A numerical tool for the impact statistics in debris disks

Bodies forming debris disks collide each other due to the presence of planets. In debris disks around stars without any type of massive companions (other stars or planets), bodies for which non-gravitational forces are negligible would follow  perfectly keplerian orbits, whose shapes and orientation are fixed in space. In such conditions, mutual orbital crossing, and consequently, mutual collisions would not be possible. In presence of one or more perturbing planets secular perturbation theory predicts departures from pure keplerian motion. The osculating elements of the bodies vary in a way depending on the orbital properties of the perturbers.

When the eccentricity of the orbit of the perturbing planet is small (as in the case of Jupiter for the Main Belt Asteroids in our Solar System) the effect of the forced eccentricity of the debris' orbits is to produce a quasi uniform circulation of the orbits (variation of pericenters and node longitudes) and limited variations of the other elements, along with a typical distribution of the instantaneous distance from the star (fig. 1). Under those conditions, classical methods for the investigation of the impact statistics (Bottke et al. 1994) are suitable for the computation of the impact rates among the Main Belt Asteroids.

Instead, the larger the eccentricity of the planet's orbit the less uniform the circulation of the debris' orbits. Furthermore, and even more importantly, in such conditions strong correlations between orientation and eccentricities of the orbits arise, entailing a completely different radial distributions of the particles (fig. 2 and 3). More general methods (Dell’Oro & Paolicchi, 1998) are able to improve the investigation of the impact statistics, but they cannot account for the effect of strong orbital element correlations in computing collision probabilities and distribution of the impact velocities.

In the last decade, a new model for the investigation of the statistics of impacts among orbiting bodies has been developed (Dell’Oro, 2017). The new numerical model is able to reproduce the exact spatial and velocity distribution of the particles forming a debris disk perturbed by one or more planets or virtually in many other complex dynamical conditions. The method is based of a completely new and independent mathematical approach, and it has been used to validate and confirm the results of the classical methods.

In particular, for what concerns the typical case of debris disks around other stars,  the new tool, unlike the previous ones, is not limited to the case of low orbital eccentricity of the perturbing planets, typical of our Solar System. Here some preliminary results are shown in comparison with analytic theories (Mustill & Wyatt, 2009), when possible.

The analytic theory is limited to the case of small orbital inclinations. Moreover it ignores the effect of the disk borders where the particles density drops to zero, and where the  particles at the internal (external) edge interact only with particles of larger (smaller) semimajor axes.  

In the case of null orbital inclinations, analytic theory (fig. 4, black solid line) overestimates systematically the mean impact speed by 10-20 % with respect to our numerical computation (fig. 4, red dots). Outside of the zero inclinations case, mean impact speed increases as dispersion of the inclinations grows, as expected, but also the variation of the impact speed as function of the distance from the star is different. The analytic theory predicts that if the perturbing planet is internal (external) to the disk the mean impact speed decreases (increases) with distance from the star. In the case of internal planet, our numerical computation shows that allowing inclinations to increase, the dependence of the average velocity with the distance from the star is reversed (fig. 4). A similar result is obtained for the external planet case.  Another difference with respect to the analytic theory concerns the effect of the eccentricity of the planet, in the sense that the mean impact speed would be proportional to the planet eccentricity. While in the case of an internal planet numerical computation confirms
this trend, with an external perturbing planet the mean impact speed results to be less than proportional to the planet eccentricity.

REFERENCES
Bottke W.F. et al. 1994, Icarus, 107, 255-268.
Dell’Oro A. 2017. Monthly Notices of the Royal Astronomical Society, 467, 4817-4840.
Dell'Oro A., Paolicchi P. 1998. Icarus, 136, 328-339.
Marzari F., Dell'Oro A. 2017. Monthly Notices of the Royal Astronomical Society, 466, 3973-3988.
Mustill A.J., Wyatt M.C. 2009. Monthly Notices of the Royal Astronomical Society, 399, 1403-1414 .

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