William R. Ward

The Formation of Planetesimals

Four stages in the accretion of planetesimals are described. The initial stage is the condensation of dust particles from the gaseous solar nebula as it cools. These dust particles settle into a thin disk which is gravitationally unstable. A first generation of planetesimals, whose radii range up to approximates 10-1 km, form from the dust disk by direct gravitational collapse to solid densities on a time scale of the order of 1 year. The resulting disk, composed of first-generation planetesimals, is still gravitationally unstable, and the planetesimals are grouped into clusters containing approximately 104 members. The contraction of these clusters is controlled by the rate at which gas drag damps their internal rotational and random kinetic energies. On a time scale of a few thousand years, the clusters contract to form a second generation of planetesimals having radii of the order of 5 km. Further coalescence of planetesimals proceeds by direct collisions which seem capable of producing growth at a rate of the order of 15 cm per year at 1 AU. The final stage of accretion during which planet-sized objects form is not considered here.

Three-Dimensional Interaction between a Planet and an Isothermal Gaseous Disk. I. Corotation and Lindblad Torques and Planet Migration

Gravitational interaction between a planet and a three-dimensional isothermal gaseous disk is studied. In the present paper we mainly examine the torque on a planet and the resultant radial migration of the planet. A planet excites density waves at Lindblad and corotation resonances and experiences a negative torque by the density waves, which causes a rapid inward migration of the planet during its formation in a protoplanetary disk. We formulate the linear wave excitation in three-dimensional isothermal disks and calculate the torques of Lindblad resonances and corotation resonances. For corotation resonances, a general torque formula is newly derived, which is also applicable to two-dimensional disks. The new formula succeeds in reproducing numerical results on the corotation torques, which do not agree with the previously well-known formula. The net torque of the inner and the outer Lindblad resonances (i.e., the differential Lindblad torque) is caused by asymmetry such as the radial pressure gradient and the scale height variation. In three-dimensional disks, the differential Lindblad torques are generally smaller than those in two-dimensional disks. Especially, the effect of a pressure gradient becomes weak. The scale height variation, which is a purely three-dimensional effect, makes the differential Lindblad torque decrease. As a result, the migration time of a planet is obtained as of the order of 106 yr for an Earth-size planet at 5 AU for a typical disk model, which is longer than the result of two-dimensional calculation by the factor of 2 or 3. The reflected waves from disk edges, which are neglected in the torque calculation, can further weaken the disk-planet interaction.

Density waves in the solar nebula: differential Lindblad torque

Abstract The differential torque between a two-dimensional fluid disc and an embedded perturber due to waves launched at its resonances is calculated. For a non-self-gravitating disc, the disturbance is in the form of pressure waves which propagate away from co-rotation and asymptotically become trailing spirals. The torque calculation includes lowest order contributions from gradients in various disc properties. The orbital drift rate of the perturber is determined and the results applied to the motion of preplanetary material in the solar nebula.

FORMATION OF THE GALILEAN SATELLITES: CONDITIONS OF ACCRETION

We examine formation conditions for the Galilean satellites in the context of models of late-stage giant planet accretion and satellite-disk interactions. We first reevaluate the current standard, in which the satellites form from a ‘‘ minimum mass subnebula ’’ disk, obtained by augmenting the mass of the current satellites to solar abundance and resulting in a disk mass containing about 2% of Jupiter’s mass. Conditions in such a massive and gas-rich disk are difficult to reconcile with both the icy compositions of Ganymede and Callisto and the protracted formation time needed to explain Callisto’s apparent incomplete differentiation. In addition, we argue that disk torques in such a gas-rich disk would cause large satellites to be lost to inward decay onto the planet. These issues have prevented us from identifying a self-consistent scenario for the formation and survival of the Galilean satellites using the standard model. We then consider an alternative, in which the satellites form in a circumplanetary accretion disk produced during the very end stages of gas accretion onto Jupiter. In this case, an amount of gas and solids of at least � 0.02 Jovian masses must still be processed through the disk during the satellite formation era, but this amount need not have been present all at once. We find that an accretion disk produced by a slow inflow of gas and solids, e.g., 2 � 10 � 7 Jovian masses per year, is most consistent with conditions needed to form the Galilean satellites, including disk temperatures low enough for ices and protracted satellite accretion times of � 10 5 yr. Such a ‘‘ gas-starved ’’ disk has an orders-of-magnitude lower gas surface density than the minimum mass subnebula (and for many cases is optically thin). Solids delivered to the disk build up over many disk viscous cycles, resulting in a greatly reduced gas-to-solids ratio during the final stages of satellite accretion. This allows for the survival of Galilean-sized satellites against disk torques over a wide range of plausible conditions.

A common mass scaling for satellite systems of gaseous planets.

