4. The elliptic three-body model
For every initial condition, we have integrated a trajectory in the three-body planar model () for yr. We have determined the frequency twice - in two intervals yr overlapping by one half, that means an offset of yr. The initial conditions were and (all angles were zero). The initial conditions for which the frequency changed relatively by more than over the given offset are marked by big crosses in Fig. 6. Small crosses correspond to the relative changes larger than and the rest, with almost constant , is left empty. The chaotic zone that appears along the low-order secondary resonances (see Fig. 2) can be compared to Murray (1986) and Wisdom (1987).
No significant chaotic behaviour exists for and even for lower eccentricities on the right side of the dotted rectangle. In the planar three-body model the high-order secondary resonances are present in this region but they do not overlap and do not generate a large-scale chaos.
We zoom now the area around the 8/1 and 9/1 secondary resonances. Figs. 7 and 8 show versus semi-major axis. The determination is done on the basis of yr numerical integrations of each initial condition (). Jupiter's eccentricity equals here. Fig. 7 is a libration-island crossing, showing positions of the separatrices at both borders of the flat, almost horizontal area. We amplified the left separatrix (inset) to see if some narrow chaotic zone appears there, but even this additional zoom did not show any irregularities of the frequency map. A discontinuity of the frequency map at a hyperbolic point crossing was observed at the 9/1 secondary resonance (Fig. 8). The zoom revealed a narrow chaotic layer in its vicinity (MLE equals there).
We placed an asteroid inside this layer with initial conditions and and numerically integrated its trajectory over 10 Myr (Fig. 9). The frequencies and were denoted by f1 and f2 in the figure. The parameters of FMA used here were the same as the ones used for an investigation of the chaotic diffusion over 10 Myr in the four-body model described in the following section. Error bars are shown in the figure of the evolution.
At approximately 6 Myr the trajectory switched from one mode to another. This behaviour resembles the one observed in a chaotic trajectory, which is originally at some relatively stable trajectory with a slowly librating resonant angle and then crosses the separatrix and appears in a region of faster circulation. The period visible in the eccentricity evolution corresponds to the period of the resonant angle of the 9/1 secondary resonance, almost 3 Myr at the beginning and slightly less than 1 Myr at the end.
Fig. 10 illustrates a sensitivity of the three-body planar model to the value of Jupiter's eccentricity (the number in the corner of each top figure) in a region close to the low-order secondary resonances. We estimated the temporal frequency variation with an offset of yr (bottom pictures) and yr. The initial eccentricity of the whole set of the initial conditions is 0.2168.
From left to right we increased Jupiter's eccentricity from 0.048 to 0.061. For the whole region is regular with an exception of a close vicinity of the separatrices of the secondary resonances (as we have seen in Fig. 8). There are some secondary resonances of higher degree between the 8/1 island on the left and the 9/1 hyperbolic point at : the small island at (resonance 26/3) and the hyperbolic-like crossing at (17/2). The secondary resonances become more apparent for and for the area at the right of 8/1 turns chaotic leaving only the small regular island, which is the enlarged 26/3 resonance. For even that one disappears.
But the motion preserves regularity already for or even for lower eccentricities, as , at . Fig. 11a-d documents this fact with . On the left-hand side of this figure, there is a set of initial conditions with , and on the right-hand side, there is another with . The bottom pictures are changes of the frequency over yr. For only the strip at is chaotic ((a) and (b)) and the region is perfectly regular for ((c) and (d)).
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998