Y .

AIAA ELECTRIC PROPULSION CONFERENCE BROADMOOR HOTEL, COLORADO SPRINGS, COLO. MARCH 11-13, 1963

I N C L I N A T I O N AND E C C E N T R I C I T Y CHANGES EFFECTED BY LOW-THRUST P R O P U L S I O N SYSTEMS

by Constance J. Golden Lockheed Missiles and Space Company Palo A l t o , C a l i f o r n i a

63039

~. * v

First publication rights reserved by AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS, 500 Fifth Ave.. New York 36. N. "

Abstracts may be published without permission if credit is given to the author and to AIAA.

. , B'CLINATiOK AND ECCE%TRICITY CHANGES

iFFECTED BY LOU'-THRUST PR0PlJI.SION SYSTEMS

BY

Constance J. Guldcn

iackhcod hiiSSilcs and Space Company*

Palo Alto, California

ABSTRACT

63039 Change in orbit inclination and ecccnlricity are often invol~eti i l l

missions performed in a s t rong central Sorcc field by low-thrust,

powol' limited propulsion systems. Thcse changes are analyncd

lor constant t h r m t acceleration using standard c~prcssiooa f rom

Cclestid mcchanlcs. Inclination change is discussed f i r s t with PC-

spcct to a pur0 binormal thrust program and a thrust program pro-

ducing mayinwm simUltmcOUS Chmgc of inclination and radius.

From gcnorai expressions for thc change in inclinatiuD per re~olu-

Lion prodirccd by cach lhrust program, the mission timc and inclil>-

ation cl>angc prcduced U u ~ i n g B lrmsfcr from one ~ i r ~ u l d r oPhit to

anolhcr arc dctcrmincd i n closed form. These expression~ lead to

a condition which shows thal t h r optimum stccring program is in-

depcndcnt of tho thrus t iicccieration applied. Thc optimirntion

C X ~ ~ C ~ E ~ O ~ S ricveloprd lor a circular orbi t transfer aim npplied to

Lhi~ specific oxamplc of thc tmnsScr to tho Z~L-lioix synchronous or-.

bit whilc enccuting B 3 0 ~ planc change.

In Lhc ~jecontl Scction, 110 programs for ch'mgin:: occentricily a c

diScUSSCd; on(, is ilcrived by mMlmiRing standard ei(prcssioos of

coleslial mc~hirn ics , whilc the thrust dircction o f thc othrr is a lmyn

ocri~endilicnlnr t o the llnc of apsides. By comparing lhc cxpmssions

dcrivcd for thr changc in ecaentricily pcr rrvolution produced b)

e n d program, thc fixed-diceciion p r o g r m is found i o he I ~ O P C ef-

frctivc t1i;uitht: f i r s t Cor ecccntricitios lmxcr tbvn . 2 6 . T i c s c m -

inxilor ilxix remains mar ly constant for the fixcrl-direction ~ ' rogram,

so thc tiinc required to retlucc any CCCcntriC orhit La a circ111ar onc

lmhy bC gircn i n ;,naiytic for".

INTRODICTIOX

Rdrrcnecs 3 through 10 tlcsignatc i l few of Ihc many reports rrhich

rcccntly have discussmi low-thrust t ransfers between circui .~~. or

oil ipt ical orbits. A eiose+rorm analytic so~ut ion for ciosc to &ti-

lnulll lor\vthrust transfers botwcen Colllanar ci~culir md clliptic

whim l ~ l s been tlOVClOPCd in rofCrCnCC 3, iiut thc samr prohicm br-

t w c n nrbilrbl'ily inclined orbits at this timc must be solved by m

ilerativc computor p r ~ r : m .

Ln ordcr to dctermine thc program atilizinS constant tliimst accclcr-

atinn which w i l l rrquira cioso to minimlim trnnsfcr timc Iletwccn ally v

GaqoJ-i two prcdctormined orbits, exprosslons for the change in inelinalmn

as a fmction of semi-major axis and the time required to make any

specified inclination chango usbig either a thrust program which is

optimum in changing Inclination and radius simultaneously, or a pro-

gram of pur(, binormal thrust are derived in Section A. Using these

erprcs.:inn;, lhr, optimum p m g a m with rcdprct to t imc can be closcly

approximiitcd anti the mission time dctcrmincd as a function of

I C C C I C T d i O l l .

