Self-gravitating Disks Research Articles

Primordial obliquities of brown dwarfs and super-Jupiters from fragmenting gravito-turbulent discs

ABSTRACT Super-Jupiters, brown dwarfs, and stars can form from the collapse of self-gravitating discs. Such discs are turbulent, with flocculent spiral arms accelerating gas to transonic speeds horizontally and vertically. Objects that fragment from gravito-turbulent discs should spin with a wide range of directions, reflecting the random orientations of their parent eddies. We show by direct numerical simulation that obliquities of newly collapsed fragments can range up to 45○. Subsequent collisions between fragments can further alter the obliquity distribution, up to 90○ or down to near-zero. The large obliquities of newly discovered super-Jupiters on wide orbits around young stars may be gravito-turbulent in origin. Obliquely spinning fragments are born on orbits that may be inclined relative to their parent discs by up to 20○, and gravitationally stir leftover material to many times the pre-fragmentation disc thickness.

Toroidal magnetic fields in self-gravitating disks around black holes

We investigate stationary models of magnetized, self-gravitating disks around black holes. The disks are assumed to rotate according to a recently introduced Keplerian rotation law. We consider different prescriptions of the toroidal magnetic field. Similarly to the purely hydrodynamical case (i.e., with no magnetic field), we observe a bifurcation in the parameter space of solutions. There are usually two branches of solutions: a branch corresponding to relatively light disks and a branch for which the disk can be more massive than the black hole. The existence of this latter branch can be explained by geometric properties of the spacetime. We investigate the influence of the magnetic field in the disk on these effects.

Circumbinary disc self-gravity governing supermassive black hole binary mergers

Abstract Understanding the interaction of massive black hole binaries with their gaseous environment is crucial since at sub-parsec scales the binary is too wide for gravitational wave emission to take over and to drive the two black holes to merge. We here investigate the interaction between a massive black hole binary and a self-gravitating circumbinary disc using 3D smoothed particle hydrodynamics simulations. We find that, when the disc self-gravity regulates the angular momentum transport, the binary semi-major axis decreases regardless the choice of disc masses and temperatures, within the range we explored. In particular, we find that the disc initial temperature (hence the disc aspect ratio) has little effect on the evolution of the binary since discs with the same mass self-regulate towards the same temperature. Initially warmer discs cause the binary to shrink on a slightly shorter timescale until the disc has reached the self-regulated equilibrium temperature. More massive discs drive the binary semi-major axis to decrease at a faster pace compared to less massive discs and result in faster binary eccentricity growth even after the initial-condition-dependent transient evolution. Finally we investigate the effect that the initial cavity size has on the binary-disc interaction and we find that, in the self-gravitating regime, an initially smaller cavity leads to a much faster binary shrinking, as expected. Our results are especially important for very massive black hole binaries such as those in the PTA band, for which gas self gravity cannot be neglected.

Growth of Massive Disks and Early Disk Fragmentation in Primordial Star Formation

Abstract Recent high-resolution simulations demonstrate that disks around primordial protostars easily fragment in the accretion phase before protostars accrete less than a solar mass. To understand why gravitational instability generally causes fragmentation so early, we develop a one-dimensional (1D) nonsteady model of the circumstellar disk that takes the mass supply from an accretion envelope into account. We also compare the model results to a three-dimensional (3D) numerical simulation performed with a code employing adaptive mesh refinement. Our model shows that the self-gravitating disk, through which the Toomre Q parameter is nearly constant at Q ∼ 1, gradually spreads as the disk is fed by the gas infalling from the envelope. We further find that the accretion rate onto the star is an order of magnitude smaller than the mass supply rate onto the disk. This discrepancy makes the disk more massive than the protostar in an early evolutionary stage. Most of the infalling gas is used to extend the outer part of the self-gravitating disk rather than transferred inward toward the star through the disk. We find that similar evolution also occurs in 3D simulations, where the disk becomes three times more massive than the star before the first fragmentation occurs. Our 1D disk model well explains the evolution of the disk-to-star mass ratio observed in the simulation. We argue that the formation of such a massive disk leads to early disk fragmentation.

