A Continuum Description of Rarefied Gas Dynamics (I)--- Derivation From Kinetic Theory

We describe an asymptotic procedure for deriving continuum equations from the kinetic theory of a simple gas. As in the works of Hilbert, of Chapman and of Enskog, we expand in the mean flight time of the constituent particles of the gas, but we do not adopt the Chapman-Enskog device of simplifying the formulae at each order by using results from previous orders. In this way, we are able to derive a new set of fluid dynamical equations from kinetic theory, as we illustrate here for the relaxation model for monatomic gases. We obtain a stress tensor that contains a dynamical pressure term (or bulk viscosity) that is process-dependent and our heat current depends on the gradients of both temperature and density. On account of these features, the equations apply to a greater range of Knudsen number (the ratio of mean free path to macroscopic scale) than do the Navier-Stokes equations, as we see in the accompanying paper. In the limit of vanishing Knudsen number, our equations reduce to the usual Navier-Stokes equations with no bulk viscosity.Comment: 16 page

Radiative Transfer in Star Formation: Testing FLD and Hybrid Methods

We perform a comparison between two radiative transfer algorithms commonly employed in hydrodynamical calculations of star formation: grey flux limited diffusion and the hybrid scheme, in addition we compare these algorithms to results from the Monte-Carlo radiative transfer code MOCASSIN. In disc like density structures the hybrid scheme performs significantly better than the FLD method in the optically thin regions, with comparable results in optically thick regions. In the case of a forming high mass star we find the FLD method significantly underestimates the radiation pressure by a factor of ~100.Comment: 4 Pages; to appear in the proceedings of 'The Labyrinth of Star Formation', Crete, 18-22 June 201

A finite volume method for a convection- diffusion equation involving a Joule term

International audienceThis work is devoted to a Finite Volume method to approximate the solution of a convection-diffusion equation involving a Joule term. We propose a way 5 to discretize this so-called "Joule effect" term in a consistent way with the non linear diffusion one, in order to ensure some maximum principle properties on the solution. We then investigate the numerical behavior of the scheme on two original benchmarks

Vanishing viscosity limits for the degenerate lake equations with Navier boundary conditions

The paper is concerned with the vanishing viscosity limit of the two-dimensional degenerate viscous lake equations when the Navier slip conditions are prescribed on the impermeable boundary of a simply connected bounded regular domain. When the initial vorticity is in the Lebesgue space $L^q$ with $2<q\le\infty$, we show the degenerate viscous lake equations possess a unique global solution and the solution converges to a corresponding weak solution of the inviscid lake equations. In the special case when the vorticity is in $L^\infty$, an explicit convergence rate is obtained

Critical Protoplanetary Core Masses in Protoplanetary Disks and the Formation of Short-Period Giant Planets

We study a solid protoplanetary core of 1-10 earth masses migrating through a disk. We suppose the core luminosity is generated as a result of planetesimal accretion and calculate the structure of the gaseous envelope assuming equilibrium. This is a good approximation when the core mass is less than the critical value, M_{crit}, above which rapid gas accretion begins. We model the structure of the protoplanetary nebula as an accretion disk with constant \alpha. We present analytic fits for the steady state relation between disk surface density and mass accretion rate as a function of radius r. We calculate M_{crit} as a function of r, gas accretion rate through the disk, and planetesimal accretion rate onto the core \dot{M}. For a fixed \dot{M}, M_{crit} increases inwards, and it decreases with \dot{M}. We find that \dot{M} onto cores migrating inwards in a time 10^3-10^5 yr at 1 AU is sufficient to prevent the attainment of M_{crit} during the migration process. Only at small radii where planetesimals no longer exist can M_{crit} be attained. At small radii, the runaway gas accretion phase may become longer than the disk lifetime if the core mass is too small. However, massive cores can be built-up through the merger of additional incoming cores on a timescale shorter than for in situ formation. Therefore, feeding zone depletion in the neighborhood of a fixed orbit may be avoided. Accordingly, we suggest that giant planets may begin to form early in the life of the protostellar disk at small radii, on a timescale that may be significantly shorter than for in situ formation. (abridged)Comment: 24 pages (including 9 figures), LaTeX, uses emulateapj.sty, to be published in ApJ, also available at http://www.ucolick.org/~ct/home.htm