The problem of reducing or increasing the eccentricity of any orbit

h a ais" been discussod in thcse ~eferences, and optimum result6

haw bccn obtained for ecccniricitics close to aero, (Rcf<:rcnces 5

and 10). In Section I3 two programs for changing eccentricity are

discussed ivhicli can be applied to any elliptical orbit. Thc consc-

w e n t eifecl of lheso programs on thc semi-major axis and the change

in eccontricity par revolution for any initial conditions are given for

each program. An ~npresslon is developed for thc timo required lo

CiTCUliirize any clliplic orbit while mainlaining a constant semi-

major _is. 'The resul ts from both ECCtionS may be applied to any

mission in order to approximato thhc optimum thrust tliroction.

in lire dovclopnient of both programs, the constant 3ccelci~ation, A ,

i s hroken d o m into three mutually perpendicular componcnis; (1) a

radinl component, Ar , whlch is parallel to the inatanlaneow direc-

tion n i the radius (positive dircction is outward): (2) a norma1 (hori-

zontnl) component. An , which is pernendicular to the radius (wsi t ive

dirrction is in the diwction of motion): (3) and il binormal compoaent,

Aw , which is normal to thc otiicr two componenb (positive direction

is dclermineri by the cross product A x h ). These components.

si lo iw wilh rclation to tho reference orbit in figure 1, may hc cx-

Pl'cssed i n terms of two angles: the angle betiwon thc normal to th?

rildiiis and thc comwnont of ihc thrust vector in tho planc of liio orbit,

P , mil tlic ?n::le between thr: thrust Vector and the plane of thc orbit,

Y .

1' n

A i A sin P cos y

An ~ A cos P cos y

A = A s h y 0

'The CWationS of motion for the variation of the elements of any

elliptic m'bit exprcssed in te rms of thc semi-major &de, a , thc

ccccntricits, e , the inclination, i , md lhc t r w anomaly, 0 , are

giircn by cquiltioils (1) through (3).

where o O + w ,

These are the basic equations referred to in the dcvolopmont of thc

t ime vilrimce of inclination a d eccentricity presented i n tho foiion

i.g S C C t i U I I S .

SECTION A INCLWATION

A l l thrust programs involving change of inclination should have, for

maximum eflectivencss, the PrOPOrtim that thc binormal thrust mag-

nitude is maximum a t the nodes and the direction of the b k x m d thruat

COmpOnent i s changed 9 0 ~ f r o m the nodes, so that the thrust direction

a t the ascending node is opposite in sign to the direction at tho do-

scending node.

requirements, two, involving Constant thrust aoceleration, are dls-

cussed in the following Section. A pure binormal thrust program,

resulting in maximum inclination change for any time period >rithout

perturbing the other orbit elements, is disoussed first. General ex-

PreSSionS are derived for the amount of inclination chango produced

per revolution and the t ime roquirod to complete any prcdetermined

Inclination change.

M the many thrust programs which meet the above

The second thrust program, a simplificatlon of the general program

developed in reference 2, Is one which produces maximum simultan-

eous change of inclination and radius. From a general expression

for the change In Indination per revolution produced by this thrust

pr'o.-ram, analytic expressions -e developed for mission t ime and

Inclination change produaed during a transfer from one drculnr orbit

to another. Thcse exprcsslon6 lead to B condition for thc optimum

steering program which Shows that tho optimum program is dctcr-

mined only by the ini t ial and flnnl orbit altitudes and the tot81 inCIInR-

tion changc dcsired, and i s Independent of the thrust nceolcration

appliod.

Although tho baslo theory derlved fo r thcse two thrust p r o p n m s >nay

be applied to thc transfor bctween any two arbitrari ly lnrllnetl orbits,

the optimizetfon exp~-cs~Ion8 dcvcloped fo r P olrcular orbit 1rnn.llcP

aye applied to the spodf l c example of tho transfer from a 500 mil?

drcular orhit lo tho 24-hour synohronous orbit whiir ~ x ~ c t i t i n g n 30'

plant chnngc.

