Self-gravitating Accretion Discs Research Articles

Effect of magnetic fields on the dynamics and gravitational wave emission of Papaloizou-Pringle instability-saturated self-gravitating accretion disks: Simulations in full GR

Effect of magnetic fields on the dynamics and gravitational wave emission of Papaloizou-Pringle instability-saturated self-gravitating accretion disks: Simulations in full GR

Numerical study of toroidal magnetic field on the self-gravitating protoplanetary disks

In this paper, we study the self-gravitating accretion disks by considering the toroidal component of magnetic field, [Formula: see text] and wind/outflow in the flow and also investigate the effect of two parameters, [Formula: see text] and [Formula: see text] corresponding to magnetic field on the latitudinal structure of such accretion disks. The cooling of the disk is parameterized simply as, [Formula: see text] (where [Formula: see text] is the internal energy and [Formula: see text] is the cooling timescale and [Formula: see text] is a free constant) and the heating rate is decomposed into two components, magnetic field and viscosity dissipations. We have shown that when the toroidal magnetic field becomes stronger, the heating process (viscous and resistivity) and the radiative cooling rate increase. Ohmic heating is much bigger than viscous heating and cooling, so we must consider the role of the magnetic field in the energy equation. Our numerical solutions show that the thickness of the disk decreases with strong toroidal component of magnetic field. The magnetic field leads to production of the outflow in the low latitude. So, by increasing the toroidal component of the magnetic field, the regions which belong to inflow decrease and the disk is cooled.

Study of the structure of the inflow/outflow region in the self-gravitating standard accretion disks

The structure of self-gravitating accretion disks is investigated. In order to describe this, we establish the quantitative changes in the θ direction by solving a set of ordinary differential equations in the spherical coordinates (r, θ, φ). The use of a spherical coordinate system causes a positive radial velocity to indicate the presence of an outflow region and a negative radial velocity to indicate an inflow region. In order to discuss the regions of inflow and outflow of self-gravitating disks, we assume that the disk is axisymmetric, that is ∂/∂ϕ=0. We also consider the disk to be steady and uniform. We use self-similar solution in the radial direction. By inserting the advection factor in the energy equation, we can distinguish standard disks from inefficient irradiation accretion disks. The effect of self-gravity of the disk is considered both in the radial and angular direction in the momentum equation. After solving the equations, the results show that the self-gravity decreases the disk’s inflow/outflow region which is observed both in gas-pressure-dominated and radiation-pressure-dominated standard accretion disks. The self-gravity can affect the angular and vertical structure and thickness of the disks. The results show that by increasing the mass of the disk, the thickness of the disk decreases. In radial direction, the self-gravity force reduces the outflow region of the disk.

Structure of the self-gravitating accretion discs in the presence of outflow

ABSTRACT Numerical simulations of self-gravitating accretion discs have shown that the evolution of such systems depends strongly on the rate at which it cools. In this work, we study the vertical structure of the self-gravitating accretion discs and also investigate the effect of the cooling rate on the latitudinal structure of such accretion discs. In the spherical coordinates, we write the hydrodynamics equations and simplify the basic equations based on the assumptions of axisymmetric and steady state. We use the self-similar method for solving the equations in the radial direction and we find proper boundary conditions. We find inflow–outflow solutions by considering the meridional component of the velocity field. In order to formulate the cooling term in energy equation, we introduce the new parameter β as a free constant that is the cooling time-scale in units of the dynamical time-scale. Our numerical solutions show that the thickness of the disc decreases with smaller β (or increasing the cooling term in energy equation) and it makes the disc colder and outflows form in the regions with lower latitude. So by increasing the cooling rate in the disc, the regions which belong to inflow decrease.

Vortex survival in 3D self-gravitating accretion discs

Large-scale, dust-trapping vortices may account for observations of asymmetric protoplanetary discs. Disc vortices are also potential sites for accelerated planetesimal formation by concentrating dust grains. However, in 3D discs vortices are subject to destructive `elliptic instabilities', which reduces their viability as dust traps. The survival of vortices in 3D accretion discs is thus an important issue to address. In this work, we perform shearing box simulations to show that disc self-gravity enhances the survival of 3D vortices, even when self-gravity is weak in the classic sense (e.g. with a Toomre $Q\simeq5$). We find a 3D, self-gravitating vortex can grow on secular timescales in spite of the elliptic instability. The vortex aspect-ratio decreases as it strengthens, which feeds the elliptic instability. The result is a 3D vortex with a turbulent core that persists for $\sim 10^{3}$ orbits. We find when gravitational and hydrodynamic stresses become comparable, the vortex may undergo episodic bursts, which we interpret as interaction between elliptic and gravitational instabilities. We estimate the distribution of dust particles in self-gravitating, turbulent vortices. Our results suggest large-scale vortices in protoplanetary discs are more easily observed at large radii.

The impact of numerical oversteepening on the fragmentation boundary in self-gravitating disks

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 functions to handle shock-capturing in Godunov type schemes can lead to an overestimation of the surface density in regions with shallow density gradients. The effect amplifies itself on timescales comparable to the dynamical timescale even at high resolutions. This is exactly the environment, where clumps are expected to form. The effect is present without, but scaled up by, self-gravity and also does not depend on cooling. Moreover it can be backtracked to a well known effect called oversteepening. If the effect is also observed in the linear case, the fragmentation limit is shifted to larger values of the critical cooling timescale.

An Analytical Model for the Evolution of the Protoplanetary Disks

Abstract We obtain a new set of analytical solutions for the evolution of a self-gravitating accretion disk by holding the Toomre parameter close to its threshold and obtaining the stress parameter from the cooling rate. In agreement with the previous numerical solutions, furthermore, the accretion rate is assumed to be independent of the disk radius. Extreme situations where the entire disk is either optically thick or optically thin are studied independently, and the obtained solutions can be used for exploring the early or the final phases of a protoplanetary disk evolution. Our solutions exhibit decay of the accretion rate as a power-law function of the age of the system, with exponents −0.75 and −1.04 for optically thick and thin cases, respectively. Our calculations permit us to explore the evolution of the snow line analytically. The location of the snow line in the optically thick regime evolves as a power-law function of time with the exponent −0.16; however, when the disk is optically thin, the location of the snow line as a function of time with the exponent −0.7 has a stronger dependence on time. This means that in an optically thin disk inward migration of the snow line is faster than an optically thick disk.

Structure of radiation-dominated gravitoturbulent quasar discs

Self-gravitating accretion discs in a gravitoturbulent state, including radiation and gas pressures, are studied using a set of new analytical solutions. While the Toomre parameter of the disc remains close to its critical value for the onset of gravitational instability, the dimensionless stress parameter is uniquely determined from the thermal energy reservoir of the disc and its cooling rate. Our solutions are applicable to the accretion discs with dynamically important radiation pressure like in the quasars discs. We show that physical quantities of a gravitoturbulent disc in the presence of radiation are significantly modified compared to solutions with only gas pressure. We show that the dimensionless stress parameter is an increasing function of the radial distance so that its steepness strongly depends on the accretion rate. In a disc without radiation its slope is 4.5, however, we show that in the presence of radiation, it varies between 2 and 4.5 depending on the accretion rate and the central mass. As for the surface density, we find a shallower profile with an exponent -2 in a disc with sub-Eddington accretion rate compared to a similar disc, but without radiation, where its surface density slope is -3 independent of the accretion rate. We then investigate gravitational stability of the disc when the stress parameter reaches to its critical value. In order to self-consistently determine the fragmentation boundary, however, it is shown that the critical value of the stress parameter is a power-law function of the ratio of gas pressure and the total pressure and its exponent is around 1.7. We also estimate the maximum mass of the central black hole using our analytical solutions.

ON THE GRAVITATIONAL STABILITY OF GRAVITO-TURBULENT ACCRETION DISKS

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 rotational support. We analyze the axisymmetric gravitational stability of viscous, non-adiabatic accretion disks with internal heating, external irradiation, and cooling in the shearing box approximation. We consider parameterized cooling functions in 2D and 3D disks, as well as radiative diffusion in 3D. We show that generally there is no critical cooling rate/viscosity below which the disk is formally stable, although interesting limits appear for unstable modes with lengthscales on the order of the disk thickness. We apply this new linear theory to protoplanetary disks subject to gravito-turbulence modeled as an effective viscosity, and cooling regulated by dust opacity. We find that viscosity renders the disk beyond $\sim 60$AU dynamically unstable on radial lengthscales a few times the local disk thickness. This is coincident with the empirical condition for disk fragmentation based on a maximum sustainable stress. We suggest turbulent stresses can play an active role in realistic disk fragmentation by removing rotational stabilization against self-gravity, and that the observed transition in behavior from gravito-turbulent to fragmenting may reflect instability of the gravito-turbulent state itself.

