POPULATION III GAMMA-RAY BURSTS AND BREAKOUT CRITERIA FOR ACCRETION-POWERED JETS

We summarize numerical results in Table 3. Here, we define the shock breakout as that the forward shock wave successfully reaches the stellar surface. It should be noted, however, that shock breakout is a minimal requirement for producing GRBs. As we shall see, even if the outflow successfully accomplishes the breakout, some models are not suitable for GRBs (see the column "Possibility of GRB production" in Table 3). We assess the possibility of GRB production by the diagnostic terminal Lorentz factor Γ_dt ≡ h × Γ profiles on the z-axis at the time of the shock breakout. (Note that the diagnostic terminal Lorentz factor is the achievable Lorentz factor after the internal energy is converted into kinetic energy.) If there are regions where Γ_dt ⩾ 100 is satisfied, we determine that the model has a potential to produce GRBs. If this condition is not satisfied, the jet is not successfully injected or does not move forward. As a result, the explosion would never become relativistic enough. Note that some models are hard to judge for the possibility of GRB production because of several reasons (see the column "Possibility of GRB production" in Table 3 where we denote these models by ▵; see also the discussion in Section 3.6). The details of these models are presented in the following subsections. One of the remarkable results in this study is that many models successfully accomplish relativistic shock breakout even though the progenitor star has an extremely large envelope such as Pop III stars. These results are the numerical verification of Suwa & Ioka (2011). We will further analyze the dependencies on the efficiency η and the opening angle θ_op in the following section. Indeed, the jet dynamics strongly depend on these key parameters.

At the beginning of the simulations, the infall starts at the innermost regions and subsequently a rarefaction wave propagates outward. The accretion rate increases with time and then the injected energy generates the forward shock wave around the polar regions. It should be noted, however, that the forward shock wave does not propagate outward into the active computational regions for a while because of the interruption by the infalling matter. If the injected energy is not enough to push the infalling stellar mantle aside, the mantle advects inward and eventually it is swallowed into a black hole. As a result, the total amount of explosion energy becomes less than the injected energy. Furthermore, the binding energy by a black hole (and also the progenitor star itself) further reduces the explosion energy. We also calculate the diagnostic energy E_dg at the time of the shock breakout in each model and show them in the fifth row of Table 3. We define the diagnostic energy E_dg as the integral of _lc (local energy density), which is the sum of the internal, kinetic, and gravitational energy densities, over the regions with positive _lc and v^r. Note that E_dg is not the isotropic energy but collimation-corrected true energy. In the weak gravitational field limit, we can define _lc as

$$\begin{eqnarray} &&\epsilon _{{\rm lc}} = T^{{\rm tt}} - \rho _{0} \Gamma c^2 + \rho _{0} \psi , \end{eqnarray} \tag{ 4 }$$

where T^tt and ψ denote the time–time component of the energy momentum tensor and gravitational potential, respectively. Note that, for this definition of diagnostic energy density, gravity is treated as an external force. This estimation is valid only when the central core object takes the dominant role for gravity. If the outer envelope plays an important role for gravitational energy, we need to take into account self-gravity. So, the term of the gravitational energy should be contributed not in the form of ρ₀ψ but $\frac{1}{2} \rho _{0} \psi$ in Equation (4). However, since we use this value for diagnosing whether or not the outflow is relativistic in this study, the severe definition of this quantity is not necessary. Thus, we use Equation (4) in the present paper. As can be seen in Table 3, we find that the diagnostic energy for all models is less than about a third of the injected energy. Thus, it implies that the central engine has to produce a larger amount of energy than that observed as GRBs.

The total amount of explosion energy would be larger than the diagnostic energy E_dg at the breakout since the central engine is able to keep operating after the shock breakout due to the continuing accretion. Indeed, as we shall see in the following subsections, some models are active at the time of the breakout. Thus, although E_dg may be lower than the typical GRB energy in some models, the outflows would gain further energy from the central engine to create GRBs after the breakout.

When the injected energy is successfully launched from the inner boundary, some portions of matter bounce back and move outward. However, we cannot see a clear collimated outflow at the beginning of the simulations even if we inject dominant kinetic outflows in a small opening angle. This is due to the fact that the injected energy is thermalized by the strong reverse shock wave in the vicinity of the inner boundary. As a result, the hot matter expands and creates a quasi-spherical forward shock wave. Nevertheless, we find that the forward shock wave is not strong enough to stop the infall of matter around the equatorial region and the matter continues to accrete through the inner boundary (see, e.g., Figures 3 and 5). The jet structure emerges when the injected energy becomes enough to push away the reverse shock wave outside the computational region. Subsequent evolution is similar to the previous studies of jet propagation. The hot cocoons cause recollimation shock waves, and strong backflows also appear in the flows and they sometimes pinch the jet. The Kelvin–Helmholtz instability also works to create rich internal structures. The jet starts to propagate and the forward shock wave eventually breaks out of the stellar surface.

