The general relativistic thin disc evolution equation

In the classical theory of thin disc accretion discs, the constraints of mass and angular momentum conservation lead to a diffusion-like equation for the turbulent evolution of the surface density. Here, we revisit this problem, extending the Newtonian analysis to the regime of Kerr geometry relevant to black holes. A diffusion-like equation once again emerges, but now with a singularity at the radius at which the effective angular momentum gradient passes through zero. The equation may be analysed using a combination of WKB, local techniques, and matched asymptotic expansions. It is shown that imposing the boundary condition of a vanishing stress tensor (more precisely the radial-azimuthal component thereof) allows smooth stable modes to exist external to the angular momentum singularity, the innermost stable circular orbit, while smoothly vanishing inside this location. The extension of the disc diffusion equation to the domain of general relativity introduces a new tool for numerical and phenomenolgical studies of accretion discs, and may prove to be a useful technique for understanding black hole X-ray transients.Comment: 7 Pages, 1 figure. Accepted for publication in MNRAS. Revised version corrects minor typos in equations (64) and (66) of original, otherwise unaltere

On the high frequency spectrum of a classical accretion disc

We derive simple and explicit expressions for the high frequency spectrum of a classical accretion disc. Both stress-free and finite stress inner boundaries are considered. A classical accretion disc spectrum with a stress-free inner boundary departs from a Wien spectrum at large $\nu$, scaling as $\nu^{2.5}$ (as opposed to $\nu^3$) times the usual exponential cut-off. If there is finite stress at the inner disc boundary, the maximum disc temperature generally occurs at this edge, even at relatively modest values of the stress. In this case, the high frequency spectrum is proportional to $\nu^2$ times the exponential cut-off. If the temperature maximum is a local hot spot, instead of an axisymmetric ring, then an interior maximum produces a $\nu^2$ prefactor while an edge maximum yields $\nu^{1.5}$. Because of beaming effects, these latter findings should pertain to a classical relativistic disc. The asymptotics are in general robust and independent of the detailed temperature profile, provided only that the liberated free energy of differential rotation is dissipated locally, and may prove useful beyond the strict domain of classical disc theory. As observations continue to improve with time, our findings suggest the possibility of using the high energy spectral component of black hole candidates as a signature prediction of classical theory, as well as an diagnostic of the stress at the inner regions of an accretion disc.Comment: 10 pages, 2 figures. To appear in MNRAS Letter

Global model of differential rotation in the Sun

The isorotation contours of the solar convective zone (SCZ) show three distinct morphologies, corresponding to two boundary layers (inner and outer), and the bulk of the interior. Previous work has shown that the thermal wind equation together with informal arguments on the nature of convection in a rotating fluid could be used to deduce the shape of the isorotation surfaces in the bulk of the SCZ with great fidelity, and that the tachocline contours could also be described by relatively simple phenomenology. In this paper, we show that the form of these surfaces can be understood more broadly as a mathematical consequence of the thermal wind equation and a narrow convective shell. The analysis does not yield the angular velocity function directly, an additional surface boundary condition is required. But much can already be deduced without constructing the entire rotation profile. The mathematics may be combined with dynamical arguments put forth in previous works to the mutual benefit of each. An important element of our approach is to regard the constant angular velocity surfaces as an independent coordinate variable for what is termed the "residual entropy," a quantity that plays a key role in the equation of thermal wind balance. The difference between the dynamics of the bulk of the SCZ and the tachocline is due to a different functional form of the residual entropy in each region. We develop a unified theory for the rotational behavior of both the SCZ and the tachocline, using the solutions for the characteristics of the thermal wind equation. These characteristics are identical to the isorotation contours in the bulk of the SCZ, but the two deviate in the tachocline. The outer layer may be treated, at least descriptively, by similar mathematical techniques, but this region probably does not obey thermal wind balance.Comment: 26 pages, 7 figures, accepted to MNRA

Convective and Rotational Stability of a Dilute Plasma

The stability of a dilute plasma to local convective and rotational disturbances is examined. A subthermal magnetic field and finite thermal conductivity along the field lines are included in the analysis. Stability criteria similar in form to the classical H{\o}iland inequalities are found, but with angular velocity gradients replacing angular momentum gradients, and temperature gradients replacing entropy gradients. These criteria are indifferent to the properties of the magnetic field and to the magnitude of the thermal conductivity. Angular velocity gradients and temperature gradients are both free energy sources; it is not surprising that they are directly relevant to the stability of the gas. Magnetic fields and thermal conductivity provide the means by which these sources can be tapped. Previous studies have generally been based upon the classical H{\o}iland criteria, which are inappropriate for magnetized, dilute astrophysical plasmas. In sharp contrast to recent claims in the literature, the new stability criteria demonstrate that marginal flow stability is not a fundamental property of accreting plasmas thought to be associated with low luminosity X-ray sources.Comment: Final version (Appendix added), 19 pages, 3 figs., AAS LaTEX macros v4.0. To appear ApJ 1 Dec 200

Ambipolar Diffusion in the Magnetorotational Instability

The effects of ambipolar diffusion on the linear stability of weakly ionised accretion discs are examined. Earlier work on this topic has focused on axial magnetic fields and perturbation wavenumbers. We consider here more general field and wavenumber geometries, and find that qualitatively new results are obtained. Provided a radial wavenumber and azimuthal field are present along with their axial counterparts, ambipolar diffusion will always be destabilising, with unstable local modes appearing at well-defined wavenumber bands. The wavenumber corresponding to the maximum growth rate need not, in general, lie along the vertical axis. Growth rates become small relative to the local angular velocity when the ion-neutral collision time exceeds the orbital time. In common with Hall electromotive forces, ambipolar diffusion destabilises both positive and negative angular velocity gradients. In at least some cases, therefore, uniformly rotating molecular cloud cores may reflect the marginally stable state of the ambipolar magnetorotational instability.Comment: Submitted to MN, 6 pages, 3 figs, MN style file v2.

When is Uniform Rotation an Energy Minimum?

A simple variational calculation is presented showing that a uniformly rotating barotropic fluid in an external potential attains a true energy minimum if and only if the rotation profile is everywhere subsonic. If regions of supersonic rotation are present, fluid variations exist that could take the sytem to states of lower energy. In any given system, these states may or may not be dynamically accessible, but their existence is important. It means that extending the degrees of freedom available to the fluid (say by weak magnetic fields) may open a path to fluid instabilities. Whether astrophysical gaseous nebula tend toward states of uniform rotation or toward more Keplerian core-disk systems appears to be largely a matter of whether the rotation profile is transonic or not. The suggestion is made that the length scale associated with coherent molecular cloud cores is related to the requirement that the cores be stable and rotate subsonically.Comment: 8 pages, AAS Tex Macros, Submitted to ApJ (Letters

On the Magnetic Prandtl Number Behavior of Accretion Disks

We investigate the behavior of the magnetic Prandtl number (ratio of microscopic viscosity to resistivity) for accretion sources. Generally this number is very small in standard accretion disk models, but can become larger than unity within $\sim 50$ Schwarzschild radii of the central mass. Recent numerical investigations suggest a marked dependence of the level of MHD turbulence on the value of the Prandtl number. Hence, black hole and neutron star accretors, i.e. compact X-ray sources, are affected. The astrophysical consequences of this could be significant, including a possible route to understanding the mysterious state changes that have long characterized these sources.Comment: 15 pages, 6 figures. Accepted for publication in the Astrophysical Journal (February 10, 2008 issue.) Minor changes from original submissio