Ultra-delayed neutrino-driven explosion of rotating massive-star collapse
Sho Fujibayashi, Koh Takahashi, Yuichiro Sekiguchi, Masaru Shibata
DDraft version February 10, 2021
Typeset using L A TEX twocolumn style in AASTeX62
Ultra-delayed neutrino-driven explosion of rotating massive-star collapse
Sho Fujibayashi, Koh Takahashi, Yuichiro Sekiguchi,
2, 3 and Masaru Shibata
1, 2 Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut), Am M¨uhlenberg 1, D-14476 Potsdam-Golm, Germany Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan Department of Physics, Toho University, Funabashi, Chiba 274-8510, Japan (Received; Revised February 10, 2021; Accepted)
ABSTRACTLong-term neutrino-radiation hydrodynamics simulations in full general relativity are performed forrotating massive stars that are evolved from He-stars with their initial masses of 20 and 32 M (cid:12) . It isshown that if the collapsing stellar core has sufficient angular momentum, the rotationally-supportedproto-neutron star (PNS) survives for seconds accompanying the formation of a massive torus of masslarger than 1 M (cid:12) . Subsequent mass accretion onto the central region produces a massive and compactcentral object, and eventually enhances the neutrino luminosity beyond 10 erg/s, resulting in a verydelayed neutrino-driven explosion in particular toward the polar direction. The kinetic energy of theexplosion can be appreciably higher than 10 erg for a massive progenitor star and compatible withthat of energetic supernovae like broad-line type-Ic supernovae. By the subsequent accretion, themassive PNS collapses eventually into a rapidly spinning black hole, which could be a central enginefor gamma-ray bursts if a massive torus surrounds it. Keywords: stars: neutron–supernovae; general–hydrodynamics–neutrinos–relativistic processes INTRODUCTIONCore-collapse supernovae (SNe) are explosive eventsthat occur at the final stage of the massive-star evolu-tion. In the typical scenario (e.g., Janka et al. 2012),after the collapse of the iron core of progenitor stars,a proto-neutron star (PNS) is first formed. Then, ashock wave is generated at the inner core of the PNSand propagates outward sweeping the material. How-ever, because of the photo-dissociation of irons, theshock is stalled in the middle of the propagation. Sub-sequently, the heating by neutrinos emitted from thePNS is believed to play a key role for supplying the en-ergy to the stalled shock (Bethe & Wilson 1985). Ifthe neutrino heating timescale becomes shorter thanthat of the matter infall from the outer envelop, thestalled shock is revived and the explosion is driven bythe neutrino heating (Janka 2001). By contrast, ifthe neutrino heating is not efficient enough, the stalledshock eventually goes back to the PNS and a black hole(BH) is formed. In particular, for high-mass progenitor
Corresponding author: Sho [email protected] stars with the zero-age main-sequence (ZAMS) mass of M ZAMS (cid:38) M (cid:12) (Woosley et al. 2002), the naive expec-tation for the final fate is the formation of a BH withoutthe shock revival.As summarized above, the key quantity for the suc-cessful explosion is the efficiency of the neutrino heat-ing (Janka 2001). In fact, many sophisticated simula-tions for core-collapse SNe have shown that the successof the SN explosion depends sensitively on the neutrinoluminosity (for the latest numerical simulations in thisfield, see, e.g., M¨uller et al. 2012; Burrows et al. 2019;Nakamura et al. 2019; Mezzacappa et al. 2020; M¨uller2020; Stockinger et al. 2020; Kuroda et al. 2020; Haradaet al. 2020; Obergaulinger & Aloy 2020a; Bollig et al.2020).In this paper, we propose a mechanism by which theneutrino luminosity of the central object is naturallyenhanced for very high-mass rotating progenitor stars.We consider a rotating progenitor core, which resultsin a PNS rapidly rotating with the rotational periodof ≤ M (cid:12) . Due to the rapid rotation, the PNSwith the rest mass (cid:38) M (cid:12) can survive for the equationof state (EOS) with which the maximum gravitational a r X i v : . [ a s t r o - ph . H E ] F e b Fujibayashi et al. mass for cold non-rotating neutron stars (NSs), M max ,is larger than 2 M (cid:12) . This appreciably increases the life-time of the PNS. In addition, due to the presence of acompact massive torus as well as the high mass of thePNS, the total neutrino luminosity is enhanced duringthe evolution of the system. Furthermore, because of theflattened geometry of these central objects, the neutrinoflux is enhanced in the polar region. As a consequence,the neutrino heating timescale of the stalled shock be-comes shorter than the infalling timescale of the matterin the polar region, leading to a bipolar explosion.By performing numerical-relativity simulations, wedemonstrate that this mechanism can indeed work for M ZAMS ≈ M (cid:12) (This corresponds to the range ofHe-core mass of M He = 20–32 M (cid:12) ). For such high-massprogenitors, the total mass of the PNS and surround-ing torus becomes also high, and hence, the neutrinoluminosity is enhanced as well. As a result, the bipolaroutflow becomes more energetic than the ordinary SNe;the resulting kinetic energy of the ejecta can be severaltimes 10 erg. Thus, this mechanism may produce aclass of energetic SNe like broad-line type-Ic SNe (see,e.g., Woosley & Bloom (2006); Cano et al. (2017) forreviews).In §
2, we summarize the progenitor models employedas the initial condition for numerical-relativity simula-tions together with a brief summary of our method forthe simulation. The results for the successful explosionare shown in §
3. Section 4 is devoted to a summaryand a discussion. MODELS AND METHODWe employ the final state of high-mass stellar evolu-tion models as the initial condition of our numerical-relativity simulations. The stellar evolution of non-rotating He-star models with their initial masses of M He = 20 and 32 M (cid:12) is calculated using the code de-scribed in Takahashi et al. (2018). For these models, M ZAMS ≈
45 and 65 M (cid:12) , respectively (Woosley et al.2002). The evolution calculation is performed until thecentral temperature reaches ≈ × K. At this stage,the central density is ≈ × g / cm for M He = 20 M (cid:12) and ≈ × g / cm for M He = 32 M (cid:12) .We then add angular momentum to the final stateof the evolved stars for numerical-relativity simulations.Specifically, the following cylindrical profile is imposedfor the angular velocity:Ω = Ω e − R /R , (1)where Ω is the angular velocity along the rotation axis( z -axis), R the cylindrical radius, and R a cut-off ra-dius. This rotational profile is somewhat different from Table 1.
List of the models and the results. t exp and t BH denote the post-bounce time at the onset of the explosionand that of the BH formation, respectively.Model M He Ω R t exp t BH E exp ( M (cid:12) ) (rad/s) (km) (s) (s) (10 erg)M20-S040 20 0.40 6000 — 1.1 —M20-S050 20 0.50 6000 3.5 4.3 3.8M20-S075 20 0.75 6000 5.2 7.2 4.6M20-S100 20 1.00 6000 6.0 9.8 3.9M20-L050 20 0.50 8500 8.3 9.1 1.5M20-S050N 20 0.50 6000 3.7 4.3 1.7M32-S050 32 0.50 5800 — 1.0 —M32-S075 32 0.75 5800 2.8 4.3 54.0M32-S100 32 1.00 5800 3.5 5.1 27.3M32-S075DD2 32 0.75 5800 2.9 4.4 59.3M32-S075N 32 0.75 5800 3.0 4.3 11.6 the one obtained in simulations