0.2377 3.574 x 10" 4 0. 0723 <10" 5 0. 0494 90 6.811 x 10 2 0. 1121 1.686 x 10" 4 0. 0341 0.0233 740 km 1. 1690 x 10 4 1.9241 2. 892 x 10" 3 0.5849 0.4003 (400 naut mi) 40 5.046 x 10 3 0.8306 1.248 x 10" 3 0. 2525 <10" 5 0. 1728 90 3.693 x 10 3 0.6078 9. 136 x 10" 4 0. 1848 0. 1264 1110 km 6.634 x 10 4 10.9197 1.641 x 10~ 2 3.3196 2.2716 (600 naut mi) 40 4. 129 x 10 4 6. 7964 1. 021 x 10~ 2 2. 0661 -4 < 10 1.4138 90 2.359 x 10 4 3.8825 5. 835 x 10" 3 1. 1803 0. 8077 1852 km 2.625 x 10 5 43.2147 6.495 x 10" 2 13. 1373 1. 803 x 10" 4 8.9898 (1000 naut mi) 40 2.088 x 10 5 34.3755 5. 166 x 10" 2 10.4502 1.434 x 10" 4 7. 1510 90 1.097 x 10 5 18.0597 2.714x 10" 2 5.4901 7.534 x 10" 5 3.7569 ionization. The factor of conversion, Relative Biological Effectiveness (RBE), yields a measure of the degree of localization, or nonuniformity, of tissue ionization. Ionization localization along the path of penetration is singularly noticeable for heavy (atomic number 6 or greater) particles. Although all atomic species through iron have regularly been observed, the biologically note- worthy heavy constituents of the primary radiation are carbon, nitrogen, oxygen, the magnesium and calcium groups, and iron. When these medium and high energy particles enter tissue, they first produce an ionization trail of great density. The high energy particles, in general, undergo nuclear disintegration during the penetration process, with a resulting large reduction in specific ioni- zation, since afterward the ionization is caused by several particles of reduced charge travelling in different directions. These primaries which have a reduced impinging energy have a signif- icant probability of being completely stopped through ionization only. This leads to extremely large specific ionizations near the ends of the paths, since the rates of energy loss increase as the particle energies decrease, down to very low energies. These thindown hits are capable of causing cell destruction. Their effects in nonreparable regions of the body, such as certain brain areas, have not yet been demonstrated. The RBE conversion from roentgen to rem ob- tained from a weighted analysis of particle type and tissue ionization characteristics between 30° and 55° latitude at the top of the atmosphere and extrapolation elsewhere, increases with increasing altitude and geomagnetic latitude, as seen in Fig. 27. This is explained by noting that at a position requiring decreased particle penetration of the magnetic field, there is a slight increase in the relative number of heavy constituents, compared with hydrogen and helium. At the same time, the heavy component energy range extends to lower values. It must be emphasized, however, that little actual biological experi- mentation has been performed to test the validity of the relation between ionization track density and the RBE for particles of large atomic number, which produce the greater fraction of the unshielded biological intensity. Shielding against cosmic radiation is not ordinarily advisable, since it requires thick- 2 nesses of aluminum greater than 25 gm/cm 2 for heavy particles, and at least 200 gm/cm 2 (400 lb /ft of shielded area) for hydrogen and helium, which have far higher penetrating power and constitute about 15 percent of the unshielded biological dose and 99 percent of the incident particle number. In fact, the biological dose increases for shielding thicknesses up to 15 2 gm/cm for the carbon, nitrogen, and oxygen 2 group, up to 10 gm/cm for magnesium, up to 2 2 6 gm/cm for calcium, and up to 5 gm/cm for iron. An estimate of the effectiveness of shielding against cosmic radiation is shown in Fig. 28 taken from Wallner and Kaufman (Ref. 50). A comparison with the curves shown in Fig. 14 shows the relatively slow decrease of dose with absorber thickness for cosmic rays as compared to other space radiations. The dose peak at 2 about 10 gm/cm is due to the increase of ionization 11-44 rate before significant numbers of particles are stopped in the absorbing material. d. Penetrating electromagnetic radiation Previous estimates of the high energy end of the solar system indicated intensities of the order -4 2 ° of 10 erg/cm -sec below 8A. Recent measure- ments indicated that during a solar flare (class 2+) this intensity increased to about 10 erg/ 2 ° cm -sec with 2 A as the lower limit of the radi- ation detected (Ref. 51). More recently, meas- urements have indicated that X-ray flashes during solar flares had energies as high as 80 kev (0. 15 A (Ref. 52). During a class 2 solar flare on 20 March 1958 an intense burst of electromagnetic energy was recorded which lasted 18 seconds (or less) (Ref. 53). This was determined to have an intensity -4 2 of 2 x 10 erg /cm -sec above 20 kev and peaking in the region of 200 to 500 kev (0. 06 to o 0. 025 A). Measurements during a class 2+ flare on 31 August 1959 indicated a peak intensity of _ n 9 4. 5 x 10 erg /cm -sec (-^20 kev) arriving at the top of the earth's atmosphere (Ref. 54). The spectrum decreases in photon count by a factor of 10 for an energy increase of about 20 kev. Although these photons are quite penetrating (the half -thickness value of aluminum for 500 kev photon is 3. cm) their intensity is so low as to produce an insignificant dose (of the order of 10 roentgen from the March 1958 event). „ In- tensity enhancements in the region of 8-20 A were also observed during the August 1959 event. In 2 this region about 1 erg/cm -sec was measured. This would result in a much greater dose than the less intense higher energy photons; their penetration is very much less. The half -thickness values are less than 10 cm of aluminum. A solar X-ray spectrum from a class 2+ flare is shown in Fig. 29 taken from Ref. 30. X-rays with energies in excess of 20 kev appear to be emitted only for short periods (a few minutes) during large flares. The X-ray dose rate to an unprotected man from a flux as shown in Fig. 29 would be about 3 rem/hr. However, since the emission lasts for much less than 1 hr we may conclude that high energy solar electromagnetic radiation will not be of concern to space flight. Saylor, et al. (Ref. 55) point out that ultraviolet light on bare skin can cause severe burns and even skin cancer. It will therefore be advisable to use windows or shutter arrangements to filter the otherwise unattenuated solar ultraviolet rays. In space there will be no warning glare of scattered light to alert the observer that his line of sight is approaching the sun. An inadvertent glance at the sun could cause temporary vision failure and ten seconds of exposure would cause permanent retinal burn. These authors conclude that pro- tection of the eyes against sunlight is a necessity. e. Radiation damage thresholds Of all the components of a space vehicle, man has the lowest threshold to damage by ionizing radiation as shown in Table 34. TABLE 34 Radiation Damage Dose Limitations People Roentgen Equivalent 2 ^ 10 (sickness) 10 (lethal) Semiconductor 10 (damage) 10 (failure) Electronics 10 8 io 10 Elastomers 10 7 10 8 Plastics 10 8 10 9 Metals 10 15 Ceramics io 17 Ref. Nucleonics Sept 1956 More detailed treatment of radiation damage mechanism are shown in Refs. 56 and 57 and in the very comprehensive Radiation Effects Information Center Series of Battelle Memorial Institute. Semiconductors are seen to be the second easiest damaged component. This is caused by the fact that their properties arise from their form of very nearly perfect single crystals. Most metals and ceramics used for structural, electrical or magnetic applications are already in a disordered polycrystalline form and their properties are only moderately changed by further disorder (ionization). It should be noted that certain types of sensing elements may give erroneous readings due to spurious signals from the Van Allen or other radiation environments. While this does not represent damage by radiation, it is neverthe- less undesirable and can be easily avoided by proper selection, design and calibration of these devices. As contrasted to actually "reading" unwanted signals from ionizing radiations in sensitive "front end" components it is known that electronic components and circuits may operate improperly while in the presence of large fluxes of ionizing radiation. Measurements made under conditions simulating a nuclear explosion in space have indi- cated that the threshold of susceptibility to these fi 7 effects is at peak dose rates of 10 to 10 roentgen per second. This again is greatly in excess of what will be encountered from the natural radiation environments. The radiation problem therefore reduces to protection of the crew. 11-45 Maximum allowable radiation doses for manned space flight have been revised upward from 25 rem considerably in the past year. Presently the Apollo maximum allowable emer- gency dosages are as shown in Column 4 of Table 35 from Ref. 58. The normal mission dosages are as shown in Column 3. These values are more meaningful than the single so- called "whole body" value used previously. TABLE 35 Radiation Dosage 5 Year Dose (rem) RBE Average Year Dose (rad) Maximum Single Acute Exposure (rad) Design Dose (rad) Skin body dose ' 0. 07 mm depth 1630 1.3 235 500 125 Skin body dose extremities, hands, etc. 3910 1.4 559 700 175 Blood forming 271 1.0 54 200 50 organism Eyes 271 2.0 27 100 25 4. Meteoroids Empirical data on meteoroids has come either from optical and radar meteor obser- vations or from impact detectors on board rockets and satellites. In the first type of ob- servation, velocity and luminous intensity history are directly measurable. The mass and density of the meteoroid is then determined using the drag equation, the shape of the light curve and the vaporization equation. Due to the variety of assumptions and dependencies in this analysis, there is a large uncertainty in flux estimates from the same type of data. The relation between meteoroid mass and visual magnitude is shown in Fig. 30 from an early survey (Ref. 59). The relation between mass and flux is shown in Fig. 31 from a later survey article (Ref. 60). The flux uncertainty is dealt with in a number of other survey articles (Refs. 61, 62 and 63), and an examination of the assumptions employed in the analysis procedure will show why it is as 3 large as 10 . The best known model of the meteoroid environment was developed by Whipple in 1957 and summarized in Table 36. The following equation fits the distribution presented by Whipple in 1957. 1 -1 9 3 x 10 m 1 where <j> is the flux/m -sec of particles with mass m grams and greater. This was revised by Whipple (Ref. 64) in 1960 to -12.6 -1. 186 10""'" m *' ' to include empirical data from rockets and satellites. A recent evalu- ation of rocket and satellite data (Ref. 65) (obtained from acoustic detectors) obtained 10 -17.0 ■1. 70 applicable between 10 to 10 gm. These distributions masses of 10 are shown in Fig. 32 taken from the last cited reference. It should be noted that meteoroid masses of greatest interest to space vehicle de- signers lie between the mass regions measured by the meteor or satellite -borne microphone techniques. Observations of meteors simulated by shaped charge firings from an Aerobee Rocket (Ref. 66) have indicated that Whipple may have underestimated meteor luminous efficiencies. This may be accounted for by a downward revision by an order of magnitude in mass (Ref. 6 7) of the 1957 flux estimate of Whipple so that 1.3 x ,„-13 - 10 m Various investigators have put forth penetration models --some based on empirical equations derived from test data and some based on theoretical con- siderations and most all giving the penetration in a thick target. Since structural skins are usually made of aluminum alloy materials, a good basis of comparison is the penetration of meteorites into aluminum. Four penetration equations were in- vestigated to obtain a comparison of the meteorite penetrations given by the different equations. These equations were: a. Whipple ' s equation This equation is given in (Ref. 63) as P "K| <^> "' E " 3 where P = penetration in a thick target K. = constant of proportionality E = meteorite energy p - target density c = heat to fusion of target material For a meteorite of diameter (d) moving at a velocity (V) cm /sec and with a meteoroid density p = 0.05 gm/cm and e = 248 cal/gm Whipple's equation is 11-46 TABLE 36 Data Concerning Meteoroids and Their Penetrating Probabilities F. L. Whipple, Ref. 5 Meteor Visual Magnitude Mass (g) Radius (u) Assumed Vel (km/ sec) KE (ergs) Pen. in Al t (cm) No. Strik- ing Earth (per day)** No. Striking 3m (Radius) Sphere (per day)*** 25.0 49,200 28 14 1.0 x 10 21.3 -- -- 1 9.95 36,200 28 3. 98 x 10 13 15.7 -- -- 2 3. 96 26,600 28 1.58 x 10 13 11.5 -- -- 3 1.58 19,600 28 6.31 x 10 12 8.48 -- -- 4 0.628 14,400 28 2.51 x 10 12 6.24 -- -- 5 0.250 10,600 28 1.00 x 10 12 4.59 2 x 10 8 2.22 x 10" 5 6 9.95 x 10" 2 7,800 28 3.98 x 10 U 3.38 5.84 x 10 8 6.48 x 10"° 7 3. 96 x 10" 2 5,740 28 1.58 x 10 11 2.48 1.47 x 10 9 1.63 x 10" 4 8 1.58 x 10~ 2 4,220 27 5.87 x 10 10 1.79 3.69 x 10 9 4.09 x 10" 4 9 6.28 x 10" 3 3,110 26 2. 17 x 10 10 1.28 9.26 x 10 9 1.03 x 10" 3 10 2.50 x 10" 3 2,290 25 7.97 x 10 9 0.917 2. 33 x 10 10 2.58 x 10" 3 11 9.95 x 10" 4 1,680 24 2.93 x 10 9 0.656 5.84 x 10 10 6.48 x 10" 3 12 3. 96 x 10" 4 1,240 23 1.07 x 10 9 0.469 1.47 x 10 11 1.63 x 10" 2 13 1.58 x 10" 4 910 22 3.89 x 10 8 0.335 3.69 x 10 11 4.09 x 10" 2 14 6.28 x 10" 5 669 21 1.41 x 10 8 0.238 9.26 x 10 11 1.03 x 10" 1 15 2.50 x 10" 5 492 20 5. 10 x 10 7 0. 170 2.33 x 10 12 2.58 x 10" 1 16 9. 95 x 10" 6 362 19 1.83 x 10 7 0. 121 5.84 x 10 12 6.48 x 10" 1 17 3. 96 x 10" 6 266 18 6.55 x 10 6 0.0859 1.47 x 10 13 1.63 18 1.58 x 10" 6 196 17 2.33 x 10 6 0.0608 3.69 x 10 13 4.09 19 6.28 x 10~ 7 144 16 8.20 x 10 5 0.0430 9.26 x 10 13 1.03 x 10 20 2.50 x 10" 7 106 15 2.87 x 10° 0.0303 14 2.33 x 10 2.58 x 10 21 9. 95 x 10 78.0 15 1. 14 x 10 5 0.0223 14 5.84 x 10 6.48 x 10 22 3.96 x 10" 8 57.4 15 4.55 x 10 4 0.0164 1.47 x 10 15 1.63 x 10 2 23 1.58 x 10" 8 39.8* 15 1.81 x 10 4 0.0121 3.69 x 10 15 4.09 x 10 2 24 6.28 x 10" 9 25. 1* 15 7.21 x 10 3 0.00884 9.26 x 10 15 1.03 x 10 3 25 2.50 x 10~ 9 15.8* 15 2.87 x 10 3 0.00653 2.33 x 10 16 2.58 x 10 3 26 9.95 x 10" 10 10.0* 15 1.14 x 10 3 0.00480 5.84 x 10 16 6.48 x 10 3 27 3.96 x 10" 10 6.30* 15 4.55 x 10 2 0.00353 17 1.47 x 10 1.63 x 10 4 28 1.58 x 10" 10 3.98* 15 1.81 x 10 2 0.00260 3.69 x 10 17 4.09 x 10 4 29 6.28 x 10" U 2.51* 15 7.21 x 10 0.00191 9.26 x 10 17 1.03 x 10 5 30 2.50 x 10" U 1.58* 15 2.87 x 10 0.00141 2.33 x 10 18 2.58 x 10 5 31 9.95 x 10" 12 1.00 15 1.14 x 10 0.00103 5.84 x 10 18 6.48 x 10 5 * Maximum radius permitted by solar light pressure. ** These No. based on entrance to atmosphere at 100 km approx *** Includes earth's shading effect of 1/2 V \U3 t P c (sp*r) 447 x 778.3 ft lb/lb for Al 11-47 where 1.08x 10" 4 V 2/3 P = penetration in thick target d = meteorite diameter V = meteorite velocity in cm /sec. Whipple's equation is theoretical and is believed to give penetration depths for hyper - velocity impacts that are too high. b. Kornhauser's equation This equation is given in (Ref. 68) as K 2<?) 1/3 <^") 0.09 where h = penetration (depth of crater) K„ = constant of proportionality T = kinetic energy of projectile E = modulus of elasticity of target material Eq = reference modulus This equation yields 0.282 x 10" 4 V 2/3 which is identical to Whipple' s except that the value of the constant is lower. c. Summer's equation This equation is an empirical equation based on experimental test data using many different projectile and target material combinations. As given in Ref. 6 9, the equation has the form of: p % 2/3 /v> 2/3 P = 2.28 U>> (V) where P = penetration in a thick target d = diameter of projectile p = density of projectile p, = density of target V = projectile velocity C = speed of sound in target material For Whipple's meteorite density of p = 0.05 3 P gm/cm , an aluminum target density of p. = 3 5 2. 8 gm/cm and C = 5. 1 x 10 cm/sec, the equation reduces to £ = 0.243 x 10" 4 V 2/3 The agreement between this constant and that of Kornhauser is noted. d. Bjork' s equation This is a theoretical equation developed by Bjork (Ref. 70) using a hydrodynamic model to explain hypervelocity impact. He derived equations for the impact of aluminum projectiles on alumi- num targets and also iron projectiles on iron targets. In Ref. 71, Bjork gives the penetration of an aluminum projectile into an aluminum target as: P = 1.09 (m v) 1/3 where P = penetration in cm m = projectile mass in gm v = impact velocity in km /sec Bjork in Ref. 72 states that the use of a correction factor of the form! — ] is subject to a great deal of conjecture as it rests on no theoretical basis. He also stated that he would favor the value of <j> = 1/ 3 and 9 = 1/3 in a general pene- tration equation such as: 1/3 equating the general and empirical relations. i no / vl/3 „ 1/3 -1/3 AA 1.09 (mv) =K 3 m i° t \^) ,09 . K 3 .,-"' (') "3 For aluminum targets, p. = 2.8 gm/cm and C = 5. 1 km/sec, Kg = 2.63. Thus we may write P = 2.63m 1/3 -1/3 Pt ft) 1/3 Then, letting "d" equal the meteorite diameter 3 in cm and its density p p = 0. 05 gm/cm yields P - 2.63 (g. d p p ) p t ^j = 0.322 V 1/3 where P = penetration = cm d = meteorite dia = cm V = meteorite velocity = km 11-48 This probably stretches Bjork' s work more where than he would care to see done but it is necessary to obtain a comparison with the other formulas. e. Engineering model Then u = sin R /R. o For purposes of evaluating meteoroid effects upon propeilant storage vessel design, the follow- ing model has been recommended (Ref. 73). (1) The integral mass flux of particles is given by ■13 ■10/9... , 2, , hits /m /sec, by $ =10 ' m particles of mass m gm and greater. Approximately 90% of the meteoroid flux is assumed to 3 have a density of 0. 05 gm/cm . The effective flux used in com- puting probability of hits is there- fore reduced by an order of magni- tude to compensate for the very low density meteoroids which will not follow the given penetration law. (2) The particle velocity (v) is 30 km/sec. (3) Penetration of impacting particles into a single thickness of steel is given by 1/3 S f = 1 - 1/2 (1 - cos u) 1 - 1 + cos (sin R /R) The integral mass flux thus becomes -14 -10/9 2 $ = 10 m hits/m sec N (> m) = 8.64 x 10 _10 m " 10/9 hits /m 2 -day Eliminating the constant meteoroid velocity (30 km/sec), and expressing the penetration law in terms of mass gives m = 101.25 as the mass in grams required to penetrate X cm of steel. With the flux and penetration expressed only by mass, it is convenient to combine the two relationships, obtaining P = 1. (mv) N (>m) = 8.64 x 10 10 (P 3 /101.25)- 10/9 (4) Aluminum is half as effective as steel in withstanding penetration. (5) The use of spaced sheets (Whipple bumpers) allows a reduction factor, B f = 5, in the total thickness required to withstand penetration. (6) Particle density, (p) is 3 gm/cu cm. (7) The area exposed to meteoroids is the total unshadowed surface area of the object. The shadowing can be ex- pressed in terms of an effective area by computing a factor to be multiplied by the actual area. This reduction factor will be in the ratio of a sphere with a conical segment removed to a sphere. The center of this sphere is the spacecraft and the conical segment is that volume intersected, as an ex- ample, by the Earth. Consider the following sketch = 1.46 x 10 hits per square meter per day capable of pene- trating P cm of steel. The reciprocal of this relation is the average number of days between penetrations. To determine the thickness re- quired so that an area of A meters is not pene- trated on the average for at least T days, P = (AT • 1.46 x 10 ■7 3/10 8. 901 7?~ (AT) 3/10 cm of steel This relationship is convenient to use for purposes of design after the effects of the time distribution of meteoroid encounters have been included. The Poisson distribution model has been used to elabo- rate on meteorite encounter probabilities. This distribution which is valid for uniform masses of low density is kt K' 1 where t is any selected interval, and -= average number of penetrations per day the probability of any number, K during time, t can be estimated. To determine the probability of no penetrations during T days (T = t) the relation reduces to is the Thus penetrations kt 0. 368 11-49 so that the probability is 0. 368 that there will be no penetrations within the average number of days between penetrations. To find the time at the end of which the probability of no penetrations is 0. 99. 0.99 = e" t/T t = -T In 0. 99 t =0. 0101T For 0. 95 and 0. 90 probabilities, the correction factors are, respectively, 0.05 and 0. 10. For example, the average time between penetrations 2 for a 93 m steel surface 2. 5 cm thick is about 1.6 x 10 days. There is a 0.368 probability that there will be no penetrations by the end of this time. For this structure, the limiting time for 0. 99 probability of no penetrations is 1. 6 x 4 4 10 days; for 0.95 probability, 8 x 10 days; and 5 for 0.90 probability, 1.6 x 10 days. Correspondingly, if the probability for no penetration of X thickness within T is 0. 36 8, then the thickness required for a 0. 99 probability of no penetrations in T days is international agreement in 1925, astronomical time is reckoned from midnight, so that the local time of day based on this origin is T = t + 12 h where t is the hour angle of the time reckoner. Because astronomers refer to two time reckoners, the sun and vernal equinox, there are two kinds of days; the solar day and the sidereal day. North celestial pole Observer's meridian Greenwich meridian (P kt at 0. 99) 10/3 pl0/3 0.0101 P, . at 0. 99 kt for 0. 90 probability. = 3. 97P P, . at 0. 90 = 1. 96X kt More generally In (prob) = - t {1.46 x 10 ) A ^TTT73 The relationships between exposed area and time, aluminum thickness and oenetration prob- ability are illustrated in Fig. 33. C. CONVERSION DATA 1. Definition of Time Standards and Conversions (Ref. 74) Time measurement may be based upon the period of motion of a stable oscillator, the decay of a radioactive isotope, or the period of any celestial body relative to the observer. The latter is the body chosen sometimes referred to as the time reckoner and a clock in most astronomical ■research. The particular day is defined to be the time span between two successive upper or lower transits of the given time reckoner across the celestial meridian of the observer. Noon is the time of upper transit (the transit in the northern celestial hemisphere). Angles measured in the equatorial plane of the celestial sphere from the observer's meridian, O, westward are called local hour angles (see following sketch). Thus O is the local hour angle of vernal equinox. Then local time of day is the hour angle of the time reckoner for days beginning at noon. Since an The sidereal day is the interval between two successive upper transits of vernal equinox. Because this time reckoner is a point on the celestial sphere, an infinite distance from the earth, the sidereal day is the period of earth rotation relative to inertial space. Because side- real time is the hour angle of vernal equinox, it is given at any instant by the right ascension of a star that is crossing the observer's meridian at that instant. The best value for the sidereal day is 86164. 091 mean solar sec. The solar day, the interval between two suc- cessive upper transits of the sun, is 3 56 longer than the sidereal day because the earth moves almost one degree each day in its orbit around the sun. Thus, the solar day is not ex- actly equal to the period of earth rotation. Also, the apparent sun (the sun we see) is not a pre- cisely uniform time reckoner because the orbit of the earth is slightly eccentric and the eliptic is inclined about 23° to the equatorial plane. Be- cause the apparent sun is a nonuniform time reckoner, the mean sun is used to measure civil time. The time unit is the average of the apparent solar days, the mean solar day and its length is defined to be 86400 mean solar sec. The differ- ence between apparent and mean solar time is called the "equation of time, " ET: ET = AT MT r M = A M " A A where AT = apparent time MT = mean solar time 11-50 t, = hour angle of apparent sun t = hour angle of mean sun Ivl A nT = right ascension of mean sun A . = right ascension of apparent sun Civil time, CT, is mean solar time measured from midnight, CT = x M + 12 h The local civil time at the Greenwich meridian is known as universal time, UT, or Greenwich mean time, GMT. The difference in local time at two places for the same physical instant is the difference in longitude, X: T l " T 2 = X 2 ' X l where \, in the astronomer's convention, is meas- ured positive westward from the Greenwich merid- ian. This equation applies for T measured in any system of local time, i.e., civil, apparent solar or sidereal times. For example. LMT = LCT = UT \ Fifteen degrees of longitude corresponds to an hour of time difference, so that for local mid- night at Greenwich, the corresponding local times at \ = 15° W and 30° W are 11:00 p.m. and 10:00 p.m., respectively. The local time increases for eastward longitude changes. Since local civil times are the same only along a given meridian, some confusion is avoided by the use of time zones. The earth is divided into 24 zones, each fifteen degrees of longitude wide. In the middle of each zone, at the "standard me- ridian, " local time differs from Greenwich time by an integral number of hours. The time read on a clock at any place, i. e. , standard time, is the local civil time of the standard meridian nearest the clock. Standard time differs in some places from zonal time where boundaries are twisted to suit geographical and political bounda- ries. Greenwich civil time is generally the system employed in astronomical almanacs. Therefore, conversions required most often are standard to GMT and GMT to standard. The conversion from a zone time to GMT is effected by dividing the longitude (in degrees) of the observation site by 15 and obtaining the nearest whole number. This value is added to the zone time for sites west of Greenwich and subtracted for sites east of Green- wich. GMT = ZT ± T5" The same rule applies for conversion of standard times, except that the irregular boundaries for the time zones must be utilized. The preceding discussions provide the basis for an appreciation of the measurement of time intervals; however, in order to relate any two events in time it is necessary to refer them to the same time reference. For earth satellite prob- lems this requires only that an epoch be selected and that the universal time be recorded at the in- stant. A record of time by days and /or seconds from this epoch thus relates all of the events. In other problems where two or more bodies are in- volved such an arbitrary solution of the time origin for one body may lead to unnecessary complexity due to the fact that all of the various time scales must be correlated each time a computation is performed. To avoid such a situation the Julian day calendar was established by the astronomers. This calendar takes the origin to be mean moon 4713 years before Christ and is a chronological and continuous time scale, i.e., days have been counted consecutively from this date to present. This practice avoids problems resulting from the nonintegral period of the earth (365. 2563835 mean solar days) and the difficulties of months of differ- ent length. On this calendar January (i. e. , mean noon January 1) 1900 is 2415020 mean solar days. The conversion of other dates in the later half of the 20th century is facilitated by Table 37 obtained from The American Ephemeris and Nautical Almanac. 2. Review of Standards of Length and Mass For many engineering purposes the conversions between units of measure need be known only to two or three significant figures. For this reason a general unawareness of the definition and use of these units has resulted and is evidenced by in- consistencies in the literature. The purpose of this section is to redefine a set of units and specify accepted conversions from this set to other com- monly used systems. a. Standard units The United States' system of mass and measures has been defined in terms of the metric system since approximately 1900; it was refined in metric terms in 1959. Therefore, care must be exercised to assure that proper standards are used for all precise computations. Before going further it is necessary to obtain an appreciation for the bases for measurement. 7 The meter was originally defined to be 1/10 part of 1 /4 of a meridian of the earth. A bar of this length was constructed and kept under standard conditions in the Archives. Since subsequent meas- urements of the earth proved this definition to be in- correct, a new international standard, the Prototype Meter, was defined to be the distance between two marks on a platinum-iridium bar at standard conditions. This bar was selected by precise measurement to have the same length as the bar in the Archives. National standards were also produced and compared to the Prototype Meter. In October 1960, at the Eleventh General Con- ference on weights and measures, the meter was redefined to be 1,6 50, 763. 73 wavelengths of the orange -red radiation of Krypton 86. However, the bar standards are also maintained because of the ease of measurement. The kilogram was originally defined to be the mass of 1000 cubic centimeters of water at its maximum density (i.e. , 4° C). However, at the time the Prototype Meter was defined, the kilo- 11-51 TABLE 37 Julian Day Numbers for the Years 1950-2000 (based on Greenwich Noon) Year Jan. 0.5 Feb. 0.5 Mar. 0.5 Apr. 0.5 May 0.5 June 0.5 July 0.5 Aug. 0.5 Sept. 0.5 Oct. 0.5 Nov. 0.5 Dec. 0.5 1950 243 3282 3313 3341 3372 3402 3433 3463 3494 3525 3555 3586 3616 1951 3647 3678 3706 3737 3767 3798 3828 3859 3890 3920 3951 3981 1952 4012 4043 4072 4103 4133 4164 4194 4225 4256 4286 4317 4347 1953 4378 4409 4437 4468 4498 4529 4559 4590 4621 4651 4682 4712 1954 4743 4774 4802 4833 4863 4894 4924 4955 4986 5016 5047 5077 1955 243 5108 5139 5167 5198 5228 5259 5289 5320 5351 5381 5412 5442 1956 5473 5504 5533 5564 5594 5625 5655 5686 5717 5747 5778 5808 1957 5839 5870 5898 5929 5959 5990 6020 6051 6082 6112 6143 6173 1958 6204 6235 6263 6294 6324 6355 6385 6416 6447 6477 6508 6538 1959 6569 6600 6628 6659 6689 6720 6750 6781 6812 6842 6873 6903 1960 243 6934 6965 6994 7025 7055 7086 7116 7147 7178 7208 7239 7269 1961 7300 7331 7359 7390 7420 7451 7481 7512 7543 7573 7604 7634 1962 7665 7696 7724 7750 7785 7816 7846 7877 7908 7938 7969 7999 1963 8030 8061 8089 8120 8150 8181 8211 8242 8273 8303 8334 8364 1964 8395 8426 8455 8486 8516 8547 8577 8608 8639 8669 8700 8730 1965 243 8761 8792 8820 8851 8881 8912 8942 8973 9004 9034 9065 9095 1966 9126 9157 9185 9216 9246 9277 9307 9338 9369 9399 9430 9460 1967 9491 9522 9550 9581 9611 9642 9672 9703 9734 9764 9795 9825 1968 9856 9887 9916 9947 9977 *0008 *0038 *0069 *0100 *0130 *0161 *0191 1969 244 0222 0253 0281 0312 0342 0373 0403 0434 0465 0495 0526 0556 1970 244 0587 0618 0646 0677 0707 0738 0768 0799 0830 0860 0891 0921 1971 0952 0983 1011 1042 1072 1103 1133 1164 1195 1225 1256 1286 1972 1317 1348 1377 1408 1438 1469 1499 1530 1561 1591 1622 1652 1973 1683 1714 1742 1773 1803 1834 1864 1895 1926 1956 1987 2017 1974 2048 2079 2107 2138 2168 2199 2229 2260 2291 2321 2352 2382 1975 244 2413 2444 2472 2503 2533 2564 2594 2625 2656 2686 2717 2747 1976 2778 2809 2838 2869 2899 2930 2960 2991 3022 3052 3083 3113 1977 3144 3175 3203 3234 3264 3295 3325 3356 3387 3417 3448 3478 1978 3509 3540 3568 3599 3629 3660 3690 3721 3752 3782 3813 3843 1979 3874 3905 3933 3964 3994 4025 4055 4086 4117 4147 4178 4208 1980 244 4239 4270 4299 4330 4360 4391 4421 4452 4483 4513 4544 45 74 1981 4605 4636 4664 4695 4725 4756 4786 4817 4848 4878 4909 4939 1982 4970 5001 5029 5060 5090 5121 5151 5182 5213 5243 5274 5304 1983 5335 5366 5394 5425 5455 5486 5516 5547 5578 5608 5639 5669 1984 5700 5731 5760 5791 5821 5852 5882 5913 5944 5974 6005 6035 1985 244 6066 6097 6125 6156 6186 6217 6247 6278 6309 6339 6370 6400 1986 6431 6462 6490 6521 6551 6582 6612 6643 6674 6704 6735 6765 1987 6796 6827 6855 6886 6916 6947 6977 7008 7039 7069 7100 7130 1988 7161 7192 7221 7252 7282 7313 7343 7374 7405 7435 7466 7496 1989 7527 7558 7586 7617 7647 7678 7708 7739 7770 7800 7831 7861 1990 244 7892 7923 7951 7982 8012 8043 8073 8104 8135 8165 8196 8226 1991 8257 8288 8316 8347 8377 8408 8438 8469 8500 8530 8561 8591 1992 8622 8653 8682 8713 8743 8774 8804 8835 8866 8896 8927 8957 1993 8988 9019 9047 9078 9108 9139 9169 9200 9231 9261 9292 9322 1994 9353 9384 9412 9443 9473 9504 9534 9565 9596 9626 9657 9687 1995 244 9718 9749 9777 9808 9838 9869 9899 9930 9961 9991 *0022 *0052 1996 245 0083 0114 0143 0174 0204 0235 0265 0296 0327 0357 0388 0418 1997 0449 0480 0508 0539 0569 0600 0630 0661 0692 0722 0753 0783 1998 0814 0845 0873 0904 0934 0965 0995 1026 1057 1087 1118 1148 1999 245 1179 1210 1238 1269 1299 1330 1360 1391 1422 1452 1483 1513 2000 245 1544 1575 1604 1635 1665 1696 1726 1757 1788 1818 1849 1879 1900 Jan 0.5 ET = Julian Day 2,415,020.0 = Greenwich Noon, January 1, 1900, a common epoch 1950 Jan 0. 5 ET = Julian Day 2, 433, 282.0 = Greenwich Noon, January 1, 1950, another common epoch and first entry in this table 11-52 gram was redefined to be the mass of the Proto- type Kilogram and, as was the case with the Prototype Meter, national standards were obtained by comparison to the Prototype Kilogram. This unit has not been changed to date though proposals have been made to base the measurement on some atomic standard. The conversion from mass to force is accomplished by the standardized con- 2 stant g„ = 9. 80665 m/sec . Effective July 1, 1959, the English speaking people defined their standards of length and mass in terms of the metric system of units. This was accomplished through the definition of an inter- national yard and an international pound. 1 yard = 0.9144 meter 1 pound (avdp) = 0.453,592,37 kilogram These two units constitute the basis for all measure with the exception of those accomplished by the U.S. Coast and Geodetic Survey which continues to use a foot defined by the old standard: The statute mile = 5280 international feet. 1 foot = 1200 3~9~3T meter 3600 = 0. 91440182 meter Of course, other units of length, area, volume, etc. , can be related by their definition to these more basic units. These second generation units (for example: statute mile, nautical mile, etc. ) are in general peculiar to particular regions and thus only a few will be discussed in the following paragraphs. The astronomical unit (AU) is defined as the mean distance from the sun to a fictitious planet whose mass and sidereal period are the same as those used by Gauss for the earth in his determina- tion of the solar gravitation constant. This defi- nition enables the astronomer to improve his knowl- edge of the scale of the solar system as more ac- curate data become available but does not require recomputation of planetary tables since angular data can be computed with an accuracy of eight or nine significant figures. The best value of this c unit is presently 149. 53 x 10 km and the mean distance from the earth to the sun is presently con- sidered to be 1. 000,000,03 AU. The nautical mile was originally defined to be one minute of arc on the earth 1 s equator. On this basis the best value of this unit appears to be ap- proximately 6087 feet. Various attempts have been made to adopt a standard length, e.g., the British nautical mile was defined to be 6080 feet and the U.S. nautical mile was defined to be 6080.20 feet. In 1954, it was agreed to standardize the nautical mile by defining it in terms of the meter. As a result, the international nautical mile was defined to be 1852 meters, or, based on the conversion between feet and meters at the time, 6076. 10333 feet. But with the redefinition of the foot (1 foot = 0. 3048 meter) as of July 1959, the nautical mile changed once again to 6 076. 11549 international feet, approximately. This value has been accepted by the National Bureau of Standards and all respon- sible agencies. The meter was previously defined; however, many units of length have been defined based on the prime unit and related by powers of 10. Ac- cordingly the following prefixes have been intro- duced and are generally recognized: 12 tera, meaning 10 9 giga, meaning 10 mega, meaning 10 3 kilo, meaning 10 2 hecto, meaning 10 deka, meaning 10 deci, meaning 10 -1 centi, meaning 10 _3 milli, meaning 10 ,.-6 micro, meaning 10 nano, meaning 10 ,„-12 pico, meaning 10 The yard = 0. 9144 meter = 3 international feet The foot = 0.3048 meter = 12 international inches The inch = 0. 0254 meter = 10 3 mils The micron = 10 meter The angstrom = 10 meter 3. Mathematical Constants u =3. 141, 592,653,6 2ir = 6.283, 185,307,2 3ir = 9.424, 777,960, 8 log 10 Tr = 0.497, 149,872,7 log it = l. 144, 729,885, 8 e =2. 718,281,828,5 log 1Q e = 0.434,294,481,9 e 2 = 7,389,056, 102 log 10 = 2.302,585,091 1/ir = 0.318,309,886,0 l/2ir = 0. 159, 154,943,0 l/3ir = 0. 106, 103,295,3 360/2ir = 57,295,779,51 11-53 1/e = 0.367,879,441,0 1/e 2 = 0. 135, 335,283, 1 4. Time Standards 1 second = 10 3.155,692,597,47 times the Besselian (tropical, solar) year at 1900. and 12 hr ephemeris time -9 1 mean solar sec >» (1 + 10 ) ephemeris seconds in 1960 sidereal day = 86, 164. 091 mean solar seconds sidereal year 365. 256,383, 5 mean solar days sidereal year = 3. 155, 814, 9x10 mean solar seconds 5. Conversion Tables Ready conversions for the more generally used units of astronomical measurements will be found in the following tables: Table 38--Length Conversions Table 3 9- -Velocity Conversions Table 40- -Acceleration Conversions Table 41--Mass Conversions Table 42 --Angular Conversions Table 43 --Time Conversions Table 44- -Force Conversions TABLE 38 Length Conversions 1 Astronomical Unit ■ 1 International Nautical Mile = 1 Statute Mile - 1 Meter - 1 International Yard ■= 1 International Foot « 1 International Inch = International International International Astronomical Units Nautical Miles 60.737, 90xlO 6 Statute Miles 92. 911. 52x 10 6 Meters 149. 5266 x 10 9 Yards Feet Inches 1 163. 524. 3 x 10 9 490. 5728x 10 9 588.687, 4X10 11 1. 238, 575x 10" 8 1 1. 150, 779, 447 1852* 2025. 371, 828 6076, 115, 485 72. 913. 385. 826 1.076, 292x 10" 8 0.868, 976, 242 1 1609. 344* 1760' 5280'' 63, 360* 0.66B, 777, 3xl0" U 0.539. 956, 803x 10~ 3 0. 621. 371, 192x 10 -3 1 1.093. 613, 298 3, 280, 839, 895 39. 370. 078, 740 0. 611, 529, BxlO -11 0.493, 736, 501 x ID -3 0. 568, 181, 818x 10' -3 0. 9144* 1 3*" 36^ 0. 203, 843, 3xl0" U 0.164, 5?8,833xl0" 3 0. 189, 393, 939x 10' -3 0. 3048*" 0. 333, 333, 333 1 12* 0. 169, 869. 4xl0~ 12 0. 137, 149, 02Bx 10" 4 0. 157, 828,282x10 -4 0.0254* 0.027, 777, 777 0.083, 333. 333 1 1 Astronomical Unit per Mean Solar Day * 1 Astronomical Unit per Sidereal Day ■ 1 International Nautitfal Mile per Hour ■ 1 Statute Mile per Hour > 1 Kilometer per Hour - 1 Meter per Second ■ 1 Foot per Second ■ TABLE 39 Velocity Conversions Astronomical Units per Mean Solar Day Astronomical Units per Sidereal Day 1.002, 737, 90 International Nautical Miles per Hour Statute Miles per Hour Kilometers Hour per 1, Meters per Second , 730,632 x 10 6 Feet per Second 1 3.364,079 x 1G 6 3.871, 313 x 10 6 6. 230, 273 x io 6 5.677,928 x IO 6 0.997,269, 57 1 3. 354, 692 x IO 6 3. 860, 743 x IO 6 6.213, 260 x 10 6 1, . 725.907 x 10 6 5.662,424 x 10 8 0.297,258, 2 x 10" 6 0.298,072, 1 x io- 6 1 1. 150,779,447 1.852* 0, . 514,444,444 1.667,809,856 0.258,310, 3 x 10" 6 0.259, 017, 5 x to" 6 0.868,976,242,6 1 1.609, 344* 0, ,447,040* 1.466,666,666 0. 160,506,6 x 10" 6 0. 160,946, 1 x io" 6 0. 539,956, 803, 4 0.621, 371, 192 1 0, ,277, 777, 777 0.911, 344,415 0.577,823, 6 x 10" 6 0. 579,405,6 x io" 6 1,943,844,491 2. 236,936,288 3.600* 1 3.260,839,895 0. 176,210,6 x 10" 6 0. 176, 602,8 x IO" 6 0. 592,483,800 0. 681, 818, 181 1.097, 280* 0, . 304B* 1 — Underlined digits are questionable. ♦ Denotes exact conversion factor. 11-54 TABLE 40 Acceleration Conversions 1 Astronomical L'nit per Solar Day - 1 Astronomical Unit per Sidereal Day 2 = 1 International Kautic Mile per Hour 2 = 1 Statute Mile per Hour 2 ■ 1 Kilometer per Hour = I Meter per Second 1 International Foot Astronomical Units Astronomical Units International Nautical Miles per Hour per Mean Solar Day per Sidereal Day 1 1.005,483, 30 0. 994, 546, 6f3 1 0. 713, 419, 4 x 10 ' 0.619,9 44,7 x 10"' 0.385,2 09,6 x 10 ' 0.049,923,97 0.015,216,62 0, 717,3 31, 1 x 10 0.623,344, 2x 10" 5 0. 387, 3 21,9 x 10 ' 0.050, 197, 70 0.015, 300, 26 .401, 700 x 10 J 1.394,056 x 10 J 0.&68, 976,242,6 0.539,956,803,4 Statute Miles per 604, 250 x 1(T 1. 150. 773,447 0, 621, 371, 192 0.699. 784.017,6 x 10 0. 805. 297, 064, 9 x 10 0.213.294. 168.6 x 10* 0. 245,245, 245, 2 x 10 4 Kilometers per Hour 2 Meters per Second International Feet per Second 2 65,716,76 2.595.989x10 20.03 0,46 2. 581.832 x 10 5 19.92 ], 23 55.. 15 8, 38 1. 852- 1.4 29, 012, 345 i 10 - " 1 4. 688, 360, 71 1 x 10*' 1.609. 344* 1.241, 777,7 78 x 10 ~ 4 4. 074, 074, 074 x 1 0"' 1 0.771,604,938. 2 x 10 2. 53 1, 5 1 2, 264 x 10" I 3.280,839,895 1.395.020,800 0.3048* TABLE 41 Mass Conversions 1 Solar Mass 1 Earth Mass 1 Moon Mass 1 Slug = I Kilogram = Solar Mass 1 3.088, 062 x 10" 3,697, 320 x 10" 7.346, Jj^x 10 5. 033, 73 x 10" 3 1 Pound (avdp) - 2.283,26x10 1 Ounce (avdp) = 1.427,04x10" 1.229,14x 10 ■ 0.244,25 x 10" 0. 167, 36 x 10" 0. 759, j_5 x 10" 0.474,47 x 10" 27, G 4E, SCO a 1 . 3 58 1 0. 198, 72 x 10" 0. 136, _16 x 10" 0. 6 17, fi3 x 10" 0. 386, 01 x 10" 23 21 Slugs 1.361, 25 x 10' 4.094. 2 x HV 5,032, 3 1 6.852, 176, 612 x 1(T' 3. 10B, 095, 016 x 10" 1.942,559,385 x 10"' Kilo gram s l. sae^ x 10 30 5. OTrj^J) x 10 24 7. 34-^0 x 10 22 14.51)3,902,876 t 0. 453, 592, 37* 0. 283.495, 231 x !0~ Pounds Ounces [avdp. <avdp) 4. 379, 70 x I0" f ° 70.075, 3 x 10 3 13. 172, fi x HI 24 210. 76 x 10 24 16. 191, x 10 22 259.06 x 10 22 32. 174, 048, 55ii 514, 784, 777, 2.204, f.22,621 35. 273, 961,94 1 16.0=-' 0. 062, 5--'' 1 -Underlined digits are questionable. * Denotes exact conversion factor. 32. 174,048,556 ft/sec TABLE 42 Angular Conversions 1 Revolution = 1 Radian = 1 Decree = 1 Minute of Arc = 1 Second of Are = 1 Aitirular- Mil = -Denotes exaet cor 1 0. 159, 154, 043 2. 777,777,777 X 10" 4. 629, 629, 629 X 10" 7. 716,049,382 x 10" -4* 1. 5625 X 10 (i. 283, 185,307 1 1.745,329,252 x 10" 2.!)08,882,086 X 10" 4.848, 136,812 x 10" 9.817,477,040 X 10" Minutes Seconi is Decrees of Arc of Ar c Angular Mills 360.* 21,600. 0* 1,296,000. 0" 6400. * 57. 295, 779, 511 3, 437,746,771 206, 264,806, 236 1018. 591,636 1 60.0* 3,600.0* 1. 666, 666,606 x 10' -2 1 60. 0* 0. 296,296, 296 2. 777,777,777 x 10' ■4 0. 016,666,666 1 4. 938,27 1. 605 X 10 5.6250 x 10" 2 * 3. 375* 202. 5" 1 11-55 TABLE 43 Time Conversions Solar Year Julian Year Mean Solar Day Sidereal Day Mean Solar Sec Siderea 1 Sec 1 Solar or Besselian Year - 1 0. 999, 978, 641 365.242, 198 366.242. 198 3. 155.692.59 X 10 ' 3. 164,332, 5 7 x 10 1 Julian Year = 1. 000,021,358 I 365. 25 366. 250,00 3. 155,760* X 10 7 3. 164,400, 16x10 1 Mean Solar Day • 2. 737,909,26 x 10 -3 2. 737,850, 787 x 10~ 3 1 1.002.737.9O 8640O* 86636.555 1 Sidereal Day • 2. 730,433,61 x 10 -3 2. 730,375, 42 x 10~ 3 0. 997, 269, 57 1 86164.091 86401)" 1 Mean Solar Sec « 3. 168,876,46 X 10 -8 3. 168,808, 78 x 10" 8 1. 157,407,40 X 10 -5 1. 160.576,27 x 10 "^ 1 1. 002,737, 90 1 Sidereal Sec = 3. 160,224,08 x 10' -8 3. 160, 156, 58 x 10" 8 1. 154.247. 18 X 10 -5 1. 157,407,40 x 10" 5 0. 997.269,57 1 Exact conversion Kg (force) TABLE 44 Force Conversions Pound (force) Newton 1 Kg Force 1 Pound 1 Newton 1 Poundal 1 Dyne 1 0.453,592,370, 1 0. 101,971,621,2 2.204,622,621 1 0.224,808,943 1.409,808, 183 x 10 3. 108,095,501 x 10 -2 1.019, 716,212 x 10 0.224,808,943 x 10 -5 9. 806,65* 4.448,221,62 1 Poundal 70. 931,635,35 32. 174,048,6 7.233,013,85 Dyne 5* 0.138,254,954 1 -5 10 7.233,013,85 x 10 9. 806,65 x 10 4.448,221,62 x 10 5 10 5 0. 138,254,954 x 10 5 1 *Exact conversion D. 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Present Standard and Model Atmospheres, and Proposed Revision of U.S. Standard Atmosphere 11-66 Defining T M Task Group IV — "" ~~~~ Related kinetic temperature Defining T M of ARDC 1959 U. S. Standard Atmosphere 196 2 Defining T M of present U.S. Standard- -ARDC 1956 700 •a 3 1000 1500 2000 Temperature (°K) 2500 3000 Fig. 3. Temperature Versus Altitude, Defining Molecular Scale Temperature and Kinetic Temperature of the Proposed Revision to the United States Standard Atmosphere II-67 3000 2800 2600 2400 2200 2000 t- 3 cd CI) 1800 a 6 CD H 1600 01 a! a (/i 1400 t. at i — H 3 01 1200 1000 800 600 400 200 ^™ Proposed U.S. standard atmosphere — - (Task Group IV) _ «_ Proposed U.S. standard atmosphere (Task Group I) — — ARDC model atmosphere 4 Ion and other gauges at WSPG (NRL) A Ion and other gauges at WSPG ~~ (USAF- Michigan) P t> Ion and other gauges at / Churchill (NRL) / X Grenades / — "^ Russian containers / — Satellite drag model day 7 _ — — Satellite drag model night / • Kallman. March 13 (computed from/ pressure scale height) / — Russian standard atmosphere, 1 J ^^ ""^ *^i / December 1960 — O Sodium cloud resonance scattering *^^^ r*' ^ - sJ* f ^^ rM • 1 „ r^ — — ■ - — — vj-. 1 PI Proposal ^ repared by T ask Group IV 1 1 1 1 1 I i I 1 1 1 1 1 1 1 1 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 Altitude (km) Fig. 4. Molecular-Scale Temperature Versus Geometric Altitude Proposed United States Standard Atmosphere Compared with United States Detailed Data, Russian Average Data, and ARDC Model Atmosphere 1959 for Altitudes Above 80 km Only 11-68 Proposed U, S, standard atmosphere < Ion and other gauges at WSPG (NRL) t> Ion and diaphragm gauges at Churchill (NRL) A Radioactive ion gauges (USAF- Michigan) o Sphere drag (USAF-Michigan and SCEL-Michigan) d Bennett mass spectrometer at Churchill (NRL) "^ Russian average of containers and rocket data at central European Russia (data 110 km and below are for summer days) f Russian satellite-borne manometer for May 16, 1958 (1300 to 1900 local time, 57° N to 65° N) V Manometer on Sputnik I X Grenade Satellite drag model, day active sun (3) Satellite drag model, night active sun (3) Satellite drag model, day quiet sun (1) Satellite drag model, night quiet sun (1) ■ ARDC model 1959 ^\ ) Jacchia to c <v Q u 0) to o 10 10 = — 10 10 10 -9 < ID" 10 t 10 -11 180 200 Altitude (km) 280 300 Proposal prepared by Task Group IV October 15, 1961 Fig. 5. Density Versus Geometric Altitude for Proposed United States Standard Atmosphere Compared with United States Detailed Data, Russian Average Data^and ARDC Model Atmosphere 1959 11-69 ARDC model atmosphere 1959 ■— - Proposed U.S. standard atmosphere -4 Ion and other gauges at WSPG (NRL) £> Ion and other gauges at Churchill (NRL) A Ion and other gauges at WSPG (USAF- Michigan) ■\ Russian average of containers for summer days mid- European Russia G" Russian satellite-borne manometer for May 16, 1958 (1300 to 1900 local time, 57° N to 65° N) x Grenade 10 10 B0 X B io s u 3 io in u a. -J 1 - -3 Is- - -4 = N 4 = -5 ^ -< » « [< ^. - -fi < 1 * 1 « 1 « > [ 1 < u 1 > > > 1 > > 1 < > 1 < - -7 - 1 4 ' < 1 , -f* ^ T^ T~! *7~7 1 = E 10 = 10 = 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 Altitude (km) Proposal prepared by Task Group IV October 15, 1961 Fig. 6. Pressure Versus Geometric Altitude for Proposed United States Standard Atmosphere Compared with United States Detailed Data, Russian Average Data and ARDC Model Atmosphere 19S9 11-70 Proposed revision to U. S. standard ARDC model atmosphere 1959 CIRA atmosphere 1961 Present U.S. standard, ARDC 1956 700 600 ■a 3 < 300 14 16 18 20 22 24 26 Molecular Weight [kg/ (kg- mol)J 28 30 Fig. 7. Molecular Weight Versus Altitude 11-71 500 400 ~ 300 0) •a 3 < 200 100 Density (gm/cm ) a. to 500 km 2500 2000 ~ 1500 0) 3 1000 500 \ L*" Miiiimum of sunspot cycle Maximum of sunspot cycle 10 •20 10 ■15 10 ■10 Density (gm/cm ) b. to 2500 km Fig. 8. Average Daytime Atmospheric Densities at the Extremes of the Sunspot Cycle 11-72 10 Air Density Sea- Level Density Fig. 9. Density of the Upper Atmosphere Obtained from the Orbits of 21 Satellites 10- a o i s 60 CD O o o to Q. S = 190 x 10 2 / m c I i -60 -40 -20 20 40 An 60 80 Fig. 10. Dependence of Atmospheric Density on &a in the Equatorial Zone (diurnal effect) 100 120 11-73 180 240 300 60 120 180 s o E 4 an ^• u o <M 3.0 O O O o — -»- A -A ^~A"" A O "0 £ A A 6 | < 20° 20" <|A 6 | <60° 12 t 16 20 24 Fig. 11a. Diurnal and Seasonal Variations in Atmospheric Density at 210 km Derived from Observations of the Satellite 1958 6 2. (The lower x-scale gives true local time, the upper Ao ■ a - o . The parameter of the curves is A 6 ■ t„ - 6 where a is right ascen- sion, 6 is declination, tt is perigee, o is sun.) 180 24 300 60 120 180 2.0 1.0 - - - - - Jr$ - - r *■ \? - 77 - 7 / V I \ i > ^B 12 t 16 20 24 Fig. lib. Variations in Atmospheric Density at 562 km Above the Earth Ellipsoid Derived from the Observations of Satellite 1959 a 1 180 240 300 60 120 180 B io - - - ft // A - Ep a __ "a, i 1 a\ 1 i |A fi | < 20° 56" > IA5 I > 20° 12 t 16 20 24 Fig. lie. Variations in Atmospheric Density at 660 km Derived from the Observations of Satellite 1958 8 2 11-74 I 1 1 1 All values are corrected to mean solar activity (solar flux of 20 -cm radiation S = 170 x 10~ 22 W/m 2 -cps) The indicator of the curve gives the true local time. 100 200 300 h (km) 500 600 700 Fig. 12. Diurnal Variations of Atmospheric Density at Altitudes from 150 to 700 km above the Earth Ellipsoid for I A6| <20° 200 +20 +30 +40 +50 Percent Density Departure of Seasonal Mean from Annual Mean Fig. 13. Model of the Seasonal Variation of Mean Density to 200 km 11-75 o c M o a (Iran) -»soo 11-76 T3 ni IB o Q 10 ^ u- -i-i HfffiifffllpiB jf^ifil fs=^^-:'4-fri"i1:.;"j"|fi^444^-^- l-:i.T=ppi 7 3rEpp|== = = = ==j=bt~ = ;-:4: r- = = . ?---. z=fr -fi - ■■ -- - ■ Mi 7|:i-7W==f=ia=?=i|=!.:! i- =i- -■; = p t -i-- =EEE- =rpEE ===== ====,=.== :=p =]= - = =: =;---- ■-■-■- i ■-!-.- ._• • i I.. 7 £|= =!47p= :]-(-:=;= -'.