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{Symplectic integrators for studying the long-term evolution
of high-eccentricity orbits}

{V.~V.~Emel'yanenko}

% {Institute, city, country}
{South Ural University, Chelyabinsk, Russia}

Discoveries of many new objects moving on high-eccentricity orbits
in the Solar system and beyond have highlighted the importance of
modeling the long-term evolution of such orbits. At present,
symplectic integrators introduced by Wisdom and Holman [1] and
Kinoshita et al. [2] are the most popular tool for studying
dynamics of solar and extrasolar system objects. Mikkola and
Tanikawa [3] and Preto and Tremaine [4] suggested a new integrator
in which the time-step depends on the potential energy of a
Hamiltonian system. In this paper we develop these methods to
handle both high-eccentricity orbits and close encounters for the
Hamiltonian of the form $$ H=H_0-H_1,$$ where $H_0$ is the
Keplerian part, $H_1$ is the perturbation part, and $H_0 \gg H_1$
in the absence of close encounters.

For the motion of a small body with infinitesimal mass in the
gravitational field of the Sun and planets, $H_0=v^2/2-\mu/r$ is
the Keplerian part, $H_1=R(q_1,q_2,q_3,t)$ is the perturbing
function, $\mu$ is a positive constant,
$r=\sqrt{q_1^2+q_2^2+q_3^2}$, $v=\sqrt{p_1^2+p_2^2+p_3^2}$, and
the conjugate canonical variables $q_1, q_2, q_3$ and $p_1, p_2,
p_3$ are Cartesian coordinates and their time derivatives. The
perturbing function $R$ depends on $t$ through planetary
coordinates. We extend the phase space, adding the canonical
variables $q_4=t$ and $p_4=-H$ [5]. Then there exists a
transformation to the new independent variable $s$ and the new
Hamiltonian $$\Gamma=\Gamma_0+\Gamma_1,$$ where $$\Gamma_0=\log
K_0, \quad \Gamma_1=-\log K_1,$$ $$
K_0=r(H_0+p_4+B_0+\frac{B_1}{r}+\frac{B_2}{r^2}), \quad
K_1=r(H_1+B_0+\frac{B_1}{r}+\frac{B_2}{r^2}),$$ $\Gamma=0$ along
the trajectory, $B_0, B_1, B_2$ are small constants, and
$$ds=\frac{K_0}{r} d t=\frac{K_1}{r} d t
=(R+B_0+\frac{B_1}{r}+\frac{B_2}{r^2}) dt. $$ Both $\Gamma_0$ and
$\Gamma_1$ are integrable. Therefore, the generalized leapfrog
scheme[1] can be applied to the Hamiltonian $\Gamma$. The step
over $s$ is constant at the symplectic integration. Thus the
time-step depends on the perturbing function $R$ and the distance
$r$. The practical choice of $B_0, B_1, B_2$ is considered for
both barycentric and heliocentric coordinate systems in details.
Although numerical tests have shown that the method is the most
stable at $B_0=0$ for sufficiently small steps, the parameter
$B_0=0$ can be used to keep the time-step within a small fraction
of the shortest period in the dynamical system. In particular, the
algorithm described above has been applied to integrations of
trans-Neptunian objects for the age of the Solar system.

The same principles can be implemented for the general $N$-body
problem. The extension of the algorithm to the Jacobi and
mixed-centre coordinates [6] has been carried out. Numerical
experiments demonstrate the efficiency of this symplectic
technique for planetary system formation problems.

This work was supported by RFBR (Grant 01-02-16006) and INTAS
(Grant 00-240).

\begin{thebibliography}{20}
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Wisdom J. and Holman M. Symplectic maps for the N-body problem.
Astron. J., 1991, {\bf 102}, 1528--1538.
\bibitem{2}
Kinoshita H., Yoshida H. and Nakai H. Symplectic integrators and
their application in dynamical astronomy. Celest. Mech. Dyn.
Astron., 1991, {\bf 50}, 59--71.
\bibitem{3}
Mikkola S. and Tanikawa K. Explicit symplectic algorithms for
time-transformed Hamiltonians. Celest. Mech. Dyn. Astron., 1999,
{\bf 74}, 287--295.
\bibitem{4}
Preto M. and Tremaine S. A class of symplectic integrators with
adaptive timestep for separable Hamiltonian systems. Astron. J.,
1999, {\bf 118}, 2532-2541.
\bibitem{5}
Szebehely V. Theory of Orbits. 1967, Academic Press, New York,
London.
\bibitem{6}
Duncan M.J., Levison H.F. and Lee M.H. A multiple time step
symplectic algorithm for integrating close encounters. Astron. J.,
1998,{\bf 116}, 2067--2077.
\end{thebibliography}
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