2.3 Fourier analysis

The procedure to extract proper eccentricity and inclination, and the corresponding proper frequencies, is a modified form of Fourier analysis, which is adapted from the synthetic theory method introduced in Carpino et al. ([1987]), and further improved by Milani ([1993,1994]). The procedure includes three steps. First, the forced secular perturbations, which have known frequencies, are removed from the filtered time series for the equinoctal elements

$$\begin{eqnarray*}&k=e \cos\varpi\\ &h=e\sin\varpi\\ &q=\tan (I/2)\cos\Omega\\ &p= \tan(I/2)\sin\Omega\ . \end{eqnarray*}$$

Given the fundamental frequencies g₅,g₆,g₇ of the precession of perihelia of Jupiter, Saturn and Uranus respectively, we remove the Fourier components with these frequencies from (k,h). Given the fundamental frequencies s₆,s₇,s₈ of the precession of the nodes of Saturn, Uranus and Neptune respectively, we remove the Fourier components with these frequencies from (q,p). Second, we compute the time series of the free arguments $\varpi_f,\Omega_f$ of the oscillation in the planes (k,h) and (q,p) respectively. This is done by computing the polar angles in these two planes, by adding multiples of $2\pi$ to obtain a continous function. Finally by a linear least squares fit we obtain an estimate of the proper frequencies g, the slope of $\varpi_f$, and s, the slope of $\Omega_f$. Third, we perform Fourier extraction of the proper mode. This can be done in two ways, as discussed in Milani ([1994]). Either the Fourier component with period $2\pi/g$ is extracted from the time series (k(t),h(t)); or the component with period $2\pi$ is extracted from the data expressed as functions of $\varpi_f$, that is $(k(\varpi_f),h(\varpi_f))$. The results would be identical if the proper oscillation was a linear one, but for non negligible eccentricity the higher order terms are important (Milani and Knezevic [1990]) and the latter algorithm leads to more stable proper elements; thus we have used it. The same applies to the inclination related plane, and we have used the extraction of the component with period $2\pi$ from $(q(\Omega_f),p(\Omega_f))$. The amplitudes of these proper modes are the proper elements e_p and $\tan(I_p/2)$; the latter is then converted to the more usual $\sin I_p$. One of the main advantages of the synthetic method is the availability of a stability test for each single set of proper elements computed in this way. In our previous work, with analytical proper elements, we needed specific numerical tests to check the stability of the results in a few supposedly representative examples; e.g., in the paper Milani and Knezevic ([1994]) we tested only 35 asteroids. In the computation of synthetic proper elements, a numerical integration over a long enough time interval [0,T] has to be performed. Given the output of the same integration, the same algorithm described in this Section can be applied to shorter time intervals $[t_j,t_j+\Delta T]$, beginning at initial times t_j, j=1,..,N, in such a way that t₁=0, $t_N=T-\Delta T$. For each of these ``running boxes'' a value of the proper elements $e_p, \sin I_p$ and of the proper frequencies is obtained, and the dispersion of these N values can be used to estimate the stability in time of the results. As an example, for the 2 Myr integrations we have used N=11 running boxes of length $\Delta T\simeq 1$ Myr; for the 10 Myr integrations, N=9 and $\Delta T\simeq 2$ Myr. The dispersion of the values of some proper element E in the different boxes can be measured by the root mean square $\sigma E$ (of the differences of the values computed in each box with respect to the value computed over the entire time span) and by the maximum difference $\Delta E$ among the values computed in all the boxes.