Precise Orbit Determination of Global Navigation Satellite System of ...

Chapter 5 Perturbation Models of IGSO,GEO and MEO Satellites Orbits
CHAPTER 5 PERTURBATION MODELS OF IGSO, GEO and MEO
SATELLITE ORBITS
The performances of IGSO, GEO and MEO satellite orbits will be affected by many perturbations such as the
non-central part of the Earths gravitational potential, gravitational effects of Sun and Moon, solid earth tidal
effects, solar radiation pressure, Albedo radiation pressure, gravitational effects of ocean tides, gravitational
effects of the planets, relativistic corrections, thermal emission of the satellite etc. Because IGSO, GEO and
MEO satellite orbits are far away from the earth, the effects of earth tides, ocean tides and earth radiation are
much smaller. Hence the non-central part of the Earth, the gravitational effects of Sun and Moon and solar
radiation pressure play a major role in the perturbations on IGSO, GEO and MEO satellite orbits. In this chapter,
the computations of these three perturbation models will be discussed. The major literatures on this topic are
referred to Soop (1994), Tscherning (1976, 1977).
5.1 The Earth Gravitational Perturbation
The earth geopotential is represented as a point mass and an expansion of spherical harmonics to represent the
nonspherical effect of the Earth mass. The geopotential can be expressed by (Rothacher, 1992)
GM
U =
r
+ GM
N
n
��
n=
2 m=
0
r
n
e
n+
1
a
P
nm
(sin ϕ )( C cos mλ
+ S sin mλ)
(5-1)
where
G Newtonian Gravitational Constant
M the Earth’s mass (GM=398.600415×10 12 m 3 s -2 )
Pnm (sin ϕ)
associated Legendre function
λ, ϕ
geographic longitude and latitude of satellite
spherical harmonic coefficients.
Cnm , Snm
nm
nm
In the equation above, the first term is the point mass part. The other terms are an expansion of spherical
harmonics to represent the nonspherical effects of the earth mass, i.e. the Earth gravitational perturbation.
The forces produced by the geopotential are computed by
N
��
n=
2 m=
0
43
n
ae
n+
m+
1
∂U
∂U
∂U
GM
( m)
grad(
U ) = ( , , ) = grad(
) + GM grad{
Pn
(sin ϕ)(
Cnmξ
m + S nmηm
)}
∂x
∂y
∂z
r
r
n
5.1.1 Computation of Legendre Polynomials
From Eq.(5-1) it can be seen that in order to compute the geopotential, the first step is computation of the
Legendre polynomials.
The Legendre polynomials Pn as a function of the independent variable x can be defined by:
n
1 d 2 n
P n ( x)
= ( x − 1)
(5-3)
n n
2 n!
dx
For a given value of x, the Legendre polynomials Eq.(5-3) can be calculated by the following recursive formula
with starting values:
P ( x)
= 1;
P ( x)
= x
0
2n
−1
pn
( x)
= xP
n
1
n−1
n −1
( x)
− P
n
n−2
( x)
if n ≥ 2
(5-2)
(5-4)