School of Physics and Astronomy, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Visiting Scientist, Department of Condensed-Matter Physics, Faculty of Physics, Weizmann Institute of Science, Rehovot 76100, Israel
Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel
In the gas pressure region, two distinct solutions are obtained. In one, the convective flux is much larger than the radiative flux and the temperature profile is close to adiabatic. The blackbody region extends over the entire gas pressure region and could also extend down to very small radial distances, so that there would be no radiation pressure region and no gas pressure electron scattering region. In the other solution, the convective flux is about a third of the total flux, the dimensionless superadiabatic temperature gradient is and all three regions of the disk exist.
In the radiation pressure region, the temperature profile is very close to adiabatic, and the disk is geometrically thin and optically thick even for super Eddington accretion rates. The fraction of the convective flux, out of the total flux, increases with the accretion rate from to close to 1, for accretion rates comparable to the Eddington limit. As a result of this variation, the radiation pressure region is secularly stable; thus all the disk solutions are secularly stable. The values of the effective -parameter are rather small: , and for the radiation pressure region and for the two solutions in the gas pressure region, respectively.
There is quite a large body of observational data suggesting the existence of accretion disks around protostars, compact stellar objects and active galactic nuclei. Naturally, accretion disk models are extensively employed to interpret these observations. However, to a large extent the models used are more a descriptive than predictive tool (Pringle 1981).
The reason for this state of affairs is the lack of a physical model for the disk turbulent viscosity, , usually parameterized in the form (Shakura & Sunyaev 1973). Here is the speed of sound, h is the disk half-width, and is a dimensionless parameter which neither value nor dependence on the physical conditions in the disk are known. Thus, often the observations are used to fit the -parameter for a given system. Usually a constant is assumed for gas pressure dominated regions of the disk. For radiation pressure dominated regions, both the former prescription and proportional to the ratio between the gas and total pressure have been suggested. Without a physical model for the turbulent viscosity, it is not clear which prescription, if any, is the correct one.
A self-consistent solution to the disk equations requires that the turbulent viscosity be determined by the physical state of the disk. In turn, the resulting turbulent viscosity will control the disk physical conditions. The determination of the turbulent viscosity requires a model for turbulence and the specification of the particular instability that generates the turbulence.
In spite the very large Reynolds numbers, Keplerian disks are stable against shear-generated turbulence, at least in the linear analysis. On the other hand, it has been pointed out that they are unstable against turbulent convection, both in radiation pressure and gas pressure regions and for various opacities (see e.g., Bisnovatyi-Kogan & Billinikov 1977, Shakura, Sunyaev & Zilitinkevich 1978, Lin & Paploizou 1980). Therefore, in this paper we implement the above proposed self-consistent scheme for the case of turbulent convection in a thin accretion disk. The turbulence model employed (Canuto, Goldman, & Hubickyj 1984, Canuto, Goldman & Chasnov 1987) provides vertically averaged values for the turbulent viscosity and for the convective flux. Thus, no attempt to resolve the vertical structure of the disk is made, and all quantities are either vertical averages or midplane values. A detailed vertical structure requires a considerably more complex turbulence model, that could yield the z-dependent values of the turbulent viscosity and of the convective flux. In this work, we focus on the more general features of the self-consistent approach and avoid the complications of a z-dependent model. We note that turbulence bulk properties such as the turbulent viscosity and the convective flux are contributed mainly by the largest eddies. The vertical extent of the latter is and thus vertical averaging is expected to represent fairly well their contribution.
We consider the following three disk-regions: radiation pressure dominated region, gas pressure dominated region with electron scattering opacity, and gas pressure dominated region with free-free absorption opacity. Since turbulent convection is taken to be the source of the disk turbulent viscosity, the convective flux plays an important role in the transfer of energy to the disk surface.
A detailed description of the model and its underlying equations can be found in Goldman & Wandel (1995). In what follows we summarize and discuss the main results.
The turbulent viscosity and the convective flux were obtained as functions of the physical parameters of the disk, which in turn are controlled by the former two. Having a model for turbulence, there is no need to resort to phenomenological parameterizations of either the viscosity (as in -disk models) or the convective flux (as in the mixing length approach).
Solutions for both radiation pressure dominated (inner) and for gas pressure dominated (outer) regions, with either electron scattering or free-free absorption opacities, were obtained. The solutions give the midplane temperature, density, the disk half-width and the ratio between the convective and total fluxes at a given distance, for given central mass and given accretion rate. The solutions depend on ---the fraction of the convective flux out of the total flux. We obtain as a function of the local physical conditions in the different regions, from an approximate estimate for the ratio of the surface to midplane temperatures. In Figure 1 we display the luminosity (in units of the Eddington luminosity) versus the electron scattering optical depth, for the various solutions.
Figure: Luminosity, in units of the Eddington luminosity, vs. the electron-scattering optical depth, for the various convective disk solutions, at a constant distance for . R: radiation pressure region, strong convection. BBS: gas pressure blackbody solution in the strong convection limit. BBM: same region for the case of moderate convection. ESM: gas pressure electron-scattering solution in the moderate convection limit.
Figure: Radiation pressure dominated region: the fraction of the convective flux as function of the luminosity, in units of the Eddington luminosity. The plot is for , and a radiation pressure comprising of the total pressure.
In the gas pressure dominated regions we find two distinct types of solutions. The first one is that of strong convection in which the convective flux is much larger than the radiative flux. The blackbody region encompasses the entire gas pressure region and could extend down to the inner disk boundary, with no radiation pressure region. The second solution is that of moderate convection, with a convective flux comprising of total flux and all disk-regions exist.
The radiation pressure solution corresponds to strong convection. In the present model the radiation pressure region can exist only for accretion rates higher than times the Eddington rate. In Figure 2 we show as function the luminosity (in units of Eddington luminosity) for this region. The values of range from up to very close to 1, for luminosity varying between to few times the Eddington luminosity, respectively. This dependence of on the accretion rate gives rise to a surface density which increases with the accretion rate. As a result, the radiation pressure region is secularly stable (as are also the gas pressure dominated solutions); see Figure 1. It is interesting to note that the trend of convection to stabilize the radiation pressure dominated region, is evident even in models in which the disk viscosity is described by an parameter and the convective flux by the mixing length approach (Milsom, Chen, & Taam 1994).
The radiation pressure region is optically thick (see Figure 1) and geometrically thin even for super Eddington luminosities. Therefore, for luminosities of the order of the Eddington luminosity, the local spectrum will be a modified blackbody.
The values of the effective -parameter in the different regions are quite small: , , and for radiation pressure region and for the two types of solutions in the gas pressure region, respectively. These small values result from the subsonic turbulent velocities and from the anisotropy of the largest eddies. This anisotropy, due to the Coriolis force, is a general feature of a three-dimensional turbulence in a rotating disk (Goldman 1991). The above values are small compared to values required to model outbursts of cataclysmic variables (see, e.g., Duschl 1989). This implies that turbulent convection cannot account for the disk viscosity in this case. A different mechanism, possibly magnetic viscosity, is required. One may speculate that turbulent convection can be the source of the disk viscosity during the quiescent state and some other mechanism provides the disk viscosity during the outburst.
For the same central mass and accretion rate, the innermost regions could be either radiation pressure dominated or gas pressure dominated blackbody (strong convection case). The values of the corresponding effective -parameter differ only by a factor of , but in the former case the disk is much thicker geometrically and much less dense than in the second case. The emitted flux is a modified blackbody while in the second case it is a blackbody.
Bisnovatyi-Kogan, G. S. & Billinikov, S. I. 1977, A&A, 99, 11
Canuto, V. M., Goldman, I., & Hubickyj, O. 1984, ApJ, 280, L5
Canuto, V. M., Goldman, I., & Chasnov, J. 1987, Phys. Fluids, 30, 3391
Duschl, W. J. 1989, A&A, 225, 105
Goldman, I. 1991, IAU Colloq. 129, Structure and Emission Properties of Accretion Disks, ed. C. Bertout, S. Collin-Souffrin, J. P. Lasota, & J.Tran Than Van ( Gif-sur Yvette: Editions frontieres), 433
Goldman, I. & Wandel, A. 1995, ApJ, 443, 187
Lin, D. N. C. & Papaloizou, J. 1980, MNRAS, 191, 37
Milsom. J. A., Chen, X., & Taam, R. E. 1994, ApJ, 421, 668
Pringle, J. E. 1981, ARA&A, 13, 137
Shakura, N. I. & Sunyaev, R. A. 1973, A&A, 24, 337
Shakura, N. I., Sunyaev, R. A., & Zilitinkevich, S. S. 1978, A&A, 62, 179