Motion of a Satellite under the influence of an oblate Earth
ASEN 3200
July 10, 2001
George H. Born
z
P
r

dm
y
x
Figure 1. Potential of an arbitrary shaped body
We wish to derive the gravitational potential function for an arbitrary shape body as shown in
figure 1. Consider a particle,  , of mass, m. The gravitational force on P due to the differential
mass dm given by Newton’s law of gravitation is
F  Gmdm

3
(1)
where
dm  dxdydz
(2)
1
and  is the density. The acceleration of dm is given by
F
i.e. the force per unit mass
m
F

 Gdm 3 .
m

a
(3)
We can express the acceleration as the gradient of a potential function,  , where

Gdm

.
(4)
This is easily shown by taking the gradient of  i.e.,
a   
 ˆ  ˆ  ˆ
i
j
k
x
y
z

 Gdm   ˆ  ˆ  ˆ
i
j
k
y
z 
 2  x

 Gdm ˆ ˆ
xi  yj  zkˆ
3

 Gdm



.
3
(5)
To obtain the potential function for the entire body we would integrate Eq. (4) over the
volume of the body (Kaula, 2000). The result is


l

R
  1   J l   Pl sin    
r  l  2  r 
l 1
l

 r l 1 Pl m sin  C l m cos m  S l m sin m
1

m 1
(6)
where the coordinates of P are now expressed in spherical coordinates r, ,   , where 
is the geocentric latitude and  is the longitude. Also, R is the equatorial radius of the primary
body and Pl m sin   is the Legendre’s Associated Functions of degree  and order m.
2
The coefficients J l , C l m and S l m are referred to as spherical harmonic coefficients.
If m = 0 the coefficients are referred to as zonal harmonics. If l  m  0 they are referred
to as tesseral harmonics, and if l  m  0 , they are called sectoral harmonics.
We will considered the effect of J 2 , the second zonal harmonic, on the orbit of a satellite.
J 2 is also known as the oblateness coefficient. Gravitationally, the earth can be modeled fairly
accurately as an ellipsoid of revolution i.e. by the J 2 harmonic. In fact the next zonal harmonic,
J 3  2.5  10 6 , is thee orders of magnitude smaller than J 2  1.08 10 3 .
+ indicates positive gravity anomaly
+
+
- indicates negative gravity anomaly
Figure 2. The Earth as described by J 2
Figure 2 illustrates the equipotential surface representing the influence of J 2 . The polar radius
of the elliptical earth is 20 km smaller than the equatorial radius. This effect is primary due to
the movement of mass to the equator cause by the centripetal force due to earth rotation.
The value of J 2 for the earth is 0.00108, for Mars J 2  0.00196 , for the Moon J 2  0.000195 ,
and for the asteroid Eros J 2  0.11 . The value of J 2 for the Moon is much lower for the Earth
and Mars because its rotation rate is much less. If we evaluate Eq. (6) for l  2, m  0,
3
i.e. J 2 , and drop the central force field contribution we obtain the perturbing potential,  p
p 
 J2  R 
2
2
  (1  3sin  ) .
r 2 r
(7)
Equation (7) may be written in terms of the Kelper elements of a satellite’s orbit by replacing  ,
the geocentric latitude, with the inclination and argument of latitude of the satellite orbit.
z
orbit
S/C
r
u

y
i

Equator
x
Figure 3. Orientation of the s/c orbit
where,
u
 argument of latitude,   
  argument of perigee

 true anomaly
i
 inclination
  right ascension of the ascending node

 geocentric latitude
From Figure (3) and spherical trig identities
sin   sin u sin i
4
Hence, Eq. (7) may be written as
p 
 J2  R 
2
2
2
  (1  3 sin u sin i )
r 2 r
2 sin 2 u  1  cos 2u
but
(8)
(9)
and  p my be written as
3 R
1 1 2 1 2

p 
  J 2   sin i  sin i cos2  2 
2rr
2
3 2

2
(10)
The differential equations which describe the variation with time of the orbit elements
are called Lagrange's Planetary Equations (Roy, 1988), and are given by
da 2  p

dt na M
de 1  e 2  p
1  e 2  p


dt na 2 e M
na 2 e 
 p
 p
di
cos i
1


2
2
2
2
dt na 1  e sin i  na 1  e sin i 
 p
d
1

dt na 2 1  e 2 sin i i
 p
d
cos i
1  e 2  p


dt
na 2 e e
na 2 1  e 2 sin i i
dM
1  e 2  p 2  p
n 2

dt
na a
na e e
5
(11)
In general the orbit element will experience secular, long and short period perturbations due to
J 2 (Brouwer, 1959). Secular perturbations grow linearly with time; long and short period
perturbations are periodic in multiples of the argument of perigee and true anomaly respectively.
The first order (i.e., to OJ 2  ) secular terms may be obtained
by averaging the short period terms out of  p by integrating over the mean anomaly from
zero to 2 , i.e.,
p 
1
2
2

p
(12)
dM
0
It has been shown by Tisserand (1889) that
1
2
and
Hence,
1
2
2
3
a
2 3 2
0  r  dM  1  e 
2

0
3
1
a
  sin 2 
2
r

3 R2
 p   3 J 2 1  e2
2 a
2
(13)
3
a
  cos 2  0 .
r
(14)
 3 2  13  12 sin 2 i  .
(15)

0
 ,  and M because
Inspection of Eqs. (11) shows that secular terms can only appear in 
 p is only dependent on a, e, and i. If the partial derivatives of  p are substituted into
Eqs. (11) the resulting secular rates are
J2R2
 3

s
2 a 2 1 e 2


2
n cos i
6
(16)
J2R2
3
 s 
2 a 2 1 e 2


2
5


n  2  sin 2 i 
2


J2R2
3
3


M s  n 
n 1  sin 2 i 
3
2
2
2 a 1  e  2 
2

Where
n
1 2
(17)
(18)
(19)
a32
and the overbar represents mean values that will be defined later.
A first order solution in J 2 may be developed by solving Lagrange’s planetary equations under
the assumption that the reference orbit is a secularly precessing ellipse. This means that a, e,
and i are held constant on the right band side of Lagrange’s equations and ,  and M vary
with linear rates given by equations (16) through (19)
Consider for example the differential equation for the semimajor axis,
da
2  p

.
dt na M
(20)
If we assume we are dealing with near-circular orbits (e < 0.002), and write Eq. (10) in terms of
the mean anomaly M, and ignore term of OeJ 2  , by using
M
r  a1  e cos M 
we obtain
2
3  R
1 1 2 1 2

p 
  J 2   sin i  sin i cos2  2 M  .
2 aa
2
3 2

7
(21)
Thus,
 p
3  R
2

  J 2 sin i sin 2  2 M  .
M
2 aa
2
(22)
Substituting Eq. (22) into Eq. (20) and assuming a secularly precessing ellipse, i.e., a , e , and i
are constant, and
2 t   M t   2 0  M 0   2 s  M s t  t0 
(23)
we can integrate Eq. (20) as follows
2

2   3   R 
da 
J 2  sin 2 i sin 2  M  dt



na  2 a  a 

(24)
2 s  M s 


da


3
J
sin
i
sin
2


M
dt .


2


2 a
 s  M s 

2



na
a t 
t
a t 
 R
t
2
2
0
(25)
0
Retaining terms of OJ 2  on the RHS yields,
2
3  R
2
a t   a t0   
  J 2 sin icos 2  M 
2 n 2a 2  a 
or
t0
t
a t   a t0   K1 cos 2 t0   M t0   K1 cos 2 t   M t 
(26)
3  R
2
 J 2 sin i
2 2 
2n a a
2
where
K1 

3 R2
J 2 sin 2 i .
2 a
If we define
a  a t0   K1 cos 2 t0   M t0 
8
(27)
then the expression for at  is generally written as
at   a  K1 cos 2 t   M t  .
(28)
Where  t  and M t  are given
 (t )     (t 0 )   s (t  t 0 )
(29)
M t   M  M (t 0 )  M s (t  t 0 )
(30)
and  s and M s are given by Eq. (17) and Eq. (18) respectively. Hence, the time history for the
semimajor axis for near circular orbits given by Eq. (28) is a sinusoid with an amplitude of
K1 and a frequency of twice per revolution. Here a , given by Eq. (27), is the mean value of the
semimajor axis and is the value used to compute n in Eq. (19).
A similar procedure may be used to solve for the time history of all six orbit elements. Note
however from Eq. (11) that  p must be carried out to terms of order e 2 J 2 to determine  t 
and M t  accurate to order J 2 ,i.e. ignoring terms of OeJ 2  . This is because of the terms
containing
1  p
.
e e
This procedure may be carried out and the solutions for all six elements obtained. Alternatively,
the method of Brouwer (1959) may be used to obtain the solutions. Note that Brouwer’s
solutions are valid for all elliptical orbits (e<1); however, they are more complex than the
equations given here for near circular orbits. The solutions are as follows
a t   a  K1 cos 2  M 
(31)
9
7
3
 3
et   e  K 2 sin 2 i  cos2  M   cos2  3M   K 2 3 cos 2 i  1cos M (32)
4
4
 2
i t   i  K 3 cos 2  M 
(33)
 t  t   K sin 2  M 
t    0  
s
0
4
(34)
 
