Global Positioning System Reference
In-Depth Information
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Compare (3.69) with (3.4). The remaining accelerations are discussed briefly below.
The most simple way to solve (3.68) and (3.69) is to carry out a simultaneous nu-
merical integration. Most of the high-quality engineering or mathematical software
packages have such integration routines available. Kaula (1966) expresses the equa-
tions of motion in terms of Kepler elements and expresses the disturbing potential in
terms of Kepler elements. Kaula (1962) gives similar expressions for the disturbing
functions of the sun and the moon.
3.1.4.1 Gravitational Field of the Earth
The acceleration of the noncentral
portion of the gravity field of the earth is given by
∂R
∂X
T
∂R
∂Y
∂R
∂Z
X
g
=
(3.70)
[67
Th
e disturbing potential
R
is
∞
n
)
C
nm
cos
m
λ
(3.71)
a
e
r
n
+
1
µ
Lin
—
-
——
No
PgE
P
nm
(
cos
λ +
S
nm
sin
m
R
=
θ
n
=
2
m
=
0
with
1
θ
m/
2
cos
2
d
(n
+
m)
d(
cos
)
(n
+
m)
cos
2
1
n
−
P
nm
(
cos
θ
=
θ −
(3.72)
)
2
n
n
!
θ
P
n
=
√
2
n
[67
+
1
P
n
(3.73)
−
1
/
2
+
(n
m)
!
P
nm
=
P
nm
(3.74)
2
(
2
n
+
1
)(n
−
m)
!
Equation (3.71) expresses the disturbing potential (as used in satellite orbital
computations) in terms of a spherical harmonic expansion. The symbol
a
e
denotes
the mean earth radius,
r
is the geocentric distance to the satellite, and
are
the spherical co-latitude and longitude of the satellite position in the earth-fixed
coordinate system, i.e.,
x
θ
and
λ
)
. The positions in the celestial system (X)
follow from (2.34).
P
nm
denotes the associated Legendre functions, which are known
mathematical functions of the latitude.
C
nm
and
S
nm
are the spherical harmonic
coefficients of degree
n
and order
m
. The bar indicates fully normalized potential
coefficients. Note that the summation in (3.71) starts at
n
=
x
(r,
θ
,
λ
0
equals the central component of the gravitational field. It can be shown that the
coefficients for
n
=
2. The term
n
=
1 are zero for coordinate systems whose origin is at the center
of mass. Equation (3.71) shows that the disturbing potential decreases exponentially
with the power of
n
. The high-order coefficients represent the detailed structure of
the disturbing potential, and, as such, the fine structure of the gravity field of the
earth. Only the coefficients of lower degree and order, say, up to degree and order
=
