Da Riva de la Cavada, Ignacio

Fraga, Enrique

Liñán Martínez, Amable

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

INSTITUTO NACIONAL DE TECNICA AEROESPACIAL 
"ESTEBAN TERRAD A S " 
THE INFLUENCE OF LAUNCHING ERRORS 
ON THE TRAJECTORY OF SPACE PKUtfiiS 
INTA REPORT I .C.I 
APPENDIX 7.III 
by 
Ignacio Da Riva 
Enrique Fraga 
Am able Linan 
INTA REPORT 1 .0 .1 (Appendix 7 . I I I ) 
THE INFLUENCE OP LAUNCHING ERRORS ON THE TRAJECTORY OP 
t i n i HI » • «i II • II i • • i. m • • • » n » • • « • • « » ii • • ii n » « « • i ii m » « • • ii • i i n 
SPAQE PROBES 
^* ^
n
^
Toduotion.-» In the following a study of launching 
errors is presented more oomplete than the one given in 
Chapter 7 of IOTA R2P0RT I.C. 1 "The Influenoe of Laun-
ohing Errors on the Trajootory of Spaoe Probes". 
The matched oonios approximation is also used here; 
however, "patohing" between the geooentrio and heliooen-
trio orbits is assumed to take plaoe at the Earth's sphere 
of influenoe (radius VL ~ 900000 Km). 
We;acknowledge the help of Dr. J. Vanderikerokhove who 
suggested this extension of our previous work, and kindly 
provided us the firat': two matrices given below* 
The nomenclature is as given in the figures. 
2. Relationships between the injection conditions and the 
orbital parameters. 
\7e give below in terms of the seven injeotion oott-
ditions : 
rin ' Vin* *in» ^in » ^in' ^in' Atin 
the six orbital parameters in a geooentrio equatorial 
inertial system s 
INJECTION AND GEOCENTRIC EQUATORIAL 
ORBITAL PARAMETERS 
a in 
The first three orbital parameters are independent of the 
seleoted ooordinate system; they are given by the following 
relational 
Af Elliptio orbit. 
OL 
e 
= r / (2 - r 72//ue ) 
t - *-
r 
1 
A© 
r 2 V2 o o s 2 r \ 
i•••••• in » — — • » • • • • • ••#•»•>•—mmmf^ I 
/ S 
2E © 
1/2 
( 2 E^ r 2 4- 2 ,u r - r ^ o o s 2 7 J +-
4. ^ * - arc - n _ / * + 2 E * r 
V ^ ( / ^ S r f V 2 oos2T E . J 7 * 
with E = V2/2 - / ^ / r ( to ta l energy) 
B. Hyperbolio orbit 
,2 
= r / ( r V 2 ^ - 2) 
f 
* . = V 1 
t - t 
p p 
/ 2 V2 ^ f r ¥ ooa2 T ^ 
" V r " /*• A /*•• / 
= _J-« f 2E r24.2>u. . r ^ c ^ o o s 2 ! J -
2EA LA • ' • ' 
A © 1 
V 2 ^ n 
£ l ± S * J E * ( 2E0 r 2 4 .2 A r - r V c o s 2 T ) ^ 
The second throe g rb i t a l parameters are dependent on the 
poordinate system, they are given "by t 
i = aro oos (sin AJ[n oos 0 , ) 
>iV = o< i n - arc tan tan A' -„ sin a. 
(X^ » arc oot j^ oos A»in oot 6^] + AUj, 
Effeot of injection errors on orbital parameters 
Differeniiation of the preceding equations permits to 
express the errors of the orbital parameters, in terms of "fehe 
errors in the injection parameters* symbolioally we oan write? 
where the matrix is given below as 2-1 
Where the expressions for the partial derivatives are 
•do* 
br 
bo* 
dv 
be* 
•57 
= 4> (r V2-2fa)c ( 4- for e l l i p s e , - for hyperbola) 
= 4. —-
2 vcV 
( rV-2^) 
j rooaf Y f_ _2
 n . 
in. iMiKunm M ••.« i r v 0 I 
bY 
2 r V 003 Y / r V 
f"re-© °< © 
r V s in T 003 Y 
i % &c* 
- 1 
2 
r Vf 
"FT - 2 
For elliptio orbits 
£*-«3 ^a Jc - * 
dr r r Y -2 LU © 
r V s in Y [ r V2oos2Y4- f^j rV2^ |U^| 
( r v 2 - 2 ^ ) 9 ^ ^ 
r V oos Y 
V & i -{**WY - ^ 14-
1 if rV ft -1 
2 2 
© f. 
