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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 contains 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 satellite 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

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

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

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'

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 elements 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 direction 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:

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.

d ^ 2 6RO dt - fill

d o (1 - ( . 2 ) JR 0 /^;2 „R. d t t . a 2 ena2 du

d i csc i ^ Ro aRc, _ - cos i ti tt ail da, t, r, 2 ^ ^Re (1s) d . /1772 8R^ cos i dt 11n2t de na g 2 sin

d 1 JRc dt na e2sini2 di

t 2 d^ 2 C 1 _ e R( d! fill da 1 neat de

6 l_ f Variation in Eccentricity; Long and Medium Period Terms g The first of the above equations yields anextremely useful result. Since we have elliminated any dependence of R 0 upon ^. we have

da = 0 (19) dt

which means that there are no long term or secular variations in the semi major axis. This means we need only to comps :.e the variations in eccentricity in order to obtain a time hLstory of periapsis. We therefore need to look at the second of equations (26) and note that again

aRo = 0 a

and therefore de _ ^ aR (20) dt nea2 aW

With reference to equation (17) we may write

aRo aRo aRo a/3 as t (21) a^; - as aW a^ aw

recalling the definitions of a, ^3 , P and Q we have

as aP -.,o aw - a^

t 7

and from equation (9) we have

dP ^^=Q.

Therefore

a s - Q r, 0 = (22)

Similarly we find a^ (23) a^,

Then equation (21) becomes

aR G 6Ro aRo /3 (24) ac. as ^^ a

The derivatives are easily found to be

(3 a) (1 + 4e2) 2r,-`as I - (25) aRG^ a 2 n 12 a , \^ ( 3 /3) (1 - e2) ^r J

Substituting (25) into (24) and then into 1.2^N v.e obtain, after some algebraic manipulations:

s 2 12 %'2 [)l CI (26) CI Cap1\ r' S

8 where

n' y =- n

The above equation represents the variation, in eccentricity due to the gravitational presence of the Sun. In the case of Venus, we have a near circular orbit for the sun and therefore we may simplify equation (26) further by setting

Equation (26) also indicates that in order to completely solve the equation we also need a time variation of Q, w and i which appear in a and 6. Before developing these equations we will make a further simplification of R. which will reduce the system to one of only 3 coupled equations.

Medium Period Terms of R..

Lorell and Anderson (Reference 2) have made a further approximation by averaging over the period of the third body about the central body. In our case this would mean eliminating dependence upon -Q, defined earlier as the longitude of the sun measured in Venus' orbital plane. Froli, equation (17) and the definitions of a and /3 we need only consider the two terms

a2 (27) 2 + 2 F32 - 1 and

a2 - k32 (28)

we consider first (27)

a2 +a 2 = cos 2 f2 cos 2 Q + sin e Q cos 2 i cos 2 Qo

+ s in' 0 s in 2 Q + cos 2 Q cos 2 s in 2 Q

+ 2 sin f2 cos 0. sin 2 0 cos i20 - 2 sin Q cos Q cos 2 i s ii ;2o cos i2a

9 1 -sin 2 Q & cos 2Q sin i +sin 2 i 1 i sin 2Q sin 2Q0- sin2Q

2 2 1 - si n r [1 - cos 20 cos 2 Q0 + sin r [sin 2i, sin 2Q

2 2 = 1- s i t i+ s in r cos 2 (Q0

Therefore

2 + 2 ^32 - 1 = 2 - s in 2 i + sin 2 i cos 2 (il (29) 2 a 4 4

In a similar manner we find that

a 2 - 3 2 = sin 2 i cos 2w + Cosa / cos 2 (Q (D - W C\^ r / (30)

+ sin 4 (2^cos 2(Qo +a_Q)

when we carry out the averaging process we find that all terms containing Po may be dropped. We may therefore redefine equation (17) as

RO = a2 rl , 2 1 3 s 2 + 15 e2 i 1 + 3 e (s n2 i cos 2] (31) 4 8 2 16

where we have let aI = i r

10 F_

q^ S 1

Variation in Eccentricity: Long Periodic Terms Only.

