Medidas de resguardo para los propietarios

H a v in g d e te r m in e d th e o rb its a n d fo rm a tio n c o n fig u ra tio n fo r th e sy ste m , th e re la tiv e p o s itio n a n d m o tio n b e tw e e n th e re c e iv in g a n d tra n s m ittin g s a te llite s is n e c e ssa ry to d e te rm in e th e tim e w h e n th e s a te llite s a re in a c o n fig u ra tio n fo r im a g in g .

T o o b ta in th e re la tiv e m o tio n b e tw e e n th e s a te llite s , th e H ill’s e q u a tio n s (C lo h e ssy - W ilts h ire e q u a tio n s ) a re u se d . T h e s o lu tio n to th e s e e q u a tio n s re la te s th e m o tio n a n d p o s itio n o f o n e s p a c e c ra ft, n a m e d th e c h a s e v e h ic le , w ith re s p e c t to th e ta rg e t v e h icle. F o r th e p u rp o s e s o f th is stu d y , th e ta rg e t v e h ic le w ill b e th e tra n s m ittin g s a te llite an d th e c h a s e v e h ic le w ill b e th e re c e iv in g s a te llite . T h e c o o rd in a te sy ste m s fo r th is sy ste m is b a s e d o n a lo c a l X Y fra m e c e n te re d o n th e ta rg e t s a te llite w ith th e x d ire c tio n b e in g th e ra d ia l, y d ire c tio n b e in g ta n g e n tia l to th e o rb it, a n d th e z d ire c tio n d e fin e d by th e rig h t- h a n d ru le , o u t o f th e X Y p la n e . T h e s e e q u a tio n s p ro v id e an a n a ly s is o f th e sy ste m b a se d on th e a s s u m p tio n o f a p e rfe c tly s p h e ric a l E a rth w ith no p e rtu rb a tio n s o r d rag .

T h e o rb it o f th e ta rg e t v e h ic le in th e in e rtia l fra m e is a ra d iu s R 0, w h ic h is c o n s ta n t d u e to th e c irc u la r o rb it. T h e o rb ita l ra te o f th e s p a c e c ra ft, m e a n m o tio n , is d e fin e d by:

* =

t e/r 3

M i

w h e re j i E is th e g ra v ita tio n a l c o n s ta n t o f th e E a rth . T h is is th e a n g u la r v e lo c ity o f th e c o o rd in a te s y s te m c e n te re d o n th e v e h ic le , w ith re s p e c t to th e in e rtia l E a rth -c e n te re d fra m e .

T h e re fo re , th e p o s itio n v e c to rs o f th e tw o sp a c e c ra ft d e fin e d in th e in e rtia l s y s te m are:

Rt

=

Rj

, , , [4-2]

R c = ( R a + x ) i + y j + z k

w h e re x, y a n d z a re th e c o o rd in a te p o s itio n o f th e c h a s e v e h ic le w ith re s p e c t to th e ta rg e t v e h ic le , a n d th e s u b s c rip ts t a n d c re la te to th e ta rg e t a n d c h a s e v e h ic le s , re sp e c tiv e ly .

T h e in e rtia l a c c e le ra tio n o f th e c h a s e v e h ic le is th e n f o u n d to be:

R c = [ x ~ 2 n y - n 2 (R 0 + x )]i + ( y + r n x - n 2y ) j + z k [4.3] T h e g ra v ita tio n a l a c c e le ra tio n o f th e c h a s e v e h ic le in th e in e rtia l fra m e is g iv e n by:

i =

j?c = _MA±vi±»±i*l

[ 4 .4 ]

R ° [( R 0 + x

T h is e q u a tio n c a n b e s im p lifie d f o r sm a ll re la tiv e m o tio n s , w h e re x2+ y2+ z2~ 0 , a n d th e g e n e ra liz e d b in o m ia l th e o re m , n e g le c tin g s e c o n d -o rd e r te rm s. T h e a c c e le ra tio n is th e n c o n sid e re d : g « - n 2[ ( R 0 - 2 x ) i + y ] + z k [4.5] In th is re d u c e d fo rm , th e tw o a c c e le ra tio n e q u a tio n s fo r th e c h a s e v e h ic le ca n b e re s o lv e d in to c o m p o n e n t fo rm : K = 8 ( x - n 2R a - 2 n y - n 2x ) i = ( R o - 2 x ) i ( y + 2 n x — n 2y ) j = —n 2y j /V - /N z k - —n z k

T h e fin a l H ill’s e q u a tio n s a re f o u n d b y s o lv in g fo r e a c h c o o rd in a te d ire c tio n :

x - 2 n y - 3 n 2x — 0

y + 2 n x = 0 [4.7]

z + n 2 z = 0

F ro m th e se g e n e ra l e q u a tio n s , th e u p d a te m a trix ca n b e fo u n d r e la tiv e to tim e a n d th e p re v io u s p o s itio n a n d v e lo c ity . T h e u p d a te m a trix o f th e in -p la n e m o tio n is d e te rm in e d to be: x ( t ) 4 — 3 c o s ( n t ) 0 s in ( n t ) l n 2(1 —c o s ( n t ) ) l n _{X o} y ( t) 6sin(rc0 - 6n t 1 2(—1 + c o s ( n t j ) ! n t 4 s i n ( n t ) / n — 3 t _y

₀

x ( t ) 3 n s \ x i ( n t ) 0 c o s ( n t ) 2sin ( n t ) _{X o} _T(0 6 n ( —1 + co s ( n t ) ) 0 — 2sin ( n t ) - 3 + 4 c o s ( n t ) _{j o} w h e re x o To K y 0

is th e m a trix o f th e p re v io u s p o s itio n a n d v e lo c ity .

