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

Chapter 6 Algorithms of Orbit Determination of IGSO, GEO and MEO Satellites
x = rcosϕcos
λ�
�
y = rcosϕsin
λ �
z = rsinϕ
�
�
we can obtain,
2
∂ U
2
∂x
e
2
2
74
(6-49)
∂ Ue
∂r
2 ∂Ue
∂ r
= ( ) +
(6-50)
2
2
∂r
∂x
∂r
∂x
e
2
Ue
2
∂ Ue
∂r
∂r
∂Ue
xy
e
2
∂r
∂x
∂y
∂r
2
Ue
2
∂ Ue
∂r
∂r
∂Ue
xz
e
2
∂r
∂x
∂z
∂r
2
Ue
2
y
e
2
∂ Ue
=
2
∂r
∂r
∂y
2 ∂Ue
+
∂r
2
Ue
2
∂ Ue
∂r
∂r
∂Ue
yz
e
2
∂r
∂y
∂z
∂r
2
Ue
2
z
e
2
∂ Ue
=
2
∂r
∂r
∂z
2 ∂Ue
+
∂r
∂
∂∂
∂
∂∂
∂
∂
∂
∂∂
∂
∂
2
∂ r
= + (6-51)
∂∂ xy
2
∂ r
= + (6-52)
∂∂ xz
2
∂ r
( ) (6-53)
2
∂y
2
∂ r
= + (6-54)
∂∂ yz
2
∂ r
( ) (6-55)
2
∂z
where, the symbol | e means the derivatives in the earth-fixed coordinate system. These derivatives will be
converted to the inertial coordinate system.
If only the J2 term in geopontential Eq.(6-48) is considered, the Eq.(6-50) to Eq.(6-55) become,
2
e
2 3 3 2
e
∂ U
∂x
2
∂ Ue
∂∂ xy
2
∂ Ue
∂∂ xz
∂ U
∂y
� GM GM ae
2 2 � 2 2 GM ae
2 2 2
= 2 −6 J 3 ϕ −1
ϕ λ 6 J
ϕ λ
3 2 2
�
�
( ) ( sin )
r r r
�
� cos cos − ( ) sin cos
r r
� GM 3 GM ae
2 2 � 2
+ − + −
�
�
J
3 3 2 ( ) ( 5sin ϕ 1)
r 2 r r
�
� sin λ
� GM GM ae
2 3 7 2 � 2 2
+ − + − +
�
� 3 J
3 3 2 ( ) ( sin ϕ) r r r 2 2 �
� sin ϕ cos λ (6-56)
e
�GM
GM ae
2 2 � 2
GM ae
2
= −3 J
ϕ ϕ λ ϕ ϕ λ
3 3 2 3 −1
2 3 J
3 2 2 2 2
�
�
( ) ( sin )
r r r
�
� cos sin − ( ) sin sin sin
r r
� GM 3 GM ae
2 2 �
− − + −
�
�
J
3 3 2 ( ) ( 5sin ϕ 1)
r 2 r r
�
� sin λcos λ
� GM GM ae
2 3 7 2 � 2
+ − + − +
�
� 3 J
3 3 2 ( ) ( sin ϕ) r r r 2 2 �
� sin ϕsinλcosλ (6-57)
e
�GM
GM ae
2 2 �
GM ae
2 2 2
= −3 J
ϕ ϕ λ ϕ ϕ ϕ λ
3 3 2 3 −1
2 6 J
3 2 2
�
�
( ) ( sin )
r r r
�
� sin cos + ( ) sin (cos −sin
) cos
r r
� GM GM ae
2 3 7 2 �
− − + − +
�
� 3 J
3 3 2 ( ) ( sin ϕ) r r r 2 2 �
� sinϕcosϕcosλ (6-58)
2
e
2 3 3 2
e
� GM GM ae
2 2 � 2 2 GM ae
2 2 2
= 2 −6 J 3 ϕ −1
ϕ λ 6 J
ϕ λ
3 2 2
�
�
( ) ( sin )
r r r
�
� cos sin − ( ) sin sin
r r
� GM 3 GM ae
2 2 � 2
+ − + −
�
�
J
3 3 2 ( ) ( 5sin ϕ 1)
r 2 r r
�
� cos λ