On Optimal Solutions to Planetesimal Growth Models

As expressions are developed which describe planetesimal collisions, it is instructive to find probability distributions for planetesimal parameters which maximize growth of the planetesimal system. Finding such optimal distributions provides clues as to the general efficiency of planet formation, and may help astronomers to determine whether young systems are likely to form planets. We consider one such collision expression (Stewart & Leinhardt 2009) and propose a simple parametric model of a planetesimal system defining its structure and the way in which collisions occur, which allows us to pose the question of finding systems in which collisions lead to maximum expected growth. We show that this leads to what is known in the optimization literature as a “standard quadratic program”. In general, the quadratic function to be minimized is non-convex, which makes the problem computationally intractable. We describe several algorithms for solving problems of this type, and present probability distributions for mass with approximately optimal growth factors. Assuming that the planetesimal velocity distribution is known, we find that there are many probability distributions of mass that are close to optimal. This might lead one to naively assume that planet formation is a relatively optimal process however, this result ignores the dependence of planetesimal velocity on mass, and further work is required to determine the effects of the coupling of mass and velocity distributions through physical processes such as aerodynamic drag, turbulence and gravitational scattering. However, this work has demonstrated that optimal solutions for planetesimal growth do not depend strongly on the initial mass distribution.