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Astron. Astrophys. 364, 887-893 (2000)

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2. Map model

Liu & 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 [FORMULA] units of time. Denote the mass of planet as µ ([FORMULA] for Jupiter). In an inertial coordinate system with the origin at the mass center of the Sun and Jupiter, the planar motion of a comet located at [FORMULA] has the following form:


where [FORMULA] and [FORMULA] are the positions of the Sun and Jupiter, respectively. They can be expressed as follows,


where [FORMULA] is the initial time. Expanding the right hand members of Eq. (1) to [FORMULA], one has


where [FORMULA], and


The initial epoch is chosen such that the comet passes through the perihelion at [FORMULA], and the initial phase angle of Jupiter motion is g. Thus [FORMULA] in the equations from (1) to (4).

For LP comets, the orbital energy K is the parameter that suffers the greatest variation during a passage in the planetary region. The (double of) orbital energy of a comet in the unperturbed motion is


where a is the semi-major axis of the comet orbit. For the near-parabolic comets, we approximate the orbits by [FORMULA], where q and f are the perihelion distances and the true anomalies of the orbits, respectively. According to the definition (5), the variation of K can be expressed as a function of time, with the help of Eq. (3) and the above parabolic-orbit approximation:


Thus the change of comet energy during one perihelion passage can be obtained by integration the above Eq. (6):


with g and q being parameters. The integral limits are set as infinite due to the parabolic-orbit approximation. The change of q during each perihelion passage is of order µ, and its effect on [FORMULA] is of order [FORMULA], therefore it can be treated as a constant during the evolution of the orbital energy of the comet (Petrosky & Broucke 1988; Liu & Sun 1994). For a given parameter q, [FORMULA] is a [FORMULA]-periodic function of g, and it is anti-symmetric with respect to [FORMULA] (Liu & Sun 1994).

The period of time between the two successive perihelion passages is [FORMULA], and the change of the phase angle g of Jupiter during this period is [FORMULA]. Combining the expression (7), one obtains a two-dimensional area-preserving map:


where the subscript n numbers the perihelion passages. This map can be applied to both the direct and retrograde comet motions.

Unlike in Liu & Sun (1994), where the function [FORMULA] in (8) is expanded into Fourier series in g, we calculate the integral (7) by numerical quadrature. This method has two advantages: 1) it avoids the problem of series truncation, as the convergence of the series is very slow when q is close to one. 2) it can be generalized to [FORMULA], which is important in the transfer of comets. While the Fourier expansion diverges in these situations, the numerical integration of (7) is still valid.

In practice, we compute the integral [FORMULA] in (7) as [FORMULA]. The discarded part [FORMULA] 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 [FORMULA], we calculate the values of [FORMULA] on [FORMULA] values of g uniformly spaced in the interval [FORMULA], and obtain [FORMULA] for other values of g by linear interpolation. Fig. 1 shows the graphs of [FORMULA] as the function of g for two different q. As one can see, [FORMULA] is continuous for [FORMULA]; while for [FORMULA], there are two values of g where [FORMULA] is discontinuous due to the comet-Jupiter collisions. The widths of these two interpolation intervals with the two collision values of g are [FORMULA], which corresponds to the length of the same order of magnitude as the equator radius of Jupiter. If a comet falls in one of these collision intervals, we assume that it collides with Jupiter and do not continue its evolution.

[FIGURE] Fig. 1. The function [FORMULA] for [FORMULA] and [FORMULA]. Each curve is plotted at 2000 discrete values of g uniformly distributed in [FORMULA].

Fig. 1 can be compared to Malyshkin & Tremaine (1999), where the function [FORMULA] 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 [FORMULA] passages, which takes 30 minutes to integrate the original system (1) up to [FORMULA] passages. Thus the use of map (8) is in this case some 3000 times faster than the integration of the differential equation system (1).

[FIGURE] Fig. 2. Evolution of the energy of a comet by the integration of the original system (1) (dash lines) and the use of the map (8) (solid lines) with the same initial values: [FORMULA] for the regular orbit (lower window) and [FORMULA] for the escape orbit (upper window).

The statistical validity of the map (8) is also verified. We take [FORMULA] comets with the same initial energy and calculate their orbits up to [FORMULA] years. Then determine the transfer probability (the average ratio of comets whose final energy [FORMULA] to [FORMULA]) 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 [FORMULA]. 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.

[FIGURE] Fig. 3. Variations of the transfer probability (lower window) and of the average transfer time (upper window) with the initial energy, obtained by integration of the original system (1) (triangles) and by the use of the map (8) (solid lines), respectively. We set [FORMULA].

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 [FORMULA]. 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.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001