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Submitted for publication in the Astrophysical JournalPreprint typeset using LATEX style emulateapj v. 8/13/10

RELATIVISTIC COLLAPSE AND EXPLOSION OF ROTATING SUPERMASSIVE STARS WITHTHERMONUCLEAR EFFECTS

Pedro J. Montero1, Hans-Thomas Janka1, and Ewald Muller1

(Dated: August 17, 2011)Submitted for publication in the Astrophysical Journal

ABSTRACT

We present results of general relativistic simulations of collapsing supermassive stars with andwithout rotation using the two-dimensional general relativistic numerical code Nada, which solvesthe Einstein equations written in the BSSN formalism and the general relativistic hydrodynamicsequations with high resolution shock capturing schemes. These numerical simulations use an equationof state which includes effects of gas pressure, and in a tabulated form those associated with radiationand the electron-positron pairs. We also take into account the effect of thermonuclear energy releasedby hydrogen and helium burning. We find that objects with a mass of ≈ 5 × 105M⊙ and an initialmetallicity greater than ZCNO ≈ 0.007 do explode if non-rotating, while the threshold metallicity foran explosion is reduced to ZCNO ≈ 0.001 for objects uniformly rotating. The critical initial metallicityfor a thermonuclear explosion increases for stars with mass ≈ 106M⊙. For those stars that do notexplode we follow the evolution beyond the phase of black hole formation. We compute the neutrinoenergy loss rates due to several processes that may be relevant during the gravitational collapse ofthese objects. The peak luminosities of neutrinos and antineutrinos of all flavors for models collapsingto a BH are Lν ∼ 1055erg/s. The total radiated energy in neutrinos varies between Eν ∼ 1056 ergsfor models collapsing to a BH, and Eν ∼ 1045 − 1046 ergs for models exploding.Subject headings: Supermassive stars

1. INTRODUCTION

There is large observational evidence of the presenceof supermassive black holes (SMBHs) in the centres ofmost nearby galaxies (Rees 1998). The dynamical evi-dence related to the orbital motion of stars in the clustersurrounding Sgr A∗ indicates the presence of a SMBHwith mass ≈ 4 × 106M⊙ (Genzel et al. 2000). In addi-tion, the observed correlation between the central blackhole masses and the stellar velocity dispersion of thebulge of the host galaxies suggests a direct connectionbetween the formation and evolution of galaxies andSMBHs (Kormendy & Gebhardt 2001).The observation of luminous quasars detected at red-

shifts higher than 6 in the Sloan Digital Sky Survey(SDSS) implies that SMBHs with masses ∼ 109M⊙,which are believed to be the engines of such powerfulquasars, were formed within the first billion years af-ter the Big Bang (e.g. Fan 2006 for a recent review).However, it is still an open question how SMBH seedsform and grow to reach such high masses in such a shortamount of time (Rees 2001).A number of different routes based on stellar dynamical

processes, hydrodynamical processes or a combination ofboth have been suggested (e.g. Volonteri 2010 for a re-cent review). One of the theoretical scenarios for SMBHseed formation is the gravitational collapse of the firstgeneration of stars (Population III stars) with massesM ∼ 100M⊙ that are expected to form in halos withvirial temperature Tvir < 104K at z ∼ 20 − 50 wherecooling by molecular hydrogen is effective. As a resultof the gravitational collapse of such Pop III stars, very

montero@mpa-garching.mpg.de1 Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-

Str. 1, D-85748 Garching, Germany;

massive BHs would form and then grow via merger andaccretion (Haiman & Loeb 2001; Yoo & Miralda-Escude2004; Alvarez et al. 2009).Another possible scenario proposes that if sufficient

primordial gas in massive halos, with mass ∼ 108M⊙, isunable to cool below Tvir & 104K, it may lead to the for-mation of a supermassive object (Bromm & Loeb 2003;Begelman et al. 2006), which would eventually collapseto form a SMBH. This route assumes that fragmenta-tion, which depends on efficient cooling, is suppressed,possibly by the presence of sufficiently strong UV radi-ation, that prevents the formation of molecular hydro-gen in an environment with metallicity smaller than agiven critical value (Santoro & Shull 2006; Omukai et al.2008). Furthermore, fragmentation may depend onthe turbulence present within the inflow of gas, andon the mechanism redistributing its angular momen-tum (Begelman & Shlosman 2009). The “bars-within-bars” mechanism (Shlosman et al. 1989; Begelman et al.2006) is a self-regulating route to redistribute angularmomentum and sustain turbulence such that the inflowof gas can proceed without fragmenting as it collapseseven in a metal-enriched environment.Depending on the rate and efficiency of the inflow-

ing mass, there may be different outcomes. A lowrate of mass accumulation would favor the formationof isentropic supermassive stars (SMSs), with mass ≥5× 104M⊙, which then would evolve as equilibrium con-figurations dominated by radiation pressure (Iben 1963;Hoyle & Fowler 1963; Fowler 1964). A different outcomecould result if the accumulation of gas is fast enough sothat the outer layers of SMSs are not thermally relaxedduring much of their lifetime, thus having an entropystratification (Begelman 2009).A more exotic mechanism that could eventually lead to

2 Montero, Janka and Muller

a SMS collapsing into a SMBH is the formation and evo-lution of supermassive dark matter stars (SDMS) (Spol-yar et al 2008). Such stars would be composed pri-marily of hydrogen and helium with only about 0.1%of their mass in the form of dark matter, however theywould shine due to dark matter annihilation. It has re-cently been pointed out that SDMSs could reach masses∼ 105M⊙ (Freese et al. 2010). Once SDMSs run out oftheir dark matter supply, they experience a contractionphase that increases their baryon density and tempera-ture, leading to an environment where nuclear burningmay become important for the subsequent stellar evolu-tion.If isentropic SMSs form, their quasi-stationary evolu-

tion of cooling and contraction will drive the stars tothe onset of a general relativistic gravitational instabil-ity leading to their gravitational collapse (Chandrasekhar1964; Fowler 1964), and possibly also to the for-mation of a SMBH. The first numerical simula-tions, within the post-Newtonian approximation, ofAppenzeller & Fricke (1972) concluded that for spheri-cal stars with masses greater than 106M⊙ thermonuclearreactions have no major effect on the collapse, while lessmassive stars exploded due to hydrogen burning. LaterShapiro & Teukolsky (1979) performed the first relativis-tic simulations of the collapse of a SMS in spherical sym-metry. They were able to follow the evolution until theformation of a BH, although their investigations did notinclude any microphysics. Fuller et al. (1986) revisitedthe work of Appenzeller & Fricke (1972) and performedsimulations of non-rotating SMSs in the range of 105-106M⊙ with post-Newtonian corrections and detailed mi-crophysics that took into account an equation of state(EOS) including electron-positron pairs, and a reactionnetwork describing hydrogen burning by the CNO cy-cle and its break-out via the rapid proton capture (rp)-process. They found that SMSs with zero initial metal-licity do not explode, while SMSs with masses larger than105M⊙ and with metallicity ZCNO ≥ 0.005 do explode.More recently Linke et al. (2001) carried out general

