Shakura-Sunyaev Viscosity Parameter Research Articles

The effect of thermal torques on AGN disc migration traps and gravitational wave populations

ABSTRACT Accretion discs in active galactic nuclei (AGNs) foster black hole (BH) formation, growth, and mergers. Stellar mass BHs migrate inwards under the influence of hydrodynamical torques unless they encounter a region where the torque flips sign. At these migration traps, BHs accumulate and merge via dynamical or gas-assisted interactions, producing high-frequency LIGO/Virgo/KAGRA (LVK) gravitational wave (GW) sources and potentially cutting off the supply of extreme mass ratio inspirals that would otherwise make low-frequency, LISA-band GWs. In this paper, we study the interplay between different types of migration torques, focusing especially on the ‘thermal torques’ generated by the thermal response of the AGN to embedded stellar-mass BHs that accrete through their own mini-discs. In contrast to previous work, we find that Type I torques cannot produce migration traps on their own, but thermal torques often do, particularly in low-mass AGN. The migration traps produced by thermal torques exist at much larger distances (∼103−5 gravitational radii) than do previously identified Type I traps, carrying implications for GW populations at multiple frequencies. Finally, we identify a bifurcation of AGN discs into two regimes: migration traps exist below a critical AGN luminosity, and do not at higher luminosities. This critical luminosity is fit as $\log _{10} L_{\rm AGN}^c = 45 {\!-\!} 0.32 \log _{10}{(\alpha /0.01)}$ where α is the Shakura–Sunyaev viscosity parameter, a range compatible with recent claims that LVK GWs are not preferentially associated with high-luminosity AGN.

3D Radiation Hydrodynamic Simulations of Gravitational Instability in AGN Accretion Disks: Effects of Radiation Pressure

We perform 3D radiation hydrodynamic local shearing-box simulations to study the outcome of gravitational instability (GI) in optically thick active galactic nuclei (AGNs) accretion disks. GI develops when the Toomre parameter Q T ≲ 1, and may lead to turbulent heating that balances radiative cooling. However, when radiative cooling is too efficient, the disk may undergo runaway gravitational fragmentation. In the fully gas-pressure-dominated case, we confirm the classical result that such a thermal balance holds when the Shakura–Sunyaev viscosity parameter (α) due to the gravitationally driven turbulence is ≲0.2, corresponding to dimensionless cooling times Ωt cool ≳ 5. As the fraction of support by radiation pressure increases, the disk becomes more prone to fragmentation, with a reduced (increased) critical value of α (Ωt cool). The effect is already significant when the radiation pressure exceeds 10% of the gas pressure, while fully radiation-pressure-dominated disks fragment at t cool ≲ 50 Ω−1. The latter translates to a maximum turbulence level α ≲ 0.02, comparable to that generated by magnetorotational instability. Our results suggest that gravitationally unstable (Q T ∼ 1) outer regions of AGN disks with significant radiation pressure (likely for high/near-Eddington accretion rates) should always fragment into stars, and perhaps black holes.

Estimating the depth of gaps opened by planets in eccentric orbit

ABSTRACT Planets can carve gaps in the surface density of protoplanetary discs. The formation of these gaps can reduce the corotation torques acting on the planets. In addition, gaps can halt the accretion of solids on to the planets as dust and pebbles can be trapped at the edge of the gap. This accumulation of dust could explain the origin of the ring-like dust structures observed using high-resolution interferometry. In this work, we provide an empirical scaling relation for the depth of the gap cleared by a planet on an eccentric orbit as a function of the planet-to-star mass ratio q, the disc aspect ratio h, Shakura–Sunyaev viscosity parameter α, and planetary eccentricity e. We construct the scaling relation using a heuristic approach: we calibrate a toy model based on the impulse approximation with 2D hydrodynamical simulations. The scaling reproduces the gap depth for moderate eccentricities (e ≤ 4h) and when the surface density contrast outside and inside the gap is ≤102. Our framework can be used as the basis of more sophisticated models aiming to predict the radial gap profile for eccentric planets.

Steady-state accretion in magnetized protoplanetary disks

Context. The transition between magnetorotational instability (MRI)-active and magnetically dead regions corresponds to a sharp change in the disk turbulence level, where pressure maxima may form, hence potentially trapping dust particles and explaining some of the observed disk substructures. Aims. We aim to provide the first building blocks toward a self-consistent approach to assess the dead zone outer edge as a viable location for dust trapping, under the framework of viscously driven accretion. Methods. We present a 1+1D global magnetically driven disk accretion model that captures the essence of the MRI-driven accretion, without resorting to 3D global nonideal magnetohydrodynamic (MHD) simulations. The gas dynamics is assumed to be solely controlled by the MRI and hydrodynamic instabilities. For given stellar and disk parameters, the Shakura–Sunyaev viscosity parameter, α, is determined self-consistently under the adopted framework from detailed considerations of the MRI with nonideal MHD effects (Ohmic resistivity and ambipolar diffusion), accounting for disk heating by stellar irradiation, nonthermal sources of ionization, and dust effects on the ionization chemistry. Additionally, the magnetic field strength is numerically constrained to maximize the MRI activity. Results. We demonstrate the use of our framework by investigating steady-state MRI-driven accretion in a fiducial protoplanetary disk model around a solar-type star. We find that the equilibrium solution displays no pressure maximum at the dead zone outer edge, except if a sufficient amount of dust particles has accumulated there before the disk reaches a steady-state accretion regime. Furthermore, the steady-state accretion solution describes a disk that displays a spatially extended long-lived inner disk gas reservoir (the dead zone) that accretes a few times 10−9 M⊙ yr−1. By conducting a detailed parameter study, we find that the extent to which the MRI can drive efficient accretion is primarily determined by the total disk gas mass, the representative grain size, the vertically integrated dust-to-gas mass ratio, and the stellar X-ray luminosity. Conclusions. A self-consistent time-dependent coupling between gas, dust, stellar evolution models, and our general framework on million-year timescales is required to fully understand the formation of dead zones and their potential to trap dust particles.

Viscous dark energy accretion activities : sonic speed, angular momentum and Mach number studies

In this present article, we study different accretion properties regarding viscous accretion of dark energy. Modified Chaplygin gas is chosen as the dark energy candidate. Viscosity is encountered with the help of Shakura–Sunyaev viscosity parameter. We study sonic speed vs radial distance curves. We compare between adiabatic and dark energy dominated cases and follow that sonic speed falls as we go nearer to the central gravitating object. As viscosity is imposed, a threshold drop in accretion sonic speed is followed. Average rate of fall in accretion sonic speed is increased with black hole’s spin. This is signifying that this kind of accretion is weakening the overall matter/energy infall. Specific angular momentum to Keplerian angular momentum ratio is found to fall as we go far from the black hole. Accretion Mach number turns high as we go towards the inner region and high wind Mach number is not allowed as we are going out. Combining, we conclude that the system weakens the feeding process of accretion.

Cooling+Heating Flows in Galaxy Clusters: Turbulent Heating, Spectral Modeling, and Cooling Efficiency

Abstract The discrepancy between expected and observed cooling rates of X-ray emitting gas has led to the cooling-flow problem at the cores of clusters of galaxies. A variety of models have been proposed to model the observed X-ray spectra and resolve the cooling-flow problem, which involves heating the cold gas through different mechanisms. As a result, realistic models of X-ray spectra of galaxy clusters need to involve both heating and cooling mechanisms. In this paper, we argue that the heating timescale is set by the magnetohydrodynamic (MHD) turbulent viscous heating for the Intracluster plasma, parameterized by the Shakura–Sunyaev viscosity parameter, α. Using a cooling+heating flow model, we show that a value of α ≃ 0.05 (with 10% scatter) provides improved fits to the X-ray spectra of cooling flow, while at the same time, predicting reasonable cooling efficiency, . Our inferred values for α based on X-ray spectra are also in line with direct measurements of turbulent pressure in simulations and observations of galaxy clusters. This simple picture unifies astrophysical accretion, as a balance of MHD turbulent heating and cooling, across more than 16 orders of magnitudes in scale, from neutron stars to galaxy clusters.

A Study of a Tilted Thin Inner Accretion Disk around a Spinning Black Hole

Abstract The inner part of a thin accretion disk around a Kerr black hole can serve as an important tool to study the physics of the strong gravity regime. A tilt in such a disk with respect to the black hole spin axis is particularly useful for this purpose, as such a tilt can have a significant effect on the observed X-ray spectral and timing features via Lense–Thirring precession. However, the inner disk has been predicted to become aligned with the spin direction of the black hole by the well-known Bardeen–Petterson effect. Here we calculate, both analytically and numerically, the radial profile of the thin accretion disk tilt angle in the viscous regime (i.e., α is the Shakura–Sunyaev viscosity parameter, H is the disk thickness, and R is the radial distance). We show that the inner disk may not be aligned at all for certain reasonable ranges of parameter values. This makes the inner accretion disk particularly promising to probe the black hole parameters, and the accretion process in the strong gravity region.

New Constraints on Turbulence and Embedded Planet Mass in the HD 163296 Disk from Planet–Disk Hydrodynamic Simulations

Abstract Recent Atacama Large Millimeter and Submillimeter Array (ALMA) observations of the protoplanetary disk around the Herbig Ae star HD 163296 revealed three depleted dust gaps at 60, 100, and 160 au in the 1.3 mm continuum as well as CO depletion in the middle and outer dust gaps. However, no CO depletion was found in the inner dust gap. To examine the planet–disk interaction model, we present results of 2D two fluid (gas + dust) hydrodynamic simulations coupled with 3D radiative transfer simulations. To fit the high gas-to-dust ratio of the first gap, we find that the Shakura–Sunyaev viscosity parameter α must be very small ( ) in the inner disk. On the other hand, a relatively large α ( ) is required to reproduce the dust surface density in the outer disk. We interpret the variation of α as an indicator of the transition from an inner dead zone to the outer magnetorotational instability (MRI) active zone. Within ∼100 au, the HD 163296 disk’s ionization level is low, and non-ideal magnetohydrodynamic effects could suppress the MRI, so the disk can be largely laminar. The disk’s ionization level gradually increases toward larger radii, and the outermost disk ( au) becomes turbulent due to MRI. Under this condition, we find that the observed dust continuum and CO gas line emissions can be reasonably fit by three half-Jovian-mass planets (0.46, 0.46, and ) at 59, 105, and 160 au, respectively.

Self-similar accretion in thin discs around near-extremal black holes

Near-maximally spinning black holes display conformal symmetry in their near-horizon region, which is therefore the locus of critical phenomena. In this paper, we revisit the Novikov-Thorne accretion thin disk model and find a new self-similar radiation-dominated solution in the extremely high spin regime. Motivated by the self-consistency of the model, we require that matter flows at the sound speed at the innermost stable circular orbit (ISCO). We observe that, when the disk pressure is dominated by radiation at the ISCO, which occurs for the best-fitting Novikov-Thorne model of GRS 1915+105, the Shakura-Sunyaev viscosity parameter can be expressed in terms of the spin, mass accretion rate and radiative efficiency. We quantitatively describe how the exact thin disk solution approaches the self-similar solution in the vicinity of the ISCO and for increasing spins.

Viscosity parameter in dissipative accretion flows with mass outflow around black holes

Numerical hydrodynamic simulation of inviscid and viscous flows have shown that significant outflows could be produced from the CENtrifugal pressure supported BOundary Layer or CENBOL of an advective disk. However, this barrier is weakened in presence of viscosity, more so, if there are explicit energy dissipations at the boundary layer itself. We study effects of viscosity and energy dissipation theoretically on the outflow rate and show that as the viscosity or energy dissipation (or both) rises, the prospect of formation of outflows is greatly reduced, thereby verifying results obtained through observations and numerical simulations. Indeed, we find that in a dissipative viscous flow, shocks in presence of outflows can be produced only if the Shakura-Sunyaev viscosity parameter {\alpha} is less than 0.2. This is a direct consequence of modification of the Rankine-Hugoniot relation across the shock in a viscous flow, when the energy dissipation and mass loss in the form of outflows from the post-shock region are included. If we ignore the effects of mass loss altogether, the standing dissipative shocks in viscous flows may occur only if {\alpha} < 0.27. These limits are tighter than the absolute limit of {\alpha} = 0.3 valid for a situation when the shock itself neither dissipates energy nor any outflow is formed. We compute typical viscosity parameters required to understand spectral and temporal properties of several black hole candidates such as GX399-4, MAXI J1659-152 and MAXI J1836-194 and find that required {\alpha} are indeed well within our prescribed limit.

