A STUDY OF PERTURBATION EFFECT ON SATELLITE
ORBIT USING COWELL’S METHOD
Muhammad Shamsul Kamal Adnan , Radzuan Razali & Md. Azlin Md. Said
School of Aerospace Engineering
Engineering Campus
University Science Malaysia
14300 Nibong Tebal
Penang, Malaysia
Tel : 60-4-5937788 ext 6501 / Fax :60-4-5941026
Abstract. Most Keplerian problems were treated as ideal or under the basic assumptions that the
motion of a body in the orbits is a result of the gravitational attraction between two bodies. This
ideal situation does not exist. Additional forces acting on any moving body must be taken into
account. These additional forces are called the perturbing forces. The perturbing forces that cause
the satellite orbit to deviate from a theoretically regular orbital motion can be divided into two
categories, conservative field forces and non-conservative perturbing forces. The conservative
perturbing forces are due to other celestial bodies such as Moon, Sun and etc. Solar pressure,
atmospheric drag, thrust and the non-homogeneity and oblateness of the Earth are the examples
of the non-conservative perturbing forces. This paper discusses the study of perturbation of a
satellite orbit due to the presence of other gravitational bodies such as Moon and Sun from the
conservative perturbing forces and from the non-conservative perturbing forces such as the nonhomogeneity and oblateness of the Earth, atmospheric drag and thrust.
To solve this perturbed problem of Keplerian orbit, Cowell’s method will be used, followed by
Runge-Kutta method to simplify the equations involved.
1.0 INTRODUCTION
A perturbation is a deviation from some
normal or expected motion. The actual
path will vary from the theoretical twobody path due to perturbations caused by
other mass bodies, such as Moon, and
additional forces not considered in
Keplerian motion, such as non-spherical
Earth. It should not be supposed that
perturbations are always small, for they
can be as large as or larger than the
primary attracting forces. For example,
ignoring the effect of the oblateness of
the Earth on an artificial satellite would
cause to completely fail in the prediction
of its position over a long period of time.
Perturbation methods are also used in
predicting the orbit of the Moon. The
objective of this paper is to present one
of the useful and well-known special
perturbation
techniques,
Cowell’s
method to solve this perturbed problem
of Keplerian Orbit.
2.0 COWELL’S METHOD
This method was discovered by P.H.
Cowell in the early 20th century.
Cowell’s method is the simplest and the
most direct perturbation of all the
perturbation’s method.
The application of Cowell’s
method is simply to write the equations
of motion of the object, including all the
perturbations and then to integrate them
step-by-step numerically. For the twobody problem with perturbations, the
equation would be
&r& + µ r = a p
(1)
r3
The Equation (1) is a second order
differential equation. Numerically, we
can change this second order differential
equation into first order differential
equation as in equation (2).
µ
r& = v
v& = a p − 3 r (2)
r
where r and v are the radius and
velocity of a satellite with respect to the
larger central body. Equation (2) will
have to be broken down into the vector
components.
µ
x& = v x
v& x = a px − 3 x
r
µ
y& = v y
v& y = a py − 3 y
(3)
r
µ
z& = v z
v& z = a pz − 3 z
r
2
2
2
r= x +y +z
3.0 PERTURBING ACCELERATIONS
They are several perturbing acceleration
found to be the cause of the perturbed
motion of a satellite orbit, i.e.,
i.
The non-homogeneity and
Oblateness of the Earth.
ii.
Third-body
Perturbing
Forces.
iii.
Solar Radiation
iv.
Earth Atmospheric Drag
v.
Spacecraft Thrust
Figure (1) indicates perturbation causes
by the third body and the effect of
earth’s equatorial buldge on the satellite.
4.0 CALCULATION
OF
THE
PERTURBING ACCELERATIONS
For a satellite with an orbit of 1600 km
and above, the effect of perturbing
accelerations from Sun and Moon cannot
be neglected.
The zonal harmonic coefficients,
Jn, due to the oblateness of the Earth,
tend to diminish each term of the series.
The comparison of these coefficients
shows that the magnitude of J2 is at least
400 times larger than the other Jn
coefficients, which can be disregarded
for engineering calculation purposes [2].
The complete equation of the
perturbing
accelerations
is
the
summation of all the perturbing forces
mentioned in the previous section and is
defined as
aptotal= aCB +aSM +aSR +aAD +aT +∆U (4)
where
aCB = acceleration due to the
center mass of the spacecraft,
aSM = perturbing accelerations due
to Sun and Moon,
aSR = perturbing acceleration due
solar radiation pressure,
aAD = perturbing acceleration due to
the Earth atmospheric drag,
aT = perturbing acceleration due to
spacecraft thruster,
? U = perturbing acceleration due to
the oblateness of the Earth.
The perturbing accelerations due to the
third body, i.e. Sun and Moon, and
oblateness of the Earth are larger than
perturbing accelerations due to the solar
radiation pressure, atmospheric drag and
spacecraft thruster [3].
Therefore, by neglecting the
perturbing accelerations due to solar
radiation pressure, spacecraft thruster
atmospheric drag, Equation (4) becomes
as
&r& + µ r = a SM + ∆U
(5)
r3
Using the Cowell’s method,
Equation (5) can be written as
(a)
(b)
Figure (1) a. Relationship of Earth, Moon and Satellite in Space.
b. Perturbation example of perturbative torque caused by earth’s equatorial
bulge.
µ
v& = − 3 r + a SM + ∆U
r
or in precise composition,
r& = v ,
r& = v
  rMSat rME
 µ M  3 − 3
&v = − µ r −   rMSat rME
r

r3
+ µ S  SSat

3
 rSSat

(6)




rSE 
− 3  + F 
rSE 

iii. Sun
iv. Earth Oblateness
- 0.2 km
- 0.014 km



(7)
The analytical formulation of the
perturbation (r and v) can be found by
applying numerical integration methods
to equation, i.e. Runge-Kutta method.
(a)
5.0 RESULTS
Runge-Kutta method is used to solve the
Cowell’s equation of perturbation, which
consists of ordinary differential equation
(ODE). To solve the ODE, the initial
conditions such as initial positions and
velocity of the satellite have to be
identified. Using the values of initial
position and velocities, the perturbed
distances are
Initial Positions
X
11335
(km)
Y
-7740
Z
941
Initial Velocity
Vx
-0.9199
(km/s)
Vy
6.859
Vz
-0.3798
Time (seconds)
3600
The results of the perturbations are:
i. Moon (moon fixed) - 0.013 km
ii. Moon (moon rotates) - 0.0083 km
(b)
(c)
(d)
Figure (2). Perturbation causes by the
major perturbing acceleration on the
satellite for one hour.
For a one day duration, the perturbation
effect can be seen in Figure (3).
(a)
(b)
6.0 CONCLUSION
The perturbed magnitudes obtained in
this paper are almost large for one hour
of duration of orbit. Therefore, in order
to have a precise calculation of a satellite
orbit, perturbing accelerations cannot be
neglected. The minor perturbing
accelerations such as the atmospheric
drag, thruster and solar radiation are
neglected in the calculation since they
are comparatively much smaller in
magnitudes in comparison with the
calculated ones. Cowell’s method is used
in calculating the perturbation because it
is the most-straight forward method and
quite accurate. Other methods, which are
more complex and better satisfaction, i.e.
Encke’s method can also be used.
REFERENCES
1.
2.
3.
(c)
4.
5.
6.
(d)
Figure (3). Perturbation causes by the
major perturbing acceleration on the
same satellite for one day.
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