of the stellar evolution,in which the angular velocity is a function of the polarradius. However, because the contribution of the ma-terial along the rotation axis to the mass and angularmomentum of the star is minor, the effect of the differ-ence in the profile from spherically symmetric one maybe also minor.For R , we choose the radius at the edge of the Si layer(L model), at which the entropy profile becomes discon-tinuous, or the 70% of this radius (S model). Equa-tion (1) implies that for R (cid:28) R , the progenitor staris approximately rigidly rotating, while for the outerregion, stellar matter rotates slowly. Such a state isreasonable if the efficiency of the angular momentumtransport in the compact central region is sufficientlyhigh. The steep cut-off of the angular velocity is alsoreasonable in the stellar evolution in the presence of theconvecting layer associated with the shell burning, inwhich the angular momentum at the bottom of the layeris transported to a large radius.Table 1 lists the models considered in this work. M20and M32 denote the models with M He = 20 and 32 M (cid:12) ,respectively. The letters “S” and “L” refer to the choiceof R and the following three-digit numbers denote thevalue of Ω in units of rad/s. We also perform simu-lations omitting the neutrino pair-annihilation heating(models M32-S075N and M20-S050N) to show that thiseffect contributes substantially to increasing the explo-sion energy (cf. Table 1).A finite-temperature EOS referred to as SFHo (Steineret al. 2013) is employed in this work except for modelM32-S075DD2, in which another EOS referred to as elayed explosion R e s t m a ss ( M (cid:12) ) M32-S075M32-S100 t pb (s)02468 R e s t m a ss ( M (cid:12) ) M20-S050M20-S075M20-S100 M PNS M τ> = M PNS + M torus Figure 1.
Evolution of the rest mass of the PNS (solidcurves) and in the optically thick region for neutrinos(dashed curves) for selected models.
Note.
Due to the defi-nition of M PNS , it becomes appreciably large before the col-lapse of the PNS, in which the torus material becomes denseand contributes to the mass. For the same reason, M PNS hasa finite value even after the BH formation for model M32-S100.
DD2 (Banik et al. 2014) is employed for comparison.With SFHo and DD2 EOSs, the maximum gravitationalmasses of non-rotating cold NS are M max ≈ .
06 and2.42 M (cid:12) , and the radii of the non-rotating NS with mass1 . M (cid:12) are 11.9 and 13.2 km, respectively. The SFHoEOS is relatively soft in the sense that the value of M max is close to 2 M (cid:12) and the radius is relatively smallas (cid:46)
12 km.With the setting listed in Table 1, the PNS formedafter the collapse is rapidly rotating and the resultingcentrifugal force plays an important role to allow therest mass of the PNS beyond 3 M (cid:12) (cf. Fig. 1 in § ≤ . / s for M He = 20 M (cid:12) and for Ω ≤ . / s for M He = 32 M (cid:12) .Numerical-relativity simulations are performed withour latest axisymmetric neutrino-radiation viscous-hydrodynamics code, for which the details are describedin Fujibayashi et al. (2017, 2020). In this paper, we donot consider the viscous effect. We employ the samegrid structure as in Fujibayashi et al. (2020) with thefinest grid spacing of 150 m. SIMULATIONS RESULTSFor all the simulations, a PNS is first formed after thestellar-core collapse. Then, the baryon rest mass of thePNS increases to M PNS = 2 . . M (cid:12) in t pb := t − t b ∼ t b denotes the time at the core bounce and we defined M PNS to be the total rest massin the region of ρ ≥ g / cm . Subsequently, M PNS exceeds 3 M (cid:12) for the models listed in Table 1. This massexceeds the maximum