All the equations involved in extending the PS phi solution to include the long periodic and second order secular effects of the zonal harmonics are presented. Topics covered include DSphi elements and relations for their conconical transformation into the PS phi elements; the solution algorithm based on the Von Zeipel method; and the elimination of long periodic terms and analytical integration of primed variables. The equations were entered into the ASOP program, checked out, and verified. Comparisons with numerical integrations show the long period theory to be accurate within several meters after 800 revolutions

Wang, K. C.

English

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https://ntrs.nasa.gov/search.jsp?R=19790010899 2020-03-21T23:37:13+00:00Z
ACM-TR-116
9
l;
(NISI-CR-160130) LONG PERIOD PERTURBATIONS 	 1179-19070
OF EARTH SATELLITE ORBITS (Analytical and
Computational Katbematics, Inc.) 27 p
BC A03/KF A01	 CSCL 22A	 Dnclas
G3/1S 16365
LONG PERIOD PERTURBATIONS OF
EARTH SATELLITE ORBITS
Itt
ANALYTICAL AND
COMPUTATIONAL
MATHEMATICS, INC.
LONG PERIOD PERTURBATIONS
OF
EARTH SATELLITE ORBITS
BY
K. C. WANG
ANALYTICAL AND COMPUTATIONAL ANALYSIS, INC.
1275 SPACE PARK DRIVE, SUITE 114
HOUSTON, TEXAS 77058
JANUARY 1979
This report was prepared for the NASA/Johnson Space :enter
under Contract #15445.
CONTENTS
^ 1
Page
5
8
11
11
13
16
19
21
23
^f
j
Section
1.0 INTRODUCTION
2.0 METHOD OF SOLUTION
2.1	 Notation
2.2	 Solution Algorithm
3.0 EQUATIONS FOR ELIMINATION OF LONG
PERIODIC TERMS AND ANALYTICAL
INTEGRATION OF PRIMED VARIABLES
*
3.1	 Generating Function S1
3.2	 Derivatives of SI
3.3	 Derivatives of F2 with Respect
to	 DSO	 Elements
4.0 CONCLUSIONS
REFERENCES
APPENDIX - COMPUTATIONAL PROCEDURE
h
S
LONG PERIOD PERTURBATIONS
OF
EARTH SATELLITE ORBITS
by
K. C. Wang
1.0 INTRODUCTION
In reference 1, Scheifele and Graf introduced a complete
first order solution for the orbital motion of a satellite
perturbed by earth oblateness. This solution was expressed
in the DSO elements. In reference 2, Bond and Scheifele
expressed the first order short period and secular J2 sol-
ution in the non-singular PSO elements. This theory was
implemented in an operational computer program named ASOP
described in reference 3. In references 4 and 5, the PSO
analytical theory was updated to include the drag effects.
In reference 6, the theory was developed to account for
the time dependent gravitational harmonics. The drag and time
dependent geupotential terms have also been included in ASOP .
Bond also extended the PSO theory to include the first
order long period terms and second order secular perturbations
due to J2 , J 3 , J4 and J5 . However, no documentation of
the equations was ever published. In reference 7, Mueller
developed a recursive theory to include the first order long
period terms and second order secular perturbations due to
zonal harmonics of any order. Mueller's theory plus the
second order J 2 theory developed by Bond have now been im-
plemented in ASOP .
r- 6 -
The purpose of this report is to document all the equations
involved in extending the PSO solution to include the long
periodic and second order secular effects of the zonal harmonics.
Dil i PACE MAW NOT FLW
1 40
2.0 METHOD OF SOLUTION
2.1 Notation
The DSO elements are a set of eight variables which have
the following description:
Angle Elements:
	
a l -	 true anomaly
	
a 2 = g	 argument of pericenter
	a 3 = h	 longitude of ascending node
	
a4 = Q	 time element
Action Elements:
	
_ 0
	
related to two body energy
1
	02 = G	 total angular momentum
	
B = H	 z-component of the angular momentum3
	
8 4 
= L	 total energy
These may be canonically transformed to the PSO elements
by the following relations:
Q1 = 0 + g + h
a2 = - 2(0 - G) uin (g + h)
a3 - -2(^ sin (h)
0 4
	
=	 V.
(1)
pl
P3 = 2(^ cos (h)
P4 = L
- 8 -
The DSO Hamiltonian for the zonal oblateness problem is
is
C 4
given by:
F = F 0 + 
cFl + E2F2
	 (2)
where
F - 4^ _ u	 (two body contributions)
o
[( 2
F 1
 = qr Xr - 3 (J 2 contribution)
N
x
F 2 
= ^ X13 A , 1-1 P^^
 (r3) (higher zonal harmonics)
31t. _ 2 J 2 lilt`
P  are the Legendre polynomials, Re is the mean equatorial
radius of earth, and J 2 and ^
n 
are oblateness coefficients.
2.2 Solution Algorithm
Von Zeipel's method of elimination of the short and long
periodic terms is used. The solution first requires the trans-
formation to eliminate the short periodic terms due to J 2 .
The generating function is assumed to be of the form
S = S o + CS 1
So give the identity transformation, S 1 is so chosen that
the new Hamiltonian is no longer a function of short period
variable ^. The Hamiltonian has the form
F'(R', 9 1 ) = FO + cFl + c 2 F2
A more thorough discussion of the elimination of short
periodic terms can be found in reference 2.
An additional transformation must be made to eliminate the
long periodic terms from F'. This transformation is defined
by the generating function
S =S0+CS1
I=
- 9 -
Again, 8 o
 gives the identify transformation, 3 1
 is chosen
such that the long period variable	 g'	 is eliminated from the
WS Hamiltonian.	 The new Hamiltonian has the form
F'1(811)
	 -a 	 Fit	 +	 Ott	 +	 C 2 
Fit
1	 20
A more thorough discussion of the elimination of the
	 J2
and higher order zonal	 perturbation long periodic terms can
be found in references 1 and 7.
The solution algorithm can be divided into three steps:
(1)	 Initialize the primed variables
Ps
	 as
cr	 k,0 =	 ak,0 +
	
ank,0	 ank,0 (3)
as	 as *
1	 ls	
-E	 +p k,0	 ak,0	
abak,O	 aQk,O	 k	 1,2,3,4
(2)	 Analytical integration of primed variables
a l	 of	 0	 + Ali
aZ = 02'o cos(A2 T) 	-	 p'2,0 sin(A2T)
a3	 o f	 cos(A 3 T) 	-	 p'3,0	 sin(A3T)3,0
a4 = a4 '0 	+ A 4 T
p l	 =	 p l ,0 (4)
P;	 =	 p 2,0	 c• os(A 2 T) 	+	 a z,0	 sin(A2T)
P; = p3,0 cos"3T)	 + a3,0	 sin(A3T)
P 4	 =	l'4 ,0
The definitions or	 All	 A 2 ,	 A 3 , A 4
 
are given
	
in section 3.0
of this report,	 The relation between time t and the new inde-
pendent variable T is given by dt = r 2 /q, the definition of 	 q
is also given in section 3.0
t "
-
40P	 .
' 
Yt
a
10 -
(3) Beek transformation
a51 +
 as
k	 k	 a pk
as	 as*
P k = Pk + e ao1 
+ 8a1k	 k
(5)
k - 1,2,3,4
r3.0 EQUATIONS FOR ELIMINATION OF LONG PERIODIC TERMS AND
ANALYTICAL INTEGRATION OF PRIMED VARIABLES
A detailed description of generating function S 1
 and deri-
vatives of S 1
 with respect to the PSO elements can be found
in Appendix F of Reference 3. In this section a detailed des-
cription of generating function S1 and derivatives of S1 with
respect to the PSO elements will be given. The derivatives
of F"2 	respect to the DSO elements will also be given.
3.1 Generating Function S1
From Reference 1 we have:
S 1	 aF1	 S	 2 8_ (2 - 3b + 6gB)e 2 b sin(2g)	 (0)
1
q
8G
S are terms related to higher order zonal perturbations.
A detailed description of S can be found in reference 7.
Now we introduce sin(g) and coo(g)
sin g) = CD (a 6 a 3 - 0 a2)
Cos (g) 
= CD (a6	 2a 7 + aa3)	 (?)
where
C	 2(
D = 3 G(G . H)
k
	
	 To write S1 in terms of PSO elements we introduce the
following abbreviations
Q=
Q 2 
	----2 (02 +a6)
a8	 (8)
2
p =	 _ 1(,J2 + oz) + u
32a 8 	 (9)
2a
e = (1-- ap) = QD	 (10)
R
a
- 12 -
G2b = 1 - Z
	
(11)
H
X	 eb#nin(g) _ Q AM + H) (0 6a 3 - a 7a 2 )	 (12)
eb il c uc(B) = 2 G 2(G + H) ( a 6 a 7 + 0 2 0 3 )	 (13)
0 - e2bnin(2g) - 2,0	 (14)
Now we have
*
sl 
= 8F1 	2 f2
2
(2 - 3b + 6gB)0]	 (15)
vei q
where
g = _ 2 (a6 + a2 	 „^) + 	 (16)
8
1	 (1?)
Pg
B 2
2	 (18)
= G 3
3Gi	 1 f Cf( ugd +p )(3- b ) +B, (19)
and
d	 ( pp )	 (20)
Let
2
T = - 1 f
aLT8-
T  = (2 - 3b + 6gB)Ta
(21)
Tc = OT 
A
T = S + T
c
then
*	 TS 1	 Jri--	 (22)
(^G q
f
lPAW1W. psum
,.	 0
as
S lk	 as	 k = 1,2,3....,8k
From now on the subscript k represents partial derivatives
with respect to the 8 PSG elements, unless otherwise specified.
*	 1
	
CNq k _
_k 	 T
• S lk	 ^F1 T 
GG
i 
+ q
	
k	 (23)
^aG t q 
• (3Fi - fk f (^+ p) (3- b ) +[3+26k - f d+ pbk
`aG !	 2	 u	 Ik
+ f(3 - b)p k + f(I - b)u (dq k + qd £ )	 (24)
^3s
	 2 
J	
3S	 `	 aS	 2	 aS
•	 Tk = rap
 - p T
	
{^ k + 1 b^ - 3Ta ()1 t^ k 	ae2 ek t Va* + 2TbX) ^k
A	 `	 \
+ ax + 2Tb ^) X k - ^q T` - 6I3T a o) q k + 6T$ gOIIk	(25)
•	 Pk = 0	 for k = 1,3,4,5,7	 "
t> 2 = -2	 «2
_ 2 _311P
p 6	 11 °6
P8 = -2	 3/)
•	 qk = 0
	 for k = 1,3,9,6,7
q 2 = i Q2
q5 a 2
lq 4 
= - 2 
(2u ^ j!?
(26)
(27)
P- 14 -
•	 G k	0	 for k
	
1,3,4 , 7,8	 (28)
G 2 = - 02
G5=1
G6-06
•	 H  = Gk	 for k = 1,2,4,5,6,8
	 (29)
H 3
	- 03
H 7	
- a7
p 	 9k
• f	 - f P + q
	