relativistic hydrodynamic simulations of the sphericallysymmetric gravitational collapse of SMSs adopting aspacetime foliation with outgoing null hypersurfaces tosolve the system of Einstein and fluid equations. Theyperformed simulations of spherical SMSs with masses inthe range of 5 × 105M⊙ − 109M⊙ using an EOS thataccounts for contributions from baryonic gases, and ina tabulated form, radiation and electron-positron pairs.They were able to follow the collapse from the onsetof the instability until the point of BH formation, andshowed that an apparent horizon enclosing about 25% ofthe stellar material was formed in all cases when simula-tions stopped.Shibata & Shapiro (2002) carried out general relativis-

tic numerical simulations in axisymmetry of the collapseof uniformly rotating SMSs to BHs. They did not takeinto account thermonuclear burning, and adopted a Γ-law EOS, P = (Γ − 1)ρǫ with adiabatic index Γ = 4/3,where P is the pressure, ρ the rest-mass density, andǫ the specific internal energy. Although their simula-tions stopped before the final equilibrium was reached,the BH growth was followed until about 60% of the masshad been swallowed by the SMBH. They estimated that

about 90% of the total mass would end up in the finalSMBH with a spin parameter of J/M2 ∼ 0.75.The gravitational collapse of differentially rotat-

ing SMSs in three dimensions was investigated bySaijo & Hawke (2009), who focused on the post-BH evo-lution, and also on the gravitational wave (GW) signal re-sulting from the newly formed SMBH and the surround-ing disk. The GW signal is expected to be emitted inthe low frequency LISA band (10−4 − 10−1 Hz).Despite the progress made, the final fate rotating isen-

tropic SMSs is still unclear. In particular, it is stillan open question for which initial metallicities hydrogenburning by the β-limited hot CNO cycle and its break-out via the 15O(α, γ)19Ne reaction (rp-process) can haltthe gravitational collapse of rotating SMSs and gener-ate enough thermal energy to lead to an explosion. Toaddress this issue, we perform a series of general relativis-tic hydrodynamic simulations with a microphysical EOSaccounting for contributions from radiation, electron-positron pairs, and baryonic matter, and taking into ac-count the net thermonuclear energy released by the nu-clear reactions involved in hydrogen burning through thepp-chain, cold and hot CNO cycles and their break-outby the rp-process, and helium burning through the 3-α reaction. The numerical simulations were carried outwith the Nada code (Montero et al. 2008), which solvesthe Einstein equations coupled to the general relativistichydrodynamics equations.Greek indices run from 0 to 3, Latin indices from 1 to 3,

and we adopt the standard convention for the summationover repeated indices. Unless otherwise stated we useunits in which c = G = 1.

2. BASIC EQUATIONS

Next we briefly describe how the system of Einsteinand hydrodynamic equations are implemented in theNada code. We refer to Montero et al. (2008) for a moredetailed description of the main equations and thoroughtesting of the code (namely single BH evolutions, shocktubes, evolutions of both spherical and rotating relativis-tic stars, gravitational collapse to a BH of a marginallystable spherical star, and simulations of a system formedby a BH surrounded by a self-gravitating torus in equi-librium).

2.1. Formulation of Einstein equations

2.1.1. BSSN formulation

We follow the 3+1 formulation in which the spacetimeis foliated into a set of non-intersecting spacelike hyper-surfaces. In this approach, the line element is written inthe following form

ds2 = −(α2 − βiβi)dt2 + 2βidx

idt+ γijdxidxj , (1)

where α, βi and γij are the lapse function, the shift three-vector, and the three-metric, respectively. The latter isdefined by

γµν = gµν + nµnν , (2)

where nµ is a timelike unit-normal vector orthogonal toa spacelike hypersurface.We make use of the BSSN formulation (Nakamura

1987; Shibata & Nakamura 1995; Baumgarte & Shapiro

Collapse of supermassive stars 3

1999) to solve the Einstein equations. Initially, a confor-mal factor φ is introduced, and the conformally relatedmetric is written as

γij = e−4φγij (3)

such that the determinant of the conformal metric, γij ,is unity and φ = ln(γ)/12, where γ = det(γij). Wealso define the conformally related traceless part of theextrinsic curvature Kij ,

Aij = e−4φAij = e−4φ

(

Kij −1

3γijK

)

, (4)

where K is the trace of the extrinsic curvature. Weevolve the conformal factor defined as χ ≡ e−4φ

(Campanelli et al. 2006), and the auxiliary variables Γi,known as the conformal connection functions, defined as

Γi ≡ γjkΓijk = −∂j γ

ij , (5)

where Γijk are the connection coefficients associated

with γij .During the evolution we also enforce the constraints

Tr(Aij) = 0 and det(γij) = 1 at every time step.We use the Cartoon method (Alcubierre et al. 2001) to

impose axisymmetry while using Cartesian coordinates.

2.1.2. Gauge choices

In addition to the BSSN spacetime variables(γij , Aij ,K, χ, Γi), there are two more quantities left un-determined, the lapse, α and the shift vector βi. We usedthe so-called “non-advective 1+log” slicing (Bona et al.1997), by dropping the advective term in the “1+log”slicing condition. In this case, the slicing condition takesthe form

∂tα = −2αK. (6)

For the shift vector, we choose the “Gamma-freezing con-dition” (Alcubierre et al. 2003) written as

∂tβi =

3

4Bi, (7)

∂tBi = ∂tΓ

i − ηBi, (8)

where η is a constant that acts as a damping term, orig-inally introduced both to prevent long term drift of themetric functions and to prevent oscillations of the shiftvector.

2.2. Formulation of the hydrodynamics equations

The general relativistic hydrodynamics equations, ex-pressed through the conservation equations for the stress-energy tensor T µν and the continuity equation are

∇µTµν = 0 , ∇µ (ρu

µ) = 0, (9)

where ρ is the rest-mass density, uµ is the fluid four-velocity and ∇ is the covariant derivative with respect tothe spacetime metric. Following Shibata (2003), the gen-eral relativistic hydrodynamics equations are written ina conservative form in cylindrical coordinates. Since theEinstein equations are solved only in the y = 0 plane withCartesian coordinates (2D), the hydrodynamic equations

are rewritten in Cartesian coordinates for y = 0. The fol-lowing definitions for the hydrodynamical variables areused

ρ∗ ≡ ρWe6φ, (10)

vi ≡ui

ut= −βi + αγij uj

hW, (11)

ui ≡ hui, (12)

e ≡e6φ

ρ∗Tµνn

µnν = hW −P

ρW, (13)

W ≡ αut, (14)

where W and h are the Lorentz factor and the specificfluid enthalpy respectively, and P is the pressure. Theconserved variables are ρ∗, Ji = ρ∗ui, E∗ = ρ∗e. Werefer to Shibata (2003) for further details.

