## 3. Numerical resultsFig. 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
corresponds to a
resonance. The sticking effect of
these resonances were studied by Malyshkin & Tremaine (1999). The
chaoticity of orbits in the phase space with
comes from the fact that, according
to the map (8), for the small , the
phase angles
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 In all of the following experiments, we take sets of orbits each with initial energy uniformly distributed in , and compute the evolution of each orbit one by one with the map (8), until it either escapes () or is transferred to . Then we make the statistics for these orbits. Note that for comet orbits with , there is also the possibility of collision with Jupiter, though it is very small ( according to our calculations). Three quantities related the transfer dynamics of the comets are determined: the transfer probability (), the average transfer time () and the average energy change of transferred comets per passage (). is defined as the ratio of transferred orbits (with final energy ) to the total number of the initial orbits, i.e. . For the transferred comets, is defined as the average number of passages needed for accomplishing the transfer, and 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 distanceIn this subsection we fix and
study the transfer dynamics with the perihelion distance
According to the fitting lines in Fig. 5, the variations of 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 and . The calculation gives, Thus the flux of transferred comets from direct orbits is 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 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
for the transferred orbits with
To explain the above exponential laws, we calculate the average
energy change per passage for the
transferred comets. The dependence of
on
The exponential dependence of on
Thus the exponential laws in Eqs. (9) and (12) for can be deduced from the Eq. (13). However, for , due to the large average energy change per passage, which is of the same order as , the condition of the diffusion approximation is no more fulfilled. ## 3.2. Dependence on the planet massTo see the influence of the planet mass on the transfer of comets,
we investigate the evolution of above
orbits with
As
With the relation of , the
variation of on Finally we present the dependence of the average transfer time
on
In the above qualitative discussion of this subsection, when the
mass of planet
© European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |