Analysis of a perturbation solution of the main Krambeck, Scott D.

Analysis of a perturbation solution of the main Krambeck, Scott D.
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1990-09
Analysis of a perturbation solution of the main
problem in artificial satellite theory
Krambeck, Scott D.
Monterey, California: Naval Postgraduate School
http://hdl.handle.net/10945/34905
NAVAL POSTGRADUATE SCHOOL
Monterey, California
NrM
DTic
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ELECTE
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OCT 3 1991
jVSTArZS 4
CJADXU'J
THESIS
ANALYSIS OF A PERTURBATION SOLUTION OF
THE MAIN PROBLEM IN
ARTIFICIAL SATELLITE TH'IEORY
by
Scott David Krambeck
September 1990
Thesis Advisor
Donald A. Danielson
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ANALYSIS OF A PERTURBATION SOLUTION OF THE MAIN PROBLEM IN
ARTIFICIAL SATELLITE THEORY
12 Personal Author(s) Scott David Krambeck
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Engneer's Thesis
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September 1990
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sition of the Department of Defense or the U.S. Government.
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rield
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Oblateness, Perturbation, First Order Solution, Numerical Solution Comparisou, Measured Satellite Data Comparison.
19 Abstract (continue on reverse 1( necessary and identify by block number)
The development of a universal solution of the main problem in artificial satellite theory has only recently been accomplished with the aid of high powered computers. The solution to this long standing problem is an analytical expression that
is sinilar in form to the two-body solution. An analysis is presented in %hich the solution is compared with the tv'.o-body
solution, a proven numerical solution, and actual measured satellite data. The solution is shoNn to be sianificantly more
accurate than the two-body solution. The theoretical accurac.y of the solution is confined. The solution compares extremely
well with a proven numerical solution for at least 41 orbits Nith a relatihe error on the order of O. The solution compares
extremely well with measured satellite data for satellites in near Earth orbits. For a satellite hi orbit at an altitude of approximately 1000 kilometers, the solution reduces the error of the twso-bod, solution by about 9 5 "o. For satellites m orbit
at semisynchronous and geosynchronous altitudes, the solution reduces the error of the two-bodN solution b) at least 50%.
The solution is free of singularities and is valid for all eccentricities and inclinations.
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Donald A. Danielson
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Analysis of a Perturbation Solution of the Main Problem in
Artificial Satellite Theory
by
Scott David Krambeck
Lieutenant, United States Navy
B.S., Iowa State University, 1982
Submitted in partial fulfillment of the
requirements for the degrees of
MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING
and
AERONAUTICAL AND ASTRONAUTICAL ENGINEER
from the
NAVAL POSTGRADUATE SCHOOL
September 1990
Author:
Scott David Krambeck
Approved by:b6
Donald A. Danielson, Thesis Advisor
Reader
Deptmnt of Aero atcsa
DEAN OF FACULTY
AND GRADUATE STUDIES
Hi
Astronautics
ABSTRACT
The development of a universal solution of the main problem in artificial satellite
theory has only recently been accomplished with the aid of high powered computers.
The solution-to this long standing problem is an analytical expression that is similar in
-form to the two-body solution. An analysis is presented in which the solution is compared with the two-body solution, a proven numerical solution, and actual measured
satellite data. The solution is shown to be significantly more accurate than the two-body
solution. The theoretical accuracy of the solution is confirmed. The solution compares
extremely well with a proven numerical solution for at least 41 orbits with a relative error
on the order of P0. The solution compares extremely well with measured satellite data
for satellites in near Earth orbits. For a satellite in orbit at an altitude of approximately
1000-kilometers, the solution reduces the error of the two-body solution by about 95%.
For satellites in orbit at semisynchronous and geosynchronous altitudes, the solution
reduces the error of the two-body solution by at least 50%. The solution is free of
singularities and is valid for all eccentricities and inclinations.
COpy
I '0
DiU i t. n
TIB r
cml
I
-i
Avallabllt
.P
!
iiiRA"
IDi
t
a a
.
-
TABLE OF CONTENTS
-1. INTRODUCTION.............................................
II.
BACKGROUND .............................................
A. ORBITAL KINEMATICS........ ............................
B. EQUATIONS OF MOTION ...................................
C. SOLUTION ...............................................
D. SIMPLIFIED SOLUTION...................................
E. THE CRITICAL INCLINATIONS.............................12
F. SPECIFIC MECHANICAL ENERGY ...........................
111.
METHOD OF ANALYSI1S....................................
A. OR'3ITAL PARAMETERS....................
...............
1. Argument of Latitude (60) ................................
I
2
2
4
7
10
14
16
16
16
2. Radius Magnitude ( r)...................................18
18
3. -Inclination (i)........................................
4. Longitude of the Ascending Node (f2 )........................18
B. ROMBERG INTEGRATION TECH-NIQUE.......................19
IV.
22
METH-OD OF COMPARISON.................................
A. NUMERICAL INTEGRATION COMPARISON..................22
1. Delta Radius Vector................ I.....................22
23
2. Earth Arc Angle ........................................
24
3. -Delta Omega, Delta Inclination, Delta Theta ...................
24
4.- Relative Errors.........................................
24
5. Track Errors...........................................
B. MEASURED DATA COMPARISON ...........................
27
V. RESULTS ..................................................
A.
NUMERICAL INTEGRATION COMvPARISON..................28
I. Delta Radius Vector Comparison............................29
2. Earth Arc Angle Comparison...............................30
iv
28
3. Delta Omega Comparison .................................
4. Delta Inclination Comparison ...............................
5. Delta Theta Comparison ..................................
6. Delta Theta Relative Error Comparison .......................
30
31
31
7. Delta Radius Relative Error Comparison .......................
8. Radial Track Error Comparison .............................
9. Along Track Error -Comparison ..............................
10. Cross Track Error Comparison .............................
B. MEASURED DATA COMPARISON ...........................
32
32
32
33
33
31
1. Near Earth Orbit Comparison ...............................
2. Semisynchronous Orbit Comparison ..........................
3. Geosynchronous Orbit Comparison ...........................
VI.
CONCLUSIONS AND RECOMMENDATIONS
34
36
38
....................
40
APPENDIX A. NUMERICAL SOLUTION COMPARISON RESULTS ...... 42
APPENDIX B. NEAR EARTH ORBIT COMPARISON RESULTS
.........
70
APPENDIX C.
SEMISYNCHRONOUS ORBIT COMPARISON RESULTS ... 83
APPENDIX D.
GEOSYNCHRONOUS ORBIT COMPARISON RESULTS
APPENDIX E.
COMPUTER PROGRAM
' ..........................
..
94
105
LIST OF REFERENCES ..........................................
126
INITIAL DISTRIBUTION LIST
128
...................................
LIST OF TABLES
Table 1. A SCHEMATIC OF ROMBERG INTEGRATION................20
21
Table 2. ROMBERG INTEGRATION..............................
vi
LIST OF FIGURES
Figure 1. Spherical coordinate system ................................
Figure 2. Orbital plane . ............................................
Figure 3. Delta radius vector and Earth arc angle .......................
2
3
23
Figure 4. Track errors . ............................................
Figure 5. Delta radius vector (1 day) .................................
Figure 6. Delta radius vector (1 day) .................................
26
43
44
Figure 7.
Figure 8.
Figure 9.
Figure 10.
Figure 11.
Figure 12.
Delta radius-vector (3 days) .................................
Earth arc angle (I day) ....................................
Earth arc angle (1 day) ....................................
45
Earth arc angle (3 days) ...................................
Delta omega (I day) . .....................................
Delta omega (3 days) .....................................
48
49
Figure 13. Delta inclination (I day) ...................................
Figure 14. Delta inclination (3 days) ..................................
Figure 15. Delta theta (I day) .......................................
Figure 16. Delta theta (1 day) .......................................
Figure 17. Delta theta (3 days) .....................................
Figure 18. Delta theta relative error (I day) .............................
Figure 19. Delta theta relative error (I day) .............................
Figure 20. Delta theta relative error (3 days) ............................
Figure 21. Delta radius relative error (1 day) ............................
Figure 22. Delta radius relative error (1 day) ............................
Figure 23. Delta radius relative error (3 days) ............................
Figure 24. Radial track error (1 day) ..................................
Figure 25. Radial track error (I day) ...................................
Figure 26. Radial track error (3 days) .................................
Figure 27. Along track error (I day) ..................................
Figure 28. Along track error (I day) ..................................
Figure 29. Along track error (3 days) ..................................
Figure 30. Cross track error (I day) ...................................
Figure 31. Cross track error (3 days) ..................................
vii
46
47
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
Figure 32. Delta radius -vector (21 days) ...............................
Figure 33. Earth arc angle (21 days) ..................................
Figure 34. Delta omega (21 days) ....................................
Figure 35. Delta omega-(21 days) ......................................
Figure 36. Delta inclination (21 days) .................................
Figure 37. Delta theta (21 days) ......................................
Figure 38. Delta theta relative error (21 days) ...........................
Figure 39. Delta radius relative error (21 days) ...........................
Figure 40. Radial track error (21 days) .................................
Figure 41. Along track error-(21 days) .................................
Figure 42. Cross track error (21 days) ..................................
Figure 43. Cross track error (21 days) .................................
Figure 44. Delta radius vector (30 days) ................................
Figure 45. Earth arc angle (30 days) ..................................
Figure 46. Delta omega (30 days) ....................................
Figure 47. Delta inclination (30 days) ..................................
Figure 48. Delta theta (30 days) ......................................
Figure 49. Delta theta relative error (30 days) ...........................
71
72
Figure 50. Delta radius relative error (30 days) ...........................
Figure 51. Radial track error (30 days) .................................
Figure 52. Along track error (30 days) .................................
Figure 53. Cross track error (30 days) .................
...............
Figure 54. Delta radius vector (28 days) ................................
Figure 55. Earth arc angle (2S days) ..................................
90
91
Figure
Figure
Figure
Figure
56.
57.
58.
59.
Figure
Figure
Figure
Figure
60.
61.
62.
63.
Delta
Delta
Delta
Delta
omega (28 days) ....................................
inclination (28 days) .................................
theta (28 days) ......................................
theta relative error (28 days) ..........................
Delta radius relative error (28 days) ..........................
Radial track error (28 days) ................................
Along track error (28 days) ................................
Cross track error (2S days) ................................
VIIi
73
74
75
76
77
78
79
80
81
82
84
85
86
87
88
89
92
93
95
96
97
93
99
100
101
102
103
104
LIST OF SYMBOLS
A TE
along track error
a
semi-major axis
ao
initial value of a
B.
rotating orthonormal base vectors
b.
fixed orthonormal base vectors ( n = 1 ,2,3
CTE
cross track error
C
Cos io
e
eccentricity
eo
initial value of e
f
function
G
universal gravitational constant ( G = 6.662 x 10- 2 km3lkg-s" )
h
angular momentum
h0
initial value of h
I
integration result
i
inclination
io
initial value of i
J
normalized
(
n = 1,2,3
)
)
J.
3 12R )
2 p;coefficient of the n zonal harmonic of a planet's gravitational potential
Al
planet mass ( Earth, Al = 5.983 19478 x 10' kg )
O
center of the planet and the coordinate system
O
order
p
serni-latus rectum (p
PO
R
= I/GAV)
initial value of p (po = ao(l - c.)
equatorial radius of a planet ( Earth, R = 637S.137 kin)
R TE
radial track error
r
radial vector from the center of a planet to the satellite
r.
radial vector from the center of a planet to the satell'te for the reference
solution
J2
( J -
= a(l - e)
ix
(p
(po
=
h 2 /GM)
r
magnitude of the radial vector from the center of a-planet to the satellite
r.
magnitude of the radial vector from the center of a planet to the satellite for
the reference solution
S
satellite
s
sin i
T
t
specific kinetic energy
time
to
u
initial value of t
p.fr
V
specific potential energy
y
variable used in the r, i, and Q equations
Z
Greenwich mean time (ZULU time)
Oright
ascension angle
fi
declination or latitude
PO
Y
initial value offil
fixed direction of the-vernal equinox
A
a finite increment
0
argument of latitude
00
initial value of 0
O
argument of latitude for the reference solution
T
0
Earth arc angle
longitude of the ascending node
-20
to
initial value of Q
argument of periapsis
too
initial value of co
x
ACKNOWLEDGMENT
Sir Issac Newton once remarked, -If I have seen a little farther than others it is because I have stood on the shoulders of giants.- In completing this endeavor there are
-several giants I wish to acknowledge.
My special thanks to Professor Don Danielson for his encouragement, inspiration,
and patience as he guided me through this project. I will be forever grateful for his
confidence and trust.
Several words of thanks go to LTC Jim Snider, USA, and LT Chris Sagovac, USN,
whose research cleared the path such that this thesis could be attempted. Their numerous and timely suggestions were extremely helpful.
I must thank John Rodell from the Colorado Center for Astrodynamic Research
(CCAR) for producing the numerical data needed in this analysis. His quick response
to my requests we'e greatly appreciated.
I must also thank CAPT Greg Petrick, USAF, and IstLT Bruce Hibbert, USAF.
from the First Satellite Control Squadron (ISCS) for supplying the measured satellite
data needed in this analysis. I extend my sincere appreciation for their help and cooperation.
I convey special gratitude to my lovely wife Susan for her support, sacrifice, and
understanding which made the completion of this thesis a reality.
.'d
I.
INTRODUCTION
With the development of any new method or theory the question always arises on
whether the approach is valid or practical for ordinary use. This is particularly true in
the prediction of satellite motion. Ever since Sir Issac Newton's discovery of the law
of universal gravitation, new methods have been developed to better predict the motion
of the heavenly bodies. Usually the method contains one or more restrictions that limits
the practical use of the solution. The goal, of course, is to develop a solution that is
valid and possesses no restrictions. Recently, such a solution has been formulated.
This analysis continues the work that was begun by Snider [Ref. 1] and Sagovac
[Ref. 2]. From their research, a higher order universal solution of the motion of an
artificial satellite around an oblate planet was developed. The solution is free of
singularities aPd is theoretical valid for all orbital parameters. The purpose and scope
of this work is to compare the solution with proven numerical solutions and actual
measured satellite data in order to determine whether the theoretical work is valid and
practical.
The first chapter summarizes the development of the theory and presents the solution in its entirety. Also-included is a somewhat less accurate simplified solution. An
explanation of the solution near the critical inclinations is presented. The chapter concludes with a discussion on the conservation of specific mechanical energy. The next
chapter describes the method of analysis and explains the type of integration routine
exercised in the evaluation. The method of comparison is presented next in which the
error parameters are described in detail. The results follow which include both a detailed
discussion and a graphic representation.
The analysis is objective in nature and designed to demonstrate both the advantages
and disadvantages of the theory and the solution. Before the solution can be applied
extensively, a general understanding of its strengths and weaknesses must be determined.
II.
BACKGROUND
A. ORBITAL KINEMATICS
A reference system for a planet in spherical coordinates ( r, a, /3) is shown in Figure
1. The radial distance (r) is measured from the center of the planet (0) to the satellite
(S). The line (0 y) is in a direction fixed with respect to an inertial coordinate system.
For Earth, the line (0 y) is in the direction of the vernal equinox. The right ascension
angle (a) is measured in the planet's equatorial plane eastward from the line (0 Y). The
declination or latitude (P) is the angle measured northward from the equator. The position vector (r) of the satellite in the spherical coordinate system is:
r = r(coscccosfl) b,+r(sincccosfl) b 2 +r(sin3) b 3
(1)
where ( b , b2, b3 ) are orthonormal base vectors fixed in the direction shown.
polar axis
equatorialplane
Figure 1.
_3
b0
Spherical coordinate system.
A reference system for a satellite in polar coordinates (r, 0) within a rotating orbital
plane is shown in Figure 2. The satellite's position and velocity vectors are contained
within the orbital plane. The argument of latitude (0) is measured in the orbital plane
2
from the ascending node to the satellite. The inclination (i) of the orbital plane is the
angle measured between the equatorial plane and the orbital plane. The longitude of the
ascending node (f2) is measured from the line (0 y) to the ascending node. The ascending node lies on the line of nodes which form the intersection of the equatorial and
orbital planes.
orbital plane
equatorial .plane
Figure 2.
S
Orbital plane.
The basis (b, , b2, b3 ) may be transformed into another orthonormal basis
(B, , B2 , B3 ) by a succession of three rotations. First the basis ( b, , b,, b3 ) is rotated
about the b, direction by the angle Q. The basis is then rotated about the new first
coordinate vector by the angle i. The final rotation is about the new third coordinate
vector by the angle 0. The position vector (r) has only one component in the rotating
basis.
r = rB1
3
(2)
The components of r in-the fixed'basis are:
r = r(cosOcosf2-sinOcosisin2) b,
+ r(cos0sin 2+-sin 0cosicos 2) b2 +r(sin0sini) b3
(3)
Equating the components of equations (1) and (3), the following relations among
the angles ( a, f ) of the spherical. coordinate system and the astronomical angles ( i,
92,0 ) can be obtained.
sinf
cosfl
sin0sini
(4)
cos 0 sec(a - a)
(5)
=
=
The velocity (drdt) of the satellite is obtained by differentiating equation (2)*with
respect to time (t) which results in:
dr
-dt
-
dr B, + r do (I + tan Ocot d )B 2
dt
dt
-dOB
(6)
Equations (2) and (6) represent the orbital kinematics of a satellite in the polar coordinate system. The position and velocity vector expressions describe the motion of a
satellite in a particular reference system and provide the information needed to develop
the equations of motion in that system. These expressions are referenced to the true,
rather than mean, orbital plane and were originally formulated by Struble
[Ref. 3,4,51.
B.
EQUATIONS OF MOTION
The motion of all objects is mathematically described by the equations of motion
that govern them. For an oblate planet, the expressions for the kinetic and potential
energies per unit mass of an orbiting satellite in spherical coordinates are respectively:
L[Or
V
) ++ 2
G
I-- +
4
22
(l - 3 sin'f)
ds(
(
(8)
In the above equations,
(M)
is the mass of the planet, (G) is the universal
gravitational constant, (R) is the equatorial radius of the planet, and (J2) is the coefficient of the second zonal harmonic of the planet's gravitational potential. The governing equations of motion can be determined by substituting equations (7) and (8) into
Lagrange's equations which are represented by:
d
dt
[(dq" 1 aq
---
(T-
(T-J'
=
0
(9)
~dt}
where:
q = a, r, and fl
Three equations result from Lagrange's equations which describe the motion of th .
satellite. The three equations are:
(
2
~
dt
d 2, r(dfl)2
dt2
=00
cos
rcos2fl
dt
)2
dt}
dt )
=
d( r2 dl+ ) ,J
d \
o,.
r2sin Pcosf (dh
-
=
ar
(11)
(12)
(12)if
From the equations of motion, two integrals result which are:
2
2ol
r Cos
dc
dt -2#-!
=
constant
T+ V = constant
(13)
(14)
Equation (13) results from integrating equation (10) and equation (14) simply states
that the specific mechanical energy of the satellite remains constant. To change the independent variable from i to 0 , equations (1) , (2) , and (6) are used in conjunction
with equation (13) and some initial conditions to form:
5
dt
r2 dos i
dO-
Ocosi 0
II +d
+tann-
-cot
-dO
(15)
'Letting u p,!r, and using equation (15), the remaining equations of motion (11)
-
(12) can be rewritten as:
di
dO
-
2 sin 0 cos 0 sin i cos 2i
2Ju
)(6
+ 2Ju sin 20 cos3 i
2
+U
d.
-3
2
, J2u
co2i
o2i [2
• 0 cos
CiCs AU Cs
2d2U4
[ --d sin
0 (3co~-1)dO
,.u -- sin 30 Cos OCos i (3 -cos2i ) /{c a + 4Jic2 sin 20 cos4i (17)
+ 4u j sin 0 cos 8 i}
d2
dO
where:
tan 0 di
sin i dt
(18)
c = cos i0
s = sin i0
3J2 R2
2 2
po
Equation (18) results from uncoupling the equations for L2 and i. The angles 0
and i can be uncoupled by applying the fact that the orbital p!ane must contain the
velocity vector. The differential equations (16) - (18) are coupled by nonlinear terms and
apparently cannot be solved analytically. If the right sides of equations (16) - (18) are
expanded in a Taylor series expansion in powers of J, the equations simplify t':
(Ii
dO
=
dO
-2.u sin 0 cos
2
3 0Ocos 0+ O(J3)
c2 0 sin i cos 3 + 4Jlusc3
sin
C2
~
6
(19)
cos 2 i
dd2u
02
U
J cos2 i
2
-4u sin2 0 cos 4 i
2
2
c2
2 zu--sin
)2-2. d
3)' •2_
si Ocos
0uco 00(d-3C
+
(-3co2.1
dO
2sin20Cos 2i}
4J?u-sin2O coi20' {u2 [3sin20 (I - 2 c6s2i)- 1]
4
+
+ 3u sin20:cos
4i
C
2
+
3'cOs 2 i)1 +
du2sin 0 cos 0 E4-'sin i+Isir20 a
U -T
(-L
\2
sin2O cos 2i
+ 0(J3 )
dO)
-
dA2
L+
(20)
_
2Juc sin20 + 4j2u2c 3 sinl0 + 0(J)
(21)
dO
Each of the neglected terms in equations-(19) - (21) are indicated by the (0) symbols.
These are terms which will be multiplied by J to the third power or higher. Equations
-(19) - (21) are identical to those used as a stai .ing point in the analysis of Eckstein, Shi
and Kevorkian [Ref. 611.
C. SOLUTION
The initial breakthrough of an analytical solution of the equations of motion represented by equations (15) , (19), (20) , and (21) was obtained by Danielson arid Snider
[Ref. 1, 71 . Further refinement of the solution was later formulated by Danielson,
Sagovac, and Snider [Ref. 2, 81 . The thlice authors, developed the solution through the
extensive use of an algebraic manipulation computer program called MACSY*1IA.
Through the use of an algorithm, MACSYMA was able to solve for the variables u (
or r ),i, £2, and dildO in terms of 0. T. solution includes all terms multiplied-by J
and excludes terms of order J and higher. in Z.
r to maintain a solution of ord.r J
when 0 1/J, the solutions also need to apprupriately include terms multiplied by J'0.
The solutions which analytically dcmonstratc .,relative accuracy of order J'0 are:
7
cosy + J
Po{1+eo
o
-i['o/S
4
eo [15(2 + e)s
+ 52
24-] sil
- 14(4 + e"+
q." -v2-
+ "TE(6 +
e2 (5S
(3S
2)cos(2y- 20)
2)]
2-(
- 2)] sin[O + wol
4)
+ eeo-,4(j 00 + w~o) + "-"Cos("0 - coo)
2
2
14) sin[.+3(0O
so-1-6s2cos22
2
+
-
+--[(5eo
2)
2
- 2eo~]
cos(y + Oo + co) + eos 2 cos(0 o + wo)
22
22
) - 00Ro) +e6
"5
o s cos(
eos
e
--
cos(y -0 0 + 3coo) +
2
0
(32 - 3 2)cosy
(2 3s2)
- cos(y-3
-2+ 2o
0+
0o )
cosCy +
200)
4'
2
+ --
~)cos~y
(4 -
- 00-
coo) +
(6-Is) cosy +
-
1-
3eos
8
+
-4
2
(2 + 5eo) 2 cos 20 + 2 (9os 8)
2
e0
cos(y- 400 + 2coo) -T(s
[2e-
-1)
S2-o 200
2
-
[eo
(2
+ o20)
(2- 3s) cos(2y +20)
+~
+
2-
-
' 20 +- (92)cs2+) osy-
[
+ 1)cos(y+2o)
(14+ 5eo)S 2 .- cos(y - 300 +
8
2y
wo)] + O(J2,
,
(2
0
y
-
2)(0- 00)
co+J{(S-
2
2
22
2
__
2
13)
-2 (5s
+
96
0
2wo
_1)cs
4-15'(5
20s Ce
S
2
)
+w)+-- 6o( (15S -13) cos(300
[5(9eo' + 34)s + 4CU- 34)s
-
o(j 2,j 30,...)
56e]} +
I= io+scj -cs20-+.-cosy+26)+-ycos(y-2)-
cA4-- 15S)
sinJ(
5s2
212(5s
-1
+
e-6s
D a
4
sin
4s
15S4~
L~
2 -4)
sin2
-e 0 - )
JO-L
2)]
(4
o[coJO-L-)
osifn(300 -sino)y-
0
)sin(o-+sin)}
62
2
4
2
1so
12(s
-
12(4
+2SsinJ(2o + w)]
020o+
0
2
+
2)] sinl[2wo
cos 200
w
~
o(
0
+)+-j-(7
2
-4
62
+-j0
eo4
3
s2)}c? ,)-L(7SJ3
)osI(o-
9
)
2
5)
2 s2
+
Ls
++ °( -s)cosy.-20)
2
+
D.
2
(26)
20eo(3 - 4S)
's2 0 0+:
2
cosO + 20)
6
2
2
+
2
-0)]+
o(J2,3o,_...)}do
2 cos(0 0 +
SIMPLIFIED SOLUTION
As shown in equations (22) - ('";),if0 is restricted such that 0 1/J, the solution
should be of order J and the nglected terms should be of order j2. For an Earth satellite, J<312 x 10- ', so for at least 100 revolutions the-relative error should be on the
order of 10-6 . If 0 is restricted such that 0 < 1 , all of the terms of order 0 in
equations (22) - (26) can be neglected if a relative e-ror of order J is desired. By neglecting terms of order JA , the solution simplifies considerably to:
r = po Il+eocos 0-oo+J(0-0o)-"--2)
e
+ JVl-"7+3s
rE
/&20
1 ---'" )
1
Lr2e-(2+5eo)s21cs20
2
+
2
'e°
(9s 2 - 4) cos(O + 300 - 2Co) + -
(6 - 7s2) cos(O + 0o - 2coo)
2
+
22
-
(6
SS)
cos(0-
22
eo
s_ cos(0
16
0
0
2
02w) co.
+ . _( 2 cos(O1,
)- 30o + 2
o)
00 + 2wo) + --- (3s2 - 2) cos(O - 200 + coo)
24
-00+o)
16 cos(O - 500 + 2o) + 7 ( -2)cs
-
2 ) cos(O-cvo)
+o)3s
-4
4 (I- 3
L
) cos(O
- 200 - coo) + "" (2-3(2
10
,
(27)
~0_(9S2
+1
+
cos(20
-8)
-
24
2(o0 ) +L-(6
IS
1-s2 cos(30
2eo'J cos(0 + 0 0) +,eoS 2 COS(0 0
1[(5eg2 - 2)s'
2o
coo)
-
_ WO)
2
+2(2 -3s2) cos(40 + 2co) + 0_- (3S 2 -2)cs2o
3eos2
e
2
- 8- .-cos( - 40 0 + w)- -T-(S+ 1)cos((0
-L ET2eO2
-
(14 + 5eo)
2
cos(0
300)+S2 cos 200
+ -L-E(6-+ 5e02S2
) - 4(1 + e02)] cos(0 +
eoS
00)
(300 - coo)J + O(J2, j20
io + scJ[-Lcos 20 + -L-cos(30 -wo) +eo
-
1
~cos(30 0 - w0 )- --- cos(00 + coo)
_+
cos(0+ w)-
2
O(J2,Jj0,.)
L
+
0)
eo sin(0 + wo)
2
-
cos 20 0
(28)
2
!o
1 sin 200 + eo sin(00
2~
-
co ) 0
0)
i(0
n(36
-
w0 )
(29)
2-sin(0 0 + coo)]
t
10
-
Jor
0 21+j[ (2-3s2) cos 20 + e(s2 _ 1) COS(0- COO)
+o(O6
co(0
0)
+
2
2
-e cs 20o
2
s
1-
CS(Is0
(30)
E. THE CRITICAL INCLINATIONS
As shown in equations (22) - (26), the solution appears to be well bounded for almost all inclinations. However, two particular inclinations immediately appear that may
produce a singularity. A possible singularity occurs when the inclination is equal to either 63.43 degrees or 116.57 degrees. These two inclinations have produced mountains
of literature and are well known as the critical inclinations. However, if the solution is
replaced by the limit as the inclination approaches either critical inclination, the solution
remains finite. More specifically, the solution at either critical inclination is as follows:
r=
po{l+eocosy+J[
12S2
+ 7 [(6
+ 5e0)s
+
22
-
3+
el_5s)
eo12
2~
4(1 + eo)]
cos(y - 00 + coo) + -- cos(300 - o0o)
g 2 -4)coso, + 360 -oo) +
(
-
7S) cosC, + 0o _ (to)
+
+ 1[(5eo2 - 2)S2 -2
2]
2
cos(y±+0
0 +coo) + eoS cos(0+Coo)
22
eos
16 cos(y - 500 + 3oo) + - eo (3s 2 -2) cos(y - 20 0 + 2coo)
-
-
2 2
eos
16cos(y - 00 +3
e2
2
+24o(3s 2 2) cos0y - 300 +3vo)
e°0 -3S) COSY + S2
^S2
3s2) cosCv - 200) + - -(2- 3
Cos 200
e°
(I 2
" (4- 5)
.-(6
24
24
-
(2-
+
)cos- + 200)
Is2)cos(y + 20) +-!- (2- 32) cos(2y + 20)
12
2 2 -(14+ 5e)s 2 1 cos
+3eos[2eo
--
8
cos0y - 400 + 2coo) -
+ "(s-8
cos
-T 0S+1coy+2&0
C
-
24 E2e02-(14+5eo) -cosO,,- 300 +oo)]
12
(31)
+
j 20[ " O-[15(2 + e2)s 4
- 14(4 + eo)S 2 + 241 sin(O + wo)
24
2 21
+
12 (15S2 - 14) sin 2cooj + o(J2 , j6,.
y = o-co+,{(-
--- 2)(O-Oo)}
2
+
2
{4 (130s 2 - 105s4 - 28) cos 2wo + -2- (15s 2 - 13) Cos 200
2
2
+ 2L (15
-~-1S
--
i=
(32)
2 -13
_ 13)c~s(Oo+oo)+--L-(15s
c(0+
)e
6 S2
E5(9eo
13) cos(30 0
-
-
coo)
+ 34)s4 + 4(9e02 - 34)s2 - 56e2j} + OJ,
io + scJ
cos 20 + -L- os + 20) +--cos(y-2)
2 Cos 200 -
-!o
-
cOs(30 0 - COO) -
cos(Oo + CO)
(33)
2
(14 - 15s2) sin 2o o + O(
+j20
+0-
=
0
+-sin
i
2 0-e
y+ o
o siny
sin 200 + co sin(Oo - coo) -
2
_,eos
o
J0,J...)
sin(y+
- coo)
___ -
2 0)--osin(
2_sin(o + oo)
-20)
(34)
2
eo s3s(30°
-
too)
-
2f j
+
s_
2,
s 2 cos
200 + - (6 - s2)l + O(J 2, J'0, ..
(2-
-S2)
+ OS
CSY
2
e°(l
s2) ' s
e°(3 - 4s2) ofv..2)
2
cos(, - 20) -:1--cos 20 0 +
6( - -cos + 20)
13
3)
(35)
+ o cos(Oo + o0) + LO2cos(30o - coo)
2 2
+ J20 °s (15214)sin2(o0 +O(J2,3O,...)dO
Clearly, equations (31) - (35) demonstrate that the solution is indeed finite for both
critical inclinations. Equations (31) - (35) are only valid for the critical inclinations and
were first developed by Sagovac [Ref. 2] . The primary purpose in developing these
equations was over a concern in computer programming. Some computers have major
problems when a denominator approaches zero, and unlike humans, will not replace a
solution with its limit. Therefore, depending upon the accuracy of a computer,
equations (22) - (26) can replace equations (31) - (35) for inclinations near the critical
inclinations. It should be noted, however, that the solution itself is -valid and bounded
for all inclinations. It is the limitation of the computer that creates the singularity.
The-simplified solution which is shown in equations (27) - (30) , is valid for all inclinations. Since all terms of order JAO have been neglected, the troublesome denominators mentioned earlier do not appear.
F. SPECIFIC MECHANICAL ENERGY
For all satellites under the influence of conservative forces, the specific mechanical
energy remains constant. Therefore, an ideal analytical check of the solution would be
to see if indeed the specific mechanical energy at any time is a constant. This simple
check was performed by Danielson, Sagovac, and Snider [Ref 1, 2, 7, 8] by substituting
equations (22) - (26) into equations (7)and (8). The substitution yields:
T +
Gjlf(l2 - eo)
po
G!IJR2 (l - 33sin 2flo) +
2[r(1o)]
O0j2, 3
)(
..
6
(36)
The first two terms on the right side of equation (36) represent the initial specific
mechanical energy. All other terms multiplied by J in equations (22) - (26) combine to
zero when substituted into equations (7) and (8). Equation (36) demonstrates that by
neglecting all terms of order J2 and higher, the specific mechanical energ at any time
14
is precisely equalto the initial specific mechanical energy. Obviously, the solution satisfies the requirement of constant specific mechanical energy.
15
'II.
A.
METHOD OF ANALYSIS
ORBITAL PARAMETERS
1. Argument of Latitude ( 0)
Figure 2 illustrates that the position of a satellite at a particular time can be
described by the argument of latitude (0) , the radius magitude (r), the inclination ('),
and the longitude of the ascending node (92) . As shown in both the solution and the
simplified solution, r, i, and i are only functions of J and the argument of latitude (0)
Since J is a constant for all planets, a simple determination of these terms is trivial once
0 is known. However, the determination of 0 is not trivial. Although it would be ideal
for all of the equations to be analytical expressions, equations (26) , (30) , and (35)
contain an integral that must be evaluated in order for 0 to be determined. Herein lies
the key to the solution. Given an elapsed time between observations, how can d be
precisely determined? Since the initial angular momentum (h0) is known, this term can
be moved to the left side of equations (26), (30), and (35) to yield equations in the form
of:
(t - to)ho
J'r2(J, 0)(l +f(J, 0))dO
(37)
00
If r was not a function of 0, an evaluation of the right side of equation (37)
could easily be conducted that would yield an analytic expression. However, r is also a
function of 0 and the only practical techni.ue in evaluating the integral is through numerical means.