3. SUPERMASSIVE STARS AND MICROPHYSICS

3.1. Properties of SMSs

Isentropic SMSs are self-gravitating equilibrium con-figurations of masses in the range of 104− 108M⊙, whichare mainly supported by radiation pressure, while thepressure of electron-positron pairs and of the baryon gasare only minor contributions to the EOS. Such configu-rations are well described by Newtonian polytropes withpolytropic index n = 3 (adiabatic index Γ = 4/3). Theratio of gas pressure to the total pressure (β) for sphericalSMSs can be written as (Fowler & Hoyle 1966)

β =Pg

Ptot≈

4.3

µ

(

M⊙

M

)1/2

, (15)

where µ is the mean molecular weight. Thus β ≈ 10−2

for M ≈ 106M⊙.Since nuclear burning timescales are too long for

M & 104M⊙, evolution of SMSs proceeds on the Kelvin-Helmholtz timescale and is driven by the loss of energyand entropy by radiation as well as loss of angular mo-mentum via mass shedding in the case of rotating con-figurations.Although corrections due to the nonrelativistic gas of

baryons and electrons and general relativistic effects aresmall, they cannot be neglected for the evolution. Firstly,gas corrections raise the adiabatic index slightly above4/3

Γ ≈4

3+

β

6+ 0(β2). (16)

Secondly, general relativistic corrections lead to theexistence of a maximum for the equilibrium mass as afunction of the central density. For spherical SMSs thismeans that for a given mass the star evolves to a criticaldensity beyond which it is dynamically unstable againstradial perturbations (Chandrasekhar 1964):

ρcrit = 1.994× 1018(

0.5

µ

)3 (M⊙

M

)7/2

gcm−3. (17)

The onset of the instability also corresponds to a criti-cal value of the adiabatic index Γcrit, i.e. configurations

4 Montero, Janka and Muller

become unstable when the adiabatic index drops belowthe critical value

Γcrit =4

3+ 1.12

2GM

Rc2. (18)

This happens when the stabilizing gas contribution tothe EOS does not raise the adiabatic index above 4/3 tocompensate for the destabilizing effect of general relativ-ity expressed by the second term on the righ-hand-sideof Eq. (18).Rotation can stabilize configurations against the ra-

dial instability. The stability of rotating SMSs withuniform rotation was analyzed by Baumgarte & Shapiro(1999a,b). They found that stars at the onset of theinstability have an equatorial radius R ≈ 640GM/c2, aspin parameter q ≡ cJ/GM2 ≈ 0.97, and a ratio of rota-tional kinetic energy to the gravitational binding energyof T/W ≈ 0.009.

3.2. Equation of State

To close the system of hydrodynamic equations (Eq. 9)we need to define the EOS. We follow a treatment whichincludes separately the baryon contribution on the onehand, and photons and electron-positron pairs contri-butions, in a tabulated form, on the other hand. Thebaryon contribution is given by the analytic expressionsfor the pressure and specific internal energy

Pb =RρT

µb, (19)

ǫb =2

3

RρT

µb, (20)

where R the universal gas constant, T the temperature,ǫb the baryon specific internal energy, and µb is the meanmolecular weight due to ions, which can be expressed asa function of the mass fractions of hydrogen (X), helium(Y ) and heavier elements (metals) (ZCNO) as

1

µb≈ X +

Y

4+

ZCNO

〈A〉, (21)

where 〈A〉 is the average atomic mass of the heavy el-ements. We assume that the composition of SMSs (ap-proximately that of primordial gas) has a mass fraction ofhydrogen X = 0.75−ZCNO and helium Y = 0.25, wherethe metallity ZCNO = 1−X−Y is an initial parameter,typically of the order of ZCNO ∼ 10−3 (see Table 1 de-tails). Thus, for the initial compositions that we considerthe mean molecular weight of baryons is µb ≈ 1.23 (i.e.corresponding to a molecular weight for both ions andelectrons of µ ≈ 0.59).Effects associated with photons and the creation of

electron-positron pairs are taken into account employ-ing a tabulated EOS. At temperatures above 109K, notall the energy is used to increase the temperature andpressure, but part of the photon energy is used to createthe rest-mass of the electron-positron pairs. As a resultof pair creation, the adiabatic index of the star decreases,which means that the stability of the star is reduced.Given the specific internal energy, ǫ and rest-mass den-

sity, ρ, as evolved by the hydrodynamic equations, it is

possible to compute the temperature T by a Newton-Raphson algorithm that solves the equation ǫ∗(ρ, T ) = ǫfor T ,

Tn+1 = Tn − (ǫ∗(ρ, Tn)− ǫ)

(

∂ǫ∗(ρ, T )

∂T

)∣

∣

∣

∣

−1

Tn

, (22)

where n is the iteration counter.

3.3. Nuclear burning

In order to avoid the small time steps, and CPU-timedemands connected with the solution of a nuclear reac-tion network coupled to the hydrodynamic evolution, weapply an approximate method to take into account thebasic effects of nuclear burning on the dynamics of thecollapsing SMSs. We compute the nuclear energy re-lease rates by hydrogen burning (through the pp-chain,cold and hot CNO cycles, and their break-out by the rp-process) and helium burning (through the 3-α reaction)as a function of rest-mass density, temperature and massfractions of hydrogen X , helium Y and CNO metallicityZCNO. These nuclear energy generation rates are addedas a source term on the right-hand-side of the evolutionequation for the conserved quantity E∗.The change rates of the energy density due to nuclear

reactions, in the fluid frame, expressed in units of [ergcm−3 s−1] are given by:

• pp-chain (Clayton 1983):

(

∂e

∂t

)

pp

= ρ(2.38× 106ρg11X2T−0.6666

6

e−33.80/T 0.33336 ), (23)

where T6 = T/106K, and g11 is given by

g11 = 1 + 0.0123T 0.33336 + 0.0109T 0.66666

6 +

0.0009T6. (24)

• 3-α (Wiescher et al. 1999):(

∂e

∂t

)

3α

= ρ(5.1× 108ρ2Y 3T−39 e−4.4/T9), (25)

where T9 = T/109K.

• Cold-CNO cycle (Shen & Bildsten 2007):

(

∂e

∂t

)

CCNO

= 4.4× 1025ρ2XZCNO

(T−2/39 e−15.231/T

1/39 +

+ 8.3× 10−5T−3/29 e−3.0057/T9). (26)

• Hot-CNO cycle (Wiescher et al. 1999):

(

∂e

∂t

)

HCNO

= 4.6× 1015ρZCNO. (27)

Collapse of supermassive stars 5

• rp-process (Wiescher et al. 1999):(

∂e

∂t

)

rp

= ρ(1.77× 1016ρY ZCNO

29.96T−3/29 e−5.85/T9). (28)

Since we follow a single fluid approach, in which wesolve only the hydrodynamics equations Eq. (9) (i.e. wedo not solve additional advection equations for the abun-dances of hydrogen, helium and metals), the elementalabundances during the time evolution are fixed. Nev-ertheless, this assumption most possibly does not af-fect significantly the estimate of the threshold metallicityneeded to produce a thermal bounce in collapsing SMSs.The average energy release through the 3-α reaction isabout 7.275 MeV for each 12C nucleus formed. Since thetotal energy due to helium burning for exploding mod-els is ∼ 1045 ergs (e.g. 9.0 × 1044 ergs for model S1.c);and even considering that this energy is released mostlyin a central region of the SMS containing 104M⊙ of itsrest-mass (Fuller et al. 1986), it is easy to show that thechange in the metallicity is of the order of 10−11. There-fore, the increase of the metallicity in models experienc-ing a thermal bounce is much smaller than the criticalmetallicities needed to trigger the explosions. Similarly,the average change in the mass fraction of hydrogen dueto the cold and hot CNO cycles is expected to be ∼ 10%for exploding models.

3.4. Recovery of the primitive variables

After each time iteration the conserved variables(i.e. ρ∗, Jx, Jy, Jz, E∗) are updated and the primitive hy-drodynamical variables (i.e. ρ, vx, vy, vz, ǫ) have to be re-covered. The recovery is done in such a way that it allowsfor the use of a general EOS of the form P = P (ρ, ǫ).We calculate a function f(P ∗) = P (ρ∗, ǫ∗) − P ∗, whereρ∗ and ǫ∗ depend only on the conserved quantities andthe pressure guess P ∗. The new pressure is computedthen iteratively by a Newton-Raphson method until thedesired convergence is achieved.

