Patrick Cassen

Melting of Io by Tidal Dissipation

The dissipation of tidal energy in Jupiters satellite Io is likely to have melted a major fraction of the mass. Consequences of a largely molten interior may be evident in pictures of Ios surface returned by Voyager I.

On the formation of protostellar disks

Abstract An analysis is presented of the hydrodynamic aspects of the growth of protostellar disks from the accretion (or collapse) of a rotating gas cloud. The size, mass, and radiative properties of protostellar disks are determined by the distribution of mass and angular momentum in the clouds from which they are formed, as well as from the dissipative processes within the disks themselves. The angular momentum of the infalling cloud is redistributed by the action of turbulent viscosity on a shear layer near the surface of the disk (downstream of the accretion shock) and on the radial shear across cylindrical surfaces parallel to the rotation axis. The fraction of gas that is fed into a central core (protostar) during accretion depends on the ratio of the rate of viscous diffusion of angular momentum to the accretion rate; rapid viscous diffusion (or a low accretion rate) promotes a large core-to-disk mass ratio. The continuum radiation spectrum of a highly viscous disk is similar to that of a steady-state accretion disk without mass addition. It is possible to construct models of the primitive solar nebula as an accretion disk, formed by the collapse of a slowly rotating protostellar cloud, and containing the minimum mass required to account for the planets. Other models with more massive disks are also possible.

The Structure and Appearance of Protostellar Accretion Disks: Limits on Disk Flaring

Vertical structure models are used to investigate the structure of protostellar, α-law, accretion disks. Conditions investigated cover a range of mass fluxes (10-9 to 10-5 M☉ yr-1), viscous efficiencies (α = 10-2 and 10-4), and stellar masses (0.5-3 M☉). Analytic formulae for midplane temperatures, optical depths, and volume and surface densities are derived and are shown to agree well with numerical results. The temperature dependence of the opacity is shown to be the crucial factor in determining radial trends. We also consider the effect on disk structure of illumination from a uniform field of radiation such as might be expected of a system immersed in a molecular cloud core or other star-forming environment: Tamb = 10, 20, and 100 K. Model results are compared to Hubble Space Telescope observations of HH30 and the Orion proplyds. Disk shape is derived in both the Rosseland mean approximation and as viewed at particular wavelengths (λλ = 0.66, 2.2, 60, 100, 350, and 1000 μm). In regions where the opacity is an increasing function of temperature (as in the molecular regions where κ ∝ T2), the disk does not flare, but decreases in relative thickness with radius under both Rosseland mean and single wavelength approximations. The radius at which the disk becomes shadowed from central object illumination depends on radial mass flow and varies from a few tenths to about 5 au over the range of mass fluxes tested. This suggests that most planet formation occurred in environments unheated by stellar radiation. Viewing the system at any single wavelength increases the apparent flaring of the disk but leaves the shadow radius essentially unchanged. External heating further enhances flaring at large radii, but, except under extreme illumination (100 K), the inner disk will shield the planet-forming regions of all but the lowest mass flux disks from radiation originating near the origin such as from the star or from an FU Orionis outburst.

Subsolidus convective cooling histories of terrestrial planets

Abstract The subsolidus convective cooling histories of terrestrial planets evolving from hot initial states are investigated with a simple analytic model which simulates the average heat transport in a vigorously convecting mantle devoid of internal heat sources. The temperature dependence of the effective viscosity of mantle rocks is the single most important factor controlling thermal history. It is responsible for the growth of the rigid lithosphere, a rheological and thermal boundary layer, and serves the function of a thermostat, regulating the rate of cooling by the negative feedback between viscosity and temperature. Except for a relatively short period of time when mantle temperature decreases rapidly during the early stages of cooling, a planet cools mainly by thickening its lithosphere; the underlying mantle temperature decreases relatively slowly. On one-plate planets, the growth of a rigid lithosphere involves an imbalance between the surface heat flux and the heat flow from the mantle; the former is always larger than the latter. Primordial heat can contribute substantially, e.g., as much as about a fourth or a third, to the present surface heat flux of a planet. For both these reasons, the radiogenic heat source content of a planet is likely to be overestimated by inferences from surface heat flow observations.

