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We investigate the Hierarchical Gravitational Fragmentation scenario throughnumerical simulations of the prestellar stages of the collapse of a marginallygravitationally unstable isothermal sphere immersed in a stronglygravitationally unstable, uniform background medium. The core developes aBonnor-Ebert (BE)-like density profile, while at the time of singularity (theprotostar) formation the envelope approaches a singular-isothermal-sphere(SIS)-like $r^-2$ density profile. However, these structures are neverhydrostatic. In this case, the central flat region is characterized by aninfall speed, while the envelope is characterized by a uniform speed. Thisimplies that the hydrostatic SIS initial condition leading to Shu's classicalinside-out solution is not expected to occur, and therefore neither should theinside-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 atime-variable mass, and evolves along the locus of the ensemble of observedprestellar cores in a plot of $M/M_{BE}$ vs. $M$, where $M$ is the core's massand $M_{BE}$ 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 evolvedynamically from the time when it first appears, even when it resembles apressure-confined, stable BE-sphere. The core can be thought of as aram-pressure confined BE-sphere, with an increasing mass due to the accretionfrom the unstable background.

Hierarchical gravitational fragmentation. I. Collapsing cores within collapsing clouds