SATELLITE MOTION AROUND AN OBLATE
PLANET: A PERTURBATION SOLUTION FOR
ALL ORBITAL PARAMETERS
D. A. Danielson, Professor
G. E. Latta, Professor
C. P. Sagovac, LT, U.S. Navy
S. D. Krambeck, LT, U.S. Navy
J. R. Snider, LTC, U.S. Army
Mathematics Department
Naval Postgraduate School
Monterey, California 93943
Abstract
The search for a universal solution of the equations of motion for a
satellite orbiting an oblate planet is a subject that has merited great
interest because of its theoretical and practical implications. Here, a
complete rst-order perturbation solution, including the eects of the
2 terms in the planet's potential, is given in terms of standard orbital
parameters. The simple formulas provide a fast method for predicting satellite orbits that is more accurate than the two-body formulas.
These predictions are shown to agree well with those of a completely
numerical code and with actual satellite data. Also, in an appendix, it
is rigorously proven that a satellite having negative mechanical energy
remains for all time within a spherical annulus with radii approximately
equal to the perigee and apogee of its initial osculating ellipse.
J
1
1 Introduction
A characteristic feature of practical orbit prediction is that the engineer may deal with
numerous satellites in a great variety of orbits. Under these circumstances analytical relations
which can quickly approximate an orbit may be far superior to large numerical programs.
While many analytical models have been developed for the articial satellite age, most are
not used in practical orbit prediction because they violate one or more of the following
principles:
The method should provide a solution that is signicantly more accurate than the
two-body solution.
The real physical eects of the orbit should be easily distinguishable in the solution.
The solution should be universal it should be valid for all orbital parameters.
The problem of predicting the motion of a satellite perturbed only by the oblateness of the
planet has received considerable attention following the rst launchings of articial satellites
about the Earth. Some of the studies of this problem by means of general perturbation
theories are listed at the end of this paper. Techniques have involved expansions in powers
p
of 2 , averaging processes, the use of spheroidal coordinates, and the edice of Hamiltonian
mechanics. It is not the intention of this present paper to compare the various methodologies
used. Suce it to say that many researchers believe a solution which embodies all of the
above principles was not achieved (e.g., see Ta).
The basic procedure used in this paper to solve the dierential equations of motion is
the perturbation technique known as the Method of Strained Coordinates. This technique
was rst applied to the title problem by Brenner, Latta, and Weiseld. Using a mean orbital
plane to specify an arbitrary orbit, they were only able to obtain a partial solution (e.g., the
eccentricity was assumed small and initial conditions were not considered).
J
2
Here we use coordinates in the true orbital plane to cast the dierential equations into a
simplied form, as was originally done by Struble.
2 Orbital Kinematics
Figure 1 shows the usual reference system of spherical coordinates (
). The radial
distance is measured from the center of the planet to the satellite . The line is in
a direction xed with respect to an inertial coordinate system. The right ascension is the
angle measured in the planet's equatorial plane eastward from the line . The declination
or latitude is the angle measured northward from the equator. The position vector r of
the satellite in the spherical coordinate system is
r r
O
S
O
O
r = (cos cos )b1 + (sin cos )b2 + (sin )b3
r
r
r
(1)
where (b1 b2 b3) are orthonormal base vectors xed in the directions shown.
We can also locate the satellite by its polar coordinates ( ) within a (possibly rotating)
orbital plane that instantaneously contains its position and velocity vectors. Here is the
argument of latitude, i.e., the angle measured in the orbital plane from the ascending node to
the satellite. The orbital plane is inclined at an angle to the equatorial plane and intersects
the equatorial plane in the line of nodes, making an angle with the line.
We introduce another orthonormal set of base vectors (B1 B2 B3) which move with the
satellite so that B1 is in the direction of the position vector r, B2 is also in the orbital plane,
and B3 = B1 B2 . The basis (b1 b2 b3) may be transformed into the basis (B1 B2 B3) by
a succession of three rotations. First the basis (b1 b2 b3) is rotated about the b3 direction
by the angle , next the basis is rotated about the new 1{direction by the angle , and
nally the basis is again rotated about the new 3{direction by the angle . The two sets of
base vectors are related by the product of the rotation matrices representing each successive
r i
O
i
3
rotation (as explained in the book by Danielson):
2
3 2
3 2
3 2
B
cos sin 0
1 0
0
cos sin 1
64 B 75 = 64 ; sin cos 0 75 6
7
6
0
cos
sin
;
sin cos 4
5
4
2
B3
0
0 1
0 ; sin cos
0
0
or
2
3 2
cos cos ; sin cos sin cos sin + sin cos cos B
1
64 B 75 = 6
4 ; sin cos ; cos cos sin ; sin sin + cos cos cos 2
sin sin ; sin cos B3
The position vector r has only one component in the rotating basis:
i
i
i
i
i
i
i
i
i
i
3 2
3
0
b1
0 75 64 b2 75 (2)
1
b3
32
3
sin sin
b1 7
7
6
cos sin 5 4 b2 5
cos
b3
i
i
i
r = B1
(3)
r
Using the rst of equations (2), we obtain the components of r in the xed basis:
r = (cos cos ; sin cos sin )b1
r
i
+ (cos sin + sin cos cos )b2 + (sin sin )b3
r
i
r
i
(4)
Equating the components of equations (1) and (4), we can obtain the following relations
among the angles ( ) of the spherical coordinate system and the astronomical angles
( ):
sin = sin sin
(5)
i
i
cos = cos sec(
The velocity r
time :
d =dt
; )
of the satellite is obtained by dierentiating (3) with respect to the
t
r = B + B1
1
d
dr
dt
dt
r
d
dt
(6)
Since the orbital plane must contain the velocity vector, we have to enforce
d
B1 B = 0
3
dt
(7)
Substitution of equations (2) into equation (7) leads to a relationship which uncouples the
equations for ( ) and ( ):
= tan
(8)
sin
i d
di
d
i
d
4
The velocity (6) can then be written
d
r= B +
1
dr
dt
r
dt
d
1 + tan cot
dt
i
di
!
d
B2
(9)
In the following part of this paper, we will obtain expressions for ( ), ( ), ( ), and
( ). The position and velocity vectors of the satellite then may be calculated from
the formulas in this section. The classical orbital elements , and are the semilatus
rectum, eccentricity, and argument of perigee of the instantaneous (osculating) conic section
determined by the position and velocity vectors. If needed, ( ), ( ), and ( ) can be
obtained from our solution ( ) and
( ):
r i dt=d p e
!
p r e ! dt=d =
p
r
GM
e
e
cos(
sin(
3 Equations of Motion
4
2
dt
d
; !) =
p
r
;1
; !) = 2
r
p
dr
!
d
The expressions for the kinetic and potential energies per unit mass of a satellite orbiting
around an oblate planet are respectively:
2
T
= 21 4
dr
!2
dt
+
r
"
2
d
!2
dt
2
+ 2 cos2
r
2
1 ; 3 sin2 d
dt
!2 3
5
(10)
#
1+ 2 2
(11)
where is the gravitational constant, is the mass of the planet, is the equatorial radius
of the planet, and 2 is the constant coecient of the spherical harmonic of degree 2 and
order 0 in the planet's gravitational eld. Substitution of these equations into Lagrange's
equations
( ; ) ; ( ; ) = 0
=
or
dq
V
=;
GM
J R
r
G
r
M
R
J
d
dt
@ T
@
V
dt
@
@q
T
V
5
q
r results in the following equations of motion:
2
d r
2 ;r
dt
d
dt
r
dt
2 d
r
!
dt
; r cos2 dt
d
d
!2
2 cos2 !2
d
=;
dt
!
d
dt
+ r2 sin cos (12)
@V
@r
=0
d
!2
=;
dt
(13)
@V
@
Initial conditions are established by requiring that at the initial time 0 the orbital parameters of the usual two-body orbit, the conic section determined by the initial position and
velocity vectors, are known. The actual orbit is then tangent to this initial instantaneous
conic section at 0 (see Figure 1). Equating the initial position and velocity vectors given by
equations (3) and (9) to the two-body expressions, we obtain
t
t
( 0) =
r t
dr
dt
d
dt
0
p
1 + 0 cos(
e
( 0) =
t
( 0) =
t
2
0
r
0 0 sin(0 ; !0 )
p0
(15)
e h
0
h
h
1 + tan 0 cot
( 0) = 0
i di
0d
i
i
( 0 ) = 0
p
(14)
0 ; !0 )
i
( 0)
(16)
(17)
(18)
Here 0 =
0 is the initial value of the satellite's specic angular momentum about the
center of the planet, and the subscript 0 on a symbol denotes that the parameter is evaluated
at the initial time 0 .
We immediately have two integrals of the equations of motion:
h
GM p
t
T
r
+ = constant
V
2 cos2 d
dt
= constant
6
(19)
(20)
Equation (19) simply states that the mechanical energy of the satellite remains constant.
Now, from equations (1) and (16)
r
2 cos2 d
dt
= r r b3 =
0 cos i0
d
(21)
h
dt
Equation (21) simply states that the component along the polar axis of the specic angular
momentum of the satellite remains constant. Inserting equations (3) and (9) into equation
(21), we obtain
!
2 cos =
1 + tan cot
(22)
0 cos 0
This allows the independent variable to be changed from to .
Letting = 0 , and using equations (5), (21), and (22), we can rewrite the remaining
equations of motion (12){(13):
dt
r
i
d
h
i
i
di
d
t
u
p =r
2
= ;2c2 sin cos 2sin cos
3
cos i + 2 sin cos
di
Ju
d
2
i
Ju
i
u
J
c
;4u
c
2
2
d
"
u
du
d
2
The terms in (24) with
du
!2
d
2
(
i
i
d
u
d
du
!
d
du
d
4
sin cos (1 ; 3 cos2 )
2
J u
u
sin cos2
u
i
J
i
c
u
sin cos (1 ; 3 cos2 ) ; 2
d
4
c
i
(24)
i
#
i
can be combined, yielding the equivalent equation
c
du
sin3 cos6
cos2 + cos2 2(1 + sin2 (7 cos2
+
=
2
2
2
+2
i
2
+
u
sin2 cos2 i ;
cos (2 + sin2 ) +
d u=d
d u
d
u
sin2 cos2 i ; 2
d u
2
i
(23)
i
i
2
2 cos
cos
2 (1 ; 3 sin2 sin2 ) + 2
2+ = 2 +
2
d u
d
du
d
!2
i
; 3))
sin2 cos2 ]
i
39
!
4J 2u sin3 cos6 i 4u2 sin cos2 i ; u du cos (2 + sin2 i) ; du 2 sin cos2 i5=
4
c
d
d
2 cos4
4
sin
1+
+4
2
Ju
i
c
7
2 2 sin4 cos8 i !
4
c
J u
(25)
Here we have introduced the shorthand notation = cos 0 , = sin 0 , = 3
c
i
s
i
J
2
2
2 20.
J R = p
4 Perturbation Procedure
The dierential equations (23){(24) are coupled by the nonlinear terms and apparently
cannot be solved analytically. If we expand the right sides of (23) and (25) in a Taylor series
expansion in powers of and retain only terms up to order 2, the equations simplify to
J
di
3
= ;2 sin cos2 sin cos + 4
Ju
d
2
i
i
2 2 sin i cos7 i
sin3 cos + O(J 3)
4
c
J u
c
cos2 + cos2 ;4 sin2 cos4 + 21 + sin2 (7 cos2
+
=
2
2
2
2
d u
d
J
i
u
i
c
+2
u
+4
J
2
J u
u
c
du
d
i
c
sin cos (1 ; 3 cos2 i) ; 2
u
du
!2
sin2 cos2
d
i
i
du
d
u
sin cos 7 cos2
i
; 5] +
u
i
du
d
!2
sin2 cos2
i
; 3)]
(27)
i
sin2 cos6 2;1 + 3 sin2 (1 ; 2 cos2 )] + 3 sin2 cos4
4
2
c
+
u
(26)
i
c
+ ( 3)
O J
Here the term in the symbols indicates that, for all suciently small , the error is less
than a constant times 3. The equations (26){(27) are identical to those used as the starting
point in the analysis of Eckstein, et al.
It is reasonable to expect that the solution for will be arbitrarily close to the two body
solution, 1 + 0 cos( ; 0), when is close to zero. This assumption is consistent with
letting
= 1 + 0 cos + 1 + 2 2 +
(28)
O
J
J
u
e
!
J
u
e
y
=
i
y
Ju
J u
:::
; !0 + J y1 + J 2 y2 + : : :
= 0+
i
2
1 + J i2 + : : :
Ji
An algorithm for the perturbation procedure is:
Let n = 1
Substitute expressions (28){(30) into the equations of motion (26){(27)
8
(29)
(30)
Equate the coecients of J n
Choose the arbitrary constants so secular terms will not arise.
Solve for the nth order solution
Satisfy the initial conditions (14){(18)
Iterate on n
The calculations were carried out with the symbolic manipulation program MACSYMA.
In this paper we only briey outline these calculations for more details see the theses of
Sagovac and Snider.
Beginning by substituting equations (28) and (30) into (26), and equating the terms
multiplied by , we obtain
J
1
= ; sin 2
di
sc
d
;
0
0
2 sin( + 2 ) + 2 sin(
sce
y
sce
y
; 2 )
(31)
A solution to this equation is
i
1
=
sc
2
cos 2 +
sce
6
0
cos( + 2 ) +
y
0
sce
2
cos(
y
; 2 ) + K1 cos(2y ; 2) + K2
(32)
The last two terms may be added because they are to lowest order homogenous solutions
to equation (30). The term multiplied by the constant 1 was added to eliminate secular
terms in 2 note that dierentiating this term with respect to produces terms multiplied
by , from equation (29). The constant 2 was added to satisfy the initial condition (17),
which implies that 1( 0 ) = 0 so
K
i
J
K
i
2
K
= ; cos 2
2
sc
0;
sce
6
0
cos(3
0 ; !0 ) ;
0
sce
2
cos( 0 + 0) ;
!
1 cos 2!0
K
Substituting equations (28){(30) and (32) into (27), and equating terms multiplied by
yields
!
2
2
2
3
5
1 (2 + 5 2 ) 2 ; 2 2] cos 2
1
2
+
=
1
;
+
;
+
1
+
1
0
0
0
2
2
4
4
d u
d
s
e
s
e
2
0 (;9s2 + 8) cos 2y + e0 (11s2 ; 6) cos(y + 2 ) +
+4
e
u
3
9
s
15 20 (3
24
e
s
e
J
2 ; 2) cos(2y + 2 )
(33)
+
" 2
e
0 (3s2 ; 2) ;
2
1
#
cos(2 ; 2 ) ; 2
2
y
c
sK
y
y
u
e
0
2
1
+ 4 ; 5s 2
dy
!
d
K
y
32+
1 =1;
2
2
s
0
2
1
2
d
d y
s
y
!
