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Astronomy & Astrophysics manuscript no. March 26, 2007(DOI: will be inserted by hand later)

General relativistic simulations of magneto-rotational

core collapse with microphysics

Pablo Cerda-Duran1,2, Jose A. Font2, and Harald Dimmelmeier1,3

1 Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany2 Departamento de Astronomıa y Astrofısica, Universidad de Valencia, 46100 Burjassot (Valencia), Spain3 Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Received date / Accepted date

Abstract. This paper presents results from axisymmetric simulations of magneto-rotational stellar core collapse toneutron stars in general relativity. These simulations are performed using a new general relativistic numerical codespecifically designed to study this astrophysical scenario. The code is an extension of an existing (and thoroughlytested) hydrodynamics code, which has been applied in the recent past to study relativistic rotational core collapse.It is based on the conformally-flat approximation of Einstein’s field equations and conservative formulationsof the magneto-hydrodynamics equations. The code has been recently upgraded to incorporate a tabulated,microphysical equation of state and an approximate deleptonization scheme. This allows us to perform the mostrealistic simulations of magneto-rotational core collapse to date, which are compared with simulations employing asimplified (hybrid) equation of state, widely used in the relativistic core collapse community. Furthermore, state-of-the-art (unmagnetized) initial models from stellar evolution are used. In general, stellar evolution models predictweak magnetic fields in the progenitors, which justifies our simplification of performing the computations underthe approach that we call the passive field approximation for the magnetic field. Our results show that for the corecollapse models with microphysics the saturation of the magnetic field cannot be reached within dynamical timescales by winding up the poloidal magnetic field into a toroidal one. We estimate the effect of other amplificationmechanisms including the magneto-rotational instability (MRI) and several types of dynamos. We conclude thatfor progenitors with astrophysically expected (i.e. weak) magnetic fields, the MRI is the only mechanism that couldamplify the magnetic field on dynamical time scales. The uncertainties about the strength of the magnetic fieldat which the MRI saturates are discussed. All our microphysical models exhibit post-bounce convective overturnin regions surrounding the inner part of the proto-neutron star. Since this has a potential impact on enhancingthe MRI, it deserves further investigation with more accurate neutrino treatment or alternative microphysicalequations of state.

Key words. Gravitation – Hydrodynamics – MHD – Methods:numerical – Relativity – Stars:supernovae:general

1. Introduction

Understanding the dynamics of the gravitational collapseof the core of massive stars leading to supernova explo-sions still remains one of the primary problems in generalrelativistic astrophysics, despite the continuous theoreticalefforts during the last few decades. This problem standsas a distinctive example of a research field where essentialprogress has been accomplished through numerical mod-elling with increasing levels of complexity in the inputphysics: hydrodynamics, gravity, magnetic fields, nuclearphysics, equation of state (EOS), neutrino transport, etc.While studies based upon Newtonian physics are highlydeveloped nowadays, state-of-the-art simulations still fail,

Send offprint requests to: Pablo Cerda-Duran,e-mail: cerda@mpa-garching.mpg.de

broadly speaking, to generate successful supernova explo-sions under generic conditions (see e.g. Buras et al. 2003;Kifonidis et al. 2006 for details on the degree of sophisti-cation achieved in present-day supernova modelling, andWoosley & Janka 2005 and references therein for a reviewon the mechanism of core collapse supernovae). The rea-sons behind those apparent failures are diverse, all havingto do with the limited knowledge of some of the underlyingkey issues such as realistic precollapse stellar models (in-cluding rotation, or the strength and distribution of mag-netic fields), the appropriate EOS, as well as numericallimitations due to the need for Boltzmann neutrino trans-port, multi-dimensional hydrodynamics, and relativisticgravity.

Aside from their assistance to understand the super-nova mechanism, numerical simulations of stellar core col-

2 Cerda-Duran et al.: Relativistic magneto-rotational core collapse

lapse are highly motivated nowadays by the prospects ofa direct detection of the gravitational waves emitted inthis scenario. In core collapse events where rotation playsa role, one of the emission mechanism for gravitationalwaves is the hydrodynamic core bounce, which generatesa burst signal. The post-bounce wave signal also exhibitslarge amplitude oscillations associated with pulsations inthe collapsed core (Zwerger & Muller 1997; Rampp et al.1998), neutrino-driven convection behind the supernovashock (Muller et al. 2004) and (possibly) rotational dy-namical instabilities (Ott et al. 2005; Ott et al. 2006a,b).However, a successful future detection of gravitational ra-diation from stellar core collapse faces the combined prob-lem of the smallness of the signal strength and of thecomplexity of the burst signal from bounce. On the otherhand, the energy released in gravitational waves is so smallthat its backreaction to the collapse dynamics is negligible,which can significantly simplify the numerical simulationof this scenario. To pave the road for a successful detec-tion through waveform templates for data analysis, suchsimulations are essential.

At birth neutron stars have intense magnetic fields(∼ 1012 – 1013 G) or in extreme cases even larger ones(∼ 1014 – 1015 G), as inferred from studies of anomalousX-ray pulsars and soft gamma-ray repeaters (Kouveliotouet al. 1998). For magnetars, the magnetic field can be sostrong as to alter the internal structure of the neutronstar (Bocquet et al. 1995). The emergence of such strongmagnetic fields in neutron stars from the initial field con-figuration in the pre-collapse stellar cores is an active andimportant field of research. Similarly, the rotation stateof the nascent proto-neutron star (PNS) is determinedby the amount and distribution of angular momentumin the core of the progenitor, which is still rather un-constrained, being only currently incorporated into stel-lar evolution codes (Heger et al. 2005). Observations ofsurface velocities imply that a large fraction of progenitorcores is rapidly rotating. The presence of intense mag-netic fields, on the other hand, may also affect rotation inthe core, as it can be spun down in the red giant phaseby magnetic torques via dynamo action which couples tothe outer layers of the star (Meier et al. 1976; Spruit &Phinney 1998; Spruit 2002; Heger et al. 2005). The lat-est numerical calculations of stellar evolution thus predictlow pre-collapse core rotation rates, which are in agree-ment with observed periods of young neutron stars in therange of ∼ 10 – 15 ms. Nevertheless, a recent estimate byWoosley & Heger (2006) indicates that ∼ 1% of all starswith M ≥ 10M⊙ could still have rapidly rotating cores,which could also be relevant for the collapsar-type gamma-ray burst scenario.

The presence of intense magnetic fields in a PNS makesmagneto-rotational core collapse simulations mandatory.The weakest point of all existing simulations to date isthe fact that both the strength and distribution of theinitial magnetic field in the core are basically unknown.If the magnetic field is initially weak, there exist severalmechanisms which may amplify it to values which can

have an impact on the dynamics, among them differentialrotation (Ω-dynamo), the magneto-rotational instability(MRI hereafter), as well as dynamo mechanisms relatedto convection or turbulence. The first of these mechanismstransforms rotational energy into magnetic energy, wind-ing up any seed poloidal field into a toroidal field. TheMRI leads to a sustained turbulent dynamo which is ableto transport angular momentum outwards, although it re-mains unclear how large the actual amplification by thisprocess can be (see below). The latter mechanisms, whichare generically called α-Ω-dynamo and will be discussedbelow, include a number of processes which can also pro-duce an amplification of the magnetic field.

Magneto-rotational core collapse simulations were firstperformed as early as in the 1970s (LeBlanc & Wilson1970; Bisnovatyi-Kogan et al. 1976; Meier et al. 1976;Muller & Hillebrandt 1979; Ohnishi 1983; Symbalisty1984), in which magneto-rotational core collapse was al-ready proposed as a plausible supernova explosion mech-anism. In recent years, an increasing number of authorshave performed axisymmetric magneto-hydrodynamic(MHD) simulations of stellar core collapse (within theso-called ideal MHD limit) employing a Newtonian treat-ment of MHD and gravity, and either a simplified equa-tion of state (Yamada & Sawai 2004; Ardeljan et al. 2005;Sawai et al. 2005) or a microphysical description of matter(Kotake et al. 2004a,b, 2005). The main implications of thepresence of strong magnetic fields in the collapse are theredistribution of the angular momentum and the appear-ance of jet-like explosions. Specific magneto-rotational ef-fects on the gravitational wave signature were first studiedin detail by Kotake et al. (2004a) and Yamada & Sawai(2004), who found differences with purely hydrodynamicmodels only for very strong initial fields (≥ 1012 G). Themost exhaustive parameter study of magneto-rotationalcore collapse to date has been carried out very recently byObergaulinger et al. (2006a,b). Their axisymmetric sim-ulations employed rotating polytropes, Newtonian hydro-dynamics and gravity (approximating general relativisticeffects via an effective relativistic gravitational potentialin their latter work), and ad-hoc initial poloidal magneticfield distributions. These authors found that for weak ini-tial fields (≤ 1011 G, which is the astrophysically most mo-tivated case) there are no differences compared to purelyhydrodynamic simulations, neither in the collapse dynam-ics nor in the resulting gravitational wave signal. However,strong initial fields (≥ 1012 G) manage to slow down thecore efficiently (leading even to retrograde rotation in thePNS) which causes qualitatively different dynamics andgravitational wave signals. For the most strongly magne-tized models Obergaulinger et al. (2006b) found highlybipolar, jet-like outflows.

Nowadays, there exists sophisticated numerical tech-nology to perform general relativistic hydrodynamicssumulations (see e.g. Font 2003). In recent years this tech-nology has been extended to general relativistic magneto-hydrodynamics (GRMHD) (Koide et al. 1999; De Villiers& Hawley 2003; Del Zanna et al. 2003; Gammie et al.

Cerda-Duran et al.: Relativistic magneto-rotational core collapse 3

2003; Duez et al. 2005; Anton et al. 2006). General rela-tivistic simulations involve the challenging computationaltask of solving Einstein’s field equations coupled to thefluid (and magneto-fluid) evolution. The first general rela-tivistic axisymmetric simulations of rotational stellar corecollapse to neutron stars were performed by Dimmelmeieret al. (2001, 2002a,b). These simulations employed simpli-fied models to describe the thermodynamics of the pro-cess, in the form of a polytropic EOS modified such thatit accounts both for the stiffening of the matter abovenuclear density as well as thermal heating by the pass-ing shock front (the so-called hybrid EOS; see Jankaet al. 1993). The inclusion of relativistic effects withinthe so-called CFC approximation results primarily in astronger gravitational pull during the contraction of thecore. Thus, higher densities than in Newtonian mod-els are reached during bounce, and the nascent PNS ismore compact. Relativistic simulations with improved dy-namics and gravitational waveforms were reported byCerda-Duran et al. (2005), who used the CFC+ frame-work, which includes second post-Newtonian correctionsto CFC. Comparisons of the CFC approach with fullygeneral relativistic simulations (employing also stable re-formulations of the Einstein equations in 3+1 form) havebeen reported by Shibata & Sekiguchi (2004), Ott et al.(2006a), and Ott et al. (2006b) in the context of axisym-metric core collapse simulations. As in the case of CFC+,the differences found in both the collapse dynamics andthe gravitational waveforms are minute, which highlightsthe suitability of CFC for performing accurate simula-tions of such scenarios without the need for solving thefull system of Einstein’s equations. Owing to the excel-lent approximation of full general relativity offered byCFC in the context of stellar core collapse, extensionsto improve the microphysics through the incorporation ofa tabulated non-zero temperature EOS and a simplifiedneutrino treatment have been recently reported by Ottet al. (2006a) and Dimmelmeier et al. (2007). These simu-lations allow a direct comparison to the models presentedin Dimmelmeier et al. (2002b), Cerda-Duran et al. (2005),and Shibata & Sekiguchi (2004), which use the same pa-rameterization of rotation but a simple hybrid EOS. Thiscomparison shows that with a microphysical treatmentthe influence of rotation on the collapse dynamics andwaveforms is significantly reduced. In particular, the mostimportant result of Dimmelmeier et al. (2007) is the su-pression of core collapse with multiple centrifugal bouncesand its associated Type II gravitational waveforms (seeDimmelmeier et al. 2002b).

On the other hand, to further improve the realism ofcore collapse simulations in general relativity, the incor-poration of magnetic fields in numerical codes via solvingthe MHD equations is also currently being undertaken(Shibata et al. 2006; Cerda-Duran & Font 2006). Thework of Shibata et al. (2006) is focused on the collapseof initially strongly magnetized cores (∼ 1012 – 1013 G).Although these values are probably astrophysically notrelevant (as the stellar evolution models of Heger et al.

2005, predict a poloidal field strength of ∼ 106 G in theprogenitor), they enable them to resolve the scales neededfor the MRI to develop, since the MRI length scale growswith the magnetic field. The results of (Shibata et al. 2006)show that the magnetic field is mainly amplified by thewind-up of the magnetic field lines by differential rotation.Consequently, the magnetic field is accumulated aroundthe inner region of the PNS, and a MHD outflow formsalong the rotation axis removing angular momentum fromthe PNS. A different approach is followed by Cerda-Duran& Font (2006). Their progenitors are chosen to be weaklymagnetized (≤ 1010 G) which is in much better agreementwith predictions from stellar evolution. Under these condi-tions the so-called “passive” magnetic field approximation(see Sect. 2.3 below) is appropriate. In addition, the nu-merical code used in that work, which utilizes sphericalcoordinates, is more suitable for core collapse simulationsthan codes based on Cartesian/cylindrical coordinates, asused e.g. by Shibata et al. (2006).