With the new definition of R o above we return now to the second of equations (18) and calculate again de /d t . Again we need only calculate DR,/ ac.,

2 n' 2 e 2 sin i sin 2cv (32) a ° 8 a

e2 1 ^Z r dt = - (1 - 2 2 n' 2 e 2 sin2 i sin 2w ena L 8 a or

2 de 15 n' e (1 _ e2)t/2 sin2 i sin 2w(33) dt g n

This equation is considerably simpler than the previous eruation (26) and it is noted that the dependence upon Q has been eliminated. However, the dependence upon i and cv is still present and we must now develop the equations for these terms.

Variation in Inclination and Argument of Periapsis

Using the new definition of R o above we return now to the third of equations (18)

aRo DRo di _ - — 1 - cos i • d nag ^ sin i an

We observe immediately that

aRo = 0 aT i a

a result that was not true before the elimination of medium peiodr terms. Also from equation (32)

Ir aR o - 15 a z n' z e z sin z i sin 2w aw 8 substituting this we obtain

di 15 n' z e sin i cos i sin 2w dt 8 n or

di - _ 15 n' z e z sin 2 i sin 2w d t 16 n ^z (34)

The equation for the variation in argument ofperiapsis requires aR o /ae and aR o /a i.

aR° = a n' z e [3 (4 - 8 sin z it + 8 (sin z i cos 2w) L //

R = a z n' z - 3 sin i cos i (1 + 322 ) + 8 e (sin i cos i cos 2w)

Substituting these into the fourth of equations (18)

dw nn,z (1 _ e z ) 1iz 4 8 s in z i + s (s in i co 2(,)) d t 8

- nrl^2 (1e - 2 ) 1,12 2 _ 4 cos z i (1 + + e z cos z i cos 2w (1 - e2) 32 g

w - -

= n2 n_ e2)1/2 (1 4 8 sin 2 i ( 3 - 5 cos 2w)

3 co s 2 i 9 e 2 cos 2 i 15 2 cos` i +— +- - e cos 2w 4 (1 _ _ e 2 ) 8 (1 -_ e2) e 2 ) 8 (1

n '2 (1 _ e 2)1/2 3 3 A sin 2 i ( 5 sin 2 w- 1) n ) 4 4 A

+ 3 cos i + 3 e 2Cos i ( 5 sin w - 1 ) 4 (1 _ e2) 4 (1 - e2)

n' 2 3 (1 - e2)1/2 1- 5 s in2 i sin 2 w+ sin 2 i 4 n

1 - 5 e 2 sin2 w - e 2 e 2 s in 2 i - sin 2 i - 5 e 2 sin2 i sin 2 CO

(1 - e2) (1 - e2)

_ 3 n'2 _ e2)1i2 e 2 - e 2 sin 2 i - sin (1 5 sin 2 w 2 i + 1 + sin 2 i 4 n 1 _ e2

1 - e 2 e 2 sin 2 i -sin2 i 1 - e 2 1 - e2 and finally

dw 3 n' 2 (1 - 2 - s in2 i) e2)1/2 1 + 5 sin w (e (35) 2 e2) • dt 2 n (1 Equations (33), (34) and (35) represent a system of three equations in e , i and w only and it is now possible to solve for a time history of the eccentricity without having to solve for t or Q. Actually one may use equation (26) instead of equation (33) by merely holding Q constant at Q = Q o . In the computer program, equation (26) was utilized instead of (33). The results will not differ significantly. It is to be noted that the above three equations are identical to those given by Lorrel and Anderson in reference 2.

13 Results Figures 1 through 5 show comparisons between the above outlined method and a precision n body Encke integration program. As can be seen from these graphs there are no significant differences between the results obtained. How- ever, it must be pointed out that the ri body program took approximately one hour to integrate the orbit for 500 days and the variation of parameters method took less than 15 seconds.