Evaluating galaxy dynamical masses from kinematics and jeans equilibrium in simulations

ABSTRACT We provide prescriptions to evaluate the dynamical mass (Mdyn) of galaxies from kinematic measurements of stars or gas using analytic considerations and the VELA suite of cosmological zoom-in simulations at z = 1–5. We find that Jeans or hydrostatic equilibrium is approximately valid for galaxies of stellar masses above M⋆ ∼ 109.5 M⊙ out to 5 effective radii (Re). When both measurements of the rotation velocity vϕ and of the radial velocity dispersion σr are available, the dynamical mass $M_{\rm dyn} \!\simeq \! G^{-1} V_{\rm c}^2 r$, can be evaluated from the Jeans equation $V_{\rm c}^2= v_\phi ^2 + \alpha \sigma _{\rm r}^2$ assuming cylindrical symmetry and a constant, isotropic σr. For spheroids, α is inversely proportional to the Sérsic index n and α ≃ 2.5 within Re, stars for the simulated galaxies. The prediction for a self-gravitating exponential disc, α = 3.36(r/Re), is invalid in the simulations, where the dominant spheroid causes a weaker gradient from α ≃ 1 at Re, gas to 4 at 5Re, gas. The correction in α for the stars due to the gradient in σr(r) is roughly balanced by the effect of the aspherical potential, while the effect of anisotropy is negligible. When only the effective projected velocity dispersion σl is available, the dynamical mass can be evaluated as $M_{\rm dyn} = K G^{-1} R_{\rm e} \sigma _{\rm l}^2$, where the virial factor K is derived from α, given the inclination and vϕ/σr. We find that the standard value K = 5 is approximately valid only when averaged over inclinations and for compact and thick discs, as it ranges from 4.5 to above 10 between edge-on and face-on projections.

A two-step gravitational cascade for the fragmentation of self-gravitating discs

ABSTRACT 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 time-scale β expressed in unit of orbital time-scale, 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 for which the disc fragments is consistent with most previous results, and is not well converged with resolution. By studying the probability density function of the fluctuations of the column density (Σ-PDF), we argue that there is no strict separation between the fragmented and the unfragmented regimes but rather a smooth transition with the probability of apparition of fragments steadily diminishing as the cooling becomes less effective. We find that the high column density part of the Σ-PDF follows a simple power law whose slope turns out to be proportional to β and we propose an explanation based on the balance between cooling and heating through gravitational stress. Our explanation suggests that a more efficient cooling requires more heating implying a larger fraction of dense material which, in the absence of characteristic scales, results in a shallower scale-free power law. We propose that the gravitational cascade proceeds in two steps, first the formation of a dense filamentary spiral pattern through a sequence of quasi-static equilibrium triggered by the viscous transport of angular momentum, and second the collapse alongside these filaments that eventually results in the formation of bounded fragments.

Particle Dynamics in 3D Self-gravitating Disks. II. Strong Gas Accretion and Thin Dust Disks

Abstract Observations suggest that protoplanetary disks have moderate accretion rates onto the central young star, especially at early stages (e.g., HL Tau), indicating moderate disk turbulence. However, recent ALMA observations suggest that dust is highly settled, implying weak turbulence. Motivated by such tension, we carry out 3D stratified local simulations of self-gravitating disks, focusing on the settling of dust particles in actively accreting disks. We find that gravitationally unstable disks can have moderately high accretion rates while maintaining a relatively thin dust disk for two reasons. First, accretion stress from the self-gravitating spirals (self-gravity stress) can be stronger than the stress from turbulence (Reynolds stress) by a factor of 5–20. Second, the strong gravity from the gas to the dust decreases the dust scale height by another factor of ∼2. Furthermore, the turbulence is slightly anisotropic, producing a larger Reynolds stress than the vertical dust diffusion coefficient. Thus, gravitoturbulent disks have unusually high vertical Schmidt numbers (Sc z ) if we scale the total accretion stress with the vertical diffusion coefficient (e.g., Sc z ∼ 10–100). The reduction of the dust scale height by the gas gravity should also operate in gravitationally stable disks (Q > 1). Gravitational forces between particles become more relevant for the concentration of intermediate dust sizes, forming dense clouds of dust. After comparing with HL Tau observations, our results suggest that self-gravity and gravity among different disk components could be crucial for solving the conflict between the protoplanetary disk accretion and dust settling, at least at the early stages.