In this paper we continue the program initiated inCerda-Duran & Font (2006) to build a numerical codewhich includes all relevant ingredients to study relativis-tic magneto-rotational stellar core collapse. To this aim wepresent here the first relativistic simulations of magneto-rotational core collapse which take into account the ef-fects of a microphysical EOS and a simplified neutrinotreatment. Those effects have been incorporated in thecode following the approach recently presented by Ottet al. (2006a) and Dimmelmeier et al. (2007). As in Cerda-Duran & Font (2006) we employ the passive magnetic fieldapproximation in the treatment of the magnetic field.

The paper is organized as follows: Sect. 2 presents abrief overview of the theoretical framework we use to per-form relativistic simulations of core collapse. Sect. 3 de-scribes how the magnetized initial models for core collapseare built and presents our sample of models. In Sect. 4we discuss aspects related to incorporating microphysicsin the core collapse models and their implementation inthe numerical code. A brief outline of our numerical ap-proach is discussed in Sect. 5. The evolution of the corecollapse initial models is discussed in Sect. 6. The mainpaper closes with a summary in Sect. 7. Relevant tests ofthe code are analyzed in Appendix A, while Appendix Bprovides an estimate for the growth rate of the Ω-dynamo.

Throughout the paper we use a spacelike signature(−, +, +, +) and units in which c = G = 1. Greek in-dices run from 0 to 3, Latin indices from 1 to 3, and weadopt the standard Einstein summation convention.

2. Theoretical framework

We adopt the 3 + 1 formalism of general relativity(Lichnerowicz 1944) to foliate the spacetime into space-like hypersurfaces. In this approach the line element reads

ds2 = −α2 dt2 + γij(dxi + βi dt)(dxj + βj dt), (1)

where α is the lapse function, βi is the shift vector, and γij

is the spatial three-metric induced in each hypersurface.

4 Cerda-Duran et al.: Relativistic magneto-rotational core collapse

Using the projection operator ⊥µν and the unit four-vector

nµ normal to each hypersurface, it is possible to build thequantities

E ≡ nµnνTµν = α2T 00, (2)

Si ≡ − ⊥µi nνTµν = − 1

α(T0i − Tijβ

j), (3)

Sij ≡ ⊥µi ⊥ν

j Tµν = Tij , (4)

which represent the total energy, the momenta, and thespatial components of the stress-energy tensor, respec-tively.

To solve the gravitational field equations we choose theADM gauge in which the three-metric can be decomposedas γij = φ4γij + hTT

ij , where φ is the conformal factor,

γij is the flat three-metric, and hTTij is the transverse and

traceless part of the three-metric. Note that this gaugechoice implies the maximal slicing condition in which thetrace K of the extrinsic curvature tensor Kij vanishes.

2.1. The CFC approximation

In our work Einstein’s field equations are formulated andsolved using the conformally flat condition (CFC here-after), introduced by Isenberg (1978) and first used in adynamical context by Wilson et al. (1996). In this approx-imation, the three-metric in the ADM gauge is assumedto be conformally flat, γij = φ4γij . Note that the sameaproximation can be realized for other gauge choices suchas the quasi-isotropic gauge or the Dirac gauge, both sup-plemented by the maximal slicing condition. Under theCFC assumption the gravitational field equations can bewritten as a system of five nonlinear elliptic equations,

∆φ = −2πφ5

(

E +KijK

ij

16π

)

, (5)

∆(αφ) = 2παφ5

(

E + 2S +7KijK

ij

16π

)

, (6)

∆βi = 16παφ4Si + 2φ10Kij∇j

(

α

φ6

)

− 1

3∇i∇kβk, (7)

where ∆ and ∇ are the Laplace and nabla operators as-sociated with the flat three-metric, and S ≡ γijSij .

2.2. General relativistic magnetohydrodynamics

The energy-momentum tensor of a magnetized perfectfluid can be written as the sum of the fluid part andthe electromagnetic field part. In the so-called ideal MHDlimit (where the fluid is a perfect conductor of infinite con-ductivity), the latter can be expressed solely in terms ofthe magnetic field bµ measured by a comoving observer.In this case the total energy-momentum tensor is given by

T µν = (ρh + b2)uµuν +

(

P +b2

2

)

gµν − bµbν , (8)

where ρ is the rest-mass density, h = 1+ ǫ+P/ρ the rela-tivistic enthalpy, ǫ the specific internal energy, P the pres-sure, and uµ the four-velocity of the fluid, while b2 = bµbµ.

We define the magnetic pressure Pmag = b2/2 and the spe-cific magnetic energy ǫmag = b2/(2ρ), whose effect on thedynamics is similar to that played by the pressure and spe-cific internal energy of the fluid, respectively. In the idealMHD limit, the electric field measured by a comoving ob-server vanishes, and Maxwell’s equations simplify. Underthis assumption the electric field four-vector Eµ can beexpressed in terms of the magnetic field four-vector Bµ,and only equations for Bi are needed. For an Eulerian ob-server, Uµ = nµ, the temporal component of the electricfield vanishes, Eµ = (0,−εijkvjBk). In this case Maxwell’sequations reduce to the divergence-free condition and theinduction equation for the magnetic field,

∇iB∗ i = 0,

∂B∗ i

∂t= ∇j(v

∗ iB∗ j − v∗ jB∗ i), (9)

where B∗ i ≡ √γBi and v∗ i ≡ αvi − βi, with vi being

the fluid’s three-velocity as measured by the Eulerian ob-server. The ratio of the determinants of the three-metricand the flat three-metric is given by γ = γ/γ. In theNewtonian limit v∗ i → vi and B∗ i → Bi, and theNewtonian induction equation and divergence constraintare recovered.

The evolution of a magnetized fluid is determined bythe conservation law of the energy-momentum, ∇µT µν =0, and by the continuity equation, ∇µJµ = 0, for therest-mass current Jµ = ρuµ. Following the procedure laidout in Anton et al. (2006), in order to cast the GRMHDequations as a hyperbolic system of conservation laws welladapted to numerical work, the conserved quantities arechosen in a way similar to the purely hydrodynamic casepresented by Banyuls et al. (1997):

D = ρW, (10)

Si = (ρh + b2)W 2vi − αbib0, (11)

τ = (ρh + b2)W 2 −(

P +b2

2

)

− α2(b0)2 − D, (12)

where W = αu0 is the Lorentz factor. With this choicethe system of conservation equations for the fluid and theinduction equation for the magnetic field can be cast as afirst-order, flux-conservative, hyperbolic system,

1√−g

[

∂√

γU

∂t+

∂√−gF i

∂xi

]

= S, (13)

with the state vector, flux vector, and source vector givenby

U = [D, Sj , τ, Bk], (14)

F i =

[

Dvi, Sj vi + δi

j

(

P +b2

2

)

− bjBi

W,

τvi +

(

P +b2

2

)

vi − αb0Bi

W, viBk − vkBi

]

, (15)

S =

[

0,1

2T µν ∂gµν

∂xj, α

(

T µ0 ∂ lnα

∂xµ− T µνΓ 0

µν

)

, 0k

]

, (16)

where δij is the Kronecker delta and Γµ

µλ are the Christoffelsymbols associated with the four-metric. We note that the

Cerda-Duran et al.: Relativistic magneto-rotational core collapse 5

above definitions contain components of the magnetic fieldmeasured by both a comoving observer and an Eulerianobserver. The two are related by

b0 =WBivi

α, bi =

Bi + αb0ui

W. (17)

The hyperbolic structure of Eq. (13) and the associatedspectral decomposition (into eigenvalues and eigenvectors)of the flux-vector Jacobians is given in Anton et al. (2006).This information is needed for numerically solving the sys-tem of equations using the class of high-resolution shock-capturing schemes that we have implemented in our code.

2.3. The passive field approximation

In the collapse of weakly magnetized stellar cores, it is afair approximation to assume that the magnetic field en-tering in the energy-momentum tensor of Eq. (8) is negli-gible when compared with the fluid part, i.e. Pmag ≪ P ,ǫmag ≪ ǫ, and that the components of the anisotropic termof T µν satisfy bµbν ≪ ρhuµuν + Pgµν . With these simpli-fications the evolution of the magnetic field, governed bythe induction equation, does not affect the dynamics of thefluid, which is governed solely by the hydrodynamics equa-tions. However, the magnetic field evolution does dependon the fluid evolution, due to the presence of the velocitycomponents in the induction equation. This “test mag-netic field” (or passive field) approximation is employedin the core collapse simulations reported in this work.

Within this approach the seven eigenvalues of theGRMHD Riemann problem (entropy, Alfven, and fast andslow magnetosonic waves) reduce to three (Cerda-Duran& Font 2006),

λi0 hydro = λi

e = λiA± = λi

s±, (18)

λi± hydro = λi

f ±, (19)

where λi0 hydro and λi

± hydro are the eigenvalues of theJacobian matrices of the purely hydrodynamics equations,as reported by Banyuls et al. (1997).

This approximation has several interesting properties.First, if we perform a simulation for a given initial mag-netic field, we can compute the result for a simulationwith the same initial magnetic field scaled by some fac-tor. To do this it is sufficient to increase or reduce thestrength of the magnetic field at any given time duringthe simulation by the same factor. The second propertyis what we call the “composition rule”. If we perform twosimulations with the same hydrodynamics but differentinitial magnetic fields, B∗ 0

1 and B∗ 02 , any linear combina-

tion B∗(t) = a B∗1(t)+b B∗

2(t) of the magnetic field at anytime, with a and b being constants, will be the solution forthe evolution of a model whose initial magnetic field is thesame linear combination, i.e. B∗ 0 = a B∗ 0

1 +b B∗ 02 . Hence,

we can make use of these properties to cover a wide rangeof magnetic field strengths and structures by performingjust a few simulations, and then constructing additionalones by means of the “composition rule”. Needless to say,

these two properties are valid only if the magnetic field re-sulting from the scaling or composition satisfies itself thepassive field approximation at all times.

2.4. Gravitational waves

The Newtonian standard quadrupole formula has been ex-tensively used in numerical simulations of astrophysicalsystems to compute the gravitational radiation and wave-forms without having to consider the full evolution of thespacetime and solving Einstein’s equations. This formulacomputes the radiative part of the spatial metric as

hquadij = PTTkl

ij

2

RQij , (20)

where PTTklij is the transverse traceless projector operator

(Thorne 1980), R is the distance to the source, Qij isthe mass quadrupole moment, and a dot denotes a timederivative. In spite of its simplicity, the particular form inwhich Eq. (20) is expressed leads to numerical difficultiesdue to the presence of second time derivatives. A way tocircumvent this problem is to eliminate all time derivativesusing the equations of motion. Following Finn (1989) andBlanchet et al. (1990) one can arrive to an expression forQkl with no explicit appearance of time derivatives. Thisis the so-called stress formula,

Qij ≈ STF

2

∫

d3x√

γD∗(

γikγjl vkvl + xk γki ∇jU)

,

(21)

where STF means the symmetric and traceless part, andD∗ ≡ √

γD. This formula has proved to be numericallymuch more accurate than the original formula and we useit in this paper to extract gravitational waveforms.

In the case of a magnetized fluid in the ideal MHD case,the gravitational radiation is also affected by the energeticcontent of the magnetic field. Kotake et al. (2004b) havederived an extension of the quadrupole formula for sucha case. In a similar way, it is possible to calculate thecorresponding stress formula (Obergaulinger et al. 2006b),which reads

Qij ≈ STF

2

∫

d3x√

γ

[

D∗(

γikγjl vkvl + xk γki ∇jU)

−γikγjl bkbl

]

. (22)

Note that in the limit of weak magnetic fields the orig-inal stress formula is recovered. We use this formula inthe magnetized core collapse simulations to calculate thecontribution of the magnetic field to the waveforms in thepassive field approximation.

3. Initial data

The structure and strength of the magnetic field in thestellar core collapse progenitors, needed as initial condi-tions of our numerical simulations, is still an open ques-tion in astrophysics. State-of-the-art models from stellar

6 Cerda-Duran et al.: Relativistic magneto-rotational core collapse

1500

500

0

2000

1000

11

10

9

8

7

6

5

40 500 1000 1500 2000

log

ρ

1500

500

0

2000

1000

11

10

9

8

7

6

5

40 500 1000 1500 2000

log

ρ

Fig. 1. Logarithm of the rest mass density (log ρ, color coded) and magnetic field lines (Bi, white curves) in a homogeneous B∗ i

field configuration (left panel) and a magnetic field generated by a circular current loop (right panel), for a typical rotating ironcore (model A1B3 in Table 1) used for the collapse simulations. The magnetic field lines have been calculated as isocontours ofthe vector potential. The axis scale is in km and the density in cgs units.

evolution including a description for the influence of themagnetic field (Heger et al. 2005), predict that the distri-bution of the magnetic field in the iron core has proba-bly a dominant toroidal component, with a strength ofabout 109 – 1010 G, and a poloidal component of onlyabout 105 – 106 G. For such weak fields (Pmag ≪ P ),the passive field approximation adopted here is likely tobe sufficient to describe the initial models considered inthis work. A second consideration is whether or not theinitial model should be an equilibrium model. In gen-eral, if one tries to construct a stationary model withoutmeridional currents and assuming an isentropic flow, theonly possible magnetic field configuration is poloidal (seeBekenstein & Oron 1979). Stationary models of magne-tized stars have been computed under these assumptionsby Bocquet et al. (1995). In the general (but still isen-tropic) case in which meridional circulation is allowed, atoroidal component of the magnetic field may exist, butthe method to calculate stationary models is far more com-plicated (Gourgoulhon & Bonazzola 1993; Ioka & Sasaki2003, 2004). When one considers ideal MHD, also purelytoroidal magnetic fields exist which maintain the circu-larity condition (Oron 2002), and therefore it is possibleto generate stationary models without meridional compo-nents. Finally, in the case that magnetic pressure does notexceed the hydrostatic pressure, Oron (2002) has shownthat stationary models with mixed toroidal and poloidalcomponent approximately accomplish the circularity con-dition.

Therefore, it makes sense to construct initial modelsfor magnetized stellar cores by simply adding an ad-hocmagnetic field to a purely hydrodynamic equilibrium con-figuration. If the condition B∗ ·∇Ω∗ = 0 is satisfied, whereΩ∗ ≡ v∗ϕ/(r sin θ) is the angular velocity of the fluid, thenthe initial magnetic field does not evolve in time either.Note that in this work we use as initial models both equi-

librium and non-equilibrium configurations for the mag-netic field, as specified in Table 2.

3.1. Magnetic field configurations

Since the numerical scheme we use to evolve the MHDequations only preserves the value of ∇ · B∗ but doesnot impose the divergence constraint of the magnetic fielditself, it is necessary to build initial configurations thatalso satisfy this condition. To do this we calculate theinitial magnetic field from a vector potential A∗, such thatB∗ = ∇×A∗, which is discretized as explained in Cerda-Duran & Font (2006).

For our code tests and core collapse simulations weuse three possible magnetic field configurations as initialconditions (or any possible combination of them):– the homogeneous “starred” magnetic field, in which B∗

is constant and parallel to the symmetry axis,– the poloidal magnetic field generated by a circular cur-rent loop of radius rmag (Jackson 1962), that can be calcu-lated from the only non-vanishing component of the vectorpotential A∗

ϕ as

A∗ϕ =

r2magB

∗0

2

∞∑

n=0

(−1)n(2n − 1)!!

2n(n + 1)!

r2n+1<

r2n+2>

P 12n+1(cos θ),

(23)

where r< = min(r, rmag), r> = max(r, rmag), and B∗0 is

the magnetic field at the center, and– a toroidal magnetic field of the form

B∗ϕ = B∗0

r2mag

r2mag − 2

, (24)

whose maximum value is reached at = rmag, where B∗0

is the initial central magnetic field and is the distanceto the axis.

Cerda-Duran et al.: Relativistic magneto-rotational core collapse 7

Note that in all three cases, we employ the “starred”magnetic field, since the divergence constraint is valid forthis quantity when computed with respect to the flat di-vergence operator. In this way we can extend any analyticprescription for the magnetic field given in flat spacetimein an easy way. Also note that in the presence of stronggravitational fields the magnetic field B is deformed withrespect to B∗ due to the curvature of the spacetime, al-though the divergence constraint is automatically fulfilled.Fig. 1 shows examples of the magnetic field structure forthe poloidal configurations of the initial magnetic field.

The initial magnetic field configuration is denoted inthe names of the models in our sample by adding a label tothe purely hydrodynamic model. For the latter we followthe notation of Dimmelmeier et al. (2002a) and Ott et al.(2006a). These models are listed in Table 1. The labelfor the magnetic field is constructed following the nota-tion of Obergaulinger et al. (2006b). We add the suffixD3M0 to denote those models with purely poloidal mag-netic field generated by a circular loop and rmag = 400 km(M0 denotes the passive field approximation) and we usethe suffix T3M0 for models with purely toroidal magneticfield and rmag = 400 km. We have also built the DT3M0model, whose magnetic field distribution is a combinationof D3M0 and T3M0 with equal magnetic field strengthsat the center. This model allows us to check the validityof the “composition rule” (see Sect. 2.3).

3.2. Hydrodynamic configurations

3.2.1. Polytropes in rotational equilibrium

For the simulation with a simplified description of matterusing the hybrid EOS (see Sect. 4.1), we construct γini =4/3 polytropes in rotational equilibrium which we obtainby using the relativistic generalization of Hachisu’s self-consistent field method by Komatsu et al. (1989)1. Theirrotation law for the specific angular momentum j is givenby

j = A2(Ωc − Ω), (25)

where A parametrizes the degree of differential rotation(stronger differentiality with decreasing A) and Ωc is thevalue of the angular velocity Ω at the center. In theNewtonian limit, this gives

Ω =A2Ωc

A2 + 2. (26)

The parameters of the selected models, which are cho-sen to be identical to some of the models considered byDimmelmeier et al. (2002a), are described in Table 1. Inaddition, as we aim at comparing our results with the re-cent numerical simulations performed by Obergaulingeret al. (2006b) in Newtonian gravity, a subset of our mod-els (those with purely poloidal magnetic field) have been

1 The adiabatic index should not be confused with the de-terminant of the spacetime three-metric, although we use thesame symbol γ (following usual practice).

Table 1. Purely hydrodynamic initial models used in the mag-netized core collapse simulations.

Model ρc A β Mg γ1

[1010 g cm−3] [km] [%] [M⊙]

A1B3G3 1.00 50.0 0.90 1.46 1.300A1B3G5 1.00 50.0 0.90 1.46 1.280A3B3G5 1.00 0.5 0.90 1.46 1.280A2B4G1 1.00 1.0 1.80 1.50 1.325A4B5G5 1.00 0.5 4.00 1.61 1.280s20A1B1 0.88 50.0 0.25 1.58 —s20A1B5 0.88 50.0 4.00 1.58 —s20A2B2 0.88 1.0 0.50 1.58 —s20A2B4 0.88 1.0 1.80 1.58 —s20A3B3 0.88 0.5 0.90 1.58 —E20A 0.42 — 0.37 2.00 —

selected as general relativistic counterparts of their mod-els. In Table 1 we also give the values for the gravitationalmass Mg (which is identical to the ADM mass MADM) andfor the initial rotation rate β = Erot/|Eb|. In the defini-tion of β we use the following expressions for the rotationalkinetic energy Erot, the gravitational binding energy Eb,and the magnetic energy Emag:

Erot =1

2

∫

d3x√

γ αvϕSϕ, (27)

Eb = Mg − Mp − Erot − Emag, (28)

Emag =1

2

∫

d3x√

γ Wb2, (29)

where Mp is the proper mass.

For these simplified initial models the gravitational col-lapse is initiated by slightly decreasing the adiabatic indexfrom its initial value to γ1 < γini = 4/3, which results ina loss of pressure support. If no pressure reduction wereimposed, the purely hydrodynamic initial models wouldremain stationary. However, even in that case the associ-ated initial magnetic field may not remain stationary (seealso Table 2 below). Only a purely toroidal initial mag-netic field would not evolve in time, while any magneticfield configuration of initial models labeled A1 would stillstay approximately stationary, since these models rotatealmost rigidly, and thus the initial magnetic field cannotwind up itself strongly.

3.2.2. Presupernova models from stellar evolution

calculations

As initial models for the simulations where we use amicrophysical EOS and deleptonization, we employ thesolar-metallicity 20 M⊙ (zero-age main sequence) modelof Woosley et al. (2002) (labeled as model s20 in Table 1).On this spherically symmetric model, which is initially notin equilibrium as it has a non-zero radial velocity profile,we impose the rotation law (25), using the same rotationnomenclature as for the previously described polytropesin equilibrium.

8 Cerda-Duran et al.: Relativistic magneto-rotational core collapse

In addition, we perform calculations with the “rotat-ing” presupernova model E20A of Heger et al. (2000),which we map onto our computational grids under theassumption of constant rotation on cylindrical shells ofconstant .

4. Treatment of matter during the evolution

In this work we improve upon preceding relativistic stellarcore collapse simulations by using an advanced descrip-tion of microphysics as presented in Ott et al. (2006a)and Dimmelmeier et al. (2007). For comparison to previ-ous results, we also perform simulations with the simpli-fied hybrid EOS (Janka et al. 1993). In the following, wedescribe both approaches for the treatment of matter.

4.1. Hybrid EOS

For calculations employing polytropes in rotational equi-librium as initial models, we utilize the hybrid EOS. Herethe pressure consists of a polytropic part, Pp = Kργ , withK = 4.897 × 1014 (in cgs units), plus a thermal part,Pth = ρǫth(γth − 1), where ǫth = ǫ − ǫp and where we setγth = 1.5. The thermal contribution is chosen to take intoaccount the rise of thermal energy due to shock heating.As ρ reaches nuclear density at ρnuc = 2.0×1014 g cm−3, γis raised to γ2 = 2.5 and K adjusted accordingly to main-tain monotonicity of P and ǫ. Due to this stiffening of theEOS the core undergoes a so-called pressure-supportedbounce. More details of the hybrid EOS can be found e.g.in Dimmelmeier et al. (2002a).

4.2. Microphysical EOS, deleptonization scheme, and

neutrino pressure

In our more realistic calculations, for which the modelss20 and E20A from stellar evolution are taken as initialmodels, we employ the tabulated non-zero temperaturenuclear EOS by Shen et al. (1998) in the variant of Mareket al. (2005) which includes baryonic, electronic, and pho-tonic pressure components. This gives the fluid pressureP (and additional thermodynamic quantities) as a func-tion of ρ, the temperature T , and the electron fraction Ye.Since the code operates with the specific internal energyǫ instead of the temperature T , we determine the corre-sponding value for T iteratively with a Newton–Raphsonscheme.

To determine the evolution of Ye, the state vector,flux vector, and source vector for the conservation equa-tions (13), as given in Eqs. (14–16) have to be augmentedby the components

DYe, DYevi, SYe

, (30)

respectively. The source term SYeis a consequence of the

electron captures during collapse, which reduces Ye. Thisdeleptonization also effectively decreases the size of thehomologously collapsing inner core, and has thus a direct

influence on the collapse dynamics and the gravitationalwave signal. Hence, it is essential to include (at least anapproximate scheme for) deleptonization during collapse.

Since multi-dimensional radiation hydrodynamics cal-culations in general relativity are not yet computationallyfeasible, in the simulations using the microphysical EOSwe make use of a a recently proposed scheme (Liebendorfer2005) where deleptonization is parametrized based ondata from detailed spherically symmetric calculations withBoltzmann neutrino transport. As in Dimmelmeier et al.(2007) we take the latest available electron capture rates(Langanke & Martınez-Pinedo 2000), which result in lowervalues for Ye in the inner core at bounce compared torecent results (Ott et al. 2006a) where standard cap-ture rates were used (Rampp & Janka 2000). FollowingLiebendorfer (2005), deleptonization is stopped at corebounce (i.e. as soon as the specific entropy s per baryonexceeds 3kB). After core bounce Ye is only passivelyadvected, neglecting any further deleptonization in thenascent PNS.

Neutrino pressure is included only in the regime whichis optically thick to neutrinos, which we define for ρ be-ing above the trapping density ρt = 2 × 1012 g cm−3.Following Liebendorfer (2005), here we approximate thecontribution of the neutrino pressure Pν as an ideal Fermigas and include the radiation stress via additional sourceterms in the momentum and energy equations for the fluid.

5. Outline of the numerical approach

The GRMHD numerical code we use in our simulationsis based on the purely hydrodynamic code described inDimmelmeier et al. (2002a,b), and on its extension dis-cussed in Cerda-Duran et al. (2005). It has been describedin detail in a previous paper (Cerda-Duran & Font 2006),which allows us to provide here only succint information.The code performs the coupled time evolution of the equa-tions governing the dynamics of the spacetime, the fluid,and the magnetic field in general relativity. The equationsare implemented in the code using spherical polar coor-dinates t, r, θ, ϕ, assuming axisymmetry with respectto the rotation axis and equatorial plane symmetry atθ = π/2.

5.1. The hydrodynamics solver

For the evolution of the matter fields we utilize a high-resolution shock-capturing (HRSC) scheme, which numer-ically integrates the subset of equations in system (13)that corresponds to the purely hydrodynamic variables(D, Si, τ). HRSC methods ensure the numerical conser-vation of physically conserved quantities and a correcttreatment of discontinuities such as shocks (see e.g. Font2003, for a review and references therein). We have im-plemented in the code various cell-reconstruction proce-dures, either second-order or third-order accurate in space,namely minmod, MC, and PHM (see Toro 1999, for def-initions). The time update of the state vector U is done

Cerda-Duran et al.: Relativistic magneto-rotational core collapse 9

using the method of lines in combination with a second-order accurate Runge–Kutta scheme. The numerical fluxesat the cell interfaces are obtained using either the HLLsingle-state solver of Harten et al. (1983) or the symmet-ric scheme of Kurganov & Tadmor (2000) (KT hereafter).Both solvers yield results with an accuracy comparableto complete Riemann solvers (with the full characteristicinformation), as shown in simulations involving purely hy-drodynamic special relativistic flows (Lucas-Serrano et al.2004) and general relativistic flows in dynamical space-times (Shibata & Font 2005). Tests of both solvers inGRMHD have been reported recently by Anton et al.(2006).

