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

Chapter 5 Perturbation Models of IGSO,GEO and MEO Satellites Orbits
The associated Legendre function for a given order m and degree n is defined by:
P
nm
m
Pn
(
m
2 m / 2 d x)
( x)
= ( 1−
x )
(5-5)
dx
With these definitions the spherical harmonics are not normalized. In order to normalize them, spherical
harmonics needs to be multiplied by
( 2n
+ 1)
if m=0; (5-6)
( n − m)!
2(
2n
+ 1)
if m≥1 (5-7)
( n + m)!
Assuming that
( m)
n
P
m
Pn
(
m
d x)
( x)
= (5-8)
dx
Eq.(5-8) can be calculated by the following recursive formula in n for given value of x and m≥1 with starting
values:
P m ( )
n
P m ( )
n
P
( m)
n
( x)
= 0
if
n < m
( x)
= 1×
3×
... × ( 2m
−1)
( x)
2n
−1
xP
n − m
if
n = m
n + m −1
( x)
− P
n − m
( m)
( m)
= n−
1
n−2
( x)
if
n > m
44
(5-9)
(5-10)
(5-11)
First, using the equations above to obtain Pnx m ( ) ( ) , then using Eq.(5-5) to calculate the associated Legendre
function Pnm; at last using Eq.(5-4) to compute Legendre polynomials.
After Legendre polynomials have been computed, the coefficients should be normalized using Eq.(5-6) and
Eq.(5-7).
5.1.2 Computation of Geopotential Perturbation
According to Eq.(5-5) and Eq.(5-8), Eq.(5-1) can be expressed by
GM
U =
r
+ GM
GM
= + GM
r
N
n
��
n=
2 m=
0
N
Assuming
x = r cosϕ
cos λ�
�
y = r cosϕ
sin λ �
z = r sin ϕ �
�
and
ξ
m
m
= r
m
m
cos
η = r cos
m
m
n
��
r
n=
2 m=
0
n
e
n+
1
a
r
ϕ cos mλ��
�
ϕ sin mλ
��
m
( m)
n
cos ϕP
(sin ϕ)(
C cos mλ
+ S sin mλ)
n
ae
n+
m+
1
P
( m)
n
ξm, ηm
can be calculated by the following recursive formulas:
ξ = ξ
m
η = ξ
m
m−1
m−1
x −η
y + η
m−1
m−1
y�
�
x�
nm
m
m
nm
m
(sin ϕ )( C r cos ϕ cos mλ
+ S r cos sin mλ)
(5-12)
nm
nm
m
(5-13)
(5-14)
(5-15)