rest mass of the non-rotating coldNSs, which is ≈ . M (cid:12) and ≈ . M (cid:12) for the SFHoand DD2 EOSs, respectively. Thus, the centrifugal force(and partly the thermal pressure) plays a key role forpreventing the collapse of the PNS to a BH for seconds.Along the rotation axis the rotational period becomes ∼ . M torus := M τ> − M PNS , where M τ> is the total rest mass in a re-gion with the average optical depth of electron neutrinosand antineutrinos ( τ ) larger than unity. We find that thetorus mass increases by the mass infall and eventuallyexceeds 1 M (cid:12) . For M He = 32 M (cid:12) models, this mass be-comes very large in a short post-bounce time. The torusinitially has a radius of ∼
200 km on the equatorial plane(see the second panel of Fig. 2 for M32-S075; the dashedcurve). During the growth of the torus, a standing ac-cretion shock with a donuts shape is formed surroundingthe PNS and torus (the second panel of Fig. 2; the dot-ted curve), and this shock expands gradually with timedue to the shock heating induced by the infalling mate-rial. Because of our choice of the initial angular-velocityprofile, the matter that accretes onto the PNS and torusat late times has smaller specific angular momenta. Be-cause of its high mass and less specific centrifugal forceat late times, the torus shrinks (its density increases;the third panel of Fig. 2), and as a result, the value of M PNS increases prior to the formation of a BH (see theupper panel of Fig 1).The shrink of the torus enhances the neutrino emission(see Fig. 3 for the increase of it in late stages), in partic-ular from the torus. The maximum neutrino luminosityis higher for the higher values of M He and could be closeto 10 erg/s as found in Sekiguchi & Shibata (2011).Because the ram pressure by the infalling material de-creases with time, such huge neutrino heating naturallyleads to the shock revival. The explosion occurs in par-ticular toward the polar direction for which the matterdensity and associated ram pressure are relatively small(see Fig. 4). The explosion occurs qualitatively in thesame manner for all the models listed in Table 1.Table 1 lists the diagnostic explosion energy, E exp .Here, the explosion energy is evaluated in the compu-tational region of (cid:46) E exp even-tually exceeds 10 erg, and for M He = 32 M (cid:12) models,it becomes higher than 10 erg, i.e., appreciably higher Fujibayashi et al.
Figure 2.
Snapshots of the rest-mass density at t pb = 0 .
05, 1.40, 2.75, and 4.40 s for model M32-S075. The solid, dashed, anddotted curves denote the surfaces of the density ρ = 10 g/cm , the neutrino optical depth τ = 1, and the shock, respectively.In the fourth panel, a black hole is formed at the center (shaded region). On all panels, the arrows display the poloidal velocityfield ( v x , v z ∼ sho.fujibayashi/share/anim den M32-S075.mp4. t pb (s)10 L ν ( e r g/ s ) M32-S075M32-S100M20-S050M20-S075M20-S100 − . . . . . . . . . t − t exp (s)10 E d i ag ( e r g ) M32-S075M32-S100M20-S050M20-S075M20-S100 M32-S075NM20-S050NM32-S075DD2M20-L050
Figure 3.
Top: Total neutrino luminosity as a function ofpost-bounce time. Bottom: Diagnostic explosion energy as afunction of t − t exp , where t exp is the explosion time definedas the time at which the explosion energy exceeds 10 erg. than the kinetic energy of typical SNe. This would bethe reflection of the neutrino luminosity by one orderof magnitude higher than in the typical SNe (on therelation between the neutrino luminosity and explosionenergy, see, e.g., Yamamoto et al. 2013). Thus, this Figure 4.
Snapshots of the rest-mass density and specificentropy at t pb = 3 .
