## 5. Masing region## 5.1. Velocity structureThe disk temperatures estimated above ( 150 K at 35 AU) correspond to isothermal speeds of sound 1 . Because (Eq. 7) and , the speed of sound is expected to increase rather slowly toward smaller disk radii ( ). This suggests that along a broad zone of disk radii shock fronts may form in the collision between the disk and the impinging clumps (Appendix A). Regarding the fact that the scale height of a standard accretion
disk varies as with
disk radius The density above the disk midplane
decreases with height Unfortunately, there is no indication about the clump densities.
But from the estimated high disk densities (Eq. 8) and the limiting
(upper) density for 22 GHz maser excitation due
to quenching ( 10 Although a rigorous treatment of the thermo- and (magneto)hydrodynamics of a clump-disk collision is required for a detailed comparison with the observations, characteristic features noticeable in the maser observations can already be obtained from a simplified model. Transforming into a coordinate system in which the collision is at normal incidence, and adopting the crude approximation of a homogeneous, stationary flow, a shock front will form parallel to the disk ("surface") plane (Stahler et al. 1994). In the polar coordinate system (Fig. 3) that is comoving with the disk center, the pre-shock gas just loses its vertical velocity component by passing through the shock front (Fig. 6). Radial and azimuthal velocity components () remain unaltered. Hence, the velocity field of the post-shock gas in this approximation is given by and (Eqs. 5), and .
If this velocity field is taken to represent the motion of the
masing gas, an upper limit on the vertical component
of the impinging clumps can be found from the
observed line of sight velocity interval covered by the
L 1287 (H ## 5.2. Shock conditionsFor diatomic (H where is the pre-shock temperature, is the Mach number, and is the shock velocity in the pre-shock gas. From the estimate of the speed of sound ( 1 ), the shock velocity ( 22 ), and the clump temperature ( 100 K), post-shock temperatures up to 9000 K can be achieved. The clump temperature has been adopted to 100 K because of the backwarming effect (see Sect. 4) that also acts on the disk approaching clumps. A minimum vertical velocity is required from the minimum temperature ( 400 K) necessary to excite 22 GHz maser emission. From the ratio of upstream ( = 100 K) and downstream temperature ( = 400 K), the minimum Mach number is found from the jump conditions in the general case, which yields = 3.97 for = 4, and hence an upstream flow velocity into the shock front of 3.97 , provided 1 . The jump condition for upstream () and downstream velocity () is given by so that 0.87 , which equals the shock velocity in the frame of the downstream flow. In this rest frame, the vertical infall velocity of the pre-shock gas equals as measured from the rest frame of the disk center. The lower limit on the vertical infall velocity of the pre-shock gas required to heat the post-shock gas to a level where it may emit 22 GHz maser radiation is therefore 3.10 . From Eqs. 4 and 5, vertical velocity components 3.10 can be obtained for impact radii 35 AU, provided 56 AU, and 1 . The shock conditions estimated above are within the requirements
for J-shocks. However, recent observations of submm
masers in some other star forming regions
(Melnick et al. 1993) indicate that shocks in a magnetized medium
(i.e., C-shocks, Draine 1980) seem required for a satisfactory
explanation of the observations. The expected rather slow shock
velocities in L 1287 (H ## 5.3. Observational constraintsFrom the rather simple model presented above, the velocity field of
the shocked gas can be compared with the position-velocity
distribution shown in Fig. 2. As was argued above, the velocity
field of the clumps' post-shock masing layers are described by
, from Eqs. 5 and
. Fig. 7 shows the line-of-sight component
of this velocity field across a thin disk. The calculated velocity
field due to the masing regions of disk-impinging-clumps shows a
remarkable agreement with the observed position-velocity distribution
of masers in L 1287 (H
Although the disk mass in L 1287 (H A rough estimate on the size scale of the clumps can be obtained, if the variability timescale of the maser emission is taken into account. In the frame of the model motivated above, the maser emission can only be maintained as long as pre-shock (clump-) gas enters the shock front. This timescale is given by the clumps size and the upstream velocity into the shock front, , which is comparable to the infall velocity . Identifying this timescale with the variability timescale of the observed maser emission ( 1 month, as estimated from the data presented in Fiebig et al. 1996), and using a representative value of 10 , the clump size would be on the order of 0.2 AU. At sufficiently small disk radii, the differential rotation of the disk could eventually disperse impinging clumps on a timescale which is shorter than the variability timescale of the masers. However, as is shown in Appendix B, this situation may only occur for very small radii on the order of 1 AU, which are not relevant for the comparison with the observational data. Provided, the (questionable) extrapolation of Larson's
density-size-law (Larson 1981) toward smaller clump sizes and higher
densities would be legitimate, clump radii of some 0.1 AU would
correspond to molecular hydrogen densities of a few 10 From the Rankine-Hugoniot jump conditions for non-dissociative shocks, the ratio of upstream to downstream velocity (Eq. 14) equals the ratio of downstream to upstream density which equals a factor of 6 for strong shocks
( 1). Taking into account that cooling in the
downstream flow increases the density even more, the density
compression can easily amount to an order of magnitude, giving a
post-shock density of order 10 ## 5.4. Pumping conditionsThe azimuthal extent of the masing region (parallel to the disk
plane) is expected to be comparable to the clump size; i.e., the
angular size of an individual maser ( 0.1 AU)
corresponds to about 10 From the optical depth and the masing layer's estimated geometrical depth along the line of sight ( 0.01 AU; besides projection effects, which only account for an additional factor of order unity), the absorption coefficient of the masing transition is approximately given by . For the purpose of an order of magnitude estimate, the hyperfine structure will be neglected, as well as effects due to cross-relaxation and the velocity field. The maximum absorption coefficient is then simply where is the population number difference at line center frequency, is the statistical weight of the (unsplitted) upper maser level, and is the nominal transition frequency. Hence, the inversion efficiency is found from which gives 1 10 © European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |