Pascale Garaud

The Effect of Internal Dissipation and Surface Irradiation on the Structure of Disks and the Location of the Snow Line around Sun-like Stars

In theory of accretion disks, angular momentum and mass transfer are associated with the generation of energy through viscous dissipation. In the construction of SED models of protostellar disks, the stellar irradiation is usually assumed to be the dominant heating source. Here we construct a new set of self-consistent analytical disk models by taking into account both sources of thermal energy and the thermal structure of the disk across the midplane. We deduce a set of general formulae for the relationship between the mass accretion rate and the surface density profile. We apply it to determine the structure of protostellar disks under a state of steady accretion and derive the radial distribution of surface density and midplane temperature. The incorporation of the viscous heating in our model reduces the disk flaring angle and leads to lower photospheric temperatures than previously thought. Around T Tauri stars, the snow line can evolve from outside 10 AU during FU Orionis outbursts, to 2 AU during the quasi-steady accretion phase, to 0.7 AU when the accretion rate falls to about 10-9 M☉ yr-1, and finally reexpand beyond 2.2 AU during the protostellar-to-debris disk transition. The nonmonotonous evolution of the snow line may lead to the observed isotopic composition of water on both Venus and Earth. We also infer the presence of a marginally opaque, isothermal region with a surface density distribution similar to that of the MSN model. With a 40% higher temperature than that in the region immediately within, this transition may lead to an upturn in the SEDs in the MIR (24-70 μm) wavelength range. The optically thin, outermost regions of the disk have a shallow surface density profile of the dust that is consistent with millimeter observations of spatially resolved disks.

FROM DUST TO PLANETESIMALS: AN IMPROVED MODEL FOR COLLISIONAL GROWTH IN PROTOPLANETARY DISKS

Planet formation occurs within the gas- and dust-rich environments of protoplanetary disks. Observations of these objects show that the growth of primordial submicron-sized particles into larger aggregates occurs at the earliest evolutionary stages of the disks. However, theoretical models of particle growth that use the Smoluchowski equation to describe collisional coagulation and fragmentation have so far failed to produce large particles while maintaining a significant population of small grains. This has generally been attributed to the existence of two barriers impeding growth due to bouncing and fragmentation of colliding particles. In this paper, we demonstrate that the importance of these barriers has been artificially inflated through the use of simplified models that do not take into account the stochastic nature of the particle motions within the gas disk. We present a new approach in which the relative velocities between two particles are described by a probability distribution function that models both deterministic motion (from the vertical settling, radial drift, and azimuthal drift) and stochastic motion (from Brownian motion and turbulence). Taking both into account can give quite different results to what has been considered recently in other studies. We demonstrate the vital effect of two ingredients for particle growth: the proper implementation of a velocity distribution function that overcomes the bouncing barrier and, in combination with mass transfer in high-mass-ratio collisions, boosts the growth of larger particles beyond the fragmentation barrier. A robust result of our simulations is the emergence of two particle populations (small and large), potentially explaining simultaneously a number of longstanding problems in protoplanetary disks, including planetesimal formation close to the central star, the presence of?millimeter- to centimeter-sized particles far out in the disk, and the persistence of ?m-sized grains for millions of years.

On the Evolution and Stability of a Protoplanetary Disk Dust Layer

In this manuscript we perform detailed analytical and numerical studies of the physical processes that result from the interaction between the dust and gas components of a protostellar accretion disk. We consider the most favorable condition for dust sedimentation in a laminar gaseous background and look at the conditions that precede the onset of gravitational and shear instability. We adopt a two-fluid formalism to examine two issues of particular interest: (1) the slow sedimentation of a laminar dust layer, for which we are able to extract self-similar solutions, and (2) the rapid growth of perturbations to that slowly settling shear flow, for which we present a linear stability analysis. We also include two fundamental physical processes that have been ignored in previous work: the specificities of the two-fluid equations compared to the single-fluid approximation, and the potentially destabilizing effect of radiative cooling by the reduction of the buoyancy of fluid motions. From these results we are able to compare the conditions for onset of gravitational and shearing instabilities for various dust-to-gas surface density ratios and nebular structures. We confirm previous findings that during the dust sedimentation, the shearing instability occurs prior to the gravitational instability unless the disk is grossly enriched up to a level where the surface density of grains is comparable to that of the gas. Finally, we discuss the implications of these results with respect to theories of planetesimal formation and observations of protoplanetary disks.

Dynamics of the solar tachocline – I. An incompressible study

Gough & McIntyre have suggested that the dynamics of the solar tachocline are dominated by the advection-diffusion balance between the differential rotation, a large-scale primordial field and baroclinicly driven meridional motions. This paper presents the first part of a study of the tachocline, in which a model of the rotation profile below the convection zone is constructed along the lines suggested by Gough & McIntyre and solved numerically. In this first part, a reduced model of the tachocline is derived in which the effects of compressibility and energy transport on the system are neglected; the meridional motions are driven instead by Ekman-Hartmann pumping. Through this simplification, the interaction of the fluid flow and the magnetic field can be isolated and is studied through non-linear numerical analysis for various field strengths and diffusivities. It is shown that there exists only a narrow range of magnetic field strengths for which the system can achieve a nearly uniform rotation. The results are discussed with respect to observations and to the limitations of this initial approach. A following paper combines the effects of realistic baroclinic driving and stratification with a model that closely follows the lines of work of Gough & McIntyre.

ASSEMBLING THE BUILDING BLOCKS OF GIANT PLANETS AROUND INTERMEDIATE-MASS STARS

We examine a physical process that leads to the efficient formation of gas giant planets around intermediate-mass stars. In the gaseous protoplanetary disks surrounding rapidly accreting intermediate-mass stars, we show that the midplane temperature (heated primarily by turbulent dissipation) can reach 1000 K out to 1 AU. The thermal ionization of this hot gas couples the disk to the magnetic field, allowing the magnetorotational instability (MRI) to generate turbulence and transport angular momentum. Further from the central star the ionization fraction decreases, decoupling the disk from the magnetic field and reducing the efficiency of angular momentum transport. As the disk evolves toward a quasi-steady state, a local maximum in the surface density and in the midplane pressure both develop at the inner edge of the MRI-dead zone, trapping inwardly migrating solid bodies. Small particles accumulate and coagulate into planetesimals which grow rapidly until they reach isolation mass. In contrast to the situation around solar-type stars, we show that the isolation mass for cores at this critical radius around the more-massive stars is large enough to promote the accretion of significant amounts of gas prior to disk depletion. Through this process, we anticipate a prolific production of gas giants at ~1 AU around intermediate-mass stars.

TURBULENT MIXING AND LAYER FORMATION IN DOUBLE-DIFFUSIVE CONVECTION: THREE-DIMENSIONAL NUMERICAL SIMULATIONS AND THEORY

Double-diffusive convection, often referred to as semi-convection in astrophysics, occurs in thermally and compositionally stratified systems which are stable according to the Ledoux-criterion but unstable according to the Schwarzchild criterion. This process has been given relatively little attention so far, and its properties remain poorly constrained. In this paper, we present and analyze a set of three-dimensional simulations of this phenomenon in a Cartesian domain under the Boussinesq approximation. We find that in some cases the double-diffusive convection saturates into a state of homogeneous turbulence, but with turbulent fluxes several orders of magnitude smaller than those expected from direct overturning convection. In other cases the system rapidly and spontaneously develops closely-packed thermo-compositional layers, which later successively merge until a single layer is left. We compare the output of our simulations with an existing theory of layer formation in the oceanographic context, and find very good agreement between the model and our results. The thermal and compositional mixing rates increase significantly during layer formation, and increase even further with each merger. We find that the heat flux through the staircase is a simple function of the layer height. We conclude by proposing a new approach to studying transport by double-diffusive convection in astrophysics.

Growth and Migration of Solids in Evolving Protostellar Disks. I. Methods and Analytical Tests

This series of papers investigates the early stages of planet formation by modeling the evolution of the gas and solid content of protostellar disks from the early T Tauri phase until complete dispersal of the gas. In this first paper, I present a new set of simplified equations modeling the growth and migration of various species of grains in a gaseous protostellar disk evolving as a result of the combined effects of viscous accretion and photoevaporation from the central star. Using the assumption that the grain-size distribution function always maintains a power-law structure approximating the average outcome of the exact coagulation/shattering equation, the model focuses on the calculation of the growth rate of the largest grains only. The coupled evolution equations for the maximum grain size, the surface density of the gas, and the surface density of solids are then presented and solved self-consistently using a standard -->1 + 1 dimensional formalism. I show that the global evolution of solids is controlled by a leaky reservoir of small grains at large radii, and propose an empirically derived evolution equation for the total mass of solids, which can be used to estimate the total heavy-element retention efficiency in the planet formation paradigm. Detailed comparisons with SED observations are presented in a following paper.

Individual and Average Behavior of Particles in a Protoplanetary Nebula

We study the interaction between gas and particles in a protoplanetary disk, using both analytical and numerical approaches. We first present analytical expressions for the trajectories of individual particles undergoing gas drag in the disk, in the asymptotic cases of very small particles (Epstein regime) and very large particles (Stokes regime). Using a Boltzmann averaging method, we obtain an analytic expression for the evolution of the average density, velocity, and dispersion of the particles as a function of distance above the midplane of the disk. Using successive moments of the Boltzmann equation, we derive the equivalent fluid equations for the average motion of the particles; these are intrinsically different in the Epstein and Stokes regimes. A simple closure of the moment equations is proposed in both regimes. These fluid equations provide much better prospects for the study of more complex problems related to protoplanetary accretion disks, since for general initial size and phase-space distributions the evolution of the average behavior of the particles can be evaluated numerically with much less computational time than that required for the numerical integration of the orbits of all individual particles. In a companion paper, for instance, we use them for the analysis of a shearing instability induced by the sedimentation of the particles. In the present work we test the adequacy of the fluid formulation against a set of idealized numerical experiments. In the Epstein regime, we study an idealized uniform initial distribution of small particles. We obtain a set of analytic solutions for the fluid equations, which are found to be in good agreement with those obtained from numerical integration of the orbits of many particles. We also verify that any initial velocity dispersion is quickly damped out by the surrounding gas on the short stopping timescale, which provides closure and justifies the description of the particles as a fluid with a linear drag force and negligible pressure. In the Stokes regime, as the large particles oscillate across the midplane with declining amplitude, their velocity dispersion remains comparable to their average speed. Their sedimentation is analogous to the cooling of a pressure-supported fluid. We propose an empirical closure scheme for the moment equations of the Stokes particles fluid and test it against idealized numerical experiments. In both cases, this method can eventually be applied to study the evolution of particle distributions in protostellar disks after additional effects such as collision, sublimation, and condensation are included.

Chemical Transport and Spontaneous Layer Formation in Fingering Convection in Astrophysics

A region of a star that is stable to convection according to the Ledoux criterion may nevertheless undergo additional mixing if the mean molecular weight increases with radius. This process is called fingering (thermohaline) convection and may account for some of the unexplained mixing in stars such as those that have been polluted by planetary infall and those burning 3He. We propose a new model for mixing by fingering convection in the parameter regime relevant for stellar (and planetary) interiors. Our theory is based on physical principles and supported by three-dimensional direct numerical simulations. We also discuss the possibility of formation of thermocompositional staircases in fingering regions, and their role in enhancing mixing. Finally, we provide a simple algorithm to implement this theory in one-dimensional stellar codes, such as KEPLER and MESA.

Dynamics of fingering convection I: Small-scale fluxes and large-scale instabilities

(Received ?? and in revised form ??) Double-diffusive instabilities are often invoked to explain enhanced transport in stablystratified fluids. The most-studied natural manifestation of this process, fingering convection, commonly occurs in the ocean’s thermocline and typically increases diapycnal mixing by two orders of magnitude over molecular diffusion. Fingering convection is also often associated with structures on much larger scales, such as thermohaline intrusions, gravity waves and thermohaline staircases. In this paper, we present an exhaustive study of the phenomenon from small to large scales. We perform the first three-dimensional simulations of the process at realistic values of the heat and salt diffusivities and provide accurate estimates of the induced turbulent transport. Our results are consistent with oceanic field measurements of diapycnal mixing in fingering regions. We then develop a generalized mean-field theory to study the stability of fingering systems to large-scale perturbations, using our calculated turbulent fluxes to parameterize small-scale transport. The theory recovers the intrusive instability, the collective instability, and the instability as limiting cases. We find that the fastest-growing large-scale mode depends sensitively on the ratio of the background gradients of temperature and salinity (the density ratio). While only intrusive modes exist at high density ratios, the collective and -instabilities dominate the system at the low density ratios where staircases are typically observed. We conclude by discussing our findings in the context of staircase formation theory.