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Recurrence relations of the inclination function
C h i n e s e Astronomy
4 (1980) 17-24
Pergamon P r e s s . P r i n t e d i n Great B r i t a i n
Acta Astr. Sinica 20 (1979) 17-24
RECURRENCE
0146-6364/80/0301-0017-$07.50/0
RELATIOk',S OF THE INCLINATIOI~!
T o n g Fu
I~.ru L i a n - d a
FUNCTION
Wang C h a n g - b i n
Purple Mountain Observatory, Academia Sinica (Received 1978 March 28)
ABSTRACT In t h i s p a p e r , t h e i n c l i n a t i o n f u n c t i o n i s e x p r e s s e d i n t e r m s o f t h e h y p e r g e o m e t r i c f u n c t i o n and t h i s has r e s u l t e d i n some s i m p l e r e c u r r e n c e r e l a t i o n s i n v o l v i n g no more t h a n t h r e e n e i g h b o u r i n g f u n c t i o n s . In d i s c u s s i n g t h e p e r t u r b a t i o n s it
by t h e E a r t h ' s g r a v i t a t i o n a l
f i e l d on a r t i f i c i a l
satellites,
i s o f t e n n e c e s s a r y t o d e v e l o p t h e p e r t u r b i n g f u n c t i o n as a P o i s s o n s e r i e s e x p r e s s e d i n t h e
orbital
elements of the satellite.
trigonometric functions.
Then, t h e o r b i t a l
More s p e c i f i c a l l y ,
inclination
I a p p e a r s as an argument i n
we have t o c o n s i d e r t h e f o l l o w i n g d e v e l o p m e n t : !
Pt.(~n~)e i'~ ~ % Fj.ect(z-ae)"
(1)
p~o
where
Plm(Z) i s
t h e a s s o c i a t e d Legendre p o l y n o m i a l , u i s t h e l a t i t u d e geocentric latitude
and X = a - £.
argument o f t h e
satellite,
~ is its
functions,
and t h e y can be e x p r e s s e d as p o l y n o m i a l s o f t r i g o n o m e t r i c f u n c t i o n s o f I .
have b e e n d i s c u s s e d by a number o f a u t h o r s Iszak
[7] and B. J e f f r e y s
[1-11].
The Flrnp i n (1) a r e t h e i n c l i n a t i o n
[8] o b t a i n e d t h e f o l l o w i n g r e s u l t
i l - " ( l 4- m ) ! F~,,,e == 2~p!(l - - p ) !
They
Using t h e method o£ t r a n s f o r m a t i o n group, i n t h e form o f a s i n g l e summation,
~i f l"~ir,21-~prw ,.3t-,,.-2p-~i~,,,-t+2p+~i . ,,---J ~i "l---i-
(2)
where
i = W--L c ~
c o s -I- , 2 C~ ~
s~
sin -~-I, 2
m!
.! (m -- .)!
max{O,l--m--2p} ~j~min{21--2p, The r e s u l t s
l--m}.
o b t a i n e d by o t h e r a u t h o r s a r e more c o m p l i c a t e d , f o r example, t h a t o f C h a l l e
a d o u b l e summation, w h i l e t h a t o f Kaula [12] i s a t r i p l e Besides the expressions for the inclination
summation.
f u n c t i o n s , many a u t h o r s have a l s o
[3] i s
18
Inclination Function
discussed
recurrence
Giacaglia
[11] d i s c u s s e d
apart
relations
f r o m some e r r o r s
neighbouring In this
hypergeometric
( i n some c a s e s
we p r o p o s e
functions relations
Giacaglia's
[5,
13,
recurrence
a s many a s 12) f o r
a method of expressing
10,
when b o t h l
14].
Recently,
andm vary.
relations
involve
them to be convenient
the inclination
However,
function
t o o many
in use. in terms of the
function. 1.
Re-writing
recurrence
in the derivation,
functions paper,
among s u c c e s s i v e
in detail
INCLINATION FUNCTION EXPRESSED AS HYPERGEOMETRIC FUNCTION
(2),
and replacing
0 2 b y (1 + c o s 2 ) / 2 ,
;'--(t+~),.
(l+co,
a n d s 2 b y (1 - c o s I ) / 2 ,
l~-I-m
Ftmp =
21_=p!(l
-
p)!
-
O
I-m-2P
(1-cos1)
'
'
" n - ' l ' ,r. llp- m - i (, -1 + c ° s I ) ' l - i P - i ( 1 - - c ° ' l ) =p-u-=-i). .¢". - - -l ,"-l i g - I - m C~.;
~
we h a v e
(s)
i or
1 +ra
_
FIrap
;'-~(1 + m ) ! (1 + co, Z) ~ - v - ( 1 -- P)I
2atp!(l
• (_
l --ra_p
~o~X)
1)t_~
1 (1 - - m ) !
d l-~ d c o s l 1-* (I +
co, I)'/-"(l
-- cos/)".
(4)
Using the Jacobi polynomials in differential form, [15],
/~: B)(~) = F ( =
, . ~ + , . 8; x)
r(~) r @ + ~)
a._~ ~ [x~+.-,O (x1-B(1- xY-O ax"
_
xy+.-B].
(5)
and applying the substitution 1 --
COS I
x = - - 2
(6)
we h a v e F(--
n, a + n,/~; d~ d cos I ~
1 -- cos/)
=
r@) (1 - - co, O ' - " ( 1 + co, O a-2"F(n + #)
(_1).
2
i(1 -- cosI)a+'-t(1
c o s I ) "+'-a]
+
(7)
Let
n~l--m,~m+ we t h e n
2p+
1 --l,c~2m+
1,
have
F(ra-- I, I-4- m-4- 1, m 4- 2 p 4 - I --I; I -- cos/) 2 ffi ( - - 0 1 - "
r ( m + 2 p + 1 - - 1) ( 1 - - c o , 0 1 - ' - " Q 2t-~F(2p + 1)
+
cos/y'-'-"
dl-m
• dqo "P-'~ [ ( 1 - - c o s l ) ' P ( 1
+
cosl)at-'P].
