Variability associated with α accretion disc theory for standard and advection-dominated discs

The α turbulent viscosity formalism for accretion discs must be interpreted as a mean field theory, modelling a steady state only on spatial or time-scales greater than those of the turbulence. The extent of the scale separation determines the relative precision error (RPE) of the predicted luminosity L _ν. Turbulence and the use of α implies that (1) field line stretching gives a magnetic pressure  α²/6 of the total pressure generally, and a one-to-one relation between α and the pressure ratio for thin discs, and (2) large turbulent scales in advection-dominated accretion flows (ADAFs) predict a lower L _ν precision than thin discs for a given observation duration and central mass. The allowed variability (or RPE) at frequency ν increases with the size of the contributing region. For X-ray binary ADAFs, the RPE ∼ 5 per cent at R  ≤ 1000 Schwarzchild radii ( R _s) for averages over  1000 s. However, current data for galaxies like NGC 4258 and M87 give RPEs in L _ν of 50–100 per cent even at R  ≤ 100  R _S. More data are required, but systematic deviations from ADAF predictions are more significant than random deviations, and may constrain properties of the turbulence, the accretion mode, the assumption of a steady state or the accretion rate.     

Outbursts of irradiated accretion discs

I solve analytically the viscous evolution of an irradiated accretion disc, as seen during outbursts of soft X-ray transients. The solutions predict steep power-law X-ray decays L _X ∼ (1 + t/t_visc)⁻⁴, changing to L _X ∼ (1 − t/t'_visc)⁴ at late times, where t _visc, t '_visc are viscous time-scales. These forms closely resemble the approximate exponential and linear decays inferred by King and Ritter in these two regimes. The decays are much steeper than for unirradiated discs because the viscosity is a function of the central accretion rate rather than of local conditions in the disc.     

Observational implications of precessing protostellar discs and jets

We consider the dynamics of a protostellar disc in a binary system where the disc is misaligned with the orbital plane of the binary, with the aim of determining the observational consequences for such systems. The disc wobbles with a period approximately equal to half the orbital period of the binary and precesses on a longer time-scale. We determine the characteristic time-scale for realignment of the disc with the orbital plane as a result of dissipation. If the dissipation is determined by a simple isotropic viscosity then we find, in line with previous studies, that the alignment time-scale is of the order of the viscous evolution time-scale. However, for typical protostellar disc parameters, if the disc tilt exceeds the opening angle of the disc, then tidally induced shearing within the disc is transonic. In general, hydrodynamic instabilities associated with the internally driven shear result in extra dissipation that is expected to drastically reduce the alignment time-scale. For large disc tilts the alignment time-scale is then comparable with the precession time-scale, while for smaller tilt angles δ , the alignment time-scale varies as (sin δ )⁻¹. We discuss the consequences of the wobbling, precession and rapid realignment for observations of protostellar jets and the implications for binary star formation mechanisms.     

Radiation mechanisms and geometry of Cygnus X-1 in the soft state

We present X-ray/ γ -ray spectra of Cyg X-1 observed during the transition from the hard to the soft state and in the soft state by ASCA , RXTE and CGRO /OSSE in 1996 May and June. The spectra consist of a dominant soft component below ∼2 keV and a power-law-like continuum extending to at least ∼800 keV. We interpret them as emission from an optically thick, cold accretion disc and from an optically thin, non-thermal corona above the disc. A fraction f ≳0.5 of total available power is dissipated in the corona.
We model the soft component by multicolour blackbody disc emission taking into account the torque-free inner-boundary condition. If the disc extends down to the minimum stable orbit, the ASCA RXTE data yield the most probable black hole mass of M _X≈10 M_⊙ and an accretion rate,     , locating Cyg X-1 in the soft state in the upper part of the stable, gas-pressure-dominated, accretion-disc solution branch.
The spectrum of the corona is well modelled by repeated Compton scattering of seed photons from the disc off electrons with a hybrid, thermal/non-thermal distribution. The electron distribution can be characterized by a Maxwellian with an equilibrium temperature of kT _e∼30–50 keV, a Thomson optical depth of τ ∼0.3 and a quasi-power-law tail. The compactness of the corona is 2≲ℓ_h≲7, and a presence of a significant population of electron–positron pairs is ruled out.
We find strong signatures of Compton reflection from a cold and ionized medium, presumably an accretion disc, with an apparent reflector solid angle, Ω/2π∼0.5–0.7. The reflected continuum is accompanied by a broad iron K α line.     

Magnetically moderated outbursts of WZ Sagittae

We argue that the quiescent value of the viscosity parameter of the accretion disc in WZ Sge may be  α_cold∼ 0.01  , in agreement with estimates of α_cold for other dwarf novae. Assuming the white dwarf in WZ Sge to be magnetic, we show that, in quiescence, material close to the white dwarf can be propelled to larger radii, depleting the inner accretion disc. The propeller therefore has the effect of stabilizing the inner disc and allowing the outer disc to accumulate mass. The outbursts of WZ Sge are then regulated by the (magnetically determined) evolution of the surface density of the outer disc at a radius close to the tidal limit. Numerical models confirm that the recurrence time can be significantly extended in this way. The outbursts are expected to be superoutbursts since the outer disc radius is forced to exceed the tidal (3:1 resonance) radius. The large, quiescent disc is expected to be massive, and to be able to supply the observed mass accretion rate during outburst. We predict that the long-term spin evolution of the white dwarf spin will involve a long cycle of spin-up and spin-down phases.     

The response of the Fe Kα line to changes in the X-ray illumination of accretion discs

X-ray reflection spectra from photoionized accretion discs in active galaxies are presented for a wide range of illumination conditions. The energy, equivalent width (EW) and flux of the Fe K α line are shown to depend strongly on the ratio of illuminating flux to disc flux,   F _x/ F _disc  , the photon index of the irradiating power law, Γ, and the incidence angle of the radiation, i . When   F _x/ F _disc≤2  a neutral Fe K α line is prominent for all but the largest values of Γ. At higher illuminating fluxes an He-like Fe K α line at 6.7 keV dominates the line complex. With a high-energy cut-off of 100 keV, the thermal ionization instability seems to suppress the ionized Fe K α line when  Γ≤1.6  . The Fe K α line flux correlates with   F _x/ F _disc  , but the dependence weakens as iron becomes fully ionized. The EW is roughly constant when   F _x/ F _disc  is low and a neutral line dominates, but then declines as the line progresses through higher ionization stages. There is a strong positive correlation between the Fe K α EW and Γ when the line energy is at 6.7 keV, and a slight negative one when it is at 6.4 keV. This is a potential observational diagnostic of the ionization state of the disc. Observations of the broad Fe K α line, which take into account any narrow component, would be able to test these predictions. Ionized Fe K α lines at 6.7 keV are predicted to be common in a simple magnetic flare geometry. A model that includes multiple ionization gradients on the disc is postulated to reconcile the results with observations.     

Hydrodynamic simulations of propagating warps and bending waves in accretion discs

We present the results of a study of propagating warp or bending waves in accretion discs. Three-dimensional hydrodynamic simulations were performed using smoothed particle hydrodynamics (SPH), and the results are compared with calculations based on the linear theory of warped discs.
We examine the response of a gaseous disc to an initially imposed warping disturbance under a variety of physical conditions. We consider primarily the physical regime in which the dimensionless viscosity parameter α < H r , where H r is the disc aspect ratio, so that bending waves are expected to propagate. We also performed calculations for disc models in which α > H r , where the warps are expected to evolve diffusively. Small-amplitude (linear) perturbations are studied in both Keplerian and slightly non-Keplerian discs, and we find that the results of the SPH calculations can be reasonably well fitted by those of the linear theory. The main results of these calculations are: (i) the warp in Keplerian discs when α < H r propagates with little dispersion, and damps at a rate expected from estimates of the code viscosity; (ii) warps evolve diffusively when α > H r ; (iii) the slightly non-Keplerian discs lead to a substantially more dispersive behaviour of the warps, which damp at a similar rate to the Keplerian case, when α < H r .
Initially imposed higher amplitude, non-linear warping disturbances were studied in Keplerian discs. The results indicate that non-linear warps can lead to the formation of shocks, and that the evolution of the warp becomes less wave-like and more diffusive in character.
This work is relevant to the study of the warped accretion discs that may occur around Kerr black holes or in misaligned binary systems, and is mainly concerned with discs in which α < H r . The results indicate that SPH can model the hydrodynamics of warped discs, even when using rather modest numbers of particles.     

Radiatively heated, protoplanetary discs with dead zones – I. Dust settling and thermal structure of discs around M stars

The irradiation of protoplanetary discs by central stars is the main heating mechanism for discs, resulting in their flared geometric structure. In a series of papers, we investigate the deep links between two-dimensional self-consistent disc structure and planetary migration in irradiated discs, focusing particularly on those around M stars. In this first paper, we analyse the thermal structure of discs that are irradiated by an M star by solving the radiative transfer equation by means of a Monte Carlo code. Our simulations of irradiated hydrostatic discs are realistic and self-consistent in that they include dust settling with multiple grain sizes  ( N = 15)  , the gravitational force of an embedded planet on the disc and the presence of a dead zone (a region with very low levels of turbulence) within it. We show that dust settling drives the temperature of the mid-plane from an   r ^−3/5  distribution (well mixed dust models) towards an   r ^−3/4  . The dead zone, meanwhile, leaves a dusty wall at its outer edge because dust settling in this region is enhanced compared to the active turbulent disc at larger disc radii. The disc heating produced by this irradiated wall provides a positive gradient region of the temperature in the dead zone in front of the wall. This is crucially important for slowing planetary migration because Lindblad torques are inversely proportional to the disc temperature. Furthermore, we show that low turbulence of the dead zone is self-consistently induced by dust settling, resulting in the Kelvin–Helmholtz instability (KHI). We show that the strength of turbulence arising from the KHI in the dead zone is  α= 10⁻⁵  .     

The stability of accretion discs with inflow driven purely by magnetic winds

A model is presented for an accretion disc in which the inflow is driven purely by the angular momentum removed in a centrifugally accelerated magnetic wind. Turbulent discs around compact stars are considered, with the required magnetic field being generated in the disc by a simple dynamo. The turbulent magnetic Prandtl number, N _p, measures the ratio of turbulent viscosity to turbulent magnetic diffusivity. Formally, the hypothetical limit   N _p→ 0  corresponds to the magnetic wind torque dominating the viscous torque, but in practice the inflow is magnetically controlled for   N _p≲ 0.1  .
The suggestion by previous authors that purely magnetic wind-driven discs may be unstable is investigated. A detailed steady solution is found which allows perturbations to the thermal balance and vertical equilibrium to be calculated, and hence the effect of perturbations to the magnetic diffusivity, η, to be assessed. For a standard parametrized form of η, the wind-driven angular momentum balance is found to be linearly unstable. An increase in the inflow rate leads to increased bending of the poloidal magnetic field and an enhanced wind mass loss rate. This increases the angular momentum loss rate which drives further inflow. There is a resultant increase in η, due to the temperature perturbation, but this does not relieve field bending sufficiently to prevent the instability.     

Alignment and precession of a black hole with a warped accretion disc

We consider the shape of an accretion disc whose outer regions are misaligned with the spin axis of a central black hole and calculate the steady state form of the warped disc in the case where the viscosity and surface densities are power laws in the distance from the central black hole. We discuss the shape of the resulting disc in both the frame of the black hole and that of the outer disc. We note that some parts of the disc and also any companion star maybe shadowed from the central regions by the warp. We compute the torque on the black hole caused by the Lense–Thirring precession, and hence compute the alignment and precession time-scales. We generalize the case with viscosity and hence surface density independent of radius to more realistic density distributions for which the surface density is a decreasing function of radius. We find that the alignment time-scale does not change greatly but the precession time-scale is more sensitive. We also determine the effect on this time-scale if we truncate the disc. For a given truncation radius, the time-scales are less affected for more sharply falling density distributions.     

