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

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5. Possible applications

5.1. Protoplanetary accretion disks

T Tauri stars have infrared spectral energy distributions [FORMULA] which can be approximated in many cases by power laws [FORMULA] with a spectral index n in the range [FORMULA]. Assuming this spectral behaviour to be due to radiation from an optically thick disk, it translates into a radial temperature distribution [FORMULA] with [FORMULA].

An optically thick non-selfgravitating accretion disk which radiates energy that is liberated through viscous dissipation, i.e., an active accretion disk shows a spectral distribution with [FORMULA] or [FORMULA]. This immediately excludes optically thick non-selfgravitating standard accretion disks as the major contributor to T Tau spectra.

Adams et al. (1988) were the first to discuss the possibility of a non-standard radial temperature distribution with [FORMULA]. Using q as a free parameter, they find that for flat spectrum sources, their best fits require disk masses that are no longer very small compared to the masses of the accreting stars. They already mention the possibility that the flatness of the spectrum and selfgravity of the disk may be related. On the other hand, at that time this indirect argument was the only evidence for large disk masses. Beckwith et al. (1990) in their survey of circumstellar disks around young stellar objects also find preferentially disk spectra that are considerably flatter than predicted by the standard optically thick disk models. For more than half of their objects they derive disk masses that correspond to the KSG and FSG cases. On the other hand, Natta (1993) proposed that flat disk spectra are the consequence of dusty envelopes engulfing a star with a standard disk around it. Recently, Chiang & Goldreich (1997) have investigated in detail non-selfgravitating passive accretion disks, i.e., disks that are heated by radiation from the star and re-radiate this energy. Depending on the details of the flaring of the disk, this can lead to considerably flatter spectra than expected from active disks.

However, in the meantime, high resolution direct observations of protostellar disks yield independent strong evidence for comparatively large disk masses. Lay et al. (1994), for instance, find a lower limit for the disk masses in HL Tau-one of the sources in Adams, Lada & Shu's sample of flat spectrum T Tauri stars-of [FORMULA].

We suggest that the flatness of the spectrum actually reflects the mass of the disk, i.e., the importance of selfgravity. For disk masses considerably smaller than [FORMULA], the standard accretion disk models apply. For disks whose masses are larger but still small compared to [FORMULA] the spectral behaviour is not altered significantly, but the disk structure and the time scale of disk evolution ([FORMULA], see Eqs. 44 and 45) change. For even more massive disks, we expect a clear trend towards flatter spectra that approach an almost constant [FORMULA] distribution if selfgravity in the disk becomes important in the radial as well as in the vertical direction.

5.2. Galactic disks

The relevance of viscosity in the evolution of galactic disks has been the subject of discussion since von Weizsäcker (1943), von Weizsäcker and Lüst (1952) first raised the issue nearly fifty years ago. They noted then that, with an eddy viscosity formulation (a [FORMULA]-disk), the time scale for evolution of typical galactic disks was comparable to the age of the universe and suggested that this might account for the difference between spiral and elliptical galaxies.

With the subsequent realization that galactic disks moved primarily under the influence of extended massive halos, interest in FSG disks waned. However, as noted above, it is possible for a massive disk to exist and evolve under the influence of viscosity, while embedded in such a halo gravitational field. Indeed, in the event that such a structure forms, it must evolve under viscous dissipation and can achieve a quasi-steady state with essentially the same mass and energy dissipation distribution as for the FSG constant velocity disk. We refer to this case as an Embedded Self-Gravitating (ESG) disk.

The time scale for viscous evolution [FORMULA] as given in Sect. 4.4 suggests a means of differentiating between the [FORMULA]- and [FORMULA]-formulations for this case. For a normal spiral galaxy with a suggested mean temperature in the gaseous disk of around [FORMULA]K and a scale height of around 300 pc, we obtain


Thus, with these parameters, little evolution would take place in a Hubble time on the [FORMULA]-hypothesis but significant evolution is predicted on the [FORMULA]-hypothesis. This problem of the viscous time scale in a selfgravitating [FORMULA] accretion disk was also noted by Shlosman & Begelman (1987), Shlosman & Begelman (1989). Shlosman et al. (1989) proposed non-axisymmetric disturbances ("bars within bars") as an alternative way of transporting angular momentum in the radial direction within a sufficiently short time scale.