(8)
Inclination Function
Substituting
19
(8) i n ( 4 ) , we have l+m_
r l . , = i ' - - (t + m)l ( 2 e ) t (1 -
•
inclination. properties
+ ~,z)
2'+'9!(l--p)l(l--m)t(m+2p--l)l F ( m - - l , t + m + 1, m + 2 p + 1--1;
E x p r e s s i o n (9) shows t h a t t h e i n c l i n a t i o n and a h y p e r g e o m e t r i c f u n c t i o n ,
¢o,Z)'-~7(1
'
'
1--eosI)
2
(9)
f u n c t i o n can be e x p r e s s e d i n t e r m s o f a p o l y n o m i a l
i n which t h e arguments a r e t h e s i n e and c o s i n e o f t h e h a l f
T h i s i s an i m p o r t a n t c o n c l u s i o n , f o r i t means t h a t we can now make u s e o f t h e of the hypergeometric function, in particular,
we can use t h e r e c u r r e n c e r e l a t i o n s
of the hypergeometric function to derive the recurrence relations I t s h o u l d be p o i n t e d out t h a t
of the inclination
(9) does n o t a p p l y when m + 2p - 1 < 0.
function.
But i f we s e t
F t = p ( 1 ) ffi I F ` ' ' ( l ) '
[( - - 1 ) l - - m l ~ l m ( l - - p ) ( ~
-- I).
(10)
then no difficulty will arise.
2. RECURRENCE RELATIONS OF THE INCLINATION FUNCTION Formula (9) r e l a t e s
the inclin
function to the hypergeometric function.
c l e a r about the r e c u r r e n c e r e l a t i o n s With
Lj M, N any
three integers,
of the latter,
is called a neighbouring function of F(a, are linearly
close
neighbours.
we can o b t a i n t h o s e among t h e f o r m e r .
the function
F(a+L,8+M, ± 1, we have t h e
v +N; x)
8, Y; z ) .
(11)
In p a r t i c u l a r ,
f o r L, /4, N e q u a l t o 0 o r
Gauss [16] p r o v e d t h a t any t h r e e n e i g h b o u r i n g f u n c t i o n s
dependent, with rational
f u n c t i o n s o f z as c o e f f i c i e n t s .
AFt + BF, 4- CF3 = i n which A, B, C a r e r a t i o n a l
Hence once we a r e
f u n c t i o n s o f x.
That i s ,
O.
(12)
Gauss a l s o gave 18 e x p l i c i t
among t h e c l o s e n e i g h b o u r s , from which t h e c o e f f i c i e n t s
linear relations
A, B, C f o r t h e l i n e a r r e l a t i o n
any t h r e e n e i g h b o u r i n g f u n c t i o n s can be d e r i v e d . To f a c i l i t a t e
d i s c u s s i o n , we r e - w r i t e
= G,.,.
(9) i n t h e form
(0, 8,
(13)
1 - oo, 5' /
where
t-M
(1 + m ) l ( 2 p ) ! (1 - - c o s I / - T ( 1
Glm9
2'+-r (l -
a~m--l, Hence we s e e t h a t ,
8=l+m+l,
p)!(l
-
m)!(m
+ co, I )
+ 2p -
Ot
y~m+2p+l--l.
I+m_~ , "
(14) (1S)
s i n c e any h y p e r g e o m e t r i c f u n c t i o n can be e x p r e s s e d i n t e r m s o f two
neighbouring functions,
any i n c l i n a t i o n
f u n c t i o n can l i k e w i s e be e x p r e s s e d ; t h u s t h e
recurrence relations. When a , 8, y and l , m, p s a t i s f y
(15), we s h a l l w r i t e
( l, m, P) O) 8 ) Y
(16)
among
20
According
Inclination
to
Function
(15), we have then
~ l,
, )L
m+l,
+I,
g+l.
--1,
g+l,
.,
l,
'
r+l
g--l,
+1,
g-l,
,)
P ) r--1
o,
r--1
(,,
(17)
,) (18)
r+l
°, ,-,)
t*, 8, 1--1,
m--l,
--1,
m+
(~+2
Y +2
1, , - - 1 ) ( 1 +
~
(1+I,
,8,
(19)
~'--2
1, m - - l ,
r
~--2,
m--b l, P ) ( I - - 1 ,
,+1)
~,
(20)
r
m--l,
(21)
P)
and 16 more such correspondences, which we s h a l l not d e t a i l here.
Expression (17) shows t h a t ,
i n order to f i n d the r e l a t i o n among Fl.rnpJ Fl, m÷l,p, Fl, m_l, p, we need only the r e l a t i o n among F(m., 8~, y~z)., E(e~+I, 8+1, y+l;x) and F(e~-I, 8-1, "y-1;x). S i m i l a r statements are implied by the other e x p r e s s i o n s .
However, the 5 l i n e a r r e l a t i o n s we r e q u i r e here are not among Gauss's 18
and must be derived anew.
After a complicated d e r i v a t i o n , we f i n d
a , V ( ~ - - 2, g, r ; z) + A , F ( ~ . 8, r ; z) + a , f ( ~ + 2, g, r ; z) - - 0
(22)
B,V(,~. # -- 2. r ; z) + B,V(,~, g, r ; . ) + g , V ( ~ , # + 2, r ; ~) ---- 0
(23)
CxF(~, ~', ~" - - 2; z) + CaF(~, g, ~'; z) "-b C3F(c% if, ~" + 2; z) = 0
(24)
D,F(g -- 1, ~ - - 1. r - 1; z) + DaV(cc, ~, r; . ) +D,F(e~+ 1 , 8 + 1, r + 1; z ) = O
(25)
E ~ F ( ~ - - 1, ~ + 1, r - - 1; z) + E,F(~, ~. r; z) + E,v(~
+ 1, # -
I , r + 1; z ) ---- 0
(26)
i n which the c o e f f i c i e n t s are Ca-
1-
r--2~+2
(~
,42
-
-- (r A3
Bl Ba '==
r)(~-
r)
+(~--fl--1)z' 1)(~-
r)(1
2~ + ( ~ -
.)
--
~(~-
r + 1)(1 -
z)
#)z),
a(g + I)(I -- z)' ~" - - 28 - - 2 + ( ~ - - 8 + l ) z ;
(~-
r)(#-
r -
I)
•y - - 2 , 8 + 2 + 4 8 - - a - -
(~-
r)(#-
l)z"
1)(1 -
z)
[r - 2# + 2 + ( # - ~ - 1)z] -
[r -
28 + ( 8 - - # ) z ] ,
~(8 + 1)(1
B3
It-
28-
--
.)'
2 + (g-~+
(r - 2)(r [r--2-(2r-a-8-
1)r(1
I).1 ' -
.)' 3)z]'
+
(#-
r -
28 -
r + 1)#(1 -
.)
2 + (8 - g + 1)z
Inclination Function
21
(27)
c,=_r(r-.-l)(r-a-1)O-.). [r--2--(2r--a--R--3)z]
+(r-a)(r-~)r(~-1)'_r[r_l_(er_~_#-D~], [r--(2r--a--#+ 1)z] c, = (r -
~ + 1)(r -
(r + 1)[r
- -
~ + 1)(r -
(2r
- -
~)(r -
~),'.
a-- # + l)z]
'
D,~l--7, D,----[(1--a--~)z--(1--r)], D3=~--~z(1--z); Y __
Et
~(r-1)O-~)
#+(~--#--l)z'
--(r--#)(#--1)(1--,)_ E'ffi[(ff--l)--(:--a--l)z
]
(r-#-l)g(1-z) [ff--(ff--~+l)z]
+ [r-2P+(P-~)z], ~(r-#)(r-#+l)~
E'=-~-[(~_I)-(~-~-I),1" Using the formulas (22)- (27) and (13)-(15),
it is not difficult to obtain
A, Ft,,.-l,v + 2aFt.=.p + A3Ft,.,+t,p == O, BxFl-i,,n.p 4- naFt, m.v 4- BsFt+t.'s.p ~ O, C,Ft...,p-* 4- C3FI.,..p 4- C3Ft. m.p+* ~ O,
D,F/-,.,,,+t,,,-, + D, Ft.r..p+D~FI+,.~.-,,v+, == O, F.iFt-l,.,-hp 4- EaFt.m,¢, 4- EsF/+b..+t,t, == O.
}
in which the coefficients are 2, = i(l + m)O -
m + 1),`.,
A2 =" m + 2 p - - l - -
2ms ~,
23 = ~,
-
=
-
is`.,
i,`.(l + ,.)(2t
2p - l )
(l + m -- 2Is2) ~ , = 2,.' ( l - - p ) ( t + ~ ) ( l + m - - 2t,')
+ `.,
(2l - 2p + 1 ) ( 1 + ra + 1) It + m + 1 - - 2(l + 1),']
+ [ 2 p - - m - - 31-- 1 + ( 2 1 + B3~--2hc ~, = 2 : :
(l--p+
1)s*],
l)(l--m+l)
[l+ra+l--2(l+l)sq' .(2/, -
1)(1 - / ,
+ 1)(,, -
1 + 21, + 1)
[m--t+2p--l--2(2,o--l--1)sq ~, .= 2s'`.2p(21 -- 2p + l ) ( m -- l + 2p + 1)
[m--l+2p--
1--2(2p--l--
1)sa]
+ 2sa`.a(2p + 1)(l -- p)(m. ,,--, l + 2p + 1) Ira-- I + 2 p + 1 -- 2(2p-- 14- l)s']
'
(2s)
22
Inclination Function
-- (m--
l +2p
l)sO,
+ 1)(m -- 1 + 2p-- 2(2p--
- 2 , ' , K ¢ + 1)(2p - 2l + 1)(m + 2p + 1 - O [m - - l + 2/, + 1 - - 2 ( 2 p - t + 1)rq '
(29)
- - c 2 ( 2 1 0 - 1)
DI == ( I - - m + 2 p - - 1 - - 21sO"
2c2p(1- m) ¢2(1 -- ra + 1)(2iO + 1) + [ 1 - - m + 2p-- 1--2Is 2] [l--m+2p+ 3--2(1+ 1)s ~] -- [ l - - m + 2 p + 1--(21+ 1)s a] D~==_ 2c'(p+l)(l--m+2)(l--m+ 1)
~ , ,=,
_
[l--
m + 2p + 3 - - 2 ( I +
~ , .= c ' ( l + m ) ( 1 + m - - 1 ) ( 2 1 [2iO~ m ~ 3 1 +
1)
~] + (1 + ra + [2p-
- -
3l--
l ) s 2]
2p-
1+21s
2c2(l - - p)(l + m) [2p- m 31 + 1 + 2Is2]
-- [2p-- m--
_
1 + (2l+
m
- -
1)(21 - - 2p + 1)c a 31 3 + 2(1 + 1)s ~] - -
l)s2],
2c~(l-- p + l) [2p--m--31--S+2(l+l)s2]"
A number o f r e c u r r e n c e r e l a t i o n s and ( 2 9 ) .
we s h a l l omit h e r e . set,
f u n c t i o n s have now b e e n g i v e n a t
(28)
which can be d e r i v e d i n t h e same manner and which
I t can be s e e n t h a t a l l t h e r e l a t i o n s
neighbouring f u n c t i o n s each. first
among t h e i n c l i n a t i o n
There a r e o t h e r s i m i l a r r e l a t i o n s
Of c o u r s e , t h e c o e f f i c i e n t s
given involve only three i n (28) and (29), a p a r t from t h e
a r e somewhat c o m p l i c a t e d . 3. THE RECURSION PROCESS
Of t h e e x p r e s s i o n s a t
(28) and ( 2 9 ) , t h e f i r s t
set,
r e c u r r e n c e i n m, h a s t h e s i m p l e s t c o e f f i c i e n t s , computations. initial starting
values.
that is,
t h e one r e l a t i n g
to
and i s t h e most c o n v e n i e n t f o r p r a c t i c a l
B e s i d e s t h e r e c u r r e n c e f o r m u l a e g i v e n i n t h e p r e c e d i n g s e c t i o n , we r e q u i r e t h e For r e c u r s i o n a l o n g m, we can t a k e t h e s e c t o r a l harmonic t e r m s
Fzl p
as t h e
point.
