It is known that an embedded massive planet will open a gap in a protoplanetary disc via angular momentum exchange with the disc material. The resulting surface density profile of the disc is investigated for one dimensional and two dimensional disc models and, in agreement with previous work, it is found that one dimensional gaps are significantly deeper than their two dimensional counterparts for the same initial conditions. We find, by applying one dimensional torque density distributions to two dimensional discs containing no planet, that the excitement of the Rossby wave instability and the formation of Rossby vortices play a critical role in setting the equilibrium depth of the gap. Being a two dimensional instability, this is absent from one dimensional simulations and does not limit the equilibrium gap depth there. We find similar gap depths between two dimensional gaps formed by torque density distributions, in which the Rossby wave instability is present, and two dimensional planet gaps, in which no Rossby wave instability is present. This can be understood if the planet gap is maintained at marginal stability, even when there is no obvious Rossby wave instability present. Further investigation shows the final equilibrium gap depth is very sensitive to the form of the applied torque density distribution, and using improved one dimensional approximations from three dimensional simulations can go even further to reducing the discrepancy between one and two dimensional models, especially for lower mass planets. This behaviour is found to be consistent across discs with varying parameters.
We revisit, via a very simplified set of equations, a linear streaming instability (technically an overstability), which is present in, and potentially important for, dusty protoplanetary disks (Youdin & Goodman 2005). The goal is a better understanding of the physical origin of such instabilities, which are notoriously subtle. Rotational dynamics seem to be essential to this type of instability, which cannot be captured by one-dimensional Cartesian models. Dust `pileups' in moving pressure maxima are an important triggering mechanism of the instability, and drag feedback of dust upon the gas …
We perform 3D SPH simulations of warped, non-coplanar gravitationally unstable discs to show that as the warp propagates through the self-gravitating disc, it heats up the disc rendering it gravitationally stable. Thus losing their spiral structure and appearing completely axisymmetric. In their youth, protoplanetary discs are expected to be massive and self-gravitating, which results in non-axisymmetric spiral structures. However recent observations of young protoplanetary discs with ALMA have revealed that discs with large-scale spiral structure are rarely observed in the midplane. Instead, …
Under the right conditions, the streaming instability between imperfectly coupled dust and gas is a powerful mechanism for planetesimal formation as it can concentrate dust grains to the point of gravitational collapse. In its simplest form, the streaming instability can be captured by analyzing the linear stability of unstratified disk models, which represent the midplane of protoplanetary disks. We extend such studies by carrying out vertically-global linear stability analyses of dust layers in protoplanetary disks. We find the dominant form of instability in stratified dust layers is one dr…
We develop simple, physically motivated models for drag-induced dust-gas streaming instabilities, which are thought to be crucial for clumping grains to form planetesimals in protoplanetary disks. The models explain, based on the physics of gaseous epicyclic motion and dust-gas drag forces, the most important features of the streaming instability and its simple generalisation, the disk settling instability. Some of the key properties explained by our models include the sudden change in the growth rate of the streaming instability when the dust-to-gas-mass ratio surpasses one, the slow growth r…
The gravitational instabilities are important to the evolution of the disks and the planet formation in the disks. We calculate the evolution of the disks which form from the collapse of the molecular cloud cores. By changing the properties of the cloud cores and the hydrodynamical viscosity parameters, we explore their effects on the properties of the gravitational instabilities. We find that the disk is unstable when the angular velocity of the molecular cloud core is larger than a critical value. The time duration of the instability increases as the angular velocity of the core increases. T…
We identify and study a number of new, rapidly growing instabilities of dust grains in protoplanetary disks, which may be important for planetesimal formation. The study is based on the recognition that dust-gas mixtures are generically unstable to a Resonant Drag Instability (RDI), whenever the gas, absent dust, supports undamped linear modes. We show that the "streaming instability" is an RDI associated with epicyclic oscillations; this provides simple interpretations for its mechanisms and accurate analytic expressions for its growth rates and fastest-growing wavelengths. We extend this ana…
The streaming instability is one of the most promising pathways to the formation of planetesimals from pebbles. Understanding how this instability operates under realistic conditions expected in protoplanetary disks is therefore crucial to assess the efficiency of planet formation. Contemporary models of protoplanetary disks show that magnetic fields are key to driving gas accretion through large-scale, laminar magnetic stresses. However, the effect of such magnetic fields on the streaming instability has not been examined in detail. To this end, we study the stability of dusty, magnetized gas…
This chapter highlights the properties of turbulence and meso-scale flow structures in protoplanetary disks and their role in the planet formation process. Here we focus on the formation of planetesimals from a gravitational collapse of a pebble cloud. Large scale and long lived flow structures - vortices and zonal flows - are a consequence of weak magneto and hydrodynamic instabilities in the pressure and entropy stratified quasi-Keplerian shear flow interacting with the fast rotation of the disk. The vortices and zonal flows on the other hand are particle traps tapping into the radial pebble…
Rossby wave instabilities (RWIs) usually lead to nonaxisymmetric vortices in protoplanetary discs and some observed sub-structures of these discs can be well explained by RWIs. We explore how the cooling influences the growth rate of unstable RWI modes in terms of the linear perturbation analysis. The cooling associated with the energy equation is treated in two different ways. The first one we adopt is a simple cooling law. The perturbed thermal state relaxes to the initial thermal state on a prescribed cooling timescale. In the second, we treat the cooling as a thermal diffusion process. The…
The streaming instability (SI) is one of the most promising candidates for triggering planetesimal formation by producing dense dust clumps that undergo gravitational collapse. Understanding how the SI operates in realistic protoplanetary disks (PPDs) is therefore crucial to assess the efficiency of planetesimal formation. Modern models of PPDs show that large-scale magnetic torques or winds can drive laminar gas accretion near the disk midplane. In a previous study, we identified a new linear dust-gas instability, the azimuthal drift SI (AdSI), applicable to such accreting disks and is powere…
Streaming instabilities in weakly ionized protoplanetary discs: the Ambipolar Streaming Instability (AmSI)
Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS（波尔多天体物理实验室，波尔多大学，法国国家科学研究中心）；Institute of Astronomy and Astrophysics, Academia Sinica（中央研究院天文及天文物理研究所）；Physics Division, National Center for Theoretical Sciences（国家理论科学中心物理组）
In protoplanetary disks the aerodynamical friction between particles and gas induces a variety of instabilities that facilitate planet formation. Of these we examine the so-called `secular gravitational instability' (SGI) in the two-fluid approximation, deriving analytical expressions for its stability criteria and growth rates. Concurrently, we present a physical explanation of the instability that shows how it manifests upon an intermediate range of lengthscales exhibiting geostrophic balance in the gas component. In contrast to a single-fluid treatment, the SGI is quenched within a critical…
We use the Fokker-Planck equation and model the dispersive dynamics of solid particles in annular protoplanetary disks whose gas component is more massive than the particle phase. We model particle--gas interactions as hard sphere collisions, determine the functional form of diffusion coefficients, and show the existence of two global unstable modes in the particle phase. These modes have spiral patterns with the azimuthal wavenumber $m=1$ and rotate slowly. We show that in ring-shaped disks, the phase space density of solid particles increases linearly in time towards an accumulation point ne…
In the innermost regions of protoplanerary discs, the solid-to-gas ratio can be increased considerably by a number of processes, including photoevaporative and particle drift. MHD disc models also suggest the existence of a dead-zone at $R\lesssim 10$ AU, where the regions close to the midplane remain laminar. In this context, we use two-fluid hydrodynamical simulations to study the interaction between a low-mass planet ($\sim 1.7 \;{\rm M_\oplus}$) on a fixed orbit and an inviscid pebble-rich disc with solid-to-gas ratio $ε\ge 0.5$. For pebbles with Stokes numbers St=0.1, 0.5, multiple dusty …
We test the effect of assumptions about stellar motion on the behavior of gravitational instabilities in protoplanetary disks around solar-type stars by performing two simulations that are identical in all respects except the treatment of the star. In one simulation, the star is assumed to remain fixed at the center of the inertial reference frame. In the other, stellar motion is handled properly by including an indirect potential in the hydrodynamic equations to model the star's reference frame as one which is accelerated by star/disk interactions. The disks in both simulations orbit a solar …
