All Figures

$\begin{figure} \par\hspace*{-0.7mm}\includegraphics[width=7.85cm,clip]{5912fg1a.eps}\vspace*{2mm} \includegraphics[width=7.75cm,clip]{5912fg1b.eps}\end{figure}$ Figure 1: Upper panel: surface helium mass fraction in stellar models (Z=0.02) at core hydrogen exhaustion, as function of the initial stellar mass. Lower panel: logarithm of the surface nitrogen abundance divided by its initial value, at the end of core hydrogen burning, for the same models as shown in the upper part.
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In the text

$\begin{figure} \par\includegraphics[width=7.65cm,clip]{5912fg2a.eps}\vspace*{0.3cm} \includegraphics[width=7.65cm,clip]{5912fg2b.eps}\end{figure}$ Figure 2: Upper panel: specific angular momentum as a function of the mass coordinate in stellar models of sequences TA1 (dashed), TA2 (dashed-dotted), TA3 (dashed-two-dotted), and TA4 (dotted), during core neon burning. The thin solid line corresponds to the specific angular momentum profile on the zero-age main sequence, which is the same for all cases. Lower panel: same as in the upper panel, but for sequences A30f0.3h (thick solid), TB1 (dashed), TB2 (dashed-dotted), and TB3 (three-dotted-dashed).
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In the text

$\begin{figure} \par\includegraphics[width=7.05cm,clip]{5912fg3a.eps}\hspace*{1cm... ....eps}\hspace*{1cm} \includegraphics[width=7.05cm,clip]{5912fg3d.eps}\end{figure}$ Figure 3: Final fate of our rotating massive star models at four different metallicities (Z= 0.004, 0.002, 0.001, & 0.00001), in the plane of initial mass and initial fraction of the Keplerian value of the equatorial rotational velocity. The solid line divides the plane into two parts, where stars evolve quasi-chemically homogeneous above the line, while they evolve into the classical core-envelope structure below the line. The dotted-dashed lines bracket the region of quasi-homogeneous evolution where the core mass, core spin and stellar radius are compatible with the collapsar model for GRB production (absent at Z=0.004). This GRB production region is divided into two parts, where GRB progenitors are WN or WC/WO types. To both sides of the GRB production region for Z = 0.002 and 0.001, black holes are expected to form inside WR stars, but the core spin is insufficient to allow GRB production. For Z = 0.00001, the pair-instability might occur to the right side of the GRB production region (see Heger et al. 2003), although the rapid rotation may shift the pair instability region to larger masses. The dashed line in the region of non-homogeneous evolution separates type II supernovae (SN II; left) and black hole (BH; right) formation, where the minimum mass for BH formation is simply assumed to be $30~{M_\odot}$ (see, however, Heger et al. 2003, for a comprehensive discussion on the issue).
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In the text

$\begin{figure} \par\includegraphics[width=8.05cm,clip]{5912fg4.eps}\end{figure}$ Figure 4: Top: Kelvin-Helmholtz time ( $t_{\rm KH}$ ) of non-rotating stars at zero-age main sequence as a function of initial mass, at different metallicities as indicated by the labels. Middle: evolutionary time for core hydrogen burning ( $t_{\rm MS}$ ) of non-rotating stars as a function of initial mass. Bottom: the ratio $t_{\rm KH}/t_{\rm MS}$ multiplied by 1000, as a function of initial mass.
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In the text

$\begin{figure} \par\includegraphics[width=7.85cm,clip]{5912fg5a.eps}\vspace*{1mm} \includegraphics[width=7.85cm,clip]{5912fg5b.eps}\end{figure}$ Figure 5: Upper panel: cumulative distribution of the fraction of the Keplerian value of the observed rotational velocity (i.e., $v\sin i$ ) of unevolved young stars in NGC 346 in small Magellanic clouds. The data are from Mokiem et al. (2006). The dotted-dashed, solid, and dashed lines are the best fits of synthesized distribution functions using three different distribution laws: beta, gamma and Maxwellian, respectively. Here we assume that stellar rotation axes are randomly oriented. Lower panel: the corresponding probability density function, as given by $P_{\rm Beta}(x; 0\le x\le1) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}(1-x)^{\beta-1}x^{\alpha-1}$ with $\alpha =1.25$ and $\beta =4.95$ (Beta distribution), $P_{\rm Gamma}(x; x\ge0) = \frac{\lambda^\nu}{\Gamma(\nu)}x^{\nu-1}\exp(-\lambda x)$ with $\lambda =9.95$ and $\nu =2$ (Gamma distribution), and $P_{\rm Maxwell}(x; x\ge0) = 4\pi x^2 (x_{\rm m})^{-3/2} \exp(-x^2/x_{\rm m}^2)$ with ${v}_{\rm m} = 0.1195$ (Maxwellian). Here $\Gamma (x)$ denotes the gamma function.
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In the text

