First order analysis and heliocentric orbit of Sunblazer space prob

Harrington, J. V.

English

NASA Technical Reports Server

FIRST-ORDER ANALYSIS OF 
SUNBLAZER ORBITS 
John V. Harrington 
Reissued as 
CSR TR-66-10 October 1966 
https://ntrs.nasa.gov/search.jsp?R=19660030042 2020-03-16T17:57:18+00:00Z
FIRST-ORDER ANALYSIS OF SUNBLAZER ORBITS 
by 
John V. Harrington 
A veh ic l e  launched from t he  earth and acce le ra t ed  t o  a 
v e l o c i t y  s l i g h t l y  greater than the terrestr ia l  escape v e l o c i t y  
w i l l  be i n j e c t e d  i n t o  a heliocentric o rb i t ,  the orb i ta l  param- 
eters being determined by the vec tor  r e l a t i o n s h i p  between the 
ear th 's  o rb i ta l  v e l o c i t y  and t h e  excess  v e l o c i t y  of t h e  vehicle 
after it has achieved terrestrial  escape. I n  the  simplest 
s i t u a t i o n  and, f o r  purposes of a solar probe t h e  most u s e f u l  
s i t u a t i o n ,  the excess  ve loc i ty  is caused t o  be e x a c t l y  oppos i te  
t o  t h e  ear th 's  ve loc i ty .  The veh ic l e  w i l l  then be i n j e c t e d  
i n t o  an o r b i t  w i t h  aphelion a t  1 A.U. and perehel ion less than 
t h i s  by an amount depending upon the  magnitude of the excess  
ve loc i ty .  The r e l a t i o n s h i p  between these i s  given by: 
where vcl is t h e  c i r c u l a r  orbi ta l  v e l o c i t y  for a 1 A.U. orbi t  
and r is  t h e  perehel ion d i s t ance  measured i n  A.U. 
P 
To achieve the excess ve loc i ty  Av, t h e  launch vehicle must 
have a t t a i n e d  a v e l o c i t y  vb a t  burnout of: 
12 I vb = + Av2 
where vegC is t h e  escape ve loc i ty  a t  t h e  burnout a l t i t u d e .  
sidereal period of a satel l i te  i n  such a h e l i o c e n t r i c  o rb i t  
The 
- 2 -  
follows d i r e c t l y  from Keppler’s t h i r d  l a w .  When t h e  u n i t s  of 
t i m e  are chosen to be yea r s  and d i s t a n c e s  are i n  A.U., the  
expression for the period has t h e  r e l a t i v e l y  simple form: 
3/2 
131 T = ( ’ i r  .p 1 
From these fundamental r e h t i o n s h i p s  t h e  data i n  T a b l e s  
I and 11 are computed as an i n d i c a t i o n  of t h e  orbi ts  achievable 
when s m a l l  payloads a r e  boosted t o  velocities s l i g h t l y  i n  excess  
of escape ve loc i ty .  I t  would seem a t  f irst  g lance  t h a t  almost 
any of these orbi ts  would be use fu l  i n  t h e  sense  t h a t  they  a l l  
w i l l  allow a t ransmission path t o  be set up between t h e  satel- 
l i t e  and t h e  earth which eventua l ly  must pene t r a t e  the i n n e r  
corona. Some f u r t h e r  ana lys i s ,  however, i n d i c a t e s  t h a t  there 
is  a p r e f e r r e d  family of orbi ts  lead ing  t o  t h e  most desirable 
experimental  s i t u a t i o n .  
Retrograde I n j e c t i o n  
If supe r io r  conjunction is made t o  occur on t h e  l i n e  of 
apses of t h e  satell i te orbi t ,  a symmetry i n  t h e  t ransmission 
path geometry before and after conjunction occurs.  Since the  
m o s t  important  measurements t o  be made by the Sunblazer veh ic l e  
are t h e  t ransmission path delay measurements as the path pene- 
trates the  corona, t he  opportuni ty  t o  repeat t h e  measurements 
w i t h  the same path geometry a f t e r  conjunct ion seems most attrac- 
t ive.  The t ransmission p a t h  length  and of fse t  have been cal- 
c u l a t e d  (F igures  l, 2 ,  3) fo r  sa te l l i tes  launched i n  a retro- 
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grade d i r e c t i o n ,  using three possible values  for launch velo- 
cities. Direct in spec t ion  of the  three sets of curves w i l l  
