The influence of launching errors on the trajectory of space probes

I N S T I T U T O N A C I O N A L DE T E C N I C A A E R O E S P A C I A L " E S T E B A N T E R R A D A S "

THE INFLUENCE OF LAUNCHING ERRORS

ON THE TRAJECTORY OF SPACE PKUtfiiS

INTA REPORT I . C . I

APPENDIX 7.III by

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 K m ) .

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 » ^ i n ' ^in' A ti n

the six orbital parameters in a geooentrio equatorial

INJECTION AND GEOCENTRIC EQUATORIAL ORBITAL P A R A M E T E R S

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 7

2

//u

e

)

t -

*-r

1

A©

r

2

V

2

o o s

2

r \

i•••••• in » — — • » • • • • • • • # • » • > • — m m m f ^ 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

o o s

2

T E . J

7

*

with E = V

2

/2 - / ^ / r ( t o t a l energy)

B. Hyperbolio orbit

,2

= r / ( r V

2

^ - 2)

f

* . = V

1

t - t

p p

/ 2 V

2

^ f r ¥ ooa

2

T ^

" V

r

"

/*•

A

/*••

/

= _J-« f 2E r

2

4.2>u. . r ^ c ^ o o s

2

! J

-2E

A

LA • ' • '

A

© 1

V

2

^

n

The second t h r o e g r b i t a l parameters are dependent on t h e

poordinate system, they are given "by t

i = aro oos ( s i n AJ[n oos 0 , )

>iV = o <i n - arc t a n tan A' -„ sin a.

(X^ » arc oot j^oos A»i n 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 V

2

-2fa)

c ( 4- f o r e l l i p s e , - for hyperbola)

= 4. — -2 v cV

( rV-2^)

j r o o a f Y f_ _2 n .

bY

2 r V 003 Y / r V

f"r

_{© °<} e

-

_©

r V s i n T 003 Y

i % &c*

- 1

2 r Vf

"FT

- 2

For elliptio orbits

£ * - « 3

^a Jc - *

d r r r Y -2 LU

©

r V s i n Y [ r V2oos2Y4- f^j r V2^ | 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^") 2 r ^ s i n 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 r V2c o s2f , .

- 1 1 / 1 1 \

e _©

ft

_©

dt

P* _ 1

2 v s i n \ ' rV 1

4-(Ub e - f r Y2o o s2Y - /C^t I 2 e2

2

-ft

© - 2

r V2o o s2 /

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) ( B24 4 E2r2s i n2r ) - 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) r2V oos Y s i n V

SY _{(2E) w}1/2 _{^ (Us 4-2Er 4-B) B}

Where E = L - f a n d B = y 4 E2r24 4 E p r - 2 E r2Y2o o s2Y '

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 s2 ^i n s i n 2 Ai n s i n i

U

Oi^

T

oos A !n oos ^ n

oos ^l n ain &la o.oa A^

. 2 . s i n i ^

^ « s i n

i n

u

_{i n} ?

s i n i

J o i n s i n " i oos A! _m

OcO^ oos 0i n3 i n 0i n s i n Aj[n

d A! n a l n 2 ^

77"

v

2

oos

2

r

. 2 . . . 2

Ji+ - W 2 - - 1V1- afasfiQl

y e £ - C^oos2Y-^2( e,

1 2

1 / rV

V

n

_& i\

2 rV o o s2 V

V ^2e 2 - ( r V2o o s2r - ^2 [

1 , rv"

-l) ( 1. rV2oos2f \ 1

Hi

©

/ i

C J C L ^ _{r V2 3in 2 V}

dr

_Vp

ffi5

v - 1

2e2- ( r V2o o s2Y - ^ )2 I

1 / r V A/'

14. . _ 2! 1 -2s2 \ /X A

rV "oos^Y • I I I "1

<3

.<5

<3

.5

0

KD

ti

qK> c 3*

3* X o

c

of

c

r o

4*r

a> 3

>•

M

1^3 ro c

4 >

_oh

sK

ro r^>

m

s

o o 3

<

<3

«*

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 . t o , 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 i n 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

GEOCENTRIC E C L I P T I C ORBITAL A N D TRANSITION PARAMETERS

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

A a ^

A e< *

At

rot

A ' ^

A ^

Au^

or, symb olio a l l y •

A

i

—

w..

e

A*

with the following expressions for the p a r t i a l d e r i v a t i v e s :

£1.

S i

di

= oos (t^U -U/£)

c*.

DA,

£_ =

s i n £. ain i - sin XI s i n i ,

cos g^cqs i^- ^ s i l ^ + 5 in £. s jn i*- cosily .cos (lu^-o^) cos i f

d-Q* = sin 1^ . ain - O ^

sin ±sr 4 ain - O .

dn,

ot oos t + ooa JX. .sin £ ,ootg i

|_oos£ .ooa i^ 4- sin £ « sin i^ . ooa -T2.1-oos (0-^ -<-^£)oosi

s i n i , . s i n (UJ - L0_ )

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

rt r = constant (/v 900000 Km)

6

oos e

t

ac

(el

- 1) 1

t a n ( Y+-, - C JP - © . , )

«*W\r

e

) "

h ~ \

₎

t r . t r .

COS (M;

>os Y _{t r .}

__ £^£

ft

a6(

rtr_VtrP

- i)

V

£ ~ ' \

2E

Y ^ t r

' *

2

{%

r

t r "

j 2

'

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

u> d <\ O) <U < - P <3 U J

. -_i

< uo

3

<J

031 SO UOl 4 } 3

/ o o

a) ^ to

o o o

llA uo

so o Uj -4-> ^ o ^o

•ir IJD

o

~ p _to U3

UJ

• O ' - O

o

o

o

/ ^ O

U_>

C5

• " O

tO u> U3 U3 Q_ 4J> A3 uo a) SO a)

so ''O

UJ t o

o

«o t o

lU

^ 5 / * 0 /"O

CD

• - 0

u ) UJ V . -f* -^o to AD

.*o t~ a>

E x p r e s s i o n s of t h e 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 s2 ( o ;g4 : 9t r ) e^. r t r s i n 6

•**<m«iM»*w*«w

t re

^dt r£ _ oos i& . o o s 2 ^ ^ ^ - ^ ) a'g ( e | - 1) 4 - r t

K o o s2 (C0£4. 0t r ) r e 2 s i n 0 ^

c 5 C £

6 . . . 2

5 i &

d

v

& c" &

ti**

t

bo

t

•cr£ _

^ e

a i

e

oos ±c • oos ( c * ^ - X I )

o o s2 (CU£~ © ^ -)

s i n i . oos (aJL + 0 . ^ )

oos ^

s i n i£ • oos (a)£4. Q . ^ )

oos ^

oos i c . s i n (cUc4« 0+_, )

c 6 u r5

003 <L

'£

•

•

c £

i . - . |

rx_{•to c t r} e s i n 0J_ £ fc &

a e ( e | - 1) 4 . r t re

bS

t a i n i £ . 003 ( o Je+ 6t l, )

5

CU, _oos

•br.

3v

_•fcr

3a,

2 Vt r a |

a

A,

oos

2

[ 2 ^ ( t

t %

) - A

£

- H

g

]

a5 ooa i . o o s ^ r ^ - ^ E - © t r J

•£ ' — V « 6 E - ^

•i-i

e.

_E

_{f r}

_^

s i n

in e

_{t r _}

3 a£4 . r t r & - a£ e£

sin Y

_txv 2% * rt r ,

e

J

2 7T

4. _ <3"Hrc

3 6 5 ' 2 5

a A,

(D Or

o o s2[ ^ ( t t r£ ) - A ~ ^ £ ]

o o s ig . oos ( Yt r- ^ 0 r * " ^-br )

£L

a

&

(eg 4- 1) *r

_{t n}

e6 rt r£

£ .

a i n ©. _•fexv

a

,-sin Y.

_{tr_ rt r&<2 a£ * r t ^ % ~ 1 )}

2 71

dt

_{t »} • » I ! • • • • • • S, 365*25 _!C,

U<

_{2 71}

it.