In terms of inflow velocities the [FORMULA]-ansatz suggests values in the range [FORMULA]km s-1 which would be exceedingly hard to measure directly. The [FORMULA]-ansatz suggests still lower values. On the other hand, it may be possible to provide limits on the viscosity through other observational constraints. For example, the build up of the 3 kpc molecular ring in our own galaxy can be interpreted as due to viscosity driven inflow in the constant velocity part of the galactic disk which ceases (or at least slows down) in the constant angular velocity inner regions Icke (1979), Däther & Biermann (1990). Similarly, several authors have suggested that the radial abundance gradients observed in our own and other disk galaxies may be due to radial motion and diffusive mixing associated with the turbulence generating the eddy viscosity Lacey & Fall (1985), Sommer-Larsen & Yoshii (1990), Köppen (1994), Edmunds & Greenhow (1995), Tsujimoto et al. (1995). According to these authors, radial inflows of around 1 km s-1 at the galactic location of the Sun are required for optimum fits to the abundance gradient data within the context of the viscous disk hypothesis. Such inflow velocities are consistent with the [FORMULA]-ansatz but could, of course, be generated also by other means (e.g., effects of bars, magnetic fields).

5.3. Ultraluminous galaxies

Recent high resolution imaging of ultraluminous galaxies in the near infrared and mm wavelengths bands shows dense gas and dust accretion disks in their galactic nuclei. The two nuclei in the merger galaxy Arp 220, for instance, have masses of the order of several [FORMULA] within radii of [FORMULA]pc Scoville (2000). Similar properties, albeit less well resolved than in Arp 220, seem to be typical for this class of galaxies Solomon et al. (1997), Downes & Solomon (1998). Most, if not all, ultraluminous galaxies seem to be merging galaxies Sanders & Mirabel (1996). In Arp 220, these gas masses are the major contributor to the dynamical mass in the two nuclei Scoville (2000), i.e., these nuclear disks are selfgravitating. Most likely this is true for the nuclear disks in other ultraluminous galaxies as well. The merger process is presumably responsible for transporting large amounts of material into the central few hundred parsecs, thus filling a mass reservoir which is then available for subsequent disk accretion to the very center.

Within the framework of [FORMULA]-disks, one finds that the viscous accretion time scale [FORMULA] increases towards larger radii as long as the surface density [FORMULA] in the disk increases with radius s not steeper than [FORMULA], which is most likely fulfilled. Then the viscous time scale at the disk's outer edge is an upper limit to its evolution time scale. For Arp 220 (disk mass [FORMULA]; outer radius [FORMULA]pc) one finds a time scale of [FORMULA] years for [FORMULA], which, in turn yields accretion rates [FORMULA]. Such rates lead to accretion luminosities [FORMULA] up to [FORMULA] erg s-1, where [FORMULA] ([FORMULA]) is the conversion efficiency of gravitational energy into radiation and c is the speed of light. Such luminosities are large enough to power even the strongest AGN and the time scales are very much shorter than the Hubble time.

Assuming that these rates can be maintained during a sizeable fraction of [FORMULA], a significant fraction of the disk's original gas mass could be accreted to much smaller radii, presumably to a black hole in the very center (some will be lost to star formation or winds). In this process the black hole gains a considerable amount of mass within a relatively short time scale. One may speculate that this is actually the process that produces the most massive black holes in the young universe. By contrast, galaxies that do not undergo mergers presumably have no way of rapidly collecting such large masses of gas within [FORMULA] pc. As a consequence, these disks are less likely to be selfgravitating and thus are likely to have longer [FORMULA]. The nuclei of such galaxies will accrete much smaller amounts of material over longer time scale, resulting in lower mass central black holes Duschl (1988a, b). An example may be our own Galactic Center.

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Online publication: June 5, 2000