From ( 2 ) , we s e e t h a t ,
when
l = m, we can o n l y have d = 0.
(2l)!
Flip ~ ~tp!(l From ( 3 0 ) , t h e f o l l o w i n g r e c u r r e n c e r e l a t i o n s
c,1_2ps2p P)!'
_
"
(30)
among t h e s e c t o r a l harmonic t e r m s a r e d e r i v e d :
1)
c~Fl_z,z_t,p '
Flit, m l ( 2 l P
1)
$2F I - - I , l--I~p--ll
(32)
Ft:p = (~l +
1)c2
Fl.l.t,+l,
(33)
F.p =
l(2l-
_
Hence we have
( l ' - - p)
(31)
- - t,)s2
Ftlp -----
l(1-
1)(21-
p(l
-
1)(2l-
-
p)
3)
s*c'Ft-,,,-2., ....
(34)
Inclination
For recursion in m, we still need F1, Z-l, p"
Function
23
an expression for the inclination function of the type
From (2), we find
F,.,_,,, = 2'p1(li(2l--_1)p)! [2pc"-'P+'; ' p - ' - ( 2 l - 2p)caz-'P-'s'P+'].
(35)
Noting (30), we have Fl, l-t,p *=" los
(36)
(p -- Is')F,,p.
or Ft, z-l,p ": ; ( 2 1 A c o m p a r i s o n o f (36) and t h e f i r s t FZ, l+I, p=0, t h e n t h e l a t t e r Fl, l,p and Fl, l+1,p, then by recurrence relations
1)cs[ F~-,,t-l,p-i-
Ft-l,t-,,p]
f o r m u l a o f (28) shows t h a t ,
also holds for m = l.
(37)
provided it
includes
Once we have t h e two i n i t i a l
values
applying the first formula of (28) and combining with the
of the sectoral
harmonic terms, all
the inclination
functions
Flmp
can be
found. When I = 0, s i n g u l a r i t i e s
w i l l a p p e a r i n (33) and ( 3 6 ) , b u t t h i s h a s no b e a r i n g on o u r
discussion.
For, an e x a m i n a t i o n o f t h e Lagranges e q u a t i o n s o f p e r t u r b a t i o n
that in this
c a s e , u s i n g I as an e l e m e n t i s i n a p p r o p r i a t e
[18] w i l l t e l l
anyway.
4. DERIVATIVE OF THE INCLINATION FUNCTION In perturbation calculations, we often require the derivative with respect to the inclination I of the inclination function.
From (13) and (14), we have
d F , . £ .= d G t . p F ( m - - l , l + m + l m + 2p + 1 - - l ; 1 - - c o s I ~ dI dI " 2
+
dI
1,.+
\
I
2
I
(38)
Because d__ F ( a , ? , r , dx
x ) ~ ~-~ F(c~ + 1 , # + r
1, r + 1" x ) . "
(39)
and ez
= G,~,,
+(,
2
- p (1 + c o s Z ) - ' ( - sinD
' 2 ") (, -
cos')-' ~ i - ' ]
(40)
we have
dFtmp ~ dl
- - 1 [1 - - 2 p -
m c o s l ] F l , , , p - - iFt,.,+1,p.
sin !
(41)
When I = m, a direct differentiation of (30) gives
dF.p
dI
~
(20 ! 2 t # ! ( / - - p)!
c'Z-2P-ls 3 p - 1 [ - (1 - - p ) s 2 + p c ' ] .
(42)
us
24
Inclination
Function
or
dFut, .ffi F~ e [ p _ dI cs Noting (36),
we
IP]. (43)
have
dFne dl
: --
iIF:,
~-~, p.
(44)
Comparing (41) and (43), we have o n l y to set F i j l + i j p = O, t h e n (41) i s a l s o a p p l i c a b l e c a s e 1 = m.
in the
REFERENCES
[ 1 ] It. R. Allan, l~rov. Itoy. 8oo., ,~288 (1965), 60. [ 2 ] G. Balmino, Teoh. [email protected]/GRGS, " S u r Certains Probl~mes et Fonctions Sp6ciales Recontr~s en G~od6ale Spatiale", Bretigny-sur-Prge, France, (1973). Challe, A. & Laeiaverie, J. J., ~stro,. Astrophys., 3 (1969), 15--28. Kaula, W. M., "Theory of Satellite Geodesy", Blalsdcll Publ. Co., Waltham, Mass, (1966), 30. [ 5 ] H. Kinoshita et al., Aaa. Tokyo Astron. Obs., 14 (1974), 14--35. [ 6 ] H. E. TxMomeaKosa, A. Mr., 49 (1972), 879--885. [ 7 ] I. G. Izsak, J. @cophy#. lies.. 69 (1964), 2621. [ 8 ] B. Jeffreys, Geophys. J., 10 (1965), 141. [ 9 ] E. Stiefel, Mathcmatische Methodcr der Himmelsmechanik und Astronautik, (1965), 341. [10] It. It. Gooding, Celcst Mech., 4 (1971), 91. [11] G. E. O. Giacaglia, Celest. Mevh., 13 (1976), 503--509. [12] W. M. Kaula, Geophysical J., 5 (1961). 104. [13] G. E. O. Giacaglia, SAO Special Iteport 352, 1973. [14] J. A. Campbell, CeIest. Mech., 6 (1972), 187--197. [3] [4]
[15] Wang Z h u - x i and Guo D u n - r e n , "Teshu Hanshu Lun" (Theory o f S p e c i a l F u n c t i o n s ) , Kexue P u b l i c a t i o n s (1965) 190. [16] Riemann, Abh. d. K. Ges. d. Wiss. zu G ~ t t i n g e n (1875). [17] Wang Z h u - x i and Guo D u n - t e n , I b i d . , 227. []8] D. Brouwer and G.M. Clemence, ' ~ e t h o d s o f C e l e s t i a l M e c h a n i c s " , Academic P r e s s , (1961), 229.
