Closed-form expressions for particle relative velocities induced by turbulence

In this note we present complete, closed-form expressions for random relative velocities between colliding particles of arbitrary size in nebula turbulence. These results are exact for very small particles (those with stopping times much shorter than the large eddy overturn time) and are also surprisingly accurate in complete generality (that is, also apply for particles with stopping times comparable to, or much longer than, the large eddy overturn time). We note that some previous studies may have adopted previous simple expressions, which we find to be in error regarding the size dependence in the large particle regime.Comment: 8 pages, accepted as Research Note by A&

The Spherically Symmetric Gravitational Collapse of a Clump of Solids in a Gas

Several mechanisms have been identified that create dense particle clumps in the solar nebula. The present work is concerned with the gravitational collapse of such clumps, idealized as being spherically symmetric. Calculations using the two-fluid model are performed (almost) up to the time when a central density singularity forms. The end result of the study is a parametrization for this time, in order that it may be compared with timescales for various disruptive effects to which clumps may be subject. An important effect is that as the clump compresses, it also compresses the gas due to drag. This increases gas pressure which retards particle collapse and leads to oscillation in the size and density of the clump. The ratio of gravitational force to gas pressure gives a two-phase Jeans parameter, $J_t$, which is the classical Jeans parameter with the sound speed replaced by an the wave speed in a coupled two-fluid medium. Its use makes the results insensitive to the initial density ratio of particles to gas as a separate parameter. An ordinary differential equation model is developed which takes the form of two coupled non-linear oscillators and reproduces key features of the simulations. Finally, a parametric study of the time to collapse is performed and a formula (fit to the simulations) is developed. In the incompressible limit $J_t \to 0$, collapse time equals sedimentation time. As $J_t$ increases, the collapse time decreases roughly linearly with $J_t$ until $J_t \gtrsim 0.4$ when it becomes approximately equal to the dynamical time

The rings of Saturn: State of current knowledge and some suggestions for future studies

The state of our current knowledge of the properties of the ring system as a whole, and of the particles individually, is assessed. Attention is primarily devoted to recent results and possibilities for exploration of the ring system by a Saturn orbiter. In particular, the infrared and microwave properties of the ring system are discussed. The behavior of the ring brightness is not well understood in the critical transition spectral region from approximately 100 micrometers to approximately 1 cm. Also, the dynamical behavior of the ring system is discussed. Recent theoretical studies show that ongoing dynamical effects continually affect the ring structure in azimuth (possibly producing the A ring brightness asymmetry) and in the vertical direction. Orbital spacecraft-based studies of the rings will offer several unique advantages and impact important cosmogonical questions. Bistatic radar studies and millimeter-wavelength spectrometer/radiometry will give particle sizes and composition limits needed to resolve the question of the density of the rings, and provide important boundary conditions on the state of Saturn's protoplanetary nebula near the time of planetary formation

The Runaway Greenhouse: A History of Water on Venus

Radiative-convective equilibrium models of planetary atmospheres are discussed for the case when the infrared opacity is due to a vapor in equilibrium with its liquid or solid phase. For a grey gas, or for a gas which absorbs at all infrared wavelengths, equilibrium is impossible when the solar constant exceeds a critical value. Equilibrium therefore requires that the condensed phase evaporates into the atmosphere. Moist adiabatic and pseudoadiabatic atmospheres in which the condensing vapor is a major atmospheric constituent are considered. This situation would apply if the solar constant were supercritical with respect to an abundant substance such as water. It is shown that the condensing gas would be a major constituent at all levels in such an atmosphere. Photodissociation of water in the primordial Venus atmosphere is discussed in this context

The Evolution of the Water Distribution in a Viscous Protoplanetary Disk

(Abridged) Astronomical observations have shown that protoplanetary disks are dynamic objects through which mass is transported and accreted by the central star. Age dating of meteorite constituents shows that their creation, evolution, and accumulation occupied several Myr, and over this time disk properties would evolve significantly. Moreover, on this timescale, solid particles decouple from the gas in the disk and their evolution follows a different path. Here we present a model which tracks how the distribution of water changes in an evolving disk as the water-bearing species experience condensation, accretion, transport, collisional destruction, and vaporization. Because solids are transported in a disk at different rates depending on their sizes, the motions will lead to water being concentrated in some regions of a disk and depleted in others. These enhancements and depletions are consistent with the conditions needed to explain some aspects of the chemistry of chondritic meteorites and formation of giant planets. The levels of concentration and depletion, as well as their locations, depend strongly on the combined effects of the gaseous disk evolution, the formation of rapidly migrating rubble, and the growth of immobile planetesimals. We present examples of evolution under a range of plausible assumptions and demonstrate how the chemical evolution of the inner region of a protoplanetary disk is intimately connected to the physical processes which occur in the outer regions.Comment: 45 pages, 7 figures, revised for publication in Icaru

