The possibility of having a high-inclination out-of-the-ecliptic probe complemented by a second probe going from Jupiter to the sun along a rectilinear path (at least for the segment from 0.3 a.u. inward to the sun) is theoretically examined. Orbit calculations of spacecraft trajectories are included

Colombo, G.

Lautman, D. A.

Pettengill, G.

English

NASA Technical Reports Server

AN ALTElbYATIVE OPTION TO THE MIAL-PROBE OUT-OF-ECLIPTIC HISSION 
VIA JUPITER SMINGBY 
6. Colombo 
and 
D. A. Lautman 
Center f o r  Astrophysics 
Harvard College Observatory and SmitJsonian Astrophysical Observatory 
Cambridge, Massachusetts 02138 
and 
G. Pettengi 1 1 
Massachusetts Ins t i t u te  o f  Technology 
Department o f  Earth and Planetary Sciences 
Cambridge, Massachusetts 02140 
We have rscently conducted a preliminary study on the f lssibi l i t ;  
o f  coinbining the out-of-ecl ipt ic (OOE) mission w i  th a solar-probe 
mission. I n  part icular,  we have been looking a t  the poss ib i l i t y  o f  
having a high-incl ination OOE probe coinplemented by a second probe 
going from Jupiter t o  the sun along a rec t i l inear  path (at  least  f o r  
the seg,nent from 0.3 a.u. inward t o  the sun). 
The sc ien t i f i c  in terest  i n  approaching close t o  the sun i s  obvious 
sir,ce i t  enhances observation o f  part ic les,  f ie lds,  and gravitat ional 
harmonics. Our par t icu lar  choice o f  patn resul ts from the associated 
s impl ic i ty  of , the spacecraft configuration needed t o  provide, fo r  
exan?le, good thermal control , a drag-f ree system, and good communica- 
t ions with the earth. 
https://ntrs.nasa.gov/search.jsp?R=19760017033 2020-03-22T15:45:03+00:00Z
A preliminaty e m r  analysis conducted by 3. D. hdemon o f  the 
Je t  Propulsion Laboratory leads to  very interest ing conclusions fop 
an e l l i p t i c a l  o r b i t  wi th  a perihelion distance of 16Ro. Assuming that  
the nong~' iv i tat imal forces are compensated by a drag-free system 
(with three degrees o f  freedm) and tha t  the  spacecraft i s  tracked 
down t o  perihelion, the quadrupole moment o f  the sun can be determined 
7 with an accuracy o f  3 parts i n  10 . Since the e s t i u t e d  value o f  J2 
ranges f m  3 x (applying Dicke's theory) t o  1 x loo7 (assuming 
r i g i d  ro ta t ion  o f  the i n t e r i o r  wi th  the observed surface), the in ter-  
es t  i n  determining t h i s  moment wi th  an accuracy o f  a t  least  1 part i n  
7 10 i s  clear. Ye remember tha t  J2 gives i fundamental constraint t o  
2 the m e n t  o f  i ne r t i a  (or the r a t i o  C/MR ) and, therefore, on the 
U 
internal  density d is t r ibu t ion  o f  th, a sun. 
As mntioned above, the resu l t  obtained by Anderson implies a 
2 three-axis drag-free system wi th an accuracy o f  m/sec . A drag- 
f ree system having t h i s  accuracy has recently been flown i n  the 
TRIAD satel ji te. I f ,  however, we choose a solar-impact 
trajectory, then by using a spinning spacecraft, a one-axis drag-free 
system c?.n be implemented tha t  requires much less complexity. I n  
fact, a spacecraft spinning abotrt an axis aligned wi th  the rec t i l inear  
path would allow 1)  gyrostabil ization (from 0.3 a.u. t o  the sun), 2) 
an easier desi gn f o r  thermal shielding, and 3) a one-degree-of-freedom 
drag-f ree system. I n  part icular, a sphere wi th  an electrostat ic 
suspension i s  f ree to move along the spin axis wtth no exchange 
o f  f o r c ~ i i  along the path. The d i  s p l a c e n t  o f  the sphere along 
the spin.axis w i l l  be sensed, causing the thruster (oriented along 
We spin axis) to compensate the nongravitational forces along the 
path t o  the desired accuracy. The spacecraft w i l l  be forced to 
fo l low the proof mass and, therefore, to  follow a purely gravita- 
t iona l  path. Transverse forces should bec'4 orders of magnitude 
smal l e r  and need not be compensated. The drag-free sys tm can be 
calibrated when the probe i s  f a r  from the sun (5 a.gC.) i n  order t o  
f i nd  the equil ibrium posi t ion along the spin axis o f  the proof mass 
