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General Disclaimer One Or More of the Following Statements May Affect General Disclaimer One or more of the Following Statements may affect this Document This document has been reproduced from the best copy furnished by the organizational source. It is being released in the interest of making available as much information as possible. This document may contain data, which exceeds the sheet parameters. It was furnished in this condition by the organizational source and is the best copy available. This document may contain tone-on-tone or color graphs, charts and/or pictures, which have been reproduced in black and white. This document is paginated as submitted by the original source. Portions of this document are not fully legible due to the historical nature of some of the material. However, it is the best reproduction available from the original submission. Produced by the NASA Center for Aerospace Information (CASI) 1-551-69-395x, PREPRINT VARIATION OF PARAMETERS-, FOR LONG P'ERI:O-Dlt- TERMS --IN LIFETIMES- OVVENUS. Oplut,tilti Kso - B E R N A R b',.,.KA U F M A R- Art , SEPTEMBER X969 , 14 10DOARD SPACE FLIGHT- CENTER 9EN1009 *YUW NT70 -J 06 911 (ACCISIN tjUmarA) O 1T (PAGEW 30 ^IAC A.... ^ ICATEOORYI r- X-531-69-395 PREPRINT VARIATION OF PARAMETERS FOR LONG PERIODIC TERMS IN PREDICTING LIFETIMES OF VENUS ORBITERS Bernard Kaufman September, 1969 Goddard Space Flight Center Mission and Trajectory Analysis Division Mission and Systems Analysis Branch M_c' ,.D!NG PAGE BLANK NOT FILMED. 3 F VARIATI ON OF PARAMETERS FOR LONG PERIODIC TERMS IN PREDICTING LIFETIMES OF VENUS ORBITERS i Bernard Kaufman ABSTRACT A very rapid variation of parameters method has been programmed to de- termine lifetimes of satellites in orbit about Venus. The perturbing function con- tains only the gravitational potential of the sun. The solar potential is expanded in a trigonometri c series and then averaged over not only the period of the satel- lite but also over the period of Venus. The equation for variation of eccentricity uses only the singly averaged function so as to include medium period effects, however, for inclination and argument of pericenter the doubly averaged function is used. It appears from the results of several sample cases that not only is this method considerably faster than a numerical integration solution of the three body problem (less than 15 seconds as opposed to one hour of machine time for an orbit study of 500 days) but is also highly accurate. Although this method contains only third body terms it should be very useful in first approximation studies. iii t r^^,R;,;. ^ U - PAGE INK K ::'a i^,iE^►EW^''1G TWA CONTENTS Page ABSTRACT ....................................... iii INTRODUCTION ................................... 1 J" SOLAR POTENTIAL FUNCTION: Ro ..................... 1 VARIATIONOF PARAMETERS ......................... 6 ^ b '_''• Vr- .iATION IN ECCENTRICITY: LONG AND MEDIUM PERIOD Tr.I:IvIS .................. 7 MEDIUM PERIOD TERMS OF Ro ....................... 9 A VARIATION ?N ECCENTRICITY: LONG PERIODIC TERMSONLY ................................... 11 VARIATION IS INCLINATION AND ARGUMENT 01' PERIAPSIS .. 11 RESULTS........................................ 14 APPENDIX A— Transformation iiom Ecliptic to VENUS' Orbital Plane ....... ........................ 29 References ....................................... 33 e v 4 .^ u:5.. VARIATION OF PARAMETERS FOR LONG PERIODIC TERMS IN PREDICTING LIFETIMES OF VENUS ORBITERS INTRODUCTION One of the determining factors in considering a Venus orbiter is the length of time the satellite will remain above the dense atmosphere of the planet. The orbit must be chosen so as to allow a sufficient lifetime to conduct tits required scientific experiments, but also to meet stringent quarantine conditions that have been imposed for Venus. In the present study, the atmosphere and oblateness of Venus is not con- sidered in the equations of motion. It is assumed that the gravitational attrac- tion of the sun is the only disturbing force acting on the satellite and Thai if the closest approach of the orbit is below a certain limit, atmospheric drag will cause the satellite to impact within a short time period. The radius of closest approach (periapsis r is defined as P ) r P - a(l - e) (1) where a is the semi major axis and a is the eccentricity of the orbit. It will be shown later that under long period perturbations due to the gravitational presence of a third body, the initial value of the semi major axis remains constant. There- fore by choosing a limiting value for r , equation (1) may be solved for a so- called critical value, e, , , of the eccentricity. A value of a larger than ec r then will be an impact orbit. The above outlined problem can of course be easily solved using a precision integration trajectory program to solve either Cowell's or Encke's equations of motion. However, past experience has shown that for small orbits, the computer time is prohibitive. This report investigates a variation of parameters solution that is intended to yield accurate but fast results. The Solar Potential Function: R. With reference to FigureA the solar potential R'0 be defined as 1 i t. r s ti SATELLITE P SUN r -. S r' VENUS 1 - r'r - µ'o (2) P r'3 where P=r' -r and P2 = (r' _ r)2 = r 12 + r 2 — 2 r r' cos S or P2=r'2 1+ r2 - r2 cosSl r ' 2 r' J Then equation (2) becomes 2 -1/2 µ ,_ 2r _rcosS (3) + --osS- - c r' 2 r' r' i 2 In this equation we may aGsume that r/r' << 1 and expand the square to include only terms of order 2. We then obtain 3 Ro 1 2 cos SI + 1 4^ z cos z S I - i co y S \ 8 \ ^z / rG Cr 22 J // 2 - i° 1 C2 cos 2 + i^2 S - 2 or z ]1 r r' R' _° 1 + r 3 - 1 (4) 0 r' r' 2 2 r r ' 2 i In the usual form of the equations of motion the gradient of the above equation ( would yield the disturbing function. However, the gradient is taken with respect to the state of the satellite and the first term of equation (4) contains only ele- ments of the sun. Therefore we have 0^°=0 and we can redefine the solar potential as 2 ^ 2 r r R0 = µOr 3 -1 2r' 3 rr' or (5) ^t, r 2 R e = [3cos2S-1] 2r'3 Lettir.g µo = n' 2 a' 3 where no is the mean motion of the sun and a' 3 is the semi-major axis we can write equation (5) as 3 i 2 '2 - ' 1 RO = a 2 [3 cos t S - 1] . (6) \e / ^a, J 3 Now r = r cos S = r ° ofr (7) r r' and r° =PCos B +Q sin 9 (8) where H is the true anomaly and cos Q cos w -sin D sin w cos i P= s i n it cos w + cos ^i s in w cos i sin i sin w (9) - cos 11 sin w - sin 0 cos w cos i Q = - sin fI sin w + cos Q cos w cos i sin i cos w P is a unit vector in the satellite ' s orbital plane from the center of Venus to pericenter, Q is a unit vector in the orbit plane perpendicular to F in the direc- tion of satellite motion. The position of the sun in Venus' orbit plane may be written as cos i^0 r °' = s in Q, (10) 0 ) Appendix A describes a venus centered coordinate system where the fundamental plane is Venus' orbital plr-ine and also develops the transformation from the ecliptic plane to the new plane. 4 t{c —t .£a `^^" -..tom--'^-.. _^ -. 3^^_'`^? ^ - ^Y p ^ -•^- -_ s I Equation (7) may now be written as ^i cos S = (P r' O ) cos 0+ (Q • r'°)sin B denoting P r'° a (11) Q r,0 — R then cos S = a cos 0 + 8 s in 6 (12) where it should be pointed out that a and 8 are independent cf 6. Substituting equation (12) into equation (6) a2 n' 2 r 2 a' 3 R O = 2 ra3 (a2 l ( ^) [cos 2 B + 2 a /3 sin B cos a + 8 2 sing 8) - 1] ` l `r 1 which after some manipulations may be written as 3 I Ro = a2 2 / ^ (^^^ a ^(3a2+3,82 -11 +2(a2-Q2)cos20+3agsin2B (13) Since we are mainly interested in long period terms it would be advantageous to eliminate any dependence of the above disturbing function on the mean anomaly t. This is done by means of the averaging process where the equation is integrated over the period of the satellite, The details of this process may be found in reference 1 and only the results are presented here. Equation (13) contains three terms which are dependent on the anomaly and the averages of these terms are: 5 r2, / \ 2 2 21 J` l a I d 1 r 2 c (Iq) o 1 2++ r cos 2Bd t - 5` (15) 2- a 2 n 2* / 2 f r 1 sir, LH3f = p 21. f (ls) o \ / Substituting; the above equations into equation (13) we obtain 1 a 2 n' 3 ( 3 3 \ 3 c- 15 c^2 1Z = 2 r /I I2^ + 2 ^^ -:I ^1+- + (i^-^) 4 (17) (a, 1 a result which is considerably easier to use than is equation (13). Variation of Parameters The development of equations for the method of variation of parameters may be found in a-tv good textbook on celestial mechanics and therefore will only be listed here. For completeness all six equations will be listed although it will be shown that not all of them are needed.
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