Hierarchical gravitational fragmentation. I. Collapsing cores within
  collapsing clouds
  We investigate the Hierarchical Gravitational Fragmentation scenario through
numerical simulations of the prestellar stages of the collapse of a marginally
gravitationally unstable isothermal sphere immersed in a strongly
gravitationally unstable, uniform background medium. The core developes a
Bonnor-Ebert (BE)-like density profile, while at the time of singularity (the
protostar) formation the envelope approaches a singular-isothermal-sphere
(SIS)-like $r^-2$ density profile. However, these structures are never
hydrostatic. In this case, the central flat region is characterized by an
infall speed, while the envelope is characterized by a uniform speed. This
implies that the hydrostatic SIS initial condition leading to Shu's classical
inside-out solution is not expected to occur, and therefore neither should the
inside-out solution. Instead, the solution collapses from the outside-in,
naturally explaining the observation of extended infall velocities. The core,
defined by the radius at which it merges with the background, has a
time-variable mass, and evolves along the locus of the ensemble of observed
prestellar cores in a plot of $M/M_{BE}$ vs. $M$, where $M$ is the core's mass
and $M_{BE}$ is the critical Bonnor-Ebert mass, spanning the range from the
"stable" to the "unstable" regimes, even though it is collapsing at all times.
We conclude that the presence of an unstable background allows a core to evolve
dynamically from the time when it first appears, even when it resembles a
pressure-confined, stable BE-sphere. The core can be thought of as a
ram-pressure confined BE-sphere, with an increasing mass due to the accretion
from the unstable background.
