In order to understand the collapse dynamics of observed low-mass starless cores, we revise the conventional stability condition of hydrostatic Bonnor-Ebert spheres to take internal motions into account. Because observed starless cores resemble Bonnor-Ebert density structures, the stability and dynamics of the starless cores are frequently analyzed by comparing to the conventional stability condition of a hydrostatic Bonnor-Ebert sphere. However, starless cores are not hydrostatic but have observed internal motions. In this study, we take gaseous spheres with a homologous internal velocity field and derive stability conditions of the spheres utilizing a virial analysis. We propose two limiting models of spontaneous gravitational collapse: the collapse of critical Bonnor-Ebert spheres and uniform density spheres. The collapse of these two limiting models are intended to provide the lower and the upper limits, respectively, of the infall speeds for a given density structure. The results of our study suggest that the stability condition sensitively depends on internal motions. A homologous inward motion with a transonic speed can reduce the critical size compared to the static Bonnor-Ebert sphere by more than a factor of two. As an application of the two limiting models of spontaneous gravitational collapse, we compare the density structures and infall speeds of the observed starless cores L63, L1544, L1689B, and L694-2 to the two limiting models. L1689B and L694-2 seem to have been perturbed to result in faster infall motions than for spontaneous gravitational collapse.
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