Global Positioning System Reference
In-Depth Information
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In Expression (3.94) we switched from the fully normalized spherical harmonic co-
efficients to regular ones and substituted the expression for the Legendre polynomial
P
2
(
cos
θ
)
. Since
Z
=
r
cos
θ
, Equation (3.94) can be rewritten as:
J
2
3
2
=
µ
a
e
r
3
Z
2
r
2
1
2
R
−
(3.95)
(X
2
Y
2
Z
2
)
1
/
2
, we can readily differentiate and compute
Re
cognizing that
r
=
+
+
th
e acceleration
X
g
as per (3.70)
X
1
r
2
=−
r
3
X
2
J
2
µ
a
e
r
5
5
Z
2
3
X
X
s
+
m
−
−
+
(3.96)
Y
1
a
e
r
5
5
Z
2
r
2
=−
r
3
Y
3
2
J
2
µ
[89
Y
+
Y
s
+
m
−
−
(3.97)
Z
1
a
e
r
5
5
Z
2
r
2
=−
r
3
Z
3
2
J
2
µ
Lin
—
0.6
——
Sho
PgE
Z
+
Z
s
+
m
−
−
(3.98)
Th
ese equations are valid in the inertial system (X) and could be integrated. The PZ90
re
ference system, however, is ECEF and rotates with the earth. It is possible to rewrite
th
ese equations in the ECEF system (x). Since the integration interval is only
15
m
in, we can neglect the change in precession, nutation, and polar motion and only
tak
e the rotation of the earth around the
z
axis into consideration. The final form of
th
e satellite's equations of motion then becomes:
±
[89
x
1
r
2
=−
r
3
x
2
J
2
µ
a
e
r
5
5
z
2
3
2
3
x
¨
−
−
+ ω
+
ω
3
˙
+¨
x
2
y
x
s
+
m
(3.99)
y
1
a
e
r
5
5
z
2
r
2
=−
r
3
y
3
2
J
2
µ
2
y
¨
−
−
+ ω
3
y
+
2
ω
3
˙
x
+¨
y
s
+
m
(3.100)
z
1
a
e
r
5
5
z
2
r
2
=−
r
3
z
3
2
J
2
µ
z
¨
−
−
+¨
z
s
+
m
(3.101)
Note that
(
z)
s
+
m
are the accelerations of the sun and the moon given in the
PZ90 frame. These values are assumed constant when integrating over the
x,
¨
y,
¨
¨
±
15 min
interval. In order to maintain consistency, the values for
ω
3
should
be adopted from GLONASS (2002). This document recommends a four-step Runge-
Kutta method for integration.
Various international observation campaigns have been conducted to establish
accurate transformation parameters between WGS84 and PZ90, with respect to the
ITRF. Efforts are continuing to include the precise GLONASS ephemeris into the IGS
products.
µ
,
a
e
,
J
2
, and
