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

Appendix A: radial dependence of the supersonic condition

The vertical component of a clump's impact velocity becomes supersonic if . From Eqs. 5 and 3, using sin ,

where has been defined. The supersonic condition can then be expressed by

where Eq. 7 has been used to express in terms of r (= ), and defining

The dependence of on the disk radius x (= ) is shown in Fig. 8. The supersonic condition is equivalent to the requirement , where is determined by the properties of the disk. For a given value of , the radial zone where disk-impinging-clumps can form a shock front is determined from the interval of those x that fulfill the supersonic condition (). The maximum is found at 1/3. If the disk is sufficiently cold to fulfill the supersonic condition at radii x ( 1) which are much larger than 1/3, it seems inevitable that disk-impinging-clumps form shock fronts at all smaller disk radii (except a very small region around disk center, see Fig. 8). For the suspected disk in L 1287 (H₂ O), is likely to be very small,

and even smaller for more reasonable disk radii 35 AU. However, it should be noticed that the details of the collisional process and the disk structure are likely to modify the above relationships.

Fig. 8. Graphical representation of the function .

Appendix B: clump dispersion due to differential disk rotation

Due to the differential rotation of the disk, a disk-impinging-clump could eventually be dispered. After some time its inner boundary (as seen from the disk center) has shifted by an azimuthal length increment as compared to its outer boundary,

where r is the radial distance to the clump's inner boundary and is the (radial) clump size. The angular velocity has been assumed to describe a Keplerian rotation,

If the clump size is assumed to be small as compared to the relevant disk radius r,

From the requirement, that a clump should not be dispered during the timescale , it follows that , which transforms to

equivalent to

For the relevant disk radii of order 10 AU considered in this paper, disk-impinging-clumps will not be dispersed on the maser variability timescale due to the differential disk rotation.

Appendix C: large scale velocity field

NH₃ observations to study the velocity field in L 1287 were performed with the 100-m telescope of the Max-Planck-Institut für Radioastronomie. Spectra of the NH₃ (1,1) and -(2,2) inversion line could be obtained using the standard MPIfR K-band maser receiver. The backend used was a 1024-channel 2-bit autocorrelator operating at a resolution of 12.21 kHz (corresponding to 0.15 at the NH₃ line frequencies). Spectra were taken in position switching mode with the reference position 3 east of the on-source position. Pointing was usually checked every one to two hours on W3(OH). The typical pointing accuracy was about 5". Calibration was performed on NGC 7027. The FWHP beam size was determined through continuum cross scans on W3(OH), NGC 7027 and 3 C 84 to 38" 2".

The overall distribution of the dense core gas, as traced by the NH₃ (1,1) inversion line, is displayed in Fig. 9. A similar mapping was carried out by Estalella et al. (1993). The emission displays an elongated structure along a southeast-northwest direction which defines a major axis. Along this axis the position-velocity diagram shows an extended velocity component toward the southeast and a more compact component toward the northwest. From the variation of the LSR-velocities along the major axis, a velocity gradient of 2.0 pc^-1 can be estimated for the southeastern component.

Fig. 9a and b. Upper panel: Map of the overall NH3 (1,1) emission represented by the velocity integrated main beam brightness temperature dv. The contours start at 3.5 K and increase by 1.5 K . The offset coordinates are centered on the position of L 1287 (H2 O); R.A. (1950) = 00h 33m 53 15, Dec (1950) = +63 12 32 1. The lines along a southeast-northwest direction indicate a defined major axis. Lower panel: Position-velocity diagram from the NH3 (1,1) main lines along the major axis. Contour lines represent 20, 35, 50, 65, 75, 80, 85, 90, and 95 % of 4.73 K maximum temperature. Notice that two velocity components overlap near the center position. The thick solid bar corresponds to the spectral channel width. The broken line indicates a velocity gradient of 2.0 pc-1.

© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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