 
 1
 e


1
2


 t    0   s t  t 0   K 5  1  sin 2 i  sin M  sin 2M   1  sin 2 i  sin 2  M 
3
2
1
2
5
2
7
3
 1

 sin 2 i  sin 2  M  
sin 2  3M   sin 2  4M  
12e
8
 4e

R
S
T
U
V
W
1
40 cos4 i
 K2 1  11cos2 i 
sin 2
8
1  5 cos2 i
(35)
 
3
1
 1

M t   M 0  M s t  t 0   K 5  1  sin 2 i  sin M  sin 2M 
2
 e

  2
7
3
 1

 sin 2 i  sin 2  M  
sin 2  3M   sin 2  4M 
12e
8
 4e

R
S
T
U
V
W
1
40 cos4 i
 K2 1  11cos2 i 
sin 2
8
1  5 cos2 i



(36)
Where
K1 
3 R2
J 2 sin 2 i
2 a
J R
K2  2  
2 a
2
2
3 R
K 3  J 2   sin 2i
8 a
2
3 R
K 4  J 2   cos i
4 a
10
(37)
K 5  3K 2 .
Given a set of osculating elements a, e, i, ,  and M at the epoch time, the computational
procedure is to first compute the mean elements a , e , i , 0 ,  0 and M 0
by evaluating Eqs. (31) through (36) at the epoch time. For example,
a  at 0   K1 cos 2 t 0   M t 0  .
(38)
Using a , compute n from Eq. (19) and then compute the secular rate from Eqs. (16), (17), and
(18). The secular rates may be computed using either mean or osculating elements since the
 
theory is only accurate to OJ 2  and using osculating elements introduces and errors of O J 2 .
2
However, a must be used to compute n in Eq. (19) to avoid introducing an error of OJ 2  in
M s .
An examination of Eqs. (31) through (36) indicates that perturbations in a , e , and i will be
short period with frequencies of twice per revolution. The right ascension of the ascending node,
 , will have the same frequency short period variation superimposed on a secular trend. Note
that both  and M contain long period terms; however, these will cancel in the argument of
latitude, +M. For near circular orbits the results given by these equations should compare to
better than a hundred meters in position with numerical integration containing the complete
effects of J 2 .
11
An expression for r (t )
The expression for the magnitude of the radius vector, r (t ) , can be developed from the
equation
r  a1 e cos E 
(39)
 e3 
E  M   e   sin M   .
8

(40)
where (Roy,1988)
We wish to have Eq. (39) accurate to first order in J 2 , i.e. terms of order e or J 2 will be retained,
and to ignore terms of O(eJ 2 ) . Hence, Eq. (40) becomes
EM .
(41)
We may ignore the second term in Eq. (40) since
e cos( M   sin M )  e cos M
 e
where
e3
8
(42)
(43)
This may be demonstrated as follows
e cos( M   sin M )  e [cos M cos( sin M )  sin M sin(  sin M )]
 e [cos M   sin 2 M ]
 e cos M
since e is O(e 2 ) . Hence,
e cos E  e cos M ,
(44)
r  a (1  e cos M ) .
(45)
and Eq. (39) becomes
12
By using the expression for a(t ) , e(t ) , and M (t ) , the expression for r (t ) can be derived i.e.,
r (t )  a(t )[ 1  e(t )cos M (t ) ]
(46)
where
a (t )  a  a
e(t )  e  e
(47)
M (t )  M  M
and a and e are the mean values of these quantities and M  M 0  M s (t  t 0 ) . Here M 0 is
the mean value of M at the epoch time and a , e and M are the periodic variations in these
elements given by Eqs. (31),(32) and (36) respectively.
Substituting Eq. (47) into (46) yields
r (t )  (a  a)[ 1  (e  e) cos( M  M ) ]
(48)
Expanding Eq. (48) and ignoring higher order terms yields,
r (t )  a ( 1  e cos M )  a  a e M sin M  a e cos M .
Note that we must include terms in e M because M contains terms proportional to
(49)
1
.
e
From Eqs. (31), (32) and (36)
a  K1 cos 2 (  M )
3
(50)