Young chondrules in CB chondrites from a giant impact in the early Solar System

Chondrules, which are the major constituent of chondritic meteorites, are believed to have formed during brief, localized, repetitive melting of dust (probably caused by shock waves) in the protoplanetary disk around the early Sun. The ages of primitive chondrules in chondritic meteorites indicate that their formation started shortly after that of the calcium-aluminium-rich inclusions (4,567.2 ± 0.7 Myr ago) and lasted for about 3 Myr, which is consistent with the dissipation timescale for protoplanetary disks around young solar-mass stars. Here we report the 207Pb–206Pb ages of chondrules in the metal-rich CB (Bencubbin-like) carbonaceous chondrites Gujba (4,562.7 ± 0.5 Myr) and Hammadah al Hamra 237 (4,562.8 ± 0.9 Myr), which formed during a single-stage, highly energetic event. Both the relatively young ages and the single-stage formation of the CB chondrules are inconsistent with formation during a nebular shock wave. We conclude that chondrules and metal grains in the CB chondrites formed from a vapour–melt plume produced by a giant impact between planetary embryos after dust in the protoplanetary disk had largely dissipated. These findings therefore provide evidence for planet-sized objects in the earliest asteroid belt, as required by current numerical simulations of planet formation in the inner Solar System.

Contribution of tidal dissipation to lunar thermal history

Abstract The possible contributions of tidal heating to lunar thermal history are investigated. Analytic determinations of tidal dissipation in a homogeneous, incompressible Moon and in a two-layer Moon with a soft core and rigid mantle are given as a function of position in the Moon and as a function of Earth-Moon separation. The most recent information on the historical values of the lunar obliquity is employed, and we present results for the constant values of orbital eccentricity of e = 0.0 and e = 0.055. For a simplified orbital evolution and a dissipation factor Q = 100, the total increase in the mean lunar temperature for the homogeneous case does not exceed several tens of degrees. For the two-layer models the local dissipation may be enhanced over that of the homogeneous Moon by a factor of 5 for a core radius of 0.5 lunar radii and by a factor of 100 for a core radius of 0.95 lunar radii. The corresponding factors for the total dissipation are 3 and 15 for the two values of core radii, respectively. We conclude that tidal contributions to lunar thermal history are probably not important. But under special circumstances the enhanced dissipation in a two-layer Moon could have led to a spectacular thermal event.

The evolution of Enceladus

Saturns small icy moon Enceladus shows evidence of several episodes of geologic resurfacing and extensional tectonism spread over much of its history. Freezing of liquid in the interior caused global expansion and fracturing of the crust. Resurfacing took place by eruption of fresh material, perhaps containing NH3, to the surface. Solid-state convection could take place in Enceladus for a crustal thickness greater than about 30 km, assuming thermal properties like those of pure H2O ice. Melting in the interior may have been made more likely by the presence of NH3, as the H2ONH3 system has a eutectic point at 173°K. Tidal dissipation seems to be the only heating mechanism capable of melting Enceladus. For the thermal properties of pure H2O, the orbital eccentricity would have to be higher by a factor of 5–7 than the present value of 0.0044 to maintain a molten interior, and may have to be greater by as much as a factor of 20 to cause melting in an initially frozen body. If the thermal conductivity is diminished by inclusion of clathrate hydrates, a significant enhancement over the present eccentricity would still be required to initiate melting, but it might be possible to maintain a molten interior and allow geologic activity with the present eccentricity. Removal of eccentricity forcing would result in rapid eccentricity damping, freezing, and cessation of tectonic activity.