*

ix

&

5OJ£

o o s2 jU

c

o o s2[ 2e( tt r £) r A e - ^ 6 1

003 i . • oos ( T ^ -GJ>£- _^

C) 3£ _ sin i£ . sin (Y^ - a>6~ 0,^ )

OOc 0 0 3 0 ,

3 ^ " r*rB-ae4 1 r^ re( e2- D e g - 1 1

sin Yt r f 2e£ ~ rt e rt sin 9^

e t

c

e £

C>C?6 sin 16 .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 r1- ir *••• nm*——****—**—*

mm3mm*mqmimmm»m**ammmm—mmm-sin Ytr |f rt "(2a£ 4- rt ) (e|-1) sin 0t e | r ^

5o£ oos i& . oos ( Y-tr£-<%i-6fec. )

^ ip oos 0^

Id

6 a±n ±t . s i n ( T-tj- - < ^ >e- © t r£)

otu

0 0 3 _&

3tt3

1 r„ 2

da,

_Vf*

2 ,Ji

r

£§*_!l

£

****"

3 a

l

(e

e -

1)

a

e 4

4- 2 r+^ a

_{ee £}

2i a2 ( e2 - 1 )

3

al/*

I

a 4- r . „ 4- U r+ T, 4- 2i>„ a - a2 ( e2- l )

a

&

e

e

« 1/2

V

^ £ * 2 ^ r £ a e a'_{6 &} 2 ( e2- 1 )

dt*

_or,, c

- 1

V^e

e

£

a

l

V

_iv

bxv^ txv. "£

z r

a - - a 2 ( e2- l )

4- a 3/2

te€4- a£

5* Relationships• ^betweenr_the t r a n s i t i o n conditione wiifcBi^ -respect

^A^j1 9. ffajfth. a n d "b^ie inJestAPAA°PAA^o n s, ^n ^^.e h e l i o c e n t r i c

or "bit.

These injeetion conditions are the following in a h e l i o c e n t r i c i n e r t i a l reference system:

r " $ » at " V " . f « i " t

t r ^ tr 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

" \ "

<5

C

" 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> rt r 2 sin 2 dt r

tan 0 = -* 2 27 "* ~ ^

tr r 4- r+T

_s

, cos o. + 2r r. cos0. cosi o< -> (t. )

x r e

®

T r

6

T r

e «•

t r 5

«-©

T r

e

r

^ - ^ + 2Vt r Te " a S£ cos Ae * 4

SenY = (V,.)""1-^/ c o s c ? c o s d sinFX + ^ + r - X ( t+ r )1

i\

•+ Vt r sin 4 sin 5 4- V cos dt r s i n | »tr ~ ^ J " ^ J I i

c CT O" <- CT © C - |

oaU (Y! c o B p - ^ cos ^ o o s f c i ^ - 2 ^ ^ ) ] + _^

4 Vt r c o s * , c o s $ c o s f c * + 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 t e r m s of o r d e r |_ r ^ / r J have b e e n :

o« = Z ( t v ) •«. ~ ± = e c o s

£

t r

s i n ! * - J (t ) ]

'© tr* (V =-=tr<£ s i n 6

t r

© t r ?

r , = r 4. r+ c o s CL s i n ; ^ - 7 ( \ „ ) ]

~r£ t rF u b Xg —© t r F j

' t C r ©

2 2 *\ \ Y. = V + 2Y. Y^ c o s <) c o s A 4- Y.

_{I e}

t r ® & t r

t r _ t r i

-er £

s i n T

CT

V

tr

ccsd'

c

s,nA

e

H'

:i

t.r

ff

-ltr

e

)l(Va + Vtr

c

°

5

4'

: c

'

; i

Vt

+

4r

a ? m

^

V

tr

V

I

COS u

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 g i v e n "by 5. 1

w i t h t h e f o l l o w i n g e x p r e s s i o n s f o r t h e d e r i v a t i v e s

7> _c>r

__iE£ _ c o s tf # s i n , 0' _ £ (t+ T, )

t i y

r

r . cos 0 . cos

t rc t r ^T€-E^Sre\

3r

_{iror _}

-DcT t r .

r . s i n s m £>

t r .

irtf=

t r .