^ •b pc< 3 rV(t--t p^") 2r^s in Y (rV^cos^ Y 4- fa ) (rV - ( V 
Sv rV - 2 r« 9 <*» - 2 f ^ ' f r ^ 
2 r^ oos Y 
^ T T T ^ N " ^ 
- (rV oos )f 
-GT I 
1
 ; rV rV
2cos2f , . 
- 1 1 / 1 1 \ 
e © ft © 
dt P* _ 1 
d r ft-e 
,-—-«-- r ^ s i i r Y I rV^oos^ Y 4- /A> ~ 
2 v s i n \ ' rV 
1 4-
(Ub e - f r Y 2 o o s 2 Y - /C^t I 2 e 2 
2 -
ft 
- 2 
© 
r V 2 o o s 2 / 
1-1 - , , 
© 
For h y p e r b o l i c o r b i t s 
5 r 2Er^~ v " "& 
( t - 1 > . 
tZEy/Z ( Lo * 2Er 4- B) 
<HPe, 3 V ( t - t ^ , ) V |"p|B4-Ltg;B2+(2Er4-B) ( B 2 4 4 E 2 r 2 s i n 2 r ) - 2 E r ^ ] 
' " I I ' I " ' 1 = " Ml " • • • • ' • ! II Ill I ' " » P ( W '«.! »'• • ••!! . . 11.11 I — . 1 . I 
bv 2E (2E) 572T7 IfV^J 
5 t ^ (2Er 4-B) r 2 V oos Y s i n V 
SY 1/2 (2E) w^ (Us 4-2Er 4-B) B 
Where E = L-fand B = y 4 E 2 r 2 4 4 E p r -2Er 2 Y 2 oos 2 Y' 
ix^ s i n A » i n s i n 0 i n s i n A! s i n i n ^ i n 8. 
}jfyn V 1 - o o s 2 ^ i n s i n 2 A i n s i n i U 
Oi^ 
T 
oos A ! n oos ^ n 
A i n s i n i, oC 
oos ^ l n ain &la o.oa A^ 
. 2 . s in i ^ 
^ « s in in 
u 
in 
? 
s in i 
J o in sin" i 
oos A! 
m 
OcO^ oos 0 i n 3 i n 0 i n s in Aj[n 
d A!
 n a l n 2 ^ 
77" 
v2 oos2 r 
. 2 . . . 2 
Ji+ - W 2 - - 1V1- afasfiQl 
ye£- C^oos2Y-^2( e, 
1 2 
1
 / rV 
V n & i\ 
2 rV oos 2 V 
V ^ 2 e 2 - ( r V 2 o o s 2 r - ^ 2 [ 
1 , rv" 
-l) ( 1. rV
2oos2f \ 1 
Hi © 
/ i 
CJCL^ r V2 3in 2 V 
dr Vp f f i 5 
v - 1 
2 e 2 - ( r V 2 o o s 2 Y - ^ ) 2 I 
1
 / r V A/' 
14. . _ 2! 1 -
2s2 \ /X A 
rV "oos^Y 
• I I I " 1 
ft & 
<3 
.<5 
<3 
.5 
0 
KD 
ti c qK> 3* 
3* X 
o 
c 
of 
c 
ro 
4*r 
a> 
3 >• 
M c 1^3 ro 4 > oh 
sK ro r^ > m
s 
o o 3 
< <3 
«* 
<I 
3 
O 
3. Transformation from an equatorial into an eoliptioal 
geooentrio and inertia! coordinate system. 
