Failure of a neutrino-driven explosion after core-collapse may lead to a thermonuclear supernova
aa r X i v : . [ a s t r o - ph . H E ] S e p D RAFT VERSION J ULY
9, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
FAILURE OF A NEUTRINO-DRIVEN EXPLOSION AFTER CORE-COLLAPSE MAY LEAD TO A THERMONUCLEARSUPERNOVA D ORON K USHNIR
AND B OAZ K ATZ Draft version July 9, 2018
ABSTRACTWe demonstrate that ∼ s after the core-collapse of a massive star, a thermonuclear explosion of the outershells is possible for some (tuned) initial density and composition profiles, assuming that the neutrinos failedto explode the star. The explosion may lead to a successful supernova, as first suggested by Burbidge et al.We perform a series of one-dimensional (1D) calculations of collapsing massive stars with simplified initialdensity profiles (similar to the results of stellar evolution calculations) and various compositions (not similar to1D stellar evolution calculations). We assume that the neutrinos escaped with a negligible effect on the outerlayers, which inevitably collapse. As the shells collapse, they compress and heat up adiabatically, enhancingthe rate of thermonuclear burning. In some cases, where significant shells of mixed helium and oxygen arepresent with pre-collapsed burning times of . s ( ≈ times the free-fall time), a thermonuclear detonationwave is ignited, which unbinds the outer layers of the star, leading to a supernova. The energy released issmall, . erg, and negligible amounts of synthesized material (including Ni) are ejected, implying thatthese 1D simulations are unlikely to represent typical core-collapse supernovae. However, they do serve asa proof of concept that the core-collapse-induced thermonuclear explosions are possible, and more realistictwo-dimensional and three-dimensional simulations are within current computational capabilities.
Subject headings: hydrodynamics — methods: numerical — supernovae: general INTRODUCTION
There is a strong evidence that type II supernovae are ex-plosions of massive stars, involving the gravitational collapseof the stars’ iron cores (Burbidge et al. 1957; Hirata et al.1987; Smartt 2009) and the ejection of the outer layers. Itis widely thought that the explosion is obtained due to thedeposition in the envelope of a small fraction ( ∼ ) of thegravitational energy ( ∼ erg) released in neutrinos fromthe core, leading to the ∼ erg observed kinetic energy ofthe ejected material (see Bethe 1990; Janka 2012; Burrows2013, for reviews). One-dimensional (1D) simulations in-dicate that the neutrinos do not deposit sufficient energy inthe envelope to produce the typical ∼ erg kinetic energy.While some two-dimensional (2D) studies indicate robustexplosions (Bruenn et al. 2013, 2014; Nakamura et al. 2014;Suwa et al. 2014) and some indicate failures or weak explo-sions (Takiwaki et al. 2014; Dolence et al. 2015), these stud-ies are affected by the assumption of rotational symmetry andby an inverse turbulent energy cascade, which, unlike manyphysical systems, tends to amplify energy on large scales.Therefore, three-dimensional (3D) studies are necessary tosatisfactorily demonstrate the neutrino mechanism, but so far3D studies have resulted in either failures or weak explo-sions (Takiwaki et al. 2014; Lentz et al. 2015; Melson et al.2015a,b).Burbidge et al. (1957) suggested a different mechanismfor the explosion during core-collapse that does not involvethe emitted neutrinos. They suggested that increased burn-ing rates due to the adiabatic heating of the outer shells asthey collapse leads to a thermonuclear explosion (see alsoHoyle & Fowler 1960; Fowler & Hoyle 1964). This has the Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540,USA Corresponding author, [email protected] Department of Particle Physics and Astrophysics, Weizmann Instituteof Science, Rehovot 76100, Israel advantage of naturally producing ∼ erg from the ther-monuclear burning (with a gain of ∼ MeV /b , where b standsfor baryon) of ∼ M ⊙ of light elements. Alternatively, afraction of MeV /b naturally explains a velocity scale ofsupernovae of thousands of km s − that is more robustlyobserved than the kinetic energy. While this mechanismcan operate only if the neutrinos failed to eject the enve-lope, it would still be possible to see the neutrinos as ob-served in SN1987A (Hirata et al. 1987). A few 1D studiessuggested that this mechanism does not lead to an explo-sion because the detonation wave is ignited in a supersonicin-falling flow (Colgate & White 1966; Woosley & Weaver1982; Bodenheimer & Woosley 1983). While these studiesare discouraging, they only demonstrate that some specificinitial stellar profiles do not lead to thermonuclear explosions,and they do not prove that thermonuclear explosions are im-possible for all profiles. We find it striking that so little efforthas been dedicated to studying this mechanism, given the rela-tively low computational requirements to examine it (see alsoBurrows 1988; Janka 2012, for a brief historical account ofhow the thermonuclear mechanism was left behind).In this paper we revisit the collapse-induced thermonuclearsupernovae mechanism. In Sections 2 we perform a series of1D calculations of collapsing massive stars with simplifiedinitial density profiles and various compositions, assumingthat the neutrinos had a negligible effect on the outer layers.We demonstrate that ∼ s after the core-collapse of a mas-sive star, a successful thermonuclear explosion of the outershells is possible for some (tuned) initial density and composi-tion profiles that includes a significant layer of He–O mixture.In Section 3 we use simple analytic arguments to explain thequalitative features of the numerical calculations. A summaryof the results and conclusions is given in Section 4.
1D SIMULATIONS
In this section we perform a series of 1D calculations ofcollapsing massive stars with simplified initial density profilesand various compositions, assuming that the neutrinos had anegligible effect on the outer layers. The initial profiles aredescribed in Section 2.1 and our numerical tools are describedin Section 2.2. In Section 2.3 we demonstrate that ∼ s afterthe core-collapse of a massive star, a successful thermonu-clear explosion of the outer shells is possible for some ini-tial density and composition profiles that include a significantlayer of He–O mixture. The ignition process in this simula-tion is analyzed in Section 2.4. In Section 2.5 we examine thesensitivity of our results to the assumed initial profile. Initial profiles
The first step is to define the pre-collapse stellar profiles.These profiles cannot be inferred from observations and re-quire the calculation of the final stages of stellar evolution,which are poorly understood (see, e.g, Smith & Arnett 2014)and are therefore uncertain. Nevertheless, there are severalphysical constraints that are likely to hold.a. The star contains a degenerate iron core with a massslightly smaller than the Chandrasekhar mass.b. The initial profile is in a hydrostatic equilibrium.c. The profile is stable with a constant or rising entropy (perunit mass) as a function of radius.d. The local thermonuclear burning time, t b , at any radius r inthe profile is much longer than the free-fall time, t ff , where t b = ε/ ˙ Q, (1) t ff = r / p GM ( r ) , (2) ε is the internal energy (per unit mass), ˙ Q is the thermonu-clear energy production rate (per unit mass), and M ( r ) isthe enclosed mass.We note that the demand for stability may be relaxed if thegrowth time of perturbations is much longer than the dynam-ical time, but this is beyond the scope of this work. Based onthese constraints, we adopt the following simple parameter-ized profile.1. A fixed mass of . M ⊙ within r < · cm is as-sumed to have already collapsed at t = 0 and is notsimulated.2. The hydrogen envelope is ignored and the temperatureis set to zero ( K in practice) at the profile’s fixedouter radius of · cm.3. To allow the shape and amplitude of the density profileto be varied, the profile is composed of two regions withan adjustable transition radius r break . The inner region · cm < r < r break has a constant entropy (per unitmass) and the outer region r break < r < · cm hasa density profile, ρ = M log πr , (3)(equal mass M log per logarithmic radius interval). Therequirement of hydrostatic equilibrium implies that the density, pressure, and temperature profiles are set (up tominor adjustments due to the composition) by two freeparameters that are chosen as the inner density, ρ i ≡ ρ ( r = 2 · cm ) , and total mass, M core . The transitionradius, r break , is adjusted accordingly.4. The composition of the explosive shell is a mixture ofhelium and oxygen. This mixture is placed at the outerparts of the profile at radii r > r base where the temper-atures are sufficiently low such that the ratio betweenthe local burning time and the free-fall time, t b /t ff , islarger than a fixed threshold t b , /t ff , . The value of r base is chosen such that this ratio is exactly t b , /t ff , .At lower radii, pure oxygen (where T < · K) andsilicon (where
T > · K) are placed, which havenegligible burning during the simulation.The above prescription has four free parameters.1. ρ i - the density at · cm.2. M core - the enclosed mass within · cm.3. r O / He - the ratio of the oxygen and helium mass frac-tions in the explosive shell.4. t b , /t ff , - the ratio between the burning time and thefree-fall time at the base of the explosive shell r base .The additional parameters r break and M log that enter the profiledescription are set by the choice of ρ i and M core .We note that significant shells of mixed He–O are notcurrently expected in non-rotating stellar evolution models.Nevertheless, stellar evolution calculations of rotating mas-sive stars generally predict the existence of a mixed He–O shell (Heger et al. 2000; Hirschi et al. 2004; Heger et al.2005; Hirschi et al. 2005; Hirschi 2007; Yusof et al. 2013). Collapse Simulations
To simulate the collapse we use the 1D, Lagrangian versionof the VULCAN code (for details; see Livne 1993), whichsolves the equations of reactive hydrodynamics with a 13 iso-tope alpha-chain reaction network (similar to the 13 isotopenetwork supplied with FLASH with slightly updated ratesfor specific reactions, especially fixing a typo for the reac-tion Si ( α, γ ) S, which reduced the reaction rate by a factor ≈ .). We use a sufficient resolution (typically ≈ km for theinitial profile) such that all of our results are converged to bet-ter than ∼ . We also use the 1D hydrodynamic FLASH4.0code with thermonuclear burning (Eulerian, adaptive meshrefinement; Fryxell et al. 2000; Dubey et al. 2009), with thesame reaction network as in VULCAN, in order to verify thatour results do not depend on the numerical scheme. Falsenumerical ignition may occur if the burning time in a cellbecomes shorter than the sound crossing time (Kushnir et al.2013). To avoid this, we modified both codes to include aburning limiter that forces the burning time in any cell to belonger than the cell’s sound crossing time by suppressing allburning rates with a constant factor whenever t sound > f t burn with f = 0 . (see Kushnir et al. 2013, for a detailed descrip-tion). The numerical convergence established below impliesthat the limiter does not modify the resulting profiles. The composition profiles of Heger et al. can be found inhttp://2sn.org/stellarevolution/