= - rt r . cos 6^ . cos i o t - £ ( t )

"| 2 j n „

o <_*> 75 <3 & <3 1 0 t O

<1

.b V u ^ D , « + J - f J

^ 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 4 J

<1 <3

v ° V - COS < ^ , „ f r* / j . \

9 r . *"- L £ '© t r . £ J

ire =

0

CA 'tre

TJ2L COS

j

* *

cos OL _-tr. - Z ( t _ ) _{© t r£}

_ t r s : = _ ^ s i n £ s i n oc - I ( *t r )

tr_ t r & £

Qdf t rc

T j tt r , "365T2 5

1 £E£ COS <y .„ . cos

_ t r tr~

D5

_tK S

'(Tr _•br* v_ _ 5i2Ji±£e

' *

2 £ t e t =—•-&&

c o s

£

?J

_H

t r .

2

v

i

3v.

t r

VA cos o* cosA_ J- V

_® £ 1 i r

Vt rV3 > • < ? A _ ^ X £

V,

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 - Y

T

oos

V

^ f e ^ I

c o s i oo s /V t > 7 j "" I

j.__tr£.J

3a. Y c o s V* s i n 1

v*.. c o s i cos A l-n^a: I - v cosi s i n r J - ^ - 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 Ar- M %fM I

x Vt r cose,. s m ^ + Vt r cos^, ^ i ^ © ^ ^ j

i ruVt\ - V cost, s i n } , 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 £ < ; tt r £ / " ' ^ ^

" " ^ " r i-n *" -^"%-i^A - VT c o s i s i n f U f r ' (

SLLSL. SK

d a + ^ . _^S- '"" +

Y (Y 4 V+ r cos J c o s A U — S s ^ K s i n J . YW ( 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 Vt r cose) cos A ) ( ^ rfi ^ 4 s i n ^ Vt r ( ^ - t a £ L )

® " £££ ^ -h

I

c08

T<r

^ f

~ £ - f e = - ( YT c o s Y ) ~ ¥ s i n T ( ~1- ) — cos a s i n

X .

t r

- COS A . COS

"([•

Cx.

-tr. '3 t r

(W)|

£^ j ' tr«_

- J .

s m

£ t

_^

U A

-•(Vj c o s p ^ l s i n ^ ^ ) ^ ^ cos^

• V

tr COB<T£

sin A

f

r ^ ^ - K t ^ ) ]

c o s

A,

t ) 1

-if

~ f - = - ( YI c o s T ^ s i n ) ; < § £ - ) f Y ^ s i n 0^ s i n Xf +

V°E + Yt r s i n J£ c o s A£j ar- Z ^ ( tt r £) 1 + Yt r J "t r cos ^

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- tpC T V A x Var

and the injection conditions into the orbit.

r t r<r ' t rcr > °*tr<r ' VI ' Xr » V ' W

These r e l a t i o n s a r e as follows

ft

_e

8

<r

f 2 VZ 2 ^ v2 C 0 S2 ^

-I - ( 7 ) ( — S I - . - - )

^ P

0

P©

t 2 ,r2 ___2

2

/

V ^

(

/ ^

2 r

t L

v

i

c 0 S

% V ^

V ^o

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 = lX

t r a ; ( r

!l&

t

_r<j

Whore the matrix |rT. ^Jis given by 6.1 with the

following expressions for the derivatives.

^

a

<r

2

P ®

5 rt^ (rtrff ^ - 2 ^ )

b aa _ 2 p/e Vl rt r c r

"^7 ""(^tr^?"

1 2

r

o

^

S ea _ VX2 cos2 V ( r-t^Yi' } ^ )

m w i , w »w •

^=t

£*

< <J

> <

b

^O

o O _{o o}

b b

A 3

o

/T5

O

« b u u _

> /-£> b A-••-> * 0 > /•o

o O O

O

o

o

^1

b

/-O ' O

<P b AD b

3

b

B

/ ^ b

3

C b ' 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

hea rt r ~ VI s i n ^ -0 0 s 1'er , rt r v? ^

—r-~— » —--*•—. and —-r are vfchose given in matrix

°Ttr<J dVj ^ r ^

2.1 when tpo. , ri n, Y±n , fin e^ , a^ and ^ are

substituted by tpc> , rt 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^ , ri n , Vin , Yin > ^ » a n ? a*1^ / ^ a r G substituted

by a)a, rt 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 ia - oos2 S.

