Morris Podolak

On the Location of the Snow Line in a Protoplanetary Disk

In a protoplanetary disk, the inner edge of the region where the temperature falls below the condensation temperature of water is referred to as the snow line. Outside the snow line, water ice increases the surface density of solids by a factor of 4. The mass of the fastest growing planetesimal (the isolation mass) scales as the surface density to the 3/2 power. It is thought that ice-enhanced surface densities are required to make the cores of the gas giants (Jupiter and Saturn) before the disk gas dissipates. Observations of our solar systems asteroid belt suggest that the snow line occurred near 2.7 AU. In this paper we revisit the theoretical determination of the snow line. In a minimum-mass disk characterized by conventional opacities and a mass accretion rate of 10-8 M☉ yr-1, the snow line lies at 1.6-1.8 AU, just past the orbit of Mars. The minimum-mass disk, with a mass of 0.02 M☉, has a lifetime of 2 million years with the assumed accretion rate. Moving the snow line past 2.7 AU requires that we increase the disk opacity, accretion rate, and/or disk mass by factors ranging up to an order of magnitude above our assumed baseline values.

Models of the giant planets

Abstract Models of the giant planets were constructed based on the assumption that the hydrogen to helium ratio is solar in these planets. This assumption, together with arguments about the condensation sequence in the primitive solar nebula, yields models with a central core of rock and possibly ice surrounded by an envelope of hydrogen, helium, methane, ammonia, and water. These last three volatiles may be individually enhanced due to condensation at the period of core formation. Jupiter was found to have a core of about 40 earth masses and a water enhancement in the atmosphere of about 7.5 times the solar value. Saturn was found to have a core of 20 earth masses and a water enhancement in the atmosphere of about 25 times the solar value. Rock plus ice constitute 75–85% of the mass of Uranus and Neptune. Temperatures in the interiors of these planets are probably above the melting points, if there is an adiabatic relation throughout the interiors. Some aspects of the sensitivities of these results to uncertainties in rotational flattening are discussed.

Interactions of planetesimals with protoplanetary atmospheres

Abstract We compute the interaction of planetesimals with the envelopes of growing giant planets that form by the “core-instability” mechanism. According to this mechanism, a core grows by the accretion of solid bodies in the solar nebula, and the growing core becomes progressively more effective in gravitationally contracting gas from the surrounding solar nebula into an envelope, until a “runaway” accretion of gas occurs. We compute the two-body trajectories of planetesimals through this envelope, including the effects of gravitational and gas-drag forces for the envelopes surrounding cores of different masses ranging from 1.1 to 16.8 Earth masses. Curves are computed which show the depth of penetration of various-sized planetesimals through these envelopes, as are cross-sections for capture of icy and rocky planetesimals. The implications for the formation of the giant planets by this sort of mechanism and its influence on their composition are discussed.

The contribution of small grains to the opacity of protoplanetary atmospheres

I compute the opacity of grains in a protoplanetary atmosphere. The grain size distribution at different levels in the atmosphere is calculated using a simple microphysical model of grain growth via collisions and destruction via vaporization at high temperatures. The Rosseland mean opacity of the resulting distribution is then computed. For most cases examined, the grain opacity is significantly lower than earlier estimates.

Evaporation from a porous cometary nucleus

In a porous cometary nucleus, ice sublimates from the volume of a surface layer rather than just from the upper boundary. Given a model for the porous medium, the equations of mass and heat transfer can be solved for any desired orbit. The temperature profile and the vapor flux as a function of depth in the upper layer of the nucleus may thus be obtained. Calculations are performed for a spherically symmetric icy nucleus in the orbit of Comet P/Halley, assuming different values of porosity and different models for the ice structure. The upper layer may be divided in two zones: in the uppermost zone, whose thickness ranges from 100 microns to about 1 mm, the vapor flux is directed outward, whereas in the lower zone, which is 1000 times thicker, the vapor flows in the opposite direction. The sublimation rate as a function of heliocentric distance depends strongly on the porosity of the nucleus and is little affected by other parameters related to the structure of the ice. This allows the determination of the porosity coefficient of a comet from observation of its water production rates at large heliocentric distances. 18 refs.

Radiogenic heating of comets by 26Al and implications for their time of formation.