$\begin{figure} \par\includegraphics[width=7.4cm,clip]{5912fg6.eps}\end{figure}$ Figure 6: The predicted number ratios of GRB progenitors over all massive stars ( $8~{M_\odot} < M < 100~{M_\odot}$ ) as a function of metallicity, obtained by folding the three different adopted distributions of ${v}_{\rm init}/{v}_{\rm K}$ as given in Fig. 5 with the results of the stellar evolution grids as displayed in Fig. 3. The connecting lines are polynomial fits.
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In the text

$\begin{figure} \par\includegraphics[width=7.55cm,clip]{5912fg7.eps}\end{figure}$ Figure 7: Ratio of GRB versus core collapse supernova rate as a function of redshift, according to our GRB progenitor models. Note that the plotted ratio is independant of the adopted star formation history.
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In the text

$\begin{figure} \par\includegraphics[width=7.25cm,clip]{5912fg8a.eps}\vspace*{3mm} \includegraphics[width=7.25cm,clip]{5912fg8b.eps}\end{figure}$ Figure 8: Upper panel: perceived supernova and GRB rate as function of redshift on an arbitrary scale, according to our GRB progenitor models, and for the specified cosmic metallicity evolution. The GRB rate is multiplied by a factor of 187.48, which is the perceived average ratio of SNe to GRBs in the universe, according to our models. Lower panel: perceived cumulative number of SNe and GRBs as function of redshift. The GRB number has been multiplied by a factor of 187.48. The Y-scale is arbitrary but the same for both curves.
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In the text

$\begin{figure} \par\includegraphics[width=7.3cm,clip]{5912fg9.eps}\end{figure}$ Figure 9: Gray-shaded contour map (in color in the electronic version) of the perceived GRB rate with an arbitrary normalization, according to our GRB progenitor models and for the specified cosmic metallicity evolution, in the plane of redshift and metallicity.
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In the text

$\begin{figure} \par\hspace{-0.7mm}\includegraphics[width=7.85cm,clip]{5912fg1a.eps}\vspace{2mm} \includegraphics[width=7.75cm,clip]{5912fg1b.eps}\end{figure}$	Figure 1: Upper panel: surface helium mass fraction in stellar models (Z=0.02) at core hydrogen exhaustion, as function of the initial stellar mass. Lower panel: logarithm of the surface nitrogen abundance divided by its initial value, at the end of core hydrogen burning, for the same models as shown in the upper part.
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$\begin{figure} \par\includegraphics[width=7.4cm,clip]{5912fg6.eps}\end{figure}$	Figure 6: The predicted number ratios of GRB progenitors over all massive stars ( $8~{M_\odot} < M < 100~{M_\odot}$ ) as a function of metallicity, obtained by folding the three different adopted distributions of ${v}_{\rm init}/{v}_{\rm K}$ as given in Fig. 5 with the results of the stellar evolution grids as displayed in Fig. 3. The connecting lines are polynomial fits.
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$\begin{figure} \par\includegraphics[width=7.55cm,clip]{5912fg7.eps}\end{figure}$	Figure 7: Ratio of GRB versus core collapse supernova rate as a function of redshift, according to our GRB progenitor models. Note that the plotted ratio is independant of the adopted star formation history.
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$\begin{figure} \par\includegraphics[width=7.3cm,clip]{5912fg9.eps}\end{figure}$	Figure 9: Gray-shaded contour map (in color in the electronic version) of the perceived GRB rate with an arbitrary normalization, according to our GRB progenitor models and for the specified cosmic metallicity evolution, in the plane of redshift and metallicity.
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