4 (1980) 17-24
Pergamon P r e s s . P r i n t e d i n Great B r i t a i n
Acta Astr. Sinica 20 (1979) 17-24
RECURRENCE
0146-6364/80/0301-0017-$07.50/0
RELATIOk',S OF THE INCLINATIOI~!
T o n g Fu
I~.ru L i a n - d a
FUNCTION
Wang C h a n g - b i n
Purple Mountain Observatory, Academia Sinica (Received 1978 March 28)
ABSTRACT In t h i s p a p e r , t h e i n c l i n a t i o n f u n c t i o n i s e x p r e s s e d i n t e r m s o f t h e h y p e r g e o m e t r i c f u n c t i o n and t h i s has r e s u l t e d i n some s i m p l e r e c u r r e n c e r e l a t i o n s i n v o l v i n g no more t h a n t h r e e n e i g h b o u r i n g f u n c t i o n s . In d i s c u s s i n g t h e p e r t u r b a t i o n s it
by t h e E a r t h ' s g r a v i t a t i o n a l
f i e l d on a r t i f i c i a l
satellites,
i s o f t e n n e c e s s a r y t o d e v e l o p t h e p e r t u r b i n g f u n c t i o n as a P o i s s o n s e r i e s e x p r e s s e d i n t h e
orbital
elements of the satellite.
trigonometric functions.
Then, t h e o r b i t a l
More s p e c i f i c a l l y ,
inclination
I a p p e a r s as an argument i n
we have t o c o n s i d e r t h e f o l l o w i n g d e v e l o p m e n t : !
Pt.(~n~)e i'~ ~ % Fj.ect(z-ae)"
(1)
p~o
where
Plm(Z) i s
t h e a s s o c i a t e d Legendre p o l y n o m i a l , u i s t h e l a t i t u d e geocentric latitude
and X = a - £.
argument o f t h e
satellite,
~ is its
functions,
and t h e y can be e x p r e s s e d as p o l y n o m i a l s o f t r i g o n o m e t r i c f u n c t i o n s o f I .
have b e e n d i s c u s s e d by a number o f a u t h o r s Iszak
[7] and B. J e f f r e y s
[1-11].
The Flrnp i n (1) a r e t h e i n c l i n a t i o n
[8] o b t a i n e d t h e f o l l o w i n g r e s u l t
i l - " ( l 4- m ) ! F~,,,e == 2~p!(l - - p ) !
They
Using t h e method o£ t r a n s f o r m a t i o n group, i n t h e form o f a s i n g l e summation,
~i f l"~ir,21-~prw ,.3t-,,.-2p-~i~,,,-t+2p+~i . ,,---J ~i "l---i-
(2)
where
i = W--L c ~
c o s -I- , 2 C~ ~
s~
sin -~-I, 2
m!
.! (m -- .)!
max{O,l--m--2p} ~j~min{21--2p, The r e s u l t s
l--m}.
o b t a i n e d by o t h e r a u t h o r s a r e more c o m p l i c a t e d , f o r example, t h a t o f C h a l l e
a d o u b l e summation, w h i l e t h a t o f Kaula [12] i s a t r i p l e Besides the expressions for the inclination
summation.
f u n c t i o n s , many a u t h o r s have a l s o
[3] i s
18
Inclination Function
discussed
recurrence
Giacaglia
[11] d i s c u s s e d
apart
relations
f r o m some e r r o r s
neighbouring In this
hypergeometric
( i n some c a s e s
we p r o p o s e
functions relations
Giacaglia's
[5,
13,
recurrence
a s many a s 12) f o r
a method of expressing
10,
when b o t h l
14].
Recently,
andm vary.
relations
involve
them to be convenient
the inclination
However,
function
t o o many
in use. in terms of the
function. 1.
Re-writing
recurrence
in the derivation,
functions paper,
among s u c c e s s i v e
in detail
INCLINATION FUNCTION EXPRESSED AS HYPERGEOMETRIC FUNCTION
(2),
and replacing
0 2 b y (1 + c o s 2 ) / 2 ,
;'--(t+~),.
(l+co,
a n d s 2 b y (1 - c o s I ) / 2 ,
l~-I-m
Ftmp =
21_=p!(l
-
p)!
-
O
I-m-2P
(1-cos1)
'
'
" n - ' l ' ,r. llp- m - i (, -1 + c ° s I ) ' l - i P - i ( 1 - - c ° ' l ) =p-u-=-i). .¢". - - -l ,"-l i g - I - m C~.;
~
we h a v e
(s)
i or
1 +ra
_
FIrap
;'-~(1 + m ) ! (1 + co, Z) ~ - v - ( 1 -- P)I
2atp!(l
• (_
l --ra_p
~o~X)
1)t_~
1 (1 - - m ) !
d l-~ d c o s l 1-* (I +
co, I)'/-"(l
-- cos/)".
(4)
Using the Jacobi polynomials in differential form, [15],
/~: B)(~) = F ( =
, . ~ + , . 8; x)
r(~) r @ + ~)
a._~ ~ [x~+.-,O (x1-B(1- xY-O ax"
_
xy+.-B].
(5)
and applying the substitution 1 --
COS I
x = - - 2
(6)
we h a v e F(--
n, a + n,/~; d~ d cos I ~
1 -- cos/)
=
r@) (1 - - co, O ' - " ( 1 + co, O a-2"F(n + #)
(_1).
2
i(1 -- cosI)a+'-t(1
c o s I ) "+'-a]
+
(7)
Let
n~l--m,~m+ we t h e n
2p+
1 --l,c~2m+
1,
have
F(ra-- I, I-4- m-4- 1, m 4- 2 p 4 - I --I; I -- cos/) 2 ffi ( - - 0 1 - "
r ( m + 2 p + 1 - - 1) ( 1 - - c o , 0 1 - ' - " Q 2t-~F(2p + 1)
+
cos/y'-'-"
dl-m
• dqo "P-'~ [ ( 1 - - c o s l ) ' P ( 1
+
cosl)at-'P].