Accretion of dust by chondrules in a MHD-turbulent solar nebula

(Abridged) Numerical magnetohydrodynamic (MHD) simulations of a turbulent solar nebula are used to study the growth of dust mantles swept up by chondrules. A small neighborhood of the solar nebula is represented by an orbiting patch of gas at a radius of 3 AU, and includes vertical stratification of the gas density. The differential rotation of the nebular gas is replaced by a shear flow. Turbulence is driven by destabilization of the flow as a result of the magnetorotational instability (MRI), whereby magnetic field lines anchored to the gas are continuously stretched by the shearing motion. A passive contaminant mimics small dust grains that are aerodynamically well coupled to the gas, and chondrules are modeled by Lagrangian particles that interact with the gas through drag. Whenever a chondrule enters a region permeated by dust, its radius grows at a rate that depends on the local dust density and the relative velocity between itself and the dust. The local dust abundance decreases accordingly. Different chondrule volume densities lead to varying depletion and rimmed-chondrule size growth times. Most of the dust sweep-up occurs within 1 gas scale height of the nebula midplane. Chondrules can reach their asymptotic radius in 10 to 800 years. The vertical variation of nebula turbulent intensity results in a moderate dependence of mean rimmed-chondrule size with nebula height. The technique used here could be combined with Monte Carlo (MC) methods that include the physics of dust compaction, in a self-consistent MHD-MC model of dust rim growth around chondrules in the solar nebula.Comment: 33 pages, 9 figures. Icarus, in pres

Scale Dependence of Multiplier Distributions for Particle Concentration, Enstrophy and Dissipation in the Inertial Range of Homogeneous Turbulence

Turbulent flows preferentially concentrate inertial particles depending on their stopping time or Stokes number, which can lead to significant spatial variations in the particle concentration. Cascade models are one way to describe this process in statistical terms. Here, we use a direct numerical simulation (DNS) dataset of homogeneous, isotropic turbulence to determine probability distribution functions (PDFs) for cascade multipliers, which determine the ratio by which a property is partitioned into sub-volumes as an eddy is envisioned to decay into smaller eddies. We present a technique for correcting effects of small particle numbers in the statistics. We determine multiplier PDFs for particle number, flow dissipation, and enstrophy, all of which are shown to be scale dependent. However, the particle multiplier PDFs collapse when scaled with an appropriately defined local Stokes number. As anticipated from earlier works, dissipation and enstrophy multiplier PDFs reach an asymptote for sufficiently small spatial scales. From the DNS measurements, we derive a cascade model that is used it to make predictions for the radial distribution function (RDF) for arbitrarily high Reynolds numbers, $Re$, finding good agreement with the asymptotic, infinite $Re$ inertial range theory of Zaichik and Alipchenkov [New Journal of Physics 11, 103018 (2009)]. We discuss implications of these results for the statistical modeling of the turbulent clustering process in the inertial range for high Reynolds numbers inaccessible to numerical simulations.Comment: 21 pages, 14 figures, accepted for publication in Physical Review

Growth of Dust as the Initial Step Toward Planet Formation

We discuss the results of laboratory measurements and theoretical models concerning the aggregation of dust in protoplanetary disks, as the initial step toward planet formation. Small particles easily stick when they collide and form aggregates with an open, often fractal structure, depending on the growth process. Larger particles are still expected to grow at collision velocities of about 1m/s. Experiments also show that, after an intermezzo of destructive velocities, high collision velocities above 10m/s on porous materials again lead to net growth of the target. Considerations of dust-gas interactions show that collision velocities for particles not too different in surface-to-mass ratio remain limited up to sizes about 1m, and growth seems to be guaranteed to reach these sizes quickly and easily. For meter sizes, coupling to nebula turbulence makes destructive processes more likely. Global aggregation models show that in a turbulent nebula, small particles are swept up too fast to be consistent with observations of disks. An extended phase may therefore exist in the nebula during which the small particle component is kept alive through collisions driven by turbulence which frustrates growth to planetesimals until conditions are more favorable for one or more reasons.Comment: Protostars and Planets V (PPV) review. 18 pages, 5 figure

Towards Initial Mass Functions for Asteroids and Kuiper Belt Objects

Our goal is to understand primary accretion of the first planetesimals. The primitive meteorite record suggests that sizeable planetesimals formed in the asteroid belt over a period longer than a million years, each composed entirely of an unusual, but homogeneous, mixture of mm-size particles. We sketch a scenario in which primary accretion of 10-100km size planetesimals proceeds directly, if sporadically, from aerodynamically-sorted mm-size particles (generically "chondrules"). These planetesimal sizes are in general agreement with the currently observed asteroid mass peak near 100km diameter, which has been identified as a "fossil" property of the pre-erosion, pre-depletion population. We extend our primary accretion theory to make predictions for outer solar system planetesimals, which may also have a preferred size in the 100km diameter range. We estimate formation rates of planetesimals and assess the conditions needed to match estimates of both asteroid and Kuiper Belt Object (KBO) formation rates. For nebula parameters that satisfy observed mass accretion rates of Myr-old protoplanetary nebulae, the scenario is roughly consistent with not only the "fossil" sizes of the asteroids, and their estimated production rates, but also with the observed spread in formation ages of chondrules in a given chondrite, and with a tolerably small radial diffusive mixing during this time between formation and accretion (the model naturally helps explain the peculiar size distribution of chondrules within such objects). The scenario also produces 10-100km diameter primary KBOs. The optimum range of parameters, however, represents a higher gas density and fractional abundance of solids, and a smaller difference between keplerian and pressure-supported orbital velocities, than "canonical" models of the solar nebula. We discuss several potential explanations for these differences.Comment: Icarus, in pres