Occurrence of instability through the protostellar accretion disks by landing of low-mass condensations

Low-mass condensations (LMCs) are observed inside the envelope of the collapsing molecular cloud cores. In this research, we investigate the effects of landing LMCs for occurrence of instability through the protostellar accretion disks. We consider some regions of the disk where duration of infalling and landing of the LMCs are shorter than the orbital period. In this way, we can consider the landing LMCs as density bumps and grooves in the azimuthal direction of an initial thin axisymmetric steady state self-gravitating protostellar accretion disk (nearly Keplerian). Using the linear effects of the bump quantities, we obtain a characteristic equation for growth/decay rate of bumps; we numerically solve it to find occurrence of instability. We also evaluate the minimum-growth-time-scale (MGTS) and the enhanced mass accretion rate. The results show that infalling and landing of the LMCs in the inner regions of the protostellar accretion disks can cause faster unstable modes and less enhanced accretion rates relative to the outer regions. Also, more fragmentation of landed LMCs in the azimuthal direction have less chance for instability, and then can produce more values of enhanced mass accretion rate.

Spiral-driven accretion in protoplanetary discs

Accretion discs are ubiquitous in the universe and it is a crucial issue to understand how angular momentum and mass are being radially transported in these objects. Here, we study the role played by non-linear spiral patterns within hydrodynamical and non self-gravitating accretion disc assuming that external disturbances such as infall onto the disc may trigger them. To do so, we computed self-similar solutions that describe discs in which a spiral wave propagates. Such solutions present both shocks and critical sonic points that we carefully analyze. For all allowed temperatures and for several spiral shocks, we calculated the wave structure. In particular we inferred the angle of the spiral patern, the stress it exerts on the disc as well as the associated flux of mass and angular momentum as a function of temperature. We quantified the rate of angular momentum transport by means of the dimensionless $\alpha$ parameter. For the thickest disc we considered (corresponding to $h/r$ values of about 1/3), we found values of $\alpha$ as high as $0.1$, and scaling with the temperature $T$ such that $\alpha \propto T^{3/2} \propto (h/r)^3$. The spiral angle scales with the temperature as $\arctan(r/h)$. The existence of these solutions suggests that perturbations occurring at disc outer boundaries, such as for example perturbations due to infall motions, can propagate deep inside the disc and therefore should not be ignored, even when considering small radii.

The Evolution of Self-Gravitating Accretion Discs

AbstractIt 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 the disc relative to the mass of the central object, and how it might depend on the presence of external irradiation. Additionally, we consider whether or not fragmentation might be stochastic, where we might expect it to occur in an actual protostellar disc, and if there is any evidence for fragmentation actually playing a role in the formation of planetary-mass bodies. Although there are still a number of outstanding issue, such as the convergence of simulations of self-gravitating discs, whether or not there is more than one mode of fragmentation, and quite what role self-gravitating discs may play in the planet-formation process, our general understanding of these systems seems quite robust.

Quantification of stochastic fragmentation of self-gravitating discs

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 fragmentation mechanism (Paardekooper 2012) cannot change the radius at which fragmentation is possible by more than ~20%, restricting direct gravitational collapse as a mechanism for giant planet formation to the outer regions of protoplanetary discs.

Dependence of fragmentation in self-gravitating accretion discs on small-scale structure

We propose a framework for understanding the fragmentation criterion for self-gravitating discs which, in contrast to studies that emphasize the ‘gravoturbulent’ nature of such discs, instead focuses on the properties of their quasi-regular spiral structures. Within this framework there are two evolutionary paths to fragmentation: (i) collapse on the free-fall time, which requires that the ratio of cooling time to dynamical time (β) < 3 and (ii) quasi-static collapse on the cooling time at a rate that is sufficiently fast that fragments are compact enough to withstand disruption when they encounter spiral features in the disc. We perform 2D grid simulations which demonstrate numerically converged fragmentation at β < 3 (in good agreement with Paardekooper, Baruteau & Meru, and others) and argue that this is a consequence of the fact that such simulations smooth the gravitational force on the scale H, the scaleheight of the disc. Such simulations thus only allow fragmentation via route (i) above since they suppress the quasi-static contraction of fragments on scales <H; the inability of fragments to contract to significantly smaller scales then renders them susceptible to disruption at the next spiral arm encounter. On the other hand, 3D simulations indeed show fragmentation at higher β via route (ii). We derive an analytic prediction of fragmentation by route (ii) when β ≲ 12, based on the requirement that fragments must contract sufficiently to withstand disruption by spiral arms. We also discuss the necessary numerical requirements on both grid-based and smoothed particle hydrodynamics codes if they are to model fragmentation via route (ii).