The effect of radiogenic heating on the thermal evolution of spherical icy bodies with radii 1 km < R < 100 km was investigated. The radioisotopes considered were 26Al, 40K, 232Th, 235U, and 238U. Except for the 26Al abundance, which was varied, the other initial abundances were kept fixed, at values derived from those of chondritic meteorites and corresponding to a gas-to-dust ratio of 1. The initial models were homogeneous and isothermal (To = 10 K) amorphous ice spheres, in a circular orbit at 10(4) AU from the Sun. The main object of this study was to examine the conditions under which the transition temperature from amorphous into cubic ice (Ta = 137 K) would be reached. It was shown that the influence of the short-lived radionuclide 26Al dominates the effect of other radioactive species for bodies of radii up to approximately 50 km. Consequently, if we require comets to retain their ice in amorphous form, as suggested by observations, an upper limit of approximately 4 x 10(-9) is obtained for the initial 26Al abundance in comets, a factor of 100 lower than that of the inclusions in the Allende meteorite. A lower limit for the formation time of comets may thus be derived. The possibility of a coexistence of molten cometary cores and extended amorphous ice mantles is ruled out. Larger icy spheres (R > 100 km) reached Ta even in the absence of 26Al, due to the decay of the other radionuclides. As a result, a crystalline core formed whose relative size depended on the composition assumed. Thus the outermost icy satellites in the solar system, which might have been formed of ice in the amorphous state, have probably undergone crystallization and may have exhibited eruptive activity when the gas trapped in the amorphous ice was released (e.g., Miranda).

Comparative models of Uranus and Neptune

Abstract Models of Uranus and Neptune are computed based on the assumption that these planets consist of three layers: a rock core, an ice shell, and an atmosphere. Uranus models require that the ice shell have a density some 10% lower than the canonical density for an ice mixture. Two Neptune models are found, one with the canonical density in the ice shell, and one with a density 20% lower. The implications of these models are discussed.

Axel dust on Saturn and Titan

Abstract The scattering and absorption properties of Axel dust were investigated by means of Mie theory. We find that a flat distribution of particle radii between 0 and 0.1 μm, and an imaginary part of the index of refraction which varies as λ−2.5 produce a good fit to the variation of Titans geometric albedo with wavelength (λ) provided that τext, the extinction optical depth of Titans atmosphere at 5000 A, is about 10. The real part of the complex index is taken to be 2.0. The model assumes that the mixing ratio of Axel dust to gas is uniform above the surface of Titan. The same set of physical properties for Axel dust also produces a good fit to Saturns albedo if τext = 0.7 at 5000 A. To match the increase in albedo shortward of 3500 A, a clear layer (containing about 7 km-am H2) is required above the Axel dust. Such a layer is also required to explain the limb brightening in the ultraviolet. These models can be used to analyze the observed equivalent widths of the visible methane bands. The analysis yields an abundance of the order of 1000 m-am CH4 in Titans atmosphere. The derived CH4/H2 mixing ratio for Saturn is about 3.5 × 10−3 or an enhancement of about 5 over the solar ratio.

Grain sedimentation in a giant gaseous protoplanet

Abstract We present a calculation of the sedimentation of grains in a giant gaseous protoplanet such as that resulting from a disk instability of the type envisioned by Boss [Boss, A.P., 1998. Earth Moon Planets 81, 19–26]. Boss [Boss, A.P., 1998. Earth Moon Planets 81, 19–26] has suggested that such protoplanets would form cores through the settling of small grains. We have tested this suggestion by following the sedimentation of small silicate grains as the protoplanet contracts and evolves. We find that during the course of the initial contraction of the protoplanet, which lasts some 4 × 10 5 years, even very small (>1 μm) silicate grains can sediment to create a core both for convective and non-convective envelopes, although the sedimentation time is substantially longer if the envelope is convective, and grains are allowed to be carried back up into the envelope by convection. Grains composed of organic material will mostly be evaporated before they get to the core region, while water ice grains will be completely evaporated. These results suggest that if giant planets are formed via the gravitational instability mechanism, a small heavy element core can be formed due to sedimentation of grains, but it will be composed almost entirely of refractory material. Including planetesimal capture, we find core masses between 1 and 10 M ⊕ , and a total high- Z enhancement of ∼40 M ⊕ . The refractories in the envelope will be mostly water vapor and organic residuals.

The photochemistry of hydrocarbons in Titan's atmosphere

Abstract Results of the far uv photolysis of methane diluted 1000 times with argon are presented for temperatures of 300 and 200°K. It was found that ethylene and acetylene reached a temperature-dependent steady state, while ethane and propane continued to rise linearly with time. The experimental ethylene column abundance in a photochemical steady state was extrapolated to the temperatures expected in the methane photolysis layer on Titan. This, together with Gilletts [(1975) Astrophys. J. 201] observations, make it possible to set a lower limit to the altitude of the top of Titans aerosol layer, below which the ethylene is protected from photolysis. We find this to be 4.8 or 7.2 scale heights above the surface, for mean temperatures in the photolysis layer of 130 and 160°K, respectively. For 130°K, the ethylene abundance above the inversion base agrees well with Gilletts observations, and leads to an aerosol cloud top coincident with the inversion base. In order to transport ethane to the cold (78°K) “surface” with a flux sufficiently large so that the rate of condensation equals the rate of production, an eddy diffusion coefficient at the surface of at least 102 cm2 sec−1 is required. The experimental results lead to an acetylene abundance in a photochemical steady state whicch is ∼2 orders of magnitude smaller than the one suggested by Strobel [(1974) Icarus 21]. Since propane is one of the major products, the possibility of observing it at 13.4 μm is suggested.