3.5. Energy loss by neutrino emission

The EOS allows us to compute the neutrino losses dueto the following processes, which become most relevantjust before BH formation:

• Pair annihilation (e+ + e− → ν + ν): most impor-tant process above 109K. Due to the large meanfree path of neutrinos in the stellar medium at thedensities of SMSs the energy loss by neutrinos canbe significant. For a 106M⊙ SMS most of the en-ergy release in the form of neutrinos originates fromthis process. The rates are computed using the fit-ting formula given by Itoh et al. (1996).

• Photo-neutrino emission (γ + e± → e± + ν + ν):dominates at low temperatures T . 4 × 108K anddensities ρ . 105gcm−3 (Itoh et al. 1996).

• Plasmon decay (γ → ν + ν): This is the least rele-vant process for the conditions encountered by themodels we have considered because its importanceincreases at higher densities than those present in

SMSs. The rates are computed using the fittingformula given by Haft et al. (1994).

4. COMPUTATIONAL SETUP

The evolution equations are integrated by the methodof lines, for which we use an optimal strongly stability-preserving (SSP) Runge-Kutta algorithm of fourth-orderwith 5 stages (Spiteri & Ruuth 2002). We use a second-order slope limiter reconstruction scheme (MC limiter) toobtain the left and right states of the primitive variablesat each cell interface, and a HLLE approximate Riemannsolver (Harten et al. 1983; Einfeldt 1988) to compute thenumerical fluxes in the x and z directions.Derivative terms in the spacetime evolution equa-

tions are represented by a fourth-order centered finite-difference approximation on a uniform Cartesian grid ex-cept for the advection terms (terms formally like βi∂iu),for which an upwind scheme is used.The computational domain is defined as 0 ≤ x ≤ L and

0 ≤ z ≤ L, where L refers to the location of the outerboundaries. We used a cell-centered Cartesian grid toavoid that the location of the BH singularity coincideswith a grid point.

4.1. Regridding

Since it is not possible to follow the gravitationalcollapse of a SMS from the early stages to the phaseof black hole formation with a uniform Cartesiangrid (the necessary fine zoning would be computation-ally too demanding), we adopt a regridding procedure(Shibata & Shapiro 2002). During the initial phase ofthe collapse we rezone the computational domain bymoving the outer boundary inward, decreasing the gridspacing while keeping the initial number of grid pointsfixed. Initially we use N×N = 400×400 grid points, andplace the outer boundary at L ≈ 1.5re where re is theequatorial radius of the star. Rezoning onto the new gridis done using a polynomial interpolation. We repeat thisprocedure 3-4 times until the collapse timescale in thecentral region is much shorter than in the outer parts.At this point, we both decrease the grid spacing and alsoincrease the number of grid points N in dependence ofthe lapse function typically as follows: N×N = 800×800if 0.8 > α > 0.6, N ×N = 1200× 1200 if 0.6 > α > 0.4,and N ×N = 1800× 1800 if α < 0.4. This procedure en-sures the error in the conservation of the total rest-massto be less than 2% on the finest computational domain.

4.2. Hydro-Excision

To deal with the spacetime singularity from the newlyformed BH we use the method of excising the mattercontent in a region within the horizon as proposed byHawke et al. (2005) once an apparent horizon (AH) isfound. This excision is done only for the hydrodynamicalvariables, and the coordinate radius of the excised regionis allowed to increase in time. On the other hand, we doneither use excision nor artificial dissipation terms forthe spacetime evolution, and solely rely on the gaugeconditions.

4.3. Definitions

Here we define some of the quantities listed in Table 1.We compute the total rest-mass M∗ and the ADM mass

6 Montero, Janka and Muller

TABLE 1Main properties of the initial models studied. From left to right the columns show: model, gravitational mass, initial

central rest-mass density, Tk/|W |, angular velocity, initial central temperature, metallicity, the fate of the star, radialkinetic energy after thermal bounce, and total neutrino energy output.

Model M ρc Tk/|W | Ω Tc Initial metallicity Fate ERK Eν

[105M⊙] [10−2g/cm3] [10−5rad/s] [107K] [10−3] [1056erg] [erg]

S1.a 5 2.4 0 0 5.8 5 BH ... 3.4× 1056

S1.b 5 2.4 0 0 5.8 6 BH .. ...S1.c 5 2.4 0 0 5.8 7 Explosion 5.5 9.4× 1045

R1.a 5 40 0.0088 2.49 13 0.5 BH ... 5.4× 1056

R1.b 5 40 0.0088 2.49 13 0.8 BH ... ...R1.c 5 40 0.0088 2.49 13 1 Explosion 1.0 ...R1.d 5 40 0.0088 2.49 13 2 Explosion 1.9 8.9× 1045

S2.a 10 0.23 0 0 2.6 30 BH ... 6.8× 1056

S2.b 10 0.23 0 0 2.6 50 Explosion 35 8.0× 1046

R2.a 10 12 0.0087 1.47 9.7 0.5 BH ... 3.1× 1056

R2.b 10 12 0.0087 1.47 9.7 0.8 BH ... ...R2.c 10 12 0.0087 1.47 9.7 1.0 BH ... ...R2.d 10 12 0.0087 1.47 9.7 1.5 Explosion 1.5 2.1× 1046

M as

M∗ = 4π

∫ L

0

xdx

∫ L

0

ρ∗dz, (29)

M = −2

∫ L

0

xdx

∫ L

0

dz

[

−2πEe5φ +eφ

8R

−e5φ

8

(

AijAij −

2

3K2

)]

, (30)

where E = nµnνTµν (nµ being the unit normal to the

hypersurface) and R is the scalar curvature associated tothe conformal metric γij .The rotational kinetic energy Tk and the gravitational

potential energy W are given by

Tk = 2π

∫ L

0

x2dx

∫ L

0

ρ∗uyΩdz, (31)

where Ω is the angular velocity.

W = M − (M∗ + Tk + Eint), (32)

where the internal energy is computed as

Eint = 4π

∫ L

0

xdx

∫ L

0

ρ∗ǫdz. (33)

In axisymmetry the AH equation becomes a nonlinearordinary differential equation for the AH shape function,h = h(θ) (Shibata 1997; Thornburg 2007). We employan AH finder that solves this ODE by a shooting methodusing ∂θh(θ = 0) = 0 and ∂θh(θ = π/2) = 0 as boundaryconditions. We define the mass of the AH as

MAH =

√

A

16π, (34)

where A is the area of the AH.

5. INITIAL MODELS

The initial SMSs are set up as isentropic objects. Allmodels are chosen such that they are gravitationally un-stable, and therefore their central rest-mass density isslightly larger than the critical central density required

for the onset of the collapse of a configuration withgiven mass and entropy. A list of the different SMSswe have considered is provided in Table 1. Models S1and S2 represent a spherically symmetric, nonrotatingSMS with gravitational mass of M = 5 × 105M⊙ andM = 1 × 106M⊙, respectively, while models R1 and R2are uniformly rotating initial models again with massesof M = 5 × 105M⊙ and M = 1 × 106M⊙, respec-tively. The rigidly and maximally rotating initial modelsR1 and R2 are computed with the Lorene code (URLhttp://www.lorene.obspm.fr). We also introduce a per-turbation to trigger the gravitational collapse by reduc-ing the pressure overall by ≈ 1.5%.In order to determine the threshold metallicity re-

quired to halt the collapse and produce an explosion wecarry out several numerical simulations for each initialmodel with different values of the initial metallicity. Theinitial metallicities along with the fate of the star aregiven in Table 1.