7
3
e  K 2 sin 2 i  cos( 2  M )  cos( 2  3M )   K 2 (3 cos 2 i  1) cos M
8
8
 4
13
(51)
M 
 3
3
1
 1

K 2  ( sin 2 i  1)  sin M  sin 2M 
2
2
 2
 e

7
3
 1
 sin 2 i  sin( 2  M ) 
sin( 2  3M )  sin( 2  4M )
12e
8
 4e
4
1 
40cos i 
 K 2 1  11cos 2 i 
 sin 2 .
8 
1  5cos 2 i 
We may ignore terms in Eq. (52) that do not contain



(52)
1
because they will be higher order when
e
substituted into Eq. (49).
Substituting Eqs. (50) through (52) into Eq. (49) and simplifying yields the desired result
r (t )  a ( 1  e cos M ) 
3 R2
1
R2
J 2 ( 3 sin 2 i  2 )  J 2
sin 2 i cos 2(  M )
4 a
4
a
(53)
Very near circular orbits
An inspection of the solutions given by Eqs. (35) and (36) reveals that they contain
eccentricity divisors. Therefore, if the epoch orbit is very near circular (say e < 0.002) both
 and M will experience vary large short period perturbations and the solutions may be
numerically unstable. However, the sum,   M , will be well behaved because the terms
with eccentricity divisors cancel. Hence, for very near circular orbits one generally replaces
the equations for e ,  , and M with the alternate expressions
h  e sin 
k  e cos
(54)
   M .
14
The solutions for these elements may be obtained by substituting the expressions for e,  and
M from Eqs. (32), (35) and (36) as follows
e sin   (e   e) sin(   )
 (e   e)(sin  cos   cos  sin  )
(55)
 e sin   e  cos    e sin 
Likewise,
e cos  e cos  e sin    e sin  .
(56)
Substituting the expressions for e and  yields the desired results
where
ht   h (t )  K 6 sin   K 7 sin 3
(57)
k t   k (t )  K 8 cos   K 7 cos 3
(58)
   M
and  and M are defined by Eqs. (29) and (30 ).
Also,
h (t )  e sin 
(59)
k (t )  e cos  ,
and e and  (t 0 ) are given by
e  h (t o ) 2  k (t 0 ) 2 .
 (t 0 ) =atan2( h(t0 ), k (t0 ) )
h (t0 ) and k (t0 ) are obtained by evaluating Eqs. (57) and (58) at the epoch time. In evaluating
these equations the epoch value of  may be used in place of  (t0 ) .
15
The equation for  (t) is given by
 (t )      M .
(60)
Substituting the expressions for  and M from Eqs. (35) and (36) yields
 (t )    K 9 sin 2 .
(61)
The constants are defined by
2
K6 
1 R 
21

