Radial, In-Track, and Cross-Track Variations Due to Perturbations in
the Initial Conditions of Near Circular Orbits
ASEN 3200
4/15/02
George H. Born
Introduction
It is often of interest to determine how two orbits differ because of small perturbations in
the initial conditions. We will develop the expressions for the first order radial, in-track,
and cross-track (RIC) variations with time between two near-circular, two-body orbits,
given differences in their initial values of a, e, i, , , and .
Variations in the Radial Direction, R
Variations in r, the satellite geocentric radius, can be determined from
r  a 1  e cos EA ,
(1)
where EA is the eccentric anomaly. Differentiating Eq. (1) using the chain rule yields
dr 
r
r
r
da  de 
dEA .
a
e
EA
(2)
Let the differential assume a small but finite magnitude so that, for example, dr  r ,
then
r  a 1  e cos EA  ae cos EA  aeEA sin EA .
(3)
We will assume a near circular orbit and generally ignore terms of O(e) and
O( i  j ) , where  is a variation in any of the orbit elements. We have retained a few
of the larger terms proportional to e  . This is because a perturbation in a changes the
mean motion which introduces a secular change in true anomaly,  .
It can be shown that (Roy, 1965)
EA    O(e).
(4)
Hence, in Eq. (3) we can use
EA  ,
and
EA  .
1
(5)
The expression for  is derived in the section on variations in the in-track direction and
is given by Eq. (17) i.e.,
(6)
  0  n(t  t0 )
where we have dropped the term, 2ecosM, and n is given by Eq(16). The first term in
Eq. (6) represents a perturbation in the initial value of  and the second term is due to a
perturbation in a which will cause a change in mean motion and hence a change in  .
Equation (3) becomes
r  a 1  e cos   ae cos  ae sin .
Hence, a small change in a yields
r  a 1  e cos   aen(t  t0 )sin  .
(7)
r  ae cos  ae sin ,
(8)
A variation in e yields
where the last term is included because a variation in e causes a variation in  which is
given by (see appendix A)
  2e sin .
(9)
Substituting Eq. (9) into Eq. (8) yields,
r  ae cos  2aee sin 2  .
(10)
In summary, Eqs. (7) and (10) plus the contribution of 0 yield the dominant variation
in the radial direction, i.e.,
R  a 1  e cos   aen(t  t0 )sin   ae cos  2aee sin 2   0ae sin  .
(11)
There is a second order contribution to R caused by a variation in true anomaly as shown
in the sketch below.
rp

r*
Here r* is the reference radius and rp is the perturbed radius. The projection of the
perturbed radius on r* is rcos. Hence, there is a contribution to R given by
2
R  r 1  cos   .
Because a perturbation in a causes a secular change in , this term becomes more
important with time. We will ignore the contribution of 0 and  due to e here
since they are small. We can write the equation for R by using Eq. (6) for , i.e.,


R  r 1  cos n  t  t0  ,
(12)
where n is given by Eq (16) and the reference orbit radius is used in Eq. (12).
Note that perturbations in i, , and  to first order do not contribute to R.
Variations in the In-Track Direction, I
Perturbations in the argument of latitude,
u    ,
will cause in-track variations in the orbit. Variations in u will be caused by direct
perturbations in  or  at the epoch time or by perturbations in a and/or e which will
cause subsequent variations in the argument of latitude since they influence .
The value of I, the in-track variation,
is given by
orbit
r
I  r u
u
 a 1  e cos M  u
equator
(13)
where,
u     .
If we ignore terms of O(e) we can write
0  M 0 .
(14)
Also
  M  2e sin M .
So
  1  2e cos M  M .
Using
3
(15)
M  M 0  n  t  t0 
yields
M  M 0  n  t  t0 
where
n
1/ 2
a3/ 2
and
n
n  3 / 2 a.
a
(16)
Hence, using Eqs. (14), (15) and (16)
  1  2e cos M   0  n  t  t0  
(17)
Also, a perturbation in e affects  (see Appendix A) and
  2e sin.
(18)
Substituting Eqs. (14), (17), and (18) into Eq. (13) and including the contribution of 
yields,


I  r   1  2e cos M   0  n  t  t0    2e sin  .
(19)
Note that
r 1  2e cos M   a 1  e cos M 1  2e cos M 
a 1  e cos M 
Hence,
I  r    2e sin    a 1  e cos M   0  n t  t0  
(20)
In addition, a perturbation in the right ascension of the ascending node, , causes both
an in-track and a cross-track variation as seen in the sketch below.
I