6. RESULTS

6.1. Collapse to BH vs. Thermonuclear explosion

First we consider a gravitationally unstable spheri-cally symmetric SMS with a gravitational mass of M =5 × 105M⊙ (S1.a, S1.b and S1.c), which corresponds toa model extensively discussed in Fuller et al. (1986),and therefore allows for a comparison with the resultspresented here. Fuller et al. (1986) found that unstablespherical SMSs withM = 5×105M⊙ and an initial metal-licity ZCNO = 2 × 10−3 collapse to a BH while modelswith an initial metallicity ZCNO = 5× 10−3 explode dueto the nuclear energy released by the hot CNO burning.They also found that the central density and tempera-ture at thermal bounce (where the collapse is reversed toan explosion) are ρc,b = 3.16 g/cm3 and Tc,b = 2.6× 108

K, respectively.The left panels in Figure 1 show the time evolution of

the central rest-mass density (upper panel) and centraltemperature (lower panel) for models S1.a, S1.c, R1.aand R1.d, i.e., non-rotating and rotating models with amass of M = 5 × 105M⊙. In particular, the solid linesrepresent the time evolution of the central density andtemperature for model S1.c (ZCNO = 7×10−3) and R1.d

Collapse of supermassive stars 7

Fig. 1.— Left upper panel shows the time evolution of the central rest-mass density for models S1 and R1 (i.e., spherical and rotatingstars with mass M = 5 × 105M⊙), and the lower left panel shows the time evolution of the central temperature. Horizontal dotted linesmark the temperature range in which nuclear energy is primarily released by the hot CNO cycle. Similarly, the time evolution of thesame quantities for model S2 and R2 (i.e., spherical and rotating stars with mass M = 1 × 106M⊙) are shown in the upper and lowerright panels. As the collapse proceeds, the central density and temperature rise rapidly, increasing the nuclear energy generation rate byhydrogen burning. If the metallicity is sufficiently high, enough energy can be liberated to produce a thermal bounce. This is the case formodels S1.c, R1.d, S2.b and R2.d shown here.

(ZCNO = 5 × 10−4). As the collapse proceeds, the cen-tral density and temperature rise rapidly, which increasesthe nuclear energy generation rate by hydrogen burning.Since the metallicity is sufficiently high, enough energycan be liberated to increase the pressure and to produce athermal bounce. This is the case for model S1.c. In Fig-ure 1 we show that a thermal bounce occurs (at approxi-mately t ∼ 7× 105 s) entirely due to the hot CNO cycle,which is the main source of thermonuclear energy at tem-peratures in the range 2 × 108K ≤ T ≤ 5 × 108K. Therest-mass density at bounce is ρc,b = 4.8 g/cm3 and thetemperature Tc,b = 3.05× 108K. These values, as well asthe threshold metallicity needed to trigger a thermonu-clear explosion (ZCNO = 7×10−3), are higher than thosefound by Fuller et al. (1986) (who found that a spher-ical nonrotating model with the same rest-mass wouldexplode, if the initial metallicity was ZCNO = 5× 10−3).On the other hand, dashed lines show the time evo-

lution of the central density and temperature for modelS1.a (ZCNO = 5 × 10−3). In this case, as well as formodel S1.b, the collapse is not halted by the energy re-lease and continues until an apparent horizon is found,indicating the formation of a BH.We note that the radial velocity profiles change contin-

uously near the time where the collapse is reversed to anexplosion due to the nuclear energy released by the hotCNO burning, and an expanding shock forms only nearthe surface of the star at a radius R ≈ 1.365 × 1013cm(i.e. R/M ≈ 180) where the rest-mass density is ≈3.5×10−6gcm−3. We show in Figure 2 the profiles of thex-component of the three-velocity vx along the x-axis(in the equatorial plane) for the nonrotating sphericalstars S1.a (dashed lines) and S1.c (solid lines) at threedifferent time slices near the time at which model S1.cexperiences a thermal bounce. Velocity profiles of modelS1.c are displayed up to the radius where a shock forms

at t ≈ 7.31 × 105s and begins to expand into the lowdensity outer layers of the SMS.The evolutionary tracks for the central density and

temperature of the rotating models R1.a and R1.d arealso shown in Figure 1. A dashed line corresponds tomodel R1.a, with an initial metallicity ZCNO = 5×10−4,which collapses to a BH. A solid line denotes model R1.dwith ZCNO = 2×10−3, which explodes due to the energyreleased by the hot CNO cycle. We find that Model R1.cwith a lower metallicity of ZCNO = 1×10−3 also explodeswhen the central temperature is the range dominated bythe hot CNO cycle.As a result of the kinetic energy stored in the rotation

of models R1.c and R1.d, the critical metallicity neededto trigger an explosion decreases significantly relative tothe non-rotating case. We observe that rotating modelswith initial metallicities up to ZCNO = 8 × 10−4 do notexplode even via the rp-process, which is dominant attemperatures above T ≈ 5 × 108 K and increases thehydrogen burning rate by 200 − 300 times relative tothe hot CNO cycle. We also note that the evolutiontime scales of the collapse and bounce phases are reducedbecause rotating models are more compact and have ahigher initial central density and temperature than thespherical ones at the onset of the gravitational instability.The right panels in Figure 1 show the time evolu-

tion of the central rest-mass density (upper panel) andthe central temperature (lower panel) for models S2.a,S2.b, R2.a and R2.d, i.e., of models with a mass ofM = 106M⊙. We find that the critical metallicity for anexplosion in the spherical case is ZCNO = 5×10−2 (modelS2.b), while model S2.a with ZCNO = 3× 10−2 collapsesto a BH. We note that the critical metallicity leading to athermonuclear explosion is higher than the critical valuefound by Fuller et al. (1986) (i.e., ZCNO = 1× 10−2) fora spherical SMS with the same mass. The initial metal-licity leading to an explosion in the rotating case (model

8 Montero, Janka and Muller

Fig. 2.— Profiles of the x-component of the three-velocity vx

along the x-axis (in the equatorial plane) for the nonrotating spher-ical stars S1.a (dashed lines) and S1.c (solid lines) at three differenttime slices near the time at which model S1.c experiences a ther-mal bounce. Velocity profiles of model S1.c are displayed up to theradius where takes place the formation of a shock that expandsinto the low density outer layers of the SMS.

