BLACK HOLE FORMATION IN FAILING CORE-COLLAPSE SUPERNOVAE

GR1D (O'Connor & Ott 2010) is a spherically symmetric GR hydrodynamics code developed for the study of stellar collapse and BH formation. It is available for download at http://stellarcollapse.org. GR1D, based on the previous work of Gourgoulhon (1991) and Romero et al. (1996), is Eulerian and uses the radial gauge—polar slicing coordinates that have the simplifying property of a vanishing shift vector. Here, we briefly outline the basics of the curvature and hydrodynamics equations and refer the reader to O'Connor & Ott (2010) for full details and derivation. We assume spacelike signature (−, + , + , +) and, unless noted otherwise, use units of G = c = M_☉ = 1. The metric of GR1D is given by the line element

$$\begin{equation} ds^2 = {-}\alpha (r,t)^2 dt^2 + X(r,t)^2 dr^2 + r^2 d\Omega ^2, \end{equation} \tag{ 1 }$$

where α(r, t) = exp(Φ(r, t)) with Φ(r, t) being the metric potential. X(r, t) = [1 − 2m(r)/r]^−1/2, where m(r) is the enclosed gravitational mass. We assume an ideal fluid with stress energy given by

$$\begin{equation} T^{\mu \nu } = \rho h u^\mu u^\nu + g^{\mu \nu } P, \end{equation} \tag{ 2 }$$

where ρ is the matter density, P is the fluid pressure and h = 1 + + P/ρ is the specific enthalpy with being the specific internal energy. u^μ in Equation (2) is the 4-velocity of the fluid, and without rotation, taken to be u^μ = (W/α, Wv^r, 0, 0), where W = [1 − v²]^−1/2 is the Lorentz factor and v = Xv^r is the physical velocity. For a given matter configuration, the Hamiltonian and momentum constraint equations give differential equations for both m(r) and Φ(r),

$$\begin{eqnarray} m(r) &=& 4\pi \int _0^r (\rho h W^2 - P + \tau _m^\nu) {r^\prime }^2 dr^\prime,\\ \Phi (r,t) &=& \int _0^rX^2\left[{m(r^\prime,t) \over {r^\prime }^2} + 4\pi r^\prime (\rho h X^2 u^{r^\prime }u^{r^\prime } + P + \tau _\Phi ^\nu)\right]dr^\prime\nonumber\\ &&+ \Phi _0, \end{eqnarray} \tag{ 3 }$$

where Φ₀ is determined by matching the metric at the star's surface to the Schwarzschild metric. The neutrino terms, τ^ν_m and τ^ν_Φ, account for trapped neutrinos and their detailed form is given in O'Connor & Ott (2010). We obtain the fluid evolution equations by expanding the local fluid rest-frame conservation laws, ∇_μT^μν = 0 and ∇_μJ^μ = 0, in the coordinates of GR1D. The conservation laws become

$$\begin{equation} \partial _t(\vec{U}) + {1 \over r^2}\partial _r\left({\alpha r^2 \over X} \vec{F}\right) = \vec{\cal {S}}\,, \end{equation} \tag{ 5 }$$

where $\vec{U} = \left(D,\ DY_e,\ S^r,\ \tau \right)$ are the conserved variables, given in terms of the primitive fluid variables ρ,  Y_e,  ,  P,  and v as

$$\begin{equation} \vec{U} = \left(\begin{array}{@{}c@{}}D\\ DY_e\\ S^r\\ \tau \end{array}\right) = \left(\begin{array}{@{}c@{}}X\rho W\\ X\rho WY_e\\ \rho h W^2 v\\ \rho h W^2 - P - D \end{array}\right)\,. \end{equation} \tag{ 6 }$$

The spatial fluxes in Equation (5) are given by

$$\begin{equation} \vec{F} = (Dv,\ DY_ev,\ S^rv+P,\ S^r-Dv), \end{equation} \tag{ 7 }$$

and the source terms

$$\begin{eqnarray} \vec{\mathcal {S}} =& \bigg [ 0,\ R^\nu _{Y_e},\ (S^rv - \tau - D)\alpha X\left(8 \pi r P + \frac{m}{r^2}\right) + \alpha P X {m \over r^2} \nonumber \\ &+ {2 \alpha P \over X r} + Q_{S^r}^{\nu,\mathrm{E}} + Q_{S^r}^{\nu,\mathrm{M}},\ Q_\tau ^{\nu,\mathrm{E}} + Q_\tau ^{\nu,\mathrm{M}} \bigg ], \end{eqnarray} \tag{ 8 }$$

where the Rs and Qs are neutrino sources and sinks which arise from the neutrino leakage scheme and neutrino pressure contributions (see O'Connor & Ott 2010 for details).

The evolution equations (Equation 5) are first spatially discritized using a finite-volume scheme (e.g., Romero et al. 1996; Font 2008). The piecewise parabolic method (Colella & Woodward 1984) is used to reconstruct the state variables at the cell interfaces and the HLLE Riemann solver (Einfeldt 1988) is employed to determine the physical fluxes through these interfaces. The evolution equations are integrated forward in time via the method of lines (Hyman 1976), using standard second-order Runge–Kutta time integration with a Courant factor of 0.5. After updating the conserved variables, they are inverted via a Newton–Raphson scheme to obtain the new fluid state variables.

In spherical symmetry, rotation can be included by assuming constant angular velocity Ω on spherical shells (shellular rotation) and including an angle-averaged centrifugal force in the radial momentum equation. This is common practice in stellar evolution codes (e.g., Heger et al. 2000) an has also been applied to Newtonian 1D stellar collapse calculations (Akiyama et al. 2003; Thompson et al. 2005). We include a GR variant of this "1.5D" rotation treatment in GR1D (1) by adding an evolution equation for the generalized specific angular momentum S_ϕ = ρhW²v_φr, (2) by including an effective centrifugal force in the equation for S^r, and (3) by modifying the expressions for the 4-velocity, the Lorentz factor, and the differential equation for the metric potential to account for rotation. Full details as well as a demonstration of conservation of angular momentum can be found in O'Connor & Ott (2010). Note that, as may be expected and was demonstrated by Ott et al. (2006), the 1.5D approximation becomes less accurate with increasing spin and quantitative results are reliable only for low rotation rates.

Neutrino effects are crucial in stellar collapse and should ideally be included via a computationally expensive GR Boltzmann transport treatment (e.g., Liebendörfer et al. 2004). However, since our aim is to perform an extensive parameter study with hundreds of simulations, we choose to resort to a less accurate, but much more computationally efficient leakage and approximate heating scheme for neutrinos. Details of this are laid out in O'Connor & Ott (2010). Here we review only its most salient features.

Before core bounce, neutrinos deleptonize the collapsing core, reducing the electron fraction Y_e and, as a consequence, the size of the homologous inner core (Bethe 1990). Liebendörfer (2005) showed that the prebounce Y_e can be parameterized as a function of density and that this parameterization varies little between progenitor stars. We follow this prescription for prebounce deleptonization and use the Y_e(ρ) fit parameters of his G15 model. Following bounce, this simple parameterization becomes inaccurate and cannot capture the effects of neutrino cooling, deleptonization, and neutrino heating. Hence, we switch to a leakage scheme that uses elements of what was laid out by Ruffert et al. (1996) and Rosswog & Liebendörfer (2003). We consider three neutrino species, ν_e, $\bar{\nu }_e$, and $\nu _x = \lbrace \nu _\mu,\bar{\nu }_\mu,\nu _\tau,\bar{\nu }_\tau \rbrace$. Neutrino pairs of all species are made in thermal processes of which we include electron–positron pair annihilation and plasmon decay (Ruffert et al. 1996). In addition, charged-current processes lead to the emission of ν_es and $\bar{\nu }_e$s. The leakage scheme provides approximate energy and number emission rates that are inserted into GR1D's evolution equations via source terms in Equation (8), $R^\nu _{Y_e},\ Q^{\nu,\mathrm{E}}_{S^r},\ \mathrm{and}\ Q^{\nu,\mathrm{E}}_{\tau }$ (O'Connor & Ott 2010).

We include neutrino heating via a parameterized charged-current heating scheme based on Janka (2001). The heating rate at radius r is

$$\begin{equation} Q^{\mathrm{heat}}_{\nu _i}(r) = f_\mathrm{heat} \frac{L_{\nu _i}(r)}{4\pi r^2} \sigma _{\mathrm{heat},\nu _i}\, {\rho \over m_u} X_i \left\langle {1 \over F_{\nu _i}} \right\rangle e^{-2\tau _{\nu _i}} \,, \end{equation} \tag{ 9 }$$

where f_heat is a scale factor that allows for artificially increased heating, $L_{\nu _i}(r)$ is the neutrino luminosity interior to r, $\tau _{\nu _i}$ is the optical depth, determined through the leakage scheme, $\sigma _\mathrm{heat,\nu _i}$ is the energy-averaged absorption cross section, and X_i is corresponding mass fraction of the neutrino reaction (X_p for $\bar{\nu }_e$ capture on protons and X_n for ν_e capture on neutrons). $\langle 1/F_{\nu _i}\rangle$ is the mean inverse flux factor which we approximate analytically as a function of the optical depth τ by comparing to the angle-dependent radiation transport calculations of Ott et al. (2008).

We include in our simulations the stabilizing effect of neutrino pressure in the optically thick PNS core via an ideal Fermi-gas approximation (Liebendörfer et al. 2005; O'Connor & Ott 2010). Leaving out this pressure contribution leads to ∼5% smaller maximum gravitational PNS masses. We also include terms due to neutrino pressure and radiation-field energy in the calculation of the gravitational mass (Equation (3)) and of the metric potential (Equation (4)). Since our leakage scheme does not treat neutrino energy separately from the internal energy of the fluid, including the energy of the neutrino gas in the former equations is not fully consistent with our present approach. This error was discovered and corrected after all simulations were performed. However, a set of test calculations showed that the error leads to an underestimate of the maximum gravitational PNS mass of only ∼2% which is well within the error of the overall leakage scheme (see also Section 4.2).

We include multiple finite-temperature nuclear EOS in this study to explore the dependence of postbounce evolution and BH formation on EOS properties. The Lattimer–Swesty (LS) EOS (Lattimer & Swesty 1991) is based on the compressible liquid-droplet model, assumes a nuclear symmetry energy S_v of 29.3 MeV, and comes in three variants with different values of the nuclear incompressibility of K_s = 180 MeV (LS180), 220 MeV (LS220), and 375 MeV (LS375). The EOS of Shen et al. (1998a, 1998b) (HShen EOS), on the other hand, is based on a relativistic mean-field model, has S_v = 36.9 MeV and K_s = 281 MeV. More details on these EOS and their implementation in GR1D is given in O'Connor & Ott (2010). The EOS tables and driver routines employed in this study are available for download at http://stellarcollapse.org.

By solving the Tolman–Oppenheimer–Volkoff (TOV) equations (Oppenheimer & Volkoff 1939) with T = 0.1 MeV and assuming neutrinoless β-equilibrium we determine the neutron star baryonic and gravitational mass–radius relationships that each of these four EOS produces and that are depicted by Figure 1. The maximum gravitational (baryonic) neutron star masses are ∼1.83 M_☉ (∼2.13 M_☉), ∼2.04 M_☉ (∼2.41 M_☉), ∼2.72 M_☉ (∼3.35 M_☉), and ∼2.24 M_☉ (∼2.61 M_☉) for LS180, LS220, LS375, and HShen, respectively. The coordinate radii of these maximum-mass stars are ∼10.1 km, ∼10.6 km, ∼12.3 km and ∼12.6 km, respectively.

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The above maximum neutron star masses hold only for nonrotating cold NSs. As we will discuss in detail in Section 4.3, the PNSs at the heart of the failing CCSNe considered in this work are much hotter. They have central temperatures of ∼10–20 MeV and tens of MeV in their outer core and mantle. Thermal effects have a significant effect on their maximum masses.

In this study, we do not consider hyperonic EOS, e.g., the hyperonic extension of the HShen EOS by Ishizuka et al. (2008), or EOS involving other phases of nuclear matter, e.g., quarks and pions Nakazato et al. (2010). Such EOS are potentially interesting in failing CCSNe, since their exotic components lead to a softening of the EOS at high density, potentially accelerating BH formation (Sumiyoshi et al. 2009). We also do not consider EOS that include QCD phase transitions that too may lead to early PNS collapse and potentially to a second bounce and neutrino burst (Sagert et al. 2009).

We make use of single-star nonrotating presupernova models from several stellar evolution studies: Woosley & Weaver (1995) (WW95), Woosley et al. (2002) (WHW02), Limongi & Chieffi (2006) (LC06A/B), and Woosley & Heger (2007) (WH07). Each of these studies evolved stars with a range of ZAMS masses at solar metallicity (Z_☉, hereafter denoted with prefix s in model names) up until the onset of core collapse. In addition to solar metallicity, WHW02 evolved stars with ultra low metallicity, 10⁻⁴ Z_☉ (denoted by prefix u) and zero metallicity (denoted by prefix z). Rotation is of relevance in stellar evolution and stellar evolutionary processes affect the rotational configuration at the presupernova stage. In order to study BH formation, BH birth properties and their impact on a potential subsequent evolution to a GRB in such spinning progenitors, we draw representative models from Heger et al. (2000) (HLW00) and from Woosley & Heger (2006) (WH06) who included rotation in essentially the same way as we do in GR1D.

In Table 1, we list key parameters for all models in our set. These include presupernova mass, iron core mass (which we define as the baryonic mass interior to Y_e = 0.495), and the bounce compactness ξ_2.5. The latter is defined as

$$\begin{equation} \xi _{M} = {M {/} M_\odot \over R(M_\mathrm{bary} = M) {/} 1000\,\mathrm{km}}\Big |_{t =t_{\mathrm{bounce}}}, \end{equation} \tag{ 10 }$$

where we set M = 2.5 M_☉. R(M_bary = 2.5 M_☉) is the radial coordinate that encloses 2.5 M_☉ at the time of core bounce. ξ_2.5 gives a measure of a progenitor's compactness at bounce. We choose M = 2.5 M_☉ as this is the relevant mass scale for BH formation. ξ_2.5 is, as we shall discuss in Section 4.4, a dimensionless variable that allows robust predictions on the postbounce dynamics and the evolution of the model toward BH formation. The evaluation of ξ_2.5 at core bounce is crucial, since this is the only physical and unambiguous point in core collapse at which one can define a zero of time and can describe the true initial conditions for postbounce evolution. Computing the same quantity at the precollapse stage leads to ambiguous results, since progenitors come out of stellar evolution codes in more or less collapsed states. Collapse washes out these initial conditions and removes ambiguities.

We point out (as is obvious from Table 1) that there is a clear correlation between iron core mass and bounce compactness. Since the effective Chandrasekhar mass increases due to thermal corrections (Burrows & Lattimer 1983; Baron & Cooperstein 1990), more massive iron cores are hotter. Hence, progenitors with greater bounce compactness result in higher-temperature PNSs.

One of the most uncertain, yet most important, variables in the evolution of massive stars is the mass-loss rate. Mass loss can vary significantly over the life of a star. Current estimates of mass loss, either theoretical or based on fits to observational data, can depend on many parameters, including mass, radius, stellar luminosity, effective surface temperature, surface hydrogen and helium abundance, and stellar metallicity (de Jager et al. 1988; Nieuwenhuijzen & de Jager 1990; Wellstein & Langer 1999; Nugis & Lamers 2000; Vink & de Koter 2005). The mass-loss rate is uncertain in both the massive O-star and in the stripped-envelope Wolf-Rayet (W-R) star stage. O-star winds are expected to be responsible for the partial or complete removal of the hydrogen envelopes of massive stars. Recent observational results suggest that the rates used in current stellar evolution models may be too high by factors of 3–10 if clumped winds are considered correctly (Bouret et al. 2005; Fullerton et al. 2006; Puls et al. 2006). With the reduced rates, W-R stars would be difficult to make in standard single-star evolution and would require binary or eruptive mass-loss scenarios (Smith et al. 2010).

In Figure 2, we plot the mass-loss-induced mapping between ZAMS mass and presupernova mass for the ensemble of nonrotating progenitors listed in Table 1. WW95 models do not include mass loss—the presupernova models of this study have a mass equal to the ZAMS mass. WHW02 and WH07 employ the mass-loss rates of Nieuwenhuijzen & de Jager (1990) and Wellstein & Langer (1999) and use significantly reduced rates for low and zero metallicity stars. The u and z models of WHW02 have almost no mass loss and their presupernova masses are very close to their ZAMS values. The solar-metallicity stars of the sWHW02 and sWH07 model sets have significant mass loss, generally scaling with ZAMS mass. The most massive stars in these model sets have presupernova masses that are a small fraction of the initial ZAMS mass. For main sequence and giant phases, Limongi & Chieffi (2006) adopt mass-loss rates following Vink et al. (2000, 2001) and de Jager et al. (1988). For W-R stars, they either use the mass-loss rates of Nugis & Lamers (2000) (hereinafter referred to as LC06A models) or Langer (1989) (LC06B models). The latter are close to the values used for solar-metallicity stars in the WHW02 and WH07 model sets. The difference in the LC06A and LC06B mass-loss rates is roughly a factor of two. This, as portrayed by Figure 2 and evident from Table 1, can significantly alter the total mass at the onset of collapse and also has a strong effect on the iron core mass and bounce compactness.

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An additional uncertainty in massive star evolution is the phenomenon of large episodic mass loss (Smith 2008). Unknowns and uncertainties in both the cause and effect of large episodic mass loss currently prevent detailed stellar evolution calculations from including this phenomenon at all.

Based on resolution studies, we employ a computational grid setup with a total of 1050 zones. Near the origin and extending out to 20 km, we employ a constant grid spacing of 80 m (250 zones). Outside of 20 km we logarithmically space the remaining 800 zones to a radius where the initial density falls to 2000 g cm⁻³. We require the high resolution near the center for late postbounce times when the postshock region becomes small (r_shock≲ 20 km) and when the PNS is close to dynamical collapse to a BH. We interpolate the various presupernova profiles (ρ, T, Y_e, v, Ω) to our grid using linear interpolation.

In simulations including 1.5D rotation, we directly use the angular velocity of the progenitor model if it was evolved with rotation or assign specific angular momentum via the rotation law

$$\begin{equation} j (r) = j_{16,\infty } \left[ 1 + \left({A_{M_\odot } \over r}\right)^2\right]^{-1} 10^{16}\ \mathrm{cm}^2\ \mathrm{s}^{-1}\,, \end{equation} \tag{ 11 }$$

where j_16,∞ is the specific angular momentum at infinity in units of 10¹⁶ cm² s⁻¹. We define $A_{M_\odot }$ to be the radius where the enclosed mass is 1 M_☉. This is a variation on the rotation law commonly used in simulations of rotating core collapse (e.g., Ott et al. 2006), where Ω(r) = j(r)r⁻² is prescribed and the differential-rotation parameter A is set to some constant radius. The advantage of prescribing j (which is conserved along Lagrangian trajectories) and choosing the value of A based on a mass coordinate is that progenitors from different groups that are evolved to different points still yield similar PNS angular momentum distributions for a given choice of j_16,∞. Equation (11) leads to roughly uniform rotation in the core inside $A_{M_\odot }$ (j(r) ∝ r²) and angular velocity Ω(r) decreasing with r² further out (j(r) = const). We note that when 1 M_☉ of material is contained within 10³ km, which is typical of many progenitors, the central rotation rate is j_16,∞ rad s⁻¹.

