Time-dependent models of the structure and stability of self-gravitating protoplanetary discs

Angular momentum transport within young massive protoplanetary discs may be dominated by self-gravity at radii where the disc is too weakly ionized to allow the development of the magneto-rotational instability. We use time-dependent one-dimensional disc models, based on a local cooling time calculation of the efficiency of transport, to study the radial structure and stability (against fragmentation) of protoplanetary discs in which self-gravity is the sole transport mechanism. We find that self-gravitating discs rapidly attain a quasi-steady state in which the surface density in the inner disc is high and the strength of turbulence very low (  α∼ 10⁻³  or less inside 5 au). Temperatures high enough to form crystalline silicates may extend out to several astronomical units at early times within these discs. None of our discs spontaneously develop regions that would be unambiguously unstable to fragmentation into substellar objects, though the outer regions (beyond 20 au) of the most massive discs are close enough to the threshold that fragmentation cannot be ruled out. We discuss how the mass accretion rates through such discs may vary with disc mass and with mass of the central star, and note that a determination of the     relation for very young systems may allow a test of the model.     

Dynamics of charged dust particles in protoplanetary discs

We study the effect of an imposed magnetic field on the motion of charged dust particles in magnetically active regions of a protoplanetary disc. Assuming a power law structure for the vertical and the toroidal components of the magnetic field for the regions beyond magnetically dead region of the disc, the radial and the vertical velocities of the charged particles, in the asymptotic case of small particles, are calculated analytically. While grains with radii smaller than a critical radius significantly are affected by the magnetic force, motion of the particles with larger radii is independent of the magnetic field. The critical radius depends on the magnetic geometry and the charge of the grains. Assuming that a grain particle has one elementary charge and the physical properties of the disc correspond to a minimum-mass solar nebula, we show that only micron-sized grains are affected by the magnetic force. Also, charge polarity determines direction of the radial velocity. For such small particles, both the radial and the vertical velocities increase due to the magnetic force.     

Effects of scattering and dust grain size on the temperature structure of protoplanetary discs: a three-layer approach

The temperature in the optically thick interior of protoplanetary discs is essential for the interpretation of millimetre observations of the discs, for the vertical structure of the discs, for models of the disc evolution and the planet formation, and for the chemistry in the discs. Since large icy grains have a large albedo even in the infrared, the effect of scattering of the diffuse radiation in the discs on the interior temperature should be examined. We have performed a series of numerical radiation transfer simulations, including isotropic scattering by grains with various typical sizes for the diffuse radiation as well as for the incident stellar radiation. We also have developed an analytic model including isotropic scattering to understand the physics concealed in the numerical results. With the analytic model, we have shown that the standard two-layer approach is valid only for grey opacity (i.e. grain size ≳10 μm) even without scattering. A three-layer interpretation is required for grain size ≲10 μm. When the grain size is 0.1–10 μm, the numerical simulations show that the isotropic scattering reduces the temperature of the disc interior. This reduction is nicely explained by the analytic three-layer model as a result of the energy loss by scatterings of the incident stellar radiation and of the warm diffuse radiation in the disc atmosphere. For grain size ≳10 μm (i.e. grey scattering), the numerical simulations show that the isotropic scattering does not affect the interior temperature. This is nicely explained by the analytic two-layer model; the energy loss by scattering in the disc atmosphere is exactly offset by the 'green-house effect' due to the scattering of the cold diffuse radiation in the interior.     

Radiatively heated, protoplanetary discs with dead zones – I. Dust settling and thermal structure of discs around M stars

The irradiation of protoplanetary discs by central stars is the main heating mechanism for discs, resulting in their flared geometric structure. In a series of papers, we investigate the deep links between two-dimensional self-consistent disc structure and planetary migration in irradiated discs, focusing particularly on those around M stars. In this first paper, we analyse the thermal structure of discs that are irradiated by an M star by solving the radiative transfer equation by means of a Monte Carlo code. Our simulations of irradiated hydrostatic discs are realistic and self-consistent in that they include dust settling with multiple grain sizes  ( N = 15)  , the gravitational force of an embedded planet on the disc and the presence of a dead zone (a region with very low levels of turbulence) within it. We show that dust settling drives the temperature of the mid-plane from an   r ^−3/5  distribution (well mixed dust models) towards an   r ^−3/4  . The dead zone, meanwhile, leaves a dusty wall at its outer edge because dust settling in this region is enhanced compared to the active turbulent disc at larger disc radii. The disc heating produced by this irradiated wall provides a positive gradient region of the temperature in the dead zone in front of the wall. This is crucially important for slowing planetary migration because Lindblad torques are inversely proportional to the disc temperature. Furthermore, we show that low turbulence of the dead zone is self-consistently induced by dust settling, resulting in the Kelvin–Helmholtz instability (KHI). We show that the strength of turbulence arising from the KHI in the dead zone is  α= 10⁻⁵  .     

