## 3. Viscosity in Thin Selfgravitating Accretion Disks## 3.1. Conditions for selfgravity in accretion disksIn the following we assume that the accretion disks are geometrically thin in the vertical direction, symmetric in the azimuthal direction, and stationary. We approximate the vertical structure by a one zone model. Then a disk model is specified by the central mass , the radial distributions of surface density , central plane temperature , and effective temperature or the radial mass flow rate . The relevant material functions are the equation of state, the opacity and the viscosity prescription. One can estimate the importance of selfgravity by comparing the respective contributions to the local gravitational accelerations in the vertical and radial directions. The vertical gravitational acceleration at the disk surface is and , for the selfgravitating and the purely Keplerian case, respectively. Selfgravitation is thus dominant in the vertical direction when where is the mass enclosed in the
disk within a radius Similar considerations lead to the condition for selfgravitation to dominate in the radial direction. Thus, we can define three regimes as follows: -
Non-selfgravitating (NSG) disks in which (i.e., the classical Shakura-Sunyaev disks) -
Keplerian selfgravitating (KSG) disks in which selfgravity is significant only in the vertical direction and which satisfy the constraint -
Fully selfgravitating (FSG) disks which satisfy
Because is a monotonically
increasing function of ## 3.2. Selfgravitating disksIn this section, we will review the structure of selfgravitating
(SG) accretion disks within the framework of the assumptions
introduced above. Compared to the standard NSG models, both the KSG
and FSG disks require modification of the equation of hydrostatic
support in the direction perpendicular to the disk. Thus, while in the
standard model the local vertical pressure gradient is balanced by the
For an SG disk, hydrostatic equilibrium in the vertical direction yields Paczynski (1978), where Since details of the thermodynamics in the Integrating the equation of conservation of angular momentum gives with the radial mass flow
rate Finally, we have for the sound velocity where is a vertically averaged mass density. On the other hand, the Jeans condition for fragmentation in the
disk into condensations of radius (see Mestel 1965) where Thus, a selfgravitating disk is on the verge of fragmenting into condensations of radius unless these are destroyed by shear motion associated with the Keplerian velocity field. Thus Paczynski (1978) and later Kozlowski et al. (1979) and Lin & Pringle (1987) proposed that the viscosity prescription was directly coupled to the above gravitational stability criterion. To solve for the dynamic and thermal structure of SG disks, a viscosity prescription has to be specified. As in the NSG case it is possible but not necessary that viscosity is limited by shock dissipation. In the absence of such dissipation, we would, as before, expect the -viscosity to apply. It is instructive, however, to follow the logic of Sect. 2.2 in the case of SG disks in which turbulence is limited by shock dissipation. ## 3.3. Viscosity in shock dissipation limited selfgravitating accretion disksFor a SG disk (whether Keplerian or not) Eq. 8 is no longer valid so the analysis of Eqs. 9 and 10 no longer applies. In physical terms, the scale height in the disk no longer reflects global properties of the disk (mass of and distance to the central star) but is set by local conditions. For the selfgravitating case we have approximately where conditions , and distinguish between the Keplerian and the FSG cases, respectively, with the KSG disks as an intermediate case. where is defined by . At the transition to the NSG regime, Eqs. 11 and 19 and the condition give as before and hence a smooth transition to the -ansatz. For the FSG regime, Eq. 19 and the condition give the simple asymptotic form and hence a viscosity of the form where is a factor of order unity. The situation is more complex for intermediate values of . The derived viscosity differs in all SG cases from the standard -ansatz, but approaches that form as . Thus when hydrodynamically induced turbulence is limited by shock dissipation, the resultant viscosity reflects local conditions and takes the standard Shakura-Sunyaev form only when the disk mass is negligible. We show below that this new prescription removes a problem previously noted by Paczynski (1978) and others, with the structure of KSG disks with -viscosity. ## 3.4. Structure of SG disks with shock limited viscosityIt follows from Eqs. 13, 14 and 15, with , that If one adopts the standard -prescription, this yields For a KSG disk, this in turn yields A similar result arises for the FSG case, albeit with a different numerical factor resulting from the solution of Poisson's equation. Thus, for a SG disk, the -ansatz
leads to the requirement of a constant temperature for all radii
The ansatz for a SG disk also requires that the disk structure satisfy In a standard NSG accretion disk the temperature is a free parameter which is determined by the energy released by the inward flow of the disk gas (), by the local viscosity, and by the respective relevant cooling mechanisms. The viscosity depends on via Eq. (1) and on the equation of hydrostatic support in the direction normal to the disk (Eq. (8), which in the non-SG case replaces Eq. (13)). In the SG case, it is the surface density
(and hence On the other hand, if one adopts the alternative prescription for
shock limited viscosity proposed in Sect. 3.3, the above problem
with constant or prescribed mid-plane temperature disappears, the
temperature once again depends on © European Southern Observatory (ESO) 2000 Online publication: June 5, 2000 |