5.2. Evolution of the magnetic field

The evolution of the magnetic field needs to be performedin a way that is different from the rest of the conservationequations, since the physical meaning of the correspond-ing conservation equation is different. Although the induc-tion equation can be written in a flux conservative way,a supplementary condition for the magnetic field has tobe given (the divergence constraint), which has to be ful-filled at each time iteration. The physical meaning of thesetwo equations is the conservation of the magnetic flux in aclose volume, in our case each numerical cell. Therefore, anappropriate numerical scheme has to be used which takesfull profit of such a conservation law. Among the numeri-cal schemes that satisfy this property (see Toth 2000, fora review), the constrained transport (CT) scheme (Evans& Hawley 1988) has proved to be adequate to performaccurate simulations of magnetized flows. Our particularimplementation of this scheme (see Cerda-Duran & Font2006, for details) has been adapted to the spherical polarcoordinates used in the code. The discretized evolutionequations for the poloidal components of the magneticfield read

∂tB∗ ri+ 1

2j =

[sin θ E∗ϕ]i+ 1

2j+ 1

2

− [sin θ E∗ϕ]i+ 1

2j− 1

2

ri+ 1

2j ∆(cos θ)j

, (31)

∂tB∗ θi j+ 1

2

= 2[r E∗

ϕ]i+ 1

2j+ 1

2

− [r E∗ϕ]i− 1

2j+ 1

2

∆r2i

, (32)

where (in vectorial form) E∗ = v∗ × B∗, and where cellcenters are located at (i j), radial interfaces at (i + 1

2j),

angular interfaces at (i j + 12), and cell corners (cell edges

along the ϕ-direction) at (i + 12

j + 12). We note that the

evolution equation for the toroidal magnetic field is analogto that used for the hydrodynamics, since in axisymmetrythis component does not play any role in the CT scheme.The previous expressions are used in the numerical codeto update the magnetic field. The only remaining aspectis to give an explicit expression for the value of E∗

i . Apractical way to calculate E∗

ϕ from the numerical fluxes in

the adjacent interfaces (Balsara & Spicer 1999) is

E∗

ϕ i+ 1

2j+ 1

2

= −1

4

[

(F r)θi j+ 1

2

+ (F r)θi+1 j+ 1

2

−(F θ)ri+ 1

2j − (F θ)r

i+ 1

2j+1

]

, (33)

where the fluxes (15) are obtained in the usual way bysolving the Riemann problem at the interfaces. The com-bination of the CT scheme and this way of computing theelectric field is called the flux-CT scheme. It is used in allnumerical simulations reported in this paper. Finally, thetime discretization of Eqs. (31) and (32) is performed inthe same way as for the fluid evolution equations.

5.3. The metric solver

The CFC metric equations are five nonlinear elliptic cou-pled Poisson-like equations which can be written in com-pact form as ∆u(x) = f(x; u(x)), where u = uk =(φ, αφ, βj), and f = fk is the vector of the respectivesources. These five scalar equations for each componentof u couple to each other via the source terms that ingeneral depend on the various components of u. We usea fix-point iteration scheme in combination with a linearPoisson solver to solve these equations. Further details onthis type of metric solver can be found in Cerda-Duranet al. (2005) and Dimmelmeier et al. (2002a).

5.4. Setup of the numerical grid

We perform all axisymmetric simulations with a resolu-tion (nr ×nθ) of 300×30 zones, except for models labeledA4B5G5 in which a resolution of 300×40 is used due to themore complex angular structure. In both cases the radialgrid is equally spaced for the first 100 points, with a gridspacing of 100 m. The remaining radial zones are logarith-mically distributed to cover the outer parts of the star andthe exterior artificial low-density atmosphere. The angu-lar grid is equally spaced and we assume equatorial sym-metry. As a consequence of our various code tests (seeAppendix A) all results discussed in Sect. 6 correspond tosimulations performed using PHM reconstruction and theHLL solver for the hydrodynamics.

6. Results

We now present the main results from our numerical sim-ulations of rotational magnetized core collapse to neutronstars. First, we note that a quantitative summary of ourfindings is reported in Table 2, to which we will referrepeteadly. The dynamics of the models we have selectedis identical to the dynamics of the unmagnetized ones,since the passive field approximation is used. Therefore,we will not describe here the morphological features of thehydrodynamics of both models with the hybrid EOS (sim-plified models hereafter) and models with the microphys-ical EOS and the deleptonization scheme (microphysicalmodels hereafter), as they have been discussed in detail in

10 Cerda-Duran et al.: Relativistic magneto-rotational core collapse

20 30 40 50 60t [ms]

10-8

10-7

10-6

β mag

, βϕ, β

polo

20 30 40 50 60t [ms]

10-8

10-7

10-6

β mag

Fig. 2. Time evolution of the magnetic energy parameters βmag

(solid line), βϕ (dashed line), and βpolo (dashed-dotted line) formodels A1B3G5-D3M0 (top panel) and A1B3G5-T3M0 (bot-tom panel).

Dimmelmeier et al. (2002a,b) and Ott et al. (2006b) as wellas Dimmelmeier et al. (2007), respectively. (It is worth toemphasize, however, the excellent agreement found in thehydrodynamical simulations performed with three inde-pendent numerical codes.) We pay more attention insteadto the magnetic field evolution. In all our simulations aninitial magnetic field strength of B∗

0 = 1010 G is consid-ered. This value is an upper limit for the T3M0 models,since the expected initial toroidal magnetic field is of thisorder (Heger et al. 2005). However, for the D3M0 models,this field strength is already at (or above) the upper endof the astrophysically expected values.

For all models we first present results for identical val-ues of B∗

0 , in a way that we can study the different effectsand compare them properly. Afterwards we present the re-sults scaled to lower, astrophysically expected values. Weanticipate that our results can change if some of the severalassumptions made in our simulations (axisymmetry andpassive field approximation) are relaxed. An estimationand discussion of these effects can be found in Sect. 6.5.

6.1. Evolution of the magnetic energy parameter

The evolution of the energy parameter for the magneticfield βmag = Emag/|Eb| can be seen in Fig. 2 for modelA1B3G5 of our sample. In order to analyze the ampli-fication of the magnetic field, we separate the effects of

20 30 40 50 60t [ms]

10-8

10-7

10-6

β mag

20 30 40 50 60t [ms]

10-9

10-8

10-7

10-6

β mag

B* 0

polo = 10

10 G

109 G

108 G

107 G

106 G

Fig. 3. Time evolution of the magnetic energy parameter βmag.The top panel shows βmag for model A1B3G5-DT3M0 (solidline) and the comparison with the composition (dot-dashedline) of models A1B3G5-D3M0 (dashed line) and A1B3G5-T3M0 (dotted line). The bottom panel shows the evolutionof βmag for the same hydrodynamic model (A1B3G5) with aninitial value of the toroidal field of B∗ 0

ϕ = 1010 G, and varyingvalues of the initial poloidal field B∗ 0

polo, from 106 to 1010 G.

the different components of the magnetic field into βϕ

for the toroidal component and βpolo = βmag − βϕ forthe poloidal component, which are also plotted in the fig-ure. As the collapse proceeds the magnetic field growsby at least two reasons: First, the radial flow compressesthe magnetic field lines, amplifying the existing poloidaland toroidal magnetic field components. Second, duringthe collapse of a rotating star differential rotation is pro-duced and increased, even for rigidly rotating initial mod-els (see e.g. Dimmelmeier et al. (2002b)). Hence, if a seedpoloidal field exists, the Ω-dynamo mechanism winds upthe poloidal field lines into a toroidal component. This(linear) amplification process generates a toroidal mag-netic field component, even from purely poloidal initialconfigurations. The toroidal component of the magneticfield is affected by the two effects, while the poloidal fieldis only amplified by radial compression of the field lines.Thus, even if the initial magnetic field configuration ispurely poloidal, the toroidal component dominates aftersome dynamical time. To study the differences in the evo-lution of the magnetic field depending on the initial mag-netic field we now describe in detail the features of model

Cerda-Duran et al.: Relativistic magneto-rotational core collapse 11

Table 2. Hydrodynamical and magnetic field properties of all models computed in this work. From left to right the columnsreport the name of the model, the stationarity properties of the initial magnetic field, the time tb of core bounce, the maximumrest-mass density ρmax, the maximum poloidal field |Bpolo|max, the maximum toroidal field |Bϕ|max, the rotational energyparameter βrot, the magnetic energy parameter βmag, its poloidal contribution βpolo, and the central angular velocity Ωc. Thetime scale τΩ of the growth of the magnetic field by the Ω-dynamo and the saturation time tsat for this process are also shown.

Model stationary tb ρmax |Bpolo|max |Bϕ|max βrot βmag βpolo Ωc τΩ tsat

[ms]h

1014 g

cm3

i

[1010 G] [1010 G] [10−2] [10−8] [10−8] [ms−1] [s] [s]

A2B4G1-D3M0 no 102.1 0.47 400 1467 15.6 8.3 0.9 0.36 85 10.3A1B3G3-D3M0 approx. 49.0 4.22 1719 2522 2.3 1.2 0.8 3.96 7 0.3A1B3G3-T3M0 yes 49.0 4.22 0 1714 2.3 0.2 0.0 3.96 — —A1B3G5-D3M0 approx. 30.2 4.57 1146 1275 0.9 0.5 0.4 3.91 21 0.7A1B3G5-T3M0 yes 30.2 4.57 0 1542 0.9 0.2 0.0 3.91 — —A1B3G5-DT3M0 approx. 30.2 4.57 1146 1537 0.9 0.6 0.4 3.91 21 0.6A3B3G5-D3M0 no 30.7 3.73 984 1672 2.3 0.6 0.4 3.75 24 1.1A4B5G5-D3M0 no 31.3 1.74 1094 2716 8.5 4.4 1.2 1.18 24 2.1A4B5G5-T3M0 yes 31.3 1.74 0 1626 8.5 0.4 0.0 1.18 — —s20A1B1-D3M0 approx. 132.1 2.69 1221 162 0.6 2.9 2.9 1.34 22 0.5s20A2B2-D3M0 no 138.8 2.75 1849 3574 5.8 7.6 3.2 3.55 7 0.5s20A2B2-T3M0 yes 138.8 2.75 0 1365 5.8 0.9 0.0 3.55 — —s20A1B5-D3M0 approx. 150.6 2.69 1100 1011 7.8 3.2 2.4 3.89 10 0.8s20A1B5-T3M0 yes 150.6 2.69 0 1447 7.8 1.2 0.0 3.89 — —E20A-D3M0 no 214.2 2.29 2343 7503 7.7 23.4 1.8 4.37 6 0.5E20A-T3M0 yes 214.2 2.29 0 2739 7.7 1.9 0.0 4.37 — —

A1B3G5 with different initial magnetic field configura-tions.

In model A1B3G5-D3M0 the initial magnetic field isentirely poloidal. The top panel of Fig. 2 shows that βϕ

(dashed line) grows much faster than βpolo, particularlyafter bounce (t ∼ 30 ms) when the radial compressionmechanism stops. We note that the magnetic field consid-ered is weak enough not to affect the dynamics, with thefinal βmag ≪ 1.

If we consider a purely toroidal magnetic field initially,as model A1B3G5-T3M0, the only amplification mecha-nism present in our simulations is the radial compression,since no poloidal field can be wound up. The bottom panelof Fig. 2 shows the behaviour of βmag for model A1B3G3-T3M0. It is important to notice that during the collapseβmag hardly grows (for other models of the T3M0 seriesit even decreases) since the radial compression is a veryinefficient mechanism to amplify the magnetic field. As aresult, for some models the final PNS is “less magnetized”than the progenitor core in the sense that βmag at bounceis smaller than before the collapse. We note that the evo-lution of this kind of purely toroidal models could changecompletely if the axisymmetry condition were removed,since in three dimensions there are mechanisms that cantransform a toroidal magnetic field into a poloidal one.Some of these mechanisms are discussed in Sect. 6.5 be-low.

To check whether the “composition rule” (seeSect. 2.3) is valid we consider next a mixed configurationof poloidal and toroidal magnetic fields at the beginning ofthe simulation (model A1B3G5-DT3M0). The top panelof Fig. 3 shows with a solid line the time evolution ofβmag for model A1B3G5-DT3M0 and with a dot-dashed

line the composition of the individual values for βmag inmodels A1B3G5-D3M0 and A1B3G5-T3M0 with identi-cal initial field strengths. (The separate evolutions for thelatter are also included in the plot as dashed and dottedlines, respectively.) The agreement of the two evolutionpaths is remarkable, which shows that the “compositionrule” works properly for our simulations. Therefore, wecan use it to obtain any desirable composition of mag-netic fields with a single hydrodynamic evolution of thetwo models D3M0 and T3M0. For the particular composi-tion showed in this model, the final value of βmag dependsvery weakly on the initial toroidal magnetic field compo-nent. In other words, the structure of the magnetic field ofthe PNS will depend almost exclusively on the radial com-pression of the initial poloidal component of the magneticfield.

Next, we consider a “composition” of these modelswith different initial magnetic field strength. We keepthe initial toroidal component fixed at a realistic value,B∗ 0

ϕ = 1010 G, and change the initial poloidal componentin a range that spans from B∗ 0

polo = 1010 G down to the

astrophysically more realistic value of 106 G. The bottompanel of Fig. 3 shows the time evolution of βmag for thesedifferent configurations. For lower values of B∗ 0

polo, theΩ-dynamo mechanism becomes increasingly slower andthe initial toroidal component becomes important for themagnetic field configuration of the PNS. In the lowest ini-tial poloidal field case analyzed, the magnetic field of thePNS is completely toroidal and depends exclusively on theinitial magnetic field configuration.