Hydrodynamical instabilities in disks around young stars depend on the thermodynamic stratification of the disk and on the local rate of thermal relaxation. Here, we map the spatial extent of unstable regions for the Vertical Shear Instability (VSI), the Convective OverStability (COS), and the amplification of vortices via the Subcritical Baroclinic Instability (SBI). We use steady state accretion disk models, including stellar irradiation, accretion heating and radiative transfer. We determine the local radial and vertical stratification and thermal relaxation rate in the disk, in dependence …
This review examines recent theoretical developments in our understanding of turbulence in cold, non-magnetically active, planetesimal forming regions of protoplanetary disks which we refer to throughout as "Ohmic zones". We give a brief background introduction to the subject of disk turbulence followed by a terse pedagogical review of the phenomenology of hydrodynamic turbulence. The equations governing the dynamics of cold astrophysical disks are given and basic flow states are described. We discuss the Solberg-Hoiland conditions required for stability, and the three recently identified turb…
We carry out three dimensional smoothed particle hydrodynamics simulations to study the impact of planet-disc interactions on a gravitationally unstable protoplanetary disc. We find that the impact of a planet on the disc's evolution can be described by three scenarios. If the planet is sufficiently massive, the spiral wakes generated by the planet dominate the evolution of the disc and gravitational instabilities are completely suppressed. If the planet's mass is too small, then gravitational instabilities are unaffected. If the planet's mass lies between these extremes, gravitational instabi…
In this paper, we model the chemical evolution of a 0.25 M$_{\odot}$ protoplanetary disc surrounding a 1 M$_{\odot}$ star that undergoes fragmentation due to self-gravity. We use Smoothed Particle Hydrodynamics including a radiative transfer scheme, along with time-dependent chemical evolution code to follow the composition of the disc and resulting fragments over approximately 4000 yrs. Initially, four quasi-stable fragments are formed, of which two are eventually disrupted by tidal torques in the disc. From the results of our chemical modelling, we identify species that are abundant in the f…
Vertical shear instability in two-moment radiation-hydrodynamical simulations of irradiated protoplanetary disks II. Secondary instabilities and stability regions
Hydrodynamic, non-magnetic instabilities can provide turbulent stress in the regions of protoplanetary discs, where the MRI can not develop. The induced motions influence the grain growth, from which formation of planetesimals begins. Thermal relaxation of the gas constrains origins of the identified hydrodynamic sources of turbulence in discs. We estimate the radiative relaxation timescale of temperature perturbations and study the dependence of this timescale on the perturbation wavelength, the location within the disc, the disc mass, and the dust-to-gas mass ratio. We then apply thermal rel…
The astonishing diversity in the observed planetary population requires theoretical efforts and advances in planet formation theories. Numerical approaches provide a method to tackle the weaknesses of current planet formation models and are an important tool to close gaps in poorly constrained areas. We present a global disk setup to model the first stages of giant planet formation via gravitational instabilities (GI) in 3D with the block-structured adaptive mesh refinement (AMR) hydrodynamics code ENZO. With this setup, we explore the impact of AMR techniques on the fragmentation and clumping…
We examine the effect of the dust size distribution on Kelvin-Helmholtz instabilities in the protoplanetary disk with dust sedimentation. With newly taking into account the dust size distribution, the growth rate of the Kelvin-Helmholtz instability is calculated using the linear stability analysis with the dust density distribution consistent with sedimentation. Dust abundance required for gravitational instabilities before the Kelvin-Helmholtz instability is derived from the linear stability analysis, and it is found that the required dust abundance significantly coincides with that estimated…
The building of planetary systems is controlled by the gas and dust dynamics of protoplanetary disks. While the gas is simultaneously accreted onto the central star and dissipated away by winds, dust grains aggregate and collapse to form planetesimals and eventually planets. This dust and gas dynamics involves instabilities, turbulence and complex non-linear interactions which ultimately control the observational appearance and the secular evolution of these disks. This chapter is dedicated to the most recent developments in our understanding of the dynamics of gaseous and dusty disks, coverin…
Small solids embedded in gaseous protoplanetary disks are subject to strong dust-gas friction. Consequently, tightly-coupled dust particles almost follow the gas flow. This near conservation of dust-to-gas ratio along streamlines is analogous to the near conservation of entropy along flows of (dust-free) gas with weak heating and cooling. We develop this thermodynamic analogy into a framework to study dusty gas dynamics in protoplanetary disks. We show that an isothermal dusty gas behaves like an adiabatic pure gas; and that finite dust-gas coupling may be regarded as an effective heating/cool…
Sharp density features in protoplanetary discs, for instance at the edge of a magnetically dead zone, have recently been proposed as effective barriers to slow down or even stop the problematically fast migration of planetary cores into their central star. Density features on a radial scale approaching the disc vertical scale height might not exist, however, since they could be Rayleigh (or more generally Solberg-Hoiland) unstable. Stability must be checked explicitly in one-dimensional viscous accretion disc models because these instabilities are artificially eliminated in the process of redu…
Recent observations of HL Tau revealed remarkably detailed structure within the system's circumstellar disc. A range of hypotheses have been proposed to explain the morphology, including, e.g., planet-disc interactions, condensation fronts, and secular gravitational instabilities. While embedded planets seem to be able to explain some of the major structure in the disc through interactions with gas and dust, the substructure, such as low-contrast rings and bands, are not so easily reproduced. Here, we show that dynamical interactions between three planets (only two of which are modelled) and a…
Spinning planetesimals in a gaseous protoplanetary disk may experience a hydrodynamical force perpendicular to their relative velocities. We examine the effect this force has on the dynamics of these objects using analytical arguments based on a simple laminar disk model and numerical integrations of the equations of motion for individual grains. We focus in particular on meter-sized boulders traditionally expected to spiral in to the central star in as little as 100 years from 1 A.U. We find that there are plausible scenarios in which this force extends the lifetime of these solids in the dis…
We fit an isothermal oscillatory density model of Uranus' protoplanetary disk to the present-day major satellites and we determine the radial scale length of the disk, the equation of state and the central density of the primordial gas, and the rotational state of the Uranian nebula. This disk does not at all look like the Jovian disk that we modeled previously. Its rotation parameter that measures centrifugal support against self-gravity is a lot smaller ($β_0=0.00507$), as is the radial scale length (only 27.6 km) and the size of the disk (only 0.60 Gm). On the other hand, the central densit…
Various instabilities have been proposed as a promising mechanism to accumulate dust. Moreover, some of them are expected to lead to the multiple-ring structure formation and the planetesimal formation in protoplanetary disks. In a turbulent gaseous disk, the growth of the instabilities and the dust accumulation are quenched by turbulent diffusion of dust grains. The diffusion process has been often modeled by a diffusion term in the continuity equation for the dust density. The dust diffusion model, however, does not guarantee the angular momentum conservation in a disk. In this study, we fir…
Recent theories suggest planetesimal formation via streaming and/or gravitational instabilities may be triggered by localized enhancements in the dust-to-gas ratio, and one hypothesis is that sufficient enhancements may be produced in the pile-up of small solid particles inspiralling under aerodynamic drag from the large mass reservoir in the outer disc. Studies of particle pile-up in static gas discs have provided partial support for this hypothesis. Here, we study the radial and temporal evolution of the dust-to-gas ratio in turbulent discs, that evolve under the action of viscosity and phot…
Instabilities of the dust layer in a protoplanetary disk are investigated. It is known that the streaming instability develops and dust density concentration occurs in a situation where the initial dust density is uniform. This work considers the effect of initial dust density gradient vertical to the midplane. Dust and gas are treated as different fluids. Pressure of dust fluid is assumed to be zero. The gas friction time is assumed to be constant. Axisymmetric two-dimensional numerical simulation was performed using the spectral method. We found that an instability develops with a growth rat…
[Abridged] Star and planet formation are the complex outcomes of gravitational collapse and angular momentum transport mediated by protostellar and protoplanetary disks. In this review we focus on the role of gravitational instability in this process. We begin with a brief overview of the observational evidence for massive disks that might be subject to gravitational instability, and then highlight the diverse ways in which the instability manifests itself in protostellar and protoplanetary disks: the generation of spiral arms, small scale turbulence-like density fluctuations, and fragmentatio…
In laboratory experiments, we heated chondritic material up to 1400K in a hydrogen atmosphere. Moessbauer spectroscopy and magnetometry reveal that, at high temperatures, metallic iron forms from silicates. The transition temperature is about 1200K after 1 h of tempering, likely decreasing to about 1000K for longer tempering. This implies that in a region of high temperatures within protoplanetary disks, inward drifting solids will generally be a reservoir of metallic iron. Magnetic aggregation of iron-rich matter then occurs within the magnetic field of the disk. However, the Curie temperatur…
We present a new instability driven by a combination of coagulation and radial drift of dust particles. We refer to this instability as ``coagulation instability" and regard it as a promising mechanism to concentrate dust particles and assist planetesimal formation in the very early stages of disk evolution. Because of dust-density dependence of collisional coagulation efficiency, dust particles efficiently (inefficiently) grow in a region of positive (negative) dust density perturbations, which lead to a small radial variation of dust sizes and as a result radial velocity perturbations. The r…
This paper shows that the eccentric debris rings seen around the stars Fomalhaut and HD 202628 are narrower than expected in the standard eccentric planet perturbation scenario (sometimes referred to as "pericenter glow"). The standard scenario posits an initially circular and narrow belt of planetesimals at semi-major axis $a$, whose eccentricity is increased to $e_f$ after the gas disc has dispersed by secular perturbations from an eccentric planet, resulting in a belt of width $2ae_f$. In a minor modification of this scenario, narrower belts can arise if the planetesimals are initially ecce…
We estimate minimum dust abundances required for secular gravitational instability (SGI) to operate at the midplane dust layer of protoplanetary disks. For SGI to be a viable process, the growth time of the instability T_grow must be shorter than the radial drift time of the dust T_drift. The growth time depends on the turbulent diffusion parameter alpha, because the modes with short wavelengths are stabilized by turbulent diffusion. Assuming that turbulence is excited via the Kelvin-Helmholtz or streaming instabilities in the dust layer, and that its strength is controlled by the energy suppl…
Layered accretion is one of the inevitable ingredients in protoplanetary disks when disk turbulence is excited by magnetorotational instabilities (MRIs). In the accretion, disk surfaces where MRIs fully operate have a high value of disk accretion rate ($\dot{M}$), while the disk midplane where MRIs are generally quenched ends up with a low value of $\dot{M}$. Significant progress on understanding MRIs has recently been made by a number of dedicated MHD simulations, which requires improvement of the classical treatment of $α$ in 1D disk models. To this end, we obtain a new expression of $α$ by …
Turbulence is the dominant source of collisional velocities for grains with a wide range of sizes in protoplanetary disks. So far, only Kolmogorov turbulence has been considered for calculating grain collisional velocities, despite the evidence that turbulence in protoplanetary disks may be non-Kolmogorov. In this work, we present calculations of grain collisional velocities for arbitrary turbulence models characterized by power-law spectra and determined by three dimensionless parameters: the slope of the kinetic energy spectrum, the slope of the auto-correlation time, and the Reynolds number…
It has been suggested that the transition between magnetorotationally active and dead zones in protoplanetary disks should be prone to the excitation of vortices via Rossby wave instability (RWI). However, the only numerical evidence for this has come from alpha disk models, where the magnetic field evolution is not followed, and the effect of turbulence is parametrized by Laplacian viscosity. We aim to establish the phenomenology of the flow in the transition in 3D resistive-magnetohydrodynamical models. We model the transition by a sharp jump in resistivity, as expected in the inner dead zon…
We fit an isothermal oscillatory density model of Saturn's protoplanetary disk to the present-day major satellites and innermost rings D/C and we determine the radial scale length of the disk, the equation of state and the central density of the primordial gas, and the rotational state of the Saturnian nebula. This disk does not look like the Jovian and Uranian disks that we modeled previously. Its power-law index is extremely steep ($k=-4.5$) and its radial extent is very narrow ($ΔR\lesssim 0.9$ Gm), its rotation parameter that measures centrifugal support against self-gravity is somewhat la…
