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Astron. Astrophys. 327, 758-770 (1997)

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5. Masing region

5.1. Velocity structure

The disk temperatures estimated above ([FORMULA] 150 K at 35 AU) correspond to isothermal speeds of sound [FORMULA] [FORMULA] 1 [FORMULA]. Because [FORMULA] [FORMULA] (Eq. 7) and [FORMULA] [FORMULA], the speed of sound is expected to increase rather slowly toward smaller disk radii ([FORMULA] [FORMULA]). 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 [FORMULA] [FORMULA] with disk radius r (Eqs. 9, 7), the polar angle [FORMULA] at impact is for all considered disk radii close to 90 [FORMULA]. Hence, the polar velocity component of a clump (Eqs. 5) is a sufficiently good representation of the velocity component at normal incidence on the disk.

The density [FORMULA] above the disk midplane decreases with height z; in the isothermal approximation according to a Gaussian, [FORMULA] [FORMULA] [FORMULA]. Taking into account that the external pressure of the infalling matter and radiation pressure within the disk tend to increase the density gradient near a scale height above/below the disk midplane, clumps may get smashed up on a dense disk at about a distance H from the midplane, if their ram pressure is much lower than the total disk pressure at H.

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 [FORMULA] maser excitation due to quenching ([FORMULA] 1011 cm -3, depending on pump and loss rates, Goldreich & Kwan 1974, Elmegreen & Morris 1979) it seems possible that maser emission could be excited in the shock-compressed layer of a clump colliding with the disk. Clump material reaching the disk will be brought to a sudden stop in the vertical direction, and fast compression due to succeeding clump gas will cause a shock front to form in the clump.

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 ([FORMULA]) remain unaltered. Hence, the velocity field of the post-shock gas in this approximation is given by [FORMULA] and [FORMULA] (Eqs. 5), and [FORMULA].

[FIGURE] Fig. 6. Sketch of the clump-disk collision. Left panel: Clump approaching the disk "surface" in the rest frame of the disk center. The radial velocity component [FORMULA] points toward the viewer. Middle panel: Magnified view of the shock region after impact in a rest frame moving at velocity [FORMULA] [FORMULA] with respect to the disk center. The upper line represents the shock front receeding with velocity [FORMULA] from the disk. The vertical velocity component [FORMULA] vanishes in the post-shock region. Right panel: The velocities [FORMULA] and [FORMULA] represent the flow velocities of upstream and downstream flow in the rest frame of the shock front.

If this velocity field is taken to represent the motion of the masing gas, an upper limit on the vertical component [FORMULA] of the impinging clumps can be found from the observed line of sight velocity interval covered by the L 1287 (H2 O) maser emission. This interval ([FORMULA] 26 [FORMULA]) is to be compared with twice the absolute value of the post-shock velocity vector, corrected for inclination i of the disk symmetry axis against the line of sight, 2 [FORMULA]. An upper limit on [FORMULA] is found from Eqs. 5 with [FORMULA] = 0, which gives [FORMULA] [FORMULA] 13 [FORMULA] / [FORMULA] = 15 [FORMULA] for i = 60 [FORMULA] and 18 [FORMULA] for i = 45 [FORMULA]. Significantly smaller inclinations seem very unlikely, taking into account that Fig. 2 indicates a rather edge-on view of the suspected disk. Since [FORMULA] is always smaller than [FORMULA] by a factor of [FORMULA] (Eqs. 5), [FORMULA] [FORMULA] [FORMULA] [FORMULA] 18 [FORMULA]. Since the vertical velocity component [FORMULA] tends to achieve higher Mach numbers for smaller disk radii (where [FORMULA] [FORMULA]), strong shocks are expected to form for at least small infall angles [FORMULA]. If pre- and post-shock gas consist primarily of molecular hydrogen, application of the Rankine-Hugoniot jump conditions for strong, non-dissociative shocks (e.g., Courant & Friedrichs 1948) yields the flow velocity of the post-shock gas to be 1/6 of the pre-shock value for diatomic molecular gas (H2) in the reference frame of a stationary shock front. Hence the shock velocity with respect to the upstream gas is 6/5 [FORMULA] [FORMULA] 22 [FORMULA]. This velocity is below the limiting shock velocity of about 25 [FORMULA], where dissociation of molecular hydrogen becomes appreciable for densities [FORMULA] [FORMULA] 104 [FORMULA] (Hollenbach & McKee 1980 ), thus justifying the assumption of non-dissociating shocks a posteriori.

5.2. Shock conditions

For diatomic (H2) gas (ratio of specific heats [FORMULA] = 7/5), the post-shock temperature [FORMULA] just behind a strong shock front is given by


where [FORMULA] is the pre-shock temperature, [FORMULA] [FORMULA] is the Mach number, and [FORMULA] is the shock velocity in the pre-shock gas. From the estimate of the speed of sound ([FORMULA] 1 [FORMULA]), the shock velocity ([FORMULA] 22 [FORMULA]), and the clump temperature ([FORMULA] 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 [FORMULA] is required from the minimum temperature ([FORMULA] 400 K) necessary to excite 22 GHz [FORMULA] maser emission. From the ratio of upstream ([FORMULA] = 100 K) and downstream temperature ([FORMULA] = 400 K), the minimum Mach number is found from the jump conditions in the general case,


which yields [FORMULA] = 3.97 for [FORMULA] = 4, and hence an upstream flow velocity into the shock front of [FORMULA] 3.97 [FORMULA], provided [FORMULA] 1 [FORMULA]. The jump condition for upstream ([FORMULA]) and downstream velocity ([FORMULA]) is given by


so that [FORMULA] 0.87 [FORMULA], 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 [FORMULA] 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 [FORMULA] maser radiation is therefore [FORMULA] [FORMULA] 3.10 [FORMULA].