The remaining computed models of our sample, includ-ing those with microphysics, behave qualitatively in a verysimilar manner, although quantitative differences can be

12 Cerda-Duran et al.: Relativistic magneto-rotational core collapse

found in the amplification of the magnetic field during thecollapse, and the amplification rates after bounce due tothe Ω-dynamo. Fig. 4 shows the evolution of the magneticenergy parameter βmag for all the simulated models withinitial purely poloidal magnetic field (label D3M0). Forall models we find the following relation between the col-lapse time and the amplification rate of the magnetic fieldafter bounce (which, however, does not hold for modelA2B4G1-D3M0, as this is the only model of our samplefor which the collapse is halted not by the stiffening of theEOS, but rather by centrifugal forces at subnuclear den-sities; cf. Dimmelmeier et al. 2002b): Models with largecollapse times, such as all microphysical models as wellas the simplified model A1B3G3-D3M0, exhibit a moreefficient amplification of the magnetic field as comparedto the rapid collapse models (G5 series). To quantify thedifferences between the models we estimate the time scaleτΩ for the amplification of the magnetic field by fitting thepost-bounce evolution of βmag to

βmag =

(

t

τΩ

)2

. (34)

The resulting values can be found in Table 2. The timescale should depend on the central angular velocity Ωc ofthe PNS, and on the strength of the poloidal magneticfield that can be wound up, which can be estimated fromβpolo. Hence, the following expression should be valid inthe most efficient scenario (see Appendix B for details):

τΩ =2

Ωc

√

βpolo

. (35)

To check this relation we plot in the top panel of Fig. 5 thevalue of the fit for τΩ versus the value from the previousanalytic expression. Apparently for all models the growthtime of the Ω-dynamo is always larger than that of themost efficient situation (solid line in the figure), and cor-responds to a fraction (30%– 90%) of the upper limit (35).This relation shows that in order to obtain higher amplifi-cation rates of the magnetic field not only strong rotationis needed, but also a sufficient compression of the poloidalmagnetic field during the collapse.

Furthermore, we also find that the value of βpolo isrelated to the collapse time in such a way that we sys-tematically observe (see the bottom panel of Fig. 5) thatlonger infall times compress more efficiently the poloidalcomponent. As the microphysical models typically featurelonger collapse times (from the onset of contraction un-til core bounce) than simulations with simplified models,they also yield higher growth rates. Note, however, thatthe collapse time of the simplified models is determined bythe freely selectable parameter γ1, which can in principlebe chosen arbitrarily close to 4/3. Such a choice, though,will lead to a collapse with multiple centrifugal bounces(as in model A2B4G1), which is a dynamical behaviorthat is strongly suppressed if more realistic microphysicsis taken into account (Dimmelmeier et al. 2007, see alsothe related discussion in Sect. 6.4).

0 50 100t − t

b [ms]

10-9

10-8

10-7

10-6

10-5

10-4

10-3

β mag

A1B3G3A1B3G5A3B3G5A4B5G5A2B4G1s20A1B1s20A1B5s20A2B2E20A

Fig. 4. Time evolution of the magnetic energy parameter βmag

for all simulated models with initial purely poloidal magneticfield (label D3M0).

1 10 100

2 Ω−1β −1/2mag polo

[s]

1

10

100

τ Ω [

s]

10 100tb [ms]

10-9

10-8

10-7

β mag

pol

o

Fig. 5. Top panel: relation between the fitted values for thegrowth time τΩ of the Ω-dynamo and the upper limit givenby 2Ω−1

PNSβ−1/2

polo . The solid line represents the upper limit, thedashed line corresponds to 50% of that, and the dotted line to10%. Bottom panel: magnetic energy parameter of the poloidalcomponent of the magnetic field at bounce versus the time ofbounce. Both microphysical models (filled circles) and simpli-fied models (open circles) are shown.

6.2. Morphology

The main qualitative differences between the various mod-els become apparent when we study the detailed structureof the magnetic field of the resulting PNS. In Fig. 6 we

Cerda-Duran et al.: Relativistic magneto-rotational core collapse 13

0 10 20 30 40 50 0 10 20 30 40 500 10 20 30 40 50

30

50

20

40

0

10

10

11

12

13

14

30

50

20

40

0

10

10

11

12

13

14

30

50

20

40

0

10

+ −lo

g (

B

/ |B

|)ϕ

polo

−2

−1

0

1

2

polo

log

|B

|

log

ρ

+ −

0 10 20 30 40 50 0 10 20 30 40 50−2

−1

0

1

2

0 10 20 30 40 50

30

50

20

40

0

10

10

11

12

13

14

30

50

20

40

0

10

10

11

12

13

14

30

50

20

40

0

10

log

|B

|

log

ρ

+ −

polo

+ϕ

polo

−lo

g (

B

/ |B

|)

Fig. 6. Configuration of the innermost region of the collapsed star at the end of the evolution for models A1B3G5-D3M0 (toppanels; t = 60 ms) and s20A1B1-D3M0 (bottom panels; t = 142.5 ms). The left panels show the rest mass density as log ρ inunits of g cm−3, overplotted by the meridional velocity field (vr, vθ) (arrows), and isocontours of the specific internal energyǫ. The center panels display the logarithm of the poloidal component of the magnetic field, log |Bpolo|, in units of G and themagnetic field lines in the r–θ plane. The right panels show Bϕ/|Bpolo|. All axes are in units of km.

show two-dimensional snapshots of selected hydrodynamicand magnetic field variables at the final time of the simula-tions for two representative models of our sample, namelymodel A1B3G5-D3M0 (top panels) and model s20A1B1-D3M0 (bottom panels). For typical simulations with ini-tial poloidal magnetic fields (D3M0 models) the result-ing PNS has two clearly distinct parts (see left panels ofFig. 6): an inner region with a size of ∼ 10 km, where nu-clear density is exceeded and which is almost rigidly rotat-ing, and a surrounding shell extending to ∼ 50 km, withsubnuclear densities and which is strongly differentiallyrotating. These two parts are also visible in the distribu-tion of the magnetic field (see center and right panels ofFig. 6). The inner region has a mixed toroidal and poloidalmagnetic field configuration, with both components hav-ing similar strength, which results in a helicoidal structurealigned with the rotation axis. As this part of the PNS isalmost rigidly rotating and practically in equilibrium, themagnetic field hardly evolves in time. On the other hand,the outer shell is differentially rotating; thus the toroidalmagnetic field component dominates and grows linearlywith time due to the Ω-dynamo mechanism.

If we compare the microphysical with the simplifiedsimulations, we find that some significant morphologicaldifferences arise. In the case of simplified models we findconvection in the PNS of only one model (A1B3G5), whileall microphysical models have remarkable convective be-havior in the region surrounding the inner part. These mo-

tions affect the magnetic field, since they twist the poloidalmagnetic field lines, generating a much more complicatedstructure of the poloidal field for those models. In partic-ular those strong meridional currents distort the magneticfield in such a way that in some regions the poloidal com-ponent changes direction with respect to the rotation axis(see e.g. bottom-right panel of Fig. 6). This produces anegative effect in the Ω-dynamo as in these regions themagnetic field will be wound up in the opposite direction.However, the overall Ω-dynamo mechanism seems not tobe affected in a significant way by these local effects. Wewill return to this important issue in the discussion of theMRI in Sect. 6.5.2.

As in the microphysical models the passing shockleaves behind an area with a negative radial entropy gradi-ent, such convective overturn motions are to be expected.On the other hand, recent simulations in Newtoniangravity including accurate Boltzmann neutrino transport(Buras et al. 2006) and using the alternative microphysicalEOS by Lattimer & Swesty (1991) show that such largeamplitude convection (which start roughly concurrentlywith the post-bounce neutrino burst) are very efficientlysuppressed. It is currently not clear whether the neglectof an appropriate neutrino treatment after core bounce orthe use of the EOS by Shen et al. (1998) is the reasonfor the strong convective post-bounce motion we observein our microphysical simulations (see also Dimmelmeieret al. 2007).

14 Cerda-Duran et al.: Relativistic magneto-rotational core collapse

30 40 50 60 70 80t [ms]

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Em

ag [

1051

erg

]

20 30 40 50 60t [ms]

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Em

ag [

1051

erg

]

50 100 150 200t [ms]

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Em

ag [

1051

erg

]

Fig. 7. Comparison of the time evolution of the magnetic energy Emag for models A1B3G3-D3M0 (left panel), A3B3G5-D3M0(center panel) and A2B4G1-D3M0 (right panel). The line styles represent simulations performed in general relativity (solidlines), a purely Newtonian treatment (dotted lines), and Newtonian hydrodynamics with an effective relativistic gravitationalpotential (dashed lines).

Model A4B5G5-D3M0 has to be discussed separately,since it has initially significantly stronger differential rota-tion and more angular momentum than the other models.As a result this model undergoes a core bounce due tocentrifugal hang-up before reaching nuclear density. Itsstructure is toroidal with an off-center maximum density.Although it exhibits stronger differential rotation at thebeginning compared to the other models, and the amplifi-cation process during collapse is thus more efficient, afterbounce its angular velocity Ω is smaller (as the PNS isless compact) and therefore the linear amplification dueto Ω-dynamo is less pronounced. The main differences inthe magnetic field structure of its PNS with respect tothe other models are that, first, the Ω-dynamo is activenot only in the high-density torus, but also in the cen-tral lower-density region, and, second, the strong merid-ional currents twist the magnetic field lines around thetorus. However, we point out that in the investigatedrange of initial rotation configurations all microphysicalmodels are significantly less influenced by rotation thanthe simplified models (like A4B5G5), and that even forrather extreme rotation such collapse dynamics, leadingto a toroidal structure, is strongly suppressed if an ad-vanced description of microphysics is used (which is inaccordance with the comprehensive parameter study byDimmelmeier et al. 2007).

In the models with initially purely toroidal field at thebeginning (T3M0 series), a poloidal field cannot emerge inaxisymmetry. Hence, the final magnetic field structure ofthe PNS consists of a stationary and entirely toroidal mag-netic configuration with the highest field strengths foundin the high density regions. As the rotational profile doesnot affect the distribution of the magnetic field, the dif-ferent regions of the PNS are not visible in the structureof the magnetic field.

6.3. Comparison with Newtonian results

In order to study the general relativistic effects in the evo-lution of the magnetic field, we choose a subset of our sim-

ulations with the hybrid EOS to represent the relativis-tic version of some of the models of Obergaulinger et al.(2006a,b). Their first paper is devoted to Newtonian sim-ulations of magneto-rotational core collapse, while in theirsecond paper an effective relativistic gravitational poten-tial was used to mimic general relativistic effects (whilestill keeping a Newtonian framework for the hydrodynam-ics). Since in contrast to their work we use the passive fieldapproximation, the comparison can only be made with thelow magnetic field models presented in that work, namelythe “M10” models. In these models the magnetic fielddoes not affect the collapse dynamics and our approxi-mation is valid. Although there are no qualitative differ-ences between Newtonian and general relativistic models(aside from those coming purely from the hydrodynamicsas described in Dimmelmeier et al. 2002a,b) some dissim-ilarities can be found in the magnetic field strength andamplification rates after core bounce.

We have studied the evolution of the total magnetic en-ergy for the various models, and plot the results in Fig. 7.In general, for a similar hydrodynamic behavior the mag-netic energy attained during the evolution is smaller inthe general relativistic case than in the Newtonian case(with either regular or effective relativistic gravitationalpotential). Since the winding up of magnetic field linesis the main mechanism responsible for the amplificationof the magnetic field, the rate of amplification is relatedto the rotation rates and to how strongly compressed thepoloidal component of the magnetic field is. Since bothhigher densities and stronger rotation are reached in thegeneral relativistic case (Dimmelmeier et al. 2002b), thelower magnetic field growth rates found for that case canbe explained by the smaller values of the poloidal magneticfield energy at core bounce. In cases such as A2B4G5-D3M0 (right panel of Fig. 7) where the dynamics differsmore noticeably with respect to the Newtonian and TOVcases, the magnetic energy can be higher in the generalrelativistic case.

Cerda-Duran et al.: Relativistic magneto-rotational core collapse 15

6.4. Gravitational waves

We calculate the gravitational wave output from all ofour simulations using the Newtonian quadrupole formulagiven in Eq. (22), which includes the magnetic terms.Thus, the quadrupole wave amplitude AE2

20 , which is re-lated to the dimensionless quadrupolar strain amplitudehquad as

hquad =1

8

√

15

πsin2 θ

AE220

R, (36)

contains the contribution AE220 mag corresponding to the

magnetic field. Here hquad is the only independent com-ponent of the radiative part hquad

ij of the spatial metricas given by Eq. (20). In order to understand how themagnetic field affects the waveforms, we also separatelycompute AE2

20 mag. The resulting waveforms for some rep-resentative models are shown in Fig. 8. As the magneticfield is very low at all times, b2 ≪ ρ, the component ofthe gravitational wave due to the magnetic field is severalorders of magnitude smaller than AE2

20 .Therefore, during the core bounce and the im-

mediate post-bounce phase, the waveforms we ob-tain are practically identical to the ones presentedfor the same model setup in Dimmelmeier et al.(2002a) (for the simplified models) and Dimmelmeieret al. (2007) (for the microphysical models), whichcan also be downloaded from a freely accessible wave-form catalog at www.mpa-garching.mpg.de/rel hydro/

wave catalog.shtml. The values for AE220 lie in the range

between about 30 cm and 3000 cm, which translates to ahquad of roughly 3 × 10−22 to 3 × 10−20 (assuming a dis-tance R = 10 kpc to the source and optimal orientationbetween the source and the detector).