Particle Dynamics in 3D Self-gravitating Disks. I. Spirals

Abstract Spiral arms are distinctive features of many circumstellar disks, observed in scattered light, which traces the disk surface; millimeter dust emission, which probes the disk midplane; as well as molecular emission. The two leading explanations for spirals are wakes generated by a massive planet and the density waves excited by disk self-gravity. We use stratified 3D hydrodynamic shearing-box simulations including dust particles and disk self-gravity to investigate how gas and dust spirals in a self-gravitating disk depend on the simulation size, the cooling efficiency, and the aerodynamic properties of particles. We find that the opening angles of spirals are universal (∼10°) and not significantly affected by the size of the computational domain, the cooling time, or the particle size. In simulations with the biggest domain, the spirals in the gaseous disk become slightly more open with a higher cooling efficiency. Small dust follows the gaseous spirals very well, while intermediate-sized dust with dimensionless stopping time (St) close to 1 is more concentrated in the spirals and shows stronger spirals. However, large dust with St > 1 also shows spirals, which is different from some previous simulations. We identify that this is due to the gravity from the gas to the dust component. We show that when St ≳ Q, the gravitational force from the gaseous spirals on the dust particles becomes stronger than the particles’ aerodynamic drag force, so that the gas significantly affects these large particles through gravitational interaction. This has important implications for both spiral observations and planetesimal formation/dynamics.

Formation of Gaps in Self-gravitating Debris Disks by Secular Resonance in a Single-planet System. I. A Simplified Model

Abstract Spatially resolved images of debris disks frequently reveal complex morphologies such as gaps, spirals, and warps. Most existing models for explaining such morphologies focus on the role of massive perturbers (i.e., planets, stellar companions), ignoring the gravitational effects of the disk itself. Here we investigate the secular interaction between an eccentric planet and a massive, external debris disk using a simple analytical model. Our framework accounts for both the gravitational coupling between the disk and the planet, as well as the disk self-gravity—with the limitation that it ignores the non-axisymmetric component of the disk (self-)gravity. We find generally that even when the disk is less massive than the planet, the system may feature secular resonances within the disk (contrary to what may be naively expected), where planetesimal eccentricities get significantly excited. Given this outcome, we propose that double-ringed debris disks, such as those around HD 107146 and HD 92945, could be the result of secular resonances with a yet-undetected planet interior to the disk. We characterize the dependence of the properties of the secular resonances (i.e., locations, timescales, and widths) on the planet and disk parameters, finding that the mechanism is robust provided the disk is massive enough. As an example, we apply our results to HD 107146 and find that this mechanism readily produces ∼20 au wide non-axisymmetric gaps. Our results may be used to set constraints on the total mass of double-ringed debris disks. We demonstrate this for HD 206893, for which we infer a disk mass of ≈170M ⊕ by considering perturbations from the known brown dwarf companion.

Jeans Instability of a Protoplanetary Circular Disk Taking into Account the Magnetic Field and Radiation in Nonextensive Tsallis Kinetics

Within the framework of Tsallis nonextensive statistics, the criteria for the Jeans gravitational instability are derived for a self-gravitating protoplanetary disk, whose substance consists of a mixture of a conducting ideal q-gas and modified radiation of a photon gas. The instability criteria are derived from the corresponding dispersion relations written for both neutral disk matter and magnetized plasma with modified blackbody radiation. The thermodynamics of a photon gas are constructed based on the nonextensive Tsallis quantum entropy, which depends on the deformation parameter. It is shown that blackbody q-radiation can stabilize the state of a nonextensive medium for a purely gaseous disk, and for an electrically conducting disk, the Jeans instability criterion is modified by the magnetic field and radiation pressure only in the transverse propagation mode of the disturbance wave.

Eigenstates of quasi-Keplerian self-gravitating particle discs

ABSTRACT Although quasi-Keplerian discs are among the most common astrophysical structures, computation of secular angular momentum transport within them routinely presents a considerable practical challenge. In this work, we investigate the secular small-inclination dynamics of a razor-thin particle disc as the continuum limit of a discrete Lagrange–Laplace secular perturbative theory and explore the analogy between the ensuing secular evolution – including non-local couplings of self-gravitating discs – and quantum mechanics. We find the ‘quantum’ Hamiltonian that describes the time evolution of the system and demonstrate the existence of a conserved inner product. The lowest-frequency normal modes are numerically approximated by performing a Wick rotation on the equations of motion. These modes are used to quantify the accuracy of a much simpler local-coupling model, revealing that it predicts the shape of the normal modes to a high degree of accuracy, especially in narrow annuli, even though it fails to predict their eigenfrequencies.