Several numerical methods could be used to evaluate the integral depending on
the speed and accuracy one desires. Since accuracy and not speed is desired in this
analysis, a Romberg integration routine was used to evaluate the integral. Since the
right side of equations (26) , (30) , and (35) are sinusoidal in nature, the Romberg
-scheme converged quickly and accurately.
16
Since 0 defines the upper limit of the integral, in order to arrive at a solution,
an initial 0 must be estimated. Once 0 is estimated, the integral can be numerically integrated and the result can be compared to the left side of equation (37). If the comparison is accurate within some predetermined error, the iteration is complete and 0 has
been determ-ined. If the comparison produces an error that is unacceptable, 0 can be
incremented either up or down and the integral can be reevaluated. Eventually, the iteration will converge and 0 will be determined. An algorithm of the iteration procedure
is as follows:
1. Estimate 0.
2. Evaluate the integral.
3. Compare the result with the left side of equation (37).
4. If outside the limit, go to (5). If within the limit, stop.
5. Increment 0 up or down as needed, go to (2).
The determination of 0 involves a combination of two errors. The first error is
contained in the numerical evaluation of the integral itself, while the second error involves the comparison of the result of the integration with the left side of equation. (37).
Unfortunately, the errors do not linearly combine, but rather multiply since the numerical evaluation of the integral is inherently nonlinear. In order to make the comparison
error meaningful, the evaluation of the integral must be made as precise as possible. In
order to avoid determining whether an error is due to computing or truncation errors,
the numerical technique used in this analysis did not rely on a step size constraint.
Therefore, the relative error, in general, can be specifically controlled. Since in this
analysis, accuracy and not speed is desired, the Romberg integration technique was utilized. The Romberg technique does not depend on any specific step size and the evaluation of the integral is determined through a converging algorithm. Also, the relative
error of the integration can be specifically controlled.
In general, the relative error
normally demanded in the integral evaluation was on the order of 1012, and the relative
error of the comparison was on the order of 10-1. Since the computer program utilized
in the analysis was written for double precision accuracy, these types of relative errors
presented no significant problems. The double precision accuracN enabled the computer
program to calculate up to sixteen digit precision.
17
2.
Radius Magnitude (r)
From equations (22), (27), and (31), it can be seen that the radius magnitude
(r) is a function or J and 0. Once 0 is known, r can be evaluated. From the appearance
of equations (22), (27), and (31), it is not obvious how r will behave as the orbit-of a
satellite -progresses. However, from observations of actual satellite motion, it is clear
that the orbit should behave elliptically with r varying from a minimum value at
periapsis to a maximum value-at apoapsis. The magnitude oLJ plays an important role
and fortunately for most planets, oblateness effects act as a pertuibation in comparison
to the main gravitational force. Therefore, a large value of J causes iargcr v riationsin
r. Since equations (22), (27) , and (31) contain a number of sine and cosine terms, a
sinusoidal behavior should be expected.
3. Inclination (i)
The solution of the inclination is shown in equations (24) and (33) , and the
inclination for the simplified solution is shown in equation (28). In general, these three
solutions are quite similar. Again, once 0 is known, i can be evaluated easily. It can
be seen from equations (24) , (28) , and (33) , that i will vary slightly from an initial
-inclination as the orbit of a satellite progresses. Also, since a number of sine and cosine
terms are present, the variation should be sinusoidal in nature. From inspection it is
clear that the magnitude of the variation is dependent upon-the magnitude of J and the
initial inclination. The variation of the inclination should not behave in a diverging
fashion, but rather in an oscillatory fashion about some arbitrary mean inclination. This
behavior is consistent with observations of actual measured satellite data. The driving
factor in all inclination variations is the magnitude of J. Since for Earth, J2 is on the
order of 10- , these variations should be quite small.
4.
Longitude of the Ascending Node ( 2)
The solution of the longitude of the ascending node (Q) is shown in equations
(25) and (34) , and the longitude of the ascending node for the simplified solution is
shown in equation (29). As expected, all solutions are quite similar. As with the case
of r and i, 92 can easily be determined once 0 is known. Unlike the behavior of r and
i, the variation of 92 is very predictable and highly meaningful. With the presence of 0
alone in equations (25), (29) , and (34) , Q possesses a linear relationship with 0 and
18
as 0 increases with time, the vaiiation of 92 from 0 should be linear. Depending upon
the initial inclination, this variation will be either positive or -negative. This type of behavior is clearly consistent with the classical behavior known as nodal regression. For
an oblate planet, nodal regression is a linear property whose magnitude and direction
depends upon the radius magnitude and inclination of the-satellite. In equations (25),
(29) , and (34) , the radius magnitude is contained in the J term. Therefore, the magnitude of the nodal regression is entirely dependent upon the magnitude of J. From the
analysis of the behavior of Q as 0 increases, the nodal regression behavior should be
extremely obvious.
B.
ROMBERG INTEGRATION TECHNIQUE
The Romberg integration technique is a powerful integration method in which arbitrary accuracy can be achieved in a relatively efficient manner. The method combines
any type of relatively inaccurate quadrature method -with a Richardson extrapolation in
order to quickly -and accurately converge on a solution. In this analysis, a simple
trapezoidal quadrature was initially used to estimate the integral and then a Richardson
-extrapolation was used to improve the integration to the desired accuracy level. The
trapezoidal quadrature first estimates the integral with a single interval. The estimate is
then improved by using 2 intervals, 4 intervals, 8 intervals, etc. For purposes of identification, the results can be labeled I.' , Io, , and so on. These results can be arranged
in column form in preparation for a Richardson extrapolation and each new member
represents the technique of halving the prior interval. The length of the column is determined by the accuracy that one desires. Once the first column is arranged, a
Richardson extrapolation can be performed by the following equation.
1n
-
I1n _ 1nh2
4 1- 1
-1
The values of 1' can be arranged in tabular form as shown in Table 1.
19
(38)
Table 1. A SCHEMATIC OF ROMBERG INTEGRATION
0
12
I4
14
14
10,
1
12
13-
421
131
1616
0l-
1616
14
To test for convergence, the value represented by I," is compared with the value
represented by Ifj . If these two- values are within some predetermined error, then 4"
becomes the evaluation of the integral. If convergence has not been reached, then another row is calculated and the process continues. An excellent example of the Romberg
integration technique is shown in Ferziger [Ref. 91 . In 'this example the following
solution of the integral is desired.
I =
fied
(39)
From elementary calculus, the exact solution is:
Iexac
= 2.718281828
(40)
The technique of Romberg integration of the integral is shown in Table 2. The relative error of I, to I is 7.81 x 10- '. The relative error of I3 to I,, is 1.97 x 10- .
As can easily be seen, the integration is converging very nicely and the error found in the
final solution is less than the error demanded within the Romberg integration scheme
itself
20
Table 2.
ROMBERG INTEGRATION
1
1.859140914
-
2
1.753931092
1.718861152
4
1.727221905
1.718318842
1.718282688
8
1.720518592
1.718284155
1.718281842
1.718281829
The advantage of the Romberg integration technique over a simple quadrature
method is obvious. The number of intervals that must be evaluated is very small and
the relative accuracy is very high. In order to attain the accuracy that the Romberg
technique delivers, the trapezoidal method would need to divide the integral into several
more intervals. This would be highly inefficient. For smooth functions, the Romberg
technique is very effective and efficient. Since equation (37) is sinusoidal in nature and
thus relatively smooth, the Romberg integration technique was used to evaluate the integral. If equation (37) had not been so well behaved, another integration technique
might have been warranted. The Romberg integration scheme is the heart of the analysis and can be found in the computer program shown in Appendix E.
21
IV.
METHOD OF COMPARISON
A. NUMERICAL INTEGRATION COMPARISON
In order to verify that-the theory is valid for practical application, the solution must
be compared with proven numerical solutions and measured satellite data. By comparing the solution with a numerical integration of the equations of motion, theoretical accuracy can be specifically determined. As shown in equations (22) - (26), theoretically
the solution is accurate to order J 20 . A numerical integration comparison will determine
whether this prediction is correct. In order to verify the solution, the following parameters will be compared.
1. Delta.radius vector (ArI)
2. Earth arc angle (T)
3. Delta omega (A92)
4. Delta inclination (Ai)
5. Delta theta (AO)
6. Delta radius relative error
(I Ar I/r)
7. Delta theta relative error (A00I)
8. Radial track error (RTE)
9. Along track error (,ITE)
10. Cross track error (CTE)
1. Delta Radius Vector
A graphical representation of the delta radius vector (IAr[) and the Earth arc
length (T) is shown in Figure 3. The delta radius vector is the magnitude of the vector
separating the solution radius vector (r) from the numerical integration radius vector
(r) . Mathematically, the delta radius vector can be expressed as:
Ar = r-r.
22
(41)
The delta radius vector describes in overall terms-the global error in the solution.
Another common name for this error is the Euclidean normed difference in ephemerides.
Although the delta radius vector provides ample information on the global error in the
solution, this error can -also be expressed by a different parameter that will be called
Earth arc angle.
Ar'
C
C,
BO
Figure 3.
2.
Delta radius vector and Earth arc angle.
Earth Arc Angle
Earth arc angle (TI) is simply the angle between the two positions if viewed from
sea level on Earth. For simplicity, the position at sea level was chosen such that the arc
angle from the center of the Earth was bisected. By using the law of sines and cosines,
the Earth arc angle is easily determined. Since satellites are tracked b instruments on
the surface of the Earth, a bet.er feel for the global error can be attained by determining
the angle between the two positions. Most satellite tracking radars possess beamwidth
23
and field of view limitations; therefore, 'V will provide useful information on whether the
solution is accurate enough fot satellite tracking radars.
3. Delta Omega, Delta Inclination, Delta Theta
A break down of the global error can be described in the errors of delta omega
(AK2) , delta inclination (Ai) , and delta theta (AO) . As mentioned previously, AQ2 will
provide an insight on the motion of the line of nodes,-and specifically-if nodal regression
is present. The change in inclination, will provide information on the movement and
stability of the -orbital plane. The parameter in which all errors are based, AO , will
provide much information on the source of the global error. It is clear that small errors
in AO will contribute significantly to the accuracy of the solution.
4. Relative Errors
The verification of the -solution will lie in the confirmation- of the relative errors.
The delta radius relative error and the delta theta relative error will demonstrate the actual accuracy of the solution. Both parameters, Ar and AO , demonstrate a theoretical
error of J2O. Therefore, the delta radius -relative error should be on the order of J 20,
while the delta theta relative error should be on the order of J 2 . Comparisons of the
relative errors between the sujution and- the numerical integrati.in solution will provide
the evidence for theoreti.a! confirmation.
5. Track -Errors
Another michod to break down the global error is in terms of track errors.
Figure 4 shows a graphical representation of radial trac., error (RTE) , along track error
(ATE) , and cross track error (CTE) . These errors car better be described by referring
also to Figures I and 2. The radial track error is the error in the radial direction or in
the B, direction. The along track error is the arc length error in the plane defined by the
solution radius vector (r) and the B. direction. Together, these errors describe errors in
three orthogonal directions or planes as compared with some reference position. The
reference position in this case is the numerical integration solution. A mathematical
derivation of these errors is as follows:
Ar = r(r + Ar, 0 + AO, i + AO, 2 +A
24
)-r(,, 0, i, 2)
(42)
Using equation (1),
Ar = (rn + Ar)(B 1 + AB1) - rnB 1
(43)
Ar = rnB 1 +ArB 1 +rnAB 1 +ArAB 1 -rnB 1
(44)
Neglecting-higher order terms,
Ar = ArB + rnAB1
(45)
AB1 = (B1 ABI)B I + (B2 . ABI)B 2 + (B3 . AB1)B3-
(46)
Continuing, defining AB1 ,
Using the rotation transformation and after performing considerable algebra,
B1 • AB1 = 0
B2 .AB
(47)
= (AO + AQ cos i)
B3 AB1 = (Ai sin 0 - A
cos 0 sin1)
(48)
(49)
Therefore, using equation (45),
Ar = (Ar)B1 + rn(AO + AK2 cos i)B2 + rn(Ai sin 0 - Af cos 0 sin )B3
(50)
From equation (50), the track errors can easily be defined.
RTE =
(51)
ATE = r(A0 + Af2 cos i)
(52)
CTE = r(Ai sin 0 - An cos 0 sin ti
(53)
25
A graphical representation of the track errors is shown in Figure 4.
ATE
~CTE
R TE
0
Figure 4.
Track errors.
Examination of equations (51) - (53) demonstrates that the track errors divide
the global error into three distinct regions. Radial track errors obviously describe errors
in the radial direction. Along track errors are similar in nature to Earth arc angle errors,
but also include errors due to nodal regression. Cross track errors describe orbital plane
errors in terms of both inclination errors and errors due to nodal regression.
In general, all these parameters should give an excellent insight into the accuracy of the solution. Also included in the numerical comparison will be the simplified
solution and the two-body solution. The simplified solution has been previously presented. The two-body solution can easily be determined by simply setting J = 0 . The
analysis of the numerical comparison wiHl demonstrate the strengths and weaknesses of
all the different solutions.
26
B. MEASURED DATA COMPARISON
In order to observe how well the solution models actual satellite motion, the solution will be compared with actual measured data from operational satellites. To
properly evaluate the solution, a wide range of orbit characteristics will be compared.
These characteristics include orbits of various altitudes, inclinations, and eccentricities.
Also included in this comparison will be the simplified solution, the two-body solution,
and if available, some particular numerical solutions.
The numerical solutions will consist of two forms. The first is a numerical solution
that only includes perturbations involving -2 , -3 , J4 , and Js . The second numerical
solution will include the following perturbation effects.
1. J2 ,J 3 ,J4 , and
Js
2. Atmospheric drag.
3. Sun gravitational effects.
4. Moon gravitational effects.
5. Solar pressure effects.
From the analysis, the accuracy of the solution and the simplified solution can be
compared to a numerical solution as well as to actual measured data. The weakness of
the two-body solution will also be demonstrated. In addition, the strengths of a well
modeled numerical solution will clearly be seen. The identical error parameters described
in the previous section will also be used in the measured data comparison. From this
comparison, the advantages and disadvantages of the solution in regard to actual satellite motion will clearly be demonstrated.
27
V.
RESULTS
A. NUMERICAL INTEGRATION COMPARISON
To verify that the theory is valid for practical applications, the solution was coinpared with a proven numerical solution. A numerical integration computer program
called UTOPIA is currently in use at :he Colorado Center for Astrodynamic Research
(CCAR) located on the campus of the University of Colorado, Boulder, Colorado.
UTOPIA is primarily used to model a wide range of perturbations and can predict satellite motion with a high degree of accuracy. The UTOPIA computer program wa, developed at the University of Texas, Austin, Texas, and is currently in use at several
universities and research centers. The solution was compared with the UTOPIA solution for a satellite with the following initial conditions:
r =
7,3S6.IS km
io = 90.03 degrees
eo
wo
00
no
= 0.003991
= 224.3S degrees
= 104.05 degrees
= 322.63 degrees
AD
= 54,205.IS km 2Is
Po =
to =
7,371.29 kin
0.00 seconds
In general, these initial conditions represent a slightly retrograde orbit of small eccentricity at an altitude of approximately 1000 kilometers. Essentially, it is a polar orbit
at an altitude where several satellites are currently in motion. From the initial conditions, the orbit should demonstrate a slight easterly nodal regression. But, since the inclination is so close to 90 degrees, some integration routines might predict zero nodi
regression. In this comparison, UTOPIA only modeled the J. perturbation; therefore,
the solution should compare well if the theo*y is valid. All error parameters depicted in
the comparison were calculated in the following manner.
A(ErrorParanzeter) = TheoreticalSoluion - UTOPIA Nmncrical Sohion (54)
2S
All solutions were compared at one hour intervals over two separate periods of time.
One comparison is for a time period of one day while the second is for a time period of
three days. The three day comparison was constructed to illustrate the effect of long
term errors while the one day comparison allows for a more detailed analysis during the
first few hours of motion. The period of rotation for the satellite is about 105.26 minutes
which equates to approximately 13.68 orbits per day. The results of the numerical solution comparison are shown in Appendix A.
1. Delta Radius Vector Comparison
The comparison of the delta radius vector is shown in Figures 5, 6, and 7 in
Appendix A. Figure 5 includes the comparison of the solution, the simplified solution,
and the two-body solution to the numerical solution. If the solution matched the numerical solution exactly, the delta radius vector would be zero. As shown in Figure 5,
the solution and the simplified solution match extremely well with the numerical solution
while the two-body solution contains gross errors.
A more detailed plot of the delta radius vector comparison is shown in Figures
6 and 7. In these figures, the two-body solution is excluded. The difference in the solution and the simplified solution can clearly be seen. The simplified solution produces
a diverging sinusoidal response about the solution. However, up to approximately four
hours of motion, the solution and the simplified solution are nearly identical. The
sinusoidal behavior of the simplified solution can be attributed to the fact that all terms
multiplied by J20 have been neglected. As 0 grows with time, these terms become significant in the solution. As shown specifically in Figure 7, the average delta radius vector of the simplified solution clearly diverges from the solution.
Figures 5, 6, and- 7 also demonstrate that for at least one day, the delta radius
vector for the solution and the two-body solution are nearly linear as a function of time.
Other comparisons will determine whether this relationship holds true and will be shown
later. As mentioned earlier, the delta radius vector is a global error. As shown in Figures 5, 6, and 7, the solution compares well globally with the numerical solution and
demonstrates a great improvement over the two-body solution.
29
2.
Earth Arc Angle Comparison
The comparison of the Earth arc angle is shown in Figures 8, 9, and 10 in Appendix A. From inspection, these plots are nearly identical in appearance to the delta
radius vector plots. This is expected since both the delta radius vector and the Earth arc
angle represent global errors. Figure 8 clearly illustrates the large error generated by the
two-body solution. The two-body solution produces unsatisfactory long term satellite
position prediction. After one day, a tracking radar would have a difficult time detecting
a satellite with a position error of over 80 degrees.
Figures 9 and 10 present much more encouraging results. Again, the simplified
solution responds in a sinusoidal behavior about the solution. After one day, the position error of the solution is only approximately 0.15 degrees. Clearly, the solution and
the simplified solution are superior to the two-body solution. Most tracking radars can
easily handle daily position errors of 0.15 degrees. In general, the solution and the simplified solution agree very well with the numerical solution.
3.
Delta Omega Comparison
The comparison of the delta omega angle is shown in Figures 11 and 12 in Appendix A. At first glance, the solution and the simplified solution in Figure 11 appear
not to agree well with the numerical solution. However, the scale of delta omega is
multiplied by 10'.
The numerical solution parallels the two-body solution nearly exactly and predicts almost no change in Q . In other words, the numerical solution predicts no nodal regression. Easterly nodal regression is represented by a positive delta
omega; therefore, it is clear that the solution and the simplified solution predict greater
nodal regression than the numerical solution.
On a larger scale, all three solutions are essentially identical. Since the initial
inclination is so close to 90 degrees, small discrepancies are not surprising. The delta
omega plot does, however, invoke confidence in the solution. Although the initial inclination is very close to 90 degrees, the solution, the simplified solution, and the numerical solution predict easterly nodal regression. This result is significant. Even initial
inclinations close to 90 degrees produce nodal regression in the correct direction for the
solution, the simplified solution, and the numerical solution.
30
4.
Delta Inclination Comparison
The comparison of the delta inclination angle is shown in Figures 13 and 14 in
Appendix A. On a larger scale, all of the solutions compare very well. On the scale
shown in Figure 10, the two-body solution and the numerical solution are nearly identical. The solution and the simplified solution oscillate about an error of approximately
-2.0 x 10- ' degrees. Obviously, this error is extremely small. In general, the solution
and the numerical solution agree very well.
An interesting aspect of the delta inclination comparison is the sinusoidal behavior of the solution and the simplified solution. This type of behavior is precisely what
was predicted in the earlier analysis. Figure 14 demonstrates that this behavior continues for even longer periods of time. On a larger scale, this type of motion would not
be detectable.
5. Delta Theta Comparison
The comparison of the delta theta angle is shown in Figures 15, 16, and 17 in
Appendix A. This comparison confirms the results found in the earlier comparisons.
The two-body solution produces very large errors, while the solution and the simplified
solution agree very well with the numerical solution. Figures 16 and 17 again illustrate
the typical sinusoidal response of the simplified solution about the solution. Since the
delta theta error produces all other errors, the excellent results found in the earlier
comparisons are now not surprising.
6. Delta Theta Relative Error Comparison
The comparison of the delta theta relative error is shown in Figures 18, 19, and
20 in Appendix A. As shown in Figure 18, the two-body solution demonstrates a relative error of 2.3J, while the relative errors of the solution and the simplified solution are
much smaller. In more detail, Figures 19 and 20 indicate that the relative error of the
solution is 2.8P. This result confirms the theoretical prediction that .ne delta theta
relative error of the solution would be on the order of P. Figures 19 and 20 illustrate
that initially the relative error of the simplified solution is also on the order of J. But
as 0 increases with time, the relative error grows in a sinusoidal fashion. This result is
expected since the simplified solution neglects all the terms multiplied by PA . In gen-
31
eral, the results shown in Figures 18, 19, and 20 confirm the theoretical relative accuracy
of delta theta that was predicted-in the earlier analysis.
7.
Delta Radius Relative' Error Comparison
The comparison of the delta radius relative error is shown in-Figures 21, 22, and
23 in Appendix A. As shown in Figure 21, the two-body solution produces a relative
error that is-linear in time-and proportional to 2.3.10. Again, the relative errors of the
solution and the simplified- solution are-magnitudes smaller.
Figures 22 and 23 present in more detail the relative errors of the solution and
the simplified solution. The relative error of the solution is very linear and proportional
to 2.8J'0. The relative error of the simplified solution is sinusoidal in nature and diverges from the solution. However, for up to four-hours of motion, the relative error
of the solution and the simplified solution are nearly indistinguishable. Again, the- results from this comparison confirm the theoretical prediction that the delta radius relative error of the solution would be on the order of 20.
8. Radial Track Error Comparison
The comparison of the radial track error is shown in Figures 24, 25, and 26 in
Appendix A. As shown in Figure 24, the two-body solution oscillates about an error
of approximately -11.0 kilometers, while the solution and the simplified solution both
produce errors that are dramatically smaller. From inspection, the two-body solution
also appears to be slowly converging as time increases.
In Figures 25 and 26, the solution and the simplified solution produce contrasting behaviors. While the solution remains relatively constant, the- simplified solution slowly diverges from zero. These two different responses continue even after three
days of motion. Again, the neglected J20 terms cause the significant divergence of the
simplified solution. Not surprisingly, the solution and the simplified solution are clearly
superior to the two-body solution.
9. Along Track Error Comparison
The comparison of the along track error is shown in Figures 27, 28, and '9 in
Appendix A. The results presented in this comparison parallel the results found in the
earlier comparisons. Since the inclination is so close to 90 degrees, the A.Q contribution
32
is negligible and the AO contribution strongly influences the responses. As a result, the
along track error comparison is practically a mirror image of the delta theta comparison.
10.
Cross Track Error Comparison
The comparison of the cross track error is shown in Figures 30 and 31 in Appendix A. The cross track error is strongly influenced by Ai and AQ. Since the twobody solution produced good results with these two parameters, it is not surprising that
the two-body solution agrees well with the numerical solution. Fortunately, the errors
produced by the solution and the simplified solution are also very small. The solution
produces a maximum cross track-error of approximately ± 0.5 kilometers after one day
of motion, and approximately ± -1.3 kilometers after three days of motion.
Clearly in this comparison, the two-body solution is superior. However, the
large errors produced by the two-body solution in the other comparisons easily overwhelm these results. In global terms, the two-body solution is no match for either the
solution or the simplified solution.
B. MEASURED DATA COMPARISON
The solution was compared with actual measured satellite data to determine the altitude band where the theory works best. The measured satellite data was obtained from
the First Satellite Control Squadron (ISCS) located at Falcon Air Force Base, Colorado.
The First Satellite Control Squadron tracks several satellites for the Air Force and was
able to supply measured data for three separate satellites. The three satellites are currently in motion and occupy orbits that are labeled Near Earth, Semisynchronous, and
Geosynchronous, respectively. All error parameters compared in the earlier numerical
comparison were also compared in this comparison using the measured data as a reference.
Included in all the comparisons were the solution, the simplified solution, the twobody solution, and two numerical solutions. The two numerical solutions were also
supplied by the First Satellite Control Squadmon and are labeled Spacom 1 and Spacom
2, respectively. The Spacom I solution includes all perturbation effects, while the
Spacom 2 solution only includes the Earth's harmonic perturbations. All error parameters in this comparison were calculated in the following manner.
33
A(ErrorParameter) = Test Solution
-
Measured Data Solution
(55)
Unfortunately, the First Satellite Control Squadron only records measured data
when an update of their numerical solution is required. Routine updates are usually
conducted after about seven days of motion. Therefore, satellite data for one month
usually consists of only four data points. Although more data points are needed for a
more detailed analysis, a long term analysis can still be conducted. The analysis of each
type of orbit will be presented separately.
1. Near Earth Orbit Comparison
The near Earth orbit comparison possesses the following initial conditions.
ro
io
eo
oo
00
0o
ho
Po
to
=
=
=
=
=
=
7,776.58 km
98.81 degrees
0.0003071
9.57 degrees
149.14 degrees
37.10 degrees
= 53,664.37 km 2/s
= 7,224.89 -km
= OOOOZ 26 July 1990
The initial conditions of this satellite represent a retrograde orbit of small eccentricity at an altitude of approximately 850 kilometers. The-period of rotation for the
satellite is about 101.89 minutes which equates to approximately 14.13 orbits per day.
From the initial conditions, J2 should be the dominant perturbation. The orbit should
demonstrate noticeable easterly nodal regression. If the theory is valid, both the solution and the simplified solution should agree well with the numerical solutions and the
measured data.
The results of the near Earth orbit comparison are shown in Figures 32 - 43 in
Appendix B. As shown in the figur,., the solution and the simplified solution agree very
well with both the Spacom I solution and the measured data. The fact that the solution
and the simplified solution produce such excellent results verifies that J, is the dominant
34
-
perturbation- for this satellite. Figures 32 - 43 also demonstrate the larger errors
produced by-the two-body solution. In almost every comparison, both the solution and
the simplified solution are far superior to the two-body solution.
One surprising result is the poor comparison produced by the Spacom 2 solution. In every comparison the Spacom 2 solution either models the two-body solution
exactly or produces results that are inferior to the two-body solution. It is clear that the
Spacorn 2 solution does-not model the Earth's harmonic forces correctly. An explanation for the poor results cannot be determined in this analysis. A detailed analysis of the
force modeling in the Spacom 2 solution must be completed in order to adequately explain the unsatisfactory results.
The delta omega comparison in Figure 34 demonstrates the easterly nodal regression produced by the solution, the simplified solution, the Spacom 1 solution, and
the measured data. The two-body solution represents zero nodal regression. Figure 35
presents the-delta omega comparison at a much smaller scale and excludes the two-body
solution. In this figure, much more detail can be observed.
There is only one comparison in which the results are mixed. The radial track
error comparison in Figure 40 indicates that the solution produces a small improvement
over the two-body solution while the simplified solution actually produces a greater error. In comparison with the along track errors, these errors are small. It is interesting,
however, that the radial track error comparison produces such mixed results. In general,
both the solution and the simplified solution produce results that are in excellent agreement with the measured data for this near Earth satellite.
35
2. Semisynchronous Orbit Comparison
The semisynchronous orbit comparison possesses the following initial conditions.
ro = 26,407.70 kin
io = 63.66 degrees
e=
0.005860
cod = 318.19 degrees
00
328.49 degrees
0=
92.13 degrees
ho = 102,892.59 krn2]s
Po = 26,559.96 km
to = OOOOZ 22 March 1990
The initial conditions of this satellite represent a direct orbit of small eccentricity
at an altitude of approximately 20,000 kilometers. The period of rotation for the satellite
is about 717.96 minutes which equates to approximately 2.01 orbits per day. An important aspect -of the orbit is that the initial inclination is very close to the critical inclination of 63.43 degrees. Although the initial inclination is not exactly that of the critical
inclination, an evaluation of the solution and the simplified solution near this important
inclination can be made. From the initial conditions, the orbit should demonstrate
substantial westerly nodal regression. Also, at this altitude, the dominance of the J2
perturbation should be diminished. Other perturbations that are not modeled should
make a considerable contribution to the errors in the comparison. If the theory is valid,
both the solution and the simplified solution should show a great improvement over the
two-body solution.
The results of the semisynchronous orbit comparison are shown in Figures 44 53 in Appendix C. As predicted earlier, the solution and simplified solution produce
results that are superior to the results produced by the two-body solution. Figures 44
and 45 present the global errors of all the solutions. In global terms, the solution and
the simplified solution reduce the error of the two-body solution by nearly one half. In
effect, the J, perturbation accounts for approximately one half the error produced by the
two-body solution. The remaining error which is represented by the solution and the
simplified solution is caused by other perturbing forces. Unfortunately, the results of the
Spacom 2 solution were not available.
36
The delta omega comparison in Figure 46 demonstrates the easterly nodal regression produced by the solution, the simplified solution, the Spacom 1 solution, and
the measured data. Again, the two-body solution represents zero nodal regression. It
is clear that at this altitude, the J2 perturbation produces -the majority of the nodal regression. The delta inclination comparison in Figure 47 indicates that the solution and
the simplified solution produce results that are not much better than the results
produced by the two-body solution. However, the error after 30 days of motion is extremely small. On a larger scale, the solutions would seem identical. Since the inclination is very near the critical inclination, these results produce more evidence in support
of the theory. Clearly, the solution and -the simplified solution are- bounded at this inclination. The delta-theta comparison in Figure 48 demonstrates that the majority of the
error produced by the solution and the simplified solution originates in the delta theta
error. It is clear that the two-body solution underestimates the value of 6 while the
solution and the simplified solution overestimate the value of 0.
The relative error comparisons are shown in Figures 49 and 50. While the delta
theta relative erroi for the solution, the simplified solution, and the two-body solution
is approximately 15.0 x 10- ', the relative error produced by the Spacom I solution is far
superior. This result is expected since the Spacom 1 solution models several more influential perturbations. The delta radius relative error comparison again demonstrates
in global terms the amount of improvement that the solution and the simplified solution
provide over that of the two-body solution.
The track error comparisons in Figures 51 - 53 produce mixed results. While
the two-body solution produces less radial and along track errors, the solution and the
simplified solution produce much less cross track error. In comparison with the along
track and cross track errors, the radial track errors are small. The poor results produced
by the solution and the simplified solution in the-along track error comparison is due
primarily to the large error in A0 . The very large error produced by the two-body solution in the cross track error comparison is due primarily to the very large error in AQ.
In summary, although the solution and the simplified solution are superior to
the two-body solution, the Spacom I solution models the satellite motion more precisely.
However, the primary reason that the solution and the simplified solution are superior
to the two-body solution is due exclusively to a better modeling of nodal regression or
the angle 2). It is clear that the solution and the simplified solution model the J2 perturbation extremely well. The Spacom I solution is expected to perform better since it
models more perturbing forces.
37
3. Geosynchronous Orbit Comparison
The geosynchronous orbit comparison possesses the following initial conditions.
ro = 42,156.57 kmio = 1.09 degrees
eo = 0.0002341
coo = 320.06 degrees
0 = 331.32 degrees
00 = 334.85 degrees
ho = 129i644.14 km 2Is
Po =
to =
42,166.25 km
OOOOZ 21 July 1990
The initial conditions of this satellite represent a direct orbit of small eccentricity
at an altitude of approximately 35,800 kilometers. The period-of rotation for the satellite
is about 1436.69 minutes which-equates to approximately 1.00 orbit per day. Since the
initial inclination is slightly greater than zero, the orbit should demonstrate westerly
nodal regression. However, since the altitude is so large, other perturbing forces that
are not modeled may influence nodal regression. At a geosynchronous altitude, the
magnitude-of other perturbing forces approach that of J2 . Since at this altitude the effect of J2 is so diminished, some comparisons of the solution, the simplified solution, and
the two-body solution may be nearly identical. As a result, the theory may not be any
better than the two-body theory for satellites in a geosynchronous orbit.
The results of the geosynchronous orbit comparison are shown in Figures 54 63 in Appendix D. The global error comparisons are shown Figures 54 and 55. In
global terms, the solution and the simplified solution produce results that are surprisingly superior to -the results produced by the two-body solution. Evidently, for this satellite, the /2 perturbation is still quite dominant. However, the other comparisons may
present a different picture. Once again, the Spacom 2 solution generates very poor results.
The delta omega comparison in Figure 56 indicates that the actual perturbing
forces produce easterly nodal regression. Conversely, the solution and the simplified
solution predict westerly nodal regression. It is obvious that other perturbing forces
38
influence the nodal regression of this satellite. Although the Spacom I solution is superior, even this accurate numerical solution has trouble predicting the value of a . The
solution and the simplified solution also produce poor results in the delta inclination
comparison in Figure 57. All solutions, except for the Spacom 1 solution, produce
identical results. Again, on a larger scale, all of the solutions would seem nearly identical. However, this detailed analysis does demonstrate a weakness in the theory. The
delta theta comparison in Figure 48 indicates that the solution and the simplified solution are inferior to all solutions including the two-body solution. Clearly, other perturbing forces are at work.
The relative error comparisons are shown in Figures 59 and 60. The delta theta
relative error comparison simply repeats the results found in the delta theta comparison.
However, the delta radius relative error comparison is much more reassuring. Again, in
global terms, the solution and the simplified solution produce better results than the
two-body solution.
The track error comparisons in Figures 61 - 63 produce mixed results. The radial track error comparison indicates that initially the Spacom I solution is inferior to
all other solutions. However, after 21 days of motion, Spacom 1 is the superior solution.
Once again, the radial track errors are small when compared to the along and cross track
errors. The clue to the favorable global results of the solution and the simplified solution
is found in the along and cross track error comparisons. The solution and the simplified
solution perform much better than the two-body solution in the along track error comparison. Although the two-body solution is superior to the solution and the simplified
solution in the cross track error comparison, the difference is small. It is clear that the
solution and the simplified solution are superior to the two-body solution due to a much
smaller along track error.
In summary, although the solution and the simplified solution are superior to
the two-body solution, other perturbing forces greatly influence the satellite's motion.
At this altitude, the solution and the simplified solution simply do not model the satellite's motion well. Other perturbing forces must be modeled at this altitude if proper
satellite position prediction is desired.
39
VI.
CONCLUSIONS AND RECOMMENDATIONS
An analysis was -conducted on a perturbation solution of the main problem in artificial satellite theory. The purpose of the analysis was to compare the solution with
proven numerical solutions and actual measured satellite data in order to determine if
the theoretical work is valid and practical. From the analysis, the-following conclusions
can be made.
1. The solution and the simplified solution are both significantly more accurate than
the two-body solution. The relative error of the two-body solution is on the order
of JO while the relative error of the solution and the simplified solution is on the
order of AG.
2. The real physical effects of the orbit are easily distinguishable in both the solution
and the simplified solution.
3. The solution and the simplified solution compare extremely well with a proven
numerical solution for at least 41 revolutions with a relative error on the order of
j20.
4. The solution and the simplified solution compare extremely well with actual measured satellite data for at least 297 revolutions at altitudes where the -2 perturbation
dominates ( e.g., near Earth orbits ). For a satellite in orbit at an altitude of around
1000 kilometers, the solution and the simplified solution reduce the error of the
two-body solution by approximately'95%.
5. The solution and the simplified solution compare less favorably with actual measured- satellite data at semisynchronous and geosynchronous altitudes. At these altitudes, however, the solution and the simplified solution reduce the error of the
two-body solution by at least 50%.
6. The solution and the simplified solution are free of singularities and are valid for
all orbital parameters.
Clearly, the solution and the simplified solution model the J2 perturbation very well.
The equations are easy to implement and can provide quick and accurate predictions of
satellite motion. However, other types of analytical solutions exist that are more accurate than the solutions described here.
One such solution was developed by Coffey and Alfriend [Ref. 10] through research that was conducted by Dep.'. [Ref. II, Coffcy and Deprit [Ref. 12], and
Alfriend and Coffcy [Ref. 13] . The solution is called the Analitic Orbit Prediction
Program generator or (AOPP). Although the program is very accurate, AOPP exten-
40
sively utilizes four different Hamiltonian transformations. As a result, the real physical
effects of the orbit are not easily distinguishable.
The beauty of the solution and the simplified solution is their similarity in form to
the well known two-body solution and the fact that a satellite's position can be easily
predicted by evaluating only one integral. Once 0 has been determined, all other orbital
parameters can be calculated easily. The structure of the solution and the simplified
solution is ideal for implementation with onboard spacecraft computers.
Before the solutions can be adapted for practical applications, more examination
and testing of the theory is required. In order to provide more confidence in the theory,
the following recommendations are suggested.
1. The solution and the simplified solution need to be compared-to a numerical integration of the equations of motion for at least 100 revolutions to confirm the theoretical accuracy for long term satellite motion.
2. The solution and the simplified solution need to be compared to several more diverse sets of actual measured satellite data.
3. To increase precision, the solution needs to include the higher order zonal harmonics of the gravitational potential ( e.g., J3 , , J, etc. ).
4. For spacecraft computer implementation, the Lagrangian coefficients of the state
transition matrix need to be determined.
For onboard spacecraft navigation, computers make use of the state transition matrix. Currently the Lagrangian coefficients of the two-body solution are the only matrix
elements that have been determined. An excellent formulation of the two-body state
transition matrix is shown by Battin [Ref. 141. Once the Lagrangian coefficients of the
solution are developed, onboard spacecraft navigation can be greatly improved.
41
APPENDIX A.
NUMERICAL SOLUTION COMPARISON RESULTS
42
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105
PROGRAM COL02
C
o
C
-C
*
*
MAIN PROGRAM
-C
*
C
-C
IMPLICIT DOUBLE PRECISION (A-I,M-Z)
CHARACTER*20 LINE
DIMENSION M(l00),MD(l00),E(' 00),W(1O0),WD(10O),OM(100),OMD(100)
DIMENSION 1(100),ID(100),F(10),FD(O0),EC(100),ECD(100),A(100)
DIMENSION R(100) ,H(100) ,N(100) ,TH( 100) ,THD( 100)
DIMENSION RF(l00) ,IF(100) ,IFD(100) ,OMF(100) ,OMFD(100),THF(100)
DIMENSION THFD( 100) ,P(lo) ,JORBIT( 100) ,DR( 100) ,DID( 100) ,DTHD(.100)
DIMENSION DOMD(100) ,RX(100) ,RY(100),RZ(100) ,RFX(-100) ,RFY(1i00)
DIMENSION RFZ(100) ,DRV(100) ,ARC(100) ,ARCD(l00),DAY(100) ,HX(100)
DIMENSION HY(100),HZ(100),VX(100),VY(100),VZ(100),DT(100),NX(100)
DIMENSION NY(100),NZ(100) ,RDV(100),EX(100),EY(100) ,EZ(100)
DIMENSION NDE(100),EDR(100),V(100),HT(100) ,RDRF(100) ,INTA(100)
DIMENSION ROMA(iCO) ,TH~'o(100) ,ATE(l00) ,CTE(100)
COMMON/OBLATE1/DAY,RXRY,RZ ,VX,VY,VZ,DT,HX,HY,HZ,Nx,NY,NZ,K,KX
COMMON/OBLATE2/RDV,R,V, FX,F,\,EZ,IU,NDE,EDR,H,N,E,P,I,OM,W,F
COMMON/OBLATE3/PI,EC,l,,Ht,'I,ER,TH,THD,RTD,MD,WD,OM"D,ID,ECD
COMMON/OBLATE4/FD,LINE, J,THF,THFD, IF,IFD,OMF,OMFD,RF, INT,ROM
COMMON/OBLATE5/RJ,DR,DID,DTHD,DOMD,ESTERR,ACTERR,TERROR,JVER
COMMON/OBLATE6/RFX, RFY ,RFZ ,ARC, ARCD , DRF,DRV,RJ2 , JN, JORBIT
COMMON/OBLATE7/ INTA ,ROMA,THJO,AT, CTE
C
10
C
20
PRINT--,'ENTER VERSION ( SOLUTION = 1, SIlf'LB
READ* ,JVER
IF(JVER. EQ. 1.OR. JVER. EQ. 2.OR.JVER. EQ. 3)THEN
GOTO 20
ELSE
GOTO 10
ENDIF
=
2, TWO BODY
PRINT*,'ENTER FIRST POINT'
READ,K
PRINT--e,'ENTER FINAL POINT'
READ*,KK
C
PRINT", 'ENTER RJ2'
READ*,RJ2
IF(RJ2. EQ. 1.ODO)TiHEN
RJ2=0. 00108263D0
ENDIF
C
LINE- - - - - - - - - - PI=3. 141592653589793238462643D0
RTD=180. ODO/PI
ER=6378. 137D0
HTS=1. 0iO/806. 812D0
MU=3. 986004 36D5
C
OPEN(UNIT=2, STATUS='OLD', FILE='/COLO2 OUT A')
106
=3
)
OPEN(UNIT=3, STATUS=&OLD', FILE='/COLO2 DS8 A')
OPENf-UNIT-=4, STATUS='OLD', FILE='/COLO2O DSS B')
C
CALL DATA
CALL ELEMENTS
CALL PINITIAL
C
J=1
WRITE(6,6000)
WRITE(6,6000)
WRITE(6,6000)
WRITE(6,6000)
'POINT = ,
'INTEGRATE COMPLETED'
'FORMULA COMPLETED'
'INERTIAL COMPLETED'
C
30
DO 30 J = K, KK
WRITE(6,6000) 'POINT =',
CALL INTEGRATE
WRITE(6,6000) 'INTEGRATE COMPLETED'
CALL FORMULA
WRITE(6,6000) 'FORMULA COMPLETED'
CALL INERTIAL
WRITE(6,6000) 'INERTIAL COMPLETED'
CONTINUE
CALL RESULTS
C
CLOSE(UNIT=-2)
CLOSE(CUNIT=-3)
CLOSEC UNIT-=4)
C
6000
C
C
C
C
C
C
C
FORMAT(3X,A,13)
STOP
END
*
SUBROUTINE DATA
SUBROUTINE DATA
IMPLICIT DOUBLE PRECISION (A-I,M-Z)
CHARACTER*20 LINE
DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM(100),OMD(100)
DIMENSION I(100),IDC100),F(100),FD(100),EC(100),ECD(100),A(100)
DIMENSION R(100) ,H(100) ,N(100) ,TH(100),THD(100)
DIMENSION RF(100),IF(100) ,IFD(100) ,OMF(100) ,OMFD(100) ,THF(100)
DIMENSION ThFPD(100),P(100) ,JORBIT(100) ,DR(-100),DID(100) ,DTHID(100)
DIMENSION DOMD(100) ,RX(100),RY(100) ,RZ(100) ,RFX(100) ,RFY(100)
DIMENSION RFZ(100) ,DRV(100),ARC(100) ,ARCD(100) ,DAY(100),HX(100)
DIMENSION HYC100),HZ(100),VX(l00),VY(100),VZ(100),DT(10o),NX(100)
DIMENSION NY(100),NZC100),RDV(J.Oo),EX(100),EY(100),Ez(1o0)
DIMENSION NDE(100) ,EDR(100) ,V(100),HT(100),RDRF(100) ,INTA(100)
DIMENSION MONTI( 100) ,DATE( 100) ,IOUR( 100) ,MIN( 100),SEC( 100)
DIMENSION ROIA( 100) ,TJ{JO( 100) ,ATiEC100) ,CTE( 100)
COXMJ ON/OBLATE1/DAY,RX,RY,RZ,VX,VY,VZ,DT,IIX,H[Y,HZ,NX,NY,NZ,K,KK
COMMON/OBLATE2/RDV,R,V,EX,EYv,EZ,M1U,NDE,EDR,H,N,E,P,I,OM,W,F
COMMON/OBL.ATE3/PI,EC,M,A,HT,ER,Th-I,THID,RTD,M]D,WD,OMID,ID,ECD
107
COMMON/OBLATE4/FD,LINE,J,THF,THFD, IF,IFD,OMF,OMFD,RF, INT,ROM
COMMiON/OBLATE5/RJ,DR,DID ,DTHD,DOMD,ESTERR,ACTERR,TERROR,JVER
COMMON/OBLATE6/RFX,RFY,RFZ ,ARC,ARCD,RDRF,DRV,RJ2,JN,JORBIT
COMMQN/OBLATE7/INTA,ROMA,THJO,ATE,CTE
C
o
C
READ IN EMPHERIS DATA
OPEN(UNIT=1I, STATUS='OLD', FILE=t /COLO2 DAT', ACTION='READ')
DO 10 J1= 1, KK
READ(1,*) MONTH(J) ,DATE(J) ,HOUR(J) ,MIN(J) ,SEC(J)
10
C
C
C
20
C
RX(J)=RX(J)/1000. ODO
RY(J)=RY(J)/1000. 0DO
RZ(J)=RZ(J)/1000. ODO
READ(1,*) VX(J) ,VY(J) ,VZ(J)
VX(J)=VX(J)/ 1000. 0DO
VY(J)=VY(J)/ 1000. ODO
VZ(J)=VZ(J)/1000. ODO
CONTINUE
CONVERT PARAMETERS
DO 20 J = 1, KK
DAY(J)=DATE(J)+((3600. ODO*HOUR(J)+
+
(60. ODO*MIN(J)+SEC(J))))/86400. ODO
DT(J)=(DAY(J)-DAY( 1))'*24. ODO*3600. ODO
CONTINUE
CLOSE(UNIT=1l)
C
RETURN
END
C
C
C
C
*
SUBROUTINE ELEMENTS
C
SUBROUTINE ELEMENTS
IMPLICIT DOUBLE PRECISION (A-I,M-Z)
CHARACTER*20 LINE
DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM(100),OMD(loo)
DIMENSION I(100),ID(100),F(100),FD(100),EC(l00),ECD(100),A(loo)
DIMENSION R(100),H(100),N(100),TH(100),THD(100)
DIMENSION RF( 100) ,IF(100),IFD( 100) ,OMF( 100) ,OMFD( 100) ,THF( 100)
DIMENSION THFD(100) ,PC100) ,JORBIT(100),DR(100) ,DID(100) ,DTHD(100)
DIMENSION DOMID(100),RX(100) ,RY(100),RZ(100) ,RFX(100),RFY(100)
DIMENSION~ RFZ( 100) ,DRY( 100) ,ARC( 100) ,ARCD( 100) ,DAY( 100) ,HX( 100)
DIMENSION HY(100),HZ(100),VX(100),VY(100),VZ(100),DT(1oo),NX(100)
DIMENSION NY(100),NZ(100),RDV(100),EX(100),EY(.Oo),EZ(100)
DIMENSION NDE(100),EDR(100) ,V(100) ,HT(100) ,RDRF(100) ,INTA(100)
DIMENSION ROMA(100) ,THJO(100) ,ATEC100) ,CTE(100)
COMMON/OBLATE1/DAY,RX,RY,RZ,VX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K,KK
COMMON/OBLATE2/RDV,R,V,EX,EY,EZ,MU,NDE,EDR,H,N,E,P, I,OM,W,F
108
COMMQN/OBLATE3/PI ,EC,M,A,HT,ER,TH,THD,RTD,MD,WD,OMD, ID,ECD
COMMION/OBLATE4/FD,LINE,J,THF,THFD, IF,IFD,OMF,OMFD,RF, INT,ROI
COMMON/OBLATE5/RJ,DR,DID ,DTHD,DOMD,ESTERR,ACTERR,TERROR,JVER
COMMON/OBLATE6/RFX,RFY,RFZ,ARC,ARCD,RDRF,DRV,RJ2,JN7,JORBIT
COIIION/OBLATE7/INTA,ROMA,THJQ,ATE ,CTE
C
C
C
CALCULATE R,V,E,H,N,P,I,OM,W,F,EC,M,HT,TH
DO 10 J = 1, KK
CALL CROSS(RX(J) ,RY(J) ,RZ(J) ,VX(J) ,VY(J) ,VZ(J) ,HX(J) ,HY(J),
+
HZ(J))
CALL CROSS(O. ODO,O. ODO,1. ODO,HX(J),HY(J),HZ(J),NX(J),NY(J),
+
NZ(J))
CALL DOT(RX(J),RY(J),RZ(J),VX(J),VY(J),VZ(J),RDV(J))
R(J)=DSQRT(RX(J)*RX(J)+RYCJ)*RYCJ)+RZ(J)*RZ(J))
V(J)=DSQRT(VX(J)*VX(J)+VY(J)*VY(J)+VZ(J)*VZ(J))
EY(J)=( (V(J)*V(J)-MU/R(J))*RY(J) -RDV(J)*VYCJ) )/MIJ
EZ(J)=((V(J)*V(J)-iU/R(J))*RZ(J)-RDV(J)*VZ(J))/MU
CALL DO(N(J),N(J),NZ/(J),EXZ(J),EY(J)EZ(J)D))
CALL DOT(EX(J),EYZ(J),EZ(J),RX(J),RY(J),RZ(J),NER(J))
H(J)=DSQRT(HXJ)*HX(J)+HY(J)*HY(J)+HZ(J)*HZ(J))
N(J)=DSQRT(NX(J)*NX(J)+NY(J)*NY(J)+NZCJ)*NZ(J))
E(J)=DSQRT(EX(J)*EX(J)+EY(J)*EY(J)+EZ(J)*EZ(J))
P(J)=H(J)*H(J)/MU
I(J)=DACOS(HZ(J)/H(J))
QM(J)=DACOS(NX(J)/N(J))
W(J)=DACQS(NDE(J)/(N(J)*E(J)))
F(J)=DACOS(EDR(J)/(E(J)*R(J)))
IF(NY(J). LE.0. ODO)THEN
ONf(J)=2. 0D0*PI-OM(J)
ENDIF
IF(EZ(J). LE. 0. ODO)THEN
ENDIF
IF(RDV(J). LE. 0.ODO)TIIEN
F(J)=2. ODO*PI-F(J)
ENDIF
EC(J)=DACOS((E(J)+DCOS(F(J)))/(1. ODO+E(J)*DCOS(F(J))))
IF(F(J). GE. PI)THEN
EC(J)=2. ODO*PI-EC(J)
ENDIF
M(J)=EC(J) -E(J)*-'DSIN(EC(J))
20
AT=(MTJU/(N(1)*N( 1)) )**(1. ODOI 3.ODO)
HT(J)=R(J)-ER.