We also emphasize that all investigated microphysicalmodels yield gravitational wave signals known as Type Iin the literature, i.e. the waveform exhibits a positive pre-bounce rise and then a large negative peak, followed by aring-down. This is to be expected, as recent studies usingthe same hydrodynamical model setup (Ott et al. 2006a;Dimmelmeier et al. 2007) have shown that the inclusionof microphysics in stellar core collapse simulations sup-presses the other signal types, which were associated tomultiple centrifugal bounce (Type II signals) or rapid col-lapse with a very small mass of the inner core (Type IIIsignals).

After bounce, the star reaches a quasi-equilibriumstate, and thus, the hydrodynamic component of the wave-form decreases. At the same time, for models D3M0, themagnetic field grows linearly with time. Such a behaviorin the magnetic field produces an increasing gravitationalwave signal. However, at the end of the simulation, themagnetic field component of the waveform is still negli-gible in comparison with the hydrodynamic component.It is expected that at later times, as the amplification ofthe magnetic field reaches saturation, the influence of themagnetic field on the waveform becomes significant, bothdue to its effect on the dynamics and also due to the contri-

20 30 40 50 60t [ms]

10-4

10-3

10-2

10-1

100

101

102

AE

2

20

[cm

]

130 140 150 160 170 180 190 200t [ms]

10-3

10-2

10-1

100

101

102

103

AE

2

20

[cm

]

Fig. 8. Absolute value of the gravitational wave amplitude AE220

(solid line) for models A1B3G5-D3M0/T3M0/DT3M0 (toppanel) and s20A1B5-D3M0/T3M0 (bottom panel). For the lowmagnetic field strengths considered, the contribution of themagnetic field to the waveform is negligible, and the signalsof series D3M0, T3M0, and DT3M0 are practically identicalto the purely hydrodynamic waveform. For clarity, the compo-nent AE2

20 mag from the magnetic field is also plotted for mod-els A1B3G5-D3M0 (top panel, dashed line), A1B3G5-T3M0(top panel, dotted line), A1B3G5-DT3M0 (top panel, dashed-dotted line), s20A1B5-D3M0 (bottom panel, dashed line), ands20A1B5-T3M0 (bottom panel, dotted line).

bution of the magnetic field to the gravitational radiationitself. We note, however, that the effect of the MRI couldadditionally lead to noticeable changes in the waveforms,provided it were able to efficiently amplify the magneticfield (see discussion in Sect. 6.5.2).

For models T3M0, on the other hand, the componentof the waveform due to the magnetic field is even smallerthan for the D3M0 models. This is a consequence of theinefficient amplification of the magnetic field via the radialcompression. After bounce, the magnetic component ofthe waveforms in models T3M0 does not grow, and henceit is not expected to dominate the waveform later in theevolution, unless other processes amplifying the magneticfield were present.

6.5. Amplification of the magnetic field

Different mechanisms that amplify the magnetic field canact during a core collapse or the subsequent evolution of

16 Cerda-Duran et al.: Relativistic magneto-rotational core collapse

the newly formed PNS. This issue is of great importance,since the evolution of the PNS during its first minute oflife until a cold NS forms can change drastically depend-ing on the initial conditions at formation. One of the mostimportant aspects is the distribution of angular momen-tum. A highly differentially rotating PNS can be subjectto various types of instabilities, such as the dynamical low-β instability, the classical bar-mode instability (which isunlikely to occur in a PNS on dynamical time scales as itrequires very high values of β), or the secular CFS insta-bility. Such instabilities are potential sources of detectablegravitational radiation. Therefore, a natural question thatarises is whether the magnetic field is going to grow suffi-ciently fast to act on the PNS dynamics by flattening therotation profiles (and therefore preventing the instabili-ties to develop), or whether, instead, the growth processmay take a few seconds, allowing the instabilities to growand the accompanying gravitational waves to become de-tectable. A number of effects can amplify the magneticfield shortly after PNS formation. They are discussed next.

6.5.1. Ω-dynamo

Within our passive field approximation we can only com-pute the amplification rates for the Ω-dynamo, for whichthe magnetic field grows linearly with time; therefore βmag

grows quadratically with time (see Appendix B). The timescale τΩ of this amplification process and the estimatedtime tsat at which the field saturation begins are given inTable 2. In the fastest case of our model sample, whichoccurs for model A1B3G3-D3M0, saturation is reached atabout 300 ms, and in most other cases, the Ω-dynamo sat-urates at times larger than 0.5 s. Note, however, that theseestimates depend on the initial magnetic field strength,which is chosen to be B∗

0 = 1010 G. For lower values ofthe magnetic field these time scales can be scaled as (seeEq. B.5)

τΩ ≈ τΩ 10

(

1010 G

B∗0

)

, tsat ≈ tsat 10

(

1010 G

B∗0

)

. (37)

We recall that stellar evolution calculations predict thatin a progenitor core the poloidal component of the mag-netic field can initially have a strength of about 106 G(Heger et al. 2005). For such an initial magnetic field thesaturation time scale becomes several hours. This makesthe Ω-dynamo a very inefficient mechanism to amplify themagnetic field during core collapse and bounce, unless theprogenitors are highly magnetized (B ≥ 1010 G) for whichthe saturation could be reached within a few dynamicaltime scales. The magnetic field at the saturation is inde-pendent of the initial magnetic field strength and of theorder of ∼ 1016 G.

6.5.2. Magneto-rotational instability

There are other magnetic field amplification processes thatour simulations cannot account for, but for which it is nev-ertheless possible to estimate the growth rates. It has been

suggested that the magneto-rotational instability couldamplify the magnetic field from arbitrary weak fields up tovalues where equipartition between the magnetic field en-ergy and the rotational kinetic energy is reached (Akiyamaet al. 2003). However, our analysis shows that in the con-text of core collapse such an amplification is still an openissue. We proceed next to describe the MRI and the un-certainties related to its effect on the amplification of themagnetic field in core collapse.

The MRI is a shear instability that generates turbu-lence and results in an amplification of the magnetic fieldin a differentially rotating magnetized plasma (Balbus &Hawley 1991, 1992), redistributing angular momentum inthe plasma. Linear analysis shows that if the magneticfield strength is very low, as in our case, the condition forthe MRI to occur, in the Newtonian limit and includingentropy effects, is (Balbus & Hawley 1991)

N2 +dΩ2

d ln< 0, (38)

where N2 is the Brunt–Vaisala frequency. For negativevalues of this frequency, N2 < 0, the system is convec-tively unstable, which enhances the MRI. On the otherhand, in convectively stable regions (N2 > 0) the MRIis suppressed. If condition (38) is satisfied and the mag-netic field has a poloidal component, then any perturba-tion grows exponentially in time. The time scale for thefastest growing unstable mode is

τMRI = 4π

∣

∣

∣

∣

N2

Ω2+

dΩ

d ln

∣

∣

∣

∣

−1

, (39)

which is independent of the magnetic field configurationand strength. Only those modes with a length scale largerthan a critical wavelength will grow (Balbus & Hawley1991). Neglecting buoyancy terms, this length scale canbe roughly estimated as λMRI ∼ 2πcA/Ω, where cA isthe Alfven speed. For the typical values attained in thenascent PNS, in which the dominant magnetic field istoroidal, the critical length scale is

λMRI ≈ 62

(

B∗ 0

1010 G

) (

1 ms−1

Ω

) (

1014 g cm−3

ρ

)1/2

m.

(40)

Note that we have scaled the length scale with the typicalmagnetic field strength B∗ 0 of the progenitor, and notwith that of the PNS itself. For the poloidal componentand realistic values of the initial magnetic field (B∗ 0 ∼106 G) this length scale is reduced by several orders ofmagnitude (λMRI polo ∼ 0.6 cm). In any case, resolving thescales needed to simulate the MRI is a challenging problemas, in the case of weak magnetic fields, the wavelength ofthe fastest growing mode (which is close to the criticallength scale) is typically much smaller than the availablegrid resolution.

Linear analysis provides tools to determine the onsetof the instability and the typical time and length scales.

Cerda-Duran et al.: Relativistic magneto-rotational core collapse 17

However, once the perturbations of the magnetic field be-come of the order of the magnetic field itself, linear anal-ysis is no longer valid (although in the weak field casethe perturbations of the fluid variables are still small).The amplification of the magnetic field due to the MRI istherefore a nonlinear effect, and can only be studied bymeans of numerical simulations. The appropriate numer-ical approach, due to the smallness of the length scalesnecessary to be resolved, are local simulations of the MRIin a shearing box. Numerical simulations of this kind inthree dimensions have been performed by Hawley et al.(1995) in the context of accretion discs. They show thatif the instability condition of linear analysis is fulfilled,then the amplification of the magnetic field proceeds bythe formation of an axisymmetric channel flow. This iswell understood, since the linear MRI solution is also asolution of the nonlinear axisymmetric MHD equations(Goodman & Xu 1994). In the ideal MHD limit, the am-plification saturates when nonaxisymmetric perturbationsdestroy the channel flow. It is important to emphasize thenecessity of performing three-dimensional simulations inthe shearing box since, in axisymmetry, the channel flowis not destroyed and any magnetic field is able to grow con-tinuously, reaching saturation only when the MRI lengthscale is of the order of the region in which the MRI ispresent (Hawley & Balbus 1992).

For a magnetic field distribution with zero mean atlarge scales, the amplification proceeds from arbitrarilyweak fields (Hawley et al. 1996) and saturates irrespec-tive of the initial magnetic field at average values ofPmag/P0 ∼ 0.01, where P0 is the initial gas pressure. If amean magnetic field is present (as in our case), the satu-rated magnetic field depends on the initial magnetic fieldstrength. In the most favorable case of a vertical mag-netic field, the total amplification by the MRI is onlyabout a factor 20 of the original field, this amplifica-tion being even smaller in the case of a purely toroidalfield (Hawley et al. 1995). On the other hand, Sano et al.(2004) have suggested that for sufficiently weak magneticfields the saturation level could be independent of the ini-tial field and equal to that in the zero-mean case. If thiswere confirmed it would mean that, for the weak magneticfield strength present in our magneto-rotational core col-lapse models (Pmag/P0 ∼ 10−8 in the PNS for progeni-tors with B∗ 0 = 1010 G), a magnetic field of ∼ 1016 Gcould be reached on time scales of τMRI. Such a strongmagnetic field would have a significant effect on the dy-namics, similar to that observed in numerical simulationswith highly magnetized progenitors (Obergaulinger et al.2006b,a; Shibata et al. 2006). In the opposite case, theMRI would fail to considerably amplify the magnetic field,and for a purely toroidal field the magnetic field shouldgrow only by a factor of about 3 according to Hawley et al.(1995).

The inclusion of more complex physics relevant for thecore collapse scenario (like radiation, diffusion, or resis-tivity) can significantly change the amplification process,since in the nonideal case the reconnection of magnetic

field lines seems to be the dominant effect in the satu-ration process of the MRI. In general, these effects acttowards lowering the values of the saturation; for reasonsof simplicity we do not consider them in this discussion(see Hawley 2005, and references therein for a detailedreview on this topic).

Furthermore, it has to be noted that all local simula-tions of the MRI have been performed in the context ofKeplerian accretion discs, and, hence, some of the underly-ing physical conditions are not valid in the case of a PNS.For example, the typical sound speed cs in those simula-tions is of the order of 10−3. Only the parametric studyperformed by Sano et al. (2004) covers a wider range ofvalues of cs ∼ 10−8 – 10−2. However, the sound speed in aPNS is higher, cs ∼ 10−1. Rotational velocities and pro-files are also very different in a disc and a PNS. Therefore,appropriate local simulations of the PNS scenario shouldeventually be performed in order to confirm the growth ofthe MRI for a weakly magnetized PNS, and to infer themagnetic field at which the instability saturates.

As the MRI involves a backreaction of the magneticfield onto the dynamics, we cannot study this effect inour simulations, as we assume the passive field approxi-mation. Furthermore, even with “active” magnetic fields,both the resolution needed to resolve the MRI lengthscale (∼ 10 m) and the requirement for three-dimensionalsimulations are not affordable with present computers.Therefore, we are limited to analyzing whether our mag-netized collapse models are susceptible to developing suchan instability, according to linear analysis estimates, leav-ing aside the issue of saturation, whose uncertainties needa deeper analysis which is beyond the scope of this work.In order to estimate how MRI could change our results ifit were taken into account properly, we determine the re-gions where condition (38) holds. Inside these regions wecalculate the time scale for the fastest growing mode. InFig. 9 we show the results for the D3M0 models A1B3G5(left panel) and s20A1B5 (right panel) at the end of thesimulation for each model. We note that since the onsetof the MRI is independent of the magnetic field strength,provided that a poloidal component exists, any combina-tion of D3M0 and T3M0 models has the same instabilityproperties as the D3M0 models.