HOW EMPTY ARE DISK GAPS OPENED BY GIANT PLANETS?

Gap clearing by giant planets has been proposed to explain the optically thin cavities observed in many protoplanetary disks. How much material remains in the gap determines not only how detectable young planets are in their birth environments, but also how strong corotation torques are, which impacts how planets can survive fast orbital migration. We determine numerically how the average surface density inside the gap, sigma_gap, depends on planet-to-star mass ratio q, Shakura-Sunyaev viscosity parameter alpha, and disk height-to-radius aspect ratio h/r. Our results are derived from our new GPU-accelerated Lagrangian hydrodynamical code PEnGUIn, and are verified by independent simulations with ZEUS90. For Jupiter-like planets, we find sigma_gap \propto q^-2.2 alpha^1.4 (h/r)^6.6, and for near brown dwarf masses, sigma_gap \propto q^-1 alpha^1.3 (h/r)^6.1. Surface density contrasts inside and outside gaps can be as large as 10^4, even when the planet does not accrete. We derive a simple analytic scaling, sigma_gap \propto q^-2 alpha^1 (h/r)^5, that compares reasonably well to empirical results, especially at low Neptune-like masses, and use discrepancies to highlight areas for progress.

Implications of a viscosity bound on black hole accretion

Motivated by the viscosity bound in gauge/gravity duality, we consider the ratio of shear viscosity (η) to entropy density (s) in black hole accretion flows. We use both an ideal gas equation of state and the QCD equation of state obtained from lattice for the fluid accreting onto a Kerr black hole. The QCD equation of state is considered since the temperature of accreting matter is expected to approach 1012 K in certain hot flows. We find that in both the cases η/s is small only for primordial black holes and several orders of magnitude larger than any known fluid for stellar and supermassive black holes. We show that a lower bound on the mass of primordial black holes leads to a lower bound on η/s and vice versa. Finally we speculate that the Shakura–Sunyaev viscosity parameter should decrease with increasing density and/or temperatures.

MRI-driven accretion on to magnetized stars: axisymmetric MHD simulations

We present the first results of a global axisymmetric simulation of accretion onto rotating magnetized stars from a turbulent, MRI-driven disk. The angular momentum is transported outward by the magnetic stress of the turbulent flow with a rate corresponding to a Shakura-Sunyaev viscosity parameter alpha\approx 0.01-0.04. The result of the disk-magnetosphere interaction depends on the orientation of the poloidal field in the disk relative to that of the star at the disk-magnetosphere boundary. If fields have the same polarity, then the magnetic flux is accumulated at the boundary and blocks the accretion which leads to the accumulation of matter at the boundary. Subsequently, this matter accretes to the star in outburst before accumulating again. Hence, the cycling, `bursty' accretion is observed. If the disc and stellar fields have opposite polarity, then the field reconnection enhances the penetration of the disk matter towards the deeper field lines of the magnetosphere. However, the magnetic stress at the boundary is lower due to the field reconnection. This decreases the accretion rate and leads to smoother accretion at a lower rate. Test simulations show that in the case of higher accretion rate corresponding to alpha=0.05-0.1, accretion is bursty in cases of both polarities. On the other hand, at much lower accretion rates corresponding to alpha < 0.01, accretion is not bursty in any of these cases. We conclude that the episodic, bursty accretion is expected during periods of higher accretion rates in the disc, and in some cases it may alternate between bursty and smooth accretion, if the disk brings the poloidal field of alternating polarity. We find that a rotating, magnetically-dominated corona forms above and below the disk, and that it slowly expands outward, driven by the magnetic force.