C
i
i

Reference
orbit
Perturbed
orbit
4
I
C
From spherical trig:
sin  C  sin  sin i.
Using the small angle approximation,
 C   sin i.
This will be the cross-track error at the equator. However, C will vary with argument of
latitude and a positive value for  will produce a negative value of C at the equator.
Hence,
C  r C cos u
C  r sin i cos u
(21)
Also, for the right spherical triangle
cos i  tan  I cot ,
or
tan  I 
cos i
.
cot 
(22)
Using the small angle approximation
 I   cos i.
(23)
Hence, the in-track contribution due to a variation in  is
I  r I
 r  cos i.
5
(24)
Cross-Track Errors, C
We have already seen that a perturbation in  yields a cross-track error given by Eq.
(21). The other orbit element which contributes to a cross-track error is the inclination.
This is illustrated in the sketch below:
C
Pertrubed
orbit
C
i
i
i
u
Reference
orbit
 C  sin 1  sin i sin u 
C  r C
C  r i sin u
 i sin u
Summary
As stated earlier, we have ignored all second order effects in  i  j . For example if we
perturb  this will affect I and C. If we then perturb i there will be a first order change
in C. In addition, there will be second order changes in both i and C. These changes will
be proportional to various products of , i,  , and  and are ignored here. The
results derived here can be summarized as follows (Note that we have substituted M for
 ):
Radial Variations:
3


R  a (1  e cos M )  ne(t  to ) sin M   r[1  cos(n  t  t0 ) ]
2


2
 ea cos M  2aee sin M  0 ae sin M .
(25)
In-Track Variations:
I  r    2e sin M   cos i   a 1  e cos M   0  n t  t0   .
(26)
Cross-Track Variations:
C  r (i sin u   sin i cos u ).
(27)
Note that we may use u    M .
6
These results also are summarized in Table 1. Figure 1 presents the total RIC
deviations due to perturbations in a reference orbit with a=6878. km, e=.001, i=75, and
    M  0. The perturbations are:
a  0.1Km, e  .0005, i      0  0.02 .
If these perturbations are applied one at a time the maximum disagreement with
STK is 0.005 km. However, from Figure 1 it can be seen that the maximum error grows
to 0.03 km in the radial direction because of interactions between the perturbations which
are ignored here.
We could replace r by a and drop all terms in Table 1 which are proportional to
e , where  represents a perturbation in any of the initial conditions. This would
have a noticeable impact on the case where we apply the perturbations one at a time, but
would have a small impact on the case where we apply all perturbations because the
interaction terms are much larger than the terms that this would eliminate.
Reference
Roy, A.E., The Foundations of Astrodynamics, Macmillan Co., New York, NY, 1965.
Acknowledgement
I thank Jason Stauch for coding the equations and generating Fig. 1.
7
Table 1
Radial, in-track and cross-track (RIC) variations corresponding to small perturbations in the kepler orbit elements for near circular orbits
a
R
e
3


a 1  e cos M   ne  t  t0  sin M  ea cos M
2

 2aee sin 2 M
r[1  cos(n(t  t0 ))]
i


0
NA
NA
NA
0 ae sin M
I
a(1  e cos M )n(t  t0 )
2er sin M
NA
r cos i
r 
a(1  e cos M )0
C
NA
NA
ir sin u
r sin i cos u
NA
NA
Where,
t 0  epoch time
a  semi-major axis
e  eccentricity
i  inclination
  right ascension of the ascending node
  argument of perigee
3n
n = - a, r  a(1  e cos M )
2a
9
0  true anomaly at epoch time t 0
M  M (t ), i.e. mean anomaly at time t

a3
  perturbed orbit element - reference orbit element
u  argument of latitude    M
n  mean motion 
range  R 2  I 2  C 2
Distance Between Unperturbed and Perturbed Satellites in RIC Coordinates
STK Generated (All Perturbations)
15
10
5
Distance (km)
0
-5
R_stk
-10
I_stk
C_stk
-15
-20
-25
-30
-35
0
6
12
18
24
30
36
42
48
Time (hours)
Difference Between Equation Generated Results and STK Generated Results
(All Perturbations)
0.1
0.08
0.06
Difference (km)
0.04
0.02
R_eqn-R_stk
I_eqn-I_stk
C_eqn-C_stk
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
6
12
18
24
30
36
42
48
Time (hrs)
Figure 1 (top panel) RIC variations due to perturbations in all initial conditions
(bottom panel) RIC differences between equations in Table 1 and STK
10
Appendix A
True anomaly variation due to eccentricity perturbation
From the expression for mean anomaly
M  n  t  t p   EA  e sin EA
(A-1)
M  0 since a  0 .
Hence,
M  0  EA  e sin EA  eEA cos EA
EA 
e sin EA
1  e cos EA
e sin EA.
e sin EA 1  e cos EA 
(A-2)
(A-3)
We must relate EA to .
From Roy (page 89)
  EA  e sin EA
(A-4)
  EA  e sin EA  eEA cos EA.
(A-5)
Using Eq. (A-3) and ignoring the term eEA yields
  2e sin EA
 2e sin   e sin EA 
 2e sin  cos  e sin EA   cos  sin  e sin EA  
(A-6)
 2e sin   e sin EA cos  
2e sin  .
Note that while a perturbation in e changes , a perturbation in  does not affect e.
11