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5Astronomy & Astrophysics manuscript no. paper November 6, 2018(DOI: will be inserted by hand later)

Exploring the relativistic regime with Newtonian hydrodynamics:

An improved effective gravitational potential for supernova

simulations

A. Marek, H. Dimmelmeier, H.-Th. Janka, E. Muller, and R. Buras

Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Straße 1, 85741 Garching, Germany

Received date / Accepted date

Abstract. We investigate the possibility approximating relativistic effects in hydrodynamical simulations of stellarcore collapse and post-bounce evolution by using a modified gravitational potential in an otherwise standardNewtonian hydrodynamic code. Different modifications of a previously introduced effective relativistic potentialare discussed. Corresponding hydrostatic solutions are compared with solutions of the TOV equations, and hy-drodynamic simulations with two different codes are compared with fully relativistic results. One code is appliedfor one- and two-dimensional calculations with a simple equation of state, and employs either the modified ef-fective relativistic potential in a Newtonian framework or solves the general relativistic field equations under theassumption of the conformal flatness condition (CFC) for the three-metric. The second code allows for full-scalesupernova runs including a microphysical equation of state and neutrino transport based on the solution of theBoltzmann equation and its moments equations. We present prescriptions for the effective relativistic potentialfor self-gravitating fluids to be used in Newtonian codes, which produce excellent agreement with fully rela-tivistic solutions in spherical symmetry, leading to significant improvements compared to previously publishedapproximations. Moreover, they also approximate qualitatively well relativistic solutions for models with rotation.

Key words. gravitation – hydrodynamics – methods: numerical – relativity – stars:supernovae:general

1. Introduction

It is a well known fact that gravity plays an importantrole during all stages of a core collapse supernova. Gravityis the driving force that at the end of the life of massivestars overcomes the pressure forces and causes the collapseof the stellar core. Furthermore, the subsequent super-nova explosion results from the fact that various processestap the enormous amount of gravitational binding energyreleased during the formation of the proto-neutron star.General relativistic effects are important for this processand cannot be neglected in quantitative models becauseof the increasing compactness of the proto-neutron star.Therefore, it is important to include a proper treatmentof general relativity or an appropriate approximation inthe numerical codes that one uses to study core collapsesupernovae.

Recently Liebendorfer et al. (2005) performed a com-parison of the results obtained with the supernova simula-tion codes Vertex and Agile-BoltzTran which bothsolve the Boltzmann transport equation for neutrinos.The Vertex code (see Rampp & Janka 2002) is based

Send offprint requests to: H.-Th. Janka,e-mail: thj@mpa-garching.mpg.de

on the Newtonian hydrodynamics code Prometheus

(Fryxell et al. 1990) and utilizes a generalised potential toapproximate relativistic gravity. The Agile-BoltzTran

code of the Oak Ridge-Basel group (Liebendorfer et al.2001, 2002, 2004, 2005) is a fully relativistic (1D) hydro-dynamics code. The comparison showed that both codesproduce qualitatively very similar results except for somesmall (but growing) quantitative differences occurring inthe late post-bounce evolution. Inspired by this compar-ison we explored improvements of the effective relativis-tic potential used by Rampp & Janka (2002) in order toachieve an even better agreement than that reported byLiebendorfer et al. (2005). To this end we tested differentvariants of approximations to relativistic gravity employ-ing two simulation codes.

On the one hand purely hydrodynamic simula-tions were performed with the code CoCoNuT ofDimmelmeier et al. (2002a, 2005) assuming spherical oraxial symmetry. This code optionally either uses aNewtonian (or alternatively an effective relativistic) grav-itational potential in a Newtonian treatment of the hydro-dynamic equations or solves the general relativistic equa-tions of fluid dynamics and the relativistic field equations.The latter are formulated under the assumption of the

2 A. Marek et al.: An improved effective potential for supernova simulations

conformal flatness condition (CFC) for the three-metric,also known as the Isenberg–Wilson–Mathews approxima-tion (Isenberg 1978; Wilson et al. 1996), which is identi-cal to solving the exact general relativistic equations inspherical symmetry. This allows for a direct comparisonof different effective relativistic potentials with a fully rel-ativistic treatment in both spherically symmetric and ax-isymmetric simulations using the same code.

On the other hand, we used the computationally ex-pensive Vertex code (Rampp & Janka 2002) for spheri-cally symmetric supernova simulations including neutrinotransport and a microphysical equation of state (EoS).The results of these calculations performed with differenteffective relativistic potentials are compared with thoseobtained with the general relativistic Agile-BoltzTran

code of the Oak Ridge-Basel group (cf. Liebendorfer et al.2005).

In this paper we discuss the results of our test calcu-lations and present improved effective relativistic poten-tials which are simple to implement without modifyingthe Newtonian equations of hydrodynamics in an exist-ing code, and which approximate the results of generalrelativistic simulations very well.