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As shown in Table 3, we find that the forward shock wave can break out if the accretion-to-jet conversion efficiency is η ≳ 10⁻⁴ (see more details in the Section 3.7). Figure 4 displays the time evolution of the radius for the forward shock wave on the z-axis for models with different efficiencies. As expected, the higher conversion efficiency generates a stronger shock wave, which quickly propagates into the stellar envelope. In the case of lower conversion efficiency models, however, it is harder to inject the jet. Even if the jet is successfully injected, it takes a longer time for the forward shock wave to reach the stellar surface than in the high-efficiency models.

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Figure 5 shows the time evolution of the accretion rate and luminosity in these models (note that we also display the results for the failed models without breakouts). At the initial phase, the higher efficiency models have slightly smaller accretion rates than the lower efficiency models because the strong outflows interrupt the infall of matter. Still, the high-efficiency models have higher jet luminosity than the low-efficiency models. As a result, the jet successfully gains energy from the central engine and accomplishes the shock breakout. It is also interesting to note that low-efficiency models often show rapid time variability of the accretion rate (see, e.g., WRef2e-4 model in the upper left panel in Figure 5). This is attributed to the fallback of the shocked envelope, which has rich internal structures. Although the envelope initially expands due to the deposited energy, they do not have enough energy to keep moving outward and they eventually fall back through the inner boundary, leading to the central engine activity.

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Figure 6 shows the map of regions with positive _lc and v^r at the time of the shock breakout. We can see that the shape of the yellow region, which contains both positive _lc and v^r (see Equation (4)), does not depend on the accretion-to-jet conversion efficiency η so much. Irrespective of η, the maximum transverse radii of the yellow region from the z-axis are ∼5 × 10⁹ cm, ∼2 × 10¹¹ cm, and ∼1.5 × 10¹² cm for the W-R, lpop3, and mpop3 models, respectively. It may imply that the outflow structure does not mainly depend on the efficiency η, as long as the shock breakout occurs (see also the next subsection). From this figure, we can also confirm that even the large efficiency jet does not expel all portions of the stellar mantle, and matter can continue to accrete onto the black hole. On the other hand, the inner parts of the envelope profiles depend on the efficiency η. As expected, a larger amount of matter is captured for the lower conversion efficiency. In addition, as discussed above, the fallback of matter causes the late time variability of the central engine.

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In Figure 7, we show Γ_dt profiles along the z-axis at the time of the shock breakout for each model. The model has the potential to produce a GRB if Γ_dt ⩾ 100 is satisfied in some region. Therefore, we use this condition to assess whether or not the model can create a GRB. We note that the reality is more complex as argued in the following. In Figure 7, the outer region tends to contain lower values of Γ_dt, even less than ∼10. The small Γ_dt is caused by the baryon pollution in the jet, most likely because of the lack of numerical resolution (see in Section 3.6). However, even if the baryon pollution were the real physical phenomenon, these outflows would create GRBs for the high-efficiency models (η = 10⁻³) because the central engine keeps operating after the shock breakout for these models (see discussions in Section 3.6). The inner fast-moving jets would eventually catch up with the slowly moving outer ejecta. Due to this energy input from the inner jets, it is quite likely that the actual terminal Lorentz factor of the outer outflows is larger than the values of the current estimation. Thus, we expect relativistic explosions in these models. However, it is less likely that the low-efficiency models such as lpop3ef-4, lpop3ef2-4, and mpop3ef2-4 can successfully gain energy from later jets. Even if the central engine keeps operating for a long time, they may not be able to accelerate outflows because the amount of the outer slow matter is large (the profile of Γ_dt deviates from the high-efficiency models) in these models. As a result, the energy of the jet is dissipated in the vicinity of the injection region without transferring the energy into the outer parts of the ejecta. According to this consideration, we regard that these low-efficiency models would end up with non-relativistic explosions and may not create GRBs (thus, we mark these models with triangles (▵) in the column "Possibility of GRB" in Table 3.) We note that if the reverse shock is stalled near the injection region at the time of the shock breakout, there is no hope for this reverse shock to go ahead afterward because the accretion rate is decreasing.