Our way of assigning rotation to precollapse models approximates well the predictions of core rotation (inner ∼ few M_☉) from stellar evolution studies (see, e.g., Ott et al. 2006 for comparison plots) and, thus, is useful for studying rotational effects on BH formation. Equation (11) does not, however, capture the rise in specific angular momentum observed at larger radii (or mass coordinate) that is important for the potential evolution toward a long GRB and seen in recent rotating progenitor models (e.g., Woosley & Heger 2006).

We begin our discussion with a detailed description of the evolution of a failing CCSN from core collapse, through bounce, and the subsequent postbounce evolution to BH formation. For this, we choose the 40 M_☉ ZAMS-mass progenitor model s40WH07. We evolve this progenitor using the LS180 EOS, do not include rotation, and use the standard setting of f_heat = 1 (see Section 2.2). In Figure 3, we show the evolution of the radial coordinate of select baryonic mass shells as a function of time and we highlight shells enclosing 0.5, 1.0, 1.5, 2.0, and 2.5 M_☉. In addition, the figure shows the shock radius and the positions of the energy-averaged ν_e and ν_x neutrinospheres as a function of time. The prebounce collapse phase (t < 0) lasts ∼450 ms. At bounce, the central value of the lapse function is α_c ∼ 0.82, and the metric function X has a maximum of ∼1.1 and peaks off-center at a baryonic mass coordinate of ∼0.6 M_☉ which roughly corresponds to the edge of the inner core. The inner core initially overshoots to a maximum central density ρ_c ∼ 5.0 × 10¹⁴ g cm⁻³, then settles at ∼3.7 × 10¹⁴ g cm⁻³. ρ_c subsequently increases as accretion adds mass to the PNS. The bounce shock forms at a baryonic mass coordinate of ∼0.6 M_☉. From there, it moves out quickly in mass, reaching a baryonic mass coordinate of ∼1.5 M_☉ at 22 ms after bounce, 2 M_☉ at ∼162 ms, and 2.25 M_☉ at ∼329 ms. In radius, the shock reaches a maximum of ∼120 km at 38 ms after bounce. There it stalls, then slowly recedes. At 10 ms after bounce, the accretion rate through the shock is ∼18 M_☉ s⁻¹ and drops to ∼2.7, ∼1.7, and ∼1.25 M_☉ s⁻¹ at 100, 200, and 300 ms after bounce, respectively. The drop in the accretion rate has little effect on the failing supernova engine. In agreement with previous work that employed a more accurate neutrino treatment (e.g., Thompson et al. 2003; Liebendörfer et al. 2005), the 1D neutrino mechanism is manifestly ineffective in driving the shock, yielding, in this model, a heating efficiency $\eta = L_\mathrm{absorbed} / L_{\nu _e + \bar{\nu }_e}$ of only ∼3% (on average). The neutrinospheres (where the energy-averaged optical depth τ = 2/3) are initially exterior to the shock but are surpassed by the latter in a matter of milliseconds after bounce, leading to the ν_e deleptonization burst. At all times, the ν_x neutrinosphere is interior to the $\bar{\nu }_e$ neutrinosphere, which in turn is slightly interior to the ν_e neutrinosphere. The mean neutrino energies also follow this order. They are the largest for ν_x and the lowest for ν_e and increase with decreasing neutrinosphere radii (e.g., Thompson et al. 2003; Sumiyoshi et al. 2007; Ott et al. 2008; Fischer et al. 2009).

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At ∼408 ms after bounce, the shock has receded to ∼20 km and the PNS has reached a baryonic (gravitational) mass of ∼2.33 M_☉ (∼2.23 M_☉). The difference between baryonic and gravitational mass, at this point in the evolution, is due to the ∼1.9 × 10⁵³ erg of energy radiated by neutrinos. At this point, dynamical PNS collapse to a BH sets in and happens on a coordinate timescale of ≲1 ms. In the rightmost part of Figure 3, we zoom in to the final 1 ms of evolution to show detail. The first signs of collapse manifest themselves in the development of a radial infall velocity profile at the PNS edge. The PNS then collapses in on itself and the central density increases by a factor of ∼3 in only ∼1 ms of coordinate time. The simulation crashes due to EOS limitations at ρ_c ∼ 4.2 × 10¹⁵ g cm⁻³ and with α_c = 0.006. At this point the peak of the metric function X = [1 − 2m(r)/r]^−1/2 is ∼4.4 at a coordinate radius of ∼6.8 km. There, the fluid velocity also peaks at ∼−0.83 c. The shock recedes by ∼8 km in the last ∼1 ms of evolution to a radial coordinate of ∼12 km. During the last ∼0.05 ms, due to the central lapse dropping to nearly zero, the evolution of the mass shells slows near the origin. This is characteristic for our choice of gauge. If the simulation were to continue, X would become singular at the event horizon that would appear after infinite coordinate time in our coordinates (Petrich et al. 1986).

The s40WH07 model discussed here is a typical example of a failing CCSN in spherical symmetry. We present the results of a large number of such models in Table 2, where for each EOS and progenitor model we show the time to BH formation as measured from bounce and the mass, both baryonic and gravitational, of the PNS when the central value of the lapse function α reaches 0.3 (roughly the point of instability). In this table, the model name describes the initial model. The metallicity is denoted by one of three letters: s, u, and z which represent solar, 10⁻⁴ solar, and zero metallicity, respectively. Following the metallicity is the ZAMS mass and the progenitor model set. In many simulations, particularly in those employing stiff EOS, a BH does not form within 3.5 s. For these simulations we include in parentheses the mass inside the shock at 3.5 s. We note that at BH formation the shock is typically at a distance of ≲20 km and there is very little mass between the shock and the PNS. The dynamical collapse to a BH happens very quickly (t ≲ 1 ms) during which very little additional accretion occurs.

The s40WW95 progenitor was considered in the BH formation studies of Liebendörfer et al. (2004), Sumiyoshi et al. (2007) (hereinafter referred to as S07), and Fischer et al. (2009) (hereinafter referred to as F09). For comparison, we perform simulations with this progenitor for both the LS180 and HShen EOS. Table 3 compares two key quantities, the time to BH formation and the maximum baryonic PNS mass, obtained with GR1D with the results obtained in the aforementioned studies.

For the LS180 EOS, we find a time to BH formation of ∼524 ms and a maximum baryonic PNS mass of ∼2.26 M_☉, which is ∼3% larger than predicted by F09. We attribute this discrepancy to the different neutrino transport methods used. GR1D's leakage scheme has the tendency to somewhat overpredict electron-type neutrino luminosities (see the discussion in O'Connor & Ott 2010), resulting in lower gravitational masses compared to full Boltzmann transport calculations. Our time to BH formation is longer by ∼100 ms or ∼20%. This disagreement is relatively larger than the baryonic mass disagreement due to the low accretion rate at late times that translates small differences in mass to large differences in time. At ∼435.5 ms, the time to BH formation of F09, our PNS has a baryonic mass of ∼2.17 M_☉, which is consistent to ∼1% with F09. We find it more difficult to reconcile our results (and those of Liebendörfer et al. (2004) and F09) with the simulations of S07. Their maximum PNS baryonic mass and the time to BH formation suggest a lower accretion rate throughout their evolution (∼2.1 M_☉ in ∼560 ms).

In the simulation run with the stiffer HShen EOS, the larger maximum PNS mass leads to a delay of BH formation until a postbounce time ∼1.129 s and we find a maximum baryonic PNS mass of ∼2.82 M_☉. The maximum PNS mass and time to BH formation of S07 again suggest an accretion rate in disagreement with F09 and our work. The results of F09 with the HShen EOS suffer from a glitch in F09's EOS table interpolation scheme which has since been fixed (T. Fischer 2010, private communication). This leads to a postbounce time to BH formation of ∼1.4 s and a maximum baryonic PNS mass of ∼3.2 M_☉. Results from more recent simulations correct this error and are presented in Table 3 (T. Fischer 2010, private communication).

The maximum PNS mass and, thus, the evolution toward BH formation, depends strongly on the EOS. This was realized early on (Burrows 1988) and has recently been investigated by S07 and F09 who compared models evolved with the LS180 and HShen EOS. Here we extend their discussion and include also the LS220 and LS375 EOS. For a given accretion history, set by progenitor structure and independent of the high-density EOS, a stiffer nuclear EOS leads to a larger postbounce time to BH formation. In Figure 4, we plot the evolution of the central density ρ_c of the s40WH07 model evolved with the four considered EOS. Each EOS leads to a characteristic maximum central density at bounce that is practically independent of progenitor model: ∼4.8 × 10¹⁴ g cm⁻³, ∼4.4 × 10¹⁴ g cm⁻³, ∼3.7 × 10¹⁴ g cm⁻³, and ∼3.4 × 10¹⁴ g cm⁻³ for the LS180, LS220, LS375, and HShen EOS, respectively. As expected, the variant using the softest nuclear EOS (LS180) shows the steepest postbounce increase in ρ_c and becomes unstable to BH formation at only ∼408 ms for this progenitor. The onset of BH formation is marked by a quick rise in the central density. This is most obvious from the ρ_c evolutions of the model variants using the stiff HShen and LS375 EOS.

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Interestingly, the nominally stiffest EOS (LS375) leads to higher central densities than the softer HShen EOS up until ∼1.1 s after bounce. We find that this is due to the HShen EOS yielding higher pressure at ρ ≲ 3 × 10¹⁴ g cm⁻³, T ∼ 10 MeV, and Y_e ∼ 0.3. This higher pressure, initially in the core and later in the outer PNS layers, maintains the PNS at a lower central density. The cold-NS mass–radius relation shown in Figure 1 also exhibits this. For a given low-mass NS, the HShen EOS predicts a lower central density. For cold NSs, this trend continues until ρ_c ∼ 5.4 × 10¹⁴ g cm⁻³. Thermal effects, which are stronger in the HShen EOS, will increase this value for hot PNSs.

We also plot in Figure 4 the evolution of the mass accretion rate $\dot{M}$ in model s40WH07 (evaluated at a radius of 200 km). Variations in the high-density EOS have no effect on $\dot{M}$ which is most sensitive to progenitor structure. Sudden drops in $\dot{M}$ occur when density discontinuities that go along with compositional interfaces advect in. An example of this can be seen in s40WH07 at ∼400 ms after bounce where $\dot{M}$ drops by ∼30% due to a density jump at a baryonic mass coordinate of ∼2.35 M_☉. Such interfaces are common features of evolved massive stars (Woosley et al. 2002) and can help jumpstart shock revival in special cases (see, e.g., the 11.2 M_☉ model of Buras et al. 2006a, and Section 4.5 of this study). In the BH-formation context, they lead to a disproportionate increase in the time to BH formation in models whose EOS permit a PNS with mass greater than the mass coordinate of the density jump.

The maximum gravitational (baryonic) PNS masses for the four models shown in Figure 4 are ∼2.23 M_☉ (∼2.33 M_☉), ∼2.31 M_☉ (∼2.45 M_☉), ∼2.68 M_☉ (∼3.02 M_☉), and ∼2.59 M_☉ (∼2.83 M_☉); and the BH formation times are ∼408 ms, ∼561 ms, ∼1.596 s, and ∼1.259 s for the LS180, LS220, LS375, and HShen EOS, respectively. The maximum cold NS gravitational masses are ∼1.83 M_☉, ∼2.04 M_☉, ∼2.72 M_☉, and ∼2.24 M_☉ for the LS180, LS220, LS375, and HShen EOS, respectively.

In models evolved with the LS180, LS220, and HShen EOS, the maximum gravitational PNS mass is larger than the maximum gravitational cold NS mass. We can understand the differences between these cold NS and PNS maximum masses by comparing the PNS structure with various TOV solutions. In Figure 5, we plot the density and temperature profiles of the s40WH07 model evolved with the HShen EOS just prior to collapse to a BH. At this time, t ∼ 1.098 s, the central lapse is α_c = 0.35, the central density is ρ_c ∼ 1.44 × 10¹⁵ g cm⁻³, T_c ∼ 42.4 MeV, and the PNS gravitational (baryonic) mass is ∼2.51 M_☉ (∼2.74 M_☉). For comparison, we include in Figure 5 three TOV solutions, all with the same central density but different temperature and Y_e profiles. Specifically, we plot the density profile assuming (1) T(r) = 0.1 MeV, this is the "cold" NS case, (2) T(r) = 42.4 MeV, which corresponds to the central temperature from the GR1D evolution, and (3) T(r) = T_GR1D, assuming the same radial temperature profile as the GR1D model. We impose neutrinoless β-equilibrium for the former two TOV solutions and, similar to the temperature, assume the Y_e profile of the GR1D model for the latter. For this comparison, GR1D is run without neutrino pressure and energy contributions, since they are also neglected in the TOV solution.

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Inside of ∼6 km, corresponding to a gravitational mass coordinate of ∼0.4 M_☉, the material is not shock heated but rather is heated only via adiabatic compression. The outer regions (∼6–11 km) of the PNS are hot compared to the inner core. This is due to accretion and compression of shock heated material onto the PNS surface. In this region, the thermal pressure support is sufficiently strong to flatten out the PNS density profile. More mass is located at larger radii compared to constant-temperature TOV solutions. This decreases PNS compactness, increasing the maximum gravitational mass. The cold-NS and the T = T_c TOV solutions have a gravitational mass of ∼2.23 M_☉ and ∼2.35 M_☉, respectively. On the other hand, the TOV solution that assumes the same T and Y_e profile as the GR1D model yields a gravitational mass of ∼2.46 M_☉, within ∼2% of the PNS gravitational mass in the full GR1D simulation. Tests in which we vary the Y_e distribution in the TOV solution with T = T(r) show that the maximum PNS mass is insensitive to variations in Y_e from the GR1D profile to neutrinoless β-equilibrium. All this leads us to the conclusion that it is thermal pressure support in the outer PNS core that is responsible for increasing the maximum stable gravitational PNS mass beyond that of a cold NS. Our finding is in agreement with the recent BH formation simulations of Sumiyoshi et al. (2007) and Fischer et al. (2009), who noted the same differences to cold TOV solutions, but did not pinpoint their precise cause. However, our result is in disagreement with Burrows (1988) who reported maximum PNS masses within a few percent of a solar mass off their cold-NS values. This could be related to Burrows's specific choice of EOS. As we discuss below, stiff nuclear EOS have a more limited response to thermal effects. Another resolution to this disagreement could be the nature of his PNS cooling simulations that were not hydrodynamic, but rather employed a Henyey relaxation approach with imposed accretion.

We find the same overall systematics of increased maximum PNS mass due to thermal pressure support for the entire set of progenitors evolved with the LS180, LS220, and HShen EOS (variations due to differences in progenitor structure are discussed in Section 4.4). In the sequence of the LS EOS, the relevance of thermal pressure support decreases with increasing stiffness. In the case of the perhaps unphysically stiff LS375 EOS, the effect of high temperatures in the outer PNS core is reversed: the PNSs in GR1D simulations become unstable to collapse at lower maximum masses than their cold counterparts. This very surprising observation is understood by considering that in GR, higher temperatures not only add thermal pressure support to the PNS, but also increase its mass energy. This results in a deeper effective potential well and, thus, is destabilizing. In the LS180, LS220, and HShen case, the added thermal pressure support is significant and dominates over the latter effect. In the very stiff LS375 EOS, the added thermal pressure component is negligible, and the destabilizing effect dominates.

Finally, we point out quantitative differences in models evolved with and without neutrino pressure in the dense neutrino-opaque core. In the s40WH07 model evolved with the HShen EOS, the difference in the maximum gravitational mass is ∼0.08 M_☉ (∼3%) and the difference in the time to BH formation is ∼160 ms (∼14%). These numbers depend on the employed EOS and progenitor model. In test calculations with a variety of progenitors and EOS, we generally find increases of the maximum PNS gravitational mass of ≲5%.

The failure of a CCSN becomes definite only when accretion pushes the PNS over its maximum mass and a BH forms. Hence, the time to BH formation is a hard upper limit to the time available for the supernova mechanism to reenergize the shock. We will demonstrate in this section that it is possible to estimate, for a given nuclear EOS, the postbounce time to BH formation in non- or slowly spinning on the basis of a single parameter, progenitor bounce compactness, ξ_2.5, which we introduced in Section 3. In Figure 6, we plot the postbounce time to BH formation (t_BH) as a function of ξ_2.5 for all nonrotating models considered in this study and listed in Table 2. The distribution of data points for each EOS can be fit with a function ∝(ξ_2.5)^−3/2. This remarkable result can be understood as follows: using Kepler's third law, consider the Newtonian free fall time to the origin for a mass element dm initially located at r_* and on a radial orbit about a point mass of M* ≫ dm,

$$\begin{equation} t_{\mathrm{ff}} = {1\over 2} \sqrt{4\pi ^2a^3 \over GM^*} = \pi \sqrt{{r_*^3 \over 8 G M^*}}\,. \end{equation} \tag{ 12 }$$

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Here, for clarity, the quantities are in cgs units. G is the gravitational constant and a is the semimajor axis equal to half of the apoapsis, r_*. Recalling the definition of ξ_2.5, if the mass element dm is located at a mass coordinate of 2.5 M_☉ and at a radial coordinate of r_*, then r_* = 2500 km/ξ_2.5, and we can write the free fall time in terms of ξ_2.5,

$$\begin{equation} t_{\mathrm{ff}}^{2.5M_\odot } = 0.241(\xi _{2.5})^{-3/2}\,\mathrm{s}. \end{equation} \tag{ 13 }$$

In Figure 6, we overplot this Newtonian free fall time for a mass element at baryonic mass coordinate 2.5 M_☉ as a function of ξ_2.5. For small ξ_2.5, the mass element begins its free fall from a large radius and, hence, takes longer to reach to origin. In general, material in outer layers of the star will not begin to fall freely until it loses pressure support. Hence, the free fall approximation is not exact (within a factor of ∼2; Burrows 1986), but describes the general behavior of t_BH very well. The deviation of data points from the free fall curve is because the maximum PNS mass is different for each model and EOS. Models evolved with the LS180 EOS have PNSs with maximum baryonic masses ranging from 2.1to2.5 M_☉. Models with low ξ_2.5 correspond to the lower end of this mass range. For these models, t_BH can be somewhat less than the free fall time of the 2.5 M_☉ mass element, because less material is needed to form a BH. The maximum baryonic PNS mass range for models using the LS220 EOS is somewhat higher, 2.3–2.6 M_☉. BH formation times for these models are more in line with the Newtonian free fall prediction. Models evolved with the LS375 and HShen EOS have PNSs that must accrete upward of ∼3 M_☉ of material before becoming unstable. This significantly increases t_BH above the free fall prediction for the ξ_2.5 characteristic mass element.

Thermal pressure support can increase the maximum gravitational PNS mass (M_g, max) as we have seen in Section 4.3 for the s40WH07 model. In Figure 7, we plot M_g, max as a function of ξ_2.5 for all nonrotating models listed in Table 2. As obvious from this figure, M_g, max depends in a predictable way not only on the EOS, but also on the bounce compactness of the presupernova model. Progenitors with high ξ_2.5, in addition to forming BHs faster, create PNSs that are stable to higher masses. This is a simple consequence of the fact that progenitors with larger ξ_2.5 have iron cores with systematically higher entropies and masses significantly above the cold Chandrasekhar mass (see Table 1 and Baron & Cooperstein 1990 and Burrows & Lattimer 1983). Adiabatic collapse leads to higher PNS temperatures after bounce in progenitors with high ξ_2.5 and, hence, more thermal support. This leads to higher maximum PNS masses. This effect can be large, up to 25% for models with large ξ_2.5 and soft EOS.