- : i ~ - : -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^ -f f * |— J ■■ 1 — | ' ' — '— L — ' 4™" -+— ' — p rhr'xF — 4*-F-*K--h- p -j-c^ ;— H- - j — = — : : p t:d— ■ |^-ri^T^--bJ# ; :J=&H^pJ53 Sp : ; r r-j E |:j : ^.^-j= t 7 " - j-7;-::= := |~Mp{lMd A: f : -;.jl _. J ;. _ . ff&f^- ... j L, f ).. . Ep r-E _ _ . .. =,t-E d- - -;: :. IE - : "'"^E: E - i E J ^pH_ .-^ -^-u. -— - ^~ ,__ = , i- ^ — ™____„_^__, — | j ,1 ,Li ,_j Jir-^ ■■< ■-j-.p-j-, ^-----444-4-^-^^ ^^§§^^8$ j^y-^^g-^^^4.-::i7^---"|Arfc-4^ Jj.i i- ■.. -- i- . - f - = -yX--- — -J=— = : -:~. - - -:-: - '- = - - = 7 : = = =:";- 7:-i--ir:7. E=fi 7= = ^i^=l=7.= =i=7777i= =i i 1 i7 t\ :-=-:=fi-\- 4 -= - : :.7==p ^ - p ~ = - = - -E [^Htf J^fflT^ ■ 1 0T r • : - ■- tHct4 |p j- i ^ |Mpi| =i= H ^M=[= - to ml Iji £ ee; jpj: }kb ? Ffet <sg -[ J-1 7 l M l : :L : j= = B -1 . zE^E^i:, ,i:3: E-=E1E to^41-EE7:-[-!-7 ■■— r— ■ T~^\t H J i 4" -'-- "!" "!" 2"j 7 -- !: j f .:-.j-;---"t: .-:..: 4: hr:^ ■r— -1 I.-H-7J i i i i i i..<j -.pi.Li- 4-"T'-r-i-i h-4- 1 i i ill-- -.- -j f \r -!> '■ pi ^ -" -k -a -z AsBr i^^-i^ ff —- H- i ]. 1 ;;i:;t:;.|i "u -p -L *-ei=E i ifi : i - 44 1 p =5--— ---:-- -:jj -^-f-" : : " "p- F-p =--=;7^=: 7. : ^={^£= = f = r^ = ^t" ~ S~ "-- - t " I -"] i--'! -:-.;--: t ; ---[- -j- j- pj- =-j^|— = - -pp -|- -1 ■ I i 3-..=: 7--- f= r f4 "■ p = "fi ^? : - 7=3= =|=;^i=fi i Ja= = =.= 4---.:-.',-'. -\-=i" ■■-:: -.;7!..-::!-: --EN.-7- -.EL -- -- : i - - p-^ -j.! [.! E=i = | ; -r- ; 7 '- ■ - p ======= = -4 E r(4 4~ E ££===:=!=-:-=:-:-•- ::-r4 -_ ippp = - ::: - z -4p =EE =E =)=: !"E 4 ^ Jj j - -1-- --^- - -.-?-- ■- . .- - -=p 1--. -|---j- — = p.= - — : ' 1-.-^:-; i i --L j ... ^4- i-'l- -=-'—-=-- "- : - : =-j 7! '--]- = i . : ;.|.- ^- ---— §;-^J -=-- b --.- ^ E-^ -- -- . =E -E- 4=E ^^^ E------ -]-: -El: ! 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Solar Proton Dose, May 10, 1959 Flare, 30-Hour Duration 11-77 100 10 -a a! <u to i o Q c o o Oh 0.1 jjlji||^^Bjlil]i::::s:i!!!i:: II=5lBBlSll=l5s i gm/cm aluminum |Si;?3ijIji||||| ^4, ^?|f ||S jpljjpiljM t- [I'I =fc= 1 -j e « f^j j+- - -p ,ii | i ■—*■■¥"'-""""+-■ Nir-M'^ - 'itij-|4J^ST| ffiiflrffflTl [4 '- w- - 444^^*44 '^ - 4444 j mrf /^\] [ NT] il^U^rfT 4j44iTM"TIT IT fTr444+""~ 10 -Rf I' "-'44" 6 nrJ rr^ 42 ^ _|- J£ ^,* . _.-^__. B „ .-^ ^^ _ki J_ _1 L .. i"X -■ — '<-/- /f ' 1— — t**^*' -^w-*" eg ^— " i ' ' i r— <- — i — i-~i n 1 — i j Jrt~~ WrjFTfri i^r^^rTT^U 1 f - frrf] T TTT"^ t HI ■ J%4 [- 1- f [-£Hf4 ^o4fffi44fP'fl " g|-g^^e!' ? ^^^«-^ = " i *.--.--^^-- = =^-T=^^^^:^:^==--^=j-t--, ^->. j^,-. ? - .^^^^^^^^HHfntt^^ff^^^r^^^S^ffl WH ifftP^P^^SPF^y =-?^44= 4^-jglgg ftr7-^ = -g^-4^^=;£---r£^^/"^^-^-fcj jjiW--— —j^ iiillillllliiilllltra rffl/l" Ml M — " T^ r^rF 4 "! r r 141 1 IfJJHI 1 1 ' ^ " " ! " ' 111 1~H~1~H 1 1 ! 1 M Jjtf-4- 1— j -t-!— | 1 1--*.- -j- -J I--.- - J -l-L.-^-i-,-; - .. 1 f _|_ W i VI i ■ 1 ■ J J_ 11 ■^_4^iT-p_^>l^MM"1 r "" T : fi"f"T "l " \~ ---"" f "4 "Ft " "I : r'r TTTTTX-li r I'I 1 "Fttzf-^PT Bp^-ErfEE '~"-l :- .: r :.' _Ei__p_E"__: l_:z.^:ti: ■:■:---■;— — ^:j—-^ ..-;-; ■ ■-- ■ .£ - .i-. __ -4^H ---i_r =__ — — -.-■ =-. .:_.- -_ B*-t4~~ : : L " : ■ """■ " ~ " ■■- — —.! —-..-...:..■—.-.. z- . !...._ :_....:. . ±- . ■ L ■":' ■ :-_ -44 = . — _-- — ^rz. -- ■■ - -■ W- ' ]~r= — TT ■ j" — — — — — Ui.^.^.^ii ~zz^^.^. — — — izz.zz^ — -^. -_:"___ ■ -.■ .-a, - -^-^-^-^--^ — [ f j~"j^ = l— _^ — — |E_ IB^ftiiiiiiiiiiiiiiiiiiiiiiJiiiiiiiiiilttiiiHi^jiiiiiiii ifa|jfilllipiiilill- : "r-p — —."_: I7_ — " = -"- .-._ --: . e_ :_e: — - — — — r.-.: — — :--. - : .-.. ■ -... .:; " : - _-_. - — — — z. : — : . — -r_ 7- :-£--- j --t ^e-— ' — — =^^^z=jz^ ==^E5Zf== = = H; === = ===^^r=^^- "3Er:--^r^^|^^^^ = zi ^ = = ^N |f|?|^ ^pfefnp ^ = = = = === = = !! === = = lllllllllllllllllllllpp^ll|lllllllllll|h^llllllllllll|l -PP _U : . :- j Tj -=r =£ L4 1 L j — j_p. _j_. ■ j r 1 '- ' ', ' ! ' 1 1 ■ i [J --^' t ---.--■-- t*^- - T ■ .- - _ ■ . -.-...;■ ■ ■ ■ 4----- -, — _-=.-- = ._ — .- -^ j--^ -_ --|- .:.-, ■■- —--.-j. --" : i --■:■:- ~- -■- ■ ■.- j . .-. j zz :_ — — — — .-- = —. -.: -* 1— -4^ ---: - : : _-_ ■ 4— : " "^= ' " ' — — -"- - ■ -}- " - — =f^L. :_. . : i--=|_.:._. .. _ -1^ 4-4--J- : 4=t^ - ~ - ^ ^ ^^^^^^ = ^^^^^^^^^^P=^^^^"^ ^^^^^^"f^l^'L^!:"!"^^^^ p. zrtj+_-J ___1 _ ;_:-= = = rz ~ -j-L—— — ~ .— ^4— zj= = = ~ — zzizr = — ttJ— :t_:-— ==z = =zrr:— zr;---;-z:— rrn=!.-lT = Ipillllllillllllpiiiiilililliljlplligijiljliliiiiiiiljra (jllllllIllB|ill||llllll E===E====M==E ======E E ==EEg=EEfl 1 _L — _L , + _i _i__ 1 1 1 1 _i 1 _i_ I ..... . _[ J_ J, 1 55 60 65 70 75 80 Magnetic Latitude of Orbital Inclination 85 90 Fig. 16. Solar Proton Dosages from February 23, 1956 Flare. 11-78 Xi O 10 - L - i — i j ; i i : ; ; ; Ml !iil Iill Ill; Hi} 111] II II jjiliilll ji ! i | : * t I '■ "hlr M; ; ~i" III III; Iill III; ;;n 1;;!' I; II jlll till !|!| — 1- -— L 1 -1 f :-j- 7- - . - " T : l 1 i 1 MM ! 1 1 i Iill : : : : ; : : : : j : : ; 1 : : fill ii;; iill \ : ! i : ! : : i Mi! ill !m -l : i iill \- : : : I iii in iiij M : ::|: ii:! :: : l illiiiii! !:!! ■f t -+■ ; -i-+tt I ; i i j ij ! l M_MM_iM- iiji. ::!: II::: :: :i iiijliiii j::; ! : !i M - : : ,_+.,__, [_^ I : : i : i lj! : !ii : ii Hill ill; 1 1 + ; : It : • i 1 1 • ■ ■ Ill Ml II 1 Ml iliilili' ! i : i 1 i i i 1 iil : in iiii j|i : iil iliiilii II llll Iill Iii: 1 "* j ■ i :. ■:. : : iill -iliilUM ||i:i;j Ii iii iiii ! ^ii I : :;:-). ::.j.::: ; : : : . ::; ::: !• ^-r!*i':T '■1 Ml . . -. j ' . ; \ \ 1 i : l M ' ■Li^iiiif !:| f ■; _ i i ! '. t"-. '. '. Ml - < ■ ; : ! i 1 ■ i •:■!;; !::::: !;j i.-l ii!i — i I — i — i . . -| ' . i . . . . ( i . \ . . . . 1 1 .. 1 H — h • - \- : : : . . , i ! _ — j—i -U-U Mllilil |i::i: |:| i" : ' ! . j . . , | . . . i . .... . *!;■ __;.._. ~> T - ~- • - - • ■ . j . , , . . j ^liii^ ■ ! ■ ■ ■ t 1 | M . 1 J , 1- • • ■ • ■ i : j ■ I 1 1 _.. -H— ----- ■-L- - . i . . ; | ^_ ■' . 1 . . \ ! ■ j] ■ -•■■ .... -\- — - i \ ■ 1 - 1 • • • ■ ' ■ 1 ' 1 1 : ' ni ; : - ■ ; ;-ri ! ~\ :..!: ' • i ' • ■ •]'! !| ' |l u ;| M --- - . . i 1 -V . j ill 'III j 1 ' 1 'I * ■ ;; [|;" ; - : - ! ■ ! : ' ... . ... | \ ■ \ i: ! i . . : . . 1 • ! ■ ■ ■ ) • j^JM ■ 1 . i 1 , . , , • t- i -. 1-1- i ; ■ i i .. ..j. . 1 i i ' " !••■ ■■ • ■ j j ■ ; • - I ' • T- -± ii J ; \\ ; . \ , ; ,... ,.|. . l! :::tm : ~ ~ ■ - ■; ; M ■ ■ ■•!.■•■■■■ ||- -f : i l + + .... . . . ... i 1 ... - 1 ! \ ■ | ■ i ! -II ■ ■: •;■■ ': ' ;: :mm. i 1 i ..... _-L|— I— - + .._; I . :. j : 1 | , 1,1. , . j . — __^i. i ■ 1 - 1 ... ! : l . i.j : M l;. ,h: .1.. 1 i "ii r ii;i -•-•- -- I ! \ i )■!■ - 1.. .!i. ,;i. .: . 1 . .... ...j ; . 1 . ; . . J. 1 - - : : - : - -■ ,..,-'. - :.. :i.. ;..- -; . ... i" — ; - !■■ U_ -■• • • ■- ;[!'■'' ... j..j .... |...|.;.. ..... jiljjii :l Ii! 2 | i - ^ L^ L^_ . . — 1 . ... ^.-4..i.!: - I I I \ : - ■'■] t:\-\ i;; 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Fraction of Solid Angle Subtended by Earth Fig. 17. Solid Angle Subtended by Earth as a Function of Altitude 200 240° 320° 0° Geodetic Longitude Fig. 18. Magnetic Dip Equator (1) from USN Hydrographic Office, 19S5 and Geocentric Magnetic Equator (2) Inclined 13° to the Equator at Longitude 290° 160° E 11-79 Fig. 19. Inner Van Allen Belt 11-80 Protons / sq- cm-sec- ster with energy > 20 mev Fig. 20. Flux of Protons at One Longitude in the Van Allen Belt 11-81 : = S1 H- tt# J444 *= ; ^H- HE I] : _: ilti * MM jT f L Htf tjt: M jEpiini *Wt "TT j li 1 ■ 1 ■' lf : ::-: 1^1 jj;j lllf ■!■■ }T- lit nf [f . . tfftf 1? jji i i ■ it — ^ i U : }f- ;Lj =« It 'Mi ■ h ■ Wi- MMffj m i *1 i il Tt -d f ' il II ■fl ! If Tftj-piit! il" jjjj ^L= Ml it Mi j J- t- IP ililil Mil i i M: i : i i li tfit 4|f" IV. MM _! :- ~ II 1 ;■ P Mi if -t llli MM* .. ,j. . it jii-!- 1 in- W\ - #!■!=■ ti*' 1 -i-_". ■III Mi M 1 ii i I : j!! HlM i;i!:! : - :tj ii ;r if M . :i iM : ;|i* ill' :] : :l! jj: (iliiiii Ml : t : I : I : i r I ! I ; lil : 'Iii 1 li;!!] i 1 i t: ; : ifi ffi^ji ! ' "i ! 1. j- •ii: HI blnhlil I Hi ^* li ;;:!" feS jUiBr itii iii: ij ; f MM [■[[ it! ; — 1 f IN i j f i ii If hi! : : . 1 '*. ^ : i ! i M ; xL HL if ffMii iil : j ■ : • f ♦ft jiihiil MM /ti ::;! i:i iii! !:i: iii: i! f ill I : • ti'f MMf Mi ' . 1 i i:'i Mi :?! ii 1 : I' • Mi' f pM ; ; [ ; H - ■ i-j ::;;!!: ijill- * M ' c o s s bop <" a. ..(. i:i K;. ::!: :!:: iii jMiii i-ii 1 - \ ; ■-rrrtirt i ; : i 1 nn rnr tt Mli !( j;|j Mi r!' ; ti j = j:!: : ■ : : ■fH-ttr ! I M ;i -III : M-t iill iili iSJiHi ; ; 1 i f 1 j t" il=:!jr li 1 i:l hjt : j : ! >-i \ r: ill;: :; !|M ■1: ;(;: i-'i iii: i : i i ii-^ Mr : : [li 111! : i i i i : WW- : iii iii: : Mi ;!;;;'! i i ii ! l !!;! :! ' ; ill! 1 •jWjiFt^ Mi i : ;i : ! :: i M : i !!|i;! ! : ' !:■:: ':i :: :.'i i; ! ; ,.,' :;;: :::: : : ' i j! | ;'| ! ;l ;ij: ii;; 1 ': ! ■ ! :;|: :i. : ; |H ; !; :i: ; i ; i i : iii ii' ; !•:• rli i :: i iii : : : ; i f r !!'! il!; i!h Mi iiilii 1 ;M :.-:; iff] vii ■'•.'■/. : : : i tr+td ii ' :: t*jf 4r-rHi- w: t:t 1 ; i i i Y\\ j i i i|;i lil; : MM i ;tt| :j:: ;:i;. filji M'Mltjt '. \ ' / \\ \ j j MM: it ■\\i\ i ! : : Mi i :: :>:• it; : i: it; :l:i ||; ; iii-i-i ■ '■'■ t § jijjM i • i'i! : i| Iji i: : ;l:: i;i: : ■li;:- M : - ■■ ; : Mi ; i:: Mi i i '■ t + jiM i t ate il; ' i-|U tlii i V til " h ■!■■ : ■i h : i" i I - ', i ! i ii : ' ;] E . .1.. iii'i" Mr Mii 2 Asm-oss- rno/suoiojd Asra-oas- mo/suojojd 11-82 Electronj/aq cm-sec-rter with enerfjr > 20, 000 er Fig. 22. Flux of Electrons in the Van Allen Belts 11-83 c id > 4J u V a. l/J >. so h U e A3^-aais-oas- ui3/suoa;oata II -84 — 10 -a u x 3 0) a, 6 o a! tf 0) to o Q c o u H u T3 nj U X J* 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. 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L., and Griffith, B. A., "Principles of Mechanics, " New York, McGraw-Hill Book Company, Inc. , 1959. Timoshenko, S. P., "Advanced Dynamics, " New York, McGraw-Hill Book Company, Inc., 1948. Vertregt, M. , "Principles of Astronautics," New York, Elsevier Publishing Company, 1960. Watson, C. , "Theoretical Astronomy, " 2nd edition, Philadelphia, J. B. Lippincott Company, 1892. Whittaker, E. T. , "Analytical Dynamics, " New York, Dover Publications, Inc. , 1944. Wintner, A. , "The Analytical Foundations of Celestial Mechanics,' Princeton University Press, Princeton, New Jersey, 1947. Introduction to Ballistic Missiles. Prepared by STL for the Air Force Ballistic Missile Divi- sion, March 1960. Vol. I --Mathematical and Physical Foundations, ASTIA No. 240177. Vol. II --Trajectory and Performance Analysis, ASTIA No. 240178. Vol. III--Design and Engineering Subsystems, ASTIA No. 240179. Vol. IV- -Guidance Techniques, ASTIA No. 240180. Notes on Space Technology. Langley Field, Vir- ginia, Langley Aeronautical Laboratory, Flight Research Division, May 1958. Ill- 40 ILLUSTRATIONS III- 41 a ■«-» o o > 3.5 — i 3.4- 3.3- 3.2- 3.1 — 3.0- 2.9— 2.8- 2.7 — 2.6- 2.5—1 2.4- 2.3- 2.2- 2.1- 2.0- 1.9- 1.8- 1.7- 1.6- 1.5- 1.0- 0.5— Note: See Fig. 15 for graphical trends and metric data •2.0 -2.1 -2.2 -2.3 -2.4 • 2.5 ■ 2.6 -2.7 •2.8 -2.9 -3.0 — 3.5 ° -4.0 -4.5 -5.0 -5.5 — 6.0 -7.0 - 8.0 -9.0 -10.0 —15.0 -20.0 -25.0 3 ■H 3 5.2 5.4 2.2 5.6 5.8 2.3 6.0 6.2 2.4 6.4 6.6 2.5 6.8 7.0 2.6 7.2 2.7 7.6 8.0 — '. 2.8 8.4 — '■ 2.9 8.8 9.2—: 3.0 9.6 10.0 0. 11.0- 12.0- 13.0- 14.0- 15.0- 20- 2 5- 30- 35- 40- 50- 60- 70- 80- 100- 2.1 3.5 4.0 4.5 5.0 6.0 - 7.0 - 8.0 - 9.0 -10.0 -11.0 -12.0 -13.0 -14.0 -15.0 -20.0 u o !? e e en F ; i;;. la. Semi major Axis as a Function of the ilnJius and Velocity at any Point (hnglish Units - sec ! : i;',s. lb and lj for Other 'InitsJ ^r,E WANK NOT FILMFT 111-43 1 I 1 1 Hi ll Hi 1 ill ill V III 11 ill VI iiii jfiiiii j i iii 1 u 1 \ A i ^ 1 M a < \ 1 M 'j , u i + a d u > w \ </ + a u a V, ' w n V > \ \ \ ,tt\u o o > > III -44 180 3.0- 2w (t - t ) M » E- . E ■ e sin E 2.0- M (rad) 40 80 120 160 200 E (deg) The Eccentric Anomaly, E (deg) 1.0- rl.O ■ 0.1 -o.6 a 0.4 W -0.2 F i jj . 2. The Relationship between Irbital Position and hecent rici tv and Tine from Perigee (Kepler's hmiation) (also see J-'i;.;. JJ) ni-4. n Fig. 3. Three-Dimensional Geometry of the Orbit Fig. 4. Geometry of the Ellipse 111-46 Directrix Fig. 5. Geometry of the Parabola Focus Fig. 6. Geometry of the Hyperbola III - 47 ■15 ^|G at o VI T3 a (1) o i— i X 01 C/3 nt o w ■a a I +-» 1000 TB ht 0. 240 •- 0.230 900-i l - 0. 220 .r0.2io 2000 if; :?ooo ■to. 35 n u h \\ 1500 -h -* + 0.30 800 Fig. 7. The Parameter \{ tfO.25 1000 Ji 1 T ::0.50 + 0.49 0.48 0.47 2500 4000 rt°- 46 ■P0.45 : +0.44 I ft i -1 .tO. 43 ttO.42 0.41 II C 1 2000 Hi tO. 40 TO. 39 0.60 0.59 0.58 -*0.57 3500-: tO. 56 3000. r0.55 n 0. 54 *t 0.53 0.52 r» 0.51 as a Function nf Semimnjor Axis n 2T (l.n'jlisli Units - see Table 9 for 'Ictric Data] 5000 Tr N Lf 0.70 II ihO.69 r*0.68 1 1*0. 1 67 ♦ 0.66 4500+- 0.65 i : -0. 64 h0. 63 tl 0.62 ?1 4000- :• 0.61 111-48 *4 II < -ilC u 5 1 0" 8 £ w 5) co T5 a as 0) si "ni 1 CO a) J 6000 T T* -0.77 -0. 76 5500 5000 0. 79 iO. 78 0. 75 -0. 74 0. 73 -•0.72 0. 71 7000 tt .*0. 88 + 0.87 + 0.86 -?0.85 6500 -■- 6000 01 0.84 r-0.83 -0.82 -■■0.81 ■1 ■i 4 0.80 8000 Tt 7500 ~i 7000 ii 0.96 ■0.95 0.94 0.93 0.92 -0.91 0.90 .5-0.89 1 9000 tt 10,000 fwl. 12 + 1.04 -1.03 -1.02 + " 8500 -- -1.01 8000 * 1. 11 trl.09 9, 500 1.00 t" :0.99 r0.98 0.97 ill 9,000* 1. 10 1. 0E 17 1.07 1. 06 fr 1. 05 Fig. 7. (continued) 111-49 o H X +-» ^«, <M o OJ rt in 01 . i= •■-i i-|n X < ii u i p -n|ci 'c? S 01 -t-t 1— 1 cd s o 1> OT w ■a « s t— I o si 2i OT tfH 0) ■a J § x +-» -C M « 11,000 15 12, OOOlg lS.OOOtF 14,000 fi n 1^1.18 I? 1.26 :• 1. 25 M ril. 17 H ■i t 1.24 Ff 1. 16 H 10,500-- 11,500 t? 1 - 15 H -*1. 14 1. 13 H F1 .* 1. 31 1. 22 ; -1.29 ■1.21 Et 1.20 lO.OOOlfi 11,000* 12,000ii 13.000.1 D I 'i 1.40 1.33 1.32 1.23 12,500*- 13,500 : ■* 1 or\ + 1. 37 1. 30 I! ft t: ±r l.2£ *"1.27 1.39 1.3E n t 1.36 LJ ;. 1. 35 * 1.34 Fig. 7. (continued) III -50 TABLE 9 Circular Velocity Period and Angular Rate (metric data; see Figs. 7 and b for English data) Velocity p |-| j5 1*7 6 )4« 6.141 8.137 B. 1 34 8.130 6.127 b ■ 1 7* 6. '20 8.117 6.114 fl. 1 in 8.107 8.104 8 . 1 00 8.097 8.094 8.090 8.0*7 Period 1 =3 1.286 1.289 1 290 1.291 1.293 1.294 1.29b 1.296 1.299 1.301 1.303 1.304 1.306 1.307 1.309 1.311 1.313 1.3M (.313 Anf. Vel. 4.390 4.384 4.878 4.872 4.866 4 . 8*0 4.634 4.846 4.842 4.636 *-830 4.824 *.8lft 4.812 4 . ftO* 4.800 4. 79« 4.788 4.7S2 *.?77 Velocity i G84 6.030 6.077 6 074 6.070 8.067 8.064 6.061 8.057 8 . 084 8.031 8.047 6.044 8.041 8.038 8. 034 6.031 8.026 8.023 8.021 Period l.*l7 1.319 1.320 1.322 1.324 1.323 '.327 1.326 1.330 1.372 1.333 1.335 1.337 1.338 1.340 1.341 1.343 1.343 1.346 1.348 Al*. V«l. 4. 771 4.763 4.739 4.733 4.747 4.7,1 4-736 *■ 730 4.724 4.7l8 * . 7l 3 4.707 4.701 *.693 4.690 4.66. 4.678 4.673 4.6*7 4.661 Velocity 8.018 6.0l3 6.012 8. 008 8.003 8.002 7.999 7.996 7.992 7.989 7.9£C. 7.9B3 7.96Q 7,970 7.973 7.970 7.967 7.964 7.9*1 7. 9(17 Period 1.330 t.331 1.333 1.334 1.336 1.338 1.339 1.361 1.363 1 . 36* ».!Ct 1 . 36P 1.369 1.371 1.372 1.374 1.376 1.377 1.379 1.381 Am V«l. 4.636 4.630 4.644 4.639 4.633 4.628 4.622 4.617 4.bl I 4 . t>03 4 . r; CO 4.594 4.569 4.383 4.578 4.572 4.56 7 4.362 4.33* 4.3B1 Velocity 7. 934 7.931 7.946 7.943 7. 942 7.9*9 7.935 7.932 7.929 7.9;6 7.9;7 7.920 7.917 7.914 7.910 7.907 7.904 7.901 7.698 7.893 Period 1.362 1-384 1.386 1.387 1.389 1.391 1.39; 1.394 1.396 1.397 1 . 7 C '' 1.400 1.402 1.404 1.403 1.407 1.409 1.410 1.412 1.414 If V«l 4.343 4.540 4.334 4.529 4.524 4.318 4.513 4.306 4.502 4.«97 *.492 4.4GU 4.48; 4.476 4.471 4-4b3 4.460 4.433 4.430 4.444 Velocity 7.892 7.889 7.B86 7.883 7.680 7.876 7.873 7.870 7.R fe 7 7.*b* 7 . S-. 1 7.35ft 7. 335 7.352 7.949 7.846 7.843 7.640 7.837 7.814 Period 1.415 1.417 1.419 1.420 1.422 1 . *24 1.423 1.427 1.429 I.43O t . * 72 l.4i« 1.435 1.437 t.4j9 1.440 1.442 1.444 1.443 1.447 Ang. Vel. *.*39 4.434 4.429 4.424 4.418 4.413 4.408 4.403 4.398 4 . 3v3 " . 3S£ 4.3&3 4.377 4.372 4.367 4.3&2 4.337 4.332 4. 34 7 4.342 VelOCtty 7.831 7 826 7.823 7.622 7.619 7.816 7.813 7.810 7.607 7.504 7.t'01 7.79& 7.793 7.792 7. 769 7.786 7.783 7.780 7.777 7.774 Period 1.449 i.43o 1.432 1.454 1.433 1.457 t.439 1 . 460 i.*i = 2 1.464 i.«l? imct 1.46" 1.470 1.472 1.474 1.476 r.*n \ . 479 i.*tti Mm- Vel. *-337 4.332 4.327 4.322 4 . J1 7 4.312 4.307 *.502 4.^7 4.292 *-iS 4.293 -.279 4. 273 4.268 4.263 4.236 4.25J 4.249 4.24* Velocity 7.771 7.768 7.763 7.763 7.760 7. 737 7.734 7.751 7.7*6 7. 743 7.742 7.739 7.73& 7.733 7.730 7. 72B 7.725 7.722 7. 7 19 7.7l6 Period I .482 1.484 1.486 1.487 1 . 489 1 . 491 1.492 1 .494 1 . 496 1 . 497 ' • *99 1.501 1 .503 1 . 504 1 . 30« 1 . 308 l . 3U9 1 .31 1 ' .513 1.614 ADf. Vel. *.239 4.234 4.229 4.225 4.220 4.213 4.210 4.205 4.101 4 . 1 96 •■ 191 4 . 1 £6 4. 182 4. 177 4. 172 4 . 1 68 4. 163 4.158 4. 154 4.149 Velocity 7.713 7.710 7.707 7 705 7.702 7.699 7.696 7.693 T.,.90 7.667 7.685 7.632 7.679 7.676 7.673 7.670 7,666 7.6«»3 7.662 7.689 Period 1.316 1.318 1.519 1.511 1.523 1.323 1.326 '.319 1.530 1.631 1.333 1.535 1.336 1.338 1.340 1.342 1.3*3 1.343 1.34' 1 . &46 AW Vel. 4. 144 4. 140 4. 133 4.131 4. 126 4. 121 4. 1 1 7 4.112 4. 109 *. < 03 4.09S 4. 094 4. 089 4.083 4.080 4 . 076 4.071 4.047 4.062 4 . OS* Velocity 7.636 7.653 7.651 7.648 7..>«3 7.6*2 7.6 39 7.63 7 7.634 7.631 7.628 7.623 7.623 7.620 7.61 7 7.614 7.612 7.609 7.606 7.6A1 Period I .350 I .352 1.554 1.533 1 .337 1.339 1.360 1 .56 2 1 .'564 1 .3(,6 1 .56 7 1.369 1.371 1.572 I .374 1.376 1.376 1.379 1.3R1 1 . o9J AdC. Vel. 4.033 4-049 4.044 4.040 4.033 4.031 4.027 4.022 4.,-il8 4 . d I 3 4.009 4.003 4. 000 3.996 3.992 3.967 3.983 3.978 3.974 3. 9?0 7.601 7.398 7.395 7.392 7.590 7.587 7.58* 7.58! 7.579 7.576 7.373 7.370 7.566 7.^65 7.362 7.360 7.337 7.334 7 . ?13 1 7-849 1 .334 1 .386 1 .388 1 .590 1 .391 1.393 1.595. 1 . 3 -:■ 7 1 .596 1 .600 1 -602 1 .603 1.605 1.6U' 1 . WJ9 l.oiu 1.612 1.614 1.616 1.617 3.966 3.96' 3.957 3.933 3.948 3.944 3. 9*c< i.vJfc. 3.y31 3.927 3.923 3.919 3.914 3.910 3.906 3.902 3.898 3.»93 3.B69 3.883 Velocity 7.546 7.343 7.341 7.338 7.533 7.53! 7.530 7.52? 7.5.25 7.522 7.*<9 7.517 7.314 7.311 7.309 7.306 7.303 7.301 7.498 7.493 Period 1.619 1.621 1.622 1.624 1.626 1.628 1 . 629 1.631 1.633 1.633 ' ■ 6Ji> 1.638 1 . o*0 1,642 1.643 1 . t>43 1.t>47 1.649 '.630 1 . 6B2 3.881 3.877 3,873 3.966 3.364 3.860 3.BS6 3.652 3.5*8 3.344 3.640 3.836 3.831 3.827 3-823 3.819 3.613 3.811 3.807 3.803 Velocity 7.493 7.490 7.487 7.485 7.482 7.480 7.477 7.*7* 7.472 7.469 7.466 7.464 7.461 7.439 7.436 7.433 7.431 7.448 7.446 7.443 2 Period 1.634 1.636 1.637 1.639 1.661 1.663 1,664 1.6-6 1 . toiiW 1 . fa 70 1 . 67 1 1.673 1.673 1.677 1.678 1.680 1.682 1.684 1.663 1.687 *" AJ*. Vel. 3-799 3.793 3. 791 3.787 3.783 3. 779 3. 77? 3.77 1 3.7 b 7 3.763 3.739 3.755 3. 751 3.748 3.744 3.740 3.736 3.732 3.728 3.724 § Velocity 7.441 7.438 7.433 7.433 7.430 7.428 7.425 7.42 3 7.420 7.417 7.413 7.412 7.410 7.407 7.405 7.402 7.400 T.397 7.394 7.392 Period 1.689 1.691 1.692 t 694 1.696 1.698 1.699 1.701 I . 703 1 . 705 1.707 1.708 1.710 1.712 1.714 1.713 1.717 1.719 1.721 1.722 ** AM Vel 3.720 3.716 3.713 3.709 3.703 3.701 3.697 3.693 J.6B9 3-686 3.682 3.678 3-674 3.670 3.6&7 3.663 3.659 3.633 3.632 3.648 o Velocity 7.3u9 7.3B7 7.384 7.382 7.379 7.377 7.374 7.372 7.369 7.367 '.■»* 7.362 7.359 7.337 '.33* 7.332 7.349 7.347 7.3*4 7.342 2 Period 1.724 1.726 1.728 t . 730 1.731 1.733 1.733 1.737 1 . 736 1.7*0 1-742 1.744 1.7*6 1.747 1.749 1.731 1.733 1.734 1.736 I - 738 p Anf. Vel. 3-6*4 3.640 3.637 3.633 3.629 3.623 3.622 3.618 3.61* 3.6| 1 3.607 3.603 3.600 3.396 3.392 3.389 3.383 3.381 3.378 3.674 § Velocity 7.339 7.337 7.334 7.332 7.329 7.327 7.324 7.32 2 7.320 7.317 7.313 7.312 7.310 7.307 7.303 7. 302 7.300 7.297 7.295 7.293 Period 1 -7f.O 1 .762 1.763 1.763 1 . 767 1.769 1.770 1 .772 1.77* 1.77* 1 . 778 1 . 779 1.781 1 . 783 1 .783 1.787 1.788 1.790 1.792 1.79* *" Am Vel 3.570 3.567 3.363 3.360 3.536 3.332 3.349 3.543 3.3*2 3. 338 J. 333 3.331 3.527 3.324 J. 320 3.317 3.513 3.310 3.306 3.303 §Veloc Perlo Am. fVelc Perl Am. Velocity Period Ar«. Vel. 7.2 90 3-499 7.2B6 1 .797 3.496 7.285 1 . 799 3.492 1,263 1 .601 3.489 7.280 1.8Q3 3.465 7. 3. 2T8 903 4£2 1.806 3.47* 3.475 7 . : 7 1 3.4L-6 7.-1. 1: 3.*t5 ?!4.':.i \'. + '*i 3! 434 ':\l?\ 1.*8 23 7. «*7 1 .624 3.444 7.249 1.826 3.441 7.247 1 .828 3.*37 7.24* I.83O 3.434 Velocity Period Am- Vel. 1 !s32 3.430 7.240 1.633 3.427 7.237 1.633 3.424 7.233 1 .837 3.420 7.233 1 . 8 J9 3.417 7. 3. 2 30 84I 7. 4 Id ?.4t* '.2-1 7.221 1 .*4S 3. 4 CO t . e-50 3. 3-"' 7 3. 39-t "■'■ ~-'A 3. 3W ' 7 . ^' 1 .637 7. ,"8* 7.207 1 .859 3. 380 7.204 1.861 3.377 7.202 1.862 3.374 7.200 1 .864 3.370 7. 197 1 .866 3.3*7 Velocity Period Am- Vel. 7. 195 1 .$68 3. 36* 7. 193 1 .870 3.36! 7. 190 1.871 3. 357 7. 188 1.873 3. 33* 7. 186 1.675 3.351 7, 3. 877 34* K 8 7 9 3. 344 1 .601 3. 341 \ . 662 3. 338 7. 174 1 .884 1. 335 7. 172 1 . 6fc6 3.331 1 .866 3. 323 1 . e*'o l. 323 7.165 1.892 3.322 7.162 1-893 3.318 7. 160 1 .895 3.313 7. 158 1.897 3.312 7.136 1 .899 3.309 7. 133 1 .901 3.30* 7.131 1 .903 3.303 Velocity Period Am- Vel. 7. 149 1 .904 3.299 7. 1*6 1 .906 3. 29b l!906 3.2 93 7. 142 1.910 3.2 90 7. 139 1.912 3.287 7. 1 . 1 '7 .914 . 2fe* 7. 135 1.915 3.250 7 . 1 3 J 1 .917 7. 1J0 1.9(9 3.27* 7. 1 28 1 .921 3.271 7. 126 1 .92 3 3.262 7. 124 3. -65 7. 121 I. 262 7. 1 19 1.926 3.259 7.117 1 .930 3.233 7.1i5 1.9J2 7. 1 12 t.93* 3.2*9 7.1 10 1.936 J. 246 7. 108 1.937 3.243 7.103 1.939 3.240 Velocity Period Am- Vel. 7. 103 3.237 7. 101 1 .943 3.23« 7.099 1.943 3. .31 K947 3.226 7. 09* 1 . 94 6 3. 225 7. 092 .9 50 ■222 7.090 1 . 95 2 3.219 7.r.:?E 1 .954 3. i 1 6 7.f*5 I . 956 7,063 1.956 7.081 1 .960 3.206 7.079 1 . 961 3.203 7.076 1 . 963 3.20C- 7.074 1 .965 3. 197 7.072 3. l'9* 7.070 1.969 3. 191 7.068 1.971 3. 188 7.063 1.973 3.183 7.063 1 .974 3. 182 '.061 1 .976 3.179 Velocity Period Am. Vel. 7.059 1 .978 3. 176 7.036 1 .980 3.173 7.054 1.982 3. 170 7.052 1.984 3.168 7.050 1 . 965 3. 163 7. .040 .987 . 162 7.0*6 1.989 .-.139 7.04 3 1.9C1 3. 156 1 . 993 i. 1-53 7 . 39 1 . 993 3. 150 7.037 1 . 997 i. 147 7.031 1 . 999 3. 1*4 7.032 2.000 3. 1 41 7.030 2.002 3. 1 38 7. 028 2.004 3. '33 7.026 2.006 3. 132 7.02* 2.008 1. 129 7.021 2.010 J. 126 2^012 3.12* 7.017 2.013 3. 121 Velocity Period Am- Vel. 7.013 2.015 3. 118 7.013 2.017 3. 1 13 7.01 1 2.019 3. 1 12 7.009 2.021 3.109 7.006 2.023 3. 106 7. 3. .0(14 .025 . 103 7.002 2-02u 3. 101 7.000 2.0*8 3.098 £.996 2.Q30 3.095 6.996 2.032 3.092 6.993 2.03* 3.089 6.991 2-036 3.086 6.989 2.038 3-083 6.997 2.O40 3.081 6.985 2.0*1 3.G78 6.983 2-0*3 3.075 6.981 2.0*5 3. 075 6.978 2.047 3,069 6.976 2.049 3.06' 6.9 74 2.001 3.U6* Velocity Period Am- Vel. 6.972 2. Q33 3.0*1 6.970 2.035 3.03B 6.966 2.056 3.053 6.966 2.038 3.053 6.96* 2.06 3.030 6. . 9b 1 . o,:2 .0*7 fa. 939 2.064 3. 0*4 2.0fa6 3.0*1 b.955 2.066 3.07? 6.933 2. 070 S.02C 6.931 2.072 3-033 6.949 2.073 3.030 6.947 2.075 3-023 t..9*3 2.077 3.025 h . 9* 3 2.079 3.^22 6.9*0 2.081 3.0l9 6.938 2.063 3.017 6.936 2.083 3.014 6.934 2.0B7 3.011 6. VJ2 2.088 3.U0S Velocity Period Am. Vel. 6.930 2.090 J. 006 6.926 2.092 3.003 6.926 2.09* 3.DO0 6.924 2.096 :.99S 6.922 2.099 : . 993 t .920 . 100 2'. 102 :.9T'0 2.9E7 . 9 1 3 2. 106 r.^64 6.911 2. 107 :.9ei 6 . f09 2. 109 2.979 6. 907 2.111 6.905 :!-"-7j 6.?03 2.H5 2.971 G.90I 2.117 2.968 6.899 2. 1 19 2.963 6-897 2. 121 2.963 6.893 2.123 2.960 6.893 2. 124 2.938 6.891 2.126 2-9B3 Velocity Period Am. Vel. 6.869 2. 128 2.932 6.887 2.130 2.930 6.884 2.132 2.9*7 6.832 2.134 2.9** c.680 :. 136 *• . 1 38 . 9 %■- '. . &76 2. 1*0 2.937 6.674 2.934 6.872 2. 1*3 2.931 6.870 2. 1*5 2.929 2.147 6.866 2.151 £,.*V>2 2. 153 2.918 6.860 i.133 2.916 6.B38 2. 157 2.913 6.856 2. 159 2.9H 6.834 2.161 2.908 6.832 2. 163 2-905 6.880 2. 1t>« 2.903 Velocity Period Am- Vel. 6.848 2.166 2.900 6.846 2.168 2.898 6.8*4 2.170 2.893 6.8*2 3.172 2.893 6.6*0 2. 174 2.390 t . ess 6.8 36 2! liO 2 . i s ;■ u.*j2 2 ■ *S0 5.830 2.16* 2.677 6.826= 2. 1&6 2.375 6.326 2. 187 6. £2* 2 . 1 6 9 2. ::7Q 6.822 2.191 2.86 7 6.820 2.193 2.663 6.816 2.193 2.862 6.616 2-197 2.860 6.814 2. 199 2.837 6.812 2.201 2.833 6.810 2.2o3 2.882 Velocity Period Am. Vel. 6.808 2.203 2.830 6.806 2.207 2. 847 6.80* 2.209 2.6*3 6.802 2.21 1 2.8*2 6.800 2.2'2 2.640 I. . 377 2-216 is . 7 9 4 2.-18 6.79 2 2. 220 2.r-J0 6.790 2-828 t.788 2.22* 2.625 6.786 2.6;3 -.' 7: ;- 2. 2 30 2 . 8 1 ■» 6.780 2.232 6.779 2.234 2.813 6.777 2.236 2.9H 6.773 2.237 2.808 6-773 .t.239 2.806 6.771 2.2*1 2.803 Velocity Period Am. Vel. 6.769 2.243 2.801 6.7<7 2.243 2.798 6.763 2.2*7 2.796 6.763 2.249 2.794 6.761 2.231 2.791 6. 2. . 759 . 25 3 t.. 737 2.255 2- 76>> 6.735 6 . 75 3 ..7&2 6. 751 2. 261 2. 779 6.749 2.26 3 2.777 6. 747 2.265 2.775 6. 7*C T.172 2.770 6.742 2. 270 2.767 6.7*0 2.272 2.765 6.738 2.274 2,763 6.736 2.276 2.760 6.73* 2.278 2.738 6.732 2.280 2.736 Velocity Period Am- Vtl. 6.73D 2.282 2.733 6.728 2.284 2.731 6.726 2.286 2.7*9 6.724 2.288 2.746 6.723 2.290 2.744 2. ;i:i 6.71v 2. 73';. 6 . 7 1 7 ii.715 i.2y8 2.753 u.713 2 . 300 2. 7J2 6.71 1 2. 702 2-730 6.70^ 2. 304 b. 707 2. J05 (,.703 2.307 -.723 6.704 2. 309 2.721 6.702 2.311 2.716 6.700 2.313 2.716 6.698 2.313 2.714 6.696 2.31? 2.712 6.694 2.319 2.709 Velocity Period Am- Vel. 6.692 2.321 2.7Q7 6.690 2.323 2.705 6.6B9 2.323 2.702 6.687 2.327 2.700 6.663 2.329 2.698 6. .663 . 331 . '=96 6.ESI 2. 333 u . f 7 2.335 t>,, 3 77 2. 337 2. =69 6.675 2.339 2.687 6.C7* 6.672 2. 3*3 2.682 L-.6 70 : . J4-. 2. '.-.60 0.668 -. 347 2.(s78 6.666 2. 349 2.675 6.66* 2.350 2.673 6.662 :-33: 2.671 6.661 2.334 2.669 6.639 2. 136 2.666 6.687 2.336 2.66* Velocity — Velocity in Kilometers per Second Period — Period in Hours Ang. Vel. Angular Velocity in Radians per Hour IH-51 TABLE 9 (continued) Anc. Vel. ■J. f,"i^ 2. J60 I'.iil 2.3^4 2. '^59 6.6*5 2.366 2.655 6.^48 2. 369 2.633 2*37.:. i.toSt S..64* 2. 372 2.649 6.t,42 2.374 2.b47 u.,,40 2 . _'"6 2.^44 6.638 2-376 2.642 6.637 2-3P0 2.6*0 6.635 2.382 2-638 6.6J3 2- 3S4 2.636 6.631 2. 38b 2.633 6.629 2. 388 2.631 2* "?90 2.629 6.626 2-392 2.627 6-624 2.394 2.625 6.622 2.396 2.623 6-620 2.39* 2.620 6.616 2.400 2 . t. 1 £ :!»o2 6 . 1 5 2.40, 6.613 2.406 2.612 6.61 ' 2. *G8 2.610 2.4|0 2-607 6.607 2-412 2.603 6.606 2.414 2.603 (j.604 2. 4I6 2.601 6.602 2.4,8 2 - 399 6.600 2.420 2.597 6.598 2.422 2.595 6.397 2.424 2-39.1 6.395 2.426 2.590 2.428 2.383 6.391 2.-29 2.5?6 6.369 2.431 2.364 6.586 2.433 2.382 6-586 2.433 2.580 6.0*4 2.437 2.676 6.5E2 2.57c *.3?0 2. 441 2.574 1-.579 2.443 2.37| c.377 2.445 2.569 6-575 2.44/ 2.567 6.373 2.449 2.5(j3 6.372 2.451 J. 363 6.370 2.433 2.361 6-366 2.453 2.339 6. 566 2.437 2.337 6-364 2.459 2.333 6.563 2.461 2.553 6.361 2.4*3 2.331 6.55) 2.463 2.345 ■i.357 2.467 2.347 6.53t 2 . < 69 2.344 6.534 2.471 2.542 6.332 2.473 2.540 6.530 2.473 2.338 6.649. 2. 477 2.536 6.3*7 2.5 3" 6.545 2.*ei 2.532 6.543 i. «9 3 2.530 6.542 2.495 2.329 6.540 2.487 2.526 6.538 2.489 2.524 6.336 2. 491 2.322 6.335 2.493 2.320 6.333 2,493 2.316 6.531 2.497 2.516 6.529 2.499 2.314 6.S28 2-301 2-312 6. 526 2.503 2.310 6,524 2.303 2.306 £.322 2.307 6.32" 2-309 2.304 5-319 2.311 2.502 6.517 2.313 2.500 6.^15 2.513 2.498 6.B14 2. 317 2.496 6.51- u.MO 2.521 2.49; ».508 2.523 ;.4*<o b.507 2.325 2.488 6.503 2.527 2.496 6.503 2.323 2.4 64 6.302 2.331 2.462 6.300 2.333 2.480 6.498 2.336 2.476 6.4 96 2.536 2-4 76 6.495 2.340 2.474 6.493 2.342 2-472 6.4-J1 2-344 2.470 6.469 2.346 2.468 6.469 2.546 2.466 6.486 2.33O 2.«64 6.484 2-332 2.462 6.463 2.534 2.460 6.461 2.336 2.459 6.47* 2.638 2-407 6.479 2 . 560 2.4?3 6.47t> 2.562 2.453 6.474 2.451 6.472 2.366 2.449 6.471 2.366 2.447 6.469 2.370 2.445 6.467 2.372 2.443 6.466 2.374 2.441 6.46 4 2.376 2.439 6.462 2.B78 2.437 6.461 2.560 2.433 6. 459 2.582 1.433 fc.457 2.394 6.453 2.58t 1.430 6.434 2.366 2.429 6.432 2.390 2.426 6.4 5c 2.392 2-424 2*594 2.422 6.447 2.396 2.*20 6.445 2.69* 2.41* 2.bOG 6.442 2.602 2.4(4 6.44Q 2.604 2.413 6.439 2.606 2.41 1 6.437 2.608 2.409 6.435 2.610 2.407 6.434 2.612 2.403 6.432 2.614 2.403 6.430 2.617 2. 401 6.429 2.M9 2.399 6.427 2-621 2.398 6-423 2.U23 2. 396 2.623 2. 394 6.422 •J.62' 2.392 6.420 2-629 2.390 f .419 2.631 2.388 6.417 2.63J 2. 396 6.415 2.635 2.385 6.414 2.637 2.383 6-412 2.639 2.381 O.410 2.641 2.379 6-409 2.643 2.377 6.407 2.6*3 2. 375 6.403 2.647 2.374 6.404 2.649 2.372 6.402 2.631 2- 370 6.400 2.653 2.368 6.399 2.635 2.366 6.397 2.63? 2.364 6.396 2.639 2.363 6.394 2 . 66 1 2.361 6. 392 2.663 2.339 6. 391 2.66b 2. 337 6.3S9 2.666 2.353 6.387 2.670 2.334 C. 396 2-l>72 2.332 6. 384 2.674 2. 330 6.382 2.676 2.348 6.381 2.678 2.346 6. 379 2.680 2.343 6.376 2.682 2. J43 6.376 2-664 2.341 6.374 2.686 2.339 6.373 2.688 2.337 6.371 2.690 2.336 6.369 2-692 2.334 6.366 2.694 2.332 6. 366 2.696 2. 330 6.365 2.698 2.329 6.363 2. TOO 2.327 6 . 3C- 1 2*. 325 6.360 2.703 2-323 6. 336 2.707 2. 321 6. 337 2. 709 2.320 6.333 2. 71 1 2.318 6.353 2.713 2.316 6.352 2.713 2.314 6.330 2.717 2.313 6. 149 2.719 2.31 1 6.347 2.721 2.309 u.345 2.7:3 2.30? 6.344 2.7;5 2.30 6 6.342 2.727 2. 304 6. 340 2.729 2.302 6.339 2.731 2. 300 6.337 2.733 2-299 6.33c 1-733 2.297 6. 334 2. 738 2.293 6.333 2. 740 2.293 6.331 2. 7*i 2-292 6.329 2.744 2-290 6.323 2.238 b.326 2-748 2.287 i=.325 2. 750 2.265 2*263 6.321 2.754 2.291 6.310 2.736 2.260 6 . 7 1 e 2.7^.6 2.276 6.317 2.760 1.276 6-313 2.762 2.273 fc.3l3 2.764 2. 27 J 6.312 2. 271 6.310 2.769 2.269 6.309 2.771 2.268 6-307 2.773 2.266 6.306 2.773 2.264 6.30* 2.777 2.263 6.302 2.779 2.261 6. 301 2.761 2.239 6. 299 2.783 2.236 6.29B 2.785 2.256 6.296 2.7g7 2. 234 6. 293 2.785 2.233 6.293 2.791 2.251 6. ;92 2.79* 2. ..49 6.290 2.796 2. 248 6.268 2.79 8 2.24^, 6. 267 2.600 2.244 6.265 2-802 2.243 6.284 2.004 2.241 6. 282 2.806 2.239 6.281 2.808 2.238 6.279 2.6*0 2.236 6.278 2.812 2.234 6.276 2.814 2.233 6.274 2.816 2.231 6.273 2-819 2.229 6.271 2.621 2.228 6.270 2.923 2.226 6.268 2.02S 2.224 6. 267 2.827 2.223 6.265 2.629 2.221 L-.264 2.631 2.219 6.262 2.633 2.216 6.261 2.9J5 6. 259 2.8 37 2.2 ( 4 6. 237 2.639 2. 21 3 6.-36 2.642 2.2' 1 6.254 2. 84* 2-210 6. 25 J 2.844 2.20* 6. 251 2.848 2.206 6.230 2.630 2.205 6.248 2.B32 2.203 6.24 7 2.834 2.201 6.243 2.936 2.200 6. 244 2.836 2. 196 6.242 2.860 2.197 6.241 2.662 2. 193 6.239 2.663 2. 193 6.238 2.667 2.192 6. 236 2.669 2. 190 6. 235 2.671 2. 169 b.233 2.873 2. 187 u.23l 2.675 2. 185 6.230 2. 184 6. 228 2.879 2 . 1 62 b.227 2.861 2. 181 6.223 2.663 2.179 6.224 2.886 2. 177 6.222 2. est 2. 176 6.221 2.890 2.174 6.219 2. 892 2.173 6.218 2.894 2.171 6.216 2.896 2. 170 6.213 2.898 2. 168 6.213 2.900 2. 166 6.212 2.902 2. 165 6.210 2.905 2. 163 6.209 2.907 2. 162 6.207 2.909 2. 160 6.206 2.91 1 2 . 1 59 2.913 2. 157 6.20 3 2-915 2. 155 6.201 2.154 fc.2 00 2. M 5 2. '32 6. i?e 2. 131 b. '97 6.193 2.i'26 6..D4 2.928 6. 192 2.930 2. 143 6.191 2.532 2.143 6. 189 2.934 2. 141 6.188 2.936 2. 140 6. 186 2.938 2. 138 6.183 2.940 2. 137 6.183 2.943 2.133 6. 162 2.943 2. 134 6. 180 2.947 2. 132 6. 179 2.94 9 2. 131 6. '76 2-951 2. 129 6. 176 2.953 2. i:s 6.17? 2.935 2. 126 6. 173 2.957 2.125 6. 1 72 2.939 2. 1 23 6. 1 70 2.962 2. 122 2-9 & 4 2.120 6. 167 2. -'66 2-119 6. 166 2.966 2.117 2.970 2. 113 6. 163 2-972 2. 114 I Velocity Period Aaf. Vel. 6. 161 2.974 2.112 6. 160 2.976 2. Ill 6. 138 2.979 2- 109 6.137 2,981 2.108 6. 133 2.983 2. 106 6. 134 2-983 2. 105 6. 133 2..S87 2-103 6.131 i.9ti> 2.102 6. '50 i.991 2.100 6. I4g i.V-jt 2.099 2*097 6. 145 2.956 2.O96 1 Velocity Period Anc. Vel. *. 132 1.017 2.0*3 «. 131 3.019 2.081 6. 129 3.021 2.080 6.128 3.023 2.078 6. 126 3.023 2. 077 6. 125 3.028 2.073 3! 030 2-Q74 6. 122 3.032 2.072 6. 121 3.034 2.071 6. 1 19 3.036 2.069 6.1 16 3.038 2.066 6. 116 3.040 2.067 g Velocity Period An*. V«l. 6.103 3. o«c 2.034 6- 102 3.062 2.032 6. 101 3.064 2.051 6.099 3.066 2.049 6.098 3.068 2.048 6.096 3.070 2.046 6. 095 3.073 2.043 6.094 3.075 2.043 6.092 3.077 2.042 6.091 3.079 2.041 6.069 3.061 2-039 6.0S6 3.083 2.036 I Velocity Period Aug. Vel. 6.073 3.103 2.025 6.074 3. 103 2.024 6.072 3. 107 2.022 6.071 3.109 2.021 6.070 3. 11 1 2.01V 6.068 3. 1 14 2.018 6.067 3. 1 16 2.017 6.065 i. 1 18 2.015 6.064 3.120 2.01* 6.063 3. 122 2.012 6.061 3. 124 2-011 6.06D 3.126 2.010 i Velocity Period Aug. Vel. 6.047 3. 146 1 . 997 6.046 3.148 1 .996 6.044 3. 130 1.993 6.043 3.152 1.993 6.042 3.133 1 .992 6,040 3. 157 1 - 990 6.039 3. 139 1.989 6.036 3. 161 1.968 6.036 3.163 1 .986 6.033 3. 163 1 .983 6.033 3. 168 1 .984 6.032 3. 170 1.982 1 Velocity Period Anc. Vel. 6.020 3. 189 1.570 6.018 3. 191 1.969 6.017 3. 194 1 .96 7 6.016 3. 196 t.ofie 6.014 3.198 1 .965 t>.U13 3.200 1 . 963 6.01 1 3.202 1.962 b.010 3.203 1 .9b1 6.009 3.207 1.939 6.007 3.209 1 .936 6.006 3.21 1 1.937 6-003 3.213 1.933 § Velocity Period Anc. Vel. 3.993 3.233 1.944 5.991 3.235 1 .942 3.990 3.237 1.94 1 3.96 8 3.23? 1.940 3.967 3.242 1 .938 5.986 3.244 1 . 937 5.964 3.246 3.96* 3.248 1 . 934 3.982 3-230 1 .933 6.980 3.233 1.932 3.979 3.255 1.930 3.97B 3.257 l.»29 § Velocity Period Anc. Vel. 3.966 3.277 1 .918 3.^64 3- 279 1.916 3.963 3.261 1.915 .*! 263 1.914 3. 360 3.2S3 5.939 3-286 1.9lt 5. V*6 3. 290 1.910 3.?36 3.95S 3-29* 1 . 90 7 5.934 3-296 t .906 3.952 3.299 1.905 3.931 3.3 1 1 .»04 i Velocity Period Anc. Vel. 3.939 3.321 ' .892 3.938 3.323 1.891 3.937 3-323 1.890 3.933 3.327 1 . 8S3 3.934 3.329 1 .86' 3.933 3.332 1.896 5. 931 3-334 1.683 3.93C 3.336 1.88J 3. 929 3. 338 1 .882 8.927 3.341 1 .661 5.926 3.343 T. 880 S.«23 3.3*3 1.878 i Velocity Period Anc. Vel. 3.^1 J 3. 3t3 1 .96 7 3.9i; 3. 367 1.S66 5. 91 1 3.369 1.863 5.909 3.371 1 .864 3. 90S 3.374 1 .862 3.907 3.376 1.861 3.903 3.376 1.860 3.904 3.360 1 .839 3.903 3.383 1.838 0.901 3.383 1.856 3.900 3.387 1 .833 3.899 J. 309 1 -854 1 Velocity Period Anf . Vel. 5, 987 3.409 1 .643 s.eafa 3.41 1 1.84? 3.695 3.4U 1 .84 1 5.684 3.416 1.639 5.882 3.418 1 .838 3.681 3-420 1.837 3.880 3.423 1.836 5.878 3.423 1.835 3.877 3.427 1 .633 0.676 3.429 1.832 3.873 3.431 1.83J 3.873 3.434 1.830 1 Velocity Period Ang. Vel. 3-3*2 3.434 1 .819 5.861 3.436 1.818 5-839 3.438 1.817 3.838 3.«60 1 .816 5.857 3.4«3 1-613 5.856 3-463 I.813 5-834 3.467 1.812 5.853 3. 4 69 1.81 1 3.632 3.472 1 -810 0.851 3.474 1 .809 3.849 3.476 1.808 3.848 3.478 1.806 i Velocity Period Ang. Vel. 5.837 3 . 499 1 .794 3.836 3.301 1.793 5.834 3.303 1.794 5.333 3.503 1.792 3.B32 3.308 1 .791 3.831 3.310 1.790 3.829 i.312 1.769 3.826 3.314 1.788 3.827 3.516 1.T87 8.826 3.0t9 1.786 5.824 3.321 1 .784 3.823 3.323 1 .783 1 Velocity Period Anc. Vel. 3.812 3.343 t.773 3.811 3-346 1.772 3.810 3.346 1.771 3.806 3.330 1 .770 3.607 3.332 1 .769 5.806 3.533 1.768 3.803 3.057 1.766 3.803 3.339 1.765 3-802 3.362 1.764 8.8OI 3.564 1.763 3.800 3.566 1.762 3.799 3-568 1.761 i Velocity Period Anc. Vel. 3.788 3.389 1.731 3.786 3-591 1.730 5.783 3.393 1.749 3.784 3.393 1 .748 3.763 3-398 1 .746 3.781 3.600 1.743 5.780 3.602 1-744 5.779 3.604 1.743 5.776 3-607 1.742 0.777 3.609 l.74» 3.775 3.61 1 1.740 3.774 3.614 1.739 3.Q0Z 2.0^3 3.004 2.0?2 3.006 -'.009 2.089 3.011 2.087 3.013 2.086 3.015 2.084 b. 1 13 3.045 J. 04V 2.C62 t . t 1 1 3.049 2.U61 6-109 3.031 2.039 6. 106 3.033 2.036 6. 1O6 3.033 2.036 6. 103 3.006 2.005 6.035 3.0B6 2.035 6.064 3.090 2.034 6.002 3.092 2.032 6.O81 3.094 2.031 6.079 3.096 2.029 6.078 3.098 2.028 6.077 3. 101 2.02* 6.057 3.131 2.00 7 6.056 3.133 2.006 6.034 3.133 2.004 6.033 3. 137 2.003 6-031 3.139 2.001 6.050 3. |*2 2.0OU 6.049 J. 1*4 l.**9 6.029 3.174 1.980 6-028 3.176 1.978 6.027 3.178 1.977 6.023 3. 161 1.973 6.024 3.183 1.974 6-022 3.183 1.973 4.021 3. 187 1 . tfi 6.003 6.002 6.001 3.999 5.998 3.997 3.993 8.*+4 3.213 3.218 3.220 3.222 3.224 3.22* 3.229 X.2>1 1 . 954 1.953 I . 951 1 . 930 I . 949 I . 947 1 . 9*6 1 . 949 3.976 3.973 3.974 3.259 3.261 1.264 1.928 1.927 1.923 3.930 3.948 3-303 3.303 1 . V02 I . 90 I 3.947 3.J07 1 -900 3.972 3.266 3.946 3.3i0 I .898 3.924 5.922 3.921 3.»20 3.3*7 3.349 3.332 3.33* 1.877 1.8/6 1.8/3 1.873 3.698 S.B96 3.893 3.391 3.394 3-396 1.833 1.631 1.830 3.822 3.323 1.782 3-89* 3.398 1.849 S.9T0 3.9*8 B.*«T 3.270 J.2T2 1.274 1 .921 1 .920 I .919 5.943 3.942 0.941 3.314 3.JI4 3.3(6 1.896 1.893 1.893 3.917 5.91* B.»I4 ».J3» 3.3*0 3. Ml 1.871 1.670 1.8*9 3.892 3.691 3.890 B.**9 3.400 3.403 3.405 J.407 1.846 1.647 1.845 1.644 3-97| 3.2*6 1.923 5.944 3-312 1.897 1.H7J 3.872 3.871 3.870 3.868 3.43* 3-438 3.440 3.4*3 1.829 1.627 1.826 1.823 5.867 3.86* 3.6*4 3.445 3.447 3.4*9 1.624 1.623 1.822 B.6»3 3-462 I -820 3.846 3,»44 3.843 3-842 3.483 3.483 3.487 3-490 1.804 1.803 I.B02 1.801 3.841 3.492 1.799 3.939 3.494 1.796 3.821 3.819 t.Bie 3.B17 3.81* 5.813 S.813 3.32b 3.330 3.332 3.534 3.33V 3.339 3.041 1.761 t.760 1.779 1.778 1.777 1.773 1.774 3.796 3.793 3.794 3.792 3.791 3.790 5.789 3.37J 3.373 3.377 3.380 J. 302 3.384 3.88* 1.739 1.738 1.736 1.733 1.734 1.733 1.752 3.772 3.618 1.737 3.767 3.766 0.765 3.627 3.629 J. 612 1.732 1.731 1.7 JO Velocity — Velocity In Kilometers per Second Period — Period in Hours Ang. Vel. — Angular Velocity in Radians per Hour III- 52 TABLE 9 (continued) £ ^ l Tl tf ^- 7 "- ^ 7M '- 759 5 - 7Hf - " : -"* —■' ''■ 7 " : 3.7-«7 f.*44 ;■■'■*: 5.--0 T..T-7 ■ . 7 J- -. ,- ^ 5.7'0 -.7:, 5.7> 3 7; 3 5 7". 5 7,3 8 Period 3.63* j. u3B 3 . b4 j V) *fl 3., .5. j..=.-: j. b M *.„.,<, j. t7 o 7....-f 5..--:? 3.6*4 ,-. t ^. 7., -.* 3.6^6 >. ?n- V 707 3" -17 3 7}b 3' 72? - Ang. Vel. -.1.497 41.445 41.393 41.341 41.190 41.23: 41.167 41. Ufa 41.035 4 1 . U 74 4 . ■ t. * 40. 93. -U.&r. 40.;^ -O.-l -,,.7/, 4,j. £& 4G..30 ,0.580 4U.S3C g Velocity ..^ u . 5 . 7I , «, <TI) 5-T09 =. 70T 5>7o4 5-T02 . 7u(l i|ii _,, T ^.,, 5^,,, 5ibl0 ^ . C| r| 5. Period 3.7:5 3.73o 3.734 3. 739 7.744 3.748 3. 753 3.757 3.~,2 :.7 L , 3.77. 7.77, 3. 730 3.7*5 7.7V9 . . 7 > j.T^ 3., 03 3.608 3.S.3 - Ang. Vel. 4U.4C0 40. *31 40. j- i 4U.331 40.18.' 40.235 »0.ltj 40.134 -4,1.-1 . ob^ 40. --'36 3^'. "if:- 39. 'J ^ •'.>--'i_i J -'.a 4. _-9.,"-ii 3'.i.'-.5 j -\ u. ' ■ 7 3V.649 39.601 35.Sfj Velocity Period 5.b:'l 5.62V 3. t9i a « Ang. Vel. 3V.505 39. "57 J ■:■.■* t 39. 362 39.3.5 39.2i-7 39.220 39. | 7J .,v.l';,_, 39.07;, 7' 7 ' . o '2 3 - . .' ^, 5 -' ! ■" 3 ' := ! S, 9 2 J-!;-i^ Jt:.'?^' Jt! 75: JB.70t 38.'b60 5W " 6 1 4 g Velocity •.<,:■! 5.6.2 5.62o 5 (-■ i 8 5..16 3 n j •■ ,-. i 5 L r,"' 5 ,,j7 «, ,,-,5 5 , n ■■ ^ ,.,,-, «• = ,._ = ..-,, ■= c <e Period 7 . ■;, 1 o 7 . 9 1 5 3.919 3.924 3 , 92 Q 3 . _i ; ~ 3 . '" 3t. -■ . "■* i 3.947 j . '^i. 1 ? . ■ .' 7 3- . -, t . ^ ?' ■!■ „ ". 3 ' .-, - 1 V ■-. -* ?" ■", e. , -'i,,^. >'■-;, .-, t " r'n ?;,., - Ang. Vel. 36 . 5.,8 ?c,.5.2 3f.47 u 39.431 3£.3S5 33.340 3.T-.194 38.149 36.204 36 . 1 5& 3J.lt.? 't . 0,,*. ?■:. . o.'4 3?'. '-■". ?.-.■', 34 --7.3:".<i '7. ..45 -7. £00 • 7 . 75 " 37.7.2 g Yf 1 ^ '-^ S.57& 5.576 5.574 5.572 ,.570 5.5.7 5. ',5 5 . ,,,3 5.=,-. -.V- 5.-5 7 5. 5 ^ 5.,^ , 5 ,o -■ ; ^ ^54C 5 544 5^ u ^,3. „ Period 4.003 -.0q& 4.013 4.01? 4.022 4.027 4.^; 4.oJt * . r - 1 ,."», 4. .■.-. ,. . t.-. -.,-,, u ».ci, ; uri 4" < : ' "4 i'07- 4"os- ;"n&- i'c^ " Ang. Vol. 3'.i.(.-3 '7.6;* 37.5so 37.536 37. 4 y2 37. 446 37.4L4 37.361 77.717 3 r . ; Tt ;'..."■ 77 . 1:7 77.1«4 37. ,1.11 ;;,,■-,, 77.m5 3 L ,!'J7; "6 9^0 3i.!f87 - u ' b4< § Velocity 5.537 5. -535 5.533 5.531 5.529 5.5;7 5.515 5.522 5. 'In 5 . 5 1 3 c .ml ' . =; 1 - =. ' 1 "■ «; -. t n - ',;? - 'n, - ■= 1 4 5 501 - 4--. 5 4 -.? Period ».o:»£ 4.13: <♦. 1 7 4.1 1 i 4.116 4.121 4. lib 4. 1 3t -. 1 35 4. 140 4. I--, ... 150 -.154 4 1 53 4 Il4 , Kv 4" 1 7 1 4 ' 1 76 4 " 1 & f 4 ' T-.B - Ang. Vel. 3l-.E:o: ru.759 3 U .71 7 3G.C75 36.633 J.i.7^0 3.^.546 36.506 36. ,65 36.423 .'o.TI 3l . :tu 3i : . 2='6 3b. .57 .^..,5 3(. . 1 '4 3^133 3g!u9I 3C ! Cj;.£i 3<,'.(Z? g Velocity --..,"5 5.493 5.4?t 5.489 5.487 5.4^5 5. 4&-. 5.481 5. 4 7^ ^.477 '..,."-4 5.47" 5 4 7n ■■,.!* 5 j,f. 5 4.4 - 4)i - - 46 n . ..^ ^ 4 . ■ « Period *. is>; «.iv ». 2u2 4.2u/ 4.212 4.itt> 4.^21 4.:^b 4.131 4.^.5 «..*u 4.145 4. /'So 4.255 4.25' 4.:„4 4' «. ( n 4 '>;, :*—'." '" - - Ang. Vel. 35.LV? 75.918 35.9*7 35.946 35.30b. 35.7i,5 35.725 35.1-94 35.b44 ^.L.114 .-' . 5 L .4 -5.5j4 35. 4f-,^ 35 444 35 . «04 = c . ! b 4 -^'3-4 '-."e*-, Velocity 5.454 5.452 5.450 5.446 5.446 5.444 5.44; -J •* Period 4.28^ 4. 29" 4. 2^8 4. 303 4. 307 4.312 4. 317 2 2 Ang. Vel. 35.li.-o 35.127 35.086 35.048 35.009 34.970 34. ?3] B g Velocity V414 5.412 3.410 5.403 5.40.; 5.404 5.402 9 w Period 4.;t:» 4. jay 4.394 4.3-19 4.4O4 4.4o ; J <.4i, 5 " Ang. Vel. 34.?.-, 3 34.355 34.31& 34.260 34.242 34.205 34.lb7 15.. .E.5 35. 2«5 35. 2jC 5. 420 =. 41 ti 5. 41f. 4.370 4. 3,-;, 4. 2&0 '4, 5de 74.46--' 34. 431 5. 3F0 5. 37fa 5. 376 3.-5 6 33^ 72; _j.3io.ft5 g Velocity 5.^4 5. 372 5. ?7t 5. 369 5. 367 5. 365 5.363 5.36 1 5. 35 > 5. 357 5 . 75 5 5. 35? 5. 351 5. 74' 5 . 34 7 -. 345 5. 344 -.34. 5. 340 " Ang. Ve). 73.6-S- 33.612 33-575 33.539 33.503 33. 4 b C 31.43Q 33.394 J3 . 3Z6 31.3U 33.2tC 33.'250 Js'.H* ■■ 7 '. f ,' :- i:'.i A i 3 1 ! 1 ;i ; J *. r 71 ? 3 ! n n* j^ior.i' Velocity 5.77k- 5.33* 5.332 5.330 5.328 5.3^6 5.324 5.J23 5.721 5.319 Period 4. 57> 4. 5 34 4. 56S' 4.5 94 4.59 9 4. 604 4. 6 09 4.614 4.619 4. 6 24 Ang. Vel. -■l.-.' T '2.695 32.360 3^.324 32.789 32.754 32.719 32.685 '2. 650 32.615 15 3 70.32 7 5b 5.15? g Velocity «,.-r. 8 ,. :3l ; S-: c, 4 ? _ : .^3 - i:?1 ,.2^9 5.297 5.235 5. .'63 5.2 8 i 5.^n 5. 27fc -.17, 5. T-. - 7. 5 ^0 5 -,' « Period 4. L .7£ 4. '^93 4.^3^. 4.^3 4.693 4 . 703 4. 70 7 4.712 *. 71 - 4. 7 27 ■.. 717 4 . 7 3 2 , . 7 _-. 7 4. T». 4.-47 4. 752 4! 757 — Ang. Vel. 32. .37 ^2.203 32.1b-' 32.135 32.101 32.067 3;. 033 32.000 3 1 . ''t >■ 31.^73 7 1 . e'^ 71-?6b 3l.Si; 31.7''9 -1.7,6 3^.737 3 1 . C 9' f Velocity -,. :6 , 5.25^ 5.258 5.25'i 5.254 5.252 5.250 5.24-9 5.24? 5.245 '..24= 5 14] 5 "■ 1 ■:, ' 7. «■, -. , .- ,-, 4 ^ ■, ,-. Period 4.777 4.7^2 4.737 4.792 4.797 4.602 4.807 4.912 4 . s 1 7 4.t21 4 . I. - 4.672 4.837 4.*4. ..^47 ,*iv 4''-57 — Ang. Vel. - 31 ■ -"■ 7 ?' ■ '35 31 . 502 3' .4b 9 7' . 436 31 .404 3. . 37. 31 . 339 31 . 30 6 31 . 274 -.1.2-1 7 1 . 1 ;, ■ 3 , . ' 77 71.1,1, : : . , 1 3 71 . ,,;,* - , . o,^ g Velocity 5,-s 5.223 5.2-2 5.120 5 . 2 1 3 5.21b 5.114 '.;I7 5.211 5.2.1.-' 5.2,7 5.-1^ 5. ."'4 5.202 5.200 5.v,-L- 5.197 2 Ang. Vel. 30. -1.1 30.469 30.85'- 30.826 30.7S'« 30.7^3 3(7.73i 3O.7O0 .'H'. ,_,<_■-■ 7'9.L-=7 -li.i..0l. 30.575 ;ri>,44 l\-'.',:'i ?\/.*zl 'C^'l 5.0! 4.;.i o Velocity 5-ISk.i 5.t8d 5.16,- 5.184 5-1S3 5 . 1 s= I 5.17s 5.177 5.1.-6- ;..I7. *-■ . 1 7 1 5.170 5. ■■>--■ 5 1,7 ' , t = «, 1-4 », ,. -. <0 Period -.-'77 4.9il 4.967 4. -.93 4 . i-k ^.0,i- 5.00a 5.ol3 5..H& 5.02" "."'.-- 5.633 5.>''J4 ..',,,1 '.! u s s!n'^ 5*&5f — Ang. Vel. 3U.29U. 30.26t. 30.235 30.205 70.174 30.144 30.113 30.033 30.:-l. 70.0.2 l"'."'''l 29. 9 U ; 19.^3. 1'"'. ,0. 2 S . ■= ?2 .i.t\; . .. :. 1 , » '* » » *0 M. « 70. H. W. 140. IN. .» 1» 140 ISO 1« o Velocity 5.155 5.15; *..t5- 5 150 5 . ■+;■ 5 1 4 ,. 5 >4= = i»f « 1 - 1 ■■ 1 ' 1 -^ e ■? «. o Period 5! 07"' 5! 084 5.. 0i9 5^o94 5! nni 5! 1 n, 5" t rr, ; .| , "- - . n -1 5! 1 ;i, ' ' 1 ■ '■ 5" i ?5 «" ■^ •' 14- - * ' «m • " ! ^ "'■ '-^ 2 Ang. Vel. 2'.b9. .9.663 I' ;■ . ,3 7 7 2 9.1,., 14 ."'.574 /'.'■^, I ■. - 1 5 2 ? . ., £ 6 .^457 ,^4*7 .■:■!."- ,"i!jo9 ;-''-40 . ■] \ 7 ■ t ■■!''i'' : ' vl'-^f -■V'- 4 Velocity 5 ,-■■ - 1 1 ■, •, t . . «■ n . . ,,„ «; 1 ■ = -11, -,,.., - ,,.. . .,-, . , c .-. c , - S Period -5- : 1 = 1 5.,;,. .;:,>.■; •;,-,; -: :J1 5_,-„; s ; : , t -j;^, <_, ^.;.^ 7 r:'_r ^;^ 7 ' ^;^! ,°-^ r^; =-'-._'^ ? - ' 1 '* -^ Ang. Vel. ."■■103 ;-".>-. 1 j .■-,,0 r i 19.022 1S.-.94 l--.. r ',5 1.-.977 ;■ ^. . --i £ .f;.. : ; .0 26.652 ..],,..j .■■_ . 795 2&!7(.-7 2 ^ ' 7 3 ") 1S.7M ■. o Velocity 5 rno ^ np, 5 ns» 5 ,-,97 5 ,-.01 ^ ,-,-;.-, = .-,-- c ,-,,■. ■= n-^ t r- T E r , , , 2 p.r.od ;:.;1 -:.88 ?:°;i ,:,fe ,:„; ^:,;; ,:,;. .:'.;," ,•,;: s:':, ;-';;; :- c ;;° ;•■ ; :■.;;- ;-^ '-^ '■," '-°'° =■«» ^.^ 2P.,;, 2e.U'4 .£,.077 28.05.J ,6.023 a - Ang. Vel. g Velocity ,.,,, .. .„ _...._._ ,.„,.. , m _,,,_,_, , , _ » Period , , B; 5.3,; ,.3»- 5. , , ,.„, ,..,, ,..,, ..... ,.,,„ .„.,. ,.,,. . . ., , ..„„ ^ * ; ^ ^-J" ^^^ -.J.| - 0.J t.0|. - Ans.V.1. ■■'.« .7.5.5 27.».. .7.5,, ,7. t6 , ,7.3.2 , ,' . 8 .V, 27. M , 27.7 8 , .7.7,, ,,. 7 ;. 27.7„ : 2,.„7 b ,,.,«, : ..^ 3 ,..,,: ,?;,„ ^^^U ^^iSTs 27!^ Q Velocity •; ,-,■* ■, m r , - . «,-,', b , , , a * r . r- .,!.■; c.,- c-l.- c-.:t =-,-^ 2 A«.V.l 27.. M 27. -« „..,. 27.383 27 . 3.2 ,7.33. 27.3,5 27. , B - 27.2,5 27.233 Ua ^. ^ ^1U ,V\11 J\^ ^ zV.ll', iV.lll ^ ^1^ § Velocity 4.991 4.990 -.93- 4. j- 1 - - ■"— = - -- ' "" " - - '- - - "■ - — Period 5.595 5. GOO 5.O.05 5. u ..„. ..... , „_, . „_.,. . u , . ^.^^ 3 ,^ u ,,,„, - , b „ . ,--. . , 7tl _ - Ang. Vel. 26.953 2b.?29 26. 902 26. 8 77 it.^l 2„.^27 2b. fe 02 26. 7 77 2 6.7 5 2 2&.721 26.70^ 26.677 26.652 26.626 26^03 26 ' 578 2e'«3 «' I i» 5:SS ;:'-2? 5:^ 5:^ ;:?3? ;:?H ;:?!; ; : ?1! 5:?:; ;:K ?:?;! ; : ; S :••» ;;- S: »; ; : - -» j-» "; : «j -- 26.1*0 26.1,6 26.052 26.066 26.04. 2d.02u 25.557 An«.vei. ,-i:;M j;:.;, =;:4i: 3 r.,«; 3 ;:.;; ,;:.ii :;•,«: ,;•;:; ^r«; ,>■;" ,..•:?; .?■?;;.'■!" ,!-? : : .^- 77 : _' : "" '•!" .5-«" '•■"' =:«^ 3 | SIST S:S ;:5» ;.;?; ; : ;is ;-», ; : ^ ; : su -?» ; : -| ; : ;y ; :S = ; : - -- ..... : . s; ..... ...=« ...» ..«,, ..502 E 2 Anf. Vel. 23.573 25.545 25.525 25.502 25.676 25.85. 25.831 25.607 25.764 25.760 ,5.737 25.7,. n.hu li.lii i!!,.;; ii'.t% 2?.j' 7 2?'SyJ '',?, 4's" □ 8 ™7?', ty *-" M •'' CS ' •• 897 .-»" .-8 9 . .- 8 » 3 .■»" .- s '0 .-See 4.887 ..88.- 4.864 ..883 ..S8, 4.886 4.878 4 877 . »7. . „. . „,. y g Period 5. .1. n.5,B 5.923 5.325 5.534 5.535 5.945 5.950 5.955 5.961 5.-I.6 5.571 5 577 5 5a . 5 o. -• Ang. Vel 2,-50' 25-.8i .5. .79 .7.4m ,5. 41.- 25-J90 ."5. -*6 ^ .r..j44 25.3,1 25. .99 25.276 25.25? 25!2il 25.208 ;','. g Velocity 4.871 4.87U 4.866 4.867 4.865 4.864 4.662 4.661 4.858 4.858 ..857 4.8,5 ,.8,4 4 85, 4 83, 4 8.9 . *.o . c.7 . ~.« . ^ g Period 6.0,0 „.0„ 6.030 6.036 6.04, 6.0.7 6.052 6.057 6.063 6 . 0„3 6.073 6.07- 1 ^"o t.Ut t.Vol '2 t"] t *,$ 1 ' *" 2 Ang. Vel. 25.05, .5.028 25.006 24.984 ,4.96. 2.. 939 24.917 24.89, 24.873 2,. 851 ;..e,<. 24.807 24.785 24:" 5 3 2. . Vt 1 2.. 7?S :,'S A'.VA £'.Hl A'.lll § Velocity ,. 8<2 .. 8 ,, 4 „,, ,. aJ8 37 . .„_„ tJ 32 Period 6.127 6.133 6. ,38 6.,,, 6.149 6.155 6.160 6.165 o.,71 6 . , 76 6. ,82 . . Tj*7 „ 19, * ,TO o''o3 ^''n' °t. ?'? ! S ?' \l *■?" E Ang. Vel. 24.6,0 24.588 24.367 24.5,5 ;,.5,3 24.502 .4.480 ,4.459 24.437 ,4.4,6 ,..3-., 24.37! 2.. 3,2 , . ! 3 3 i!.:,." ^ijse 24.'2..7 .VM, £.l£ 2^203 § of'^ 17 "■ S ". 4 "■?'? *■?" "■ 8 '° "■ S ° 8 '- 807 *- 806 •' 804 *- 303 .- 8UI "■ B0LI .- 7 "' 9 '•'" .-'96 ..79, 4.753 . 752 4 790 . 789 4 788 f. Period 6. ,36 ...., t .,47 6.2,2 6. ,56 6.2 t 3 6- 26 5 6.274 ».,79 6.2 8 5 ,.„ 6 . 2 9, 6.3o, 6.307 6 7,2 6 3,6 6 323 ' , ' , ,! ?'-!„ - Ar*.V.l. 24. ,82 24. ,6, 2.. ,40 24. ,,5 ,4.098 2.. 07, 2..056 24.035 „.,„. .3.9.., 23.573 .3.2-2 22.97, 23.9,0 J.lUo 2 5:»65 23.I" iVAil 2^* 23\'l° S S'^' 7 '• 7Bt 4 - 78 ' *' 78J *- 782 .■ r8 ' 4 - 77 ' .- 778 4 - 777 «- 77 5 4.774 4.773 ..77, ,.770 4 76b 4 767 . 7 66 . 76. . ,6 3 . „ ■. . ,-„ J. Period 6.3.5 6.350 6.356 6.36, 6.367 6.37, 6.376 6.383 6.389 6.394 „ . . 6.4D5 6.4, l.<,iZ „ I,, 1 ,"j V,%i t'' , T.l Vi"i E AD,. V.l. 23.766 23.7.6 23.72, 23.705 23. ,6. ,3.66< 23.644 23.6,3 2 3 . t 3 „.«. 23.56? 23.54, 23.522 23.502 A.Ui iV.ltl iV.." 23.' "2 23.'402 2V III 1 S!^ ::s; : : :s : : :a : : : 7 ? : : n; ::i^ :::.; .::;? ::is ::;s ::':s j : ;:; s.-ss? ::;s. s:s; ::?s j-;j 7 tin :-s; :-:g E An,. V.l. 23.362 23.3.2 23.322 23.303 23.283 23.2.3 23.2,3 .3.22. „.,_,. 23., ». 23. ,65 23. , .5 23., 2, 23.106 2 3 . ot 2 Y.£l 2^.11] 23: 28 23.'„08 22 . «1 1 ^Lf j:S ;:jj! ;:JS : : S ::?S? ::J|S ::';: -;: - ; -s ::^ US £:.'!! i:a; j : ;ii <: 7 :i : : jy ;-- ;■- ;•- -4 Ang. Vel. --.^0 .2.950 2.. 931 22.9.. 22. =92 22. 67j 22.854 .2.,:35 .2.B16 22.796 22. 7 77 2 2. 758 2;. 739 22.720 2 2.701 ;2.6«2 2^66* 22^44 2:^26 22 . 607 Velocity Velocity in Kilometers per Second Period Period in Hours Ang. Vel. — .Angular Velocity in Radians per Day III -53 TABLE 9 (continued) a »..««, ..,oo ,.?o. ..?« ..70, ..,o, -,, ;: ..^ .,;; -.,5 -. r; ..»;; -..o, ..-o ^ ..<-« ...^ ..»« ..,*. ;.«| ••!£ 8 Period t.L-To o.ob. 6.667 '•«'- .■;■;" .''" "'. . ',,: --'.W -; .7> -2..F, 2.-,- - : .-'- -2.7.3 :..:-? 22.308 22.290 22. .72 22.23! 22.233 S Ar«. V«l. 22.533 22.36? ...350. -.33. ...,3 ....-4 ...4. ...45. _...-* ... g viocirr ..6bo ..i7> ..»" •.»:* •■■.■>■ «■»•-: *-!; *-^; ;•»,■? *•■" ;•-,; *- s r ;•;•' ":?'■:- l'.ti r.:5il ;:S" «:«; ::>« *:.£ 8 Period 6.788 6.793 l. 7» '-BO. ^..tl °-f.o ;•;■;' v:',, - '£,7 a - \" : 0,7 ', >,. 21 541 LMl ; , . 9.5 2, .927 2 1 . -'05 2 , . 89 , 2I.87! ..632 4.631 7. Dili 7.007 ™*. ,?-:ss 4-l°7 3 t-2;;, 2^: :':;'.. .-k':? .;:?.: 2?:?n ^:,;: 2?:K 2';::''; :,:^ .'^'.l ; \:^: 2kc , c S 2;:,,, 2 I ;:,7 J 2,:,,6 *:.». .,:«■ " Ang. Vel f Velocity ,.654 4. £.53 4.b52 4.65. 4.649 4.^48 *.b*7 P«rlO(l b.^OO ii. 305 6.911 6.917 ■ ' - - - ■ 2 Ang.Vel. 21.855 21.837 21. Bio 21. SO* I g VelOCit, 4.629 4.628 ..6,7 4.,26 ..,24 4.623 4.«Z 4.«i. 4.,,, 4.6,8 ;..; ;-.,. ..M* *.««3 4.bJ| - *' J 4.-»g* 4.60. « 607 4 606 § Period 7.o«3 7.0,8 7. os, 7.030 7.w 7.0,. 7.0,. 7..,5 ; 7.,** _,.o., ; .c _7.u ,.ue. .. .. , ■« j; J; ^ 21 ;* 2 ,,^ 3 Ang. Vel. 21.50* 21.466 21.465 .1.452 21.435 21.-17 .1.400 .I.i6. ^I.'ob .M« *,.-.i -i.-i« -'■- - ■ - - ''-O- _ v.lrwltv . ~. 2, *„■> .-.-.- * -i-ji i mn J tcia - 5<;-t 4 5-.,, 4 5'-'5 4.t94 4.-92 *.5 l J' -.59.J 4.5S9 4. -38 4.386 4.565 4.584 4.593 4.661 8 2JSP Vfi ,'t?' • VT.i *1.° 'il!. "'iii 7:,66 7 W2 7.177 7. ,57 7., B <, 7.19. 7.200 7. 2 06 7.2,, 7.2,7 7.223 7. 2 29 7.2,4 S Anj!^.l. 2':!« 2,'Mi :<'Ml :,:" :,:.''• 21.07, 2, .06, 2,.n.. 2, .027 21.0,0 20.™. 20.9?T 20.-76, 20.9.. 20.927 20 . 9 , , 20.89. 20. 8 76 2 0.86, 20.9.3 §V.lnrlrv . -co . 379 . 378 4 37" 4 375 4.37* 4. 573 4.37. 4.57, 4.565 4.5..L- 4. So? 4.56(. 4.365 4.504 4 . 3o2 4.36, 4.560 4.33? 4.539 SSr 7 V'!S I' "J i'-3, I,-' '"i' ' '.; ,27. 7-6O 7.2S6 7.29, 7.297 7.30, 7.309 7.31. 7. 320 7.326 7.332 7. 337 7.343 7.J49 3 2?™. «:!« 2^:1" d:iV, ,1:^ .oMi z».r4 :l:."l ,1:^. 20:;,, 20.^, .o. B o5 ; o.6, s . . U . (J . 20.0,., 20.6.00 20.3,. 2 .56 t 20. ,52 20.330 20..20 § V.lOCltf 4 336 4 335 . 334 < 55- ..552 ..350 ,.549 4.3." ..54? ..5.6 4.5.3 ..347 ..5.. 4.5-, 4.540 ..339 4.538 4.336 4.335 4.334 D..4nd 7 - 333 7*360 7''66 7 372 7.376 7.383 7.3S9 7.7-5 7..0, 7.406 7..I2 7.4, & 7.4^. 7.429 T.4J3 7.44, 7.447 7. ,53 7.458 7.464 2 ^ TO. 2^304 20 1 J" 23:.'° 20:;5* :0...O 20..;. 20..0 E 20. -,2 20.:7 O LO.S^O 20.3.5 20.325 20.31. 20.297 20.28, 20.266 20.250 20.23. 20.2,9 20.203 3 S PerST 7'.7o W?7 W,'l '^' "".^ V Hi VM- V.IX V.l\Z VM -\Vil 'M -.V,l ':VA ?:U7 ;:,;? V.lll *:l'd ;:?;. V.Hi, i I JS!«i. 2J:is? 2 :?72 2J:t?6 .0:;., .;.:tJ 2,:,,o 1:^ 20.07- . u .o„3 .o.o. E .- L ,. c ,, ; ,0.0,7 20.002 ,9.. JB6 ,9.97, ,,.956 ,9.,.o ,9.923 ,9.9,0 ,,..9. 3 Ang. Vel. g g Velocity 4.510 4 . 5o& •»- 5 07 9 « Period T.s&tr, 7.5?i 7.5 C ^ S 2 Ang. Vel. i n .*7? 19.8^, iy.&"v o Velocity 4. ,87 4.48b 1.49? S Period 7. 702 7.70& 7. 71 4 S Ang. Vel. i957? i?-5i>4 lv.54S o Velocity 4.464 4.4(,.t h.*62 5 Period 7.61? 7.625 7.831 S Ang. Vel. 19.286 19.1-71 19.257 o Velocity ..■■95 ■ . & '2. 1 f .(bS3 7^667 7. £73 7is7v> 13.&3& ?! «..£5 I9.^2'3 7'. 1,90 19.608 7.696 1 -E- . 594 >.472 '.779 4.471 7.7S4 1 i',37 3 4.470 7 . 790 l^. 358 4.469 7. 79fc 19.344 4.4t6 7,«01 19. 329 *.,6 7 7.8U7 19.315 4.46? 7.81 i 1 9 . 300 >,*50 : . 3 95 4.4*.9 7.501 I9.f.£5 4.44S 7 . 907 1SJ.071 ,.♦47 7.513 19.057 4. 445 7.^1? 7.V25 1-J.Oit 4.4,3 7.931 19.014 -,..» -.-... -.^.- 4.422 4.421 025 8.031 &.0J7 B.043 6.U49 § ^!°?.i. iJ^oo ,l:ik , s :9,: ,»:-.3 3 .b:^:. „:9>f, ,*:9,; i.:™: , 6 . 8 o t , E . 8 74 ,:..,o ,,.*.< u. 8! . ,=.»,* , s .9 . .3.7,0 ,8.777 ,..,.3 ,..7.9 ,..7„ O V.lOClly , ,; 4.4,9 ...U 4.4,7 ..4,6 4.4,3 4.4,4 4.4,3 4.4,. ..4,1 ...,0 4.409 ...07 ...06 4.405 4.404 4.403 4.402 4.40, 4.400 S P.«Od" »:U° 8.0„i 6.0.7 S.073 3.076 3 . 094 8. '0 6 . o9, . . , 02 . . ,06 , . , ,; 8 ; 20 6. , 2, 6 . , 3 ; | . , « 8 . ,4. 6. ,50 6 . , 36 62 . * Ang. Vel. 18.721 16-706 1 8- § Velocity Period <n Ang. Vel. . Ii» B.14, S.13U C.13U O.ltii O.I ISO .530 18.517 18.503 16. ,90 18.476 18.46J 4.284 4.383 4.382 4 . J6 1 4 . JeO 4.379 6. 257 8. 263 8. 269 8.273 6. 281 8. 287 !$.263 18.250 16-237 16.223 16. 210 18.197 a Velocity < 176 *377 4 37,. 4 17, 4.'T7 4,17: *.371 -.770 4 . J-. f *.3*Z A. XI 4.7b- 4.Jb5 4.?64 *.36I 4.362 4.361 4.JS0 4.359 4.356 I 21Sl l'*Z\ t-ll t'%-, 8' U I : MI ^ 323 ^.-'2- 8.?:5 :>.J-> S.3,7 8.35' c.3?9 e.^5 *.371 ^.377 e.i^J 8.JB5 8.395 8.4 1 6-4Q7 S S^el. .-i*- il^Ti .S:;^ .I:;,3 .*..3: -:.--, . E ..0 L .S.,:i ,= .060 li.O^ 1 & .05, ,S. 4l .6.0.8 1».C5 18.002 ,7.989 -7.976 -7.963 w.WO ,7.938 10 . M. 30 40 » tO 70 » M 100 HO 1M 130. 1*0. 150 1*0. 170 l»- '», § Velocity 4, 357 4, 3?<_- 4, 355 ..•% -. 3-.J ■.."-: 2 -. 7* I ■*. - -. :~i -.:■*' ■• - -"-'■.■. *. J, 5 •> ■ i44 ,. 3< Period ?. ,1 7 5. *1S' 8. "-5 -:.-. 31 :- .-,.'" ::.-•-.' i.. +*' ' . ■< " " ■ -r ' - - "l"' t . ,7 " - . «7:- ^-4^5 *-"^ « Ang. Vel. iT.,-25 t7. 9i: 17. a;--. 17.fi... 1~..74 1, ".-..I I'.t*- '.'. : ; .. W...-' t 7 . .- 1 1.. 7 -.■■:■ 17.. '£5 . - . t g Velocity „..■-.,.; 4.33^ 4.3.'- *.'■;: -.??. 4 . .' : I -.-K -■■- *-?Z* -..3.7 *.J2t. 4.325 -.3i* 4.3. S Period &.53J 8.539 6.5,5 :.55l ,.*■■= 7 ::.'..? :.5'. r ' :.T* o . '^ > >."■'> ■"■V:'- & . b 00 S.^'--: 8.^ « Ang. Vel. 17.672 ,7.6-59 17.6-7 1 7 . ,.. 3 . I 7 - ,_ 2 2 ,;. -u? IT./ 7 1 . ° ■■ '■-' - l- -lO 1 7 . ,4 , I , . 5 . .. 1 .■ . 5-J 1 ...1 542 ,91 '47 e. 503 4. 340 e.5o<> 1 7. 7 22 4.3 39 S.515 1 7 . 70*j ,.338 4.337 8.521 6.327 17.697 ,7.684 122 u. 6 24 4.320 8.630 V.Vil ,.3.6 4.3.7 8.642 8.6*8 .202 -...'CI 4.300 4.2-9:' 4.296 4.297 . 7 7 c - -■>■. 7,-. 8. 751 6. ^56 8.764 6. 770 .:55 W.243 17. 2?> .7.2,9 17.207 17.195 * Period e.n.54 s.6t.o f.t«- \-'>j- ;-'.^' 7r ■;''•"'" ~^\V %''S'Z 'i'\'-. J - ,-'\% ,-'»n. ,"-''■'--] ,^' ' 7 I 17 S Ang. Vel. ' 7. 4 25 17.412 1 7. -t.iri I ,■ . J,-. ■ 7. ■ 7... '7,'n 1 ''. ■ ■- !■..-..■' 1 . V ■ . 1 ■ - -' -■ ■ - g Velocity ,.;^t. 4.295 ,..'.'. *.2">3 -..:'- 4.2-.1 h.-9i_ h_-.'J 4.;eJ=. •.:.' J . : .. < ^.-S'. «.-!.-4 4-tj'J 4.;d2 S Ang.Vel. i?!ipj i^m i^il'. i^m: ttI'i^ i^^', i^ii: wiiut- iv!^, ,7.l-t, ,7.o. : 5 .7.05? it.l'*i it. 029 i7. i7 O Velocity 4 --v, 4 -7- 4 ;7- ■* 21' - 172 4.271 4.270 *.:'-■: *..*V 4.2b7 4.JLL 4.2'.5 «.._«.■♦ *.:t. ,.262 «.i-.l 4.ib0 4.259 4.256 4.258 S iu.i~i .,' rig fi'?,-m e'.«.ri i ?1» i,^:' ?.v-9 e.97; i.V<*\ 8.-:-7 t . "'5 3 i. .<*,.■ a.':\\ >.S'7_ c.?78 8.9fc4 a-990 8.99b 9. 002 9.008 9.015 _ Period n Ang. Vel 6.762 16.751 16.739 ,6.7;e Velocity 4 ~- 4 25r 4 25^ 4 -5- 4.153 4 . 2-2 •.15.1 *.25t' 4..-V «. . 2-f 4.^,7 4.;*u ^.2,-. 4.24- 4. 243 4.2,2 ,.24, 4.240 4.239 4. 2 3 6 X P.Hnd =.7"-'., *j'q^ 9'033 <■ ' "' ^O," 9.052 9054 3.064 9. 070 9.07,: '.0^2 I.Of:* ' . 0V* 9.101 9-107 9.1,3 9.119 9.125 9. ,32 9.138 3 Ang!v.l. l,:^? .6I703 U^L U.t8J K.^t ,6.660 1,.64, i,. S H tt .^G ,-=.6.= I6.6O3 '-^2 ..>.5^l 16.570 16.558 16.3.7 l6 .336 .b,323 16.5.* 16.B02 o Velocity . -.7-- , -ic * 2'5 4 M, * -.3* 4 13 4 2'"' 4 23. 4.230 ,.221 4.-.2S «-'27 h.226 4.225 4. 2 24 4.223 4.222 4.22. 4.220 4.2.9 § pT,S mm 9'tso *' v "/T ' *:;'* * 7^ >iT .'^ 9.^3 ..^ ?.:c %:i: ',*.* 5. 2 24 9.231 9.237 ^..243 9.249 9.233 9.2*2 « PerlOa ,'•]** ,;*!'" l: 1",, ,,;■,., , ;, II7 ,r; ,7r- i.. 4^ 112.4 14 li.4Li3 lb.3','1 16.3fQ It. 2b* U . 338 1S.347 i b . 336 1 r, . 326 .6.31! .6.304 16.293 16.262 _ Ang. Vel. j o Velocity 3 * Period E n Am. Vel. 20! 4. 202 4.202 367 9.374 9. jgO .098 U. OS? lb. 077 8 Velocity 4 -no 4 .99 4 1*& 4. I ■/? -..,9c 4. , -15 ,. 14 4. IV I 4. 1 ■•: 4. 191 4. 1.0 «. t&v 4. 169 4.18= ,.187 ,. 1st, «, 1 85 4.164 4.183 4 . 1 82 « Period ^ * U,ft ,,4 05 4.11 " 41.- * - ■■■> 4 4'n - 4 ■ . . ,*2 ''.4,3 ?.4 e 5 9.4iil 9.4-7 v. 473 9.480 9.486 9.492 9.49b 9.305 9. 51 I § Ang.vel. '6.^5 ,c'.:5,5 iJj! icio;: , ■: ! o 1 7 .^'-^ i5>?: i-ijft ,-..-.?i i-.: : t,(t i^.-- .5.?ji t-.o: e 15.91B .5.-07 ,5.9.? i5.ee, T5.5TL ,5. 8U 5 ,3.855 a Velocity' ...i*. 4 IE0 4 1 7 4 4.173 -.17-; 4.I77 ,.176 4. .75 4.174 ,.f? 4. it; 4.171 -..170 4., u 9 -.168 4-U.6 4.167 4.166 4.165 4.164 S Period '• 5T7 <■■ 52^ « *3'< 9.5 '1- ^.5.2 9.-4? 9.535 9.5-,. ? . 5o 7 9.57, ^.^0 -..336 9.592 D.599 9.^03 9.411 9.618 9.624 9.630 9- 636 S Ang.Vel. 15.843 ,5.834 .5.^24 < 5 . 8 « 3 15.SCJ t 5.-v3 .5.782 15.77; ,5.7^2 ,5.75, 15.7-1 ,5.771 ,5.720 15.-, Q ,5.700 .5.69r : ,5.679 13.669 13.639 IB. 649 § Velocity 4 „;, 4. 162 . 161 * lb 'J 4. I«9 4.15? 4. 158 4.137 4. 15.2. 4. ,55 4. 154 *. 15J ,.1\ 4,151 -. ]5J 4. 1,9 ..!«? 4 . 1 48 4. ,47 4 . I 46 Period 9.643 9.649 9.655 9.,_62 9. & bS ,'.C7, <.. U t 9.6&7 9.6-->? *.«9^ "'-70: 9 . 7 1 2 i.7l£ 9.7.5 .. . 7 3 1 9.737 9.744 9.730 9.736 9.762 Ang. Vel. Velocity 15.477 ,3.467 ,5.457 .0. Velocity , ,4? 4.144 4.14-t ,'42 4.141 4.141 4.140 4.U9 4..3t *.1j7 4.1?.. 4.135- 4.13- 4. .33 -.133 4.132 4.131 4.130 4.129 4.12B Period 9 7 6 1 9.775 ft 79J n 788 9. 794 J.eoo 9.e07 9.81 3 9. J,9 --^ 8:r. 9.t.'2 9.8 38 9.^45 9.63. 9.637 9.efa4 9. 67Q 9.876 9. 883 9.88* Ang.Vel. .5-4 37 ,5.427 15., 17 .5., 7 ,5.39? .5.387 15.377 15.3„7 .5.357 ,5.:,' 1 5 . 32 7 .5.327 15.318 13.308 !5.296 15.288 .5.278 .3.268 |5.259 15-249 Velocity 4, 27 4. ,26 4. 125 4. 123 4. .24 4.123 4.122 4. ,2, 4. ,20 ,.119 -.1,8 4. 118 4.H7 4. ,16 ».,|5 *-114 4. ,13 4, I, 2 4.111 4. 111 Period Mt, * %S! * 4& v »1, 9; 9 2. 9.92" 9.933 9.9,0 -.946 9. 9 5 3 &.?59 9.9b5 9.972 9.978 9.984 9.991 9.99? 10.003 .0.0.0 .0.016 Anc Vel l3.1'9 ,3.:29 15.22u 15.210 .5.200 1 5 . , 90 15.16. 15. ,71 ,5.161 ,5.152 ,5.14. 13.13. 15.123 I5.11j 15.107 ,3.094 15.06, 15.075 13.065 15.085 Velocity 4 ,10 4 10'. 4 ,03 , ,07 ,.106 4.t0* 4.1OS *.,0- ,.10I 4. '02 4.101 4.10U 4.099 4.098 4.698 4.097 4.096 4.093 4.094 4.093 1 Period lu 022 10.0^' '0'.O35 '0.042 10.048 10.034 10-061 .0.067 .0.07,- , C . 080 1 0-086 -u.0^3 1 D. 0«>° I'i.lOf lO.Hi 1O.U8 10.12? 10.131 .0..37 10.U4 1 Ang.Vel. 15. 04b 13.036 15.02-' .5.017 I5.00e ,4.998 ,4.969 14.979 1,-970 14.9,2.0 1 4 . 95 1 14.9,1 .4.932 14. --.22 ,4.913 14.903 l,.894 14.863 14.875 14.666 Velocity 4 n-:.^ 4 09 '■ 4 091 4 090 4.089 4.088 4-087 4.066 4.0*6 4.065 4.084 4.083 4.082 4.081 4.080 4.080 4.079 4.078 4.077 4.076 Period I0'l5fi ,n 157 to' 161 io'k-9 ,f..i7< ; - in. ,82 10.169 iQ.1^-5 ir..;n. ,n.206 1O.21* 10.22, '0.227 m.,.33 10...0 .0-246 10.253 10.259 .0.266 10.272 Ang. Vel. 14.357 14.64? Velocity — Velocity in Kilometers per Second Period — Period in Hours Ang. Vel. — Angular Velocity in Radians per Day *.;-£..- 14^73 |4.7 U 3 '4.75* 14.745 1».73i> 14.726 14.717 I4.70& 14.699 ,4.690 1,.68r. 111-54 TABLE 9 (continued) Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. Velocity Period Ang. Vel. >72 4.071 to* 10.311 >?5 14.626 4.058 4 10.4 14 10 14.481 '4 4.05-i 10.426 ' 14.463 ' 4.039 10.556 ■ 14.286 ' 4.009 10.797 t J. 967 3.993 10.927 tJ.eoo 3.977 1 1 . 039 13.636 T-.946 11.323 13-318 3.931 I 1 .456 13. 163 3.913 I 1.390 13.01 t 4.008 4. 007 ID. 803 10. 810 ' 13.959 1 3. v50 ■ 3.992 3. 591 10.934 1Q.-:'4 1 ' 13-792 13, 7=>3 3. 976 3. 9 7u 1 1 . 065 1 1 .072 ' 13.628 13.620 ■ 3.961 3.960 1 1 . 197 11 .204 13.467 13.435 3,945 3.^44 3.' I .330 ti. 33& » i .. 3.310 13.3Q2 13.: 3. 930 3. 925 3.^ i.03' 4.038 i.5(>2 l0.5fc5 ..277 14.269 1.022 4.021 t.692 10,699 1.103 14.095 . .ooe. 4.003 1.623 10.829 1.933 13.923 4.070 10.317 1 1*.61b I 4.U33 10.446 I 14.43C I 4. on 10-573 ' 14.259 ' 4.020 10. 703 14.086 4.063 0.32! 4.t 07 3.991 0.9-7 T.lT. ■ 3.973 1 .079 3.61 2 3.959 1.211 J.*31 ;.-?90 J. 939 ■.954 10.960 J.76"7 IJ.739 3.974 3.973 1 .093 11 .091 1.603 13.393 3.988 10.967 13.730 3. 971 1 1.099 13.397 3.9i3 1.914 1.396 11.603 3.004 12.996 3.900 3.699 1 I .730 11.737 12. B55 12.648 3.8S3 3.684 l 1.863 ' 1 .872 ■ 12.710 12.702 3.913 1.610 2 .969 3.B71 1 I .993 12.374 3. 637 12. 129 12.43 3 3. 670 3.870 3.869 2. 000 12.007 12.013 2. 567 12.539 12.332 3.836 3.833 2. 133 12. M2 2.4^6 12.419 3.834 12.149 12.412 1,3*'. 3.942 . *30 11.336 J. 266 13.279 1.928 3.927 .463 11.489 !. 132 13. 123 ! . a 1 2 3.912 .616 11.623 I. 961 12.974 1.898 3.897 .750 11.737 ■.S3' 12.826 S.S83 3.982 .365 11 .892 i.eee 12.681 1.866 3.867 i.G20 12.027 1.5*3' 12.338 1.834 3.833 !. 136 12. '63 i.403 12.396 4.020 0.712 4.078 4.003 10.842 I3.9UB 3.967 10.973 13.742 3.972 1 1 .103 13.379 J.93fa 11.237 13.420 3.94! 1 1.370 13.263 3.926 13.1 17 3.911 1 1.630 12.966 3.896 1 1 . 76* 1 2.019 3.861 It. 891 ' 12.674 3.8*7 12.034 12-331 3-832 12.169 12.392 3.910 1 1.636 12.939 3.866 12.040 12.524 3.632 12. 176 12.363 4.0b9 4.068 10. J30 10.336 14.598 14.589 4.032 4.0B1 10.439 to. 463 14.418 14.409 4.035 4.034 IO.38& 1O.B93 14-242 14. 233 4.019 4.0i8 10.718 1U.723 14.069 14.061 4.003 4.002 10.649 10.633 1 3.900 13.891 3.987 3.986 10.980 10.986 I 3.734 13.726 3.971 3.970 11.112 11.118 13.371 13.563 3.953 3.934 11.244 1 1.250 13.412 13.40* 3.940 3.939 II .376 I 1.383 13.233 13. 24B 3.923 3.92* 11.309 I 1.816 13. 102 13.094 3.909 3.909 1 .64 3 11 .600 2.932 12.944 3-895 j.o-., I 1.777 1 I .784 12.8U4 12.797 3. 660 3.879 1 1 .<»12 11 .9i9 12.639 12.652 3-863 3.865 12.047 12.054 12.317 12.510 3.831 3.850 12.183 12. 190 12. 378 12.371 4.0o7 10.343 14.380 4.03* 10.601 14.224 4.017 10.731 14,03 2 4.001 10.862 13.S83 3.983 10.993 13.717 4.55 J 4.04a 10.7JS 10.7*. 1 4.04' 14. U J' 4.000 3.9 C - 10.868 '0.87; 13.873 U.Sb, 4.014 10.737 t4.o' e 3.999 10.882 I t3.858 1 3.934 1 I . 237 13.396 3.938 I 1 . 390 13.240 3.984 1 1 . ooo 13.709 3. 966 13.547 3-933 13.36B 3.9'7 ' I . 396 1 7. 23: 3.923 3. 9.2 11.327 I'.!P 13.087 13.079 3.908 3.9o7 I 1.637 1 1 .663 12.937 12.T-29 3.B»3 7.8=': 3.878 11 .923 12.6*3 3.8(4 12.061 12.303 3. -'63 1 . 00b ■ 3.701 ■ 3. "32 1.27 D i. 380 ?.967 I 1 . 144 I 3.331 3.951 I 1.277 1 3. 372 3.9 30 t 1.410 13.217 3.921 r 1.5*3 13.064 3.906 I 1 .677 12,914 7.982 J. 981 11.019 1 1 .0:6 13. 683 1 J.b77 3.96k. 3.965 11.131 ' 1 . 1-8 13.32 3 1 3.3)3 '.93' 3.930 1-263 11.290 3.3b* ) 3. 357 3. "35 3.9?4 11.416 1i ,*;r >S.2l<> 1 ?.2Q> 3.920 3.919 11.54^ 1-.536 I 3. 057 1 3- 049 3.-;03 3.904 1 1 .683 1 1 .690 i:,o(17 I 2.900 10. 51 1 10.51 '4. 3*7 1*. '3i *.029 4.021 3.996 3.9^5 10.90) 10.908 I3.*33 1 3.813 3. "60 3.979 1.032 11.03"? i.666 13.660 3.963 3. 964 1.16* I I . 1 7l 3-507 I3.»99 «.04< 4.0*3 10.523 10.530 1*. 37.0 1*. 321 4.027 4.026 10. '.-53 IO.06O 1*. 155 14. 1»C- 3.9^5 3.994 10,91 4 10. 921 17. 61' 13-808 3.979 3.978 1 1 .046 l1 .052 1 3.b52 13.6*4 3-96 3 3-962 1 1 . )78 1 1 . \9i 13.«?i ,3.*e3 3.9*7 1 1.310 1 3. 3 .3)7 13. 323 3. 933 3. c '32 3.0*1 I 3.034 3.903 3.9U3 1.697 11.70 3 2.69; I 2. 985 760 12.733 "2.7*. 2.0*8 IT. 074 3.S73 3.075 1 1 .93? 1 1 .959 I2. u '6 i2.bfl9 7.S* 7 3.846 1.-.22* 12.231 1 2. 336 )2. 729 3.87« 3.873 1.96U 11,973 2.602 12.593 3. "*5 3.84, 12.137 12.2* '2.3:3 1:. 31' 3.917 3.916 1 1 .57C 11 .587 1 3.026 13. 019 3.902 3.90' 1 1 .710 11.717 1 2.877 12. 670 3.697 3.96i? 11.8*3 H.85' 12.731 12-72* 3.873 3.87: 1 1 .960 1 1 .986 12.368 )2.5e' .8*4 3.6*3 . 25 ' 1 2 . 236 . 3C9 12. 702 © Velocity 8 Period S Ang. Vel. g Velocity g Period K Ang. Vel. g Velocity 9 period ** Ang. Vel. velocity 1 Period E Ang. Vel. Velocity 3 Period fi Ang. Vel. 9 Velocity 8 Period S Ang. Vel. 8 Velocity Period A Ang. Vel. o Velocity 9 Period « Ang. Vel. g Velocity 8 Period « Ang. Vel. e Velocity S Period «S Ang. Vel. Velocity 8 Period S Ang. Vel. c Velocity « Period 8 Ang. Vel. o Velocity 3 Period 8 Ang. Vel. g Velocity * Period » Ann. Vel. g Velocity 8 Period « Ang. Vel. 3.828 2.401 12.408 2. 1*0 12. 133 3.81* 3.61J 2.338 12.343 2.027 12.020 3.800 3. EDO 2.676 12.683 I .897 1 1 .B90 3.787 3.786 2.A1* 12. en 1.76P 11.762 3.773 3.772 2.932 12.939 l.<;*3 11.636 3.627 3. *27 J. 760 3.091 1.319 3.746 3.23' 1.397 3.739 3.096 1.313 3.733 1 J.37G 1 1.272 3.7V9 1 2.66V 1 1 . 884 I. 76-. 12.629 11.736 3.772 12.946 11.630 3.739 t J . 1 ri?/ 1 1.307 3.745 13.2*3 1 i.3a5 3.732 1 3. 395 13. 639 1 *. 666 3.312 12.531 ! 12.. "-0 7 1 3.796 2.696 1 1 .877 1 3.7*3 2.83* 1 1 .74" 1 3.771 2.-77 3 1 1.624 1 3.736 .501 3.682 3.661 3.936 13.943 0.62' 10.813 3.670 3.669 ••078 14.0B3 0.71 I 10.706 3.657 3.637 *.22l 14.228 0.60* 10.398 ' 3.636 *.233 0.593 3.6'>8 14, 100 1 10.693 1 3.635 14.243 1 10.348 1 3.839 '2.?99 '2.261 3.823 t 2 . 4 Jl 1 2 - ' 26 3.611 12.572 1 I . 99* 3.836 3.S37 12.'30b \?.1H 12. 23« l2.24o 3.783 12. 8*8 It. 737 '.810 3.S09 12.579 12.580. t 1 .968 I I . c, 8l 3.796 3.79^ 12.717 12.72* ' It. 656 It. 851 1 3.763 3.7a2 17.83- 12. 6«2 ' 1 1. 77o ' 1.7.'* ' 3-736 13.126 1 1.408 3. 7 JO 13.406 1 1 - 2*9 1 .973 H .968 i.795 2.731 1 1.84 5 1 3.7*2 3.74- ■ 3.27J I 3.; 80 1 I. 361 1 1. 355 3.729 3.729 13.413 1 3.4iii 1 1l.?*3 ".277 , 3.692 13.029 1 10.904 1 3.679 13.971 1 1O.T9J 3.654 I * . 237 to. 577 3.-.-9t 3.6*0 11.83*. 13.843 ir..89« 10,893 3.678 3.676 i].97i 13.985 10-788 '0.7R2 3-*66 3.663 t4.l2i 14.t28 10.679 10.674 3.634 3.633 4.264 '4.271 0.372 10.3C6 3.793 3.793 12.743 12. ?32 1 1.832 1 t-S:b 3.780 3.77V 12.88 3 12. "90 1 1.703 11 .699 3.7*6 '.76,- 13.022 1 3.029 I 1.500 1 ' .57* '.73* 3.734 3.147 1 J. 164 I . «70 11.464 3.7*1 3.740 3.2?7 13.294 1 .349 11 .343 3.728 3.727 3-»;7 13.*3* 1-23' 11.223 3.713 3.714 13.568 13.573 11.114 1 t . 1 09 3.7"2 3.702 13.709 13.716 1 t . 000 10. 994 3.U90 3.689 13.B30 13.030 10.68/ 10.8B2 3.677 3.676 1 3.993 U.OOO 1 ft. 777 10.7TI 3. u »5 3.66* 14. 133 14. U2 ■0.66B 10.663 3-652 3.682 4.276 14.786 0.361 '0.356 . 819 .778 . 8-J7 . 693 !.*}.3 3.6*2 1 i. 333 12. 3^-0 1 r.207 1 ;. 200 3.,31'j 3.618 1 2.*9J 12.497 12.073 12.0'.-7 '.SO* 3.804 1 :. u r7 12-63* ■ ' 1.942 I t .93" ' 3.791 3.791 1.1.763 12.772 ' .813 11 .gi)7 j . & 1 a 3. 8i7 :.504 12.311 r.ObO 12.033 11,929 11. '•■22 3.790 3.78! 12. 779 12. 78( 1 1 . 800 ' 1 . 79- .816 3.816 3.81S .318 12.124 12.331 '.''47 12.0*0 12.03* .802 3.802 3.901 .653 12-662 12.669 .9'6 11 .909 1 1 .903 3.789 J.78B 3.78? 12.793 12.800 12.607 1 1.767 I 1.781 1 I .773 3.776 r.TT 1 2.924 i:.9Ji 1 1.667 1 I .661 3.774 12.938 1 1 . 633 ' '. 76 T 3. 762 3.762 J . 76 1 3.760 1.056 13.063 13.07C 13.077 13.084 .530 1i.5*3 H.5J7 11.331 it.SVJ ■.73^ !. '73 .4*6 3.740 3.7;9 13.301 13.308 1 1.337 1 1 .332 3.727 3.7:t> 13.441 13. 44S ■ 1 1.219 11.213 ' 3.71* 3- 713 13.3*2 ' J -58 n 11. 103 n .097 ' 3.701 3.700 13. 723 13. 730 ' 10.989 TO. 983 ) 3.(99 3.6?8 13.863 '3.872 1 10.B76 10.871 I 3.676 3.673 t*.007 14.01* 1 10.766 t0.7,:o I 3.663 3.663 14. 130 )4. 137 I 10.637 10.632 1 J. 631 3.631 14.293 14.300 1 10.331 10.543 1 3.731 3.730 13, 182 1 3. I=r9 I 1.44C 1 1 .434 ■ 3.738 3.737 i 13. 322 13.329 1 1.320 I I .314 3.723 3.724 i 13.4K.2 I 3.4fc.9 ' M . 202 I I . 196 3.712 J,7H . 1 3.603 13.&I0 1 1 .086 1 1 .090 i 3.ij99 3.696 ■ 1 3.74* 13.731 ' 10.972 10.966 3.686 3. C86 1 1 3.88b 13.893 10.860 10.83* 3.674 3.673 14.028 14.033 10.749 10.7*4 3.662 3.661 14.171 14.17B 10.641 10.636 3.649 7.649 14.314 1*. J2' 10.335 1C.329 3.750 3.7*9 3.7*8 T. 19C '3. 20 J 13.210 1.426 11. 4 22 11.415 3.737 3.736 3.733 3.336 13. 3*3 13.330 1 . 308 1 I. 302 1 1 .296 J. 746 3.747 13.217 13.224 11.404 1 I.4U3 3.733 3.734 13.337 13.36* 1 1 . 290 1 1 . 2B4 T.72J 3.722 3.722 3.721 13.483 13.490 13.497 |3.80* 11. 190 1 t. 184 1 1.178 1 1 . 172 11 . 167 3-7)1 3.710 3.709 3.709 3-706 13.617 13.624 13.631 13.630 13.6*3 11.07* 11.068 11.063 11.037 11.051 3.699 3.697 3.697 3.696 3.693 13.738 13.763 13.772 13.780 13.787 10.960 10-933 10.9*9 10.9*3 10.930 3.6?3 3.665 3.68* 3.683 3.6B3 3.900 13.907 13.91* 13.921 13.929 0-8*9 10.643 to. 837 10.832 10-826 3.673 3.672 3.672 3.671 3.670 4.0*2 14.030 14.037 14.064 14.071 0.739 10.733 10.728 10.722 10.717 3. 660 3. 660 3 . 659 3. 639 3 . 658 14.183 14.192 14.2D0 I*. 207 14-214 10.630 10.623 10.620 10.614 10.609 3.6*8 3.6*7 3.646 3.6*6 14.336 14.343 14.730 t*.3?7 10-319 10.314 10.308 10.B03 Velocity Velocity in Kilometers per Second Period — Period in Hours Ang. Vel. — Angular Velocity in Radians per Day III- 55 TABLE 9 (continued) g vew I I 1 I ii 11 f I ! Velocity period Ang. VeL Velocity Period Aag. VeL velocity Period Aag. Val, Velocity Period Abb. Vel. Velocity Period Ang. Vel, Velocity Period Aag. Vel. Velocity Ptrlod Am. vei. Velocity Period Aug. Vel. Velocity Partot* . lei. Velocity Period Ang. Vel. Velocity Period Aag. Vel. Velocity Period :. vei. Velocity Period Aag- Vel. Velocity Period Ang. Vel. 3. u 45 J. 042 3.63y 14.34.4 (4.400 14. *H i 10.498 10.472 10.4*6 i J. 386 3.383 3.3B0 |3.G89 i3.*23 13. 162 ' 9.49* 9.970 9.9** 3.633 J.6 31' 14T308 |4.*l44 10.394 10. 368 3.374 '.371 13-233 is. 272 9.696 9.8T4 3.613 3.612 3.609 (•.723 14.761 1*. 798 10.241 10. 21ft (0. 191 3.329 J. 327 J. 324 3.321 3.316 15.823 13.662 13.999 13.936 13.973 9.529 9.307 9.483 9.463 9.441 3.3*3 9.397 3.5(0 3.307 3. 603 6.093 16. 1?2 \r. 160 9.373 9.333 9.3J2 3.473 3.473 3.470 3.468 (6.372 16.610 *6.6*8 16.683 9.099 9.079 9.058 3.033 3.424 3.421 3.419 17.331 17. 369 17.406 8.701 8.682 6.663 3.373 3.372 3.370 16. |01 18. WO 18. 1 79 8.331 6. 313 8. 293 3.32B 3.323 3.323 (9.483 16.922 19.961 ' 7.986 7.969 7.933 3.282 3.280 3.279 (9.673 (9.7)3 19.733 ' 7.664 7.649 7.633 3.239 3.237 3.235 3.232 20.478 20.518 20.331 20.399 7-364 7.349 7. 3J3 7.321 3.416 J. 41* 17.446 17.4 ? 4 8.644 a.t.23 3.4U6 J. 40* 3.402 17.599 l7.u38 17.676 8.368 6.330 8.331 3.273 19.833 7.603 3.263 3-262 19-993 20.133 7.3*2 7.327 e VtfOCltJ 3.(97 3.(93 3.193 3.191 3.189 3.(87 3.183 3.(83 3.101 3 8 P»rt«* 21.291 21.3J2 7i.3TJ 7T.<T* 21-453 2 l.4\j6 2(.53 S Tt 1.579 27 . 6 7D ;T II *"g Xml 7.082 7.069 7.033 7.042 7.028 7.015 7.002 u.98£ 6 . c. 7 5 6 bfcl e Vale 3 peri 3 Aag. 3.041 3.039 3.038 3.0?6 24.736 24.779 24.622 24 . ' a 5 6.096 6.086 6.073 i..0b5 3.010 3.008 3.006 3.003 3.003 23.514 23.336 23.602 23. (,43 25.68-1 3. 910 3.«00 3.890 5.880 5.870 3-001 3.000 25.732 23.77fc 5.960 3.650 3.032 3. 031 3.029 i*.9bl 2*. 99* 2». Tij6 0.0** 6.0*3 ft. 02? 2.998 2. 9"6 2. 9^5 '5.82U 23.663 23. V07 3.8*0 3.831 t.?2l 3.606 3.603 J. 600 3.397 3.393 4.834 14.870 14.906 I4.94J 14.979' Ij.u» 0.166 (0.141 10.116 10.092 10.067 10.043 (0.018 t 13. Oil IB. 002 3.337 3.534 13.*33 13-492 3.332 3.349 3.346 3.543 3.340 3.33B 3. SJS 3.632 ._ 3.329 13.366 13.603 13.6*0 13.676 (5.7(3 I S. 750 1B.?»B 734 9.7(1 9 688 9.663 9.642 9.6(9 9.397 9.574 9.B«2 3.502 3.499 4 97 3.4 9* 3.491 3. *69 3.466 234 16.272 lb. 309 (6.347 16.38* 16.42; 289 9.267 9.2*6 9.223 9.20* 9.183 3.483 3.481 3.478 16.459 t6.497 16.B34 9.162 9.141 9.120 3.463 3.462 3-*b0 3,*57 3. ,55 3.4B2 16.723 16.761 lb. 799 16.836 16.87* 16.91? 9.017 6. '97 B.977 S.937 6.936 8.°16 3.439 3. (7. 102 17. 9.817 8. *J7 3.*34 3.432 3.429 3.424 140 17.178 17.217 l7.23f 17.283 798 8.77B 8.739 8.739 t. 720 3.367 3.363 3.363 3.360 3. 35B 3.336 3.353 18 218 18.;-37 18.296 '8-335 '8.373 (8.4'i 18-452 8.2?7 8.260 8-2*2 8.223 8.2^7 & . 1 90 8-173 3.3H 3.309 3.307 3. 3*-9 3. 197 3.3'.'* 3.392 3.389 3-387 3.384 3.382 3.380 3- 3^7 17.713 17.753 17.??2 17.630 I7. e e9 i7.9q8 17.94* 17.983 18. o2* l6-0*J 8.512 6.494 8.476 8.457 6.439 8.421 8.403 8.J83 8.3*7 8-349 3.351 3.348 3. J*6 3.34* 1 . 34 1 3.339 *. 337 3.334 3.332 3.330 16.491 (8.330 18.369 16.608 16.647 18.686 18.723 18.763 18.8O* (8-8*3 6.133 8.136 8.121 8.104 B.Q87 8.070 8-033 8.036 8.0(9 8.0Q3 .302 3.300 3.298 3.296 3.293 3.291 .317 19.33' 19.^96 19.436 19.476 19.313 .606 7.790 7.77* 7.739 7.743 7.727 3.289 3.287 3- 2*4 9.333 19.593 1§.6J3 7.711 7.696 7.680 3.230 3.228 3-226 3.224 3.222 3.220 20-440 20.680 20.721 20.761 20.8u2 20-843 7. 306 7. -92 7. 2 7& "> . 26 1 >'. 2«V 7. 233 3.260 3.238 3.236 3.234 3. 252 3.249 3.24 7 3.243 3.243 3.2*1 20.075 20-K3 20. 133 20. (96 20-236 20.276 20.316 70.357 70. JTT 20-437 7.512 7.497 7.*B2 7.407 7.432 7.437 7.422 7.408 7.393 7.378 3.218 3.216 3.213 3.211 3.209 3.207 3.205 3-203 3.201 3.199 20.683 20.9?* 20-963 21.003 21.046 21-08* 2l.'2e !(.'*& 21-210 21-2B0 7. i21 7. 207 7. 193 7. 1 79 7. 163 7. 131 7. 1 37 7. 1 24 7. HO 7.09* .173 3. (73 3. 171 .743 21.763 21.826 i .933 6.922 6.909 3. 165 21.930 ; 6.870 3-137 3. 133 3. 133 3.131 3.149 3. '47 3. 1 *5 3. 143 3. 14! 3. 1 39 22.1** 22.157 22.199 22.240 22.282 22.121 22.363 22.40C- 22.4*6 22.490 6.819 6. 806 6-T93 6.780 6.768 b. 7»3 <,. 74J 6.730 6.7'8 6. 7o3 3.118 3-1(6 3.1(4 r.lU 3.110 3.10O 3.1 r, 7 *.|Q^ 3.103 3-'0( 22.930 22.992 23.03* 23.07b 23. lie 23.160 23.202 23.2*3 23-267 23.329 6.371 6.339 U.347 6.333 6.323 6.311 6.499 6.487 6.476 6.464 3.08( 3.079 3.077 3.073 3.073 3.0 72 3.07Q 3.068 3.066 3.064 23.793 23.837 23.880 23.922 23.9b3 2*. 008 24.050 24.u93 24.136 24.176 6-337 6.326 6.313 6.304 6.292 6.2<ji 6.27Q 6.259 6-2*8 6.137 3. 1 37 3. 133 3. 133 3.13! 3. 129 3. 1 28 3. 1 26 3.12* 3-122 3. '20 22.332 22. 5?3 22.613 22. 6^7 ;;.699 22. 7*0 22.7*2 22. B2* 22.866 22-*08 6.693 6.660 6.668 6.656 6.643 6.631 6.6(9 6.607 6.593 6.383 3.099 3-097 3.095 3.09* 3.092 3.090 3.086 3.086 3. 084 3.083 23.37' 23.*'3 2j.*3b 23.496 21-3*0 23-383 23.623 23.6*7 2J.T1O 23-732 6.432 6.441 6.*2'i 6. 41 7 b.406 6. 394 6. 383 6.37( 6. 360 6. 349 3.062 3.061 3.039 3.037 3.053 3.03* 3.032 3.030 3.048 3.046 24.221 2*-2b4 2*. 306 24.3*9 24.392 2* . * 33 2*.47e 24.321 24.564 24.607 6.226 6-213 6.204 b.l93 6.182 6- (71 6.161 4.130 6.139 6.128 3.027 3-023 3.024 J. 022 3.020 3-Ote 3.017 3.015 23.081 25.12* 23.167 25.211 23.234 25.297 23.J41 23.3*4 6.012 6.002 5-992 3.981 3.971 3.9&1 3-93i 3.941 3. Oil 3.012 13.428 20.471 5.930 3.920 2.993 2.99i 2.990 2.988 2.986 2.985 2.983 2.981 2.980 2.97fl 23.951 23.99* 2o.Q3e 26.082 26.126 26.170 26.213 26.237 26.30' 26.3*3 5.6H 5.801 5.791 5.7*2 S".T72 5.762 3.TJ3 3. 743 5.73J 3.724 Valoctty Ptrlod Ang. Val. Valoclty Parlod Val. Velocity Parlod . Val. Valoctty Parlod S Vsl ° S Pari * Ang. 2.976 2.975 2.973 2.971 2.970 2.965 2.9ml 26.389 26.4J3 26.477 26,321 26.563 2l-.'>0$ 2, : .^« 3. 7(4 3-703 3.695 5. 68iS 3.676 5.6<-7 5.6"' 2.944 2.942 2.V4Q 2.939 i.937 2.936 2.9,'* 27.274 27.3(8 27.363 27.407 27.432 27 . 49t, 27.3*1 3.329 3-320 5.3d 3.302 5.493 3.*84 5.*7- 2.912 2.9(1 2.909 2.908 2.906 2.90* 2.903 X?.-?l 3 2J._23ft 28.303 28.3*8 2*".39? 2e.438 26. B Period 28.(68 28.2(3 28, 5 Ang. Val. r»J V »3 i v^*? _<>. ?• ;■ 5.tJS' 3 . ■": 30 2 . ytO 2i..e30 3 . ,;20 2.^38 26. 874 3-61 1 i . 9 S 3 .■''.3*1' 5.466 3.438 2.929 27.-5 75 3-*49 2.92? 27.720 5.440 2.926 27. 764 3.4!' 2.901 r6.*63 2.900 28.328 2.B98 29. B7? 2.6V7 26.';I9 2.B93 ' 2.935 2 . 55 3 2 . 932 I - 93c 2 . 9* B 2 . 9* 7 2 . 943 | 2i.9bJ 27. CO? 27.032 27.096 27.140 27.ifl3 27.229 5.393 3,38* 5.37* 3.363 3.336 3.34? 8.8JI 3.*23 3.* 9*3 2.922 2.920 2.918 2. '.'< 7 2.9(5 2.914 83* 27.999 27.943 27.988 28.033 28.078 2t.'23 03 5.396 3. 388 3.3?» 3. J7I «. 36! 5. *1( 5.703 S.->7e 5. i£9 3. . §Velc Pari Ang. 9 Valo 9 Pari » Aug. e Valo g Part 8 Aag. | § Valoctty Period !. Val. Valoctty Period t. Val. Velocity Parlod Ang. Val. Velocity Parlod Ang. Val. Valoclty Parlod Aag. Val. Valoctty Bd Val. Valoctty Parlod Val. Valoclty Period Ang. Vel. 8 *•** 8 perl X Ang. §V.lc Pari •» Aag. Valoclty Parlod Valoctty Parlod Ang. Val. 2.882 2.680 I.8?9 2.877 2.87c. 2.874 2.87J 2.871 2.970 2 . Bf.8 29.072 29.117 29.163 29.208 29.254 29.2^9 2". 3*T 29.3^0 29.«36 :y.4P1 3.187 3.179 3.l?1 3.163 3.153 3.147 5. "39 3.131 5.123 3.(13 2.852 2.831 2.8*9 2.846 2.846 2.843 2.643 2.6*2 2-6*1 2.839 29.983 30.031 JO-077 30.(23 30.(69 30.2lS 3Q.26i 30.307 30.333 30.399 3.02* 3.021 3.014 3.006 4.T.98 4.9">1 «.9b? *.97b *."68 *.9^1 2.823 2.622 2.821 2.8 l 9 2.818 2.816 2.813 2.814 2.812 2.811 30.907 30.934 31.000 31. 0*7 31.093 3|.(*0 31.166 31-233 jl.279 3|.326 4.879 4.872 4.6T4 *.83"7 4.630 4. »4 J 4.635 4.8.6 4.921 *.rf* 2.79* 2.?94 2.7J3 2.792 2.790 2.7e9 2.7B7 2.786 2.783 2.763 jl-839 31.886 31.933 31.980 32.027 32.07* 32. '21 J2.lf8 32.21? 32.262 4.736 4. 779 4.772 4.713 4.706 4.70? 4.695 4.089 4.681 4.674 ^.769 2.7*7 i.T«6 2.76T 2.7&T ^.7^2 2.76 J2.780 32-828 32.875 32.922 32-970 33-017 33. Ob. 4.600 4.394 4.387 4.380 4.374 4.5b7 4.*6 2. iZ r > 2.758 2.757 33. I (2 33. 159 33-207 4.554 4. 346 4.541 3 2.732 2.731 3 3*. 113 3*- '61 2.742 2.74( 2.7*0 2.739 2.737 2.736 33.730 33.778 33.626 33.874 33.922 33-9 ( ,9 4.47f 4.4*4 4.4~58 4.432 4.4*5 4 . * 39 2.7|7 2.716 2.714 2-713 2-712 2.711 J4.690 34.738 34.786 34.934 34.882 34.931 4.J47 4.341 4.335 4.329 4.323 4.3)7 2.692 2.691 2.690 2.688 2.687 2.686 2.685 2.684 2.(82 2.681 33.638 33.706 33.755 35.804 35.832 33.901 33.930 33.998 36.1M7 36.096 4.229 4.223 4.2(8 4.212 4.206 4.200 4.(93 4.189 4.183 4.178 2.735 2.7J I4-OI7 3*.0t 4.433 4.*: 2.709 2. 70S 2. 707 7. 706 34.979 33.<i27 33.076 33.124 4.311 4. 305 4.2 a 9 4.293 2.668 2.6*7 2.666 2.664 2.66J 36.*34 36.684 36.733 36.7B2 J6.831 4. 1 16 4.171 4. (05 4. 100 4.094 2.309 -'1.372 , 4.807 2.782 32. 309 ; *.6£7 2.753 33-234 : 4.535 2.7 30 34.20? . 4 . 4fiB 2.8*5 29.573 5.099 2.63b 0.491 4.946 2-808 2.894 2.892 2.891 2-839 2.666 2.886 2-885 2. 66 J 2&.709 2e.73* 28.800 28-8*3 28.890 26.935 28.961 29.026 5.253 3.24* 3.236 3.228 5.220 5.2IT 5.203 g.'95' I.864 2.66< 2.86 1 2. 859 2.856 2.837 2.833 2-8B4 -n .-.*» 5n .-,-.. ,r, ,.!-, -«.7 Sfe 29.601 29,6*7 29.093 29-93* .066 5.060 3.032 3.043 B-0T7 3.091 5.083 3.076 5 30 '.8 35 1.337 ..938 1.633 2.832 2.031 .383 30.630 3o-*-"?6 1.931 4.923 4.9i6 2.829 2.82*8 2.626 2.B29 30.722 30.768 30.B13 30-86i 4.906 4.901 4.g94 4 . 86C 2.807 2.805 .800 2. 7^8 U.257 4.40: 2.?03 13. 221 4.261 2.30 1 M .559 3' 4.778 . 2.803 2.801 2.800 2.796 2.797 11.699 31.7*6 31 .7« 4.757 4.T50 4.7*1 .778 2.777 2.773 2.774 2.773 2.771 2.770 ■.*30 32.497 32.344 32.391 32.639 32.686 32-733 .6*7 4.640 4.634 4.627 4.620 4.614 4.607 .75? 2.730 2.7*9 2.748 2.746 2.745 2-"»44 .397 33.44* 33-«92 33. 34o 33. 3B7 33.*33 33-6iJ .515 4.509 4.502 4.496 4.4*0 4.4WT 4.4T7 .726 2.72* 2.723 2.722 2.771 2. 719 2.M6 .333 34.4QI 34.449 34.497 34.545 34.393 34.641 .J^O 4.3F4 4.T77 4.3Ti 4.3F3 4.J59 «. J5j 2. 697 2.b9o 2.693 2.693 3 33.51^ 33.360 30.609 2 4.24b 4.241 4.233 ■.3tb 35.415 33. 36. U3 36. 19* 36. 2*3 3".2^ 2.662 2. 661 2.660 2.636 2. 637 !b-BB0 3b. 929 36.V7B 3?.0«B 37.077 4.0»9 4. OB 3 4.07B 4.073 4.067 2.6*4 2.643 2.642 2.641 2.640 2.639 2.637 2.636 2.633 2.634 n. te 20 3?.*T0 J7.?I9 37.769 37.616 37.B*8 37.917 37.96/ 3B. ol 7 36. 0*6 4.008 4.003 3.998 3.993 3.987 3-982 3.977 3.g72 3.9&7 3.9*1 2-622 2.620 2.619 2.618 2-617 2.616 2.6'3 2.61* 2-613 2. oil 38.614 36.664 18.714 36.764 36.814 38.864 38.914 38.964 39.0(5 39.063 3.905 3.900 3.893 3.890 3.883 3.880 j.Kl 3. 670 3. 865 3.860 2.399 2.398 2.397 2.596 2-393 2-394 2.593 2.592 2.590 2.889 39.*|7 39.6*8 39.718 39.769 3^.819 39.969 39.9*0 39.t)70 40-021 40-071 3.80* 3.801 3.797 3.792 3.787 3.782 3.777 3.773 3. 7btf 3.76? 2.6J3 39. lit 3 . -.3C 2.632 J.931 2.^73 2.I-T4 r.b7j 2.672 2.„70 2.669 3-- . 340 3fe.?-39 ?C-.*J8 36.487 36.336 3*. 385 4. 150 *. 1*« *. 1 36 *. 133 *. 127 * . 1 22 2.631 2.630 2.b49 2.b4fi 2.b«7 2. 6*6 J7.i23 3^-373 3V«22 37.*72 37-521 J?.37' *. 040 4.033 4.0 30 4.024 4.019 4. On 2.ui8 2.6i7 ;.62b 2,623 2 . o2« 2.623 3^.'i5 36--'o3 38.415 J6.4t5 3B.5i5 38.365 J. ■-> 3b r. 9]1 J. "25 J.".0 J.<*13 3. 9i0 .;0' 2. uOC 2-605 2. L-04 2.bPJ 2 . bO ' 2-600 '.2i*3 V3.;i6 3-}. 3 fa( , 3y.4l to 39,466 1<-.517 39.3*7 .840 J.tJ6 (.331 3.626 3.B21 J. Bib 3. 611 2. 5&6 3.?58 2.587 •0. '73 • 3.754 .385 i.*.i34 2. 3^! 2.58; 1.27* 40. 3.3 40- '7y *o. 42b ' .744 J. 7,0 3.735 3.730 2.^60 2.B79 40.32? 40. 378 3.721 3.71* Velocity — Velocity in Kilometers per Second Period — Period in Hours Ang. Vel. — Angular Velocity In Radians per Day IH-56 a) o W ■a c ai JS i -*-■ X! 3? s 420 246 50 -:- 24700 400 ■-.- 380 - ■-- 24750 :r 24800 360 - *T 24850 340 if 24900 320 -r i 300 280 .;■ r- 24950 260 240 220 ■ 200 - 180 " 160 :i 140 -!! 120 100 -: 25000 25050 ■■- 25100 25150 25200 ■-• 25250 25300 25350 ■■•25550 25450 25500 ■ 25600 rr 23650 23700 760 740 f- 23850 720 z 700 *- 680 660 t 640 620 600 580+: 560 540-::. :i "24100 25400 520 f : . 24400 500 480 ■!! 460- 440 ■:! "24550 ^25650 1180 1160 23900 23950 24000 24050 •H .If it 22900 ■--' 22950 23000 -•24450 24150 24200 24250 24300 24350 980 880 r-24500 24600 24650 820 ■22750 t- 23100 23150 r- 23200 23250 i- 23300 23350 23400 tt23450 '-•23500 rr 21650 it 21700 21750 ft 21800 21950 r + 22000 23050 U 22050 ■tl T- 22100 i- 23650 22150 ■■* 22200 -t 22250 ;-22300 22350 22400 ■■- 22550 22600 22650 Fig. 8. Velocity of a Satellite in a Circular Orbit as a Function of Altitude (Lim-lish Unit - see Table 9 for Metric Data) III- 57 3 o o o r— ( > o rH o O ni o en T3 C cS s: bjD •r-l «r 20650 2250 ::t 20700 t- 20750 2200 ft! 20800 20850 2150 34,20900 2100 2050 2000 :■- 1950 1900 -: ■:-21400 2450 '> 1850 '!■ 1800 ! a — 19650 20950 21000 21050 21100 4J-21150 1-21200 21250 ! ; -21300 21350 2900-:! 2850-;; 2800t 2750- 19700 19750 3700 4- 3650. 3600* St 19800 19850 3550& 18850 19900 2700.. 2650 f : 2600 |: 19950 ■U 20000 H20200 2550 2500 ■'■.'. T20350 :-■ 21450 2400 -13 1*21500 21550 2350 - 1700IJ21600 -^•21650 2300 - :: 20050 20150 ■20400 ;t20550 f~20600 {:.i 3500! t- 3450 t 0? 20100 335 ° 3300 .: 3250 20250 20300 3200*19300 8900 3*18950 19000 *4 19050 19100 -7 19150 I 19200 H. 3150 3100 I .;?20450 "H'20500 3050llf 19500 3000 19350 19400 19450 19550 ffi-20650 2950tti 19650 Fig. 8. (continued) 4550J 4500f 4450! 4400.: 4350 4300 17650 17700 17750 17800 17850 17900 17950 *18000 4250 42 00 T-1 4150 4100 T T f 11 -ft 19250 4050 t- 3950r : 3900 + 38504; 18050 M18100 ii P18150 Ii 18200 18250 4000* 18300 r*18350 |t 18400 "4-18450 it 18500 .il I! 18550 " : '19600 3750 if 186 00 ^18650 III-5{ I o w ■a c n! u o > U o a i 5600 ■16700 16750 5500ffl. 16800 5400 :;■; :g- 16850 16900 5300:: 5200 ■'■'-. 5100 16650 16950 t 17000 17050 £ 17100 17150 :|17200 6200J 5000 i-17250 4900 4800- 4700 4600 ■£■17300 17350 ;£I7400 17450 B17500 17550 17600 ^17650 6900 ; 6800::: 6700:i: 6600 :;; 6500: 64001 :. 6300 : : : 6100:;: 6000' 5900 5800 '■:- 5700:: 15650 15700 15750 15800 15850 15900 15950 16000 16050 16100 16150 16200 16250 16300 16350 16400 16450 ft 16500 16550 16660 14650 8400ii 1470 o 8300 ft 14750 8200-1 J14800 8100 8000 7900 7800 7700 76001^ m 14850 14.900 14950 15000 15050 15100 ■#15150 15200 7400 ■•■■ 7500 ■!* 15250 7300 ■;; 15350 .ft 15400 7200 .;"15450 8800 ■ ■■ ^^so 1530C ■""15500 7100 # ■15550 7000 -ffi-15600 15650 10000 flj 9900 9800 9700 9600 9500 -ed 9400 9300 9200 9100 9000 8900 14300 14350 8700 8600 8500 14500 14550 14600 14650 Fig. 8. (continued) III- 59 Semiperimeter, S/r CD <1 C < a! U c u 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Shape Factor, W Fig. 9. Parameters of Lambert's Theorem 0. 8 0. 8 1.0 III -60 1. -0.6 0. 1 0.2 0.3 0.4 Mean Anomaly, n At J s Fig. 10a. Lambert's Theorem (case 1) 111-61 0. 6 0. 8 1.0 U HHHmrumffM 0.1 0.2 0. 3 0. 4 0. 5 0. 6 Mean Anomaly, n„ At Fig. 10b. Lambert's Theorem (case 2) 111-62 1.0 f 0. 7 a a 0. 8 0.9 1.0 Fig. 11a. Solution for Eccentricity 111-63 0.2 0. 1 0.2 0.3 0.4 0.5 O.fi Eccentricity e 0. 7 0.8 0. 9 1.0 Fig. lib. Solution for Eccentricity 111-6 4 20.0— j 19.0- 1B.0— 17.0— r ' a (1 + e) r * a U _ e P a 1 + p r 1 - p P ■2. -3. 16.0— - 4.0 15.0 — -5.0 14.0 — 13.0- E 12.0- 11.0- X 10.0- a. o.o — 8.0- 6.0- 5.0- 4.0— 3.0— 1.0 0. 2 0.4 0. 6 0.8 1.0 — a.o Fig. 12. Solution for Apogee and Perigee Radii 111-65 ■} cm o t~ in w o I I I I I I I I I I I I I I I I I I l l I I I I I I I I 1 I I I I I I I I I I III! i i i i i I I I I I I I UTUUU II 1 tuttut 1 mum / 1 // Umttt :t::: : I 7, mutt t J 1 // mutt t" J 1 / uttttt r tuttt 1 tutu P uttuL u tutu r ^~ ' tuttt 44 " Utttt 44 " T ttttt 44 " *\J*- ttutt ttt //m i ttt tut tut ' tt / n: / / / / (Sap) e 'X^BUiouy 3nj X III -66 180 :;:ni . o_. o . o o o o p P P P 0. 1 0.2 Eccentricity, e 0. 4 Fig. 13b. True Anomaly as a Function of r/a, e and y III -67 180 >I2 - r/a = „ „ ooooooooopopo op pp • - -J -J °"» ? a 5 jj, ^Mooaa ii. o 1 »J.g- = V2 - r/a Fig. 13c. True Anomaly as a Function of r/a, e, and y 111-68 (Sap) e "as^Tjad uioa; a^uy ibjju33 J I I I I I I I I I I I L (JJap) a '/-[Buiouy OTJIuaaaa 111-69 Radius (ft or m x 10 ) 14 12 10 8 6 4 2.5 2.5 4 6 8 10 12 14 2.1 2.1 Q-(7) c Fig. 15. Q-Parameter as a Function of Orbital Semimajor Axis and Radius III -70 2.0 1.8 l.G 1. 4 1.2 1.0 0. 8 0. 4 0.2 1. 2 1. 1 1.0 0, 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 True Anomaly, (deg) Fig. 16. Relationship Between Radius, Eccentricity and Central Angle from Perigee in an Elliptic Orbit 111-71 1 IV \\ v \ ' A L A \ ) \\ \ \v \ \ \ i K e^- 9 w A A Y \ A s \\ \\\ V A \ \ <j Z'9 \ A A v \ , \ \ \* o A v A \ M x CD \ A v V V' — i v A u \ s" 7 ? ^ \ \\ k ' gTTr N \ \ \ \, u s o \ \ o*g m r^ > sp? Note: The values for the lines of constant r It are equal to the values of the abscissa r It at the points where a p the two intercept. s-* 2> o O'fr t SL-E ire" c, u a u (X u n) U o Hre <n + ni| a. sp? ii w o / Z'Z / / / p < < ( O'Z r 1 U A cb O O iH -tA "-H ^ II - Sn ni ff^ y OS'I o < i. ( D O o D 4) 4) s- (Sap) X. 'aiSuy H^a mSHJ IBOoq III- 72 (gap) a^Suv m^d imHti j I i i I i I l l I ll i l i iIiimIiiiiIi.iiIiiJ I m c <D In ■O -M -3 ••-1 ;d M a En SO 4) tl 0) o <t-i W S £ (, E x ^<~luj) a;i.j(j[3j\ f ■ ■ < ' '/' T ' V ' f ■ ' ■ ' ■ ''i ■ ' ■'■ i'i 'i ^i m to r- co e» O m o ir (01 * sri J) K1100I3A III -73 >l> (dog) Fig. 19. Q-Pa and rameter as a licccntricity Function of Local Flight Path Angle III -74 I- 45 30 20 10 -20 -30 # e = 0.5 Pi fffl "I 1 :i li !!!! 1 [jit n i :; it •:!• if 1 ijt i$ )m IF 0.25 liii jlP •r 1 l!:i I it II !=ii r b ti- ll liii III M ill ;;j| til tilt illl 'III h 1^1 jji ill ■ : !l H:: li- lit? & a ill; l' ! ill! 1 i=ii fe Ill; ill! ill '■■■ 90 180 270 360 8 (deg) Fig. No. -i~ii ERNo. — — Job No. I- 1*4 I'* Drawn P.U. — -30 --35 I- -40 Fig. 20. The Solution for Local Flight Path Angle III -75 180 1.0 2.0 Mean Anomaly, M (rad) 3.0 Time from perigee: (t - t ) = — M P n where - is tabulated as a function of a in Fig. 7 and Table 9. Fig. 21. Index for Figs. 22a Through 22i' (circled numbers in field designate areas covered by corresponding figure numbers) 111-76 Eccentricity O '-C CM CC Tf* O00 Tl* O tD EN033tC-fMO<C(Oi"^ O CO UJ't MO CO «3 -^ CN O 00 CO -r CM cjj cc cc t~- r- c-co co to in mcif ^tt^nnnnn NNNNNHHrtHnoooo oooooooo o o oooooooooooooooooooocooooo o 0.1 0.2 0.3 0.4 0.5 0.6 Mean Anomaly (rad) 0.7 0.8 0.9 Fig. 22a. Mean Anomaly as a Function of liccentricity and Central Angle from Perigee III - 7 7 Eccentricity O CO fg 03 rt" C CO C-J CO cc t- t— cc co co 10 in «y -r o cm c oooooooooo o © © CO -Cf C] C cc to 1> CN O cc c co co co crj c-j cm cm cn c\) •— 0.08 0.05 0.10 Mean Anomaly (rad) Fig. 22a' Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee III -78 100 80 I 60 o K 'r -r r-i c :c ■- t mo :c trs .-. ro cc r; ;c 33 r- l- t- t- t~ er. ir> 1'a t'l.'Il! t'k'ity !■■] O m uo i7 ic -r o o .' ' - f : / ■ / o o o / / ■:/■ /■ '/: / 7 7 /:■ / ;/" / • / / 1 •' '' -' / / / / ■ J / ■;■'■/ •/-■/, : ■ . , , , / 7 / • / ■■/ / / / r....; I if... -J.:. J. ... J.. . 7- /■!■/-:/-. o.4o ? 7! / .'! / 7 / ■ f / : I !/■ / /: / 7 / 7,/ 7 7 /• : 7 /■//:// / / / .•' ./ 7 / ■/■ / 7: / / 7 /• / 7 A 38 : < / / / ■/ 7 ; / / / ; / . i \l / ' / / 7 / 7 / / / ; ///// //-/■/■://■■/■/ //!// 7 ■/■///■■/ //- 36 1 '■ r ■ -i ' i ''■•'■•■/ /•■ 4-../..;./-..-^- /--/■■■/••■/ / '.•' '~7 ■■/ - /■ 7 ■,, :!■!!; ! ! : ' : / I / : / / /' ' / / 7 / : / / /: ' I t ■■' ' ' / .' / ' / / t ! ' f f ■' '' '■ / A * i ' i 777 W ;-t f ;>U : , i,: ;iul /'/' /:/ /V /77// / ////// / /=/ / /"" /■/7 7 //■/•/ / ' / ■/ 7 7 A/7 //■/■ / 7/' / /-7 i A//// /■/ /■//////// /:///:/ / / / / / ,o.2o t ^ ; / < V •» / / -/ /■■•■/■-,• - h ■+-)-■+ 7--r ■■-/■■* ■•/- -/ -7--/-/ /■-.-'■■ <0.1« ' .' ' ' / / ; ///// / ///;// / / / ■ / / / ; ' ;/ / / / " M7 i ////7//7///!///7/7///7^7/ / : 7; :; 777/ ; J if f / f? //■/■/// / /;// // 7"-'" / \ \ I m 7/ / ./ / /■ / / / / ■■'!]//■ I I I / / > ■/] / / / / 7 J - l - ;;■;;■// ■' '' ////// ; 1 1 / / // /V 7 / //: / 7uo illl 11: ! If /. /:/ ; 7 / / ,'./:/. 7}/./;./ A/ / ./ /./.1/..7- / / -7 777;/'////// / / / /;/ // / f// / 7 / / / 7 / / 7 7/ Jo.om 777/7///F II I//./ 7/V//// ///■■/ A// '//A - 04 ■ 77 77//7//;//7////L//7 /;//■/ ////// ,• ^ J o.i 0.2 o.:? 0.4 o.r> / 7 77 7/ /■/ / / / / 7 7 o.c 0.7 0.8 0, o.:? 0.4 o.r> o.( Mean Anomaly (rad) Fig. 22b. Mean Anomaly as a Function of Kccentricity and Central Ancle from Perigee III -79 Eccentricity 150 140 .. 130 "3b U 120 100 no ..— .- 0.3 0.4 0.5 0.6 Mean Anomaly (rad) Fig. 22c. Mean Anomaly as a Function of Eccentricity and Central Angle from I'erigee III -80 T-i r-i ") ~i --J rry-7 / /// / / /■////.'// 7'/7777r / 77777V7T / ///'/// ' ■ V///V ///////// A '////////7////A ■ / /// / / 777Y/77777'77777 //' -■' // /// A '/ S////7 // ///////////////// ///// . ■ ■' / / A A / - / //A / AA //A //// / / '■'/ // ■' - ''/A / '/////// ''/////// / ///y/////////y / ^/////// ■ : / / / • / / ,' f / / i / / / / / -' / .' , / * ' ■ r / y .- ' / / /,-.';• ■■////>■., A/ A/// // '// A I ■ / ' / / -V - ,'' / //>'///.. ' ' /' - / ,■ / 7. '■-'/v •■■■;'/: ///A A 1 / / / / ■" / , 7 A A /•■ / /. / / , ,• / 1 ' / ■ ,- ■' r Fig. 22d. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee 111-81 .^CC'lUrUM! y "# :m 140,. 9 OS JO T M 1.0 in ir; m o o y- -7- ;'-' y ,r s / ■*■/. ,//// „ . y / , s / / ye ■ ■ / '//"', '-■■/.- v///////^y^//////AZo // / / ^ - ' ' ■///sy//////////Y'' / /A / -4 / //* . .'. / / / / /■ / / s / ■ / . / ,.' /' /////y///^///^/////////V/////^/////^ / //////////y////////////Ay/y/A^//^- n ■: // / / / / / •' / // / / / / // / / / '/ ////////'//// /// •- > v, '/'A ■-/. ■'/////<////// ■y///////////////o :> ' / / /// / // V • ■'///// ////////// //.//. / ', / , /',•/.■/ .///// //y/////^'///// //a <//////////7//////. l // ////////////// ////// /y / /////////////// • iwit / / / / / / W///// / // , ' ,// ' ,/ ' / / / t / / ,■ / '' ,■■■ // / / // 7 / ,■■/// / ' //////////// / ///A'// ■" - f ' ■' ' '///// ///// ////. 'J- 10 ,■' / 08 0.06 / / / ■ . ' / / / ,■ ' / /'/ .'/'/ ■'////>//#//Ay^ 7'A///u//-/////////////yfy//////7" "' ■ ///////, '' / ' ' ■ ■ : / ■ .' / ////// / / / / , /' / / / / / / / / / // / '-' / / 90 0.8 0. 1! 1.0 1.1 1.2 1.3 1.4 1.3 Mean Anomaly (rad) Fig. 22e. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee III -82 .80 170l 160 i 150 140 130 0.5 0.6 o. a o. 9 l.o i.i Mean Anomaly (rad) i. :•! . 4 Fig. 22f. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee III -83 ,ceun; rielty !)0 1.6 ] . a 1. 9 2.0 2. 1 2. 2 Mean Anomaly (rad) 2. 3 2. 4 Fig. 22g. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee HI- 84 10. 00 0. 8(i 1.7 1.8 1.9 2.0 Mean Anomaly (rad) Fig. 22h. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee III -85 2.30 2.40 2.50 Mean Anomaly (rad) 2.60 2.70 2.80 2.90 00 3. 10 Fig. 22i. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee 111-86 Mean Anomaly (rad) 3.0 3. 1 -- .1 80 ■|17!/ im I7G 175 ■- 17-1 173 172 171 . 1 70 160 Kit! 0. 10 Fig. 22i'. Mean Anomaly as a Function of Eccentricity and Central Angle from Perigee III -87 CHAPTER IV PERTURBATIONS Prepared by: G. E. Townsend, Jr. Martin Company (Baltimore) Aerospace Mechanics Department March 1963 Page Symbols IV- 1 A. Introduction IV-2 B. Special Perturbations IV-2 C. Methods for Numerical Integration IV-7 D. General Perturbations IV- 14 E. References IV-50 F. Bibliography IV-5 2 Illustrations IV-59 LIST OF ILLUSTRATIONS Figure Title Page 1 Comparison of Perturbation Magnitudes (for equinoctial lunar conjunction) IV-61 2 Solution for the Secular Precession Rate as a Function of Orbital Inclination and Semi- parameter IV-62 3 Change in the Mean Anomaly Due to Earth's Oblateness IV-63 4 Solution for the Secular Regression Rate as a Function of Orbital Inclination and the Semi- parameter IV-64 5 Change in the Anomalistic Period Due to the Earth's Oblateness IV-65 6 Change in the Nodal Period Due to the Earth's Oblateness (for small eccentricities) IV-66 7 The Variations of the Radial Distance as Functions of the True Anomaly e and IV-6 7 8 Maximum Radial Perturbation Due to Attraction of the Sun and Moon IV-68 9 Satellite Orbit Geometry IV-69 10 Effects of Solar Activity on Echo I 11 Apogee and Perigee Heights on Echo I (40-day interval) IV-70 12 Minimum Perigee Height as a Function of Days from Launch, Showing Effect of Oblateness, Drag, and Lunisolar Perturba- tions IV-71 13 Minimum Perigee Height of Satellite as a Function of Days from Launch (8 to 14 hr, expanded scale) IV-72 IV-ii LIST OF ILLUSTRATIONS (continued) Figure Title Page 14 Comparison of Approximate and Exact Solutions of Satellite Motions IV- 73 15 Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Neglecting Oblateness IV- 73 16 Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Neglecting Moon, Sun IV- 74 17 Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Sun and Moon 90° Out of Phase IV- 74 18 Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Changing Orbit Size IV-75 19 Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Change in Inclination IV-75 20 Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Change in Argument of Perigee IV- 76 21 Minimum Perigee Height of Satellite as a Function of Days of Launch for About a 10-Year Period # IV-76 IV-iii IV. PERTURBATIONS A a B C, 1 J n L M SYMBOLS Right ascension, area Semimajor axis Ballistic coefficient C,.A/2m Drag coefficient Eccentric anomaly Eccentricity [6, Universal gravitation constant |_6.670 (1 ± 0.0007) x 10" 8 cm 3 /kg-sec 2 ] Magnitude of the angular momentum per unit mass; step size in numerical integration Orbital inclination Coefficients of the zonal harmonics in the Vinti potential Latitude Mean anomaly n(t - I.) - E - e sin E Mass Mean motion = 2tiIt Vw^ Semilatus rectum = a( 1 - e ) Perigee radius = a(l - e) also quantity in Encke's equation - 1 R, S W Vehicle coordinates R - r, S normal to R in the plane of instantaneous mo- tion (S • V - positive number), W com- pletes the set Radius dr ar t u V x, y, z r, r p r c 1 :<%< 2 "> d r/dt" Time Potential function Velocity vector Equatorial Cartesian coordinates Angular coordinates of perturbing mass cos _1 (r • V) - 90° -/u/2a = energy per unit mass Dimensionless parameter - 1 True anomaly = GM - masses' gravitational constant Disturbing potential Perturbed 4> in Anthony and Eosdick theory (x - x i ), (y - y J, (z - y, ) in Encke' s solution--also vehicle-centered coordi- nates Orbital period Time of perigee passage- Central angle measured from the as- cending node Right ascension of the ascending node Rotational rate of the earth, 1 revolution every 8(i, lf>4.001 mean solar seconds Argument of perigee IV- 1 A. INTRODUCTION The Keplerian relations, as discussed in Chapter III, give convenient approximations for use in preliminary orbit computations. However, in order to obtain precise earth satellite orbits, the various perturbing factors which give rise to accelerations (in addition to that of the central force field) and cause the motion to deviate from pure conic form must be considered. These per- turbative accelerations may be due to the mass asymmetry of the earth, the gravitational attrac- tion of other bodies, atmospheric drag, electro- magnetic drag, radiation pressure, thrust, or may be required to account for relativity effects. These factors affect the motion of the satellite to a varying degree depending on the shape and mass of the satellite and the type of trajectory. Special perturbation methods involve the formulation of the differential equations of mo- tion in such a manner that the computation of an orbit is achieved by numerical integration. The perturbation method to be used is determined by the type of problem that is under consideration. Similarly, all combinations of integration tech- niques and perturbation methods are not equally suited to the solution of a particular problem, even though the use of such combinations is pos- sible. Because numerical integration is subject to the inevitable accumulation of errors which eventually destroy the validity of the results, special perturbation methods are restricted to the prediction of earth satellite orbits for times dependent upon the desired accuracy, the for- mulation of the problem and the number of digits carried in the computations. One source of error in the numerical integra- tion process is roundoff error, resulting from the limited number of digits which can be carried in computation. The roundoff error is not reduced by double-precision computation where tabulated values to be interpolated at each integration step are known to less than single-precision accuracy. This error obviously increases with the number of computations, which in turn increases with decreased integration step size. Roundoff propa- gates through the numerical integration so that, assuming a normal error distribution, the absolute error incurred in double integration is (the product of the number of steps and 3/ 2 the original roundoff) r ~3" r + F (1) where F is the sum of the accelerations due to the various perturbing forces. If F = 0, there are no perturbations and the motion is Keplerian. If the position coordinates of the vehicle and the perturbation accelerations are given in rec- tangular equatorial coordinates, Eq (1) can be written: x y°° (2) Z x i here ) x. is the sum of the perturbation ac- celerations. These terms are discussed in the following paragraphs. a. Vinti potential If the earth were homogeneous in concentric spherical shells, its potential would be that of a point mass. The effects of the flattening of the poles and lack of symmetry about the equator, however, manifest themselves as perturbative forces on satellites in the vicinity of the earth. The acceleration due to the oblateness of the earth can be written in a simple form attributable to J. Vinti of the National Bureau of Standards: yx 3 (?)' r ',M£)'i(>-'£ + J, R\ 4 5 3+42 ~2~ r 63 r + J 5(f) 7 4 1-693^ + 630^-105 r ~1 r (3) A second source of error is truncation. This error arises because of the finite polynomial approximations in the integration formulas. Since the terms in the polynomials involve powers or differences of the integration interval, the trun- cation error can be reduced by choosing a smaller integration step. Therefore, increasing the num- ber of integration steps decreases the truncation error, but increases the roundoff error. B. SPECIAL PERTURBATIONS 1. Perturbative Forces The equation of motion of a perturbed orbit is of the form: 1X2. -^ r + J , ^\V\l + j,m -= 693 6 z ~6" r r 3 5* r 1 + 10 z 35 z r r f) 4 | (- 15 + 70 V 63 7, £) 5 a("-3»4 + °«4 IV-2 where J. are the harmonic coefficients. Since the earth is almost spherically symmetric, the J i are all small compared to 1 (see Chapter II). b. Perturbative terms due to remote bodies The perturbative terms due to remote bodies which can be considered as point masses can be written directly from the integrals for the n-body problem as developed in Moulton (Ref. 1) and in other texts on celestial mechanics. 7 ■%<%■?) " \ / y Ai * y =2 V (4) i=l \ r Ai z. l ~7 where r A - is the distance from the satellite to the ith body and r. is the radius from the center of the earth to the ith perturbing body. For the case of an earth satellite, lunar and solar attractions are the major sources of perturbations for short term orbits. The order of magnitude of these perturbing forces may be observed in Fig. 1. (Subsequent discussions appear in Section C of this Chapter. ) c. Thrust If thrust is applied, it may also be handled as a perturbation. The general procedure, how- ever, for large thrust-to-mass ratios is to treat the thrust periods in a different fashion by con- sidering the vector sum of the thrust and central force terms as defining the reference trajectory rather than the central force term alone. Since the thrust vector is determined by the maneuver requirements and the guidance law to be utilized, no analytic solutions are available for this ref- erence trajectory; thus, numerical integration is necessary. Indeed, no single form of the per- turbing acceleration can be written other than its resolution in terms of generalized vectorial com- T T T ponents; for example: — — , — £- and z mm m ' d. Atmospheric lift and drag (Ref. 2) x -- D 2 s 2 J|*t'Y <v) a(H) Y (or) vl| - H< A + - A< Wv) o- (H) Y M -^ j^- \(r x l) sin g * <i* ML H} cos £ > x -» y, z (5) where the vehicle velocity relative to a rotating atmosphere with cross winds is given by v x = x +y n e + q (cos a sin 4>' cos j3 + sin a sin /3) v y = y - x" e + q (sin a sin <t> • cos £ - cos a sin £) v z = z - q cos <t»' cos /3 where A «= constant fitted to the Mach number variation of the drag coefficient with a mean sonic speed = 1 A * Initial projected frontal area of o the vehicle, m B -D„ v- f(r) = H m ■ Q - q r ■ s >= constant fitted to Mach number variation of the drag coefficient with a mean sonic speed /C T reference (hypersonic continuum) value of the drag coefficient (0. 92 for a sphere, 1. 5 for a typical entry capsule) lift coefficient local sonic speed in terms of sur- face circular satellite speed C D A 0<>0 V C0 2/2 So m O 2 " D ctv(ct) acceleration of gravity at unit dis- tance (surface of earth) altitude above an oblate earth « r - ' + + fsinV +^- (I - Jj sin 2 2<t» where the flattening f ■ 2^-3 (units of earth radii) mass of space vehicle (kg) unit vector in the orbit plane perpen- dicular to the line of apsides speed of the cross wind measured in a system rotating with earth's angular rate (units of surface circular satellite speed V CQ ) radius from the geocenter to the vehicle speed of the vehicle with respect to an inertia! frame, directed along Q IV-3 d-3. v co x,y, z = a - - V(v) = V(a) = Q e = M' = £ = p <T = <f,. = Surface speed for circular orbit -- 7905. 258 m/sec equatorial coordinates in units of equatorial earth radii right ascension of the vehicle (radians) azimuth of the direction from which the wind is coming C (v/C )/C„ , the drag coefficient D s u o variation with Mach number C„ (<r)/C n , the drag coefficient varia- D L> tion in the transitional regime constant relating to the rotational rate of the earth, 0.058834470 m Q /m bank angle 3 atmospheric density, kg/m "sea level" atmospheric density, 1.225 kg/m 3 P p geocentric latitude, radians e. Radiation pressure A body in the region of the earth is subjected to solar radiation pressure amounting to about 4. 5 x 10~ 5 dyne/cm 2 , the order of the force being the same for complete absorption and specular reflection of the radiation. Radiation pressure is an important source of perturbations for satel- lites with area -to -mass ratios greater than about 25 cm 2 /gm. The effects of radiation pressure on lifetime are discussed in Chapter V and also in Section C-7 of this chapter. The rectangular coordinates (X-axis toward vernal equinox) of the accelerations are: x = f cos A v = f cos i sin A •> z = f sin i sin A @ (6) where: i = inclination of the ecliptic to the equator, ° 23.4349° A = mean right ascension of the sun during the computation 4. 5 x 10" (=)■ f. Electromagnetic forces As a satellite moves through a partly ionized medium, the incident flux of electrons on the satellite surface is larger than the ion flux, so that the satellite acquires a negative potential. On the day side of the earth, this effect is op- posed by the photoejection of electrons. Jastrow (Ref. 3) estimates that the satellite potential may approach -60 volts on the day side and will not be greater than -10 volts on the night side. In addition to the potential acquired by ionic collision, the motion of a conducting satellite through the magnetic field of the earth causes the satellite to acquire a potential gradient which is proportional to the strength of the magnetic field and the velocity of the satellite. The inter- action of the electric currents thus induced in the satellite skin with the magnetic field causes a magnetic drag to act upon the satellite; this drag is proportional to the cube of the satellite dimen- sions. If these forces are found not to be negligible, they can be included directly by the use of Max- well 1 s equations or indirectly by use of an at- mospheric model which takes the effects into ac- count. g. The effects of relativity Perturbations caused by relativity are of the V 2 order a = — E- = -Ay , where c is the speed of c re light. Since a is a very small quantity and any measurable deviations occur only after a long period of time, relativistic effects can usually be ignored in the case of earth satellites. A mod- ification of Newton's law as a consequence of the theory of relativity can be found in Danby (Ref. 4). Substitution of these perturbative accelera- tions (a through g) in Eq (2) yields the complete equation of motion. 2. Special Perturbation Methods Three special perturbation methods currently used for computing earth satellite orbits will now be discussed with an evaluation of the main ad- vantages and disadvantages of each. a. Cowell's method In Cowell's method, the total acceleration, central as well as perturbative, acting on a satellite is integrated directly by one of the numerical integration techniques (Section B of this chapter). The equations of motion which must be integrated twice to obtain position co- ordinates are: MX r £ ° x °i' x -y. z - T- These equations are symmetrical in the rec- tangular coordinates and are simple in form; they apply to elliptic parabolic and hyperbolic orbits, and require no conversion from one co- ordinate system to another. IV-4 A disadvantage of the method is the large number of places which must be carried because of the large central force term to prevent loss of significance for the small perturbations. Also, since the total acceleration, which is subject to fairly rapid changes, is being integrated, it is necessary to use a smaller integration step to maintain a given accuracy. This requires an increase in the number of integration steps and the inherent roundoff error accumulation. De- tection of small perturbation effects such as those caused by radiation pressure may be im- possible due to roundoff and truncation errors. Cowell's method is especially useful when the perturbation forces, such as thrust, are of the same order as the central force. b. Encke's method In the Encke method, only the deviations of the actual motion from a reference orbit, which is assumed to be reasonably close to the actual orbit, are integrated. Usually a two-body ref- erence orbit is used since the position at any time on this orbit can be determined analytically. How- ever, more complicated reference orbits such as Garfinkel's solution (Ref. 5), which is known analytically and which incorporates some of the oblateness effects in the earth's gravitational potential, might be used on an earth satellite nrbit. Let x, y, z denote the actual position of the satellite and x , y , z the position on a Keplerian e J e e reference orbit. Because of the possible loss of significance in subtracting nearly equal quantities in Eq (10), it is necessary to rewrite Eq (10) in better compu- tational form. Substitute Eq (9) into the defining equation for 2 4- 2 + 2 x + y + z = (x e + ?) 2 + (y e + r,) 2 + (z e + ?) 2 = r e 2 + 2 [C(x e + i?)+ n(y e + in) + ?(z e+ Jf)] Define q to be: q = -V [ ?(x e + 7 5) + T ><y e + £' l) + ?(z e + i e>] So that Eq (13) becomes 2 (11) (12) (13) fc) = 1 + 2q or m (1 + 2q) 3/2 (14) (15) Encke's series, using a binomial expansion, is defined by: The equations of motion in an inertial frame of reference are then: ^ + I * i e r x -y, z x -» y , z e J e e (7) (8) Let the deviations from the reference orbit be ?, n, ? so that: 1 y - y e (9) Differentiation of Eq (9) and substitution of Eqs (7) and (8) into the result yield: x -*■ y, z for f -* r\, ? I *i < 10) x ~3" r i 3-| - O + L x i -m 1 - (1 + 2q) -3/2 CO I ( _ 1)k -l(2l lii )) q k k=l -1/2 < q < 1/2 2 k (kl)' fq (16) Substitution of Eq (16) into Eq (10) yields Encke's formula: f = J±^ (fqx - €> + £ *i (17) This equation, which employs series expansion, yields more accurate deviations when the terms are small. When the terms exceed a certain limit, a process of rectification is initiated, that is, a new reference orbit is computed. The limits on q needed for rectification are estab- lished as: n+1 AC Vi (18) where Af is the allowable error in | and a . is the coefficient of the first neglected term of the Encke series. IV-5 In contrast to Cowell's method, only the dif- ferential accelerations due to perturbations are integrated to obtain deviations from a two-body orbit. These deviations are then added onto the coordinates of the satellite as found from the two-body orbit to obtain the actual position of the satellite. Since the deviations are much smaller and, therefore, need not be determined as ac- curately, it is possible to maintain a given ac- curacy with larger integrating steps. As a con- sequence of the larger integrating steps, there is less danger of serious roundoff accumulation. Moreover, the integration errors affect only the least significant figures in the deviations and, when added to the much larger positions deter- mined from the reference orbit, should have a less serious effect on the overall accuracy. Al- though the roundoff error is less, Encke 1 s method involves expressions that are much more complicated and often less symmetric than Cowell's simple formulas. In addition, both the necessity of solving the two-body formulas at every step and the possible need for rectification introduce additional sources of error. In the former case, the frequency of rectification af- fects the attainable accuracy and also introduces small errors in the determination of the mean anomaly M. For the case of nearly parabolic orbits, errors in the use of the two-body formu- las in an unaltered form are especially critical. This is due to the fact that when the eccentricity e ~ 1, and the eccentric anomaly E is small, can- cellation errors arise in forming the radial dis- tance r = a (1 - e cos E) and the mean anomaly M = E - e sin E. In addition, small division er- 2 rors will be introduced in forming p/a = (1 - e ). The Encke method is especially suited to problems in which the perturbative accelerations are not large and have their major effect over a limited portion of the orbit, e. g. , lunar and in- terplanetary orbits except microthrust or long- thrust trajectories. c. Variation-of-parameters method The variation-of-parameters or variation-of- elements method differs from the Encke method in that there is a continuous set of elements for the reference orbit. The reference motion of the satellite can be represented by a set of param- eters that, in the absence of perturbative forces, would remain constant with time. The perturbed motion of a satellite may thus be described by a conic section, the elements of which change con- tinuously. The variable Keplerian orbit is tan- gent to the actual orbit at all times, and the ve- locity at any time is the same in both orbits. This reference orbit thus osculates with the ac- tual orbit. The variations in the elements used to describe the osculating conic can be integrated numerically to solve for the motion. Any set of six independent constants can be utilized for this purpose though it is conventional to use the geometrical set a, e, T , u, f2 and i. Lagrange's planetary equations, which specify the variations for this set of parameters, are derived in Section C of this chapter. It is also possible to choose a different form for the reference motion. As in Encke 's method, Garfinkel's solution which includes part of the perturbative forces caused by the nonspherical shape of the earth might be employed. If the drag force predominates, as in the case of entry, a rectilinear gravity-free drag orbit as applied by Baker (Ref. 6) can be used instead. Many variation-of-parameters methods have been proposed including those of Hansen, Strb'mgren, Oppolzer, Merton and Herrick. These methods differ in the choice of elements or parameters and of the independent variable. Of these, the parameters suggested by Herrick (Ref. 7) will be briefly described here. Let x , y be rectangular coordinate axes in the instantaneous orbit plane with x the axis along the perigee radius as shown. Let P be the unit vector in the orbit plane in the di- rection of perigee, Q be the unit vector perpen- dicular to P in the direction of motion along the y -axis and W be the unit vector normal to the orbit plane in a right-hand system. The parameters selected by Herrick for or- bits of moderate eccentricity are vectors A(t) and B(t), the mean anomaly M and the mean motion n. The vectors A and B are defined by: A = eP B = e y p Q M = n(t - t Q ) where a = semimajor axis e = eccentricity p = semilatus rectum k = VgM IV-6 The differential equations in the parameters have the form: A = A Q + k g \ A' dt s B = B Q + k e \ B 1 dt s n(t) n +k eJ n ' dt t M(t) = M Q + n Q (t - t Q ) + k g CC n' dt dt t k 4 + k \ M dt X and the perturbative variations A', B\ n 1 , M' are defined as: D = e ya sin E -* B H = e x = -r • A CO Y^Td' = r • F = xF + yF + zF * x J y z »T ■»£•*■»(£ '.♦£',♦&'.) H' = 2 DD 1 - r 2 dD 1 ~3T ^ r—W dH' _ r-D' r ^L - E£ H' - F H ^' - 'T - S V7A.=r- dD' dr "ar D' - F D e J~pV = A • B' = A B' + A B' + A B ' T x x y y z z ja~M> = yp~v' -2D 1 . _ 3 n a dD 1 The Herrick elements must be related to the rectangular coordinates and to the usual elliptic elements because the perturbative forces F are given in rectangular coordinates. It is thus necessary to go through the two -body formulas at every step, as in the Encke method, and through some complicated conversions as well. The essential characteristic of this method is that the integration is carried out on parameters which are much more slowly changing functions of time than rectangular coordinates. Since they vary slowly, the error accumulation from the calculation of the derivative is, for a long time, far beyond the eighth significant digit of the initial calculation. Thus, it is expected that truncation error would appear only for very large intervals and much larger integrating steps can be taken for a given accuracy. Since in this method a system of first order equations is being integrated, there is less danger of round- off error accumulation. A disadvantage is that the programming and numerical analysis in- volved in this method are the most complicated of the three methods discussed. Because of this, the computing time per integration step is at least twice as long as for a Cowell method. The Herrick formulas given here lead to special difficulties on low eccentricity orbits because of small division problems. Similar difficulties arise with other variation-of -parameter methods for low inclination orbits, as well as for hyper- bolic and parabolic orbits. Such cases all re- quire special consideration, thus detracting from the usefulness of parameter methods as basic integration tools. A new method due to Pines (Ref. 8) is apparently suitable for all earth satellite orbits. The variation of parameters method is primarily applicable to missions in which small perturbations act throughout the orbit, e.g., microthrust transfer. C. METHODS FOR NUMERICAL INTEGRATION (REF. 9) Of the factors affecting the choice of an in- tegration method for space trajectory calcula- tions, the two most important are speed and ac- curacy. Other factors, such as storage require- ments, complexity, and flexibility, are of sec- ondary importance with most modern computers such as the IBM 7090. A good integration sub- routine should have the following features: (1) It should permit as large a step -size as possible. Thus, higher order methods should generally be given preference over lower order methods. (2) It should allow for the automatic selection of the largest possible integrating step for a required accuracy. The procedure for increasing or decreasing the step- size should be reasonably simple and reasonably fast. (3) It should be reasonably economical in computing time. (4) It should be stable; that is, errors in- troduced in the computation from any source should not grow exponentially. (5) It should not be overly sensitive to the growth of roundoff errors, and every effort should be made to reduce roundoff error accumulation. Some of the more commonly used integration methods are compared in detail on the basis of these criteria. IV-7 1. Single Step Methods Of the various Runge-Kutta methods the Gill variation is most popular. It was devised to re- duce the storage requirements and to inhibit roundoff error growth. There seems to be little reason to choose the Gill variation over the standard fourth order method when modern com- puters are available, because the storage savings are insignificant and the roundoff error control can be achieved more simply and more effectively by double precision accumulation of the dependent variables. The process of double precision accumulation can be used with any integration method. It is extremely effective in inhibiting roundoff error growth and very inexpensive in machine time. The process consists simply of carrying all de- pendent variables in double precision, computing the derivatives and the increment in single pre- cision, and adding this precision increment to the double precision dependent variables. For integrating a single equation of the form Y' = dy/dt = f(t, y), the formulas for the standard Runge-Kutta fourth order method are = hf, (v y n ) hf t_ + n hf = hf (v h 2"* + h, y + y + •'n 2 T y + k c ■'n 3 ) (19) (continued) y n+ l=W( k l + 2k 2 +2k 3 + k 4) where h denotes the integration step -size and n denotes the integration step. Runge-Kutta methods are stable, follow the solution curves well, have a relatively small truncation error among fourth order methods, and do not require any special starting proce- dure. However, (1) They tend to require more computing time, since four derivative evaluations per step must be made compared to one or two for other multistep methods. (2) The usual fourth order methods restrict the step -size for a required accuracy. (3) There is no simple way to determine the local truncation error and, as a conse- quence, it is difficult to decide on the optimum step-size for a required accu- racy. Various suggestions have been made for over- coming this deficiency. The same trajectory could be integrated twice: first with step -size h and then with step -size h/2. The difference between the two values at a time t can then be used to decide whether the step -size should be increased or decreased. This process involves three times as much computing and, therefore, cannot be seriously considered. The simplest method, proposed by Aeronutronic, is to integrate over two intervals of length h and then to re- compute the dependent variable using Simpson's rule. (s) y n + 1 y n h — (v' , + 4y' + y' 3 \ y n+ 1 y n ■'n The difference between this value and that obtained by the Runge-Kutta method at time t , , is then used as a criterion. This pro- n + 1 cedure is relatively simple and inexpensive, but there is no mathematical justification for it. Any decision to change the step -size based on it might be erroneous. Other single step methods include several attributable to Heun, the improved polygon or Euler-Cauchy method, and a method employed by C. Bowie and incorporated in many Martin programs. Bowie's method is outlined below. A *0 = x h/2 x + x 5" "h/2 y + y ( h 5" 'h/2 , • h •• h x + X 2" + X T 'h/2 = y + yo2 + y o T = x Q + x Q h y h =y + y h X Q h y h = y o + y o h Step A = f h/2 x h/2* J h/2 e h/2' A ^T .- h 2 ^0-2- S h f h' y h = g h i h/2 x + 2-4 { 5x + yh/2 = h + -k { 5y o + x h/2 " X hJ y n /2 -'4} x h/2 = x + x o!? + 9Tr ( 7x + 6x h/2 " x h) h h 2 y h/2 = y o + y o ? + m ( 7 y<> + 6 y h/2 - yh) h { X + 4x h/2 +x h} *h X + !T IV-8 ^ = yo + ^{^o + 4 y h /2 + > ; h} and the Adams -Moulton formulas are x h x + x y h =y + .y h + ^|x 0+ 2x h/2 J ,2 ^o h+ T-{yo +2 y'h/2} Step B x h/2 = f h/2* y h/2 = g h/2' x h = f h' y h = g h x h = x O + !r{ x + 4x h/2 + x h} ^h = y o + !r {yo + 4 y h /2 + y'h} x h =x + x h+^{x + 2x h/2 | hi \ y h = yo + yo h + ^\y + 2 y h i2J If the functions f, g do not actually involve x, y it is clear that x, ,„, y . need never be com- puted and that x, , y, need only be computed at the point they occur for the last time in the above list. It will be noted that the process as described above involves two iterations and requires that the functions f, g be evaluated five times. If further iterations are desired, one simply goes back to the point marked "A" when he completes all the steps of the preceding page. Note that Steps "A" and "B" are identical, though the formulas immediately following them are not. If the number of iterations are continued un- til there is no (sensible) change, the solution is exact on the assumption that '£ and y vary quad- ratically over each interval. Since this assump- tion is exactly realized only in trivial cases (for which it would be unreasonable to use any step- wise method), the optimum procedure seems to be to do only the two iterations as the list of steps implies. Put another way: when the over- all accuracy is not sufficient, it is better to shorten the time interval than to increase the number of iterations beyond two per interval. 2. Fourth Order Multistep Prediction -Correct Method ' ~~~~ Of this type, for a first order system y' = f (t, y) are the Milne and Adams -Moulton methods. The Milne formulas are: ^ y (p > = y ^n+l y n - 14 +4J1 3 3 5 v (K 2y n - 2 ) h-V(i) y n+i = y n -i + t (y n +i + 4 y; + yn -1) h v >(20) - m y (i) Vi y (c } 'n + ^( 55y „ -^a-l^n -9y' •'n 3) , 251 , 5 v , , + 720- h y ^ = y + J n + y' •'n ^(9y n+1 +i9y n+1 -5y n \(21) i) ~TZd 19 , 5 v . . h y (r|) For these methods, as well as for all multi- step methods, special formulas must be used to obtain starting values at the beginning of the in- tegration and wherever it is desired to double or halve. A Runge-Kutta method is the most con- venient for obtaining these starting values. The difference between the predicted and corrected values provides a good estimate of the local truncation error and this estimate can then be used to decide on whether to increase or reduce the step-size. The Milne method has a somewhat smaller local truncation error, but for some equations it may be unstable (i. e. , errors introduced into the computation will grow exponentially) and, while some techniques have been suggested to eliminate this instability, it is probably advisable to avoid the use of the Milne method. The Adams -Moulton formulas are uncondi- tionally stable and lead to a fast and reasonably accurate method. Its principal disadvantage is its low order of accuracy which restricts the integration step -size. 3. Higher Order Multistep Methods Variation-of -parameter methods lead to systems of equations which are essentially first - order in form as contrasted to Cowell and Encke methods which lead to systems of second order equations. For second order systems, special integration methods are available. Before considering these, the Adams back- ward difference method applicable to first order systems must be mentioned. If the sys- tem has the form y' = f(t, y), the Adams formulas are 'n+ N 1 = y n + h I °kv k f (22) k=0 .k . where V is the backward difference operator defined by V k f =V k_1 f - V k_1 f • V°f =f n n n-1 n n The first few values of a\ are (1, 1/2, 5/12, 3/8, 251/720, 95/288) for k = 0, 1, 2, 3, 4, 5. If Nth differences are retained, the principal part of the local truncation error is 0(h ). If Nth differences are retained, then N + 1 consecutive values of y. must be available, and IV-9 these must be supplied by some independent method. This Adams formula is of the open type and, therefore, not as accurate as a closed type formula of the same order would be. How- ever, it involves only one derivative evaluation per step and this, combined with the smaller truncation error, leads to a very fast, stable integration method for first order systems. The Adams method can be modified for second order systems. Thus, if the system to d 2 be solved has the form y" = — £ = f(t, y, y'L dr the method consists of applying the formulas IN y n+ i = y' n + h I °k v k f k=0 n N ) (23) k=0 The first six values of a , are the same as those given above, while the first six values of j3^ are (1/2, 1/6, 1/8, 19/180, 3/32, 863/10080). In contrast to the straight use of differences as exemplified by the Adams method the Gauss - Jackson method makes use of a summation process. The formulas may be expressed in terms of differences or in terms of ordinates. In ordinate form, predicted values for y at time t = t are given by the equations n-1 < = h2 (" f n + I C k f k) x k=0 ' H="(v.,4! dA ) (24) where the first sums 'f , ,„ and the second n-1 1 1 sums "f are defined by the recurrence relations 'f = f + 'f n-1/2 n-1 n-3/2 "f = 'f + "f *n n-1/2 n-r (25) Using these predicted values, y , d/dt(y n >, and the attractions f may be computed from the equations. The following corrector formulas can then be used to obtain improved values for v d / dt <y n > ^ y c n = h2 k + I c k f k \ k=l ' (26) The coefficients c, , d, , c,, d , depend upon the number of differences retained. For n = 11, the coefficients are given in Ref. 10. With a single precision machine, it is recommended that eight differences be retained in these for- mulas. The starting values as well as the first and second sums must be supplied by an in- dependent method. The difference between the predicted and corrected values can be used to decide whether to double or halve the step-size. A convenient method for starting or changing the step-size is the Runge -Kutta method, but, since this is a lower order method, several Runge - Kutta steps will have to be taken for each Gauss - Jackson step. The Gauss -Jackson second -sum method is strongly recommended for use in either Encke or Cowell programs. For comparable accuracy, it will allow step-sizes larger by factors of four or more than any of the fourth order methods. The overall savings in computing time will not be nearly so large, however, because per step computing time is somewhat greater and because the procedure for starting and changing the in- terval is quite expensive. As compared with unsummed methods of comparable accuracy, the Gauss -Jackson method has the very important advantage that roundoff error growth is inhibited. It can be shown that, in unsummed methods 3 /2 roundoff error growth is proportional to N , where N is the number of integration steps com- 1 /2 pared with N for summed methods. The Gauss -Jackson method is particularly suitable on orbits where infrequent changes in the step- size are necessary. Frequent changes in the step-size will result not only in increased com- puting time but in decreased accuracy as well. Finally mentioned is a higher order method, associated with the name of Obrechkoff, which makes use of higher derivatives. A two-point predictor -corrector version as applied to a first order system y 1 = f(t, y) makes use of the for- mulas y (p) =y y n+l y n-l 2h K - 3y n-l) 4" K + 7y n -i) ^K-K-x) . 13h vii , t , + 63W y (l) >(27) (c) , y n + l=y n+ £ 2 (y n +i + y n ) -Jo- (y n +i - y'n) T2D (' n+1 + y s; h vn 100, 800 y <£) where the higher order primes mean the higher order derivative of y with respect to t. The dis- advantage of this method is that the higher deriv- atives of the dependent variable must be available. Thus, to use these formulas, the first order sys- tem would have to be differentiated two times. IV- 10 Moreover, as the force terms in the equations of motion change, these higher derivatives will also have to be changed. Thus, in spite of the favorable truncation error, this method cannot be recom- mended as a general purpose subroutine for space trajectory computations. However, the method appears clearly tailored to the; lunar trajectory problem (Kef. 11). 4. Special Second Order Equations of the Form y^fdUyr The free -flight equations in the absence of thrust or drag forces can be written in the form y" = f(t, y) with missing first derivative terms. Some formulas which take advantage of this form have been proposed. The following special Runge-Kutta method, for example, requires only three derivative evaluations per step and, thus, results in a saving of about 25 percent over the standard Runge -Kutta formulas: k. = hf(t , v ) ^ 1 n • n I k 2 =hf k„ = hfYt + h, y + hy' + } ± k„\ .5 \ n J n J n 2 2 / y n +i =y n +h [y n+ l/6(k 1 + 2k 2 )] (28) y n+1 =y n+ l/6(k 1 + 4k 2+ k 3 ). J A predictor -corrector method (due to Milne and Stormer) adapted to this form makes use of the formulas h 2 ^ y n+l = ^ + >n-2 " y n-A + X (5f n + 2f n-l , - , . , 1 I h + ;,t n-2 ) + W y vl (l) - n-1 12 n+1 n n-1 n vi , , MU y (ti) - (29) J These formulas appear to achieve a local trun- cation error of 0(h ) while retaining only four ordinatcs, compared with an 0(h' ) error for other fourth order methods. However, this advantage is illusory since the overall error is 4 still 0(h ) as in fourth order methods. In ad- dition ttiese formulas are somewhat unstable rel- ative to roundoff error propagation. In practice there appears to be little to recommend the Milne - Stormer method. The characteristics of these various integra- tion routines are summarized in Table 1. 5. Evalu ation of Integration Methods The more important integration methods in general tisage will be evaluated below as they arc utilized with the various special perturbation formulations. a. Cowell method For the Cowell method, the choice of an in- tegrating routine is very important because of the greater danger of loss of significance due to roundoff error accumulation. The Gauss- TARLE 1 Comparison Criteria Method of Numerical Integration Truncation Error Ease of Changing Step -Size Speed Stability Roundoff Error Accumulation Single Step Methods Runge -Kutta h 5 * Slow Stable Satisfactory Runge-Kutta Gill h 5 * Slow Stable Satisfactory Bowie h 3 Trivial {step -size varied by error con- trol) Fast Stable Satisfactory Fourth Order Multistep Predictor-corrector M ilne h 5 Excellent Very fast Unstable Poor Adam s-Moul ton h 5 Excellent Very fast Unconditionally stable Satisfactory Higher Order Multistep Adams Backward Difference Arbitrary Good Very fast Moderately stable Satisfactory Gauss -Jackson** Arbitrary Awkward and expensive Fast Stable Excellent Obrechkoff h 7 Excellent *** Stable Satisfactory Special Second Order Equations [y" = f(t, v)] Special. Runge -Kutta i, 5 * Slow Stable Satisfactory Milne -Stormer h 6 Excellenl Very fast Moderately stable Poor *R-K (single step} triv: *G;iuss -Jackson is for *Speed of Obrechkoff <j, Lai to change stepb, very difficult to determine proper f second order equations. epends on complexity of the higher order derivatives re IV-ll Jackson method of integration is recommended for Cowell programs because it allows larger step-sizes and because it inhibits roundoff error growth. b. Encke method For the Encke method, the choice of an in- tegration method is less important relative to accuracy. There is some advantage in computing time, however, in choosing a single step method which will allow frequent changes in step-size without the necessity of going through an expen- sive restart procedure. For lunar flights, it has been found that the Obrechkoff method is es- pecially useful in reducing computing time, but this method does not appear to be easily adaptable to other types of orbits or to other formulations. Although the Gauss -Jackson method is recom- mended in Encke programs, its advantages over other methods are not as great as in Cowell pro- grams. c. Variation-of -parameters method For variation-of -parameters methods, the Adams backward difference formulas are re' commended among higher order methods and the Adams -Moulton formulas among lower order methods. In general, multistep integration methods which allow for automatic adjustment of the size based on an error criterion are preferred. With any integration method, the process of double precision accumulation of the dependent variables should be used to prevent excessive roundoff error growth. Summary of Studies on Special Perturbation Methods In order to provide the mission analyst with a set of guide lines in determining the best integra- tion methods for various special perturbation methods used in computing precise satellite tra- jectories, it is useful to examine the results ob- tained by others in the industry. This section is intended to show the interrelation of the mission, formulation of the problem, and method of inte- gration so that the most efficient, accurate, and economical balance is achieved. Several serious questions, which must be carefully considered by the mission analyst, are raised in connection with the balance between the type of orbit and the scheme of integration. a. Aeronutronic report (Refs. 12 and 13) The Cowell, Encke and Herrick methods are compared for the following problems: a selenoidal satellite which is physically unstable, but for which an analytic solution is known; a low thrust trajectory; a high thrust trajectory and a ballistic lunar trajectory. In all cases the integration is carried out with a Runge-Kutta method with variable step-size adjustment. Their conclusions are: (1) For the Cowell method, the effect of roundoff error is felt very quickly -- within a few hundred steps. (2) Overall, the Encke and Herrick methods are computationally more efficient than the Cowell method. (3) On ballistic lunar trajectories, the Encke method is best. The Cowell method requires almost ten times as many integrating steps as the Encke method and three times as many as the Herrick method. (4) On continuous low thrust trajectories, the Herrick method is superior. (5) On trajectories where high thrust corrective maneuvers are introduced, the Cowell method is superior. Although the trend of the conclusions in this study is probably correct, there are serious questions as to the validity of the conclusions on the degree of superiority of the perturbation methods. For one thing the method of integra- tion (Runge-Kutta) favors the perturbation meth- od. For the Cowell method, the choice of in- tegration method is much more important, as indicated earlier. Experience has shown that roundoff error effects are not nearly so critical as concluded here. Both the use of the Gauss - Jackson integration method and double precision accumulation make roundoff error much less serious for the Cowell method than indicated here. The evidence presented, moreover, is not conclusive relative to accuracy. The nu- merical results, for example, are not given at corresponding times, and no accurate standard for comparison is available except for the un- stable selenoidal satellite. The selenoidal satel- lite is by no means typical of the earth satellite problems and any generalizations of results based on a study of this orbit must certainly be viewed with skepticism. b. Republic Aviation report (Ref. 14) The orbit selected is that of a vehicle moving in the gravitational field of two fixed centers. An analytic solution in terms of elliptic functions is available for this orbit so that an accurate standard is thus available. This study compares the Encke, Cowell and Herrick methods with two different integration routines: a fourth order Runge-Kutta method and a sixth order Adams method. The conclusions of this study are: (1) The Encke method was superior to the others in both accuracy and machine time. For an integration over a 100- hr period the Encke method required 0. 5 min, the Herrick method 2. 5 min and the Cowell method 3.5 min. All of those programs used the same in- tegration method and the results were comparable as to accuracy. (2) The Herrick method is superior to the Cowell method relative to attainable IV-12 accuracy and slightly better relative to computing time. (3) An integral of the motion, such as the energy integral or a component of the angular momentum, is a poor positive test of accuracy. (4) The Adams method is considerably faster than the Runge-Kutta method by a factor of almost three. (5) Double precision accumulation is very effective in reducing errors due to roundoff. (6) The largest error in the Encke and Herrick methods arises from errors in solving the two -body formulas, particularly as such errors affect the mean anomaly calculation. The conclusions of this study appear to be well grounded. The only serious consideration is that the orbit selected is quite specialized and that no strong perturbations such as those due to oblateness or thrust are considered. Thus the extent to which these results can be assumed typical for satellite orbits is in some doubt. c. Experiments at STL The relative efficiency of the special per- turbation methods is a function of (1) the type of orbit and (2) the method of integration. A given integration subroutine may favor one of the methods over another, so that the use of the same subroutine for all methods does not con- stitute a fair test. In general there appears to be no doubt that the Encke method is computationally the most efficient on ballistic lunar trajectories. For comparable accuracy, however, the advantage in computing time is probably on the order of two or three, rather than ten as is sometimes quoted, when any of the standard integration subroutines are used. There is no doubt that the Cowell method requires much greater care to ensure that roundoff errors do not become a serious factor in the accuracy. However, effective methods are available to curb roundoff error growth. When these are used, the Cowell method is still a very useful tool for many space computations. None of the orbits considered in the reports by Aeronutronic and Republic Aviation appear to be applicable to the earth satellite problem in which a small but significant force, such as that of oblateness, is continuously applied. To obtain information about the comparative performance of these special perturbation meth- ods on earth satellite orbits, a numerical study was recently completed at STL. An idealized orbit was selected for the study with initial ele- ments: a = 1.5 earth radii 0.2 45° co = M Q = 155 m in 800 mi 3200 mi e i Q period of the un- perturbed orbit perigee distance apogee distance The only perturbation force considered was that due to the second harmonic in the earth's gra- vitational potential (J,)- An accurate standard against which to check the programs was pro- vided by a double precision Cowell program. The double precision program yielded results on the unperturbed orbit (J„ = 0) which agreed with the known analytic solution to a few digits in the eighth significant figure. For the per- turbed orbit, the results provided by the standard are correct to at least seven significant figures. Single precision floating point programs for the Cowell, Encke and Herrick methods were run on an IBM 7090 and compared with the double precision standard. Great care was used to en- sure that all physical constants and initial con- ditions were identical in all programs. The in- tegration was performed over 64 revolutions with output at 20-min intervals. Table 2 gives the method of integration used, the local trunca- tion error criterion, the number of integration steps required, the computing time for 64 revo- lutions, and the maximum error in the distance Ar over the 64 revolutions. For each method several runs were made with successively tighter error criteria, and the most accurate of these was selected for the comparison. While the Cowell method required almost twice as many integrating steps, overall computing time was only slightly greater than the Encke method and, moreover, the accuracy was somewhat bet- ter. The Herrick method gave the best accuracy. The relatively large computing time required by the Herrick method is partially accounted for by the fact that the Adams -Moulton formulas (fourth order) are of lower order than the Gauss - Jackson formulas (sixth order). Since the latter will allow integrating steps perhaps twice as large for the same accuracy, the adjusted com- puted time would be comparable to that for the Cowell method. A more detailed comparison of achievable accuracy is contained in Table 3 where the maxi- mum errors in the distance r, the mean anomaly M, the semimajor axis a, and energy integral E are given on the 20th, 40th and 64th revolutions. It is clear that the Herrick method consistently yields the most accurate results and the Encke method yields the worst results. For all meth- ods, there is a strong correlation between mean anomaly errors and position errors, indicating that the error is largely along the path of the motion. This conclusion also follows from the energy integral errors which are seen to be rela- tively constant and much smaller than the position errors. It may also be concluded that the con- stancy of the energy integral is a poor positive test of accuracy in the position coordinates. The IV-13 TABLE 2 Numerical Results --Special Perturbation Methods Formulation Method of Integration Error Criterion Number of Steps Computing Time (min) Maximum (ft) Ar Cowell Gauss -Jackson IxlO" 10 10, 200 5.75 800 En eke Gauss -Jackson 7X10" 10 6395 5.31 1700 Herrick Adams -Moulton 5X10" 10 7000 11.45 400 TABLE 3 Maximum Error --Special Perturbation Methods Method Cowell Encke Herrick Revolution 20 40 64 20 40 64 20 40 64 Ar x 10 6 (er) 1.2 2. 2 4.0 2. 2 6 8.4 0.2 0.8 2 AM x 10 3 (cleg) 0.3 0.6 1 1 2 2.7 0.1 0.2 0.6 Aa x 10 7 (er) 1.6 1.4 1 3 3.5 3 2. 2 2.2 2. 2 AE x 10 9 1 1 1 4 6 9 2 2 2 I min / error in the semimajor axis is also seen to be smaller than the position errors, indicating that the geometry of the orbit is much more accurately determined than position in the orbit. Although these results show that the Herrick method yields the most accurate results and the Encke method takes the least computing time, the order of magnitude of the difference is not suffi- cient to lead to a clear preference for any one method. Some improvement in the Encke and Herrick results could probably be obtained by even more careful analysis of the two-body formula computations. The Encke method, for example, is quite sensitive to the frequency of rectification and some improvement might be obtained by experimenting with rectification. There appears to be little reason to prefer either the Encke or the Herrick methods on earth satellite orbits of moderate eccentricity particularly, since they are considerably more complicated and require much more careful numerical analysis. In addition, special difficul- ties will arise in limiting type orbits (low eccen- tricity, high eccentricity, critical inclination) which do not arise when the Cowell method is used. D. GENERAL PERTURBATIONS Chapter III presented the discussion of motion about point mass (or a spherically symmetric mass). Although that discussion is revealing, it does not in general constitute a solution to the problem because the assumptions utilized prevent the solution from behaving as it should for the true gravitational field. In the preceding sections of this chapter, discussions have been presented which circumvent these limitations; however, in the process much generality has been lost since nothing can be said for trajectories beyond the neighborhood of the numerically obtained trajec- tory and nothing can be said about the long-term behavior of the orbit. (Before proceeding, it must be added in defense of numerical integra- tion that the solutions thus obtained are valid to a very high order of approximation. ) For these reasons it is desired that analytic expressions be presented which can be utilized to describe the motion of a satellite to varying orders of approxi- mation. The approach taken here will be first to discuss the variation of the orbital elements and secondly, the first order secular or cumulative perturbations which can be added as linear func- tions of time or as discrete corrections to the two- body solution to improve the fit of the resulting motion. Then as a third step, the various general perturbation theories (i.e., approximate analytic IV-14 solutions for the perturbed motion obtained by series expansion) which present second order secular and periodic effects will be discussed. The advantages and disadvantages of this ap- proach are summarized at this point. Advantages of general perturbation methods (1) They are very fast both because no step-by-step integration is necessary to obtain the elements at a given time and because the computing time per point is very small (on the order of 1 sec per point on an IBM 704). (2) The accuracy of the computation is limited only by the order to which the expansion is carried out, and not by the accumulation of roundoff and trun- cation errors. (3) They can maintain reasonable accuracy over many hundreds of revolutions. (4) They allow for a clearer interpreta- tion of the sources of the perturba- tion forces and the qualitative nature of an orbit. Disadvantages of general perturbation meth- ods are: (1) Nonoonservative forces, such as drag, are not easily included in the theory. No simple and adequate theory has yet been prepared which includes such forces in a form suitable for numerical computation. (2) The effect of other forces, such as luni -solar perturbations and radiation pressure, are difficult to incorporate since they involve substantial amounts of new analysis and checkout. (3) '['he series expansions are very com- plicated, and programs based upon them are complicated to write and difficult to check out even for a first order theory. (4) There is a serious degradation in ac- curacy for special types of orbits in- cluding the important case of nearly circular orbits (e - 0), highly ellipti- cal orbits (e - 1) and orbits near the critical inclination (i - 63.4°). (5) Although agreement with observations does confirm practical convergence, no mathematical proof of convergence has yet been given for any of the general perturbation methods, nor are any estimates of the error in the trun- cated series available. Finally, these discussions will be followed by those of atmospheric effects and extra-ter- restrial effects. 1. Rat es of Chan ge of Satellite Orbital Elements Cau sed by a Pertur bing Force (HefT 15) The instantaneous rates of change of satellite orbital elements caused by a perturbing force, as given, for example, by Moulton (Ref. 1, pp 404 and 405) are derived from astronomical perturbation theory involving tedious mathemati- cal transformations. The purpose of this de- velopment is to give a simplified derivation of the same equations by using only elementary principles of mechanics. It is hoped that this approach will make the equations more meaning- ful and the discussions which follow later in the chapter more readily appreciated. Consider a satellite of mass m moving in the inverse square force field of the earth. Its or- bit is a Kepler ellipse (Ref. 1, Chapter V) specified by the following orbital elements a, e, h, w, i and M n (see following sketch). The location of the satellite in its orbit is given by the angular position 4> which is measured in the orbital plane from the node. The angular dis- tance of the satellite from perigee is called Un- true anomaly, 9. Therefore, + (30) The radial ciistance, r, from the center of the earth to the satellite is given by P 1 + e cos (31) The satellite's energy per unit mass, c, and its angular momentum per unit mass, h, are related to the orbital elements by the equations and 2a r 2 u p = na f[ (32) (3 3) where: u = CM (the product of the gravitational constant and the earth's mass) and a dot over a quantity indicates a time rate and (34) Now suppose that a perturbing force K acts on the satellite. The orbit will no longer be a Kepler ellipse, but at every instant we can associate an "instantaneous osculating ellipse" with the new- orbit by choosing the Kepler orbit corresponding to the instantaneous radius and velocity vectors of the satellite and to the potential energy, - ti , of the satellite in the gravitational field of the spherical earth. This is the orbit the satellite would follow if the perturbing force were re- moved at that instant. The true orbit can thus be specified completely by a series of elements of the instantaneous osculating ellipse. There- fore, the set of differential equations which shows how these elements change with time is equivalent IV-15 Z-axis \ Y-axis to the Newton or LaGrange set involving the co- ordinates and their rate of change with time. With this discussion as background, the rates of change of the orbital elements a, e, n, u and i will now be derived. Following Moulton (Ref. 1, p 402), the per- turbing acceleration, F'/m, may be resolved into a component R along the radius vector (meas- ured positive away from the center of the earth), a transverse component S in the instantaneous plane of the orbit (measured positive when making an angle less than 90 deg with the velocity vector V), and a component W normal to the in- stantaneous plane (measured positive when making an angle less than 90 deg with the north pole or z -axis). Let the unit vectors along the three direc- tions be denoted by n ,, n and n w . That is, dt m (37) /here V is the instantaneous velocity vector, = fn r + r6n v = e ^n>rn g ). (38) Now from the definition of the instantaneous os- culating ellipse, it is clear that its velocity vector is the same as the instantaneous velocity vector of the actual orbit. Therefore 6 and ^ in Eq (38) may be evaluated from Kqs (31) and (33) to obtain V = {l^J /re sin - + ^ \ (3 V 1 + e cos r s/ 9) m (Rn , + Sn + Wn (35) To find the rate of change of the semimajor axis, a, refer to Eq (32) for the relationship to the energy da dT 2a 2 dc_ u at (36) Forming the dot product with F/m and substi- tuting the resulting expression for ^- in Eq (36) yields da dt 2e sin nVl 2 la Vl -e 2 (40) The energy change (per unit mass) may be found from the definition of the work done on the satel- lite by the perturbing force. da which is the expression given for- -yp by Moulton (lief. 16). IV-16 To derive the changes in the other orbital elements, it is necessary to know thc^rate at which the angular momentum vector h (per unit mass) changes. This rate of change of h is then known to be equal to the summation of the ex- ternal moments acting on the satellite. dh dt 1 (r x F) rn x (Rn + Sn + Wn ) r r s w rSn - rWn w s (41) The rate of change of h can also be written as (42) dh dh -» , . da -» dt dt w dt s where da is the angle through which the angular momentum vector is rotated in time dt. There- fore, dh -rr- = rS dt (43) Now, the eccentricity of the orbit may be ex- pressed in terms of a and h through Eqs (33) and (34) which yield hi Ida 1/2 (1 - p/a) 1/2 By differentiating, the following is obtained h L dh __ h da\ 2uae l dT a 3T J Vl -e 2 /„ dh J] 2 da\ de 2na e (45) dh Upon substituting Eqs (40) and (43) for -rr- and da -rr- , Eq (45) takes the final form, de ar '1 - e sin 9 R + fl 2 na e a 2 (l e ) S. (46) and da dT rW h (44) The motion of the node is the same as the motion of the projection of FT' on the equatorial plane (see the following sketch). Let the sub- script p denote the projection of any vector on the equatorial plane. Then it can be seen that Y-axis (*)'(£-) Node X-axis IV-17 h = projection of h on the equatorial p plane. -> -» 2£- ) = projection of -rr- on the equatorial 'p plane. ( h ar%) = " rW [><- c ° s <> cos 1 sin P - sin <j) cos p) +j (-sin 4> sin P + cos <)> coo i cos P)J- ft) * ( g|- y = the component P -( dh\ 3*7 p which is normal to h . Thus, upon performing the cross product, Eq (47) becomes dP dt rW sin $ 4V^J (48) dP YT (&) v (» £ »«,) (47) The change in the orbital inclination is re- lated to the change in the node. This can be seen by referring to the following sketch in which two positions of the node, P,, and P., are shown with Afi = P 1 - P Q and Ai h ~ V But h = h sin i (i sin P - j cos P) where i and j are unit vectors along the X- and Y-axes, respectively, and By spherical trigonometry, it can be shown that sin Ai = sin i. cos i„ - sin i„ cos i. 1,, : — [cos i.-. sin 4>_ (1 - cos AP) :os <t>„ sin Ap] . Y-axis X-axis IV-18 Differentiating and taking the limit as AS2 >0, the following is obtained di sin i , d£l tt - —■ — r cos A at sin 6 dt - Therefore, di rW cos , 2 VT (49) (50) The change in the argument of perigee, to, arises from two sources. One is the motion of perigee caused by the forces in the orbital plane tending to rotate the ellipse in its plane. The other change occurs because cj is measured from the moving node (see preceding sketch). To evaluate the latter changes, assume that the in-plane perturbing forces are zero. Then the change in u equals the change in $. According to the relations in a spherical triangle, cos An cos >„ + sin Afi sin 4>„ cos i„. Differentiating and taking the limit as An -> 0, yields d<j) _ /du\ dT " \-3T) W . dn cosl ar -r sin 4> cot i VT w, (51) where the subscript W means that this is the change in oo contributed by the nodal motion which is caused by the component of the per- turbing acceleration, W, normal to the orbital plane. The change caused by the in -plane com- The R, S ponents, R and S, is denoted by I pre- ) effect of these in -plane forces is to change the instantaneous velocity vector which must, at every instant, remain tangent to the instantaneous osculating ellipse. This ellipse will therefore have a changing perigee position. The resulting rate of change of the argument of perigee will clearly be (5?) R, S dO ar- (52) d9 Here -rr- , the rate of change of the true anomaly caused by the perturbing force, must not be con- fused with 5 which is the rate of change of 9 in H 9 an unperturbed Kepler orbit. To evaluate -tt- , refer to the following sketch. After the force m (Rn +- Sn ) has been ap- plied for the time dt, the velocity vector is changed from V* to V*+ dv, the true anomaly from 8 to 9 + d9 and the angle v, between n -> s and V, is changed from y to y + dv. The ex- pression for y is obtained from the angular momentum, h = rV cos y. Since h that 2 .22 r 8 and V = (r + r cos y 1 + 3" W 1/2 -1/2 it follows dr Computing -rg from Eq (31) yields (Rn + Sn ) r s Old perigee New perigee IV-19 and cos V sin "V 1 + e cos 6 Vl + e 2 + 2 e cos 6 e sin 9 Vl + e + 2e cos 9 (53) (54) Differentiating Eq (54) with respect to time and using Eq (52), it is found that (£) R, s L 1+e + 2ecos9 e (e+ cos 9) sin 9 de dt d\ "at (55) V 1 + e +2ecos9 If N is the component of the force normal to V, Ndt dv = ~v — But and N = R cos y - S sin y, V = h Vl + e 2 + 2e cos e r 1 + e cos 9 Th erefore dt | r(l + e cos 6) )h(l + e 2 + 2e cos 0) • [R(l + e cos 6) - (e sinG)s]j" (56) de Equation (56), along with Eq (46) for -ry-, yields lar) r, VTT [- (cos e) r S nae (57) + sin 9(1+ ,— - — - B ) si 1+e cos 9 J The total rate of change of the argument of perigee is dw _ /duA ,/du\ dt \dt/W \dWR,: (58) The final element, mean anomaly at epoch, which provides the position of the satellite at any time also has a time rate. This relation- ship is obtained directly from Kepler's equation a = M„ = E - e sin E - nt IV-20 and can be found by using the equations already obtained for -nr and -n- , with the relationship dt dt ^ between E and given by „ cos E - e 1 - e c os E sin E :os E Vi 2 sin 9 = y ] - e 1 - e < The result is da __ 1 /2r _ 1 - e 2 dT na \ a e cos ej r . d - e 2 ) nae 1 + L a( rr (sin 9) S 1 - e^)J - t dn ar (59) Note is made at this point that the last term has been omitted in Moulton, Ref. 1, p 405. This completes the set of equations for the orbital elements. The remaining 5 are sum- marized below for reference: da ar 2e sin 9 n x 2a 11 - e rt + nVl - de dt _V^ e sin 9 na ,V7T7" a 2 (l e ) dn "ar di_ dt dqj dt r sin <fy W VT r cos <j) VT w r sin <b cot i 2 J7 2~ 1 1 1 - e - W - V7 2 e cos 'L_L±_ fl+T-r ^sinOS nae \ 1 + e cos / _, (60) If at this point we introduce a disturbing function rather than the four components, we can put these equations in the Lagrangian form R 3F o _ 1 9" b = F 7f (61) W = r sin <(> 7T J da = _2_ dj~._ dT na dcr~ \ de dt dt£ dt dM dt di_ dt dS2 tl - e' na e M - e — dcr d" duj J V," 2~ Tl - e d~ cot 1 ~ 2~~ na e de 2 »n 2~ na Tl - e d~ Ji 2 d~ _ 1 na da a- >(62) + n : v7 e sin i 2 VT na Tl 3" "BT e sin l J 2. First Order Secular Perturbations For an oblate body having axial symmetry, the gravitational potential at any extension point may be represented by Vinti's potential (Chapter II). If for the present analysis we neglect terms with coefficients the order of J J (i.e., J, 3' J .... ) we can write the work 4 function (minus the potential) as: 2 U but since 1 1 r 1 + 1 + ''4 (?) (3 sin L /R\ 2 ,, . 2. . 2 I — J (3 sin i sin 1) (63) <f -1) + u is a periodic quantity, sin if 1 2 2 cos 2<J> has a nonperiodic part •=•. Thus, the potential ,1 will produce secular changes in the orbital elements as well as periodic changes. Before the magnitude of this change can be evaluated, however, the constant part of the function (a/r)' must be evaluated. Following the method of Dr. Krause (Ref. 16) we have: (f)' '0 ~2~ + C ; . cos M + C„ cos 2M + . + c cos n M n where 1 f 2rr a n - n = T ] Q < F > cos n M d M The C are simple functions of the eccentricity as may be seen in the expansions of Chapter III. Thus, ^0 T 1 ,2tt (p ) dM 2tt ~ (1 - e 2 )" 3/2 K (1 + e cosO) dO 2tt = (1 -e 2 ) 3 ' 2 and secular secular r r2 '2 ~TT a (1 - o-) -3/2 (1 - 3/2 sin - i) (6 4) (65) At this point we refer to the Lagrangian equations of Section I)-l of this chapter and con- clude that the secular variations in the elements are expressible to the first order in J as: A a = A e = (66) (67) A co = 3tt J 2 n±\ (2-5/2 sin 2 i) (rad/rev) (6 8) 3 ° ' A M nt 2" J 2 (rad/rev) e 2 (1 - 3/2 sin 2 i) A i = A n = -3t J, (6 9) (70) («)' cos i (rad/rev) (71) The physical significance for the fact that the secular variations in a, e and i are zero may be seen by looking at the potential function itself. The fact that J„, J, and J. are small implies that to a first approximation the orbit will be nearly elliptical. Although one cannot assign an un- ambiguous major axis or eccentricity to the per- turbed satellite orbit, the experience of astrono- mers has shown that it is convenient to refer the motion to an osculating ellipse. This is the orbit in which the satellite would move if at some instant the perturbing terms were to vanisli (J„ = J, = J, = 0) leaving the satellite under the at- traction of the "spherical" earth. Hence the IV-21 actual position and velocity vector at each point define the osculating ellipse in terms of a set of elements a, e, and i, where a and e are the semimajor axis and eccentricity and i is the inclination of the plane of the ellipse to the equator. The major axis a may be specified in terms of the energy E, associated with the osculating ellipse. When J„, J„ and J. are set equal to zero to calculate E, only the potential energy is altered, and it can be seen that unless r exhibits a secular (nonperiodic) variation, which is not possible here since we are dealing with bound orbits, only periodic variations in E can occur. Hence there can be only periodic variations in a. 2 Although p, i. e. , a (1 - c ), is a constant of the motion, the total angular momentum h is not constant, because the equatorial bulge produces a nonradial component of force. But by the same arguments as above, the torque, and hence h, can exhibit only periodic variations. Further, since at each equatorial crossing the momenta are related by (h cos i) The perturbed anomalistic period can be evaluated from the average angular rate using the method of Kozai (Ref. 18) and a relation 2 3 analogous to n a = u. -2-3 - L 3 T /R\ ,. n a = »=» {1 -^ J^-J (1 3 . 2 .. J~. 2 2 sin l) T 1 - e where n = perturbed mean angular rate a = mean value of the semimajor axis 2 a jl - 3/2 J .(?) (1 - 3/2 sin i) VT N constant. iu = effective gravitational constant as sensed by the satellite in its orbit. This process yields where N means node, it follows that the orbit inclination i behaves similarly. The same may be said for the orbit eccentricity, since the equation for eccentricity depends explicitly only on I hi and a. It is noted at this point that since 3 of the 6 elements vary, the satellite periods will vary. The plural of period was intentionally utilized at this point because of the manner in which three distinct periods are defined (Ref. 17). Anomalistic period is defined as the time from one perigee to the next. In that time the elliptic angles (true, mean, and eccentric anomaly) increase by 360°, while the central angle ji increases by more or less than 360°, depending on whether the apsidal notation is against or in the direction of satellite motion. Nodal period, also called synodic or draconic period, is defined as the time from one ascending node to the next. In that time the central angle ji increases by 360°, since $ is measured from the instantaneous position of the ascending node. The satellite does not, except at an orbit in- clination of 90°, return to the same relative position in inertial space after one nodal period due to the regression of the nodes. Sidereal period is defined as the time for the satellite to return to the same relative position in inertial space. In that time the satellite central angle as measured from a fixed reference, which is not to be confused with the central angle as measured from the ascending node, increases by 360°. In artificial satellite theory, the sidereal period is less important than the other two periods, it is rarely used, and it will not be discussed any further. 2?r 2tt ,-*3/2 — (a) 3 J 2 R e a 2 (l - e") 2.3/1" 3 cos i_ -1 '(72) For a near -polar orbit the anomalistic period is longer than the unperturbed period, while for a near -equatorial orbit the anomalistic period is shorter. At inclination angles of i„ - 54. 7° and in - 125.3°, 3 cos \q = 1, and hence the anomalistic period equals the unperturbed period. Physically this is due to a combination of the mass distribution of the earth and the apsidal rotation at these inclination angles. The perturbed nodal period, however, has been subject to much more confusion since the results of many of the authors are in conflict. Upon review of this work, however, it is felt that to the order J„ the results of King Hele (Ref. 19) and Struble (Ref. 20) are the most pre- ferable for small eccentricities. (Additional discussions and proofs appear in Ref. 17.) This result is: T = 27T n #~!> - ».(*.) 7 cos i - 1 (73) These two period expressions (Eqs 72 and 73) may be seen to differ in both magnitude and in the algebraic sign of the corrective term. This IV-22 apparent discrepancy is due to the fact that the perigee is moving. Thus at the time the perigee has rotated through 360° the number of nodal and anomalistic periods should differ by 1. Equations (68), (69), (71), (72) and (73) are presented in graphical form as Figs. 2, 3, 4, 5 and 6, respectively. 3. Higher Order Oblateness Perturbation The errors inherent in numerical integration are not conducive to accurate computation of orbits over long time intervals. For this rea- son, general perturbations (analytic approxi- mate solutions for the perturbed motion obtained by series expansions) are more useful in mis- sions of long duration. a. Oblateness of the earth The potential function of the earth can be accurately expressed as an infinite series of zonal harmonics. U = -=- L k = 2 ©' P R (sin L) where P, (sin L) is the Legendre polynomial of order k, given by p k (x) - "IT- ^k (x2 " 1)k K 2 k k : dx k Phis is the form of the potential function given by Vinti. The recommended values of the co- efficients J. and several expansions are given in Chapter II. The potential function determines the motion of a small body in the earth's field by au Jx >y, z. The classic approach of the general perturbations method is the analytic integration of one of the sets of equations for variation of parameters, i.e., a set similar to that of Section C-l (this chapter) with the perturbing function H defined by (1) Libration, min. q. < q. < max q. i — n — n (i = 1, 2, 3). (2) Circulation, -„ < q. < „ . These two possible regions are shown in the following sketch. Libration region Circulation region Element value In the neighborhood of the so-called critical in- clination, the elements which become in- determinant merely leave the circulation region and enter the libration region. Since the theory isn't prepared to handle points of this type along with the more regular points, it ceases to apply in this region. This behavior is no reflection on the theory in general, since other approaches can be utilized in these neighborhoods. In the latter cases (i. e. , e = or i = 0) the problem is one of indeterminacy in one or more of the elements being utilized to describe the motion. More specifically, the angle u cannot be utilized for e = because of the fact that the line of apsides cannot be located. Similarly, the nodal angle Q becomes meaningless if the plane of motion is the primary plane of reference. Special sets of elements have been developed however, which may be utilized effectively for very low eccentricity orbit. These sets will not be discussed. One set of solutions obtained using this method including J„ and J , terms in secular perturbations, J„ to J r terms in long period perturbations and J„ terms in short period perturbations, is presented below. This form is exactly analogous to those referenced pre- viously; however, there are differences in the notation and in the coefficients Secular terms This approach has been taken by several authors [Brouwer (Ref. 21), Kozai (Ref. 18), Garfinkel (Ref. 22), Izsak (Ref. 23) and Krause (Ref. 16) to name a few] . The method results in easily visualized perturbations since the variables are geometric quantities. However, because of a failing peculiar to the method of analysis, the equations exhibit singularities in certain elements in the vicinity of the "critical inclination," i.e., i = 6 3.4° and for i = or e = 0. In the first case a physical explanation exists in that since the momenta of the canonical equations are bounded, the system is conditionally periodic. This situation admits 2 possibilities: M 1 + | jj a o * a o ■ <-i + 3cos 2 i > ^^^yr^f .[l0 + 16 Vl - e 2 - 25 e 2 2 + (-60 - 96 yi - el: + 90 e 2 ) cos 2 i + (130 + 144 \l - continued IV-23 25eJ)cos 4 i ]- 1 f J^/f 7 ^ M t = ii J 2© 2(1 - e o 2 > 3/2 ^- llcOB21 '- 30 cos 2 i Q + 35 cos 4 i Q )i + M Q (74) V?'tf '.(£)-'* s a Q Ta Q j4 2 2 . , COS 1„ ) 40 cos i Q Y 5 J 4 /R_Y - - 2 , , 2 . r 16 J_ IpJ 1 - 5 cos i/ 2 fe) (1 ' e o) 3/2 x 4 8 cos i 1-3 cos i n — u 1-5 cos" 1 i r sin 2 a, 1 - ep + 126 e Q ) cos i Q + (430 + 360 Vl - e 2 - 45 e 2 ) cos 4 i Q j (i) 4 [l2 + 9e 2 + (-144 os 4 ljj 128 J 4 - 126 e 2 ) cos 2 i Q + (196 + 189 e 2 ) c J (1 - e ) 3/2 + -i ^4 — — sin i„ cos u 2 J„ p„ e„ 0s *l = Y T2 J 2© 2 ^0 2 ] (79) 2 + e 2 - 11 (2 + 3 e 2 ) cos 2 i Q 4 . cos 1 40 (2 + 5 e^) , ,,„ 2 6 . 400 e„ cos i n 1 c 2 . / 9 \2 1 - 5 cos i Q ^ _ 5 cQS 2 r, , 2 ■ V 11-5 cos i 0/ / = J* '^\2 5 4 32 J r 2 2 2 2 + e Q - 3 (2 + 3 e Q ) cos i Q (75) 9 e^) cos i Q + (-40 - 36 vr 4 . 2 COS X (2+5 e^) -^ 1-5 cos i r on 2 6 . 80 e n cos i , sin 2^ ;i - 5 cos i„) , . . 2 . \ (80) 1 J 3 R / Slnl e C ° S l o ) co= .. sin i„ / s 2 J 2 P \ e "0 5 e 2 )cos 3 i ]-i|j 4 (^) 4 (2 + 3e 2 ) (3 U i = ) 1 6 J 2 V R\ 2 — ) e„ cos i n Pn/ ° ° 80 cos i„ 11 + g— L 1-5 cos i. - 7 cos i n ) cos i Q > + fi Q b. Long period terms (76) 1-11 cos i r 40 cos 1-5 cos 4 • -l T 2 . J 16 J, :OS l„ J 2 4 © •» " .;>[ 4 . COS 1, 1 - 3 cos i . 2 . l-o COS l r 1 J 3 R ,. 2. . . tt -r— — (1 _ e„ ) sin 1- sin to 2 J 2 p n s e e f I 2 (1 - e Q ) tan i Q cos 2w s (77) (78) 200 cos i (l - 5 cos i„j 5 J 4/R\ 2 2 2 ,21 16 J^fe) 6 0— 2 4 16 cos ig 40 cos ig 1 s 2 ■ 9 2 1-5 COS lr, /, - „„' ; \ J (1 - D COS 1„J J A ^_R e COSi cos , 2 J 2 P g Sin Ig sin 2" (81) b. Short period terms 3 J 2T U!- Sin2i 0) {t) P 2 z a n j .3 -3/2" (l-e<) + MJ sin 2 i cos 2 (6 + o> s + u) e )J. (82) IV-24 (1 - e Q 2 ) "P 2 e r I'.lfh'-'li ■3/2 U-e 2 ) "'" [ + 3sin 2 i (^) (1 - e Q ) |cos2(6+(. s +u ( ) "l J 2© Sin2i [3 e Q cos O + 2c s + 2 + e cos (38 + 2w + 2co f ) > (83) V^ ^ (0 cos 4 o sin [ o [ 3 co s 2 (6 + co + u, ) s * + 3 e cos (9 + 2^ + 2" { ) + e n cos (39 + 2co s+ 2^)] " = " -7 Jo f— ) cos i n [6 (9 - M - M, p 4 2 Vp„7 [ s * + e sin 8) - 3 sin 2 (9 + u + co { ) - 3 e Q sin (9+2" + 2" ( ) - e,, sin (38 +2" +2%) s 1 J 9 3/2 M p = " — 8T J (84) (85) "0 2 + 3 cos i„) + 3 sin i r '©'I"-' sin (9 + 2" + 2co { ) -0' r sin 6 -(-^) (1 - e„ 2 ) - ^ + If + sin (39 + 2c + 2" (1 - e o> T "p = * - J !■ .. -(0^" '0 + 3 cos 2 i fl ) Q (1 -e 2 )+^+l '0' ' r + 3 sin 2 i r sin (8 + 2 1" + 2", sin 2 <<>-(?) (1 - e 2 )- Ji+ 1 ( + sin (39 + 2^, + 2^, (^) «-«M + i ^(^j j 6 '" 1 + 5 cos 2 i Q ) (9 - M s - M f + e Q sin 9) + (3 _ 5 cos 2 y [3 sin 2 (8 + " g + " f ) + 3 e n sin (0+2" + 2co ) + e n sin (39 s l + 2u> + 2co f ) J} (87) where E - e„ sin E = M + M, , s l tan ■£ , E - e - tan ^ Z The solutions for the perturbed elements are then X = X + X . + X sip where x = a, e, i, ", £2, M. These expressions provide all of the in- formation necessary to describe the motion of a 2 satellite to the order J„ . However, there exist requirements in many studies for the perturbed expressions for r and 4> , (§ = + uj). This in- formation can be obtained from the equations presented above; however, the procedure is lengthy and unnecessary in view of some of the work quoted in (lief. 18) by Kozai. This ref- erence gives r and $ to the order J„ r =r + 2 J 2 R p (1 " 2 Sln ° -1-^(1 - Vl - e 2 ) cos 9 + r 1 1 1 - e* + 4 Jn R 2 - sin 2 i cos 2 (9 + co) 4 2 p ►o*! J 2 (?) {H« (>-W')[£('-^-V : (88) in i 1(6 - M + e sin 9) 1 - e /sin *M 1 - e sin 29 (i I sin 2 i) e sin (9 + 2u) - U 7 2 1 e 2 - -r^sin il sin 2 ( 9 + co) - -^ cos i sin (39 12 / + 2co) (89) where r n and 4 > n are values computed from mean orbital elements. IV-25 Oblateness of the central body tends to make a twisted space curve out of the satellite orbit. It is customary to map this orbit as a plane curve on the orbital plane which contains at any instant the satellite radius and velocity vectors. In this plane one may either approximate the trajectory by an osculating ellipse (the astronomical ap- proach) or try to assume the actual equation of the plane curve to the desired accuracy. This latter approach is the one taken by R. Struble (Refs. 20 and 24). Another significant difference is that in this work some of the conventional orbital elements become variables to the order J . Struble in this reference derives per- turbations based on the following model i=u=— |l+e cos (? - ") - J 2 c + J 2 dj fr , e, w, c, d variablej . In the solution obtained, the short period pertur- bations are isolated in the c and d variables, while r , e and <» have only long period oscillations (with a secular variation in «). The independent variable ? is related to the central angle from the node, <t>, but provides simpler solutions than <t>. In particular. $ = <t> when J 2 = 0. The solutions for some of the elements, accurate to the second order, are included below. Note is made of a shorthand notation employing a set of inter- mediate variables n 2 • • • ^6 and w l and v 2 - These terms are presented following the equations for the terms c and d defined in Eq (90). (90) 1 + e cos (3> CO) J 2 c J 2 2 d 3 . & 3 sin (91) r sin e* dA where A is the right ascension and 6* = 90 - L. = e ' 2 J 2 e 9 - 1 (fJ (5 cos 2 i Q -lXl^ cos 2o + 4-1.. sin 4") 4 4 (92) cos i„ - 1) 9 T 2 + n 5 4 J 2 ? + + ! J 2 (-^\ (5 cos 2 i Q - D" 1 (n 6 sin 2c -^ t| 4 sin 4u) (93) i=i + (94) I J Q?\ sin 2 i Q [e cos (<t> + u) + cos 2<i> +| cos (3? - to)J + l! J 2 *1 (^) 4sin21 J 2 (t) Sin 2 l ^ C ° s2 '<> " J ) -14 + 15 sin 2 i Q - 5 -j(G - 7 sin 2 i )J x l 00 + 32 14 + 15 sin 2 i Q - 5 -j(6 - 7 sin 2 i Q )| cos 2<- 1 (95) ♦ = * + 1 J 2 (t) [ 4 e C ° s2 L Sin ^ " W) + 2e (1 - 2 cos i„) sin (4> + u ) + (1 - 3 cos i n ) sin 2? + „ e (1 4 cos 2 i () ) sin (3$ -<•>)] +| J 2 V 2 (96) Now adopting the shorthand notation d - - 35 ii. D l TS T7T J 2 The short period terms c, d can be written 2> i(^) sin l (2 +^-) cos 2$ + e cos (3^ - u ) + 4- cos (4 ^ - 2 U ) 3 2 cos 2 1 2 © ^2 . 4 + UP-) e (2-3 sin 2 i ) cos (2? - 2 J (97) [m D i sin2l o -(* + h D 1 )sin 4 l ]co S (2?-4 (1) ) continued IV-26 /45 _. , 281\ . 4. ( , 4 l/3 n 1\ \T D l + -W sln 'of +e j(7 D l-¥) /3 , 15 „ \ .2. U + TT D lJ sln l (fff + Tff D l/ sin *() j J cos ^ " 2u ' 1 far, 8 "\ < 2 - a-/37 3 _ \ . 4. 7 [W D 1 -7J sln l + (T7-7 D lJ sln l J(n D i " w) sin2i o "(if D i + T-) sin4i o[J cos 2? 1 f 2 (3 .2. , /23 _. 41\ . 4. "7 L C i^ Sm 1 + (TB- D l-5?j Sin l D x ^g. sin 4 i J cos (2? + 2 J 3 |S°i-i)-(J^°0- taai o D i' ain4l ojj + e + e + e 1 ~ff ,191 15 cos (3?- 3 J 1 "8" e (3D 1 - |) sin 2 i - (§ + J D x ) sln\ j + e 3 \<1™ D, - ^-) sin 2 l cos (3 $~ - u ) .33 , 41 „ > , 4. 1 ff e ^H D 1 "i- )sin4i n^ +e 3 1 sin i r { TU + ix D i )sin l cos (3"$ + io ) Jp J3 sinVos2io |] -^[ e4 !! D i-4 + ^ D i )sin2i o D i> sln4i o|] + f 9 + 15 1 T5" cos (4 <(> - 4 U ) (^D.-Dsln 2 ^ -(§ + ^ D l )sin \ + e4 t ( W D l-W )Sin % ^2- + TO D l )sl A) cos (4<f - 2 U ) 1 T3" |sin 2 i + (|D 1+ |) S ln 4 i , 2 \ 17 .2. . .19 _ , 41. . 4, + e •(- -m- sin i„ + (,--- D. + Txr )sin i- 7Z' , 4 169 n .4, + e W D i sin l T ^TB" "l ^T4T'' cos 4^ "T5" 1 "2T 2 « . 2. 2 S e ^ sin i Q cos i Q cos(4^ + 2 u ) 3 1,243 „ '7577^1 -T4T )ain *c ,35 , 81 _. . . 4. W + M D l )sln *<) cos (5^ - 3 J 1 "2T T7 ,33 35, sln^l. + (^D.- ")sln*l l ^TU U 1 "14T j- 3 ) ,1 n ± 73, ,4, + e i- ( mr D i + MT )sln i cos (5^ - u ) - 1 I" 1 3,2, 2. ^ e sin i Q cos i Q 1 "2T 1 T5- 3*3 .4, e TC sin i 63 cos (5^ + u) cos (5? + 3 U ) \ZSU D l sin *o ( TOr D i + T6- )sln4i o cos (,6~§ - 4 U ) 1 f" 2 (1 .2. , ,53^ 23, . 4. / W L C |^4- Sln l Q +{ m D l-T& )sin *()[ "3T , 4 \ 123 . 4. + e <D, , , „„ sin 1. '\ TTSTT j. ! [ 2 ( 3 ,, + T5" [_ e |3T (4 1 T 3 (287 ■if [ e I 147 sin i - 5 sln cos (6^ - 2J cos(6?+2u,) 'Ml , 2, sin i. _,_ ,13- ^ 845, . 4, + %ff D l + -OTl> Sln ^ 3 . 4, 1l4T sIn ! cos (7? - 3cj ) cos (7^ - a,) "iff P j "ffff [ e4 j D l isin 4 l [J cos(8?-4J + [e 2 |(3D 1+ i^ . 4 ),1 , 33 „ , , 2. + e {(rTr +T-P- D^sin i 33 )sin 2 i Q -(*| +^.D 1 )sin 4 - L - < 79 + 3 Dj^Js^IqI cos 2o e4 1D 1 T2ff aln '* 1 COS 4(; (98) IV-27 Finally the pseudo variables r\ „ . . . r\ and v, and v„ can be defined in terms of the true variables. n 2 - j(3 Dl -i) + ( ^-l5 Di)sln 2 lo + ^ Dl -^i)sin\ + e^ [(» Dl -l, ^ ,53 45 „ . . 2, , ,45 _. 95* , 4. + fe* ^ D i-i ) - { ^ D i )sln \ + ( 9 + 45 D l> **\ ]j (99) "3= !<7 + ^ D l )sln2i 0- ( ^4 D l )sIn4l (100) S D i> sln \J -e 2 4D 1+ ^) S ln 2 l + ( 1 J- B . r, 4 - je 2 [^sin 2 i -||sin 4 l ]j _ 5 ,24 _ .. , ,151 93 _ > . 2. n 5 |(^D 1 -4) + (- TT -^D 1 )sin i Q _, ,21_ 229, .4. , 2 [",11 9 „ . + ( -2- D i--2T )aln i o + e [^nr-Tr !* (101) -4 D i + § )sln2l o + ( ¥ D i-^ )sin41 „I ■ 7 "M 9 1 7 Q 4(102) *!«" i4 + " D l ,sini 0- ( T + 7 D l )slnl + e [< 14 "7 D 1 D^sln^jj ,,,1119,, 158 . . 2. )+( ^B- D l"T- )sin l . ,289 29 |g. D x (6 - 7 sin i Q ) cos (? - 3 J (103) + ^r (36-89 sin 2 i ) cos (? - J ■3T e [ 3D 1 (4 + e 2 ) - 28(6-7sin 2 i ) - 7e 2 (2 - 3sin 2 i Q ) cos ($ + to) e 3 2 + ^-sin i Q D x cos (^ + 3 J 7sin 2 l ) - 7(4-5sln 2 i ) + ^-(9 -25 sin 2 i Q ) cos (2?- 2 J + T¥ [ 2D 1< 6 + e 2 3Dj (6 - 7sin 2 i )-^.(2-3sin 2 i ) j cos 2? + TC [6D lS in 2 i -(2-sln 2 i )] cos (2? +2 J ■23? |28 (2 - sin 2 i ) + 9D X (4 +e 2 ) (6 - 7 sin 2 i Q ) - 21e 2 (2-3sln 2 i ) cos (3^ - u ) "2T D 1 (4 + e 2 ) sin 2 i Q - 2 (3 - 2sin i )J cos (3?+ J 2 + e 7 (10 -9 sin i Q ) + 18D 1 (6 - 7sin i Q ) cos (4'$' - 2 U ) T34" [ 18D 1 (2 + 3e 2 )sin 2 i n -6(3+sin 2 i n ) - e"(12-7s in 2 i Q )] cos 4<iT + ^ r D 1 (6-7sin i Q ) cos(5?-3 u ) - 20 2 J [ 27D 1 (3 + sin 2 i )J (4 + e 2 )sin 2 i r cos (5^ - u) 18D 1 sin 2 i - (2 + sin 2 i Q ) cos (6?- 2J + |^. D x sin 2 i Q cos (7?- 3 J ']l (104) and j[-D ie 3 ^(6-14sin 2 i sinff - 3 u )+[j. j(^ D !-i + 7 sin i Q ) + (4 #-? D i )sin2l o + ( 9D i-^ sin \ + e cos i Q U-J2 D 1 - j) + (|-|D l)s ln 2 i j sin Of - oj) + [. j(3-J Dl ) 53, + (|d i -^.)sin"l + (ll-3D 1 )sin'*l c + e ' |4-^ D l) + 4 D l"F )sln2l + (J-|D 1 )sin 4 i j] sin(?+ u ) - [e 3 D 1 ^ sln 2 i (l-2sin 2 i )n 8 in(?+3 u ) + 7 L e | ( T D l--6- )+( H¥-2¥ D l )sin ^ + (|D 1 -|^.)sin 4 i |] sln(2?-2j IV-28 + (g-|D l)si n% + e^ |(l-9 Di> , ,69 _ 91.2. + ( T¥ D 1 -3TT sin l , ,155 15 _ . .4. + Hir--4- D i )sin x , ,163 3 „ , .2. + ( T4T"F D l )sln r sin 2^ + ^. e 2 J- 1 + ( tV D l _ Ul ) sin \ J sin (2? + 2u) e 2 iD 1 ^(4-Hsin 2 i + 7sin 4 i )|J sin (3«7 - 3,J ,25 7 lr 2TG" 7 D l)s in 4 i j] + e 3 j^-^D^ + ( H D i "lr )sin2i o ,17 4 + ( 7 - F Djjsin i ( sin(3^ -<J + [• (l + |D 1 )sin 2 i + 4 Di + n )s . n 4. o J 2. , 3 5 ,2 '1 oin 1 ( TC"2T sln : 0' sin(3^+ M ) 2 5 / 5 , 9 _. , , ,129 _ , 91 > .2. 6 <"<T7 + T4- D l ) + ( 3^ D l + T^4- )sln l ,27 59 . . 4. ( T7J D 1 + W sln 'o + 1 l" 1 - f 1 D + 5 sin(4^ - 2 U ) , . 2. , ,5 „ 1 . . 4. )sin i +(jD fH ) S m i Q + e 2)1 , 65 , 3 T " ( T^ + T D 1> , ,15 _. ^ 41 . .4. + ( TTT D 1 + W Sln >0 . 2. sin 4<jT e " 1D i ( 'l¥ + W sln2i o 11 . 4. , 4TP sm 1 ) J sin (5?- 3,,,) +3. e J ( ! D l + ^ sin2i + (TO D l-A> s AS 5 T7 -e 3 D^in^^-^sin 2 ^) 2 1 , 1 3 _. . . 2 e Jttt - Onr + T D,)sin 1 sin(5^ - c) ( T6- D i--m )sin4i T7 " V TB" F^l ; sin(6? - 2 U ) 1 T 3 L 7 [_ e Pr 2 1 2 "7" | c ]"l sin ^m^ 7 " 8sin i ^ sin(7^-3 M ) (105) In these equations ^ Q , i„„ and e„ are inte- gration constants and as before the singularity at i = 63.4° occurs. However, Struble notes that for this inclination the motion is given by the simple pendulum equation and concludes, as was done earlier, that an oscillation occurs in the element ^ . Still a third approach, though somewhat more similar to the second than the first, to predicting the motions of a satellite has been developed by Anthony and Fosdick (Ref. 25). This work, based upon the method of Lindstedt, is the re- sult of series expansions for all variables in power series of the small parameter J„. Since the higher order coefficients (J , etc.) are neglected, these series are truncated following terms of the order J . This being the case, each of the variables may be represented as u (£)+ 3/2 J 2 Ul (I) 4> = P (£)+ 3/2 J 2 P. (i) \ (106) )' = (90 - L) = 7T/2+ 3/2 3 2 6 l 'd) J where the new variable i is defined by 4> =4(1+ 3/2 J 2 4,^ i> . = constant to eliminate secular variations in u and u = 1/r (for Kepler ian orbit) IV -29 Now starting the solution _for the motion at an apse (i.e., at a point where r = 0), the equations of motion were found to be as follows: General First -Order Results (Arbitrary 4 Q ) * = 4 , 3J 2 / R\' (2-3 sin 2 i) (given 4> n , use this equation to find 4 Q (107) in 2i R . 2 i T~(h) r sinl ° J sin 2i 4c' - 2ti sin 4 cos (4 - 4 Q ) + (3 + 2ti) cos i„ sin (4 - 4 Q ) - t] sin 4 cos 2(4 - i Q ) - n cos 4 sin 2 (£ " ^o* - 3(4 - 4 ) [cos 4 (4 - 4 > - sin 4 Q sin (4 - 4q)J \ (108) P = r> = r Q V Q I 1 °2 4c (Ji) 2 ^ , [«, + 4r]) cos 24 Q - 3ti cos 24 q cos (4 - 4 Q ) + 3n sin 24 Q sin (4 - 4 Q ) - 3 cos 24 Q cos 2(4 - 4 Q ) + 3 sin 24 Q sin 2(4 - 4 Q ) - n cos 24q cos 3(4 - 4g) + n sin 24 Q sin 3(4 - 4 >]| (109) l + n cos (4 - 4 Q ) T 2 2 (2X 16^ W (HO) 1 + n r \ T~+ r, cos (4 - 4^1 J 2 /R.X 2 _K ) ^ V r 0/ (1+ ri) [l+n cos (4 -4 )] T ) ,r 2 + J 2 /RV M ll 1 + T| + 2T| cos (4 - 4q) (112) V / y i + ^ 2 + 2 2tj cos (4 - 4 Q ) |(iL) 2 Ml {<i*„» 2 [I*, 2 ♦ ^»- 2 + 2) 7 cos (? - f )] r 1 where i i (113) -36 - 18r, L. = {24 + 12r, 2 + (sin i) [-3 + (24 + 32r| + 3r, 2 ) cos 2 4q]| + i- 24 - 8r, 2 + (sin 2 i) [(-20 - 27r, + 4n 2 ) cos 24 Q + 36 + 12ri 2 ]} cos (4 - 4 Q ) + |- [8 + 15n + 16r, 2 ] (sin 2 i) sin 24J sin (4 - 4 Q ) + j- 4n 2 + [6r, 2 + (-4 - 6n 2 ) cos 24 Q ] sin i> cos 2( 4 - 4 Q ) + |(4 + 6ti 2 ) (sin 2 i) sin 24 Q | sin 2 (4 - 4 Q ) - J5n (sin 2 i) cos 24 Q 1 cos 3(4 - 4 Q ) + |5n (sin 2 i) sin 24 Q } sin 3( 4 - 4 Q ) - L 2 (sin 2 i) cos 24 Q | cos 4(4 - 4 Q ) + It, 2 (sin 2 i) sin 24 Q | sin 4(4 - 4 Q ) (114) M 1 = !l6(3 - 3ti - r, 3 ) + (sin 2 i) [24(- 3 + 3n + n 3 ) + 8(3 - r, - 6T, 2 - 3^) cos 2£^ty + J4(-12 + 12t! - 4n 2 + 3ii 3 ) + (sin 2 i) [6(12 - 12t, + 4iq 2 - 3r, 3 ) + (-40 - 18r| + 8n 2 3 + 12^°) cos 24f,]^ cos (4 -4 ) 5 2 2n 3, , . 2 ^0]} + i- (16 + 66t! + 32t + 6r, 3 ) (sin 2 i) sin 24 A sin (4 - 4 Q ) + |l6r, 2 + (sin 2 i) [-24T, 2 + (16 + 24t! 2 ) cos 24 Q ]| cos 2(4 - i Q ) - /(16 + 24r, 2 ) (sin 2 i) sin 24 1 sin 2(4 - 4 ) + {4n 3 + (sin 2 i) [- 6 n 3 + (26n + 9t| 3 ) cos 24 ]| cos 3(4 - 4 Q ) - |(26t, + 9r, 3 ) (sin 2 i) sin 24 Q l sin 3(4 - 4 Q ) + |l6Ti 2 (sin 2 i) cos 24 | cos 4(? - 5 Q ) continued IV-30 - il6-n 2 (sin 2 i) sin 24 Q } sin 4(4 - 4 ) 3ti (sin i) cos 2| Q J. cos 5(5 - 4 Q ) 3T) 3 (sin 2 i) sin 2| | sin 5(4 - 4 Q )- (115) + 1 vS (116) (117) Under the assumption that the trajectory is nearly circular these equations can be simplified to yield Nearly Circular Orbits (Arbitrary 4 Q ) 3J r 1 + ®' (2-3 sin i) 4 (given <$>., use this equation to find 4q) (118) 3J„ 4 ( JL] sin 2 i I cos 4 n si" (£ " 4 n ) 2 4 ^r 0/ / ) (4 - 4 Q ) [cos 4 cos (4 - 4 ) sin 4 sin (4- 4 )]j (119) [cos 24 Q I 3J„ / R \ 2 , P = r„V h - —A l — \ sin i 00 I 4 VO/ - cos 24 Q cos 2(4 - 4 ) + sin 24 Q sin 2(4 - 4q)]| u = _L r 1 - T! 1 - cos (4 " 4 > + ^(^) 2 j 6 [1 - cos (4 - 4 )] + (sin 2 i) [- (9 - 6 cos 24 Q ) + (9 - 5 cos 24 Q ) cos (4 " 4 ) - 2 (sin 24 Q ) sin (4 - 4 Q ) - (cos 24 Q ) cos 2(4 - 4 ) + (sin 24 Q ) sin 2(4 - 4 Q )] 1 + n ll - cos (4 - 4 >} g) 2 {e [1 - cos (4 - 4 )] + (sin 2 i) [-(9-6 cos 24 Q ) + (9 - 5 cos 24 Q ) cos (4 - 4 ) - 2(sin 24 Q ) sin (4 - 4 Q ) + (120) (121) u 2 T (cos 24 Q ) cos 2(4 - 4 ) + (sin 24 Q ) sin 2(4 - 4 )]j : V Q 2 ll - 2 n |l - cos (4 - 4 )} + J 2 {lS i 3 [l " C ° S ^ _ ^ + (sin 2 i) |(~ 2" + 2" cos 2 ^0/ + (I " I cos 24 o) cos (e " l ) (122) - (sin 24 Q ) sin (4 - 4q) + (cos 24 Q ) cos 2(4 - 4 ) (sin 24 Q ) sin 2(4 - 4 V = V, ~2" + (sin i) 1 - r, |l - COS (4 " 4q)| Q 2 | 3 [l -cos (4 -4 )] |-| + |cos24 ) + (I " I cos 2 ^o) cos (l " e o ) - (sin 24„) sin (4 - 4J + (cos 24 Q ) cos 2(4 - 4 Q ) (123) (sin 24 Q ) sin 2(4 - 4 4 (124) The solution obtained using these equations exhibits no singularity at the "critical inclination" and indeed is well behaved at every point. For this reason this set of equations, though not pre- cise, seems well suited to analytic studies involv- ing computer programs. 4. A nalytic Comparison of General Perturba- tTbns Formulations' Recently several analytical methods of deter- mining the oblateness perturbations have been published (Refs. 18 and 23 to 28) in which basically different mathematical approaches are employed. These approaches include: (1) The classical approach of general perturbation theory in celestial me- chanics, using the concept of an oscu- lating ellipse and solving for the varia- tions in orbital elements. (2) Integrating the equations of satellite motion by seeking a solution in the form _ = — [l + e cos (^ - u) - J 2 c + J 2 dJ continued IV-31 where c and d are unknown functions in terms of short -period perturba- tions (to be determined by the integra- tion process), while r„, e and to ex- hibit only long -period perturbations. (3) Direct approximate integration of the equations of motion with oblateness perturbations, solving directly for the instantaneous coordinates of the body in orbital motion. Depending on the variables and mathematical tools used, the final solutions of various authors are seemingly different and physical interpreta- tions of certain important variables are some- times hard to visualize. The transformations between the different sets of variables employed in the literature have not been obtained previ- ously. Due to these facts a somewhat bitter contro- versy has arisen about the merits of classical celestial mechanics (Refs. 20, 23 and 29) for the solutions of near -circular orbits. The present analysis, which was made by J. Kork (Ref. 30) compares the solutions obtained by all the above mentioned authors for nearly circular orbits within the first order accuracy in the oblateness parameter J (i. e., neglecting J , J J terms). a. Kozai's formulation (Refs. 18 and 26) Upon a change in the notation utilized by Kozai to that utilized by Vinti and upon changing the symbols to be consistent with those presented in Chapter III, the first order perturbation in posi- tion may be written <r,r = a - 1 \l J 2 (a?) i 1 -| sin2i ) [d-r 51 — r ' e ) cos 8 + - a 5 * = | J 2 \ J 2 (aV) fi 1 - e J sin i cos 2 (8 + gj )! '(125a) |(2 - g- sin i) (9 - M + e sin 9) . ,, 3 . 2 .. T 2 „ 2 + (1 - 2 sin i) [^ U - e Vl - e 2 ) sin 6 + I (1 - Vl - e 2 ) sin 29J /I 5 . 2 A . . „ , lj-jr sm ll e sin (0 + 2o> ) G- 7 2 \ T-Tj- sin ij sin 2(0 + tu) e 2 p- cos i sin (30 + 2oj)! (125b) and the secular perturbations in the orbital ele- ments are 2 '0 2~ J 2 ® ' (= o 5 • 2 .I, 2 - ;y sin i) t (126a) r, '"0-7 J 2 (f) M = M Q + nt n = n + 2 J 2 nt cos i (126b) (126c) (B)\ (i.j.,.',)^: . 3 . 2 A J, 2~ 1 - ir sin ij Tl - e (126d) where w Q , n and M„ are the mean values at the epoch, i.e., the initial values of the osculating elements from which the periodic perturbations have been subtracted. There are no first order secular perturba- tions of the semimajor axis, a, of the eccentricity, e, and of the inclination, i. The mean value of a (i.e., a) is given by Kozai in terms of the unperturbed semimajor axis a n , as * = a \-l*2 (^ (l-|sin 2 i)^ (127) Notice that the classical relationship n„ 2 a 3 = p, becomes in these variables -2-3 3 n— =„ 11 -JJ 2 (5) 3.2 ly sm 1 ,) i '1 - e' (128) The value of the mean semimajor axis, a, has been already used in the derivations of Eq (5). If the eccentricity, e, of the orbit is a small quantity of the first order or less, Eqs (125) can be reduced to the simple form given below (Ref. 26). 6r 1 — /r\^ 2 = ¥ a J 2 \a ) sin i cos 2>t 1 - ■ 2 . = * a ( sin i cos 2\ (129a) ■3 /-p \ /l 7 9 \ H = ' 2" J 2 U ) \2" " IT sin V sin (l 7 . 2 .\ . -- t I j - Yz Sln l / sin 2x - ( where (within a first order accuracy) \ = M + u .2 2\ 129b) _ 3 c " 2 J 2 (?)'•*', ©' Since E is a small quantity, and since the relation- ship between M and is (Ref. 31 ) M = - 2e sin + . . . it can be shown that for small eccentricities, i.e., e = 0( f ) IV-32 1 + € cos 2\ ~ 1 + e cos 2 (6 + u ) + 4t sin sin 2(9 + u ) ss 1 + « cos 2<J> and similarity 1 + e sin 2\ = 1 + e sin 2<\> Thus Eqs 129a and b can be written also as (130a) (130b) 1 - • 2 . or = £ a e sin i cos 24> (17 2 \ 2 ' Y2 sin sin 2< t> (131) Finally, the expression for the instantaneous ra- dius vector in near -circular orbits can be written as e Q cos (<(> - go) i', ®' sin i cos 2*] (132) From Eqs (126) and (130a) it can be seen that for small eccentricities the average angle from node to perigee co can be approximated for one revolution by its initial value, oj-. Kozai's solution for near-circular orbits con- sists basically of two independent components varying about a mean radius, a. These com- ponents are: 1 2 (1) An oblateness term, j e sin i cos 2$ which has a period of ir (double periodic within one full revolution) and depends mainly on the shape of earth seen by the satellite vehicle (i. e. oblateness parameter J„ and inclination of the or- bit, i) but is independent of the orbital eccentricity, e, and nodal angle to perigee, u . The oblateness term de- pends also on the semimajor axis 2 through the term s = =- J„ ©' (2) An elliptical term, e„ cos (t '0 ) de- pending only on the geometrical prop- erties of the orbit, e_ and oj _ but being s completely independent of the oblateness of the planet or the orbital inclination. It is obvious from the mathematical form of Eq (132) that depending on the relative size of the oblateness and ellipticity terms, in connection with proper phase shifts between the two, two, three or four "apses" can be obtained during a single revolution (i.e. points where r = 0). This fact will be graphically illustrated in the discussion of Izsak's work. b. Struble's formulation If only terms to the first order in J are re- tained, Struble's main results, periodic in ra- dius, can be presented in the following form (Ref. 24, p 93). - = — [l + e cos (JjT - w ) - J 2 c] (133a) I = J£ P„ r T cos2i + ¥r + | J 2 ) ( 2 " 3sin2i o) (133b) 4e cos i sin (<j> - co ) 2 _ + 2e (1 - 2 cos i Q ) sin ($ + u ) 2 — + (1 - 3 cos i ) sin 2 <(> 2 2 — 1 + ■§■ e (1 - 4 cos i ) sin (3<j> -to) (133c) where 4 {^f sin2i l + cos2 < e cos (3<t> - cj)+ -r cos (44> - 2oj ) i 3e „ + — *- cos 2o i© ! e 2 (2 3 sin i Q ) cos (2<j> - 2cj ) (133d) dA P — P m = r sin 6'-gy- = angular momentum about the polar axis G 1 = 90° - L (133e) In Ref. 32 it is shown that the angular mo- mentum orbital plane is given by h = r (0 + oj + cos i 6) = yfp (134) From Eqs (133) and (134) it can be shown that 2 P. . 2 . p = JJTp cos i or - = ti cos rl — _- m » ^ p 2 (135) For small eccentricities of the order J,. 1 + e cos (<)> - oj)« 1 + e cos ( $ - u ) (136) at least for one revolution. Similarly all terms 2 containing e , J e, etc can be neglected. Using Eqs (135) and (136) the results given in Eqs. (133) can be simplified to read 1 t /rV ■ 2 . „ J tJ, ( — I sin icos 2* ¥ 2 \ r 0/ J r = r n |1 - e cos (9 - u ) I «■ 1 (137a) IV-33 r Q = P II i-nm (2 3 sin (137b) ■>] Furthermore it should be noted that for small eccentricities fe) p = a(l -e ) = a (138) Remembering this approximation and comparing Eq (137b) with Eq (127) similarily Eq (137a) with Eq (132) it becomes obvious that for e = O (J 2 ) the first order results of Struble are identical with Kozai's formulation and the constant r Q is given simply by the mean semimajor axis: r Q = a (139) Izsak's formulation (Ref. 23) The instantaneous radius is given by Izsak as follows T 12 r = a* 1 - e cos (<t> - w ) + j e cos 2(ob - &> ) I'. (?)' sin i cos 2w + •■] where H- 2 4' 2 (i)V^" 2 '3 w = (1 + E ') 9 + o, (140) (| =a constant for the motion of the perigee of the order J„ For e =0(J 2 ) tne solution for one revolution is simply [' r = a* 1 - e cos (4> - u) + \ J 2 (!) Sin2 ic ° S 2 *1 (141) Comparing Eq (141) with Eq (132) it is seen that Izsak's solution can be also reduced to the form given by Kozai and the parameter a* is simply a* = a. An interesting feature of Ref. 23 is a set which represents parametric families of curves obtained by solving Eq (141) of this study nu- merically for various values of e Q (0. , 0. 00012, 0.00030, 0.00049) and for three particular cases of o n (0°, 45°, 90°). The curves show clearly the possibilities of 2, 3 and 4 "apses" (i.e. points where r = 0) during one revolution, depending on the relative sizes of ellipticity terms with respect to the oblateness terms and also on certain phase shifts between them. These figures have been reproduced and are presented for convenience as Fig. 7. d. Equations derived by Anthony and Fosdick The form of the resulting equations in Ref. 25 is completely different from the results obtained by the authors considered previously. In Ref. 28 the equations of motion in spherical coordinates are integrated directly and certain new variables are introduced, which do not have a simple phys- ically intuitive connection with the variables used previously. There may exist some doubt, how the initial value, 4 Q , of the "independent variable for which the first -order analytical results for r and V are periodic" compares with the classical V„ 2 j , and how the analog of eccentricity t| = V may depend on the classical eccentricity, e. These transformations are far from obvious, thus, they are derived in this section by reducing Anthony's solution to an analytical form similar to Kozai's results and then comparing the coef- ficients term -by-term. The equations for arbitrary near -circular or- bits are given as Eqs (118) through (124) assuming n = 6(JJ. Certain terms in these equations can be simplified by using the equality cos 2| Q cos 2(| - l ) - sin 2i Q sin 2<£, - Z Q ) = cos 2£ (143) / R \2 _ Next, using the previous notation £ = J I — j = | JL] the expressions for r and V can be \ r 0/ written as follows 3 J 2" J 2 jl +T1 [j i - cos (i - e )J + c cos a - £ ) - ^-t sin 2 i [(-9+6 cos 2£ Q ) 4- (9 - 5 cos 2i Q ) cos (| - £ Q ) - 2 sin 2| Q sin (i - i Q ) - cos 2£j? V = V Q < 1 -ti + r| cos (i - i Q ) ' i 2 + £ - « cos (i - i ) + j £ sin i • _(-! + ! cos2 ^o) + (I - | cos 2£ J cos (| - £ ) - sin 2| Q sin (% - | Q ) + cos 2£ (144a) where £ (2 - 3 sin i) (144b) (144c) Notice, that in Eqs (144a) and (144b) the sine and cosine terms appear combined with a small con- IV-34 stant of the form a. cos |, where | = (1 - a„) 4>. Since for the nearly circular orbit considered here both a. and a„ are of the order s , it follows by a reasoning similar to Eqs (130a) and (130b) that 1 + a. cos 4 = 1 + a i cos 4>, etc. (145) Equation (145) indicates that for the purposes of this analysis it does not make a noticeable dif- ference, if during any single revolution 4 is simply visualized as the central angle from the ascending node, c)>. Next, collecting the cosine and sine terms in Eq (144a) r = r Q (1 + A ) i - a x cos (4 - e > 1 2 + A„ sin (4 - 4 Q ) + ,- e sin i cos 24 (146) where 3 2 2 A„ = t| - c + tj c sin i - e sin i cos 24 r 3 2 5 2 + =- € sin i - *- e sin i cos 24 n 1 . 2 . . ot o- € sin i sin 2|„ By trigonometry A 1 cos x + A sin x V^l 2 + k 2 C0S x + tan Thus Eq (146) becomes r = r Q (1 + A Q ) Va x 2 + a 2 2 cos (^ -e c 1 2 + ct„) + £- « sin i cos 24 (147) where *o = tan_1 Wj Kozai's form of radius, given by Eq (132) can be written as follows r = a 1 - e cos ( $ - u n ) . 1 .2 + 7t ( sin l c b os 2$ (148) By comparing Eq (147) with Eq (148), while re- membering that within the first order accuracy 4 ~ <J>, the following important transformation equations can be derived by equating the corre- sponding coefficients of two Fourier series ex- pansions of the same function cj>. Thus, Anthony's variables are related to Kozai's formulation by the following equations: 1 + ,, 3 2 i + p- e sin i + continued sin i cos 24 n (149a) ,3 . 2 . * + ^ i sm i 5 . 2 . n£ 7t s sin i cos 24 r + (tv « sin is in 2| ) 2 1/2 (149b) tan (- "! 4 e sin i sin 24 o) 2 . e sin i cos 2t »)■'] (149c) The inverse transformation equations for t) and r„ can also be obtained from Eqs (149a) and (149b) to be: (I ■ 2 • • «\' i sin i sin 2| ) 1/2 3 2 + € • j! sin i 5 . 2 . „, p- e sin i cos 2| (150a) 3 . 2 . Tj- £ sin i . 2 . sin i 1 ( 2 A ■ 2 • ■ nt V) 1 1 - le - f =■ e sin l sin 24 0/ / y + e sin i cos 24 n (150b) /2 ,1 • 2 . „, + =■ e sin i cos 24 n 4 Q = 4 Q <u . i. e. «) (150c) Unfortunately, Eq (149c) is transcendental and the third transformation must be found by nu- merical successive approximations. Character- istic solution curves for Eq (150c) can be obtained by the following procedure: (1) For a given e, i, e solve for various values of u. by assuming values for 4 in steps of 10°, for example. (2) Plot the data and obtain a value of 4 n corresponding to the given oj„. For step (1) it is advantageous to write Eq (149c) in the following form 4 Q - tan 1 . 2 . . „, =- e sin i sin 2£,(-| If " (3* sin i sin 2 ^o) J 72" (151) Note: If in Eq (151) the eccentricity becomes smaller than a critical value e* = ^ sin i, the values of 4 n IV-35 can no longer be picked arbitrarily. This fact is illustrated by assuming e = in Eq (149b) and observing that the required value of £, Q = 0°, 90°, 180°, 270°. Physically this means that for e = the "apoapsis" always occurs at the equatorial crossings ( ^ = 0°, 180°) and "periapsis" always occurs at the maximum latitude (£ Q = 90°, 270°), there being four "apsidal" points during one revolution. It is noted once again that cj Q gives the loca- tion of the minimum point of the eccentrical com- ponents of orbital radius, while 4 Q , gives the ex- treme of the radius. Finally, it should be remarked that the state- ment made in Ref. 28 "e = \r\\ for an elliptical orbit" is misleading since it is true only for the non- oblate case, while in general e = e (ti, e, i, i Q ) and must be computed by Eq (149b). Only for large eccentricities is the approximation e = |r|| valid for rough engineering estimates. e. General comparisons It was shown above that to the order J 2 in oblateness all the methods considered are identi- cal at least in the case of nearly circular orbits. Mathematically, Kozai's formulations for the instantaneous radius, Eq (132), and secular per- turbations, Eqs (126) are generally the simplest to use. However, if for any fixed orbit the or- bital injection conditions are desired, the results of Anthony and Fosdick merit investigation. It was thus shown that the classical method of oscula- ting ellipses is still valid for nearly circular or- bits and that it provides a somewhat clearer ge- ometrical interpretation of end results. node and the disturbing body, and r be the central angle between perigee and the disturbing body. Also, let i be the angle between the ve- hicle orbit plane and the plane containing the origin, perigee and the disturbing body. Perigee — Y Disturbing body The deviations in the elements are derived in a system based on this latter plane. In this system, n = 0. u> =0 and i is the inclination. The P P P solutions obtained for the perigee system are then transformed into the solutions in the original X, Y, Z system. The solutions are: M, <M r sin T cos T sin i sin 9 p P P P r d (1-e) 3 (1+ e cos 9)* + 2e 2 ) e - 3 (l-9e 2 - 2e 4 ) cos 9 (13 4 2 e (l-6e ) cos r (1 + e) sin V sin i cos i M r d e 2 (1+ e cos d) 6 (1 5. Sola r and Lunar Perturbations The problems of defining the changes in the motion of an earth satellite due to the presence of distant gravitating masses and the discussion of the stability of an orbit are of necessity closely related. This relationship exists because the two analyses differ only in the time intervals consid- ered and the fact that forces other than those pro- duced by external masses (for example atmospheric drag) must be included in the discussion of sta- bility. For this reason much of the material presented in the following paragraphs is applicable to subsequent discussions. Analytic expressions for the perturbations due to the gravitational attraction of a third body may be derived by techniques similar to those used in the oblateness derivations. This approach has been taken by Penzo (Ref. 33) with the result that one set of equations for the variations in the or- bital elements may be obtained. This solution is outlined below: Choose geocentric coordinates with the X-Y plane being the orbit plane of the disturbing body. Let r be the central angle between the ascending + 3e cos 9) i d 3r 3 (1+ 4e 2 ) sin r cos T sin l p E + C; ^n„ r d (l-er v>~ (152) 9 9 r (1+ e) sin r cos i sin 9 " p P P r d (1 - e) (1 +e cos 6) 3e + 3(l + e 2 ) cos 9 + e (1+ 2e 2 ) cos 9 u , r ( 1 + e) *in T cos T (1 + 3e cos 0) M d p v P_ FL r ,e 2 (1 + e cos 9) 3 u , 3r 3 (1+ e) sin r cos i d P P P E + C r d (l-e) ^ Q (153) IV-36 ^p = -° os i p AQ p r 3 (1+e) 3 sin 2 r cos i p P P_ r d 2e 4 (l + ecose) 3 + 3e (4 - e 2 ) cos 6 + 12e 2 cos 2 9 5e r (1+e) sin 6 p r, e 3 (1-e) 2 (1+e cos 8 ) A 44 e 2 + 13 e 4 - 2 e 6 ) + 3 e (4 - 25 e 2 (6 + 3e 4 ) cos 6 + e 2 (8-37e 2 4 2 + 2 e ) cos 2 2 2 (cos T - sin r cos 1 ) p P P + e f(2 + e 2 ) + 3e( 1 + e 2 ) cos 6 + e 2 (1+ 2e 2 ) cos 2 ej (1-3 sin 2 r cos 2 1 )| M d 15 ^pr p e sin2 T cos 1 p p 2r d (l - e) 3 yT E + C e (156) where n , and r . are the gravitational constant and orbital radius (assumed constant) of the disturbing body, respectively, and the C^C^, etc., are con- stants of integration, i.e. , they are functions of the initial conditions. The transformations to the elements in the X, Y, Z system are Al = — , — r (cos a sin i sin l L p sin a cos i cos V) A 1 p P P sin a sin i sin r P J (157) An = 3r (1 + e) (4cos z r . s in z T cos i -P r d (1-e) f- + C, (154) cos v sin I sin 1 sin r cos i (cos a sin i P P P - sin acos i cos r ' ) - sin 2 i cos i p sin r p J A i p + (sin 2 isini cos T - sin 2 i sin 2 T p cos i sina)AJ2 p (158) sin a cos u sin i r 2 (sin a sin r cos isini 3 . L P F sin i cos V ) Afi o 2 2 2a p r.e 2 (1+ ecos 6) 2 + sin V cos i (cos a slni P P sin acos i p cos r) M ] + A < 3e sin 2 r cos i sin 6 cos 6 L p p -6e (cos 2 T - sin 2 T cos 2 1 ) cos 6 . 3 cos 2 T + 3(l+e 2 ) sin 2 T cos 2 i p - e 2 ]+ C a Ae = P Aa (155) 2ea ea T p. j p 8) ^OT> r sin 2 F cos i sin 6 r , P £ P,. [e(2e 4 2r (1 -e) 3 (l + ecos9) 3 9e .2 o _4^ Q j.„/o _ Q „ 2 _ a ^ 4, » ™= 2 e + 3(2 - 9e -3e ) cos 9 +e(2 -9e -8e ) cos ej M d ykP r (1 + e) T —7 P '" ' w (cos 2 r -sin 2 T cos 2 i) (1 r, e (1+e cos 6) d + 3 ecos 8) where sincr sin to sin i sin r The assumptions in the derivation of these solu- tions are that r d >> r and that the disturbing body does not move during the interval of varia- tion. Thus, in order to solve for the perturbed mo- tion of a satellite it would be necessary to compute the perturbations (for some small time, say 1 period) due to each body being considered, resolve these perturbations into a common coordinate sys- tem, add the resultant motions, adjust the orbital elements and then continue the computation. This is obviously a lengthy procedure and is not intended to be performed by hand. Another approach to perturbations has been reported by Geyling (Ref. 34), who presents the effects of these remote bodies in terms of varia- tions in the position of the satellite in cartesian coordinates. Only circular satellite orbits, how- ever, are considered. IV-37 Choose X, Y, Z axes such that the orbit plane of the disturbing body is the X-Y plane, the X axis being in the direction of the satellite ' s ascending node. The deviations from the nominal trajectory will be given in the £,, r|, ? system, which moves with the position in the nominal or- bit, i is radial, and r\ is in the direction of mo- tion. Kft 3 "d r c ~s 73 T 2 sin 2 i sin 4, - f sin 2 i cos 4, _ (1 - cost) s in i . . - 2A(X+TJ sin ^ + 4>) - (! + co s i) sin i , , - y x. ( \ - n — sin &* + kg cos <|> + k sin 4, -4>>J (162) where r = radius of the circular nominal orbit, c and the k' s are constants to be evaluated from initial conditions. These solutions are indetermi- nate for X. = 0, ±1/2, ±3/2, ±1. However, for X = 0, i.e., for a stationary disturbing body, the particular solutions are The position of the disturbing body in the X-Y plane is given by the central angle $" = $7 +• \ f where <jT is an initial value at t = f = and \ is the ratio of the angular velocity of the disturbing body to that of the vehicle. Geyling's solutions are t 3 "d r c r d ■n- (2 cos i - sin i) + ^. sin i cos 2 <j> - 2 sin icos2<j> t ' 3 " d 2 ,„ 2 . . 2 .. g- (2 cos 1 - sin 1) 4 2 + 3- sin i cos 2 4> + 4\ - 1 sin i cos 2<J> (X + 2) (1 - cos if <\ + l)(2\+l) (2\ + 3) *2 cos 2 C$ + <(>) (X- 2) (1 +cosl)' _„„,. ., + {k- 1) (2V- 1) (3X.- 3) cos2 <♦ " * ) 9 2 + -j (1-cosi) cos 2(<)> + 4> ) 2 9 + y (l + cosi) cos 2(c)) - J) 3 "d r c 11 = " » 7T- 4 /o 2. ,2, •it (2 cos i - sin i + 3 sin 2 i cos 2 $ Q )f - ^1 sin 2 i sin 2<t> (163) + k + k sin 4 + k cos . (160) 3 4 (2 cos 2 i -sin 2 i)f 11 sin 2 i sin 2 4. - 2 3l " t sln2 4T X(4x 2 -1) 2 _ (4x + 12\+ 11) (1 - cos t) 4 (\+l) 2 (2\+ 1) (2x-t-3) + (4X 2 - 12X.+ 11) (1 + cosi) 2 4(X- 1)^ (2X - 1) (2X- 3) + k 4 + k 5 4 + k g sin 4. -t k ? cos 4, sin 2 (4> +<(>)' sin 2 {<)> - <(>) (161) 11 2 ~~ Yj- (1 - cos i) sin 2(<)> + <t> ) j^(l +cosi) 2 sin2((t> - 4> ) (164) 1 + cos i) sin i cos (<j> - 2 4> n ) - (1 - cos i) sin I cos (4> + 24> n ) - sin 2 i cos 4> : + •= sin 2 i sin 4> + -p (1 _ cos i) sin i sin (4> + 2 $ Q ) - I (1 + cos i) sin i sin (<j> - 2^ Q )> (165) IV-38 Again, if more than one disturbing body is considered, it is necessary to consider them in- dependently, compute the resultant displacements r|, |, £ in the respective coordinate systems, re- solve the displacement vectors and add. Despite the limitation imposed by the assumption of cir- cular orbits, this approach affords a simple means of computing realistic coordinate variations for many satellite orbits. The magnitude of these radial perturbations for near earth circular orbits can be seen in Fig. 8. This data is based on the work of Blitzer (Ref. 35). Another approximate method for computing the effects of external masses on the orbit of an earth satellite has been reported by M. Moe (Ref. 36). This work is outlined below: First consider the perturbations of a satellite orbit due to a disturbing body assumed to be in the X-Y plane. The geometry is shown in Fig. 9. The orbit will be described in terms of the oscu- m , n. lating ellipse whose elements are a, e, cj, and i, and expressions will be derived to com- pute the approximate changes in the elements during one revolution of the satellite. The param- eters i, to, n, and Tare taken relative to the dis- turbing body plane. For an earth satellite, this is either the ecliptic or the earth-moon plane. Now, if the equations for the variation of ele- ments of Section C-l of this chapter are utilized together with the components of R, S and W, the approximate changes in the elements can be evalu- ated. Moulton (Ref. 1, p 340) gives the form of these forces. Under the assumption that the ratio of orbital radius to the distance to the disturbing body is small these components may be expanded in powers of r /a, and all but first order terms can be neglected. This procedure yields: R = K d r (1 + 3 cos 2 F ) S = 6 K.r [cos T sin (cj + 6) - sin T cos (oj + B) cos i] cos r W = -6K.r cos F sin i sin T where K d = M d /2a d = MH a , = assumed constant, d Letting c stand for any orbital element and A e for the change in that element after one revolution of the satellite (from perigee to perigee), we have t = 2ir/n t = o 2m a. = ^ dT dt = I -- de (i66) de dt ar as = o where t is time measured from perigee passage of the satellite. Since Ae is supposed to be small compared to i , it is permissible to approxi- mate all variables in the equations for element variations for dc /dt by the values they would have in the unperturbed orbit, and to approximate dt/ d6 by its relationship to the conservation of angu- lar momentum, h dt _ r 2 ao ' IT 2 J 2 where h = na Tl - e is assumed constant. Since the angular velocity of the satellite is usu- ally large compared to the angular velocity of the disturbing body, we may further assume that T is constant during the time the satellite takes to complete one revolution. Then integrals of the type in Eq. (166) can be evaluated easily. The results are Aa = (167) Aq = 15 Hira 4 e Vl - e <sin 2 T cos 2uCOS o 9 2 ) sin 2 u (cos T - sin T cos i)| where q = r = a (1 - e) £ e = -- Aq a n (168) (169) Ai = - 3 Hra (2 sin 2 r sin i [l - e 2 (1 2 yT^ l - 5 cos 2 oj)J + 5 e 2 sin T sin 2 u sin 2 ij (170) £ fi = 1 3 HlTa {5 e 2 sin 2 T sin 2 u 2 £~7 x 9 r 2 2 + 4 sin T cos i 1(1 - e ) cos to + 4 (1 +4 e 2 ) sin 2 tuj} (171) 2 I 2 Au=- cos i Afi + 6 Hira Vl - e < 5 sin 2 r sin cj cos <<j cos i 2 2 2 - 1 +3 sin T cos i - (5 sin to 9 2 2 \ - 4) (cos T - sin r cos i)| (172) where M H D GM D ^E " 2 ^I^ Here, M„ and M„ are the masses of the earth and the disturbing body, a D is the average distance to the disturbing body. IV-39 G is the universal gravitational constant and n is the satellite's mean angular motion. For the moon as the disturbing body H = H m = 0.68736 x 10" 18 (naut mi)" 3 = 10.8207 x 10" 20 km" 3 = 2. 80763 x 10"° (earth radii)" If the disturbing body is the sun, then Recall from Eq. (157) that Aq = A ]sin 2 T cos 2 c cos i ' / 2 2 - sin 2 u> (cos r - sin r c where os 2 i)j A =15 HTra 4 e Vl - e 2 Using trigonometric identities, the expression for Aq can be written in the following form: H = H g = 0.31584 x 10" 18 (naut mi)" 3 4.97207 x 10" 20 km" 3 1.29010 x 10"° (earth radii) -3 Note that H m = 2. 17631 H g , but remember that the fundamental planes are different for the two perturbations. Assuming that the other variables (a, e, i, and u.) remain constant during one period, Aq can be integrated from to it (the period of T) to give the approximate total change. Dividing by tr gives the average change in q for one revolution of the satellite. Similarly, formulas for the average change in the other parameters can be determined to be: Aq Ae 7. 5 Hrra e - Aq a ^sec if A " „„ = 6 Hira 3 Vl - e< ■ 2 .. sin i) 2 2 e sin 2 u> sin i (173) (174) 2 . , 5 sin (i) , 2 1 + 7 (e - 2(l-e' :: ) (175) Ai An -3.75 Hrra" 5 , 2 . _ . „ ., (e sin 2 uj sin 2 1) i« '1 - e' -3 Hita cos (176) vT i [d-e 2 ) 2 COS CJ (1 a 2 \ ■ 2 4 e ) sin »] (177) where the subscript sec means secular. To com- pute the changes per unit time, divide by the period of the satellite in the specified time units. Note also that H and a must be in units consistent with those used for q. The above expressions indicate the secular trend in the various parameters due to a disturb- ing body, for example, the moon. To illustrate the meaning and importance of these formulas, it is helpful to return to the complete formula for the perturbation of perigee distance q. Aq = Aq + Aq x ^per ^sec where subscript per means periodic Aq per = A L sin 2 r cos 2 w cos i 1 2 1 - ^ cos 2 T sin 2 u (1 + cos i) and Aq - - >r H sec 2 1 2 A sin 2 (jj sin i. Thus Aq can be expressed as the sum of two terms; the first of which is a periodic function of r , and the second is independent of r . This nonperiodic or secular term is precisely A a J M sec which was previously derived. The effect indicated by the periodic term (Aq ) can be better understood if its form is changed as follows Aq per = AB * sin 2 r cos « - cos 2 r sin a) = AB sin (2 V - a) where Tcos i + 1 . 2 „ . 4 . . sin 2 oi. sin i and a = ± cos - 1 cos 2 u cos i ~B" with the minus sign holding if sin 2 w is negative. The formulas for A u, A i, and At. can each be expressed in a similar form, and in each case the secular terms have already been derived. Since the forms of the periodic terms are not important for most purposes, they will not be given. From this point the method of computation parallels Penzo's. 6. Drag Perturbation of a Satellite Orbit The effect of air drag on the osculating orbital elements of a satellite can be determined using the approach outlined by Moe and discussed under solar lunar perturbation. The effect on each ele- ment is expressed as the change in that element in one orbital revolution. That is, if the elements at a certain perigee are a, e, i, co, and fi, then IV-40 the elements at the following perigee will be changed by the amounts A a, Ae, Ai, Aw, and Aft (Refs. 37 and 38). a. Perturbation equations and the drag force To obtain expressions for these changes, start with Eqs. (178) through (181), relating the time derivatives of the orbital elements to the components of a general perturbing force. A particular form of these equations, given by Moulton (Ref. 1, pp. 404 to 405) and Moe (Kef. 39), is da dT de at df di ar 2e sinO R + 2a Yl vT (178a) sin na R + r S dft _ r sin (0 + u) Vl - e 2 2 na e 2 i, 2 \ a (1 - e ) r (178b) V (178c) na iT r cos (0 + o.) W (178d) VI ^ and where w = -B p (r) V V sin 3 B C D A 2m m = mass of the satellite (179c) C = drag coefficient A = effective area of the satellite r = radius vector from the center of the _ earth to the satellite p(r) = density of the atmosphere at r V = velocity of satellite relative to the atmosphere V„ = velocity of satellite relative to inertial space V = velocity of the atmosphere relative to inertial space 3 = the angle between V and the plane of the orbit b. Assumptions and approximations ar r sin (0 + tu) cot i yT w t 1 - e 1 + e cos e cos R lae r ) sin 9 S ()78e) R is the component along the radius vector (measured positive away from the center of the earth), S is the transverse component in the in- stantaneous plane of the orbit (measured positive when making an angle less than 90° with the satellite's velocity vector), and W is the com- ponent normal to the instantaneous plane (meas- ured positive when making an angle less than 90° with the north pole). When the disturbing force is caused by air drag, the perturbing acceleration is 1 (r) V' C D A B p (r) V which has the components, R = -B p (r) V V e sin Yl + e 2 + 2e cos 6 S = -B p (r) V -V cos /3 a V Q (1 + e cos 9) Vl + e 2 + 2e cos G (179a) (179b) Equations (168a), (168b) and (168c) can also be expressed in terms of the eccentric anomaly E, instead of the true anomaly Q. This step is desirable since the integration of Eqs. (167a) through (16 7e) over an orbital revolution can be most easily carried out by using E as the variable of integration (limits to 2tt). To facilitate the integration, the following assumptions and ap- proximations are made: (1) The density, p (r), is spherically sym- metric. It is assumed to change ex- ponentially above perigee height, i.e. , -(h - h n )/H P (r) = Pp e p (180) where p is the density at perigee. It is a function of the height, h , of peri- gee above the surface of the earth. II is the scale height at perigee altitude and h is the height of the satellite above the surface of the earth. (2) In integrating the effect of the perturbing force over one revolution, the satellite is assumed to move along the unperturbec Kepler orbit. This is a good approxima- tion because the perturbation has little effect on the orbit over one revolution. This is not true during the last few revolutions of the lifetime. Other methods must be used to determine the effect of air drag during that short time. (3) The integrand is expanded in the quanti- ty e (1 - cos E) (which is always small IV-41 wherever the perturbing force is im- portant). Only the most important terms of the series are integrated. (4) The entire atmosphere rotates at a uniform angular rate equal to the rate of rotation of the earth about its axis. Several investigators (Refs. 40 and 41) have carried out integrations using variants of the above approximations. Sterne (Ref. 41), for example, in addition to treating the problem with a spherically symmetric atmosphere, also made a more refined analysis taking account of the atmosphere ' s flattening. However, for altitudes above 200 naut mi or 3 70 km, the neglect of the diurnal bulge causes errors, which overshadow the improvement obtained by considering atmos- pheric flattening. This was shown by Wyatt (Ref. 42). Moreover, fluctuation in the density of the atmosphere causes uncertainties large enough that highly refined expressions for the changes in orbital elements are not warranted for most pur- poses. c. Approximate changes in osculating orbital elements Given below are methods useful in simplified programs, based on approximations (1), (2), (3) and (4). Most of the results were obtained in series form, but only the dominant terms are given here. For higher order terms see Sterne's paper (Ref. 41). The case of ae/H > 2. When the parameter ae/H x 2, the changes in the orbital elements per revolution are ae/H Aa = -Q Ae = -Q 1 + 1 - 8e + 3e 8c (1 - e ) (181a) (v) l _ (3 + 4e - 3e ) Be (1 - e ) J (181b) Ai = -D(l - e) 2 |cos 2 u, + p-sc- [«(j-H) _l i/if* , 9e + 6e - 151 2 + |4f + n 1 cos OJ (1 - er (181c) AQ = -D(l - e) 2 Jl +^ 9e + 6e - 15 4f* (1 - ef A oj = - A£2 cos i > sin to COS CO (181d) (181e) where „ OT , 2 , (1 + e) Q = 2B p a f — - P 2 (1 - e) m (!-) 1/2 2U f = 1 - -— (1 - e) n /l-eN l/2 f* 1 - e f-fc D = 2ttB — ap f l/2 (2irc)" I/ ' 2 n ^p n = angular rate of rotation of the earth's e atmosphere in inertial space (2it in approximately 24 hr) It might also be useful to know how the radius of perigee, q, changes in a revolution; q is simply related to a and e through the equation q = a (1 - e) Thus, the change in q, when ae/H > 2, is A « - -Q (l^l) 2c < 181f) and the change in the period can be found from the change in a through the relation At/t (I) Aa/c The case of ae/il < 2. When the parameter ae/H_2, the appropriate changes are ,3/2 (1 - 2e)I Q (c) + 2e I, G (182a) (l-e) 1/2 L l (c >] + | [l Q (c) + I 2 (c)l | (182b) Ai =-K j ["l o (c) - I 2 (c)] + (cos 2 oj) [l 2 (c) - 2e I x (c)j( sin i (182c) A Q = -K [l 2 (c) - 2e Ij (c)] sin a cos oj A< -An cos i (182d) (182e) and (c)] (182f) 3e) Ij (c) - | I 2 where C D A 2 -c G = 2tt a p f e m *p IV-42 K C D A Q , — a p n 'p VF, and I is the Bessel function of imaginary argu- ment and nth order. The secular time rate of change of the elements may be obtained by dividing Eqs. (181a) through (18 If) and Eqs. (182a) through (1821) by the Kepler- period, .3/2, r - 2-na'' From Eqs. (181) and (182) it can be seen that the rotation of the earth's atmosphere relative to the satellite affects the inclination, node, and argument of perigee of the orbit. If there were no atmospheric rotation lo ^ =0), only the semi- major axis and eccentricity (hence the height of perigee) would be affected. The orbital parameters most sensitive to drag are the heights of apogee and perigee, the period, and the eccentricity. The reason for this sensitivity is primarily the fact that V relative to the atmosphere is not vastly different than V rela- tive to space. Thus, the perturbing force is nearly planar and therefore affects semimajor axes and eccentricity. The procedure for evaluating the effects due to drag is now clear: First the element variations are computed, then the elements are adjusted and the process continued. If a sufficiently small in- terval of time is utilized for the stepping proce- dure, say 1 revolution for satellites above ap- proximately 180 km, then the element changes will be sufficiently small so that they may be added to those produced by the sun, moon, ablate - ness, etc., to produce a first order approximation to the total solution. Numerical data and discus- sions of the planar effects are presented in Chap- ter V (Satellite Lifetime). Thus, graphical data will not be included at this point. Data for the non- planar parameters will not be prepared because of the fact that too many parameters are involved to make such a presentation meaningful. Rather it is suggested that these effects be evaluated for each orbit. d. Contribution of random drag fluctuations to error in predicted time of nodal cross- ing of a satellite, assuming perfect initial elements* If the period is known to be exactly P(0) during the zeroth revolution, then the period will be pre- dicted to be P'(n) during the nth revolution. This prediction will be based on the average rate of change of period during the preceding revolutions. But suppose there are random fluctuations about the average change in period. Let these random fluctuations be p,, p„, . . ., p., . . ., p N . Then after N revolutions the period will actually be N P(N) = P'(N) + I y- i *This subsection was included as Appendix E, Special Derivations" in Flight Performance Handbook for Orbital Operation, STL report prepared under Contract NAS 8-863. The time of nodal crossing will be predicted to be .N t'(N) = t(0) + ") P'(n) n = l while the actual time of nodal crossing will be N N (o) + y t(N) = t(0) + x P'(n) + ) r(n) n=l n=l where n r(n)s I p y The error, E(N), in the prediction is N N n E(N) = -£ r(n)--l £ P f n=l n=l j=l This double sum can be written out explicitly as E(N) = - T( Pl ) + ( Pl + p 2 ) + . . . + (p 1 + p 2 +. . . +p N )]. Rearranging terms, we obtain E(N) = - Tn Pi + (N - 1) p 2 + . . . + p N ~J . (183) Case a: Fluctuations Indepen dent from Revo- lution to Revolution? If each p. is independent and has the standard deviation F, then the standard deviation of E(N) is , , „ / N \^ 2 G„„„(N) = E(N) „ =lF ^ ~ 2 n=l = f[n(N + 1) (2N + l)/6] 1/2 . (184) Case b: Fluctuation s Correlated over 25 Revolutions? On the~bther hand, suppose th a t the random drag fluctuations are perfectly cor- related over intervals of 25 revolutions, but in- dependent from one interval to the next. A 25- revolution interval is chosen because it is the usual smoothing interval in published orbits. We begin with Eq (183). Since the accelerations are assumed to be correlated over intervals of 25 revolutions, p q+l = P q+2 P q = P A = P q+25 = P B p q+26 P q+27 ' = P q+50 = P C IV-43 The fluctuations in acceleration about the smoothed value are illustrated in the following sketch. '////////ti n • i i q + 25 n = Revolution number 50 The possible values of q range from 1 to 25. In the absence of particular information, all values of q will be assigned equal weights. When n = 1, p = p.. When n = 2, p will equal p. if 2 < q < 25, and p = p if q = 1. When n = 3, p = p . if 3 < q < 25, and p = p„ if q = 1 or 2, etc. The equal weighting of the 25 values of q can be expressed by averaging over the ensemble of possible values, that is J l J A (1/25) (24 p A + p B ) (1/25) (23 p A + 2 p B ) > 25 =(l/25)(p A+ 24p B ) J 26 _ H B 3 27 = (1/25) (24 p B + p c ) p 50 = (1/25) (p B + 24 p c ), etc. The timing error, averaged over the ensemble of possible values of q, is found by substituting these p. 's into Eq (184). ETnT = - [Np A + (N - 1) (24 p A + p B )/25 + (N - 2) (23 p A + 2 p B )/25 + ... + (N - 24) (p A + 24 p B )/25 + (N - 25) p B + (N - 26) (24 p R + p c )/25 + ... + (N - 49) (p B + 24 p c )/25 + (N - 50) p c + (N - 51) (24 p c + p D )/25 + ...J , for all (N - k)>0 . . . (185) Collecting coefficients of p., p R , and p„ Let ETNT = - (p /25) [25 N + 24 (N - 1) + ... + (N - 24)] - (p R /25) [(N - 1) + 2(N - 2) + . . . + 24(N - 24) + 25(N - 25) + 24(N - 26) + . . . + (N - 49)] - ( P( .,/25) [(N - 26) + 2(N - 27) + . . . + 24(N - 49) for all (N - k) -. . . . a(N) = [25 N + 24(N - 1) + . . . + (N - 24)] . b(N) = [ (N - 1) + 2(N - 2) + . . . + 24(N - 24) + 25(N - 25) + 24(N - 26) + . . . + (N - 49)] c(N) = [(N - 26) + 2(N - 27) + . . . 24(N - 49) + 25 (N - 50) + 24(N - 51) + . . . + (N - 74)] d(N) = [ (N - 51) + 2(N - 52) + . . . + 25(N - 75) + . . .] e(N) etc. , for all (N - k) -> 0. If the standard deviation of p. is <t, and each p. is ind ependent, then the standard deviation of e7nT is K (N)h |eTMt1 =(a/25) Ta 2 (N) rms L Jrms L 1/2 + b 2 (N) + c 2 (N) + ...1 (186) In case N <_ 25, a(n), b(n), and c(N) are calcu- lated as b(N) = (N - 1) + 2(N - 2) + . . . + 24(N - 24), N-l for all (N - k) -> and for N s_ 25 N-l N-l I'-I = ^ q(N - q) = N q = l 1 1 = N 2 (N - l)/2 - N(N - 1) (2N - l)/6 b(N) = [N(N - l)/2] [N - (2N - l)/3] . for N < 25 a(N) = 25(N + N - 1 + ... + 1) - b(N) IV-44 a(N) = 25 N (N + l)/2 - b(N). for N < 25 c(N) = 0. for N < 25. In case N is greater than 25, the contribution of the first 25 terms in Eq (185) to b(N) is 24 24 24 b l (N) = A q(N - q) = N^ q -^ q^ q=l 1 1 b (N) = 100 (3 N - 49), for N > 25. a(N) is then given by a(N) = 25(N + N - 1 + . . . + N - 24) - b^N) a(N) = 025 (N - 12) - b^N), for N -, 25. We define b„(N) to be the contribution to b(N) of all those terms of the second 25 terms in Eq (185) for which the quantity N - k is positive. For N ^ 25, b 2 (N) = 0, and for N ^ 26, b 2 (N) is given by b 2 (N) = a(N - 25), for N -, 26. b(N) is given by b(N) = b^N) + b 2 (N). The quantities c(N), d(N), etc., are given by c(N) = 0, for N - 26 c(N) = b(N - 25), for N ^ 27 d(N) = 0, for N <_ 51 d(N) = b(N - 50), for Nn 52 etc. Compari son of Case a and Case b . The limits of the equations for correlated and uncorrected errors will now be calculated, to show how the two cases are related. For uncorrected errors (Case a), take the limit of Eq (184). lim F [N(N+ 1) (2 N + 1)/6] 1/2 = F(N /3) (187) For correlated errors (Case b), take the limit of Eq (186) im (a/ 25) |[625 (N - 12) - 100 (3N -49)] + [lOO (3 N - 49) + 625(N - 37) - 100 (3 K - 124)1 2 + J continued + [lOO (3 N - 124) + 625 (N - 62) - 100 (3 N - 199)J + ... \ '" = lim a { [13 (N - 8)1 2 + [25 (N - 25)] 2 + [25(N-50)] 2 + ...} ^ 2 . Let N = 25 M, where M is an integer. Then the above limit becomes lim (25) 2 ct { M 2 + (M - l) 2 + (M - 2) 2 . . . + l 2 - M 2 + [(13/25) (M - 8/25)] 2 } = lim (25)' M -, „ ■{ M (M + 1) (2 M + l)/6 1/2 - M 2 + [(13/25) (M - 8/25)] 2 | = lim (25) 2 a { M (M + 1) (2 M + I) I s\ l M ->,*, ■ <■ ' ? , 1/2 3 1/2 = (25T a (M 13) -= 5(j (N /3) /2 (188) Thus, the limits (5) and (6) for correlated and uncor related errors approach the same asymp- totic form for large N. This makes it possible to evaluate the constant F, which must equal 5a. The relationship F = 5a corresponds exactly to the situation in the theory of errors, in which the standard deviation of the mean of k indepen- dent observations equals the standard deviation of one observation divided by the square root of k. The asymptotic form Eq (188) is a convenient approximation to represent the error contributed by random fluctuations, when the initial elements are perfect. The satellite accelerations, i.e., the rate of change of the period published to July 1961, furnish no evidence for choosing be- tween Case a and Case b, because they are smoothed over intervals of 25 revolutions. 7. Radiation Pressure Above a height of 500 naut mi or 926 km, radiation pressure usually has a greater effect on the orbit of an artificial satellite than air drag (though for ordinary satellites, the effects of radiation and drag both are very small). How- ever, both effects are significant for balloon satellites since the area-to-mass ratio is large. (The area-to-mass ratio of the Echo I balloon satellite was 600 times that of Vanguard I.) At first glance it may appear that it is possible to handle this force as was done in the previous sections. However, this is not the case because of the fact that the earth affords a shield from the sun 1 s rays during a portion of the orbit. This shadow effect is investigated in detail in Chapter XIII. Kozai (Ref. 43) has integrated the pertur- bations of first order over one revolution, in terms of the eccentric anomaly, E. The satellite leaves the shadow when E equals Ej, and enters the shadow when E equals Eg. (Reradiation from the earth is ignored. ) IV-45 The perturbations over one revolution are given by 6 a = 2a 3 F (S cos E + T Vl - e 2 sin E) 6 e = a 2 F Vl - e 2 I S Vl - e 2 cos 2E + T (-2e sin E + ^ sin 2E) *! C TdE p (189) ^2 (190) 5i = a'F W vT ■[{«'* e ) sin E e 4~ + VT unless other values are written; S and T are the expressions of S (6) and T(6), in which <J> is re- placed by io ; that is, S = S(0), T = T(0). (195) If the satellite does not enter the shadow dur- ing one revolution, the terms depending explicitly on E vanish, and, in particular, 8a vanishes. In the expressions of 6oo and 5fi, indirect effects. of the solar radiation pressure through (1) and U must be considered as d5w do. . , doj c . ( dto t -n- = -j- iet-jr 6i+-j- 5a, dt de di da d5ft dS2 5 , dn . . ^ dft , -37- = j— 5e + 3-- 5l + j— ° a - at de di da (196) - j co s 2E) sin u 3 2 e f W COS o; dE (191) sin i SQ = a ^H + e ) sin E T sin 2E > sin w - Vl - e (cos E (192) t cos 2E) cos oj E E tj- e \ W sin u dE ' J J 5(i> = - cos i 512 + a F vTT^ S(esinE + ^ sin 2E) cos 2E. + T Vl - e 2 (e cos E if SdE (193) ATT 6M = - | C ^- dM - Vl - e 2 5u The disturbing functions S(0), T(6), and W S(6) = - cos k- cos y cos (X n - 4> - SI) ■ 2 i . 2 ( . . sm y sm 2 cos * n n " * ' k- sin i sin e jcos ( \„ - <t> ) W cos (-X Q - ^ )| • 2 i 2 « . . sin *■ cos n- cos (ft - x - <t>) 5s -5 sin g- cos (-Xq - <j> -ft), (19V) sin i cos ,,- sin ( \., - £2) sin i sin „- sin (x„ + ft) cos 1 sin « sin X (198) where \„ is the longitude of the sun, and e is the obliquity. The expression of T(6) is obtained if cos in S(6) is replaced by sin except for the trig- onometrical terms with an argument i, e, i/2, or e /2. - Si - e cos i 6ft 2a 2 F is (1 + e 2 ) sin E - I sin 2e| T Vl - e 2 (cos E - I cos2E) SdE (194) where the limits of integration are E. and E„ The conventional symbols are used for the orbital elements: a is the major axis, e the ec- centricity, i the inclination, ft the node, 10 the argument of perigee, M the mean anomaly, and 9 the true anomaly. In addition, and + 00 a (1 - e"); n 2 a 3 F S(9), n 2 a 3 F T(6), and n 2 a 3 F W are three components of the disturbing force due to the solar radiation pressure in the direction of the IV-46 radius vector of the satellite, in the direction perpendicular to it in the orbital plane, and in the normal to the orbital plane; and F is a product of the mass area ratio, solar radiation pressure, and a reciprocal of GM. The smallness of the effect of radiation pres- sure on an ordinary satellite is illustrated by the orbit of Vanguard I (Rets. 44, 45 and 46). Radiation pressure periodically changes its height of perigee by about one mile. The effect of rad- iation pressure on the period is obscured by the fluctuations in air drag. Both radiation pressure anil air drag would have had very small effects on a conventional satellite at the original perigee height of Echo I, but both effects were magnified by the area-to-mass ratio, which was, 600 times that of Vanguard I. The consequent large effects on the rate of change of period are shown in Fig. 10, which originally appeared in Ref. 45. The correlation of air drag with the decimeter solar flux is also shown to persist to this great height (see Chapter II). Note also in Fig. 10 that radiation pressure sometimes lias no effect on the period. This occurs when the whole orbit is [E„ = E-, + 2 i in the expression for -Sa of Eq (194).] in sunlight. L u„ - u. The radiation pressure sometimes acts to in- crease the period. Echo I was the first satellite for which this was observed (Ref. 45). It was also the first satellite for which the eccentricity was observed to increase. This can be clearly seen from the increasing distance between peri- gee and apogee in Fig. 11, which is modified from the NASA Satellite Situation Report of July 18, 1961, though for most satellites the eccentricity has decreased during the lifetime. Detailed behavior of a satellite due to this per- turbation cannot be tabulated in a parametric form due to the large number of factors affecting the solution. These factors include longitude of the nodes, orbital inclination, position of the earth in its orbit and semimajor axis and eccen- tricity of the orbit. Thus, it is necessary to ob- tain a particular solution for the perturbed rates of the elements given a set of desired elements, then incorporate them in a numerical manner with the rates produced by other forces. The analyst is urged to consult a growing body of literature for this perturbative influence. Some of these references have been collected and presented as Refs. 1, 34, and 43 through 57. 8. Satellite Sta bility The study of satellite stability concerns the long term orbital behavior of artificial satellites. It attempts to provide the mission analyst with answers to such questions as: How will the various orbital elements change? What will be the magni- tude of these changes'? Will their pattern be highly erratic or regular? Will there be a change in the pattern from erratic to regular or vice versa 1 ? In order to answer these and other questions it is necessary to combine the perturbing forces acting upon the satellite orbit and their effect upon the various orbital elements of interest for a particu- lar mission. This section discusses some approximate methods for dealing with satellite stability problems. The formulas and methods given here can be used to: (1) construct approximate computer programs, which arc much faster and cheaper than ''exact" programs; (2) solve some satellite stability problems without the need for a high speed computer; (3) help in gaining more insight into the behavior of satellites. Section C2 of this chapter discussed the ap- proximate method of IY1. Moe and presented most of the formulas which will be used in this sec- tion. The following discussions present some of the results obtained using this method. Although only earth satellite results are given here, these methods have also been used extensively for lunar satellites and can be applied to orbits about other planets. Part 2 illustrates a method for computing satellite trajectories by hand. Care must be taken not to use the methods of this section on orbits which are physically too large, in which case the approximations for luni -solar perturbations break down. While definite rules cannot be laid down, Table 4 should prove helpful. The table lists ttie various bodies and the approximate upper limits where "very good, " "good, " and "fair" results can be obtained. The parameter used is the period of the satellite in days. TABLE 4 Validity of the "Approximate" Method as a Function of Orbital Period (days) Ear tli Moon Mars Venus Mercury Very Good 2. 0. 5 45. 15. Good 60. Fair 90. 5. 25. 8. 35. 10. A special case arises for very remote earth satellites which do not pass near the moon. These may also be treated by approximate meth- ods and in these cases some orbits with periods as long as 45 days can be studied. For this class of orbits the effects of the moon are ignored and the sun is treated as the only disturbing body. Another class of orbits for which the methods of this section are not very helpful is the very near eartli orbit where drag and oblateness perturbations are predominant. Accurately predicting the future history of an artificial satellite is difficult and expensive. Fortunately approximate methods often give good results. This section discusses approximate methods which have been extensively used for terrestrial and lunar satellite orbits. It is convenient to consider the stability of the orbit of an earth satellite as a two -body problem with perturbations introduced by the sun, moon, IV-47 earth shape, drag and radiation pressure. These effects must be analyzed separately and then combined. This procedure is accomplished only after allowing for the fact that the various equa- tions refer to different planes; the results can then be summed to yield the new orbit. The process can then be repeated. Performing this operation by slide rule or desk calculator is very slow and requires about 8 hr to compute the change for one revolution, or 1 man -year for 1 month of the satellite's orbit. However, the combined equations can be eval- uated on a high speed computer such as the IBM 7090 at the rate of about 5 rev/ sec. Subsequent paragraphs of this section discuss results ob- tained in the latter manner. When high speed computers are not available, good results can be obtained by using the secular terms to estimate the results over many revolu- tions. This method is illustrated in Part 2. but the period is one -half year and the amplitude is about 200 naut mi or 370 km. Figure 12 is a graph of perigee height versus time. Note that the moon waves are shown only for the first 100 days. The rest of the curve shows the envelope of minimum perigee height. This simplification is adopted for all similar graphs in this section. Note also that the moon waves should be just a sequence of separate points plotted at 1.73-day intervals since perigee is reached only once each revolution of the satellite whose period was 1 . 73 days. Now consider the combined secular effect caused by the sun and moon. This is given by the following formula which is derived in Section C5 of this chapter. A q K* sin 2o.- sin i + A sin 2t, 2 . (199) Part 1: Sample Results by "Approximate" Method. Early in 1961, a~study (Ref. 58) was made at STL to determine the lifetimes of earth satellites in highly eccentric orbits. The project was the Eccentric Geophysical Observatory (EGO). Some of the results of this study will be used to illustrate the approximate method and the general problem of orbital stability. The experimental objectives of Project EGO made it desirable to keep perigee height as low as possible consistent with lifetime require- ments. A graph of the suggested nominal an- swering these requirements is shown in Fig. 12. This graph will be discussed in detail since it illustrates most of the important features of this type of orbit. The initial conditions in terms of equatorial spherical coordinates are given in the figure. These were the suggested burnout con- ditions of the missile which were to inject the satellite into orbit. The resulting orbital param- eters in terms of equatorial coordinates are as follows: where a = 32, 879 naut mi = 60,892 km e = 0. 891057 i = 31. 289° a U = 41. 796° a = 135.617 Launch time = 3 hr 30 min GMT Launch date = 1 April 1963 The most important parameter in the EGO study is perigee height or equivalently perigee distance q and to the first order, the only per- turbations affecting q are caused by the sun and the moon. The periodic term for the lunar per- turbations of q may be written as q per A B sin (2T + m m m *m> and ( v are as given in Section m fo where A , B , in m 05 of this chapter. Therefore the moon causes the satellite's perigee to alternately rise and fall. The period is one-half the moon's sidereal period or a little less than fourteen days. The amplitude for EGO-type satellites is about 40 naut mi or 74 km. The sun has a similar effect A = 15 II tt a 4 e VI - e 2 m m f> and 15 H ira e s V7 Recall that H and H are positive constants, m s ' Note that the subscripts m and c indicate moon plane and ecliptic plane parameters. Equatorial parameters will be indicated by the subscript a in the following discussions. Initially, the nominal orbit had equatorial parameters i =31.29°, V = 41. 80° and ' a a co =135.62°, and gj =94.68°, i =20.30°, a m t Si =87.47°, and oj = 85. 69°, respectively. At the end of 402 days, the orbit parameters take on the values: a = 32, 793 naut mi or 60, 733 km, e= 0.8893, i =37.58°, Si = 8. 55°, c, =181.38°, = 16. 11°, = 187.07°, i = 14. 75", and o. = 167. 96°. Note that the secular trend is now nearly 0, which is again shown in Fig. 12. At the end of 554 days, the orbit parameters are: a = 32, 779 naut mi or 60, 707 km, e = 0. 8902, „ = 195.01°, i a = 36.87°, U a = -1.65 i = 16.77°, w m 214. 50°, i = 13. 45° and oj = 198.43°. The secular trend is now negative. Now a brief discussion will be given of the other figures in this section. In the initial EGO study (Ref. 58), the burnout conditions of the missile were given. The only variation per- mitted was in time of launch. A series of satel- lite lifetime runs (Ref. 59) were made on the IBM 7090 with 1 April 1963 as launch day. The first run was at hr GMT, the next at 2 hr and so forth to 24 hr. The results are illustrated in Fig. 13. At first glance, it is surprising that merely changing the launch time would have such a large effect on the satellite's future history. This IV-48 behavior results since changing the launch time of day changes the satellite's nodal longitude (U). At h, V -10.849. From then on Q i , and m increases by 30.083° for each 2 hr added to the launch time. This, of course, is due to the earth rotating 360. 996° in 24 mean solar hours. Changing ii does two important things. First, it changes the phase of the sun and moon desig- nated by T and Y . For EGO -type satellites, the moon's periodic effect is only about 40 naut mi or 74 km in amplitude and hence is not too critical. The sun's periodic effect, however, is very important. Secondly, changing il changes the ecliptic and moon plane parameters of the orbit and hence changes the secular trend of the satellite. The secular trend is large and posi- tive for the 8-, 10-, 12-, and 14-hr orbits. In Fig. 14 comparison is made between ap- proximate results as obtained from the Satellite Lifetime Program (Ref. 59) and results obtained by integrating the equations of motion in a way that is essentially exact. Note that the agree - ment is good. Figure 15 illustrates how oblateness indirectly affects perigee height even though its direct effect is zero to first order. It does this by changing the equatorial inclination i and the nodal longitude ii a . This in turn changes the ecliptic and moon-plane parameters i , This then changes the secular effect as is shown. In Fig. 16 the effect of leaving out the effects of sun or moon is demonstrated. Here the nomi- nal graph is shown in comparison with the same orbit computed with the sun only and with the moon only. Note especially the difference in secular trend. The effect of making various changes in the initial parameters of the nominal orbit is shown in Figs. 17, 18, 19 and 20. The graph of the 6-hr orbit for a period of 10 yr is shown in Fig. 21. This orbit illustrates an important phenomenon. From the secular trend in perigee distance given by Eq (185) it follows that A q depends mainly on the incli- sec J nation and argument of perigee. The inclination does not change very rapidly; however, the argu- ment of perigee is perturbed very much by oblate- ness and to a lesser extent by luni-solar effects. As i increases, oblateness perturbations get smaller (0 < i < 03. 7') and as a result cj and o. change slowly. Thus the secular term can be nearly constant over a long period of time. If this happened when the secular trend was down, the satellite would probably expire. This effect also explains the short life of most lunar satel- lites (Ref. 58). Part 2: Hand Calculation of an Earth Satellite Orbit! The detailed revolutum by revolution ap - proximate calculation of a satellite orbit is too slow and tedious to be practical by hand. However, the process can be accelerated by treating the periodic and secular terms separately. To illustrate this method, part of the tra- jectory of the EGO Nominal will be calculated (see Fig. 12). Consider first the periodic term for the lunar perturbations (given in Section C2 of this chapter). Aq , .. = A B sin (2T , - a ) per(mt) m m mt m where il = 0.68736 x 10 ~ 18 (naut mi)" :i was evaluated in Part 2. A = 15. 3 naut mi = 28. 3 km m H = 0. 961 m a = -170. 64° m (Note that the minus sign is taken when sin 2gj is negative. ) m h The parameter Y , denotes the angular mt h position of the moon measured from the satel- lite's ascending node at time t (see Fig. 9). This parameter is given by the following formula. r = (t - t ) n - !.■ mt m m mt where t = time the moon was at its ascending equatorial node n = moon's angular- rate = m t m i. ! , = satellite's moon-plane ascending node measured from the moon's equatorial node t = time . If time is measured in days, and angles in degrees and if the initial time t., = then t = -6.9658 davs (ephemeris) m ^ n = 13. 176°/day m J V = 67.58° m t = (initially) T = 24. 14° mo T = 24. 14 + 13. 176° mt where t is measured in days. Substituting the computed values of A , B , and „ gives m m m b Aq , . = 14. 7 sin (2 T +170.60) per(mt) mt = 14.7 sin (218. 92 + 26. 352 1). IV-49 The period of the satellite once again is 1.73 days. Hence the periodic term alone indicates that the moon's gravitational field will push the satellite down for four revolutions. The satellite will then be at a minimum height as far as the periodic effect of the moon is concerned. From then on this periodic motion can be ignored (see Fig. 12). Evaluating Aq , ,■. for time t = 0, t = 1. 73, b T>er(mt) t = 3.46, and t = 5.19 days, and then summing gives the initial downward push by the moon to be 36. 2 naut mi or 67. km. Consider now the periodic term of the sun's perturbation in perigee distance as measured from the center of the earth (q) Aq , ,, = A B sin (2 T . - a ) ^per(et) t t etc 1 / 2 Aq = - «- [ A sin 2uj sin i + M sec 2 V m m m + A sin 2 .. Aq = +0.0319 naut mi /rev. = +0. 0591 km/ rev sec Assuming the various parameters are relatively invariant during the first 164.35 days, the secular rise in perigee height for this period can be com- puted as v Aq = l^Lj^L (0.0319) = 3.0 naut mi or 'sec 1. 73 5 . 6 km . The combined periodic and secular results indi- cate that perigee height should have decreased by 3 6.2 + 21.0 -3.0 =54. 2 naut mi or 100. 4 km. where A = 7. 03 naut mi = 13 km £ B = 0. 961 £ a =171. 38°. € The parameter r is given by r = (t - t ) n -si • 6 t £6 6 t This checks reasonably well with the results shown in Fig. 12. Better results could be obtained by summing the secular perturbations over perhaps 20- or 50- day intervals and taking into account changes in the parameters e, i , u. , i and <~ (in such com - 1 m m £ e putations the periodic terms in these parameters are not important). The main difficulty here would be in converting solar and lunar perturba- tions into changes in the equatorial parameters. t = - 11.4258 days £ n =0. 9856° /day S. , = 87. 47° when t = € t r e0 = -76.21". Thus r = - 76.21 + 0. 9856 t° £ t whore t is measured in days. Combining the above equations gives Aq , . = 6. 59 sin (2 r , - 171. 38) per (ft) et = 6. 59 sm (36. 20 + 1. 9712 t). Note that the sun's periodic effect is initially upward. But after about 146 days, this upward move is cancelled. The satellite than has about 18. 4 days or eleven revolutions to reach a min- imum, i: valuation Aq . at time t = 147.05, n per (ft) t = 148.78, t = 150.05 1 , • ■ • , t = 164.35- that is, once each revolution from time t = 147.05 to t = 164.35--and summing yields the net downward push of the sun as 21 naut mi or 39 km. The satellite will then be at a minimum height as far as the periodic effect of the sun is concerned. From then on this periodic motion can be ignored (see Fig. 12). Now consider the combined secular effects of the sun and moon on perigee distance q: Using this method with, say, 50-day steps should yield results of fair accuracy for many satellite orbits. For example, the hr, 2 hr, 8 hr, 10 hr, 12 hr and 14 hr would be quite easy to compute by hand (see Fig. 13). Hand com- putation of the orbit of a lunar satellite is also easy because the moon's equator is very close to the ecliptic, and because the sun's effect is very small compared with the effect of earth. F. 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F. , "Perturbations of Elliptic Orbits by Atmospheric Contact," Journal of the British Interplanetary Society, Vol. 16, April 1958, pp 368 to 379. (Also Journal of the British Interplanetary Society, Vol. 17, No. 1, January-February 1959, pp 14 to 20.) O'Keefe, J. A. , "An Application of Jacobi's Integral to the Mo- tion of an Earth Satellite, " Astronomical Journal, Vol. 62, October 1957, pp 265 and 266. "Determination of the Earth's Gravitational Field, " Science, Vol. 131, 1960, pp 607 and G08. Zonal Harmonics of the Earth's Gravitational Field and the Basic Hypothesis of Geodesy," Journal of Geophysical Research, Vol. 64, No. 12, December 1959, p 2389. O'Keefe, J. A. , and Batchlor, C. D. , "Perturba- tions of a Close Satellite by the Equatorial El- lipticity of the Earth, " Astronomical Journal, Vol. 62, 1957, p 183. O'Keefe, J. A. , Eckels, A. , and Squires, R. , "Vanguard Measurements Give Pear-Shaped Component of Earth's Figure, " Science, Vol. 129, No. 3348, 1959, pp 565 and 566. O'Keefe, J. A. , et al. , "The Gravitational Field of the Earth, " Astronomical Journal, Vol. 64, No. 7, September 1959, pp 245 to 253. (Also, American Rocket Society, Preprint No. 873-59. ) Okhotsimskij, D. E. , Eneyev, T. M. , and Taratynova, G. P. , "Determination of the Life of an Artificial Earth Satellite and an In- vestigation of the Secular Perturbations," Uspekhi Fiz. Nauk, Vol. 63, No. la, 1957. Orlov, A. A. , ''Formulas for Computing Perturba- tions in Elliptical and Hyperbolic Motion, " USSR, Vestnik Moskovskogo Gosudarstvenngo Universiteta, Seria Fizikii Astronomicheskii, No. 2, March-April 1960, pp 51 to 60. (Ab- stracted in Physics Express, Vol. 3, No. 3, November 1960, pp 28 and 29.) Parkinson, R. , Jones, H. , and Shapiro, I. , "The Effects of Solar Radiation Pressure on Earth Satellite Orbits, " Science, Vol. 131, I960, pp 920 and 921. Parkyn, D. G. , "Satellite Orbits in an Oblate Atmosphere, " Journal of Geophysical Research, Vol. 65, No. 1, January 1960, pp 9 to 17. Penzo, P. , "Analytical Perturbation Expressions for' the Restricted Three-Body Problem," AFBMD-STL-Aerospace 5th Symposium on IV-56 Ballistic Missiles and Space Technology, Space Trajectories Session, 1960. Petty, C. M. , and Breakwell, J. V. , "Satellite Orbits About a Planet with Rotational Sym- metry, " Journal of the Franklin Institute, Vol. 270, No. 4, October 1960, pp 259 to 282. Pines, S. , "Variation of Parameters for Elliptic and Near Circular Orbits, " Astronomical Journal, Vol. 66, No. 3, February 1961. 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IV-58 ILLUSTRATIONS IV-59 o c '3 cr S-, o CO ■a c he a c o JO a, in .° o o U o (J = jCjJABjg -[8A9t B3S) SSB]/M 1T U H • ia d SOJOjI "«>Cli>tf4€ «**GE BLANK NOT FILWFt IV-61 90- 80- 70- 60- i> T) M) •^ ^ c o rt a O S3 t— i , i at u 40 O 30- 20- 10- 0^ 3ttJ 2 (— ) (2 - | sin 2 i)rad/rev J 2 = 2/3 J 20 40 60 80 100 Inclination (deg) ,6,000 2.0 ,8, 000 X ,9,000 3.0/X/10, 000 Of 12,000 14,000 + ">^ .16,000 > 4.0/ '$ 5.0^/18,000 6.0^ 7.0' r-24 -22 -20 -If -16 -14 — -12 % T3 ■10 .3 a)" PS ■8 § 41 O 0) Li P. Li 3 O -6 -4 -2 --2 --4 C--6 Fig. 2. Solution for the Secular Precession Rate as a Function of Orbital Inclination and Semiparameter IV-62 AM = 3 M 2~ J 2 Tf 1 -e' (1 - 3/2 sin i) ~ 2.0 90- 80- 70- 60- <D TJ C rso O m C r— 1 o c •—4 r— 1 til ■■-J 40 JJ In 30- 20- 10- 0- T3 a! S-i 1.0 -1.0 -2.0 ; t&itl 11 Willi 1 r: ::ip::±I -+l : ft i + " T 1 1 1 f Mri>lT" 3 :±fff; g R/p = i; -If "+^"iiffinffli ffij:: ft Trfflii" 1 trt : tg'i(|m|fflj lis- if .';•■ -f -f$$ffi^ffi f ± jll: _i: TTrrmi / - tt "H " ■T- : f fffj|[j R/P - -l/i TrHrrr -ffiff : ::t: II '+ ; 1-2.0 10 20 30 40 50 60 70 80 90 Inclination, deg 1.0 <* % 2.0 -1.0 ^-1.0 Fig. 3. Change in the Mean Anomaly Due to the Earth's Oblateness IV-63 1. 5 1—6,000 2.0- o <a CO -6,200 -6,400 -6,600 6,800 7,000 B -7,500 "^ 2.5- -8,000 g .-8,500 3.0- -9, 000 3.5- 4.0- -9,500 -10,000 5.0-" 6.0- 7.0-i 10.0- --11,000 -12,000 13,000 14,000 15,000 10 20 30 40 50 60 70 80 90 Inclination (deg) d U a a; co = 3 Mf) J 2 = 2/3 J cos i rad/ rev Fig. 4. Solution for the Secular Regression Rate as a Function of Orbital Inclination and the Semiparameter IV -64 — ^=-= (^- - l\ 1 = 3 J ( n ) 2 ( 3cos 2 i- l \ ;VT~7 U 'VT7" Mpj ^ — 5 — > 90- 80- 70- 60- 3 c 50- o c O 40- 30- 20- 10- 0- o t- <l |t- -0.5 20 40 60 80 100 120 Inclination (deg) 1.0 1.5 r 9t "to*/*' 3.0 4.0 p-1 ~-0 9 ro 8 ~-0 7 ~—o. 6 r°- 5 _ r°- 4 E-o. r-o. 3 7 o 2 * ^-0. n! 1 < |"- r-o r--° . 1 r--° .2 r-0 3 r—0 4 ~--0 5 t--0 6 Fig. 5. Change in the Anomalistic Period Due to the Earth's Oblater IV-65 90-. 80- 70- 60- bfl I 5 °" 40- 30- 20- 10- 0- ^=(f-.v- 2 (D 2 K^) n \ n / o x < It- 20 40 60 80 Inclination (deg) 1.0 r-2.5 -2. — 1.5 --1.0 li- — 0. 5 •— +0. 5 Fig. 6. Change in the Nodal Period Due to the Earth's Oblateness (for small eccentricities) IV-66 e = 0.00049 e = 0.00030 e = 0.00012 e = 0.0 Fig. 7. The Variation 135 180 225 270 True Anomaly (deg) 315' 360 True Ano m SeaX Radial ^^ " FUnCU ° nS ° f th ° IV-67 Circular Orbit Radius in 10 km (1 ft 50 60 70 b|o B .^ c o XI t< 3 £ a* a S £ 3 _ s ~ ■l-« nl s :--! i— Circular Orbit Radius (ft x 10 ) Fig. 8. Maximum Radial Perturbation Due to Attraction of the Sun and Moon IV-68 Ascending node of satellite orbit To disturbing body Fig. 9. Satellite Orbit Geometry IV-69 a> ~ M 73 c ° o m ■£ ^ Oft, >> nt T) o « -C ID ■"-■ 73 o 0.5 0.4 0. 3 0. 2 0.1 240 2 20 200 180 160 140 120 100 Attributed to: . . . Air drag ooo Solar radiation pressure „o°°o ' 00 oOooooooooo**.° . ..'»»' Ad°° o°°°o°Oo • — oooooooooo -L- 1_ 1 1 I I I L 37, 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 306" 310 Aug 12, 1960 Sep Oct Nov = T Dec Jan 1961 Fig. 10. Effects of Solar Activity on Echo I p a_ <u 0) OD U D ft, T3 C a) 0) <D O a < 01 X 8U0- 700- 600- iApogee 500- 1 Perigee |||||||l|||||l |Wffl 400- 300- 200- Aug 12 1960 Sep 21 Oct Dec Jan Feb 31 10 19 28 1961 Apr 9 May 19 Jun 28 1400 1300 1200 1100 1000 900 800 700 600 500 400 Aug 7 a <u .— ^ b£ b u X ft, in T) oo -" a> ttl Hi •r-> to O a (1, -"I* 3 <~ c Fig. 11. Apogee and Perigee Heights on Echo I (40 -day interval) IV-70 (un\ gcj8 -\ = mi jncu i) X3 C o r al o XI c tj t 3 a! ^ >— * fl) s k o !h s- a) o U) >-, nl C n 3 C4— u TI a o c a! r> Wi c ai fa Q £ CD hi ai £ X) o <D n hr t~ o <ii a) Oh ^ F fa M F C •< c o _ .c <5 (/J CO fa (itu }neu) }q3f3H asSi-tad uitilutuii/\[ IV-71 000 -■6000 5000 6 ^ ^ ^ h r, y: 4000 M (M <u ic X CO 0) -* <u Ul 1- 0) s CU 3 3000 E s 'c cd C 2000 1000 100 200 300 400 500 600 700 800 900 1000 Time from Launch (days) Fig. 13. Minimum Perigee Height of Satellite as a Function of Days from Launch (8 to 14 hr, expanded scale) IV-72 J3 ■an % X a 01 60 380 340 300 260 220 180 140 100 20 30 40 50 60 70 Time from Launch (mean solar days) Fig. 14. Comparison of Approximate and Exact Solutions of Satellite Motions 1200 J5 M 'a X 0) 0) Fig. 15. 300 400 500 600 Time from Launch (days) Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Neglecting Oblateness IV-73 .a n X <u V u 01 £ 3 1600 150 200 250 300 350 400 450 500 550 600 650 700 750 Time from Launch (days) Fig. 16. Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Neglecting Moon, Sun 50 100 150 200 250 300 350 400 450 500 550 600 Time from Launch (days) 650 700 750 800 850 900 Fig. 17. Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Sun and Moon 90° Out of Phase IV-74 ii;itl600 1500 3 a J3 K 0) 01 Oj0 £ 3 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 Time from Launch (days) Fig. 18. Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Changing Orbit Size (A r = 100 mm ' <D g X M CD OJ coco 3 ^ .S c max 6000 naut mi) 3 CP M 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 Time from Launch (days) E hi) ^ CI) E X ^ CM in hn 00 i< ^H CU eu " E E -i fa 3 a c 2^ Fig. 19. Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Change in Inclination IV- 75 T5 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 Time from Launch (days) 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 a 11 •rH t-l H (£'' ■r-t a s S 2 §C Fig. 20. Minimum Perigee Height of Satellite as a Function of Days from Launch, Showing Effect of Change in Argument of Perigee 1200 1600 2000 2400 2800 Time from Launch (days) 3200 6 J3 m U " s s U ■as Fig. 21. Minimum Perigee Height of Satellite as a Function of Days from Launch for About a 10- Year Period IV-76 CHAPTER V SATELLITE LIFETIMES Prepared by: G. E. Townsend, Jr. Martin Company (Baltimore) Aerospace Mechanics Department March 1963 Page Symbols y_4 A. Introduction y_2 B. The Drag Force y-2 C. Two-Dimensional Atmospheric Perturbations . . . v-8 D. Three -Dimensional Atmospheric Perturbations . . V-21 E. The Effects of Density Variability v-25 F. References V-30 G. Bibliography V -30 Illustrations V-35 LIST OF ILLUSTRATIONS Figure Title Page 1. Drag Coefficient for a Sphere at 120 km Versus M V-37 2. Cone Drag Coefficient, Diffuse Reflection V-37 3. Drag Coefficient for a Rich Circular Cylinder with Axis Normal to the Stream at 120 km Versus M V-37 00 4. Comparison of Drag Coefficient of a Trans- verse Cylinder for Specular and Diffuse Reflection V-38 5. Cone Drag Coefficient, Comparison of Free Molecular and Continuum Flow Theory; a- = V-38 6a. ARDC 1959 Model Atmosphere V-39 6b. ARDC 195 9 Model Atmosphere V-40 7a. Logarithmic Slope of Air Density Curve V-41 7b. Logarithmic Slope of 1959 ARDC Atmosphere V-41 8. Values of True Anomaly as a Function of Eccentricity for Which pip (h ) = Constant (exponential fit to ARDC 1959 atmosphere) V-42 9. Nondimensional Drag Decay Parameters for Elliptic Satellite Orbits V-43 10. Decay Parameters P and P for Elliptic Orbits V-44 11a. Apogee Decay Rate Versus Perigee Altitude V-45 lib. Perigee Decay Rate Versus Perigee Altitude (Part I) V-46 lie. Perigee Decay Rate Versus Perigee Altitude (Part II) V-47 V-ii LIST OF ILLUSTRATIONS (continued) Figure Title Page 12a. Apogee Decay Rate Versus Perigee Altitude V-48 12b. Perigee Decay Rate Versus Perigee Altitude (Part I) V-49 12c. Perigee Decay Rate Versus Perigee Altitude (Part II) v " 50 13. Satellite Lifetimes in Elliptic Orbits V-51 V-52 14. Generalized Orbital Decay Curves for Air Drag 15. Comparison of Errors in Orbital Prediction for Correlated and Uncorrelated Atmospheric Density Fluctuation V-5 3 16a. The Ratio of the rms Error in Orbital Pre- diction Caused by Sinusoidal Drag Variations to the Amplitude of the Sinusoidal Variation V-54 16b. The Ratio of the rms Error in Orbital Pre- diction Caused by Sinusoidal Drag Variations to the Amplitude of the Sinusoidal Variation V-54 16c. The Ratio of the rms Error in Orbital Pre- diction Caused by Sinusoidal Drag Variations to the Amplitude of the Sinusoidal Variation V-5 4 16d. The Ratio of the rms Error in Orbital Pre- diction Caused by Sinusoidal Drag Variations to the Amplitude of the Sinusoidal Variation V-54 17. The Ratio of the rms Error in Orbital Pre- diction Caused by Random Drag Fluctuation from Period to Period V-55 18. The Ratio of the Error in Orbital Prediction Caused by Smoothed Observational Errors to the rms Error of a Single Observation V-55 V-iii CHAPTER V. SATELLITE LIFETIMES A a B C D C L c /c P v D E E T erf (x) Fj (z, e ), F 2 (z. € ) G 1 (z, e ). G 2 (z, E ) H h I n (z) K K N f , m, n M M P + , P" P R R r r S SYMBOLS Area Semimajor axis Ballistic coefficient C A/ 2m Drag coefficient Lift coefficient Ratio of specific heats Drag force Eccentric anomaly Total energy Error function of argument x Nondimensional decay parameters Nondimensional decay parameters Angular momentum per unit mass Altitude Modified Bessel function of n tn order Orbital inclination; smoothing interval Inverse of square of most probable velocity, negative log slope of atmos- pheric density Knudsen number Direction cosines Mach number Molecular speed ratio Mass Disturbing force normal to velocity in the plane of motion; number of revolutions since epoch; number of molecules hitting satellite surface. Drag parameters for low eccentricity Semilatus rectum Universal gas constant; radius of the earth; radial component of disturbing force Reynolds number Radius Circumferential component of dis- turbing force T T L t V W x, y, z Z a r(n) 7 \ p (J n a e to Subscripts c i o P r Tangential component of disturbing force; temperature Lifetime Time Velocity Component of the disturbing force normal to plane of motion Position coordinates in Cartesian coordinates Lifetime parameter Kae Angle of attack Emissivity of surface Gamma function Flight path angle Eccentricity (to differentiate from base of natural logs) True anomaly; 1/2 angle of a cone Mean free path Gravitational constant for the earth = GM Yaw angle Atmospheric density Stefan -Boltzmann constant; statistical variance; ratio p/p n Orbital period Right ascension of the ascending node Rotational rate of the earth's atmos- phere Argument of perigee Circular Initial incident Original Perigee Relative Wall; surface V-l A. INTRODUCTION For most of the low altitude orbits for satel- lite payloads it is either interesting or necessary to study the effects of the atmospheric perturba- tions on the orbital elements of the satellite and on the lifetime. (Some material of this sort is in Chapter IV; however, the scope of the previ- ous discussion of this subject is not adequate for the present task. ) Many analytic approximations to these effects are presented in the literature; however, in obtaining these solutions approxima- tions have been made which at times drastically restrict the validity of the results. For this reason, it is the purpose of this chapter to present not only the information but also higher order solutions to the nonlinear equations of motion for the effects of atmospheric drag. The combina- tion of these effects with those due to gravitational accelerations, etc. , will not be discussed beyond the statement that such a process requires the simultaneous utilization of special perturbations and general perturbation techniques as discussed in Chapter IV. (The present analysis, of course, falls into the latter category. ) As a matter of fact, special perturbations will be utilized even in this study in the integration of the analytically determined decay rates. It is believed that this approach is inherently more accurate than those utilizing either general or special perturbation techniques alone. It should be noted in support of this statement, that even though numerical integration of the equations of motion has become increasingly popular with the advent of faster digital computers, special perturbations have three definite limitations: (1) Loss of numerical accuracy, if long integration times are involved (hun- dreds or thousands of revolutions). (2) Long running times even with IBM 7090. or 7094. B. THE DRAG FORCE As a preface to the discussion of atmospheric perturbations, certain phenomena and techniques must be presented. These discussions will be divided into three general areas: (1) Gaseous flow regimes. (2) The force exerted by the atmosphere on the vehicle. (3) Tumbling satellites. Each of these areas will be divided in turn into discussions of the factors necessary in subse- quent discussions. In particular they are slanted C D A toward the evaluation of the quantity ->>— - which will be designated the ballistic coefficient. 1 . Gaseous Flow Regimes The work in the field of aerodynamics has been divided into investigations in four general regions or flight regimes: (1) Continuum flow. (2) Slip flow. (3) Transition flow. (4) Free molecule flow. These regimes are defined in terms of the Knudsen number: K X mean free path N " J characteristic length of body IT C P _M for small R N (Ref. 1) C v N (3) Lack of general trends, since only iso- lated particular cases are solved. As an additional step to enhance the value of the results, the analysis will be conducted, where possible, carrying the density as a parameter. Thus, the final result of the study will be of value for all atmospheres. This advantage is quite significant due to the fact that the atmospheric models are constantly changing and the fact that there are seasonal and other variations (discussed in Chapter II). In order to develop an appreciation of the ma- terial and methods of analysis, this chapter will be presented in three basic parts: (1) The drag force. (2) Two-dimensional atmospheric perturba- tions. (3) Three-dimensional atmospheric perturba- tions. M for large R •N where c /c = ratio of specific heats P v M = Mach number V JIgRT v R = Reynolds number Though there is overlap of the regions, and though no truly definitive numerical values of K^ for these regions exist, generally accepted values for the four flight regimes are: V-2 Continuum flow--K < 0.01. Slip flow--0. 01 < K < 0. 1. Transition flow--0. 1 < K < 10. Free molecule flow--10 < K . N These flow regimes are illustrated in the fol- lowing sketch (Ref. 1): l ' if ! Ill [ i 1 i / 1 1 ' ■ II 4rf i 1 / 1 // / n 1 / ; ir 1 / / / L± ^ 1 / 1 1 I i 1 i I 1 As / /i i n 1 . -t ! 1 /^ ,„ 2 i — ^ / / yf V Tr N / --a, ^ _^ /, -V S <i- ? -W Reynolds Number It is noted that in addition to the defining lines men- tioned above, a second set of lines denoting alti- tude is also included on this figure. It is also noted that for any satellite above the altitude of 100 stat mi (161 km), the flow is always free molecule and that free molecule flow could be considered to extend down to as low as 75 stat mi (121 km) without introducing significant errors in the analysis. Since this region (121 to 161 km) is the lowest possible altitude for even moderate durations in orbit, the entire lifetime analysis can be conducted, based on the assumption of free molecule flow. This assumption, however, makes it necessary in subsequent calculations to stop the decay analysis or integration at the afore- mentioned altitude of 120 km (-=400, 000 ft). At this altitude the mean free path is 20.49 ft (6.25 meters); thus the Knudsen number for all but ex- tremely large vehicles is such that the analyses will be valid. 2 . The Forc e Ex erted by the Atmosphere on the Vehicle ~ In order to determine the drag coefficients analytically it is necessary to study the mech- anism by which the force is exerted on the satel- lite. This step will be accomplished in the fol- lowing analyses utilizing the work reported in Ref. 2 as the basis for the discussions. Let x 1 , y' and z' be the velocity components of a molecule of gas relative to the mean velocity of the gas. In addition, assume that the distri- bution of these velocities is normal--i.e. , that the number of molecules with velocities in the region x to x + dx, etc. , is dN = N 3/2 + z' 2 )] dx' dy 1 dz' exp \-K (x' 2 +y' 2 where N K = the number of molecules per unit volume = the reciprocal of the square of the most probable velocity - 2RT R T = universal gas constant = absolute temperature These molecules impact on a surface whose velocity components in the same coordinate sys- tem are IV, mV, nV (I, m and n being the direc- tion cosines for V). Thus, the velocity relative to the surface is x = x' - IV y = y' - mV z = z' - nV and the distribution of the impacting molecules with veloc ities x + I V to x + I V + dx , etc . , is : dN = N (f) expf-K [(x+*V> 2 + (y + mV) 2 + (z + nV) 2 ]| dx dy dz It is noted at this point that while either positive or negative values of y and z are permissible, only negative values of x will yield impacts; thus the total number of particles of all velocities hitting the surface is 3/2 N - 00 _ 00 _ eo -K[(x +iV) 2 + (y +mV) 2 + (z +nV) 2 ]}xdz N r -I 2 v 2 K ^ttK N S.V + -^— [l+erf (IV fio] where erf (I V (K) = — I *v(k" ds At this point it is possible to relate the number of particles hitting the plate to the mass and hence to the momentum transferred. The force acting on the surface is the integral of the momenta V-3 imparted by the molecules for all possible veloc- ities. Assuming for the moment that complete energy transfer is made and that the direction cosines of the stream are t ' , m'and n 1 , this pressure on the surface is: Sphere (A = tt r ) Specular C_ = erf (M ) D °° 2 + ■ M -^1 2 M J - P o/ „ 00 oo + m' y + n' z) exp {- K [ (x + IV)' + (y + mV) 2 + (z + nV) 2 ] } d z + I (i I ' + mm' + L2K V ,,] •[l + erf (I VVK)] > This estimate is not correct, however, because of the molecules impacting the surface. Some are reflected specularly (i.e. , according to Snell's law), while the others are temporarily absorbed and reflected diffusely (i.e. , in random directions) at a later time. For specular reflec- tion, the effective pressure is thus. eff 2p while for diffuse reflection, the equation remains unaltered. Thus, the two types of reflection bracket the actual process and the true force can be written (2 ' p incident P reflected where f is the fraction of the total molecules which is diffusely reflected. (Experiment indicates the value lies in the range . 9 < f < 1 . . ) At this point attention is turned to the computation of the drag and lift coefficients, defined as follows: C D A = r d . 5Jp D dA P V P V C L A = T 77T ■o- P V k dA P V Since dA is a function of geometry and orientation, these coefficients can be defined for various shapes. The succeeding paragraphs present data for C D both for specular and diffuse reflection (see Ref. 2). Note is made that the surface temperature, which is calculable as a function of the same set of variables, has been included in the diffuse re- sults. The derivations are in themselves not unique or necessary for this discussion; thus, only the final forms will be presented. Additional material may be found in the reference and in the literature. -M TT 1 + 2 M (la) Diffuse C D = C D specular K 3 M . )' T i (lb) where T is the surface temperature obtained w by iterating the following equation: 8/k P u T 3 + 1 " w \ 3 p R $ M = speed ratio )---A w».': v J2RT T. = temperature of incident stream P = surface emissivity = Stefan-Boltzmann constant pN surface -M 2 + erf(Mj (M^^-) for a monatomic atmosphere of oxygen and nitrogen in the shadow. Since the properties of the atmosphere are integrally associated with this evaluation of these coefficients only specific data can be generated for C . An example of the application is pre- sented in Fig. 1. This figure, obtained from Ref. 2, presents C^ as a function of M and for r L) °° an altitude of 120 km. Though computations for this figure were made with atmospheric data available in 1949, the variations which are shown are representative and the limiting values, which are rapidly approached, valid for this reference altitude. Data for other altitudes must be gen- erated as needed. Flat plate at angle of attack a to the flow (A = ab) For this body configuration the drag coefficients vary according to the following equations: V-4 2 2 . • 2 -M sin a „ , „ 4 sin a «. Specular C = e M [^ 00 ' ' 2 s in a . . ■ 3 \ „ — + 4 sin o\ M ' I erf (M sin a) (2a) tvt 2 ■ 2 -M sin » Diffuse C D M + 2 sin a (1 + ^-^A erf (M sin cr) 1 2 M Z > I it sin TvT T. (2b) where T is obtained from w 1 ? S 1 4 V^ + | R T. - ^ R T 2 2 i 2 w Cone with axis parallel to flow (A = tt r ) Specular C T 2 sin 6 2 2 -M sin 9 M 3 1 +(' 1 ^— + 2 sin 2 9j [1 + erf (M^sinG)] (3a) Diffuse C T 1 M sin o nr J^ 2 M w i 2 2 -M sin 1 + 2 M n9 [T 2 M where T... is obtained from 3 w 1 + erf (M sin 9) (3b) " T 4 Jtr K (3 a T 3 p R w -)--i K M » 2 and where 9 is the half angle of the cone. These results can be extended to nonzero incidence angles by utilizing the flat-plate results mentioned earlier. Such calculations are presented graphically in Fig. 2 (Ref. 3). Right circular cylinder with axis perpendicu- Tar to flow (A = 2 r"T) Specular C D M I M = M 2n r |2n + 3 (-D 1 n ! T(n + 2) M L m 2n r (-D 1 2n + 1 n=0 n ! 1' (n + 2) n =0 M 2n (-1)' 2n + 3 T (n + 3) 2 r 2n + 1 T(n + 2) (4a) Diffuse C 1 D M n =0 >(^S M n~! V(n + 1) 3/2 4 M T. M + M M 2n r (-1) *fL) n=0 T(n + 2) (4b) where T is computed from w ^ tt (3 a [k ' p~tt 3 5 2 U + M m 2n r' 2 (-1)' n + l\ T(n + 1) r-^ IV! > (-1)" - /_, n n r ^ T(n + 2) Figure 3 presents data comparable to that discussed in conjunction with the sphere. Of particular interest is the fact that this coefficient approaches a limit which is not unlike that of the sphere. 2 Circular -arc ogive (A = tt r ) This figure is constructed by rotating an arc of a circle about its chord then cutting the body of revolution perpendicular to the axis at its mid- point. The angle of the nose (29) analogous to the half angle of the cone is utilized to describe the shape. Specular C D 1 ([4 1 - cos 9 S [J 2 2 -M Z 9 Z 1 M (1 - cos 9) 29' cos 9 + + erf (M 9) 4 M "3" + 2TT "3" 2 M M (5a) V-5 Diffuse C + erf (M 6) (1 -cos 6) -M 2 „2 2 M , V q 4 M 1 + Q -1^- (l + erf (M el) 12 M 4 6M » -M 2 n 2 + e 1 6 M 12 M (5b) where T is obtained from w 4 pN PjT N L i V 2 +|rT. 4rT 2 2 i 2 w To provide a feel for the validity of these re- sults, tests have been performed (Refs. 3 and 4) and data prepared for the transverse right circular cylinder. The results of these tests are shown in Figs. 4 and 5. These figures depict the varia- tion in the critical region for molecular speed ratios in the vicinity of 0. 7 to 2. 5. The agree- ment between these data and the theoretical values is observed to be very good. Also noted is the tendency for the results to agree better at higher values of the speed ratio with the specular reflec- tion theory than with the diffuse theory and vice- versa at the lower speeds. 3. Tumbling Satellites Side Now approximating the effective drag coefficient based on one of the surfaces (say A ) * A 2 Cr~, = C n cos a cos y + C„ -,r— cos ~ sin a D D : D 2 A l A 3 A 4 + C„ tt — cos a sin ~ + C„ -* — sin a sin U 3 A 1 U 4 A l where a and H are uniformly randomly selected variates always lying in the range to it/ 2 C ' is the affective drag coefficient for the body A is the reference area for the nth geo- metrical shape Since the distributions of a and ~ are known 2 'D (the joint density function is (-1 ), it is desired to determine the distribution of the function C T This is accomplished as follows: g (C D *. a) = f [a, y (C D *. a)\ 9 C D The preceding discussions have presented data for bodies fixed relative to the flow field. However, in most satellite applications this is not the case. The first class of such exceptions consists of those satellites which by design orient themselves relative to the earth or space in order to perform some mission. The time history of attitude for this vehicle is thus known, and a time history of the drag coefficient can be constructed. The second class of vehicles consists of those which tumble in both time and space, thus com- plicating their aerodynamic description. One path around this impasse is to describe the param- eters statistically and assume that they are inde- pendently distributed. This approach, while not rigorous for either class of exception, provides a convenient means of computation for the latter case and an approximate method for long time intervals in the former case. Consider the fol- lowing sketches. but H (C n , a) must be obtained from ?_ = a. cos H + a„ sin tr = a cos (k - w) where a, = C„ cos a + C„ ■* — sin a 1 D 1 u 2 1 A 3 A 4 a 2 = C D 3 Ay COSff + C D 4 Ay sina a„ cos w = a 1 a„ sin w = a„ tan (ag/aj) 2 j. 2 a l +a 2 thus also .-1 /' C D a c Top D + w - a 1 sin - + a_ cos H -1 V-6 or, 3 C D 1 / C D , cos I ] -f w a 3 + a„ cos ■1/' C D + w 2 2 2 a l + a 2 = ^5 cos Q + C fi cos a sin " + C„ sin a 2 2 ? a 2 " a l = ^8 cos " ° + Cq cos » sin « + C 1Q sin a At this point it is noted that the area A can be 2 9 1 selected so that a > a "; thus, since a and -- are always between and tt/ 2 the function defined is everywhere positive in every term. Thus, the absolute value signs can be dropped and thus a 3 " C D / a n + ~D la„/ a„ [ a~37 + a„ a,/ 2 „ * a 3 -C„ ,a r MS) 7~2 a 3 " C D ( a 2 2 " a l 2 ) 8 (C D :! < °> = (I) a 3 " C D The distribution of C 'is obtained at this point D by integrating g (C ' , a) with respect to a over the range to tt/2. First, however, it is nec- essary to replace a in the joint density function. g(c D ) > d. cos i=0 . do Z"V -l .1 » sin i=0 (6) This function may be approximated analytically upon studying the behavior or integrated numer- ically. Analytic integration, however, does not appear attractive. It is noted that for the special case of 2-D analysis this problem is circumvented, since ^integration is not required. For this case g (C ''") is obtained directly to be: « «v ■ 2 , 2 a l +a 2 [*1 2 +a 2 -C * 2 (a 2 -a 2 ' + a 2 C D \ a 2 a l where a 2 = C 2 a l C D1 « < C D*' *»■(!) 2 , 2 a l +a 2 2 lx 9 2 a l = * C D1 oos <* + C D2 A - " sin a ' a l 2+a 2 2 - C D* h"- ai 2 A, = C D3 m A 2' A 4* C D2 and ^n4 do not a PP ear m tnis form for the reason that only a 2-D analysis is made. Thus, if the vehicle is tumbling in a known plane this much simpler solution can be utilized. - (C, cos a + C 9 sin a)' 2 2 9 9 Cj cos a + 2 Cj C 2 cos a sin a + C 2 sin a 2, 3 4 r *2 = (C D3 A^~ C0S Q, + C D4 Ay sin a) ' = (C„ cos a + C. sin a-)^ The density function is known or at least de- finable for the 3-D case and known analytically for the 2-D case, the problem turns to one of evaluating the moments of the distribution. These moments may be obtained directly from the mo- ment generating function in the following manner: m(t) fc tu(x. X ) n f(x, x n> C 3 cos a+ 2 C 3 C 4 cos a sin a + C. sin 2 a TT dx i V-7 4m(t) dt where t=0 fi' = the mean 9 2 a = /j' - /n' = the variance Substitution for this problem into the previous formula yields: ir 77 2 * * r m(t) h. cos a cos ~. where h. + h„ cos a sin y. + h, sin a cos 2 ^ + h. sin » sin ~; Id »d^ 1. 2, 3, 4 But this problem, like the first, is not easily integrable. Thus, a numerical evaluation is sug- gested for each case of interest. In fact, even for the 2-D case, in which m(t) -) e 't '1 dC, where C 2 +C 2 C D1 D3 *! C 2 -C 2 ^Dl D3 C 2 + C 2 U D1 D3 A 3 *~1 A 3 an analytic form is not readily available. Since the mean is not available in analytic form, little can be said relative to the best value of C 'D A 1 in the general problem. Many investigators avoid this problem by using the approximation derived from consideration of a spherical satel- lite. C D* A = °D h (A surface> sphere surface of sphere projected area of sphere -1 sphere A surface Though this may seem to be a crude approximation, there are many cases in which it is reasonable. In fact, Ref. 5 reports an investigation in which a body randomly tumbling (about three principal axes) is analyzed and in which the author concludes that for convex surfaces the average drag on a surface element in random orientation is the same as that on a sphere of equal area. This work thus lends credibility to the previous assumption and provides a numerical value which can be utilized as an initial estimate in the numerical calculations outlined previously. C. TWO-DIMENSIONAL ATMOSPHERIC PERTURBATIONS (REF. 6) The motion of a point mass in a nonrotating atmosphere surrounding a central force is given by the following set of simultaneous differential equations -^ - B p r V (7) Ji(r 2 9) = - BpVrl where V ' ? .2 (rer + r Id = earth's gravitational constant 9 = ^- = angular velocity (rad/sec) C D A 2m ballistic coefficient (8) It is noted that this set of equations is nonlinear and that a solution can be obtained only by nu- merical integration. This fact is somewhat dis- concerting, since these equations neglect atmos- pheric rotation, which introduces considerations of a third dimension and complicates the analysis further by entering the equations explicitly in the drag term. This latter factor results in the re- placement of V as defined previously with V = velocity relative to the atmosphere r V + V . atm Thus, if analytic approximations are desired, it becomes necessary to divide the problem into two phases--a perturbed orbit phase and an aerody- namic entry phase. In the first phase, a region is considered where the orbit is determined by the inverse square gravity field and only small per- turbations are caused by the relatively small drag forces. In the entry phase, the aerodynamic forces (lift, drag, etc.) become the important factors influencing the trajectory of the satellite and grav- ity forces become less important. This last phase is by far the more complicated, and fortunately for a lifetime study it can be neglected, since rel- atively short periods of time are spent at the alti- tudes where drag forces become dominant. Thus, the present problem is the analysis of only the first phase. References 7 through 20 present a portion of the pertinent literature and will be discussed as the presentation progresses. 1. Near-Circular Orbits (approximate solution) To initiate these discussions, consider the decay of a circular orbit. The energy loss due to drag during one revolution, Ae , is given by the loss in total energy AE D E - !•' Tl T2 v2 , c 1 ~7" j" V c2 r (9) Using the equation for circular velocity and letting Ar = r 2 -r-j. ,uA r AE D 2r l r 2 (10) The energy loss per unit mass due to drag is also equal to the drag force per unit mass integrated over a full revolution AE D D ds (11) Assuming small altitude losses during each single revolution Equation (16) shows that the decay rate for this special case is a linear function of the ballistic coefficient. This fact will be utilized in much of the future work in order to restrict the number of variables in the analysis. Equation (16) is not directly integrable because of the odd fashion in which the true density varies. However, if the density is assumed to vary exponentially with altitude, approximate lifetimes for circular orbits can be obtained: \ «" "J f dr 2Bp Q e IsTW "^ (17) where r.. = the final radius = R + 120 km r + r f p Q = the density at the -5-^ (see Figs. 6a and 6b) K = the negative of the logarithmic density slope (see Figs. 7a and 7b). (Note: This data is for the 1959 ARDC Atmos- phere. Data for the U.S. Standard 1962 Atmos- phere is presented in Chapter II. Either can be utilized if the lifetimes are adjusted, as will be discussed on p V-20.) Thus * E D~8^-(h) 2. (-i r, + r. (12) r l + r 2 where g = an average radius for the revolution. Now using the approximation that the circular velocity is averaged approximately as V 2 = .__ c r . + r„ 2/u (13) Eqs (12) and (13) and the relation — = flpV 2 yield m Mr J AE^ = 2tt/uB p (14) A r 2 If — — << 1, then r r„ - r and Eq (10) with *- i J- £ cLV Eq (14) results in the decay rate of the orbital altitude per revolution A r rev 4TrBp r av av (15) A' This decay rate can be converted to —5- by sec J considering that the orbital period for this per- turbed circle is t = 2tt av Thus AT - 2B p J/nr av f av (16) f -K r , e dr jTBp c F let x = Kr 2xdx = 2 x K dr or dr = ^- dx Thus 1 |T~r dx r e r -Kr , e dr 2 C -x" \ e dx fl[ erf (f^f) " erf (po)" and -Kr 2 {HB Pq - erf ( ^7 f j erf if^o) (18) The disadvantage of utilizing this form for the com- plete lifetime is that the density does not vary exponentially, and thus the approximation becomes poorer as the difference in r and r. becomes large. This deficiency can be circumvented through the simple expedient of breaking the true radial incre- ment into several subdivisions and evaluating the times required to descend through each interval. These times can then be summed to yield the life- time. Computations utilizing this philosophy will yield accurate estimates provided that the intervals are no larger than 50 stat mi or 80 km. V-9 The case of even slightly elliptic orbits must be treated in a different fashion since the assump- tions made in generating circular orbit lifetimes are not valid for other orbits. Thus, it is neces- sary to consider the equations of variation of ele- ments derived in Chapter IV or to approximate the motion in some other fashion. If the latter approach is taken, one possible avenue of investigation is to linearize the equations of motion by expanding the variables in Taylor series and retaining only first- order terms. This approach is valid only for small variations in the parameters. One such in- vestigation is reported in Ref. 12. The author utilizes a small parameter /3 1 defined as H' =Bp Q r (19) All orbital parameters are expressed as power series of /3, considering only the first order terms AV Ar rev (23) Now, from the first two relationships in Eq (22), exactly the same relationship follows: Ay rev V c A r rev This implies that for a first order approximation in B p„ r„ the speed at any given altitude remains exactly equal to the circular speed during the drag decay of a circular orbit. And, from Eq (21) for n = 2it the corresponding angle 9 is obtained as = 2tt + 6tt B p„ r„ (24) r +P ' r l ^ V = V Q + 0" Vj H = H + P H l J (20) where 2 ' H = r is the angular momentum per unit mass (to differentiate from h = altitude). Substituting Eq (20) into the differential equations, Eq (7), the following relationships are obtained 1 + Bp Q r M 4cose 0+ |8^ r = r r V = V 1 +2Bp Q r (sine Q V H where e o H r l + Bp r (-2sine 0+ e ) 1 - B ' 3 o r o 9 o (21) V t c Expressions for these quantities on a per revo- lution basis are next obtained from the differences in Eq (21) evaluated at the limits 6 = and 2tt: Ar rev 47TB p Q r — = 2T7Bp n r„V rev c ^ = -2Bp„r n rev But, for circular orbits V - J — c I r (22) and dV c o— J— » giving the following condition: Equation (24) indicates that the line of apsides is advancing by the amount 6tt B P r (rad) (25) Since the equation for the change in the radius per revolution is the same as that for the circular orbit. The lifetime of this slightly elliptic orbit will be the same as that presented earlier. Ac- tually, as will be shown later, the lifetime is slightly longer, but a quantitative analysis is left until subsequent paragraphs. These subsequent discussions will concern the behavior of these and other more elliptic orbits. 2. Elliptic Orbits (approximate solution) The type of expansion outlined for near-cir- cular orbits can also be utilized for elliptic orbits as was shown in Ref. 12. This reference pre- sented power series expansions for decay rates in elliptic orbits utilizing the small parameter =B P (h P o )r po (26) where p(h ) » air density at perigee radius l P o Initial perigee radius. Next, a density ratio is defined ^0 "P/P (h p0 ) ' For these orbits Eq (7) becomes rfc) « -- £<r 2 e) /3a. rV r 0*. pO r6V (27) r pO , and Using a change of variables u neglecting higher order terms in 3, the power series expansions assume the fol- lowing form: u »u Q + |8u 1 V - v + p v 1 H - H n + H, (28) V-10 Now the ratio of the Initial speed at the perigee radius to the circular speed at r is defined as V pO (29) and the corresponding eccentricity is expressed as 2 c 2 -i m i (30) An exponential atmosphere is assumed in the form -K(r - r J a °*^p7 pO' (31) The differential equations given by Eq (27) are then solved for the two cases below: Case I : near- circular orbits Case II: eccentric orbits Case I -- near -circular orbits. The solutions derived by Kef. 12 are summarized below. First, the orbit parameters: H - r „ V n pO pO 1 " B e<\> r po + ! (Kr P o<> 2 - T^po* > [l - K : > 3 ] PO" + sin 6 Tkt Q 6 (l - K pO L" pO V pO" 2 7 Kr pO« ■)1 . Ko'f n + Bl n 30V_i-xjJ (32a) — -*+' H |l-2Bp(h)>f + '\\\l r„ n 1+e cos y )| ^ v p 1 + « cos t) j l p0 -Kr p0 « + |(Kr p0 ,) 2 -fV(Kr p0 O 3 T4T (Kr p0 € ' e cose l 1 -* 15 ^' + I< Kr pO° 2 + TS< Kr pO«> 3 J ]sin (Kr„«) pO 23" (1 - Kr Q «) sin 2£ (Kr Qt) * 4r™ sin 39 (32b) Second, the decay rates obtained from the above equations : M.- 2 ,Bp(h p0 )V p0 r p0 2 [« pO + l (Kr po £ ) 2 -TV( Kr P o e n (33a) Ar 2 rif-'le = 2nf r (e - 0)* - 4TrB ^ h p0 )r p0 - Kr P o' + ^ Kr P o e)2 -Tli< Kr po< (33b) Ar a "rev" * r (e - 3tt)" r (e » ir) 4lTB ^ h pO )r pO (-) 1 - Kr _ t pO + !< Kr P o<> 2 55 tv \ 3 Ttt (Kr pO () (33c) Note that for « « both Eqs (33b) and (33c) reduce to the circular decay rate given previously by Eq (22). The given series expansions are adequate only for small values of Kr . s, the upper limit being suggested asKr n e <0.5. Reference 12 gives the following table, indicating the upper limits of eccentricity for various altitudes from sea level satisfying this condition: (km) (stat mi) K (ft" 1 ) (m" 1 ) 161 322 483 100 200 300 9.3 x 10" 6 30.5 x 10" 6 5.1 x 10" 6 16.7 x 10" 6 3. 65 x 10~ 6 12. x 10~ 6 0.0025 0.0045 0.0061 (1 stat mi = 1. 609 km; 1 ft = 0. 3048 meter) Case II-- elliptic orbits. For values of Kr _ « >1, terms up to the seventh power were carried. The resulting series expansions are shown below. 1 - e p° ( Cl e H * r n V n pO pO + Y C n + l slnne J (34a) n=l 1 + i r _ 1 + c cos pO 1 - - Kr .€ Bp(h n )r n e P° "pO 7 pO 1 + « cos t) 2C 1 6 - C 2 6 coe e + C* sin I |-C 3 sin 26 - ^C 4 sin36-^.C 5 sin4e ■^ Cg sin 56 + ^ C ? sin 66 + ® sin 76 (34b) V-ll where c i- 1+ i< Kr po e)2+ ir< Kr po <)4 C 2" Kr pO' + i (Kr pO° 3+ TTO (Kr pO €)5 + W32 < Kr po £)7 + -"- C 3"F (Kr p0 4)2+ ^ (Kr pO° 4 + WT? (Kr pO £)6 + --- C 4-^ (Kr pO €)3+ T^ (Kr pO° 5 C 5 " TTO< Kr p0 6)4 + T53F (Kr P O t)6 + ••• c 6 ' mm (Kr po* )5 + W?w (Kr po° 7 C 7"T3CTIT (Kr pO e)6 + -" C 8 " 2,358, 720 ^pO** + '" C*- -2C 1 + C 2+ |C 3+ JC 4+ ^C 5 T2" L 6 "37 7 T7^8 •" The accuracy of the series solution 1b limited to a region near the perigee, due to expansion of a n aroung the perigee point. Therefore a limiting central angle, e llm » was designated, such that „ p . < 0. 01 for 6 < e., . The limiting angle is p(h ) — — lim given as Kr P For ,, p ■ < 0. 1 the constant 4. 60 is replaced by «v ~ 2. 30. Figure 8 presents 9 plotted versus the orbital eccentricity for two values of density ratios and two initial perigee altitudes. Since the air density has decreased to 1% of the perigee value at a central angle of ^ lim > the following assumptions can be made: (1) The drag effects are negligible for the arc BCD. (2) All the drag takes place in the region DAB. (3) A symmetry exists about the line AOC (i.e., Drag DA - Drag^). lim B \ l im lim + 2it Therefore, the change of orbital radius at a cen- tral angle ltm is expressed as 7§i ' r B' " r B " r(e iim + 2W) " r(e ilm> From Eq (34b) (35a) Ar , \_ ^ (h P )r p0 e 2 - Kr p0 6 I 1 + <= cos u c„ t; cos e + . . 2C 1 e lim - e lim But -M Aa From the chain rule *'-£)*■♦©" (35b) (36a) (36b) and from Eqs (36a) and (36b) it can be shown that the following orbital parameters can be obtained from Eq (35b): Aa* (1 V C0Be) Ar (1 - O (1 - cost) Ah - 2(1 + ^ cose) 2 A „ a (1 - e) (1 - cosG) (37a) (37b) Equations (37a) and (37b) are based on the assumption that Ah » Ah . Thus the apogee decay rates can a p be obtained by the expansion of a small parameter method by Eqs (35b) and (37b). For perigee decay rates no information is given by this solution. 3. Variation of Elements As was noted in the previous paragraphs, a second method of solution for the effects of drag is available in the form of the equations for varia- tion of elements. These equations will be utilized in the investigations of elliptic orbits which follow. V-12 Since the interest in this discussion is in the solution for the lifetime of a satellite in a nonro- tating atmosphere, the disturbing acceleration will be due to drag and will act along the velocity vector that is tangent to the path. Thus, since R = (1 + € cos 6) T | 1 + e + 2« cos (e sin 9) T J 1 + e + 2e cos (s sin 9) N 1 + e + 2« cos 9 (1 + c cos 9) N 2 " 1 + « + 2e cos 9 where S = circumferential disturbance Ft = radial disturbances T = the tangential acceleration N = the normal acceleration s « = the eccentricity to differentiate from the base of natural logarithms The equations of variations of constants can be written as da ar de ar ar V777T 2e cos 9 n ^1 AH (cos 6 + e) la -y 1 + e +2* cos 2 sin 6 ^1 + *' + 2( cos 9 £ = - [2(1 -e 2 )(l + e 2 + e cos 9) sin el [na « (1 2 1 + 6 cos 9) (1 + e + 2e cos 9) dfi ar o , di ar /2]" 1 (38) Consider a slowly decaying elliptical orbit as shown on the sketch. Take points 1 and 2 as shown in the sketch in such a manner that the angle from perigee is constant. Then Q^ = 9 g , r J > r 2 and p < p From the basic equations of elliptic orbits V a 1 + 2c cos 6 + (39) From Eq (38) 2 B p sin 9 <y£. f 1 + {2 + 2 2 C COS 6 ) The ratio u>/a>„ becomes (40) 1 P l 6 2 / a 2\ l/2 / 1 + t j + 2 *l cose A l/2 2 " "2 € 1 V a J \ 1 + .* + 2._ cos 6 J Then for the first order of eccentricity 1/2 f\ '2 P 2 £ 2_( a J,\ / 1 + { l cos6 l \ lUi/ V + '2 cos °l/ (41) 1 + « . cos 8 But, t-t — i -i- „ 1 1 + e 2 cos B 2 where 2tt T D y~ mean angular velocity drag deceleration. From Eq (38) it follows that for a nonrotating atmosphere, drag does not cause any variations in the inclination or the nodal position of the orbit. Aerodynamic drag will, however, cause a forward rotation of the perigee in the orbital plane, as was shown quantitatively in Eq (25). An appreciation of the reason for this advance can be obtained from the following qualitative analysis. 2 P-[ < 1, — < 1 and — < 1 p 2 1 Therefore < 1 and the perigee advances due to air drag as was stated. This advance does not affect the lifetime of the satellite to the order of approximation of this analysis; however, since the atmosphere is not considered to rotate, den- sity need not be considered to vary with posi- tion around the earth. Thus, the orientation of the orbit while it changes does not change the de- cay history (again, to this order of approximation). For this reason, attention can be focused on the change of the three elements in the plane of the V-13 orbit (a, e and a). Further, since a relates posi- tion in the orbit as a function of time and not a change in the size or shape of the orbit, the ele- ments of primary concern are a and t. Variations in both of these elements are discussed in the fol- lowing paragraphs. However, before these dis- cussions it is desirable to relate the change in altitude of apogee and perigee to the changes in the elements a and e. The altitude variations during one revolution are quite large for elliptic orbits with high eccen- tricity, and therefore it is necessary to pick certain reference points during one revolution, for which the altitude, air density and decay rate can be found more easily. Since this geometry of a two- dimensional ellipse is completely determined by the perigee and apogee altitudes, and since air drag occurs primarily in the vicinity of perigee, apogee and perigee radii will be utilized as the reference points. These radii are expressed in terms of the semimajor axis and eccentricity as (42) Assuming an orbit with a very high eccentricity, the significant part of air drag takes place near the perigee and the maximum variations of orbital parameters can be found approximately by setting cos 9 a 1.0. Equations (38) become da de dT • 2 (1 + «) T ->j n^TTT^ . 2^TJ T (46) and the ratio of a to c is found as da ar 7 t^- r d< dT (47) Substituting Eq (47) into Eq (45) yields da dh a ar-° / 2a 1 dc ar dh -ar- dc a oT - a dc ar ar ) (47a) Now, orbital altitude is given by h. = r. - R , & J i i e where R is the radius of the equivalent spheri- cal earth. Therefore the partial derivatives be- dh. dr. i l come, since -* — = -s- — 8x 9x 1 + c 8h a 8h a "5i~ And from the chain rule for derivatives 9h 8h ) (4 3) dh 8h a . a "dT" "BIT dh 8h P - P "dT" 8"a~ da ar 8h dc "ST- ar A 8h da p ar "si - ar dc (4 4) Substituting Eqs (43) into Eqs (44) yields Equations (47a) indicate that orbits with large ec- centricities tend to become more circular during the drag decay process. For highly elliptic orbits the perigee decay rate is zero for a first approx- imation and in all cases it is considerably smaller than the apogee decay rate, as proven by numeri- cal integrations (Ref. 10). Now continuing, using the expression for drag deceleration T -R - m BpV (48) Equations (38) become 2 da 2a n ,.3 -rr = BpV dt p ^ ^ =-2pV (cos G + € ) Substituting for V and 6 from (49) V U~ 2 e cos e + e ' 1 (50a) dh , , •> a ,, , , da , de ■ar " (1 + <) ar + a ar dh P aT~ ,, , da dc (1 - €) ar " a ar (45) Thus, after the time derivatives of semimajor axis and eccentricity are determined from the Lagrange planetary equations, the time rates of the perigee and apogee altitudes can be found by substitution. The instantaneous orbital alti- tudes can be determined by integrations of Eq (45) either by numerical or analytical expres- sions. dt ae (50b) 1 the equations for the variation of elements can be expressed as derivatives with respect to the cen- tral angle 9. At this point it should be noted that Eq (50b) applies rigorously only if angular mo- [vp = na I 1 ■ mentum is conserved, i.e., r 9 = i up = na" In Ref. 17 the correct expression is given in terms of the osculating elements as + u + COS 1 dt [mp (51) V-14 However, as seen from Eq (25) Acq AW 37rBp„ r_ (rad/rad). But since 1 » - , Eq (50b) is justified for the e present analysis. Thus, Eqs (49) become da - o 2 p„ (1 + 2t cose + Q ■fig ^ a a P rj — 3/2 (52a) (1 + c cos 6) ^-- 2aBp(l - « 2 ) (cos e + t) (1 + 2e cos + O 1/2 >- (1 + c cos 0) T (52b) Next, the functions of the central angle are expressed as functions of the eccentric anomaly by the following relationships : r = a(l -■ e cos E) ' 2 sin 9 * cos 8 V - t sin E 1-6 COS cos E - € E 1 - < cos E v !-.> dE T^ e cos E (53) d6 Substituting Eq (53) into Eq (52) and using the approx- imate symmetry relationship of drag decay functions 2* I fd6=2 I f de o o The decays per revolution are found by the follow- ing integrals: 3/2 Aa _ ,.2, rev -4a ! b Po r £. n±±^^\ dE U J P (1-c cos E) T/2 (54a) ^^aBp^l-E 2 )]^ illljW'* EdE p (l-ecosE) 1 '^ (54b) Note that Eqs (54) basically involve the application of the Krylov and Bogoliuboff averaging method (Refs. 13 and 14), by which approximate differential equa- tions are obtained for the variation of orbital elements by averaging the original equations over one full revolution (i. e. , E = to E » 2n). This removes all trigonometric terms from Eqs (54) and is actually equivalent to a conservation of energy approach (Ref. 14, p. 238). The fraction in Eqs (54) can be expressed in a simplified form by employing power series ex- pansions as: Aa A 2 a C — » -4a Bp„ \ rev ^0 J '0 L 2e cos E + (continued) j. 32 2_ , 3 3^74 4 + tj- € cos E+ e cos E+j( cos E dE u J 1 2 1 3 (55a) cos E + e cos E 3 4 + 7 € cos E+ T € cos E + - € cos E 5" dE (55b) In general, the density function -2- is empiri- p cally found (see atmospheric models) and cannot be expressed in a simple exact analytical form. Thus, the analytic integration of Eqs (55) is not possi- ble. Numerical integrations of Eqs (54) or (55) can be performed on a high speed digital com- puter, however. If this step is to be taken, the density is related to eccentric anomaly in two steps: (1) (2) Altitude: - R e Density: tables. Defining S = 1 + 2 e cos E + £ e 2 h=r-R g = a(l-€ cos E) p(h) from atmospheric density (56) 2 ~ cos E and dropping terms higher than the second power of eccentricity (Ref. 12) has numerically com- puted the function of the integrand in Eq (55a) for Explorer IV, considering both Smithsonian 1957-2 and ARDC 1959 model atmospheres. The most important conclusion from this study and related studies performed elsewhere is that even for orbits of relatively small eccentricities (Explorer IV had e = 0. 14). The most significant portion of the drag perturbation takes place in the vicinity of perigee in a region where |E| < 40° . Utilizing this conclusion (not the limit on IE I) and approximating the density in this region by an exponential, Eqs (55) can be put in an integra- ble form. Let -K(h-h p ) J (57a) where K is the negative logarithmic slope given in Figs. 7a and 7b. Equation (57a) implies a straight line variation of p versus h on a semilog paper, which does not exist for any altitude range. Nevertheless, for a relatively small region, say 50,000 ft (15 km) around the perigee point, this approximation is valid to a very high order if an instantaneous value of K is selected. Using relationships r = a(l - e cos E) and r = a(l - e), Eq (57b) can be written as -Kat Kae cos E e (57b) Now substituting Eq (57b) into (55a, b) yields V-15 Aa . 2 D -Ka< = - 4a Bp n e rev K ( + . . . ) dE ^S- « -4aBp n (l-t 2 )e rev r e Ka e cosE (1 + 2( cQs E (58a) -Ka* f e Ka.cosE (cog E 3 2 3 74 3 5 jll 6 , F' ,3 2J 3,7 4, TF' + <r cos E + . . . ) dE (58b) 1 3 3 7 4 a 2 = T « " F « " F « The integrals above could be evaluated in the form of modified Bessel functions of imaginary argument, if the brackets contained a series of sine terms. Therefore, at this point a further crucial approximation is introduced. It is as- sumed that significant drag exists only near the perigee. This assumption breaks down for very small eccentricities (i. e. , as e -*0), but the va- lidity of it is good for moderately elliptic orbits. 2 n Assuming that sin E << 1 then cos E can be written as an infinite series of sines for odd n or as a finite polynomial in sines for n even. The first five sine expansions are aa follows: isin 4 E --^-sin 6 E cos E 1 ? - 1 - -j- sin E 5 ■ 8 „ - T5¥ sin E 2 ^ cos E = 1 - sin 2 E 3 ^ cos E -1-2- sin 2 E 3 . 4_ , 1 , 6„ Tj- sin E +-t£ sin E > (59) + TM sin E + 4 cos E 2 sin 2 E + sin 4 E 5„ , 5 . 2„ , 15 ■ 4 ,-, cos E = 1 - w- sin E + — tt- sin E TB sin E T2F sin E + Substituting Eq (59) into Eqs (58a, b) the fol- lowing expressions are obtained: Aa rev -4a 2 Bp *-s z cos E . . 2 _ e (o n - a- sin E 4 fi fi a„ sin E - a, sin E - a . sin E-. . . )dE — * -4aBp rev ^ 0'- S z cos E lo (60a) sin 2 E 2 sin 4 E - /3 3 sin 6 E - |3 4 sin 8 E-. . JdE (60b) where z h Kac and the constants a., P. are power series in terms 4 of eccentricity, up to « , as follows: 1 13 a 3 = F * " TF « 56 3 3 (61a) a , , 12131415 16 ^0 cl + £ "T e 'I' ~ F e "F 6 " TG < " « .i +£+ ic 2 + ^c 4 + ... 'l 'J IF 1 5 2 13 33 4 3 2 " F " TB ' " 2" ( " "BT* a ! 3 2 19 4, ?3 * TB ~ "F2 { + TZB e + • • " 5 13 2 A 27 4 ^ P4 " T78 ~ IW € + TTj^ e + --- .„... (61b) j It is noted that Eqs (60a, b) conform to the modified Bessel functions of imaginary argument, which can be written as ( l 2\* n« _ . . \Z I ( z cos E , 2p 1 (z) = -i '- \ e sin v P , , U ,U J „ r(p + i) r (i) EdE (62) where: p = (1, 2, 3 ---) r (n + 1) = h r (n) and <h"F The integrals in Eqs (60a, b) can now be expressed in terms of Bessel functions as zcos E ,_ t / \ e dE = « I„ (z) C Z CO J e s C z CO J e sE , 2_, ._. « l l lz) sin E dE = — — zcosE . 4^ .„ 3 « I 2 (z) e sin E dE = 9 — \ (63a) "E • ^h*. 3 " 5,tI 3 (z ) sin E dE = — — 75 f z J eZc ° sE sin 8 EdE = 3-5-7n I (z) V-16 NOTE: For modified Bessel functions I Q (0) = 1 and I 2 (0) = I 3 (0) = . . . =1 (0) = 0, so that for z = 0, Eqs (63a) are seemingly indeterminate for p > 2. The limiting values, however, can actually be found to be finite: lim z_^0 I p (z) 1 2 P (p!) (63b) Now in terms of modified Bessel functions the integrals of the orbital decay rates can be ex- pressed as: I n ffI i (z) 3*I 2 (z) 3- 5 it I 3 (z) 3-5-7«I 4 (z) "4 T ■••(64) (and a similar equation involving j3. ). Thus, both Aa and Ae can be expressed as scries of the same form but differing coefficients. How- ever, the computation of these changes is unnec- essarily complex due to the fact that higher order modified Bessel functions can be reduced to a linear combination of orders zero and one (I n (z) and IjCi)) by the use of the reduction formula 2p W z)=I P-i (z) --ir r p (z) (6 5) 840a 4 (z +6) I x (z) (6 8a) F 2 (z, O = e" 3/3, 60/3. 105/3 4 (z + 24) z 6/3 ...]l (z) 2 15^ 3 (z i +8) (68b) 840^ 4 (z^ +6) Note is made that Ref. 16 tabulates e Z I„(z), e l^z). Note also that the following asymptotic series are given in Ref. 16, p. 271 for large z: e- Z I (z) (2ttz) 1/2 1 + l 2 -3 2 1! 8z 2 ! (8z) ,l 2 -3 2 .5 2 + 1 2 .3 2 .5 2 .7 2 Z~ + , , ,„_ v 4 + ' 3 ! (8z)" 4! (8z) (69a) J Uz) (2ttz) 12 ) !.- 1-3 9 1-3-5 l!8z 2! (8zT The reduction formulas up to the order four I 2 (z) =I Q (z) -f Ij(z) r 3 (z) I 4 (z) 1 + 2 I l l z / (z) -f~ V z > }(66) '^)V-' 'h?) Ij(z) Now using Eqs (66) the decay rates of elements can be written in the final form for elliptic orbits l 2 - 3 2 - 5- 7 l 2 -3 2 -5 2 -7-9 3 ! (8z) J 4! (8zf (69b) Note is made at this point that decay rates as predicted by these formulas have been checked against the numerically determined rates and ' agreement shown to be good for the cases of mod- erate eccentricity. In no case, however, should the method be employed for eccentricities less than approximately 0.03 since the assumptions made previously restrict the range of applicability of the method. The value 0. 03 was determined numerically. Aa rev Ae rev 4* a ^Bp F x (z, O 4naB P() F 2 (z, O (67a) (67b) where the following nondimensional functions are used- F : (z,«)= e" 60c 105c? 4 (z + 24) z 6 a, I (z) 2 15c 3 (z +8) z z + (continued) Now, noting that a = _ p , Eqs (67a, b) can be written in the following form: da ar de ar 2Bp pr r OWT (70) But, since (-2Bp \^ r p ) is simply the decay rate for a circular orbit at initial perigee altitude, 'dr \ 3F/ = 0' * ne equations can be rewritten as dS "f ^P] (1 «)" 1/2 F (Via) V-17 de 1 f dr p ar a \ "ar, (i - •) -1/2 £ = (71b) From Eqs (45) and (71 ) the final decay rates are obtained dh a ( dr p ar - \~w> dh /dr ir Tar (i-o" 1/2 Gl (1 - «) G, (72) 0. 03<€<0. 4 where 'dr P -ar -2Bp ^ ^r }j « (1 + f) F x + (1 OF 1 -F 2 (nondimens ional ) (nondimension&l) At this point it should be noted that the functions G. and G„, although they are relatively complicated, are nondimensional and need be computed only once. In the present study these nondimensional drag de- cay parameters for elliptic satellite orbits were 4 hand computed, carrying terms up to € . The re- sulting parametric curves are presented in Fig. 9. Thus, the upper limit on c, e < 0. 