Figures 1 through 3 are of similar size orbits with only the inclinations being changed. The apparent secular variation in eccentricity appears in all three graphs, however for an inclination of 5 0 , this variation is very small. Figures 4 and 5 show trajectories that impact the planet after 500 and 217 days respec- tively. Impact is defined as that point where the Venusian atmosphere will drastically alter the orbit such that virtual impact will shortly occur. For this study an altitude of 200 km was used for impact. .4

Figure 6 through 9 .re for orbits with periapses of 6575 km and apoapses of 46025 km. Here the 1--,rameters that were varied are inclination and argument of pericenter. All four of these orbits impact in less than one terrestial year.

Figures 10 through 14 represent an orbit with periapsis radius of 7550 km and apoapsis radius of 46050 km. Again the parameters that were varied are inclination and argument of pericenter. The first three of these orbits have not yet impacted after 800 days but all indications are that they will. The last two will impact within a very short time after 800 days.

14 r

_-- -_--- r „ ^ ^- ,..- 1 _:; ..,..- :. tom, ^:;tisg,^`B•. ^, ^ -. '-^^^` - -,^^'^=` ^-, _- -=

a ^ S

0 i

W ^Y aa. > J N r

CD cr u O a In d Wi 1 U > a ^' ^, a y (D O O E W) J O o n rA ,r co W ON ^-E Z Q ^^--• • . O W) Q 2 °OO Q WY i N1^.M to ^^ U T N N 11 N N 11 N 11 11 11 o • I I - V; u ^ 7 0 N N a a n o

^ Q W CL W r r c 0 N7 Z D O W I^ O > 1

^ 10 N 0

N LL N

N E

w R ro iD I ;i t0 11 1 10 (wi) snion SlSdVI83d

15 r

!j^ }

a N

OD W O N N rW O W W d cr ^_ a Pit n J G+ ^`c O g 1j Y m Qf ^.^ ^ ^N O p^ O 2 g Oj• . .^^`► 1AlDy) i ^ W Nr7b i Q^7 tDR ^ ^ V u n n n x u u n x Z 0 0 ._ G^ 3 ._ C; 3 0 0 > W

I ^ n I ^ O I ^ d' to a r N C ]L tt W 61 W d

T n j C Z d C N O O W^ O

t D CN u y

rn u L V

3 0

_ n Q V pyoO^(( f7 ^ (^ ^ ^ {^ 10 f^ 1^ 3 (wm) Snim sisdWW3d

16 I

f, -

o ^ pp^ i

o` N q 7 Myt,^ mayy•^o

T 4^alt

a W 5 h ^ l ll N

4 r

m0 .0" z

;, i' s Y

a_ W

U I S J N f^ ^ j

N^ hb O ^^•^ ^^ I i a o• ._ C: a _C3

f rJ ^O

N 2 O Y ^W L

^y 3 CLC > ^ Z C Q O N O 2 ^ W ^ O 1 I cn O u d

_rn L v v

w S i (w4) snom s►sdVW3d

17 ,IJ

X3..1 N I 5^ > !:: a^J N J a nr ^ J Z Wu W>

W j N N i

W r ui^ _cl a _C: a !2' e c o N IL O - mZ ao H L Q u W a W d 5 W r >; c

I I ` ZW ^ oO VI '^ u

_rn LL N N

^ 1O ^ fo 1'^ 1D ^ IY fV (uq) snmu wdww3d

x're

3 d de

6590 s

6550

6510

6470 YE

6430 a0 rr r^

Q- 6390 a cr w Q- 6350

6310

6270

4 8 12 16 20 24 28 32 WEEKS ^.f ^f«T.of MOION ^ TPPJfROP ♦ ul^LYflf OI VIfION :RAMC. %I O of a!M fY fMl^ NO. If Y) Figure 5—Variation in Periapsis Radius.

f. S

G_ 3 Y 3

l.9

6590

a = 26300 6550 e=.75 i = 70° w = 42° ECLIPTIC PLANE 6510 a= !80* i = 70°830515 VENUS' ORBITAL w = 38°506616PLANE 6470 R= 104°74359 E Y rp = 6575 km ro = 46025 km 0 6430 Q cr V) d 6390 a o: w d 6350

