Viscosity Prescription Research Articles

Notes on the general relativistic viscous ringed disk evolution

Abstract A ringed accretion disk (RAD) models a cluster of axi-symmetric co-rotating and/or counter-rotating tori orbiting in the equatorial plane of a central Kerr super-massive Black Hole. We discuss the time evolution of such a ringed disk within the general relativity framework. Our analysis presents a study of the evolving RAD properties using a thin-disk scheme and solving a diffusion-like evolution equation for a RAD in the Kerr spacetime. In the first stage of evolution there is the inter–disk interaction where the individual rings spread inwardly and outwardly, levelling the structure and forming a single distribution with maximum density determined by the initial spread of the component rings. Timescales are dependent on viscosity prescriptions. The early time luminosity, dominated by the dynamics of the inner ringed structure, shows a clear mark of the inner ringed structure. The RAD eventually reaches a single disk phase, building accretion to the inner edge regulated by the inner edge boundary conditions. The late–time luminosity associated to the ringed disk follows a power law decline for the final single disk. In the sideline of this analysis we also considered a modified prescription mimicking an effective turbulent viscosity in the early phases of the rings evolutions.

Long-term evolution of binary orbits induced by circumbinary disks

Circumbinary disks are found in a variety of astrophysical scenarios, spanning binary star formation to accreting supermassive black hole binaries. Depending on the characteristics of the system, the interaction with a circumbinary disk can either damp or excite the binary’s eccentricity and can also widen or shrink the orbit. To predict the outcome of the long-term disk-binary interaction, we present a new formalism based on the results of recent suites of hydrodynamic simulations, which resolve the complex geometry of the gas in the vicinity of the binary and fully account for the gravitational and accretion forces. We released a python package, spindler, that implements our model. We show that – under the assumed thin disk model with a fixed thickness and viscosity prescription – accretion onto the binary depletes the disk mass before inducing a significant change in the orbital separation or the mass ratio, unless the mass reservoir feeding the disk is comparable to the mass of the binary. This finding implies that, in most scenarios, an interaction with a circumbinary disk is not an efficient mechanism to shrink the orbit of the binary. However, the interaction can excite the eccentricity up to an equilibrium value, and induce a statistical correlation between the mass ratio and eccentricity, as long as the mass of the disk is at least a few percent of the mass of the binary. We consider the applicability of our model to a variety of astrophysical scenarios: during star formation, in evolved stellar binaries, triples, and in supermassive black hole binaries. We discuss the theoretical and observational implications of our predictions.

Hydrodynamical simulations of circumbinary accretion: balance between heating and cooling

ABSTRACT Hydrodynamical interaction in circumbinary discs (CBDs) plays a crucial role in various astrophysical systems, ranging from young stellar binaries to supermassive black hole binaries in galactic centres. Most previous simulations of binary-disc systems have adopted locally isothermal equation of state. In this study, we use the grid-based code Athena++ to conduct a suite of two-dimensional viscous hydrodynamical simulations of circumbinary accretion on a Cartesian grid, resolving the central cavity of the binary. The gas thermodynamics is treated by thermal relaxation towards an equilibrium temperature (based on the constant − β cooling ansatz, where β is the cooling time in units of the local Keplerian time). Focusing on equal mass, circular binaries in CBDs with (equilibrium) disc aspect ratio H/R = 0.1, we find that the cooling of the disc gas significantly influences the binary orbital evolution, accretion variability, and CBD morphology, and the effect depends sensitively on the disc viscosity prescriptions. When adopting a constant kinematic viscosity, a finite cooling time (β ≳ 0.1) leads to a binary inspiral as opposed to an outspiral and the CBD cavity becomes more symmetric. When adopting a dynamically varying α-viscosity, binary inspiral only occurs within a narrow range of cooling time (corresponding to β around 0.5).