STAR FORMATION IN A QUASAR DISK

Using a version of the ZEUS code, we carry out two-dimensional simulations of self-gravitating shearing sheets, with application to QSO accretion disks at a few thousand Schwarzschild radii, corresponding to a few hundredths of a parsec for a 10^8 solar-mass black hole. Radiation pressure and optically thick radiative cooling are implemented via vertical averages. We determine dimensionless versions of the maximum surface density, accretion rate, and effective viscosity that can be sustained by density-wave turbulence without fragmentation. Where fragments do form, we study the final masses that result. The maximum Shakura-Sunyaev viscosity parameter is approximately 0.4. Fragmentation occurs when the cooling time is less than about twice the shearing time, as found by Gammie and others, but can also occur at very long cooling times in sheets that are strongly radiation-pressure dominated. For accretion at the Eddington rate onto a 10^8 solar-mass black hole, fragmentation occurs beyond four thousand Schwarzschild radii, r_s. Near this radius, initial fragment masses are several hundred suns, consistent with estimates from linear stability; final masses after merging increase with the size of the sheet, reaching several thousand suns in our largest simulations. With increasing black-hole mass at a fixed Eddington ratio, self-gravity prevails to smaller multiples of r_s, where radiation pressure is more important and the cooling time is longer compared to the dynamical time; nevertheless, fragmentation can occur and produces larger initial fragment masses. We also find energy conservation is likely to be a challenge for all eulerian codes in self-gravitating regimes where radiation pressure dominates.

PLANETESIMAL AND PROTOPLANET DYNAMICS IN A TURBULENT PROTOPLANETARY DISK: IDEAL UNSTRATIFIED DISKS

The dynamics of planetesimals and planetary cores may be strongly influenced by density perturbations driven by magneto-rotational turbulence in their natal protoplanetary gas disks. Using the local shearing box approximation, we perform numerical simulations of planetesimals moving as massless particles in a turbulent, magnetized, unstratified gas disk. Our fiducial disk model shows turbulent accretion characterized by a Shakura-Sunyaev viscosity parameter of α ~ 10^-2, with rms density perturbations of ~10%. We measure the statistical evolution of particle orbital properties in our simulations including mean radius, eccentricity, and velocity dispersion. We confirm random walk growth in time of all three properties, the first time that this has been done with direct orbital integration in a local model. We find that the growth rate increases with the box size used at least up to boxes of eight scale heights in horizontal size. However, even our largest boxes show velocity dispersions sufficiently low that collisional destruction of planetesimals should be unimportant in the inner disk throughout its lifetime. Our direct integrations agree with earlier torque measurements showing that type I migration dominates over diffusive migration by stochastic torques for most objects in the planetary core and terrestrial planet mass range. Diffusive migration remains important for objects in the mass range of kilometer-sized planetesimals. Discrepancies in the derived magnitude of turbulence between local and global simulations of magneto-rotationally unstable disks remains an open issue, with important consequences for planet formation scenarios. (Less)

Two temperature viscous accretion flows around rotating black holes: Description of under-fed systems to ultra-luminous X-ray sources

We discuss two temperature accretion disk flows around rotating black holes. As we know that to explain observed hard X-rays the choice of Keplerian angular momentum profile is not unique, we consider the sub-Keplerian regime of the disk. Without any strict knowledge of the magnetic field structure, we assume the cooling mechanism is dominated by bremsstrahlung process. We show that in a range of Shakura–Sunyaev viscosity parameter 0.2 ≳ α ≳ 0.0005 , flow behavior varies widely, particularly by means of the size of disk, efficiency of cooling and corresponding temperatures of ions and electrons. We also show that the disk around a rotating black hole is hotter compared to that around a Schwarzschild black hole, rendering a larger difference between ion and electron temperatures in the former case. With all the theoretical solutions in hand, finally we reproduce the observed luminosities ( L) of two extreme cases—the under-fed AGNs and quasars (e.g. Sgr A ∗ ) with L ≳ 10 33 erg / s to ultra-luminous X-ray sources with L ∼ 10 41 erg/s, at different combinations of mass accretion rate, ratio of specific heats, Shakura–Sunyaev viscosity parameter and Kerr parameter, and conclude that Sgr A ∗ may be an intermediate spinning black hole.

DYNAMICAL STRUCTURE OF VISCOUS ACCRETION DISKS WITH SHOCKS

We develop and discuss global accretion solutions for viscous advection-dominated accretion flow (ADAF) disks containing centrifugally supported isothermal shock waves. The fact that such shocks can exist at all in ADAF disks is a new result. Interestingly, we find that isothermal shocks can form even when the level of viscous dissipation is relatively high. In order to better understand this phenomenon, we explore all possible combinations of the fundamental flow parameters, such as specific energy, specific angular momentum, and viscosity, to obtain the complete family of global solutions. This procedure allows us to identify the region of the parameter space where isothermal shocks can exist in viscous ADAF disks. The allowed region is maximized in the inviscid case, and it shrinks as the level of viscous dissipation increases. Adopting the canonical value γ = 1.5 for the ratio of specific heats, we find that the shock region disappears completely when the Shakura-Sunyaev viscosity parameter α exceeds the critical value ~0.27. This establishes for the first time that steady ADAF disks containing shocks can exist even for relatively high levels of viscous dissipation. If an isothermal shock is present in the disk, it would have important implications for the acceleration of energetic particles that can escape to power the relativistic jets commonly observed around underfed, radio-loud black holes. In two specific applications, we confirm that the kinetic luminosity lost from the disk at the isothermal shock location is sufficient to power the observed relativistic outflows in M87 and Sgr A*.

Particle Acceleration and the Formation of Relativistic Outflows in Viscous Accretion Disks with Shocks

In this Letter, we present a new self-consistent theory for the production of the relativistic outflows observed from radio-loud black hole candidates and active galaxies as a result of particle acceleration in hot, viscous accretion disks containing standing, centrifugally-supported isothermal shocks. This is the first work to obtain the structure of such disks for a relatively large value of the Shakura-Sunyaev viscosity parameter ($\alpha=0.1$), and to consider the implications of the shock for the acceleration of relativistic particles in viscous disks. In our approach, the hydrodynamics and the particle acceleration are coupled and the solutions are obtained self-consistently based on a rigorous mathematical method. We find that particle acceleration in the vicinity of the shock can provide enough energy to power the observed relativistic jet in M87.

Viscous effects on the interaction between the coplanar decretion disc and the neutron star in Be/X-ray binaries

We study the viscous effects on the interaction between the coplanar Be-star disc and the neutron star in Be/X-ray binaries, using a three-dimensional, smoothed particle hydrodynamics code. For simplicity, we assume the Be disc to be isothermal at the temperature of half the stellar effective temperature. In order to mimic the gas ejection process from the Be star, we inject particles with the Keplerian rotation velocity at a radius just outside the star. Both Be star and neutron star are treated as point masses. We find that the Be-star disc is effectively truncated if the Shakura-Sunyaev viscosity parameter alpha_SS >> 1, which confirms the previous semi-analytical result. In the truncated disc, the material decreted from the Be star accumulates, so that the disc becomes denser more rapidly than if around an isolated Be star. The resonant truncation of the Be disc results in a significant reduction of the amount of gas captured by the neutron star and a strong dependence of the mass capture rate on the orbital phase. We also find that an eccentric mode is excited in the Be disc through direct driving due to a one-armed bar potential of the binary. The strength of the mode becomes greater in the case of a smaller viscosity. In a high-resolution simulation with alpha_SS=0.1, the eccentric mode is found to precess in a prograde sense. The mass capture rate by the neutron star modulates as the mode precesses.

Accretion disc outbursts: a new version of an old model

We have developed 1D time-dependent numerical models of accretion discs, using an adaptive grid technique and an implicit numerical scheme, in which the disc size is allowed to vary with time. The code fully resolves the cooling and heating fronts propagating in the disc. We show that models in which the radius of the outer edge of the disc is fixed produce incorrect results, from which probably incorrect conclusions about the viscosity law have been inferred. In particular we show that outside-in outbursts are possible when a standard bimodal behaviour of the Shakura–Sunyaev viscosity parameter α is used. We also discuss to what extent insufficient grid resolution has limited the predictive power of previous models. We find that the global properties (magnitudes, etc.) of transient discs can be addressed by codes using a high, but reasonable, number of fixed grid points. However, the study of the detailed physical properties of the transition fronts generally requires resolutions which are out of reach of fixed grid codes. It appears that most time-dependent models of accretion discs published in the literature have been limited by resolution effects, improper outer boundary conditions, or both.