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

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3. Clumpy infall

Although the physical processes that establish and maintain small scale clumping are far from being understood, it seems likely that clumping continues to small and high density eddies (Stutzki & Güsten 1990) which may at least partly account for the infalling material onto the accretion disk around the deeply embedded YSO in L 1287. Clearly, some mechanism is required to either prevent dispersion of the individual clumps or to restore density gradients during their infall motion (at least shortly before impact).

On the large scale size of molecular cloud cores, the clumpy structure has been attributed, for example, to shocks arising from collisions of magnetosonic waves excited by virialized motions (Elmegreen 1990), turbulent motions excited by stellar winds (Norman & Silk 1980), or ejected protostellar material (Myers et al. 1988 ).

Pretending some mechanism to account for the clumpy structure of the collapsing matter to be present, the infall flow can be followed provided the initial conditions are specified.

3.1. Cloud features

Appreciable infall motion is expected, when the flow velocity becomes supersonic. This is expected near the sonic point of the flow, i.e., at a distance

[EQUATION]

from the protostar, where [FORMULA] is the time-dependent accumulated mass inside [FORMULA] and [FORMULA] is the speed of sound at the clumps' location, considered to be constant. A spherically symmetric mass distribution is implicitely assumed for this equation. This assumption is probably not correct regarding the asymmetry introduced by the molecular outflow and taking into account that the bulk mass - on the size scale of [FORMULA] - typically resides in a few large, massive clumps (Blitz 1987, Loren 1989, Lada 1990, Stutzki & Güsten 1990). However, from the dynamical timescale of the outflow ([FORMULA] 104 yr, Yang et al. 1991, 1995), it might be suspected that an appreciable protostellar mass has already formed more than 104 yr ago. Hence, at least on that timescale a central gravitational potential due to protostellar and accretion disk mass may have been a reasonable approximation for radii much larger than the disk radius but smaller than [FORMULA]. Assuming that the accumulated mass [FORMULA] equals one solar mass, and taking into account that the gas temperature derived from NH3 observations ([FORMULA] K, Estalella et al. 1993) was constant ([FORMULA] = 0.27 [FORMULA]), the sonic point is expected at a radius of about 0.06 pc. The free fall time for [FORMULA] then amounts to 2 [FORMULA] 105 yrs. Material that has started with appreciable infall motion 2 [FORMULA] 105 yrs ago probably has not been influenced significantly by the molecular outflow during the last 104 yrs.

Due to the overlap of cloud velocity components in L 1287 (Appendix C), the derived LSR-velocity gradient toward L 1287 (H2 O) has a rather large uncertainty. However, the southeastern component extends far enough to allow an extrapolation. The estimated gradient of about 2.0 [FORMULA] pc-1 transforms into an angular velocity [FORMULA] of 6.5 [FORMULA] s-1, which compares well with values found in other clouds (Goldsmith & Arquilla 1985, Goodman et al. 1993). Notice, that the orientation of the velocity gradient is consistent with the rotation of the suspected accretion disk as suggested earlier (Fiebig et al. 1996).

The maximum angular momentum per unit mass for matter at distances [FORMULA] from the protostar, [FORMULA] [FORMULA] [FORMULA] 2.2 [FORMULA] 1021 cm2 s-1 ([FORMULA] / [FORMULA])2 leads to a rotational energy per unit mass [FORMULA] [FORMULA] 7.2 [FORMULA] 107 cm2 s-2 ([FORMULA] / [FORMULA])2 which is smaller than the gravitational potential energy, [FORMULA] [FORMULA] 7.2 [FORMULA] 108 cm2 s-2 for a low mass protostar (neglect disk mass). Hence, when clumps reach an appreciable infall velocity, their rotational velocity components are rather small.

These estimates implicitely presume that [FORMULA] increases with time but is independent of direction. However, the actual collapse rate depends on the mass distribution of the clumps, which may represent large scale structures that deviate from spherical symmetry (Foster & Chevalier 1993, Hartmann et al. 1994).

3.2. Infall trajectories

If the total energy per unit mass of each clump far from the protostar (at [FORMULA] [FORMULA]) is small compared to the absolute value of the gravitational and translational kinetic energy per unit mass of each clump at distances of the accretion disk size ([FORMULA] [FORMULA]), as will be assumed, the clumpy infalling flow essentially moves on zero-energy free-fall trajectories, i.e., parabolas. In spherical coordinates, centered on the protostar position, the trajectories are described by (Cassen & Moosman 1981)

[EQUATION]

where [FORMULA] is the angle between the accretion disk rotational axis (taken identical to the cloud rotation axis) and the trajectory plane, and [FORMULA] is the polar angle between the rotational axis and the radius vector toward the mass element under consideration (Fig. 3). Although the protostellar mass is time-dependent, it only varies on the evolutionary time-scale, which is long compared to the orbit time-scale of the clumps. For the present discussion, it will be treated as constant.

[FIGURE] Fig. 3. Trajectory of a clump approaching the accretion disk (according to Cassen & Moosman 1981); [FORMULA] is the angle between the disk/cloud rotation axis and the trajectory plane.

The function [FORMULA] characterizes the angular dependence of the (conserved) specific angular momentum of matter approaching the disk at an angle [FORMULA]: [FORMULA], where [FORMULA] is the specific angular momentum near sonic points in the disk plane (where [FORMULA] = [FORMULA] /2). In the case of a constant angular velocity [FORMULA] of the cloud (at least up to [FORMULA]), [FORMULA] [FORMULA], so that [FORMULA] [FORMULA]. The temporal dependences implicitely assume that [FORMULA] does not change within [FORMULA] [FORMULA]; otherwise the inflow motion would not be nearly radial. However, [FORMULA] and [FORMULA] increase with time as more and more matter starts to proceed toward the protostar/disk system ([FORMULA] increases with time). The rigid cloud rotation required for constant [FORMULA], [FORMULA] may indeed be a reasonable assumption, taking into account that magnetic breaking can eliminate differential rotation even on large spatial scales within rather short timescales (Mouschovias & Paleologou 1980). The influence of magnetic fields could in fact play a crucial r [FORMULA] le on the velocity field of the inflowing matter (Galli & Shu 1993) and the formation of the masing layers (Sect. 5.1). However, in order to keep the free parameters as small as possible, the ad hoc assumption of a negligible magnetic field strength in all relevant processes is adopted. Only if the results to be obtained show obvious deficiencies by a comparison with the observational results can a magnetic field be taken into account. This, of course, does not exclude, that future observations of L 1287 (H2 O) or a more detailed modeling of the available observational results will force to include magnetic fields.

The radius [FORMULA] where a clump would impact on an infinitely thin disk, i.e., at [FORMULA] = [FORMULA] /2, is given by

[EQUATION]

where

[EQUATION]

is the disk radius, i.e., the radius in the disk plane where infalling clumps arrive with specific angular momentum appropriate for centrifugal balance. If [FORMULA] is identified with [FORMULA], the accretion disk around the embedded protostar in L 1287 would have a radius of [FORMULA] 4500 AU ([FORMULA] / [FORMULA])3. This value is somewhat higher than the sizes of circumstellar disks detected around T Tau stars (Sargent & Beckwith 1987, 1991, Dutrey et al. 1994). Since the deeply embedded YSO associated with L 1287 (H2 O) is very likely to be in an earlier evolutionary state than optically revealed T Tau stars, the above disk radius will be considered as an upper limit for the accretion disk around the YSO in L 1287.

If, generally, [FORMULA] is a monotonically increasing function, as justified by Cassen & Moosman for non-intersecting streamlines, clumps approaching the disk from all directions of equal [FORMULA] would reach the (infinitely) thin disk at a ring of radius [FORMULA], which becomes larger for increasing [FORMULA]. Taking into account that the zero-energy free-fall trajectory is only an approximation, the streamlines may indeed intersect, but the relative collisional velocities are expected to be rather small. However, the ring where clumps finally reach the disk could then have a significant breadth.

Cassen & Moosman obtained expressions for the velocity components at disk encounter ([FORMULA] = [FORMULA] /2),

[EQUATION]

[EQUATION]

[EQUATION]

From these equations, Eq. 2, and the assumption of a constant cloud angular velocity ([FORMULA] [FORMULA], infall trajectories and velocity components along the line of sight toward an observer for clumps in the disk plane can easily be constructed.

Fig. 4 shows projected sample trajectories onto an infinitely thin disk for four different angles [FORMULA]. The rotation axis is inclined 60 [FORMULA] against the line of sight.

[FIGURE] Fig. 4. Projected sample trajectories of matter approaching an infinitely thin disk. The indicated finite vertical extent of the outer disk edge has no physical relevance, and is only intended for a three-dimensional impression (the northeastern part of the disk should appear closer than southwestern part). The outer disk edge was truncated at 35 AU (lower limit of outer disk radius). Vectors indicate the rotational axes and the sense of rotation. The angles [FORMULA] are given in the upper left box corners, respectively. A central mass of 1 [FORMULA] as well as an inclination angle of 60 [FORMULA] of the rotation axis against the line of sight was arbitrarily adopted. Notice that the lower angular momentum material ([FORMULA] = 30 [FORMULA], 150 [FORMULA]) approaches the disk on orbits which are closer to the rotational axis and impinge on the disk at smaller disk radii.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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