We assume that neutrinos emitted during the collapse of theinner core do not lead to an explosion and escape with a neg-ligible effect on the outer layers. We also neglect the gravita-tional mass loss from the neutrino emission (which may leadto a very weak explosion with kinetic energy ∼ erg if thethermonuclear explosion fails as well; Lovegrove & Woosley2013; Piro 2013). The layers below r = 2 · cm areassumed to have already collapsed, and the initial pressurewithin this radius is set to zero. The pressure at the simu-lation inner boundary, r = 10 cm, is held at zero through-out the simulation. The mass of material that (freely) flowsthrough the boundary is added to the original collapsed massof . M ⊙ and is taken into account in the gravitational field.The results are insensitive to the details of the collapse of theinner parts due to the supersonic flow near the boundary thatdoes not allow information to propagate outward to the outershells where thermonuclear burning takes place. To verifythis, we experimented with other schemes for the collapse ofthe inner parts (e.g., the inner numerical node constrained tofree-fall motion until crossing r = 10 cm), and found negli-gible effects on our results.For most of the range of the possible values of the freeparameters ρ i , M core , r He / O , and t b , /t ff , , the thermonuclearburning does not release sufficient energy to unbind the star.However, there is a range of profiles with reasonable parame-ters for which successful explosions occur. Before discussingthe full set of simulations that were performed (Section 2.5),we describe in Sections 2.3 and 2.4 one successful explosion.The fact that some 1D profiles lead to successful explosionsserves as a proof of concept for the possibility of collapse-induced thermonuclear supernovae. Example of a successful explosion
A pre-collapse profile that leads to a successful explo-sion is shown in Figure 1. The parameters for this profileare ρ i = 1 . · g cm − , M core = 10 M ⊙ (leading to r break ≈ . · cm and M log ≈ . M ⊙ ), t b , /t ff , ≈ . and a mixture of helium and oxygen with equal mass frac-tions X O = X He = 0 . ( r O / He = 1 ). To achieve therequired t b , /t ff , , the base of the He–O mixture is set to r base ≈ . · cm, with an enclosed mass of m ≈ . M ⊙ (leading to t b , ≈ s). The obtained density, temperature,and enclosed mass profiles are similar to the pre-collapse pro-files of a M ⊙ star, calculated by Roni Waldman with theMESA stellar evolution code (Paxton et al. 2011), which areshown for comparison. The main differences between the pro-files are the existence and location of the He–O mixture.The dynamical evolution of the collapse, as calculated withVULCAN, is shown in Figures 2 (snapshots from the simula-tion) and 3 (energy evolution) for the initial conditions of Fig-ure 1. A rarefaction wave propagates from the center of thestar outward (evident as a velocity break appearing in panel(a) of Figure 2 at m ≈ . M ⊙ ). Each element begins tofall inward as soon as the rarefaction wave reaches it. As itfalls, each element is first slightly rarefied and then increas-ingly compressed. The velocity of the collapsing material in-creases and at some point the flow becomes supersonic. Forexample, s after the collapse the sonic point is located at m ≈ . M ⊙ . Sound waves cannot cross the sonic pointoutward, which is the cause for the low sensitivity to the ex-act inner boundary conditions, as explained above. As thebase of the He–O shell is compressed and heated up adiabati-cally, the rate of thermonuclear burning is enhanced (which is r [cm] Si O He-O1 . M ⊙ not simulated ρ ∝ r − s = const. log ( ρ [g cm − ]) T [10 K] M [ M ⊙ ]log ( t b /t ff )MESA F IG . 1.— Pre-collapse profile (density, temperature, enclosed mass, andburning to free-fall time ratio, t b /t ff ) that leads to a successful explosion.The parameters for this profile are ρ i = 1 . · g cm − , M core = 10 M ⊙ (leading to r break ≈ . · cm and M log ≈ . M ⊙ ), t b , /t ff , ≈ . ,and r O / He = 1 (see Section 2.1 for details). A fixed mass of . M ⊙ within r < · cm is assumed to have already collapsed at t = 0 and is notsimulated. The transition radius, r break ≈ . · cm, between constantentropy (per unit mass) and a density profile ρ ∝ r − is indicated with a bluecircle. The base of the He–O mixture is at r base ≈ . · cm. At lowerradii, pure oxygen (where T < · K) and silicon (where
T > · K)are placed. For comparison, the pre-collapse profiles of a M ⊙ star, calcu-lated by Roni Waldman with the MESA stellar evolution code (Paxton et al.2011), are shown (dashed gray). the cause of the small density jump in panel (a) of Figure 2 at m ≈ . M ⊙ ), and causes an ignition of a detonation waveat t ≈ s, as described in detail below. The ignition processtakes place at a subsonic region (i.e., outward from the sonicpoint). An ignition of a detonation in a subsonic region oc-curred for all simulations in which a successful explosion wasobtained.The detonation wave propagates outward (panel (b) of Fig-ure 2 at m ≈ . M ⊙ ), producing thermonuclear energy ata rate of few × erg s − (Figure 3). The pressure builtfrom the accumulating thermonuclear energy manages to haltthe inward collapse and cause an expansion that leads to anoutward motion. Once the detonation wave reaches outer lay-ers with densities ρ . g cm − it decays and transitions toa hydrodynamic shock that continues to propagate outwards(panel (c) of Figure 2 at m ≈ . M ⊙ ). Note that the com-position above the transition radius has a negligible effect onour results (and could be pure He, for example) as no furtherburning occurs. In this example, the shock reaches the stellaredge at t ≈ s (Figure 3), and the resulting ejecta has a massof ≈ . M ⊙ and a kinetic energy of ≈ erg. It is evidentin Figure 3 that the potential energy of the burning shells andthe mass external to them are of the same order as the releasedthermonuclear energy. The small kinetic energy of the ejectais only a small fraction of the released thermonuclear energyof erg. Furthermore, no post-collapse synthesized mate-rial is ejected. The properties of the ejecta may change if ahydrogen envelope is added. Ignition of a detonation m [ M ⊙ ] head of rarefaction wave t = 12 s ρ [10 g cm − ] v [10 cm s − ]2 × X He − (a) m [ M ⊙ ] head of rarefaction wavedetonation wave t = 20 s ρ [10 g cm − ] v [10 cm s − ]2 × X He − (b) m [ M ⊙ ] head of rarefaction waveshock wave t = 28 s ρ [10 g cm − ] v [10 cm s − ]2 × X He − (c)F IG . 2.— Dynamical evolution of the collapse, as calculated with VUL-CAN, for the initial conditions of Figure 1. Each panel shows profiles (den-sity, velocity, and X He ) for a snapshot of the simulation. Panel (a) t = 12 s;panel (b) t = 20 s; panel (c) t = 28 s. Note that the scale of the density is g cm − in panels (a) and (b), and is g cm − in panel (c). Time [s] [ e r g ] , [ e r g s − ] shock breakout ˙ E burn E burn E tot( ε tot > , v > − E tot( m > m shock) F IG . 3.— Energy evolution during the collapse, as calculated with VUL-CAN, for the initial conditions of Figure 1. The rate of thermonuclear en-ergy production, ˙ E burn , is shown in red, and the accumulated thermonuclearenergy produced, E burn , is shown in blue. The total energy, including thegravitational, internal (not including potential thermonuclear), and kineticenergy of mass elements with positive velocity and positive total energy, E tot ( ε tot > , v > , is shown in black. The negative of the total en-ergy of mass elements outward from the outgoing shock wave (or detonationwave), − E tot ( m > m shock ) , is shown in green. Note that the last quantity isdefined only after ignition, at t ≈ s. The ignition process in the example above is shown in Fig-ure 4. For material near the base of the He–O shell, the col-lapse leads to a burning time that is comparable to the free-fall time at a radius of ≈ . · cm, and He is efficientlyconsumed, leading to an ignition of a detonation. The condi-tion for the formation of a detonation wave is that the thermalenergy increases significantly (thereby increasing the burn-ing rate) in a timescale shorter than the time it takes to hy-drodynamically distribute the resulting excess pressure. Thelatter timescale is given by the sound crossing time ∆ r/c s of the burning region, where c s is the speed of sound and ∆ r ∼ ˙ Q/ ( d ˙ Q/dr ) is the length scale of the burning region.In case of a well defined burning wave, propagating with aphase velocity v ϕ , and ε ≈ Q , this condition reduces to theZel’dovich criterion (Zel’dovich 1980), v ϕ > c s . The igni-tion condition is met at the time t ≈ s, shown in panel (a),where the scale of the burning region is ∆ r ≈ · cm,the typical speed of sound there is ≈ · cm s − , and theburning rate is ˙ Q/ε ∼ > s − . Note that at earlier times ˙ Q is significantly smaller, while ∆ r is slightly larger, such thatthe ignition criterion is not met. Once the ignition criterionis met, significant thermonuclear energy is deposited locally,which increases the temperature and leads to a faster burn-ing rate. This runaway process leads to the formation of ashock that is powered by the fast burning in its post shockedregion, i.e., a detonation wave, as seen in t = 17 . s. Be-cause of the increased temperature and burning rate, the scaleof the burning region decreases substantially, leading to thewell known small length scale of thermonuclear detonationwaves (Khokhlov 1989). However, this small length scale isirrelevant to the ignition process (contrary to what is com-monly believed, e.g., Khokhlov 1989) which is determined atearlier times as explained here.We are now in a position to estimate the numerical resolu-tion required to resolve the ignition process. As seen in thesnapshot at t = 17 . s, in which the ignition criterion ismet, the scale of the burning region is ∆ r ≈ km, imply-ing that a resolution of ∆ r ∼ km is sufficient to resolvethe ignition process. Indeed, the VULCAN simulation is pre-formed with this resolution and ˙ Q/ε is converged to ∼ .A series of FLASH simulations with increasing resolutions ispresented in panel (b) of Figure 4. As can be seen the (inverse)burning time, ˙ Q/ε , is converged to a good approximation forresolutions ∆ r ≈ km. This demonstrates that a modest res-olution (that can be easily achieved in a full star simulation)is sufficient to resolve the ignition in this case. Note that at aslightly lower resolution ( ∆ r ≈ km) an ignition of a det-onation is still obtained, although at a slightly different timeand location. At much lower resolutions ( ∆ r ≈ km) anignition of a detonation is not obtained. Successful 1D explosions require tuning
In this section we examine the sensitivity of our results tothe assumed initial profile. The asymptotic kinetic energyof the ejecta as a function of M log is shown in Figure 5.For M core = 10 , r O / He = 1 and t b , /t ff , = 10 , asymp-totic kinetic energy of ∼ erg is obtained for . M ⊙ ∼ 825 s FLASH , ∆ r = 114 kmFLASH , ∆ r = 57 kmFLASH , ∆ r = 29 kmFLASH , ∆ r = 14 kmVULCAN , ∆ r ≃ 60 kmVULCAN , ∆ r ≃ 30 km (b)F IG . 4.— Ignition process for the initial conditions of Figure 1. Panel (a):profiles (temperature, He mass fraction, and burning rate) from a VULCANsimulation at t = 17 . , . , . s. The ignition of a detonation oc-curs when the burning rate becomes higher than the inverse sound cross-ing time c s / ∆ r ∼ s − which is shown as a dashed black line, where ∆ r ∼ · cm is the burning region size (as evident in the figure) and c s ≈ · cm s − is the speed of sound (see the text for more details). Theformation of the shock is clearly seen in t = 17 . s. Panel (b): Snapshotof ˙ Q/ε at t = 17 . s, in which the ignition criterion is met. VULCANsimulations (black) and FLASH simulations (red) are compared at differentresolutions (for VULCAN, the actual resolution within the plotted region isgiven). life of ≈ s, which is too long. One can imagine beginningwith a roughly equal number of protons and heavier nuclei,but such a mixture is not energetic enough (see below). An-other requirement is that the released thermonuclear energyis high enough to overcome the binding energy of the star,which requires a MeV /b yield (see Section 3). Mixtures thatdo not contain significant fractions of He cannot fulfill this re-quirement, as the energy per baryon decreases for heavier nu-clei. These mixtures also typically ignite at high temperatures, M log [ M ⊙ ] E k i n [ e r g ] M ⊙ , r O/He = 1 , t b, /t ff , = 210 M ⊙ , r O/He = 1 , t b, /t ff , = 1010 M ⊙ , r O/He = 1 , t b, /t ff , = 10010 M ⊙ , r O/He = 3 / , t b, /t ff , = 1010 M ⊙ , r O/He = 2 / , t b, /t ff , = 108 M ⊙ , r O/He = 1 , t b, /t ff , = 106 M ⊙ , r O/He = 1 , t b, /t ff , = 104 M ⊙ , r O/He = 1 , t b, /t ff , = 10 F IG . 5.