  Low-mass stars are generally understood to form by the gravitational collapse
of the dense molecular clouds known as starless cores. Continuum observations
have not been able to distinguish among the several different hypotheses that
describe the collapse because the predicted density distributions are the
almost the same, as they are for all spherical self-gravitating clouds.
However, the predicted contraction velocities are different enough that the
models can be discriminated by comparing the velocities at large and small
radii. This can be done by observing at least two different molecular line
transitions that are excited at different densities. For example, the spectral
lines of the H2O (110 - 101) and C18O (1-0) have critical densities for
collisional de-excitation that differ by 5 orders of magnitude. We compare
observations of these lines from the contracting starless core L1544 against
the spectra predicted for several different hypothetical models of contraction
including the Larson-Penston flow, the inside-out collapse of the singular
isothermal sphere, the quasi-equilibrium contraction of an unstable
Bonnor-Ebert sphere, and the non-equilibrium collapse of an over-dense
Bonnor-Ebert sphere. Only the model of the unstable quasi-equilibrium
Bonnor-Ebert sphere is able to produce the observed shapes of both spectral
lines. This model allows us to interpret other observations of molecular lines
in L1544 to find that the inward velocities seen in observations of CS(2-1) and
N2H+ are located within the starless core itself, in particular in the region
where the density profile follows an inverse square law. If these conclusions
were to hold in the analysis of other starless cores, this would imply that the
formation of hydrostatic clouds within the turbulent interstellar medium is not
only possible but not exceptional and may be an evolutionary phase in low-mass
star formation.