2
2 ;5s
e0
4
!
!
1 ; 2 (2 + 5 2) + 2 2] cos 2
+ 1 + 12
0
0
s
e
e
2
2 ; 8) cos 2y + e0 (;11s2 + 6) cos(y + 2 ) + e0 (;3s2 + 2) cos(2y + 2 )
24
+ 0 (3 2 ; 2) ; 2
8
+ 5 cos(
" 2
e
y
u
y e
e
K
+ 0 (9
12
y
s =
s
u
cos +
y
dy =d
y
c
+
sin
8
In the above equation, the cos and sin terms would produce secular terms sin and
cos in 1. The choice 1 = 5 2 2 ; 2 will eliminate these possibilities. Integrating
yields
!
2
5
; 2 ( ; 0 ) + 3 sin(2 ; 2 ) + sin 2 0 ]
(34)
1=
2
The term multiplied by 3 was added to eliminate secular terms in 2. The constant terms
in (34) were added to satisfy the initial condition ( 0) = 0 ; 0.
A solution to lowest order of equation (33) is then
sK
sK
s
K
1
#
c
y
cos(2
y
; 2 ) ;
2
24
sK
c
2
+
K
(35)
4 cos(y ; 2 )
; 0 + !0 ) + K6 sin(y ; 0 + !0 )
The term multiplied by 4 was added to eliminate secular terms in 2. The terms multiplied
by 5 and 6 were added to satisfy the initial conditions (14){(16).
With all terms in place to deal with secular terms, the calculations are continued by
substituting equations (28){(30), (32), (34), and (35) into (26) and equating terms multiplied
by 2:
"
2 (15 2 ; 14) #
2
0
= 1 + 24(5
sin(2 ; 2 ) +
(36)
2 ; 4)
K
K
u
K
J
di
K
d
sce
s
y
s
:::
We have for brevity only indicated on the right side of equation (36) the term that would
produce secular terms in 2. Removal of this term by making its coecient zero determines
1 . Equation (36) is then integrated to determine 2 .
Continuing the procedure by equating the terms multiplied by 2 in the expansion of
equation (27) determines 2, 3 , and 4 . Final values of all the constants are listed in
Appendix I.
i
K
i
J
y
K
K
10
Knowing the solution for ( ), we can determine ( ) by integrating equation (8) and
applying the initial condition (18). The angle , which increases continuously from an initial
value 0 , may be related to the time by numerically integrating (22).
i t
5 Solution
Here we assemble the complete solution:
r
=
0
p =
2
1 + 0 cos + 1 ; 3 +
2
e
2
J
!
2
1 ; 5 + 1 (;(2 + 5 20 ) 2 + 2 20) cos 2
4
12
s
e
24
s
2
s
y
e
e
s
e
s
s
s
h
2 2 (15s2 ; 14) sin J
0
2
i
h
(5 2 ; 4) sin 2
6(5 2 ; 4)
e s
+
e
24
+ 0 (3 2 ; 2) cos(2 ; 2 )
8
h
2 ) 4 ; 14(4 + 2 ) 2 + 24] sin J (5
15(2
+
0
0
0
2
+
2
12(5 ; 4)
e
s
2
2 ; 8) cos 2y + e0 (;11s2 + 6) cos(y + 2) + e0 (;3s2 + 2) cos(2y + 2 )
+ 120 (9
e
y
2
e0
s
s
J
0; 2
!
i
2 ; 4)
(5
s
sin + 0]
2 ; 4)
i
2
2 2
e s
; 30 + 3!0 ) ; 0
2 2
e s
; 0
16 cos(
s
+ 0 (3 2 ; 2) cos(
24
+ 40 (3 2 ; 2) cos(
!
y
; 0 + 3!0 )
cos( ; 5 0 + 3 0)
16
3 0 2 cos( ; 4 + 2 )
; 2 0 + 2 0) ;
0
0
8
1
0
; ( 2 + 1) cos( + 2 0 ) + (;2 + 5 20 ) 2 ; 2 20 ] cos( + 0 + 0 )
4
8
+ 14 (6 + 5 20) 2 ; 4(1 + 20)] cos( ; 0 + 0)
1 ;(14 + 5 2) 2 + 2 2] cos( ; 3 + )
+ 24
0
0
0
0
e
y
s
y
e
s
e
s
y
e
s
e s
!
!
e
e
e
s
e
y
y
y
y
s
(37)
!
!
e
y
!
!
!
2
2
e0
0
2
+ (9s ; 4) cos(y + 30 ; !0) + (;7s2 + 6) cos(y + 0 ; !0)
48
8
2
e0
+ (;5s2 + 4) cos(y ; 0 ; !0)
e
16
+ 40 (2
e
+
0
2 ; 1) cos(y + 2 ) + e0 (;3s2 + 1) cos(y ; 2 ) + e0 (;3s2 + 2) cos y
0
0
s
2
2 cos( + ! ) + e0 s
0
0
e s
3
4
cos(3
0 ; !0 ) + s
11
2 cos 2 ]
0
4
+
0
(
p O J
2
3
J )
=
y
2
5s
; !0 + J
2
!
;2
(
; 0 )
h
i
(
2
(;75s6 + 260s4 ; 296s2 + 112) sin J2 (5s2 ; 4)
J e0
+ 24(5s2 ; 4)
(5s2 ; 4)
+J s2 (;15s2 + 14)(15s2 ; 13) cos 2!
2
)
0
+ J 2
e s2
0
2 (15
s
2
+ 06 (15 2 ; 13) cos(3 0 ; 0) + 2 (15 2 ; 13) cos 2
1
2
4
2
2
2
+ 5(9 0 + 34) + 4(9 0 ; 34) ; 56 0 ] + ( 2
96
e s
s
e
i
=
i
s
0 + scJ
(
e
s
s
e
e
y
O J
h
e
"
cJ
+ 0 sin(
e
0;+
0 ; !0 ) ;
1 sin 2
2
0 ; !0 ) ;
0
6 sin(3
e
0
3
J 0 ; !0 ) ;
)
e
0
6
!
O J
sin( + 2 ) ;
y
#
0
t
=
1Z
+
0
t
0
h
r
0
2
1+
J
2
2
3
sin(
h
cos 2
1 (; 2 + 6) + (
12
k
O J
cos 2 + 0 (
2 ; 1)
12
e
s
2 ; 4)
)
(39)
; 2 ) ;
J
0; 2
!
s
1 sin 2
2
0
!
i
s
y
i
(5
J
0; 2
J + cJ 2 ;e0 s2 cos(0 + !0) ; e0 s
(;3s2 + 2)
2
0
e
2 sin( 0 + 0)
e
h
2
e
;s2 cos 20 + 0 (7s2 ; 4) +
24
h
!
2(15s4 ; 45s2 + 28) sin J (5s2 ; 4)
2
2
0
+
12(5s2 ; 4)
(5s2 ; 4)
(38)
i
cJ e
+J s2 (15s2 ; 14) cos 2!0
)
2 cos( 0 + 0 ) + (
e
; e0 sin y +
i
0
s
s
2 ; 4)
s
2 ; 13) cos( + ! )
0
0
(5 2 ; 4) sin 2
12(5 2 ; 4)
e
y
(5
2 (;15s2 + 14) sin J
0
2
+ 0 cos( ; 2 ) +
2
1
0
; cos 2 0 ; cos(3
2
6
J
0; 2
!
1 cos 2 + 0 cos( + 2 )
2
6
e
= 0 +
s
!
h
cos 2
3
J )
s
2
3 cos(3
2
i
(5
2 ; 4)
0 ; !0 )
(40)
+
2
0
2
0 (;2s + 1)
cos( + 2 ) +
cos( ; 2 )
6
2
i h
i
h
2 (15 2 ; 14) sin J (5 2 ; 4) sin 2 ; J (5 2 ; 4)
0
2
2
12(5 2 ; 4)
cos +
y
2 + 3)
0 (;4s
e
e s
y
e
s
s
!
y
s
(41)
s
+s2 ; 1 +
2
0
2
2 cos 2 0 + 6 cos(3
s
e s
0 ; !0 ) +
0
2
2 cos( 0 + 0)
e s
!
d
+
2
0 O(J 2
h0
p
2
J )
In obtaining the equations (37){(41), use has been made of trigonometric formulas
to simplify terms containing the factor 5 2 ; 4 in the denominator. In the form given,
these terms can clearly be seen to approach a nite limit at the \critical inclination"
q
;1 4 5 = 63 260 or 116 340 . Hence the solution is actually valid for all values
0 = sin
q
of 0 . If j 0 ; sin;1 4 5j , the formulas (37){(41) can still be used by letting 5 2 ; 4 = ,
q
or the limiting forms for 0 ! sin;1 4 5 can be used.
To check the solution, we can see if the specic mechanical energy (18) of the satellite
remains constant. Substitution of the solution (36){(37) into equation (10) plus (11) yields
s
i
=
i
i
=
< J
s
=
i
T
+ =;
V
GM
J
(1 ; 20) ;
20
e
2
GM J R
p
2 (1 ; 3 sin2 )
0
2r03
+
GM
0
p
( 2)
O J
The right side is easily recognized as the value of the specic mechanical energy at the initial
time 0 .
As a further check on the solution, we can see if it reduces to our previous results for
equatorial and polar orbits, obtained by completely separate derivations (Danielson and
Snider, 1989). Setting 0 = 0 and using the independent variable measured from the line
, we nd that equations (37){(41) reduce to equations (18){(22) of our previous paper.
Setting 0 = 2 and using the expansion cos( + ) cos ; sin , we nd that
equations (37){(41) reduce to equations (38){(41) of our previous paper.
Comparing the terms in the -symbols, we see that the relative error in equation (41)
may be greater than that of equations (37){(40). Since the underlined terms in equations
(37){(40) are of this same order of magnitude, we can drop the underlined terms except
when (37){(38) are used to calculate in equation (41). The relative error of our solution
t
i
O
i
=
y
O
r
13
Jk
y
Jk
y
will then still be of order ( ; 0 ) 2 .
If we retain only the two-body solution, the relative error terms will be of the order
( ; 0 ) . Here the error in our solution, as compared to the exact solution of the equations
of motion, should be of the order times the error in the two-body solution (for an Earth
satellite
0015).
J
J
J
J < :
6 Comparison of Perturbation, Two-Body, Numerical, and Measured Solutions
In this section we compare the preceding perturbation solution, the two-body solution, a
completely numerical solution of the dierential equations, and actual measured satellite
data for more comparisons see the thesis of Krambeck. The dierence between the position
vector r determined by the numerical integration code or measured data and the position
vector rref calculated from our perturbation solution or the two-body solution is the error
r:
r = r ; rref
If the errors ( ) in the orbital parameters (
r from equation (4) and the linear approximation
r
i
r i
) are small, we can estimate
r r + r + r + r @
@r
@
r
@
@
It is customary to decompose r into components (
@i
i
(42)
@
@
B B2 B3):
1 2 3 ) along the moving triad ( 1
r = 1 B1 + 2 B2 + 3 B3
The component 1 is called the radial error, 2 is the down track error, and 3 is the cross
track error. Applying (42) to equation (4), and expressing the base vectors (b1 b2 b3 ) in
terms of (B1 B2 B3), we obtain the following approximations:
1 r
2 r( + cos i)
14
3 r(sin i ; cos sin i)
(43)
We obtained the numerical integration code UTOPIA from the Colorado Center for
Astrodynamics Research located on the campus of the University of Colorado. The code
was specialized to the dierential equations used in this paper. We compared the solutions
for an earth satellite with the following initial conditions:
r
0
= 7 386 18 km
e
0
= 003991
0
= 104 05
0
= 224 38
0
= 90 03
!
i
:
:
:
:
:
0 = 322 63
:
0
t
= 0
These initial conditions represent an essentially polar orbit at an altitude of approximately
1000 kilometers and period about 1 43 hours. For this satellite the perturbation and numerical
orbits match extremely well while the two-body orbit is grossly erroneous. The magnitude of
the error in r is shown in Figure 2. Note that the relative error in our perturbation solution
is 2 8 2( ; 0 ), and that this error is 1 1 times the error in the two-body solution.
We obtained measured satellite data from the First Satellite Control Squadron located
at Falcon Air Force Base, Colorado. A near earth satellite processed the following initial
conditions:
: J
: J
0
= 7 776 58 km
0
= 0003071
0
= 149 14
0
= 9 57
0
= 98 81
r
e
!
i
:
:
:
:
:
15
0 = 37 10
:
= 0000Z 26 July 1990
0
t
Again, the perturbation orbit is far superior to the two-body orbit. The radial, down track,
and cross track errors ( 1 2 3) are shown in Figure 3. Note that although the perturbation
solution produces only a small improvement in the radial error, this error is negligible in
comparison to the down track error.
7 Conclusions
Our solution embodies the principles outlined in the introduction. The relative error of our
solution is of order ( ; 0 ) 2 , which is a factor of times the relative error of the two-body
solution our solution loses its validity after an angular change ( ; 0 ) of order 1 2, which
is a factor of J1 longer than the interval of validity of the two-body solution. Secondly, our
solution is in terms of classical orbital elements no transformation to an alternative nonphysical set of elements is required. Finally, our solution is free of singularities for all values
of the initial orbital parameters, including elliptic, parabolic, and hyperbolic orbits.
Our formulas should agree closely with satellite orbits whose dominant perturbation is
the planet's oblateness. Of course, the eects of higher-order terms in these expansions,
higher-order terms in the planet's potential, and of other perturbation forces may also be
important. The formulas will have to be amended to include these additional eects.
J
J
=J
APPENDIX I: Values of the Constants K1{K6
1=
K
2 = ; cos 2
2
K
sc
0;
sce
6
0
cos(3
3
K
=
2 (;15s2 + 14)
0
24(5s2 ; 4)
cse
0 ; !0 ) ;
sce
2
0
cos( 0 + 0) +
!
2
6
4
2
e0 (;75s + 260s ; 296s + 112)
48(5s2 ; 4)2
16
2 (15s2 ; 14)
0
cos 2!0
24(5s2 ; 4)
cse
K
K
5
2 + 2)
15(
0
4= 0
2
e
s
0 ; 2!0 ) +
e
e
= 120 (;9 2 + 8) cos(2
s
4 ; 14(e2 + 4)s2 + 24]
0
2
24(5s ; 4)
e
2
0 (3s2 ; 2) cos(4 ; 2! )
0
0
24
2
e
e0
;(e0 s2 + K4 ) cos(0 + !0 ) + (s2 ; 2) cos(30 ; !0 ) + 0 (;3s2 + 2) cos 2!0
8
;
K
6
1 5(2 ; 2 ) 2 + 2 2] cos 2 + 1 (15 2 + 18)
0
0
0
12
12 0
e
s
e
2
e
8
s
2 ; (e2 + 1)
0
2
= + 0 (6 2 ; 5) sin(2 0 ; 2 0) + 0 (;3 2 + 1) sin(4 0 ; 2 0)
6
12
+ 21 0 (; 2 + 1) + 2 4] sin( 0 + 0 ) + 20 (3 2 ; 2) sin( 0 ; 0)
2
+ 0 (;7 2 + 2) sin(3 0 ; 0) + 0 (; 2 + 1) sin 2 0 + 1 ;(5 20 + 2) 2 + 2 20] sin 2
8
4
6
e
s
e
s
e
e
!
K
s
s
e
!
e
!
!
s
s
!
!
e
s
e
0
APPENDIX II: Rigorous Bounds on the Orbit
It follows from (10){(12) that
+ = 12
T
V
dr
!2
dt
+2
r
2
d r
2 ;
dt
2 +
GM
2
3
4r
GM J R
r
2
(1 ; 3 sin2 )
This can be rewritten in the form
d
dr
2 !2 3
4r2 dr 5 = 4(T
dt
+ ) +2
V
r
+
GM
2
GM J R
r
2
2
(3 sin2
; 1)
from whence it follows that
d
dr
2 !2 3
4r2 dr 5 4(T
dt
+ ) +2
V
r
GM
+2
2
GM J R
r
2
2
Integrating from ( 0 ) to ( ) yields
r t
2
r
dr
dt
r t
!2
2(T
+ ) 2+2
V
r
GM r
;
2
2
GM J R
<
T
V
r
GM r
17
; h20 +
r
It follows that
0 2( + ) 2 + 2
2
; h20 1 ;
3
2
p0 r0
J R
2
]
3
2
2
0 2
p0 r0
h J R
(44)
When +
0, the quadratic polynomial on the right side of (44) has the roots (exact
values can be found from the quadratic formula)
T
V
<
min
r
= 1 +0 1 + ( 2)]
p
e
0
max
O J
r
= 1 ;0 1 + ( 2)]
p
e
0
O J
Hence a satellite having negative mechanical energy remains for all time within the spherical
annulus min
max . Since the position vector is bounded, we can invoke the recurrence
theorem i.e., the satellite will come as close as desired to its initial position in a suciently
long period of time (as shown by Poincare). Furthermore, we are guaranteed of the validity
of supressing secular terms to describe the orbit via perturbation analysis.
r
< r < r
Acknowledgements
John Rodell from the Colorado Center for Astrodynamics Research produced the numerical
data for the comparison shown in Figure 2. Capt. Greg Petrick and 1st Lt. Bruce
Hibert from the First Satellite Control Squadron supplied the measured satellite data shown
in Figure 3. Misprints in equations (24){(26) and the formula for 6 in our earlier work
(Danielson, Sagovac, Snider, 1990) were pointed out by Professor Clyde Scandrett. The
research was supported by the Naval Postgraduate school.
K
References
Brenner, J. L., Latta, G. E., and Weiseld, M., 1959, \A New Coordinate System for Satellite
Orbit Theory," Stanford Research Institute Project No. SU-2587, AFMDC TR 59-27, ASTIA
AD-216455.
Brenner, J. L., and Latta, G. E., 1960, \The Theory of Satellite Orbits, Based on a New
Coordinate System," Proc. the Royal Society, A, Vol. 258, pp. 470{485.
Danielson, D. A., 1992, Vectors and Tensors in Engineering and Physics, Addison-Wesley.
Danielson, D. A., and Snider, J. R., 1989, \Satellite Motion Around an Oblate Earth: A
Perturbation Procedure for all Orbital Parameters: Part 1 | Equatorial and Polar Orbits,"
Proc. AAS/AIAA Astrodynamics Conf., held in Stowe, Vermont.
Danielson, D. A., Sagovac, C. P., and Snider, J. R., 1990, \Satellite Motion Around an
Oblate Planet: A Perturbation Solution for all Orbital Parameters: Part II | Orbits for all
Inclinations," Proc. AAS/AIAA Astrodynamics Conf., held in Portland, Oregon.
18
Eckstein, M. C., Shi, Y. Y., and Kevorkian, J., 1966, \Satellite Motion for all Inclinations
Around an Oblate Planet," Proc. Symp. No. 25, International Astronomical Union, Academic Press, pp. 291{322, equations 61.
Krambeck, S. D., 1990, \Analysis of a Perturbation Solution of the Main Problem in Articial
Satellite Theory," M.S. Thesis, Naval Postgraduate School.
Poincare, H., 1967, \New Methods of Celestial Mechanics" (translation), Vol. III, NASA
TT F-452.
Sagovac, C. P., 1990, \A Perturbation Solution of the Main Problem in Articial Satellite
Theory," M.S. thesis, Naval Postgraduate School.
Snider, J. R., 1989, \Satellite Motion Around an Oblate Planet: A Perturbation Solution
for all Orbital Parameters," Ph.D. thesis, Naval Postgraduate School.
Struble, R. A., 1960, \A Geometrical Derivation of the Satellite Equations," J. Mathematical
Analysis and Applications, Vol. 1, pp. 300{307.
Struble, R. A., 1961, \The Geometry of the Orbits of Articial Satellites," Arch. Rational
Mech. Analysis, Vol. 7, pp. 87{104.
Struble, R. A., 1961, \An Application of the Method of Averaging in the Theory of Satellite
Motion," J. Mathematics and Mechanics, Vol. 10, pp. 691{704.
Ta, L. G., 1985, Celestial Mechanics, John Wiley and Sons, p. 340.
Weiseld, M., 1960, \An Approximation of Polar Orbits of Near Satellites Around an Oblate
Earth," Stanford Research Institute Report 3163-3(T).
Other Papers on this Subject
Arsenault, J. L., Enright, J. D., and Pursell, C., 1964, \General Perturbation Techniques
for Satellite Orbit Prediction Study," Technical Documentary Report No. AL-TDR-64-70,
Vols. I and II.
Brouwer, D., 1959, \Solution of the Problem of Articial Satellite Theory Without Drag,"
Astron. J., Vol. 64, pp. 378{397.
Cefola, P., and McClain, W. D., 1978, \A Recursive Formulation of the Short-Periodic Perturbations in Equinoctial Variables," AIAA/AAS Astrodynamics Conf., Palo Alto, Report
No. 78-1383.
Coey, S. L., Deprit, A., and Miller, B., 1986, \The Critical Inclination in Articial Satellite
Theory," Celestial Mechanics, Vol. 39, pp. 365{406.
deLafontaine, J., and Hughes, P. C., 1987, \A Semianalytic Theory for Orbital Decay Predictions," J. Astron. Sci., Vol. 35, No. 3, pp. 245{286.
Garnkel, B., 1959, \The Orbit of a Satellite of an Oblate Planet," Astron. J., Vol. 64, pp.
353{367.
Hoots, F. R., \Theory of the Motion of an Articial Earth Satellite," 1981, Celestial Mechanics, Vol. 23, pp. 307{363.
19
Hughes, S., \The Critical Inclination: Another Look," 1981, Celestial Mechanics, Vol. 25,
pp. 235{266.
Jupp, A. H., 1980, \The Critical Inclination Problem with Small Eccentricity,", Celestial
Mechanics, Vol. 21, pp. 361{393.
King-Hele, D. G., 1958, \The Eect of the Earth's Oblateness on the Orbit of a Near
Satellite," Proc. Roy. Soc. (London), Ser. A, Vol. 247, pp. 49{72.
Kozai, Y., 1962, \Second-Order Solution of Articial Satellite Theory Without Air Drag,"
Astron. J., Vol. 67, pp. 446{461.
Lubowe, A., 1969, \How Critical is the Critical Inclination," Celestial Mechanics, Vol. 1,
pp. 6{10.
Morrison, A. G., 1977, \An Analytical Trajectory Algorithm Including Earth Oblateness,"
J. Astron. Sci., Vol. 25, No. 1, pp. 35{62.
Petty, C. M., and Breakwell, J. V., 1960, \Satellite Orbits About a Planet with Rotational
Symmetry," J. Franklin Institute, Vol. 270, pp. 259{282.
Small, H. W., 1963, \Satellite Motion Around an Oblate Planet," AIAA Astrodynamics
Conf., held at Yale Univ., Paper No. 63-393.
Stellmacher, I., 1981, \Periodic Orbits Around an Oblate Spheroid," Celestial Mechanics,
Vol. 23, pp. 145{158.
Sterne, T. E., 1957, \The Gravitational Orbit of a Satellite of an Oblate Planet," Astron.
J., Vol. 63, No. 1255, pp. 28{40.
Vinti, J. P., 1961, \Theory of an Accurate Intermediary Orbit for Satellite Astronomy," J.
Res. Nat. Bur. Stand. B., Vol. 65, pp. 169{204.
Zaropoulos, B., 1987, \The Motion of a Satellite in an Axi-Symmetric Gravitational Field,"
Astrophysics and Space Science, Vol. 139, pp. 111{128.
Zare, K., 1983, \The Possible Motions of a Satellite About an Oblate Planet," Celestial
Mechanics, Vol. 30, pp. 49{58.
20
Figure 1: Orbital geometry.
Figure 2: Comparison of perturbation, two-body, and numerical orbits.
Figure 3: Comparison of perturbation, two-body, and measured orbits.
21