6310

6270 IMPACT I VENUSIAN YEAR

6230 I I I I I I L I I -^f :1 4 8 12 16 20 24 28 32

WEEKS NASA G$« .eo5 ss^ON a -A111iOev ^N^L reie OivieiON N— eel OAT. faw er ew1 nor NO 1]w Figure 6 —Variation in Periapsis Radius. — —

i 's t s

'^:,+,+. •_ -,3,.: •' .v -', '+ •^. =. pT>• . as ^ ,. ^ ^ ,d . - 'r .: tom• ;:. - ^f!" :• __ -.^._r,^ ^ -: ° = r"?^._^,{.--..-- _. - - _. _ .. -^ i--_ ^ --- - - 1

6500

6 5701 a = 26300 6550 e = •75 6530 3 = 60° v+ = 45" ECLIPTIC PLANE 6510 Sc = 180° 6490 i = 80°850511 w= 41°.221422 VENUS ORBITAL PLANE 6470 aZ = 105!43776

E 6450 r.= 657 5 km :430 0 a 6410

Fn C390 a Cr 6370 w CL 6350

6330

6 310

6290 6270

6250 C-230 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

y.f. l TIM WEEKS I,RfiIOM • TR VECTORY MI^LYfIf DI V IfIOM •II^NCM NI DATE •J# KOT I.O 11Of Figure 7—Variation in Periapsis Radius. RM,+•I,

21 o' w z tea^ a J a .a 1 'o: J ^ vo a a m W} x o=^ W Z Z O a ^ smm a c) J D d z Z U_ w W > > H a ^^ v

U (D Oi w E Y ^ Lo N^ rte. f` IT N " O Lr) N f ° O^ N O u ^ o o O 0.° tD to O^ D O (D N r to to ^' tD d u u u n u u n i`a io a 3 G U) Y v w n- 3 C c 0 D o 0

N D _T LL

O O O_ 0 O O O O ^ M t'n r fn N N 5 to d d r1r) to ILD 0 tD tD (D (D (D (D (D (D ( w i) smovd sisdd183d

M A

22 -OW

-Y'l

N I

i^I i^ o T :r fit

D T

O Qf

OD C\j

C\j 0c O O 0 > C\j

LM cD LL

0 0 C) 0 C) 0 0 0 0 U') r • rf) LC) (4) a) a) K) C\j C\j U') rr) LO w (D (D (D (D (D ( w N) sniov8 SISdVI83d

23 W f

LO w In w 11 z N4— LLJ co :D z uj 0 c > ID 0 cli 0 O

Ul)

OCZ) zui o LL t 0

Lo 0 Ir z Le) D < w Z LLJ 1 0) L'i —2^ Z 15 C'i Lf) — 11) E D uJN q 0 ,1 1 E Z uit, Cj O0 OD

OF— 0 O

0 N

§ ^ 8 E ^ wIsco N (wm) sn14v8 siSdvi83d

I-

c a

O sa^iY

11 Z N N ^^^ a w i^Yi r r OD 2 m ZW a a N N V J ZU m a w w V > H a W M1 "I- N vr- o N M E Y W 'n ^[ ° m W N Q O

NF w tDOO

r W ^

v o r- o

N U < ti N Wa a r W aZ D NY d ^ N W z 0 3 c c > m o N yl - O N^ >O

r d a v T U-

c M a wr v Z ^^ a N7 O Z In W > N

O ° 8 8 8 8 8 8° 8 8 8 ID I'D Lo ( WTI) SnldVb SiSdVi83d

26 a^

hoar

O Q' N .N

CL O

Y W d

CC O

(WM) snim8 SISdV183d

. 1 g^^o us t

N aN 0

m` a.

