## 3. Clumpy infallAlthough 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 featuresAppreciable 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 from the protostar, where is the
time-dependent accumulated mass inside and
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 - 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 ( 10 Due to the overlap of cloud velocity components in L 1287 (Appendix
C), the derived LSR-velocity gradient toward L 1287 (H The maximum angular momentum per unit mass for matter at distances
from the protostar,
2.2
10 These estimates implicitely presume that 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 trajectoriesIf the total energy per unit mass of each clump far from the protostar (at ) 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 ( ), 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) where is the angle between the accretion disk rotational axis (taken identical to the cloud rotation axis) and the trajectory plane, and 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.
The function characterizes the angular
dependence of the (conserved) specific angular momentum of matter
approaching the disk at an angle :
, where is the specific
angular momentum near sonic points in the disk plane (where
= /2). In the case of a
constant angular velocity of the cloud (at
least up to ),
, so that
. The temporal dependences implicitely assume
that does not change within
; otherwise the inflow
motion would not be nearly radial. However, and
increase with time as more and more matter
starts to proceed toward the protostar/disk system
( increases with time). The rigid cloud rotation
required for constant ,
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 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 The radius where a clump would impact on an infinitely thin disk, i.e., at = /2, is given by where 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 is identified with
, the accretion disk around the embedded
protostar in L 1287 would have a radius of
4500 AU ( / ) If, generally, is a monotonically increasing function, as justified by Cassen & Moosman for non-intersecting streamlines, clumps approaching the disk from all directions of equal would reach the (infinitely) thin disk at a ring of radius , which becomes larger for increasing . 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 ( = /2), From these equations, Eq. 2, and the assumption of a constant cloud angular velocity ( , 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 . The rotation axis is inclined 60 against the line of sight.
© European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |