Page 1

arXiv:1108.3090v1 [astro-ph.CO] 15 Aug 2011

Submitted for publication in the Astrophysical Journal

Preprint typeset using LATEX style emulateapj v. 8/13/10

RELATIVISTIC COLLAPSE AND EXPLOSION OF ROTATING SUPERMASSIVE STARS WITH

THERMONUCLEAR EFFECTS

Pedro J. Montero1, Hans-Thomas Janka1, and Ewald M¨ uller1

(Dated: August 17, 2011)

Submitted for publication in the Astrophysical Journal

ABSTRACT

We present results of general relativistic simulations of collapsing supermassive stars with and

without rotation using the two-dimensional general relativistic numerical code Nada, which solves

the Einstein equations written in the BSSN formalism and the general relativistic hydrodynamics

equations with high resolution shock capturing schemes. These numerical simulations use an equation

of state which includes effects of gas pressure, and in a tabulated form those associated with radiation

and the electron-positron pairs. We also take into account the effect of thermonuclear energy released

by hydrogen and helium burning. We find that objects with a mass of ≈ 5 × 105M⊙and an initial

metallicity greater than ZCNO≈ 0.007 do explode if non-rotating, while the threshold metallicity for

an explosion is reduced to ZCNO≈ 0.001 for objects uniformly rotating. The critical initial metallicity

for a thermonuclear explosion increases for stars with mass ≈ 106M⊙. For those stars that do not

explode we follow the evolution beyond the phase of black hole formation. We compute the neutrino

energy loss rates due to several processes that may be relevant during the gravitational collapse of

these objects. The peak luminosities of neutrinos and antineutrinos of all flavors for models collapsing

to a BH are Lν∼ 1055erg/s. The total radiated energy in neutrinos varies between Eν ∼ 1056ergs

for models collapsing to a BH, and Eν∼ 1045− 1046ergs for models exploding.

Subject headings: Supermassive stars

1. INTRODUCTION

There is large observational evidence of the presence

of supermassive black holes (SMBHs) in the centres of

most nearby galaxies (Rees 1998). The dynamical evi-

dence related to the orbital motion of stars in the cluster

surrounding Sgr A∗indicates the presence of a SMBH

with mass ≈ 4 × 106M⊙ (Genzel et al. 2000). In addi-

tion, the observed correlation between the central black

hole masses and the stellar velocity dispersion of the

bulge of the host galaxies suggests a direct connection

between the formation and evolution of galaxies and

SMBHs (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 powerful

quasars, 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 seeds

form and grow to reach such high masses in such a short

amount of time (Rees 2001).

A number of different routes based on stellar dynamical

processes, hydrodynamical processes or a combination of

both have been suggested (e.g. Volonteri 2010 for a re-

cent review). One of the theoretical scenarios for SMBH

seed formation is the gravitational collapse of the first

generation of stars (Population III stars) with masses

M ∼ 100M⊙ that are expected to form in halos with

virial temperature Tvir < 104K at z ∼ 20 − 50 where

cooling by molecular hydrogen is effective. As a result

of the gravitational collapse of such Pop III stars, very

montero@mpa-garching.mpg.de

1Max-Planck-Institut f¨ ur Astrophysik, Karl-Schwarzschild-

Str. 1, D-85748 Garching, Germany;

massive BHs would form and then grow via merger and

accretion (Haiman & Loeb 2001; Yoo & Miralda-Escud´ e

2004; Alvarez et al. 2009).

Another possible scenario proposes that if sufficient

primordial gas in massive halos, with mass ∼ 108M⊙, is

unable 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 collapse

to 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 a

given critical value (Santoro & Shull 2006; Omukai et al.

2008).Furthermore, fragmentation may depend on

the turbulence present within the inflow of gas, and

on 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 angular

momentum and sustain turbulence such that the inflow

of gas can proceed without fragmenting as it collapses

even in a metal-enriched environment.

Depending on the rate and efficiency of the inflow-

ing mass, there may be different outcomes.

rate of mass accumulation would favor the formation

of 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 outcome

could result if the accumulation of gas is fast enough so

that the outer layers of SMSs are not thermally relaxed

during much of their lifetime, thus having an entropy

stratification (Begelman 2009).

A more exotic mechanism that could eventually lead to

A low

Page 2

2Montero, Janka and M¨ uller

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 they

would 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 of

their dark matter supply, they experience a contraction

phase that increases their baryon density and tempera-

ture, leading to an environment where nuclear burning

may 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 to

the onset of a general relativistic gravitational instabil-

ity leading to their gravitational collapse (Chandrasekhar

1964; Fowler 1964), and possibly also to the for-

mation of a SMBH. The first numerical simula-

tions, within the post-Newtonian approximation, of

Appenzeller & Fricke (1972) concluded that for spheri-

cal stars with masses greater than 106M⊙thermonuclear

reactions have no major effect on the collapse, while less

massive stars exploded due to hydrogen burning. Later

Shapiro & 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 the

formation of a BH, although their investigations did not

include any microphysics. Fuller et al. (1986) revisited

the work of Appenzeller & Fricke (1972) and performed

simulations 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 reaction

network 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 than

105M⊙and with metallicity ZCNO≥ 0.005 do explode.

More recently Linke et al. (2001) carried out general

relativistic hydrodynamic simulations of the spherically

symmetric gravitational collapse of SMSs adopting a

spacetime foliation with outgoing null hypersurfaces to

solve the system of Einstein and fluid equations. They

performed simulations of spherical SMSs with masses in

the range of 5 × 105M⊙− 109M⊙ using an EOS that

accounts for contributions from baryonic gases, and in

a tabulated form, radiation and electron-positron pairs.

They were able to follow the collapse from the onset

of the instability until the point of BH formation, and

showed that an apparent horizon enclosing about 25% of

the 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 collapse

of uniformly rotating SMSs to BHs. They did not take

into 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, and

ǫ the specific internal energy.

tions stopped before the final equilibrium was reached,

the BH growth was followed until about 60% of the mass

had been swallowed by the SMBH. They estimated that

Although their simula-

about 90% of the total mass would end up in the final

SMBH with a spin parameter of J/M2∼ 0.75.

The gravitational collapse of differentially rotat-

ing SMSs in three dimensions was investigated by

Saijo & Hawke (2009), who focused on the post-BH evo-

lution, and also on the gravitationalwave (GW) signal re-

sulting from the newly formed SMBH and the surround-

ing disk. The GW signal is expected to be emitted in

the low frequency LISA band (10−4− 10−1Hz).

Despite the progress made, the final fate rotating isen-

tropic SMSs is still unclear.

an open question for which initial metallicities hydrogen

burning by the β-limited hot CNO cycle and its break-

out via the15O(α,γ)19Ne reaction (rp-process) can halt

the gravitational collapse of rotating SMSs and gener-

ate enough thermal energy to lead to an explosion. To

address this issue, we perform a series of general relativis-

tic hydrodynamic simulations with a microphysical EOS

accounting 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 the

pp-chain, cold and hot CNO cycles and their break-out

by the rp-process, and helium burning through the 3-

α reaction. The numerical simulations were carried out

with the Nada code (Montero et al. 2008), which solves

the Einstein equations coupled to the general relativistic

hydrodynamics equations.

Greek indices run from 0 to 3, Latin indices from 1 to 3,

and we adopt the standard convention for the summation

over repeated indices. Unless otherwise stated we use

units in which c = G = 1.

In particular, it is still

2. BASIC EQUATIONS

Next we briefly describe how the system of Einstein

and hydrodynamic equations are implemented in the

Nada code. We refer to Montero et al. (2008) for a more

detailed description of the main equations and thorough

testing of the code (namely single BH evolutions, shock

tubes, evolutions of both spherical and rotating relativis-

tic stars, gravitational collapse to a BH of a marginally

stable spherical star, and simulations of a system formed

by 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 spacetime

is foliated into a set of non-intersecting spacelike hyper-

surfaces. In this approach, the line element is written in

the following form

ds2= −(α2− βiβi)dt2+ 2βidxidt + γijdxidxj,(1)

where α, βiand γijare the lapse function, the shift three-

vector, and the three-metric, respectively. The latter is

defined by

γµν= gµν+ nµnν, (2)

where nµis a timelike unit-normal vector orthogonal to

a spacelike hypersurface.

We make use of the BSSN formulation (Nakamura

1987; Shibata & Nakamura 1995; Baumgarte & Shapiro

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Collapse of supermassive stars3

1999) to solve the Einstein equations. Initially, a confor-

mal factor φ is introduced, and the conformally related

metric 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).

also define the conformally related traceless part of the

extrinsic curvature Kij,

We

˜Aij= e−4φAij= e−4φ

?

Kij−1

3γijK

?

,(4)

where K is the trace of the extrinsic curvature.

evolve the conformal factor defined as χ

(Campanelli et al. 2006), and the auxiliary variables˜Γi,

known as the conformal connection functions, defined as

We

e−4φ

≡

˜Γi≡ ˜ γjk˜Γi

jk= −∂j˜ γij,(5)

where˜Γi

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.

jkare the connection coefficients associated

2.1.2. Gauge choices

Inadditiontothe BSSNspacetimevariables

(˜ γij,˜Aij,K,χ,˜Γi), there are two more quantities left un-

determined, the lapse, α and the shift vector βi. We used

the 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 takes

the 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= ∂t˜Γi− ηBi, (8)

where η is a constant that acts as a damping term, orig-

inally introduced both to prevent long term drift of the

metric functions and to prevent oscillations of the shift

vector.

2.2. Formulation of the hydrodynamics equations

The general relativistic hydrodynamics equations, ex-

pressed through the conservation equations for the stress-

energy tensor Tµνand the continuity equation are

∇µTµν= 0 ,∇µ(ρuµ) = 0,(9)

where ρ is the rest-mass density, uµis the fluid four-

velocity and ∇ is the covariant derivative with respect to

the spacetime metric. Following Shibata (2003), the gen-

eral relativistic hydrodynamics equations are written in

a conservative form in cylindrical coordinates. Since the

Einstein equations are solved only in the y = 0 plane with

Cartesian coordinates (2D), the hydrodynamic equations

are rewritten in Cartesian coordinates for y = 0. The fol-

lowing definitions for the hydrodynamical variables are

used

ρ∗≡ ρWe6φ,(10)

vi≡ui

ut= −βi+ αγijˆ uj

hW,

(11)

ˆ ui≡ hui,(12)

ˆ e ≡e6φ

ρ∗

Tµνnµnν= hW −

P

ρW,

(13)

W ≡ αut,(14)

where W and h are the Lorentz factor and the specific

fluid enthalpy respectively, and P is the pressure. The

conserved variables are ρ∗, Ji = ρ∗ˆ ui, E∗ = ρ∗ˆ e. We

refer 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⊙, which

are mainly supported by radiation pressure, while the

pressure of electron-positron pairs and of the baryon gas

are only minor contributions to the EOS. Such configu-

rations are well described by Newtonian polytropes with

polytropic index n = 3 (adiabatic index Γ = 4/3). The

ratio of gas pressure to the total pressure (β) for spherical

SMSs 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 energy

and 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 are

small, they cannot be neglected for the evolution. Firstly,

gas corrections raise the adiabatic index slightly above

4/3

Γ ≈4

3+β

6+ 0(β2).(16)

Secondly, general relativistic corrections lead to the

existence of a maximum for the equilibrium mass as a

function of the central density. For spherical SMSs this

means that for a given mass the star evolves to a critical

density beyond which it is dynamically unstable against

radial 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

Page 4

4Montero, Janka and M¨ uller

become unstable when the adiabatic index drops below

the critical value

Γcrit=4

3+ 1.122GM

Rc2.(18)

This happens when the stabilizing gas contribution to

the EOS does not raise the adiabatic index above 4/3 to

compensate for the destabilizing effect of general relativ-

ity expressed by the second term on the righ-hand-side

of Eq. (18).

Rotation can stabilize configurations against the ra-

dial instability. The stability of rotating SMSs with

uniform rotation was analyzed by Baumgarte & Shapiro

(1999a,b). They found that stars at the onset of the

instability have an equatorial radius R ≈ 640GM/c2, a

spin parameter q ≡ cJ/GM2≈ 0.97, and a ratio of rota-

tional kinetic energy to the gravitational binding energy

of 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 which

includes separately the baryon contribution on the one

hand, and photons and electron-positron pairs contri-

butions, in a tabulated form, on the other hand. The

baryon contribution is given by the analytic expressions

for 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,

ǫbthe baryon specific internal energy, and µbis the mean

molecular weight due to ions, which can be expressed as

a 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 of

hydrogen X = 0.75−ZCNOand helium Y = 0.25, where

the 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 consider

the mean molecular weight of baryons is µb≈ 1.23 (i.e.

corresponding to a molecular weight for both ions and

electrons 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, not

all the energy is used to increase the temperature and

pressure, but part of the photon energy is used to create

the rest-mass of the electron-positron pairs. As a result

of 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) = ǫ

for 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-time

demands connected with the solution of a nuclear reac-

tion network coupled to the hydrodynamic evolution, we

apply an approximate method to take into account the

basic effects of nuclear burning on the dynamics of the

collapsing 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 mass

fractions of hydrogen X, helium Y and CNO metallicity

ZCNO. These nuclear energy generation rates are added

as a source term on the right-hand-side of the evolution

equation for the conserved quantity E∗.

The change rates of the energy density due to nuclear

reactions, in the fluid frame, expressed in units of [erg

cm−3s−1] are given by:

• pp-chain (Clayton 1983):

?∂e

∂t

?

pp

= ρ(2.38 × 106ρg11X2T−0.6666

6

e−33.80/T0.3333

6

), (23)

where T6= T/106K, and g11is given by

g11= 1 + 0.0123T0.3333

0.0009T6.

6

+ 0.0109T0.66666

6

+

(24)

• 3-α (Wiescher et al. 1999):

?∂e

where T9= T/109K.

∂t

?

3α

= ρ(5.1 × 108ρ2Y3T−3

9 e−4.4/T9), (25)

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

?∂e

∂t

?

CCNO

= 4.4 × 1025ρ2XZCNO

(T−2/3

9

+ 8.3 × 10−5T−3/2

e−15.231/T1/3

9

+

9

e−3.0057/T9).(26)

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

?∂e

∂t

?

HCNO

= 4.6 × 1015ρZCNO.(27)

Page 5

Collapse of supermassive stars5

• rp-process (Wiescher et al. 1999):

?∂e

∂t

?

rp

= ρ(1.77 × 1016ρY ZCNO

29.96T−3/2

9

e−5.85/T9).(28)

Since we follow a single fluid approach, in which we

solve only the hydrodynamics equations Eq. (9) (i.e. we

do not solve additional advection equations for the abun-

dances of hydrogen, helium and metals), the elemental

abundances during the time evolution are fixed. Nev-

ertheless, this assumption most possibly does not af-

fect significantly the estimate of the threshold metallicity

needed to produce a thermal bounce in collapsing SMSs.