The orbital parameters a£ ., e& , tp , i„ , J\t , 
and cu, measured in an eoliptioal geooentrio and inertial 
coordinate system are related to the orbital parameters 
a^ , e .to , L , J~l and cu measured in an equatorial 
geooentrio and inertial coordinate system, by the following 
relati onships. 
a£ = a^ 
e
£ = ^ 
tp - tp 
oos i = oos £ , oos i 4- sin £ . sin i , . oos-H^ 
oos £ . sin i~, . o o s / i - sin £ - oos i _ 
sin i 
c 
oos £ . s in i^ - s i n£ i oos i_ . oos-Q 
COS OJ - O J ( ^ " ^ ) = • • • i i • •'• •'• 
sin x, 
The errors A £ in system 6 are, therefore, related to 
the errors in system ex by a set of equations, which oan be 
written conveniently, in matrix form : 
GEOCENTRIC ECLIPTIC ORBITAL AND TRANSITION PARAMETERS 
N 
3 
.2-
A a £ 
A e £ 
p£ 
A L£ 
A n £ 
A L J £ 
0 
0 
0 
0 
0 
0 
o 
TTnT 
/Lul 1 1 
Aa^ 
A e < * 
At 
rot 
A ' ^ 
A
^ 
Au^ 
or, symb olio a l ly • 
Ai — w..e A* 
with the following expressions for the p a r t i a l der ivat ives: 
£1. 
S i 
di 
= oos (t^U -U/£) 
c*. 
DA, 
£_ = 
sin £. ain i - sin XI 
s in i , 
cos g^cqs i^- ^ s i l ^+ 5 in £. s jn i*- cosily .cos (lu^-o^) cos i f 
sin i £ . sin -Q 
d-Q* = sin 1^ . ain -O^ 
sin ±sr 4 ain -O. dn, oos t + ooa JX. .sin £ ,ootg i ot 
|_oos£ .ooa i^ 4- sin £ « sin i^ . ooa -T2.1-oos (0-^ -<-^ £)oosi 
s in i , . s i n (UJ - L0_ ) 
an. 
s in£ , cos L^;^in-floi, — c o s ^ ^ " ^ , ) c ° t 9 L^ • -sin 6 . sin "L^ • s i n J~2 & 
s i n 1- . a i n (OJQ, - CO- ) 
4* HQlationghipa between the orbital parameters and transition 
point oonditiona* 
The following parameters determine the trans ition point in 
tho Hart's sphere of influenoe and the velooity of the probe 
there, aa well as the transit time- s 
rtre dtr- trc vtr
 Af % *trc 
These are given in terms of the orbital parameters t 
by the following relations for hyperbolio orbits. 
sin 8^. = sin i . sin (cu 4- (?tr) 
c »— 
tan C^^. --^r) = °o3 ic» t a n (w£ + 0-tr) 
Vtr 
rtr = constant (/v 900000 Km) 
6 
oos e t 
ac (el - 1) 1 
ain <j « s i n i g • oos ( f t r - CA) - © t r ) 
t a n ( Y+-, - C J P - © . , ) 
«*W\re ) " h ~ \ ) t r . t r . COS (M; 
>os Y 
ft a6( 
t r . 
__ £^£ 
rtr_VtrP 
- i) 
V£ ~ ' \ 2E Y ^ t r ' *
2
 {% r t r " j2 ' 
L ( a +2E r t r ^j^fE =4£ * 2 ft*tr " '* 
,3/2 
« ; 
1/2 
Ii ( Lt/ e ) 
' © 8 
© 
( © A / 2 
Where E = " J = V (C// a (e_ - 1) 
2 * g ' I $.e. *=• 
Effect of orbital parameters errors on transition oonditions. 
Differentation of prooeding equations permits to express 
the errors in the transition parameters in terms of the errors in 
the orbital parameters; symbolioallyj 
A 
trc \ \ 'tr. 
A 
where ^V is given by 4.1. 
u> 
d 
<\ 
O) 
 
^ 
o 
^o 
•ir IJD 
o 
~ p U3 to 
UJ 
• O ' -O 
o 
o 
o 
/ ^ O 
U_> 
C5 
• " O 
tO 
u> 
U3 
U3 
Q_ 
4J> 
A3 
uo 
a) 
SO 
a) 
so ' 'O 
UJ to 
o 
«o to 
lU 
^ 5 
/*0 /"O 
CD 
• - 0 
u) 
UJ 
V. 