(8)
Inclination Function
Substituting
19
(8) i n ( 4 ) , we have l+m_
r l . , = i ' - - (t + m)l ( 2 e ) t (1 -
•
inclination. properties
+ ~,z)
2'+'9!(l--p)l(l--m)t(m+2p--l)l F ( m - - l , t + m + 1, m + 2 p + 1--1;
E x p r e s s i o n (9) shows t h a t t h e i n c l i n a t i o n and a h y p e r g e o m e t r i c f u n c t i o n ,
¢o,Z)'-~7(1
'
'
1--eosI)
2
(9)
f u n c t i o n can be e x p r e s s e d i n t e r m s o f a p o l y n o m i a l
i n which t h e arguments a r e t h e s i n e and c o s i n e o f t h e h a l f
T h i s i s an i m p o r t a n t c o n c l u s i o n , f o r i t means t h a t we can now make u s e o f t h e of the hypergeometric function, in particular,
we can use t h e r e c u r r e n c e r e l a t i o n s
of the hypergeometric function to derive the recurrence relations I t s h o u l d be p o i n t e d out t h a t
of the inclination
(9) does n o t a p p l y when m + 2p - 1 < 0.
function.
But i f we s e t
F t = p ( 1 ) ffi I F ` ' ' ( l ) '
[( - - 1 ) l - - m l ~ l m ( l - - p ) ( ~
-- I).
(10)
then no difficulty will arise.
2. RECURRENCE RELATIONS OF THE INCLINATION FUNCTION Formula (9) r e l a t e s
the inclin
function to the hypergeometric function.
c l e a r about the r e c u r r e n c e r e l a t i o n s With
Lj M, N any
three integers,
of the latter,
is called a neighbouring function of F(a, are linearly
close
neighbours.
we can o b t a i n t h o s e among t h e f o r m e r .
the function
F(a+L,8+M, ± 1, we have t h e
v +N; x)
8, Y; z ) .
(11)
In p a r t i c u l a r ,
f o r L, /4, N e q u a l t o 0 o r
Gauss [16] p r o v e d t h a t any t h r e e n e i g h b o u r i n g f u n c t i o n s
dependent, with rational
f u n c t i o n s o f z as c o e f f i c i e n t s .
AFt + BF, 4- CF3 = i n which A, B, C a r e r a t i o n a l
Hence once we a r e
f u n c t i o n s o f x.
That i s ,
O.
(12)
Gauss a l s o gave 18 e x p l i c i t
among t h e c l o s e n e i g h b o u r s , from which t h e c o e f f i c i e n t s
linear relations
A, B, C f o r t h e l i n e a r r e l a t i o n
any t h r e e n e i g h b o u r i n g f u n c t i o n s can be d e r i v e d . To f a c i l i t a t e
d i s c u s s i o n , we r e - w r i t e
= G,.,.
(9) i n t h e form
(0, 8,
(13)
1 - oo, 5' /
where
t-M
(1 + m ) l ( 2 p ) ! (1 - - c o s I / - T ( 1
Glm9
2'+-r (l -
a~m--l, Hence we s e e t h a t ,
8=l+m+l,
p)!(l
-
m)!(m
+ co, I )
+ 2p -
Ot
y~m+2p+l--l.
I+m_~ , "
(14) (1S)
s i n c e any h y p e r g e o m e t r i c f u n c t i o n can be e x p r e s s e d i n t e r m s o f two
neighbouring functions,
any i n c l i n a t i o n
f u n c t i o n can l i k e w i s e be e x p r e s s e d ; t h u s t h e
recurrence relations. When a , 8, y and l , m, p s a t i s f y
(15), we s h a l l w r i t e
( l, m, P) O) 8 ) Y
(16)
among
20
According
Inclination
to
Function
(15), we have then
~ l,
, )L
m+l,
+I,
g+l.
--1,
g+l,
.,
l,
'
r+l
g--l,
+1,
g-l,
,)
P ) r--1
o,
r--1
(,,
(17)
,) (18)
r+l
°, ,-,)
t*, 8, 1--1,
m--l,
--1,
m+
(~+2
Y +2
1, , - - 1 ) ( 1 +
~
(1+I,
,8,
(19)
~'--2
1, m - - l ,
r
~--2,
m--b l, P ) ( I - - 1 ,
,+1)
~,
(20)
r
m--l,
(21)
P)
and 16 more such correspondences, which we s h a l l not d e t a i l here.
Expression (17) shows t h a t ,
i n order to f i n d the r e l a t i o n among Fl.rnpJ Fl, m÷l,p, Fl, m_l, p, we need only the r e l a t i o n among F(m., 8~, y~z)., E(e~+I, 8+1, y+l;x) and F(e~-I, 8-1, "y-1;x). S i m i l a r statements are implied by the other e x p r e s s i o n s .
However, the 5 l i n e a r r e l a t i o n s we r e q u i r e here are not among Gauss's 18
and must be derived anew.
After a complicated d e r i v a t i o n , we f i n d
a , V ( ~ - - 2, g, r ; z) + A , F ( ~ . 8, r ; z) + a , f ( ~ + 2, g, r ; z) - - 0
(22)
B,V(,~. # -- 2. r ; z) + B,V(,~, g, r ; . ) + g , V ( ~ , # + 2, r ; ~) ---- 0
(23)
CxF(~, ~', ~" - - 2; z) + CaF(~, g, ~'; z) "-b C3F(c% if, ~" + 2; z) = 0
(24)
D,F(g -- 1, ~ - - 1. r - 1; z) + DaV(cc, ~, r; . ) +D,F(e~+ 1 , 8 + 1, r + 1; z ) = O
(25)
E ~ F ( ~ - - 1, ~ + 1, r - - 1; z) + E,F(~, ~. r; z) + E,v(~
+ 1, # -
I , r + 1; z ) ---- 0
(26)
i n which the c o e f f i c i e n t s are Ca-
1-
r--2~+2
(~
,42
-
-- (r A3
Bl Ba '==
r)(~-
r)
+(~--fl--1)z' 1)(~-
r)(1
2~ + ( ~ -
.)
