1 Introduction

One important aim of star formation theory, which is still an unsolved problem, is to provide the initial conditions for stellar evolution, i.e., the masses, radii and the internal structure of young stars as soon as they can be considered to be in hydrostatic equilibrium for the first time.

Figure 1: Early stellar evolution in the Hertzsprung-Russell diagram: Comparison of results for the collapse of a $1~{M_\odot}$, "Bonnor-Ebert'' sphere (thick line) and a classical pre-main sequence track ( thin line). Both are for mixing length convection, with $\alpha _{\rm ML}=1.4$. The classical PMS-track (thin line on the left) is the $1~{M_\odot}$, "MLT Alexander'', case after D'Antona & Mazzitelli (1994). The thick line for the cloud collapse gives the effective temperature as defined by $L = 4 \pi r_\tau ^2 \sigma T_{\rm eff}^4$. $r_\tau$ is the radius at which the Rosseland mean optical depth, $\tau _{\rm Ross}$ as integrated inward from the outer boundary of the cloud equals 2/3. Alternative temperatures are given for the cloud collapse results: the dashed line is the temperature defined by $L = 4 \pi r^2 \sigma T^4$, at the first point encountered inward starting from the outer boundary of the cloud, i.e., based on the temperature radius as discussed by Baschek et al. (1991). The dotted line gives the temperature at $\tau _{\rm Ross} = 2/3$, an estimate for the photospheric temperature. Note that the photospheres obtained in the collapse calculations are convective at this point and $T=T_{\rm eff}$ is not guaranteed. The two diamonds indicate zero age, defined here as the moment when $\tau _{\rm Ross} = 2/3$ for the first time. Temperatures before zero age (on the dash dotted curve) are central temperatures of the still transparent cloud. The alternative temperatures are given here to illustrate the uncertainties involved in putting a collapsing cloud fragment into the HRD and to emphasise different physical aspects.

Following the evolution of a collapsing cloud fragment requires self-gravity, a description of the (supersonic, compressible) motion of the cloud gas, radiative transfer in extended, non-planeparallel media of low, moderate and very high optical depth, and, sooner or later in the non-isothermal phase, a time-dependent treatment of convection.

We account for all of the mentioned processes that are of importance in clouds and stars. However, we assume spherical symmetry and the grey Eddington approximation for radiative transfer to keep the problem as simple as possible, since this is the first look at the problem of cloud collapse, star formation and early stellar evolution that is based on the solution of one set of equations for all evolutionary stages. In the next section we assemble this system of equations that contains all the physics described above.

To discuss the results and illustrate the differences we focus on the comparison with classical, i.e., fully hydrostatic results that are obtained with model equations (stellar evolution equations) being as close as possible to the "stellar structure limit'' of our set of equations. Comparison of our results with studies that include more physical processes (e.g., disc accretion or frequency dependent photospheric radiative transfer) can then be made by using existing intercomparisons of different hydrostatic studies to our hydrostatic reference study.

To relate our results to quasi-hydrostatic stellar evolution calculations on the pre-main sequence we chose the calculations by D'Antona & Mazzitelli (1994), DM94, as our reference for the following reasons: (1), the photospheric radiative transfer in DM94 is an almost accurate limit of our grey radiative transfer in the Eddington approximation in its "stellar structure limit''. We thereby exclude differences between our calculations and the quasi-hydrostatic calculations which might be due to effects that are specific to the details of the non-grey treatment of the photospheres. (2), the low temperature, molecular opacities for DM94, "Alexander'' sets of PMS-tracks are the same as used here. (3), standard mixing length convection, our "stellar structure limit'' for convection, is one of the convection theories used by DM94.