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Astron. Astrophys. 320, 672-680 (1997)

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5. Chaotic diffusion in the four-body model

Fig. 12a-d illustrates a disappearance of the continuous frequency map in the full, four-body planar model. The top figures are, once again, the frequency [FORMULA] versus semi-major axis and the bottom figures are the frequency changes over [FORMULA] yr. Initial eccentricities are equal to 0.3. The planar three-body model with fixed Jupiter's orbit ([FORMULA]) is on the left and the planar four-body model with Saturn is on the right. The frequency map is smooth in the three-body model (a). Only slight discontinuities appear at some places due to the high-order secondary resonances and the corresponding frequency changes suggest a presence of chaos near their separatrices (b). On the other hand, the frequency map is irregular in the four-body model (c) and a significant diffusion exists in the whole investigated interval (d). Other similar experiments were done in different regions on the plane [FORMULA] and [FORMULA]. All of them showed the significant chaotic diffusion in the four-body model.

[FIGURE] Fig. 12. The continuous frequency map vanishes in the four-body model at [FORMULA]

We have studied several intermediate three-body models including only some basic frequencies in Jupiter's orbit. The chaotic diffusion was already present in a wide area of the phase space, when the two main secular frequencies [FORMULA] and [FORMULA] were incorporated in Jupiter's eccentricity and longitude of perihelion, but it was roughly by one order slower than in the four-body model. The diffusion was significantly accelerated, roughly to the same rate as in the four-body model, when we added the short-periodic terms as well. These short-periodic terms, which are present due to the 2/1 and 5/2 quasi-resonances between Jupiter's and Saturn's mean longitudes, seem to play a crucial role in the diffusion acceleration (Ferraz-Mello 1996).

In order to investigate the possible diffusive effect on longer time intervals, we have made numerical integrations with various initial conditions over 10 Myr. Each integration was started with [FORMULA] and [FORMULA]. The initial positions of Jupiter and Saturn were projected into the reference plane and turned so that initially [FORMULA]. The frequencies were determined using [FORMULA] Myr and shifting the interval with a relatively small offset of [FORMULA] yr along the integration.

In Fig. 13 we show an interesting case of a trajectory initially situated near the libration centers: [FORMULA] (3.27789 AU) and [FORMULA]. An amplitude of the libration suddenly increased (in this case at 3 Myr), what was observed in several other examples in the neighbouring region.

[FIGURE] Fig. 13. The trajectory with an increasing amplitude of libration

In Fig. 14 we sum up the results of 16 integrated examples over 10 Myr and plot their diffusion trajectories in the frequency space of [FORMULA] and [FORMULA]. The thin lines in the figure are the secondary resonances with [FORMULA] from [FORMULA] to [FORMULA]. The strong low-eccentricity chaos is located roughly under the 6/1 line (Fig. 2 and Fig. 6). The numbers correspond to each particular initial condition. The diffusion rate ranges from almost zero for the example 8 to values leading to a notable transfer in the frequency space as it can be seen, for instance, in the examples 3 and 10. In the case 3 ([FORMULA] ([FORMULA] AU)), an asteroid crosses the secondary resonances 9/1, 8/1 and 7/1, and almost reaches the low-eccentricity chaotic region. Its eccentricity decreases from initial 0.2 to 0.15. The case 10 is initially placed at [FORMULA] and [FORMULA] ([FORMULA] AU). The [FORMULA] variations are accompanied by changes of the amplitude of the semi-major axis oscillations, which decreases to roughly one half of the initial value and the trajectory gets near the libration centers. The moderate diffusion observed in the cases seen approximately along the line connecting the cases 1 and 11 in the figure corresponds to initial conditions far from the librations centers at [FORMULA] ([FORMULA] AU) and [FORMULA] ([FORMULA] AU).

[FIGURE] Fig. 14. Chaotic diffusion in the frequency space

Since each integration over 10 Myr needs several computer hours at a workstation, we have not yet been able to extend the set of initial conditions to a more representative sample. But, at least as a qualitative estimate of the diffusion effect over 1 Gyr, one may suppose, as known for diffusive processes, that the mean displacement depends on the square root of time. Thus, if the phase-space were roughly homogeneous, the extent of each trajectory in Fig. 14 would increase by one order. However, longer numerical integrations showed a complexity of the phase-space and an existence of the slow-diffusion barriers, which often confine a trajectory with a fast diffusion to a small bounded area for a long time.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998