--
~(~-
r + 1)(1 -
z)
#)z),
a(g + I)(I -- z)' ~" - - 28 - - 2 + ( ~ - - 8 + l ) z ;
(~-
r)(#-
r -
I)
•y - - 2 , 8 + 2 + 4 8 - - a - -
(~-
r)(#-
l)z"
1)(1 -
z)
[r - 2# + 2 + ( # - ~ - 1)z] -
[r -
28 + ( 8 - - # ) z ] ,
~(8 + 1)(1
B3
It-
28-
--
.)'
2 + (g-~+
(r - 2)(r [r--2-(2r-a-8-
1)r(1
I).1 ' -
.)' 3)z]'
+
(#-
r -
28 -
r + 1)#(1 -
.)
2 + (8 - g + 1)z
Inclination Function
21
(27)
c,=_r(r-.-l)(r-a-1)O-.). [r--2--(2r--a--R--3)z]
+(r-a)(r-~)r(~-1)'_r[r_l_(er_~_#-D~], [r--(2r--a--#+ 1)z] c, = (r -
~ + 1)(r -
(r + 1)[r
- -
~ + 1)(r -
(2r
- -
~)(r -
~),'.
a-- # + l)z]
'
D,~l--7, D,----[(1--a--~)z--(1--r)], D3=~--~z(1--z); Y __
Et
~(r-1)O-~)
#+(~--#--l)z'
--(r--#)(#--1)(1--,)_ E'ffi[(ff--l)--(:--a--l)z
]
(r-#-l)g(1-z) [ff--(ff--~+l)z]
+ [r-2P+(P-~)z], ~(r-#)(r-#+l)~
E'=-~-[(~_I)-(~-~-I),1" Using the formulas (22)- (27) and (13)-(15),
it is not difficult to obtain
A, Ft,,.-l,v + 2aFt.=.p + A3Ft,.,+t,p == O, BxFl-i,,n.p 4- naFt, m.v 4- BsFt+t.'s.p ~ O, C,Ft...,p-* 4- C3FI.,..p 4- C3Ft. m.p+* ~ O,
D,F/-,.,,,+t,,,-, + D, Ft.r..p+D~FI+,.~.-,,v+, == O, F.iFt-l,.,-hp 4- EaFt.m,¢, 4- EsF/+b..+t,t, == O.
}
in which the coefficients are 2, = i(l + m)O -
m + 1),`.,
A2 =" m + 2 p - - l - -
2ms ~,
23 = ~,
-
=
-
is`.,
i,`.(l + ,.)(2t
2p - l )
(l + m -- 2Is2) ~ , = 2,.' ( l - - p ) ( t + ~ ) ( l + m - - 2t,')
+ `.,
(2l - 2p + 1 ) ( 1 + ra + 1) It + m + 1 - - 2(l + 1),']
+ [ 2 p - - m - - 31-- 1 + ( 2 1 + B3~--2hc ~, = 2 : :
(l--p+
1)s*],
l)(l--m+l)
[l+ra+l--2(l+l)sq' .(2/, -
1)(1 - / ,
+ 1)(,, -
1 + 21, + 1)
[m--t+2p--l--2(2,o--l--1)sq ~, .= 2s'`.2p(21 -- 2p + l ) ( m -- l + 2p + 1)
[m--l+2p--
1--2(2p--l--
1)sa]
+ 2sa`.a(2p + 1)(l -- p)(m. ,,--, l + 2p + 1) Ira-- I + 2 p + 1 -- 2(2p-- 14- l)s']
'
(2s)
22
Inclination Function
-- (m--
l +2p
l)sO,
+ 1)(m -- 1 + 2p-- 2(2p--
- 2 , ' , K ¢ + 1)(2p - 2l + 1)(m + 2p + 1 - O [m - - l + 2/, + 1 - - 2 ( 2 p - t + 1)rq '
(29)
- - c 2 ( 2 1 0 - 1)
DI == ( I - - m + 2 p - - 1 - - 21sO"
2c2p(1- m) ¢2(1 -- ra + 1)(2iO + 1) + [ 1 - - m + 2p-- 1--2Is 2] [l--m+2p+ 3--2(1+ 1)s ~] -- [ l - - m + 2 p + 1--(21+ 1)s a] D~==_ 2c'(p+l)(l--m+2)(l--m+ 1)
~ , ,=,
_
[l--
m + 2p + 3 - - 2 ( I +
~ , .= c ' ( l + m ) ( 1 + m - - 1 ) ( 2 1 [2iO~ m ~ 3 1 +
1)
~] + (1 + ra + [2p-
- -
3l--
l ) s 2]
2p-
1+21s
2c2(l - - p)(l + m) [2p- m 31 + 1 + 2Is2]
-- [2p-- m--
_
1 + (2l+
m
- -
1)(21 - - 2p + 1)c a 31 3 + 2(1 + 1)s ~] - -
l)s2],
2c~(l-- p + l) [2p--m--31--S+2(l+l)s2]"
A number o f r e c u r r e n c e r e l a t i o n s and ( 2 9 ) .
we s h a l l omit h e r e . set,
f u n c t i o n s have now b e e n g i v e n a t
(28)
which can be d e r i v e d i n t h e same manner and which
I t can be s e e n t h a t a l l t h e r e l a t i o n s
neighbouring f u n c t i o n s each. first
among t h e i n c l i n a t i o n
There a r e o t h e r s i m i l a r r e l a t i o n s
Of c o u r s e , t h e c o e f f i c i e n t s
given involve only three i n (28) and (29), a p a r t from t h e
a r e somewhat c o m p l i c a t e d . 3. THE RECURSION PROCESS
Of t h e e x p r e s s i o n s a t
(28) and ( 2 9 ) , t h e f i r s t
set,
r e c u r r e n c e i n m, h a s t h e s i m p l e s t c o e f f i c i e n t s , computations. initial starting
values.
that is,
t h e one r e l a t i n g
to
and i s t h e most c o n v e n i e n t f o r p r a c t i c a l
B e s i d e s t h e r e c u r r e n c e f o r m u l a e g i v e n i n t h e p r e c e d i n g s e c t i o n , we r e q u i r e t h e For r e c u r s i o n a l o n g m, we can t a k e t h e s e c t o r a l harmonic t e r m s
Fzl p
as t h e
point.