Our analysis of all computed models shows that dur-ing the infall phase the MRI is either not possible or thetypical time scales involved are much larger (i.e. > 10 s)than the duration of the collapse itself. Therefore the in-stability can affect neither the dynamics nor the magneticfield strength in that phase. After core bounce, however,three regions are susceptible to develop the MRI: the re-gions surrounding the inner part of the PNS, the innerregion of the PNS itself (which is in approximate equilib-rium after core bounce), and the region in the vicinity ofthe propagating shock wave. Important differences appearwhen comparing microphysical and simplified models.

A general feature of the microphysical models is thepost-bounce convective overturn in regions surroundingthe inner part (see Sect. 6.2). This property, which is ex-

18 Cerda-Duran et al.: Relativistic magneto-rotational core collapse

15

10

0

5

20

0 5 10 15 20−1

0

1

2

3

log

τ MR

I

15

10

0

5

20

0 5 10 15 20−1

0

1

2

3

log

τ MR

I

Fig. 9. Time scales for the magneto-rotational instability in the magneto-rotational collapse models A1B3G5-D3M0 (left panel)and s20A1B5-D3M0 (right panel). Colour coded is the logarithm of τMRI for the fastest growing mode in ms. Arrows indicatethe velocity field. Axes labels are in units of km.

plained by the presence of regions where N2 < 0, is muchless prominent in the simplified models (except in modelA1B3G5). Thus, the presence of such convectively unsta-ble regions in the microphysical models enhances the ap-pearance of MRI just outside the inner region as com-pared to the simplified models (see Fig. 9). An exceptionis model s20A1B1-D3M0, where this MRI-unstable regiondisappears shortly after bounce. The time scale for thefastest growing mode in this region is about 10 ms, corre-sponding to a dynamical time scale. However, as alreadydiscussed in Sect. 6.2, it is not yet clear whether the ap-pearance of the strong convective overturn in the micro-physical models vanishes if a more accurate neutrino treat-ment is included or if an alternative microphysical EOS isused. Therefore, further investigations of this issue, whichwill also clarify the impact on the MRI, are essential.

In the inner region there are also differences betweenmicrophysical and simplified models. For all investigatedmicrophysical models, the inner region is always convec-tively unstable, which makes it MRI-unstable. In con-trast, not all simplified models satisfy the criterion for theMRI in the inner region (only models A1B3G3-D3M0 andA1B3G5-D3M0, the latter plotted in Fig. 9), and in onecase (model A3B3G5-D3M0), the instability is not pos-sible anymore after several 10 ms. In all cases where theMRI could appear in the inner region, the time scale forthe fastest growing mode is also dynamical, between 1 and10 ms.

The final region to consider is the one around the shockfront. In none of the simplified models the MRI conditionis fulfilled here, while in all microphysical models it is al-ways satisfied. In the latter case, the time scales involvedare larger than 1 s, and thus the MRI is not expectedto play an important role in these regions on dynamicaltime scales. The condition for the MRI to grow is fulfilledin this region because the value of N2 is small (thoughstill positive). Small, scattered regions behind the shock

front can also fulfill the MRI condition with time scalesof ∼ 10 ms, although it is difficult to establish if the in-stability will develop in those regions without performing“active” magnetic field simulations.

As a result of this analysis, for collapse progenitorswith a magnetic field smaller than 1010 G (hence includ-ing astrophysically more relevant initial values of 106 G),we infer that perturbations of the magnetic field are go-ing to grow exponentially on dynamical time scales andwill reach saturation in the unstable regions mentionedabove. However, the value of the magnetic field at whichsaturation appears is still unknown, which is the key is-sue in order to establish the effects of the MRI, if any, onthe dynamics. Nevertheless, even if the MRI were unableto considerably amplify the magnetic field, at late timesduring the evolution of the PNS it could still play a majorrole, provided other amplification mechanisms were ableto increase the magnetic field to significant larger values(see below). In such a situation the MRI could have an im-pact on the dynamics by transporting angular momentumoutwards and driving the PNS towards rigid rotation.

6.5.3. Dynamo mechanisms

The wind-up process of the magnetic field (Ω-dynamo)discussed before is a mechanism that works by transform-ing the poloidal magnetic field into a toroidal field and ex-tracting energy from differential rotation. In axisymmetrythis process amplifies the magnetic field linearly with timeas long as differential rotation exists. If the axisymmetrycondition is relaxed, however, a number of instabilities ofthe toroidal field can transform the toroidal magnetic fieldback into a poloidal magnetic field. This feedback then“closes” the dynamo process.

The first group of instabilities are those related to con-vective unstable regions, neutron-finger instabilities and,in general, turbulence. In these cases the α-effect is the one

Cerda-Duran et al.: Relativistic magneto-rotational core collapse 19

which closes the dynamo in the α-Ω-dynamo (Thompson& Duncan 1993). Computations of this effect (Bonannoet al. 2005) suggest that even for a rapidly rotating PNSwith a period around 1 ms (i.e. comparable to the mod-els presented here), the time scale for the growth of themagnetic field is ∼ 1 s. Therefore, this effect is probablynot important after core bounce on dynamical time scales.However, for larger time scales (i.e. several seconds), andif the MRI is not efficient enough, this mechanism willmost likely amplify the magnetic field, leading to mag-netic braking of the PNS within a few seconds.

There are also types of instabilities that can act instably stratified regions. Spruit (1999) has proposed theTayler instability (Tayler 1973) as a mechanism to closethe dynamo. This dynamo has been confirmed in numeri-cal simulations by Braithwaite (2006a,b). The growth rateof this kink-type instability is of the order of the Alfventime scale,

τT =2π

ΩA

(Ω ≪ ΩA), τT =2πΩ

Ω2A

(Ω ≫ ΩA), (41)

where ΩA = cA/R and R is the typical size of the regionconsidered. One of the features of this dynamo is that themagnetic field saturates as the growth rate reaches thewind-up growth rate, i.e. τT ≈ τΩ. The saturation valuecan be low enough not to flatten the rotational profiles. Ifwe consider the typical toroidal magnetic field at bounceto be 1013 G (as in the T3M0 models) with a typical den-sity in the PNS of ρ = 2 × 1014 g cm−3 and a typical sizeof the inner region of R ∼ 10 km, then the time scale forthe growth of the Tayler instability is strongly increasedby rotation:

τT ≈ 3

(

1010 G

B∗ 0

)2 (

R

10 km

)2 (

Ωc

1 ms−1

)

hr. (42)

This means that for a typical progenitor with a toroidalmagnetic field of B∗ 0

ϕ ∼ 1010 G, the Tayler instability isgoing to be very inefficient in the amplification of the mag-netic field. However, if the magnetic field grows by meansof other mechanisms, the importance of this dynamo effectcould also increase. Note that a growth by two orders ofmagnitude in the magnetic field lowers the time scale forthe instability to several seconds, making it comparableto the α-Ω-dynamo.

7. Conclusions

In this paper we have presented numerical simulations ofthe collapse of rotating magnetized stellar cores in theCFC approximation of general relativity, as well as testsassessing our numerical approach for solving the ideal gen-eral relativistic magneto-hydrodynamics (GRMHD) equa-tions.

As initial models we have set up (either fully or nearly)stationary configurations of weakly magnetized stars ingeneral relativity, with either toroidal or poloidal (or both)magnetic field components. We have used the “test” pas-sive field approximation for evolving these initial models,

for which the magnetic pressure in all cases considered isseveral orders of magnitude smaller than the fluid pres-sure.

We have performed tests to check the accuracy andconvergence properties for the GRMHD extension of ourcode. For magnetic field quantities we have found an orderof convergence above 1 in all of the performed tests. Theseresults are consistent with the second-order accuracy (inspace and time) of our numerical scheme, reduced to firstorder only at shocks and local extrema. The errors in allof the cases in which the theoretical solution is known arebelow 0.1%, except at shocks, which are correctly capturedwithin only few numerical cells thanks to the use of high-resolution shock-capturing schemes.

For the simulations of magnetized core collapse, wehave considered cases with magnetic fields which are ini-tially either purely poloidal (series D3M0), purely toroidal(series T3M0), or a combination of both. The D3M0 mod-els are a general relativistic extension of a subset of thecases evolved in fully coupled MHD by Obergaulingeret al. (2006a,b), who used a Newtonian formulation (ap-proximating general relativistic effects to some extent inthe latter work). One of our aims has been to comparethe dynamics and gravitational waveforms with their re-sults. No qualitative differences have been found in themodels studied, while quantitatively the strength of themagnetic field at bounce and after the collapse are consis-tently smaller in general relativity.

We have also compared simulations of models withimproved microphysics (employing a tabulated non-zerotemperature equation of state (EOS) and an approxi-mate but effective deleptonization scheme) with the sim-ple (though still widely used) analytic hybrid EOS. Theresults show that the microphysical models (i) lead toa more complex structure of the poloidal magnetic fielddue to convective motions surrounding the inner region ofthe PNS, and (ii) exhibit a wind-up of the magnetic field(Ω-dynamo) that is more efficient than in the simplifiedmodels for comparable rotation rates, which is due to thelarger compression of the poloidal component during thecollapse. We have shown that the latter issue is related tothe larger duration of the collapse until core bounce in themicrophysical models. Simplified models with such largecollapse times (and identical initial rotation profiles) willinevitably undergo multiple centrifugal bounces, a behav-ior that has recently shown to be an artifact of the neglectof microphysics (Dimmelmeier et al. 2007).

As we have adopted the passive field approximationand the investigated magnetic fields are weak in all phasesof the collapse, the waveforms of the gravitational radia-tion emitted by all our models are practically identical tothe corresponding ones in a purely hydrodynamical sim-ulation (Ott et al. 2006a; Dimmelmeier et al. 2007), withthe contribution due to magnetic fields being several or-ders of magnitude smaller than the total signal ampli-tude. However, if the MRI could become dominant forthe dynamics in the post-bounce phase, in a fully coupledGRMHD simulation we would expect a clear imprint of

20 Cerda-Duran et al.: Relativistic magneto-rotational core collapse

such an instability on the signal waveform. As expected,for the microphysical models we obtain gravitational wavesignals exclusively of Type I, as all other waveform types(in particular the Type II signals associated with multi-ple centrifugal bounces) are suppressed if more realisticmicrophysics is taken into account.

For astrophysically expected configurations of themagnetic field (Heger et al. 2005), where the initialtoroidal component is much larger than the poloidal one,we have obtained a topology of the magnetic field in thePNS that is purely toroidal due to the radial compres-sion of the initial toroidal component. In this case thetime scale for the Ω-dynamo is very long (several hours).For progenitors with stronger poloidal magnetic fields, wehave found that a core/shell structure is formed. Inside thecore, where nuclear density is exceeded, a mixed configu-ration of a poloidal and a toroidal magnetic field yields ahelicoidal configuration of the field lines. In the surround-ing shell (which extends several 10 km) the poloidal mag-netic field lines are wound up due to differential rotation(Ω-dynamo), and shortly after core bounce the magneticfield is dominated by the toroidal component. The growthtime scale for the toroidal component due to this processis, in the best case scenario, of several 100 ms.

We have also estimated the growth times for severalother instabilities that could appear if the passive fieldapproximation or the restriction to axisymmetry are re-moved. Among these the MRI is apparently the fastestgrowing instability, although it remains unclear if it is go-ing to amplify the magnetic field sufficiently (from theinitially weak field values) to affect the dynamics at all. Inaddition, we have found that the inclusion of microphysicscould enhance the MRI, since the regions surrounding theinner part of the PNS become convectively unstable, re-sulting in a growth time of ∼ 10 ms for the MRI. However,the influence of our simplified neutrino treatment or thechoice of the microphysical EOS must still be investigated.

In the event that the MRI were unable to sufficientlyamplify the magnetic field in the PNS (which is stillan open issue), the main amplification mechanism wouldprobably be the α-Ω-dynamo, which can amplify the mag-netic field to values where the magnetic energy is inequipartition with the rotational kinetic energy on a timescale of, at least, several seconds. The study of this effectis well beyond the goals of the work presented in this pa-per, since the required time scales are much longer thanthose affordable with current numerical hydrodynamicalcodes. Moreover, the underlying physics necessary to beincluded (like neutrino transport, diffusion, radiation, andcooling) is far more complex. However, in the light of theresults presented here, in which astrophysically expectedvalues for the magnetic field have been adopted, and ifthe MRI is ineffective, we can speculate about a scenarioin which after core bounce the magnetic field does notgrow significantly during one (or maybe several) seconds,and therefore differential rotation generated in the col-lapse could persist. This “one-second-window” would pro-vide sufficient time for several instabilities to develop in

the PNS. Such instabilities are promising sources of grav-itational waves.

The restriction to the passive magnetic field approx-imation in studying magneto-rotational core collapse ofweakly magnetized progenitors can be justified if the MRIis indeed inefficient, since none of the other estimatedmechanisms seem to be able to amplify the magneticfield significantly on dynamical time scales. Otherwise,an “active” magnetic field approach becomes necessary.However, it has to be stressed that the use of active mag-netic fields alone for core collapse simulations will prob-ably not be sufficient to model all the effects amplifyingthe magnetic field, since the numerical resolution neededto correctly describe them (probably less than 10 m) isnot affordable in current numerical simulations. In addi-tion most of the prospectively relevent effects have to beinvestigated in three dimensions, which makes the compu-tational task even more challenging.