Dynamical thin disks

Thin disk accretion is often modeled in highly dynamical settings using the two-dimensional equations of viscous hydrodynamics, with viscosity representing unresolved turbulence. These equations are supposed to arise after vertical integration of the full three-dimensional equations of hydrodynamics, under the assumption of a geometrically thin disk with mirror symmetry about the midplane. However, in the dynamical context, vertical dynamics are neglected by incorrectly assuming instantaneous vertical hydrostatic equilibrium. The resulting errors in the local disk height couple to the horizontal dynamics through the $\ensuremath{\alpha}$-viscosity prescription and disk height-dependent gravitational softening models. Furthermore, the viscous terms in the horizontal equations are only complete if they are inserted after vertical integration, as if the system is actually two-dimensional. Since turbulence breaks mirror symmetry, it is more physically correct to insert a turbulence model at the three-dimensional level and impose mirror symmetry only on average. Thus, some viscous terms are usually missing. With these considerations in mind, we revisit the vertical integration procedure, restricting ourselves to the regime of a Newtonian, non-self-gravitating disk. We obtain six evolution equations with only horizontal dependence, which determine the local vertical position and velocity of the disk surface, in addition to the usual surface densities of mass, horizontal momentum, and energy. This ``2.5-dimensional'' formulation opens the door to efficiently study vertical oscillations of thin disks in dynamical settings, and to improve the treatment of unresolved turbulence. As a demonstration, by including viscous stress at the three-dimensional level, we recover missing viscous terms which involve the vertical variables. We also propose a resummation of the vertically integrated gravitational force, which has a strikingly similar radial profile to a gravitational softening model advocated for in protoplanetary disk studies.

Structure and evolution of a tidally heated star

Context. The shearing motion of tidal flows that are excited in non-equilibrium binary stars transform kinetic energy into heat via a process referred to as tidal heating. Aims. We aim to explore the way tidal heating affects the stellar structure. Methods. We used the TIDES code, which solves the equations of motion of the three-dimensional (3D) grid of volume elements that conform multiple layers of a rotating binary star to obtain an instantaneous value for the angular velocity, ω″, as a function of position in the presence of gravitational, centrifugal, Coriolis, gas pressure, and viscous forces. The released energy, Ė, was computed using a prescription for turbulent viscosity that depends on the instantaneous velocity gradients. The Ė values for each radius were injected into a MESA stellar structure calculation. The method is illustrated for a 1.0 + 0.8 M⊙ binary system, with an orbital period of P = 1.44 d and departures from synchronous rotation of 5% and 10%. Results. Heated models have a larger radius and surface luminosity, a smaller surface convection zone, and lower nuclear reaction rates than the equivalent standard stellar models, and their evolutionary tracks extend to higher temperatures. The magnitude of these effects depends on the amount of injected energy, which, for a fixed set of stellar, rotation and orbital parameters, depends on the perturbed star’s density structure and turbulent viscosity. Conclusions. Tidal heating offers a possible alternative for describing phenomena such as bloated or overluminous binary components, age discrepancies, and aspherical mass ejection, as well as the extended main sequence turnoff in clusters. However, establishing its actual role requires 3D stellar structure models commensurate with the nonspherically symmetric properties of tidal perturbations.

A survey of disc thickness and viscosity in circumbinary accretion: Binary evolution, variability, and disc morphology

