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

Chapter 8 Geostationary Orbit Determination And Prediction During Satellite Maneuvers
where p = a 1−e
2
Clearly, from the results Eq.(8-22) and Eq.(8-23), the short periodic variations of the orbital parameters under
the influence of the non-spherical earth gravitation are removed, the remaining is secular and long periodic
variation terms, i.e.
� 3 J2
35 J4
�6
3 2 � 3 2 1 3 �
∆Ωs =−� 2 cosi − 4 � cosi+ sin icos i�( 1 + e + e ) �
(8-24)
�2
p 8 p �7
2 � 2 3 �
� J3
� 15 2 3 � 35 J4
�3
2 3 1 2 3 2 3 ��
∆is = cos i�−
3 �(
sin i− ) ecosω�+
4 � e sin i( − sin i)sin 2ω
+ e sin isin
4ω
��
(8-25)
� p � 8 2 � 8 p �2
7 2
8
��
General forms of Eq. (8-24) and Eq.(8-25) can be written as,
∆Ω s
�3
J 35 J �6
3 �
�
=−� i − � i+ i i�1+ e + e �n t−T �2
p 8 p �7
2 � �
3
2
4
2 2 1 3
2 cos 4 cos sin cos ( ) ( 0)
(8-26)
2 3
� J3
� 15 2 3 � 35 J4
�3
2 3 1 2 3 2 3 ��
∆is = cos i�−
3 �(
sin i− ) ecosω�+
4 � e sin i( − sin i) sin 2ω
+ e sin isin 4ω��n(
t−T0) � p � 8 2 � 8 p �2
7 2
8
��
(8-27)
Since J2=-0.1083×10 -2 , J3=0.2532×10 -5 , J4=0.1620×10 -5 , J3 and J4 are much smaller than J2, and for
geostationary satellite, e≈0 and i≈0, therefore in Eq. (8-26) and Eq.(8-27), the terms related to J3 and J4 can be
neglected, Eq.(8-26) and Eq.(8-27) become
3 J2
∆Ωs =−( 2 cos int ) ( −T0
)
(8-28)
2 p
∆is = 0 (8-29)
Eq.(8-28) shows the secular variations of the longitude of the ascending node Ω of the satellite orbit are
produced by geopotential zonal terms, especially J2. This effect causes the ascending node to drift toward west if
ο ο
0 ≤i ≤ 90 . From Eq.(8-29) the zonal terms have no direct influence on orbital inclination i .
8.1.2 Solar and Lunar Attractions
Because of the great distance between the Earth and the Sun and the Earth and the Moon, the non-spherical parts
of the Sun and the Moon don’t need to be taken into account. Due to this reason, in the perturbation models, the
Sun and the Moon can be considered as point masses. The solar and lunar attractions on the satellite can be
expressed in the inertial coordinate system (the center of the earth as the origin of the system) as follows
dr
dt r r
2ϖ
2 ϖ ϖ ϖ
µ ϖ ri−r ri
2 =− 3 + � µ i[
ϖ ϖ3
− 3]
| ri−r| r
i=
1
i
(8-30)
where
ϖ
r
ϖ
r1 ϖ
r2 position vector of satellite
position vector of Sun
position vector of Moon
µ , µ 1, µ 2 the gravity constants of the Earth, the Sun and the Moon respectively.
According to Eq.(8-30) the perturbation functions of the Sun and the Moon on satellite can be respectively
written as
1 rs⋅r Rs = GsMs( − 3 )
| rs−r| rs
= GM s s(
( x − x) 1
+ ( y − y) + ( z −z)
xx s + yy s + zz s
−
/
( xs + ys + zs)
)
2 2 2 2 2 2 3 2 (8-31)
s s s
rmr Rm = GmMm r − r r
−
1 ⋅
( )
3
| |
m
m
99