7 . 1 . Relationships between the e r r o r s i n the p o s i t i o n and velooity of the probe at a r r i v a l and the e r r o r s i n the heUooentrio orbit parameters(vith r-j- fixed)

We give below the r e l a t i o n between the p r o b e ' s p o s i t i o n and velooity at the monent of a r r i v a l at the t a r g e t j

and the h e l i o o e n t r i o o r b i t a l parameters :

These r e l a t i o n s are as follows : s i n 6T = s i n i g . s i n ( COa+ 0-r )

t a n (otT - £l„) = oos i - t a n ( cOa 4- 9T )

a<T ^

r = . constant

oos 9 =

&0> ( 1

"

9

^

}

_ j .

-t a n (&l-t;*v - X I ) = oos i ^ o t g (Yl ~ O J - 6T )

VT- ff a *cr ** '<r

'cr

s i n <L = s i n x _ . c o s ( YT - W - 6T )

VT ° »o~ 'cr

'a-1 tT — tp - ——

•a »a 2E r 4-Z u» r._ — 0 *• — •—» a x i l ; -yamj yxauaienat* t

ro T<r y ^ Z E <- V V 7t / 4*2EJ ^

where

E = -

*- VvfV

₁

- ^

2a Y a \ ©

<7

The e r r o r s i n t h e p o s i t i o n and v e l o o i t y of t h e probe a t

a r r i v a l ( w i t h TT c o n s t a n t ) a r e g i v e n i n t e r m s of t h e e r r o r s i n

t h e h e l i o o e n t r i o o r b i t a l p a r a m e t e r s by

where t h e m a t r i x I u i j - , T<J|l s g i v e n by 8 . 1 , w i t h t h e e x p r e s s i o n s

g i v e n below f o r t h e 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*- , JT o ( 1dT ( r, Vj ana Oy

are t h e same a s t h o s e given i n m a t r i x 4 . 1 i f we s u b s t i t u t e

rt r6» ^ t r g . ^ t r g . V , fc and a£ f eg , t pe , i£ , J ^ , U ^

"by ,

X V Cs . . . .

p

and.the heliooentrio orbit being eliptio, the oombination e - 1

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 ^

3

b

/"O CD

o o

yg

b b b

/'O o "15

b b

/ T 5

o o O _o

0 0

£

£

£

A

z

<

b

ft) AD b b > 1 b t < b b 0 b

i

0 > > <3 $

A

b b ^

• 5

b so b. a >

1

>*> • 0 >• < b b b > 1. 1 -p - P 1 < b

- ^ 0

b

b

The remaining derivatives are as follows :

CW V _T

a c o s iqWC* V T g ~, (V

• a .

0~ o* '<T

tO- 2 VT s i n YTff_

V l - « 5

ein'

j

(r

T

-co-e

r

) r

T

c

V

1 - e; e s i n 0T

bcxv

[c oos i^. oos (^Y_{ii urn••»••• PW" • » w » i i j > imimmnvmmmwmmmmmmmmm} T-Xy

s i n2( YT- O L - 6 , . )

o1 UV

e„

rrVT s i n YT

ft

o ^cr 1 - e

cr

1

e2 s i n 8T

(J 'CT

( 1 4- arr - a-, e - )

CT a< T

?W\/

T<r _ oos

2 (c*vT - AT) s i n Vr t a n ( YT - 6 O _ ~ 0T )

' 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 n2 ( YT - CO - 0 . )

&*tffc

d»a,

J t T

b a

a

f£ -

8

%

a

c r

+ 3

4

+ 2

4 <

1

~*c?>

8 a5

3/2

s i n

V ^ ^ - r ^ a ^ l - e ^ )

JL

2

a

cJ

3/2

V

(a^ e }2 - ( a _ . - r )2

f% v

( 2

v 5

/2

s

e _<J

V4^afV^f

a

-^V

1

-

_#

1/2

(fW)

g Tqr.