## 4. Relevant disk featuresDue to the temporal increasing radius of the
infall region, the specific angular momentum
increases with time, provided the cloud angular velocity can be
considered constant, as was reasoned above. Within the frame of the
expansion-wave collapse model proposed by Shu (1977), the specific
angular momentum is given by =
, where = 0.975, and
which yields 1.4 10 A rough estimate of the temperature and density in the disk can be obtained from the structure equations of a standard accretion disk (e.g., Frank et al. 1992). Those equations require the specification of the Rosseland opacity, which - for a protostellar disk - is assumed to be dominated by the dust opacity at the relevant radii of order 10 AU. For temperatures below about 150 K, the opacities were calculated for small graphite and silicate grains using optical grain properties according to Draine (1985), and physical grain properties and abundances according to Lenzuni et al. (1995). Fig. 5 shows the calculated Rosseland opacities.
Those opacities can roughly be approximated by a power law,
, with
= 2 and 2
10 Using this power law description for the Rosseland opacity, the structure equations for a standard accretion disk yield for the temperature at the disk midplane equivalent to where is the Shakura-Sunyaev viscosity
parameter, From the estimated age of the disk ( 10 This value appears somewhat high, but it should be recalled that the mass accretion rate was only estimated as an upper limit, and the viscosity parameter is completely unknown. Nevertheless, the effect of "backwarming", i.e., the scattering of photons from the protostar and the inner disk in the surrounding envelope material, can heat up the outer regions of the disk ( 40 AU) to temperatures of about 100 K (Butner et al. 1994). The disk structure equations also allow to estimate the midplane density according to which transforms into a molecular hydrogen density Notice, that the density strongly increases toward smaller disk radii. This is still true, even if smaller viscosity parameters are adopted (e.g., Lin & Pringle 1990). An estimate of the scale height This scale height is to be considered rather an upper limit since the disk structure equations used do not take into account the external pressure of the infalling gas exerted onto the disk. The trajectories of infalling matter described above are valid only in the limit of a central gravitational potential, i.e., negligible disk mass (as compared to the mass of the central object). The disk mass can be found by straightforward integration of the disk structure equations. For simplicity, an isothermal stratification of the vertical disk structure is assumed, = (e.g., Frank et al. 1992 ). Integration of the vertical density dependence yields the surface (column-) density and integration along the radial direction gives the disk mass, equivalent to The radial integration assumed that the outer disk boundary is much larger than the inner disk boundary. Notice that - for any given outer disk radius - the disk mass does not depend on the mass of the central object. Nevertheless, regarding the evolution of the star forming region, itself does depend on the central mass according to Eq. 4. The assumption of negligible disk mass (Sect. 3), ( 1 ), is valid only for a sufficiently small outer disk boundary ( is scaled with its lower limit in Eq. 11). It seems worth to stress that the estimated disk parameter values can only be considered as rough estimates, regarding all the uncertainties, especially those of the viscosity parameter, and stringent approximations (e.g., the stationary state of the disk, the isothermal stratification, the constant values in space and time for , , ). © European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |