vf-
A V,m= GM"e I Flmp(i) X
Glpq(e)S*lmpq(a>, M, Q, 0
with
s*
•cosy
and
y (l 2p)a> (l 2p q)M m(Q Q)
0 is the rotation of the earth. The F,mp(iare func
tions of the orbital inclination and the Glpq(e) are
functions of the eccentricity. Complete expressions
for their evaluation are given by Kaula [4].
It is well known that the principal perturbations in
the orbit are caused by the earth's oblateness and
that they are secular for the mean anomaly M, the
argument of perigee co, and the longitude of the
ascending node 12. The other elements a, e, i (semi-
major axis, eccentricity and inclination) are, to a
first approximation, constant so that in equation (2)
all the important time dependencies are contained
in the function S*.
If we substitute the perturbing potential AVlm into
the Lagrangian equations of motion and integrate
them, assuming that only S* is time dependent, we
obtain expressions for the orbital elements that are
a summation of terms containing the 51* or the inte
gral of S* with respect to its argument. The periods
of these perturbations can be calculated and only
their amplitudes, which are proportional to the
C,m and Slm, are considered as unknown. Thus if
we observe the satellite's motion in detail we can
carry out an analysis for the amplitudes of the va
rious perturbations with known period in order to
determine the C and S.
The question immediately raised is that in the ex
pansion (2) we have several summations to infinity;
a clearly impossible situation and we have to trun
cate the expansion somewhere in order to give a
manageable solution. But this would tend to distort
the terms unless there is some "natural selection"
of those terms that are significant. Such a selection
is indeed possible thanks to the nature of the orbital
theory, of the observational data and of the earth's
gravity field.
In the first instance Glpq(e) is proportional to e'"1
and since e is usually quite small we need only sum
q from about —10 to +10. Secondly, we are
measuring the earth's gravity field at the satellite
height and there is a term ae/ain equations (1) and
(2) that decreases with increasing That is, the
higher the degree of the perturbing potential the
less will be the magnitude of the perturbation in the
orbit. Thirdly, the amplitudes of the harmonic
coefficients, Clm, Slm, tend to decrease with in
creasing degree. If we define the average amplitude
of harmonics of degree by
21 1 m 0
X (CL+Sfj
we find that the V, decrease according to an approx
imate rule of 10~5//2 [1].
The fourth characteristic concerns the spectrum of
frequencies in S*lmpq (to, M, Q, 0) or the y. The rates
of change of aand Q are quite small compared
with the mean motion of the satellite, n, and with
the earth's rotation, 0, so that the frequency of S*
is governed mainly by M (k n) and 1 rev/day).
The mean motion for close earth satellites is usually
of between 10 and 16 revolutions/day so that if the
order m is less than n, the minimum frequency of S
is approximately m revolutions/day. That is, har
monics in the potential such as C4-2, C6 2, etc.,
give rise to a series of perturbations whose lowest
frequency is about 2 revolutions/day.
Harmonics such as C12>12, C14i)2, etc. give rise
to a series of perturbations whose lowest frequency
is about 12 revolutions/day. That is, the higher the
order of the harmonic, the lower will be the maxi
mum period in the series of perturbations caused by
the harmonic. With optical and laser observations
we are usually limited to studying only those per
turbations that have periods longer than about the
duration of one orbital revolution. This restriction
is caused by the difficulty of obtaining good orbital
coverage with a limited number of stations and with
tracking systems that are limited to observations
during clear nights only.
We make use of the above characteristics to deter-
I I
Cl p-0 q= - c
(2)
Gin,
I m even
Slm
- ~Slm-
l m odd
- G im_
l m even
•cosy
l m odd
ngt 72
43
