Secular perturbations and statistics of asteroid collisions

A significant part of what we know about the past evolution of the small bodies in our Solar System relies on the solution of the classical problem of modeling the statistics of the mutual impacts.Regardless of their are Main Belt Asteroids, Edgeworth-Kuiper belt objects or bodies forming debris disks around other stars in different phases of evolution, the determination of the frequencies of the mutual collisions and the distribution of the impact speed is a common problem and those parameters are fundamental for the modeling of their evolution and interpretation of the observational data. In the present contribution we aims to discuss some theoretical issues and advancement related to this problem.

The parameters of impact (frequency, relative velocity, impact geometry, and so on) are all fundamental quantities underpinning many models of global evolution of small bodies populations. Such models aim to reproduce the evolution in time of the size distribution of the debris, the production of collisional families (like the dynamical asteroid families in the Main Belt, but not only) and the production of dust. The latter feature is related to important observables, and in some cases the only observable in terms of infrared excess revealing the presence of collisionally active debris disks around other stars. For those reasons, statistics of the parameters of impact require a case-by-case careful model calibration.

A consolidated approach in Solar System studies is to try to compute the impact probabilities starting from the orbital elements of the bodies under investigation. Very few analytic or semi-analytic theories to solve the problem have been developed, including the  most general one developed so far (Dell'Oro 2017). In any case all such methods require as input a list or a distribution of orbits, from which the probabilities of collision are derived. Beyond the validity of the dynamical hypotheses underlying the different methods, the choice of orbits to use proves to be a critical point. In the case of debris disks around other stars, the forced choice is to assume simplified hypotheses about the distribution of the semimajor axes, taking into account the presence of perturbing planets.  In the case of our Solar System, the abundance of data about the orbits of small bodies can be misleading. First of all, orbits catalogs are affected by different type of observational bias.  A crude use of the available data entails implicit assumptions about the orbital distributions that can be wrong (especially with respect to the distribution of physical dimensions).

Another problem concerns the type of orbits to be used.  Minor Planet Center archive provides osculating elements the validity of which is limited to the present epoch.  AstDyS archive includes also proper elements, quasi-constant of motion describing the long-term essential dynamical features of the asteroids (aside from the influence of the non-gravitational forces), fundamental parameters for the identification of the asteroid families. On the other hand, the use of proper semimajor axes, proper eccentricities and proper inclinations alone do not enlighten all the features of the asteroid motion, affecting the computation of the statistical parameters of impact.  The question if it is preferable to use the osculating elements, as a representative snapshot of the dynamical status of the system, rather than proper elements as "mean" values of the previous one, has not been investigated yet.

Moreover, because of the influence of the perturbing planets and depending on the values of the forcing elements, the presence of secular perturbations does not always allow us to ignore subtle correlations among the variations of the orbital elements. The most classical approach consists in approximating the orbital evolution assuming semimajor axes, eccentricities and inclination fixed, while node longitudes and arguments of perihelion are set uniformly distributed (Wetherill 1967, Bottke 1994). A first attempt to overcome these limitations showed the importance to take into account the non-uniform orientation of the orbits (Dell'Oro & Paolicchi 1998), but that model was not yet able to take into account the correlation between eccentricity and longitude of perihelion, or between inclination and longitude of the nodes caused by the secular perturbations. Only recently a new mathematical approach has been developed in order to address this problem (Dell'Oro 2017).

In this presentation will we discuss the impact of the observational bias on the computation of the velocities and probability of collision among asteroids, the  different output obtained using osculating or proper elements, and we show some examples of the effects of the secular perturbations in our Solar System, also in comparison to what happen in more dynamically "hot" environments.

References

• Bottke W.F. et al. 1994, Icarus, 107, 255.
• Dell’Oro A. 2017. Monthly Notices of the Royal Astronomical Society, 467, 4817.
• Dell'Oro A., Paolicchi P. 1998. Icarus, 136, 328.
• Wetherill G.W. 1967. Journal of Geophysical Research, 72, 2429.