From Eqs. 4 and 5, vertical velocity components [FORMULA] 3.10 [FORMULA] can be obtained for impact radii [FORMULA] 35 AU, provided [FORMULA] 56 AU, and [FORMULA] 1 [FORMULA].

The shock conditions estimated above are within the requirements for J-shocks. However, recent observations of submm [FORMULA] 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 (H2 O) are consistent with C-shocks, so that a more detailed investigation is required to clarify the question, whether J- or C-shocks prevail in L 1287 (H2 O).

5.3. Observational constraints

From 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 [FORMULA], [FORMULA] from Eqs. 5 and [FORMULA]. 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 [FORMULA] masers in L 1287 (H2 O) for a central mass of 3 [FORMULA]. An inclination angle of 60 [FORMULA] of the disk rotation axis against the line of sight was adopted in each case; in agreement with the estimated inclination angle of the outflow direction (Yang et al. 1991). A corresponding position angle of 45 [FORMULA] was chosen to approximately fit the projected orientation of the outflow. No attempt was made to optimize the agreement between measured data and model calculations.

[FIGURE] Fig. 7. Calculated LSR-velocity distribution for masing gas of clumps impinging onto an infinitely thin disk (grey shaded areas). The disk rotation axes are inclined by 60 [FORMULA] against the line of sight. The indicated finite vertical extent of the outer disk edge has no physical relevance, and is only intended for a three-dimensional impression. The adopted mass of the corresponding central object is indicated in the upper left box corners, respectively. Velocities have been binned into six equally spaced intervals, according to Fig. 2. The LSR-velocity of the central object is at -17 [FORMULA]. The outer disk edges were truncated at 35 AU. The data displayed in Fig. 2 were overlayed after shifting manually the offset coordinates by (-0:0002, +0:0002) in order to achieve a reasonable agreement. Recall that the error bars indicate one-channel errors. The white areas in the disks represent loci where calculated LSR-velocities fall either below -30 [FORMULA] or above -4 [FORMULA]. Notice that the 3 [FORMULA] case shows very good agreement.

Although the disk mass in L 1287 (H2 O) cannot be reasonably derived, the agreement of the masers' position-velocity distribution with the derived velocity field shown in Fig. 7 is consistent with a negligible disk mass, as compared to the central protostar. A negligible disk mass would support the conclusions drawn by Terebey et al. (1993), who found that the typical disk mass for YSOs in the embedded phase cannot be significantly higher than for those in the T Tau phase.

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, [FORMULA], which is comparable to the infall velocity [FORMULA]. Identifying this timescale with the variability timescale of the observed maser emission ([FORMULA] 1 month, as estimated from the data presented in Fiebig et al. 1996), and using a representative value of [FORMULA] [FORMULA] 10 [FORMULA], the clump size would be on the order of [FORMULA] 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 109 [FORMULA]. Such a clump density (before impact) would be in agreement with the density requirements for 22 GHz [FORMULA] maser emission, if the density compression factor is within about two orders of magnitude ([FORMULA] 1011 [FORMULA], see above).

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 ([FORMULA] 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 1010 [FORMULA]. Hence, the maximum vertical height of the masing layer is expected be an order of magnitude smaller than the clumps size, i.e., about 0.01 AU.

5.4. Pumping conditions

The 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 ([FORMULA] 0.1 AU) corresponds to about 10-4 arcsec at the distance of L 1287. Taking into account, that the typical observed maser flux densities range on the 0.1 Jy level (Fiebig et al. 1996), the maser (source) brightness temperature would be on the order of 1010 K. Although the maser background brightness temperature is unknown, the environment discussed in Sect. 4 suggests that a brightness temperature on the order of 100 K might be a reasonable assumption. As there is indication for unsaturated maser emission (Fiebig 1996b), the corresponding optical depth in the maser would be [FORMULA] = [FORMULA] K / 100 K) [FORMULA] 18. Notice that higher background temperatures result in lower optical depths.

From the optical depth [FORMULA] and the masing layer's estimated geometrical depth along the line of sight [FORMULA] ([FORMULA] 0.01 AU; besides projection effects, which only account for an additional factor of order unity), the absorption coefficient [FORMULA] of the masing transition is approximately given by [FORMULA] [FORMULA].

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 [FORMULA] is the population number difference at line center frequency, [FORMULA] is the statistical weight of the (unsplitted) upper maser level, and [FORMULA] is the nominal transition frequency. Hence, the inversion efficiency [FORMULA] [FORMULA] is found from


which gives 1 [FORMULA] 10-7, if [FORMULA] [FORMULA] and [FORMULA] 1010 [FORMULA] is the density in the masing layer and [FORMULA] [FORMULA] (Takahashi et al. 1985) is the abundance of water molecules with respect to molecular hydrogen. Since the inversion efficiency depends linearly on the difference of the ratios of pump and loss rates ([FORMULA], [FORMULA]) for the upper and lower maser level, [FORMULA] [FORMULA], it is a usable variable characterizing various pumping schemes at equal temperatures. An inversion efficiency of 10-7 indeed compares well with rather conservative estimates of pump and loss rates (e.g., Goldreich & Keeley 1972), and seems even easily attainable in the light of recent more detailed investigations (e.g., Anderson & Watson 1993).

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998