C o n ~ t n n t Binormd Thrust Accrloration

FOP constant binormnl zwcclCrntlon, thr out-of-pinnc tlmiirt nngli y

oqunlr * 9 , whrrr thc s i p of y 18 chnnged 90' from rnch nodc, 1. e . ,

thc dirrotion 01 i l u ~ ~ s t I s the snmu DR thc a i m of 009 U.

thrust we to r I s conRtnntly normal to tho plnnc 01 thr Ol'hlt, Al. innd

n a m IC?", nnti by cquntions (1) nnd (a). thr ro~wntr l r i ty iinrl 'icmi-

mnJor ax16 wi l l i i l w n y ~ rmmnin cunnlmt. Under thld t h r w t pramnm,

the nodc will o~c l l ln t c i l i ~ h t l y pCF rcvolutlOn nbOUl l !R DFiRinnl pWl-

2

Slllct' t h P

tion. I I o w c ~ ~ c r , slnm clnsslo c r l o ~ t l n l mcchanloi show^ thnt whcn

thc ablnlencas of tho cnrth IR consldcrod no#llglblc, the ncl rhnngr 111

thc nodal p0sItlon wi l l bo Poro for any OOmPlCtC ICYOlUlion.

bc comidurcd almost w n s t m l lor purpo"c6 of Intosmlion.

oblatCnC?x 01 thr cnrth I n not co"sldorcd nogllgiblc, lhr rnlntinli Of

thc node is dcpcnrlcnt upon thr nnRio 01 incli~>ntlon; howovrr, the nod?

may still bo consldcred con~ lnn l during onc P C V d U t i o t l .

0 mny

bmcv thc

OCLbQi- 2 The foilowing cxprcssion i s obtninod from equation (3) by sctting

.1 :/ z 2

Since the semi-major axis, ccccntricihi and t b w t accclcration are

constant, this expression map bo r e w i t t e n as,

where the negatiic sign in front 01 the integral indicates that s i n y

is negativc us u varies from 1 2 2 to , so that the integral is posi-

tive. N t o r integration the expression becomes,

Ai a' - = A - f(e, w) L-0". p (41

where

1 2 2 2 . 2 z t O co6 0 - 4 e W+2e4~in2w-3e4~in20c052w 2 ( I + e sin wj (1 - e sin uj2

(1 + el 1 -sin a 1

If w oqua~s n or n , i . e . , tho nsconding or descending node IS ai

thc perloontrle point, thon fic, u) I s inaxi~ni;..ed lo r any ecccntricity,

anti tho chuigr in inclination glvcn I n oquation (4) LE m u l n i l ~ . ~ t i for

any ollipticnl OYbit.

~iiorofore, for optimum L.OSUNS, thc following is obtained from

equntion (41,

a Ai N A Y f ( e , 0) , (81

w h o m II ib thc aumbor of orbit Twolu l lom. and

OII (7)

\ -

resul ts coincided with the ~ d u e s obtzllned from qua t ion (5). Thus

tho t ime requirod to complete any inclination change using constant

binormal acceleration can he determinod by cxpre6Sion ( I ) .

Simultaneous Change of Inclination and Radius

Thc thrust pragran, developed in reference 2 o m be appliod to obtain

maximum simultaneous change of both inclination and radius for

constant tlwust accclcration. It has a very simple form in this case

if all impressed and perturbational forces acting on the Yehick, ex-

cluding the thrust, are neglected, and if the position of thc vchiclc

st t =. 0 is assumed to be a t the ascending node of the ollipticai orbit.

Under those conditions the program in reference 2 is reduccd lo the

followiw,

u

Aw ~ A sin yo cos u

1/2 A T ~A ( - 1 s in2y0 00s'~) , (8)

where y o is thc value of the out-of-plane angle at the vscending

(and descending) noie. Since sin y ~~ sin y cos u, the out-of-plane

thrust angle y lee16 the argument of thc latitude by 9 0 ~ and the

binormal thrust component i8 maximum a t both nodes.

By substitution of A sin y o cos u for Aw,expression (9) is obtained

from equation (3).

Since the accelerations under consideration are of the order of lo-*

g's, and the orbit tranefem are within a strong gravitational field,

a small-thrust appmximation may be mado in which a and c are

conaidcrctl OSSOntially Constant per revolution. The error introduced

undcr these CirCUmStanCoS will not bc largor than 1% for mifisiom out

to the symchronous orbit altitude. In this thrust program the node

oscillbtcs slightly about ita initial position, but for the same re&9ons

discusscd above, w Is considered Constant per revolution.

~4

'Themfore from equation (9),

After integration, this OXpreSSiOn becomes,

2 re". = A sin y o ";;" g(e, w) (10)

where

Tho change in inclination i s maximized, as before, bj

equal to zero. Then, v

- li 2 Ai = A sin y -g(e, a) , re". O P

i setting

GOLWN-3 whore

If the orbit is circular, thcn

Figure 5 of Cquation ( U ) Shows lhat choosing the most cccentrir

orbit posaible for any fixed semi-n~ajor axis is tho most effeclim

way to change inclination, for g(e , 0) , which r i s e s sharply fov high

occentricitics, i s direotiy proportional to the change in inclination

required, a i d for a f i x ~ d somi-major axis, tho poriod for a m revolii-

tion Is indepondcnt of eccentricity.

t ra tes tho effect of altitude upon inclination chango for an accelcrhtion

of

corresponding chsnge in inclination for any other y o , simply multi-

ply tho valw on figure 6 by sin y o . For any mission Starting from a circular Orbit, the time rcquircd to

complete the transfer 10 another oircular orbit while executing a plane

change, using the thrust program spcoified above in equation (a), can

be found in analytic form. Firs t , the inciination change pmduecd

during any transfer using this program is detcrniincd by using cquntion 1 (13) and equation (1) whore c .= 0 ,

For tangcntial thrust direction and nearly circular orbit, cquation

Figure G of qua t ion (11) illus-

6's and with y o chosen as 90* for simplicity. To find the

(I) reduces to:

Tho change i n semi-major axis per revolution C M then be found by

integrating thc above derivative from 0 to 2 K Lo produco,

(14)

whom E ( s h yo) is a complete elliptic intogral of the second kind.

Taking the qnotient of equatlon (13) with equation (14), and replacing

this ratio of the increments by a derivative, yields,

'For thrust ncccler3tiona on the order of 6 '6 , a constant tan- gential thrust program maintains an essentially circular orbit during Spiral-out. Using an average tangential thrust of 0.76 x 1 W 4 g's , the computer program showed that an cccentrieity of 0.00646 was the maximum value obtained at tho synchronous orbit when the Ini- tial circular orbit was at 500 statute miles. Tho occontriclty O R C ~ I - IateS with p e r i d 2(r, so by adjusting the start ing point on the initial orbit, a minimum eccentricity of approximately 0.002 can be ob- tainod at the 24-hour grbit. If the identical thrust program IF uscd with an neceleration of 0 . 5 x 10-2 g's, thc final Orbit has an occon- tr lcity of 0.45. t ransfer elements are l inear with accolcration. Above this lovol any mission near tho "escape radius" of lhe omtrzl body is very SenSltlve to changcs in acceleration.

For aCCeIeration8 of g ' s o r Iowcr, the orbit

If y, , varies during thc transfer, then the above dorivalivc cannot

ubirnlly bc integrated in CIDSE~ form, so Y is ~ s u m c d constant

throyrrhout the transfer, ( i . e . , y o , the magnitude of thc out-of-

plane thrust anglc at tho nodes, is maintained constant for cach orbit

rcxdution), and cxpression (15) may then be integrated to pmdUCo a

genera1 exprc~sion for inclination as B function of altitude and a

specific constant y o thrust program.

The time required to complete this inclination change while trans-

ferr ing i rom onc circular orbit to another can be found Using thc

snmc Sncthod shown above for the Same constant y o thrust pmgrilm

The derivation requires thc following formula from claaSical celestial

mechanics,

dt (i - e cos E) Ti= P

By integration, thc change of time per revolution may be approxi.

mnlCtl by

Substitution of this exprcss io~~ for altitude intc expression (18) and

integrntion produec

2(1 - exp [b(i, - i)] ) t = 1/2 ' (20)

A b sin yo(>)

where

Equation (20) may 21.0 bc written 6s

If the CxpreSSion for inclination, givon in equation (16 ) . is substituted

into cqriation (20). the following expression for time as a function of

initial md final altitudes result%

6 a D E N - d ~ 1

1~ (22) 2 A E (sinyo) (ZA~)~"

while thc timc required to p~'oducc any inclinetion cheng-e is invcrso-

ly proportional to the thrust a x c l e m t i o n , thc inclination change pro-

ducod while t rmsfcr r ing f rom one altitudo to mothcr , expressed by 'd ( 1 0 , is not dependent upon the thrust acceleration. but is only dc-

pendcnt upon the y o thrurit direction program selected and tho ratio

of the final to initial altitudes.

For any niissioii Starting from a circular orbit about a c e n t i d body

where the thrust dircction i s exprcssed by equation (a), th,, m n i

mission time will be minimined if thc condition on y o , dcrivod from

thc :above Pquatiolm and exprcssed in cqaation (24). is satisfied. I.?t

thc initial and final ~ l t i l u d e s (semi-major axes) be specified as :L

and a . Thc thrusl acceleration is Constantly A , md it is thc

total inclination change dcsired. Tho total time to compicte thc

transfer is then the time required to spiral-out to the fiiiai aititudc.

while simultano~usly changing inclination ~ccord ing to thc y o thrnat

program selected. added to the timc required to comp1oto thc i n d i n -

ation change using pur0 binormal thrust. Using exprcssioim (22). (I).

and ( I G ) derived &ove,

Total t imc ~ time (spiral-out) plus time (binormal thrust)

Since y o is the only variable, total time will be minimizcd lf ils LJ dwivht iw with reapect to y o is s e t equal Lo zero, o r equix,;knllg,

if i t s dcriiwtivc with respect to s in Y is set cqual to nero,

Setting d ~ ~ ~ ~ ~ l ~ ~ 0 and simplifying produce tho required con

dition as long as sin y f 0 .

where F ~ F(s inyo) and E ~- E(sin yo) are both oompletc eiliptic

integrals of tho f i r s t and second kind rcspectiveiy. Thc optimum

thrust program is thus independent of the acceleration and depends

only on thc speclfiod initial and final altitudes.

If iT , the total inclination change desirod, i s I C S B than the ~nclinn-

tion change produced while spiraling-out undcr the optimum q o pro-

gram detcrmined from oqjuation (24), then cquatione (IG), (211, and

(22) may bc uscd to determine the optimum y o and a. fo r thc

t ransfer , where a

gram is initialed.

"01" represents the altitudo wherc the y o pro-

Applioation to Estiblishmcnt of a Synchmnous Orbit

The allow analysis may be applied to determine the optimum program

for the cStabliShmefit of a 24-hour circular equatorial orbit nl 22,300

Statute inilcs above the ear th Starling from a 500 statute m i l c circuIar

Orbit. A 30" orbit inclination change w u postulidcd to Silnlilatc n

L/

. - launoh from Cape Cnnaveral. The applloation of three poaslble TABLE 1 6.W-5

thrust p r u g r a m ~ to this misslon La inveatlgated where the thrust

soooleration is OOnIitant at 0.76 I IO4 g*s 80 that the theoretical

results may be aompwed to the results o b t a h d by computer.

The three programs me: L4

1.

ooplanar transfer to the 22.300 mUe orhit.

11.

Constant b l w r m a l thrust to change Inclination siter tangential

The thrust direction discussed above which changee Inclination

68.1. .46425 (26.67 6,000 10.6' ,34456 (19.17

10,000 78.7' ,23156 (13.37

1,000

.I2501 (7.21 i 15.000 1 0 4 . I !

the t h e r q u l r e d to transfer from a 600 mUe clroular orblt (4,485

mlles) to the s y n o h r ~ n o ~ s orblt (26,285 mlles) la given by equation (22)

aa 61.8 days. The value obtalned by the computer, w i q identioal

conditions, was within .5% of thfs value of 67.8 days. Once the

22,300 mile a l t i M e Is reached for Lhe BynChr~Doull orbit, P pure

binormal thrust pmgram produces a 3 0 ) orblt inellnation change.

uluatlon (7) for zero eecentrlolty of the orbit is applled,

in 19 3 adddlonal days for the inollnation change. The total mission

tMB Of 107.1 days 1s l u t e d In Table 3 lor WmpBrlaon with the other

program tlmes.

P*Ogram II.

resulting

A second pll~gram to nooomplish the *annfer from the 800 mile orbit

to the sywhranoua orblt involvell maldng iome or all of the inclina-

tion change while #pimlhg-out to the final orbit. The thrust program

expressed in eqoation (8). whem yo Is Wnstant throughout (he

transfer, is wed. The optlmum constant y o thrust p r o g r m to

t ranefsr to the synchronous ofbit and complete II plane change wp8

determined far several initlal altltvdes using enpresslon (24). The

plane change completed during aplral-out WM then computed from

qua t ion (16). The followlng tabla summarlees the results.

ii 'AS deso rbed in Foemote 1, the aDEentricity of the flnal arbl t ia

neegllglble for pura tangential pomIere.tlon8 of 10-4 g's or lower. U for som@ rewon, the eccentricity is mot negligible, then B pro- gram for remwhg the m w b d eoccntrlelty, dlsoussed in section R, may be used.

7 , ohangee during the splral-out: 8lnw the change in inollnation La

proportional to the mquare of the semi-major pxis, y o should be amopll'at low altltudea md new 90' at hlgh altitudes. The problem of

determlnhg the best y o - f(a) ha8 net yet been ocmpletely solved:

however. the m e a d used for program n e m be ajsplied to several

intermediate trMSfer'B, whose sum is & total transfer desired, Sd

the Bum of theae re8ulte can be used to appl'uximate the total tlme

required for the complete mission wing various function* for y o . Once the total mlsalon 1Mes are Imam. tho bext function for y o is

determincd ?ram those Iwertlgetsd.

In order to cdculate the total transfer tunc slni lncl in~l ion change,

$he S p ~ a I - o u t to the synchronous orblt Is divided inlo %wen Oter-

mediate trmsf~rs, PB ehown in the tsble below:

TABLE 2

Transfer bit is1 d flnal Average semi-malor UIB semi-majar WBB

7,410

8,618

10,003

2

3

4

5

8

7

I , , I"'

22,854

10,923

14.784

18,381

26.282

12,864

,I .,."

Given nfunctlon y o = f(a) , the werage y o for a i e h Intemnedlate

transfer la datermlned by taklng the value OB y,, st the aversgo

semi-majar ads. The average yo for each transfer 1s then wed

In equatlon (16), In order to d e b r m h the Incllnatlon charge mads

dur(rg that Intermedlak transfer, urd In squaten (U), In order to

d e t e r m h e the time rsqolrad for t b t transfer. The num of these

Lnollnatlon ohangem a d times f d for saob tverage value of Y,

approximates the Md time rsqulrsd lo apfrrl-out to the fine1 altl-

M e ard the total ln~linatlon change prduced durlng thle Bplral-out

Mth the spcUled yo progrm.

The tlms required to m&e the remalnlng Inclination ohnnge uehg

oonntantblnarmal thrust , as before. Is found from equatlon ('7).

This additional tlme lt; addad to the total time for aplrnl-out, found

shove. to glve the told mlsalon time for each funotlon for Y , wed.

The optlmvm program, mong those Investlgated. 11 shown In flgvres

7 and 8 and H L I c ~ or, Table 1, dow Mth B progr:, ' in whlch Y o

changes witti altitudc so that &I1 the deslred Inelin' Lon chmge 19

made whlle spiraliw-out.

Although the r(.euit6 are very scouratc for p;.,>gr,.v:%

progrem for Y

uelng "averaged" YGLIUO?. The best one m u s t bv cI1 , . , ,:o\>- p...~ iiig

the total mlsslon trmes of several "raasanablp" lui ' j . , , !,I .:,

.md 11, the

B R P function of altltudc civ. oiilp 1 1 ; ,, ,-rroxInmtad

TIRI

! .. .

. . . , ,.

. ....

- *

InfQlrre 8, a l o q wlth the constant yo programs the Gauar- M o t k

used In progum Ill-B. In whlch all the lnoltnatlon Ohma 1s made

while aplrallng-out.

m r e ia a s a v u of lT .2 days in the total mlaslon time by m&W

most of the tncllwtlon cbmm whlle splral@g-out under the y o - 68'

program. E y o L. varied - In program m-A. a VIO OB of approxi-

mstely 6 daya la made. An Indepedent appmimltlm. b r m d OD

computer rsaulte wx3 described In reference 1. prsdlote that the

mlsilon will require 88.7 days under program Ui-A, Instead of the

84.1 days predicted above. In 0 1 t h owe, a reduotlon of t lme, urd

of fuel e x p e d l t w e , rsaulte U the varying Y , pro t r am I# employed.

Whether this nnvlag of tlme la slgnlfleant snavph to jwtlfy the &led

syyatem progrmmlag oomplsxlty. whbh would be required In d e r

to car ry out this vuylng t h i t dlnotlon proprun, will d e p d upon

other factors lfluenclnp aaoh mtsilrm.

For L constant thrust w c s l s n ~ n do. 76 x IO4 e'#, prcgram U-A

requirem 88.8 day. to oomplate the transfer to the aynohronous orblt.

Unlnp P 30 KW SNAP 8 elsotr1o.l FaMr p w k w .ad .II lnUlal vehlole

weight of 7.800 pmdr. the ~ m p e l l m t wulred lor thL. tr-afer 1s

3438 pound..

J

All transfer8 oonsIddsrsd In this analysla started f w m YI In1tl.l

cIrouIar orblt. An Inlt ld elllptlcal orblt may b. mom praot lod to

d cetsbllsh, and may roduw total tranafer time. .hOa the IMlInUlon

'hangs per revolution InOreuem with Lnoraulne eoo.ntrIoIty. How-

ever.. thla transfer ham a d h e n analyzed.

SECTION B ECCENTRICITY

hluny mleslona utllb- low-thnut prcpulilon Sylbml rngulre that

the flnal orblt &ut romi o1ntrd bcdy h o l rou lu . A# de8orib.d In

lootnota 1 of Bdotlon *. U th tnltlrl orbit l a o l r o u l u .ad U lcw enough

Lhrwl accoleratloru IN urd, e pura t w m t l d h l p r q r u n pro-

,Iiiow amly B negllslblr Inorour In roo~nirbI!+. nowvr, 11) moat

T M B U , lhu ocosntrlclty II not Mellglbla ud LD Ufaotlva p r q r m IO

laiuasiiuy f o w m w e tha mow# rootntrlol!+. TlU p r q r m should

ulro be covarnlblo, 10 that by ohuylw t& thrwt mgh, P , by 180', the m e t nffeotlva sqrm for lnorruw roomtrlolty la

PPOilUOttd.

By mulmi r lna e t a d u d oxpriiiloly from caloat1.l mechwlos, M

wtlrnuni P c n K r m la dwnlawd lor ohhnytq eooontrlolei bawver,

t b soml.-mnlor anb vubm durbq this 9 w m for nan-o I rmIu

orbits, Anethsr proerm, In Whloh tha wlnul dlraotloa LI I IW.yi

llxad wlth relpsot to 1NrtI.l 19.08, r l v r i a l o n (0 opttmum niul t .

lor .I1 Oooontrloltlal md mllnblnI t& Inlt1.l ieml-maJor ulr through-

out tha tronnlor. By acmpu!q th W N W ~ M d r i l v d for UU o h u y .

In eccenlrtolty wr ~avOIUnoII p roduod by twh p r q r m , lb fhad

dlr~lct lon piorrun la found b br mom tffaotlw Uun th !bat pro-

grim for sooontrlofbi l w r r Uun ,a@,

J

-7 For thrusts of the order of

thrust approxlmation, where the orbltal elements. L md e , may

be coqsldeied almost constant ps

tlon otl toalinatlon change. Thus equdlon (28) m

give the change In eccsntriclty per rcvohtion as,

g'8 or smaller, the aame small- The change in semi-major .xis per rsvolution Is rxpreased for both

pmgrams md the time requlred to reduce nny eccentric orbit to a

ciroular one fs glvan In uldytlo form. This expression 1s quite

useful when determining $he time neece88ary to establish s ~11'cUIar " pwking orbit.

nechaolcs, the following form for

pect to time Is glven. vhers the

oul-of-plane Ulrvst angle, f , La eem.

The opUmum steering program for changing ecoentrlolty is then

fowd by soltlng the partial derivative of thls expression ari

to the steering angle. B , equal lo cero.

solvlng and reducing 4 (g) = 0 I

prnd"OfS,

l/Z (26)

For e - 0 , thle evpreaslon bemmes tan B - $ tan E , which Is the

expression d efeerence 10 direction

1 - e ' ) s toE Z E O B E - e - E - e t a n # = f 2

program given by equation (26) may also be written a function of

e * the range angle.

p = e - n .

In order to dstermlne how muah ohawe In eseontriclty C M be made

per revolulion d e r thla program, the h e express iom for me B

and ein ,S me Irubstlhzted into equatlon (25). afhzr the Independent

a VZ ' ( 1 - 3 a c o s E + 3 m a E - e e o s ' E ) d E . (29)

I h e abwe Integral may bs reduced to 8 hyperelliptic Integral ualng

reference E? d then evaluated. For ~alcdations made In this *eo-

tion, the V ~ W S of ~ n t e ~ a l .

shown in flgure 10 we used. Equation (29) may now be rewritten more

aimply LLI,

It h.8 been tvggsstad in previous seurcea. e.g., rrferenca 10. tbpt L

thrust program which La flxed with respect to Inertial lpant prwldes

ne- op(lmum change of eceentrlcity p r wvdution. lhls p r o g r m .

in which tb thrust dlreotion is always mrmal to the llne of apsides.

can be cxpreseed Simply UI @ = e - I end B = e for Incresstog ecoentrloity. Thls Wed dtrectlon program

Waa compared tc Program 1 descrlhed above because. although It 1s

not optimum for reducing socentriclty for e'@ near cero, it ~a very

close to optlmum for 1-r eccantrlcit tes and the semi-major -1s

nmilaa newly c - t ~ t for s w h evolution.

for deoresding NoentriolCy

In order to lmrestigate the properu.3s of the B = e -I thrust pro-

gram, the thrust direction glveo by

slob + i s t o e m d a o s # -*ease,

where the positive elgn 111 t&n for Iacresding eocontricity uld the

negative sun for decreasing eceentrloity, La substituted into the fol-

lowing expression derived from equation (2). where the Independent

~mlable Is now E Insteed of 9 ,

g = A t ( i - # 2 [ ( z e o s 2 E - + f m ~ e - e ) c o s a + ( r . s ~ f / 2

. *in E sin B] (39

After subetllutlon and almpllficatlon. (31) becomes.

end e to be consldered nearly constant per revohtlon. Therefore, ble h.8 been c h v g e d from t to E . u

2 1/2 '/Z after Integration, j. A (I - e') [(l -

1/2 (281 rw. = A $ ( l -e2) (3.)

(34)

0.0 0.1 0 . 3 0.5 0.8

0 fmt 0 feet

41.8 feet -1l8.6 IWt M1.6 feet +lU.6 leet

-18,Ofwt 4 . 7 isat - a m feet -6a.a isst

orbit with eccentricity e, to a circular orbit with the same semi- the pericentric point

- -9r_, -, -- - - lqegration and simplification then produce,

as an expression for the time required to reduce an lnltlal elliptic E eccentric anomaly of the Lnstantaneous el l iwe measured from

L

major axis. The time required to reduce an inltial eccentric orbit

to B circular one using a constant thrust acceleration of 10 8'8.

determined by equation (38) and verified by computer, is shown in

figure 14. The curves are almost linear for eccentrtclties smaller

than .4.

-4

Missions in the near future wlll probably require simple constant

thrust programs instead of the constant acceleration programs dis-

cussed in thfa report. However. &B long as the accelerations are of

the order of lo4 g's and the final acceleration does not exceed the

initial acceleration by more than 15% tk average acceleration sub-

stituted for A in the formulas derived above will give results withln

1% accuracy.

Rendezvous missions may also be analyzed using the formulas de-

rived in the two sectione above for change of both or either the in-

clination and eccentricity. For any given rendezvous, a selected

thrust direction program will transfer either we vehicle to the orbit

of the other, or will transfer both vehicles to a commcn intermediate

orbit. Fine control of low-thmt propulsion sgstems may be maintained either by a self-contained programmer or ground to

vehicle command link. Also, the number of orbit revolutions re-

quired to transfer from one orbit to another in a strong gravitational

field Is large, thus there may be many contacta to any one ground

station. The mmeuverabili.?y of low-thrust systems and the capabil-

ity of repeated and extremely fine adjustment to the tbrvet program

allow fine control of the rendezvous and make low-thrust propulsion

attractive Ior rendezvous missions.

LIST OF SYMBOLS

a

A

An component of acceleration due to thrust in piane

semi-major axis measured from center of

thrust acceleration; thrwt per unit m-8

normal

to the radius

Ar component of acceleration due to thrust parallel to the inatan-

tanems direction of the radius

AT component of acceleration due to h t in plane of orbit in /

F thrust

g

i

gravitational acceleration at earth's surface

inclination of orbit plane to equatorial plane of the central body

in radians

mean motion of vebble In orbit = w/a3)"' n

N number of revolutions

r distance from center of mass M to vehicle S

t time

T time for one orbital revolution

u argument of the latitude measured from the ascending d e to

the vehicle 8. u = 6 + w

LI angle in plane of orbit measured poeitlvely from the normal io

the radius vector to the normal projection of the velocity vector

onto the instantaneoue orbit plane

angle in plane of orhit measured positively from the normal to

the radius veotor to the normal projection of the thrust vector

onto the iwtantaneous orbit plane

angle b e h e n the thrust vector and ita normal projection onto

the Instantanem orbit plane

8

y

6 1) true anomaly of the ellipse measured frcm the periceahic

point to the vehiole 8

angular distance In radians measured from the pericentric

point

2)

P Gravitational constant of central force fleld = QM

where Q = Universal gravitational constant

M = Mass of central body

Ped = 1.401528 x 10" ft3/sec2

PmWn = 1.733 x 1014 ft3/sec2

w longihde of the perlcentric point measured from the d e in the

direction of motion L

8ubscrfpta

o initial value

tangential direction

Au component of acceleration due to thrust perpendicular to the

plane of the orbit

REFERENCES

oiblt

R - Radlru 8 - Posltlon Of VsNElo XY plane 1. tho equatorial plane

of the central body

O o 0.1 O.R os o* a 01 a7 am os LCCCNTRICITY

P

3

LJ

d

I . ' .

d

4

L

1

..."