Semi-analytical homologous solutions of the gravo-magnetic contraction

We propose an extension of the semi-analytical solutions derived by Lin et al. (1965) describing the two-dimensional homologous collapse of a self-gravitating rotating cloud having uniform density and spheroidal shape, which includes magnetic field (with important restrictions) and thermal pressure. The evolution of the cloud is reduced to three time dependent ordinary equations allowing to conduct a quick and preliminary investigation of the cloud dynamics during the precollapse phase, for a wide range of parameters. We apply our model to the collapse of a rotating and magnetized oblate and prolate isothermal core. Hydrodynamical numerical simulations are performed and comparison with the semi-analytical solutions is discussed. Under the assumption that all cores are similar, an apparent cloud axis ratio distribution is calculated from the sequence of successive evolutionary states for each of a large set of initial conditions. The comparison with the observational distribution of the starless dense cores belonging to the catalog of Jijina et al. (1999) shows a good agreement for the rotating and initially prolate cores (aspect ratio $\simeq 0.5$) permeated by an helical magnetic field ($\simeq 17-20 \mu$G for a density of $\simeq 10^4$ cm$^{-3}$).