In the following plots we report contour plots of the
period of precession in \omega, of sigma for the integration with Jupiter
and Saturn and the normal version of SWIFT_WHM, and the orbital elements
of saturn for this integration.
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In this table we report the average value and the standard
deviation of the difference in \sigma between the integration with Jupiter
and Saturn, and those that used solutions from the Bretagnon model.
For SEC we intend a solution that only has secular terms in the eccentricity,
\omega, inclination, \Omega of Saturn. a is assumed constant.
In A21_SEC we take 21 terms for the variation in a of Saturn (only
terms due to either Jupiter or Saturn), plus the secular terms, while in
GI_SEC we took 8 terms for e and 6 for i associated with
the great-inequality, but a constant. Finally, in A_GI_SEC
we combined everything together. The following plots report
the results of our simulations for \sigma and the orbital
elements according to the Bretagnon model. Results in the table are
multiplied by 10^-4. In the third row we report the results
of a \Chi-square test between the values of \sigma of the integration with
Jupiter and Saturn, and the various Bretagnon solutions. To compute
\Chi-square we used the following method: we subtracted the values
of \sigma before the filtering from those after the filtering, and compute
the relative standard deviation. Then, if we call \sigma(1)_ij the
filtered values of \sigma for the traditional simulation and \sigma(2)_ij
the same for the Bretagnon one, we have:
\Chi-square=Sum(\sigma(1)_ij-\sigma(2)_ij)**2/(std1**2+std2**2)
The lower the value of \Chi-square, the better the fit.
The fourth Bretagnon simulation shows the best \Chi-square value.
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SEC
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A21-SEC
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GI-SEC
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A21-GI-SEC
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