## 2. Map modelLiu & Sun (1994) derived a map which describes the evolution of comets on near-parabolic orbits in the framework of the planar circular restricted three-body problem (the Sun-planet-comet). In our work, we generalize this map in order to study the motion of comets on both planet-crossing and non-crossing orbits. Let us take the distance between the Sun and Jupiter (i.e., 5.2AU)
as the unit of length, the total mass of the Sun and Jupiter as the
unit of mass, and the period of Jupiter's circular motion as
units of time. Denote the mass of
planet as where and are the positions of the Sun and Jupiter, respectively. They can be expressed as follows, where is the initial time. Expanding the right hand members of Eq. (1) to , one has The initial epoch is chosen such that the comet passes through the
perihelion at , and the initial phase
angle of Jupiter motion is For LP comets, the orbital energy where Thus the change of comet energy during one perihelion passage can be obtained by integration the above Eq. (6): with The period of time between the two successive perihelion passages
is , and the change of the phase
angle where the subscript Unlike in Liu & Sun (1994), where the function
in (8) is expanded into Fourier
series in In practice, we compute the integral
in (7) as
. The discarded part
corresponds to the situation where
the comet is more than 100 AU away from Jupiter, and thus the
perturbations by Jupiter can be safely neglected. Considering the
periodic properties of , we calculate
the values of on
values of
Fig. 1 can be compared to Malyshkin & Tremaine (1999), where the function is interpolated (called the Kicked function) from discrete values obtained by integration of the original Hamiltonian system. The maps in their work and in this paper are nearly equivalent, and should have almost the same computational efficiency. To test the validity and the computational speed of the map (8), we calculate the orbits with the same initial configurations by the map (8) and by integration of the original system (1). The Runge-Kutta-Fehlberg 7(8) method is used in our integration. Fig. 2 shows that the final qualitative results (escape, remain or transfer) of the evolution of individual orbits in the original system (1) is preserved by the map (8). However, for the orbit presented in the lower window of Fig. 2, the use of map (8) takes only 1 minute for the orbit evolving up to passages, which takes 30 minutes to integrate the original system (1) up to passages. Thus the use of map (8) is in this case some 3000 times faster than the integration of the differential equation system (1).
The statistical validity of the map (8) is also verified. We take comets with the same initial energy and calculate their orbits up to years. Then determine the transfer probability (the average ratio of comets whose final energy to ) and the average transfer time (number of passages preceeding the transfer). Fig. 3 shows the variations of transfer probability and averaged transfer time for different initial energies and . As one can see, the results obtained by the integration of the original system (1) coincide with those obtained by the map (8) quite well.
Since the use of parabolic orbits to approximate the orbits of the comets, the map is not valid when the comets are in orbits with very small eccentricities. According to our numerical experiments, the minimum eccentricity that the map (8) is still valid is . Since the LP comets we studied in the paper all have large eccentricities (at least 0.7) because of their large semi-major axes and small perihelion distances, this limitation does not affect our results. © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |