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Astron. Astrophys. 344, 639-646 (1999)

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1. Introduction

Accretion discs play an important role in astrophysics. Whenever material with some angular momentum falls towards a central object a disc is generally formed.

The basic philosophy behind the disc accretion model is the following: The matter first forms a disk shaped annular structure around the accreting object at the circularization radii appropriate to the specific angular momentum of the incoming matter. The thickness of the disc H is very small relative to the radius [FORMULA]. The material in the disc rotates at near Keplerian velocities and hence there is shear between two adjacent layers. Viscosity, the exact nature of which is still under discussion, operates in discs, transports angular momentum outwards, and leads to a [FORMULA] flux of matter in the radial direction. The viscosity generates heat which is assumed to be radiated in the perpendicular direction to the surface of the disc (the z direction). So while matter flows slowly inwards in the radial direction, energy dissipated by the inflowing matter is radiated in the perpendicular direction. Clearly, the structure of the disc depends crucially on how efficiently the energy dissipated by viscous heating can escape from the disc. If the vertical structure is such that it cannot cope with the dissipated energy, then it will heat up. If heating does lead to a configuration where the energy loss (cooling) is equal the energy gain (heating), a steady state disc can form. However, if no such equality exists, the matter will expand (or contract), and lead to the formation of a wind (or collapse). In the classical disc problem, the rate at which the disc must carry mass, namely [FORMULA], and the specific angular momentum are fixed by the outside. The disc must then adjust to the required structure which is consistent with these constraints, if such a structure can be found.

The disc is basically a 2D structure and all thermodynamic quantities vary with the cylindrical radial coordinate [FORMULA] and z. Obviously, modeling in 2D is much more complicated than in 1D, and hence a frequently encountered approach to modeling the disc behavior and structure is to integrate the disc equations in the vertical z direction and ignore completely the disc energy loss mechanism along with the complicated radiative transfer in the vertical direction. One hopes that `the disc will converge to the vertical structure needed to carry out the dissipated energy and radiate it out'. The purpose of this paper is to examine if such a state exists, namely: if for any accretion rate and opacity law, is there a corresponding vertical structure?

Viscous energy dissipation removes angular momentum and generates heat. Under certain conditions it is possible that the heat generated in the process cannot be removed by the radiation and as a consequence the gas does not cool and cannot be accreted. The gas then moves away from the z=0 plane and no disc structure forms. Such a possibility can be seen from the following example. Consider a constant temperature disc, namely [FORMULA] but not [FORMULA]. Next assume the standard assumption that the heating is proportional to the gas pressure P while the cooling is via free-free emission which is proportional to [FORMULA]. Then clearly, for sufficiently low pressures always heating will win and the gas will heat until one of the above assumptions breaks down!

The structure of discs in the vertical direction must allow for the viscous energy released in each volume element. If the disc is in thermal equilibrium, the energy balance dictates an [FORMULA] constraint which the hydrostatic equilibrium equation must satisfy. It is this condition which affects so dramatically the structure of the disc in the vertical direction. Shaviv & Wehrse (1986,SW) integrated the vertical structure equations including the radiation field and encountered numerical problems in convergence. They solved the problem by assuming an artificial cutoff in the viscous energy dissipation and in this way were able to obtain disc models and continuum spectra of classical nova disc with high accretion rate. Later Adam et al. (1987) showed how the problem encountered by SW is actually a thermal instability which leads to the formation of corona above the disc.

Czerny & King 1989a,b expanded the idea of SW (1986) and considered optically thin layer above an optically thick disc. They confirmed the results of SW and showed that viscous dissipation in the outer atmospheres of unilluminated accretion disc produces hot corona and thermally driven winds. Czerny & King 1989a also obtained a critical accretion rate above which a full hydrostatic equilibrium does not exist. The limiting accretion rate corresponds to [FORMULA] where [FORMULA] is the critical accretion rate which leads to the Eddington luminosity and [FORMULA] is the viscosity constant. Here we treat the optically thick regime and show that the flow starts already in the optically thick region.

Dumont et al. (1991) (here after DCKL) discussed the structure of optically thin accretion disc. In particular, they investigated in detail the effects NLTE has on the vertically integrated structure equations. DCKL obtained the energy balance of the disc from equating the cooling to the required energy dissipation for a given accretion rate. The basic equations used by DCKL are the vertical height averaged equations, namely they assumed a height H given by:


where [FORMULA] is the speed of sound and a surface density which is given by


where [FORMULA] is the total accretion rate and the kinematic viscosity [FORMULA] is given by


Thus, the first difference between the present treatment and that of DCKL is that DCKL considered averaged equations while here we use the local equation. The second difference is in the energy balance. Given the cooling expressed in terms of the vertical averaged equations, DCKL equate the averaged cooling to the required energy loss for a given accretion rate and thus obtain that:


where [FORMULA] is the vertically averaged number density of hydrogen nuclei and [FORMULA] the vertically averaged cooling rate. The advantage of this approach is that in this way the details of the dissipation mechanism and in particular the assumption expressed in Eq. 3 disappears from the equations. In short, DCKL investigated the radial structure while we here explore the vertical structure. The present work is a further investigation of the instability in the thermal balance of the vertical structure of the disc for low [FORMULA] systems. We show that it determines the fate of the entire disc structure.

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999