S/2001 J10 is the only jovian satellite
currently known to be in the Kozai resonance. Its state, however,
is a peculiar one, since, according to its value of mean inclination (145.5
degrees) (and \Theta (=0.66273), this satellites should have been in a
circulating orbit according to the secular theory. Apart from
the interesting dynamical problem, this poses problems in setting up the
initial conditions for our simulations. For S/2000 S5 the values
of eccentricities were determined from the inclination, by using the average
value of \Theta of the satellite. In this case, these values are
out of the range of the values of eccentricities actually covered by the
real satellite. Moreover, since e=sqrt(1-\Theta/cosI^2), for
i
< 144.496o the eccentricity is not even defined.
To overcome this problem, we first
perfomed a 1000 yr integration of the satellite, and, as done for S/2000S5,
found the values of e, i, \Omega, \omega, and M for which the satellite
has maximum eccentricity. These values were:
e
i
\Omega
\omega
M
0.267563194036
147.659674005971 38.154140213176
91.358438337935 188.349378514542
Then, for each bin of 0.1o
in
inclination, we computed the averaged value of the eccentricity.
We then fitted a nth-order polinomial to the data, and found that the second
order approximation is the best in term of the compromise between accuracy
of the fitting and ability to extrapolate the function to values of inclinations
out of the original range of values for S/2001 J10. Since our
simulations show that the limit between libration and circulation is at
~148.3o, and the range covered by S/2001 J10 is between 143
and 147.6o, that explains the need to have a function to extrapolate
the relationship between e and i for such values of i.
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For our low-resolution survey we
choose particles with the average value of a of S/2001 J10 (0.12885 AU),
values of inclination between 142 and 151o, the corresponding
values of eccentricity from the second order fit, and the values of the
angles above reported. The next two pictures show the set of
initial conditions in the inc-omega space and in the x-omega space.
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A similar procedure has been applied for the region of the phase space closer to J10 itself. Only, in this case, we fitted only a region of inclination as close as possible to the satellite itself (147.65 to 147.45o), so as to minimize the extrapolation errors. We still used a second order polynomial in inclinations. There are 25 steps in omega of 2.5o, starting from 61.36o, and 16 steps of 0.02o in inclination, starting from 147.51o. The eccentricity was derived from this expression of the inclination:
e(j)=1e+4*(0.00027803759708*inc(j)**2-0.08203372187051*inc(j)+6.05093963342187)
In the following figures we report the initial conditions
for the region around J10.
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Finally, we need to consider the region of the lower boundary
of the libration region. We essentially followed a procedure similar
to the one used for S/2000S5. Here we just report the results
and initial conditions for the simulations. The initial value for
the inclinatio was of 148o, with a step of 0.02o. The distribution
in omega is essentially equal to the case of the J10 simulation.
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