80 s for model M32-S075. The arrowsdisplay the poloidal velocity field ( v x , v z ∼ sho.fujibayashi/share/anim den s zfull M32-S075.mp4. neutrino-driven energy injection could be a substantialfraction of the energy injection for broad-line type-IcSNe with a bipolar outflow (Maeda et al. 2002; Maeda& Nomoto 2003; Mazzali et al. 2005; Maeda et al. 2008).We note that the energy deposition rate to the outflowis (cid:38) erg/s for M He = 32 M (cid:12) models. Thus, Niwith mass of 10 − –10 − M (cid:12) is likely to be synthesizedin the ejecta (Tominaga et al. 2007, Wanajo et al. inpreparation). elayed explosion M PNS to the criticalmass for the gravitational collapse to a BH.We note that for model M32-S100, the massive torusremains, maintaining a high neutrino luminosity ( (cid:38) erg/s), after the BH formation (see Figs. 1 and 3).As a result, the neutrino-driven outflow is still presentafter the BH formation. However, the neutrino luminos-ity is not enhanced significantly and explosion energy isrelatively low for this model, in spite of the formationof a massive torus. The reason for this is that the for-mation timescale of the torus is as long as its neutrinocooling timescale. Thus to enhance the neutrino lumi-nosity far beyond 10 erg/s, the torus has to be formedbefore the sufficient neutrino cooling occurs. Achievingsuch a situation, it is advantageous to have a progenitorwith a compact core that have angular momentum suf-ficiently large in its inner region with a steep cut off ata radius.For the high-mass progenitors employed in this work,a BH is eventually formed due to the continuous matteraccretion onto the PNS, in particular from the equatorialdirection. Since the central object gains a large amountof the angular momentum from the rotating progenitor,the BH at the formation is rapidly spinning with the di-mensionless spin (cid:38) . (cid:46) g / cm after the BH formation (seethe fourth panel of Fig. 2). SUMMARY AND DISCUSSIONSThis article proposes a new mechanism for driving en-ergetic SNe like broad-line type-Ic SNe by the neutrinoheating. The model supposes that the progenitor starsof the SNe have high mass and rotation rapid enough toform a rapidly rotating PNS surrounded by a high-masstorus. The resulting PNS can survive for seconds dueto the strong centrifugal-force support, and in addition,due to the presence of a high-mass torus surrounding it,the total neutrino luminosity can be quite high at sev-eral seconds after the core bounce. Then, the neutrinoheating drives a high-energy SNe, in particular towardthe polar direction. In the successful explosion, the totalrest mass of the central object become high enough (i.e., (cid:38) M (cid:12) ) and the explosion is significantly delayed, typ-ically for seconds, after the core bounce (thus to studythis model, a long-term general relativistic simulation is required). In this case, the explosion occurs in a bipolarmanner via the neutrino heating.In the chosen angular-velocity profiles, the materialinitially located in large radii has small angular mo-menta. Thus, except for M20-L050, the torus mass isnot very large after the BH formation, and hence, it isunlikely to subsequently cause long-term energetic phe-nomena powered by the accretion of the material in atorus onto a BH. By contrast, if the outer material haslarger specific angular momenta than the material in thecentral region, which would be a reasonable assumptionconsidering more realistic stellar evolution, a massiveaccretion torus may be formed after the BH formation.In such a case, a further activity of the system may beexpected. Since the polar outflow found in this papertogether with the BH formation produces a low-densityfunnel along the rotation axis, a remnant composed of arapidly spinning BH and a high-mass torus sounds suit-able for launching an ultra-relativistic jet, i.e., gamma-ray bursts (GRBs; Woosley 1993; MacFadyen et al. 2001;Woosley & Bloom 2006; Cano et al. 2017), in the pres-ence of an energy injection. We here note that the mech-anism for launching the ultra-relativistic jet is not nec-essarily the same as that for inducing the bipolar out-flow; e.g., a magnetohydrodynamics process may be thesource for GRBs; see, e.g., Piran (2004).The speculation here suggests that the presence orabsence of the activity after the bipolar explosion coulddepend on the angular momentum distribution of theprogenitor stars, and this may explain a variety of theactivity duration of the central engine and a variety ofthe high-energy events associated with the broad-linetype Ic SNe (Woosley & Heger 2006; Margutti et al.2014; Lazzati et al. 2012) (see also Nakar (2015) onthe importance