e*exp(i\varpi)
decomposition:
1 Ceres: fake case
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The objectif of this work is to retrieve the first frequency
associated with the librating oscillations in \omega of satellites trapped
in the Kozai resonance. This web-page is intended as a summary of
the work done on this subject. By all means, it is not supposed
to be a rigorous work on the subject. At the moment it is so divided:
These are just some simple facts about Fourier transform.
(1)
f=360*3600/P(yr)
With \Delta T the sampling time. As an example, with a sampling time of 1000 years, the maximum frequency retrievable would be of:
fN=(360*3600)/(2*1000) = 648 "/yr.
Higher frequencies are not completely lost. Given a frequency
at fN+\Delta f, the FFT would retrieve it and alias
it as a lower frequency at fN-\Delta f.
That
is the reason why we want to consider using orbital elements filtered with
a low-pass filter...
In our case:
\Delta f=(360*3600)/(32768*1000)=0.03955"/yr
To check my code I have run three
tests. First, I have generated a fake signal mimicking the
behavior of e*exp(i\varpi) for Ceres (i.e., I generated a file with
time, e*cos(\varpi), e*sin(\varpi)). I used the values of the first
10 frequencies, amplitudes, and phases as in Sidlichovsky and Nesvorny
(1997). These values are reported in the following table:
e*exp(i\varpi)
decomposition:
1 Ceres: fake case
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In the figure there are the results of the analysis of the fake signal with my code. The data sampling was of 120,000 days (328.549011 yrs) and I used 32768 data points. As can be seen, the code does a good job in retrieving all the 10 frequencies, and preserve the ratio among amplitudes.
The next logical step was to check
how the code performed on the output of a real integration. I therefore
integrated 1 Ceres, 2 Pallas, and 4 Vesta under the influence of the 4
giant planets with SWIFT_WHM and checked if I were able to retrieve the
frequencies found by Sidlichovsky and Nesvorny 1997 for a similar integration.
I output both e*exp(\varpi) and sin(I/2)*exp(\Omega), made
the FFT and confronted the results with those found by Sidlichovsky and
Nesvorny 1997 (10 frequencies for 1 Ceres, first frequency for 2
Pallas and 4 Vesta). In the following table I report the first 10
frequencies from the decomposition of sin(I/2)*exp(\Omega) as from
Sidlichovsky and Nesvorny 1997, while the figure shows my results.
The next two figures show my results for the analysis of e*exp(\varpi)
for 1 Ceres, and 4 Vesta. In all the considered cases
there was good agreement with the published values.
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The conclusion is that the code does a good job in retrieving the main frequency. Therefore it can be used to obtain an estimate of chaotic diffusion as in Robutel and Laskar 2001:
(4) \sigma=1-f(2)/f(1)
Where f(1) is the frequency determined in the first interval, and f(2)is the frequency determined in the second interval. But we should be aware that the precision with which the exact value of the amplitude is retrieved depends on the resolution (i.e., the number of data points used). This may be a problem if we want to use Mitchencko and Ferraz-Mello Spectral Analysis Method. But, overall, it looks very promising...
I performed an integration with
the four giant planets and four irregular satellites: S/2000S5, S6, S2,
and S3. The integration length was 100 Myr, and the time step
was of 1000 yrs (which means a Nyquist frequency of 648 "/yr). Once
again, I have used 32768 data points, that gives a resolution of 0.03955"/yr.
The first figure reports the FFT of e*exp(\varpi) for the case of
S/2000S5, the second a blow-up of the region between -100 and 100 "/yr
(where the planetary frequencies are suppose to be found), and the third
figure the output of the fmft code of Nesvorny', used to retrieve the first
20 frequencies in the same interval -100-100 "/yr. There is a very
good agreement between the second and third figure. The orbital
elements used for this computation were output referred to the fixed ecliptical
plane of the Earth.
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The next three figures report analogous results for S/2000S6,
S2, S3.
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Since in this work we are particularly interested in studying
the behavior of the two Kozai resonators, D. Nesvorny' suggested to use
e*exp(\omega)
for
the cases of S/2000S5 and S6, and predicted that their spectra should
be dominated by the frequency associated with the libration in \omega.
The
next two figures reports the FFT of e*exp(\omega) for these two
satellites. Also in this case, the elements are referred to the ecliptical
plane of the Earth.
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The previous figures were obtained using orbital elements referred to the initial ecliptical plane of the Earth. However, quantities like the inclination and \Omega should be computed with respect to the Laplace plane of Saturn, which at that distances from the planet corresponds to the instantenous ecliptical plane of the planet. We generated element referred to to this plane for the four integrated satellites and then applied the FFT to these new elements.
In the following two figure I show
the results of the FFT of e*exp(\varpi) for the case of S/2000S5,
with a blow-up of the region where the planetary frequencies are located.
Results for S/2000S6, S2, and S3 follows. Note the differences with
the previous cases where the element referred to the ecliptical plane of
the Earth were used.
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Finally, we report the spectra of e*exp(\omega)
for
the cases of S/2000S5 and S6, computed using the element referred
to the instanteneous ecliptical plane of Saturn. As expected, they
are identical to those obtained for the case of Earth's ecliptical plane.
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