Джинсовская неустойчивость протопланетного околозвездного диска с учетом магнитного поля и излучения в неэкстенсивной кинетике Тсаллиса

Within the framework of the Tsallis non-extensive statistics, a derivation of the Jeans gravitational instability criteria is given for a self-gravitating protoplanetary disk, the substance of which consists of a mixture of a conducting ideal q-gas and modified radiation of a photon gas. The instability criteria are derived from the corresponding dispersion relations written for both neutral disk matter and magnetized plasma with modified blackbody radiation. The thermodynamics of a photon gas is constructed based on the nonextensive quantum entropy Tsallis, which depends on the deformation parameter. It is shown that blackbody radiation can stabilize the state of a nonextensive medium for a purely gaseous disk, and for an electrically conducting disk, the Jeans instability criterion is modified by a magnetic field and radiation pressure only in the transverse regime of propagation of the disturbance wave.

Nonlinear cosmogony of the spiral galaxy bulges

Abstract In this paper, we develop an early idea of one of the authors (Nuritdinov 1992a,b), who was the first to propose the mechanism of instability of the warp perturbation mode on the background of a nonstationary disk. For this aim, we have studied a model of a nonlinearly non-stationary self-gravitating disk with an anisotropic velocity diagram. The model has a composite nature, or rather, it is a superposition of isotropic and anisotropic states of the disk. In the general case, it is obtained a nonstationary analogue of the dispersion equation of this composite model. We have also investigated the behavior of the domed perturbation mode, the instability of which leads to the formation of a classical bulge in the central region of the disk. In addition, we considered the critical diagrams of the dependence of the virial ratio on the rotation rate of the system for various values of the superposition parameter and the corresponding diagrams for the increments of instability.

Global Spiral Density Wave Modes in Protoplanetary Disks: Morphology of Spiral Arms

Abstract We analyze two-armed global spiral density wave modes generated by gravitational instability in razor-thin, nonviscous, self-gravitating protoplanetary disks to understand the dependence of spiral arm morphology (pitch angle α and amplitude) on various disk conditions. The morphologies of the resulting spiral density wave modes closely resemble observations. Their pitch angles and pattern speeds are insensitive to the boundary conditions adopted. Gaussian disks exhibit more tightly wound spirals (smaller pitch angle) than power-law disks under the same conditions. We find that at a fixed disk-to-star mass ratio (M d /M *), the pitch angle increases with average Toomre’s stability parameter ( ) or average disk aspect ratio ( ). For a given , density wave modes with higher M d /M * have larger pitch angles, while the behavior reverses for a given . The interdependence between pitch angle and disk properties can be roughly approximated by , where c s is the sound speed. Our gravitational instability-excited spiral density waves can be distinguished from planet-launched spirals: (1) massive cool disks have spiral pitch angle falling with radius, while low-mass hot disks have spiral pitch angle rising with radius; (2) the profile of spiral amplitude presents several dips and bumps. We propose that gravitational instability-excited density waves can serve as an alternative scenario to explain the observed spiral arms in self-gravitating protoplanetary disks.

Hiding Signatures of Gravitational Instability in Protoplanetary Disks with Planets

Abstract We carry out three-dimensional smoothed particle hydrodynamics simulations to show that a migrating giant planet strongly suppresses the spiral structure in self-gravitating disks. We present mock Atacama Large Millimeter/submillimeter Array (ALMA) continuum observations that show that in the absence of a planet, spiral arms due to gravitational instability are easily observed. Whereas in the presence of a giant planet, the spiral structures are suppressed by the migrating planet resulting in a largely axisymmetric disk with a ring and gap structure. Our modeling of the gas kinematics shows that the planet’s presence could be inferred, for example, using optically thin 13C16O. Our results show that it is not necessary to limit the gas mass of disks by assuming high dust-to-gas mass ratios in order to explain a lack of spiral features that would otherwise be expected in high-mass disks.

