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Astron. Astrophys. 320, 672-680 (1997)

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2. The technique

The equations of motion of the restricted four-body model (Sun - Jupiter - Saturn - asteroid) are


where G is the gravitational constant, [FORMULA] is the mass of the Sun, [FORMULA] and [FORMULA] are the masses of Jupiter and Saturn, [FORMULA], [FORMULA] and [FORMULA] are the heliocentric position vectors of the asteroid, Jupiter and Saturn.

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 [FORMULA] is the Cartesian coordinate at step n, [FORMULA] is a corresponding component of the acceleration computed by Eq. (2), h is a fixed stepsize and [FORMULA], [FORMULA] are coefficients of the method. We used the symmetric method with [FORMULA] (SMU12), which has [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. The coefficients were chosen so that a 13th-order polynomial is integrated exactly.

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 [FORMULA] in semi-major axis and [FORMULA] in mean longitude after 1 Myr for eccentricity [FORMULA] and orbit at the 2/1 resonance (a stepsize-period ratio of [FORMULA]) 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 [FORMULA], suppression [FORMULA], limits of passband and stopband 0.005 and 0.05, respectively. The filter B has, similarly, ripple [FORMULA], suppression [FORMULA], passband 0.05 and stopband 0.15. See Press et al. (1992) for a definition of these characteristic parameters.

Data w with a certain sampling on the filter input are replaced by [FORMULA] on the output following the convolution relation,


where [FORMULA] are coefficients of the filter with length [FORMULA]. The discrete Fourier transform of this relation gives


where [FORMULA], [FORMULA] and [FORMULA] denote transforms of [FORMULA], [FORMULA] and [FORMULA]. Fig. 1 shows the Fourier transforms [FORMULA] of both filters.

[FIGURE] Fig. 1. Fourier transforms of the filters A and B described in Quinn et al. (1991), the thin line is [FORMULA] of B where the ripple as well as the limit of passband are visible (sampling frequency [FORMULA] is defined by [FORMULA], where [FORMULA] is the sampling interval)

As a result we get the low-band limited signal [FORMULA], 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 [FORMULA] of the critical argument [FORMULA] (characteristic period of several hundreds of years) and the circulation frequency [FORMULA] of the longitude of perihelion [FORMULA] (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 [FORMULA] and [FORMULA] on the interval [FORMULA] by


where [FORMULA] is the Hanning window. If [FORMULA] is singly periodic, [FORMULA], where [FORMULA] is a complex amplitude and [FORMULA] is a frequency, then a modulus of the function [FORMULA] defined by


has a maximum at [FORMULA] and [FORMULA]. Thus, having the values of f regularly spaced over the [FORMULA] -interval as an output of numerical simulation, we numerically compute the maximum of [FORMULA] and obtain the frequency [FORMULA].

In a general case, where f includes an infinite number of periodic terms, we perform Laskar's iterative process, which is stopped when a desired number of frequencies is obtained or if, at a certain step, the new frequency falls closer than [FORMULA] to any already determined one. As a result, we get n frequencies separated by more than [FORMULA]. Amplitudes [FORMULA] obtained by the final projection complete the reconstruction [FORMULA] of the function f: [FORMULA].

This representation is not exact and an error, which originates from overlapping of different terms in [FORMULA], is estimated in the following way. We sample the function [FORMULA] and reconstruct its representation [FORMULA]. The differences between frequencies [FORMULA] and amplitudes [FORMULA] can be considered to be the overlap errors of the representation [FORMULA] (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 [FORMULA], where [FORMULA] is a maximum of the spectra between [FORMULA] and [FORMULA] 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 [FORMULA]. The [FORMULA] can be approximated by [FORMULA] near [FORMULA], where [FORMULA], and the condition [FORMULA] 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 [FORMULA] 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 T and an offset (the time interval separating two frequency determinations): If, for certain T, the estimated error is larger than or comparable with the observed frequency change over the given offset, the time T must be enlarged and/or longer offset should be used, and only if the error is apparently smaller than the frequency change, the chaoticity of the trajectory is affirmed. The frequencies and amplitudes should be fixed in the case of a regular trajectory and sufficiently long T.

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 [FORMULA] is calculated as the strongest peak in the spectrum of [FORMULA] (e is eccentricity). The frequency [FORMULA] is evaluated in a more complicated way since it is not easy to define any phase-space variable in which the spectral peak [FORMULA] always dominates (Michtchenko & Ferraz-Mello 1996). We determine several peaks in spectra and choose the one which is the nearest to the libration frequency of [FORMULA] in the three-body circular model.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998