3. Self-regulation in self-gravitating accretion disks
Before proceeding further, we should make here a digression and try to better explain the physical justification at the basis of the regulation prescription set by Eq. (8), which is the distinctive property of the class of models addressed in this paper. This discussion expands and clarifies a brief description provided earlier (Bertin 1997).
3.1. The mechanism
We start by noting that, when self-gravity dominates, processes associated with the Jeans instability basically determine the relevant scales of inhomogeneity of the system under consideration. On the other hand, it is well known (Toomre 1964) that in the plane of a thin, self-gravitating, rotating disk the tendency toward collapse driven by self-gravity, which is naturally expected to be balanced locally only at small wavelengths by pressure forces, can be fully healed by rotation also at small wavenumbers. As a result, if the disk is sufficiently warm, it can be locally stable against all axisymmetric disturbances. The condition of marginal stability thus ensured is usually cast in the form , where Q is proportional to the effective thermal speed of the disk.
Therefore, in galactic dynamics it has long been recognized that an initially cold disk is subject to rapid evolution, on the dynamical timescale. For a stellar disk, the Jeans instability induces fast overall heating of the disk up to levels of the effective thermal speed for which the instability is removed, basically following the above criterion (). The heating rate is thus very sensitive to the instantaneous value of Q. Therefore, even in the absence of dissipation, for relatively low values of Q the collective Jeans instability provides a mechanism to stir the system and to heat it. However, a collisionless non-dissipative disk would be unable to evolve in the opposite direction, i.e. to cool if initially hot (), and, because of other residual processes (such as tidal interactions), may still be subject to some perennial heating even when Jeans instability is ineffective. In turn, if efficient dissipation is available (such as that associated with the interstellar medium in galaxy disks), a competing mechanism can cool an initially hot disk down toward conditions of marginal Jeans instability.
These important dynamical ingredients have been studied from various points of view, giving rise to interesting scenarios, also by means of numerical experiments (see Miller et al. 1970, Quirk 1971, Quirk & Tinsley 1973, Sellwoood & Carlberg 1984, Ostriker 1985), so that eventually the possibility of an actual self-regulation in self-gravitating disks has been formulated and explored. Accordingly, the two competitive mechanisms, of dynamical heating and dissipative cooling, can set up a kind of dynamical thermostat. In particular, it has been shown that self-regulation is very important in determining the conditions for the establishment of spiral structure in galaxies (Bertin and Lin 1996; and references therein). The general concept has also found successful application to the dynamics of the interstellar medium, even when dynamically decoupled from the stellar component, in relation to the interpretation of the observed star formation rates (Quirk 1972; Kennicutt 1989). Note that the most delicate aspect of the regulation process is the cooling mechanism; in galaxy disks, an important contribution to cooling is thus provided by cloud-cloud inelastic collisions that dissipate energy on a fast timescale.
A final remark is in order. As shown by the case of galaxy disks, the dynamics involved may be extremely complex, so that there is little hope of providing a simple "ideal" energy equation for the description of a coupled disk of stars (of different populations) and gas (in different forms and phases). This point has an additional important consequence. If we try to describe such an inherently complex system by means of an idealized one-component model, we should be ready to introduce the use of some effective quantities (in particular, of an effective thermal speed; see also the use of the term "effective" after Eq. (5) and at the beginning of this subsection).
3.2. Viability of the mechanism for some accretion disks
It should be emphasized that the main strength of the self-regulation scenario described above is rooted in semi-empirical arguments. The actual data from many galaxy disks (among which the Milky Way Galaxy) and from planetary rings show that conditions of marginal stability () often occur and can indeed be established in systems subject to complex dynamical processes. For accretion disks, there may already be some empirical indications in this direction, to the extent that the application of standard models to some observed systems points to very cold disks, with Q well below unity (e.g., see Kumar 1999, as briefly mentioned in Sect. 6). Additional clues in the same direction also derive from numerical experiments; in the context of protostellar disks, numerical simulations show that disks formed under conditions where self-gravity is important include wide regions characterized by a constant Q-profile (Pickett et al. 1997).
An accretion disk may be subject to efficient cooling by a variety of mechanisms, depending on the physical conditions that characterize the specific astrophysical system under consideration. From this point of view, the cooling necessary for the establishment of self-regulation may occur efficiently already via the radiative processes included in "standard" models (see the general discussion of the outer disk by Bardou et al. 1998, which we will summarize in Sect. 5.1). In reality, the dynamics of matter slowly accreting in a disk can be significantly more complex. Cold systems, such as a protogalactic disk, a protostellar disk, or the outer parts of the disk in an AGN, may have a composite and complex structure. They may include dust, gas clouds, and other particulate objects with a whole variety of sizes and "temperatures". Much like for the HI component of the interstellar medium, the main contribution to the effective temperature of the disk might be from the turbulent speed of an otherwise cold medium. On the one hand, for these systems it may be hard or even impossible to write out a simple "ideal" energy transport equation. On the other hand, such a complex environment is likely to possess all the desired cooling and heating mechanisms that cooperate in self-regulation. In this respect, one is thus encouraged to bypass the problem of defining a representative set of equations for energy transport, and to use instead the semiempirical prescription of Eq. (8). Somewhat in a similar way, our inability to derive from first principles a satisfactory set of equations for momentum transport is often taken to justify the adoption of the -prescription of Eq. (5). These phenomenological prescriptions have several limitations, but may still work as a useful guide to our efforts and provide interesting models to be compared with the observations.
To be sure, some types of accretion disk, or some regions inside accretion disks (for example, very close to the center; see Sect. 5.1), may lack the physical ingredients invoked above. In fact, there is no reason to claim that self-gravity must always be important. Therefore, we will study the structure of self-regulated accretion disks, as a viable class of astrophysical systems, while we do recognize that warmer, non-regulated disks may exist and are likely to be basically free from the effects associated with the self-gravity of the disk.
3.3. The impact of Q on momentum and energy transport
The self-regulation mechanism has been demonstrated by considering a simplified set of equations (Bertin 1991) where efficient cooling is included and the role of self-gravity is modeled by means of a heating term with an analytic expression (inversely proportional to a high power of Q) meant to incorporate the results of dynamical studies that show that heating is indeed very sensitive to the value of Q. The main features of this formula, with its threshold at , aimed at representing the "thermal evolution" of the disk, are somewhat analogous to the heuristic characterization of the viscosity dependence on Q adopted by Lin & Pringle (1990) in the parallel problem of constructing the momentum transport equations when self-gravity is important.
In our discussion of self-regulated accretion disks, we actually have no doubt that self-gravity is likely to have an important impact on viscosity, and this is still tacitly incorporated in the prescription. This impact is even more obvious if one recalls that a self-gravitating disk can be subject to non-axisymmetric instabilities, which are bound to contribute significantly to angular momentum transport. Our class of axisymmetric, steady-state accretion models represents only one approximate idealization of the actual system that we are addressing. Given the indications of several dynamical studies (in addition to those of Lin & Pringle, see, for example, Laughlin & Bodenheimer 1994, Laughlin & Róyczka 1996), a more complete analysis should thus include one further relation between and Q. In general, this might practically require that the phenomenological prescription (5) be used with a parameter varying with radius. In reality, we believe that the proper way to include the relevant physical effects, especially those associated with non-axisymmetric instabilities, would be through some global constraint. Until such global description remains not available, the assumption of a free, constant may provide a first approximation, best applicable when Q is self-regulated. This choice can be physically consistent a posteriori , at least for the self-similar solution of Sect. 2.2.
© European Southern Observatory (ESO) 1999
Online publication: October 4, 1999