-f* 
-^ o 
to 
AD 
.*o a> t~ 
<3 
to 
t_ 
< 
V . 
^ 
< 
to 
- < 
<l 
t o 
<SO 
< 
Expressions of the p a r t i a l d e r i v a t i v e s 
l o < t r £ oos i g . oo32(o< i . r - ^ e ) 1 - ~c 
d a £ o o s 2 ( o ; g 4 : 9 t r ) e^ . r t r s i n 6 
•**<m«iM»*w*«w 
t r e 
^
d t r £ _ oos i & . o o s 2 ^ ^ ^ - ^ ) a'g ( e | - 1) 4 - r t 
K oos2 (C0£4. 0 t r ) r e 2 s i n 0 ^ 
c 5 C £ 
6 . . . 2 
5 i & 
dv 
& c
" & 
ti**t 
bot 
•cr£ _ 
^ e 
a ie 
oos ±c • oos ( c*^ - X I ) 
oos2 (CU£~ © ^ -) 
s in i . oos (aJL + 0 . ^ ) 
oos ^ 
s i n i £ • oos (a)£4. Q . ^ ) 
oos ^ 
oos i c . s in (cUc4« 0+_, ) c 6 u r 5 
003 <L 
'£ 
• 
• 
c
 £ 
i . - . | 
r x e s i n 0J_ 
•to c t r 
£ fc & 
a e ( e | - 1) 4 . r t r e 
r t r e °a s i n ^tr6 
bS t a i n i £ . 003 ( o J e + 6 t l , ) 
5 CU, oos 
•br. 
3v 
•fcr 
3a, 2 V t r a | 
a A, oos2[ 2^ ( t t % ) - A £ - H g ] 
a 5 ooa i . o o s ^ r ^ - ^ E - © t r J 
•£ ' — V « 6 E - ^ 
•i-i 
e. E f r ^ in e s i n tr_ 
3 a £ 4 . r t r & - a £ e £ 
sin Y txv 2% * r t r , e J 
2 7T 
4. _ <3"Hrc 
365 '25 
a A, 
(D Or 
o o s 2 [ ^ ( t t r £ ) - A ~ ^ £ ] 
oos i g . oos ( Y t r -^0r*" ^-br ) 
£L 
a& (eg 4- 1) *r t n 
e 6 r t r £ 
£ . 
a i n ©. 
•fexv 
a ,-
sin Y. tr_ r t r & < 2 a £ * r t ^ % ~ 1 ) 
2 71 dt t » 
• » I ! • • • • • • S, 
365*25 !C, 
U< 2 71 
d tp 365,25 
it. * 
ix& 
5OJ£ 
o o s 2 jU 
c 
o o s 2 [ 2 e ( t t r £ ) r A e - ^ 6 1 
003 i . • oos ( T ^ -GJ>£-
^ 
C) 3 £ _ sin i £ . sin (Y^ - a>6~ 0,^ ) 
OOc 003 0 , 
3 ^ " r*rB-ae4 1 r ^ r e ( e 2 - D eg- 1 1 
sin Y t r f 2e£ ~ r t e r t sin 9^ 
e t c e £ 
C>C?6 sin 1 6 .sin ( Ytl>£ - cu£ - e ^ j 
— — — — i i •! • 11 —WMWI I / i npi< i M- - • • rir • • HI i ii • irnwrt WT - IT T a r rr-mm ~i -- n u n r 1 - ir *••• nm*——****—**—* mm3mm*mqmimmm»m**ammmm—mmm-
sin Ytr |f r t "(2a£ 4- r t ) (e|-1) sin 0 t e | r ^ 
5o£ oos i & . oos ( Y-tr£-<%i-6fec. ) 
^ i p oos 0^ 
v* 
Id 6 a±n ±t . s i n ( T-tj- -<^> e -©tr £ ) 
otu 0 0 3 & 
3tt3 r„ 2 1 
da, Vf* 
2 ,Ji 
r£§*_!l£ ****" 3 a l (ee - 1) 
ae 4 4- 2 r + ^ a 2 i a 2 ( e 2 -1) ee £ 
3 al/* I a 4- r . „ 4- Ur + T , 4- 2i>„ a - a
2
 ( e 2 - l ) 
a& ee 
« 1/2 
V ^ £ * 2 ^ r £ a e a'2 ( e 2 -1 ) 6 & 
dt* 
or,, 
c 
- 1 
V^e 
e£ a l 
V ivz r bxv^ txv. "£ a- - a 2 ( e 2 - l ) 
4- a 3/2 
te
€
4- a £ 
V ^ * 2 ^ - ae (98- l ) 
5* Relationships• ^betweenr_the t rans i t ion conditione wiifcBi^ -respect 
^A^j19. ffajfth. a n d "b^ie inJestAPAA°PAA^ons, ^n ^^.e hel iocentr ic 
or "bit. 
These injeetion conditions are the following in a hel iocentr ic 
ine r t i a l reference system: 
r " $ » at " V " . f « i " t 
t r ^ t r f i r tro- I ° ^ t r ^ 
That are given in terms of the transition conditions with 
respect to ^ geocentric ecliptic inertial reference system: 
r » £ " oi " V " \ " <5C " t 
tr tr& trp tr; £ C trg 
by the '^following relations: 
tr r© cos 2 (t ) 4- cos o( cos ^ 
<2~ rZT ® trc tr_ tr,-tr^ g £ c 
2 j> rtr2 sin 2 dtr 
tan 0 = -* 2 27 "* ~ ^ 
tr r 4- r+T, cos o. + 2r r. cos0. cosi o< -> (t. ) s x r e ® T r 6 T r e «• t r 5 «-© T r e 
r 
^ - ^ + 2V t r T e " a S£ cos Ae * 4 
*tr = *tre 
TRANSITION AND HELIOCENTRIC ORBITAL PARAMETERS 
SenY = (V,.)""1-^/ cosc?cosd sinFX + ^+r - X ( t + r )1 i\ 
•+ Vtr sin 4 sin 5 4- V cos dtr s i n | »tr ~ ^ J "^ J I i 
c CT O" <- CT © C - | 
oaU (Y! coBp-^ cos ^oosfci^- 2 ^ ^ ) ] + _^ 
4 V t r c o s * , cos $ cosfc* + l - y ( t t ) ] 
6 'J 
However, taking into account that r Jr is a smell number 
(^ 0.9A50) V7e shall use the following appoximate relations in 
- 2 
which terms of o rde r |_ r ^ / r J have been: 
o« = Z ( t v ) •«. ~ ± = e cos £ t r s i n ! * - J (t ) ] 
'© tr* 
(V =-=tr<£ s i n 6 
t r © t r ? 
r , = r 4. r + cos CL s i n ; ^ - 7 ( \ „ ) ] 
~
r£ t r F u bXg —© t r F j ' tCr © 
2 2 *\ \ 
Y. = V + 2Y. Y^ cos <) cos A 4- Y. I e t r ® & t r 
t r _ t r i -
er £ 
s i n T CT 
Vtrccsd'cs,nAeH':it.rff-ltre)l(Va + Vtr c°54' : c ' ; iVt+4r a ? m^V tr 
VI 
COS u (7 Yj- cos y 
Effect of_ transition errors on injection error in the heliocen-
tric orbit. 
Differentiation of the preceding equations permits to express 
the errors in the injection conditions in terms of the errors in -
transition conditions. 
A tr^cr w trs,tr<r HI 
where VL 
-tre t r<j i s given "by 5. 1 
with the following expressions for the derivatives 
7> c>r 
__iE£ _
 c o s tf # s i n , 0 ' _ £ (t+T, ) 
t i y 
r 
r . cos 0 . cos 
t r c t r ^T€-
E
^Sre\ 
3r iror _ 
-DcT 
t r . 
r . s in s m £> t r . 
irtf= 
tr . 