Self-gravitating razor-thin discs around black holes via multi-hole seeds

We construct self-gravitating razor-thin discs of counterrotating matter around Schwarzschild black holes (BHs) by applying the ‘displace, cut, and reflect’ method to known seed solutions representing multi-holes. All but one of the sources of the seed solution generate the surrounding annular disc, whereas the remaining BH is mapped onto a Schwarzschild BH which lies at the disc centre after the transformation. The discs are infinite in extent, have annular character, and are linearly stable up to the innermost stable circular orbit (ISCO) of the system. The spacetime is asymptotically flat, having finite Arnowitt–Deser–Misner mass. Moreover, all energy conditions for the disc are satisfied for radii larger than the ISCO radius; the method, however, produces counterrotating streams with superluminal velocities in the vicinity of the central BH. We also comment on charged discs around extremal Reissner–Nordström BHs constructed from a Majumdar–Papapetrou N-BH seed solution. These simple examples can be extended to more general ‘BH + disc’ solutions, obtained by the same method from seeds of the type ‘BH + arbitrary axisymmetric source’. A natural follow-up of this work would be to construct discs around Reissner–Nordström BHs with arbitrary charge-to-mass ratio and around Kerr BHs.

The observational impact of dust trapping in self-gravitating discs

ABSTRACT We present a 3D semi-analytical 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 (Atacama Large Millimeter/submillimeter Array) 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 the grain fragmentation threshold may satisfy this criterion; thus, we predict that signatures of gravitational instability may be detectable in discs of lower mass than has previously been suggested. For example, we find that discs with disc-to-star mass ratios as low as 0.10 are capable of driving observable spiral arms. Substructure becomes challenging to detect in discs where no grain growth has occurred or in which grain growth has proceeded well beyond the grain fragmentation threshold. We demonstrate how we can use our model to retrieve information about dust trapping and grain growth through multiwavelength observations of discs, and using estimates of the opacity spectral index. Applying our disc model to the Elias 27, WaOph 6, and IM Lup systems, we find gravitational instability to be a plausible explanation for the observed substructure in all three discs, if sufficient grain growth has indeed occurred.

Orbital evolution of Saturn’s satellites due to the interaction between the moons and the massive rings

Context. Saturn’s mid-sized moons (satellites) have a puzzling orbital configuration with trapping in mean-motion resonances with every-other pairs (Mimas-Tethys 4:2 and Enceladus-Dione 2:1). To reproduce their current orbital configuration on the basis of a recent model of satellite formation from a hypothetical ancient massive ring, adjacent pairs must pass first-order mean-motion resonances without being trapped. Aims. The trapping could be avoided by fast orbital migration and/or excitation of the satellite’s eccentricity caused by gravitational interactions between the satellites and the rings (the disk), which are still unknown. In our research we investigate the satellite orbital evolution due to interactions with the disk through full N-body simulations. Methods. We performed global high-resolution N-body simulations of a self-gravitating particle disk interacting with a single satellite. We used N ∼ 105 particles for the disk. Gravitational forces of all the particles and their inelastic collisions are taken into account. Results. Dense short-wavelength wake structure is created by the disk self-gravity and a few global spiral arms are induced by the satellite. The self-gravity wakes regulate the orbital evolution of the satellite, which has been considered as a disk spreading mechanism, but not as a driver for the orbital evolution. Conclusions. The self-gravity wake torque to the satellite is so effective that the satellite migration is much faster than was predicted with the spiral arm torque. It provides a possible model to avoid the resonance capture of adjacent satellite pairs and establish the current orbital configuration of Saturn’s mid-sized satellites.

Dynamics of the Parker–Jeans Instability of Gaseous Disks Including the Effect of Cosmic Rays

Abstract The effect of cosmic rays on the Parker–Jeans instability in magnetized self-gravitating gaseous disks is studied using three-dimensional magnetohydrodynamic simulations with cosmic rays taken as a massless fluid with notable pressure. Cosmic-ray diffusion along the magnetic field is taken into account in the simulation. The initial configuration of the disk is a magnetized cold gas slab sandwiched between hot coronae (on top and bottom). We show that cosmic rays play an important role in the formation of filaments or clumps in some parameter regimes. In a certain range of the thickness of the gas slab, the cosmic-ray diffusion coefficient plays a decisive role in determining whether the filaments lie along or perpendicular to the magnetic field. We also consider the effect of rotation on the system.

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.