2 A model for protostellar collapse and the pre-main sequence

2.1 Flow equations

We use the equations of radiation hydrodynamics (RHD) in the grey Eddington approximation and with spherical symmetry (Castor 1972; Mihalas & Mihalas 1984) in their integral form (Winkler & Norman 1986). Convective energy transfer and mixing is included by using a new, time-dependent convection model derived from the model of Kuhfuß (1987). It closely approximates standard mixing length theory in a local, static limit and has been successfully tested in calculations of the Sun and RR Lyrae pulsations (Wuchterl & Feuchtinger 1998; Feuchtinger 1999a,b). The convection model is used with the standard parameters given in Wuchterl & Feuchtinger (1998). The few cases where the mixing length parameter $\alpha _{\rm ML}$ is changed are explicitly emphasised. In all other cases the standard value $\alpha _{\rm ML} = 1.5$ is used, which results in a solar model that is in accord with standard solar models and the Sun for our "stellar structure limit''. The equations and boundary conditions are summarized in Table .1, in the Appendix. Detailed equations of state (Wuchterl 1989, 1990) and opacities for a (proto) solar composition are used (X = 0.77, Y = 0.213, Z = 0.017, $X_{\rm D} = 3.85\times 10^{-5}$). The equations of state have been compared to the MHD equations of state (Mihalas et al. 1988) and agree to better than the table-interpolation errors (cf. Götz 1993). Opacities include the contribution of dust particles with "interstellar'' properties (Yorke 1979, 1980a), Alexander & Ferguson (1994) values in the molecular range and Weiss et al. (1990) Los Alamos high temperature opacities (compilation by Götz 1993). Convective mixing of deuterium and deuterium burning (with reaction rates from Caughlan & Fowler 1988) are included as outlined in the next section.

2.2 Deuterium burning

During early stellar evolution the first nuclear reaction that contributes significantly to the energy budget of a young star is the D burning reaction $\rm {}^2H(p,\gamma){}^3He$ (e.g., Kippenhahn & Weigert 1990). The energy release of $5.5 \rm ~ MeV$ per reaction contributes as a source $\epsilon_{\rm nuc}^{\rm D}$ to our internal energy equation (cf. Table .1). The proton partial density, $\varrho _{\rm p}$, is approximated by the hydrogen partial density, $X \rm\varrho$. Once D becomes significantly depleted, it has to be budgeted by the D balance equation that accounts for D destruction and transfer from regions of different D abundance due to advection and the D flux due to convective mixing, $j_{\rm D}$, (see, Table .1, Eq. (3)). Rate coefficients, $\left<\sigma v\right>$ for $\rm {}^2H(p,\gamma){}^3He$ are taken from Caughlan & Fowler (1988). D burning via $\rm {}^2H(p,\gamma){}^3He$ is the only nuclear reaction that we take into account. As a consequence, our study is limited to evolutionary phases before the onset of hydrogen burning and does not address the evolution of abundances of trace species like $\rm {}^7Li$ and $\rm Be$.

It is assured that our extensions to the system of stellar structure equations do have a fixed relation to standard stellar structure and stellar hydrodynamics. After requiring a correct solar convection zone in the "stellar structure limit'', the equations we use do not contain any additional free parameters and the formation and evolution of a star is determined by the properties of the initial cloud fragment alone. The star formation theory is fully specified with the specification of the initial cloud state and the boundary conditions. See Appendix A for the equations.