To date, most simulations of the chemistry in protoplanetary discs have used 1+1D or 2D axisymmetric $α$-disc models to determine chemical compositions within young systems. This assumption is inappropriate for non-axisymmetric, gravitationally unstable discs, which may be a significant stage in early protoplanetary disc evolution. Using 3D radiative hydrodynamics, we have modelled the physical and chemical evolution of a 0.17 M$_{\odot}$ self-gravitating disc over a period of 2000 yr. The 0.8 M$_{\odot}$ central protostar is likely to evolve into a solar-like star, and hence this Class 0 or e…
We present a 3D hydrodynamics study of gravitational instabilities (GIs) in a 0.14 M$_{\odot}$ circumbinary protoplanetary disc orbiting a 1 M$_{\odot}$ star and a 0.02 M$_{\odot}$ brown dwarf companion. We examine the thermodynamical state of the disc and determine the strengths of GI-induced density waves, nonaxisymmetric density structures, mass inflow and outflow, and gravitational torques. Results are compared with a parallel simulation of a protoplanetary disc without the brown dwarf binary companion. Simulations are performed using CHYMERA, a radiative 3D hydrodynamics code. The onset o…
Protoplanetary disks are quasi-steady structures whose evolution and dispersal determine the environment for planet formation. I review the theory of protoplanetary disk evolution and its connection to observations. Substantial progress has been made in elucidating the physics of potential angular momentum transport processes - including self-gravity, magnetorotational instability, baroclinic instabilities, and magnetic braking - and in developing testable models for disk dispersal via photoevaporation. The relative importance of these processes depends upon the initial mass, size and magnetiz…
Protoplanetary disks often appear as multiple concentric rings in dust continuum emission maps and scattered light images. These features are often associated with possible young planets in these disks. Many non-planetary explanations have also been suggested, including snow lines, dead zones and secular gravitational instabilities in the dust. In this paper we suggest another potential origin. The presence of copious amounts of dust tends to strongly reduce the conductivity of the gas, thereby inhibiting the magneto-rotational instability, and thus reducing the turbulence in the disk. From vi…
It is expected that a pressure bump can be formed at the inner edge of a dead-zone, and where vortices can develop through the Rossby Wave Instability (RWI). It has been suggested that self-gravity can significantly affect the evolution of such vortices. We present the results of 2D hydrodynamical simulations of the evolution of vortices forming at a pressure bump in self-gravitating discs with Toomre parameter in the range $4-30$. We consider isothermal plus non-isothermal disc models that employ either the classical $β$ prescription or a more realistic treatment for cooling. The main aim is …
Protoplanetary discs should exhibit a weak vertical variation in their rotation profiles. Typically this `vertical shear' issues from a baroclinic effect driven by the central star's radiation field, but it might also arise during the launching of a magnetocentrifugal wind. As a consequence, protoplanetary discs are subject to a hydrodynamical instability, the `vertical shear instability' (VSI), whose breakdown into turbulence could transport a moderate amount of angular momentum and facilitate, or interfere with, the process of planet formation. Magnetic fields may suppress the VSI, however, …
Protoplanets may be born into dust-rich environments if planetesimals formed through streaming or gravitational instabilities, or if the protoplanetary disc is undergoing mass loss due to disc winds or photoevaporation. Motivated by this possibility, we explore the interaction between low mass planets and dusty protoplanetary discs with focus on disc-planet torques. We implement Lin & Youdin's newly developed, purely hydrodynamic model of dusty gas into the PLUTO code to simulate dusty protoplanetary discs with an embedded planet. We find that for imperfectly coupled dust and high metallicity,…
I attempt to summarize our knowledge of planet formation in evolving protoplanetary discs. I first review the physics of disc evolution and dispersal. For most of the disc lifetime evolution is driven by accretion and photoevaporation, and I discuss how the interplay between these processes shapes protoplanetary discs. I also discuss the observations that we use to test these models, and the major uncertainties that remain. I will then move on to consider planet formation and migration in evolving discs, and discuss how observations of both discs and planets can be used to inform our understan…