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Figure 8 shows the time evolution of the forward shock wave on the z-axis for models with different opening angles of the jet at the injection site. Generally, the forward shock wave propagates fast and succeeds in breaking out if the opening angle is small. It is mainly attributed to two reasons: one reason is that the isotropic flux is increased by the small cross-sectional area of the injected jet, and the other is the increment of the accretion rate. A wide opening angle tends to interrupt the accretions and reduce the jet luminosity (see Figure 9).

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Note that the overall dynamics for very wide opening angle jets (e.g., θ_op = 36 and 45°) are more complex than the collimated jets. As shown in Figure 8, the jet breakout time for θ_op = 45° models are (slightly) earlier than θ_op = 36° models in all types of progenitors. It is because the fallback process plays an important role for wider jets by enhancing the accretion rate. Indeed, for WRop45, the accretion rate reaches $\dot{M} \sim 1 \,M_{\odot }{\rm \,s^{-1}}$ at t = 45 s, and leads to the strong outflows from the inner boundary and finally to the shock breakout. Thus, the late time activity of the central engine is possible for a wide opening angle due to the strong fallback accretions, even though the mean accretion rate decreases with time in the late phase. (But these wide opening angle jets are not suitable to produce relativistic outflows; see below.)

In Figure 10, we display the time evolution of the forward shock wave for different angle radial rays from the symmetry axis. For reference models (left panels in this figure), the forward shock velocity decreases with increasing angle. Particularly, at the time of the shock breakout, we find that the forward shock waves along the off-axis radial ray are still deep inside the star, which means that the outflow is well collimated. On the other hand, for θ_op = 36° models, almost all models experience quasi-spherical evolutions. Although the shock wave on the z-axis is slightly faster than the off-axis shock waves, it is much more spherical than the reference model at the breakout. As a result, it is expected that the wide opening angle models never create GRBs in view of the spherical morphology of the outflows.

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Figure 11 is the same as Figure 6, but for models with different opening angles. We do not find any clear differences for narrow opening angle jet models (θ_op = 3° and 6°), so we do not display these models in this figure. As shown in this figure, the wider jet tends to expel a larger outer envelope, and clearly these profiles are different from each other. Also, we again see that very wide angle cases are very complicated. Interestingly, some fraction of stellar mantle around the equatorial region may never fall back to the black hole (see the θ_op ⩾ 36° models). From what has been discussed above, we can conclude that the outflow profile is mainly determined by the opening angle of the jet θ_op rather than the accretion-to-jet conversion efficiency η (see also Figure 6 and discussions in Section 3.2).

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The diagnostic terminal Lorentz factor profiles along the z-axis for different opening angle models are shown in Figure 12. For small opening angle models (θ_op ⩽ 9°), the diagnostic terminal Lorentz factor is very much not different (slightly larger value for wider opening angle). However, the profile of very wide models is completely different from narrow jet cases and the terminal Lorentz factor for the wide angle models is quite low. Thus, they are no longer capable of producing GRBs. Note that, although the jet of mpop3op36 model is successfully injected around the inner boundary (see in Figure 12), this outflow may not create GRBs since the outer ejecta is completely non-relativistic, Γ_dt ∼ 1, and the outflow configuration is not collimated. As a result, a small opening angle θ_op ≲ 20° is necessary for relativistic shock breakout and GRBs with the case η ∼ 10⁻³. Note that if the efficiency becomes lower than η = 10⁻³, the opening angle needs to be smaller than the current value, because it is harder for the low-efficiency jet to push aside the infall matter than the high-efficiency jet (see Section 3.7 for the analytic estimation of the breakout criteria).

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In this section, we discuss how the injection Lorentz factor and the jet injection timing affect the evolution of the jet. In Figure 13, we display time evolutions of some key quantities, such as the forward shock wave on the z-axis, the mass accretion rate, and the jet luminosity. As we can see, the injection Lorentz factor does not change the qualitative feature of the jet evolutions for all quantities as long as the final coasting Lorentz factor is the same (see the left panels in Figure 13).

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On the other hand, the jet dynamics depend on the timing of the jet injection (see the right panels in Figure 13). We can see that late injection leads to a slightly faster evolution than early injection, and also that the time evolution of the accretion rate and the jet luminosity are quite different in the early phase. This is because the accretion rate is already high at the time of the jet injection for the late injection. As a result, the strong jet is injected and evolves faster than the early injection case. It should be noted that although the late jet injection slightly interrupts the infall of the material (see the middle and right panels of Figure 13), it does not stop the accretion of all the infalling material. We also speculate that a wide opening angle would suppress the infall of matter more strongly, so that the collimated outflows are preferred for GRBs even in the late injection cases.

It is also interesting to investigate the diagnostic terminal Lorentz factor Γ_dt profile for the different timings of jet injection as displayed in Figure 14. As we can see in this figure, the late jet injection tends to have a higher diagnostic terminal Lorentz factor than the early jet injection. This is also because the jet luminosity for the later injection case is stronger than the earlier jet (see the bottom panel of Figure 13). The strong forward shock waves propagate outward and the outer envelope obtains a large amount of energy from them. Besides, we also find that the late jet injection tends to have a smaller amount of baryon mass in the jet since a portion of envelope matter has already been swallowed in the black hole. As a result, the outflow easily achieves relativistic velocity after the breakout (see Table 4 and Section 3.6 for more details). We again must caution that the artificial baryon pollution affects the distribution of the diagnostic terminal Lorentz factor and the amount of baryon mass. The detailed discussions of these issues are described in Section 3.6. Note also that too late an injection may not be suitable for GRBs, since the accretion rate becomes very low and centrifugal bounce takes place at a later time (see Nagakura et al. 2011). Although we do not know exactly the starting time of the central engine, we speculate that the most plausible starting timing for the central engine is the time when the accretion disk is formed around a black hole. It clearly depends on the angular momentum distribution of the progenitor star, so we will investigate this dependence by performing the rotational collapse of progenitor stars in a forthcoming paper (H. Nagakura et al. 2012, in preparation).

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Based on the above results, we construct representative models, which are expected to create GRBs in each progenitor. As we have seen, the favorable conditions for relativistic shock breakout are high conversion efficiency, small opening angle, and late jet injection. We summarize these parameters for representative models in Table 2. We denote these representative models as WRrepr for the W-R progenitor, lpop3repr for the light Pop III star, and mpop3repr for the massive Pop III star. The accretion-to-jet conversion efficiency is set as η = 10⁻² which corresponds to nearly the maximum conversion efficiency for a Schwarzschild black hole (see Section 2). The half-opening angle is set as θ_op = 9°. We start to inject the jet when the mass accretion rate becomes the largest for each progenitor. We also note that we carry out simulations with higher spatial resolutions (seven-level AMR) in order to suppress the artificial baryon pollutions (see in Section 3.6 for more details).

We summarize numerical results in Table 3. As you can see in this table, the forward shock wave propagates faster than the reference model. We also find that both the total amount of injection energy and diagnostic energy at the breakout time are larger than those of reference models. Figure 15 shows the Γ_dt distribution along the jet axis at the shock breakout for each model. As can be seen in this figure, every model produces more relativistic breakout than the corresponding reference models. In particular, it is interesting to note that mpop3repr succeeds with highly relativistic breakout in spite of a large massive envelope. Note that the outer ejecta for WRrepr and lpop3repr is still mildly relativistic (Γ_dt ∼ 30). This is most likely caused by the numerical baryon contamination. However, even if the baryon pollution were real, these outflows could accelerate relativistically and create GRBs for the same reason discussed in Section 3.2 (see also Section 3.6).

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In this subsection, we give some important cautions for the results presented in this paper. Although our simplification of numerical methods does not change the essence of our new findings, there are some technical limitations for the current numerical simulations. Here, we point out several limitations on the present work with some additional simulation results.

3.6.1. Baryon Pollution

Although our numerical code succeeds in capturing strong shock waves and complex turbulence in relativistic outflows, numerical diffusion is inevitably inherent in it. As a result, numerical diffusion potentially induces artificial baryon pollution which leads to a lower diagnostic terminal Lorentz factor in the jet. As we have already mentioned, since Γ_dt is important in determining whether or not the jet would eventually create a strong relativistic outflow, we need to know how the numerical resolution affects Γ_dt along the jet axis.

Figure 16 shows the comparison of Γ_dt distribution along the jet axis for models with different spatial resolutions. As you can see in this figure, models with higher resolution exhibit a larger Γ_dt. This confirms that higher resolution reduces the baryon pollution and shows the higher value of Γ_dt. In fact, the amount of baryon mass around the jet axis at the breakout time is decreasing with higher resolutions (see Table 4). The baryon mass contained in the jet is roughly estimated as

$$\begin{equation} M_{b} = 2 \pi (1 - \cos \theta _{{\rm op}}) \int _{r_{{\rm in}}}^{r_{{\rm out}}} \rho _z (r) r^2 dr, \end{equation} \tag{ 5 }$$

where ρ_z(r) denotes the density profile along the z-axis. We also estimate the average terminal Lorentz factor Γ_f for the jet as

$$\begin{equation} \Gamma _{f} = \frac{E_{j}}{M_{b} c^2 }, \end{equation} \tag{ 6 }$$

where E_j denotes the total injected energy after the breakout. We estimate E_j as

$$\begin{equation} E_{j} = \eta M_{{\rm bound}} c^2, \end{equation} \tag{ 7 }$$

where M_bound denotes the total mass of gravitationally bound matter (see also Equation (4) for the definition of bound matter). Here, we assume that all of energy from the central engine is successfully transferred to outgoing ejecta. As shown in Table 4, the outflow for WRref can accelerate Γ_f ∼ 60 although this model is heavily influenced by the baryon pollution. Thus, even if baryon pollution were occurring in reality, this model could create relativistic outflow. In addition, we would like to emphasize that the macroscopic jet evolution is not sensitive to the numerical resolution (see Figure 17).

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We also point out that the two-dimensional axisymmetric setup affects baryon pollution. In our simulations, the baryon pollution near the pole is mainly attributed by the numerical diffusion, since the meridian velocity near the pole is almost zero due to the axisymmetric property. However, in reality, non-axisymmetric motions and finite meridian velocity may happen in the jet region. In this case, there is a possibility that the cocoon or strong back flows thrust into jets, then a lot of baryons could mix with the jet matter. Note also that the dense "plug" at the head of the jet is an artifact of the axisymmetric property. In reality, the non-axisymmetric motions of jet would cause dispersion of the dense "plug" (Zhang et al. 2004).

3.6.2. Dependence on Inner Boundary Location

One of the other drawbacks in the present study is the choice of inner boundaries in current models. For all simulations, the inner boundary is located well outside the central core of the star. As a result, the mass accretion rate for each model would be different from the actual accretion rate in the vicinity of the black hole. In addition, the impact of feedback would also be changed. In order to remove these uncertainties, we demonstrate some numerical simulations for different positions of the inner boundary.

Figure 18 shows the dependence on the inner boundaries for the evolution of the forward shock wave on the z-axis and the jet luminosities. As shown in these panels, models with a smaller inner boundary imply that the forward shock wave propagates faster. Since the density is higher at the inner radius, the accretion rate consequently becomes larger so that the jet luminosity also becomes larger. In these panels, we find that the model lpop3in5e9 has a peak accretion rate of about 10 times that of model lpop3ref, while W-R and mpop3 simulations exhibit only a factor of two or three amplification, when the inner boundary is smaller by a factor of two. We also note that, for the mpop3 models, the initial enclosed mass at the inner boundary is 414 M_☉ and it corresponds to nearly half the total mass of the star. Thus, the location of the inner boundary substantially affects the jet dynamics especially for lpop3 and mpop3. In order to obtain more quantitative arguments for the actual jet luminosity and breakout time, we need simulations with a smaller inner boundary, which are beyond the scope of this paper.

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It should be noted, however, that the effect of location of the inner boundary is quite systematic: the forward shock wave becomes fast and the luminosity becomes large if we put the inner boundary on a small radius. This is probably because the density is initially higher near the boundary. Consequently the total mass accretion rate increases and therefore the jet luminosity becomes large with a strong forward shock wave. According to these results, it is a robust claim that the shock breakout is possible if the accretion-to-jet conversion efficiency satisfies the condition η ≳ 10⁻⁴ (see more details in Section 3.7).

3.6.3. The Time Delay between Infall and Injection

3.6.4. Long-term Simulations

In the current study, we investigate the jet propagation inside the stellar mantle and discuss the possibility of GRBs. However, in order to know how these outflows accelerate in the interstellar medium (ISM), it is necessary to carry out simulations with longer duration and a large spatial region. Although our simulations have a limited duration and spatial extent, we perform extended numerical simulations for the W-R reference model for the purpose of qualitative understanding of the outflow properties. We extend the computational boundary to r ∼ 5R_star and carry out the simulation until the forward shock wave reaches the outer boundary. We cover 1500 uniform radial grids in this extended spatial region, i.e., the total number of our radial grids are 2000 (500 + 1500). The grid width is the same as the outermost radial width of previous calculations. The outer boundary in this calculation is located at r_out = 2.34 × 10¹¹ cm. In this simulation, the three-level AMR technique is employed. Thus, the radial resolution in the extended region corresponds to Δr = 4.27 × 10⁷ cm. The meridian grid width is the same as previous calculations.