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In a successful CCSN, the shock is reenergized before enough material can accrete onto the PNS to make it unstable. While fully self-consistent spherically symmetric simulations generally fail to explode in all but a few very low mass progenitors (cf. Kitaura et al. 2006; Burrows et al. 2007a), one can explode any star by the 1D neutrino mechanism by artificially increasing the energy deposition in the postshock region. Without such an increase, all of our simulations fail to explode. Our parameterized heating (f_heat in Equation 9) allows us to explore "how much" neutrino heating is needed to explode a given model (in 1D). By comparison with results from previous self-consistent radiation hydrodynamics simulations we can then estimate whether a given progenitor and EOS combination is more likely to lead to an explosion or BH formation.

Our method for driving explosions is similar to Murphy & Burrows (2008), but has the advantage of being proportional to the neutrino luminosity obtained from the neutrino leakage scheme and therefore conserves energy. We iteratively determine the critical value of f_heat to within 1% to what is needed for a successful explosion for a large subset of our models and the LS180, LS220, and HShen EOS. Of particular interest in this analysis is the time-averaged heating efficiency of the critical model, $\bar{\eta }^\mathrm{crit}_\mathrm{heat}$. We define $\bar{\eta }_\mathrm{heat}$ as

$$\begin{equation} \bar{\eta }_\mathrm{heat} = \overline{\int _{\mathrm{gain}} \dot{q}_\nu ^+\, dV \bigg / \left(L_{\nu _e} + L_{\bar{\nu }_e}\right)_{r_\mathrm{gain}}}, \end{equation} \tag{ 14 }$$

where $\dot{q}_\nu ^+$ is the net energy deposition rate and the neutrino luminosities are taken at the gain radius. We perform the time average between bounce and explosion, the latter time defined as when the postshock region assumes positive velocities and accretion onto the PNS ceases. $\bar{\eta }^\mathrm{crit}_\mathrm{heat}$ is a useful quantity because it characterizes how much of the available luminosity must be redeposited on average to explode a given progenitor. This is rather independent of transport scheme and code. For example, for the 15 M_☉ ZAMS solar-metallicity progenitor of Woosley & Weaver (1995) we find $\bar{\eta }^\mathrm{crit}_\mathrm{heat} \sim 0.13$. Buras et al. (2006b) who also artificially exploded this progenitor in 1D, though with much more sophisticated neutrino transport, find¹ an average heating efficiency of 0.1–0.15 which is consistent with our result. Note, however, that Marek & Janka (2009) observed in the same progenitor the onset of a self-consistent neutrino-driven explosion in 2D at an average heating efficiency of ∼0.07. This indicates a dependence of $\bar{\eta }^\mathrm{crit}_\mathrm{heat}$ on dimensionality, should be kept in mind, and is consistent with recent work that suggest that dimensionality may be the key to successful neutrino-driven explosions (Murphy & Burrows 2008; Nordhaus et al. 2010).

Since GR1D's leakage/heating scheme is only a rough approximation to true neutrino transport, and because our simulations assume spherical symmetry, we cannot make very robust quantitative predictions for any one particular model, but rather study the collective trends exhibited by the entire set of 62 progenitors that we consider here. In Figure 8, as a function of bounce compactness ξ_2.5, we plot $\bar{\eta }_\mathrm{heat}^\mathrm{crit}$ for all considered models and EOS. The data are summarized in Table 4. We can divide the results into two general regimes: models with ξ_2.5 ≲ 0.45 and those with ξ_2.5 ≳ 0.45.

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Table 4. Explosion Properties

MZAMS	ξ2.5	fcritheat	$\bar{\eta }^\mathrm{crit}_\mathrm{heat}$	MZAMS	ξ2.5	fcritheat	$\bar{\eta }^\mathrm{crit}_\mathrm{heat}$
(M☉)				(M☉)
sWH07 LS220				sWH07 LS180
14	0.128	1.17	0.158	15	0.182	1.16	0.193
15	0.182	1.17	0.172	21	0.143	1.32	0.144
16	0.150	1.33	0.134	23	0.452	1.18	0.192
17	0.169	1.32	0.146	24	0.409	1.16	0.163
18	0.195	1.17	0.188	25	0.334	1.13	0.158
19	0.177	1.24	0.146	27	0.258	1.18	0.153
20	0.288	1.15	0.176	30	0.219	1.16	0.179
21	0.143	1.34	0.133	35	0.369	1.14	0.164
22	0.292	1.15	0.181	40	0.599	1.32	0.266
23	0.453	1.17	0.165	45	0.556	1.26	0.245
24	0.410	1.15	0.163	50	0.221	1.18	0.172
25	0.334	1.14	0.185	80	0.210	1.22	0.142
26	0.235	1.21	0.142	sWH07 HShen
27	0.258	1.20	0.152	15	0.182	1.30	0.245
28	0.274	1.16	0.163	21	0.143	1.50	0.135
29	0.225	1.25	0.138	23	0.447	1.27	0.245
30	0.219	1.18	0.163	24	0.406	1.31	0.245
31	0.219	1.21	0.144	25	0.333	1.49	0.217
32	0.255	1.17	0.166	27	0.258	1.52	0.186
33	0.287	1.15	0.162	30	0.218	1.32	0.194
35	0.369	1.13	0.166	35	0.367	1.37	0.167
40	0.600	1.30	0.259	40	0.581	1.22	0.245
45	0.557	1.25	0.228	45	0.542	1.24	0.240
50	0.221	1.19	0.170	50	0.221	1.41	0.218
55	0.238	1.24	0.129	80	0.210	1.50	0.226
60	0.175	1.29	0.142
70	0.234	1.21	0.161	sWW95 LS180
80	0.210	1.24	0.143	15	0.088	1.33	0.130
100	0.247	1.15	0.175
120	0.172	1.25	0.152
uWHW02 LS220
20	0.338	1.13	0.155	40	0.721	1.44	0.297
25	0.223	1.16	0.168	45	0.656	1.22	0.267
30	0.326	1.13	0.156	50	0.575	1.09	0.174
35	0.666	1.37	0.284	60	0.624	1.12	0.133

Notes. f^crit_heat corresponds to the critical value needed to cause a successful explosion in GR1D. $\bar{\eta }^\mathrm{crit}_\mathrm{heat}$ is the associated critical average heating efficiency defined in Equation (14).

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For many models with ξ_2.5 ≲ 0.45, oscillations in the shock position are ubiquitous near the transition from failing to exploding supernovae in 1D (cf. Murphy & Burrows 2008; Buras et al. 2006b; Fernández & Thompson 2009). For both the LS180 and LS220 EOS, the $\bar{\eta }_\mathrm{heat}^\mathrm{crit}$ required for an explosion, modulo noise, is roughly constant and ∼0.16 on average for low ξ_2.5 models. Hence, explosion is the likely outcome of core collapse for progenitors with ξ_2.5 ≲ 0.45 if the nuclear EOS is similar to the LS180 or LS220 case.

The noise in the $\bar{\eta }_\mathrm{heat}^\mathrm{crit}$ distribution (absolute variations by up to ∼10%) is in part a consequence of variations in postbounce dynamics, such as the number and duration of pre-explosion oscillations. Compositional interfaces in some progenitor models, where jumps in the density lead to jumps in the accretion rate, also affect individual models leading to variations in $\bar{\eta }_\mathrm{heat}^\mathrm{crit}$. For the LS180 and LS220 EOS, any differences in $\bar{\eta }_\mathrm{heat}^\mathrm{crit}$ with choice of EOS are indistinguishable given the noise in the data.

For progenitors with ξ_2.5 ≳ 0.45, the $\bar{\eta }_\mathrm{heat}$ required to cause an explosion increases with ξ_2.5 when run with the LS180 or LS220 EOS. Progenitors in this regime have tremendous postbounce accretion rates, accumulating ≳2 M_☉ of baryonic material behind the shock within the first ∼200 ms after bounce. Without explosion, they form BHs within ≲0.8 s (with the LS180 and LS220 EOS). Hence, a very high heating efficiency of $\bar{\eta }_\mathrm{heat} \gtrsim 0.23\hbox{--}0.27$ is necessary to drive an explosion at early times against the huge ram pressure of accretion. It appears unlikely, even when multi-dimensional dynamics are factored in, that progenitors with ξ_2.5 ≳ 0.45 can be exploded via the neutrino mechanism. The most likely outcome of core collapse in such stars is BH formation.

We draw the reader's attention to two outliers in the uWHW02 data set included in Figure 8, the u50WHW02 and u60WHW02 progenitors. These models have high ξ_2.5, but feature compositional interfaces where the density drops by ∼50%. These are located at a mass coordinate of 1.82 M_☉ and 2.22 M_☉ in u50WHW02 and u60WHW02, respectively. When such an interface advects through the shock, the accretion rate drops suddenly, but the core neutrino luminosity remains large and an explosion is immediately launched. This results in a small value of f^crit_heat and, therefore, in a low required $\bar{\eta }_\mathrm{heat}$. This demonstrates that the single parameter ξ_2.5 is not always sufficient to predict a progenitor's fate.

In models with ξ_2.5 ≲ 0.45 and calculated using the HShen EOS, both $\bar{\eta }_\mathrm{heat}^\mathrm{crit}$ and f^crit_heat are systematically higher than with the LS180 and LS220 EOS and explosion is less likely. Furthermore, the qualitative behavior of our simulations is different with the HShen EOS. In many models with subcritical f_heat and $\bar{\eta }_\mathrm{heat}$, the shock is revived and begins to propagate to large radii of $\cal {O}$(10³–10⁴ km), but the material behind it fails to achieve positive velocities. Hence, accretion onto the PNS is slowed but does not cease. High values of f_heat are needed to avoid this and achieve full explosions. We caution the reader that this regime may not be well modeled by our neutrino treatment. Nevertheless, our results suggest that systematically higher f^crit_heat and $\bar{\eta }_\mathrm{heat}^\mathrm{crit}$ are required to explode models with the HShen EOS, even at low ξ_2.5. In contrast to models using the LS180 or LS220 EOS, models with ξ_2.5≳ 0.45 with the HShen EOS require roughly constant $\bar{\eta }_\mathrm{heat}$ to explode. Since the HShen EOS can support a high maximum mass, the PNS can withstand BH formation longer and explosions may set in at later postbounce times when the accretion rate has dropped sufficiently.