J 2    6  sin 2 i 
4 a 
2

K7 
7 R
J 2   sin 2 i
8 a
K8 
1  R   15 2 
J 2    6  sin i 
4 a 
2

2
(62)
2
2


3 R
K 9  J 2   3  5 cos 2 i .
8 a
The values of e ,  , and M may be determined from
e  h2  k 2
(63)
  a tan 2h, k 
(64)
M    .
(65)
Secular Rates of O(J22)
The secular rates of O(J22) from Brouwer (1959) are given here for completeness. These terms
may simply be added to the secular rates given by Eqs. (16) – (18). However, mean elements
must be used to evaluate the O(J2) secular rates to avoid introducing errors of O(J22). The mean
16
value of the semimajor axis must always be used to evaluate n . The second order secular rates
are given by
3
2
2
 s ( J 2 )  nK10 [(5  12 K11  9 K11 ) cos i
8
2
 (35  36 K11  5 K11 ) cos3 i ]
 s ( J 22 ) 
3
2
nK10 [35  24 K11  25 K11  (90  192 K11
32
2
2
 126 K11 ) cos 2 i  (385  360 K11  45 K11 ) cos 4 i ]
M s (J2 ) 
2
(66)
3
2
nK10 K11[15  16 K11  25 K11  (30  96 K11
32
2
2
 90 K11 ) cos 2 i  (105  144 K11  25 K11 ) cos 4 i ]
where
FJ FR I 1 I
 G GJ
H2 Ha K(1  e ) J
K
2
2
K10
2
2 2
(67)
K11  1  e 2
While mean values are shown in these equations, epoch values may be used since they would
introduce errors of O(J23).
Results
Results of comparing the analytical solutions for the classical and nonsingular elements with
numerical integration for one day are presented in Figs. 4 - 15. Figures 4 - 7 present results for a
satellite in the Quikscat orbit, Figs. 8 - 11 show the results for a satellite in the ICESat orbit and
Figs. 12 – 15 are results for Quikscat and ICESat including secular rates of O(J22). Difference
plots are analytical minus numerical integration values. Quikscat is in a sun synchronous, 800
km orbit while the ICESat orbit is more nearly polar with an altitude of about 600 km. Figure 4
17
shows the time history for the classical elements of Quikscat as computed from Eqs. (31) - (36).
The history for r(t) is computed using Eq. (53). Figure 5 shows the difference between the
analytical results and those obtained by numerical integration of the orbit perturbed by J 2. The
RMS of the differences also is shown.
Figures 6 and 7 present the analogous results for Quikscat using the nonsingular elements of Eqs.
(57), (58), and (60). The figures show results for the classical elements that have been computed
from the nonsingular elements. The history of r(t) is again computed from Eq. (53). Note that
the elements computed from the nonsingular formulation are more accurate than those computed
using classical elements. In particular e, , M and r are significantly more accurate when
computed from the nonsingular elements. This is primarily due to the fact that the mean value of
eccentricity is more accurate when computed from h and k. Furthermore, the nonsingular
formulation does not require dealing with eccentricity divisors. Figures 8 –11 present the
analogous results for the ICESat orbit.
Figures 12 and 13 present the results for the Quikscat nonsingular elements including the secular
rates of O(J22) given by Eq. (66). Figures 14 and 15 present the same results for ICESat. Note
that only the node is noticeably effected by including the second order effects. The RMS error in
the node is reduced by an order of magnitude. Errors in the argument of perigee and mean
anomaly are dominated by short period terms and including the second order secular terms does
not change the error RMS.
18
Reducing the epoch values of eccentricity for the Quikscat and ICESat orbits by an order of
magnitude reduces the differences between the analytical and numerical integration results by a
factor of two or more. Hence, periodic errors in the analytical solutions of O(eJ2) are larger than
periodic errors of O(J22) for the orbits evaluated here.
The theory presented here is only valid for near circular orbits (e < .002) perturbed by J 2. It
could be extended to include terms of O(eJ2) in order to handle more elliptical orbits
Acknowledgement
I thank Yoola Hwang for coding the theory and generating the numerical results for this memo.
References
1. Kaula, W., Theory of Satellite Geodesy, Dover, Mincola N.Y., 2000.
2. Roy A.E., Orbital Motion, Adam Hilger, Philadelphia, 1988.
3. Tisserand F., Traite de mecanique Celeste (Vol#1), Gauthier-Villars, Paris, 1889.
4. Brouwer D., “Solution of the Problem of Artificial Satellite Theory Without Drag”, The
Astronomical Journal, Vol 64, No. 1274, pp 378-397, Nov. 1959.
19
Figure 4. QUIKSCAT analytical solutions with classical elements
Figure 5. Difference between QUIKSCAT analytical solutions and numerical integration for classical elements
20
Figure 6. QUIKSCAT analytical solutions with nonsingular elements
Figure 7. Difference between QUIKSCAT analytical solutions and numerical integration for nonsingular elements
21
Figure 8. ICESat analytical solutions with classical elements
Figure 9. Difference between ICESat analytical solutions and numerical integration for classical elements
22
Figure 10. ICESat analytical solutions with nonsingular elements
Figure 11. Difference between ICESat analytical solutions and numerical integration for nonsingular elements
23
Figure 12. QUIKSCAT analytical solutions of
O( J 22 ) with nonsingular elements
Figure 13. Difference Between QUIKSCAT analytical solution of
for nonsingular elements
24
O( J 22 ) and numerical Integration
Figure 14. ICESat analytical solutions of
O( J 22 ) with nonsingular elements
Figure 15. Difference Between ICESat analytical solution of
for nonsingular elements
25
O( J 22 ) and numerical Integration