The average energy release through the 3-α reaction is

about 7.275 MeV for each12C nucleus formed. Since the

total energy due to helium burning for exploding mod-

els is ∼ 1045ergs (e.g. 9.0 × 1044ergs for model S1.c);

and even considering that this energy is released mostly

in a central region of the SMS containing 104M⊙of its

rest-mass (Fuller et al. 1986), it is easy to show that the

change 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 critical

metallicities needed to trigger the explosions. Similarly,

the average change in the mass fraction of hydrogen due

to 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 allows

for the use of a general EOS of the form P = P(ρ,ǫ).

We calculate a function f(P∗) = P(ρ∗,ǫ∗) − P∗, where

ρ∗and ǫ∗depend only on the conserved quantities and

the pressure guess P∗. The new pressure is computed

then iteratively by a Newton-Raphson method until the

desired convergence is achieved.

3.5. Energy loss by neutrino emission

The EOS allows us to compute the neutrino losses due

to the following processes, which become most relevant

just before BH formation:

• Pair annihilation (e++ e−→ ¯ ν + ν): most impor-

tant process above 109K. Due to the large mean

free path of neutrinos in the stellar medium at the

densities of SMSs the energy loss by neutrinos can

be significant. For a 106M⊙SMS most of the en-

ergy release in the form of neutrinos originates from

this 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 and

densities ρ ? 105gcm−3(Itoh et al. 1996).

• Plasmon decay (γ → ¯ ν + ν): This is the least rele-

vant process for the conditions encountered by the

models we have considered because its importance

increases at higher densities than those present in

SMSs. The rates are computed using the fitting

formula given by Haft et al. (1994).

4. COMPUTATIONAL SETUP

The evolution equations are integrated by the method

of lines, for which we use an optimal strongly stability-

preserving (SSP) Runge-Kutta algorithm of fourth-order

with 5 stages (Spiteri & Ruuth 2002). We use a second-

order slope limiter reconstruction scheme (MC limiter) to

obtain the left and right states of the primitive variables

at each cell interface, and a HLLE approximate Riemann

solver (Harten et al. 1983; Einfeldt 1988) to compute the

numerical 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 outer

boundaries. We used a cell-centered Cartesian grid to

avoid that the location of the BH singularity coincides

with a grid point.

4.1. Regridding

Since it is not possible to follow the gravitational

collapse of a SMS from the early stages to the phase

of black hole formation with a uniform Cartesian

grid (the necessary fine zoning would be computation-

ally too demanding), we adopt a regridding procedure

(Shibata & Shapiro 2002). During the initial phase of

the collapse we rezone the computational domain by

moving the outer boundary inward, decreasing the grid

spacing while keeping the initial number of grid points

fixed. Initially we use N×N = 400×400 grid points, and

place the outer boundary at L ≈ 1.5re where re is the

equatorial radius of the star. Rezoning onto the new grid

is done using a polynomial interpolation. We repeat this

procedure 3-4 times until the collapse timescale in the

central region is much shorter than in the outer parts.

At this point, we both decrease the grid spacing and also

increase the number of grid points N in dependence of

the lapse function typically as follows: N×N = 800×800

if 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-mass

to be less than 2% on the finest computational domain.

4.2. Hydro-Excision

To deal with the spacetime singularity from the newly

formed BH we use the method of excising the matter

content in a region within the horizon as proposed by

Hawke et al. (2005) once an apparent horizon (AH) is

found. This excision is done only for the hydrodynamical

variables, and the coordinate radius of the excised region

is allowed to increase in time. On the other hand, we do

neither use excision nor artificial dissipation terms for

the spacetime evolution, and solely rely on the gauge

conditions.

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

Page 6

6Montero, Janka and M¨ uller

TABLE 1

Main 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, radial

kinetic energy after thermal bounce, and total neutrino energy output.

ModelMρc

Tk/|W|ΩTc

Initial metallicity

[10−3]

5

6

7

0.5

0.8

1

2

30

50

0.5

0.8

1.0

1.5

FateERK

[1056erg]

...

..

5.5

...

...

1.0

1.9

...

35

...

...

...

1.5

Eν

[erg][105M⊙]

5

5

5

5

5

5

5

10

10

10

10

10

10

[10−2g/cm3]

2.4

2.4

2.4

40

40

40

40

0.23

0.23

12

12

12

12

[10−5rad/s]

0

0

0

2.49

2.49

2.49

2.49

0

0

1.47

1.47

1.47

1.47

[107K]

5.8

5.8

5.8

13

13

13

13

2.6

2.6

9.7

9.7

9.7

9.7

S1.a

S1.b

S1.c

R1.a

R1.b

R1.c

R1.d

S2.a

S2.b

R2.a

R2.b

R2.c

R2.d

0

0

0

BH

BH

3.4 × 1056

...

