The FAM analysis of the high-resolution survey showed
us the presence of five possible secondary resonances.
Here we try to understand the causes of these resonances.
In the first plot we report the periods of our test particles determined
with FAM versus the initial values of x=1-e2. The
black lines report the positions of the secondary resonances identified
by FAM, and the green lines show the periods associated with the commensurability
between periods of precession/libration and the Great-Inequality.
We report the positions of the first four resonances of order 1.
The 2:1 resonance is out of the range of \omega periods.
We are consciuous that the commensurability between Great_inequality and
\omega are not real resonances, since they do not respect d'Alembert rule.
Nevertheless, we think this is a useful plot, since gives a first-order
estimate of the position of the real resonances.
In the second plot we show how the frequency of largest amplitude changes
with x, for each particle. Regions of frequency scattering
or region where the frequency is constant are associated with secondary
resonances. The dotted vertical lines are associated with the transition
region from circulation to libration.
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The first (and strongest) resonance is associated with the pericentric secular resonance. Here we show a plot of the resonant argument for a particle in this resonance.
As can be seen in the first figure. at least other four regions of high \sigma values exist. Resonance 2 seems to be connected to a 1:1 commensurability between \omega and the Great Inequality. This resonance has an argument of:
The first figure below report a plot of the resonant argument
for a particle in this resonance. To be sure that the resonance
is connected to the Great Inequality, we performed two other simulations
in which the posiotion of Jupiter was modified such that the Great-Inequality
period was of 810 and 950 yrs, respectively. Then, we computed
the expected position of the resonance (vertical lines in the figure) and
compared these values with the results of numerical simulations.
The good agreement between expected and measured positions seem to confirm
our interpretation of this secondary resonance.
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Two other resonances are in the circulating regions. According to our results, their argument (plotted in the first two figures below) should be:
4(\varpi-\Omega)+3(2\lambda_J-5\lambda_S)+9\Omega_S
And,
3(\varpi-\Omega+2\lambda_J-5\lambda_S+3\Omega_S)+(\Omega-\Omega_S)
We repeated the analysis done for the 1:1 resonance also
for these two other resonances, and the results of our numerical simulations
seem to confirm our interpretation (see third figure).
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Finally, we investigate the region of librating behavior. In the following figure the vertical dotted line show the region of transition. By 4:3 and 3:2 we mean resonances of argument:
3(\varpi-\Omega)+4(2\lambda_J-5\lambda_S)+12\Omega_S
And,
2(\varpi-\Omega)+3(2\lambda_J-5\lambda_S)+9\Omega_S
Unfortunately, for librating particle plotting the resonant
argument is not an easy task, since \omega oscillates around \pm 90o,
instead of going from 0 to 360o. However, the position
of the two resonances computed with these two resonant arguments seem to
be in good agreement with regions of high frequency dispersion. Also,
when P_GI=810 years, the 1:1 resonance appear in the librating region.,
and this produces a highly chaotic behavior (see first panel of the figure).
