4 Global results

To facilitate statistical comparison with observations of star formation and to have values for the factors of order unity in scaling arguments we give some quantities averaged over the star formation time for each fragment in Tables 1 and 2. Ages at $M_\tau(t)=0.99~ M_{\rm final}$, $t_{\rm0.99M}$ are given in units of Myr and the initial free-fall time; mean mass-accretion rates ${<\dot{M}>} = M_\tau(t) / {\rm Age}(t)$ are given in units of $M_\odot /\rm yr$ and $M_{\rm final} / t_{\rm ff}$; M₀ is the fragment mass. For the luminosities the median values <L> are given in solar units.

 

 

Table 1: Duration of accretion and averages of accretion rates and luminosities for the accretion time lasting from zero age to the end of significant mass accretion. For the end of significant mass accretion we use the moment when the young star has reached an optically thick mass $M_\tau$ of 99% of its final mass. The different $1~{M_\odot}$ cases are for different mixing length parameters.

Accretion time duration and averages
M0	$\rm t_{0.99 M}$		$<\dot{M}>$		<L>	Remark
$M_\odot$	$\rm Myr$	$t_{\rm ff}$	$M_\odot /\rm yr$	$M/t_{\rm ff}$	$L_\odot$
1	0.55	3.23	$1.79 \times 10^{-6}$	0.30	13.1	RC1M
1	0.60	3.54	$1.65 \times 10^{-6}$	0.28	14.1	RCML15
1	0.55	3.26	$1.79 \times 10^{-6}$	0.30	12.8	RCML14 $\alpha _{\rm ML}=1.4$
1	0.54	3.18	$1.84 \times 10^{-6}$	0.31	11.9	RCML12 $\alpha_{\rm ML}=1.2$
0.05	0.038	4.51	$1.30 \times 10^{-6}$	0.22	1.02	RC005M
0.1	0.074	4.34	$1.35 \times 10^{-6}$	0.23	2.21	RC01M
0.5	0.32	3.81	$1.56 \times 10^{-6}$	0.26	7.58	RC05M
1	0.55	3.23	$1.79 \times 10^{-6}$	0.30	13.0	RC1M
2	1.28	3.64	$1.54 \times 10^{-6}$	0.27	25.1	RC2M

Quantities are averaged over a time starting at zero age and ending at the end of significant mass-accretion. The end of significant mass accretion is defined as the instant at which 99% of the final mass has been accreted, more precisely when $M_\tau = 0.99~ M_{\rm final}$, cf. the respective thin line in Fig. 4. To indicate the sensitivity on the definition of the end of accretion we give a few averages obtained for the $1~{M_\odot}$ case for periods ending with $M_\tau$ closer to the final mass in Table 2.

 

 

Table 2: Duration of accretion and averages of accretion rates and luminosities for the accretion time lasting from zero age to the end of significant mass accretion. For the end of significant mass accretion we use different fractions $M_\tau /M_{\rm final}$ to show how averages change for the $1~{M_\odot}$ case. The mean mass accretion rate for the $10~{M_\odot}$ fragment corresponds to $0.89 M/t_{\rm ff}$ in the first 0.7 Myr ($\sim$ $0.4 ~ t_{\rm ff}$).

Accretion time and averages for different end of accretion
and a $10~{M_\odot}$ fragment.
$M_\tau$	$\rm Age$	$<\dot{M}>$	M0	$M_{\rm final}$	Remark
$M_\odot$	$\rm Myr$	$M_\odot$/yr	$M_\odot$	$M_\odot$
0.99	0.60	$1.67 \times 10^{-6}$	1	1	RCML15
0.99	0.55	$1.80 \times 10^{-6}$	1	1	RC1M
0.999	0.82	$1.22 \times 10^{-6}$	1	1	RC1M
0.9999	1.08	$9.26 \times 10^{-7}$	1	1	RC1M
0.99999	1.29	$7.78 \times 10^{-7}$	1	1	RC1M
	0.70	$7.11 \times 10^{-7}$	10	4.46	RC10M

To indicate the effects due to the time-dependence of the accretion flow and, in particular, due to the temporal variation of $\dot{M}$, we split the star formation time into 4 phases, based on $M_\tau /M_{\rm final}$. By this procedure we obtain 4 different populations corresponding to different stages of the accretion process. For each of the 4 phases we give averaged quantities, for fragment masses, M₀, of 0.05, 0.1, 0.5, 1 and $2~ M_\odot$, in Table 3.