VERY LOW ENERGY SUPERNOVAE FROM NEUTRINO MASS LOSS

Hydrodynamics and radiation transport in the region immediately surrounding a protoneutron star or black hole is complex and difficult to model. A realistic simulation of the neutrino transport alone is well beyond the scope of this paper and the outcome would depend upon the chosen neutron star EOS and the dimensionality of the calculation. Since we are only interested in the temporal evolution of the central gravitational potential, however, reasonably good results can be obtained from parameterized calculations with a qualitative description of the neutrino energy loss. The inner boundary of our simulation is placed, in CASTRO, at the outer edge of the pre-collapse iron core. Matter interior to this boundary is treated as a point having variable gravitational mass. Initially, this mass is equal to the baryonic mass of the interior matter, i.e., the iron core mass. As time passes, that gravitational mass decreases due to neutrino emission and, if no mass were added, it would eventually decrease to the known cold neutron star value for a given baryonic mass and EOS (Prakash et al. 1997). However, appreciable mass does accrete, leading in the more interesting cases to the formation of a black hole. During the accretion phase, neutrinos continue to carry away effective mass. Since we neglect rotation, we assume that after the black hole forms, matter and energy flow into the black hole without any more emission.

An upper limit to the mass lost is given by the gravitational binding energy of the maximum stable neutron star mass, the "Tolman–Oppenheimer–Volkoff (TOV) limit" (Oppenheimer & Volkoff 1939). Current best limits place this parameter in the range 2.0–2.5 M_☉ of gravitational mass (Akmal et al. 1998; Demorest et al. 2010). The characteristic time scale for this loss, which is assumed to occur exponentially, is given by the cooling timescale parameter τ_c, taken here to be 3 s (e.g., Burrows & Lattimer 1987). For this simple case, the mass-loss rate is

$$\begin{equation} \dot{M}_{\rm G} = \frac{{\rm BE}}{\tau _c} e^{-t/\tau _c}. \end{equation} \tag{ 2 }$$

This equation gives an upper bound on the mass loss and therefore on the strength of the transient produced. This model is referred to as the "maximum mass loss" case.

Somewhat more realistically, we can take into account the time for the critical mass to accrete and follow the neutrino energy losses that occur during that time. For each bit of mass that accretes, some fraction of its gravitational binding energy is radiated away promptly, but more is emitted with a delay since the neutrinos must diffuse out. Once the black hole has formed, the neutrino-emitting material falls within the event horizon much faster than the neutrino diffusion timescale, and any energy that has not been released by the time of collapse ends up in the black hole (Prakash et al. 1997; Burrows 1988). The protoneutron star is also extremely hot and thus behaves in a slightly different manner from cold neutron stars. We represent the protoneutron star as a low-entropy core surrounded by a hot envelope and track three masses: M_Gh, M_B, and M_th.

M_B is the baryonic mass, i.e., all mass that either was inside the inner boundary at the time of collapse or subsequently accretes through the inner boundary. M_Gh is the effective gravitational mass of the hot neutron star, which is not, in general, equivalent to M_o, the gravitational mass of a cold neutron star with the same baryon content. This is the mass that defines the gravitational potential in the outer part of the star. M_th accounts for the difference in mass–energy stored in this hot neutron star as opposed to a cold one, due to both its higher internal energy and inflated radius. This extra internal energy diffuses away on the cooling timescale τ_c. The gravitational mass M_Gh of the hot neutron star is then

$$\begin{eqnarray} M_{{\rm Gh}} &= M_o + M_{{\rm th}} \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &= M_B - {\rm BE}_c + M_{{\rm th}}, \end{eqnarray} \tag{ 4 }$$

where BE_c is the cold neutron star binding energy. As matter accretes, a fraction of the subsequent change in binding energy is assumed to be trapped and added to M_th while the remaining fraction 1 − is radiated promptly as neutrinos. The trapped energy diffuses out on a cooling timescale τ_c. The evolution of M_th is thus given by

$$\begin{equation} \dot{M}_{{\rm th}} = -\frac{M_{{\rm th}}}{\tau _c} + \epsilon \, \frac{d \, {\rm BE}_c}{dM_B} \ \dot{M}_B \end{equation} \tag{ 5 }$$

and the initial condition M_th = BE_c. The derivative d BE_c/dM_B is evaluated from the binding energy equation (Equation (1)). Once all M_th has diffused away, i.e., the neutron star has cooled, Equation (4) reduces to the standard cold neutron star equation M_G = M_B − BE_c. Assuming rapid virialization of the accreted material implies that the parameter should not exceed ∼0.5. As the cooling timescale at the surface is likely less than the timescale for the full protoneutron star, is likely to be much lower. In the limit that the cooling timescale were much shorter than the accretion timescale, would go effectively to zero. For our models we adopt = 0.1. In Section 3.3, we test the effects of varying this parameter and find them to be small.

The emission halts when the neutron star has accreted past the TOV maximum mass limit. However this limiting mass must be adjusted for the neutron star's evolving heat content. Therefore, we wait until the cold core of the neutron star (M_Gh − M_th) has accreted past the TOV limit before we assume the core has become a black hole and shut off the neutrino emission. From this point on, the central object behaves as a purely gravitational point mass that absorbs all of $\dot{M}_B$. This is referred to as the "full loss" model.

Figure 1 shows an example of the full model for RSG15, TOV = 2.5 M_☉. The blue curve shows M_B, the total baryonic mass that has entered the core region (pre-collapse core plus accretion); it is higher than the black curve showing M_Gh, the effective gravitational mass of the core. The red curve shows M_th, which decays exponentially as the protoneutron star cools. The green curve shows the cumulative mass loss in the simulation. Even though the core in this case lives for almost 24 s as a neutron star, most of the mass loss occurs during the first 5 s.

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The two presupernova models used for this paper were red supergiants with a ZAMS mass of 15 M_☉ (RSG15), shown in Figure 2, and 25 M_☉ (RSG25), shown in Figure 3, both of solar metallicity. They are taken from the survey of Woosley & Heger (2007). Both stars shed substantial amounts of mass before reaching the end of their lives. The 15 M_☉ star represents a more common supernova progenitor, while the 25 M_☉ model has a more compact structure and may be more likely to make a black hole promptly. The observed upper limit on the progenitor masses of standard core-collapse supernovae (CCSNe) found to date is 18 M_☉ (e.g., Smartt 2009), suggesting that higher mass progenitors are more likely to collapse in a non-traditional way, or that these progenitors are preferentially obscured. Model RSG15 has a helium core of 4.27 M_☉ that extends to 3.568 × 10¹⁰ cm. RSG25 has a helium core of 8.20 M_☉ that extends to 1.807 × 10¹⁰ cm. Both red supergiants have an extended, tenuously-bound hydrogen envelope extending from the helium core out to ∼5 × 10¹³ cm that is easily ejected. The net binding energy of the envelope external to what is nominally the helium core (where the hydrogen mass fraction declines to 1%) is 1.1 × 10⁴⁸ erg, but a short distance out at 4.48 M_☉ this declines to 10⁴⁷ erg, a value that characterizes most of the hydrogen envelope. These values include internal energy but not the energy potentially available from recombination. For the 25 M_☉ model, the net binding energy external to the helium core is 6.4 × 10⁴⁸ erg, but this declines to 10⁴⁷ erg when the interior mass increases from 8.20 M_☉ to 8.36 M_☉ and a still smaller value characterizes most of the envelope. We note that the recombination of 10 M_☉ of hydrogen (13.6 eV per atom) would release 2.6 × 10⁴⁷ erg.

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Some additional parameters of both models can be found in Table 1. Also given is the compactness parameter, ξ_2.5, as defined by O'Connor & Ott (2011), but computed at the time when the star is moved to CASTRO rather than at the time of core bounce. A star with a higher compactness parameter has a denser region surrounding its core; for our purposes, this means it will accrete to the TOV limit faster and therefore potentially lose less mass as neutrinos.

Because of the extended size of the red supergiant progenitors, we do not carry the entire star on our simulation grid in CASTRO. Instead, we model only the helium core and base of the hydrogen envelope.

Without an outgoing shock or mass loss from neutrinos, the inner layers of the star should collapse directly into the black hole. This scenario provides an excellent check of the fidelity of our simulation. As Figure 4 shows, our model accurately reproduces this behavior. Dark (purple) colors on the plot indicate early times, shading to light (red) colors at late times. Initially, the bulk of the star is in hydrostatic equilibrium (zero velocity) with a small portion near the core showing high infall velocities. As the collapse continues, more and more of the stellar material acquires negative velocities. Without an outgoing shock to counter this motion, it eventually falls into the core. If the simulation is run for long enough, then all mass will disappear from the grid. Physically, this scenario corresponds to prompt black hole formation, when the core collapses without passing through an intermediate protoneutron star stage. If the outer layers of the star do not have sufficient angular momentum to form a disk when they reach the black hole, then the entire star will disappear without producing a transient—the unnova case as defined by Kochanek et al. (2008).

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Figure 5 shows the growth of the simulated point mass in RSG15. The black curves show the model with no neutrino mass loss; in this case, the point mass corresponds to M_B. The dashed line represents the growth of the point mass while the dot-dashed line shows the mass on the simulated grid. Over time this point mass grows as the grid mass declines, with most of the change occurring in the first 15 s. The solid line shows the sum of the point mass and grid mass, i.e., the total mass represented in the simulation; in the no-mass-loss case, it is constant throughout. This confirms that our simulation is reproducing the collapse accurately, without spurious shocks or unphysical mass loss.

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The set of blue curves in Figure 5 shows the evolution using the simplified maximum mass loss model. In this case, the point mass corresponds to M_G with losses defined by Equation (2), in which the core loses the total binding energy appropriate to a TOV-limit neutron star on a time scale τ_c, regardless of the amount of mass flowing in from the collapsing star. This model therefore provides an upper bound on possible transients, as the core cannot lose more mass than the binding energy of the largest possible neutron star. The amount of mass lost in each case is listed in Table 2. Following the blue curves in Figure 5, as the collapse begins the point mass growth (dashed) is noticeably suppressed for the first 5 s as neutrino losses balance accreted mass. The overall mass in the simulation (solid) drops accordingly, then becomes constant as mass loss ceases.

This mass decrement was sufficient to produce an outgoing shock in the inner layers of the 15 M_☉ presupernova star. The shock's evolution is seen in Figure 6 (purple at early times, shading to red at late times). The shock grows in speed as it leaves the helium core, and succeeds in reaching the base of the hydrogen envelope. Of interest is the fact that the shock strength varied noticeably with the choice of neutron star upper mass limit. The approximate shock strengths at 10¹¹ cm for our six different choices of TOV limit are listed in Table 2.

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The red curves in Figure 5 show the mass evolution for the full model for neutrino losses described in Section 2.1. In this case, the dashed line showing the point mass corresponds to M_Gh as given by Equation (4). This model takes into account the thermal mass loss and ties the cessation of neutrino losses to the amount of material that has been accreted by the core, rather than switching it off after a predetermined timescale as in the upper bound model. As can be seen in Figure 5, this model (red) loses mass over a longer timescale than the maximum loss model (blue), continuing until the point mass reaches the TOV limit, in this case 2.5 M_☉, after which the total mass becomes constant. We expect equal or less mass loss in this case as compared to the maximum loss model. In the case where the TOV limit is high enough that the neutron star lives for longer than the cooling timescale, the core loses close to the maximum possible amount of mass; in the case where it does not, however, mass loss is suppressed as neutrinos that would have been emitted instead end up inside the black hole. The amount of mass lost in each case is listed in Table 3.

Though the overall mass decrement in the full model cases is lower than in the maximum loss case, it is still sufficient to produce an outgoing shock. Figure 7 shows the shock evolution for RSG15, TOV = 2.5 M_☉. The approximate shock strengths at 10¹¹ cm for our six different choices of TOV limit are listed in Table 3. The six shocks created in RSG15 are shown in Figure 8.

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We also tested variations in the parameter , which controls the fraction of binding energy trapped as thermal mass. Changes in have a small but real effect on the total mass loss, depending on the amount of accreted mass. The more mass accreted, the more important will be. As higher TOV limit models tend to accrete longer, has a higher impact here. A lower epsilon leads to a higher mass loss as less of the binding energy is temporarily trapped as thermal mass. We tested the range < 0.5, identified as the physically reasonable range of this parameter. For the case where the TOV limit is 2.5 M_☉, the most sensitive, a change of 0.05 in in RSG25 resulted in approximately a 0.011 M_☉ change in the overall mass loss. In the extreme case = 0.5, this will make a TOV = 2.5 M_☉ model look like a TOV ∼ 2.35 M_☉ model. For lower TOV limits, the effect of varying was smaller. Thus, although a change in will affect the mass decrement and by extension the shock strength, the results are robust.

The kinetic energy of the shock that reaches the hydrogen envelope is strongly influenced by the size of the presupernova core and the overlying stellar structure through which it must travel, as can be seen by the differences in kinetic energy between RSG25 and RSG15 in the maximum mass loss model. This model loses the same amount of mass in the same time regardless of stellar structure, but the final shock in RSG25 models is significantly weaker than in RSG15, having propagated through a much heavier carbon–oxygen and helium core before reaching the hydrogen envelope.

Additional effects come into play when using the full neutrino model based on the speed at which the core accretes. RSG25 has a denser inner structure (higher compactness) and a more massive iron core of 1.83 M_☉, compared to 1.63 M_☉. As it therefore accretes faster than RSG15 and is already closer to the TOV limit, the core spends significantly less time in the protoneutron star state; consequently, the same choice of parameters in the larger star causes less mass loss. The full mass loss model in RSG25 also shows systematically lower mass decrements than the maximum mass loss model in all cases, indicating that even with a high TOV limit and a relatively long-lived protoneutron star, not all the binding energy is being emitted before collapse to a black hole. The six shocks produced in RSG25 are shown in Figure 9.

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