==Introduction==

Various properties of the “Bonnor-Ebert” (BE) sphere, a hydrostatic sphere in pressure equilibrium with its ambient environment, make it a good candidate for numerical modeling of protostellar collapse. First, as a candidate star forming structure is envisaged as gravitationally bound and unstable, it is easy to imagine a protostar evolving from an initially hydrostatic configuration. Indeed, spherical clumps have been observed in or near hydrostatic equilibrium, such as the Bok Globule, B68 (Myers). Second, the stability criterion against gravitational collapse has been worked out analytically. Third, pushing the sphere out of the stability regime with various physical perturbations illuminate collapse characteristics. Such features of the collapse may help advance single star formation theory as well as provide clues to observational astronomers in identifying potential star forming sites.

While the collapse of a BE sphere has been studied extensively over the years, the literature reveals studies of the BE sphere in precarious and unphysical conditions. The BE sphere has largely been modeled as residing in low density ambient mediums, so as to seemingly isolate the collapse of the sphere from the environment. {As BE spheres are differentiated by both a truncation radius and an external pressure bounding the sphere, naturally these earlier studies forced pressure equilibrium at the spheres outer boundary}. In this way, previous models have been physically unrealistic; any actual collapsing cloud of gas would not be discontinuous from the ambient medium. However, modeling the cloud in this way did reveal a feature of the collapse that was not expected.

Shu’s model, the other models, F&C even went far with the different xis - still no inside out. But while this was helpful, the next step needed to be taken. (And were taken - review Myers and Hannabelle) That was the aim of this paper. We have found that the collapse acquires a “crushing solution”, in the matched ambient limit. In contrast, when the limit of light ambient material (on the order of previous studies) was approached, the outside-in collapse was recovered. The difference in these collapses can be understood in terms of an accretion time -scale. In a light ambient, the material accretes, and the sphere re-equilibrates, until it can’t hold itself up anymore under the “weight”. In a higher density ambient, the accretion time is < equilibration time, so the sphere doesn’t have time to equilibrate. Thus the sphere succumbs to a compression induced collapse.

Check F&C did various different xi’s

Although modeling the collapse in this way does illuminate unique features of the BE collapse, it inadvertently constrains the model from being realistic. If the BE sphere might be considered as an early basic model of star formation, it should be examined in a more physically plausible setting. That is, the BE sphere should be tied to its parent cloud as it would be in nature, by being modeled with no discontinuous boundary between BE sphere and parent clump. The collapse properties of such a simulation would be more physically accurate to any actual situation involving the collapse of a protostellar BE sphere.