Implications of mean field accretion disc theory for vorticity and magnetic field growth

In addition to the scalar Shakura–Sunyaev α _ss turbulent viscosity transport term used in simple analytic accretion disc modelling, a pseudo-scalar transport term also arises. The essence of this term can be captured even in simple models for which vertical averaging is interpreted as integration over a half-thickness and each hemisphere is separately studied. The additional term highlights a complementarity between mean field magnetic dynamo theory and accretion disc theory treated as a mean field theory. Such pseudo-scalar terms have been studied, and can lead to large-scale magnetic field and vorticity growth. Here it is shown that vorticity can grow even in the simplest azimuthal and half-height integrated disc model, for which mean quantities depend only on radius. The simplest vorticity growth solutions seem to have scales and vortex survival times consistent with those required for facilitating planet formation. In addition, it is shown that, when the magnetic back-reaction is included to lowest order, the pseudo-scalar driving the magnetic field growth and that driving the vorticity growth will behave differently with respect to shearing and non-shearing flows: the former pseudo‐scalar can more easily reverse sign in the two cases.     

Super-Eddington accretion discs around Kerr black holes

We calculate the structure of the accretion disc around a rapidly rotating black hole with a super-Eddington accretion rate. The luminosity and height of the disc are reduced by the advection effect. In the case of large viscosity parameter, α>0.03, the accretion flow deviates strongly from thermodynamic equilibrium and overheats in the central region. With increasing accretion rate, the flow temperature steeply increases, reaches maximum, and then falls off. The maximum is achieved in the advection-dominated regime of accretion. The maximum temperature in the disc around a massive black hole of M =10⁸ M⊙ with α=0.3 is of order 3×10⁸ K. The discs with large accretion rates can emit X-rays in quasars as well as in galactic black hole candidates.     

Viscosity alpha in rotating spherical shear flows with an external magnetic field

We study the turbulent behaviour induced by the magnetic shear instability for a magnetized, incompressible fluid in a spherical shell. A differential rotation that is decreasing outwards but hydrodynamically stable according to the Rayleigh criterion is prescribed, and an external, uniform magnetic field is imposed parallel to the rotation axis. Our main concern in this paper is the fully global treatment of this magnetohydrodynamical system, so we focus particular attention on the influence of the boundary conditions. Non-linear, steady solutions are presented for stress-free as well as for rigid boundary conditions for one specific model with a fixed strength of the external magnetic field and a fixed differential rotation rate. We calculate the eddy viscosity ν_T and the viscosity alpha α_SS resulting from the total stress tensor. These turbulence parameters turn out to differ drastically depending on the boundary conditions for the flow. An investigation of the radial structure of the viscosity alpha (whilst varying the differential rotation law) shows that the enhanced generation of turbulence takes place mainly in the boundary layers of the shell.     

Negative Reynolds stress generation by accretion disc convection

The phenomenon of negative viscosity-alpha in convectively unstable Keplerian accretion discs is discussed. The convection is considered as a random flow with an axisymmetric mesoscale pattern. Its correlation tensor is computed with a time-averaging procedure using Kley's 2D hydrocode. There is a distinct anisotropy between the turbulence intensities in the radial and azimuthal directions, i.e. the radial velocity rms dominates the azimuthal one. As a consequence, an extra term in the expression for the turbulent transport of angular momentum appears which does not vanish for rigid rotation ('Λ-effect'). It is negative ('inwards transport') and even seems to dominate the positive contribution of the eddy viscosity representing outwards transport of angular momentum. For a turbulence model close to that of the mixing-length theory, the rotational influence on the anisotropy of the turbulence intensities,     , and the covariance  〈 u ' _R u ' _φ 〉  – representing the angular momentum transport – is computed and compared with the accretion disc simulations. Indeed, the negative angular momentum transport can be explained with the observed dominance of the radial turbulence intensity. If, on the other hand, in turbulence fields the azimuthal intensity would dominate or the turbulence is even isotropic, then we always find a positive transport of the angular momentum.     

Turbulence and its parameterization in accretion discs

Accretion disc turbulence is investigated in the framework of the shearing box approximation. The turbulence is either driven by the magneto‐rotational instability or, in the non‐magnetic case, by an explicit and artificial forcing term in the momentum equation. Unlike the magnetic case, where most of the dissipation occurs in the disc corona, in the forced hydrodynamic case most of the dissipation occurs near the midplane. In the hydrodynamic case evidence is presented for the stochastic excitation of epicycles. When the vertical and radial epicyclic frequencies are different (modeling the properties around rotating black holes), the beat frequency between these two frequencies appear to show up as a peak in the temporal power spectrum in some cases. Finally, the full turbulent resistivity tensor is determined and it is found that, if the turbulence is driven by a forcing term, the signs of its off‐diagonal components are such that this effect would not be capable of dynamo action by the shear–current effect. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Damped harmonic oscillator interpretation of the soft-state power spectra of Cyg X-1

We develop a model of an accretion disc in which the variability induced at a given radius is governed by a damped harmonic oscillator at the corresponding epicyclic frequency. That variability induces both linear and non-linear responses in the locally emitted radiation. The total observed variability of a source is the sum of these contributions over the disc radius weighted by the energy dissipation rate at each radius. It is shown that this simple model, which effectively has only three parameters including the normalization, can explain the range of the power spectra observed from Cyg X-1 in the soft state. Although a degeneracy between the black hole mass and the strength of the damping does not allow a unique determination of the mass, we can still constrain it to  ≲(16–20) M_⊙  . We also show that our model preserves the observed linear rms–flux relationship even in the presence of the non-linear flux response.     

Simulations of superhumps and superoutbursts

We numerically study the tidal instability of accretion discs in close binary systems using a two-dimensional SPH code. We find that the precession rate of tidally unstable, eccentric discs does not only depend upon the binary mass ratio q . Although the (prograde) disc precession rate increases with the strength of the tidal potential, we find that increasing the shear viscosity ν also has a significant prograde effect. Increasing the disc temperature has a retrograde impact upon the precession rate.   We find that motion relative to the binary potential results in superhump-like, periodic luminosity variations in the outer reaches of an eccentric disc. The nature and location of the luminosity modulation are functions of ν. Light curves most similar to observations are obtained for ν values appropriate for a dwarf nova in outburst.   We investigate the thermal–tidal instability model for superoutburst. A dwarf nova outburst is simulated by instantaneously increasing ν, which causes a rapid radial expansion of the disc. Should the disc encounter the 3: 1 eccentric inner Lindblad resonance and become tidally unstable, then tidal torques become much more efficient at removing angular momentum from the disc. The disc then shrinks and M _d increases. The resulting increase in disc luminosity is found to be consistent with the excess luminosity of a superoutburst.     

The helicity constraint in turbulent dynamos with shear

The evolution of magnetic fields is studied using simulations of forced helical turbulence with strong imposed shear. After some initial exponential growth, the magnetic field develops a large-scale travelling wave pattern. The resulting field structure possesses magnetic helicity, which is conserved in a periodic box by the ideal magnetohydrodynamics equations and can hence only change on a resistive time-scale. This strongly constrains the growth time of the large-scale magnetic field, but less strongly constrains the length of the cycle period. Comparing this with the case without shear, the time-scale for large-scale field amplification is shortened by a factor Q , which depends on the relative importance of shear and helical turbulence, and which also controls the ratio of toroidal to poloidal field. The results of the simulations can be reproduced qualitatively and quantitatively with a mean-field α Ω-dynamo model with alpha-effect and turbulent magnetic diffusivity coefficients that are less strongly quenched than in the corresponding α ²-dynamo.