ABSTRACTMuch of the parameter space relevant to the evolution of astrophysical circumbinary accretion discs remains unexplored. We have carried out a suite of circumbinary disc simulations surveying both disc thickness and kinematic viscosity, using both constant-ν and constant-α prescriptions. We focus primarily on disc aspect ratios between 0.1 and 0.033, and on viscosities between ν = 0.0005 and ν = 0.008 (in units of binary semimajor axis and orbital frequency), and specialize to circular equal-mass binaries. Both factors strongly influence the evolution of the binary semimajor axis: at ν = 0.0005, inspirals occur at aspect ratios ≲ 0.059, while at ν = 0.004 inspirals occur only at aspect ratios ≲ 0.04. Inspirals occur largely because of the increasingly strong negative torque on the binary by streams of material which lag the binary, with negligible contributions from resonant torques excited in the circumbinary disc. We find that reductions in accretion rate occur when simulations are initialized too far from the eventual quasi-steady state driven by interaction with the binary, rather than being intrinsically linked to the disc aspect ratio. We find not only that the cavity size increases as viscosity is decreased, but that thinner circumbinary discs become more eccentric. Our results suggest that supermassive black hole binaries should be driven, more rapidly than previous estimates, from ∼parsec separations to distances where gravitational waves drive their inspiral, potentially reducing the number of binaries observable by pulsar timing arrays.

A Self-similar Solution of Hot Accretion Flow: The Role of the Kinematic Viscosity Coefficient

Abstract We investigate the dependency of the inflow-wind structure of a hot accretion flow on the kinematic viscosity coefficient. In this regard, we propose a model for the kinematic viscosity coefficient to mimic the behavior of the magnetorotational instability that would be maximal at the rotation axis. Then, we compare our model with two other prescriptions from numerical simulations of the accretion flow. We solve two-dimensional hydrodynamic equations of hot accretion flows in the presence of thermal conduction. The self-similar approach is also adopted in the radial direction. We calculate the properties of the inflow and the wind such as velocity, density, and angular momentum for three models of the kinematic viscosity prescription. On inspection, we find that in our suggested model the wind is less efficient at extracting the angular momentum outward where the self-similar solutions are applied than it is in two other models. The solutions obtained in this paper might be applicable to hydrodynamical numerical simulations of hot accretion flows.

X-ray Dips in AGN and Microquasars – Collapse Timescales of Inner Accretion Disc

Abstract The temporal behaviour of X-rays from some AGN and microquasars is thought to arise from the rapid collapse of the hot, inner parts of their accretion discs. The collapse can occur over the radial infall timescale of the inner accretion disc. However, estimates of this timescale are hindered by a lack of knowledge of the operative viscosity in the collisionless plasma comprising the inner disc. We use published simulation results for cosmic ray diffusion through turbulent magnetic fields to arrive at a viscosity prescription appropriate to hot accretion discs. We construct simplified disc models using this viscosity prescription and estimate disc collapse timescales for 3C 120, 3C 111, and GRS 1915+105. The Shakura-Sunyaev α parameter resulting from our model ranges from 0.02 to 0.08. Our inner disc collapse timescale estimates agree well with those of the observed X-ray dips. We find that the collapse timescale is most sensitive to the outer radius of the hot accretion disc.

The g-mode spectrum of reactive neutron star cores

Abstract We discuss the impact of nuclear reactions on the spectrum of gravity g-modes of a mature neutron star, demonstrating the anticipated disappearance of these modes when the timescale associated with the oscillations is longer than that of nuclear reactions. This is the expected result, but different aspects of the demonstration may be relevant for related problems in neutron star astrophysics. In particular, we develop the framework required for an explicit implementation of finite-time nuclear reactions in neutron star oscillation problems and demonstrate how this formulation connects with the usual bulk viscosity prescription. We also discuss implications of the absence of very high order g-modes for problems of astrophysical relevance.