R2.d) is more than an order of magnitude smaller than inthe spherical case. As for the models with a smaller grav-itational mass, the thermal bounce takes place when thephysical conditions in the central region of the star allowfor the release of energy by hydrogen burning throughthe hot CNO cycle. Overall, the dynamics of the moremassive models indicates that the critical initial metal-licity required to produce an explosion increases with therest-mass of the star.Figure 3 shows the total nuclear energy generation rate

in erg/s for the exploding models as a function of timeduring the late stages of the collapse just before and af-ter bounce. The main contribution to the nuclear energygeneration is due to hydrogen burning by the hot CNOcycle. The peak values of the energy generation rateat bounce lie between several 1051erg/s for the rotatingmodels (R1.d and R2.d), and ≈ 1052 − 1053erg/s for thespherical models (S1.c and S2.b). As expected the max-imum nuclear energy generation rate needed to producean explosion is lower in the rotating models. Moreover,as the explosions are due to the energy release by hy-drogen burning via the hot CNO cycle, the ejecta wouldmostly be composed of 4He.As a result of the thermal bounce, the kinetic energy

rises until most of the energy of the explosion is in theform of kinetic energy. We list in the second but lastcolumn of Table 1 the radial kinetic energy after thermalbounce, which ranges between ERK = 1.0× 1055ergs forthe rotating star R1.c, and ERK = 3.5× 1057ergs for thespherical star S2.b.

6.2. Photon luminosity

Due to the lack of resolution at the surface of the star,it becomes difficult to compute accurately the photo-sphere and its effective temperature from the criterionthat the optical depth is τ = 2/3. Therefore, in order to

Fig. 3.— Nuclear energy generation rate in erg/s for the explod-ing models (S1.c, R1.d, R1.c, S2.b and R2.d) as a function of timenear the bounce. The contribution to the nuclear energy gener-ation is mainly due to hydrogen burning by the hot CNO cycle.The peak values of the energy generation rate at bounce lie be-tween ≈ 1051[erg/s] for the rotating models (R1.d and R2.d), and≈ 1052 − 1053[erg/s] for the spherical models (S1.c and S2.b).

estimate the photon luminosity produced in associationwith the thermonuclear explosion, we make use of thefact that within the diffusion approximation the radia-tion flux is given by

Fγ = −c

3κesρ∇U, (35)

where U is the energy density of the radiation, and κes isthe opacity due to electron Thompson scattering, whichis the main source of opacity in SMSs. The photon lumi-nosity in terms of the temperature gradient and for thespherically symmetric case can be written as

Lγ = −16πacr2T 3

3κesρ

∂T

∂r, (36)

where a is the radiation constant, and c the speed of light.As can be seen in the last panel of Figure 4 the distribu-tion of matter becomes spherically symmetric during thephase of expansion after the thermonuclear explosion. Inthis figure (Fig. 4) we show the isodensity contours forthe rotating model R1.d. The frames have been taken atthe initial time (left figure), at t = 0.83 × 105s (centralfigure) just after the thermal bounce (at t = 0.78×105s),and at t = 2.0× 105 s when the radius of the expandingmatter is roughly 4 times the radius of the star at theonset of the collapse.The photon luminosity computed using Eq.(36) for

model R1.d is displayed in Figure 5, where we also in-dicate with a dashed vertical line the time at which thethermal bounce takes place. The photon luminosity be-fore the thermal bounce is computed at radii inside thestar unaffected by the local dynamics of the low den-sity outer layers which is caused by the initial pressureperturbation and by the interaction between the surfaceof the SMS and the artificial atmosphere. Once the ex-

Collapse of supermassive stars 9

Fig. 4.— Isodensity contours of the logarithm of the rest-mass density (in g/cm3) for the rotating model R1.d. The frames have beentaken at the initial time (left figure), at t = 0.83 × 105 s (central figure) just after a thermal bounce takes place, and at t = 2.0 × 105 swhen the radius of the expanding matter is roughly 4 times the radius of the star at the onset of the collapse.

Fig. 5.— Logarithm of the photon luminosity of model R1.din units of erg/s as a function of time. The vertical dashed lineindicates the time at which the thermal bounce takes place.

panding shock forms near the surface, the photon lumi-nosity is computed near the surface of the star. Thelightcurve shows that, during the initial phase, the lu-minosity is roughly equal to the Eddington luminosity≈ 5 × 1043erg/s until the thermal bounce. Then, thephoton luminosity becomes super-Eddington when theexpanding shock reaches the outer layers of the star andreaches a value of Lγ ≈ 1 × 1045erg/s. This value ofthe photon luminosity after the bounce is within a fewpercent difference with respect to the photon luminos-ity Fuller et al. (1986) found for a nonrotating SMS ofsame rest-mass. The photon luminosity remains super-Eddington during the phase of rapid expansion that fol-lows the thermal bounce. We compute the photon lu-minosity until the surface of the star reaches the outerboundary of the computational domain ≈ 1.0 × 105

after the bounce. Beyond that point, the luminosityis expected to decrease, and then rise to a plateau of∼ 1045erg/s due to the recombination of hydrogen (see

Fuller et al. 1986 for a nonrotating star).

6.3. Collapse to BH and neutrino emission

The outcome of the evolution of models that do notgenerate enough nuclear energy during the contractionphase to halt the collapse is the formation of a BH.The evolutionary tracks for the central density and tem-perature of some of these models are also shown inFigure 1. The central density typically increases upto ρc ∼ 107gcm−3 and the central temperature up toTc ∼ 1010K just before the formation of an apparenthorizon.Three isodensity contours for the rotating model R1.a

collapsing to a BH are shown in Figure 6, which displaythe flattening of the star as the collapse proceeds. Theframes have been taken at the initial time (left panel),at t = 0.83 × 105s (central panel) approximately whenmodel R1.d with higher metallicity experiences a thermalbounce and, at t = 1.127× 105s, where a BH has alreadyformed and its AH has a mass of 50% of the total initialmass.At the temperatures reached during the late stages of

the gravitational collapse (in fact at T ≥ 5 × 108K) themost efficient process for hydrogen burning is the break-out from the hot CNO cycle via the 15O(α, γ)19Ne re-action. Nevertheless, we find that models which do notrelease enough nuclear energy by the hot CNO cycle tohalt their collapse to a BH, are not able to produce athermal explosion due to the energy liberated by the15O(α, γ)19Ne reaction. We note that above 109K, notall the liberated energy is used to increase the temper-ature and pressure, but is partially used to create therest-mass of the electron-positron pairs. As a result ofpair creation, the adiabatic index of the star decreases,which means the stability of the star is reduced. More-over, due to the presence of e± pairs, neutrino energylosses grow dramatically.Figure 7 shows (solid lines) the time evolution of the

redshifted neutrino luminosities of four models collaps-ing to a BH (S1.a, R1.a, S2.a, and R2.a), and (dashedlines) of four models experiencing a thermal bounce(S1.c, R1.d, S2.b, and R2.d). The change of the slopeof the neutrino luminosities at ∼ 1043erg/s denotes thetransition from photo-neutrino emission to the pair an-

10 Montero, Janka and Muller

Fig. 6.— Isodensity contours of the logarithm of the rest-mass density (in g/cm3) for the rotating model R1.a. The frames have beentaken at the initial time (left panel), at t = 0.83 × 105 s (central panel), at t = 1.127 × 105 s, where a BH has already formed and itsapparent horizon encloses a mass of 50% of the total initial gravitational mass.

