Observations of systems hosting close in ($<1$ AU) giant planets and brown dwarfs ($M\gtrsim7$ M$_{\rm Jup}$) find an excess of binary star companions, indicating that stellar multiplicity may play an important role in their formation. There is now increasing evidence that some of these objects may have formed via fragmentation in gravitationally unstable discs. We present a suite of 3D smoothed particle hydrodynamics (SPH) simulations of binary star systems with circumprimary self-gravitating discs, which include a realistic approximation to radiation transport, and extensively explore the companion's orbital parameter space for configurations which may trigger fragmentation. We identify a "sweet spot" where intermediate separation binary companions ($100$ AU $\lesssim a\lesssim400$ AU) can cause a marginally stable disc to fragment. The exact range of ideal binary separations is a function of the companion's eccentricity, inclination and mass. Heating is balanced by efficient cooling, and fragmentation occurs inside a spiral mode driven by the companion. Short separation, disc penetrating binary encounters ($a\lesssim100$ AU) are prohibitive to fragmentation, as mass stripping and disc heating quench any instability. This is also true of binary companions with high orbital eccentricities ($e\gtrsim0.75$). Wide separation companions ($a\gtrsim500$ AU) have little effect on the disc properties for the setup parameters considered here. The sweet spot found is consistent with the range of binary separations which display an excess of close in giant planets and brown dwarfs. Hence we suggest that fragmentation triggered by a binary companion may contribute to the formation of these substellar objects.
Using 2D smoothed particle hydrodynamics, we investigate the distribution of wait times between strong shocks in a turbulent, self-gravitating accretion disc. We show the resulting distributions do not depend strongly on the cooling time or resolution of the disc and that they are consistent with the predictions of earlier work (Young & Clarke 2015; Cossins et al. 2009, 2010). We use the distribution of wait times between shocks to estimate the likelihood of stochastic fragmentation by gradual contraction of shear-resistant clumps on the cooling time scale. We conclude that the stochastic frag…
Circumstellar discs are thought to be self-gravitating at very early times. If the disc is relatively cool, extended and accreting sufficiently rapidly, it can fragment into bound objects of order a few Jupiter masses and upwards. Given that the fragment's initial angular momentum is non-zero, and it will continue to accrete angular momentum from the surrounding circumstellar disc, we should expect that the fragment will also possess a relatively massive disc at early times. Therefore, we can ask: is disc fragmentation a hierarchical process? Or, can a disc fragment go on to produce its own se…
As a formation route for objects such as giant planets and low-mass stars in protostellar discs (as well as stars in AGN discs), theories of self-gravitating disc fragmentation need to be able to predict the initial masses of fragments. We describe a means by which the local Jeans mass inside the spiral structure of a self-gravitating disc can be estimated. If such a self-gravitating disc satisfies the criteria for disc fragmentation, this estimate provides a lower limit for the initial mass of any fragments formed. We apply this approach to a series of self-gravitating protostellar disc model…
We investigate structure of self-gravitating disks, their fragmentation and evolution of the fragments (the clumps) using both analytic approach and three-dimensional radiation hydrodynamics simulations starting from molecular cores. The simulations show that non-local radiative transfer determines disk temperature. We find the disk structure is well described by an analytical model of quasi-steady self-gravitating disk with radial radiative transfer. Because the radiative process is not local and radiation from the interstellar medium cannot be ignored, the local radiative cooling would not b…
We attempt to verify recent claims (made using semi-analytic models) that for the collapse of spherical homogeneous molecular clouds, fragmentation of the self-gravitating disc that subsequently forms can be predicted using the cloud's initial angular momentum alone. In effect, this condition is equivalent to requiring the resulting disc be sufficiently extended in order to fragment, in line with studies of isolated discs. We use smoothed particle hydrodynamics with hybrid radiative transfer to investigate this claim, confirming that in general, homogeneous spherical molecular clouds will prod…
When a self-gravitating disc is subject to irradiation, its propensity to fragmentation will be affected. The strength of self-gravitating disc stresses is expected to dictate disc fragmentation: as the strength of these torques typically decrease with increasing sound speed, it is reasonable to assume, to first-order, that disc fragmentation is suppressed when compared to the non-irradiated case, although previous work has shown that the details are complicated by the source of the irradiation. We expand on previous analysis of the Jeans mass inside spiral structures in self-gravitating discs…
It is likely that most protostellar systems undergo a brief phase where the protostellar disc is self-gravitating. If these discs are prone to fragmentation, then they are able to rapidly form objects that are initially of several Jupiter masses and larger. The fate of these disc fragments (and the fate of planetary bodies formed afterwards via core accretion) depends sensitively not only on the fragment's interaction with the disc, but with its neighbouring fragments. We return to and revise our population synthesis model of self-gravitating disc fragmentation and tidal downsizing. Amongst ot…
We investigate the role of magnetic fields in the fragmentation of self-gravitating discs using 3D global ideal magnetohydrodynamic simulations performed with the "phantom" smoothed particle hydrodynamics code. For initially toroidal fields, we find two regimes. In the first, where the cooling time is greater than five times the dynamical time, magnetic fields reduce spiral density wave amplitudes, which in turn suppresses fragmentation. This is the case even if the magnetic pressure is only a tenth of the thermal pressure. The second regime occurs when the cooling time is sufficiently short t…
We carry out three dimensional radiation hydrodynamical simulations of gravitationally unstable discs to explore the movement of mass in a disc following its initial fragmentation. We find that the radial velocity of the gas in some parts of the disc increases by up to a factor of approximately 10 after the disc fragments, compared to before. While the movement of mass occurs in both the inward and outward directions, the inwards movement can cause the inner spirals of a self-gravitating disc to become sufficiently dense such that they can potentially fragment. This suggests that the dynamical…
We carry out a resolution study on the fragmentation boundary of self-gravitating discs. We perform three-dimensional Smoothed Particle Hydrodynamics simulations of discs to determine whether the critical value of the cooling timescale in units of the orbital timescale, beta_{crit}, converges with increasing resolution. Using particle numbers ranging from 31,250 to 16 million (the highest resolution simulations to date) we do not find convergence. Instead, fragmentation occurs for longer cooling timescales as the resolution is increased. These results suggest that at the very least, the critic…
We carry out a resolution study on the fragmentation boundary of self-gravitating discs. We perform three-dimensional Smoothed Particle Hydrodynamics (SPH) simulations of discs to determine whether the critical value of the cooling timescale in units of the orbital timescale, beta_{crit}, converges with increasing resolution. Using particle numbers ranging from 31,250 to 16 million (the highest resolution simulations to date) we do not find convergence. Instead, fragmentation occurs for longer cooling timescales as the resolution is increased. These results certainly suggest that beta_{crit} i…
Recent simulations of self-gravitating accretion discs, carried out using a three-dimensional Smoothed Particle Hydrodynamics (SPH) code by Meru and Bate, have been interpreted as implying that three-dimensional global discs fragment much more easily than would be expected from a two-dimensional local model. Subsequently, global and local two-dimensional models have been shown to display similar fragmentation properties, leaving it unclear whether the three-dimensional results reflect a physical effect or a numerical problem associated with the treatment of cooling or artificial viscosity in S…
Massive circumstellar disks are prone to gravitational instabilities, which trigger the formation of spiral arms that can fragment into bound clumps under the right conditions. Two dimensional simulations of self-gravitating disks are useful starting points for studying fragmentation, allowing for high-resolution simulations of thin disks. However, convergence issues can arise in 2D from various sources. One of these sources is the 2D approximation of self-gravity, which exaggerates the effect of self-gravity on small scales when the potential is not smoothed to account for the assumed vertica…
We investigate the fragmentation criterion in massive self-gravitating discs. We present new analysis of the fragmentation conditions which we test by carrying out global three-dimensional numerical simulations. Whilst previous work has placed emphasis on the cooling timescale in units of the orbital timescale, β, we find that at a given radius the surface mass density (i.e. disc mass and profile) and star mass also play a crucial role in determining whether a disc fragments or not as well as where in the disc fragments form. We find that for shallow surface mass density profiles (p<2, where Σ…
Recently, the gravitational instability (GI) model of giant planet and brown dwarf formation has been revisited and recast into what is often referred to as the "tidal downsizing" hypothesis. The fragmentation of self-gravitating protostellar discs into gravitationally bound embryos - with masses of a few to tens of Jupiter masses, at semi major axes above 30 - 40 au - is followed by grain sedimentation inside the embryo, radial migration towards the central star and tidal disruption of the embryo's upper layers. The properties of the resultant object depends sensitively on the timescales upon…
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…
It is quite likely that self-gravity will play an important role in the evolution of accretion discs, in particular those around young stars, and those around supermassive black holes. We summarise, here, our current understanding of the evolution of such discs, focussing more on discs in young stellar system, than on discs in active galactic nuclei. We consider the conditions under which such discs may fragment to form bound objects, and when they might, instead, be expected to settle into a quasi-steady, self-regulated state. We also discuss how this understanding may depend on the mass of t…
Context. It is still an open issue whether a self-gravitating accretion disk fragments. There are many different physical and numerical explanations for fragmentation, but simulations often show a non-convergent behavior for ever better resolution. Aims. We investigate the influence of different numerical limiters in Godunov type schemes on the fragmentation boundary in self- gravitating disks. Methods. We compare the linear and non-linear outcome in two-dimensional shearingsheet simulations using the VANLEER and the SUPERBEE limiter. Results. We show that choosing inappropriate limiting funct…
It has recently been suggested that in the presence of driven turbulence discs may be much less stable against gravitational collapse than their non turbulent analogs, due to stochastic density fluctuations in turbulent flows. This mode of fragmentation would be especially important for gas giant planet formation. Here we argue, however, that stochastic density fluctuations due to turbulence do not enhance gravitational instability and disc fragmentation in the long cooling time limit appropriate for planet forming discs. These fluctuations evolve adiabatically and dissipate away by decompress…
Angular momentum transport within young massive protoplanetary discs may be dominated by self-gravity at radii where the disk 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 di…
Protoplanetary disks fragment due to gravitational instability when there is enough mass for self-gravitation, described by the Toomre parameter, and when heat can be lost at a rate comparable to the local dynamical timescale, described by t_c=beta Omega^-1. Simulations of self-gravitating disks show that the cooling parameter has a rough critical value at beta_crit=3. When below beta_crit, gas overdensities will contract under their own gravity and fragment into bound objects while otherwise maintaining a steady state of gravitoturbulence. However, previous studies of the critical cooling par…
Fragmentation of protoplanetary disks due to gravitational instabilities is a candidate of a formation mechanism of binary stars, brown dwarfs, and gaseous giant planets. The condition for the fragmentation has been thought that the disk cooling timescale is comparable to its dynamical timescale. However, some numerical simulations suggest that the fragmentation does not occur even if the cooling time is small enough, or the fragmentation can occur even when the cooling is inefficient. To reveal a realistic condition for fragmentation of self-gravitating disks, we perform two-dimensional numer…
Self-gravity becomes competitive as an angular momentum transport process in accretion discs at large radii, where the temperature is low enough that external irradiation likely contributes to the thermal balance. Irradiation is known to weaken the strength of disc self-gravity, and can suppress it entirely if the disc is maintained above the threshold for linear instability. However, its impact on the susceptibility of the disc to fragmentation is less clear. We use two-dimensional numerical simulations to investigate the evolution of self-gravitating discs as a function of the local cooling …
Triggered Fragmentation in Self-Gravitating Protoplanetary discs: Cooling, Mass Movement and Instability
Centre for Exoplanets and Habitability, University of Warwick（行星宜居中心，华威大学）；Department of Physics, University of Warwick（物理系，华威大学）；School of Physics and Astronomy, Monash University（物理与天文学学院，蒙纳士大学）
We study numerical convergence in local two-dimensional hydrodynamical simulations of self-gravitating accretion discs with a simple cooling law. It is well-known that there exists a steady gravito-turbulent state, in which cooling is balanced by dissipation of weak shocks, with a net outward transport of angular momentum. Previous results indicated that if cooling is too fast (typical time scale 3/Omega, where Omega is the local angular velocity), this steady state can not be maintained and the disc will fragment into gravitationally bound clumps. We show that, in the two-dimensional local ap…
We carry out simulations of gravitationally unstable discs using a Smoothed Particle Hydrodynamics (SPH) code and a grid-based hydrodynamics code, FARGO, to understand the previous non-convergent results reported by Meru & Bate (2011a). We obtain evidence that convergence with increasing resolution occurs with both SPH and FARGO and in both cases we find that the critical cooling timescale is larger than previously thought. We show that SPH has a first-order convergence rate while FARGO converges with a second-order rate. We show that the convergence of the critical cooling timescale for fragm…
We present a 3D semi-analytic model of self-gravitating discs, and include a prescription for dust trapping in the disc spiral arms. Using Monte-Carlo radiative transfer we produce synthetic ALMA observations of these discs. In doing so we demonstrate that our model is capable of producing observational predictions, and able to model real image data of potentially self-gravitating discs. For a disc to generate spiral structure that would be observable with ALMA requires that the disc's dust mass budget is dominated by millimetre and centimetre-sized grains. Discs in which grains have grown to …
Self-gravitating discs are believed to play an important role in astrophysics in particular regarding the star and planet formation process. In this context, discs subject to an idealized cooling process, characterized by a cooling timescale $β$ expressed in unit of orbital timescale, have been extensively studied. We take advantage of the Riemann solver and the 3D Godunov scheme implemented in the code Ramses to perform high resolution simulations, complementing previous studies that have used smoothed particle hydrodynamics (SPH) or 2D grid codes. We observe that the critical value of $β$ fo…
The gravitational instability model of planet/brown dwarf formation proposes that protostellar discs can fragment into objects with masses above a few Jupiter masses at large semimajor axis. Tidal downsizing may reduce both the object mass and semimajor axis. However, most studies of tidal downsizing end when the protostellar disc disperses, while the system is embedded in its parent star-forming region. To compare disc fragment descendants with exoplanet and brown dwarf observations, the subsequent dynamical evolution must be explored. We carry out N-Body integrations of fragment-fragment sca…
Stimulated by recent results by Meru and Bate (2010a,b), we revisit the issue of resolution requirements for simulating self-gravitating accretion discs with Smoothed Particle Hydrodynamics (SPH). We show that the results by Meru and Bate (2010a) are consistent with those of Meru and Bate (2010b) if they are both interpreted as driven by resolution effects, therefore implying that the resolution criterion for cooling gaseous discs is a function of the imposed cooling rate. We discuss two possible numerical origins of such dependence, which are both consistent with the limited number of availab…
There have been several recent detections of candidate Keplerian discs around massive young protostars. Given the relatively large disc-to-star mass ratios in these systems, and their young ages, it is worth investigating their propensity to becoming self-gravitating. To this end, we compute self-consistent, semi-analytic models of putative self-gravitating discs for five candidate disc systems. Our aim is not to fit exactly the observations, but to demonstrate that the expected dust continuum emission from marginally unstable self-gravitating discs can be quite weak, due to high optical depth…
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…
Using local three dimensional radiation hydrodynamics simulations, the nonlinear outcome of gravitational instability in an irradiated protoplanetary disc is investigated in a parameter space of the surface density $Σ$ and the radius $r$. Starting from laminar flow, axisymmetric self-gravitating density waves grow first. Their self-gravitating degree becomes larger when $Σ$ is larger or the cooling time is shorter at larger radii. The density waves eventually collapse owing to non-axisymmetric instability, which results in either fragmentation or gravito-turbulence after a transient phase. The…
An interaction between a star-disc system and another star will perturb the disc, possibly resulting in a significant modification of the disc structure and its properties. It is still unclear if such an encounter can trigger fragmentation of the disc to form brown dwarfs or gas giant planets. This paper details high resolution Smoothed Particle Hydrodynamics (SPH) simulations investigating the influence of stellar encounters on disc dynamics. Star-star encounters (where the primary has a self-gravitating, marginally stable protostellar disc, and the secondary has no disc) were simulated with …
We study the numerical convergence of hydrodynamical simulations of self-gravitating accretion discs, in which a simple cooling law is balanced by shock heating. It is well-known that there exists a critical cooling time scale for which shock heating can no longer compensate for the energy losses, at which point the disc fragments. The numerical convergence of previous results of this critical cooling time scale was questioned recently using Smoothed Particle Hydrodynamics (SPH). We employ a two-dimensional grid-based code to study this problem, and find that for smooth initial conditions, fra…
Duschl et al. (2000) have shown that the standard model for geometrically thin accretion disks ($α$-disks) leads to inconsistency if self-gravity play a role. This problem arise from parametrization of viscosity in terms of local sound velocity and vertical disks scale hight. The $β$-viscosity prescription was introduced by Duschl et al. (2000), which has been derived from rotating shear flow experiment ($ν=βΩR^2$). Following the Duschl et al. (2000) suggestion for a $β$-prescription for viscosity, we apply this model for a thin self-gravitating disk around newborn stars. Our result is quite d…
We carry out global three-dimensional radiation hydrodynamical simulations of self-gravitating accretion discs to determine if, and under what conditions, a disc may fragment to form giant planets. We explore the parameter space (in terms of the disc opacity, temperature and size) and include the effect of stellar irradiation. We find that the disc opacity plays a vital role in determining whether a disc fragments. Specifically, opacities that are smaller than interstellar Rosseland mean values promote fragmentation (even at small radii, R < 25AU) since low opacities allow a disc to cool quick…
The long-term evolution of a circumstellar disk starting from its formation and ending in the T Tauri phase was simulated numerically with the purpose of studying the evolution of dust in the disk with distinct values of viscous α-parameter and dust fragmentation velocity v_frag. We solved numerical hydrodynamics equations in the thin-disk limit, which are modified to include a dust component consisting of two parts: sub-micron-sized dust and grown dust with a maximum radius a_r. The former is strictly coupled to the gas, while the latter interacts with the gas via friction. The conversion of …
Modern graphics processing units (GPUs) lead to a major increase in the performance of the computation of astrophysical simulations. Owing to the different nature of GPU architecture compared to traditional central processing units (CPUs) such as x86 architecture, existing numerical codes cannot be easily migrated to run on GPU. Here, we present a new implementation of the numerical method smooth particle hydrodynamics (SPH) using CUDA and the first astrophysical application of the new code: the collision between Ceres-sized objects. The new code allows for a tremendous increase in speed of as…
We investigate the evolution of grains composed of an ice shell surrounding an olivine core as they pass through a spiral shock in a protoplanetary disk. We use published three-dimensional radiation-hydrodynamics simulations of massive self-gravitating protoplanetary disks to extract the thermodynamics of spiral shocks in the region between 10 and 20 AU from the central star. As the density wave passes, it heats the grains, causing them to lose their ice shell and resulting in a lowering of the grain opacity. In addition, since grains of different sizes will have slightly different temperature…
We have conducted the first comprehensive numerical investigation of the relative velocity distribution of dust particles in self-gravitating protoplanetary discs with a view to assessing the viability of planetesimal formation via direct collapse in such environments. The viability depends crucially on the large sizes that are preferentially collected in pressure maxima produced by transient spiral features (Stokes numbers, $St \sim 1$); growth to these size scales requires that collision velocities remain low enough that grain growth is not reversed by fragmentation. We show that, for a sing…
The observation of massive exoplanets at large separation from their host star, like in the HR 8799 system, challenges theories of planet formation. A possible formation mechanism involves the fragmentation of massive self-gravitating discs into clumps. While the conditions for fragmentation have been extensively studied, little is known of the subsequent evolution of these giant planet embryos, in particular their expected orbital migration. Assuming a single planet has formed by fragmentation, we investigate its interaction with the gravitoturbulent disc it is embedded in. Two-dimensional hy…
We have investigated the properties of gravito-turbulent discs in 3D using high-resolution shearing-box simulations. For large enough domain sizes, $L_y \gtrsim 60H$, the disc settles down into a quasi-steady state, showing no long term trends in properties or variation with box size. For smaller boxes, we find that the azimuthal wavelength of the dominant spiral modes are limited to the domain size. This is associated with a bursty behaviour that differs from the quasi-steady dynamics at larger sizes. We point out that a similar transition may be expected in global simulations at the point wh…
Many protoplanetary discs are self-gravitating early in their lives. If they fragment under their own gravity, they form bound gaseous clumps which may evolve to become giant planets. Today, the fraction of discs that undergo fragmentation, and the frequency of conditions that may lead to giant planet formation via gravitational instability, is still unknown. We perform a population synthesis of discs from formation to dispersal. In varying the infall radius, we study the relationship of the early disc size with fragmentation. Furthermore, we investigate how stellar accretion heating affects t…
In the early stages of a protoplanetary disk, when its mass is a significant fraction of its star's, turbulence generated by gravitational instability (GI) should feature significantly in the disk's evolution. At the same time, the disk may be sufficiently ionised for magnetic fields to play some role in the dynamics. Though usually neglected, the impact of magnetism on the GI may be critical, with consequences for several processes: the efficiency of accretion, spiral structure formation, fragmentation, and the dynamics of solids. In this paper, we report on global three-dimensional magnetohy…
In this paper we consider the effects of opacity regimes on the stability of self-gravitating protoplanetary discs to fragmentation into bound objects. Using a self-consistent 1-D viscous disc model, we show that the ratio of local cooling to dynamical timescales Omega*tcool has a strong dependence on the local temperature. We investigate the effects of temperature-dependent cooling functions on the disc's gravitational stability through controlled numerical experiments using an SPH code. We find that such cooling functions raise the susceptibility of discs to fragmentation through the influen…
Low mass, self-gravitating accretion disks admit quasi-steady,`gravito-turbulent' states in which cooling balances turbulent viscous heating. However, numerical simulations show that gravito-turbulence cannot be sustained beyond dynamical timescales when the cooling rate or corresponding turbulent viscosity is too large. The result is disk fragmentation. We motivate and quantify an interpretation of disk fragmentation as the inability to maintain gravito-turbulence due to formal secondary instabilities driven by: 1) cooling, which reduces pressure support; and/or 2) viscosity, which reduces ro…
The young star Elias 2-27 has recently been observed to posses a massive circumstellar disc with two prominent large-scale spiral arms. In this Letter we perform three-dimensional Smoothed Particle Hydrodynamics simulations, radiative transfer modelling, synthetic ALMA imaging and an unsharped masking technique to explore three possibilities for the origin of the observed structures -- an undetected companion either internal or external to the spirals, and a self-gravitating disc. We find that a gravitationally unstable disc and a disc with an external companion can produce morphology that is …
Radiation Hydrodynamics of Self-gravitating Protoplanetary Disks II. Accretion, Migration, Spin, and Internal Structure of GI Fragments
Institute for Advanced Study, Tsinghua University（清华大学高等研究院）；Shanghai Astronomical Observatory, Chinese Academy of Sciences（中国科学院上海天文台）；Department of Astronomy, Tsinghua University（清华大学天文系）