Figure 19 shows some snapshots for the diagnostic terminal Lorentz factor distribution along jet axis. As we can see in this figure, the outflow propagates into this ISM and the inner ejecta with high Γ_dt gradually progresses outward with time. This is attributed to the continuous energy injection of the central engine. According to these results, even if baryon pollution were real, the outflows would continuously propagate into the ISM because it is being pushed by high Lorentz factor ejecta in the back. Note also that, since this simulation is contaminated by the artificial baryon pollution, the outflow is more relativistic in reality than this simulation. The encouraging tendencies increase the possibility of GRBs being produced in a wider range of scenarios, but future long-duration simulations are needed.

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In summary of this subsection, in order to judge whether the outflows become relativistic or not in reality, we need to simulate them with (1) high resolution in order not to affect the baryon pollution, (2) smaller inner boundary, and (3) long duration and large spatial range. Here, we emphasize that the results of the current study give conservative claims for the possibility of relativistic jet breakout, which is the minimum requirement for producing GRBs. In fact, we show that the accretion-powered jet succeeds in breaking out relativistically from massive Pop III progenitors if the adequate conditions are satisfied.

In this subsection, we compare our numerical results with the analytic estimate of the shock breakout in Suwa & Ioka (2011) and Appendix A.2. Since setups are rather different between the numerical study and the analytic one, we concentrate on the parameter dependence of the successful relativistic shock breakout. They derived the shock breakout time (t_b; Equation (12) in their paper) and the free-fall timescale of the envelope materials, i.e., the duration of the accretion-powered jet (t_ff; Equation (15)). When t_b is shorter than t_ff, the jet successfully arrives at the stellar surface and the relativistic shock breakout takes place. Employing Equations (A10) and (A11) in Appendix A.2, we can derive the criteria for the successful relativistic shock breakout as

$$\begin{equation} \frac{t_\mathrm{ff}}{t_\mathrm{b}}\sim \lambda \left(\frac{\eta }{10^{-4}}\right)^{\frac{3}{9-2n}}\left(\frac{\theta _{{\rm op}}}{5^\circ }\right)^{-\frac{6}{9-2n}}\gtrsim 1, \end{equation} \tag{ 8 }$$

where n (≈2.6 for mpop3 and W-R and ≈2.1 for lpop3) is an index for the density profile of the stellar envelope,

$$\begin{equation} \rho (r) \propto \left(\frac{R_*}{r}-1\right)^n, \end{equation} \tag{ 9 }$$

with the stellar radius R_* (Matzner & McKee 1999). The prefactor λ depends on the stellar radius, the core mass, and the envelope mass, and is found to be typically around λ ∼ 3 for mpop3 according to Suwa & Ioka (2011) and Appendix A.2. In this paper, we find that λ ≈ 2 is suitable for explaining the numerical results of mpop3. The reason of the difference in λ is that some of the assumptions in Suwa & Ioka (2011) are violated, that is, they assumed a conical jet propagation (i.e., θ_op is independent of the radius; see Ioka et al. 2011) and neglected the negative feedback from the cocoon on the accretion rate. The breakout condition is also not exactly the same between the analytic and numerical calculations. Nevertheless, it is encouraging that both results coincide within a numerical factor.

In Figure 20, we present η − θ_op diagrams to compare our numerical results with the analytic criteria in Equation (8) (black solid line). The circles, triangles, and crosses in this figure correspond to the column "Possibility of GRB production" in Table 3. We use λ = 2 for all progenitor models. As we can see from Figure 20, the analytical criteria in Equation (8) and Suwa & Ioka (2011) are useful to forecast the penetrability of the relativistic jet and the possible GRB production in massive star models. We can predict the parameter space where GRBs could occur in the η − θ_op plane once we calibrate λ with numerical simulations. In addition, we find that the analytical critical curve can be approximated by a much simpler form for all current progenitors (W-R, mpop3, lpop3) as

$$\begin{equation} \eta \gtrsim 10^{-4} \left(\frac{\theta _{{\rm op}}}{8^{\circ }}\right)^2. \end{equation} \tag{ 10 }$$

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Note that a red supergiant (RSG) satisfies the breakout criteria in Equation (8) with λ ∼ 6.8. Actually a GRB jet might be associated with an RSG, which has not been observed so far. A GRB from an RSG would be very dim and long because the breakout time is long due to the expanded envelope and the accretion rate is low at the late time. Such dim GRBs might even dominate as the sensitivity is improved. Alternatively, another condition could prevent the GRB production. First, the velocity of the jet head should be faster than that of the cocoon, v_h > v_c (Matzner 2003; Suwa & Ioka 2011). In some parameter space, an RSG does not satisfy this second criterion because of its shallow envelope and the outflow becomes almost spherical (Suwa & Ioka 2011). Second, the central engine of an RSG might not rotate fast enough for the jet production. The detailed discussion will be presented in a forthcoming paper.

Using Equation (9) of Suwa & Ioka (2011), we can calculate the stellar radius R_* as

$$\begin{eqnarray} R_* &=& 10^{13} \,\mathrm{cm} \left(\frac{10^{\frac{1.2}{3-n}}}{10^{3}}\right) \left(\frac{0.4}{3-n}\right)^{-\frac{1}{3-n}} \left(\frac{R_c}{10^{10} \,\mathrm{cm}}\right)\nonumber\\ && \times \left(\frac{M_c}{400 \,M_\odot }\right)^{-\frac{1}{3-n}} \left(\frac{M_\mathrm{env}}{500 \,M_\odot }\right)^{\frac{1}{3-n}}, \end{eqnarray} \tag{ A1 }$$

where we calibrate the overall factor to reproduce the mpop3 radius with n = 2.6. This equation is a general form of Equation (10) of Suwa & Ioka (2011). From Equation (A1), we can estimate the stellar radii of W-R, lpop3, and RSG as 8 × 10¹⁰ cm, 7 × 10¹¹ cm, and 2 × 10¹³ cm, respectively, which reproduce well the actual values within a factor of two to three. In these estimations, we employ following parameters:

• 1.  
  W-R: n = 2.6, M_c = 11 M_☉, M_env = 2 M_☉, and R_c = 10¹⁰ cm.
• 2.  
  lpop3: n = 2.1, M_c = 15 M_☉, M_env = 25 M_☉, and R_c = 10¹⁰ cm.
• 3.  
  RSG: n = 1.5, M_c = 4 M_☉, M_env = 8 M_☉, and R_c = 10¹² cm.

In this section, we derive an analytic expression for the breakout criteria. Note that we employ typical sets of parameters to normalize physical quantities (e.g., η, θ_op). In order to apply this to the individual progenitor, one should insert the values from Appendix A.1.

Here, we approximate the density profile of the envelope as

$$\begin{equation} \rho (r)=\rho _1\left(\frac{R_*}{r}\right)^n, \end{equation} \tag{ A2 }$$

in which the term of unity in Equation (9) is neglected for simplicity. This approximation is valid for r ≪ R_*. The free-fall timescale of a mass shell at radius r and the enclosed mass M_r is given by

$$\begin{equation} t_{{\rm ff}}=\sqrt{\frac{r^3}{{\it GM}_r}}, \end{equation} \tag{ A3 }$$

where G is the gravitational constant. The mass accretion rate is given by

$$\begin{eqnarray} &&\dot{M}=\frac{{\it dM}_r/dr}{dt_{{\rm ff}}/dr}\approx \frac{8\pi }{3}\frac{r^3}{t_{{\rm ff}}}\rho _0\left(\frac{R_0}{r}\right)^{n}, \end{eqnarray} \tag{ A4 }$$

where R₀ is the arbitrary characteristic radius and ρ₀ = ρ₁(R_*/R₀)ⁿ. Here we neglect the derivative of M_r with respect to r for dt_ff/dr. Using Equation (A4), the jet luminosity emitted from the central object is written as

$$\begin{eqnarray} L_j &=& \eta \dot{M} c^2= 7.5\times 10^{45} \,\mathrm{erg\, s^{-1}} \left(\frac{\eta }{10^{-4}}\right) \left(\frac{t_{{\rm ff}}}{1 \,\mathrm{s}}\right)^{-1}\nonumber\\ && \times \left(\frac{r}{10^{11} \,\mathrm{cm}}\right)^{3} \left(\frac{R_0}{r}\right)^{n} \left(\frac{\rho _0}{10^{-5} \mathrm{\,g\, cm^{-3}}}\right), \end{eqnarray} \tag{ A5 }$$

where t_ff corresponds to the jet breakout time at R₀. On the other hand, the necessary luminosity for outgoing jet propagation (Suwa & Ioka 2011) is given by

$$\begin{eqnarray} L_{{\rm iso}}\frac{\theta _{{\rm op}}^2}{2} &=& 1.3\times 10^{48} \mathrm{\,erg \,s^{-1}} \left(\frac{R_0}{10^{11} \,\mathrm{cm}}\right)^{4}\nonumber\\ && \times \left(\frac{\rho _0}{10^{-5} \mathrm{\,g \,cm^{-3}}}\right) \left(\frac{\theta _{{\rm op}}}{5^\circ }\right)^{2} \left(\frac{t_{{\rm ff}}}{1 \,\mathrm{s}}\right)^{-2}. \quad \end{eqnarray} \tag{ A6 }$$

