## 2. The techniqueThe equations of motion of the restricted four-body model (Sun - Jupiter - Saturn - asteroid) are where We have performed an extensive number of numerical integrations of Eq. (2) (system Sun-Jupiter-Saturn has been propagated by a parallel integration of the three-body model) using the symmetric multistep method of Quinlan & Tremaine (1990). Although slower than the step-variable integrator RA-15 of Everhart (1985), especially at higher eccentricities, this method is of better precision due to the reduced error propagation. The integrator has the form where is the Cartesian coordinate at step
We have tested a precision of the integrator in the two-body Keplerian problem. A comparison with the Störmer method (Cohen et al. 1973) has shown better stability properties of the symmetric method in higher eccentricities. For example, a step of 10 days leads to the reasonable relative errors in semi-major axis and in mean longitude after 1 Myr for eccentricity and orbit at the 2/1 resonance (a stepsize-period ratio of ) with SMU12 while the Störmer method is unstable there. The Runge-Kutta method of the 4th-order with small stepsize has been used to start SMU12. This is as accurate as the starting iterator of Cohen et al. (1973). An effective procedure of a memory management requires a digital low-pass filter. We used two digital filters called A and B which are described in Quinn et al. (1991). The filter A has ripple , suppression , limits of passband and stopband 0.005 and 0.05, respectively. The filter B has, similarly, ripple , suppression , passband 0.05 and stopband 0.15. See Press et al. (1992) for a definition of these characteristic parameters. Data where are coefficients of the filter with length . The discrete Fourier transform of this relation gives where , and denote transforms of , and . Fig. 1 shows the Fourier transforms of both filters.
As a result we get the low-band limited signal , which allows us to increase the sampling. The typical value was a decimation by the factor 6, from the initial spacing of 2 yr to 12 yr. It means, when working with the filter A, that the frequencies smaller than 40 yr were removed and the frequencies larger than 400 yr were retained. For the filter B, which had to be applied sequentially two times, these values were 40 yr and 120 yr. The basic frequencies, which appear in the spectrum of asteroid's osculating elements, are the libration frequency of the critical argument (characteristic period of several hundreds of years) and the circulation frequency of the longitude of perihelion (thousands to tens of thousand years). See Michtchenko & Ferraz-Mello (1995) for an overview of this subject. The frequencies were determined by Laskar's technique (Laskar et al. 1992). One defines a scalar product of two functions and on the interval by where is the Hanning window. If is singly periodic, , where is a complex amplitude and is a frequency, then a modulus of the function defined by has a maximum at and .
Thus, having the values of In a general case, where This representation is not exact and an error, which originates from overlapping of different terms in , is estimated in the following way. We sample the function and reconstruct its representation . The differences between frequencies and amplitudes can be considered to be the overlap errors of the representation (Laskar et al. 1992). Our extension of Laskar's error estimation is the following. A final error is calculated as composed from the overlap error and an error 'due to residuals' left in the given peak. As an error of the amplitude, we take , where is a maximum of the spectra between and after the subtraction of all determined terms. In order to estimate the frequency error due to the residuals, we imagine a sum of the function and some unknown function with the maximum value . The can be approximated by near , where , and the condition gives the maximum frequency distance where a false absolute maximum can appear: This expression can be seen as an upper estimate of the error due to residuals. Thus, the final frequency error is computed as a maximum of and Eq. (9). Now, for a chaotic trajectory, where a change of frequency
determined for different time periods is expected, the error
estimation gives us the following criterion for adjusting of The error analysis is important since it is necessary to reduce the timespan of integrations as much as possible in order to be able to extend the frequency calculations to a large number of initial conditions. The frequency is calculated as the strongest
peak in the spectrum of ( © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |