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Journal of Multidisciplinary Engineering Science and Technology (JMEST)
ISSN: 2458-9403
Vol. 1 Issue 4, November - 2014
www.jmest.org
JMESTN42353750 13550
Fixed Point Iteration Computation Of Nominal
Mean Motion And Semi Major Axis Of Artificial
Satellite Orbiting An Oblate Earth
Ozuomba Simeon
Department of Electrical/Electronic and Computer Engineering
University of Uyo
Akwa Ibom State, Nigeria
simeonoz@yahoo.com
Abstract— In this paper, fixed point iteration
method for the computation of the semi major axis and
the nominal mean motion of an artificial satellite
orbiting an oblate earth is presented. The algorithm and
the flowchart, along with relevant analytical
expressions are presented. FPI computation of the semi
major axis and the nominal mean motion for the case
study orbit was performed with required estimation
error, . The results of the FPI iteration show
that the algorithm converged at the second (2nd) cycle
with an estimation error of -8.74134E-08 km which is
less than the specified required estimation error of
km. At the convergence cycle, the semi major
axis of the orbit is 27338.632605314 km and the
nominal mean motion ( ) is 0.000139670 rad/s.
Meanwhile, the mean motion (n) is 0.000139683 rad/s.
and the difference between n and (that is, n - ) is
1.2345640E-08 rad/s. Also, the results show that the
orbit semi major axis without the impact of oblate earth
is 27332.341001289 km whereas the orbit semi major
axis with the impact of oblate earth is 27338.632605314
which gives a difference of (27332.341001289-
27338.632605314) of -6.291604025 km . Accordingly,
with the oblate earth, there is an increase in the semi
major axis of the orbit when compared to the case of
non-oblate earth. Additional effect is that the mean
motion of the satellite orbiting an oblate earth is higher
or faster than the nominal mean motion of the satellite.
Keywords—Fixed Point Iteration, Perturbed
Orbit, Artificial Satellite, Oblate Earth, Nominal
Mean Motion, Anomalistic Period, Orbital Semi
Major Axis, Seeded Secant Iteration, Anomalistic
Period, Iterative Root Finding Scheme
I. INTRODUCTION
Today, the earth is orbited by numerous artificial satellites
deployed for diverse applications ranging from
telecommunication, remote sensing, astronomical purposes,
weather forecasting, climatic studies, radar applications,
military applications and many others
[1,2,3,4,5,6,7,8,9,10,11]. The motion of each of the
satellites in their orbits is governed by planetary motion
laws postulated by Kepler [12,13,14,15,16,17,18]. The
orbital motion laws by Kepler are based on ideal case
where the earth is assumed to be a perfect sphere with
homogeneous density [19,20,21,22,23]. However, the earth
is neither spherical nor homogeneous in mass or density.
Rather, the shape of the earth has bulge at the equator and
flattening at the poles [24,25,26,27,28,29,30,31]. As such,
the shape of the earth is better modeled as oblate spheroid
[32,33,34,35,36]. The oblateness of the earth introduces
some complex motion of the satellites in their orbits. Such
orbits with the complex motions are referred to as perturbed
orbit.
Analysis of the mean motion of satellites on perturbed orbit
is quite complex; it requires iterative approach to determine
the nominal mean motion or the semi major axis of the
perturbed orbit when the anomalistic orbit period is given.
Consequently, in this paper, fixed point iteration (FPI)
method [37,38,39,40] is adapted for the computation of the
semi major axis and the nominal mean motion of an
artificial satellite orbiting an oblate earth. The FPI
flowchart is presented along with the algorithm and
requisite analytical expressions associated with the FPI
algorithm. Furthermore, sample perturbed orbit parameters
are used for numerical examples.