From ( 2 ) , we s e e t h a t ,
when
l = m, we can o n l y have d = 0.
(2l)!
Flip ~ ~tp!(l From ( 3 0 ) , t h e f o l l o w i n g r e c u r r e n c e r e l a t i o n s
c,1_2ps2p P)!'
_
"
(30)
among t h e s e c t o r a l harmonic t e r m s a r e d e r i v e d :
1)
c~Fl_z,z_t,p '
Flit, m l ( 2 l P
1)
$2F I - - I , l--I~p--ll
(32)
Ft:p = (~l +
1)c2
Fl.l.t,+l,
(33)
F.p =
l(2l-
_
Hence we have
( l ' - - p)
(31)
- - t,)s2
Ftlp -----
l(1-
1)(21-
p(l
-
1)(2l-
-
p)
3)
s*c'Ft-,,,-2., ....
(34)
Inclination
For recursion in m, we still need F1, Z-l, p"
Function
23
an expression for the inclination function of the type
From (2), we find
F,.,_,,, = 2'p1(li(2l--_1)p)! [2pc"-'P+'; ' p - ' - ( 2 l - 2p)caz-'P-'s'P+'].
(35)
Noting (30), we have Fl, l-t,p *=" los
(36)
(p -- Is')F,,p.
or Ft, z-l,p ": ; ( 2 1 A c o m p a r i s o n o f (36) and t h e f i r s t FZ, l+I, p=0, t h e n t h e l a t t e r Fl, l,p and Fl, l+1,p, then by recurrence relations
1)cs[ F~-,,t-l,p-i-
Ft-l,t-,,p]
f o r m u l a o f (28) shows t h a t ,
also holds for m = l.
(37)
provided it
includes
Once we have t h e two i n i t i a l
values
applying the first formula of (28) and combining with the
of the sectoral
harmonic terms, all
the inclination
functions
Flmp
can be
found. When I = 0, s i n g u l a r i t i e s
w i l l a p p e a r i n (33) and ( 3 6 ) , b u t t h i s h a s no b e a r i n g on o u r
discussion.
For, an e x a m i n a t i o n o f t h e Lagranges e q u a t i o n s o f p e r t u r b a t i o n
that in this
c a s e , u s i n g I as an e l e m e n t i s i n a p p r o p r i a t e
[18] w i l l t e l l
anyway.
4. DERIVATIVE OF THE INCLINATION FUNCTION In perturbation calculations, we often require the derivative with respect to the inclination I of the inclination function.
From (13) and (14), we have
d F , . £ .= d G t . p F ( m - - l , l + m + l m + 2p + 1 - - l ; 1 - - c o s I ~ dI dI " 2
+
dI
1,.+
\
I
2
I
(38)
Because d__ F ( a , ? , r , dx
x ) ~ ~-~ F(c~ + 1 , # + r
1, r + 1" x ) . "
(39)
and ez
= G,~,,
+(,
2
- p (1 + c o s Z ) - ' ( - sinD
' 2 ") (, -
cos')-' ~ i - ' ]
(40)
we have
dFtmp ~ dl
- - 1 [1 - - 2 p -
m c o s l ] F l , , , p - - iFt,.,+1,p.
sin !
(41)
When I = m, a direct differentiation of (30) gives
dF.p
dI
~
(20 ! 2 t # ! ( / - - p)!
c'Z-2P-ls 3 p - 1 [ - (1 - - p ) s 2 + p c ' ] .
(42)
us
24
Inclination
Function
or
dFut, .ffi F~ e [ p _ dI cs Noting (36),
we
IP]. (43)
have
dFne dl
: --
iIF:,
~-~, p.
(44)
Comparing (41) and (43), we have o n l y to set F i j l + i j p = O, t h e n (41) i s a l s o a p p l i c a b l e c a s e 1 = m.
in the
REFERENCES
[ 1 ] It. R. Allan, l~rov. Itoy. 8oo., ,~288 (1965), 60. [ 2 ] G. Balmino, Teoh. [email protected]/GRGS, " S u r Certains Probl~mes et Fonctions Sp6ciales Recontr~s en G~od6ale Spatiale", Bretigny-sur-Prge, France, (1973). Challe, A. & Laeiaverie, J. J., ~stro,. Astrophys., 3 (1969), 15--28. Kaula, W. M., "Theory of Satellite Geodesy", Blalsdcll Publ. Co., Waltham, Mass, (1966), 30. [ 5 ] H. Kinoshita et al., Aaa. Tokyo Astron. Obs., 14 (1974), 14--35. [ 6 ] H. E. TxMomeaKosa, A. Mr., 49 (1972), 879--885. [ 7 ] I. G. Izsak, J. @cophy#. lies.. 69 (1964), 2621. [ 8 ] B. Jeffreys, Geophys. J., 10 (1965), 141. [ 9 ] E. Stiefel, Mathcmatische Methodcr der Himmelsmechanik und Astronautik, (1965), 341. [10] It. It. Gooding, Celcst Mech., 4 (1971), 91. [11] G. E. O. Giacaglia, Celest. Mevh., 13 (1976), 503--509. [12] W. M. Kaula, Geophysical J., 5 (1961). 104. [13] G. E. O. Giacaglia, SAO Special Iteport 352, 1973. [14] J. A. Campbell, CeIest. Mech., 6 (1972), 187--197. [3] [4]
[15] Wang Z h u - x i and Guo D u n - r e n , "Teshu Hanshu Lun" (Theory o f S p e c i a l F u n c t i o n s ) , Kexue P u b l i c a t i o n s (1965) 190. [16] Riemann, Abh. d. K. Ges. d. Wiss. zu G ~ t t i n g e n (1875). [17] Wang Z h u - x i and Guo D u n - t e n , I b i d . , 227. []8] D. Brouwer and G.M. Clemence, ' ~ e t h o d s o f C e l e s t i a l M e c h a n i c s " , Academic P r e s s , (1961), 229.