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

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4. Selfgravitating [FORMULA]-disks

4.1. General observations

A general analysis of SG disks is complex and beyond the scope of this paper. In this section, we examine the structure of [FORMULA]-disks, in which the turbulence is subsonic at all radii. Before doing so, we make the following general observations.

First, with the [FORMULA]-viscosity prescription, Eqs. 14 and 23 give

[EQUATION]

and

[EQUATION]

Thus the SG [FORMULA]-disks recover the thermostat property of the standard disk, namely that the temperature and scale height can adjust to accommodate (radiate away) the energy input to the system from viscous dissipation and inward motion.

Second, we note that if the disk matter distribution is clumpy (e.g., clouds within a low density smoothed out distribution) then there may be a formal connection between the [FORMULA]- and [FORMULA]-prescriptions. Since in the [FORMULA]-formulation the clump velocities are of order [FORMULA] shock heating will tend to heat the low density inter-clump gas until its sound speed [FORMULA]. The inter-clump gas then has a scale height [FORMULA], the scale of the clumpy disk, and will hence be roughly a spherical structure. At this point the [FORMULA]- and [FORMULA]-prescriptions look formally identical but the scale height and sound speed now refer to a more or less spherical background distribution of hot gas in which a disk structure of cloudy clumps is imbedded.

4.2. Keplerian selfgravitating [FORMULA]-disks (KSG)

For the particular case of a KSG [FORMULA]-disk we have from Eq. 29

[EQUATION]

For the SG [FORMULA]-disk it follows immediately from Eq. (28) and from mass conservation in the disk that the radial inflow velocity [FORMULA] is given by

[EQUATION]

For the KSG [FORMULA]-disk, we then have

[EQUATION]

Thus at each radius the inward velocity is the same fraction of the local orbital velocity. From Eq. (31) this, in fact, holds for any SG [FORMULA]-disk in which the angular velocity is a power law function of s with adjustment only to the numerical factor in Eq. (32). If [FORMULA] satisfies the constraint (3), then the approximation of centrifugal balance in the radial direction remains well justified.

Under these conditions the dissipation per unit area of a SG [FORMULA]-disk is given by

[EQUATION]

where [FORMULA] is the Stefan-Boltzmann constant. For an optically thick KSG disk this yields the same radial dependence of [FORMULA] as for the standard disk, namely

[EQUATION]

This temperature dependence which is identical to that of the standard model then leads to the well known energy distribution for an optically thick standard disk of [FORMULA]. This also implies that-as long as the disks are not fully selfgravitating-it is hard to distinguish between an [FORMULA]- and a [FORMULA]-disk model observationally.

4.3. Fully selfgravitating [FORMULA]-disks (FSG)

We turn now to the case of the fully selfgravitating (FSG) [FORMULA]-disk, in which the disk mass is sufficiently great that it dominates the gravitational terms in the hydrostatic support equation in both the radial and vertical directions. While there are many potential solutions for the FSG disk structure, one is well known in both mathematical and observational terms, namely the constant velocity ([FORMULA]) disk. Within such a disk structure we have simultaneous solutions to the equation of radial hydrostatic equilibrium and Poisson's equation of the form

[EQUATION]

Toomre (1963), Mestel (1963). For the FSG disk, Eq. (28) then leads to

[EQUATION]

which has the same radial dependence as the structural solution shown in Eq. (35). Thus Eq. (36) may be viewed as giving the rate of mass flow through the disk for a FSG [FORMULA]-disk with constant rotational velocity [FORMULA]. Finally, the equation of continuity provides a constraint if the structure is to maintain a basically steady state structure. For the [FORMULA] const. disk, this yields

[EQUATION]

Thus the constant velocity disk represents a steady state solution in regions sufficiently far from the inner and outer boundaries of the [FORMULA]-disk.

It is then possible, in the spirit of the discussion of Eqs. (33) and (34), to calculate the energy dissipation rate per unit area D for the constant velocity [FORMULA]-disk. We then find

[EQUATION]

so that

[EQUATION]

The flux density, [FORMULA], emitted by an optically thick, constant velocity [FORMULA]-disk is then given by

[EQUATION]

In reality, a sufficiently massive disk may be expected to have an inner Keplerian (standard) zone, a Keplerian selfgravitating zone (KSG), and a fully selfgravitating zone (FSG). We should therefore expect a smooth transition in the spectral energy distribution from the [FORMULA] spectrum of the inner two zones to the [FORMULA] spectrum arising at longer wavelengths from the FSG zone. The transition frequency [FORMULA] may be derived by solving Eqs. (34) and (39) for s, determining a value of transition temperature [FORMULA] and setting

[EQUATION]

One could turn this argument around and argue that, if no other components contribute to the spectrum, the flatness of the [FORMULA] distribution is a measure for the importance of selfgravity and thus for the relative mass of the accretion disk as compared to the central accreting object. This, of course, applies only to the optically thick case which may not arise frequently in strongly clumped disks.

4.4. Time scales

The evolution of accretion disks can be described by a set of time scales. For our purposes, the dynamical and the viscous time scale are of particular interest.

The dynamical time scale [FORMULA] is given by

[EQUATION]

While this formulation applies to all cases, selfgravitating or not, it is only in the non-SG and in the KSG cases that [FORMULA] is given by the mass of the central accretor and by the radius. In the FSG case, [FORMULA] is determined by solving Poisson's equation.

The time scale of viscous evolution [FORMULA] is given by

[EQUATION]

In the standard non-SG and geometrically thin ([FORMULA]) case ([FORMULA] disks), this leads to

[EQUATION]

In KSG and FSG disks ([FORMULA]-disks), [FORMULA] is given by

[EQUATION]

With [FORMULA] and [FORMULA] (Eq. 3) under all circumstances [FORMULA]. In the SG cases the ratio between the two time scales decouples from the disk structure. In all cases the models are self-consistent in assuming basic hydrostatic equilibrium in the vertical direction.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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