— Sensitivity of the asymptotic kinetic energy of the ejecta to theprogenitor parameters (see Section 2.1 for the description of the initial pro-files and the free parameters). The dependence on the density normalizationat the outer radii r > r break , M log = dM ( r ) /d ln( r ) , is shown on the x -axis, while the different curves probe the dependence on the other parameters, M core , r O / He , and t b , /t ff , as indicated in the legend. when the material is already too deep in the potential well ofthe star, and the thermonuclear energy has a higher bindingenergy to overcome. We did not obtain explosions for purehelium shells, since the triple alpha reaction weakly dependson temperature for the relevant temperature range and the ρ dependence of the reaction is not steep enough to allow igni-tion after a small amount of infall. These considerations leaveonly He–C and He–O as viable mixtures. It is hard to deter-mine in advance which mixture is better, but our detailed sim-ulations show that explosions can be obtained only for He–Omixtures. The reason is probably that for He–O mixtures thesteep increase of the reaction rate with temperature happensat a slightly lower temperature than for He–C mixtures.In summary, the required profiles are tuned and require thepresence of a mixture of He and O with burning times of . s ( ≈ times the free-fall time) prior to collapse, whichis not currently expected in stellar evolution models. Whilethe uncertainties involved with the pre-collapse final evolu-tion stages of the star do not allow us to determine whetherthe initial conditions that lead to explosions in 1D are possi-ble, they are probably unlikely. APPROXIMATE ANALYTIC TREATMENT The numerical experiments described above imply that anexplosion is possible, but only for a narrow range of initialprofiles. In this section we attempt to provide an approximateanalytic explanation for these results.In successful explosions, a detonation wave is formed ina collapsing shell that propagates faster than the infall speedand manages to propagate out. As the wave traverses the pro-genitor, thermonuclear energy of order MeV /b is released and(mostly) accumulated. At some point the wave reaches radiiwhere the density is too low to support it and the thermonu-clear burning is halted. In successful explosions, this energyis greater than the potential energy of the traversed shells andthe mass external to them, which is of the order of erg.It is implied that there are two basic requirements for a suc-cessful thermonuclear explosion in a collapsing star, 1. an ignition of a detonation needs to occur at a suffi-ciently large radius so that the detonation wave propa-gates faster than the in-falling material; and2. the detonation wave should traverse a significantamount of mass ( & M ⊙ ) before it fades out in order toallow ∼ erg to be released.The hydrodynamical collapse is analyzed in section 3.1.This allows the derivation of approximate conditions for theformation of an outgoing detonation wave. In addition, it isshown that at any given time, the amount of in-falling massthat is compressed to a density significantly higher than its ini-tial (Lagrangian) density is very small ≪ M ⊙ . This impliesthat in successful explosions, most of the contributing ther-monuclear burning occurs in regions that have not sufferedsignificant collapse. The approximate conditions for the ther-monuclear burning of a significant amount of mass can there-fore be found by analyzing the structure of the initial profile,assuming that a detonation wave traverses it. This is done insection 3.2. Collapse In order to study the collapse of a mass element, we makethe following approximations about the profile in its neigh-borhood: (1) ρ is a power-law in radius ρ ∝ r − δ ρ , (2) theaccumulated mass is independent of radius, and (3) adiabaticcompression is described by a constant adiabatic index γ . Un-der these reasonable approximations, the flow is described bya self-similar solution that is found in the appendix. Note thatwhile the self-similar solution assumes these assumptions tohold throughout the profile, the evolution of any given masselement is not sensitive to the profile at distant radii and thusthe results are approximately correct for general profiles. Thecompression of a mass element as a function of its radius isshown in panel (a) of Figure 6 for various values of the power-law index δ ρ and the adiabatic index γ . As can be seen, asthe radius decreases, the density first decreases and then in-creases, approaching a compression of ∼ at r/r = 0 . .For comparison, the compression of the m = 3 M ⊙ mass el-ement from the simulation of Section 2.3 is shown. The com-position of this mass element is pure oxygen and negligibleburning occurs during the compression. As can be seen inthe figure, the self-similar solutions agree with the numericalcompression to an accuracy of ∼ .The compression at small radii can be obtained as follows.Consider two adjacent mass elements that start at r and areinitially separated by dr . The rarefaction wave reaches thetwo elements at slightly different times separated by dt . Asthe rarefaction wave moves at the speed of sound c s we have(using Equation (A3)) dr = c s dt = s γGM ( δ ρ + 1) r dt . (4)As the elements fall they reach each radius r at slightly dif-ferent times separated by dt = dr/v , where dr is their instan-taneous separation and v is their velocity. At small radii, theelements approach free-fall and therefore v = (2 GM/r ) / so that dr = vdt = r GMr dt. (5) −1 r/r ρ / ρ γ = 4 / , δ ρ = 25 / , / , / , / , / , m = 3 M ⊙ of Section 2.30 . r /r ) / (a) m/m falling ρ / ρ γ = 4 / , δ ρ = 25 / , / , / , / , / , t = 12 s of Section 2.3 (b)F IG . 6.— Self-similar collapse. Panel (a) shows the compression of a La-grangian element as a function of its radius (normalized to its original radius)for different values of the adiabatic index γ and the density power-law index δ ρ . For comparison, the compression of the m = 3 M ⊙ mass element fromthe simulation of Section 2.3 is shown. Panel (b) shows the compression asa function of mass at a given time, normalized to the total mass of fallingmaterial. For comparison, the compression at t = 12 s from the simulationof Section 2.3 is shown. The asymptotic compression is thus given by ρρ = r dr r dr = r γ δ ρ + 1) dt dt ( r/r ) − / . (6)At small radii, dt approaches a constant dt f –the time differ-ence between the arrival at r = 0 , and we can approximate ρρ = r γ δ ρ + 1) (cid:18) dt f dt (cid:19) − ( r/r ) − / . (7)The values of dt f /dt for different choices of δ ρ and γ aregiven in Table 2. As can be seen, dt f /dt ≈ . We thereforeexpect that at small r , ρρ ≈ . r/r ) − / , (8) which is consistent with the results shown in panel (a) of Fig-ure 6.The amount of time spent at small radii is short and thusthe mass at any given time that is significantly compressed issmall, as seen in panel (b) of Figure 6, which shows the com-pression as a function of mass at a given time, normalized tothe total mass of falling material (i.e., with negative velocity), m falling . For comparison, the compression at t = 12 s from thesimulation of Section 2.3 is shown. As can be seen in the fig-ure, the self-similar solutions agree with the numerical com-pression to an accuracy of ∼ . Note that the small densityjump at m/m falling ≈ . is caused by a small amount ofthermonuclear burning that is not present in the self-similarsolutions.As the density of a falling mass element becomes higher,the temperature rises due to the adiabatic compression. InSection 3.2 we show that the pre-collapse electron-to-photonnumber ratio is of order unity. We next show that under suchconditions the photon-to-electron density remains practicallyconstant during the adiabatic compression, allowing the tem-perature to be easily calculated. The equation of state is ap-proximated by p = n e T + a R T / n e + n γ ) T, (9)where n e = ρ m p , n γ = a R T , (10)are the electron and photon densities, respectively, and a R isthe blackbody radiation constant.During adiabatic compression we have d (cid:18) en e (cid:19) = − pd (cid:18) n e (cid:19) = pn e dn e n e , (11)where e is the energy density and is given by e = 32 n e T + a R T = 3 (cid:18) n e + n γ (cid:19) T. (12)Using the fact that ( n γ /n e ) n e = a R T / , we find dTT = dx γ x γ + dn e n e , where x γ ≡ n γ n e . (13)Equations (9), and (11)–(13) can be used to obtain dn e n e = dx γ x γ + 8 dx γ , (14)implying that during adiabatic compression we have n γ n e exp (cid:18) n γ n e (cid:19) ∝ n e ∝ ρ. (15)In order to change the photon-to-electron ratio from . to ( to ), n e needs to be compressed by a factor of about ( ), implying that for n γ /n e & . , it is approximatelyconstant for a very wide range of compressions. The temper-ature follows T ∝ n / γ ∝ ( n e /n γ ) / n / e , and is thereforeproportional to ρ / to an excellent approximation. Once theelement reaches radii much smaller than its initial radius, wecan use Equation (8) to find T ≈ . r/r ) − / T . (16)At r ∼ . r the temperature rises by about a factor of ,typically allowing for a much higher burning rate. At suffi-ciently fast burning rates, significant energy can be releasedon a timescale that is shorter than the sound crossing time andthe free-fall time and a detonation wave forms.Once a detonation wave forms, its velocity with respect tothe local rest-frame must be larger than the infall velocity sothat it propagates out. Assuming that the energy release ismuch larger than the thermal energy, the shock velocity isgiven by the Chapman–Jouguet velocity v s = p Q ( γ − ≈ . · (cid:18) Q MeV /b (cid:19) / cm s − , (17)where the appropriate γ ≈ / was used. In order for v s > v ff = (2 GM/r ) / , the detonation must be ignited ata sufficiently large radius r det > GMQ ( γ − ≈ · (cid:18) Q MeV /b (cid:19) − M M ⊙ cm . (18)Given that compression requires a change in radius of at least , the material that can ignite the detonation must initiallybe at a radius greater than about cm. This constraint issatisfied in the successful numerical explosions described inSection 2.2. Explosion We next derive the constraints on the initial profile requiredso that a considerable mass is traversed by the detonationwave before it fails. The detonation wave requires high den-sities to propagate so that the released thermonuclear energy(per unit volume) ∼ Qρ is sufficiently high to increase the tem-perature to values of T > T c ∼ K, where the burning isfaster than the free-fall time. The threshold density ρ det isroughly given by ρ det ∼ a R T c /Q ≈ (cid:18) T c K (cid:19) (cid:18) Q MeV /b (cid:19) − g cm − . (19)As explained above (panel (b) of Figure 6), the amount ofmaterial that is significantly compressed at any given time isvery small. The initial profile must therefore contain & M ⊙ of explosive material at high densities ρ & ρ det . In addition,the mass available for the explosion must initially have a lowtemperature T < T c to avoid fast burning prior to the collapse.As we next show, the amount of mass with a high density andlow temperature is tightly constrained by the requirement ofa hydrostatic equilibrium. This is the basic reason for the finetuning required in Section 2.5.For simplicity consider a density profile with an inner coremass M in within r in = 2 · cm and a uniform mass perlogarithmic radius interval at larger radii, ρ = M log πr . (20)The enclosed mass within a radius r is M ( r ) = M in + M log ln( r/r in ) (21)and assuming hydrostatic equilibrium the pressure is given by p = Z ∞ r GM ( r ′ ) ρ ( r ′ ) r ′ dr ′ = G ( M + M log / M log πr . (22) Using these approximations, the temperature can be readilyfound at each radius, given M in and M log . As in Section 2, weadopt an inner mass of M in = 1 . M ⊙ . The temperature anddensity profiles for a range of density normalizations, M log ,are shown in Figure 7 at radii where ρ > g cm − . Ascan be seen in the figure, the range of radii where there aresufficiently low temperatures T < K and sufficiently highdensities ρ > g cm − is narrow and smaller for higherdensity normalizations. The amount of mass that satisfies thisconstraint reaches about . M ⊙ for the high density normal-ization M log = 4 M ⊙ , significantly limiting the amount ofavailable thermonuclear energy, and is lower for smaller M log .For comparison, the profile from Figure 1 is also shown. Ascan be seen, while the profile is shallower (mostly due to thedeviation from ρ ∝ r − ), the simple model is confined be-tween the normalization range of M log = 0 . – M ⊙ in therange · cm < r < · cm. This explains the nar-row range of parameters that allows successful explosions,which was obtained in Section 2.5. In order to understandthe origin of this tight constraint, the temperature is plotted asa function of the density in panel (b) of Figure 7. As can beseen the temperatures and densities are related by T ∝ ρ / with little dependence on M log (also holds for the profile fromFigure 1). This reflects the fact that for massive extendedstars in hydrostatic equilibrium, the ratio of the photon density n γ = a R T / to the electron number density n e ≈ ρ/ (2 m p ) is of order unity, as we next demonstrate.To a good approximation, Equation (22) can be written as p = GM ρ r . (23)Using Equations (23), (20), and (10), we find p n e = πG M m p M log . (24)Using the equation of state, (9), we find p n e = 3 a R n γ n e (cid:18) n γ n e (cid:19) . (25)Equating Equations (25) and (24) we find n γ n e (cid:18) n γ n e (cid:19) ≈ . M M log M (26)where M ch ≈ . (cid:18) ~ cG (cid:19) / µ − e ≈ . a R G ) − / µ − e (27)is the Chandrasekhar mass and µ e = 2 m p .Since M & M log , M ch , the right hand side is larger thanunity. For < M / ( M log M ) < , we have . < ( n γ /n e ) / < . .The temperature can be expressed as T = (cid:18) ρ m p a R (cid:19) / (cid:18) n γ n e (cid:19) / ≈ . · (cid:18) ρ g cm − (cid:19) / (cid:18) n γ n e (cid:19) / K (28) r [cm] T [ K ] M log = 0 . M ⊙ M ⊙ M ⊙ M ⊙ Figure 1 pro fi le (a) ρ [g cm − ] T [ K ] M log = 0 . M ⊙ M ⊙ M ⊙ M ⊙ Figure 1 pro fi le T ∝ ρ / (b)F IG . 7.— Simple model. For M log /M ⊙ = 0 . , , , the mass that sat-isfies the requirements T < K and ρ > g cm − is found to beapproximately . , . , , . M ⊙ , respectively. The profile from Figure 1is shown for comparison, with ≈ . M ⊙ , which satisfies the above require-ments. and for ( n γ /n e ) / ≈ , agrees with the values presented inpanel (b) of Figure 7.Equation (28) demonstrates that there is a tight region forwhich the density can be high ρ & g cm − while the tem-perature is low T . K . Moreover, by relating the temper-ature to the radius, T = 1 . · (cid:16) r cm (cid:17) − (cid:18) M log M ⊙ (cid:19) / (cid:18) n γ n e (cid:19) / K , (29)we see that the small range in temperature corresponds to asmall range in radii and therefore a limited amount of massavailable for thermonuclear burning by detonation. DISCUSSION In this paper we revisited the collapse-induced thermonu-clear supernovae mechanism. In Section 2 we performed a series of 1D calculations of collapsing massive stars with sim-plified initial density profiles and various compositions, as-suming that the neutrinos had a negligible effect on the outerlayers. We demonstrate that ∼ s after the core-collapse ofa massive star, a successful thermonuclear explosion of theouter shells is possible for some initial density and composi-tion profiles that include a significant layer of He–O mixture.There are several challenges in associating these simula-tions with observed supernovae.1. Post-collapse synthesized material, and in particular Ni is not released in the simulations.2. The obtained kinetic energies of the ejecta are limited to . erg, which is not sufficient for explaining typicalobserved type II supernovae.3. The required profiles are tuned (Section 2.5) and re-quire the presence of a mixture of He and O with burn-ing times of . s ( ≈ times the free-fall time) priorto collapse, which is not currently expected in stellarevolution models.In Section 3 we used simple arguments to demonstrate thatfor a general family of profiles, satisfying some reasonableconstrains, strong explosions may only be possible for a nar-row range of density amplitudes. The detonation wave re-quires high densities of & g cm − to propagate. Whilethe elements are compressed adiabatically as they fall, only asmall mass is significantly compressed at any given time (seepanel (b) of Figure 6). A successful explosion thus requires asignificant mass of explosive material M & M ⊙ to be presentin the initial profile at high densities. The high required den-sities are contrasted by the requirement for low initial tem-peratures T . K so that the pre-collapse burning rate ismuch slower than the free-fall time. Indeed, hydrostatic equi-librium requires a roughly equal number of photons and elec-trons where significant mass is present (see Equation (26)),implying that high densities ρ & g cm − require hightemperatures T & · K (see Equation (28)).While the 1D collapse scenarios studied here are thereforeunlikely to represent the majority of observed type II super-novae, they serve as a proof of concept that core-collapse-induced thermonuclear explosions are possible. In fact, as faras we know, these are the first set of 1D simulations, basedon first-principles physics, where a supernova is convincinglydemonstrated to occur following core-collapse. The crucialingredient in this scenario is the ignition of a detonation wave,which is fully resolved here for the first time (Section 2.4).Further studies are required to examine whether more realisticsimulations (in particular multi-dimensional) may lead to ex-plosions that better agree with observations and stellar evolu-tion constraints. Unlike neutrino driven explosions, which re-quire the solution of nonthermal transport equations, 3D sim-ulations of this thermonuclear mechanism are possible withcurrent computational capabilities.An interesting property of the core-collapse-induced ther-monuclear explosions reported here is the fact that the poten-tial energy of the star canceled most of the released thermonu-clear energy. This means that even a small increase in thereleased thermonuclear energy can significantly increase theobtained kinetic energy of the ejecta. To demonstrate this, wererun the set of simulations with M core = 10 M ⊙ , r O / He = 1 and t b , /t ff , = 10 , with increased available thermonuclearenergy per unit mass. To achieve this, we change the binding0 M log [ M ⊙ ] E k i n [ e r g ] f Q = 1 f Q = 1 . f Q = 1 . f Q = 1 . f Q = 1 . f Q = 1 . f Q = 1 . F IG . 8.