Acknowledgements. This research has been supported by theSpanish Ministerio de Educacion y Ciencia (grant AYA2004-08067-C03-01), by the DFG (SFB/Transregio 7 and SFB 375),by the DAAD and IKY (IKYDA German–Greek researchtravel grant), and by a Marie Curie Intra-European Fellowshipwithin the 6th European Community Framework Programme(IEF 040464). It is a pleasure to thank C. D. Ott, A. Marek,and H.-T. Janka for their contributions related to the im-proved microphysics, L. Anton for many useful discussions insetting up the numerical scheme for the magnetic field evolu-tion, M. Obergaulinger for his help on understanding the MRI,and E. Muller as well as N. Stergioulas for useful comments.

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Appendix A: Code tests

Here we discuss several tests we have designed in order tocheck the accuracy of our numerical code when solving theinduction equation with the numerical methods describedin this paper (see also Cerda-Duran & Font 2006). The“toroidal test” is set up for assessing the ability of the codeto maintain various magnetic field configurations in equi-librium (labelled TTA and TTB) and to correctly com-pute the amplification of the toroidal magnetic field as itis wound up by a rotating fluid (TTC). On the other hand,the “poloidal test” (PT) is designed to check whetherthe code can correctly compute the compression of thepoloidal magnetic field in a spherical collapse. Finally, thestrong spherical explosion test checks that the code is ableto handle the presence of radial shocks. We refer the in-terested reader to Cerda-Duran & Font (2006) for detailson the setup of the toroidal and poloidal tests as well asthe diagnostics we use to compute the errors and order ofconvergence of our numerical schemes.

A.1. Toroidal tests

Fig. A.1 shows the global error σ in the toroidal mag-netic field Bϕ for the three tests TTA, TTB, and TTCagainst 1/f and the corresponding fits to a power law.Here f denotes the factor which specifies the increase inresolution from a coarse reference grid (see Cerda-Duran& Font 2006, for details). The resulting convergence or-der of each numerical scheme (minmod, MC, and PHM)as well as the errors for the highest resolution grid canbe found in Table A.1. Our results show that (i) the or-der of convergence and the error is almost independentof the cell reconstruction scheme employed, (ii) the orderof convergence for the TTC test is smaller than for theTTA and TTB tests, and (iii) the order of convergence forthe tests TTA and TTB is N > 2, and hence higher thanthe theoretical expectation (which is second order, sinceit is limited by the order of the time discretization, forwhich we use a conservative, second order Runge–Kuttascheme).

0.1 11/f

10-5

10-4

10-3

σ (B

ϕ )

0.1 11/f

10-6

10-5

10-4

σ (B

ϕ )

Fig.A.1. Global error σ in the toroidal magnetic field Bϕ aftera time evolution of 1 ms as a function of 1/f for a sequence ofmodels with grid resolutions 80× 10 (f = 1), 160× 20 (f = 2)and 320 × 40 (f = 4). The top panel shows the error for theTTC test using different reconstruction schemes and the corre-sponding best fits to a power law: minmod (open circles, dottedline), MC (filled squares, dashed line), and PHM (filled circles,solid line). The bottom panel shows the error and respectivefits using the PHM reconstruction for the TTA test (open cir-cles, dotted line), TTB (filled circles, dashed line) and TTC(open squares, solid line).

Table A.1. Convergence order N for the tests performed(TTA, TTB, TTC, and PT) and for different reconstructionprocedures (minmod, MC, and PHM). The error σ320×40 forthe higher resolution grid is also given.

Test Reconstruction scheme N σ320×40

TTA minmod 2.45 1.2 × 10−6

TTA MC 2.16 2.4 × 10−6

TTA PHM 2.46 2.1 × 10−6

TTB minmod 2.64 7.7 × 10−6

TTB MC 2.53 1.2 × 10−5

TTC PHM 2.85 0.5 × 10−6

TTC minmod 1.38 4.0 × 10−5

TTC MC 1.48 7.0 × 10−5

TTC PHM 1.39 3.5 × 10−5

PT minmod 1.41 8.3 × 10−4

PT MC 1.11 8.6 × 10−4

PT PHM 1.17 8.6 × 10−4

The numbers reported in Table A.1 demonstate thatwe obtain similar results in all three tests for linear recon-

Cerda-Duran et al.: Relativistic magneto-rotational core collapse 23

30 40 60502010 70

3.02.52.01.51.00.50.0

2

8

6

4

Fig.A.2. Local order of convergence (color coded) for the TTAtest after a total time evolution of 1 ms. White color is usedfor values ≥ 3.0. The horizontal and vertical axes represent thenumber of cells of the reference grid in the radial and angulardirection, respectively.

0 10 20t

10-6

10-5

10-4

10-3

10-2

σ (r

D* /B

*θ )

minmod 320 × 40minmod 160 × 20minmod 80 × 10MC 320 × 30MC 160 × 20MC 80 × 10PHM 340 × 40PHM 160 × 20PHM 80 × 10

Fig.A.3. Global error on r D∗/B∗ θ for the poloidal test (PT)as a function of time for three different grid resolutions: 80×10,160 × 20, and 320 × 40, and for four different reconstructionschemes (minmod, MC, and PHM).

struction schemes (minmod and MC) and for the thirdorder reconstruction scheme (PHM), as the order of thescheme is limited by the second order discretization intime and by the linear interpolation of the cell-centeredmagnetic fluxes (which is a consequence of using a stag-gered grid in the flux-CT scheme for the magnetic field).To understand these results we note that in test TTC thereis a component of the magnetic field, B∗ϕ, which growslinearly in time, while in tests TTA and TTB no compo-nents evolve. Hence, the order of convergence for the latteris higher than for test TTC. This can be explained by in-vestigating the local order of convergence, i.e. the orderobtained when computing the error σij in each numericalcell instead of the global error σ. The results for test TTAare displayed in Fig. A.2 (similar plots can be obtainedfor the other two cases). At some particular grid zonesthe order of convergence is larger than two, while at mostlocations it remains around two.

A.2. Poloidal test

As mentioned before the setup and specifications of thepoloidal test are described in detail in Cerda-Duran &Font (2006). Here we simply focus on showing the com-parison and performance of the various numerical schemesemployed in our simulations. (Note that in Cerda-Duran& Font 2006, only the minmod scheme was assessed.)Fig. A.3 shows the evolution of the error in the quan-

5

4

0

2

6

3

1

0 2 3 5 61 4

4

6

8

10

2

W

Fig.A.4. Spherical explosion test at t = 4. The Lorentz factorW is color coded, and magnetic field lines are overplotted.

tity r D∗/B∗ θ at the equatorial plane (which is a quan-tity that should not change with time with respect to aLagrangian coordinate system) during the spherical col-lapse of a 4/3-polytrope for different r, θ grid resolutions(80 × 10, 160 × 20, and 320 × 40), equally-spaced in theangular direction and logarithmically spaced in the radialdirection. Table A.1 gives again numbers for the error andthe order of convergence of the various schemes computedat the end of the simulation (t = 20 ms). In all cases theerrors are below 1%, even for the coarsest grid, and theorder of convergence is higher than 1 (the presence of lo-cal extrema in the radial profiles of some hydrodynamicalvariables explains the reduction of the theoretical orderas a built-in feature of total-variation diminishing numer-ical schemes). Comparisons between the HLL approximateRiemann solver and the KT symmetric scheme yield al-most identical results (in agreement with Lucas-Serranoet al. 2004; Shibata & Font 2005; Anton et al. 2006).

A.3. Strong spherical explosion

Explosions are among the most demanding tests for multi-dimensional codes as they show the ability of numericalschemes to handle shocks. Since the majority of existingMHD codes are written in Cartesian coordinates, the mostcommon test is the cylindrical explosion. For relativisticMHD codes the setup of Komissarov (1999) for this testhas been used by other authors (Del Zanna et al. 2003;Leismann et al. 2005) to compare different codes. However,in spherical coordinates it is not possible to impose thesymmetries needed for this test. The most natural choiceis thus the spherical explosion. Kossl et al. (1990) per-formed this test in the case of Newtonian MHD. To ourknowledge, no spherical explosions tests have been per-formed in relativistic MHD. Therefore, we have designedsuch a spherical explosion test in which the initial jump

24 Cerda-Duran et al.: Relativistic magneto-rotational core collapse

conditions in the variables are the same as for the testby Komissarov (1999). In this way a relativistic shock isformed which does not occur in the Newtonian case ofKossl et al. (1990).

Our test setup consists of an initial explosion zone withP = 1.0 and ρ = 10−2 for r < 0.8, surrounded by an am-bient gas with P = 3× 10−5 and ρ = 10−4. The explosionregion joins the ambient medium by matching an expo-nential decline in a transition region region 0.8 < r < 1.0.The velocities are initially zero, and the magnetic fieldis homogeneous and parallel to the symmetry axis. Thebackground spacetime is considered to be flat. The initaldata are evolved using an ideal gas EOS with adiabaticindex γ = 4/3. We use an evenly spaced grid with a max-imum radius of r = 6.0. We perform the test for threeresolutions (80× 10, 160× 20, and 320× 40) for all recon-struction schemes.

Fig. A.4 shows the Lorentz factor W at t = 4.0. Astrong spherical shock has formed, propagating close tothe speed of light, and as a consequence the magnetic fieldlines are compressed in the direction perpendicular to theaxis. The results for this test are qualitatively compara-ble to the weakly magnetized case in Komissarov (1999).Fig. A.5 shows radial profiles for P and Bθ along the equa-torial plane at the end of the simulation, using variousreconstruction schemes. These plots are qualitatively sim-ilar to those of the cylindrical explossion (see e.g. Fig. B.4in Leismann et al. 2005). All numerical schemes exhibitfirst order convergence with increasing resolution, as is ex-pected to happen at shocks. The MC and PHM schemesyield very similar results, while minmod shows slightlylower values.

Appendix B: Estimation of the growth rates of

the Ω-dynamo

To compute the characteristic time scales on which the Ω-dynamo mechanism amplifies the magnetic field one hasto study how the wind-up proceeds. Let us consider a sta-tionary rotating configuration with no meridional flows,v∗ r = v∗ θ = 0 and v∗ϕ = Ω∗(r, θ) r sin θ, where Ω∗(r, θ)stands for the rotation law. Under these conditions andin the passive field approximation, the induction equationcan be integrated analytically. The solution shows that thepoloidal component of the magnetic field remains constantand the toroidal component grows linearly with time as

B∗ϕ(t) = B∗ϕ(t = 0) + t r sin θ B∗ · ∇Ω∗. (B.1)

This equation specifies the toroidal magnetic field at anygiven time, provided that the poloidal component is con-stant and the angular velocity profile is fixed. The rotationprofiles of the final PNS can be approximated in all ourmodels by the rotation law (25) (Villain et al. 2004). Sincethe angular velocity (at least in the Newtonian limit (26)which is good enough for this estimate) depends only onthe distance to the axis, Eq. (B.1) can be written as

B∗ϕ(t) = B∗ 0

ϕ + B∗

∂Ω∗

∂t ≈ B∗

∂Ω∗

∂t, (B.2)

0 0.2 0.4 0.6 0.8 1r − c t

0.000

0.001

0.002

P

0 0.2 0.4 0.6 0.8 1r − c t

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Bθ

Fig.A.5. Results for the spherical test explosion at t = 4. Theplots show profiles for the fluid pressure P (top panel) and themagnetic field component Bθ (bottom panel) along the equa-torial plane using different reconstruction schemes: minmod(dotted line), MC (dashed line), and PHM (solid line). Thegrid resolution is 320 × 40.

where the latter expression holds for times t ≫B∗ 0

ϕ /B∗ (∂Ω∗/∂)

−1, which is ∼ 1 ms in our simu-

lations. If we use this expression to compute the mag-netic energy, and the fact that the maximum value of∂Ω∗/∂|max = −Ω∗

c/2, then

Emag ϕ =

∫

d3xB∗

2

2

(

∂Ω∗

∂

)2

t2 ≤ Emag Ω∗

c2

4t2.

(B.3)

Therefore, an estimate for the upper limit of the amplifi-cation of the magnetic energy parameter is

βmag ≈ βϕ ≤ βΩ∗

c2

4t2 ≤ βpolo

Ω∗c2

4t2 =

(

t

τΩ

)2

, (B.4)

where we have defined the time scale for amplification ofthe magnetic field by the Ω-dynamo as

τΩ =2

Ω∗c

√

βpolo

. (B.5)

This gives us the characteristic time scale in which βmag

reaches a value of 1; therefore, the saturation time tsatshould be a fraction of this time. As the Ω-dynamo oper-ates by transforming rotational energy into magnetic en-ergy, the maximum energy can be extracted by the mag-netic field is the one that is contained in the differential

Cerda-Duran et al.: Relativistic magneto-rotational core collapse 25

rotation of the core. In accordance with numerical simu-lations using strong magnetic fields (Obergaulinger et al.2006b) we estimate this amount to be 10% of the total ro-tational energy. We also assume that the evolution of themagnetic energy parameter is given by Eq. (B.4) and thatthe energy is conserved, i.e. βrot(t) = βrot(t0) − βmag(t).