of the density profile of the pre-collapseprogenitor). Our numerical results also match with thespeculation that the rapidly rotating massive stars arelikely to be the progenitors for the energetic type-Ic SNeand GRBs (Yoon & Langer 2005; Fryer & Heger 2005;Woosley & Heger 2006; Aguilera-Dena et al. 2018).The explosion energy is beyond the typical explosionenergy of SNe ∼ erg / s, and even larger than 10 ergfor some models. Thus, this mechanism could provide(at least a part of) energy-injection for energetic SNe likebroad-line type-Ic SNe. Furthermore, a rapidly spin-ning BH is formed. If it is surrounded by a massivetorus eventually, such a system could drive a relativis-tic jet by subtracting the rotational kinetic energy ofthe BH (Blandford & Znajek 1977). The relativistic jetcould not only drive a GRB but also be the additionalenergy injection for the SN explosion. Thus, this model Fujibayashi et al. could be also used as a scenario for the association ofbroad-line type-Ic SNe and GRBs.Recent magneto-radiation-hydrodynamics simulationsby Obergaulinger & Aloy (2017, 2020a,b) also haveshown that in the presence of a rapid rotation, a high-mass progenitor star can explode by the combinationof the neutrino heating, rotation, and magnetic-field ef-fects. Our result is similar to theirs, but our work showsthat an energetic explosion can occur purely by the neu-trino heating effect even in the absence of magnetoro-tational effects for the progenitor stars which are moremassive than that employed in Obergaulinger & Aloy(2017, 2020a,b). The only required condition for thisis the sufficiently rapid rotation inside the stellar core,and thus, this is a robust mechanism for the explosionof massive stars.In this paper, we present only models that show theexplosion. For low angular-momentum models, the PNScollapses to a BH before the explosion. Thus, for theexplosion, the progenitor stars need to have a sufficientangular momentum. The details on the non-explosionmodels and approximate criterion for the explosion willbe systematically studied in the future.There are several issues to quantitatively improve thepresent work. First, our treatment for the neutrino-radiation transfer is currently based on a gray leak-age scheme. Obviously, simulations with a better ra-diation transfer code are needed. The present work isbased on axisymmetric simulations. Because the torus ismassive, non-axisymmetric deformation is likely to takeplace in reality (e.g., Shibata & Sekiguchi 2005; Shiba-gaki et al. 2020). This may cause an angular momentumtransport in the torus and the accretion onto the PNSmay be enhanced leading to earlier collapse to a BH. The angular momentum transport can also be enhancedby magneto-hydrodynamics (MHD) effects such as themagneto-rotational instability (M¨osta et al. 2014). Al-ternatively, MHD effects may help earlier explosion ifthe magnetic field is amplified by MHD instabilities.Thus, in the future, we need to perform more sophis-ticated simulations, although the present work providesthe first step toward more detailed studies.The non-axisymmetric deformation of the massivetorus could also lead to the burst emission of gravita-tional waves. Our latest study shows that if an one-armed spiral deformation mode grows in a dynamicaltimescale comparable to the typical rotational periodof the torus, the degree of the non-axisymmetric den-sity fluctuation can be 10–20% of the torus mass (Shi-bata et al. 2021). In such deformation, the maximumamplitude of burst-type gravitational waves at the hy-pothetical distance to the source of 100 Mpc can be ∼ − with the typical frequency of 0.7–0.8 kHz for M PNS ≈ M (cid:12) with the comparable torus mass (Shibataet al. 2021). Such gravitational waves are the interestingsources for the third-generation gravitational-wave de-tectors such as Einstein Telescope (Punturo et al. 2010)and Cosmic Explorer (Abbott et al. 2017). Thus, in thefuture, high-energy supernovae with the bipolar outflowmay be explored not only by electromagnetic telescopesbut also by the gravitational-wave detectors.We thank T. Kuroda, K. Maeda, N. Tominaga, andS. Wanajo for useful discussions. This work was inpart supported by Grant-in-Aid for Scientific Research(Grant Nos. JP20H00158) of Japanese MEXT/JSPS.Numerical computations were performed on Sakura andCobra at Max Planck Computing and Data Facility andXC50 at National Astronomical Observatory of Japan.REFERENCES Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017,Classical and Quantum Gravity, 34, 044001,doi: 10.1088/1361-6382/aa51f4Aguilera-Dena, D. R., Langer, N., Moriya, T. 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