The Thermal Regulation of Gravitational Instabilities in Protoplanetary Disks

We present a series of high-resolution, three-dimensional hydrodynamics simulations of a gravitationally unstable solar nebula model. The influences of both azimuthal grid resolution and the treatment of thermal processes on the origin and evolution of gravitational instabilities are investigated. In the first set of simulations, we vary the azimuthal resolution for a locally isothermal simulation, doubling and quadrupling the resolution used in a previous study; the largest number of grid points is (256, 256, 64) in cylindrical coordinates (r, , z). At this resolution, the disk breaks apart into a dozen short-lived condensations. Although our previous calculations underresolved the number and growth rate of clumps in the disk, the overall qualitative, but fundamental, conclusion remains: fragmentation under the locally isothermal condition in numerical simulations does not in itself lead to the survival of clumps to become gaseous giant protoplanets. Since local isothermality represents an extreme assumption about thermal processes in the disk, we also present several extended simulations in which heating from an artificial viscosity scheme and cooling from a simple volumetric cooling function are applied to two different models of the solar nebula. The models are differentiated primarily by disk temperature: a high-Q model generated directly by our self-consistent field equilibrium code and a low-Q model generated by cooling the high-Q model in a two-dimensional version of our hydrodynamics code. Here, high-Q and low-Q refer to the minimum values of the Toomre stability parameter Q in each disk, Qmin = 1.8 and 0.9, respectively. Previous simulations, by ourselves as well as others, have focused on initial states that are already gravitationally unstable, i.e., models similar to the low-Q model. This paper presents for the first time the numerical evolution of an essentially stable initial equilibrium state (the high-Q model) to a severely unstable one by cooling. The additional heating and cooling are applied to each model over the outer half of the disk or the entire disk. The models are subject to the rapid growth of a four-armed spiral instability; the subsequent evolution of the models depends on the thermal behavior of the disk. The cooling function tends to overwhelm the heating included in our artificial viscosity prescription, and as a result the spiral structure strengthens. The spiral disturbances transport mass at prodigious rates during the early nonlinear stages of development and significantly alter the disks vertical surface. Although dense condensations of material can appear, their character depends on the extent of the volumetric cooling in the disk. In the simulation of the high-Q model with heating and cooling applied throughout the disk, thin, dense rings form at radii ranging from 1 to 3 AU and steadily increase in mass; later companion formation may occur in these rings as cooling drives them toward instability. When heating and cooling are applied only over the outer radial half of the disk, however, a succession of single condensations appears near 5 AU. Each clump has roughly the mass of Saturn, and some survive a complete orbit. Since the clumps form near the artificial boundary in the treatment of the disk gas physics, the production of a clump in this case is a numerical artifact. Nevertheless, radially abrupt transitions in disk gas characteristics, for example, in opacity, might mimic the artificial boundary effects in our simulations and favor the production of stable companions in actual protostellar and protoplanetary disks. The ultimate survival of condensations as eventual stellar or substellar companions to the central star is still largely an open question.

Tidal dissipation, orbital evolution, and the nature of Saturn's inner satellites

Abstract Estimates of tidal damping times of the orbital eccentricities of Saturns inner satellites place constraints on some satellite rigidities and dissipation functions Q . These constraints favor rock-like rather than ice-like properties for Mimas and probably Dione. Photometric and other observational data are consistent with relatively higher densities for these two satellites, but require lower densities for Tethys, Enceladus, and Rhea. This leads to a nonmonotonic density distribution for Saturns inner satellites, apparently determined by different mass fractions of rocky materials. In spite of the consequences of tidal dissipation for the orbital eccentricity decay and implications for satellite compositions, tidal heating is not an important contributor to the thermal history of any Saturnian satellite.

On the evolution of gravitationally unstable protostellar disks

The formation of single stars from rotating interstellar clouds generally requires that the angular momentum of the cloud be redistributed in such a way that a major fraction of the mass (eventually residing in the star) retains only a minor fraction of the angular momentum, most of the latter being deposited in a circumstellar disk. Gravitational instabilities in a growing disk would promote such a redistribution by the action of self-excited density waves. We have attempted to quantify this process by conducting two-dimensional, N-body simulations of the nonlinear development of long-wavelength instabilities predicted to occur early in the formation stage of the disk, under conditions of controlled energy loss