Finally, as an interesting aside, we point out the evolution of the u75WHW02 progenitor evolved with the LS220 EOS. This model has a bounce compactness of ∼1.15 and, in the absence of an explosion, forms a BH ∼0.285 s after bounce (with the LS220 EOS). This progenitor has a compositional interface at which the density drops by ∼50% that is located at a baryonic mass coordinate of ∼2.5 M_☉. This is very close to the maximum mass of the u75WHW02 PNS (with the LS220 EOS) and well above the maximum cold NS (baryonic) mass. The model can be made to explode with f^crit_heat = 1.35 with a corresponding $\bar{\eta }_\mathrm{heat}^\mathrm{crit} = 0.287$. The resulting PNS has a baryonic (gravitational) mass of ∼2.54 M_☉ (2.44 M_☉). Interestingly, within ∼100 ms after the launch of the explosion, cooling of the outer PNS layers removes sufficient thermal pressure, rendering the PNS unstable to collapse and BH formation. This scenario will necessarily occur within the cooling phase for any PNS that is initially thermally supported above the maximum cold NS baryonic mass and is another avenue to BH formation. In our simulations, this condition is also met only in very few other models with very high ξ_2.5 and fairly soft EOS, such as the 23, 40, and 45 M_☉ progenitors from the sWH07 series using the LS180 EOS. In order to fully investigate this BH formation channel, a more sophisticated neutrino treatment is required that allows accurate long-term modeling of PNS cooling (Pons et al. 1999), since, in general, the Kelvin–Helmholtz cooling phase of PNS is $\cal {O}$(10–100 s).

4.6.1. ZAMS Mass and Metallicity

Having established the systematic dependence of core collapse and BH formation on progenitor bounce compactness in Section 4.4, we now go further and attempt to connect to the conditions at ZAMS. Doing this is difficult, and, given the current state and limitations of stellar evolution theory and modeling, can be done only approximately. In general, presupernova stellar structure will depend not only on initial conditions at ZAMS (mass, metallicity, rotation), but also on particular evolution history and physics (binary effects, [rotational] mixing, magnetic fields, nuclear reaction rates, and mass loss; cf. Woosley et al. 2002). While keeping this in mind, we limit ourselves in the following to the exploration of single-star, nonrotating progenitors without magnetic fields. We focus on collapse models run with the LS220 EOS, but the general trends with EOS observed in the previous sections extend to here.

In the top panel of Figure 9, we plot the bounce compactness ξ_2.5 as a function of progenitor ZAMS mass M_ZAMS for a range of progenitors from multiple stellar evolutionary studies. Even within a given model set, the M_ZAMS–ξ_2.5 mapping is highly non-monotonic. At the low end of M_ZAMS covered by Figure 9, where mass loss has little influence even in progenitors of solar metallicity, variations in ξ_2.5 are due predominantly to particularities in late burning stages, caused, e.g., by convective versus radiative core burning and/or differences in shell burning episodes (Woosley et al. 2002). At the high ZAMS-mass end, ξ_2.5 is determined by a competition of mass loss and rapidity of nuclear-burning evolution.

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The bottom panel of Figure 9 depicts the time to BH formation t_BH in a failing CCSN as a function of M_ZAMS for the sWH07 solar-metallicity progenitors of Woosley & Heger (2007) and the uWHW02 10⁻⁴ solar-metallicity models of Woosley et al. (2002). Models of very low ξ_2.5 that require more than 3.5 s to make a BH are omitted. As demonstrated in Section 4.4, t_BH scales ∝(ξ_2.5)^−3/2 and, hence, progenitors that form BH the fastest and are (generally, cf. Section 4.5) the hardest to explode are those with high values of ξ_2.5. In the low-metallicity uWHW02 series whose progenitors experience only minuscule mass loss, BHs form within ≲1 s of bounce for M_ZAMS ≳ 30 M_☉ and the high bounce compactness ξ_2.5 ≳ 0.45 makes a successful shock revival rather unlikely (Section 4.5). Hence, the most likely outcome of core collapse is BH formation in these progenitors. This may also be the case for uWHW02 progenitors in the ZAMS mass range from ∼20to25 M_☉. The sWH07 progenitors have high ξ_2.5 and form BHs rapidly only in the M_ZAMS ranges ∼23–25 M_☉ and ∼35–45 M_☉. At higher ZAMS masses, strong O-star mass loss leads to an early removal of the hydrogen envelope. Subsequent mass loss in the W-R phase leads to bare, low-mass, low-compactness carbon–oxygen cores in the most massive progenitors that are unlikely to make BHs.

4.6.2. Variations with Mass-loss Prescriptions

Rotation, if sufficiently rapid, alters the CCSN dynamics via centrifugal support. This important effect is captured by GR1D's 1.5D rotation treatment, albeit approximately. Initially, centrifugal support acts to slow the collapse of the inner core, delaying core bounce. At bounce, lower peak densities are reached, the hydrodynamic shock forms at a larger radius, and its enclosed mass is larger. Conservation of angular momentum spins up the core from precollapse angular velocities that may be of order rad s⁻¹ to ${\cal {O}}$ (1000 rad s⁻¹) as the core, initially with $r \sim {\cal {O}}$ (1000 km), collapses to a PNS with $r \sim {\cal {O}}$ (30 km). During the postbounce evolution, the spinning PNS is stabilized at lower densities, is less compact, generally colder, and has a softer neutrino spectrum than a non-spinning counterpart (Ott et al. 2008, and references therein).

4.7.1. Models with Parameterized Rotation

We investigate the effect of rotation in failing CCSNe by assigning specific angular momentum profiles to the uWHW02 model set (see Section 3) via Equation (11). This rotation law approximates what is generally found in stellar evolution calculations that account for rotation (Heger et al. 2000; see Ott et al. 2006 for comparison plots). The inner iron core (∼1 M_☉) is rotating almost uniformly. Outside of this core, the angular velocity drops roughly ∝r⁻². In Table 5, we summarize key parameters of our rotating model set. Among them is T/|W|, the ratio of rotational kinetic energy to gravitational binding energy. It is particularly indicative of the dynamical relevance of rotation.

Table 5. Properties of Rotating Models

Model	Ωc, inita	Jb	T/|W|initc	T/|W|bounced	tBH	Mb,max	Mg,max	JPBHe	T/|W|PBHf	a*PBHg	$\overline{L_\nu }$h
	(rad s−1)	(1049 erg s)	(%)	(%)	(s)	(M☉)	(M☉)	(1049 erg s)	(%)		(100 B s−1)
u30WHW02LS180J1.0	1.34	9.97	0.048	3.32	0.990	2.24	2.11	1.61	5.06	0.41	2.07
u30WHW02LS180J2.0	2.69	19.94	0.193	13.44	1.223	2.33	2.20	3.48	20.08	0.81	1.58
u40WHW02LS180J0.5	0.57	8.07	0.006	0.87	0.371	2.35	2.25	0.86	1.16	0.19	4.99
u40WHW02LS180J1.0	1.14	16.14	0.025	3.50	0.381	2.37	2.27	1.75	4.66	0.39	4.77
u40WHW02LS180J1.5	1.71	24.21	0.057	7.96	0.398	2.41	2.30	2.69	10.41	0.58	4.41
u40WHW02LS180J2.0	2.28	32.29	0.101	14.26	0.421	2.45	2.35	3.69	18.09	0.76	3.95
u40WHW02LS180J2.5	2.85	40.36	0.157	21.91	0.455	2.52	2.41	4.81	27.70	0.94	3.46
u40WHW02LS180J3.0	3.42	48.43	0.226	24.92	0.519	2.63	2.52	6.19	38.94	[1.11]	2.93
u50WHW02LS180J1.0	1.31	22.51	0.020	3.05	0.588	2.30	2.18	1.66	4.79	0.40	3.30
u50WHW02LS180J2.0	2.62	45.02	0.079	12.58	0.662	2.38	2.26	3.54	18.80	0.78	2.73
u60WHW02LS180J1.0	1.05	30.03	0.013	3.00	0.453	2.38	2.28	1.72	4.36	0.38	3.64
u60WHW02LS180J2.0	2.10	60.07	0.052	12.46	0.540	2.44	2.34	3.60	17.17	0.75	2.92
u30WHW02LS220J1.0	1.34	9.97	0.048	3.30	1.419	2.38	2.20	1.75	4.97	0.41	1.89
u30WHW02LS220J2.0	2.69	19.94	0.193	13.43	1.697	2.45	2.29	3.77	19.89	0.82	1.41
u40WHW02LS220J0.5	0.57	8.07	0.006	0.86	0.455	2.47	2.34	0.94	1.16	0.19	5.19
u40WHW02LS220J1.0	1.14	16.14	0.025	3.48	0.462	2.49	2.36	1.89	4.61	0.39	4.95
u40WHW02LS220J1.5	1.71	24.22	0.057	7.94	0.474	2.51	2.39	2.89	10.22	0.58	4.58
u40WHW02LS220J2.0	2.28	32.29	0.100	14.29	0.488	2.55	2.42	3.93	17.78	0.76	4.09
u40WHW02LS220J2.5	2.85	40.36	0.157	22.13	0.508	2.59	2.48	5.00	26.58	0.93	3.57
u40WHW02LS220J3.0	3.42	48.43	0.226	24.57	0.549	2.67	2.56	6.36	38.01	[1.10]	2.99
u50WHW02LS220J1.0	1.31	22.51	0.020	3.04	0.725	2.43	2.28	1.82	4.73	0.40	3.45
u50WHW02LS220J2.0	2.62	45.03	0.079	12.60	0.777	2.49	2.35	3.81	18.51	0.78	2.86
u60WHW02LS220J1.0	1.05	30.03	0.013	2.99	0.602	2.48	2.35	1.84	4.31	0.38	3.48
u60WHW02LS220J2.0	2.10	60.07	0.052	12.49	0.664	2.53	2.41	3.80	16.83	0.74	2.87
u30WHW02LS375J1.0	1.34	9.97	0.048	3.24	...	(2.81)	(2.50)	(2.30)	(5.14)	(0.42)	(1.35)
u30WHW02LS375J2.0	2.69	19.94	0.192	13.32	...	(2.82)	(2.55)	(4.55)	(18.73)	(0.79)	(1.15)
u40WHW02LS375J0.5	0.57	8.07	0.006	0.85	0.941	3.02	2.71	1.27	1.10	0.20	5.53
u40WHW02LS375J1.0	1.14	16.14	0.025	3.44	0.926	3.02	2.72	2.52	4.31	0.39	5.34
u40WHW02LS375J1.5	1.71	24.22	0.057	7.88	0.904	3.01	2.74	3.77	9.50	0.57	5.02
u40WHW02LS375J2.0	2.28	32.29	0.100	14.26	0.873	2.99	2.75	5.02	16.52	0.75	4.59
u40WHW02LS375J2.5	2.85	40.36	0.157	22.36	0.845	2.99	2.78	6.07	23.52	0.89	4.11
u40WHW02LS375J3.0	3.42	48.43	0.226	23.43	0.788	2.96	2.79	7.46	35.38	[1.09]	3.45
u50WHW02LS375J1.0	1.31	22.51	0.020	2.99	1.346	3.02	2.69	2.56	4.53	0.40	4.07
u50WHW02LS375J2.0	2.62	45.03	0.079	12.54	1.296	2.99	2.72	5.09	17.49	0.78	3.44
u60WHW02LS375J1.0	1.05	30.04	0.013	2.95	1.330	3.00	2.71	2.44	4.08	0.38	3.62
u60WHW02LS375J2.0	2.10	60.07	0.052	12.47	1.284	2.98	2.74	4.93	16.01	0.75	3.09
u30WHW02HShenJ1.0	1.34	9.96	0.049	3.03	3.335	2.75	2.49	2.00	3.81	0.37	1.25
u30WHW02HShenJ2.0	2.69	19.92	0.195	12.45	...	(2.79)	(2.56)	(4.48)	(18.10)	(0.78)	(1.03)
u40WHW02HShenJ0.5	0.57	8.07	0.006	0.79	0.854	2.88	2.64	1.17	1.06	0.19	4.44
u40WHW02HShenJ1.0	1.14	16.13	0.025	3.21	0.873	2.90	2.67	2.36	4.13	0.38	4.28
u40WHW02HShenJ1.5	1.71	24.20	0.057	7.33	0.901	2.93	2.71	3.64	9.35	0.56	4.03
u40WHW02HShenJ2.0	2.28	32.26	0.101	13.27	0.932	2.97	2.76	5.00	16.51	0.74	3.69
u40WHW02HShenJ2.5	2.85	40.33	0.157	20.79	0.959	3.01	2.82	6.36	25.20	0.91	3.28
u40WHW02HShenJ3.0	3.42	48.39	0.226	25.09	0.999	3.06	2.88	7.83	35.14	[1.07]	2.80
u50WHW02HShenJ1.0	1.31	22.49	0.020	2.78	1.195	2.85	2.61	2.35	4.45	0.39	3.29
u50WHW02HShenJ2.0	2.62	44.99	0.080	11.62	1.266	2.93	2.71	4.93	17.23	0.76	2.86
u60WHW02HShenJ1.0	1.05	30.01	0.013	2.72	1.197	2.88	2.65	2.31	4.05	0.37	3.02
u60WHW02HShenJ2.0	2.10	60.02	0.052	11.48	1.273	2.94	2.73	4.75	15.38	0.72	2.65
m35OCWH06LS180	1.98	780.36	0.281	8.73	0.749	2.33	2.22	2.64	11.68	0.61	2.37
m35OCWH06LS220	1.98	780.38	0.281	8.71	0.972	2.43	2.29	2.69	10.18	0.58	2.33
m35OCWH06LS375	1.98	780.42	0.281	8.66	2.194	3.01	2.69	3.47	8.34	0.54	2.36
m35OCWH06HShen	1.98	779.77	0.281	8.00	1.907	2.84	2.60	3.30	8.85	0.55	2.01
E20HLW00LS180	3.13	18.15	0.242	17.59	1.126	2.25	2.12	2.77	14.87	0.70	1.82
E20HLW00LS220	3.13	18.15	0.242	17.63	1.666	2.41	2.25	3.14	14.79	0.70	1.66
E25HLW00LS180	1.83	8.05	0.089	5.92	1.103	2.23	2.08	1.76	6.44	0.46	1.97
E25HLW00LS220	1.83	8.05	0.088	5.90	1.703	2.37	2.18	1.83	5.76	0.44	1.68

Notes. The u series of presupernova models in this table are taken from the Z_☉ = 10⁻⁴ model set of Woosley et al. (2002). We imposed a rotation law via Equation (11), the value of j_16, ∞ is given in the model name following the letter J. The m35OC presupernova model is taken from Woosley & Heger (2006) and both the E20 and E25 models are from Heger et al. (2000), these model are evolved with rotation. In simulations where a BH did not form within 3.5 s we give, in parenthesis, the values at this time. ^aInitial central angular velocity of star. ^bTotal angular momentum of star. ^cInitial T/|W| of the star. ^dT/|W| of the star at bounce. ^eAngular momentum of protoblack hole (PBH) when α_c = 0.3. ^fT/|W| of the star when α_c = 0.3. ^gDimensionless spin of the PBH when α_c = 0.3. Unphysical values for a BH are shown in braces [  ⋅⋅⋅  ]. ^hTotal neutrino luminosity averaged over postbounce time.

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In the right panel of Figure 10, we plot the central density evolution of model u40WHW02 run with the LS180 EOS for j_16,∞ ranging from 0 to 3.25 in increments of 0.25. While we choose the u40WHW02 model here, the results are generic and apply to all progenitors. $A_{M_\odot }$, of Equation (11), is 936 km for this model and the initial central rotation rate is 1.14 × j_16,∞ rad s⁻¹. The nonrotating model takes ∼433 ms to reach bounce and a further ∼369 ms before the PNS becomes unstable to collapse to a BH with a gravitational mass of 2.24 M_☉. For the j_16,∞ = 1, 2, and 3 models, respectively, the times to bounce are 11 ms, 47 ms, and 125 ms greater than in the nonrotating case. Their times to BH formation t_BH are 12 ms, 52 ms, and 150 ms longer than in the nonrotating case. The maximum gravitational PNS masses M_g, max are 0.03 M_☉, 0.09 M_☉, and 0.28 M_☉ greater. We find that the time to bounce, time to BH formation, and maximum gravitational PNS mass increase above the nonrotating values proportional to ∼(j_16,∞)². The increase in t_BH is due almost entirely to the increase in M_g, max, since the accretion rate is not significantly affected by rotation.

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The lower temperatures and densities of rotating PNSs lead to systematically lower mean neutrino energies and total radiated energy from the PNS core (time-averaged total luminosities are summarized in Table 5). Fryer & Heger (2000) and Ott et al. (2008), who considered similarly rapidly rotating models, also see this effect. There is a clear trend toward lower L_ν with increasing j_16,∞ and for a given model and at a given time, with increasing j_16,∞, less gravitational binding energy has been carried away by neutrinos and M_g is larger. Given essentially unaltered accretion rates, one may expect earlier PNS collapse and BH formation. This, however, is not the case in models run with the LS180, LS220, and HShen EOS, since the centrifugally increased M_g,max systematically outweighs the increased gravitational mass due to the lowered neutrino emission. For these EOS, the time to BH formation is delayed by rotation. For models run with the extremely stiff LS357 EOS the situation is different. For them, the centrifugal support provided by rotation is too weak to significantly enhance M_g, max and, hence, BHs form faster with increasing j_16,∞.

In the left panel of Figure 10, we plot the T/|W| evolution for the rotating u40WHW02 model series run with the LS180 EOS. During collapse, gravitational binding energy is transferred to rotational energy, increasing the value of T/|W|. Similar to how the central density overshoots its new equilibrium, T/|W| also exhibits a local maximum at bounce. Continued accretion and contraction of the PNS increases T/|W| throughout the postbounce evolution for all models. Initially very rapidly spinning models experience core bounce under the strong influence of centrifugal effects, leading to reduced compactness and T/|W| at bounce. These qualitative features are in good agreement with what was found by previous extensive parameter studies of rotating core collapse (Ott et al. 2006; Dimmelmeier et al. 2008). Quantitatively, we find and summarize in Table 5 that models with j_16,∞ ≲ 1.5 yield T/|W| ≲ 0.14 throughout their entire evolution. Models with j_16,∞ ≳ 2.25 have T/|W| ≳ 0.14 during their entire postbounce evolution. Models that have j_16,∞ ≲ 2.25 have T/|W| ≲ 0.27 at all times. Models with j_16,∞ ≳ 2.5 reach T/|W| ≳ 0.27 before BH formation. When considering these numbers, it is important to keep in mind that GR1D's 1.5D approach to rotation has the tendency to overestimate T/|W| in rapidly spinning models. Ott et al. (2006) found model-dependent differences in T/|W| of $\mathcal {O}\,(10\%)$ between 1.5D and 2D. In addition, GR1D's neutrino leakage scheme also tends to lead to somewhat more compact PNS cores and consequently higher T/|W| than would be expected from full neutrino transport calculations.