RJ=3. OD0*RJ2*ER*-ER/( 2.ODO*P( 1:P( 1))
TH( 5)=F( J)+W( J)
THD( J)=TH( J)*RTD
IF(THD(J).GT. 360. ODO)THEN
TIID(J)=THD(J) -360. ODO
GOTO 20
END IF
THC J)=THD(5) /RTD
C
109
C
C
CONVERT ORBITAL ELEMENTS TO DEGREES
MD(J)=M(J)"RTD
WD(J)=W(J)"RTD
OMD( J)=OM(J)*RTD
ID(J)=I(J)*RTD
ECD(J)=EC(J)*RTD
FD(J)=F(J)RTD
THD(J)--TH(J)*RTD
10
CONTINUE
RETURN
END
C
C
C
C
C
C
C
C
C
C
A
*
*
SUBROUTINE CROSS
*
SUBROUTINE CROSS(AX,AY,AZ,BX,BY,BZ,CX,CY,CZ)
IhPLICIT DOUBLE PRECISION (A-I,M-Z)
CALCULATE THE CROSS PRODUCT OF 'WO VECTORS A AND B
CX=AY-*BZ-AZ*BY
CY=AZ*BX-AX*:BZ
CZ=AX*BY-AY*BX
C
RETURN
END
C
C
C
C
*
SUBROUTINE DOT
C
SUBROUTINE DOT(AX,AY,AZ,BX,BY,BZ,ADB)
IMPLICIT DOUBLE PRECISION (A-I,M-Z)
C
C
C
CALCULATE THE DOT PRODUCT OF TWO VECTORS A AND B
ADB=AXC*BX+AY--BY+AZ*BZ
C
RETURN
END
C
C
C
-
C
C
SUBROUTINE PINITIAL
**
C
C
SUBROUTINE PINITIAL
IMPLICIT DOUBLE PRECISION (A-I,M-Z)
CIIARACTER*:20 LINE
110
DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM(100),OMD(100)
DIMENSION I(100),ID(100),F(100),FD(100),EC(100),ECD(100),A(100)
DIMENSION R(100),H(100),N(100),TH(100),THD(100)
DIMENSION RF(100) ,IF(100) ,IFD(100) ,OMF(100) ,OMFD(100) ,THF(100)
DIMENSION THFD(100) ,P(100) ,JORBIT(100) ,DR(100) ,DID(100),-DTHD(100)
DIMENSION DOMID(100),RX(100),RY(100),RZ(100),RFX(100),RFY(100)
DIMENSION RFZ(100),DRV( 100) ,ARC( 100) ,ARCD( 100) ,DAY( 100) ,HX( 100)
DIMENSION HY(100),HZ(100),VX(100),VY(100),VZ(100),DT(100),NX(100)
DIMENSION NY(100),NZ(100),RDV(100),EX(100),EY(100),EZ(100)
DIMENSION NDE(100) ,EDR(100) ,V(100) ,HT(100) ,RDRF(100) ,INTA(100)
DIMENSION ROMA( 100) ,THJO( 100) ,ATE( 100) ,CTE( 100)
COMMION/OBLATE1/DAY,RX,RY,RZ,VX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K,KK
COMMfON/OBLATE2/RDV,R,V,EX,EY,EZ,MU,NDE,EDR,H,N,E,P, I,OM1,W,F
COMMON/OBLATE3/PI ,EC,M,A,HT,ER,TH,THD,RTD,MD,WD,OMD,ID,ECD
COMMON/OBLATE4/FD ,LINE, J ,THF ,THFD,IF,IFD ,OMF ,OMFD ,RF, INT ,ROM
COMMON/OBLATE5/RJ ,DR,DID ,DTHD,DOMD,ESTERR,ACTERR,TERROR,JVER
COMMON/OBLATE6/RFX,RFY,RFZ ,ARC ,ARCD,RDRF,DRV,RJ2,JN,JORBIT
COMMON/OBLATE7/INTA,ROMA ,THJO,ATE ,GTE
C
C
C
PRINT INITIAL ORBITAL ELEMENTS
+
+
+
+
+
+
+
+
C
6000
6100
6200
C
WRITE(6,'(/)')
WRITE(2,'(/)')
WRITE(6,6000) 'ORBITAL ELEMENTS'
WRITE(2,6000) 'ORBITAL ELEMENTS'
WRITE(6,6100) LINE
WRITE(2,6100) LINE
WRITE(6,6200) :M = ',MD(1),'N = ',N(1),'EI ',= )
'w = 'WD(l), OM = 'OMD(1),'
, D=)
'EC = ,ECD(l),'A =,IA(1), IR =
Rl)
'HT = 'HT(l),'H
= ',H(1),'F = 'F~)
'TH = ,TIID(1)
WRITE(2,6200) 'M = ',MD(1),'N = ',N(1),'E = ,l)
1w = IWD(l),lOM = 'OMD(l),'
=i II~)
'EC = ',ECD(l),'A =',A(l), 'R = ,~)
'HT = ,HT(l),'H = ',H(.1),'F
FDl)
'TH = 'THD(l)
FORMAT(3X,A)
FORMAT(3X,A20/)
FORMAT(13(3X,A5,D18. 10/)!)
RETURN
END
C
C
C
C
o
C
C
*
*
*
SUBROUTINE INTEGRATE
SUBROUTINE INTEGRATE
IMPLICIT DOUBLE PRECISION (A-I,M-Z)
CHARACTER*20 LINE
DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM(100),OMD(100)
DIMENSION I(100),ID(100),F(100),FD(100),EC(100),ECD(100),A(100)
III
DIMENSION R(100) ,H(100) ,N(100) ,TH( 100) ,THD( 100)
DIMENSION RF(100) ,IF(100) ,IFD(100) ,OMfF(100),OMFD(100) ,THF(100)
DIMENSION THFD(.'0),P(100),JORBIT(100),DR(100),DID(100),DTHD(100)
DIMENSION DOME
'0),RX( 100) ,RY( 100) ,RZ( 100) ,RFX( 100) ,RFY( 100)
DIMENSION RFZ(O
),DRV(100),ARC(100),ARCD(100),DAY(100),HX(100)
DIMENSION HY(10,j),HZ(100),VX(100),VY(100),VZ(10Q),DT(100),NX(100)
DIMENSION NY(100),NZ(100),RDV(100),EX(100),EY(100),EZ(100)
DIMENSION NDE(100),EDR(100),V(100),HT(100),RDRF(100),INTA(100)
DIMENSION ROMAC 100) ,THJO( 100) ,ATE( 100) ,CTE( 100)
COIMON/OBLATE1/DAY,RX,RY,RZ,'YX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K5 KK
COMMON/OBLATE2/RDV,R,V,EX,EY,EZ,MU,NDE,EDR,H,N,E,P, I,0MW,F
COMMON/OBLATE3/PI,EC,M,A,HT,ER,TH,THD,RTD,MD,WD,OMD,ID,ECD
COMMON/OBLATE4/FD,LINE,J,THF,THFD,IF,IFD,OMIF,OMFD,RF,INT,ROM
COMMION/OBLATE5/RJ,DR,DID ,DTHD ,DOMD ,ESTERR ,ACTERR ,TERROR, JVER
COMMION/OBLATE6/RFX ,RFY ,RFZ ,ARC ,ARCD ,RD,DRV,RJ2,JN, JORB IT
COMMON/OBLATE7/INTA,ROMA,THJO,ATE,CTE
C
C
C
EQUATE INITIAL VALUES
THF( 1)=TH( 1)
THFD( 1)=THD( 1)
IF(1)=I( 1)
IFD( 1)=ID( 1)
OMF( 1)=OM( 1)
OMFD(1)=OMD(1)
RF(1)=R(1)
C
C
C
ESTIMATE UPPER BOUND OF THETA
THF(J)=TH(J)+(J-1)*0. 57D0*2. ODO*PI
C
C
C
C
C
C
10
C
30
20
ROM00080
INITIALIZE
(1X1O)-12
ESTERR=0. 00000000000 iDO
INT=DT(J)*H( 1)
DTHF=0. 1745329251994
(1X1O)-10
TERROR=0. 000000000 iDO
CALL ROMBERG
ACTERR=-INT-ROM
IF(ACTERR. LT.0. ODO)THEN
THF(J)=THF(J) -DTHF
GOTO 10
END IF
TEMPTHF=-THF( J)
GOTO 20
ROM00190
CALL ROMBERG
ACTERR=INT-ROM
IF(ACTERR. GE. 0. ODO)THEN
IF((ACTERR/INT). LE. ESTERR)THEN
GOTO 40
ELSE
TEMPTHF=-THF( J)
112
THF( J) THF( J)+DTHF
GOTO 30
ENDIF
ELSE
DTHF=DTHF/2. ODO
THF( J)TEMPTHF+DTHF
GOTO 30
END IF
C
40
INTA(J)=INT
ROMA(-J)=ROM
RETURN
END
C
o
o
C
o
SUBROUTINE ROMBERG*
C
*
C
C
SUBROUTINE ROMBERG
ROM00190
IMPLICIT DOUBLE PRECISION (A-I,Mi-Z)
CHARACTER*20 LINE
DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM1(100),OMD(100)
DIMENSION I(100),ID(100),F(100),,FD(100),EC(100),ECD(100),A(100)
DIMENSION R(1OO),H(1O0),N(100),T{ 100),THD(10O)
DIMENSION RF(100) ,IF(100) ,IFD(100) ,OMF(100) ,OMFD(100) ,THF(100)
DIMENSION THFD( 100),P( 100) ,JORBIT( 100) ,DR( 100) ,DID( 100) ,DTHD( 100)
DIMENSION DOMD(100),RX(100),RY(100),RZ(100),RFX(100),RFY(100)
DIMENSION RFZ( 100) ,DRV( 100) ,ARC( 100) ,ARCD( 100),DAY( 100) ,HX( 100)
DIMENSION HY(100),HZ(100),VX(100),VY(100),VZ(100),DT(100),NX(100)
DIMENSION NY(100),NZ(100),RDV(100),EX(100),EY(100),EZ(100)
DIMENSION NDE( 100) ,EDR( 100)-,V(100) ,HT( 100) ,RDRF( 100) ,INTA( 100)
DIMENSION ROMAC 100) ,THJO( 100) ,ATE( 100) ,CTE( 100)
COMMION/OBLATE1/DAY,RX,RY,RZ,VX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K,KK
COMMON/OBLATE2/RDV,R,V,EX,EY.,EZ,MU,NDE,EDR,H,N,E,P,I,OM,Wi,F
COMMON/OBLATE3/PI ,EC,M,A,HT,ER,TH,THD,RTD,MD,WD,OMD, ID,ECD
COMMON/OBLATE4/FD,LINE,J,THF,THFD,IF,IFD,OMIF,OMFD,RF,INT,ROM
COMMON/OBLATE5 /RJ ,DR ,DID, DTHD ,DOMD ,ESTERR ,ACTERR ,TERROR, JVER
COMIMON/OBLATE6/RFX ,RFY,RFZ ,ARC, AROD ,RDRF ,DRV ,RJ2 ,JN, JORB IT
COMMON/OBLATE7/INTA,ROMA,THJO,ATE ,CTE
C
C
C
C
C
C
C
EXTERNAL FUNC
R0M00460
INITIALIZE VARIABLES
R0M00560
HS=THF(J) -THF( 1)
FUNCA=FUNC(RJ,A( 1),1(1) ,E(1) ,W(1) ,TH( 1),THF(1) ,JVER)
FUNCB=FUNC(RJ,A( 1),1(1) ,E(1) ,W(1) ,TH(1) ,THF(J) ,JVER)
P(1)=HS*(FUNCA+FUNCB)/2. ODO
JM=1
ROM005 70
BEGIN THE ROMBERG LOOP.
ROM005 80
R0M00590
ROM00610
ROM00620
DO 10 JN = 1, 100
OLDINT=P( 1)
ROM00630
ROM00640
113
20
30
10
40
C
C
C*
C
HH=HS
SS=O.ODO
TT=-THF( 1)+HH/2. ODO
DO 20 L = 1, JM
T=Tr
SS=SS+FUNC(RJ,A(l),I(1),E(i),W(l),TH(1),T,JVER)
fTT-HHCONTINUE
SUM=HH*SS
P(JN-I1)=(P(JN)+SUM)/2. 0DO
D=1. 0Db
DO 30 JK = JN, 1, -1
D=4. 0D0*D
P(JK)=P(JK+1)+(P(JK+1)-P(JK))/(D-.-ODO)
CONTINUE
-ERROR=(PC1) -OLDINT)
IF(JN. GE. 10)THEN
IF (DABS(ERROR/OLDINT). LB.TERROR)THEN
GOTO 40
ENDIF
ENDIF
HS=HS/2. ODO
JM=JM*2
CONTINUE
ROM=P(1):
RETURN
END
R0M00670
R0M00680
R0M00690
R0M00720
R0M00730
R0M00740
ROM00 750
R0M00760
R0M00780
ROM00800
R0M00830
R0M00840*
R0M00850
ROIIO0860
R0M00870
ROM00900
R0M00930
R01100940
R0M00950
ROM009 60
R0M00970
FUNCTION FUNC
C
*
C
C
FUNCTION FUNC(RJ,A1,11,E1,W1,TH1l,THFJ,JVER)
IMPLICIT DOUBLE PRECISION (A-I,O-Z)
C
EXTERNAL RADIUS
C
S=DSIN( II)
S2=S*S
S4=52*52
56=S4*52
E2=E 1*E 1
C
RAD=RADIUS(RJ,A1,11,E1,Wl,THI,THFJ,JVER)
C
C
C
F ( SOLUTION)
IF(JVER. EQ. 1)THEN
C
Y1=112. ODO-75. ODO*S6+260. OD0*S4-296. ODO*S2
Y2=RJ*THFJ*(2. 5D0*S2-2. 0Db)
Y3=2.ODO*Wl-Y2
Y4=24. ODO*(5. 0D0*S2-4.ODO)*(5. 0DO*S2-4. 0Db)
Y5=E2*S2*( 14.0Db- 15. ODO*S2)*( 15. OD0*S2- 13. 0Db)
114
ROM00030
Y6=9. ODO*E2+34. ODO
Y9=12. ODO*(5. ODO*S2-4. ODO)
Y12=9. ODO*E2-34. ODO
C
YF=(2. 5DO*S2-2. 0DO)*(THFJ-TH1)+E2*Y1*DSIN(Y2)*DC0S(Y3)/Y4
C
+
+
+
+
YS=Y5*DCOS( 2.ODO*W1)/(2. ODO*Y9)+
E1*S2*( 15. ODO*S2-13. ODO)*DCOS(TH1+W1)/2. ODO+
E1*S2*( 15. ODO*S2-13. ODO)*DCOS(3. ODO*TH1-W1)/6. ODO+
S2*( 15. ODO*S2-13. ODO)*DCOS(2. ODO*TH1)/2. ODO+
(5.ODO*Y6*S4+4. ODO*Y12*S2-56. ODO*E2)/96. ODO
C
Y=THFJ-Wl+RJ*YF+RJ*RJ*THFJ*YS
C
F=(2. ODO-3. ODO*S2)*DCOS(2. ODO*THFJ),/2. ODO+
El*(S2-1)*DCOS(Y)+E1*(3. QDO-4. ODO*S2)*DCOS(Y+2. ODO*THFJ)/6. ODO+
E1'(1. ODO-2. ODO*S2)*DCOS(Y-2. ODO*THFJ)/2. ODO+S2-1. ODO+
E2*S2*( 15. ODO*S2-14. ODO)*DSIN(Y2)*DSIN(Y3)/Y9+
S2*DCOS( 2.ODO*TH1)/2. ODO+E1*S2*DCOS(3. ODO*TH1-W1)/6. ODO+
+
E1*S2*DCOS(TH1+W1)/2. ODO
+
+
+
+
C
END IF
C
o
C
F ( SIMPLIFIED SOLUTION)
IF(JVER. EQ. 2)THEN
CF=(2. ODO-3. ODO*S2)*DCOS(2. ODO*THFJ)/2. ODO+
+- E1*(S2-1)*DCOS(THFJ-W1)+
E1*(3. ODQ-4. ODO*S2)*DCOS(3. ODO*THFJ-Wl)/6. ODO+
+
+ El*(1. ODO-2. ODO*~S2)*DCOS(THFJ+W1)/2. ODO+S2-1. ODO+
S2*DCOS(2. ODO*TH1)/2. ODO+E1*S2*DCOS( 3.ODO*TH1-Wl)/6. ODO+
+
+- E1*S2*DCOS(TH1+1?1)/2. ODO
C
ENDIF
C
C
C
F ( TWO BODY SOLUTION)
IF(JVER. EQ. 3)THEN
F=O. ODO
ENDIF
C
C
C
FUNCTION
FUNC=RAD*RAD*(1. ODO+RJ*F)-
C
RETURN
END
C
C
C
o
C
C
C
FUNCTION RADIUS
FUNCTION RADIUS(RJ,A1,I1,E1,W1,TI1,TFJ,JVER)
115
IMPLICIT DOUBLE PRECISION (A-I,O-Z)
C
C
C
CALCULATE E, SINE, AND COSINE FUNCTIONS
S=DSIN(I1)
S2=S*S
S4=S2*S2
S6=S4*S2
C=DCOS( Il)
C2=C*C
SC=DSIN( I1)*DCOS( Il)
E2=El*El
PO=A1*(1. ODO-E2)
C
C
C
RADIUS BOTTOM ( SOLUTION)
IF(JVER. EQ. 1)THEN
C
Y1=112. ODO-75. ODO*S6+260. ODO*S4-296. ODO*S2
Y2=RJ*TFJ*c(2:-5D0*52-2. ODO)
Y3=2. ODO*W1-Y2
Y4=24. ODO*(5. ODO*S2-4. ODO)*(5. ODO*S2-4. ODO)
Y5=E2*S2*( 14. ODO-iS. ODO*S2)*( 15. ODO*S2-13. ODO)
Y6=9. ODO*E2+34. ODO
Y9=12. ODO*(5. ODO*S2-4. ODO)
C
Y11=15. ODO*(2. ODO+E2)*S4-14. ODO*(4. ODO+E2)*S2+24. ODO
Y12=9. ODO*E2-34. ODO
YF=(2. 5DO*S2-2. ODO)*(TFJ-TH1)+E2*Y1*DSIN(Y2)*DCOS(Y3)/Y4
C
+
+
+
+
yS=y5*DCOS( 2.ODO*W1)/( 2.ODO*Y9)+
E1*S2*( 15. ODO*S2-13. ODO)'*DCOS(TH1+W1)/2. ODO+
E1*S2*( 15. ODO*S2-13. ODO)*DCOS(3. QDO*THI-W1)/6. ODO+
S2*( 15. ODO*S2-13. ODO)*DCOS(2. ODQ*TH1)/2. ODO+
(5.ODO*Y6*S4+4.-ODO*Y12*S2-56. ODO*E2)/96. ODO
C
Y=TFJ-W1+RJ*YF+RJ*rRJ*TFJ*YS
C
+
+
+
+
+
+
RF1=1. ODO-1. 5D0*S2+E2*(1. ODO-1. 25D0*S2)((2. ODO+5. ODO*E2)*S2-2. ODO*E2)*DCOS(2. ODO*TFJ)/12. ODO+
E2*(9. ODO*S2-8. ODO)*DCOS(2. ODO*Y)/12. ODO+
E1*(-11. ODO*S2+6. ODO)*DCOS(Y+2. ODO*TFJ)/24. ODO+
E2*( -3. ODO*S2+2. ODO)*DCOS(2. ODO*Y+2. ODO*TFJ)/24. ODO+
E2*(3. ODO*S2-2. ODO)*DCOS(2. ODO*Y-2. ODO*TFJ)/8. ODO+
E1*Yl1*DSIN(Y2)*DSIN(TFJ+Wl)/Y9
+
+
+
+
+
+
RF2=E2*S2*( 15. ODO*S2-14. ODO)*DSIN(Y2)*DSIN(Y3)/(O. 5D0*Y9)E2*S2*DCOS(Y-TH1+3. ODO*W1)/16. ODO+
E2*(3. ODO*S2-2. ODO)*DCOS(Y-3. ODO*TH1+3. ODO*W1)/24. DOE2*S2*DCOS( Y-5. ODOI-TH1+3. ,ODO*W1) /16. ODO+
El*(3. ODO*S2-2. ODO)*DCOS(Y-2. ODO*TH1+2. ODO*W1)/4. ODO3. ODO*E1*S2*DCOS(Y-4. ODO*TH1+2. ODO*W)/8. ODOE1*(S2+1. ODO)*DCOS(Y+2. ODO*'W1)/4. ODO
-C
C
116
ROM00030
+
+
+
+
+
+
RF3=((5. ODO*E2-2. ODO)*S2-2. ODO*E2)*DCOS(Y+TH1+W1)/8. ODO+
((5. ODO*E2+6. ODO)*S2-4. ODO*(E2+1. ODO))*DCOS(Y-THI1W1)/4. ODO+
(2.ODQ*E2-S2*(5. ODO*E2+14. ODO))*DCOS(Y-3. ODO*TH1+W1)f 24. ODO+
E2*(9. ODO*S2-4. ODO)*DCOS(Y+3. ODO*TH1-W1)/48. DO+
E2*(6. ODO-7. ODO*S2)*DCOS(Y+TH1-W1)/8.ODO+
E2*(4..ODO-5. ODO*S2)*DCOS(Y-TH1-W1)/16. ODO+
E1*(2. ODO*S2-1. ODO)*DCOS(Y+2. ODO*THI)14. ODO
+
+
+
RF4=El*(1. ODQ-3.'ODO*S2):*DCOS(Y-2. ODO*TH1)14. ODO+
E1*1(2. ODQ-3. ODO*S2)*DCOS(Y)/4. ODO+
E1*S2*DCOS(TH1+Wl)+S2i"DCOS(2. ODO*TH1)+
E1*S2*DCOS(k3. ODO*THI-W1)/3. ODO
c
c
RFB=1. ODO+E 1*DCOS( Y)+RJ*(RF1+R2+P3+RF4)
END IF
C
o
C
RADIUS BOTTOM ( SIMPLIFIED SOLUTION)
IF(JVER. EQ. 2)THEN
c
+
+
+
+
+
+
+
+
+
+
+
+
RF1=1. ODO-1. 5D0*S2+E2*(1. ODQ-1. 25D0*S2)((2. ODO+5. ODO*E2)*S2-2. ODO*E2)*DCOS(2. ODO*TFJ)/12. ODO+
E2*(9. ODO*S2-8. ODO)*DCOS(2. ODO*(TFJ-Wi))/12. ODO+
E1*( -11. ODO*S2±6. ODO )*DCOS( 3. ODO*TFJ-W1 ) /24. ODO+
E2*( -3. ODO*S2+2. ODO)*DCOS(4. ODO*TFJ-2. ODO*W)/24. ODO+
E2*(3. ODQ*S2-2. ODO)*DCOS(2. QDO*W1)/8. ODOE2*S2*DCOS(TFJ-TH1+2. ODO*W1)/16. ODO
RF2=E2*(3. ODO*S2-2. ODO)*DCOS(TFJ-3. ODO*TH1+2. ODO*W1)/24. DOE2*S2*DCOS(TFJ-5. ODO*TH1+2. ODO*W1)/16. ODO+
El*(3. ODO*S2-2. ODO)*DCOS(TFJ-2. ODO*TH1+W1l)/4. ODO3.ODO*E1*S2*DCOS(TFJ-4. ODO*TH1+Wl)/8. aDOE1*(S2+1. ODO)*DCOS(TFJ+W1)/4. ODO+
((5. ODO*E2-2. ODO)*S2-2. ODO*E2)*DCOS(TFJ+TH1)/8. ODO+
((5. ODO*E2+6. ODO)*S2-4. ODO*(E2+1. ODO))*DCOS(TFJ-THI)/4. DO
C
RF3=(2. ODO*E2-S2*(5. ODO*E2+14. ODO))*DCOS(TFJ-3. DO*TH1)/24. DO+
E2*(9. ODO*S2-4. ODO)*DCOS(TFJ+3. ODO*TH1-2. ODO*W1)/48. DO+
E2*(6. ODO-7. ODO*S2)*DCOS(TFJ+THI-2. ODO*Wl)/8. ODO+
E2'*(4. ODO-5. ODO*S2)*DCOS(TFJ-TH1-2. ODO*Wl)/16. ODOI
El*(2. ODO*S2-1. ODO)*DCOS(TFJ+2. ODO*THI-W1)/4. ODO+
E1*(1. ODO-3. ODO*S2)*DCOS(TFJ-2. ODO*TH1-W1)/4. ODO+
+
El*(2. ODO-3. ODO*S2)*DCOS(TFJ-Wl)/4. ODO
+
+
+
+
+
C
+
RF4=E1*S2*DCOS(TH1+Wl)+S2*DCOS(2. ODO*TH1)+
El*S2*DCOS(3. ODO*THl-W1)/3. ODO
+
RFB=E1*DCOS(TFJ-W1+RJ*(TFJ-TH1)*( 2. 5DO*S2-2. ODO) )+
1. ODO+RJ*(RFI+RF2+RF3+RF4)
C
*
C
ENDIF
C
C
C
RADIUS BOTTOM ( TWO BODY SOLUTION)
IF(JVER. EQ. 3)TIIEN
117
RFBl.ODO+E1*DCOS(TFJ-W1)
ENDIF
C
C
RADIUS=PO/RFB
C
RETURN
END
R0M00410
R0M00420
C
o
SUBROUTINE FORMULA
o
C'
C
SUBROUTINE FORMULA
IMPLICIT DOUBLE PRECISION (A-I,M-Z)
CHARACTER*20 LINE
DIMENSION M(1OIMD(100),E(100),W(100),WD(1O0),OM(100),OMID(1o0)
DIMENSION I(100)_ ID(100),F(100),FD(JO0),EC(100),ECD(100),A(100)
DIMENSION R(100),H(100),N(100),TH(100),THD(100)
DIMENSION RF(100)4TF(100),IFD(100),OMF(100),OMFD(100),THF(100)
DIMENSION THFD(1OQ) ,P(100) ,JORBIT( 100) ,DR( 100) ,DID( 100) ,DTHD( 100)
DIMENSION DOMD( 100) ,RX( 100) ,RY( 100) ,RZ( 100) ,RFX( 100) ,RFY( 100)
DIMENSION RFZ(100) ,DRV(100) ,ARC(100) ,ARCD(100) ,DAY(100) ,HX(100)
DIMENSION HY(100),-HZ(100),VX(100),VY(100),VZ(100),DT(100),NX(100)
DIMENSION NY(100) ,NZ( 100) ,RDV(100) ,EX(100) ,EY(100) ,EZ(100)
DIMENSION NDE(100) ,EDR(100) ,V(100) ,HT(i0),RDRF(100) ,INTA(100)
DIMENSION ROMA( 100) ,THJO( 100) ,ATE( 100) ,CTE( 100)
COMMON/OBLATE1/DAY,RX,RY,RZ,VX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K,KK
COMMON/OBLATE2/RDV,R,V,EX,EY,EZ,MIU,NDE,EDR-,H,N,E,P, I,OMI,W,F
COMMON/OBLATE3/PI,EC,M1,A,HT,ER,TH,THD,RTD,MD,WD,OMD,ID,ECD
COMMION/OBLATE4/FD,LINE,J,THF,THFD, IF,IFD,OMF,OMFD,RF, INT,ROM
COMMON/OBLATE5/RJ,DR,DID ,DTHD,DOMD,ESTERR,ACTERR,TERROR,JVER
COMMON/OBLATE6/RFX, RFY ,RFZ ,ARC, AROD ,RDRF ,DRV ,RJ2 ,JN, JORB IT
COMMON/OBLATE7/INTA,ROMA,THJO ,ATE , TE
C
C
C
C
-EXTERNAL RADIUS
CALCULATE E, SINE, AND COSINE FUNCTIONS
S=DSIN(I(1))
S2=S*S
S4=S 2*S2
S6=S4*S2
C=DCOS(I(1))
C2=C*C
SC=DSIN(1(l) )*DCOS(I( 1))
E2=E( 1)*E( 1)
C
C
FORMULA ( SOLUTION)
IF(JVER. EQ. 1)THEN
C
Yl112. ODO-75. ODO*S6+260. 0D0*S4-296. ODO*S2
118
-Y2=RJ*T~i(J)*(2. rDO*S2-2. ODO)
Y3=2.ODO*W(1) -Y2Y4=24. ODO*(5. qDO*S2- 4. ODO)*(5. ODO*S2-4. ODO)
Y.5 =E2*S2*( 14. ODO- 15. ODO*'S2)*( 15. ODO*S2-13. ODO)
Y6=9.0OD*E2+34.0OD
Y77i5. ODO*S4-45. ODO*S2+28. ODO
Y8=6.ODO*(5. ODO*S2-4. ODO)*(5. ODO*S2-4. ODO)
Y9=12. ODO*(5. ODO*S2-4. ODO)
'C
+
Y1O=(6. ODO-S2)/12. ODO-E(1)*S2*DCOS(3. ODO*TH(1)-W(1))/3. ODO-S2*DCOS(2.ODO*TH(l))+E2*(7. ODO*S2-4. ODO)/24. ODO
1!12=9. ODO*E2-34. ODO
C
YF=(2. 5D0*S2-2. ODO)*(THF(J)-TH(l))+E2*Y1*DSIN(Y2)*DCOS(Y3)/Y4
YS=Y5*DCO S(2:ODO*W( 1))/(2..ODO*Y9)+
+
E(1)*S2*(15. ODO*S2-13. ODO)*DCOS(TH(1)+W(l))/2. ODO+
+
E(1)*S2*(15. ODO*S2.13. ODO)*DCOS(3. ODO*TH(1)-W(1))/6. ODOI
+
S2*(15. ODO*S2-13. ODO)*DCOS(2. ODO*TH(1l))/2. ODO+
4- '(5. ODO*Y6*S4+4. ODO*YL12*S2-56. ODO*E2)/96. ODO
,Y=TfHF( J) -W(1) +RJ*rYF+RJ*RJ*THF( J) *YS
C
C
CALCULATE INCLINATION ( SOLUTION)
IF(J)=I( 1)+SC*~RJr,(DCOS(2. ODO*THF(J))/2. ODO+
E(1)*DCOS(Y+2. ODO*THF(J))/6.DO+E(1)*DCOS(Y-2.DO*THF(J))/2.DO+
E2*( 14. DO-15.-DO*S2)*DSIN(Y2)*DSIN(Y3)/(12. DQ*(5. DO*S2-4. DO))DCOS(2. ODO*TH(1))/2. ODO-E(1)*DCOS(3. ODO*TH(1)-W(1))/6. ODO+
E(1)*DCOS(TH(i)+W(1))/2. ODO)
+
+
+
C
C
C
CALCULATE LONGITUDE OF THE ASCENDING NODE ( SOLUTION)
RJ2=RJ*RJ
C
OMF(J)=O1( 1)+C*RJ*(TH(1)-TIIF(J)+DSIN(2. ODO*THF(J))/2. ODOE(1l)*DSIN(Y)+E(1)*DSIN(Y+2. ODO*THF(J))/6. ODOE(1)*DSIN(Y-2. ODO*THF(J))/2. ODO-DSIN(2. ODO*TH(l))/2. ODO+
E(1y*DSIN(TII(1)-Wi(l))-E(1)*DSIN(3. ODO*TH(l)-W(1))/6. ODOE(1)*DSIN(TH(1)+W (1))/2. ODO+E2*Y7*DSIN(Y2)*DCOS(Y3)/Y8)+
+
C*RJ2*THF(J)*(E2*S2*( 15. ODO*S2-14. ODQ)*DCOS( 2.DO*W( 1)) /Y9+
E(1y*S2*DCOS(TH(1)+Ii(1))+YlO)
+
+
+
+
C
ENDIF
C
C
C
FORMULA ( SIMPLIFIED SOLUTION)
IF(JVER. EQ. 2)THEN
C
C
C
CALCULATE INCLINATION ( SIMPLIFIED SOLUTION)
IF(J)I( 1)+SC*RJ*c(DCOS(2. ODO*THF(J))/2. ODO+
E(1)*DCOS(3. ODO*THF(J)-Wi(1))/6. ODO+
E(1)*DCOS(THIF(J)+W(1))/2. ODO-DCOS(2. ODO*TH(1))/2. ODO+
E(1)*DCOS(3. ODQ*Ti(1)-W?(1))/6. ODO+
+
119
+
C
C
E(:1,)*DCOS(TH(1)+W(l))/2. ODO)
CALCULATE LONGITUDE OF TlHE ASCENDING NODE-( SIMPLIFIED SOLUTION)
O1M(J)=OM(1)+ C*RJ*(THCI)'-THF(J)+DSIN(2. ODO*T}{F(J))/2. ODOE(1)*DSIN -THF(J)-W(1l))+E(1)*DSIN(3.OD0*THF(J)-W(l))/6. 0D0+
E(1)*DSIN(THF(J)+W(l))f". ODO-DSIN(2. ODO*TH(l))/2. 0D0+
+ E(1)*DSIN(TH(1)-W(l))-E(1y*DSIN(3. ODO*TH(1)-W(1l))/6. ODO+
E(1)*DSIN(TH(1)+W(l))/2. ODO)
+
C
END IF
C
C
c
FORMULA ( TWMO BODY SOLUTION)
IF(JVER. EQ. 3,)THEN
C
CALCULATE INCLINATION
TW
l~O BODY SOLUTION)
C
IF(J)1( 1)
C
C
C
CALCULATE LONGITUDE OF THE ASCENDING NODE CTWO BODY SOLUTION)
OMF(J)0OM( 1)
C
ENDIF
-C
C
C
CALCULATE RADIUS ( SOLUTION, SIMPLIFIED, OR TWO0 BODY)
RF(j)=RADIUS(RJ,A(1) ,I(1) ,E(1) ,W(1) ,TH(1) ,THF(J) ,JVER)
C
C
C
10
20
C
C
C
CONVERT ANGLES TO DEGREES
OLIFD( J)=OMF(J)*RTD
IFD( J)1F( J)*RTD
THFD( J)=THF( J)*RTD
JORBIT( J)=O
IF(THFDCJ). GT. 360. 0D0)THEN
THFD(J)=-THFD(j)36O. ODO
JORBIT( J)=JORBI'T.(J)+1
GOTO 10
END IF
THJO(J)=JORBIT(J)*2. OD0'*PI+TH(J) -TiI( 1)
IF(OMFD(J). GT. 360. 0D0)T[EN
OMFD(J)=OMFD(J) -360. ODO
GOTO 20
END IF
THF(J)=THFDCJ) /RTD
OMF( J)=OMFD(J) /RTD
CALCULATE DELTAS
DRCJ)=RFCJ) -R(J)
DID(J)=IFD(J) -ID(J)
DTHD(J)=TIIFD(J) -TIID(J)
IF(DABS(DTHD(j)). GE. 180. ODO)THEN
IF(DTHD(J). LT, 0. ODO)THEN
120
DTHD(J)=DTHD(J)+360. ODOELSE
DTHD(J)=DTHD(J) -360. ODO
ENDIF
ENDIF
DOMD(J)OMFD(J) -OMD(J-)
C,
RETURN
END
C
C
o
o
o
C
C
C
SUBROUTINE INERTIAL
*
C
SUBROUTINE INERTIAL
IMPLICIT DOUBLE PRECISION (A-I ,M-Z)
CHARACTER*20 LINE
DIMENSION M(100),M D(100),E(100),W(100),WD(1OO),OM(100),OMD(100)
DIMENSION I(100),ID(100),FX100),FD(100),EC(100),ECD(100),A(100)
