ICARUS 129, 384–400 (1997) ARTICLE NO. IS975777 Sources of Planetary Rotation: Mapping Planetesimals’ Contributions to Angular Momentum Richard Greenberg and Mark Fischer Lunar and Planetary Laboratory, University of Arizona, Tucson, Arizona 85721 E-mail: greenberg@lpl.arizona.edu and G. B. Valsecchi and A. Carusi Istituto di Astrofisica Spaziale CNR, Reparto di Planetologia, Rome, Italy Received November 4, 1996; revised May 9, 1997 important. Monte-Carlo numerical simulations reported in the latter two references yielded the expected planetary A systematic study of the motion backward and forward in time from a uniform distribution of bodies entering a planet’s spin rate for various assumed planetesimal distributions. Hill sphere yields a mapping of the contribution to the planet’s Numerical solutions are required because planetesimals’ rotation from planetesimals as a function of their source in trajectories are controlled by both the gravity of the Sun heliocentric orbital element space. The mapping allows deter- and the planet. Tanikawa et al. (1991) performed numerical mination of the contribution to rotation from any assumed integrations of planetesimals’ orbits near a planetary em- distribution of heliocentric planetesimal orbits. For example, it bryo to help lay the groundwork for understanding the can systematically reproduce earlier results from Monte-Carlo processes that control rotation. The value of such studies, studies of contributions to rotational angular momentum. With our method of calculation, contributions to planetary rotation especially for very low-velocity encounters, is to under- can be interpreted in terms of the arrival geometries at the stand the trajectories of planetesimals as they make the planet’s Hill sphere and the subsequent two-body motion inside transition from heliocentric (approximately two-body) or- the Hill sphere leading to impact. Prograde rotation is strongly bits to planetocentric (approximately two-body) orbits. By favored if a planet grows in a relatively quiescent population one definition, this transition occurs formally at the Tisser- of planetesimals with accretion nibbling at the edges of its and radius and near the Hill radius (the size of the gravita- feeding zone. However, if the impacting population were domi- tional well) around a planet. nated by large bodies with high relative velocity, the direction In this paper, we review and interpret the results of and magnitude of rotation would be random.  1997 Academic Press Tanikawa et al. We then extend numerical integrations backward away from the Hill sphere to map (in heliocentric I. INTRODUCTION orbital space) the sources of planet-hitting planetesimals in such a way as to identify the sources of planetesimals If a planetary embryo grew by accreting a swarm of that make various contributions to planetary rotation. Our planetesimals, with all bodies on circular orbits, and with method produces a map of the contribution of planetesi- negligible mutual gravitation, it seems it should rotate in mals to planetary rotation as a function of parameters the retrograde direction, because of Keplerian shear: Im- (a, e) in initial heliocentric orbit. For example, our map pactors orbiting the Sun faster than the planet would hit shows the specific regions of heliocentric space that con- the sunward side and impactors orbiting slower would hit tribute maximum prograde angular momentum. the anti-Sun side. In reality, however, conditions were dif- Our results are similar to those found by Lissauer and ferent: The impactors probably had eccentric orbits; grow- Kary (1991) and Dones and Tremaine (1993). In their ing planets did have gravity; and the population of im- Monte-Carlo studies, assumed heliocentric populations of pactors may not have been distributed continuously, either planetesimals were randomly sampled, and each represen- spatially or in size. Giuli (1968a,b), Harris (1977), Lissauer tative body tracked forward in time by numerical integra- and Kary (1991), Dones and Tremaine (1993), and Lissauer tion of trajectories up to planetary impact. Like those stud- ies, our systematic mapping confirms the result of Giuli et al. (1997) showed that all of these effects were probably 384 0019-1035/97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.