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

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3. Numerical results

Fig. 4 shows a phase diagram of map (8), which has both regular and chaotic orbits. The islands correspond to the locations where a comet is in the exterior mean motion resonance with Jupiter. For example, the first resonance visible at [FORMULA] corresponds to a [FORMULA] resonance. The sticking effect of these resonances were studied by Malyshkin & Tremaine (1999). The chaoticity of orbits in the phase space with [FORMULA] comes from the fact that, according to the map (8), for the small [FORMULA], the phase angles g of two consecutive perihelion passages are almost independent.

[FIGURE] Fig. 4. Phase diagram of the map (8) with [FORMULA] and [FORMULA].

As we intend to study the possibility of orbital transfer of comets with near zero energy, we concentrate on cases when a comet obtains enough (negative) energy from the planet to change its orbit from a LP one to a SP one. The relation between the energy K and the period P in our units is [FORMULA] yr [FORMULA]. The boundary of period between LP and SP comet is 200 year, which corresponds to the energy of about [FORMULA]. Since the Oort Cloud is supposed to be located at [FORMULA] AU, we study comets with the initial energy of -0.001.

In all of the following experiments, we take sets of [FORMULA] orbits each with initial energy [FORMULA] uniformly distributed in [FORMULA], and compute the evolution of each orbit one by one with the map (8), until it either escapes ([FORMULA]) or is transferred to [FORMULA]. Then we make the statistics for these [FORMULA] orbits. Note that for comet orbits with [FORMULA], there is also the possibility of collision with Jupiter, though it is very small ([FORMULA] according to our calculations). Three quantities related the transfer dynamics of the comets are determined: the transfer probability ([FORMULA]), the average transfer time ([FORMULA]) and the average energy change of transferred comets per passage ([FORMULA]). [FORMULA] is defined as the ratio of transferred orbits (with final energy [FORMULA]) to the total number of the initial orbits, i.e. [FORMULA]. For the transferred comets, [FORMULA] is defined as the average number of passages needed for accomplishing the transfer, and [FORMULA] is defined as the root-mean-square average of the energy change per perihelion passage. We study the dependence of these quantities on the perihelion distance of the comet orbit (which is assumed constant during the energy evolution) and the planet mass.

3.1. Dependence on perihelion distance

In this subsection we fix [FORMULA] and study the transfer dynamics with the perihelion distance q varying from 0.2 to 1.6. Fig. 5 shows the variation of [FORMULA] with the perihelion distance q. We note that, for [FORMULA], [FORMULA] has values ranging from 0.09 to 0.13 for comets on direct orbits and it is about 0.07 for comets on retrograde orbits. In both cases the variation of [FORMULA] with q is small. In contrast, for comets with perihelion distances [FORMULA], the variation of [FORMULA] with q is large. [FORMULA] decreases from 0.11 to 0.01 as q increases from 1 to 1.6 for direct motion, and it decreases drastically for retrograde motions ([FORMULA] at [FORMULA]). Thus the transfer of comets with [FORMULA] on retrograde orbits should be negligible. For comets on direct motion with [FORMULA], only those with [FORMULA] can be transferred efficiently.

[FIGURE] Fig. 5. Dependence of the transfer probability [FORMULA] on q with [FORMULA]. The fitting lines have the form of [FORMULA], [FORMULA] for [FORMULA] and [FORMULA], [FORMULA] for [FORMULA], respectively. The unit of q is 5.2AU. Note the different scales between the two windows.

According to the fitting lines in Fig. 5, the variations of [FORMULA] obey exponential laws with the form of


where the subscripts "d" denote for comets in direct orbits and "r" for those on retrograde orbits, respectively. Assuming these exponential laws, we define the integral of transfer probability:


and similarly for [FORMULA] and [FORMULA]. The calculation gives,


Thus the flux of transferred comets from direct orbits is [FORMULA] times as large as that from retrograde orbits, this may be the reason why most of the observed HFCs are on the direct orbits (Fernández 1999). Moreover, the flux of transferred comets from Jupiter-crossing orbits is about [FORMULA] times as large as that from non-crossing orbits. This phenomenon is noticed for the first time, as far as we know.

The variation of the average transfer time [FORMULA] for the transferred orbits with q is shown in Fig. 6. We find that the variation of [FORMULA] is smaller for [FORMULA] than that for [FORMULA]. For [FORMULA], [FORMULA] varies with q very slowly and equals to 10-25 passages for the direct motion and to about 60 passages for the retrograde motion. For [FORMULA], the number of passages needed for transfer from direct motion increase from 10 to 3000 as q increases from 1 to 1.6. For comets in direct motion, the dependence of [FORMULA] on q can be roughly fitted with the exponential laws:


[FIGURE] Fig. 6. Dependence of the averaged transfer time [FORMULA] on q, with [FORMULA]. The fitting lines have the form of [FORMULA] and [FORMULA] for [FORMULA] and [FORMULA], respectively.

To explain the above exponential laws, we calculate the average energy change per passage [FORMULA] for the transferred comets. The dependence of [FORMULA] on q is shown in Fig. 7. We notice that the values of [FORMULA] for [FORMULA] are large, sometimes even [FORMULA]. This is because the encounters for [FORMULA] can be much closer than those with [FORMULA]. From the graph of [FORMULA] (Fig. 1), one can also see that the energy exchange of the comet with [FORMULA] is much larger than those with [FORMULA]. For the direct motions [FORMULA] varies with q exponentially with the form


[FIGURE] Fig. 7. Dependence of the average energy change [FORMULA] on q when [FORMULA]. The fitting lines have the form [FORMULA] and [FORMULA] for [FORMULA] and [FORMULA], respectively.

The exponential dependence of [FORMULA] on q gives an explanation of the exponential forms of Eqs. (9) and (12). In fact, according to Fig. 7, the typical values of [FORMULA] is about [FORMULA] for [FORMULA] in the direct motions, which is much less than the energy decrement required for a comet with near zero initial energy to reach [FORMULA]. Thus the evolution of energy for [FORMULA] obeys the diffusion approximation. According to the diffusion approximation, the transfer probability and average transfer time for a comet with initial energy near zero to reach energy [FORMULA] obey (Fernández & Gallardo 1994)


Thus the exponential laws in Eqs. (9) and (12) for [FORMULA] can be deduced from the Eq. (13). However, for [FORMULA], due to the large average energy change per passage, which is of the same order as [FORMULA], the condition of the diffusion approximation is no more fulfilled.

3.2. Dependence on the planet mass

To see the influence of the planet mass on the transfer of comets, we investigate the evolution of above [FORMULA] orbits with µ varying from [FORMULA] to [FORMULA]. The variations of the transfer probability [FORMULA] with µ are shown in Fig. 8. One can see that the probabilities for [FORMULA] are larger than those for [FORMULA] as long as [FORMULA]. This is the case for our Solar System, where the largest perturbation for comet motions comes from Jupiter ([FORMULA]).

[FIGURE] Fig. 8. Dependence of transfer probability [FORMULA] on µ with different q. The straight line in the up-left corner has a slope [FORMULA]. [FORMULA] stands for the perihelion distance of comets in the direct motion, [FORMULA] stands for that in the retrograde motion.

As µ increases, we find approximately that [FORMULA] in the cases of (a) [FORMULA] for [FORMULA] and (b) [FORMULA] for [FORMULA]. However, in the case of [FORMULA] with [FORMULA], this relationship does not hold (Fig. 8). This is caused by the differences in the average energy change [FORMULA]. To demonstrate this we show the dependence of [FORMULA] on µ (Fig. 9), where we find [FORMULA] in all the studied cases. This is expected since Eq. (7) shows that the change of energy is proportional to µ.

[FIGURE] Fig. 9. Dependence of the average energy change [FORMULA] on µ with different q. The straight line in the up-left corner has a slope [FORMULA]. The dotted line shows the critical value [FORMULA] below which the diffusion approximation holds.

With the relation of [FORMULA], the variation of [FORMULA] on µ can be explained. For the situations (a) and (b) listed above, the average energy changes are small so that the diffusion approximation holds, with [FORMULA], which obeys the relation (14). For the other situations, the diffusion approximation is not fulfilled due to the large energy change, thus the relation (14) does not hold since [FORMULA] does not hold.

Finally we present the dependence of the average transfer time [FORMULA] on µ (Fig. 10). For the situations (a) and (b) listed above but with [FORMULA] for [FORMULA], one can identify roughly a power-law with the form of [FORMULA], which is consistent with the relation (14). In the other cases, when conditions of the diffusion approximation is not fulfilled, the relation [FORMULA] is no more obeyed.

[FIGURE] Fig. 10. Dependence of the averaged transfer time [FORMULA] on µ with different q. The straight line in the down-left corner has the slop [FORMULA].

In the above qualitative discussion of this subsection, when the mass of planet µ is changed, the energy boundary between LP comets and HFCs is fixed as [FORMULA]. However, when the only perturbing planet is changed (e.g. Saturn, Uranus or Neptune), [FORMULA] should be also changed due to the different orbital periods of the planets. We calculate the transfer probabilities in the cases of different planet pertubations as in the last subsection and Fig. 5. The results are showed in Table 1. As we can see, only Saturn contribute by a non-negligible amount to the transferred comets while the contributions of Uranus and Neptune are negligible.


Table 1. Integral transfer probability for different planet perturbations

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Online publication: January 29, 2001