 - r t r . cos 6^ . cos i o t - £ ( t ) "| 2 jn „ 
tr,. t r <S t r ^ j 365.25 
o 
<_*> 
75 
<3 
& 
<3 
1 0 
t O 
<1 
.b 
V 
u 
^D 
, « 
+ J 
- fJ 
^ O 
£ 
.** 
o 
• o 
.«o 
/ o 
•Jl 
t V 
o cc> 
«0 t v - M 
rs> 
-Q 
ra 
C=> 
c=> 
2 ^ i! >< o 
J? 
c=> 
,*o 
,W 
r=> C2> 
>o M 
Co res 
«o 
o 
."^ 
M 
it if 
C3 
- o 
fa 
£> 
rf* ( O & O 
tf 
/ 
O 
M 
r,*» 
-o 
«>> 
*> 
r=> 
-1 
* 
0> 
c?» 
<? 
F* 
v
-o 
W 
,<JL> 
A> 
bi 
cz> 
<u 
<p 
j ? 
<3 
4J 
<1 <3 
£ • J 
v ° V - COS <^,„ f r* / j . \ 
9 r . *"- L £ '© tr . £ J 
ire = 
0 CA 
'tre 
TJ2L COS j 
* * 
cos OL 
-tr. 
- Z ( t _ ) 
© t r £ 
_ t r s :
 = _ ^ s in £ s in oc - I ( * t r ) 
tr_ t r &
 £ Qdf 
t r c 
T j t t r , "365T2 5 
1 £E£ COS <y .„ . cos 
_ t r 
tr~ 
D5 tK S 
'(Tr v_ _ 5i2Ji±£e 
•br* ' * 
2£tet =—•-&& cos £ 
?J t r . H 
2vi 
3v. t r 
VA cos o* cosA_ J- V 
_® £ 1 i r 
V t rV3> • < ? A _ ^ X £ 
V, 
c 
I doc. £ - 1 - V^ COS V . s i n JJ5 
T+„ cosi o ° s A £ b : ! I *\__ Ti--
• t r £ 
^ . Y + r c o s y ^ ~ ' £ l / _ ; r / 
A A E U - YT oos V ^ f e ^ I c o s i o o s /Vt>7 j "" I 
j.__tr£.J 
3a . Y c o s V* s i n 1 
v*.. cosi cos A l-n^a: I - v cosi s inrJ-^-f r | 
oM, V c o s Y" sin L_ 
dur (v. =03 r sin ip-^loof 4 0 0 S A * 7 / Or<r 1 \ 
"^"tr ^sinA*- - £ (*tr > i 4 V c o s V s i n t I BYM J 
r . \ A V cose) c o s A A r - M % f M I 
x Vtr cose,. sm^+ V tr cos^, ^ i ^ © ^ ^ j 
i ruVt\ - V cost, sin}, TJT") r 
^ - ^ . - ^ - V w c o s o cos/VA^yr-" ' ^ „_ i 
' ^ C _ = (V- c o s T s m v ) y - b r C O b i £ < ; t t r £ / " ' ^ ^ 
" " ^ " r i-n *" -^"%-i^A - VT c o s i s i n f U f r ' ( 
" 3 6 5 , 2 5 " J 
SLLSL. SK 
d a + ^ . _^S- '"" + Y (Y 4 V+ r cos J c o s A U — S s ^ K s in J . Y W ( W ^ - - ) 
9- tr-. v I c o s t 
2r«- Yfl4 V^ r cos dF cos A£ 
£>ot t r Vx cos r VOL t r £ 
<Ofc 
y«) t r £ 
(V 4 V t r cose) cos A) ( ^ r f i ^ 4 s i n ^ V t r ( ^ - t a £ L ) 
® " £££ ^ -h 
I c08T<r 
^ f 
~£-fe=-(YT c o s Y ) ~ ¥ s i n T ( ~ 1 - ) — cos a s i n X . 
t r 
- COS A . COS "([• Cx. -
tr. 
(W)| - J . 
'3 t r £ ^ j ' tr«_ sm £ t 
_^ 
U A -
•(Vj c o s p ^ l s i n ^ ^ ) ^ ^ cos^ 
• Vtr COB<T£ sin Af r ^ ^ - K t ^ ) ] 
cos A, 
t )1 -if 
~ f - = - (Y I c o s T ^ s i n ) ; < § £ - ) f Y ^ s i n 0^ s i n X f + 
V
°
E
 + Y t r s i n J £ c o s A £ j a r - Z ^ ( t t r £ ) 1 + Y t r J " t r cos ^ 
i» p* -^ 
6. Relationships between in the injection conditions and orbital 
parameters of the heliocentric orbit. 
Y/o give below the relation between the orbital parameters 
in a heliocentric ecliptic inertial reference system: 
acr eo- tpCT V A x Var 
and the injection conditions into the orbit. 
r t r<r ' t rcr > °*tr<r ' V I ' Xr » V ' W 
These r e l a t i o n s are as follows 
ft e 
8<r 
f
 2 V Z 2 ^ v 2 C 0 S 2 ^ 
-I - (
 7 ) ( — S I - . - - ) 
^ P0 P© 
t 
2 ,r2 ___2 2 
/ V ^ ( / ^ 2 r t L v i c 0 S % V ^ 
V o^ 
0 2 rtr 
cos (ctt - ^ ) = ctg iT. ion £i 
°" Bln<ytr t p ^ 
sin (OV- A!)^ = .-£-
^ ^ sin ^ 
The errors in the heliocentric orbital parameters may "be 
expressed symbolically in terms o'f the orr'ors in the injection 
conditions by 
K i = lXtra;(r!l&t r<j 
Whore the matrix |rT. ^Jis given by 6.1 with the 
following expressions for the derivatives. 
^
a<r 2P® 
5rt^ (rtrff ^ - 2^) 
b a a _ 2 p/e Vl r t r c r 
"^7 ""(^ tr^ ?"1 2 r o ^ 
S e a _ VX2 cos2 V ( r-t^Yi' } ^ ) 
m w i , w »w • 
^
r t r a f© ecr P e 
=^t £* < <J > 
/-£> 
b 
A-
••-> 
* 0 
> 
/•o 
o O O 
O 
o 
o 
^1 b 
/-O ' O 
 
/ O 
•Jh> 
--0 
5 
b 
c 
£> 
- ^ 
t_ 
* ; : > 
b 
-a_ 
+* 
• O 
£ 
•4-3 
V . 
"O 
o 
b 
"^ O 
/ o 
V 
-•-
i _ 
SQ> 
\s 
< 
^ 
-o 3 
<3 
3" 
hea rtr~ VI s i n ^ - 0 0 s 1'er , r t r v? ^ 
—r-~— » —--*•—. and —-r are vfchose given in matrix 
°
Ttr<J dVj ^ r ^ 
2.1 when tpo. , r i n , Y±n , fin e^ , a^ and ^ are 
substituted by tpc> , r t r ^ , VI? X& ' . °tr ' a<? » a n d / ^ 
<3 
d ^ ctg IQ. 
d^cr _ "tan O ^ 
§i<r sin2 ig. . sin ( c*tr -A) 
-.-.-—
 t —~ and -—— are those given in matrix 2.1 
when cO^ , r i n , Vin , Yin > ^ » a n ? a*1^ /^ a r G substituted 
by a)a, r t r c r , 7X , Xc . ^ . V a ^ ^ 
b cOo- oos Str^ 
dotr 
V
 <? tr^. 
OS i 0 C % OOS 0-fc2» • o 
^ sin i ^ Vain2 i a - oos2 S. t r a 
7 . 1 . Relationships between the errors i n the posi t ion and velooity 
of the probe at a r r iva l and the errors in the heUooentrio 
orbit parameters(vith r-j- fixed) 
We give below the r e l a t ion between the probe's posi t ion and 
velooity at the monent of a r r iva l at the ta rge t j 
and the heliooentrio o rb i ta l parameters : 
These re l a t ions are as follows : 
s in 6T = s in i g . s in ( COa+ 0-r ) 
tan (otT - £l„) = oos i - tan ( cOa 4- 9T ) 
a<T ^ 
r = . constant 
oos 9 = &0> ( 1 " 9 ^ } _ j . 
><7 ^ y ° -
t a n (<*v -XI ) = oos i ^o tg (Yl ~ O J - 6 T ) VT- ff a *cr ** '<r 
'cr 
s i n <L = s i n x _ . c o s (Y T - W - 6 T ) VT ° »o~ 'cr 
'a-
1 
t T — t p - —— 
•a »a 2E r 4-Z u» r._ — 0 *• — •—» a x i l ; -yamj yxauaienat* t To- r o T<r y^ZE <- V V 7 t / 4*2EJ ^ 
where 
E = - *- VvfV1 - ^ 2a Y a \ © 
<7 
The e r r o r s i n the p o s i t i o n and v e l o o i t y of t he probe a t 
a r r i v a l (with TT cons tant ) are given i n terms of t h e e r r o r s i n 
the h e l i o o e n t r i o o r b i t a l parameters by 
where the matrix I u i j - , T<J| l s given by 8 . 1 , with the express ions 
given below for the d e r i v a t i v e s . 
The p a r t i a l d e r i v a t i v e s of f*- , J T o ( 1 d T ( r , Vj ana Oy 
are the same as those given i n matr ix 4.1 i f we s u b s t i t u t e 
r t r 6 » ^ t r g . ^ t r g . V , fc and a £ f e g , t p e , i £ , J ^ , U ^ 
"by , 
X V Cs . . . . 
p 
and.the heliooentrio orbit being eliptio, the oombination e - 1 
by 1 - e^ . • 
b 
C5 
<J 
b 
<D 
<3 
cfe 
_»^ 
<3 
b 
. 0 
<\ 
h 3 
<J 
3 
< 
o 
^ 
b b 
3 ^ 
A } *^D 
b 
o 
/ I D 
b ^ 3b 
/"O 
CD 
o o 
yg 
b b b 
/'O o "15 
b b 
/T5 
o o O o 
0 
0 
£ 
£ 
£ 
A 
z 
 
1 
b 
t 
 
> 
<3 
$ 
A 
b 
b ^ 
• 5 
b 
so 
b. 
a 
> 
1 
>*> 
• 0 
>• 
 
1. 
1 
-p 
-P 
1 
< 
b 
-^0 
b 
b 
1 
The remaining derivatives are as follows : 
CW V T 
a c o s i q W C * V T g ~ , ( V 
• a . 
0~ o* '<T 
tO- 2 VT s i n YTff_ 
Vl-«5 
ein'j(rT -co-e r ) rT c 
V 1 - e; 
e s i n 0 T 
bcxv [c oos i^. oos (^YT-Xy 
ii urn••»••• PW" • » w » i i j > imimmnvmmmwmmmmmmmmm 
s i n 2 ( Y T - O L - 6 , . ) 
o1 U V 
e„ 
r r V T s i n YT 
ft o ^cr 
1 - e 
cr 
1 
e 2 s i n 8 T 
(J 'CT 
( 1 4- arr - a-, e - ) CT a <T 
?W\/ T<r _ oos2 (c*vT - A T ) s i n Vr t a n ( YT -6O_~0 T ) 
' a c 'a Tcr 
doo a 
oos i^- oos (ot\zT -^W) 
M» wmmatmmmmmmi •* timmm** %n I I I I I i i i i — i n m i H n n * * 
s i n 2 ( YT - CO - 0 . ) 
&*tffc 
d»a, 
J t T 
b a 
a f£ - 8 % a c r + 3 4 + 2 4 <1~*c?> 
8 a 5 
3/2 
s i n 
V ^ ^ - r ^ a ^ l - e ^ ) 
JL 
2 cr "or 
acJ 
3/2 
V (a^ e } 2 - (a_.- r ) 2 
f% v ( 2 v 5/2 
s e <J V4^afV^fa-^V1- # 
1/2 (fW) g Tqr. 
^%(jQ(J)2 - ( V - * ! ^

The influence of launching errors on the trajectory of space probes

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http://oa.upm.es/2160/2/LINAN_MONO_1964_01a.pdf

The influence of launching errors on the trajectory of space probes

Abstract

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