nihilation dominated region. The peak luminosities inall form of neutrino for models collapsing to a BH areLν ∼ 1055erg/s. Neutrino luminosities can be that im-portant because the densities in the core prior to BHformation are ρc ∼ 107gcm−3, and therefore neutrinoscan escape. The peak neutrino luminosities lie betweenthe luminosities found by Linke et al. (2001) for the col-lapse of spherical SMS, and those found byWoosley et al.(1986) (who only took into account the luminosity in theform of electron antineutrino). The maximum luminos-ity decreases slightly as the rest-mass of the initial modelincreases, which was already observed by Linke et al.(2001). In addition, we find that the peak of the red-shifted neutrino luminosity does not seem to be very sen-sitive to the initial rotation rate of the star. We also notethat the luminosity of model R1.a reflects the effects ofhydrogen burning at Lν ∼ 1043erg/s.The total energy output in the form of neutrinos is

listed in the last column of Table 1 for several models.The total radiated energies vary between Eν ∼ 1056 ergsfor models collapsing to a BH, and Eν ∼ 1045 − 1046

ergs for exploding models. These results are in reason-able agreement with previous calculations. For instance,Woosley et al. (1986) obtained that the total energy out-put in the form of electron antineutrinos for a sphericalSMS with a mass 5 × 105M⊙ and zero initial metallic-ity was 2.6 × 1056ergs, although their simulations ne-glected general relativistic effects which are importantto compute accurately the relativistic redshifts. On theother hand, Linke et al. (2001), by means of relativisticone-dimensional simulations, found a total radiated en-ergy in form of neutrinos of about 3 × 1056ergs for thesame initial model, and about 1 × 1056ergs when red-shifts were taken into account. In order to compare withthe results of Linke et al. (2001), we computed the red-shifted total energy output for Model S1.a, having thesame rest-mass, until approximately the same evolutionstage as Linke et al. (2001) did (i.e. when the differen-tial neutrino luminosity dLν/dr ∼ 4 × 1045erg/s/cm).We find that the total energy released in neutrinos is1.1× 1056ergs.The neutrino luminosities for models experiencing a

thermonuclear explosion (dashed lines in Fig. 7) peak atmuch lower values Lν ∼ 1042 − 1043erg/s, and decrease

Fig. 7.— Time evolution of the redshifted neutrino luminositiesfor models R1.a, S1.a, R2.a, and S2.a all collapsing to a BH; and formodels R1.d, S1.c, R2.d, and S2.b experiencing a thermal bounce.The time is measured relative to the collapse timescale of eachmodel, R1, S1, R2 and S2, with t0 ≈ (1, 7, 0.2, 6) in units of 105s.

due to the expansion and disruption of the star after thebounce.

6.4. Implications for gravitational wave emission

The axisymmetric gravitational collapse of rotatingSMSs with uniform rotation is expected to emit a burstof gravitational waves (Saijo et al. 2002; Saijo & Hawke2009) with a frequency within the LISA low frequencyband (10−4 − 10−1Hz). Although through the simula-tions presented here we could not investigate the devel-opment of nonaxisymmetric features in our axisymmet-ric models that could also lead to the emission of GWs,Saijo & Hawke (2009) have shown that the three dimen-sional collapse of rotating stars proceeds in a approxi-mately axisymmetric manner.In an axisymmetric spacetime, the ×-mode vanishes

and the +-mode of gravitational waves with l = 2computed using the quadrupole formula is written as

Collapse of supermassive stars 11

(Shibata & Sekiguchi 2003)

hquad+ =

Ixx(tret)− Izz(tret)

rsin2θ, (37)

where Iij refers to the second time derivative of thequadrupole moment. The gravitational wave quadrupoleamplitude is A2(t) = Ixx(tret) − Izz(tret). FollowingShibata & Sekiguchi (2003) we compute the second timederivative of the quadrupole moment by finite differenc-ing the numerical results for the first time derivative ofIij obtained by

Iij =

∫

ρ∗(

vixj + xivj)

d3x. (38)

We calculate the characteristic gravitational wavestrain (Flanagan & Hughes 1998) as

hchar(f) =

√

2

π

G

c31

D2

dE(f)

df, (39)

where D is the distance of the source, and dE(f)/dfthe spectral energy density of the gravitational radiationgiven by

dE(f)

df=

c3

G

(2πf)2

16π

∣

∣

∣A2(f)

∣

∣

∣

2

, (40)

with

A2(f) =

∫

A2(t)e2πiftdt. (41)

We have calculated the quadrupole gravitational waveemission for the rotating model R1.a collapsing to a BH.We plot in Figure 8 the characteristic gravitational wavestrain (Eq.39) for this model assuming that the sourceis located at a distance of 50 Gpc (i.e., z ≈ 11) , to-

gether with the design noise spectrum h(f) =√

fSh(f)of the LISA detector (Larson et al. 2000). We find that,in agreement with Saijo et al. (2002), Saijo & Hawke(2009) and Fryer & New (2011), the burst of gravita-tional waves due to the collapse of a rotating SMS couldbe detected at a distance of 50 Gpc and at a frequencywhich approximately takes the form (Saijo et al. 2002)

fburst ∼ 3× 10−3

(

106M⊙

M

)(

5M

R

)3/2

[Hz], (42)

where R/M is a characteristic mean radius during blackhole formation (typically set to R/M = 5).Furthermore, Kiuchi et al. (2011) have recently in-

vestigated, by means of three-dimensional generalrelativistic numerical simulations of equilibrium toriorbiting BHs, the development of the nonaxisym-metric Papaloizou-Pringle instability in such systems(Papaloizou & Pringle 1984), and have found that a non-axisymmetric instability associated with the m = 1 modegrows for a wide range of self-gravitating tori orbitingBHs, leading to the emission of quasiperiodic GWs. Inparticular, Kiuchi et al. (2011) have pointed out thatthe emission of quasiperiodic GWs from the torus result-ing after the formation of a SMBH via the collapse of aSMS could be well above the noise sensitivity curve ofLISA for sources located at a distance of 10Gpc. The

Fig. 8.— Characteristic gravitational wave strain for model R1.aassuming that the source is located at a distance of 50 Gpc, to-

gether with the design noise spectrum h(f) =√

fSh(f) for LISAdetector.

development of the nonaxisymmetric Papaloizou-Pringleinstability during the evolution of such tori would lead tothe emission of quasiperiodic GWs with peak amplitude∼ 10−18−10−19 and frequency∼ 10−3Hz and maintainedduring the accretion timescale ∼ 105s.

6.5. Conclusions

We have presented results of general relativistic simu-lations of collapsing supermassive stars using the two-dimensional general relativistic numerical code Nada,which solves the Einstein equations written in the BSSNformalism and the general relativistic hydrodynamicequations with high resolution shock capturing schemes.These numerical simulations have used an EOS that in-cludes the effects of gas pressure, and tabulated thoseassociated with radiation pressure and electron-positronpairs. We have also taken into account the effects ofthermonuclear energy release by hydrogen and heliumburning. In particular, we have investigated the effectsof hydrogen burning by the β-limited hot CNO cycle andits breakout via the 15O(α, γ)19Ne reaction (rp-process)on the gravitational collapse of nonrotating and rotatingSMSs with non-zero metallicity.We have found that objects with a mass of ≈ 5×105M⊙

and an initial metallicity greater than ZCNO ≈ 0.007explode if non-rotating, while the threshold metallicityfor an explosion is reduced to ZCNO ≈ 0.001 for ob-jects which are uniformly rotating. The critical initialmetallicity for a thermal explosion increases for starswith a mass of ≈ 106M⊙. The most important con-tribution to the nuclear energy generation is due to thehot CNO cycle. The peak values of the nuclear energygeneration rate at bounce range from ∼ 1051erg/s for ro-tating models (R1.d and R2.d), to ∼ 1052 − 1053erg/sfor spherical models (S1.c and S2.b). After the ther-mal bounce, the radial kinetic energy of the explosionrises until most of the energy is kinetic, with valuesranging from EK ∼ 1056ergs for rotating stars, to up