9.4 × 1045

5.4 × 1056

...

...

8.9 × 1045

6.8 × 1056

8.0 × 1046

3.1 × 1056

...

...

2.1 × 1046

Explosion

BH

BH

Explosion

Explosion

BH

Explosion

BH

BH

BH

Explosion

0.0088

0.0088

0.0088

0.0088

0

0

0.0087

0.0087

0.0087

0.0087

M as

M∗= 4π

?L

0

xdx

?L

0

ρ∗dz,(29)

M = −2

?L

0

xdx

?L

−e5φ

0

dz

?

˜Aij˜Aij−2

−2πEe5φ+eφ

8

??

˜R

8

?

3K2

,(30)

where E = nµnνTµν(nµbeing the unit normal to the

hypersurface) and˜R is the scalar curvature associated to

the conformal metric ˜ γij.

The rotational kinetic energy Tkand the gravitational

potential energy W are given by

Tk= 2π

?L

0

x2dx

?L

0

ρ∗ˆ 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 nonlinear

ordinary differential equation for the AH shape function,

h = h(θ) (Shibata 1997; Thornburg 2007). We employ

an AH finder that solves this ODE by a shooting method

using ∂θh(θ = 0) = 0 and ∂θh(θ = π/2) = 0 as boundary

conditions. 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. All

models are chosen such that they are gravitationally un-

stable, and therefore their central rest-mass density is

slightly larger than the critical central density required

for the onset of the collapse of a configuration with

given mass and entropy. A list of the different SMSs

we have considered is provided in Table 1. Models S1

and S2 represent a spherically symmetric, nonrotating

SMS with gravitational mass of M = 5 × 105M⊙ and

M = 1 × 106M⊙, respectively, while models R1 and R2

are uniformly rotating initial models again with masses

of M = 5 × 105M⊙ and M = 1 × 106M⊙, respec-

tively. The rigidly and maximally rotating initial models

R1 and R2 are computed with the Lorene code (URL

http://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 we

carry out several numerical simulations for each initial

model with different values of the initial metallicity. The

initial metallicities along with the fate of the star are

given 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 to

a model extensively discussed in

and therefore allows for a comparison with the results

presented here. Fuller et al. (1986) found that unstable

spherical SMSs with M = 5×105M⊙and an initial metal-

licity ZCNO= 2 × 10−3collapse to a BH while models

with an initial metallicity ZCNO= 5×10−3explode due

to 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 to

an explosion) are ρc,b= 3.16 g/cm3and 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 central

temperature (lower panel) for models S1.a, S1.c, R1.a

and R1.d, i.e., non-rotating and rotating models with a

mass of M = 5 × 105M⊙. In particular, the solid lines

represent the time evolution of the central density and

temperature for model S1.c (ZCNO= 7×10−3) and R1.d

Fuller et al. (1986),

Page 7

Collapse of supermassive stars7

Fig. 1.— Left upper panel shows the time evolution of the central rest-mass density for models S1 and R1 (i.e., spherical and rotating

stars with mass M = 5 × 105M⊙), and the lower left panel shows the time evolution of the central temperature. Horizontal dotted lines

mark the temperature range in which nuclear energy is primarily released by the hot CNO cycle. Similarly, the time evolution of the

same quantities for model S2 and R2 (i.e., spherical and rotating stars with mass M = 1 × 106M⊙) are shown in the upper and lower

right panels. As the collapse proceeds, the central density and temperature rise rapidly, increasing the nuclear energy generation rate by

hydrogen burning. If the metallicity is sufficiently high, enough energy can be liberated to produce a thermal bounce. This is the case for

models 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 increases

the nuclear energy generation rate by hydrogen burning.

Since the metallicity is sufficiently high, enough energy

can be liberated to increase the pressure and to produce a

thermal bounce. This is the case for model S1.c. In Fig-

ure 1 we show that a thermal bounce occurs (at approxi-

mately t ∼ 7×105s) 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. The

rest-mass density at bounce is ρc,b= 4.8 g/cm3and the

temperature Tc,b= 3.05×108K. These values, as well as

the threshold metallicity needed to trigger a thermonu-

clear explosion (ZCNO= 7×10−3), are higher than those

found by Fuller et al. (1986) (who found that a spher-

ical nonrotating model with the same rest-mass would

explode, 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 model

S1.a (ZCNO = 5 × 10−3). In this case, as well as for

model 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 an

explosion due to the nuclear energy released by the hot

CNO burning, and an expanding shock forms only near

the 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 the

x-component of the three-velocity vxalong the x-axis

(in the equatorial plane) for the nonrotating spherical

stars S1.a (dashed lines) and S1.c (solid lines) at three

different time slices near the time at which model S1.c

experiences a thermal bounce. Velocity profiles of model

S1.c are displayed up to the radius where a shock forms

at t ≈ 7.31 × 105s and begins to expand into the low

density outer layers of the SMS.

