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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Symplectic integrators for studying the long­term
evolution of high­eccentricity orbits
V. V. Emel'yanenko
South Ural University, Chelyabinsk, Russia
Discoveries of many new objects moving on high­eccentricity 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 tools 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 time­step
depends on the potential energy of a Hamiltonian system. In this paper we develop
these methods to handle both high­eccentricity orbits and close encounters for
the Hamiltonian of the form
H = H 0
\Gamma H 1 ;
where H 0
is the Keplerian part, H 1
is the perturbation part, and H 0
AE H 1
in
the absence of close encounters.
For the motion of a small body with infinitesimal mass in the gravita­
tional field of the Sun and planets, H 0
= v 2 =2 \Gamma ¯=r is the Keplerian part,
H 1 = R(q 1 ; q 2 ; q 3 ; t) is the perturbing function, ¯ is a positive constant, r =
q
q 2
1
+ q 2
2
+ q 2
3
, v =
q
p 2
1
+ p 2
2
+ p 2
3
, 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 perturb­
ing function R depends on t through planetary coordinates. We extend the phase
space, adding the canonical variables q 4 = t and p 4 = \GammaH [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 ; \Gamma 1 = \Gamma log K 1 ;
K 0
= r(H 0
+ p 4
+ B 0
+ B 1
r
+ B 2
r 2
); K 1
= r(H 1
+B 0
+ B 1
r
+ B 2
r 2
);
61

\Gamma = 0 along the trajectory, B 0 ; B 1 ; B 2 are small constants, and
ds = K 0
r
dt = K 1
r
dt = (R +B 0
+ B 1
r
+ 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 trans­Neptunian objects for the age of
the Solar system.
The same principles can be implemented for the general N­body 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 01­02­16006) and INTAS (Grant
00­240).
References
1. Wisdom J., Holman M. Symplectic maps for the N--body problem. Astron.
J., 1991, 102, 1528--1538.
2. Kinoshita H., Yoshida H., Nakai H. Symplectic integrators and their appli­
cation in dynamical astronomy. Celest. Mech. & Dyn. Astron., 1991, 50,
59--71.
3. Mikkola S., Tanikawa K. Explicit symplectic algorithms for time­transformed
Hamiltonians. Celest. Mech. & Dyn. Astron., 1999, 74, 287--295.
4. Preto M., Tremaine S. A class of symplectic integrators with adaptive
timestep for separable Hamiltonian systems. Astron. J., 1999, 118, 2532--
2541.
5. Szebehely V. Theory of Orbits. 1967, Academic Press, New York, London.
6. Duncan M. J., Levison H. F., Lee M. H. A multiple time step symplectic
algorithm for integrating close encounters. Astron. J., 1998, 116, 2067--2077.
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