The paper is organised as follows: In Sect. 2 we describethe effective relativistic potentials used in this investiga-tion. In Sect. 3 we briefly discuss the most important fea-tures of the numerical codes we used for our hydrodynamicsimulations. In Sect. 4 we present the results obtained byapplying the effective relativistic potentials in sphericallysymmetric simulations of supernova core collapse and toneutron star models, while Sect. 5 is devoted to a dis-cussion of the multi-dimensional simulations of rotationalsupernova core collapse. Finally, in Sect. 6 we summariseour findings and draw some conclusions.

Throughout the article, we use geometrised units withc = G = 1.

2. Effective relativistic potential

Approximating the effects of general relativistic gravity ina Newtonian hydrodynamics code may be attempted byusing an effective relativistic gravitational potential Φeff

which mimics the deeper gravitational well of the rela-tivistic case. In the following Sects. 2.1 and 2.2, several ofthese effective relativistic potentials will be discussed.

2.1. TOV potential for a self-gravitating fluid

For a self-gravitating fluid it is desirable that aneffective relativistic potential reproduces the solutionof hydrostatic equilibrium according to the Tolman–Oppenheimer–Volkoff (TOV) equation. With this require-ment in mind and comparing the relativistic equation ofmotion (cf. van Riper 1979; Baron et al. 1989) with itsNewtonian analogon, Rampp & Janka (2002) rearrangedthe relativistic terms into an effective relativistic poten-tial (see Kippenhahn & Weigert 1990, for the hydrostatic,neutrino-less case).

Thus for spherically symmetric simulations using aNewtonian hydrodynamics code the idea is to replace theNewtonian gravitational potential

Φ(r) = −4π

∫ ∞

0

dr′ r′2ρ

|r − r′| (1)

by the TOV potential

ΦTOV(r) = −4π

∫ ∞

r

dr′

r′2

(mTOV

4π+ r′3(P + pν)

)

× 1

Γ2

(ρ+ e+ P

ρ

), (2)

to obtain the effective relativistic potential Φeff as

Φeff = ΦTOV. (3)

Here ρ is the rest-mass density, e = ρǫ is the internalenergy density with ǫ being the specific internal energy,and P is the gas pressure. The TOV mass is given by

mTOV(r) = 4π

∫ r

0

dr′ r′2(ρ+ e+ E +

vF

Γ

), (4)

where pν , E, and F are the neutrino pressure, the neu-trino energy density, and the neutrino flux, respectively(Baron et al. 1989; Rampp & Janka 2002).

The fluid velocity v is identified with the local radialvelocity calculated by the Newtonian code and the metricfunction Γ is given by

Γ =

√1 + v2 − 2mTOV

r. (5)

The velocity-dependent terms were added for a closermatch with the general relativistic form of the equation ofmotion (van Riper 1979; Baron et al. 1989). In the treat-ment of neutrino transport general relativistic redshiftand time dilation effects are included, but for reasons ofconsistency with the Newtonian hydrodynamics part ofthe code the distinction between coordinate and properradius is ignored in the relativistic transport equations(for details, see Sect. 3.7.2 of Rampp & Janka 2002). Thequality of this approach was ascertained by a comparisonwith fully relativistic calculations (Rampp & Janka 2002;Liebendorfer et al. 2005).

In order to calculate the effective relativistic poten-tial for multi-dimensional flows we substitute the “spheri-cal contribution”Φ(r) to the multi-dimensional Newtoniangravitational potential

Φ(r, θ, ϕ) = −∫

V

dr′ dθ′ dϕ′ r′2 sin θ′ρ

|r − r′| (6)

by the TOV potentialΦTOV:

Φeff = Φ−Φ +ΦTOV. (7)

Here Φ(r) and ΦTOV are calculated according to Eqs. (1)and (2), respectively, however with the hydrodynamicquantities ρ, e, P , v and the neutrino quantities E, F , pνbeing replaced by their corresponding angularly averagedvalues. Note that v here refers to the radial component ofthe velocity, only.

A. Marek et al.: An improved effective potential for supernova simulations 3

2.2. Modifications of the TOV potential

In a recent comparison Liebendorfer et al. (2005) foundthat gravity as described by the TOV potential in Eq. (3)overrates the relativistic effects, because in combinationwith Newtonian kinematics it tends to overestimate theinfall velocities and to underestimate the flow inertia inthe pre-shock region. Thus, supposedly via the nonlineardependence of Φeff on e and P the compactness of theproto-neutron star is overestimated, with this tendencyincreasing at later times after core bounce. Consequently,the neutrino luminosities and the mean energies of theemitted neutrinos are larger than in the correspondingrelativistic simulation.

In order to reduce these discrepancies – without sac-rificing the simplicity of Newtonian dynamics – we testedseveral modifications of the TOV potential, Eqs. (2),which all act to weaken it. In particular, we studied thefollowing variations:

Case A: In the integrand of Eq. (4) a factor Γ, Eq. (5)is added. Since Γ < 1 this reduces the gravitationalTOV mass used in the potential.