The evolutionary tracks for the central density and

temperature of the rotating models R1.a and R1.d are

also shown in Figure 1. A dashed line corresponds to

model R1.a, with an initial metallicity ZCNO= 5×10−4,

which collapses to a BH. A solid line denotes model R1.d

with ZCNO= 2×10−3, which explodes due to the energy

released by the hot CNO cycle. We find that Model R1.c

with a lower metallicity of ZCNO= 1×10−3also explodes

when the central temperature is the range dominated by

the hot CNO cycle.

As a result of the kinetic energy stored in the rotation

of models R1.c and R1.d, the critical metallicity needed

to trigger an explosion decreases significantly relative to

the non-rotating case. We observe that rotating models

with initial metallicities up to ZCNO= 8 × 10−4do not

explode even via the rp-process, which is dominant at

temperatures above T ≈ 5 × 108K and increases the

hydrogen burning rate by 200 − 300 times relative to

the hot CNO cycle.We also note that the evolution

time scales of the collapse and bounce phases are reduced

because rotating models are more compact and have a

higher initial central density and temperature than the

spherical 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) and

the central temperature (lower panel) for models S2.a,

S2.b, R2.a and R2.d, i.e., of models with a mass of

M = 106M⊙. We find that the critical metallicity for an

explosion in the spherical case is ZCNO= 5×10−2(model

S2.b), while model S2.a with ZCNO= 3×10−2collapses

to a BH. We note that the critical metallicity leading to a

thermonuclear explosion is higher than the critical value

found by Fuller et al. (1986) (i.e., ZCNO= 1×10−2) for

a spherical SMS with the same mass. The initial metal-

licity leading to an explosion in the rotating case (model

Page 8

8Montero, Janka and M¨ uller

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 different

time slices near the time at which model S1.c experiences a ther-

mal bounce. Velocity profiles of model S1.c are displayed up to the

radius where takes place the formation of a shock that expands

into the low density outer layers of the SMS.

R2.d) is more than an order of magnitude smaller than in

the spherical case. As for the models with a smaller grav-

itational mass, the thermal bounce takes place when the

physical conditions in the central region of the star allow

for the release of energy by hydrogen burning through

the hot CNO cycle. Overall, the dynamics of the more

massive models indicates that the critical initial metal-

licity required to produce an explosion increases with the

rest-mass of the star.

Figure 3 shows the total nuclear energy generation rate

in erg/s for the exploding models as a function of time

during the late stages of the collapse just before and af-

ter bounce. The main contribution to the nuclear energy

generation is due to hydrogen burning by the hot CNO

cycle.The peak values of the energy generation rate

at bounce lie between several 1051erg/s for the rotating

models (R1.d and R2.d), and ≈ 1052− 1053erg/s for the

spherical models (S1.c and S2.b). As expected the max-

imum nuclear energy generation rate needed to produce

an 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 would

mostly be composed of4He.

As a result of the thermal bounce, the kinetic energy

rises until most of the energy of the explosion is in the

form of kinetic energy. We list in the second but last

column of Table 1 the radial kinetic energy after thermal

bounce, which ranges between ERK= 1.0 × 1055ergs for

the rotating star R1.c, and ERK= 3.5×1057ergs for the

spherical 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 criterion

that 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 time

near 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 association

with the thermonuclear explosion, we make use of the

fact 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 κesis

the opacity due to electron Thompson scattering, which

is the main source of opacity in SMSs. The photon lumi-

nosity in terms of the temperature gradient and for the

spherically symmetric case can be written as

Lγ= −16πacr2T3

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 the

phase of expansion after the thermonuclear explosion. In

this figure (Fig. 4) we show the isodensity contours for

the rotating model R1.d. The frames have been taken at

the initial time (left figure), at t = 0.83 × 105s (central

figure) just after the thermal bounce (at t = 0.78×105s),

and at t = 2.0 × 105s when the radius of the expanding

matter is roughly 4 times the radius of the star at the

onset 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 the

thermal bounce takes place. The photon luminosity be-

fore the thermal bounce is computed at radii inside the

star unaffected by the local dynamics of the low den-

sity outer layers which is caused by the initial pressure

perturbation and by the interaction between the surface

of the SMS and the artificial atmosphere. Once the ex-

Page 9

Collapse of supermassive stars9

Fig. 4.— Isodensity contours of the logarithm of the rest-mass density (in g/cm3) for the rotating model R1.d. The frames have been

taken at the initial time (left figure), at t = 0.83 × 105s (central figure) just after a thermal bounce takes place, and at t = 2.0 × 105s

when 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.d

in units of erg/s as a function of time. The vertical dashed line

indicates 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. The

lightcurve shows that, during the initial phase, the lu-

minosity is roughly equal to the Eddington luminosity

≈ 5 × 1043erg/s until the thermal bounce. Then, the

photon luminosity becomes super-Eddington when the

expanding shock reaches the outer layers of the star and

reaches a value of Lγ ≈ 1 × 1045erg/s. This value of

the photon luminosity after the bounce is within a few

percent difference with respect to the photon luminos-

ity Fuller et al. (1986) found for a nonrotating SMS of

same 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 outer

boundary of the computational domain ≈ 1.0 × 105

after the bounce. Beyond that point, the luminosity

is 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 not

generate enough nuclear energy during the contraction

phase 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 in

Figure 1. The central density typically increases up

to ρc ∼ 107gcm−3and the central temperature up to

Tc ∼ 1010K just before the formation of an apparent

horizon.

Three isodensity contours for the rotating model R1.a

collapsing to a BH are shown in Figure 6, which display

the flattening of the star as the collapse proceeds. The

frames have been taken at the initial time (left panel),

at t = 0.83 × 105s (central panel) approximately when

model R1.d with higher metallicity experiences a thermal

bounce and, at t = 1.127×105s, where a BH has already

formed and its AH has a mass of 50% of the total initial

mass.

At the temperatures reached during the late stages of

the gravitational collapse (in fact at T ≥ 5 × 108K) the

most efficient process for hydrogen burning is the break-

out from the hot CNO cycle via the15O(α,γ)19Ne re-

action. Nevertheless, we find that models which do not

release enough nuclear energy by the hot CNO cycle to

halt their collapse to a BH, are not able to produce a

thermal explosion due to the energy liberated by the

15O(α,γ)19Ne reaction. We note that above 109K, not

all the liberated energy is used to increase the temper-

ature and pressure, but is partially used to create the

rest-mass of the electron-positron pairs. As a result of

pair 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 energy

losses 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 (dashed

lines) of four models experiencing a thermal bounce

(S1.c, R1.d, S2.b, and R2.d). The change of the slope

of the neutrino luminosities at ∼ 1043erg/s denotes the

transition from photo-neutrino emission to the pair an-

Page 10

10Montero, Janka and M¨ uller

Fig. 6.— Isodensity contours of the logarithm of the rest-mass density (in g/cm3) for the rotating model R1.a. The frames have been

taken at the initial time (left panel), at t = 0.83 × 105s (central panel), at t = 1.127 × 105s, where a BH has already formed and its

apparent horizon encloses a mass of 50% of the total initial gravitational mass.

nihilation dominated region. The peak luminosities in

all form of neutrino for models collapsing to a BH are

Lν ∼ 1055erg/s. Neutrino luminosities can be that im-

portant because the densities in the core prior to BH

formation are ρc ∼ 107gcm−3, and therefore neutrinos

can escape. The peak neutrino luminosities lie between

the luminosities found by Linke et al. (2001) for the col-

lapse of spherical SMS, and those found by Woosley et al.

(1986) (who only took into account the luminosity in the

form of electron antineutrino). The maximum luminos-

ity decreases slightly as the rest-mass of the initial model

increases, which was already observed by

(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 note

that the luminosity of model R1.a reflects the effects of

hydrogen 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∼ 1056ergs

for 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 spherical

SMS with a mass 5 × 105M⊙ and zero initial metallic-

ity was 2.6 × 1056ergs, although their simulations ne-

glected general relativistic effects which are important

to compute accurately the relativistic redshifts. On the

other hand, Linke et al. (2001), by means of relativistic

one-dimensional simulations, found a total radiated en-

ergy in form of neutrinos of about 3 × 1056ergs for the

same initial model, and about 1 × 1056ergs when red-

shifts were taken into account. In order to compare with

the results of Linke et al. (2001), we computed the red-

shifted total energy output for Model S1.a, having the

same rest-mass, until approximately the same evolution

stage 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 is

1.1 × 1056ergs.

The neutrino luminosities for models experiencing a

thermonuclear explosion (dashed lines in Fig. 7) peak at

much lower values Lν ∼ 1042− 1043erg/s, and decrease

Linke et al.

Fig. 7.— Time evolution of the redshifted neutrino luminosities

for models R1.a, S1.a, R2.a, and S2.a all collapsing to a BH; and for

models R1.d, S1.c, R2.d, and S2.b experiencing a thermal bounce.

The time is measured relative to the collapse timescale of each

model, 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 the

bounce.

6.4. Implications for gravitational wave emission

The axisymmetric gravitational collapse of rotating

SMSs with uniform rotation is expected to emit a burst

of gravitational waves (Saijo et al. 2002; Saijo & Hawke

2009) with a frequency within the LISA low frequency

band (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 = 2

computed using the quadrupole formula is written as

Page 11

Collapse of supermassive stars11

(Shibata & Sekiguchi 2003)

hquad

+

=

¨Ixx(tret) −¨Izz(tret)

r

sin2θ, (37)

where¨Iij refers to the second time derivative of the

quadrupole moment. The gravitational wave quadrupole

amplitude is A2(t) =¨Ixx(tret) −¨Izz(tret).

Shibata & Sekiguchi (2003) we compute the second time

derivative of the quadrupole moment by finite differenc-

ing the numerical results for the first time derivative of

Iijobtained by

Following

˙Iij=

?

