Self-gravitating Disks Research Articles

Effects of radiative transfer on the structure of self-gravitating discs, their fragmentation and the evolution of the fragments

We investigate the structure of self-gravitating discs, their fragmentation and the evolution of the fragments (the clumps) using both an analytic approach and three-dimensional radiation hydrodynamics simulations starting from molecular cores. The simulations show that non-local radiative transfer determines the disc temperature. We find the disc structure is well described by an analytical model of a quasi-steady self-gravitating disc with radial radiative transfer. Because the radiative process is not local and radiation from the interstellar medium cannot be ignored, the local radiative cooling is not balanced with the viscous heating in a massive disc around a low-mass star. In our simulations, there are cases in which the disc does not fragment even though it satisfies the fragmentation criterion based on disc cooling time (Q ∼ 1 and Ωtcool ∼ 1). This indicates that, at least, the criterion is not a sufficient condition for fragmentation. We determine the parameter range for the host cloud core in which disc fragmentation occurs. In addition, we show that the temperature evolution of the centre of the clump is close to that of typical first cores, and that the minimum initial mass of clumps is about a few Jupiter masses.

Metal loading of giant gas planets

One of many challenges in forming giant gas planets via Gravitational disc Instability model (GI) is an inefficient radiative cooling of the pre-collapse fragments. Since fragment contraction times are as long at $10^5 -10^7$ years, the fragments may be tidally destroyed sooner than they contract into gas giant planets. Here we explore the role of "pebble accretion" onto the pre-collapse giant planets and find an unexpected result. Despite larger dust opacity at higher metallicities, addition of metals actually accelerates -- rather than slows down -- collapse of high opacity, relatively low mass giant gas planets ($M_p$ below a few Jupiter masses). A simple analytical theory that explains this result exactly in idealised simplified cases is presented. The theory shows that planets with the central temperature in the range between $\sim$ 1000 to 2000K are especially sensitive to pebble accretion: addition of just $\sim 5$ to 10 % of metals by weight is sufficient to cause their collapse. These results show that dust grain physics and dynamics is essential for an accurate modelling of self-gravitating disc fragments and their near environments in the outer massive and cold protoplanetary discs.

A two-dimensional study for cooling and self-gravitating accretion discs

The importance of cooling for the structure and evolution of self-gravitating accretion discs has been confirmed through the use of direct numerical simulations. In this paper, we present a two-dimensional study for self-gravitating accretion discs, to investigate the influence of the cooling rate on the latitudinal structure of such accretion discs. The disc is cooled using a simple parametrization for the cooling function, (de/dt)cool=−e/tcool with e as the internal energy and tcool as the cooling timescale. The cooling timescale is in units of the dynamical timescale, tdyn[=Ω−1], is Ωtcool=β, where β is a free parameter. The mechanism of energy dissipation is assumed to be turbulent viscosity in the disc and an α-prescription is applied for the kinematic coefficient of viscosity. To study the gravitational stability of the self-gravitating disc, we use the Toomre parameter. We obtain the radial dependence of the physical variables through the use of a self-similar method and we numerically solve the equations to obtain the latitudinal dependence of the physical variables. The solutions show that the radial velocity is smaller than the Keplerian rotational velocity; however, the disc, dependent on the values of parameters α and β and only near the zone close to the equatorial plane, can rotate in a super-Keplerian manner. With the magnitude of both parameters α and β, the disc thickness increases due to the increase of the vertical pressure gradient. The dependence of the gas density on the parameters α and β indicates two zones in the accretion disc. In the first zone near the equatorial plane, the mass density decreases by increasing these parameters. However, in the second zone, the regions with higher latitude, the mass density increases with the magnitude of parameters α and β.

How to detect the signatures of self-gravitating circumstellar discs with the Atacama Large Millimeter/sub-millimeter Array

In this paper, we present simulated Atacama Large Millimeter/sub-millimeter Array (ALMA) observations of self-gravitating circumstellar discs with different properties in size, mass and inclination, located in four of the most extensively studied and surveyed star-forming regions. Starting from a smoothed particle hydrodynamics simulation and representative dust opacities, we have initially constructed maps of the expected emission at sub-mm wavelengths of a large sample of discs with different properties. We have then simulated realistic observations of discs as they may appear with ALMA using the Common Astronomy Software Application ALMA simulator. We find that, with a proper combination of antenna configuration and integration time, the spiral structure characteristic of self-gravitating discs is readily detectable by ALMA over a wide range of wavelengths at distances comparable to TW Hydrae (∼50舁pc), Taurus-Auriga and Ophiucus (∼140舁pc) star-forming regions. However, for discs located in Orion complex (∼400舁pc) only the largest discs in our sample (outer radius of 100 au) show a spatially resolved structure while the smaller ones (outer radius of 25 au) are characterized by a spiral structure that is not conclusively detectable with ALMA.

FEATHERING INSTABILITY OF SPIRAL ARMS. II. PARAMETER STUDY

We report the results of a parameter study of the feathering stability in the galactic spiral arms. A two-dimensional razor-thin magnetized self-gravitating gas disk with an imposed two-armed stellar spiral structure is considered. Using the formulation developed previously by Lee and Shu, a linear stability analysis of the spiral shock is performed in a localized Cartesian geometry. Results of the parameter study of the base state with a spiral shock are also presented. The single-mode feathering instability that leads to growing perturbations may explain the feathering phenomenon found in nearby spiral galaxies. The self-gravity of the gas, characterized by its average surface density, is an important parameter which 1) shifts the spiral shock further downstream; 2) increases the growth rate and decreases the characteristic spacing of the feathering structure due to the instability. On the other hand, while the magnetic field suppresses the velocity fluctuation associated with the feathers, it does not strongly affect their growth rate. Using a set of typical parameters of the grand-design spiral galaxy M51 at 2 kpc from the center, the spacing of the feathers with the maximum growth rate is found to be 530 pc, which agrees with the previous observational studies.

The statistical mechanics of self-gravitating Keplerian discs

We describe the dynamics and thermodynamics of collisionless particle discs orbiting a massive central body, in the case where the disc mass is small compared to the central mass, the self-gravity of the disc dominates the non-Keplerian force, and the spread in semi-major axes is small. We show that with plausible approximations such discs have logarithmic two-body interactions and a compact phase space, and therefore exhibit thermodynamics that are simpler than most other gravitating systems, which require a confining box and artificial softening of the potential at small scales to be thermodynamically well-behaved. We solve for the microcanonical axisymmetric thermal equilibria and demonstrate the existence of a symmetry-breaking bifurcation into lopsided equilibria. We discuss the relation between thermal and dynamical instability in these systems and draw connections to astrophysical settings, as well as to the wider subject of the statistical mechanics of particles with logarithmic long-range interactions, such as point vortices in two-dimensional fluids.

LINEAR STABILITY OF MAGNETIZED MASSIVE PROTOPLANETARY DISKS

Magneto-rotational instability (MRI) and gravitational instability (GI) are the two principle routes to turbulent angular momentum transport in accretion disks. Protoplanetary disks may develop both. This paper aims to reinvigorate interest in the study of magnetized massive protoplanetary disks, starting from the basic issue of stability. The local linear stability of a self-gravitating, uniformly magnetized, differentially rotating, three-dimensional stratified disk subject to axisymmetric perturbations is calculated numerically. The formulation includes resistivity. It is found that the reduction in the disk thickness by self-gravity can decrease MRI growth rates; the MRI becomes global in the vertical direction, and MRI modes with small radial length scales are stabilized. The maximum vertical field strength that permits the MRI in a strongly self-gravitating polytropic disk with polytropic index $\Gamma=1$ is estimated to be $B_{z,\mathrm{max}} \simeq c_{s0}\Omega\sqrt{\mu_0/16\pi G} $, where $c_{s0}$ is the midplane sound speed and $\Omega$ is the angular velocity. In massive disks with layered resistivity, the MRI is not well-localized to regions where the Elsasser number exceeds unity. For MRI modes with radial length scales on the order of the disk thickness, self-gravity can enhance density perturbations, an effect that becomes significant in the presence of a strong toroidal field, and which depends on the symmetry of the underlying MRI mode. In gravitationally unstable disks where GI and MRI growth rates are comparable, the character of unstable modes can transition smoothly between MRI and GI. Implications for non-linear simulations are discussed briefly.

Nonaxisymmetric instabilities in self-gravitating disks. II. Linear and quasi-linear analyses

We studied global nonaxisymmetric hydrodynamic instabilities in an extensive collection of hot, self-gravitating polytropic disk systems, systems that covered a wide expanse of the parameter space relevant to protostellar and protoplanetary systems. We examined equilibrium disk models varying three parameters: the ratio of the inner to outer equatorial radii, the ratio of star mass to disk mass, and the rotation law exponent q. We took the polytropic index n=1.5 and examined the exponents q=1.5 and 2, and the transitional one q=1.75. For each of these sets of parameters, we examined models with inner to outer radius ratios from 0.1 to 0.75, and star mass to disk mass ratios from 0 to 103. We numerically calculated the growth rates and oscillation frequencies of low-order nonaxisymmetric disk modes, modes with azimuthal dependence ∝e imϕ . Low-m modes are found to dominate with the character and strength of instability strongly dependent on disk self-gravity. Representatives of each mode type are examined in detail, and torques and mass transport rates are calculated.

GRAVITO-TURBULENT DISKS IN THREE DIMENSIONS: TURBULENT VELOCITIES VERSUS DEPTH

Characterizing turbulence in protoplanetary disks is crucial for understanding how they accrete and spawn planets. Recent measurements of spectral line broadening promise to diagnose turbulence, with different lines probing different depths. We use 3D local hydrodynamic simulations of cooling, self-gravitating disks to resolve how motions driven by "gravito-turbulence" vary with height. We find that gravito-turbulence is practically as vigorous at altitude as at depth: even though gas at altitude is much too rarefied to be itself self-gravitating, it is strongly forced by self-gravitating overdensities at the midplane. The long-range nature of gravity means that turbulent velocities are nearly uniform vertically, increasing by just a factor of 2 from midplane to surface, even as the density ranges over nearly three orders of magnitude. The insensitivity of gravito-turbulence to height contrasts with the behavior of disks afflicted by the magnetorotational instability (MRI); in the latter case, non-circular velocities increase by at least a factor of 15 from midplane to surface, with various non-ideal effects only magnifying this factor. The distinct vertical profiles of gravito-turbulence vs. MRI turbulence may be used in conjunction with measurements of non-thermal linewidths at various depths to identify the source of transport in protoplanetary disks.

Planetesimal formation in self-gravitating discs – the effects of particle self-gravity and back-reaction

We study particle dynamics in self-gravitating gaseous discs with a simple cooling law prescription via two-dimensional simulations in the shearing sheet approximation. It is well known that structures arising in the gaseous component of the disc due to a gravitational instability can have a significant effect on the evolution of dust particles. Previous results have shown that spiral density waves can be highly efficient at collecting dust particles, creating significant local over-densities of particles. The degree of such concentrations has been shown to be dependent on two parameters: the size of the dust particles and the rate of gas cooling. We expand on these findings, including the self-gravity of dust particles, to see how these particle over-densities evolve. We use the PENCIL CODE to solve the local shearing sheet equations for gas on a fixed grid together with the equations of motion for solids coupled to the gas through an aerodynamic drag force. We find that the enhancements in the surface density of particles in spiral density wave crests can reach levels high enough to allow the solid component of the disc to collapse under its own self-gravity. This produces many gravitationally bound collections of particles within the spiral structure. The total mass contained in bound structures appears nearly independent of the cooling time, suggesting that the formation of planetesimals through dust particle trapping by self-gravitating density waves may be possible at a larger range of radii within a disc than previously thought. So, density waves due to gravitational instabilities in the early stages of star formation may provide excellent sites for the rapid formation of many large, planetesimal-sized objects.

The density and temperature dependence of the cooling timescale for fragmentation of self-gravitating disks

The purpose of this paper is to explore the influences of cooling timescale on fragmentation of self-gravitating protoplanetary disks. We assume the cooling timescale, expressed in terms of the dynamical timescale Ω tcool, has a power-law dependence on temperature and density, Ω tcool ∝ Σ−aT−b, where a and b are constants. We use this cooling timescale in a simple prescription for the cooling rate, du/dt = −u/tcool, where u is the internal energy. We perform our simulations using the smoothed particle hydrodynamics method. The simulations demonstrate that the disk is very sensitive to the cooling timescale, which depends on density and temperature. Under such a cooling timescale, the disk becomes gravitationally unstable and clumps form in the disk. This property even occurs for cooling timescales which are much longer than the critical cooling timescale, Ω tcool ≳ 7. We show that by adding the dependence of a cooling timescale on temperature and density, the number of clumps increases and the clumps can also form at smaller radii. The simulations imply that the sensitivity of a cooling timescale to density is more than to temperature, because even for a small dependence of the cooling timescale on density, clumps can still form in the disk. However, when the cooling timescale has a large dependence on temperature, clumps form in the disk. We also consider the effects of artificial viscosity parameters on fragmentation conditions. This consideration is performed in two cases, where Ω tcool is a constant and Ω tcool is a function of density and temperature. The simulations consider both cases, and results show the artificial viscosity parameters have rather similar effects. For example, using too small of values for linear and quadratic terms in artificial viscosity can suppress the gravitational instability and consequently the efficiency of the clump formation process decreases. This property is consistent with recent simulations of self-gravitating disks. We perform simulations with and without the Balsara form of artificial viscosity. We find that in the cooling and self-gravitating disks without the Balsara switch, the clumps can form more easily than those with the Balsara switch. Moreover, in both cases where the Balsara switch is present or absent, the simulations show that the cooling timescale strongly depends on density and temperature.

Gap formation in a self-gravitating disk and the associated migration of the embedded giant planet

We present the results of our recent study on the interactions between a giant planet and a self-gravitating gas disk. We investigate how the disk's self-gravity affects the gap formation process and the migration of the giant planet. Two series of 1-D and 2-D hydrodynamic simulations are performed. We select several surface densities and focus on the gravitationally stable region. To obtain more reliable gravity torques exerted on the planet, a refined treatment of the disk's gravity is adopted in the vicinity of the planet. Our results indicate that the net effect of the disk's self-gravity on the gap formation process depends on the surface density of the disk. We notice that there are two critical values, ΣI and ΣII. When the surface density of the disk is lower than the first one, Σ0 < ΣI, the effect of self-gravity suppresses the formation of a gap. When Σ0 > ΣI, the self-gravity of the gas tends to benefit the gap formation process and enlarges the width/depth of the gap. According to our 1-D and 2-D simulations, we estimate the first critical surface density to be ΣI ≈ 0.8 MMSN. This effect increases until the surface density reaches the second critical value ΣII. When Σ0 > ΣII, the gravitational turbulence in the disk becomes dominant and the gap formation process is suppressed again. Our 2-D simulations show that this critical surface density is around 3.5 MMSN. We also study the associated orbital evolution of a giant planet. Under the effect of the disk's self-gravity, the migration rate of the giant planet increases when the disk is dominated by gravitational turbulence. We show that the migration timescale correlates with the effective viscosity and can be up to 104 yr.

Migration of massive black hole binaries in self-gravitating discs: retrograde versus prograde

We study the interplay between mass transfer, accretion and gravitational torques onto a black hole binary migrating in a self-gravitating, retrograde circumbinary disc. A direct comparison with an identical prograde disc shows that: (i) because of the absence of resonances, the cavity size is a factor a(1+e) smaller for retrograde discs; (ii) nonetheless the shrinkage of a circular binary semi--major axis, a, is identical in both cases; (iii) a circular binary in a retrograde disc remains circular while eccentric binaries grow more eccentric. For non-circular binaries, we measure the orbital decay rates and the eccentricity growth rates to be exponential as long as the binary orbits in the plane of its disc. Additionally, for these co-planar systems, we find that interaction (~ non--zero torque) stems only from the cavity edge plus a(1+e) in the disc, i.e. for dynamical purposes, the disc can be treated as a annulus of small radial extent. We find that simple 'dust' models in which the binary- disc interaction is purely gravitational can account for all main numerical results, both for prograde and retrograde discs. Furthermore, we discuss the possibility of an instability occurring for highly eccentric binaries causing it to leave the disc plane, secularly tilt and converge to a prograde system. Our results suggest that there are two stable configurations for binaries in self-gravitating discs: the special circular retrograde case and an eccentric (e~ 0.6) prograde configuration as a stable attractor.

Convergence of simulations of self-gravitating accretion discs – II. Sensitivity to the implementation of radiative cooling and artificial viscosity

Recently it has been suggested that the fragmentation boundary in Smoothed Particle Hydrodynamic (SPH) and FARGO simulations of self-gravitating accretion discs with beta-cooling do not converge as resolution is increased. Furthermore, this recent work suggests that by carefully optimising the artificial viscosity parameters in these codes it can be shown that fragmentation may occur for much longer cooling times than earlier work suggests. If correct, this result is intriguing as it suggests that gas giant planets could form, via direct gravitational collapse, reasonably close to their parent stars. This result is, however, slightly surprising and there have been a number of recent studies suggesting that the result is likely an indication of a numerical problem with the simulations. One suggestion, in particular, is that the SPH results are influenced by the manner in which the cooling is implemented. We extend this work here and show that if the cooling is implemented in a manner that removes a known numerical artefact in the shock regions, the fragmentation boundary converges to a value consistent with earlier work and that fragmentation is unlikely for the long cooling times suggested by this recent work. We also investigate the optimisation of the artificial viscosity parameters and show that the values that appear optimal are likely introducing numerical problems in both the SPH and FARGO simulations. We therefore conclude that earlier predictions for the cooling times required for fragmentation are likely correct and that, as suggested by this earlier work, fragmentation cannot occur in the inner parts (r < 50 au) of typical protostellar discs.

Radiative feedback from protoplanets in self-gravitating protoplanetary discs

It is well known that massive protoplanetary disc are gravitationally unstable beyond tens of AU from their parent star. The eventual fate of the self-gravitating gas clumps born in the disc is currently not understood, although the range of uncertainty is well known. If clumps migrate inward rapidly, they are tidally disrupted, which may leave behind giant or terrestrial like planets. On the other hand, if clumps migrate less rapidly, they tend to accrete gas, becoming proto brown dwarfs or low mass companions to the parent star. Here we argue that radiative feedback of contracting clumps (protoplanets) on their discs is an important effect that has been overlooked in previous calculations. We show analytically that temperature in clump's vicinity may be high enough to support a quasi-static atmosphere if the clump mass is below a critical value, $M_{\rm cr} \sim 6$ Jupiter masses ($M_J$). This may arrest further gas accretion onto the clump and thus promote formation of planets rather than low mass companions. We use numerical simulations to evaluate these analytical conclusions by studying migration and accretion of gas clumps as a function of their initial mass, $M_i$. Simulations neglecting the radiative preheating effect show that gas clumps with mass less than $\sim 2 M_J$ migrate inward rapidly; more massive clumps result in low mass companions. In contrast, simulations that include radiative preheating from the clump show that clumps as massive as $8 M_J$ migrate inward rapidly and end up tidally disrupted in the inner disc. We conclude that, with other parameters being equal, previous simulations neglecting radiative feedback from self-gravitating clumps over-estimated the population of brown dwarfs and low mass stellar companions and under-estimated the population of planets.

Orbital migration of giant planets induced by gravitationally unstable gaps: the effect of planet mass

It has been established that self-gravitating disc–satellite interaction can lead to the formation of a gravitationally unstable gap. Such an instability may significantly affect the orbital migration of gap-opening perturbers in self-gravitating discs. In this paper, we extend the two-dimensional hydrodynamic simulations of Lin & Papaloizou to investigate the role of the perturber or planet mass on the gravitational stability of gaps and its impact on orbital migration. We consider giant planets with planet-to-star mass ratio q ≡ Mp/M* ∈ [0.3, 3.0] × 10−3 (so that q = 10−3 corresponds to a Jupiter mass planet if M* = M⊙), in a self-gravitating disc with disc-to-star mass ratio Md/M* = 0.08, aspect ratio h = 0.05 and Keplerian Toomre parameter Qk0 = 1.5 at 2.5 times the planet's initial orbital radius. These planet masses correspond to |$\tilde{q}\in [0.9,\,1.7]$|⁠, where |$\tilde{q}$| is the ratio of the planet Hill radius to the local disc scale-height. Fixed-orbit simulations show that all planet masses we consider open gravitationally unstable gaps, but the instability is stronger and develops sooner with increasing planet mass. The disc-on-planet torques typically become more positive with increasing planet mass. In freely migrating simulations, we observe faster outward migration with increasing planet mass, but only for planet masses capable of opening unstable gaps early on. For q = 0.0003 (⁠|$\tilde{q}=0.9$|⁠), the planet undergoes rapid inward type III migration before it can open a gap. For q = 0.0013 (⁠|$\tilde{q}=1.5$|⁠) we find it is possible to balance the tendency for inward migration by the positive torques due to an unstable gap, but only for a few 10 s of orbital periods. We find the unstable outer gap edge can trigger outward type III migration, sending the planet to twice its initial orbital radius on dynamical time-scales. We briefly discuss the importance of our results in the context of giant planet formation on wide orbits through disc fragmentation.

THE FORMATION OF MASSIVE POPULATION III STARS IN THE PRESENCE OF TURBULENCE

Population III stars forming in the infant universe at z=30 heralded the end of the cosmic dark ages. They are presumed to be assembled in so-called minihaloes with virial temperatures of a few thousand K where collapse is triggered by molecular hydrogen cooling. A central question concerns their final masses, and whether fragmentation occurs during their formation. While studies employing Lagrangian codes suggest fragmentation via a self-gravitating disk, recent high resolution simulations indicated that disk formation is suppressed. Here we report the first high-resolution large-eddy simulations performed with the Eulerian grid-based code Enzo following the evolution beyond the formation of the first peak, to investigate the accretion of the central massive clump and potential fragmentation. For a total of 3 halos, we see that a disk forms around the first clump. The central clump reaches $\sim10$ solar masses after 40 years, while subsequent accretion is expected at a rate of $10^{-2}$ solar masses per year. In one of these halos, additional clumps form as a result of fragmentation which proceeds at larger scales. We note that subgrid-scale turbulence yields relevant contributions to the stability of the protostellar disks. We conclude that the first protostar may reach masses up to $\rm 40-100 M_{\odot}$, which are only limited by the effect of radiative feedback.

The possibility of a self-gravitating disc around L1527 IRS?

Recent observations of the Class 0 protostar L1527 IRS have revealed a rotationally supported disc with an outer radius of at least 100 au. Measurements of the integrated flux at 870 microns suggest a disc mass that is too low for gravitational instability to govern angular momentum transport. However, if parts of the disc are optically thick at sub-mm wavelengths, the sub-mm fluxes will underestimate the disc mass, and the disc's actual mass may be substantially larger, potentially sufficient to be self-gravitating. We investigate this possibility using simple self-gravitating disc models. To match the observed mass accretion rates requires a disc-to-star mass ratio of at least ~0.5, which produces sub-mm fluxes that are similar to those observed for L1527 IRS in the absence of irradiation from the envelope or central star. If irradiation is significant, then the predicted fluxes exceed the observed fluxes by around an order of magnitude. Our model also indicates that the stresses produced by the gravitational instability are low enough to prevent disc fragmentation. As such, we conclude that observations do not rule out the possibility that the disc around L1527 IRS is self-gravitating, but it is more likely that despite being a very young system, this disc may already have left the self-gravitating phase.

Simulated observations of young gravitationally unstable protoplanetary discs

The formation and earliest stages of protoplanetary discs remain poorly constrained by observations. ALMA will soon revolutionise this field. Therefore, it is important to provide predictions which will be valuable for the interpretation of future high sensitivity and high angular resolution observations. Here we present simulated ALMA observations based on radiative transfer modelling of a relatively massive (0.39 M_solar) self-gravitating disc embedded in a 10 M_solar dense core, with structure similar to the pre-stellar core L1544. We focus on simple species and conclude that C17O 3-2, HCO+ 3-2, OCS 26-25 and H2CO 404-303 lines can be used to probe the disc structure and kinematics at all scales.

Gravitational instability of planetary gaps and its effect on orbital migration

AbstractGap formation by giant planets in self-gravitating disks may lead to a gravitational edge instability (GEI). We demonstrate this GEI with global 3D and 2D self-gravitating disk-planet simulations using the ZEUS, PLUTO and FARGO hydrodynamic codes. High resolution 2D simulations show that an unstable outer gap edge can lead to outward migration. Our results have important implications for theories of giant planet formation in massive disks.