12 Montero, Janka and Muller

EK ∼ 1057ergs for the spherical star S2.b. The neutrinoluminosities for models experiencing a thermal bouncepeak at Lν ∼ 1042erg/s.The photon luminosity roughly equal to the Edding-

ton luminosity during the initial phase of contraction.Then, after the thermal bounce, the photon luminositybecomes super-Eddington with a value of about Lγ ≈1×1045erg/s during the phase of rapid expansion that fol-lows the thermal bounce. For those stars that do not ex-plode we have followed the evolution beyond the phase ofblack hole formation and computed the neutrino energyloss. The peak neutrino luminosities are Lν ∼ 1055erg/s.SMSs with masses less than ≈ 106M⊙ could have

formed in massive halos with Tvir & 104K. Although theamount of metals that was present in such environmentsat the time when SMS might have formed is unclear,it seems possible that the metallicities could have beensmaller than the critical metallicities required to reverse

the gravitational collapse of a SMS into an explosion. Ifso, the final fate of the gravitational collapse of rotatingSMSs would be the formation of a SMBH and a torus.In a follow-up paper, we aim to investigate in detail thedynamics of such systems (collapsing of SMS to a BH-torus system) in 3D, focusing on the post-BH evolutionand the development of nonaxisymmetric features thatcould emit detectable gravitational radiation.

We thank B. Muller and P. Cerda-Duran for use-ful discussions. Work supported by the DeutscheForschungsgesellschaft (DFG) through its TransregionalCenters SFB/TR 7 “Gravitational Wave Astronomy”,and SFB/TR 27 “Neutrinos and Beyond”, and the Clus-ter of Excellence EXC153 “Origin and Structure of theUniverse”.

REFERENCES

Alcubierre M., Brandt S., Brugmann B., Holz D., Seidel E.,Takahashi R. & Thornburg J. 2001, Int. J. Mod. Phys. D10

Alcubierre M., Brugmann B., Diener P., Koppitz M., Pollney D.,Seidel E. & Takahashi R. 2003, Phys. Rev. D, 67, 084023

Alvarez M. A., Wise J. H. & Abel T. 2009, ApJ, 701, L133Appenzeller & Fricke 1972Baumgarte T. W. & Shapiro S. L. 1999,ApJ, 526, 937Baumgarte T. W. & Shapiro S. L. 1999,ApJ, 526, 941Baumgarte T. W. & Shapiro S. L. 1999,Phys. Rev. D, 59, 024007Begelman M. C. 2009,MNRAS, 402, 673Begelman M. C.& Shlosman I. 2009,ApJ, 702, L5Begelman M. C., Volonteri M. & Rees M. J. 2006,MNRAS, 370,

289Bona C., Masso J., Seidel E. and Stela J. 1997,Phys. Rev. D, 56,

3405Bromm V. & Loeb A. 2003, ApJ, 596, 34Campanelli M., Lousto C. O., Marronetti P. & Zlochower Y.

2006, Phys. Rev. D, 96, 111101Chandrasekhar S. 1964, ApJ, 140Clayton D.D. 1983, in Principles of Stellar Evolution and

Nucleosynthesis, Univ. Chicago Press, 366Einfeldt B. 1988, SIAM J. Numer. Anal., 25, 294Fan, X. 2006, New Astronomy Reviews, 50, 665Flanagan, E. E. & Hughes S.A 1998, prd, 57, 4535Freese K., Ilie C., Spolyar D., Valluri M. & Bodenheimer P.

2010,ApJ, 716,1397Fryer C. L. & New K. C. B. 2011, Living Reviews in Relativity,

14, 1Fowler W. S. 1964, Rev.Mod.Phys., 36, 545Fowler W. S. & Hoyle F. 1964, Rev.Mod.Phys., 36, 545Fuller G. M., Woosley S. E. & Weaver T.A. 1986, ApJ307, 675Genzel R., Picho C., Eckart A., Gerhard O.E. & Ott T.

2000,MNRAS, 317, 348Haft M., Raffelt G. & Weiss A. 1994,ApJ, 425, 222Haiman Z. & Loeb A. 2001,ApJ, 552, 459Harten A., Lax P. D., & van Leer B. 1983, SIAM Rev., 25, 35Hawke I., Loffler F. & Nerozzi A. 2005, Phys. Rev. D, 71, 104006Hoyle F. & Fowler W. A. 1963, MNRAS, 125, 169Iben I. 1963, ApJ,138, 1090Itoh N., Hayashi H., Nishikawa A. & Kohyama Y. 1996, ApJS,

102, 441Kormendy J. & Gebhardt K. 2001, AIP Conference Proceedings:

20th Texas Symposium, 586, 363

Kiuchi K., Shibata M., Montero P. J. & Font J. A. 2011,Phys. Rev. Lett., 106, 251102

Larson S. L., Hiscock W. A. & Hellings R. W.2000,Phys. Rev. D, 62, 062001

Linke F., Font J. A., Janka H.-T., Muller E. & Papadopoulos P.2001, A&A, 376, 568

Montero P. J. , Font J. A. & Shibata M. 2008, Phys. Rev. D, 78,064037

Nakamura T.,Oohara K. & Kojima Y. 1987, Prog. Theor. Phys.Suppl.,90, 1

New K. C. B. & Shapiro S. L. 2001, ApJ, 548,439Omukai K., Schneider R. & Haiman Z. 2008,ApJ, 686, 801Papaloizou J. C. B. & Pringle J. E. 1984,MNRAS,208, 721Rees M. J. 1998, in Black Holes and Relativistic Stars, ed. R. M.

Wald, Univ. Chicago Press, 79Rees M. J. 2001, in Black Holes in Binaries and Galactic Nuclei,

ed. L. Kaper, E. P. J. van de Heurel, and P.A. Woudt, NewYork Springer, 351

Saijo M., Baumgarte T.H., Shapiro S.L.& Shibata M. 2002, ApJ,569, 349

Saijo M. & Hawke I. 2009, Phys. Rev. D, 80, 064001Santoro F. & Shull M. J. 2006, ApJ, 643, 26Shapiro S. L. & Teukolsky S. A. 1979, ApJ, 234, L177Shibata M., Phys. Rev. D, 55, 2002Shibata M. 2003, Phys. Rev. D, 67, 024033Shibata M. & Sekiguchi Y. I. 2003,Phys. Rev. D, 68, 104020Shibata M. & Shapiro S. L. 2002,ApJ, 572, L39Shibata M. & Nakamura T. 1995,Phys. Rev. D, 52, 5428Shen K. & Bildsten L. 2007, ApJ, 660, 1444Shlosman I., Frank J. & Begelman M. C. 1989, Nature, 338, 45Spiteri R. J. & Ruuth S. J. 2002, SIAM J. Num. Analysis, 40, 2,

469Spolyar D., Freese K. & Gondolo P. 2008,Phys. Rev. Lett., 100,

051101Thornburg J. 2007, Living Rev. Relativity 10, 743Volonteri M. 2010, A&ARv, 18, 279Yoo J. & Miralda-Escude J. 2004,ApJ, 614, L25Wiescher M., Gorres J. & Schatz H., J. Phys. G 1999, 25, R133Woosley S. E., Wilson J. R. & Mayle R. 1986, ApJ302, 19