[4.8]

A s c a n be s e e n f r o m th e H ill’s e q u a tio n s , th e o u t-o f-p la n e m o tio n , z , is d e c o u p le d w ith th e in -p la n e m o tio n , x a n d y.

T h e re fo re , th e s o lu tio n o f o u t- o f-p la n e m o tio n is:

~ z ( t ) c o s ( n t ) s i n ( n r ) / n ~z 0 z ( t ) — n s i n ( n t) c o s ( n t ) _{j o}

[4.9]

w h e re is a g a in th e p re v io u s p o s itio n a n d v e lo c ity .

S p e c ific to th is p ro je c t, th e d e c o u p le d n a tu r e o f th e in -p la n e to o u t-o f-p la n e m o tio n g re a tly s im p lifie s th e p ro b le m o f d e te r m in in g th e re la tiv e p o s itio n o f th e tra n s m ittin g s a te llite to th e r e c e iv in g sa te llite . T h e o rb its o f b o th s a te llite s are c irc u la r o rb its w ith a ra d iu s o f 7 0 7 8 k m . T h e o n ly v a ria tio n b e tw e e n th e tw o o rb its is th e in c lin a tio n o f

a p p ro x im a te 0.5° o f th e tra n s m ittin g s a te llite ’s o rb it. T h is, th e re fo re , g iv e n u s in itia l c o n d itio n s in th e in -p la n e m o tio n o f a z e ro m a trix , s ta tin g th a t th e re la tiv e to th e re c e iv in g s a te llite , th e in -p la n e m o tio n o f th e tra n s m ittin g s a te llite is id e n tic a l. T h e o n ly re la tiv e m o tio n o f th e o n e s a te llite to th e o th e r is o u t o f p la n e .

T h e o u t-o f-p la n e m o tio n o f th e tra n s m ittin g s a te llite w ith re s p e c t to th e re c e iv in g s a te llite in th e tim e d o m a in can b e s im p lifie d to:

z ( t ) = A. cos(/zr - (b)

\ v [4.10]

z ( t ) — A s m ( n t - ( j ) )

w h e re A n a n d A are c o e ffic ie n ts o f th e sy ste m , b a s e d u p o n th e m a x im u m d is ta n c e b e tw e e n th e tra n s m ittin g a n d r e c e iv in g s a te llite s d u rin g th e o rb it (A n= A x 11) a n d (j) is th e p h a s e d iffe re n c e o f th e s e p a ra tio n o f th e s a te llite s a t s ta rt tim e .

W h e n th e s e e q u a tio n s a re a p p lie d to th e p r o p o s e d c o n s te lla tio n , it is p o s s ib le to d e te rm in e th e re la tiv e p o s itio n a n d v e lo c ity . T h e a s s u m p tio n s u s e d to sim p lify th e e q u a tio n s a re a p e r fe c tly s p h e ric a l E a rth a n d n o p e r tu r b in g fo rc e s . T h is sim u la tio n w a s ru n , s ta rtin g a t th e c ro s s in g o f th e e q u a to ria l p la n e . In th is c a se , th e c ro s s in g o f th e e q u a to ria l p la n e c o in c id e s w ith th e in te rs e c tio n o f th e tr a n s m itte r a n d re c e iv e r o rb it. T h e re la tiv e p o s itio n o f th e s a te llite s is sh o w n by:

F ig u re 4 .1 2 R e la tiv e P o s itio n

T h e r e la tiv e p o s itio n b e tw e e n th e s a te llite s m u s t b e g re a te r th a n 6 0 k m f o r im a g in g to o c c u r. T h is s e p a ra tio n ta k e s p la c e a t tw o in te rv a ls : 9 9 0 -2 0 2 0 se c o n d s a n d 4 0 2 0 -5 0 5 0 s e c o n d s f r o m th e s ta rt o f th e s im u la tio n . In th e c o n fig u ra tio n o f th e sim u la tio n , th is w o u ld b e m a d e to c o in c id e w ith th e p o la r re g io n s o f th e ea rth .

T h e r e la tiv e v e lo c ity is:

F ig u re 4 .1 3 R e la tiv e V e lo c ity

A s c a n b e se e n fro m th e s e g ra p h s, as th e fo rm a tio n a p p ro a c h e s th e m a x im u m se p a ra tio n th e re la tiv e v e lo c ity slo w s to z e ro . N e e d in g a p p ro x im a te ly 4 s e c o n d s fo r im ag in g , it is o p tim u m , fo r im a g e q u a lity , to im a g e w h e n th e re la tiv e v e lo c itie s a re slo w e st. T h is o cc u rs a t 1 5 1 0 -1 5 2 5 a n d 4 5 4 0 -1 4 5 5 , w h e re th e re la tiv e v e lo c ity is 25 c m /s g o in g to ze ro an d b a c k to 25 c m /s b y th e e n d o f th e in te rv a l. A g a in , th is c o rre s p o n d s to th e P o la r R e g io n s.

In document Tesis para optar al grado de Licenciado en Ciencias Jurídicas y Sociales CARLOS MAXIMILIANO FISCHER GILSON (página 76-79)