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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Two trends in the development of numerical
algorithms of celestial mechanics
T. V. Bordovitsyna, V. A. Avdushev, A. M. Chernitsov
Research Institute of Applied Mathematics and Mechanics, Tomsk, Russia
The motion of a material particle with the mass m in the gravitational field
of the central body with the mass M under the action of conservative forces F i
with the potential function V i and non--conservative forces P in a rectangular
coordinate frame related to the central body M , can be described by equations
d 2 x
dt 2
+ ¯x
r 3
= \Gamma
@V
@x + F; (1)
with initial conditions x 0 = x(t 0 ); —
x 0 = —
x(t 0 ). Here x = (x 1 ; x 2 ; x 3 ) T is the
position vector, t is physical time, r = jxj, ¯ = k 2 (M + m), k 2 is the universal
gravitational constant, V = V (x; t); is a perturbed function of potential forces
and F is the vector of the acceleration due to the forces which have no potential.
As is well known, Equations (1) are singular in vicinities of the central and
perturbing masses. In process of numerical integration these non--uniformities
require a regular change of the size of the integration step. It involves the loss of
accuracy of numerical solution and wasteful expenses of computer time.
Besides, the solutions of the equations (1) are instable in Lyapunov sense even
in the case of the unperturbed motion. This instability intensifies the influence
of truncation and round off errors in the process of numerical integration.
The first trend in the development of numerical algorithms of celestial me­
chanics is related with the construction of transformations enabling completely
or partly to avoid the singularity mentioned above.
Initial conditions x 0
= x(t 0
); —
x 0
= —
x(t 0
) of Equations (1) are determined by
the region of possible motion R 0
:
In classical way, under the assumption that the law of distribution of errors
of observations is close to the normal one, the initial regions of possible motion
R 0 are dedermined by the LSM­evaluations of vector of initial parameters q 0 =
fx 0
= x(t 0
); —
x 0
= —
x(t 0
)g and by the covariance matrix of its errors “
D 0
,
R 0 : N
i
“
q 0 ; k 2 “
D 0
j
; k = 1; 2; 3;
31

where k is the gain factor of the LSM­evaluations of the covariance matrix of
errors in initial parameters.
In the case, when the law of distribution of errors of observations greatly differs
from the normal one, one has to search other ways to assign initial domains of
the object motion.
In any case the orbital evolution of celestial body should be considered as
evolution of the domain of the body possible motion.
Construction of the algorithms for determining the evolution of the possible
motion domains represents the second trend in the development of numericsl
algorithms of celestial mechanics.
This paper presents a brief summary of the results in the development of both
trends obtained by the authors during last several years.
We discuss new Encke­type algorithms in regularising and stabilising vari­
ables. The algorithms do not contain the equations for fast variables and display
high efficiency in numerical simulating the motion of special asteroids and plan­
etary satellites.
The problem of numerical investigation of close encounters of small bodies
with large planets is analysed.
New algorithms for determining initial domains of possible motions are con­
sidered. The analysis of using linear and non--linear algorithms for determining
evolutions of the domains of possible motions are given. Several interesting nu­
merical examples are given. These examples show that the main merit of the
nonlinear method is the fact that evaluations obtained on its basis are much
more profound and give a greater amount of interesting information of motion.
New results and the results that have been partially published in [1] are
presented.
References
1. Bordovitsyna T. V., Avdyushev V. A., Chernitsov A. M. New Trends in
Numerical Simulation of the Motion of Small Bodies of the Solar System.
Cel. Mech. and Dyn. Astr., 2001, 80, 227--247.
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