II. METHODOLOGY
A satellite orbit with earth geocentric gravitational
constant, and semi major axis, a , then the nominal mean
motion , is given as;
(1)
Also, for an artificial satellite orbiting an oblate earth with
anomalistic period , P, the mean motion, is given as;
(2)
The mean motion is related to the nominal mean motion,
as follows;
(3)
Then, the semi major axis (a) can be expressed in terms of n
and as follows;
(4)
(5)
Journal of Multidisciplinary Engineering Science and Technology (JMEST)
ISSN: 2458-9403
Vol. 1 Issue 4, November - 2014
www.jmest.org
JMESTN42353750 13551
Let f() and g() denote functions of a in cycle k which
are given as follows;
=
(6)
(7)
The algorithm and the flowchart (Figure 1) for the fixed
point iteration (FPI) used in the computation of the semi
major axis and the nominal mean motion of artificial
satellite orbiting an oblate earth are given as follows;
Step 1: The initial parameters;
Step 1.1: Counter for the cycle, k = 0
Step 1.2: Set the tolerance error,
Step 1.3:
(8)
Step 2 Compute the next root,
(9)
Step 3:
(10)
Step 4:
Step 4.1:
Step 4.1.1:
If then
Step 4.1.2:
Step 4.2:
If however, , then
Step 4.2.1:
Step 4.2.2:
Step 4.2.3:
Repeat step 2 to step 4
Figure 1: The flowchart for the Fixed Point Iteration (FPI) computation of the semi major axis and the nominal mean motion
of artificial satellite orbiting an oblate earth
Start
Input
NO
YES
Output k,
,
,
Stop
Journal of Multidisciplinary Engineering Science and Technology (JMEST)
ISSN: 2458-9403
Vol. 1 Issue 4, November - 2014
www.jmest.org
JMESTN42353750 13552
III RESULTS AND DISCUSSION
A sample numerical computation was performed for the
orbit of a hypothetical artificial satellite orbiting an oblate
earth, where the orbit parameters are as follows:
i. Eccentricity (e) = 0.0025;
ii. Earth Geocentric Gravitational Constant (μ) =
3.986005 x
iii. Inclination Angle (i) = 0 degree
iv. Anomalistic Period (P) = 12.5 hours
v. Constant (K1) = 66,063.1704
FPI computation of the semi major axis and the nominal
mean motion for the case study orbit was performed with
required estimation error, . The results of the
FPI iteration show that the algorithm converged at the
second (2nd) cycle with an estimation error of -8.74134E-
08 km which is less than the specified required estimation
error of km. At the convergence cycle, the semi
major axis of the orbit is 27338.632605314 km and the
nominal mean motion ( ) is 0.000139670 rad/s.
Meanwhile, the mean motion (n) is 0.000139683 rad/s. and
the difference between n and (that is, n - ) is
1.2345640E-08 rad/s. Also, the results show that the orbit
semi major axis without the impact of oblate earth is
27332.341001289 km whereas the orbit semi major axis
with the impact of oblate earth is 27338.632605314 which
gives a difference of (27332.341001289-27338.632605314)
of -6.291604025 km . Accordingly, with the oblate earth,
the semi major axis is larger and hence, in order to
maintain the given anomalistic period , the mean motion
(n) has to be larger than the nominal mean motion (). In
this case, the difference, n - is 1.2345640E-08 rad/s.
That means, the case study artificial satellite orbiting an
oblate earth has men motion that is faster (by a value of
1.2345640E-08 rad/s) than what it would have been if the
earth is not oblate.
Table 1 The result of the FPI computation of the semi major axis and the nominal mean motion for the case study orbit of an
artificial satellite orbiting an oblate earth (the required estimation error, )
Semi major axis
g() (km)
Estimation
Error
Nominal mean
motion (no)
Mean motion (n )
Difference between n
and
Cycle
(km)
(km)
f() (km)
(rad/s)
n (rad/s)
n- (rad/s)
0
27332.341001289
27338.633347090
-6.29235E+00
0.000139718
0.000139683
-3.5882947E-08
1
27338.633347090
27338.632605314
7.41776E-04
0.000139670
0.000139683
1.2351325E-08
2
27338.632605314
27338.632605402
-8.74134E-08
0.000139670
0.000139683
1.2345640E-08
3
27338.632605402
27338.632605402
0.00000E+00
0.000139670
0.000139683
1.2345641E-08
4
27338.632605402
27338.632605402
0.00000E+00
0.000139670
0.000139683
1.2345641E-08
5
27338.632605402
27338.632605402
0.00000E+00
0.000139670
0.000139683
1.2345641E-08
6
27338.632605402
27338.632605402
0.00000E+00
0.000139670
0.000139683
1.2345641E-08
IV. CONCLUSION
Fixed Point Iteration (FPI) computation of the semi major
axis and the nominal mean motion of an artificial satellite
orbiting an oblate earth is presented. The algorithm and the
flowchart, along with relevant analytical expressions are
presented. A hypothetical artificial satellite orbiting an
oblate earth was used as a case study. The results showed
that the algorithm converged at the 2nd iteration. Also, the
oblate earth causes an increase in the semi major axis of the
orbit when compared to the case of non-oblate earth.
Additional effect is that the mean motion of the satellite
orbiting an oblate earth is higher or faster than the nominal
mean motion of the satellite.
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