DIMENSION R(100),H(100),-.;N(100) ,TH(100) ,THD(100)
DIMENSION RF(100) ,IF(100) ,IFD(100) ,OMF(100) ,OMFD(100) ,THIF(100)
DIMENSION THFD( 100) ,PGLOO) ,JORBIT( 100) ,DR( 100) ,DID( 100) ,DTHD( 100)
DIMENSION DOMD(100) ,RX(100) ,RY(100) ,RZ(100) ,RFX(100) ,RFY(100)
DIMENSION RFZ(100) ,DRV(100) ,ARC(100),ARCD(100) ,DAY(100) ,HX(100)
DIMENSION HY(100),HZ(100),VX(100),VY(100),VZ(100),DT(100),,NX(100)
DIMENSION NY(100),NZ(100),RDV(100),EX(100),EY(100),EZ(100)
DIMENSION NDE(100) ,EDR(100),V(100) ,HT(100) ,RDRF(100) ,INTA(100)
DIMENSION EARC( 100) ,EARCD( 100) ,PDR( 100)
DIMEN-SION ROMiA(QO) ,THJO(100) ,ATE(-100) ,CTE(100)
COMMION/OBLATE1/DAY,RX,RY,RZ,VX,VIY,VZ,DT,HX,HY,HZ,N.,NY,NZ,K,KK
COMON/OBLATE2/RDV,R,V,EX,EY,EZ,MU,NDE,EDR,H,N,E,P,I,OM,W,F
COMMION/OBLATE3/PI,EC,M,A,HT,ER,TH,THD,RTD,MD,WD,OMD, ID,ECD
COMMON/OBLATE4/FD,LINE,J,THF,THFD, IF,'IFD,OMF,OMIFD,RF, INT,ROM
COMMON/OBLATE5/RJ,DR,DID ,DTHD,DOMD,ESTERR ,ACTERR,TERROR,JVER
COMMON/OBLATE6/RFX ,RFY,RFZ ,ARC, ARCD ,RDRF,DR,RJ2,J,JORBIT
COMMfON/OBLATE7/INTA,ROMA,THJO ,ATE ,CTE
COMMON/SPECIAL/EARC ,EARCD,PDR,ENG,ENGF
C
C
C
CALCULATE INITIAL ENERGY
ENG=V( 1)*V( 1)/2. ODO-MU/R( 1)
ENGF=V( 1)*V(1)/2. ODO-IIU/RF( 1)
C
C
C
CALCULATE INERTIAL COORDINATES
RFX(J)=RF(J)*(DCOS(THF(J) )*DCOS(OMF(J)-) DSIN(THF(J))*DCOS(IF(J))*DSIN(OMF(J)))
RFY(J)=RF(J)*(DCOS(THF(J) )*DSIN(ON1FCJ) )+
+
DSIN(THF(J) )*DCOS( IF(J) )*DCOS(OMF(J)))
RFZ(J)=RF(J)*(DSIN(THF(J) )*DSIN( IF(J)))
+
C
C
C
CALCULATE DR VECTOR
DRV(J)=DSQRT( (RFX(J) -RX(J) )*(RFX(J) -RX(J) )+
121
+
+
(RFY(J)--RY(J))*(RFY(J)-RY(J))+
(RPZ(J)-RZ(J))*(RFZ(J)-RZ(J)))
,PDR(J)=DRV(J)/R(J)
C
C
CALCULATE ANGLE ERROR
C
CALL DOT(RX(J),RY(J),RZ(J),RFX(J),RFY(J),RFZ(J),RDRF(J))
ARC(J)F=DACOS(RDRF(J)/(R(J)*RF(J)))
CC=RF( J)
CCP=R( J)
BB=ER
AA=DSQRT(BB*BB+CC*CC-2. ODO*BB*CC*DCOS(ARC(J)/2. ODO))
AAP=DSQRT(BB*BB+CCP*CCP-2. ODO*BB*CCP*VDCOS(ARC(J)/2. Oj)0)),
CCA=PI-DASIN(CC*DSIN(ARC(J) /2.ODO)/AA)
CCPA=PI-DASIN(CCP*DSIN(ARC(J)/2. ODO) fAAP)
EARC(J)=2. ODO*PI-CCA-CCPA
ARCD(J)=ARC(J)*RTD
EARCD(J)=EARC( J)*RTD
C
C
C
CALCULATE DOWNRANGE AND CROSSRANGE ERRORS
ATE(J)=R(J)'*(DTHD(-J)/RTD+DCOS( 1(J) )*DOMD(J)/RTD)
CTE(J)=R(J)*(DSIN(TH(J) )*DID(J)/RTD+
DCOS(TH(J)Y)*DSIN(I(J))*DOMD(J)/RTD)
C
RETURN
END
C
C
C*
C
C
C
*
SUBROUTINE RESULTS
JJJJJ..L.J....J4JJJJ.*......J
C
SUBROUTINE RESULTS
IMPLICIT DOUBLE PRECISION (A-I,M-Z)
CHARACTER*20 LINE
CHARACTER-*11 VERS ION
DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM(100),OMD(loO)
DIMENSION I(100),ID(100),F(100),FD(100),EC(1oo),ECD(1oo),A(100)
DIMENSION R(100) ,H(100) ,N(100) ,TH(100) ,THD(100)
DIMENSION RF(100) ,IF(100),IFD(100) ,OMF(100) ,OMFD(100) ,THF(100)
DIMENSION THFD(100) ,P(100) ,JORBIT(100) ,DR(100) ,DID(100) ,DTHD(100)
DIMENSION DOMD(100) ,RX(100) ,RY(100),RZ(100) ,RFX(100) ,RFY(100)
DIMENSION RFZ( 100) ,DRV( 100) ,ARC( 100) ,ARCD( 100) ,DAY( 100) ,HX( 100)
DIMENSION HYC100),HZ(100),VX(100),VY(100),VZ(100),DT(100),NX(100)
DIMENSION NY(100),NZ(1CO),RDV(100),EX(100),EY(.Oo),EZ(100)
DIMENSION NDE(100),EDR(100) ,V(100),HT(100) ,RDRF(100),INTA(100)
DIMENSION EAROC 100) ,EARCD( 100) ,PDR( 100)DIMENSION ROMA(100),THJO(100),ATE(100) ,CTE(100)
COMMON/OBLATEI/DAY,RX,RY,RZ,VX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K,KK
COMMON/OBLATE2/RDV,R,V,EX,EY,EZ,MU,NDE,EDR,H,N,E,P,I ,OM,W,F
COMMION/OBLATE3/PI,EC,M,A,IIT,ER,TH,TIID,RTD,MID,WD,OMD,ID,ECD
COMMON/OBLATE4/FD,LINE,J,THIF,THFD, IF,IFD,OMF,OMFD,RF, INT,ROM,,
COMMON/OBLATE5/RJ,DR,DID,DTHD, DOMD,ESTERR,ACTERR,TERROR,JVER
122
COMMON/OBLATE6/RFX,RFY,RFZ,ARC,ARCD,RDRF,DRV,RJ2-,JN,jORBITT
COMMON/OBLATE7/INTA,ROMA,THJO,ATE,CTE
COMMON/SPECIAL/EARO ,EARCD, PDR,ENG,ENGF
C
DR( 1)O.ODO
DID(1)=0.ODO
DTHD( 1)=0. ODO
DOMD( 1)=0. ODO
DRV( 1)0. ODO
ARCD( 1)=O. ODO
EARCD( 1)=0. ODO
PDR( 1)=0. ODO
THJO( 1)0. ODO
ATE(1)=0. ODO
CTE(1)=0.ODO
C
IF(JVER.EQ. 1)THEN
VERSION' SOLUTION
ELSE IF(JVER. EQ. 2)THEN
VERSION='SIMPLIFIED'
ELSEIF(JVER. EQ. 3)THEN
VERSION=' SECULAR
ENDIF
IF(RJ. EQ.0. ODO)THEN
VERSION='TWO BODY
ENDIF
C
C
OUTPUT RESULTS FOR DISSPLA
VC
IF(JVER. EQ. 1)THEN
WRITE(3,3000) K
WRITE(3,3100) RJ
ENDIF
If
C
10
C
C
C
DO 10 J = 1, KK
WRITE(3,3100)
WRITE(3,3100)
WRITE(3,3100)
WRITE(4,3100)
CONTINUE
DR(J) ,DID(J) ,DTHD(J)
DOiM-D(J),DRV(J) ,EARCD(J)
PDR(J),ATE(J),CTE(J)
THJO(J)
PRINT RESULTS
WRITE(6,(/)')
WRITE(2,'(/)')
WRITE(6,6000) 'RESULTS'
WRITE(2,6000) 'RESULTS'
WRITE(6,6100) LINE
WRITE(2,6100) LINE
WRITE(6,6200) 'J = IRJ
WRITE(2,6200) 'J = ',RJ
WRITE(6,6300) 'VERSION = 'VERSION
WRITE(2,6300) 'VERSION = 'VERSION
WRITE(6,6100) LINE
WRITE(2,6100) LINE
C
123
DO 20 J = K, KK
C
+
+
WRITE(6,6400) 'POINT =
'ROMBERG
WRIITE(2,6400) 'POINT =
'ROMBERG
',J,'ORBIT = ',JORBIT(J),
ITERATIONS = ,JN
IPj,'ORBIT = ,JORBIT(J),
ITERATIONS = 'JN
C
WRITE(6,6500) 'R
WRITE(2,6500) 'R
=
=
WRITE(6,6500) 'I
WRITE(2,6500) 'I
T ( ,5 0 T
WRITE(6,6500) 'ITH
=
WRITE(6,6500) '0OM
WRITE(2,6500) '014
=
=
',R(J),'RF
'R(J): RF
'RF(J),:DR
',RF(J),'DR
=
=
=
=
',DR(J)
'pDR(J)
C
CR
'ID(J),:IF
',IFD(J).,'DI = ',DID(J)
,IFD(J),'DI = ',DID(J)
, H ( ) : HF = : T F ( ) : T
TD J
= ',THD(J),'THF = 'THFD(J), DITH = ',DTHDCJ)
='ID(J),'IF
=
=
C
',OMD(J),'OMF
',OMD(J),'OMF
=
'OMFD(J),'D014f=
=,OMFD(J),'DOM
=
',DOMD(J'pr
',DOMD(J,%
C
W'RITE(6,6500) 'RX = ',RX(J),'RY
WRITE(2,6500) 'RX = 'RX(J),'RY
=
=
'RYCJ),'RZ
= ',RZ(J)
,RY(J),'RZ = ',RZ(J)
C
WRITE(6,6500) 'RFX = ',RFX(J),'RFY
W'RITE(2,6500) 'RFX = ',RFX(J),'RFY
=',RFY(J),'-RFZ
W'RITE(6,6500) 'DRy =
'EARC=
WRITE(2,6500) 'DRV =
'EARC=
',DRV(J),'PDR
',EARCD(J)
',DRV(J),'PDR
',EARCD(J)
=',PDR(J),
WRITE(6,6500) 'RITE
'CITE
WRITE(2,6500) 'RITE
'CITE
'DR(J),'ATE = ',ATE(J),
',CTE(J)
'DR(J),'ATE = ',ATE(J),
',CTE(J)
=',RFY(J),
C
+
+
=',PDR(J),
C
+
+
=
=
=
=
C
WRITE(6,6600) 'INT = 'INTA(J),'ROM = ',ROMA(J)
WRITE(2,6600) 'INT = 'INTA(J),'ROM = ',ROMA(J)
C
20
C
CONTINUE
WRITE(6,6500) 'EG = ',ENG,'EGF = 'ENGF
WRITE(2,6500) 'EG = ',ENG,'EGF = ,ENGF
C
UWRITE(6,'(/)')
WRITEC2, '(I)')
C
3000
3100
C
6000
6100
6200
6300
6400
6500
FOR14AT(3X,13)
FOR.%AT(3(3X,D18. 10))
FORKA4T(3X,A)
FORM.AT(3X,A20//)
FOR14AT(3X,A,F8. 6)
FORNAT(3X,A,Al1//)
FORMAC2(3X,AB,13/) ,3X,A21,I3//)
FORMAT(3(3X,A6,F23. 15/))
124
= ',RFZ(J)
RFZ = ',RFZ(J)
6600
C
FORMAT(3(3X,A6,F23. 8/))
RETURN
END
C
125
LIST OF REFERENCES
1. Snider, J.R., Satellite Motion Around An Oblate Planet: A PerturbationSolution
for All Orbital Parameters, Ph.D. Dissertation, Naval Postgraduate School,
Monterey, California, June, 1989.
2. Sagovac, C. P., A PerturbationSolution of the Main Problem in Artificial Satellite
Theory, Master's Thesis, Naval Postgraduate School, Monterey, California, June,
1990.
3. Struble, R. A., "A Geometrical Derivation of the: Satellite Equations," Journal of
Mathematical Analysis and Applications, Volume 1, 1960, pp. 300-307.
4. Struble, R. A., "The Geometry of the Orbits of Artificial Satellites," Architectural
Rational and Mechanical Analysis, Volume 7, 1961, pp. 87-104.
5. Struble, R. A., "An Application of the Method of Averaging in the Theory of Satellite Motion," Journal of Mathematics and Mechanics, Volume 10, 1961, pp.
691-704.
6. Eckstein, M. C., Shi, Y. Y., and Kevorkian, J., "Satellite Motion for All' Inclinations Around an Oblate Planet," Proceedings of Symposium No. 25, International
Astronomical Union, Academic Press, 1966, pp. 291-322, equations 61.
7. Danielson, D. A. and Snider, J.R., "Satellite Motion Around an Oblate Earth: A
Perturbation Solution for All Orbital Parameters: Part I - Equatorial and Polar
Orbits," Proceedingsof the AAS/AIAA Astrodynamics Conference, Stowe, Vermont,
August, 1989.
8. Danielson, D. A., Sagovac, C. P., and Snider, J. R., "Satellite Motion Around an
Oblate Earth: A Perturbation Solution for All, Orbital Parameters: Part II - Orbits
for All Inclinations," Proceedings of the AAS/AIAA Astrodynamics Conference,
Portland, Oregon, August, 1990.
9. Ferziger, J. H., Numerical Methods for Engineering Application, John Wiley & Sons,
New York, New York, 1981, pp. 32-37.
10. Coffey, S. L. and Alfriend, K. T., "An Analytic Orbit Prediction Program Generator," Journal of Guidance, Control, and Dynamics, Volume 7, September-October,
1984, pp. 575-581.
11. Deprit, A., "The Elimination of Parallax in Satellite Theory," Celestial Mechanics,
Volume 24, June, 1981, pp. 111-153.
12. Coffey, S. L. and Deprit, A., "A Third Order Solution to the Main Problem in
Satellite Theory," Journal of Guidance, Control, and Dynamics, Volume 5, JulyAugust, 1982, pp. 366-371.
13. Alfriend, K. T. and Coffey, S. L., "Elimination of the Perigee in the Satellite
Problem," Celestial Mechanics, Volume 32, February, 1984, pp. 163-172.
126
14. -Battin, R. H., An Introduction to the Mathematics and Methods of Astrodynamics,
American Institute of Aeronautics and Astronautics, New York, New York, 1987,
pp. 128-130, pp. 450-470.
127
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