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{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
unconservative forces $P$ in a rectangular coordinate frame connected with
the central body $M$, can be written with such equations
$$
\frac{d^2 x}{dt^2}+\frac{\mu x}{r^3}=-\frac{\partial V}{\partial x}+F, \eqno(1)
$$
with initial conditions
$x_0=x(t_0), \dot{x}_0=\dot{x}(t_0)$
Here $x=(x_1,x_2,x_3)^T$ is the position
vector, $t$ is physical time, $r=|x|$,
$\mu=k^2(M+m)$, $k^2$is the universal gravitational constant, $V=V(x,t), $
is a perturbed function of potential forces and $F$ is the
vector of the accelerations due to the forces which have no potential.

As is well known, Equations (1) are singular in
vicinties of the central and perturbing masses. In process of numerical integration
these nonuniformities require a regular change of integrating step size.
It involves losses in accuracy of numerical solution and wasteful expenses of
computer time.

Besides, the solutions of the equations (1) are unstable in
Ljapunov sense even in the case of unperturbed motion. This unstability
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
mechanics is connected with construction of transformations,
allowing completely or partly to avoid the singularity mentioned above.

Initial conditions $x_0=x(t_0), \dot{x}_0=\dot{x}(t_0)$ of Equtions (1) are
determined by the region of possible motions $R_0.$

In classical way, when it is expected that law of distribution
errors of observations is close to normal, initial regions of possible
motions $R_0$ are defined by LSM-evaluations of vector of initial parameters
$q_0=\{x_0=x(t_0), \dot{x}_0=\dot{x}(t_0)\}$ and by covariance matrix of its errors
$\hat D_0$,
$$R_{0} \colon N\left(\hat q_0, k^2\hat D_0\right),\quad k=1,2,3,$$
where $k$ is the gain factor of LSM-evaluations 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 object motions.

In any case the orbital evolution of celestial body has to be considered as
evolution of the domain of the body possible
motion.

Constraction of algorithms for determining evolution of the
possible motion domains forms the second trend in the development of numericsl
algorithms of celestial mechanics.

A brief summary of the results in the development both trends obtained by the
authors during last same years are given in this report.

We discuss new Encke-type algorithms in regularising and stabilising variables.
The algorithms do not contain the equations
for quick variables and display high efficiency in numerical simulating
the motion of special asteroids and planetary satellites.

The problem of numerical investigation of close encounters of small
bodies with large planets is analysised.

New algorithms for determining initial domains of possible motions
are considered. The analysis of using linear and non-linear
algorithms for determining evolutions of the domains of possible
motions are given. Several interesting numerical examples are given.
These examples show that the main value of the nonlinear method is the fact
that evaluations obtained on its base are much more profound and give a
greater amount of interesting information on motion.


New results and the results, which have been partially published
in \cite {1}, are presented.

\begin{thebibliography}{20}
\bibitem{1}
Bordovitsyna, T.V., Avdyushev, V.A. and Chernitsov, A.M.
New Trends in Numerical Simulation of the Motion of Small Bodies
of the Solar System, Cel. Mech. and Dyn. Astr., 2001, 80/3-4, pp. 227-247.

\end{thebibliography}
\end{document}