Since we are interested in the long-term behavior of the frequency associated with the \omega precession of S/2000S5 (676 yrs) we need to find a way to eliminate shorter period frequencies in our integrations.  This can be achieved by performing integrations with on-line low pass filters.  A detailed description of these filters can be found in Carpino et al. 1991.   Here we just report a description of a few parameters useful for our integrations.  There are two filters, A and B.   Both filters have a stop band, in which frequencies are almost completely stopped, and a pass band in which frequencies are altered.   There are five parameters of interest for our integrations.

    r is the fractional distorsion of the signal within the passband.

    s is the fraction of original signal allowed to pass in the stop band.

    x_pdetermine the limit of the pass band.  In particular, given data with a sampling time of 1 year, and a value of x_p of 0.005, the passband would be starting at:

           P_p=(0.005/1yr)^-1 = 200 yrs => 6480 "/yr

    x_sdetermine the stop band.

    K is the decimation factor, which tells what data should be passed to succesive filtering processes.   For instance, if K =10, then only one every ten data points will be kept.

As mentioned, there are two filters.   These are the parameters for the two filters:

A:    r_A=3.5*10^-5;    s_A=10^-5;    x_p,A=0.005;    x_S,A=0.05

B:     r_B<10^-9;            s_B<10^-9;     x_p,B=0.05;       x_S,B=0.15

In our case we are dealing with satellites with periods P of ~2.6 yrs.  This is the fastest period, and,  to avoid aliasing this frequency into higher period frequencies, we should choose a sampling time at least <1/2 P.   Let's use a sampling time of 1.333 years and apply filter A to the output.   The passband for this filter would be of:

          P_p=(0.005/1.333)^-1 = 266.66 yrs => 4861.2 "/yr

And the stop band would be of:

           P_s=(0.05/1.333)^-1 = 26.66 yrs => 48612.2 "/yr

This means that a signal with a period shorter than 26.66 yrs would be stopped.   We do not therefore need to sample the signal every 1.333 yrs, the period associated with the Nyquist frequency for 26.66 yrs would be appropiate.  Therefore:

           P_N=26.66/2=13.33 yrs

This implies a decimation factor of:

           K_A=P_N/\Delta t=13.33/1.333=10

In our case, for S/2000S5 the period of \omega precession is 676 yrs, and reaches a minimum of ~400 yrs for a particle in the librating point.   A Nyquist frequency of 26.66 yrs is still too low, something closer to 300 yrs would be more appropiate.    Further filtering can be used.    If we apply filter B, which has a stronger suppression band than filter A, than the pass band would be:

          P_p=(0.05/13.33)^-1 = 266.66 yrs => 4861.2 "/yr

And the stop band would be:

          P_s=(0.15/13.33)^-1 = 88.88 yrs => 14580 "/yr

With a stop band of 88.88 yrs the Nyquist period would be of 44.44 yrs.    Since the sampling period after one filtering with A was of K_A*\Delta t =10*1.333=13.33, it then follows that the decimation factor for the B filter would be:

           K_B=44.44/13.33=3.3~3

I.e., we keep one every three points.  With the two filtering, we have a total decimation factor of:
 

           K=K_A*K_B=10*3=30

Finally, the filtered elements are output starting with 0.8619959E+06.
To summarize, these are the parameters needed for filter.in:

2
AB
10 3
80 0.024d0 10.0d0
80 0.10d0 20.0d0
487.d0
bin.filter.dat
T       ! write real*8 bin.dat output of osculating elements? [T|F] (useful for precise restart)

And in the figure there are the results of  two simulations with filtered and unfiltered elements.  The final sampling time in both cases was of 40 yrs, and the simulations integration period was of 4 Myr.   The time step was of 9.86175 days.
 
 

Unfiltered elements	Unfiltered elements
Filtered elements	Filtered elements

As can clearly be seen from the two figures, frequencies with period smaller than 266.66 yrs have been attenuated, and those with period shorter than 88.88 yrs have been eliminated.   Note how this reflects in the plots of \omega:  filtered elements do not show the "bumps" associated with short-period oscillations.