i n  the gravi ty f i e l d  o f  the spacecraft. 
During the 3.5da.y~ tha t  the spacecraft w i l l  spend i n  going fm 
0.25 a.u. t o  0.01 a.u. and closer, the eartksun-probe geometry w i l l  
permit the earth t o  be i n  the beam o f  the 0.2-*diameter antenna 
pointing para l le l  to  the spacecraft spin axis. The Jupiter-swingby 
technique has enough f l e x i b i l i t y  t o  enable the mission t o  be timed 
so that the earth-spacecraft l i n e  remains wi th in  a few degrees of 
the direct ion o f  the spacecraft track. A 20-cnr dish mounted on the 
s2acecrafi operating i n  the X band has a beamwidth o f  8' and a gain 
o f  27 do.  By using a 61-18 dish on the earth, th is  w i l l  allow a 
i i - a ~ ~ i ~ i i ~ s i o n  dsia rat2 f r o ~ n  56 t o  103 bits/sec, evsn wi th  the  nois2 
of the sur, i n  the backgrotind. Doppler tracking using two frequencies 
i n  the X b 6 n ~  (8 and 12 GHz) should y i e l d  a relat ive-veloci ty rrieasure- 
;;rnt accuracy of the order o f  crnlsec (1 ) with 60-sec iniegra- 
ticn t-;me. 
Fro& Jupiter inward Lo 0.5 a. u. , tile spacecraft w i  1 l operate 
i n  the Pioneer mode, performing selected experiments related t o  the 
so;ar stereoscopic and missions. The two-f requency radio- 
science experiment w i l l  allow the integrated electron content t o  be 
deterigf ned a t  each instant, and perhaps a wei ghteci component o f  the 
in2egrated magnetic f i e l d  (from Faraday rotation). 
From 0.5 a.u. inward, however, the mission w i  11 become m r e  sun- 
oriental. The spacecraft spin axis w i l l  be directed sunward a t  
th is  tire, and the drag-free servo system w i l l  be activated. S a w  
relevant' probc parameters inside 0.5 a.u. are given i n  Table 1. 
Table I .  Probe parameters. 
Solar Distance Time to 4% Velocity Equi 1 i b r i  urn 
(a*u. (Rs) (days (km/sec) Temperature 
(OK)  
The temperatures shwn i n  the table are surface temperatures 
related t o  a properly designed re f lec t i ve  heat shield tha t  covers 
the "front" side o f  the spacecraft; the internal  temperatures o f  
the spcecra f t  will not necessarily be so high. Furthermore, the 
l a s t  few solar r a d i i  are traversed i n  less than an hour, during 
which time, thermal equil ibrium w i l l  not be established. It i s  
en t i re ly  possible that  the system w i l l  survive t o  2 R,! I f  i t  does, 
a straightforward calculation shows that  tracking t o  a doppler 
accuracy o f  cm/sec over the half-hour in terva l  required f o r  
the spacecraft t o  fa1 1 from 3 t o  2R would permit J2 t o  be deter- 
rninrd t o  an accuracy o f  100~. Since a realistic estimate o f  the mg- 
n i  tude o f  J2 i s  about lom7, as i s  sham i n  the Append i x  , an extremely 
valuable resu l t  would be guaranteed. 
Obviously, many detai 1s o f  engineering design and sc ien t i f i c  
agplicabil  i t y  rsrnain t o  be worked out f o r  th is  mission. But the 
preliminary e f f o r t  so f a r  expended appears more than su f f i c ien t  to 
warrant t h i s  fur ther  pursuit. 
J2 OF THE SUN 
The value o f  J2 can be in fe r red  f o r  a r o ta t i ng  a x i a l l y  sym- 
metr ic  bocly by means o f  the f i r s b o r d e r  formula (Jef f r ies,  1970) 
3 - z J 2 - f g z  I", s (1 ) 
2 where f i s  the f la t ten ing  and m = u Re/ge i s  the r a t i o  o f  cent r i -  
fiigal force a t  the equator t o  g rav i t y  a t  the equator. The assump- 
t ions made are tha t  the grav i ta t iona l  potent ia l  i s  given by i t s  
f i r s t  two terms only, 
vg = P [I - J2 (3' P2 ( s i n  (jj , 
acd tha t  the surface i s  r o ta t i ng  uniformly so t ha t  the cent r i fuga l  
force can be derived from the potent ia l  
Then the actual surface w i  11 be a leve l  surface o f  the potent ia l  
i f  the surface i s  no t  ro ta t ing  uniformly, we can modify equation 
(i) ay assuniing that, a t  ally la t i tude,  the surface w i l l  be perpen- 
dicular  t o  t he  r e s ~ l i  tant  o f  the g rav i t y  force given by the  gradient 
2 o f  s(;uaiion (2 )  and the centr i fugal  force equal t o  w ( ~ ) r  c o s ~  and 
directed away froii) the axis o f  ro ta t ion.  Assuming tha t  the shape o f  
tne surface i s -g iven  by r = RE ( 1 - Y ) ,  we f ind, t o  f i r s t  order, 
We assume tha t  the angular vc loci  ty o f  the s ~ n ' s  surface can be 
2 approximated by a(() = wo - u2 s i n  $ and obtain 
where mo a u 2 RJg,. Fur the sun, we have m = 2.14 x lo-', 4 - 0 
14.4 '/day, and u2 4.5°/day. so the second t e r n  i n  equation (5) 
l i e s  between 7.7 x 1 0 ' ~  and 10.7 x 10-', depending on whether the 
d i  f fe ren t i  a1 ro ta t i on  i s  included o r  not. The best determination 
o f  the f l a t ten ing  o f  the sun ( H i 1  1 ,  1974) i s  f = (9.6 t 6.5) x 10'~. 
It i s  c lear  t h a t  J2 cannot be derived w i th  ary accuracy f r o m  equdtion 
(5) ,  since i t  i s  the di f ference between two not  very we1 l-known quan- 
t i t i e s  o f  nearly equal magnitude. 
I f  the sun i s  ro ta t ing  uniformly and i f  the densi.ty d i s t r i bu t i on  
i s  known, J2 cae be d i r e c t l y  calculated. Following an analysis by 
Sterne (19396), we define the "apsidal motion coe f f i c ien t  ," 
where sS i s  the value a t  the surface o f  the var iable Q, which i s  zero 
a t  r = 0 and which sa t i s f i es  Radeau's equation: 
I n  equation ( 6 ) ,  p i s  the d e n s ~ t y  a t  r and p m  i s  the mean density 
i n t e r i o r  t o  r. Then, 
where m has been previously defined. The coe f f i c ien t  k depends 
so le ly  on the d i s t r i bu t i on  o f  mass w i t h i n  the star, ranging from 
zero f o r  a completely concentrated s t a r  t o  314 f o r  a homogeneous 
star .  Values o f  k have been calculated (Motz, 1952) f o r  so lar  
rrodels by Schwarzschild (1946) and by Epstein (1951). Motz obtained 
k = 0.00585 and 0.00599, which leads t o  J2 = 8.3 x and 8.5 x 
respe-tively. Calculating J2 f o r  three l a t e r  so lar  n~odels, we found 
J2 = 1.56 x lo-' f o r  a zero-age sun (Schwarzschi ld,  1958)and J2 = 
1.41 x and 1.20 x loo7 f o r  two models o f  the present sun 
(Wey~ann, 1957, and Sears, 1964). A1 though we do not a t  present have 
deta i led calculat ions o f  l a t e r  so la r  models, we note t ha t  a recent one 
(Hoyle, 15?5),proposed t o  explain the low neutr ino emission from the 
3 sdn , has the unusually 1 ow central  densf t y  o f  75 g/cm . The r a t i o  o f  
central  t o  mean density i s  then 53.2, which i s  qu i te  close t o  54.2, the 
r a t i o  of central  t o  mean density o f  the "standard model ," a polytrope 
o f  index 3. Russell (1928) found k = 0.0144 for  a polytrope o f  index 
3, so we consider 2 x t o  be a redsonable upper l i m i t  t o  the value 
o f  d2 ftir a uniformly ro ta t ing  sun. 
I: i s  o f  i n te res t  t o  consider a lower bound t o  k and, hence, t o  J2. 
The mast concentrated s ta r  w i th  a given central  density i s  the general- 
ized Roche model, which consists o f  a homogeneous core, w i th  a density 
equal t o  the central  density contafning a1 1 the s ta r ' s  mass, and an 
envelope w i th  in f in i tes ima l  
density. Radeau' s equation can then be solved analytically (Sterne, 
1939b) to obtain 
Uith current estimates of the central density o f  the sun ranging from 
about 75 t o  about 150 g/an3, we f ind  the lower l i m i t  of J2 to be 
between 1.4 x loo8 and 4.5 x loo9. 
References 
-- 
Epstcin, I. 1951, Ap.J., 114, 4380. 
Hi l l ,  H.A., e t  a l ,  1974, Phys. Rev. Letters, 33, 25, 1497. 
Hoyle, F., 1975, Ap.J. (Letters), 197, L127. 
J e f f r i e s ,  H. r 1970, The E a r t i ,  F i f t h  ed., Cambridge, p. 171. 
!4otz, L. ,  1952, Ap.J., 115, 562. 
2ussel1, H. N. , 1928, M.N., 88, 641. 
Schwartschild, M., 1946, Ap.J., 104, 203. 
Schwarzschi 1 d, M., 1958, Structure and Evolution of the Stars ,  
Princeton. 
Sears, R. L. , 1964, Ap. J .  , 140, 477. 
S term, T. E. , 1939a, tl. N. , 99, 451. 
Sterne, T.E., 1939b, M.N., 99, 670. 
Weymann , R. , 1957, Ap. J .  , 126, 208. 


An alternative option to the dual-probe out-of-ecliptic mission via Jupiter swingby

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