ALBERT

Description: One fundamental controversy about terrestrial planet and asteroid formation is the discrepancy between meteoritical evidence for high temperatures (1500 to 2000 K) in the inner solar nebula, and much lower theoretical temperature predictions on the basis of models of viscous accretion disks that neglect compressional heating of infalling gas. It is shown here that rigorous numerical calculations of the collapse of a rotating, three-dimensional presolar nebula are capable of producing temperatures on the order of 1500 K in the asteroid region (2.5 astronomical units), in either nearly axisymmetric or strongly nonaxisymmetric nebula models. The latter models may permit significant thermal cycling of solid components in the early inner solar nebula.

Description: Roche showed that equilibrium is impossible for a small fluid body synchronously orbiting a primary within a critical radius now termed the Roche limit. Tidal disruption of orbitally unbound bodies is a potentially important process for planetary formation through collisional accumulation, because the area of the Roche limit is considerably larger then the physical cross section of a protoplanet. Several previous studies were made of dynamical tidal disruption and different models of disruption were proposed. Because of the limitation of these analytical models, we have used a smoothed particle hydrodynamics (SPH) code to model the tidal disruption process. The code is basically the same as the one used to model giant impacts; we simply choose impact parameters large enough to avoid collisions. The primary and secondary both have iron cores and silicate mantles, and are initially isothermal at a molten temperature. The conclusions based on the analytical and numerical models are summarized.

Description: The dynamic problem of the tidal disruption of a rocky planetismal was solved by a direct integration of the fully three-dimensional, nonlinear equations of motion. The hypothesis that any object that passes within the Roche limit is disrupted was disproven. A time dependent solution was performed numerically, treating the planetismal as a fluid with a Murnaghan equation of state in the solid regions and zero pressure otherwise. Calculations show that a rocky body which passes by the Earth on a parabolic orbit with a perigee within the Roche limit is not tidally disrupted. Objects on hyperbolic orbits would experience even less tidal disruption. The results herein do not apply to bodies with very low viscosity. It is shown, however, that tidal disruption can be ruled out as a mechanism for reducing planetismal masses. Mechanisms for forming the Moon which rely upon tidal disruption are unlikely to be correct.

Description: A theory for the formation of the Moon which involves the dynamic fission of a rapidly rotating protoplanet, which might then result in the formation of the Earth and the Moon is discussed. The fission hypothesis was originally based on analytic, linearized models of the growth of asymmetry in homogenous bodies. The fully nonlinear evolution of the dynamic instability in inviscid, compressible bodies was calculated by numerical techniques. It was found that the dynamic instability degenerates into the ejection of a ring of matter with a substantial fraction of the mass, leaving behind a central body with most of the mass. The linearized analytical approach and the numerical approach were used to show that dynamic fission probably does not occur in rocky protoplanets. The numerical calculations are performed with a fully three dimensional hydrodynamical code, which allows the nonlinear, time evolution of the instability to be followed. Sequences of uniformly rotating equilibria were constructed and are used as the initial models for the fission calculations. An initially imposed asymmetry consisting of a 10% binary perturbation in the density was found to disappear on the rotational period time scale. No dynamic instability occurred. This result are verified by including the velocity dissipation terms in the linearized analysis of the stability of a Maclaurin spheroid: the dynamic instability disappears when the simulated viscous dissipation terms are included. It is concluded that any rocky body, even with considerable partial melt or a molten core, should be stable to dynamic fission; any rotational instability that occurs can only result in equatorial mass loss.

Description: Analytical and numerical approaches are taken to consider if a rapidly rotating, viscous protoearth would have lost mass by a fission process and thereby given birth to the moon. The fast rotation is assumed as the source of the instability in the dissipative liquid protoearth. Governing hydrodynamic equations are defined for the evolution of the protoearth. Account is taken of viscous dissipation, the pressure equation of state for the atmospheric material sent on a ballistic trajectory, and the effective viscosity. The results indicate that dynamic fission was probably not the process by which the protomoon came into existence.

Description: The paper considers six different categories of models for the formation of the moon within the context of the general theory of terrestrial planet formation by accumulation of planetesimals. These categories are: (1) rotational fission, (2) precipitation fission, (3) intact capture, (4) disintegrative capture, (5) binary accretion, and (6) giant impact accretion. It appears that the only plausible mechanism proposed thus far involves the formation of the moon following a giant impact that ejects portions of the differentiated earth's mantle and parts of the impacting body into circumterrestrial orbit.

Description: Given a solar nebular surrounding the early protosun, containing dust grains that have already undergone growth through collisions to about centimeter-size, the question of the formation of the terrestrial and giant planets is considered. In contrast to the usual approach of emphasizing how well a problem is understood, the uncertainties and areas where more work needs to be done will be accentuated. Also, the emphasis will be on the dynamics of planetary formation, because profound problems still exist in this area, and because it seems most logical to concentrate first on the dynamical questions involved with assembling the planets before putting too much effort into the detailed chemical and geological consequences of certain formation mechanisms.

Description: A self-consistent numerical model is developed for the tidal disruption of a solid planetesimal. The planetesimal is treated as a highly viscous, slightly compressible fluid whose disturbed parts are an inviscid, pressureless fluid undergoing distortion and disruption. The distortions were constrained to being symmetrical above and below the equatorial plane. The tidal potential is expanded in terms of Legendre polynomials, which eliminates the center of mass acceleration effects, permitting definition of equations of motion in a noninertial frame. Consideration is given to viscous dissipation and to characteristics of the solid-atmosphere boundary. The model is applied to sample cases in one, two and three dimensions.

Description: Constraints were previously placed on various properties of a Mercurian liquid core for compatibility with Mercury's escape from the stable spin-orbit resonance with the spin angular velocity equal to twice the orbital mean motion (the 2n resonance), under the assumption that the planet's obliquity was nearly zero at the time of resonance passage. Capture probabilities at arbitrary nonzero obliquities for the 2n resonance are determined for the cases where the core is strongly or weakly coupled to the mantle. It is found that the capture probabilities are reduced to below unity for all core-fluid viscosities in the weak-coupling limit, but are almost unchanged in the strong-coupling limit. The reduction in capture probability is attributed to reduction of the mantle's spin angular velocity by the core-mantle interaction, which would also reduce the obliquity to negligibly small values before the 2n resonance was even reached. It is concluded that the constraints on the core may still be maintained since Mercury most likely passed through the 2n resonance with nearly zero obliquity.

Description: The escape of Mercury from the stable spin-orbit resonance in which the spin angular velocity is twice the orbital mean motion (2n) requires that the kinematic viscosity of a molten core with a laminar boundary layer be comparable to that of water (0.01 sq cm/s) and the tidal Q be less than about 100. If the boundary layer is turbulent, escape from the resonance is only consistent with a liquid core of low viscosity if the critical Reynolds number for the onset of turbulence is above about 500, the moment difference (B - A)/C is below about 0.00001, and the tidal dissipation factor Q is less than about 40. These conclusions depend on the assumptions that Mercury's obliquity was near 0 deg at the time of resonance passage, that the liquid core was not stably stratified at the time at which Mercury passed through the resonance, that a turbulent boundary layer can be characterized by a turbulent or eddy viscosity coefficient, and that the most important coupling between core and mantle is a viscous coupling at a smooth spherical boundary.