— Asymptotic kinetic energy of the ejecta as a function of M log for M core = 10 M ⊙ , r O / He = 1 and t b , /t ff , = 10 , with artificially increasedavailable thermonuclear energy per unit mass. The legend indicates f Q , theincrease difference in binding energy between the initial composition andthe final composition (assumed to be pure silicon), achieved by artificiallychanging the binding energy of helium. energy of helium, such that the difference in binding energybetween the initial composition and the final composition (as-sumed to be pure silicon) increased by a factor f Q . The resultsshown in Figure 8 indicate that f Q = 1 . is enough to increasethe kinetic energy of the ejecta to ∼ erg (which includespost-collapse synthesized material).One physical process that cannot be treated in 1D andthat may play an important role is rotation. In fact, pre-liminary 2D calculations that include rotation (not reportedhere) indicate that stronger explosions are possible for awider range of initial conditions (Kushnir 2015). In addi-tion, post-collapse synthesized material is ejected. This isdifferent from the results of a previous study that includedrotation where an ignition of a detonation wave was not ob-tained (Bodenheimer & Woosley 1983), with the main differ-ence likely being the presence of He–O mixtures in Kushnir(2015).We thank R. Waldman and E. Livne for useful discussions.D. K. gratefully acknowledges support from Martin A. andHelen Chooljian Founders’ Circle and from the Friends of theInstitute for Advanced Study. FLASH was in part developedby the DOE NNSA-ASC OASCR Flash Center at the Univer-sity of Chicago. Computations were performed at IAS cluster. REFERENCESBethe, H. A. 1990, Reviews of Modern Physics, 62, 801Bodenheimer, P., & Woosley, S. E. 1983, ApJ, 269, 281Bruenn, S. W., Mezzacappa, A., Hix, W. 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The initial pressure profile can be calculated and is given by, p = Z ∞ r ρ GMr dr = 1 δ ρ + 1 GMr ρ , (A2)while the speed of sound is, c s = r γp ρ = r γδ ρ + 1 r GMr , (A3)1where γ ≈ / is the adiabatic index. The refraction wave moves at the speed of sound and reaches a given mass element r at atime t = Z r dr c s = 23 r c s = 23 s δ ρ + 1 γGM r / . (A4)Since there is no scale in this problem, the hydrodynamic collapse is self-similar and the position as a function of time can begiven as r ( r , t ) = r R ( s ) ρ ( r , t ) = ρ Q ( s ) p = p Q − γ (A5)where s = tt ∝ tr − / , (A6)and R ( s ) and Q ( s ) are functions that depend on s alone (note that in this section Q has nothing to do with the available thermonu-clear energy). By employing conservation of mass and momentum, we next derive two first order ordinary differential equationsof the form R ′ = f ( R, Q, s ) and Q ′ = g ( R, Q, s ) , where prime denotes a derivative with respect to s . These equations can beeasily integrated numerically to obtain R and Q and therefore the entire collapsing flow.Mass conservation implies that ρr dr = ρ r dr . (A7)To relate dr to dr note that dsdr = − sr , (A8)and drdr = R − sR ′ , (A9)where derivatives with respect to r or r are taken at a constant time t . Using equations (A5) and (A9) we have Q = r drr dr = R ( R − sR ′ ) , (A10)from which we obtain an equation for R ′ , R ′ = 23 R − QsR . (A11)Momentum conservation implies that ¨ r = − ρ dpdr − GMr . (A12)It is straightforward to find the following expressions for each of the three terms in Equation (A12), ¨ r = r t R ′′ = 94 γδ ρ + 1 GMr R ′′ , (A13) ρ dpdr = GMr R δ ρ + 1 (cid:20) sγQ − ( γ +1) Q ′ − ( δ ρ + 1) Q − γ (cid:21) , (A14)and GMr = 1 R GMr , (A15)resulting in γδ ρ + 1 R ′′ = R Q − γ − R − sγ R δ ρ + 1 Q − ( γ +1) Q ′ . (A16)By differentiating Equation (A10) with respect to s , we get Q ′ = 2 RR ′ ( R − sR ′ ) + R ( R ′ − R ′ − sR ′′ ) . (A17)Equations (A16) and (A17) involve R, R ′ , R ′′ , Q, Q ′ and s and can be used to express Q ′ in terms of Q, R, R ′ and s , Q ′ = R ′ s (cid:16) QR − R (cid:17) + δ ρ +1 γ (1 − R Q − γ )( s − sR Q − ( γ +1) ) . (A18)2At s = 1 we have R (1) = Q (1) = 1 by construction. Equations (A11) and (A18) can be integrated from s = 1 up to a point s c where R ( s c ) = 0 . Note that at s = 1 , the denominator in Equation (A18) vanishes and we need to start the integration fromsome value s close to with appropriate asymptotic conditions. The self-consistency of these equations implies that for s closeto unity, s = 1 + ds , we have R (1 + ds ) = 1 + O ( ds ) Q (1 + ds ) = 1 + 2 + ( δ ρ + 1) γ + 1 ds + O ( ds ) . (A19)The obtained values of s c are given in Table 2. s c can be used to find the time it takes for a mass element that started at r to getto zero, t f = s c t , (A20)from which we have dt f dt = s c . (A21)At a given time t , the original location of each element can be related to s by t/t = s , implying that r ∝ s − / (A22)and dm ∝ r dr ρ ∝ d ( r − δ ρ ) ∝ d ( s − δ ρ / ) (A23)where for δ ρ = 3 we have dm ∝ d log( s ) instead. At small values of r , where s ≈ s c ≈ , we have dm ∝ ( s c − s ) . Thecompression factor is ρ/ρ ∝ ( r/r ) − / ∝ ( s c − s ) − , explaining the fact that high compression is only possible for a smallamount of mass, as seen in Figure 6.3 TABLE 1T HE SIMULATIONS IN WHICH THE ASYMPTOTIC KINETIC ENERGY OF THE EJECTA IS LARGER THAN · ERG M core [ M ⊙ ] ρ in [10 g cm − ] M log [ M ⊙ ] r O / He t b , /t ff , r base [10 cm ] E kin [10 erg ] 10 0.9 3.57 1 2 1.81 0.7210 1.0 3.43 1 2 1.96 1.410 1.1 3.31 1 2 2.09 2.110 1.2 3.21 1 2 2.20 2.510 1.3 3.12 1 2 2.28 2.610 1.4 3.04 1 2 2.35 2.710 1.5 2.97 1 2 2.40 2.510 1.6 2.91 1 2 2.43 2.310 1.7 2.86 1 2 2.46 1.910 1.8 2.81 1 2 2.48 1.510 1.9 2.77 1 2 2.49 1.110 2.0 2.73 1 2 2.49 0.7810 0.9 3.57 1 10 2.25 0.9810 1.0 3.43 1 10 2.41 1.510 1.1 3.31 1 10 2.53 1.910 1.2 3.21 1 10 2.62 2.010 1.3 3.12 1 10 2.69 2.010 1.4 3.04 1 10 2.74 1.810 1.5 2.97 1 10 2.77 1.510 1.6 2.91 1 10 2.79 1.110 1.7 2.86 1 10 2.80 0.7710 1.1 3.28 3/2 10 2.52 0.6110 1.2 3.18 3/2 10 2.61 0.6510 1.3 3.10 3/2 10 2.67 0.6110 1.4 3.02 3/2 10 2.72 0.5110 1.0 3.46 2/3 10 2.39 0.5710 1.1 3.34 2/3 10 2.52 0.7410 1.2 3.23 2/3 10 2.62 0.7910 1.3 3.14 2/3 10 2.69 0.7310 1.4 3.06 2/3 10 2.75 0.628 1.0 2.61 1 10 1.99 0.688 1.1 2.51 1 10 2.11 1.18 1.2 2.43 1 10 2.20 1.48 1.3 2.36 1 10 2.27 1.68 1.4 2.30 1 10 2.32 1.78 1.5 2.25 1 10 2.36 1.78 1.6 2.20 1 10 2.39 1.68 1.7 2.16 1 10 2.41 1.48 1.8 2.12 1 10 2.43 1.28 1.9 2.09 1 10 2.43 0.958 2.0 2.05 1 10 2.43 0.758 2.1 2.02 1 10 2.43 0.586 1.2 1.65 1 10 1.80 0.636 1.3 1.60 1 10 1.86 0.836 1.4 1.56 1 10 1.91 0.996 1.5 1.52 1 10 1.95 1.16 1.6 1.49 1 10 1.98 1.26 1.7 1.46 1 10 2.00 1.26 1.8 1.43 1 10 2.02 1.26 1.9 1.41 1 10 2.03 1.16 2.0 1.39 1 10 2.03 0.986 2.1 1.37 1 10 2.03 0.866 2.2 1.35 1 10 2.03 0.746 2.3 1.33 1 10 2.02 0.634 1.7 0.778 1 10 1.57 0.554 1.9 0.751 1 10 1.58 0.574 2.1 0.729 1 10 1.58 0.50TABLE 2T IME TO REACH ORIGIN INSELF - SIMILAR COLLAPSE , s c = t f /t = dt f /dt .E QUATIONS (A4),(A20) AND (A21) γ = 4 / . / δ ρ = 2 s c = 2 .4