Geoscience Reference
In-Depth Information
As usual,
a
denotes
da/dt
, etc. The derivation of these equations may be
found in any textbook on celestial mechanics, e.g., Plummer (1918: p. 151),
Brouwer and Clemence (1961: p. 301), and Seeber (2003: Sect. 3.2.1.3), who
uses the symbols
K
1
,K
2
,K
3
instead of
W, S, R
.
7.3
Determination of zonal harmonics
The effect of the zonal harmonics on satellite orbits is much greater than that
of the tesseral harmonics. Only zonal harmonics (
J
2
,J
3
,J
4
, ...
) will give
observable variations of the orbital elements themselves. The tesseral har-
monics cause oscillatory disturbances that rapidly change their sign, whereas
the effect of the zonal harmonics is cumulative. For this reason, we consider
first the effect of zonal harmonics, that is, the effect of those independent of
longitude
λ
. Hence we set
V
=
GM
r
+
R,
(7-11)
where the
perturbing potential
a
e
r
n
+1
∞
GM
a
e
R
=
−
J
n
P
n
(cos
ϑ
)
(7-12)
n
=2
is a function of
r
and
ϑ
only. Note that the main difference between the
perturbing potential
R
of celestial mechanics and the disturbing potential
T
of physical geodesy is that
R
, but not
T
, also incorporates the effect
of the flattening through
J
2
. There are also other perturbing forces acting
on a satellite, such as the resistance of the atmosphere (atmospheric drag),
radiation pressure exerted by the sunlight, etc. These nongravitational per-
turbances must be taken into account separately and will not be considered
here.
Note that the equatorial radius of the earth (the semimajor axis of the
terrestrial ellipsoid) has been denoted by
a
e
, in order to distinguish it from
a
, which now denotes the semimajor axis of the orbital ellipse. This notation
will be used in what follows.
Since
S
is the component of the perturbing force along the radius vector,
we have
S
=
∂R
∂r
.
(7-13)
The components of the perturbing force along the meridian and the prime
vertical are
1
r
∂R
∂ϑ
1
r
sin
ϑ
∂R
∂λ
.
−
and
(7-14)