From boulders to planetary systems

In this chapter, we will give a brief overview on our current theoretical understanding how planets form from the solid material in circumstellar disks in the core accretion-gas capture model. This chapter will not be as concise and complete as a review on this matter, yet will serve as an introductory text to generate interest in the subject. Students are referred to comprehensive text books and some important reviews.This chapter will discuss “dusty storms”, e.g. the dust transport in turbulent protoplanetary disks, followed by the latest model of planetesimal formation, e.g. gravoturbulent planetesimal formation, which deals with particle concentration in turbulence and N-body simulations thereof. We also briefly describe the core accretion-gas capture process and talk about nascent planets, e.g. the observability of planet–disk interaction concluding with the migration of young planets and the final arrangement of planetary systems.     

Evolution of protoplanetary discs driven by the MRI, self-gravity and hydrodynamical turbulence

We study the viscous evolution of protoplanetary discs driven by the combined action of magnetohydrodynamic turbulence, resulting from the magneto-rotational instability (MRI), self-gravity torques, parametrized in terms of an effective viscosity and an additional viscous agent of unspecified origin. The distribution of torques driving the evolution of the disc is calculated by analysing where in the disc the MRI develops and, to incorporate the effect of self-gravity, calculating the Toomre parameter. We find that, generally, discs rapidly evolve towards a configuration where the intermediate regions, from a fraction of an au to a few au, are stable against the MRI due to their low-ionization degree. As an additional source of viscosity is assumed to operate in those regions, subsequent evolution of the disc is eruptive. Brief episodes of high mass accretion ensue as the criterion for the development of the MRI is met in the low-ionization region. The radial distribution of mass and temperature in the disc differs considerably from disc models with constant α parameter or layered accretion models, with potentially important consequences on the process of planet formation.     

The vertical structure of planet-induced gaps in protoplanetary discs

Giant planets embedded in circumstellar discs are expected to open gaps in these discs. We examine the vertical structure of the gap edges. We find that the planet excites spiral arms with significant (Mach number of a half) vertical motion of the gas, and discuss the implications of these motions. In particular, the spiral arms will make the edge appear 'puffed up' relative to the bulk of the disc. Infrared observations (sensitive to dust) would be dominated by the light from the thick inner edge of the disc. Submillimetre observations (sensitive to gas velocities) would appear to be hot in 'turbulent' motions (actually the ordered motion caused by the passage of the spiral arms), but cold in chemistry. Resolved submillimetre maps of circumstellar discs might even be able to detect the spiral arms directly.     

Three-dimensional calculations of high- and low-mass planets embedded in protoplanetary discs

We analyse the non-linear, three-dimensional response of a gaseous, viscous protoplanetary disc to the presence of a planet of mass ranging from 1 Earth mass (1 M_⊕) to 1 Jupiter mass (1 M_J) by using the zeus hydrodynamics code. We determine the gas flow pattern, and the accretion and migration rates of the planet. The planet is assumed to be in a fixed circular orbit about the central star. It is also assumed to be able to accrete gas without expansion on the scale of its Roche radius. Only planets with masses   M _p≳ 0.1 M_J  produce significant perturbations in the surface density of the disc. The flow within the Roche lobe of the planet is fully three-dimensional. Gas streams generally enter the Roche lobe close to the disc mid-plane, but produce much weaker shocks than the streams in two-dimensional models. The streams supply material to a circumplanetary disc that rotates in the same sense as the orbit of the planet. Much of the mass supply to the circumplanetary disc comes from non-coplanar flow. The accretion rate peaks with a planet mass of approximately 0.1 M_J and is highly efficient, occurring at the local viscous rate. The migration time-scales for planets of mass less than 0.1 M_J, based on torques from disc material outside the Roche lobes of the planets, are in excellent agreement with the linear theory of type I (non-gap) migration for three-dimensional discs. The transition from type I to type II (gap) migration is smooth, with changes in migration times of about a factor of 2. Starting with a core which can undergo runaway growth, a planet can gain up to a few M_J with little migration. Planets with final masses of the order of 10 M_J would undergo large migration, which makes formation and survival difficult.     

Migration of giant planets in planetesimal discs

Planets orbiting a planetesimal circumstellar disc can migrate inward from their initial positions because of dynamical friction between planets and planetesimals. The migration rate depends on the disc mass and on its time evolution. Planets that are embedded in long-lived planetesimal discs, having total mass of 10⁻⁴– 0.01 M_⊙ , can migrate inward a large distance and can survive only if the inner disc is truncated or as a result of tidal interaction with the star. In this case the semimajor axis, a , of the planetary orbit is less than 0.1 au. Orbits with larger a are obtained for smaller values of the disc mass or for a rapid evolution (depletion) of the disc. This model may explain not only several of the orbital features of the giant planets that have been discovered in recent years orbiting nearby stars, but also the metallicity enhancement found in several stars associated with short-period planets.     

Episodic accretion in magnetically layered protoplanetary discs

We study protoplanetary disc evolution assuming that angular momentum transport is driven by gravitational instability at large radii, and magnetohydrodynamic (MHD) turbulence in the hot inner regions. At radii of the order of 1 au such discs develop a magnetically layered structure, with accretion occurring in an ionized surface layer overlying quiescent gas that is too cool to sustain MHD turbulence. We show that layered discs are subject to a limit cycle instability, in which accretion on to the protostar occurs in ∼10⁴-yr bursts with ̇ ∼10⁻⁵ M_⊙ yr⁻¹, separated by quiescent intervals lasting ∼10⁵ yr where ̇ ≈10⁻⁸ M_⊙ yr⁻¹. Such bursts could lead to repeated episodes of strong mass outflow in young stellar objects. The transition to this episodic mode of accretion occurs at an early epoch ( t ≪1 Myr), and the model therefore predicts that many young pre-main-sequence stars should have low rates of accretion through the inner disc. At ages of a few Myr, the discs are up to an order of magnitude more massive than the minimum-mass solar nebula, with most of the mass locked up in the quiescent layer of the disc at r ∼1 au. The predicted rate of low-mass planetary migration is reduced at the outer edge of the layered disc, which could lead to an enhanced probability of giant planet formation at radii of 1–3 au.     

The inner magnetized regions of circumstellar accretion discs

In this lecture, I will briefly address several phenomena expected when magnetic fields are present in the innermost regions of circumstellar accretion discs: (i) the magneto-rotational instability and related “dead zones”; (ii) the formation of magnetically-driven jets and the observational constraints derived from Classical T Tauri stars; (iii) the magnetic star–disc interactions and their expected role in the stellar spin down.It should be noted that the magnetic fields invoked here are organized large scale magnetic fields, not turbulent small scale ones. I will therefore first argue why one can safely expect these fields to be present in circumstellar accretion discs. Objects devoid of such large scale fields would not be able to drive jets. A global picture is thus gradually emerging where the magnetic flux is an important control parameter of the star formation process as a whole. High angular resolution technics, by probing the innermost circumstellar disc regions should provide valuable constraints.     

The evolution of galactic discs

We use recent observations of high-redshift galaxies to study the evolution of galactic discs over the redshift range 0 <  z ≲1. The data are inconsistent with models in which discs were already assembled at z  = 1 and have evolved only in luminosity since that time. Assuming that disc properties change with redshift as powers of 1 +   z and analysing the observations assuming an Einstein–de Sitter universe, we find that for given rotation speed, disc scalelength decreases with z as ∼ (1 +  z )⁻¹, total B -band mass-to-light ratio decreases with z as ∼ (1 +  z )⁻¹, and disc luminosity (again in B ) depends only weakly on z . These scalings are consistent with current data on the evolution of disc galaxy abundance as a function of size and luminosity. Both the scalings and the abundance evolution are close to the predictions of hierarchical models for galaxy formation. If different cosmogonies are compared, the observed evolution in disc size and disc abundance favours a flat low-Ω₀ universe over an Einstein–de Sitter universe.     

Numerical simulations of type I planetary migration in non-turbulent magnetized discs

Using 2D magnetohydrodynamic (MHD) numerical simulations performed with two different finite-difference Eulerian codes, we analyse the effect that a toroidal magnetic field has on low-mass planet migration in non-turbulent protoplanetary discs. The presence of the magnetic field modifies the waves that can propagate in the disc. In agreement with a recent linear analysis, 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     planet, when the ratio β 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 _φ  is uniform in the disc, reduced migration when   B _φ  decreases as   r ⁻¹  and outward migration when   B _φ  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.     

Gravitationally unstable protoplanetary discs

The possibility that protoplanetary gaseous discs are dynamically unstable to axisymmetric and non-axisymmetric gravity perturbations (e.g. those produced by spontaneous disturbances) with characteristic scales larger than the vertical scale height is discussed analytically, using a local Wentzel–Kramers–Brillouin (WKB) approach. It is shown that such discs might be clumpy, and these gravitationally bound clumps may later collapse to become giant planets ('hot Jupiters'). The chief aim in this paper is to underscore a fact of vital importance for application in the planetary formation process: gravitationally unstable non-axisymmetric (spiral) perturbations can effectively transport both the angular momentum and the mass in a spatially inhomogeneous disc.     

A submillimeter view of protoplanetary dust disks

We present some results from our submillimeter single-dish and aperture synthesis imaging surveys of protoplanetary disks using the JCMT, CSO, and Submillimeter Array (SMA) on Mauna Kea, Hawaii. Employing a simple disk model, we simultaneously fit the spectral energy distributions and spatially resolved submillimeter continuum emission from our SMA survey to constrain disk structure properties, including surface density profiles and sizes. The typical disk structure we infer is consistent with a fiducial accretion disk model with a viscosity parameter α≈0.01. Combined with a large, multiwavelength single-dish survey of similar disks, we show how these observations provide evidence for significant grain growth and rapid evolution in the outer regions of disks, perhaps due to an internal photoevaporation process. In addition, we discuss SMA observations of the disks in the Orion Trapezium (proplyds) in the context of disk evolution in a more extreme environment.     

The migration and growth of protoplanets in protostellar discs

We investigate the gravitational interaction of a Jovian-mass protoplanet with a gaseous disc with aspect ratio and kinematic viscosity expected for the protoplanetary disc from which it formed. Different disc surface density distributions are investigated. We focus on the tidal interaction with the disc with the consequent gap formation and orbital migration of the protoplanet. Non-linear two-dimensional hydrodynamic simulations are employed using three independent numerical codes.
A principal result is that the direction of the orbital migration is always inwards and such that the protoplanet reaches the central star in a near-circular orbit after a characteristic viscous time‐scale of ∼10⁴ initial orbital periods. This is found to be independent of whether the protoplanet is allowed to accrete mass or not. Inward migration is helped by the disappearance of the inner disc, and therefore the positive torque it would exert, because of accretion on to the central star. Maximally accreting protoplanets reach about 4 Jovian masses on reaching the neighbourhood of the central star. Our results indicate that a realistic upper limit for the masses of closely orbiting giant planets is ∼5 Jupiter masses, if they originate in protoplanetary discs similar to the minimum-mass solar nebula. This is because of the reduced accretion rates obtained for planets of increasing mass.
Assuming that some process such as termination of the inner disc through a magnetospheric cavity stops the migration, the range of masses estimated for a number of close orbiting giant planets as well as their inward orbital migration can be accounted for by consideration of disc–protoplanet interactions during the late stages of giant planet formation.     

The photoevaporation of discs around young stars in massive clusters

We present models in which the photoevaporation of discs around young stars by an external ultraviolet source (as computed by Adams et al.) is coupled with the internal viscous evolution of the discs. These models are applied to the case of the Orion Nebula Cluster (ONC), where the presence of a strong ultraviolet field from the central OB stars, together with a detailed census of circumstellar discs and photoevaporative flows, is well established. In particular we investigate the constraints that are placed on the initial disc properties in the ONC by the twin requirement that most stars possess a disc on a scale of a few astronomical unit (au), but that only a minority (<20 per cent) are resolved by Hubble Space Telescope ( HST ) at a scale of 50 au. We find that these requirements place very weak constraints on the initial radius distribution of circumstellar discs: the resulting size distribution readily forgets the initial radius distribution, owing to the strong positive dependence of the photoevaporation rate on disc radius. Instead, the scarcity of large discs reflects the relative scarcity of initially massive discs (with mass  >0.1 M_⊙  ). The ubiquity of discs on a small scale, on the other hand, mainly constrains the time-span over which the discs have been exposed to the ultraviolet field (<2 Myr). We argue that the discs that are resolved by HST represent a population of discs in which self-gravity was important at the time that the dominant central OB star switched on, but that, according to our models, self-gravity is unlikely to be important in these discs at the present time. We discuss the implications of our results for the so-called proplyd lifetime problem.     

Gravitational interaction between a planet and an optically thin disc

We investigate the gravitational interaction between a planet and an optically thin protoplanetary disc, performing local three-dimensional hydrodynamical simulations. In the present study, we take account of radiative energy transfer in optically thin discs. Before the stage of planetary accretion, dust opacity is expected to decrease significantly because of grain growth and planetesimal formation. Thus, it would be reasonable to consider optically thin discs in the disc–planet interaction. Furthermore, we focus on small planets that can neither capture disc gas nor open a disc gap. The one-sided torque exerted on a planet by an optically thin disc is examined for various values of the disc optical thickness (<1). In optically thin discs, the temperature behind the density waves is lower than the unperturbed value because of radiative cooling. Heating due to shock dissipation is less effective than radiative cooling. Because of radiative cooling, the density distribution around the planet is not axisymmetric, which exerts an additional torque on the planet. The torque enhancement becomes maximum when the cooling time is comparable with the Keplerian period. The enhancement is significant for low-mass planets. For planets with  3 M_⊕  , the additional one-sided torque can be 40 per cent of the torque in the isothermal case. The radiative cooling is expected to change the differential torque and the migration speed of planets, too.