Testing the disk-corona interplay in radiatively-efficient broad-line AGN

The correlation observed between monochromatic X-ray and UV luminosities in radiatively-efficient active galactic nuclei (AGN) lacks a clear theoretical explanation despite being used for many applications. Such a correlation, with its small intrinsic scatter and its slope that is smaller than unity in log space, represents the compelling evidence that a mechanism regulating the energetic interaction between the accretion disk and the X-ray corona must be in place. This ensures that going from fainter to brighter sources the coronal emission increases less than the disk emission. We discuss here a self-consistently coupled disk-corona model that can identify this regulating mechanism in terms of modified viscosity prescriptions in the accretion disk. The model predicts a lower fraction of accretion power dissipated in the corona for higher accretion states. We then present a quantitative observational test of the model using a reference sample of broad-line AGN and modeling the disk-corona emission for each source in the LX − LUV plane. We used the slope, normalization, and scatter of the observed relation to constrain the parameters of the theoretical model. For non-spinning black holes and static coronae, we find that the accretion prescriptions that match the observed slope of the LX − LUV relation produce X-rays that are too weak with respect to the normalization of the observed relation. Instead, considering moderately-outflowing Comptonizing coronae and/or a more realistic high-spinning black hole population significantly relax the tension between the strength of the observed and modeled X-ray emission, while also predicting very low intrinsic scatter in the LX − LUV relation. In particular, this latter scenario traces a known selection effect of flux-limited samples that preferentially select high-spinning, hence brighter, sources.

Theoretical model of hydrodynamic jet formation from accretion disks with turbulent viscosity

We develop the theoretical model for the analytic description of hydrodynamic jets from protostellar disks employing the Beltrami-Bernoulli flow configuration of disk-jet structure. For this purpose we extend the standard turbulent viscosity prescription and derive several classes of analytic solutions using the flow parametrization in self-similar variables. Derived solutions describe the disk-jet structure, where for the first time jet properties are analytically linked with the properties of the accretion disk flow. The ratio of the jet ejection and disk accretion velocities is controlled by the turbulence parameter, while the ejection velocity increases with the decrease of local sound velocity and the jet launching radius. Derived solutions can be used to analyze the astrophysical jets from protostellar accretion disks and link the properties of outflows with the local observational properties of accretion disk flows.

Orbital decay in binaries containing post-main-sequence stars

The orbital decay of binaries containing a primary sub-giant or red giant star and a stellar or substellar companion is investigated. The tide raised in the primary by the companion leads to an exchange of angular momentum between the orbit and the stellar spin, causing the orbit to contract and the primary to spin up. The rate of orbital decay is computed including both the equilibrium tide, damped by turbulent viscosity in the convective envelope, and the dynamical tide, assumed to be a traveling internal-gravity wave in the radiative core. For close binaries, the tidal forcing period is expected to be much shorter than the eddy turnover timescale in the convective envelope, and the prescription for "reduced" viscosity is an important consideration. The dynamical tide tends to dominate for the closest orbits, while the equilibrium tide dominates for more distant orbits, with the crossover point depending on the stellar mass. The spin up of the primary to synchronous rotation occurs for sufficiently massive secondaries, and this greatly slows the orbital decay until the Darwin instability occurs. A parameter survey is presented for orbital decay as a function of primary and secondary mass, as well as turbulent viscosity prescription. These results are summarized with analytic formulae and numerical results for the age-dependent critical separation, $a_{\rm crit}$, inside of which orbital decay is rapid, and few systems are expected to be observed. The calculations of $a_{\rm crit}$ are compared with APOGEE binaries, as well as solar mass exoplanet host stars.

Low-density, radiatively inefficient rotating-accretion flow on to a black hole

We study low-density axisymmetric accretion flows onto black holes (BHs) with two-dimensional hydrodynamical simulations, adopting the $\alpha$-viscosity prescription. When the gas angular momentum is low enough to form a rotationally supported disk within the Bondi radius ($R_{\rm B}$), we find a global steady accretion solution. The solution consists of a rotational equilibrium distribution at $r\sim R_{\rm B}$, where the density follows $\rho \propto (1+R_{\rm B}/r)^{3/2}$, surrounding a geometrically thick and optically thin accretion disk at the centrifugal radius, where thermal energy generated by viscosity is transported via strong convection. Physical properties of the inner solution agree with those expected in convection-dominated accretion flows (CDAF; $\rho \propto r^{-1/2}$). In the inner CDAF solution, the gas inflow rate decreases towards the center due to convection ($\dot{M}\propto r$), and the net accretion rate (including both inflows and outflows) is strongly suppressed by several orders of magnitude from the Bondi accretion rate $\dot{M}_{\rm B}$ The net accretion rate depends on the viscous strength, following $\dot{M}/\dot{M}_{\rm B}\propto (\alpha/0.01)^{0.6}$. This solution holds for low accretion rates of $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}< 10^{-3}$ having minimal radiation cooling, where $\dot{M}_{\rm Edd}$ is the Eddington rate. In a hot plasma at the bottom ($r<10^{-3}~R_{\rm B}$), thermal conduction would dominate the convective energy flux. Since suppression of the accretion by convection ceases, the final BH feeding rate is found to be $\dot{M}/\dot{M}_{\rm B} \sim 10^{-3}-10^{-2}$. This rate is as low as $\dot{M}/\dot{M}_{\rm Edd} \sim 10^{-7}-10^{-6}$ inferred for SgrA$^*$ and the nuclear BHs in M31 and M87, and can explain the low luminosities in these sources, without invoking any feedback mechanism.

Stability of radiation-pressure-dominated magnetized discs under a delayed viscosity prescription

Stability of radiation-pressure-dominated magnetized discs under a delayed viscosity prescription

Modified viscosity in accretion disks

Black holes (BHs) surrounded by accretion disks are present in the Universe at different scales of masses, from microquasars up to the active galactic nuclei (AGNs). The current picture remains ad hoc due to the complexity of the magnetic field action. In addition, accretion disks at high Eddington rates can be radiation-pressure dominated and, according to some of the heating prescriptions, thermally unstable. The observational verification of their resulting variability patterns may shed light on both the role of radiation pressure and magnetic fields. We compute the structure and time evolution of an accretion disk, using the code GLADIS (which models the global accretion disk instability). We supplement this model with a modified viscosity prescription, which can to some extent describe the magnetisation of the disk. We study the results for a large grid of models, to cover the whole parameter space, and we derive conclusions separately for different scales of black hole masses, which are characteristic for various types of cosmic sources. We show the dependencies between the flare or outburst duration, its amplitude, and period, on the accretion rate and viscosity scaling. We show that if the heating rate in the accretion disk grows more rapidly with the total pressure and temperature, the instability results in longer and sharper flares. In general, we confirm that the disks around the supermassive black holes are more radiation-pressure dominated and present relatively brighter bursts. Our method can also be used as an independent tool for the black hole mass determination, which we confront now for the intermediate black hole in the source HLX-1.

Accretion and wind dynamics in tidal disruption events

AbstractWe have constructed self similar models of time dependent and non relativistic accretion disks in both sub and super-Eddington phase of TDEs with wind outflows for a general viscosity prescription which is a function of surface density of the disk Σd and radius r. The physical parameters are black hole (BH) mass M•, specific orbital energy E and angular momentum J, star mass M⋆ and radius R⋆. We have considered an accretion disk where matter is lost due to accretion by black hole and out flowing wind (in case of super-Eddington) and added through fallback of the disrupted debris. We have simulated the light curve profiles in various spectral bands and fit them to the observations to determine the above mentioned physical parameters.

On the orbital evolution of a pair of giant planets in mean motion resonance

Pairs of extrasolar giant planets in a mean motion commensurability are common with 2:1 resonance occurring most frequently. Disc-planet interaction provides a mechanism for their origin. However, the time scale on which this could operate in particular cases is unclear. We perform 2D and 3D numerical simulations of pairs of giant planets in a protoplanetary disc as they form and maintain a mean motion commensurability. We consider systems with current parameters similar to those of HD 155358, 24 Sextantis and HD 60532, and disc models of varying mass, decreasing mass corresponding to increasing age. For the lowest mass discs, systems with planets in the Jovian mass range migrate inwards maintaining a 2:1 commensurability. Systems with the inner planet currently at around 1 au from the central star could have originated at a few au and migrated inwards on a time scale comparable to protoplanetary disc lifetimes. Systems of larger mass planets such as HD 60532 attain 3:1 resonance as observed. For a given mass accretion rate, results are insensitive to the disc model for the range of viscosity prescriptions adopted, there being good agreement between 2D and 3D simulations. However, in a higher mass disc a pair of Jovian mass planets passes through 2:1 resonance before attaining a temporary phase lasting a few thousand orbits in an unstable 5:3 resonance prior to undergoing a scattering. Thus finding systems in this commensurability is unlikely.

