2.4 Semimajor axis and Lyapunov exponents

The other proper element is proper semimajor axis a_p, which in principle is obtained by removing the short periodic perturbations from the time series of the osculating a. A rigorous definition would be somewhat more complicated than just the average of a(t), because of second order perturbations which contain both constant and long periodic terms (Milani et al. [1987]). However, the second order perturbations are very small and in practice they are seldom the main source of time variation of the proper semimajor axis: chaotic effects are dominant in most cases. For this reason we have not attempted to perform second order corrections, and we have computed a_p as the mean of the values of a(t) as it came out of the digital filter; that is, we are taking a mean of the already smoothed data. Stability tests are done again by the running box method. There is a third proper frequency, f_p, which is the slope of the angle $\lambda=\ell + \varpi$. Note that in our numerical integration code the output for $\lambda$ includes the principal value in $[0, 2\pi]$ and the number of revolutions completed since the initial conditions, thus the value of f_p can be obtained by a simple linear fit to the filtered $\lambda$; the filter also preserves the number of revolutions. Stability of f_p is tested by the running box method. In principle we can define a ``mean mean motion'' n_p=f_p-g as the proper frequency of the mean anomaly $\ell$; however, the frequency n_p is not related to a_p by Kepler's third law. The mean anomaly contains not only perturbations acting through changes in a, but also because of the accumulated effect of the perturbations in a (this is the classical double integration argument). In practice large deviations of $\lambda$ from $f_p*t + \lambda_0$, let us say several revolutions, are a strong indication of chaotic motion (Milani et al. [1997]), in particular of jumps between different chaotic states corresponding to different values of a_p. For this reason we also use, as indicator of irregular motion, the root mean square and the maximum difference of the residuals of filtered $\lambda$ with respect to the linear fit. A well known indicator of chaotic motion is the maximum Lyapunov Characteristic Exponent (LCE). Although the LCE is rigorously defined as a limit for $t\to \infty$, an indicator can be obtained by a finite integration of both the equation of motion and the corresponding variational equation. Following Milani and Nobili ([1992], we compute an approximation of the LCE as the best fit slope of the function of time $\gamma(t)= \log [D(t)/D(0)]$, where D(t) is the length of a variation vector (with initial conditions selected at random). Renormalisation of the variations vector needs to be applied when it becomes too large, to avoid numerical instability and overflow. This method typically allows to detect a positive LCE after 7-8 Lyapunov times T_L (T_L is the inverse of the LCE). The approximate values of the LCE computed in this way are reliable as order of magnitude, unless a real state transition between two chaotic regions takes place; the exact values are not very significant, changes by $20\%$ when the integration time span is extended are typical.