Self-similar evolution of self-gravitating viscous accretion discs

A new one-dimensional, dynamical model is proposed for geometrically thin, self-gravitating viscous accretion discs. The vertically integrated equations are simplified using the slow accretion limit and the monopole approximation with a time-dependent central point mass to account for self-gravity and accretion. It is shown that the system of partial differential equations can be reduced to a single non-linear advection diffusion equation which describes the time evolution of angular velocity. In order to solve the equation three different turbulent viscosity prescriptions are considered. It is shown that for these parametrizations the differential equation allows for similarity transformations depending only on a single non-dimensional parameter. A detailed analysis of the similarity solutions reveals that this parameter is the initial power law exponent of the angular velocity distribution at large radii. The radial dependence of the self-similar solutions is in most cases given by broken power laws. At small radii the rotation law always becomes Keplerian with respect to the current central point mass. In the outer regions the power law exponent of the rotation law deviates from the Keplerian value and approaches asymptotically the value determined by the initial condition. It is shown that accretion discs with flatter rotation laws at large radii yield higher accretion rates. The methods are applied to self-gravitating accretion discs in active galactic nuclei. Fully self-gravitating discs are found to evolve faster than nearly Keplerian discs. The implications on supermassive black hole formation and Quasar evolution are discussed.

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.

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.

Planetesimal formation in self-gravitating discs

We study particle dynamics in local two-dimensional simulations of self-gravitating accretion discs with a simple cooling law. It is well known that the structure which arises 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 using global simulations indicate that spiral density waves are highly efficient at collecting dust particles, creating significant local over-densities which may be able to undergo gravitational collapse. We expand on these findings, using a range of cooling times to mimic the conditions at a large range of radii within the disc. Here we use the Pencil Code to solve the 2D local shearing sheet equations for gas on a fixed grid together with the equations of motion for solids coupled to the gas solely through aerodynamic drag force. We find that spiral density waves can create significant enhancements in the surface density of solids, equivalent to 1-10cm sized particles in a disc following the profiles of Clarke (2009) around a solar mass star, causing it to reach concentrations several orders of magnitude larger than the particles mean surface density. We also study the velocity dispersion of the particles, finding that the spiral structure can result in the particle velocities becoming highly ordered, having a narrow velocity dispersion. This implies low relative velocities between particles, which in turn suggests that collisions are typically low energy, lessening the likelihood of grain destruction. Both these findings suggest that the density waves that arise due to gravitational instabilities in the early stages of star formation provide excellent sites for the formation of large, planetesimal-sized objects.

The Cosmogony of Super-Massive Black Holes

We summarize our recent results on the evolution of super-massive black holes in the cores of galaxies. Our models are based on the viscous, Eddington-limited evolution of self-gravitating accretion disks. We find that, within this framework, one can explain the growth time scales and the luminosities of individual objects as well as the variation of these quantities for AGN with different black hole masses.

Anomalous gravitomagnetic viscosity in accretion disks

We propose a viscosity prescription in a self-gravitating system based on the self-generated gravitomagnetic field. It has been proved that the self-generated gravitomagnetic field with very low frequency is modulational unstable, leading to a self-collapse, and resulting in the enhancement of perturbed matter density and gravitomagnetic field in a small region; the interactions between highly spatially intermittent gravitomagnetic structures and the macrofluid with differential rotation are responsible for the anomalous gravitomagnetic viscosity. The result indicates that, the gravitomagnetic viscosity would be nearly 108 stronger than the molecular viscosity, which maybe useful for investigating the structures and instabilities in the self-gravitating accretion disks; provide an explanation for the transportion of angular momentum and the formation of gravitomagnetic jets.