4. This figure shows G 2> the perigee parameter, to be independent of e to a high order of approxi- mation though there is a variation of G 2 with the parameter Z. This behavior is not the case with G. , the apogee parameter, the reason for this behavior being that apogee decays much more ra- pidly than perigee for an elliptic orbit. Special attention is also drawn to the curves denoting low eccentricities. These curves will be discussed in subsequent paragraphs. dr a ST dr P -ar K P \ -C , ., b . ?0 K|/-°)e-C(a->> Po (75) But since V r « ^r (1 + t), Eq (92) can be (3L / written as dr a -ar 1 + e '(-0 dr /dr \ r~ + 1 (76a) (76b) where (50 n --"poiFT i P + =e €=0 C (a + 5-) e- C (a-|) L n (nlf (1M (2!T n»0 C> ^2- £0 wr<n + i) 1 + 2(1!) V (77) 3(2.' ) 2 (c/2r kr t P J 4. The Case of Small Eccentricities Since the Bessel function expansions of the previous section are not valid for eccentricities below 0.03, an alternate approach will be applied in this region. This approach was developed by Perkins (Ref. 8) and again assumes an exponential -kAr atmospheric model p = p Q e .In this analysis a nondimensional parameter C and a drag constant K are defined to be T /V„\2 "I kr t . c -*pHvl]-rfr (l + €) K^g, C D A w "o r o 2Bp n r p. (73) (74) Using Laplace transformations, the decay rates are found as and The nondimensional parameters P and P of Eq (76) are plotted in Fig. 10. The trends of the curves are noted to be the same as those ob- tained by numerical integrations. Figure 10 is, of course, limited to small eccen- tricities, as can be seen from the following ex- ample: Assume: h . = 85 stat mi = 448, 800 ft = 136, 794 meters Pi Pi 2. 135, 170 x 10 7 ft = 6. 507998 x 10 6 meters 0.02 V-lf Solution From Fig. 7a: k p = 1.98 x 10" 5 /ft = 6.50 x 10" 5 /meter = 7. 15 x 1(T 12 slug/ft 3 - 3. 684 x 10 kg/meter° (from Chapter II) . dt / = = " 2B Po^ r P -7. 84 fps = -2. 39 mps From Eq (73): kr C = IT P = 8. 24 From Fig. 10: P + = 2.73, P" = 0.0088 From Eq (76a): r & = (_jJPJ j i + £ p+ 6 = = 2.16 fps = 0.658 mps dr /Or \ From Eq (76b): r = -P J 1 + 6 P" = 0.070 fps = 0.021 mps Consider the same example for a slightly larger 6. If £ = 0. 04, then C = 16. 1 and x = 64 Proper convergence of Eq (77) now requires an extremely large number of terms (at least 25) thus making the solution impractical. Thus, since Perkins' methods and the Bessel method are applicable in different regions and since the solutions have the same form, i. e. , ^dr f^) .„ f 1 + e P € < 0. 03 dr I dt if 1 + e G 1 (z) t > 0.03 and similarly for r p . Perkins' parameters P + and P , can thus be considered to be analytic extensions of the parameters G y and G.-,. This fact was noted to be responsible for the low eccentricity curves of Fig. 9. J 5. Apogee and Perigee Decay Rates and Satellite Lifetimes The previous Subsections C-3 and -4 have pre- sented in nondimensional form equations and graphi- cal data for r & and r However, before determin- ing an estimate of the lifetime of a satellite it is necessary to dimensionalize the various param- eters. This has been done in Figs. 11a, b, c and 12a, b, c, which present apogee and perigee decay rates both in English and metric units for altitudes in the range 75 to 400 stat mi (120 to 640 km) and eccentricities from to 0. 4. It is noted that there are bumps on these curves. These irregularities are the direct result of similar behavior for the density slope of the ARDC 1959 atmosphere. Correction of this data for atmospheric variation will be discussed in Subsection C-6. Changes resulting from changes in the model atmosphere (e.g. , 59 ARDC to 62 U.S. Standard) require recomputation of Fiss 11 12, 13 and 14. S ' ' These decay rates must be integrated to yield the lifetime. As was mentioned earlier, this portion of the analysis will be conducted numeri- cally. The reason for this step is simple --it is not desired to introduce further approximation, which could materially affect the accuracy of study. To be sure, approximations have been made to this point; however, the validity of each has been well founded. If a further assumption were made to obtain an integrable form, the accuracy would suffer materially and the attention to detail exhibited earlier would be for naught. Some have argued that since the atmosphere is not known and since the other approximations have been made, such core is unnecessary. While this is true to a degree, a philosophy such as this will never yield good estimates even as the various density variability factors become known, while the philosophy of this section will reflect such improvements. is The integration procedure for this computation At. < Ah a> .1 where (Ah ) is the j-th apogee altitude increment ffl.' is the apogee decay rate at this altitude thus reentry At. j = ° This integration is very simple and can be rapidly performed even for small values of (A h ) type of integration also admits several refinements involving the use of iteration and average decay rates rather than instantaneous rates. However, if the step size is sufficiently small this is not necessary. The correct value of (A h ) is deter- mined by the repetition of the same integration until the values of T^ for successive values agree to within a prescribed error. This step size need not be the same for all orbits, but for orbits of similar a and e , the step sizes generally are the same (a value of 500 ft or 150 meters was utilized). The results of this integration are presented in Figs. 13 and 14 in both English and metric units V-19 for a value of B = 1 ft' slug or 0.6365 x 10 -2 meters kg Decay histories for typical satellites were added in dotted lines in order to indicate the changes in eccentricity and perigee altitude as functions of time. Lifetimes for all other values of B are obtained via the approximation L, B T L 2 * 2 observed 2 P p~a (G L + G 2 This approach compensates for a variety of sins since the nature of the body in question, the mass, the nature of the tumble, and even variations in the density of the atmosphere are factors included in the correction. TABLE 1 Comparison of Satellite Lifetime Estimates Effective B<- L r The basis for this approximation is that the decay rates were all noted to be linear functions of B. Thus, since B is a constant, it does not affect the integration, and as a result lifetime is inversely proportional to B. This behavior is true in free molecular flow; however, as B is made signifi- cantly larger or as the altitude is decreased, the vehicle leaves the free molecule region, and the assumptions of this chapter deteriorate. Thus, the simpler conversion must not be used indiscrim- inately. If there is a question as to the regime of flight, specific data should be prepared. Other- wise the conversion is justifiable. Though much has been written on the variation of lifetime with eccentricity, it is noted that these figures show the extreme sensitivity of this param- eter even for small eccentricities. This sensitivity explains why satellites with the same total energy per unit mass (i.e., same a) do not necessarily have the same lifetime. 6. Comparison with Satellite Data In the final analysis, the value of a computational technique such as this must be assessed in terms of its ability to predict phenomena correctly. Thus, the actual lifetimes of several satellites will be checked in order to provide this information. First the value of B to be utilized must be com- puted for initial determinations of lifetime or for preliminary estimates. The value of B must be computed based on estimates made earlier in the discussion of free molecular flow. However, once the initial tracking data from the satellite is available, a more accurate method is available. This method is based on the formulas developed for the change in the element a. Name Sputnik 1 Sputnik 11 Sputnik III Explorer 111 Explorer IV Score Discoverer 1 Discoverer II Discoverer V Discoverer VI Discoverer VII Discoverer VU1 Discoverer XI Discoverer XIII Discoverer XIV Discoverer XV Discoverer XV11 (ftVslug) 0. 69 1.00 1. 13 3. 69 1. 55 2.98 -1 . 5 1. 50 1. -16 1. 13 1.53 1. 38 1.65 1.04 1. 30 1. 50 0.95 (in /kg) 0. 44 X 10' 0.64 0. 72 0. 38 1.91 0. 95 0. 95 0. 93 0.72 0. 97 0. 88 1.05 0. 66 0. 83 0. 95 0.61 Estimated Lifetimes (days) 145 84 46 9 12. 6 11.0 14 100 87 24 51 Actual Lifetimes (Rel. 15) ( days) 92 162 202 693 93 455 34 19 109 11 97 29 (.Computed from the satellite data of the initial decay rates of semimajor axis. (1 ft 2 /slug = 0.6365 x 10" 2 m y'ke) Since effective ballistic coefficient is considered the more accurate, it was used in the construction of the following table. Two things in Table 1 are important and should be noted. First, the values of B gff as computed from the orbital decay during the first few orbital revolutions are not in all cases in good agreement with the values predicted theoretically. Consider the following examples: B eff B t , „ theo Satellite (ft /slug) Agreement Remarks 0.69 0.603 Good Sputnik I Neglecting antennas r + r a p Explorer III 3.69 3.71 Good Random tumbling r + r a P h + h a p a = T~ *~ = ~~ " T Thus, if a is known, an effective ballistic coeffi- cient B ff can be found by utilizing the computed h and n for B = 1 (rather than the observed a p values). Thus 2 a 'observed eff (fl a + ^ F theoretical Explorer IV 1.55 3.21 Poor Random tumbling This being the case, it is necessary to update the knowledge of B as data becomes available in order to obtain reasonable lifetime estimates. The second point is that the agreement between the computed data and the true data is good. To pro- vide an appreciation of the level of improvement, several previous works in the field were reviewed (Refs. 7, 9, 10, 11, 12 and 15). Data for these references are not included here because of the V-20 fact that different atmospheric models and differ- ent data for the satellites have been assumed and different corrective procedures (i. e. , B „„) utilized in the correction of the results. As a general rule the estimates obtained here are superior to these works, though there were cases for which other curves were more accurate. Since this was expected, the relative value of the approach was determined by a root mean square estimate of the errors in the predicted lifetimes. (The results included here produced approximately 13% error, while those of the literature varied from approximately 15% to 3 5%. ) atm sini sin(B + u) A s A , w sini cos (8+ u) ! cos i ; ° r A A-, r U I cos i S - sin i cos (0 + u) WJ Secondly, the vehicle velocity V=rR+r9S thus This improvement in the agreement seems very significant. However, the magnitude of the final error is still large. The reason for this large error lies in the fact that the method does not provide for atmospheric rotation, for density variability for variations in B, or for the oblate nature of the atmosphere. This being the case, subsequent paragraphs will be devoted to refining the previous work. D. THREE-DIMENSIONAL ATMOSPHERIC PERTURBATIONS Due to the fact that the atmosphere rotates, the velocity of the vehicle relative to the atmo- sphere will not be the velocity of the vehicle rela- tive to space. Thus, the drag force will not lie in the plane of unperturbed motion and each of the six elements or constants of integration will be affected rather than just the three considered previously. Since the equations for variation in the elliptic constants have previously been de- veloped, it thus remains to describe the perturb- ing force and discuss the resulting motion. 1. The Perturbing Force The drag acceleration which acts on the vehicle n 2 A — = -B p V V m r r where V r =< V - V atm> V . =fi xr atm e This acceleration must now be resolved into com- ponents in order to permit evaluation of the re- sultant motion. The specific set of components to be utilized is the set R, S, W discussed in Chapter IV. /\ R is measured along the radius S is measured in the general direction of motion perpendicular to R /\ W completes the right handed set. First, the atmospheric velocity V = r R + (r i r r Q cos i) S e and + r f2 sin i cos (9 + u) W e |v I 2 = r 2 +(r9) 2 - 2r 2 G fi cosi+(rQ cosi) 2 r I g e tT"r n si L e sin i cos (6 + u>l] V 2 - 2HVi cos i + r 2 Q. Tcos i e e u 2 2 "I + sin i cos (9 + u)J V 2 - 2Hf2 cos i+ r 2 Q 2 fl sin i sin (8 + u)J where H= the angular momentum per unit mass This result was also obtained by Sterne (Ref. 18) and Kalil (Refs. 19 and 20). Now at this point the function |v | must be expressed in terms of the eccentric anomaly in order to facilitate inte- gration with respect to time. V M 1 + € cos E a 1 - t cos E a 2 (1 - 2c cos E + i 2 cos E) thus 2 _ n 1 + e cos E V a 1-6 cos E 2 "eP' € ■ !-« cosE "5 cosl l+ £ cosE + %_ U ; « c° s gf . gin 2 . sin 2 + u)) 2 (1 + e cos E) n n = (ii/a But, as was noted by Sterne, U : /n can be no larger than approximately 1/15 for earth satellites; thus V can be obtained in an approximate sense r by the binomial expansion of the quantity within the braces by neglecting terms of the order V-21 (n l /n) . This step appears justifiable in view of the fact that there is such a large uncertainty in the atmospheric density at any time and in the aerodynamic characteristics of the vehicle. Under this assumption, V can be expressed as 1 + £ cos E 1 - € cos E n Ji-t 1 - £ cos E 1 + e cosE This equation shows that to the order of corrective ^ 2 1 or 45fT i /i r terms smaller than approximately 7 f y-rl the effect of the earth's rotation is a simple func- tion of the inclination and of time. The form of this corrective term being sufficiently simple, the subsequent integration of the equations of motion appears attractive. Now, the drag acceleration is: d€ ^- = jj cos <j> [R sin 9 + S (cos 6 + cos E)] di _ r cos (6 + w) dt 2 . W n a sin i cos <j> df2 _ r sin (9 + u) dT -W n a sin i cos <b 2 2 du _ a cos 4> cos 8 R - r sin 6 (2 + € cos 9) , dT 2 • X n a sin d> cos 2 n a cos if tan i W d€'_ 2r R , „ . 2 4> d (u + «) dT T + ' i sin -2 at aJ (l-£CosE) 3 1 -C 1-6 cos E 1 + e cosE e sin E R {(' + [ il - € c 2 - « o cos i J — (1 - £ cos E) 2 j S + Q e sin i cos (9 + u) ( ^- (1 - e cos E) 2 W where C = U || 1 - £ cos i where , „ , . 2 idfi + 2 cos <)> sin -p -TT- 1/2 sin 4> = (1 - £ ) as is customary in some of the astronomical texts £' = mean longitude at the epoch R, S , W = the components of the disturbing acceleration At this point it is noted that since But cos (6 + u) = cos 9 cos w - sin sin u n (t - t Q ) = E - £ sin E, E = 1 ~ W 1 - £ cos E cos E sin E I 1 1 - £ cos E 1 - £ COS E Thus the final form of the drag acceleration is R.= _ Bp A< l + *cosE a (1 -ecosE) 3 [-c^ £ cosE £ cosE| I . „ g 1 ' £ sinE R .{{: \-(u e sin i I — (1 - £ cos E)J (( a " (1 - £ cos E) 2 ] S £ cos E)l ((cos E - e) cos u / (sin E » 1 - £ ) sin u) W 2. The Change in the Orbit At this point it is necessary to refer to equa- tions for the time variations of the orbital elements (Eqs (60), Chapter IV) or to the form utilized by Sterne and presented in Plummer (Ref. 21): gr- = - [R tan <)> sin 9 + S sec $ (1 + £ cos 9)] Also from Chapter III, cos E - £ cos 9 = r sin 9 .rr. £ sin E 1 - £ cos E Thus the expressions for the changes in the orbital elements obtained by substituting for R, S and W can be transformed into functions of the independent variable E and its time rate E. Integration for the secular change in each element would then be possible (utilizing the limits for E of to 2tt) if the density could also be expressed as a function of the variable E. As was noted in previous sections of this chapter, the density of the true atmosphere does not vary exponentially with altitude. However, as was also noted for small variations in the altitude the approximation is valid. Selecting once again the perigee altitude as the reference for the approximation (since the largest portion of the drag force occurs near perigee), the den- sity can be written as P = P, -K (h - hp) V-22 a 4 where p„ = density at perigee h = a (1 - e cos E) - R [l , . 2 . . 2 . ,i - f sin l sin (9 + a>)J (1 - - R [l - f sin 2 i sin 2 u] h = a P h - h = at (1 - cos E) + R e f sin 2 i[sin 2 (e+u>) . 2 sin «3 R„ v e = earth's equatorial radius Thus the approximate density is P = Pq expQ-Z (1 - cos E) + q (sin (9 + u) . 2 u)] where Z was previously defined to be Kae, and where q = K R f sin i n e At this point Sterne presents a Taylor expansion of p in the form Since the angle w is approximately constant during any single revolution, the q. can be treated as approximate constants when integrating over one revolution, without the introduction of appreciable error. It is noted that according to the remainder theorem for alternating series, a series whose terms are alternately positive and negative, and such that their absolute values form a monotone null sequence, is convergent (this is the case here for the series expansion of the atmospheric density). This being the case, the absolute value of the remainder after n terms of such a series does not exceed the absolute value of the (n + 1) st term. Hence, the relative error introduced in the series expansion of the atmospheric density by retaining only terms through q D is A p K krrji ex p w Thus, by retaining terms through q , the relative error in p is 3. 4% at altitudes of 100 naut mi (185 km) where q ■* 0. 5, and only 0. 16% at altitudes of 200 naut mi (370 km) where q -v. 0. 2. Upon substitution of this density model into the equations of variation of constants and perform- ing the integration, Sterne reported the following secular changes in the elements: P = P Q e e -Z Z cos E 7 ±$— (sin 2 ( 9 + U ) - sin 2 u) -e i = -Z Z cos E ; e q' ■ 2m _ sin E m = 2m ,. „,2m (1 - ( cos E) In the series, the terms which are odd functions of 9 are also odd functions of E and may be ig- nored since they will not contribute to the com- plete integral for the secular changes in the elements. Using the even part of the series through terms in q , which gives the series ac- curately to about 1 part in 1000 for the altitudes in which this study is concerned, Kalil obtained q = 1 2 n 9 q 1 = (1 - c ) (-q cos 2u> + \ gin^ 2u) ^3 (I-, 2 ) 2 2 q ~2~ 4 ~| + Trx sin 2(j (I-, 2 ) 3 c 4u - SL cos 2u sin 2u 3 J_ 2 cos 2u + %- cos 2u sin 2u 4 1 2"4" 2u + ^j- cos 2u sin 2u - ^ sin 4 .4 4 4 i YZ 2.4 (1 - c ) 5L cos 2u - % sin 2 4u + %? sin 4 2u 24 16 24 (Aa) sec = - 2B ^ a (1 +0 3/2 (1 - c) 9f n T72 1 - 1 + € 'ote l+ A + 128Z (A«> sec = -2B (1 - «<) rl . ( ^ 1 - C 1 - e 1 + e 4e C p oQk (Ai) 8Z B 1 (3 + 4. N + ^ - + t 4tC \ + 1 '^ ini(1 - (2)(1 - c fe) -Po/^k + cos 2 u |"l - gL. A 5 + 4, N + 4t 5 + 6 A + ... (AC!) sec = "I "e sin 2 " (1 " e2 > (1 - C f^|) apfT •^l-i z (15-4e 1^^+46 N) + (Au) sec = - cosi < An > S ec (Ae,) sec = (1 - cos i)(^) sec V-23 or (AM) sec where t l = 1 - 8c N 4« r+ 8q Ni^T? B„ (1 -C)' 4 1 33 2 iKa 3q r (6 - 5C) (1 - C ) 2 + ^^ (10+ 17C> t , , 8 6 2 (1 + 5€ 2 ) ,16 eN (5e 2 - 1) ^ 32 ,2 M 2 2 d (1 - £ V 1-6 T' + -| J*! (7 - lOd + 6C 2 ) N = -i|.q (1+lOe + 8€ N) + i^-q 2 (1 + 4€) 1 + C 1 -C+(+«C These results are believed valid for all of the cases for which Z > 2 to the order of q 2 and represent the solution well for such cases. However, if Z < 2 a more general solution is necessary. This solution suggested in Sterne's paper (carried out for the element a) is reported for the elements a and e by Kalil. The results are shown below. 5 (Ar) sec = -GTrrBad - C) 2 p Q e" Z £ A^ (Z) (Aa) = -4tt Ba 2 (1 -C) 2 p n e sec A I (Z) n n (Ae) ■kBa (1 - c 2 ) P Q e" Z B I (Z) n n n=0 irhere the constants evaluated for small eccentric- ities (i. e. , e << 1) are presented below: ^0 A n = 1 + , 2 (j 2 + \) A 1 = 2j,-i z ( j 2 4) + | l + e 2 (j 2 + 4j + I> A 2 =2q 1 |(j + l)-3^q 1 (r + 4 j + ^) + 3- 2 A, = 6| q 9 (j+ 2)+ 15-4 "3 "Z 4 2 15q ^"l 3 " 105 q. * 9 4S 2e <j+ 3) + ^- (j + 12j + ^) A 5 = 210 q 4 -^ (j + 4) B Q = e (2C+ 1) B l = ^- C ) 2 -^f + ^a< 3 - 2C) 3q, 97 ~2~Ka + € (5 - 4C) 15q, (1 -C) 2 --[^(7 - ■ 2 ^C+6C 2 ) + 6 2 (^- ~^L 30C+21C ) Ka M 105, _, + 73 ^(9-14C + 8C ) q 3 2 B c = -4(105^) -^ - 33C+ 21C 2 + — i (105) -I Z' (1 l + c "Ka (9 29 2,89 C+8CT) + e {=£-- 56C) K = negative log density slope The symbols C, Z, e and q. are the same in this set of equations as previously defined. The re- duction formulas discussed earlier can also be utilized, to relate all of the higher order Bessel functions to the fundamental functions I„ (Z) and I 1 (Z). This step simplifies the numerical evalua- tion of the time history of the decay; however, it only serves to make the functional form of the resultant equations more complex. For this reason the equations are left in their present form. This set of equations is believed valid for satellite orbits extending down to approximately 180 km with errors less than several percent. Thus, if the inclination of the orbit were to be specified, the equations could be integrated numerically to yield realistic lifetime and decay histories for the vehicle as was done in the discussion of the nonrotating atmosphere. The possibility of being able to construct a family of lifetime figures for various inclinations is also noted, though to date this has not been accom- plished. Indeed, this step does not appear at- tractive for general computations because the procedure would result in an error source when data is applied for values of B other than that utilized in the construction of the figures. Thus, the most attractive procedure involves the numeri- cal integration of the decay rates for each satellite of interest. This approach, though more cumber- some, will be more numerically exact and should result in errors approaching an order of magnitude less than those obtained with the nonrotating at- mospheric analysis. V-24 Though numerical data is not presented, several general observations will be made. First, the equations show that the effect of the atmospheric rotation is to decrease inclination for all orbits (inclination defined 0" < i < 180°). Secondly, the effect is to decrease the rate at which a and t vary for i < 90° and increase the rate i > 90° . Thirdly, rotation produces secular regression and precession of the osculating ellipse. Numerical computations reported by Sterne substantiate not only these general trends but also to a good degree, the numerical values of the perturbed elements. This being the case, the theory as evinced by the equations of this section is believed to represent the best theoretical esti- mate of the behavior of the vehicle. E. THE EFFECTS OF DENSITY VARIABILITY (Ref. 22) To this point the approximations made in the discussion of atmospheric effects have been re- fined to include oblateness and rotation. Still no mention has been made of the effects of density variability. If the time intervals are large and the altitudes sufficiently high that the forces are not extremely large, the density variability effects will tend to null out due to the fact that the model atmosphere approximates average conditions. These cases are treated in previous discussions to varying degrees of approximation. However, if the time intervals are short or the densities more significant, the effect of variability will be more pronounced, and the equation should be integrated with the estimated density rather then with the model density. One approach to the problem of analysis of this latter case was shown in Chapter IV-C-6-d, which discusses random drag fluctuations. The following paragraphs (Ref. 22) extend this approach and provide some numerical data which is of general interest. The parameter of these discussions is the time of nodal crossing, a readily observable and easily computed quantity; the other parameters, be they orbital elements or position and velocity, should be checked as time permits. One such investiga- tion is reported in Ref. 23. 1. Errors in the Time of Nodal Crossing due to Drag Fluctuations Alone The contribution of random drag fluctuations to the rms error in predicted time of nodal crossing depends on the correlation function of the random fluctuations, which is unknown. Upper and lower bounds, however, can be constructed. These bounds on the random error are given in Fig. 15. In the upper bound, the random drag fluctuations are assumed independent from one revolution to the next. In the lower bound, the random fluctuations are assumed perfectly cor- related over intervals of 25 revolutions, but un- corrected from interval to interval. The curves actually show the ratio of the standard deviation of the prediction to the standard deviation of the random fluctuation, a, which is calculated from observations smoothed over intervals of 25 revolutions. The estimation of a is thus necessary to trans- late the data of this figure to errors in the pre- dicted time. No completely satisfactory method is available to perform this function; however, observations of satellites with perigees in the range 220 to 650 km indicate that a (in minutes/ revolution) is given by the empirical equation = 2. 2 x 10 h (78) where h is the height of perigee in km, and t is the smoothed rate of change of period (unperturbed by sinusoidal and random fluctuations) in minutes per revolution. For orbiting satellites the smoothed rate of change of period, t, can be determined from observations. For satellites not yet launched, the values obtained from the previous discussions can be used as an estimate for the smoothed rate of change of period. A simple approximation for the prediction error caused by both of the assumed random drag fluctuations is dashed in between the two bounds in Fig. 15. It is 1/2 G rms < N >/ CT= 5 < N ' 3 > (79) where G (N) is the rms error in the predicted rms time of nodal crossing (in minutes), N revolutions after the orbit was perfectly known. Equation (79) is asymptotic to both bounds and all three curves derived in Chapter IV. The contribution of a different assumption (i. e. , of a sinusoidal drag variation) to the error in the time of nodal crossing is given by (80) H (N) = (2)" 1/2 A (k) 2 rms - (kNr/2 1 - cos (kN) kN - sin(kN) 2 1/2 where: A the rms sinusoidal prediction error (in minutes) for arbitrary initial phase of the sinusoidal drag I I -3 (81) 1.8 h D x 10 (empirically P ' ' determined for same conditions as cr, Eq (78)). perigee altitude(km) (1.61 t) 10~ 4 the period in minutes Thus the sinusoidal and random errors can be combined to give the rms error in timing of an orbital prediction when the initial elements are perfect: V-25 (N) = G 2 (N) + H 2 (N) rms rms 1/2 (82) Now, if the local speed of nadir point is V n , and changes only slightly during the N periods over which the prediction is made, then the correspond- ing positional error tangential to the projection of the orbit on the earth is X (N) v o A r n (N) (83) 2. Errors in Orbital Predictions When the Elements and Rate of Change of Period are Obtained by Smoothing Observations - In the preceding simplified formulas, a perfect knowledge of the orbit at the initial time, or epoch, has been assumed. In actual orbital predictions, the elements at the epoch and the rate of change of period are usually found by some smoothing pro- cedure, using data containing observational errors. (Discussions of the errors made by various satellite tracking devices appear in Chapter XI.) Thus, to be rigorous these error sources must also be in- cluded in the analysis. Suppose that the rate of change of period is cal- culated from M(< i) "measured" times of nodal crossing, which are uniformly distributed through- out an interval of i revolutions. Assume that there are three independent causes of fluctuations in the "measured" time of nodal crossing: (1) A 27-day sinusoidal variation in the rate of change of period (2) A random fluctuation in the rate of change of period, which is independent from revolution to revolution (3) A measurement error introduced by the tracking device. Of course, only (3) can be regarded as an error of measurement, but (1) and (2) will contribute an error to the smoothed values of the period and the rate of change of period. The errors will be given as a function of the number of revolutions N, after the epoch. The epoch is taken to be at the center of the smoothing intervals. (1) The contribution of the smoothed sinu- soidal drag variation to the rms error in an orbital prediction which runs for N revolutions from the epoch is S(N) A where cos kN F 2i TT f 2 + f) (*) 1/2 (84) 64 . /ki\ 737 sin It) i k cos (£) N .2 (i + 2)\ T2~ = sin kN - kN + 8N £ i ( i + 2) k] T i • .2,2] .[cos^-l+i-^ and A is given by Eq (81), i is the smoothing in- _4 terval in revolutions, and k = 1.61 x 10 t, where t is the period in minutes. As the smoothing interval, i, approaches zero, Eq (84) approaches Eq (80), which represents the sinusoidal error when there is no smoothing. The quantity S(N)/A is graphed in Figs. 16a through 16d. (2) The contribution of the smoothed random fluctuation to the rms error in orbital prediction is R(N) *iv + 2 & - *6(¥) 3 - (?) 2 64 /N (f) 2U J 1/2 for i >> 1 where a is given by Eq (78). (85) Equation (85) should be compared with its unsmoothed counterpart, Eq(79). The quantity R(N)/(5(j) is graphed in Fig. 17. The contribution of smoothed measurement errors to the rms error in the predicted time of the Nth nodal crossing is O(N) C7 Q (M) - 1/2 (if 2 ( (ir M (M +2)" 1 + (16/9) (M +2) 2 /M 2 + 256 N 4 + 16 (Ni) 2 I M (M + 2)" 1 •■0 1 + 2)" 2 J + 32 Ni (i) 2 /(3M) - 4N 2 (M + 2)" 1 j - (8/3) (M + 2)/M - 2M (M (86) where all the observations are assumed to have the same standard deviation, cr n , and M is the number of observations in a smoothing interval of i revolutions. The quantity 0(N)/a n is graphed in Fig. 18. The observational errors, a n , made by various tracking devices are given in Chapter XI. In order to have the error given by Eq (86) in minutes of time, it is necessary to use <r n , the error of a single observation in minutes of time. Angular errors, A 9 (in radians), can be approxi- mately converted to timing errors, <j Q (in minutes) V-26 by (' * sM AO (87) where h is the height of the satellite, and R is the radius of the earth, and V Q is the local speed of the nadir point in units of length per minute. Doppler errors are more difficult to convert to errors in timing. They are subject to refraction and azimuth uncertainties, and it is difficult to tell how many independent observations are made in one pass. In addition, refraction and oscillator instability can create biases as large as the random errors of observations, and these biases cannot be reduced by smoothing observations from a pass over a single station. The observational error in minutes for one independent doppler observation is approximately ,(t.-t.) . Ar f l (r. - r.) (88) where the range rate changes from an initial value of r. to a final value r f during the time (t - t.), in minutes, that a doppler signal is being measured by the station. The range-rate error in a doppler observation is A f. For a typical case, (t - t.) is 10 minutes, and (r. - f f ) is 20, 000 feet per second (or 6100 mps). /o^ There ls an im P° rta nt difference between Eq (87) on the one hand, and Eq (88) on the other Equation (87) is applicable to each individual ' observation, hence to the average of a group of observations. Equation (88) only represent average conditions, so they only apply to the average of a group of observations, such as would be used with Eq (86). The errors are given as a function of the number of revolutions after the epoch assumed to be at the center of the smoothing interval. Now assum- ing that the observational, sinusoidal, and random errors are independent, they can be combined to give (N) =|[0(N)] 2 + [S(N)] : -oV/2 [K(N)] : (89) where E rms (N) is the standard deviation of the predicted time of the Nth nodal crossing after the epoch, when the elements and rate of change of period are obtained by smoothing observations. E rms (N ^ re P re sents the error tangential to the orbit of the satellite projected on the celestial sphere. Errors at right angles to the orbit are usually an order of magnitude smaller. Errors in actual predictions issued by the Vanguard Computing Center, NASA Computing Center, Smithsonian Astrophysical Observatory, and Naval Weapons Laboratory are compared with the theoretical model in Tables 2 and 3. Table 2 contains the errors in one to two-week predictions made near the peak of the sunspot cycle. Table 3 shows the errors in predictions half-way between sunspot maximum and sunspot minimum. In the tables, N is the number of revolutions predicted, beginning at the center of the smoothing interval. The smoothed rate of change of period is f (minutes per revolution). The root-mean-square prediction error F i^n /• ■ ^ > • ' " r ms (N) (in minutes), includes the contributions of observational errors and drag fluctuations The theoretical prediction error caused by observational errors alone is designated by O(N). TABLE k Prediction Errors Near Peak of Sunspot Cycle Satellite Dates No. of Predictions -T (Min/ Rev) N (Rev) O(N) (Min) E rms (N > Actual (Min) Theoretical (Min) Explorer IV 1958 8 2. 15 x 10" 3 165 0.024 3.2 3.7 Sputnik III 1958 7 1. 32 x 10" 3 220 0. 01 3. 3 1.9 Vanguard I Fall, 1958 20 5.5 x 10" 5 154 0. 056 0.25 0.22 Vanguard I Summer, 1959 11 2. 1 x 10" 5 154 0. 056 0. 13 0.097 Vanguard I Winter, 1959 to 1960 7 6.5 x 10~ 6 154 0.056 0.062 0.061 Atlas -Score Dec. 1958 to Jan. 1959 1=:= 2. 2 x 10" 2 271 0.3 67.0 74.0 **™ g ^ ol ? set T ati °? has ™> statistical significance. This case is included merely to show how large the error can be when the rate of change of period is large. g V-27 TABLE 3 Prediction Errors Half-Way Between Sunspot Maximum and Minimum Satellite Tiros II Vanguard I Transit III-B Echo I Dates Dec. 1960 to May 1961 Oct. 1960 to May 1961 Feb. to Mar. 1961 Oct. to Dec. 1960 No. of Predictions 12 12 10 (Min/Rev) 3. 7 x 10 7.4 x 10" 1.05 x 10 6. 8 x 10 N (Rev) TABLE 4 Errors in Individual Orbital Predictions for Vanguard I Number of Pass 2309 2986 2836 2234 2459 2535 3173 1934 2911 2610 Errors (seconds of time) +37 -25 +21 -21 + 17 -16 + 14 -14 + 12 -12 Number of Pass 2159 1708 2685 2009 1633 2384 2760 2084 1858 1783 Errors (seconds of time) -12 -12 -11 - 9 - 7 + 6 - 3 + 2 + 2 + 1 rms 15 seconds = 0.25 minutes It is interesting to note that observational errors were the principal cause of errors in orbital predictions for only one of the cases shown, that of Vanguard I with its perigee in darkness (Winter 1959-1960). In all the cases, the pre- diction errors attributable to observational errors were smaller than the total error for Vanguard I in darkness. If the errors in predictions had been caused mainly by observational errors, then the prediction errors would have been independent of the smoothed rate of change of period. A de- tailed discussion of the theory and the method of calculation is given in Ref. 21. Theoretical calculations of the errors in orbital predictions by the methods described above are subject to uncertainties because of variations in methods of fitting, spin of nonspherical satel- lites, and sampling errors as well as uncertain- ties in the estimates of the smoothing intervals. The uncertainty in the theoretical rms error is approximately +100 to -50 percent. All of the examples in Tables 2 and 3 were within these 250 150 22 145 O(N) (Min) Actual Theoretical (Min) (Min) 0.08 0.06 0. 04 0.04 E (N) rms 0. 12 0. 12 0. 74 4.4 0.0E 0.06 0. 50 3.3 bounds. Deviations from the theoretical model have tended to be on the high side so far (1958 to 1961). During the two years near sunspot mini- mum, the percentage variations of the decimeter solar flux (which is correlated with atmospheric density) are only one-third as large as during the rest of the sunspot cycle, so the deviations from the theoretical model can be expected to be on the low side during 196 3 and 1964. E (N) in Tables 2 and 3 is, of course, rms a root-mean-square error. The error in an individual prediction can be larger or smaller than the root-mean-square value, and can be positive or negative. The distribution function appears to be normal. Table 4 shows the individual errors in twenty predictions made for Vanguard I when its perigee was in sunlight (Fall, 1958). 3 Err ors in Orbital Predictions When the Rate of Change Period is Calculated from a~ S tandard Atmosphere ~~ The usual way of making satellite orbital predictions is to compute the elements and rate of change of period at the epoch by smoothing all the observations made during a certain time in- terval (usually a few days). This orbit is then projected forward in time. All of the predictions listed in Tables 2 and 3, with the possible ex- ception of the predictions for Transit III-B, were made by this method. The theory appropriate to this method of making predictions has been de- scribed above. The theory for the case in which the rate of change of period is derived from a standard atmosphere will now be described. Such a method might be used when there are not enough observations to determine the rate of change of period. In this case, the error can be separated into three parts, described under the following headings: (1) The error in the period and the time of nodal crossing. (2) The error caused by computing the rate of change of period from a standard atmosphere. V-28 (3) The error caused by the sinusoidal and random drag fluctuations. (1) If the period and the time of nodal crossing at the epoch are obtained by a single orbital fit over N revolutions containing M independent ob- servations, then the errors in the period, X~t (in minutes), and time, At (in minutes), caused by observational errors, are At M ■1/2 (90) where the errors in predictions contributed by the time of nodal crossing, the period, and the rate of change of period are At, NAt, and 2 (N /2) t, respectively. If the coupling among the period and the time of nodal crossing (which should not cause much error) is ignored, then the root mean square error in a prediction made with a standard atmosphere, N revolutions after the epoch, is approximately and At 4a i M •1/2 (91) where a is the error of a single independent ob- servation (in minutes of time) and may be obtained from the observational errors in angular and dop- pler units by Eqs (87) and (88), respectively. In the case of precision doppler observations, an alternative method of calculating the period is feasible but is not recommended, because it pro- duces large errors in the period. This method is to compute independent values of the elements from each pass of doppler data recorded by a station, and average all the sets of elements de- rived during i revolutions. The errors in period and timing (caused by observational errors) pro- duced by this method are roughly At ff (M) •1/2 (90a) E * (N) rms (At) 2 + (NA?) 2 + G (N) rms ft-"* (93) H (N) rms 1/2 where the epoch is taken to be the center of the smoothing interval employed in calculating the period and time of nodal crossing. Equation (93) applies in cases in which a standard atmosphere is used for calculating the rate of change of period. The error E * (N) is tangential to the orbit of the satellite projected on the celestial sphere. The error at right angles to the orbit is usually smaller. and 4. Example At CT (w) 1/2 t„ - t. f l (91a) Problem: where (t f - t.) is the time interval during which a single station is recording doppler data during a pass. (2) The rate of change of period f can be ap - proximately calculated by using the theory of drag perturbations in Chapter IV and one of the stand ard atmospheres described in Chapter II. This method is not precise and a certain amount of error is thus inserted. However, the magnitude of this error can not be described analytically and must thus be accepted. (3) The errors caused by sinusoidal and ran- dom drag fluctuations are given by Eqs (80) and (79), respectively. The reason for using models which do not include smoothing is that f is ob- tained from a standard atmosphere. Now that the three factors have been discussed, the predicted time of nodal crossing can be written in the following form: t (N) = t + Nt + (i-y (92) Calculate the root-mean-square error in an orbital prediction for Explorer IV, 165 revolutions from the center of the smoothing interval. The period at the time of interest was 109 minutes, and the heights of perigee and apogee were 142 and 1190 naut mi or 263 and 2200 km, respectively. The smoothing interval is estimated to be i = 100 revolutions, the number of observations, M = 25, and the prediction interval, N = 165. The smoothed -3 rate of change of period, r = -2. 15 x 10 min/rev, and the observational error is es- timated to have been 0. 7 milliradian. The elements and rate of change of period were derived by smoothing observations. Solution: The errors given by Eqs (84) through (89) are appropriate. The average height of the satellite h, was 666 naut mi or 1232 km and the approximate speed of the nadir point was V n 2 2tt R /P = 198 naut mi per minute or 367 km/min, so Eq (87) gives for the average error of an observation, a„ = 2 x 10 minutes. From Fig. 18, 0(N)/a Q = 12, so the con- tribution of observational errors to the error V-29 .0. in an orbital prediction is 2.4 x 10 -2 minutes. The normalized random error, R(N)/(5a) is 3 1.6 x 10 , from Fig. 17. According to Eq (78), a is 3. 7 x 10 minutes per revolution. Therefore, the prediction error caused by random fluctuations is 2.95 minutes. The normalized sinusoidal error is S(N)/A = 7. 5 x 3 10 , interpolating between Figs. 16b and 16c. According to Eq (81), A is 3. 06 x lO -4 minutes per revolution. Therefore, the prediction error caused by the sinusoidal variation is 2. 3 minutes. Combining the three errors by Eq (89), the theoretical error of prediction is 3.7 minutes. For comparison, the root- mean-square error of eight predictions issued by the Vanguard Computing Center was 3. 2 minutes. F. REFERENCES 12. 13. 14. 15. 16. 17. 1. Eckert, E. and Drake, R., "Heat and Mass Transfer, " McGraw-Hill Publishing Company (New York ), 1959. 2. Ashley, H. , "Applications of the Theory of Free Molecule Flow to Aeronautics, " IAS Journal, Vol. 16, 1949, pp 95 to 105. 3. Stalder, J. R. and Zurick, V. J., "Theoretical Aerodynamic Characteristics of Bodies in a Free Molecule Flow Field, " NASA TN2423 1951. 4. Stalder, J. R., et al. , "A Comparison of Theory and Experiment for High Speed Free Molecule Flow, " NASA TN 2244, 1950. 5. 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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: -"- :__j\^ TSJT- ■;■:::: . _. ::' ... : i>:-: :: ' / : !..L 1 4 I :-;-!--]■ ..... :.l:rr. -. — ■> ::: 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 -| ] ! 111 >nn----ff TN-i- -^ Psi-l-J-l-- - - fe !.ij~! ill ti'li — 1 ■■■■ -- - SU -j Xl-. : :- ■ -3 ^.-. ^fc['-j.-r-= ■4-1-1- i- T'-'M. ■- -i-^l=iX-l- ! = ;; : ?t J:; ;■[. ra|r;-- - -= : ; ~. '- Si^\ - ! - -!-£\i- - - ^TETtX^ '■■\- "- z X E =;s;=X ^i = - -■ '-■ X ^- 7i:;^X= -!'iX-X-;-|; zi:.^;-j. .j : ; : | • i i • ■ ■ V " |- - ^ — S : i- f --; T Si^ ^ " \^ =-= i S ; I 1 ^ : ^L' : / \ 1 1 002 ^-i- .- I - 1 : ! \ _ . : ; - : ^ V i ; - t— Si" : - ■- ---=- -\t = ' - :'- \ I — :- :-- ^c : - - - : " =rVi^;= : ^ ^t: 7 :■ -- - \ --— iiiU= i =J= - S - ■ !E = :-L !r -\--\~~- ~ T - ; ;^S3=;= ^= r =Vi = = -; ; t"f^iBj="H-i-!=N ji- 1 iN. 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"! -■ i ; I j ! .-.. - ! - - - ■■ V ■ - -■{-'■ = ' -'■-!-- = ■- - - i ' - | M ! = ; i : i ! -i-t~ " ' " ■ -' !. Lj !_.::...:-;-!:.!--:..-; z - : - j :.:-;-<-" - -i-\ -i- ! !--|-|::- vi;|-!- I j"!" " X-X-- X. ■ ■ • : i " IT- T; _ " ; - :; - -i i i-i_-.i-|.-| |; ;;E=:^:=:_ - (-= =j = i- i . | ■ "1 \ !--!-;-L!^^N^kH--^Sj ;Xi ' J < . - i"-[--- — r-- - 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 ^^ X N _!..._._.. .-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 U4.-V.uLL,i t jy 5 ,i.J l 400 ■J^' ■■■) 500 1 I 600 ■.un-f 1 ,-.-! 1 — r»5 3I"*" i:p f; ~~~frT|?~g" 3"? i**^*^! /^^^ I . - Jf ■ ' :■ /^ ^ >^/ - f-^- - '_; ; ;;; r ^.ii==ii=i u^WmyA -Sf^ ^ -7^- i — 7^^~y^~ l yf S ==:~=E=^=jt=?=--y^ — ^irr '_;_Ut.. i rfJjFT 1 " j- -T--/'- y S 1 / ^ 1* s'= = ;=f_ : -|— . _^^..f L T -'.__ -_ z - = = "=-ii^p=S-S j .- j-jF =■-- ;^ i^= --: ; - r j Jj r . >j + . , JT L : ;^ r ■ ■! J^ #:i = '■■'- ! 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L? =EJiEf - - ' 1" "" " (""'"' ^ V / / '-'-f^-fr /t^ -1—1-'- J yy y^i -4=p~ " ;_;-;■; - -;~|- ; ; . >" ~< ■■ ^C2*, F? -F / Y *wy^-fy 4. ■ | --/-.-3T-4 /f - ~_ f =- ^17^^^= pj=. = ^4= dg = £t --i .1^ r. -=^f?~i -'- 1 ^K/' ^ ^*~^ — t-M j-iU ;;-.=->.■>[:; jeL^=j= = g Bi ^hsjjFfir =^^-j- 5EF ^p ^jlt - jrj* ~\-i^p- >^ (^ p4r ; "" >*T^ -^-H-l-- ^BiBl^ IsgfcpBi f y hj=J^ ^p*^ Hi - i^l r :. : F [>»_H: - U "i^* g«=0.7 [?g |9 y~ z f~~n~ *fi ^"^ -^^jg 1 ^" - 5 - ; -^%i=v%?i -- 2 = = EgjsF aPp =a 3z=FFa!E ~[jf^ s= gk* =f-fi ^=F-5?M= ^=ff-i-|- Fsfe r_ j£ r /\^ :: y/J J' ~ E -Jl &:^ Hi ^?E ,y / s E 0.4 — -* V / y.- r ?" ~ J2p 3=-^ y\-~~yt- EyF:u r \jyk jri "";■■> *F -" j ; | 0.3 "ia :: ^ t -"b "f \'\Jr: - ":!;!" r-^^-y^-^--^ fSM .//■K u-t-jy- < ■pr. -j/T-~~ ^-- — - . 1 ■ - ■ 3 j 0.2 T^^-L.//^) \jf i : <fl 4-1-1- 1 -^ -H- j S^^SiSBB saSi "— .=s sfj? -gS =|q ?ifS3^E|E-- it_i t ' -- T- : 1 o.ie 1 0.K S|r5^||ln5pl ^ fMpftg 1 i§B^p-8iS^ vyir«i?jirj 1 ^ ii Fl^'^EM Ml ttsa^eiB "T"1™"1 F— T~ 1 0.0S fy- jP\'< yfy .'j? ■'' W=W=m B^ffl B ■\\ i ■ "fTT = ^i=j=P5r a*^ 1L " 0.0 y^ /~J f / =JE L it ="r|i ^ / iH- j ..JjF = 0.0' iB^/^mm^f f~yf E£ EjEJEiEi- ~" l "|- t'" -iv 1" r " im^" jS ?TjH "[> " "" r //t/ / i?&- -j- .:.:■. -■•::: - o.o; ' frtfhtfWH/--- f- j\'ia - i ^-/ ' y y -| _j-J... T .. |-_ - : r S:: \ : ':'}"' ;: 1 X MjR /\/l > 'Jr >> ' "T" 2 ? o.o; f / 1" 1 "j - L |SS| I|iB||| 8 £, jJ^a afe. i^T.if^ ■j- ] irr 'j [ [ " p 1 0.0 i^S^S 1 0.0( Iz=3=ijl!; l^.€-[Et-.::r -: = :"H"' [ "■ : !^ ^Pfe^^""''! |J^ _. = ■- ^4= :4_.. 4^ = ? h] ■ i ! 1 4 ; "F :; "Pi")" - ~ i£e Exi£ r?Lj ^-^i^Ll ^ j- - ^. ^^1-^ -!---= = o ={^j^ t =pr / --■ = =7??:" :~ T H = ^1= - ' ]-- ^ - - i " I: '-:~r 1 - 1 I - ■ f= t-JEr" ^^fcj-^h&i: jfejJrJ Of: E^F! Flr^E F ^y< .. ^_ =|E;F_j.-rF - - -=!■■ - ; F = |=- F 7 =-|=j- - _:- f": - \--\-- :-j- / / s y r 7^-- , /7 F3§r-|?^r^^FF) = hl- z z zzvz^z - '-!.-! -;F:Z I:.F;F ":* = T ■—— — jr : ■" --rfz j-:z = i^ %f^ ^E^ ^§ iE.E=^j-:jE = =N Tt^:i^Hi ■j LIE-: :-- ■- ^ //" Xn/s^ =Vp l^! T : .-F!::i .ii-JFj:-:;: ; ■: zE h F- FF u f-f :■; : j : : ^U^. J . _ . -,_.-, — F^ 4-- ■- -f ^y ""*" ■' -FF •;;■__ -(■■ 1 - |-- — ; • ! ' "i , . ; " jlmau ■ET*fiBB- MMW,«»Mi isriMBSBSS sasis 5!^^=:* =§£ nEism iasstsiesi MSHmpE -^ = i = Efitt=3 E / r~/H~- -- Illllfci; B lllliBlHi £ = ARDC 1959 = iFFFl _C D A 2m = initial decay = 1.0 ft 2 eccentr historie /slug =0.63 icity s 7 x 10 2 /kg -=jr=fcF rftr±^=:-/j-|r/=rp-" = :-:■=■ -:p -." ^a Ife- i\z^- U-; l- TFnr t it; i — F --:.+M IEEE.-..t 1 • ■ . ■ - ^^t' r 4 tJ ^i^^ z — ■: fH- --■::- ^J__ J_ \ \f ! f\ \ i \ F l ii_ II l/l i f\ \i 1 1 lit F -i 1 F 1 j ■ iii 10" « 50 100 150 200 250 300 Initial Perigee Altitude (stat mi) 350 400 Fig. 13. Satellite Lifetimes in Elliptic Orbits V-51 Altitude (km) 1000 900 800 700 600 3 0) 0) o 500 400 300 200 100 i-1500 -1400 ;— 1300 100 200 300 400 Perigee Altitude (stat mi) Fig. 14. Generalized Orbital Decay Curves for Air Drag V-52 1U.UUU ■+- ~ ~4m —— -r— L ! _ .. - -4—1— ' : ;:.- i 1 (7 ' -^ — ' --..:.- :- " " ■ ^7-" ' :-- : w h.: - 'J ! : : ' 7 ,7 : : :. 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"i'i : -- i-,i 1 1 1 1 j . . . ..- .^- ■ 1 1 ' 1 1 : 1 . ■ H ! : ' - 1 ; '.': ii'; ' . . . 1 ■ ■ Z "I \\:X ■'•■'- L :• ^li r /| ;_• -1— j — — r- — - 3 i~i : /, . jSjH- |;|; : 7') :: ' ■ r ,.. ...... Z f ' i J : l M :~ Jf i ' : .. 1 . ■ ■ 7- —b— ■:■■-■ — r- 'i:j ' - ■-t- -+- O t: -: r : '/ff f : i : :: :i: ■'.-■■: t . : j - : cc .' !; ■ s, •:t- : i; 'f N 3 /3) l/; ..)::■■. 1: :• ■: : : :;i,. ;- - 5 ( - TO - 7':.'l"i.:: .1 ' .:.■ z ^ ij iiu. (ASYMPTdT t. .. . . o (- o io ft BOTH CUR\ /ES) ■ " '["", ::::};::: :: -(-•■ ..-■ Q . , ■ '■ , r+ — ;;L i~ ■ 1" ! ■ 1I7 - -i — ' / / : [ 1 . 1 ■. : . T ] / : : 7 77 , ... " ^ C : / , . » S : il.il :l:h-L t 7i' b Hi; r,. ... [: ■:;- ; . . " COR RELAI tU !■ ■ -: ; : - :-.-;:i: : .i I .. ;~: ■ \ R" | i 1 ■' ; : 1 ■ : : * J_ . -i ' : .:. :■ t-i -!; -1 ■ : - |:-7 ;i! : :- : f - si; .... 1 .: ■ : :"7 .::■ - :. Jji :•• ■■:!■: kfe- +H-- w :H.: : i : ;■ ^ '■ -- i •i-.K: . 1 «5 *r~- .7; i::! 1 !: \*$- 'HHil ?jl; .;;• :- : : ■ = :-=: . ; J.;:-.: : ;:;'. :il' : ' t : 'f : " t i;;i|p-Ez; iii! 'ri^:: i; i: : -:;:; :; - ; " f': f. ~ ■- ■ ' 1 / t ill' iii; -r -.; p. •;-}Ht ■■!:■.•■■ ■-J1 ':'::: f:f Br. :.:;:: ; ;.::i: : ■ ;,, ;;;-i f: .,::::::: 10 100 1000 N = NUMBER OF REVOLUTIONS SINCE ORBIT WAS PERFECTLY KNOWN Fig. 15. Comparison of Errors in Orbital Prediction for Correlated and Uncorrelated Atmospheric Density Fluctuation V-5 3 100 200 300 400 5O0 600 N; NUMBER OF REVOLUTIONS AFTER THE EPOCH i (1^115 MINUTES1I 100 ?00 300 400 500 600 N=NUMBER OF REVOLUTIONS AFTER THE EPOCH hig. 16a. The Ratio of the mis Error in Orbital Pre- diction Caused by Sinusoidal Drag Variations to the Amplitude of the Sinusoidal Variation Fig. 16c. The Ratio of the rms Error in Orbital Pre- diction Caused by Sinusoidal Drag Variations to the Amplitude of the Sinusoidal Variation io» i i SMOOTHING NTERVAL <C' n // |(T= 100 MINUTES)! / A L t _ - ._ - — 100 200 300 400 500 N-NLJMBER OF REVOLUTIONS AFTER THE EPOCH ,c? -- h-~ SMOOTHING INTERVAL _■■ , // 1 \ it A / ft / / ii i i / ft f 1 1 in 1 1 | (T=I40 MINUTES) 1 1 1 ll II — J; / A / i\ / ' / i i 100 200 300 400 500 N;NUMBER OF REVOLUTIONS AFTER THE EPOCH Fig. 16b. The Ratio of the rms Error in Orbital Pre- diction Caused by Sinusoidal Drag Variations to the Amplitude of the Sinusoidal Variation Fig. 16d. The Ratio of the rms Error in Orbital Pre- diction Caused by Sinusoidal Drag Variations to the Amplitude of the Sinusoidal Variation V-54 - — --■ . SMOOTHING NTERV4L i= 12 -- i= 48 ^^, i= 80 i= 160 "^»" i=240 s^^ 1 — in 7/ I — ■ . 1 - I r j 00 200 3O0 400 WO N = NUMBER OF REVOLUTIONS AFTER THE EPOCH Fig. 17. The Ratio of the rms Error in Orbital Prediction Caused by Random Drag Fluctuation from Period to Period 100 200 300 400 500 N= NUMBER OF REVOLUTIONS AFTER THE EPOCH Fig. 18. The Ratio of the Error in Orbital Prediction Caused by Smoothed Observational Errors to the rms Error of a Single Observation V-55