c 0

_rn LL

I w a1 Sniovd sisdvw3d

28 APPENDIX A

TRANSFORMATION FROM ECLIPTIC TO VENUS' ORBITAL PLANE

Zeci nRRIT OF CATPI i irc

Ysci

x^ N i

Figure A-1—Venus Coordinate System

Defir.a:

i 4 = inclination of Venus' orbital plane to ecliptic

Q 9 = right ascension of ascending node of Venus' orbit upon the ecliptic

0 = right ascension of ascending node of satellite orbit upon ecliptic a Of

_ r = argument of periapsis of satellite orbit referenced to ecliptic

N f = ascending node of satellite referenced to ecliptic

039 = right ascension of ascending node of satellite orbit upon orbital plane of Venus

w q ::gument of periapsis of satellite orbit referenced to orbital plane 4 = of Venus

N = ascending ne-de of satellite referenced to orbital plane of Venus. 9

Let r be the Venus centered position of the Sun in mean ecliptic and equii.;Ox of Date Jan. 1, 1972 at OI"

cos cos v,

c, = s in cos ¢o (A1)

s i n'0

where at the above date

f2G = 1750 27S6; 0 - 760427.751766

Let

T = x ecl (i 9 ) z ecl (i2 Q ) (A.2)

denote a rotation first about the z axis of the ecliptic through the angle ri, fol- lowed by a rotation about the new x axis of the ecli ptic through the angle i o . ,f is then the transformation matrix from the ecl i i,iic plane to the orbital plane of Venus with the x axis in the direction of the ascending node of Venus' orbit upon the ecliptic and the Y axis 90 0 in the direction of planetary motion.

30 a

V 0 °

ul ^mm

N a 0 d CL

c 0 0 0 I 0 d

_rn LL

(wM) snion SiSd0 dad

^y i

Then r4 c , the position of the Sun in Venus' orbital plane at the above date, is

/Cos 51,, r4 a = T c0s = fj in X (A3) \\ 0 0) also

sin Q" Q' (1972) = tan - 1 0 cos ^C

I .S To obtaino t at an- time t we must have then

Q ct - Q" (1972) + 1.6 At (A4)

where L t is days from Jan. 1, Ohl, 1972. Equation (A4) is needed if the form of the disturbing function in equation (17) is used.

The orbital elements of the satellite referenced to Venus' orbital plane may now be found as follows:

The position r E of the satellite referenced to the ecliptic is

tE =P' cos B t Q' sin 6 where

COs W - sin sin Cos 15F SE Coss Q. w.

P' = s in COs !I)sc + Cos sin CJ Cos i" SE ,LSE SE

sin i sin wsE

(A5) F- COs Qsin w - sin QSE Cos WSE Cos i

SE W Q , - I - Sir QSE s in Q1 + COs D"n COs S COs 1SE '

Lsin i, cos CASE

and 6 is the true anomaly of the satellite. Then r', the position of the satellite referenced to Venus' orbital plane is

P = T r' = (T P') cos 6 + (T Q') s in 6

Letting T P' = P; T Q' = Q

r° = P cos 6 + Q sin 6 (A6)

where P and Q are defined as above in (A5) with the appropriate change in notation to signify that the reference plane is now the orbital plane of Venus. Using the notation

(Pi

Q _ ^Q1`Q2

^Q 3

and PxQ=R-R2 R3

we have

( P3) cis = Tan-1 (A7) Q3

= cos -1 (P1 Q, - P 2 Q1) := (R3 ) (A 8) is4 and

2 Q; 3Q ` " 1 S2 -Tan-1 =Tan (A9) s q p1 Q3 _ P3 Q - 1

32 Finally we write the full e%oansion for

cos ^ 4 sin Q. 0

T = -s in Q Y cos i 4 cos Q 9 cos i Q sin i4 (A10)

sin Q 9 sin i Y - cos Q 9 sin i 4 cos iQ

REFERENCES

1. Kaufman, B., "A Semi Analytic Method of Predicting the variation in Peri- apsis of a '.Mars Orbiter," GSFC Document X-551-69-41, February 1969.

2. Williams, R. R., Lorell, J., "The Theory of Long-Term Behavior of Arti- ficial Satellite Orbits Due to Third-Body Perturbations," JPL Technical Report No. 32-916, February 15, 1966.

33 I^s