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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Precision increase for the theory of relative motion of
a satellite's pair
K. V. Kholshevnikov
Astronomical Institute of St.Petersburg State University,
and Institute of Applied Astronomy, St.Petersburg, Russia
Modern higher--precise theories of motion of special Artificial Earth Satel­
lites (geodetical, navigational) consider all forces which one may describe with at
least 30% precision, including so exotical ones as a reaction of photon flux due to
radiowave--interchange with ground stations. Accuracy may be increased drasti­
cally when we confine ourselves by the description of relative motion of a pair of
satellites close to one another. Here we regard similar satellites moving along the
same track, i.e. the set of subsatellites points. The physical reason for accuracy
increase is rather simple: both satellites undergo the same perturbations with a
little shift in time only.
Let pass to more strict formulations. Denote F 1
forces per unit mass depend­
ing on position and velocity of a satellite with respect to the solid Earth, but
not on time explicitly. Such are attraction of the solid Earth in Newtonian and
relativistic approach, Lorentz force, main part of the air drag. Let F 2 are all
other forces which we consider as disturbing ones. At the first step we may ne­
glect them. Then in the frame related with the solid Earth we have differential
equations of motion of the form
Ё r = F 1
+ J ; (1)
J being the sum of centrifugal and Coriolis forces of inertia independent of t as
well.
Denote r 1
(t) a solution of (1). Due to the autonomy of (1) r 2
(t) = r 1
(t + Ь)
is also a solution for any constant shift Ь . Solutions r k (t) correspond to a pair of
satellites moving along the same track. The vector r(t) = r 2
(t) \Gamma r 1
(t) describes
relative position of the second AES with respect to the first one.
Note that the centrifugal force and especially Coriolis one are not very small
in comparison with the main part of the Earth's attraction. That is why one uses
inertial frame as a rule. Denoting corresponding vectors by Greek letters we have
ae k
(t) = A(t)r k
(t) ; (2)
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A being orthogonal matrix describing the Earth's rotation.
Finally,
ae(t) = A \Gamma1 (Ь) [ae 1 (t + Ь) \Gamma ae 1 (t)] +
i
A \Gamma1 (Ь) \Gamma E
j
ae 1 (t) ; (3)
E being a 3 \Theta 3 unit matrix.
In case Ь is small so are the differences ae 1
(t + Ь) \Gamma ae 1
(t) and A \Gamma1 (Ь) \Gamma E .
For testing the smallness it is better to use dimensionless quantities nЬ and n 0 Ь
equal to multiplied by Ь satellite's mean motion and angular velocity of the
Earth's rotation, respectively.
The expression in brackets in the formula (3) represents a small difference of
close quantities. To avoid the loss of accuracy we may use Lie -- series represen­
tation in powers of Ь for the solutions of similar to (1) differential equations in
the inertial space. In our case Lie -- series in powers of Ь begins with the first
degree term. As to the second term of the sum (3), the difference of corresponding
matrices has maximal modulus of eigenvalues equal to 2 sin(n 0 Ь=2) and is also
small.
We see that errors in ae(t) diminish by a factor of nЬ or n 0 Ь in comparison
with the errors in ae 1 (t).
As to numerical estimations of corresponding errors they depend on the value
of Ь and, what is more important, on the satellite's orbit we choose. For close
to the Earth's surface AES nЬ plays the main role whereas for higher ones n 0 Ь
does. As an example, for an orbit with the semi--major axis a = 7400 km and
eccentricity e = 0:01 we have n = 1:0 \Delta 10 \Gamma3 s \Gamma1 and n 0 = 0:73 \Delta 10 \Gamma4 s \Gamma1 . Choosing
Ь = 25 s we have nЬ = 0:025 and simultaneously we guarantee that the distance
between satellites lies in the interval from 160 to 200 km.
Note that for satellites higher than stationary ones the approach we deal with
looses its advantages since the luni--solar disturbing forces become considerable.
Meanwhile they depend explicitly on time.
This work was partly supported by the RFBR grant No. 02­02­17516 and the
Leading Scientific School grant No. 00­15­96775.
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