The Solar Systems outer planets that contain hydrogen gas all host systems of multiple moons, which notably each contain a similar fraction of their respective planets mass (∼10-4). This mass fraction is two to three orders of magnitude smaller than that of the largest satellites of the solid planets (such as the Earths Moon), and its common value for gas planets has been puzzling. Here we model satellite growth and loss as a forming giant planet accumulates gas and rock-ice solids from solar orbit. We find that the mass fraction of its satellite system is regulated to ∼10-4 by a balance of two competing processes: the supply of inflowing material to the satellites, and satellite loss through orbital decay driven by the gas. We show that the overall properties of the satellite systems of Jupiter, Saturn and Uranus arise naturally, and suggest that similar processes could limit the largest moons of extrasolar Jupiter-mass planets to Moon-to-Mars size.

THREE-DIMENSIONAL INTERACTION BETWEEN A PLANET AND AN ISOTHERMAL GASEOUS DISK. II. ECCENTRICITY WAVES AND BENDING WAVES

We perform linear calculations to investigate three-dimensional density waves excited by planets on elliptical and inclined orbits in isothermal protoplanetary disks. We consider small planets that have no disk gap around their orbits. Eccentricities and inclinations of planets are assumed to be smaller than the disk aspect ratio. This is reasonable for planets with no disk gap. The density wave excited by a planet with nonzero small eccentricity e and inclination i is decomposed into three components: the waves by a planet with e = i = 0, the eccentricity waves, and the bending waves. The eccentricity waves are related to the noncircular motion of the planet, while the bending waves are excited by the motion normal to the equatorial plane. In our formulation, these waves are described by the same wave equations, and only the perturbing potentials are different. We numerically solve the wave equations and calculate the force exerted on the planet by the waves. The force is not parallel to the velocity of the epicycle motion. From the force obtained, we also find the evolution rates in the eccentricity, the inclination, and the longitudes of the perihelion and the ascending node. The characteristic evolution time of these orbital elements is about 300(r/1 AU)2 yr for Earth-sized planets in the minimum-mass nebula disk. Eccentricity damping is caused by eccentricity waves, while inclination damping is due to bending waves for planets with small eccentricities and inclinations and with no disk gap. This means that to lowest order there is no coupling between the evolutions of the eccentricity and the inclination.

Large-Scale Variations in the Obliquity of Mars

Large-scale variations in the obliquity of the planet Mars are produced by a coupling between the motion of its orbit plane due to the gravitational perturbations of the other planets and the precession of its spin axis which results from the solar torque exerted on the equatorial bulge of the planet. The obliquity oscillates on a time scale of approximately 1.2 x 105 years. The amplitude of this oscillation itself varies periodically on a time scale of 1.2 X 106 years. The present-day obliquity is approximately 25.1 degrees. The maximum possible variation is from about 14.9 to 35.5 degrees. Signtificant climatic effects must be associated with the phenomenon.

Survival of Planetary Systems

Gravitational interactions (i.e., disk tides) between a newly formed protoplanet and its precursor disk give rise to a net torque that drains angular momentum from the protoplanets orbit. As a result, protoplanetary objects suffer orbital decay as the disk attempts to destroy the very system it spawns. Survival of a planetary system may be a rather uncertain outcome, and the fraction of circumstellar disks that produce an extant system could be significantly less than unity. Newly discovered close stellar companions may be circumstantial evidence of such large-scale orbit migrations. A scheme for in situ accretion of such objects is outlined that is consonant with a strong tidal influence.

Periodic Insolation Variations on Mars

Previously unrecognized insolation variations on Mars are a consequence of periodic variations in eccentricity, first established by the theory of Brouwer and Van Woerkom (1950). Such annual insolation variations, characterized by both 95,000-year and 2,000,000-year periodicities, may actually be recorded in newly discovered layered deposits in the polar regions of Mars. An additional north-south variation in seasonal insolation, but not average annual insolation, exists with 51,000-year and 2,000,000-year periodicities.

Solar nebula dispersal and the stability of the planetary system: I. Scanning secular resonance theory

Secular resonances in the early solar system are studied in an effort to establish constraints on the time scale and/or method of solar nebula dispersal. Simplified nebula models and dispersal routines are employed to approximate changes in an assumed axisymmetric nebula potential. These changes, in turn, drive an evolutionary sequence of Laplace-Lagrange solutions for the secular variations of the solar system. A general feature of these sequences is a sweep of one or more giant planet resonances through the inner solar system. Their effect is rate dependent; in the linearized models considered, characteristic dispersal times ≤O(104−5 years) are required to avoid the generation of terrestrial eccentricities and inclinations in excess of observed values. These times are short compared to typical estimates of the accretion time scales [i.e., ∼O(107−9 years)] and may provide an important boundary condition for developing models of nebula dispersal and solar system formation in general.