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Astron. Astrophys. 357, 1123-1132 (2000)

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3. Viscosity in Thin Selfgravitating Accretion Disks

3.1. Conditions for selfgravity in accretion disks

In 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 [FORMULA], the radial distributions of surface density [FORMULA], central plane temperature [FORMULA], and effective temperature [FORMULA] or the radial mass flow rate [FORMULA]. 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 [FORMULA] and [FORMULA], for the selfgravitating and the purely Keplerian case, respectively. Selfgravitation is thus dominant in the vertical direction when

[EQUATION]

where [FORMULA] is the mass enclosed in the disk within a radius s and is given approximately by [FORMULA]. Typical numbers for [FORMULA] are in the range [FORMULA].

Similar considerations lead to the condition

[EQUATION]

for selfgravitation to dominate in the radial direction. Thus, we can define three regimes as follows:

  • Non-selfgravitating (NSG) disks in which [FORMULA] [FORMULA] (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 [FORMULA] [FORMULA]

  • Fully selfgravitating (FSG) disks which satisfy [FORMULA]

Because [FORMULA] is a monotonically increasing function of s, all three regimes will arise in sufficiently massive, thin ([FORMULA] disks.

3.2. Selfgravitating disks

In 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 z component of the gravitational force due to the central object, in the SG case we have balance between two local forces, namely the pressure force and the gravitational force due to the disk's local mass. In the KSG case, in the radial direction centrifugal forces are still balanced by gravity from a central mass (Keplerian approximation), while in the fully selfgravitating case we have to solve Poisson's equation for the rotation law in the disk.

For an SG disk, hydrostatic equilibrium in the vertical direction yields

[EQUATION]

Paczynski (1978), where P is the pressure in the central plane ([FORMULA]), [FORMULA] is the surface mass density integrated in the z direction, and G is the gravitational constant.

Since details of the thermodynamics in the z direction are of no particular relevance to our argument, we shall assume the disk to be isothermal in the vertical direction.

Integrating the equation of conservation of angular momentum gives

[EQUATION]

with the radial mass flow rate 2 [FORMULA], the rotational frequency [FORMULA], its radial derivative [FORMULA], and a quantity [FORMULA] allowing for the integration constant or, equivalently, for the inner boundary condition. For a detailed discussion of [FORMULA] see, e.g., Duschl & Tscharnuter (1991), Popham & Narayan (1995) and Donea & Biermann (1996). For simplicity, we set the boundary condition [FORMULA] in the subsequent discussion. This does not alter the essence of our argument, and only changes details close to the disk's inner radial boundary, since the product [FORMULA] increases with s. In fact Eq. 14 applies in the general case (i.e., NSG and SG); for Keplerian disks (NSG and KSG), we may write [FORMULA].

Finally, we have for the sound velocity

[EQUATION]

Eqs. 13 and 15 give

[EQUATION]

where [FORMULA] is a vertically averaged mass density.

On the other hand, the Jeans condition for fragmentation in the disk into condensations of radius R is

[EQUATION]

(see Mestel 1965) where q is factor of order unity.

Thus, a selfgravitating disk is on the verge of fragmenting into condensations of radius [FORMULA] 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 [FORMULA]-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 disks

For 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

[EQUATION]

where conditions [FORMULA], and [FORMULA] distinguish between the Keplerian and the FSG cases, respectively, with the KSG disks as an intermediate case.

From Eq. 7

[EQUATION]

where [FORMULA] is defined by [FORMULA].

At the transition to the NSG regime, Eqs. 11 and 19 and the condition [FORMULA] give as before

[EQUATION]

and hence a smooth transition to the [FORMULA]-ansatz.

For the FSG regime, Eq. 19 and the condition [FORMULA] give the simple asymptotic form

[EQUATION]

and hence a viscosity of the form

[EQUATION]

where [FORMULA] is a factor of order unity.

The situation is more complex for intermediate values of [FORMULA]. The derived viscosity differs in all SG cases from the standard [FORMULA]-ansatz, but approaches that form as [FORMULA]. 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 [FORMULA]-viscosity.

3.4. Structure of SG disks with shock limited viscosity

It follows from Eqs. 13, 14 and 15, with [FORMULA], that

[EQUATION]

If one adopts the standard [FORMULA]-prescription, this yields

[EQUATION]

For a KSG disk, this in turn yields

[EQUATION]

or

[EQUATION]

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 [FORMULA]-ansatz leads to the requirement of a constant temperature for all radii s (or, if [FORMULA] in Eq. 14, the temperature is prescribed as a function of s), independent of thermodynamics. While the exact____ constancy of the temperature may very well be an artefact of our simplified one-zone approximation for the vertical structure, there is no reason to expect that proper vertical integration of the structure will change this fundamentally.

The [FORMULA] ansatz for a SG disk also requires that the disk structure satisfy

[EQUATION]

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 ([FORMULA]), by the local viscosity, and by the respective relevant cooling mechanisms. The viscosity depends on [FORMULA] 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 [FORMULA] (and hence h) which must adjust in order to radiate the energy deposited by viscous dissipation and provided by the inward flowing material. While detailed solutions are beyond the scope of this paper, they clearly exist formally. On the other hand, the normal thermostat mechanism does not operate, at least in the steady state. Indeed in certain circumstances, the condition of constant mid-plane temperature appears to be inconsistent with the basic thermodynamic requirement that the average gas temperature in the disk exceed that of the black body temperature required to radiate away the energy dissipated by viscous stresses (see Appendix). It is therefore doubtful whether a physically plausible and stable quasi-steady state solution exists.

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 h, and the normal thermostat can operate. While this does not prove the validity of the shock limited viscosity prescription given in Sect. 3.3, it is certainly an interesting consequence.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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