Accretion of Planetesimals within a Gaseous Ring

Abstract

An analytical study is made of the accretion of planetesimals by a planetary embryo within the framework of a modem Laplacian theory for the formation of the planetary system. The equation of motion of the particle, which is initially comoving ahead of (or behind) the growing planet on the same circular Keplerian orbit about the Sun, is examined both in the presence and abseflce of a gaseous torus which is also centred on the same mean orbit. The gas density in the torus is taken to be uniform and the drag exerted on the particle is assumed to vary as the square of the relative velocity, corresponding to motion at high Reynolds number. It is found that the gas acts as a damper to the coriolis acceleration due to the Sun in the rotating frame of reference of the embryo, which tends to pull the particle off the mean circular orbit, thus preventing accretion. In the absence of the gaseous drag, less than 1 % of particles lying well inside the so-called sphere of gravitational influence of the embryo are accreted, whilst if the gas drag is included nearly all of these particles are captured. In all instances the accreting particles impart a spin angular momentum to the embryo which is prograde with the orbital motion. The actual spin rate decreases with increasing gas drag and is found to be lowest for the innermost planets Mercury and Venus, where the gas density is greatest. A more detailed numerical study is probably required to determine the rotational period of larger planets and planetary cores which possess an outer atmosphere, not included in the p$ent study, and where nonlinear effects in the particle's equation of motion cannot be ignored.' 'r.'