A study with spherical symmetry can certainly not address all aspects of the star formation process, as magnetic fields and angular momentum are ignored, but it has the advantage that the process can be studied by keeping close to basic physical principles-at least as close as stellar evolution theory, with the notorious problem of the convection model.

Based on the concept of "maximum deduction at minimum assumptions'', spherical symmetry provides a useful limiting and reference case. With the symmetry assumption we are able to keep a close and explicit relation to the fundamental conservation laws and other basic physical principles. The essential modeling assumption that goes into this work then is the ever present problem of the convection model. But we have tried not to add further ambiguity that is not already present in the modeling of stellar evolution. As a consequence of our convection model the system of equations we use represents the structure of the Sun correctly. The number of free parameters in any study of star formation cannot be made smaller than in ours unless a parameter-free description of convection becomes available.

The energetics of the star formation process can be addressed rather completely, most notably because all relevant energy transfer processes, i.e., radiation, convection, advection and conduction can be described accurately. Very likely star formation is most efficient with zero angular momentum and hence a lower limit for star formation time and upper limits for luminositites and accretion-rates are obtained.

Table 3: Duration and averages of accretion rates and luminosities for different accretion phases of cloud fragments of 0.05, 0.1, 0.5, 1 and $2~ M_\odot$ . The phases are defined by ranges in optical thick mass $M_\tau$ in units of the final mass $M_{\rm final}$ , in the second column. For each phase and fragment mass, M₀, duration, $\Delta t = \min({\rm Age})-\max({\rm Age})$ , mean mass-accretion rates, ${<\dot{M}>} = (\min(M_\tau)-\max(M_\tau))/\Delta t$ , and the median value for the luminosities, <L>, is given.
Averages for 4 accretion phases separated by $M_\tau/M_{\rm final}=0,0.8,0.9,0.99$ and 0.999 (cf. Fig. 4).

M₀ 0.05 $M_\odot$ 0.1 $M_\odot$ 0.5 $M_\odot$ 1 $M_\odot$ 2 $M_\odot$

$M_\tau$ $\Delta t$ $<\dot{M}>$ <L> $\Delta t$ $<\dot{M}>$ <L> $\Delta t$ $<\dot{M}>$ <L> $\Delta t$ $<\dot{M}>$ <L> $\Delta t$ $<\dot{M}>$ <L>

$[M_{\rm final}]$ $\rm kyr$ $M_\odot$ /yr $L_\odot$ $\rm kyr$ $M_\odot$ /yr $L_\odot$ $\rm kyr$ $M_\odot$ /yr $L_\odot$ $\rm kyr$ $M_\odot$ /yr $L_\odot$ $\rm kyr$ $M_\odot$ /yr $L_\odot$

$\rm0$ $\le$ 0.8 16 $2.5 \times 10^{-6}$ 1.28 30 $2.7\times 10^{-6}$ 2.92 126 $3.2\times 10^{-6}$ 12.4 189 $4.2 \times 10^{-6}$ 19.5 496 $3.2\times 10^{-6}$ 34.1

$\rm I$ (0.8,0.9] 6 $8.1 \times 10^{-7}$ 1.19 11 $8.8\times 10^{-7}$ 2.63 52 $9.8\times 10^{-7}$ 8.21 85 $1.2 \times 10^{-6}$ 15.3 200 $1.0\times 10^{-6}$ 27.3

$\rm II$ (0.9,0.99] 16 $2.7 \times 10^{-7}$ 0.51 32 $2.7 \times 10^{-7}$ 0.97 144 $3.0\times 10^{-7}$ 4.60 280 $3.2 \times 10^{-7}$ 8.22 592 $3.0\times 10^{-7}$ 13.9

$\rm III$ (0.99,0.999] 15 $2.9 \times 10^{-8}$ 0.21 28 $3.0\times 10^{-8}$ 0.33 139 $3.2\times 10^{-8}$ 3.03 264 $3.4 \times 10^{-8}$ 4.69 554 $3.3\times 10^{-8}$ 8.42

With our assumptions we can describe the pre-main sequence evolution as a consequence of the star formation process, more precisely as a result of the collapse of Bonnor-Ebert spheres. All "stellar'' masses appear at locations in the HRD that are different from the ones predicted by classical, fully hydrostatic calculations. In particular, they do not start at the top of a classical, initially fully convective track or at the birthline. Those differences cannot be attributed to differences in energy transfer treatment or differences in microphysics since we have constructed our model equations in a way that assures a "stellar structure'' limit, that is approached during pre-main sequence contraction and that coincides with the assumptions made in our classical fully hydrostatic reference-study.