show t h a t  t h e  path symmetry is  best achieved for Case 2 i n  which 
t h e  launch v e l o c i t y  i s  of t h e  order of 40,000 feet per second. 
I t  is i n s t r u c t i v e  to examine f u r t h e r  t h e  condi t ions  under which 
t h i s  symmetry is  achieved t o  determine what  other o rb i t s  may 
also be acceptable .  
If oo = 2n rad ian  pe r  y e a r  is the  mean angular  v e l o c i t y  
2n of the  ear th  about the sun and os = - is t h e  mean angular  
v e l o c i t y  of the s a t e l l i t e ,  then t h e  relative angular  v e l o c i t y  
w 
s a t i s f y  the r e l a t i o n :  
= (us - oo) m u l t i p l i e d  by tn, the  t i m e  t o  conjunct ion,  must r 
[41 (us - w o ) t n  = nn; n = 1, 3, 5 ,  . . . 
where n is  an odd i n t e g e r  and is  a measure of the number of 
synodic half-periods between launch and conjunction. 
d i t i o n  t h a t  conjunction occur on the l i n e  of apses requ i r e s  
t h a t  the earth be on t h i s  l i n e  a t  t h a t  t h e  so that :  
The con- 
1 
z (51 t m = m *  
where m i s  any i n t e g e r  and is a measure of t h e  number of h a l f -  
e a r t h  r o t a t i o n s  (half-years)  between launch and conjunct ion.  
By e l imina t ing  tm i n  [ 4 ]  and (51 and so lv ing  for ts, we  f i n d  
t h a t  the family of sa t e l l i t e  per iods  which w i l l  s a t i s f y  t h e  
desired symmetry condi t ions  is given simply by: 
m = 
r61 tS n + m yrs 
. 
- 6 -  
I n  Table I11 are l i s t e d  t h e  p re fe r r ed  sa te l l i t e  per iods  
for t he  f i r s t  few values  of n and m. 
are n o t  achievable s i n c e  t h i s  corresponds t o  r = 0. 
Periods below 0.353 years 
P 
O f  t h e  f i r s t  t h r e e  of these  poss ib l e  o r b i t s ,  t h e  one f o r  
which n = 1, m = 2 appears t he  m o s t  a t t r a c t i v e  s i n c e  the  neces- 
s a r y  launch v e l o c i t y  (39 .4  K f t / s e c )  seems p r a c t i c a l l y  achievable  
and t h e  t i m e  t o  conjunction ( 1 . 0  y r )  is n o t  too long (Table I V )  . 
Direct I n j e c t i o n  -
When t h e  s a t e l l i t e  is launched so t h a t  i t s  excess  v e l o c i t y  
adds t o  t h e  e a r t h ' s  o r b i t a l  v e l o c i t y ,  it i s  i n j e c t e d  i n t o  an 
o r b i t  with perehel ion a t  1 A . U .  and aphelion several t i m e s  
g r e a t e r  than t h i s .  The condi t ions  for symmetry may be set  down 
i n  much t h e  same manner a s  before r equ i r ing  ( w o  - us) tT = n r ,  
b u t  now t h e  time t o  conjunction tv must equal  k h a l f  per iods  
of t h e  slowest orbi ter  w h i c h  is now t h e  satel l i te .  The r e s u l t -  
a n t  expression for  permissible  s a t e l l i t e  per iods is  then: 
(71 t * = 7  + Y ears 
From t h e  t a b u l a t i o n  of p r e f e r r e d  s a t e l l i t e  per iods  i n  
Table V, it is evident  t h a t  t h e  earliest conjunction t i m e  is 
obta ined  for  n = k = 1, and t h i s  is  a l s o  achievable  by a reason- 
a b l e  launch v e l o c i t y  (Table VI). 
E f f e c t  of I n j e c t i o n  Errors 
I f  w e  adopt as an ob jec t ive  t h e  r e t rog rade  i n j e c t i o n  of  
t h e  sa te l l i t e  i n t o  a h e l i o c e n t r i c  orbi t  having a perehe l ion  a t  
TABLE 111. Orbital Periods for Superior 
Conjunction on Line of Apses 
. 
,. 
E a r t h  
H a l f  Periods 
m 
1 
2 
3 
4 
5 
6 
7 
- 
Satel l i te  Period 
Years 
n = l  
0 . 5 0  
0 . 6 6 7  
0 . 7 5  
0 . 8 0  
0 . 8 3 3  
0 . 8 5 5  
0 . 8 7 5  
n = 3  
( 0 . 2 5 )  
0 . 4 0  
0 . 5 0  
0 . 5 5 5  
0 . 6 2 5  
0 . 6 6 7  
0 . 7 0 0  
- n = 5  
( 0 . 1 6 7 )  
( 0 . 2 8 6 )  
0 . 3 7 5  
0 . 4 4 4  
0 . 5 0  
0 . 5 5 5  
0 . 5 8 3  
Time to 
Conjunction 
m t* = 2 
0.5 yrs. 
1.0 
1.5 
2 . 0  
2 . 5  
3 . 0  
3 . 5  
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0 .53  A.U., then t h e  s e n s i t i v i t y  of t h e  desired o r b i t  t o  o r i en -  
t a t i o n  errors i n  t h e  excess  v e l o c i t y  vec tor  may be s tudied .  
An e l l i p t i c a l  o rb i t  is charac te r ized  by t h e  following equat ions  
where 8 is the argument of perehel ion,  B is  t h e  anqle  between 
t h e  i n j e c t i o n  ve loc i ty  vec tor  and t h e  d i r e c t i o n  orthogonal t o  
t h e  r ad ius  vec tor ;  k = 
c i t y  t o  the  c i r c u l a r  ve loc i ty  at t h a t  r ad ius ;  e i s  t h e  eccen- 
t r i c i t y  of the  o r b i t ,  and a i s  t h e  semi-major ax i s .  
is the  r a t io  o f  t h e  i n j e c t i o n  velo- 
V 
C 
181 - k2 s i n  8 cos tan * - (k2-1) cos2g-1 
a 1 
r 2 - k i  - =  
Since the i n j e c t i o n  v e l o c i t y  v is  obtained by adding the  
very s m a l l  excess v e l o c i t y  BV t o  the r e l a t i v e l y  l a r g e  ear th ' s  
o rb i t a l  ve loc i ty  vc, it is evident  t h a t  the  angle 8 cannot be 
very large. 
t h a t  t h e  excess  v e l o c i t y  misses the  p re fe r r ed  d i r e c t i o n  by 90° 
i s  only 0.15 rad ians  o r  9'. Thus t a n  e for a l l  cases  of i n t e r e s t  
t o  us is  given by: 
I n  fact ,  i t s  maximum value i n  t h e  un l ike ly  case 
k2 s i n  B 
k2 - 1 t a n  8 % 
and 
. 
- 1 2  - 
r 131 e2 'L (k2 - 1)2+ s i n 2  8 
Consider t h e  case where A V  lies i n  t h e  e c l i p t i c  b u t  is 
disp laced  f r o m  i t s  intended o r i e n t a t i o n  by an angle a. Then 
[141 v = vc - hV cos a 
COS a k 2 * 1 - 2 -  AV 
vC 
Av sin a AV ' vg - AV COS a for  < 1 
C C 
Using these approximations i n  [ 1 2 ] ,  w e  f i n d  
r171 tan  a AV 
C tan e - F-) 
AV o r  for  - = 0.15 
vC 
A 30° error i n  launch angle  w i l l  then produce a r o t a t i o n  i n  t h e  
argument of perehel ion of 9' and w i l l  no t  be serious. 
I f  t h e  excess  v e l o c i t y  conta ins  a component orthogonal t o  
t h e  p lane  of t h e  e c l i p t i c ,  an i n c l i n a t i o n  of t h e  sa te l l i te ' s  
orb i ta l  p lane  relative t o  the  e c l i p t i c  w i l l  r e s u l t .  
tude of t h i s  i n c l i n a t i o n  is small and f o r  t h e  30° misalignment 
The magni- 
used earlier would be 84 7 * 5  or 5' a t  most. T h i s  w i l l  n o t  be 
s e r i o u s  provided s u p e r i o r  conjunction s t i l l  occurs on or close 
t o  t h e  l i n e  of apses. 
t h e  e c l i p t i c ,  and the transmission pa th  a t  t h e  t i m e  of conjunc- 
A t  conjunction then both bodies l i e  i n  
c 
.. 
13 - 
t i o n  must s t i l l  pene t r a t e  t h e  solar  d i s k .  
tage of causing conjunction t o  occur on the l i n e  of apses  i s  
t h a t  the  effect of any i n c l i n a t i o n  of t h e  s a t e l l i t e ’ s  o rb i ta l  
p lane  on e s t a b l i s h i n g  some minimum path o f f s e t  i s  minimized. 
Thus a second advan- 
Conc l u s  i o n s  
The m o s t  desirable o r b i t  for t h e  Sunblazer h e l i o c e n t r i c  
sa te l l i t e  has  t he  following characteristics: 
S i d e r e a l  Period 
Perehelion 
Aphelion 
Eccen t r i c i ty  
Tine t o  Superior 
Con junct ion 
Launch Velocity for 
Burnout a t  l o 6  f t  
Excess Velocity 
0.667 yea r s  
0.53 A.U. 
1.0 A.U. 
0.31 
1 .0  years 
39.4 K ft/sec 
16.4 K ft/sec i n  a 
d i r e c t i o n  op- 
posi te  the  
earth’s o rb i t a l  
v e l o c i t y  
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First-order Analysis of Sunblazer Orbits

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