4. Relevant disk features
Due 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 t is the time measured from the beginning of the collapse (Cassen & Moosman 1981). From the largest extent of the observed maser positions (Fiebig et al. 1996), a lower limit on the disk radius is 35 AU. Thus, a lower limit on the age of the protostar can be given using Eq. 4 and the parameter values estimated in Sect. 3,
which yields 1.4 105 yrs ( / ). However, even for the rather upper value for the disk radius of 4500 AU (discussion following Eq. 4), the timescale increases only to 4.6 105 yrs ( / ). An age of some 105 yrs agrees with the lower limit of T Tau star ages, which range between 105 and 107 yrs (Beckwith et al. 1990). Hence, the protostellar object in L 1287 is likely to reach the end of its embedded/accretion-dominated phase.
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 10-4 g-1 cm2 K-2.
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
where is the Shakura-Sunyaev viscosity parameter, k is the Boltzmann constant, is the Stefan-Boltzmann constant, is the proton mass, is the mean molecular mass number, and is the mass accretion rate. The derivation of Eq. 7 implicitely assumed that the disk is geometrically thin, but optically thick, and that the vertical hydrostatic pressure is dominated by gas pressure. The internal thermal energy is due to viscous heating. The disk matter approximately rotates on Keplerian orbits, where the disk mass is assumed negligibly small compared to the protostellar mass, and considered disk radii r are large compared to the protostellar radius.
From the estimated age of the disk ( 105 yrs) an average mass accretion rate of yr-1 can be assumed, provided that the protostar is not too massive. Unfortunately, the viscosity parameter is completely unknown for protostellar disks. The collisions between infalling clumps and the protostellar disk certainly lead to local disk heating and, subsequently, disturbances in the disk flow, and thus will be a cause of viscosity. However, a detailed investigation of the effect of disk-impinging-clumps on the disk viscosity is beyond the scope of the present work. For example, adopting 0.1, the estimated temperature at the (outer) disk radius of 35 AU gives 150 K for a 1 protostar.
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 H at r can also be derived from the disk structure equations,
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,
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