A heuristic model for MRI turbulent stresses in Hall MHD

Although the Shakura-Sunyaev $\alpha$ viscosity prescription has been highly successful in characterizing myriad astrophysical environments, it has proven to be partly inadequate in modelling turbulent stresses driven by the MRI. Hence, we adopt the approach employed by \citet{GIO03}, but in the context of Hall magnetohydrodynamics (MHD), to study MRI turbulence. We utilize the exact evolution equations for the stresses, and the non-linear terms are closed through the invocation of dimensional analysis and physical considerations. We demonstrate that the inclusion of the Hall term leads to non-trivial results, including the modification of the Reynolds and Maxwell stresses, as well as the (asymptotic) non-equipartition between the kinetic and magnetic energies; the latter issue is also addressed via the analysis of non-linear waves. The asymptotic ratio of the kinetic and magnetic energies is shown to be \emph{independent} of the choice of initial conditions, but it is governed by the \emph{Hall parameter}. We contrast our model with the Kazantsev prescription from small-scale dynamo theory, and the Hall term does not contribute in the latter approach, illustrating the limitations of the Kazantsev formalism. We indicate potential astrophysical applications of our model, including the solar wind where a lack of equipartition has been observed.

Hydrodynamical wind on vertically self-gravitating ADAFs in the presence of toroidal magnetic field

We present the effect of a hydrodynamical wind on the structure and the surface temperature of a vertically self-gravitating magnetized ADAFs using self-similar solutions. Also a model for an axisymmetric, steady-state, vertically self-gravitating hot accretion flow threaded by a toroidal magnetic field has been formulated. The model is based on $\alpha-$prescription for turbulence viscosity. It is found that the thickness and radial velocity of the disc are reduced significantly as wind gets stronger. In particular, the solutions indicated that the wind and advection have the same effects on the structure of the disc. We also find that in the optically thin ADAF becomes hotter by including the wind parameter and the self-gravity parameter.

Effects of turbulent viscosity on a rotating gas ring around a black hole: Results of numerical simulations

AbstractIn this paper, we present the time evolution of a rotationally axisymmetric gas ring around a non rotating black hole using two dimensional grid‐based hydrodynamics simulations. We show the way in which angular momentum transport is included in simulations of non‐selfgravitating accretion of matter towards a black hole. We use the Shakura‐Sunyaev α viscosity prescription to estimate the turbulent viscosity for all major viscous stress tensors. We investigate how a gas ring which is initially assumed to rotate with Keplerian angular velocity is accreted onto a black hole and hence forms an accretion disc in the presence of turbulent viscosity. We show that a centrifugally pressure supported sub‐Keplerian flow with shocks forms when the ring starts to disperse with inclusion of relatively small amount of viscosity. But, if the viscosity is above the critical value, the shock disappears altogether and the whole disc becomes Keplerian which is subsonic everywhere except in a region close to the horizon, where it supersonically enters to the black hole. We discovered a multiple valued Mach number solution and the corresponding density distributions that connect matter (a) from the initial Keplerian gas ring to a sub‐Keplerian disc with shocks in presence of small amount of viscosity and (b) from the sub‐Keplerian flow to a Keplerian disc in presence of huge amount of viscosity. We calculate the temporal variations of the magnitude of various time scales which ensure us about the stability of the flow. (© 2015 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)