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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
An improved analytical technique for accurate
calculation of satellite motion perturbations due to
the Moon/Sun/planets
S. M. Kudryavtsev
Sternberg Astronomical Institute, Moscow, Russia
An improved analytical technique for accurate calculation of satellite orbit
perturbations caused by attraction of the third body (the Moon, the Sun, plan­
ets) is presented. The work expands previous studies [1--3] of the author on de­
velopment of accurate analytical theory of satellite motion. The theory is based
on Poincare method of small parameter and enables one to calculate all pertur­
bations proportional up to and inclusive the 5th­order of the parameter.
One of the major tasks arising in analytical methods of calculating effects
of the Sun, the Moon, and planets attraction in satellite motion is an accurate
representation of the third body orbits. In the simplest case the latter are assumed
to be Keplerian ellipses with constant (or some secularly precessing) elements.
More advanced studies employ some known well--developed analytical expansions
of spherical coordinates of the Moon, the Sun, and planets and transform them
to relevant series representing the perturbing functions of the bodies. However,
the accuracy of this approach is obviously limited by accuracy of the original
analytical expressions for the third body coordinates. So, some modern analytical
theories use precise numerical planetary/lunar ephemerides of DE--series [4] as
either an additional (see, e.g. [5]) or the main (in work [6]) source of the third
body coordinates, but do that over a relatively short time interval (a few days
only). In the present study we use DE--ephemerides for expanding the perturbing
function due to the Moon, the Sun, and planets over long--term intervals (up to
two thousand years).
After some simple modification of results obtained in [7, 8], the perturbing
function due to the third body attraction can be presented as follows:
R =
1
P
l=2
l
P
m=0
l
P
p=0
1
P
q=\Gamma1
i a
Rs
j l ¯
F lmp (i)X l;l\Gamma2p
l\Gamma2p+q (e)\Theta
\Theta
\Gamma ¯
A lm cos / lmpq + ¯
B lm sin / lmpq
\Delta
112

where a; e;
i;\Omega ; ú; – are osculating Keplerian elements of a satellite orbit, ¯
F lmp is
the normalized inclination function, X l;l\Gamma2p
l\Gamma2p+q is the Hansen coefficient,
/ lmpq = (l \Gamma 2p + q)– \Gamma qú + (m + 2p \Gamma
l)\Omega ;
¯
A lm =
8 !
:
¯
C lm if l \Gamma m even
\Gamma ¯
S lm if l \Gamma m odd;
¯
B lm =
8 !
:
¯
S lm if l \Gamma m even
¯
C lm if l \Gamma m odd;
¯
C lm = 1
2l + 1
X
j
¯ j
R s
/
R s
r j
! l+1
¯
P lm (sin ffi j ) cos mff j ;
¯
S lm = 1
2l + 1
X
j
¯ j
R s
/
R s
r j
! l+1
¯
P lm (sin ffi j ) sin mff j ;
and ¯ j ; r j ; ff j ; ffi j are, respectively, the gravitational parameter, geocentric dis­
tance, right ascension and declination of the j th perturbing body (referred to
the Earth mean equator and equinox of a fixed epoch, e.g. of J2000); ¯
P lm is the
normalized associated Legendre function; R s is an arbitrary scaling parameter
(in our study we chose the latter equal to 43000 km to have the ratio a=R s be­
ing less than 1 for all Earth's artificial satellites at orbits extending up to the
geostationary orbit, inclusively).
It is seen that the coefficients ¯
C lm ; ¯
S lm accumulate all information about
instant positions of the perturbing bodies. By using the latest long--term
ephemerides DE/LE--406 we calculated numerical values for those coefficients
over the two thousand year interval [1000AD, 3000AD] with a sampling step 1
day. As the perturbing bodies we considered the Moon, the Sun, Venus, Jupiter,
Mars, and Saturn. Then we made a spectral analysis of the calculated series by
using the improved technique [9]. The feature of this technique is that the final
expansions are Poisson series whith the arguments being high--degree polynomials
of time as opposed to the classical Fourier analysis where arguments are always
linear functions. The frequencies of the series are linear combinations of the fun­
damental arguments of motion of the Moon, the Sun, and planets. It results in the
essential improvement in accuracy of the final series. In the spectrum of ¯
C lm ; ¯
S lm
we took into account all ``waves'' of amplitude increasing the absolute level of
10 \Gamma6 m 2 =sec 2 (the corresponding relative limit is about 10 \Gamma8 ). Expansions of the
coefficients of up to degree l = 8 have been made. The total number of terms in
the final expansion of the perturbing function over two thousand years is about
30000. The exact number of terms to be taken into account when calculating the
third body perturbations depends on the satellite altitude and prediction time
span.
The final series for ¯
C lm ; ¯
S lm are well adapted for the use in analytical calcu­
lation of the third body perturbations of satellite motion by means of computer.
113

We employ this technique in our analytical method for prediction of both high--
altitude and low--altitude Earth satellites orbits, and the obtained results are
presented as well.
The work is supported in part by a grant No. 02­02­16887 from the Russian
Foundation for Basic Research.
References
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Motion of a Satellite in Orbit around a Non--Spherical Planet, Cel. Mech. &
Dyn. Astron., 1995, 61, 207--215.
2. Kudryavtsev S. M. Accurate Analytical Calculation of Effects of Rotations
of the Central Planet on a Satellite Orbit, Cel. Mech. & Dyn. Astron., 1997,
67, 131--144.
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1998, IOM 312, F--98--048, Pasadena.
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