Equating Equations (A5) and (A6), we get

$$\begin{eqnarray} 5.8\times 10^{-3} &=& \left(\frac{\eta }{10^{-4}}\right)^{-1} \left(\frac{t_{{\rm ff}}}{1 \,\mathrm{s}}\right)^{-1} \left(\frac{R_0}{10^{11} \,\mathrm{cm}}\right)^{4}\nonumber\\ && \times \left(\frac{r}{10^{11} \,\mathrm{cm}}\right)^{-3} \left(\frac{r}{R_0}\right)^{n} \left(\frac{\theta _{{\rm op}}}{5^\circ }\right)^{2}. \quad \end{eqnarray} \tag{ A7 }$$

By deleting r from Equation (A7) using Equation (A3) (i.e., r ∼ 10¹¹ cm(M_r/250 M_☉)^1/3(t_ff/170 s)^2/3), we can estimate the jet breakout time (i.e., the jet arrival time at R₀) as

$$\begin{eqnarray} t_\mathrm{b}(r=R_0) &\sim & 170 \,\mathrm{s} \left(\frac{\eta }{10^{-4}}\right)^{-\frac{3}{9-2n}} \left(\frac{\theta _{{\rm op}}}{5^\circ }\right)^{\frac{6}{9-2n}}\nonumber\\ && \times \left(\frac{R_0}{10^{11} \,\mathrm{cm}}\right)^{\frac{3(4-n)}{9-2n}} \left(\frac{M_{R_0}}{250 \,M_\odot }\right)^{-\frac{3-n}{9-2n}}. \quad\quad \end{eqnarray} \tag{ A8 }$$

On the other hand, the free-fall time in Equation (A3) is

$$\begin{equation} t_\mathrm{ff}(r=R_0)\sim 170 \,\mathrm{s} \left(\frac{R_0}{10^{11} \,\mathrm{cm}}\right)^{\frac{3}{2}} \left(\frac{M_{R_0}}{250 \,M_\odot }\right)^{-\frac{1}{2}}. \end{equation} \tag{ A9 }$$

Here, we choose R₀ ≈ 0.3R_*, beyond which the density profile is not a power law but decreases rapidly due to the term of unity in Equation (9). The jet head accelerates due to decreasing ram pressure of the ambient matter beyond this radius. In addition, the mass accretion rate by the mass shell at r ≳ 0.3R_* decreases rapidly. Therefore, we consider the critical condition for the successful jet breakout using R₀ ≈ 0.3R_* in the following.

By combining Equations (A8) and (A9), we obtain the criterion for the successful jet break out, which is that the free-fall timescale should be longer than the breakout timescale, as

$$\begin{eqnarray} \frac{t_\mathrm{ff}}{t_\mathrm{b}} &\sim & 1.0 \left(\frac{\eta }{10^{-4}}\right)^{\frac{3}{9-2n}} \left(\frac{\theta _{{\rm op}}}{5^\circ }\right)^{-\frac{6}{9-2n}} \left(\frac{0.3 R_*}{10^{11} \,\mathrm{cm}}\right)^{\frac{3}{2(9-2n)}}\nonumber\\ && \times \left(\frac{M_{0.3 R_*}}{250 \,M_\odot }\right)^{-\frac{3}{2(9-2n)}} \gtrsim 1. \end{eqnarray} \tag{ A10 }$$

According to the notation of Equation (8), we define λ as

$$\begin{equation} \lambda \equiv \left(\frac{0.3R_*}{10^{11} \,\mathrm{cm}}\right)^{\frac{3}{2(9-2n)}} \left(\frac{M_{0.3R_*}}{250 \,M_\odot }\right)^{-\frac{3}{2(9-2n)}}. \end{equation} \tag{ A11 }$$

For $M_{0.3R_*}$, we replace this term with M_c  +  0.4M_env to estimate the specific values (see Suwa & Ioka 2011). Using values in Appendix A.1, we find that λ ∼ 1.5, 2.6, 2.7, and 6.8 for W-R, lpop3, mpop3, and RSG, respectively.