Spiral-driven accretion in protoplanetary discs - I. 2D models

We numerically investigate the dynamics of a 2D non-magnetised protoplanetary disc surrounded by an inflow coming from an external envelope. We find that the accretion shock between the disc and the inflow is unstable, leading to the generation of large-amplitude spiral density waves. These spiral waves propagate over long distances, down to radii at least ten times smaller than the accretion shock radius. We measure spiral-driven outward angular momentum transport with 1e-4 1e-8 Msun/yr. We conclude that the interaction of the disc with its envelope leads to long-lived spiral density waves and radial angular momentum transport with rates that cannot be neglected in young non-magnetised protostellar discs.Comment: 4 pages, 4 figures, accepted in A&A Letter

A simple toy model of the advective-acoustic instability. II. Numerical simulations

The physical processes involved in the advective-acoustic instability are investigated with 2D numerical simulations. Simple toy models, developped in a companion paper, are used to describe the coupling between acoustic and entropy/vorticity waves, produced either by a stationary shock or by the deceleration of the flow. Using two Eulerian codes based on different second order upwind schemes, we confirm the results of the perturbative analysis. The numerical convergence with respect to the computation mesh size is studied with 1D simulations. We demonstrate that the numerical accuracy of the quantities which depend on the physics of the shock is limited to a linear convergence. We argue that this property is likely to be true for most current numerical schemes dealing with SASI in the core-collapse problem, and could be solved by the use of advanced techniques for the numerical treatment of the shock. We propose a strategy to choose the mesh size for an accurate treatment of the advective-acoustic coupling in future numerical simulations.Comment: 9 pages, 10 figures, ApJ in press, new Sect. 5 and Fig.

Dynamics of an Alfven surface in core collapse supernovae

We investigate the dynamics of an Alfven surface (where the Alfven speed equals the advection velocity) in the context of core collapse supernovae during the phase of accretion on the proto-neutron star. Such a surface should exist even for weak magnetic fields because the advection velocity decreases to zero at the center of the collapsing core. In this decelerated flow, Alfven waves created by the standing accretion shock instability (SASI) or convection accumulate and amplify while approaching the Alfven surface. We study this amplification using one dimensional MHD simulations with explicit physical dissipation. In the linear regime, the amplification continues until the Alfven wavelength becomes as small as the dissipative scale. A pressure feedback that increases the pressure in the upstream flow is created via a non linear coupling. We derive analytic formulae for the maximum amplification and the non linear coupling and check them with numerical simulations to a very good accuracy. We also characterize the non linear saturation of this amplification when compression effects become important, leading to either a change of the velocity gradient, or a steepening of the Alfven wave. Applying these results to core collapse supernovae shows that the amplification can be fast enough to affect the dynamics, if the magnetic field is strong enough for the Alfven surface to lie in the region of strong velocity gradient just above the neutrinosphere. This requires the presence of a strong magnetic field in the progenitor star, which would correspond to the formation of a magnetar under the assumption of magnetic flux conservation. An extrapolation of our analytic formula (taking into account the nonlinear saturation) suggests that the Alfven wave could reach an amplitude of B ~ 10^15 G, and that the pressure feedback could significantly contribute to the pressure below the shock.Comment: 18 pages, 14 figures, accepted for publication in ApJ. Added a discussion of the energy budget in subsection 7.

Midplane sedimentation of large solid bodies in turbulent protoplanetary discs

We study the vertical settling of solid bodies in a turbulent protoplanetary disc. We consider the situation when the coupling to the gas is weak or equivalently when the particle stopping time tau_{st} due to friction with the gas is long compared to the orbital timescale Omega^{-1}. An analytical model, which takes into account the stochastic nature of the sedimentation process using a Fokker-Planck equation for the particle distribution function in phase space, is used to obtain the vertical scale height of the solid layer as a function of the vertical component of the turbulent gas velocity correlation function and the particle stopping time. This is found to be of the same form as the relation obtained for strongly coupled particles in previous work. We compare the predictions of this model with results obtained from local shearing box MHD simulations of solid particles embedded in a vertically stratified disc in which there is turbulence driven by the MRI. We find that the ratio of the dust disc thickness to the gas disc thickness satifies H_d/H=0.08 (Omega tau_{st})^{-1/2}, which is in very good agreement with the analytical model. By discussing the conditions for gravitational instability in the outer regions of protoplanetary discs in which there is a similar level of turbulence, we find that bodies in the size range 50 to 600 metres can aggregate to form Kuiper belt-like objects with characteristic radii ranging from tens to hundreds of kilometres.Comment: 8 pages, 4 figures, accepted in MNRA

Numerical simulations of type I planetary migration in nonturbulent magnetized discs

Using 2D MHD numerical simulations performed with two different finite difference Eulerian codes, we analyze the effect that a toroidal magnetic field has on low mass planet migration in nonturbulent protoplanetary discs. The presence of the magnetic field modifies the waves that can propagate in the disc. In agreement with a recent linear analysis (Terquem 2003), we find that two magnetic resonances develop on both sides of the planet orbit, which contribute to a significant global torque. In order to measure the torque exerted by the disc on the planet, we perform simulations in which the latter is either fixed on a circular orbit or allowed to migrate. For a 5 earth mass planet, when the ratio \beta between the square of the sound speed and that of the Alfven speed at the location of the planet is equal to 2, we find inward migration when the magnetic field B_{\phi} is uniform in the disc, reduced migration when B_{\phi} decreases as r^{-1} and outward migration when B_{\phi} decreases as r^{-2}. These results are in agreement with predictions from the linear analysis. Taken as a whole, our results confirm that even a subthermal stable field can stop inward migration of an earth--like planet.Comment: 17 pages, 12 figures, accepted in MNRAS. A version with full resolution, colour figures is available at http://www.maths.qmul.ac.uk/~rpn/preprints

On the aerodynamic redistribution of chondrite components in protoplanetary disks

Despite being all roughly of solar composition, primitive meteorites (chondrites) present a diversity in their chemical, isotopic and petrographic properties, and in particular a first-order dichotomy between carbonaceous and non-carbonaceous chondrites. We investigate here analytically the dynamics of their components (chondrules, refractory inclusions, metal/sulfide and matrix grains) in protoplanetary disks prior to their incorporation in chondrite parent bodies. We find the dynamics of the solids, subject to gas drag, to be essentially controlled by the "gas-solid decoupling parameter" $S\equiv \textrm{St}/\alpha$, the ratio of the dimensionless stopping time to the turbulence parameter. The decoupling of the solid particles relative to the gas is significant when $S$ exceeds unity. $S$ is expected to increase with time and heliocentric distance. On the basis of (i) abundance of refractory inclusions (ii) proportion of matrix (iii) lithophile element abundances and (iv) oxygen isotopic composition of chondrules, we propose that non-matrix chondritic components had $S1$ when the other chondrites accreted. This suggests that accretion of carbonaceous chondrites predated on average that of the other chondrites and that refractory inclusions are genetically related to their host carbonaceous chondrites.Comment: 13 pages, 6 figures, accepted to Icaru

The ionization fraction in alpha-models of protoplanetary disks

We calculate the ionization fraction of protostellar alpha disks, taking into account vertical temperature structure, and the possible presence of trace metal atoms. Both thermal and X-ray ionization are considered. Previous investigations of layered disks used radial power-law models with isothermal vertical structure. But alpha models are used to model accretion, and the present work is a step towards a self-consistent treatment. The extent of the magnetically uncoupled (``dead'') zone depends sensitively on alpha, on the assumed accretion rate, and on the critical magnetic Reynolds number, below which MHD turbulence cannot be self-sustained. Its extent is extremely model-dependent. It is also shown that a tiny fraction of the cosmic abundance of metal atoms can dramatically affect the ionization balance. Gravitational instabilities are an unpromising source of transport, except in the early stages of disk formation.Comment: 25 pages including 8 figures, Latex in the MN style - Accepted by MNRA

Kinematic Dynamos using Constrained Transport with High Order Godunov Schemes and Adaptive Mesh Refinement

We propose to extend the well-known MUSCL-Hancock scheme for Euler equations to the induction equation modeling the magnetic field evolution in kinematic dynamo problems. The scheme is based on an integral form of the underlying conservation law which, in our formulation, results in a ``finite-surface'' scheme for the induction equation. This naturally leads to the well-known ``constrained transport'' method, with additional continuity requirement on the magnetic field representation. The second ingredient in the MUSCL scheme is the predictor step that ensures second order accuracy both in space and time. We explore specific constraints that the mathematical properties of the induction equations place on this predictor step, showing that three possible variants can be considered. We show that the most aggressive formulations (referred to as C-MUSCL and U-MUSCL) reach the same level of accuracy as the other one (referred to as Runge-Kutta), at a lower computational cost. More interestingly, these two schemes are compatible with the Adaptive Mesh Refinement (AMR) framework. It has been implemented in the AMR code RAMSES. It offers a novel and efficient implementation of a second order scheme for the induction equation. We have tested it by solving two kinematic dynamo problems in the low diffusion limit. The construction of this scheme for the induction equation constitutes a step towards solving the full MHD set of equations using an extension of our current methodology.Comment: 40 pages, 10 figures, accepted in Journal of Computational Physics. A version with full resolution is available at http://www.damtp.cam.ac.uk/user/fromang/publi/TFD.pd