(30)
It
• bk = _ 2H ( " k - G k )	 (31)
G
•	 dk=2 Guy _p k 	(32)
•	 Bk = ' 3 (4Hk 	Ll Gk )	 (33)
G
•	 ek = - 2a 8 "p k
	for k = 1,2,3,4,5,6,`7
2	 (34)
e8 = - 2a 8 pp - 2U1)
32(G + II)	 _	 Q
•	 Xk a (a60 s - 0702)	 2G	 Qk 4 Gk + 2(G + H) ( Gk + Hk)
k =	 4, 5, 8	 (35)
X _ 0"	 - a a )F2 +H) IQ - Q G+ --Q	(G +H )2	 <	 7 2	 2G	 2	 (', 2 2(G + H)	 2	 2
2G
2 6+H)
X3
 = (0603 - v7v2	 2G	 IQ 
3 - u G3+ 2(GQ + N) (G3 + H3)
.4
2 FG -+I I
+ ^6	 2G
15 -
X6 = (a6a 3
 - a7G2) 2 2G
+ 
H 
IQ6 g G6 + 2 G + H) (G6 + H6)
a 3Q 2(
+	 2G
	2(G + HT 	 QX 7 - ( a 6a3 - a 7a 2 )--- G^	 Q7 - -q G7 + 2(-	 )(G7 + H1)
a 2Q 2(G —+H
2G
2(
^k (a6a7 + 0
2a 3 )	 2G	 IQk - G Gk + 2(^+ H)(Gk + Gk)
k = 1,4,5,8	 (36)
2(G + H)
*2 = (a 6 a 7 + a2a 3 )	 2G	 IQ 2 - -9G G2 + 2(G+ H) (G2 + H2)
a 3Q 2(
+	 2G
2(
'^3 = (a 6a 7 + a 2 a 3 )	 2G	 IQ3 - G G3 + 2((G + H) (G 3 + H3)
a 2Qr2(G + 11)
+	 2G
i-	 ^6 = (^^ 6 « 7 + a 2 cJ 3 ) — 2 ---Y Q6 - ^ G6 + 2 (—G^'H)(G6 + H6)J
a 7Q 2( ry + H)
+ - 2G
	
- (a a + c a ) 2(__G + H) 	 _q G + (G7 + H7)
a 6Q 2(
+	 2G
I	 i --
F
- 16 -
The partial derivatives of 8 with respect to p b , e2
and X can be found in Reference 3.
3.3 Derivative of F2 with Respect to DSO Elements
From Reference 1, one can find that
2	
(3?)F2	 ^ s-r, 6 ♦ ri^tt
2
6	 e (- 3b 2 + 24b - 8) + 18 92 - 	 d 2 + U,
	
(38)
9	 /
(60b 2 - 96b + 32) - 131)(24o 2 + 36)
H is the Hamiltonian of higher harmonics, see Reference T
for detailed description. Because the new Hamiltonian is a
function of only action DSO elements, from now on the sub-
script k represents partial derivative with respect to
those DSO action elements.
6k	 12 C e 2q - e 2 g k )(- 3b 2 + 24b - 8)] + e
2 
C -6b + 24)bkJ
q	 q
bq It
	
2	 2	 2+ q-b (kq - 2 ) -	
(dk (
P + u) + d( ekp 	 a pk (39)
p
L	 2
+ uk ) (60b - 96b + 32) + d(p + u)(120b bk
 - 96bk)
- (24v 2+ :36) (B 1) + iilli k ) - 24v2Bb
where
$1 = 0
6H2B 2
 = - G4
(40)
^	 43	 3G
B 4	0
di
d2 1.
d3 0.
d4 - U(2L)-3/2
p l = - 2(P)4
vt
p2= -pl
'	 p3=0.
P 4 - -2(pu)j1(2L)-3/2
2
e l = 2L- u pl
2
e 2 = 2-e,
e3 = 0.
e4 = - 2 ( p + Lp4)
q l = -0.5
q 2 = 1.0
q 3 = 0
q 4 = - 0.5U(2L)-	 3/2
b l
 = 0
b 2 = G(G)2
b 3
 = ...
	
Z. (Y
b 4 = 0.
(41)
(42)
(43)
(44)
I
(45)
L l
 = 0
L 2
 = 0
L 3 = 0
	 (46)
L4 = 1
-a.-,..,•aw ^ ,--rte"°"-ar^^a,_'E-_ :'_'°' ^^-..^...^ . ,.,. _ -,--..z_ . 4^	 r	 _ _	
-,,
-/
^t
- 
18 -
F2k	 288 (2fk6 + fd k ) + 9 (	 k8k - 9 N )	 (47
/^ 	 3	 A 2 30
Hk 	 3p pk + De2ek + 8b b 
where
2
f 1 = u ( Up + 2q 3u
_
p)
2
f2 = - u ( up + 2q up )
f,l = 0
1 ) +4	 ^^ ful ► )C	 — 3/2 ^ ul	 1(208 1
(48)
(49)
Now the abbreviations A 1 , A 2 , A 3 , A4 in the expressions
of analytical integration will be given:
2
A4 = 2 f4 (b - 2/3) + u(2L) -3/2 + 288 f (2f 46 + U )
2	 q
+ q 
(H4 - 
q4 H)
	 (50)
2	 2	 q
A3 = 2 fb 3
 + 288 f26 3 + y (H3 - q3)	 (51)
2
A2 = 2 [f ._(b - 2/3) + fb 2^ + 288 f (2f 2 6 + f6 2 )
'
+ q ( H2 _ q2 9) + A 
3 (52)
A i = 1 + 2 f l (b - 2/3) +
 288 	 (21' 1 6 + r(S	 )
2
+ 9 (1 - 91 li) + A2	
(53)
q
u^
i
3
i
r ^•	
b	
^°
4.0 CONCLUSIONS
The equations described in this report have been imple-
mented into the ASOP program. The program has been checked
out and verified with results documented in reference 8. Com-
parisons with numerical integrations show the long period
theory to be accurate to within several meters after 800
revolutions. The extension of ASOP to include the long
period terms, allows the solution to maintain a high degree
of accuracy even for extremely long prediction intervals.
ru PAai ^K No.
REFERENCES
1. Scheifele, G. and Graf, 0.: "Analytical Satellite Theories
Based on a New Set of Canonical Elements". AIAA paper
No. 74-838, Aug., 1974.
2. Bond, V.R.: "An Analytical Singularity-free Solution to
the J 2 Perturbation Problem", NASA/Johnson Space Center
Report JSC-13/28, 1977.
3. Starke, S.E.: "An Analytical Satellite Orbit Predictor
(ASOP)", NASA/Johnson Space Center Report JAS-13094,
1977.
4. Scheifele, G., Mueller, A. and Starke, S.: "A Singularity
Free Analytical Solution of Artificial Satellite Motion
with Drag", ACM Technical Report TR-103, March, 1977.
5. Mueller, A.C.: "An Atmospheric Density Model for Applica-
tion in Analytical Satellite Theories", ACM Technical
Report TR-107, Nov., 1977.
F	 6. Mueller, A.C.: "Perturbations of Non-Resonant Satellite
Orbits due to a Rotating Earth", ACM Technical Report
TR-112, June, 1978.
7. Mueller, A.C.: "Recursive Analytical Solution Describing
Artificial Satellite Motion Perturbed by an Arbitrary
Number of Zonal Terms", presented at the 1977 AAS/AIAA
Astrodynamics Confe> >nce, 1977.
8. Wang, K.C.: "Check Out and Verification of Long Period
Perturbations", ACM Memo. 189, Dec., 1978.
14
'S
f
- 23 -
A
PAGE BUWK WT FILWW
APPENDIX
COIIMPUTATIONAL PROCEDURE
The computational procedure for elimination of long peri-
odic terms and analytical integration of primed variables are
described below. First subroutine LONGPP(NN) (long period
perturbations) is called with parameter 0
	
it will return
initialized primed variable. During the procedure subroutine
DETERM is called to compute terms related to the higher order
harmonics. Then subroutine will be called again with para-
meter 1 , this time it will return the partial derivatives
of primed Hamiltonian with respect to the DSO elements.
Daring the procedure subroutine FPRIME is called to compute
derivatives of higher order harmonics. The sequence of com-
putation will be given below. The left column gives the
quantity to be computed, and the right column references the
equation number in the text.
LONGPP(0)
Computating Sequence	 From Equation
(20)
(18)
(10)
(19)
(12)
(13)
(14)
subroutine DETERM
f 
	 (30)
b 
	 (31)
d 
	 (32)
j
d
B
e
aF1
aG
X
e
Sk
- 24 -
r	 ,
d
B
e
Bk
dk
Pk
2
e 
q 
H 
d k
F"2k
A1'A21A31A4
subroutine FPRIME
(39)
(47)
(50) - (53)
(20)
(18)
(10)
(40)
(41)
(42)
(43)
(44)
Computating Sequence (continued)
B 
2
e 
BF1
BG k
Xk
*k
T 
Slk
a1 ( 0 ), p'(0)
From Equation
(continued)
(33)
(34)
(24)
(35)
(36)
(25)
(23)
(3)
LONGPP(1)


Long period perturbations of earth satellite orbits

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