The origin of the systematic component of planetary rotation: I. Planet on a circular orbit

Abstract

The rotation rate of a planet which accretes from small, solid planetesimals is computed as a function of the eccentricities of the planetesimals' orbits. Planets acquire spin angular momentum from the relative motion of planetesimals at impact. The calculations presented herein assume that the planet is spherical and that its orbit is circular. The planetesimals orbit in or very near the plane of the planet's orbit and are uniformly distributed in periapse longitude. Unless otherwise stated, the planetesimal distribution is also assumed to be uniform in semimajor axis. The stochastic effects of accretion of large planetesimals are ignored. The spin angular momentum produced by a particular collision can be either positive or negative; due to symmetries these contributions nearly cancel. Thus, in order to obtain meaningful nonzero results, perturbation expansions must be carried through to second order in the ratio of the planet's radius to the amplitude of the radial excursions of the planetesimals, and numerical simulations require the calculation of a very large number of impacts. We find that nongravitating planets accreted from the disk of planetesimals described above rotate very slowly in the prograde direction; their spin period is longer than their orbital period. When the planet's gravity is included in a three-body treatment of the problem, the rotation rate is a function of e H, the ratio of the amplitude of the planetesimals' radial (eccentric) motion to the radius of the planet's Hill Sphere, and r H, the ratio of the planet's radius to the radius of its Hill Sphere. The number of rotations per orbit generally increases as r H decreases. For e H ≲ 1 (nearly circular initial orbits), the planet rotates in the retrograde direction. Rotation is prograde for larger eccentricities. For small r H, the rotation rate peaks at e H ≈1.2 and decreases to the nongravitating limit as e H → ∞. The maximum rates of systematic prograde rotation are comparable to those observed in our Solar System; however, rapid prograde rotation only occurs for a very narrow range of e H values. In simulations which start with a Rayleigh distribution of planetesimal eccentricities, we find that the maximum prograde rotation rates are much slower than those observed in the Solar System, suggesting that other factors are important. Material accreted from near the outer edges of a planet's feeding zone tends to provide the planet with positive spin angular momentum. If the surface mass density in these regions is greater than that in the rest of the planet's accretion zone, due, e.g., to the expansion of a growing planet's feeding range into parts of the disk not depleted by previous accretion, then the observed spin periods of the planets could be accounted for. Alternatively, the stochastic component of rotation due to impacts of large planetesimals may overwhelm the ordered prograde component of rotation for solid planets, and/or unmodeled effects, such as the orbital eccentricity of a growing planet, might be important in determining planetary rotation rates.