The systematics of T/|W| depicted by Figure 10 (left) and listed in Table 5, albeit only approximate due to GR1D's 1.5D treatment of rotation, shed interesting light on the potential role of nonaxisymmetric rotational instabilities during the evolution of failing CCSNe. Of course, due to its 1.5D nature, GR1D cannot track the development of such multi-dimensional dynamics. Analytic theory and to some extent 3D computational modeling have identified multiple instabilities that may lead to triaxial deformation of PNSs, redistribution of angular momentum, and to the radiation of rotational energy and angular momentum via gravitational waves (see Stergioulas 2003 and Ott 2009 for reviews). A global dynamical instability sets in for T/|W| ≳ 0.27 (Chandrasekhar 1969), leading to a lowest-order m = 2 "bar" deformation. Global secular instability, driven by viscosity or GW back-reaction sets in at T/|W| ≳ 0.14 (Chandrasekhar 1970; Friedman & Schutz 1978). Finally, dynamical shear instabilities, arising as a result of differential rotation, may lead to partial or global nonaxisymmetric deformation at even lower values of T/|W| (≳0.05; e.g., Saijo et al. 2003; Ott et al. 2007; Scheidegger et al. 2008; Corvino et al. 2010, and references therein). In nature, and in full 3D simulations, these instabilities, through gravitational radiation or redistribution of angular momentum, will effectively and robustly prevent T/|W| from surpassing the corresponding T/|W| threshold. The growth times of dynamical instabilities are short, $\cal {O}$(ms). Secular instabilities grow on timescales set by the driving process and are typically $\cal {O}$(s) (Lai & Shapiro 1995). The low-T/|W| shear instabilities in PNSs appear to grow on intermediate timescales of $\cal {O}$(10–100 ms) (e.g., Ott et al. 2007; Scheidegger et al. 2008).

In Figure 11, we plot the value of T/|W| (left panel) and the dimensionless spin of the protoblack hole (PBH), a*_PBH = J_PBH/(M²_g, PBH) (right panel) at the onset of BH formation (when α_c = 0.3) for the same values of j_16,∞ used in Figure 10. Assuming that the entire PNS is promptly swallowed once the horizon appears, a*_PBH corresponds to the BH birth spin.³ We again show results for model u40WHW02, but for all four EOS. The data are also presented in Table 5 for these and other models. T/|W| at BH formation scales ∝(j_16,∞)²: T/|W|_PBH is ∼0.05, ∼0.1, ∼0.2, and ∼0.3 at j_16,∞ of ∼1, ∼1.5, ∼2.2, and ∼2.75, respectively. a*_PBH scales linearly with j_16,∞, reaching a maximally Kerr value of a*_PBH ∼ 1 at j_16,∞ ∼ 2.75. T/|W|_PBH and a*_PBH vary little with EOS.

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A disturbing fact depicted by Figure 11 is that our 1.5D simulations predict BH birth spins a* ≳ 1 for j_16,∞ ≳ 2.75. In Kerr theory, such BHs cannot exist with a horizon. They would instead be naked singularities, violating the cosmic censorship conjecture (Penrose 1969). However, when comparing right and left panels of Figure 11, one notes that all models achieving a* ≳ 1 are predicted to reach T/|W| in excess of 0.27. Hence, in nature and in a 3D simulation, these PNS will be dynamically nonaxisymmetrically unstable and angular momentum redistribution and gravitational radiation will limit their T/|W| robustly below ∼0.27, corresponding to a* ≲ 0.9. Rotational instabilities at lower values of T/|W| may also be relevant. Dynamical shear instabilities have timescales significantly less than the time to BH formation. Secular rotational instabilities may be relevant if the true nuclear EOS allows for a large maximum PNS mass as more time is needed to accrete the necessary material to form a BH (see Section 4.3). Large t_BH is also possible if ξ_2.5 is small (see Section 4.4) therefore allowing secular instabilities to grow. In all rotating models considered here (see Table 5), PNSs stable against the dynamical rotational instability with T/|W| ≲ 0.25–0.27 throughout their postbounce evolution have a*_PBH ≲ 0.9. Similarly, PNSs with T/|W| ≲ 0.14–0.16, the threshold for secular instability, have a*_PBH ≲ 0.6–0.7. If low-T/|W| instabilities are effective at limiting T/|W| in PNSs on short timescales, nascent BH spins may be limited to low values (a* ≲ 0.4 for a T/|W| instability threshold of ∼0.05).

4.7.2. Rotating Progenitors and the Connection to Long GRBs

The rotation law of Equation (11) qualitatively follows the predicted angular velocity distribution in the inner ∼1–3 M_☉ of presupernova models evolved with rotation. However, Equation (11) asymptotes to constant specific angular momentum j and cannot capture jumps and secular increase of j in overlying mass shells (e.g., Heger et al. 2000).

Here, we consider three different supernova progenitors evolved with rotation that have the potential of forming BHs soon after bounce. Models E20 and E25 are rapidly rotating unmagnetized solar-metallicity presupernova models of a 20 and 25 M_☉ ZAMS stars from Heger et al. (2000). Model m35OC of Woosley & Heger (2006) with M_ZAMS = 35 M_☉ has 10% solar-metallicity, reduced mass loss, and magnetic fields. These presupernova models have initial central angular velocities of ∼3.13, ∼1.83, and ∼1.98 rad s⁻¹ and values of ξ_2.5 of ∼0.319, ∼0.294, and ∼0.456 for the E20, E25, and m35OC models, respectively. Due to their moderate ξ_2.5, we perform collapse simulations of models E20 and E25 with the LS180 and LS220 EOS. Model m35OC is calculated with all four EOS. The progenitor characteristics are summarized in Tables 1 and 5.

For a given EOS, due to very similar ξ_2.5., the evolutions of E20 and E25 are alike. They form BHs in ∼1.1 s with the LS180 EOS and in ∼1.7 s with the LS220 EOS. Model E20 is more rapidly spinning. Its T/|W| peaks at ∼0.18 at bounce and settles down to a nearly constant value of ∼0.15 throughout the postbounce evolution. Its a^⋆_PBH is ∼0.7 for both the LS180 and LS220 EOS. Model E25 reaches T/|W| ∼ 0.06 at bounce and ∼0.065 at BH formation with a^⋆_PBH ∼ 0.45 for both EOS.

The core of the m35OC model is sufficiently compact to form a BH soon after bounce if no explosion is launched (e.g., via magneto-rotational explosion; Dessart et al. 2008). The nascent BH forms at a time of ∼0.75 s, ∼0.97 s, ∼2.19 s, and ∼1.91 s for the LS180, LS220, LS375, and HShen EOS, respectively. The initial gravitational (baryonic) BH masses are ∼2.22 (∼2.33) M_☉, ∼2.29 (∼2.43) M_☉, ∼2.69 (∼3.01) M_☉, and ∼2.60 (∼2.84) M_☉ for the LS180, LS220, LS375, and HShen EOS, respectively. The BHs are modestly rapidly spinning with a*_PBH of ∼0.61, ∼0.58, ∼0.54, and ∼0.55 for the LS180, LS220, LS375, and HShen EOS, respectively. For all EOS, the PNS, during the accretion phase, has a modest T/|W| of ≲0.12.

Once a BH is formed, material from the stellar mantle will continue to accrete at high rates. Accretion will only be slowed once material with sufficiently high specific angular momentum reaches small radii and becomes centrifugally supported, forming an accretion disk. This is the crucial prerequisite for the collapsar scenario for long GRB central engines to work (Woosley 1993). Models E20 and E25 lost much of their initial mass and angular momentum during their evolution to the presupernova stage and, therefore, there is too little angular momentum in their outer regions to allow for a long-term accretion disk.

The situation is different for the m35OC progenitor. Its particular evolution prevented dramatic loss of mass and angular momentum while keeping its envelope radius small. In Figure 12, we show the specific angular momentum distribution of the m35OC progenitor as a function of enclosed baryonic mass. We also include graphs of the j_ISCO, the specific angular momentum required for a stable orbit at the innermost stable circular orbit (ISCO) of a Kerr hole with mass M and spin a^⋆ (Bardeen et al. 1972). Curves for a* = 0 (Schwarzschild), a* = 1 (maximally Kerr), and a* = J(M)/M² are shown. We also plot the value of a* that a BH of baryonic mass M formed from the m35OC progenitor would have. Figure 12 is independent of the detailed collapse evolution, assuming that no angular momentum is radiated by neutrinos and/or gravitational waves, or ejected. However, we note that due to emission of neutrinos before BH formation, the enclosed gravitational mass (entering into the calculation of a*) will be smaller by up to ∼0.2–0.4 M_☉ than the baryonic mass given in the figure. This leads to slightly underpredicted values of a* for small M. Since relatively little energy is emitted in neutrinos after BH formation, the relative discrepancy between gravitational and baryonic mass decreases with growing BH mass.

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Figure 12 can be interpreted as follows. If the CCSN mechanism fails to reenergize the shock and the PNS collapses to a BH of mass M, then its initial angular momentum J and spin a^⋆ will be set by the enclosed angular momentum and gravitational mass. Initially, hyperaccretion will increase both J and M, but a^⋆ may increase or decrease, depending on the angular momentum of the accreted matter. Accretion will slow down and a disk will form once infalling material has specific angular momentum j greater than j_ISCO. In model m35OC, this occurs between a BH mass coordinate of ∼7.6 M_☉ (for a mass element with equatorial j) and ∼10.1 M_☉ (for a mass element with angle-averaged j). These points are marked in Figure 12 with a (★) and (▪), respectively. Using accretion simulations with GR1D setup to include an inflow inner boundary condition we find that with the m35OC model, the accretion time for 7.6 M_☉ and 10.1 M_☉ is ∼10.2 s and ∼14.3 s from the onset of collapse, respectively. These times are roughly twice the free fall time, since material in outer regions in hydrostatic equilibrium for a sound travel time (Burrows 1986). Once the disk has formed, accretion will continue via processes that will transport angular momentum out and mass in. A collapsar central engine may begin its operation (Woosley 1993; MacFadyen & Woosley 1999) with a central BH of M ∼ 8 M_☉, a* ∼ 0.75, and an ISCO radius of ∼40 km.