Geoscience Reference
In-Depth Information
1
2
r
2
dv
, this law may be formulated mathematically as
r
2
dv
dt
v
is
=
GM a
(1
−
e
2
)
,
(7-7)
where the constant has already been given its proper value.
Kepler's third law
reads
n
2
a
3
=
GM ,
(7-8)
where the satellite mass has been neglected and where
n
=
2
π
P
(7-9)
is the “mean motion” (mean angular velocity) of the satellite,
P
being its
period.
So far we have assumed that all
J
n
,
C
nm
,and
S
nm
in (7-1) are zero. This
is not true because of the irregularities of the earth's gravitational field, even
though these coecients are small. Therefore, the satellite is subject to small
perturbing forces. We may still consider the satellite orbit as an ellipse, but
then the parameters of this ellipse, the orbital elements, will no longer be
constant but will change slowly. At each instant, this
osculating ellipse
will
be slightly different. It is defined as follows. Imagine that at the instant
under consideration all perturbing forces suddenly vanish. Then the satellite
will continue its motion along an exact ellipse; this is the osculating ellipse.
If we resolve the total perturbing force into rectangular components
S
,
T
,and
W
,where
S
is directed along the radius vector,
W
is normal to the
orbital plane, and
T
is normal to
S
and
W
- note that this notation follows
astronomical usage; there is no relation to the geodetic use of
T
and
W
for
potentials! -, then the time rate of change of the orbital parameters can be
expressed in terms of these components:
a
=
2
a
2
b
a
GM
eS
sin
v
+
p
r
T
,
a
GM
S
sin
v
+
r
+
p
r
T
,
e
=
b
a
cos
v
+
er
p
a
GM
W
cos(
ω
+
v
)
,
ı
=
r
b
a
GM
W
˙
Ω=
r
b
sin(
ω
+
v
)
sin
i
,
a
GM
p
W
sin(
ω
+
v
)cot
i
.
(7-10)
ω
=
b
a
1
e
S
cos
v
+
r
+
p
r
−
T
sin
v
−
ep