**Table 3:** Duration and averages of accretion rates and luminosities for different accretion phases of cloud fragments of 0.05, 0.1, 0.5, 1 and $2~ M_\odot$ . The phases are defined by ranges in optical thick mass $M_\tau$ in units of the final mass $M_{\rm final}$ , in the second column. For each phase and fragment mass, M₀, duration, $\Delta t = \min({\rm Age})-\max({\rm Age})$ , mean mass-accretion rates, ${<\dot{M}>} = (\min(M_\tau)-\max(M_\tau))/\Delta t$ , and the median value for the luminosities, <L>, is given.
Averages for 4 accretion phases separated by $M_\tau/M_{\rm final}=0,0.8,0.9,0.99$ and 0.999 (cf. Fig. 4).
M₀		0.05 $M_\odot$	0.1 $M_\odot$	0.5 $M_\odot$	1 $M_\odot$	2 $M_\odot$
	$M_\tau$	$\Delta t$	$<\dot{M}>$	<L>	$\Delta t$	$<\dot{M}>$	<L>	$\Delta t$	$<\dot{M}>$	<L>	$\Delta t$	$<\dot{M}>$	<L>	$\Delta t$	$<\dot{M}>$	<L>
	$[M_{\rm final}]$	$\rm kyr$	$M_\odot$ /yr	$L_\odot$	$\rm kyr$	$M_\odot$ /yr	$L_\odot$	$\rm kyr$	$M_\odot$ /yr	$L_\odot$	$\rm kyr$	$M_\odot$ /yr	$L_\odot$	$\rm kyr$	$M_\odot$ /yr	$L_\odot$
$\rm0$	$\le$ 0.8	16	$2.5 \times 10^{-6}$	1.28	30	$2.7\times 10^{-6}$	2.92	126	$3.2\times 10^{-6}$	12.4	189	$4.2 \times 10^{-6}$	19.5	496	$3.2\times 10^{-6}$	34.1
$\rm I$	(0.8,0.9]	6	$8.1 \times 10^{-7}$	1.19	11	$8.8\times 10^{-7}$	2.63	52	$9.8\times 10^{-7}$	8.21	85	$1.2 \times 10^{-6}$	15.3	200	$1.0\times 10^{-6}$	27.3
$\rm II$	(0.9,0.99]	16	$2.7 \times 10^{-7}$	0.51	32	$2.7 \times 10^{-7}$	0.97	144	$3.0\times 10^{-7}$	4.60	280	$3.2 \times 10^{-7}$	8.22	592	$3.0\times 10^{-7}$	13.9
$\rm III$	(0.99,0.999]	15	$2.9 \times 10^{-8}$	0.21	28	$3.0\times 10^{-8}$	0.33	139	$3.2\times 10^{-8}$	3.03	264	$3.4 \times 10^{-8}$	4.69	554	$3.3\times 10^{-8}$	8.42

The ages obtained with collapse models differ significantly, i.e., up to 1 Myr, from the ages derived from classical calculations. This is only in part due to the fact that the time needed for collapse and accretion are not accounted for in classical PMS calculations. The primary reason is that the contraction behaviour of the young stars resulting from the collapse differs from the one obtained for the usually assumed fully convective, isentropic, or polytropic initial structures. The fact that deuterium is almost completely burnt during the main accretion phase, i.e., before the PMS is reached for all "truly'' stellar masses calculated here (the $0.1~{M_\odot}$ case is the exception) also contributes to the difference. Hence calculations of early stellar evolution depend on the star formation process. Initial conditions are not forgotten in general. Another way to summarize this is that the star formation time, in our study about 3-4 free-fall times in all cases, is not sufficiently different from the Kelvin-Helmholtz time-scale of young objects to separate initial energy deposition into the star during the accretion phase from energy losses that control the contraction to the main sequence. As a consequence, protostellar collapse determines the PMS evolution.

This effect is enhanced by the off-centre ingnition of D burning and its interaction with the thermal structure. Unlike during most of stellar evolution it is not only controlled by the existing thermal structure but helps also to conserve the global thermal structure outside thermal equilibrium. This differs from the usual behaviour close to complete equilibrium, e.g., of main sequence stars, where nuclear reactions are adjusted to the thermal needs of the star and re-establish global thermal equilibrium.

Our study re-emphasises the fact that star formation is an intrinsically time dependent process, with variations in all flow quantities. We have tried to tentatively group the young population into four different evolutionary phases based on the fraction of the residual circumstellar mass around them. We gave characteristic quantities for those phases showing that mass accretion rates vary significantly over the main accretion phase. Accretion rates are typically a few $10^{-6}~ M_\odot/$ yr in the earliest phase, a few $10^{-7}~ M_\odot/$ yr for the last 10% and a few $10^{-8}~ M_\odot/$ yr for the accretion of the last percent of the final mass. Those values are not too far from what is currently inferred from observations (e.g., Brown & Chandler 1999). Also our star formation times of 0.07 to $1.3~\rm Myr$ for 0.1 to $2~ M_\odot$ are consistent with estimates from a comparison of T Tauri star- and embedded source counts (Kenyon et al. 1990). Our median luminosities between about 0.3 to 10 $L_\odot$ , depending on accretion phases, for cloud fragment masses from 0.01 to $0.5 ~{M_\odot}$ , make it much harder to construct a "Class I luminosity problem'' than with the assumed constant, canonical mass accretion rate of $10^{-5}~M_\odot/\rm yr$ , that is in conflict with observations (Brown & Chandler 1999). Let us finally summarize our main points: (1) the star formation processes, i.e., the protostellar collapse translates initial cloud conditions into initial stellar thermal structure, (2) pre-main sequence evolution depends qualitatively and quantitatively on the collapse flows; (3) D burning ignites and mostly occurs during the accretion phase for cloud fragments of 0.5, 1, 2, and $10~{M_\odot}$ ; (4) we confirm that protostars and PMS objects have an outer convective shell and a central radiative zone; (5) the radiative core lasts at least to a (new) age of 1.5 Myr for cloud fragments of 0.5, 1, and $2~ M_\odot$ and we expect off-centre ignition of hydrogen burning; (6) collapse pre-main sequence tracks are initially cooler than classical tracks. Later they cross the Hayashi line rapidly and start their quasi-classical decent along a mixed radiative/convective contraction track. After the final stellar photosphere becomes visible they appear to the left of the Hayashi lines. Typically at 0.05 dex hotter effective temperatures than the corresponding classical, fully hydrostatic pre-main sequence model; (7) already during the early pre-main sequence phase the solar mass case is roughly homologous to the Sun with its radiative core rather than to a fully convective star on the Hayashi track; (8) collapse ages below 2 Myr are up to a Myr older than classical ages for the same luminosity.

5 Conclusions