ρ∗

?vixj+ xivj?d3x.(38)

We calculate the characteristic gravitational wave

strain (Flanagan & Hughes 1998) as

hchar(f) =

?

2

π

G

c3

1

D2

dE(f)

df

,(39)

where D is the distance of the source, and dE(f)/df

the spectral energy density of the gravitational radiation

given by

dE(f)

dfG16π

=c3

(2πf)2

???˜A2(f)

???

2

,(40)

with

˜A2(f) =

?

A2(t)e2πiftdt.(41)

We have calculated the quadrupole gravitational wave

emission for the rotating model R1.a collapsing to a BH.

We plot in Figure 8 the characteristic gravitational wave

strain (Eq.39) for this model assuming that the source

is located at a distance of 50 Gpc (i.e., z ≈ 11) , to-

gether with the design noise spectrum h(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 could

be detected at a distance of 50 Gpc and at a frequency

which approximately takes the form (Saijo et al. 2002)

?fSh(f)

fburst∼ 3 × 10−3

?106M⊙

M

??5M

R

?3/2

[Hz],(42)

where R/M is a characteristic mean radius during black

hole formation (typically set to R/M = 5).

Furthermore, Kiuchi et al. (2011) have recently in-

vestigated,by means of three-dimensional general

relativistic numerical simulations of equilibrium tori

orbiting 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 mode

grows for a wide range of self-gravitating tori orbiting

BHs, leading to the emission of quasiperiodic GWs. In

particular,Kiuchi et al. (2011) have pointed out that

the emission of quasiperiodic GWs from the torus result-

ing after the formation of a SMBH via the collapse of a

SMS could be well above the noise sensitivity curve of

LISA for sources located at a distance of 10Gpc. The

Fig. 8.— Characteristic gravitational wave strain for model R1.a

assuming that the source is located at a distance of 50 Gpc, to-

gether with the design noise spectrum h(f) =

detector.

?fSh(f) for LISA

development of the nonaxisymmetric Papaloizou-Pringle

instability during the evolution of such tori would lead to

the emission of quasiperiodic GWs with peak amplitude

∼ 10−18−10−19and frequency ∼ 10−3Hz and maintained

during 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 BSSN

formalism and the general relativistic hydrodynamic

equations with high resolution shock capturing schemes.

These numerical simulations have used an EOS that in-

cludes the effects of gas pressure, and tabulated those

associated with radiation pressure and electron-positron

pairs. We have also taken into account the effects of

thermonuclear energy release by hydrogen and helium

burning. In particular, we have investigated the effects

of hydrogen burning by the β-limited hot CNO cycle and

its breakout via the15O(α,γ)19Ne reaction (rp-process)

on the gravitational collapse of nonrotating and rotating

SMSs with non-zero metallicity.

We have found that objects with a mass of ≈ 5×105M⊙

and an initial metallicity greater than ZCNO ≈ 0.007

explode if non-rotating, while the threshold metallicity

for an explosion is reduced to ZCNO ≈ 0.001 for ob-

jects which are uniformly rotating. The critical initial

metallicity for a thermal explosion increases for stars

with a mass of ≈ 106M⊙.

tribution to the nuclear energy generation is due to the

hot CNO cycle. The peak values of the nuclear energy

generation rate at bounce range from ∼ 1051erg/s for ro-

tating models (R1.d and R2.d), to ∼ 1052− 1053erg/s

for spherical models (S1.c and S2.b).

mal bounce, the radial kinetic energy of the explosion

rises until most of the energy is kinetic, with values

ranging from EK ∼ 1056ergs for rotating stars, to up

The most important con-

After the ther-

Page 12

12Montero, Janka and M¨ uller

EK∼ 1057ergs for the spherical star S2.b. The neutrino

luminosities for models experiencing a thermal bounce

peak 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 luminosity

becomes 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 of

black hole formation and computed the neutrino energy

loss. The peak neutrino luminosities are Lν∼ 1055erg/s.

SMSs with masses less than ≈ 106M⊙ could have

formed in massive halos with Tvir? 104K. Although the

amount of metals that was present in such environments

at the time when SMS might have formed is unclear,

it seems possible that the metallicities could have been

smaller than the critical metallicities required to reverse

the gravitational collapse of a SMS into an explosion. If

so, the final fate of the gravitational collapse of rotating

SMSs would be the formation of a SMBH and a torus.

In a follow-up paper, we aim to investigate in detail the

dynamics of such systems (collapsing of SMS to a BH-

torus system) in 3D, focusing on the post-BH evolution

and the development of nonaxisymmetric features that

could emit detectable gravitational radiation.

We thank B. M¨ uller and P. Cerd´ a-Dur´ an for use-

ful discussions. Work supported by the Deutsche

Forschungsgesellschaft (DFG) through its Transregional

Centers SFB/TR 7 “Gravitational Wave Astronomy”,

and SFB/TR 27 “Neutrinos and Beyond”, and the Clus-

ter of Excellence EXC153 “Origin and Structure of the

Universe”.

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