Case B: In Eq. (4) the internal gas energy density andthe neutrino terms are set to zero, e = E = F = 0,which again decreases the gravitational TOV mass.

Case C: In Eq. (2) the internal gas energy is set to zero,e = 0, which directly weakens the TOV potential.

Case D: In the equation for the TOV potential, Eq. (2),mTOV is replaced by 1

2(mTOV+mg). Here a Newtonian

gravitational mass is defined as mg = mr − mb withthe rest mass mr = 4π

∫ r

0dr′ r′2ρ and the mass equiv-

alent of the binding energy mb = 2π|∫ r

0dr′ r′2ρΦ|. As

mg < mTOV, the strength of the potential is reduced.

Case E: Both in the equation for the TOV potential,Eq. (2), and the equation for the TOV mass, Eq. (4),we set e = 0.

Case F: In the equation for the TOV potential, Eq. (2),we set Γ = 1. As Γ < 1 otherwise, this weakens thepotential.

Case G: In the expression for Γ, Eq. (5), the velocity isset to zero, v = 0. Hence, Γ−2 increases in Eq. (2). Thismodification is used to also test a potential which iseven stronger than the unmodified TOV potential.

In addition to these cases with a modified version of theTOV potential, we use the following notations:

Case N: This denotes the purely Newtonian runs with“regular” Newtonian potential.

Case R: This is the “reference” case with the TOVpotential as defined by Eq. (2).

Case GR: This case refers to fully relativistic simula-tions with either the code CoCoNuT or the Agile-

BoltzTran neutrino radiation-hydrodynamics codeof the Oak Ridge-Basel collaboration.

Note that setting the internal energy density e to zeroin Case B is unambiguous when a simple EoS is used andthe particle rest masses are conserved. In general, how-ever, particles can be created and destroyed, or boundstates can be formed (e.g., in pair annihilation processesor nuclear reactions, respectively). Then only the sum ofthe rest mass energy and internal energy per nucleon –both appear in Eqs. (2, 4) only combined in form of the“relativistic energy” per unit of mass, (ρ + e)/ρ – is welldefined, but not the individual parts. Therefore there ex-ists ambiguity with respect to which contribution to theenergy is set to zero. In order to assess a possible sensitiv-ity of the core collapse results to this ambiguity, we triedtwo different variants of Case B in our Vertex simula-tions with microphysical EoS. On the one hand we usede = E = F = 0 in Eq. (4), with e being the internal energydensity plus an energy normalization given by the EoS ofLattimer & Swesty (1991), e = (ρ + e) − ρ (mn −∆)/mu

(where mu = 1.66× 10−24 g is the atomic mass unit, mn

the neutron rest mass, and ∆ = 8.8MeV). On the otherhand we tested e∗ = E = F = 0 with e∗ being e withoutthis energy normalization, i.e. e∗ = (ρ + e) − ρmn/mu.Nucleons are then assumed to contribute to the TOVmass, Eq. (4), with the vacuum rest mass of the neutron,increasing the mass integral and reducing Γ (Eq. (5) rela-tive to the first case, thus making the effective relativisticpotential a bit stronger again. In order to compensate forthis we also set pν = 0 in the TOV potential, Eq. (2).Both variants are found to yield extremely similar resultsand we therefore will discuss only one of them (the firstvariant) as Case B for the Vertex simulations.

Ideally, a Newtonian simulation with an effective rela-tivistic potential not only yields a solution of the TOVstructure equations for an equilibrium state (as doesCase R), but in addition closely reproduces the resultsfrom a relativistic simulation (Case GR) during a dynamicevolution. Applying the modifications of the TOV poten-tial listed above, we find that Cases A to D yield improvedresults as compared to Case R, while Cases E to G eitherweaken the potential too much or are very close to Case R.These findings are detailed in Sect. 4 and 5.

2.3. Theoretical motivation

There are (at least) two basic requirements which ap-pear desirable for an effective relativistic potential in aNewtonian simulation. Firstly, the far field limit of thefully relativistic treatment should be approximated rea-sonably well in order to follow the long-term accretion ofthe neutron star and the associated growth of its baryonicmass. Secondly, the hydrostatic structure of the neutronstar should well fit the solution of the TOV equations.

The second point will be discussed in detail in Sect. 4.1.A closer consideration of the first point suggests the mod-ified effective relativistic potential of Case A as promis-ing, and in fact it turns out to be the most preferablechoice concerning consistency and quality of the results.

4 A. Marek et al.: An improved effective potential for supernova simulations

The other cases listed in Sect. 2.2 are mostly ad hoc mod-ifications of the original effective relativistic potential ofEqs. (2)–(4) (Case R) with the aim to reduce its strength,which was found to overestimate the effects of gravitycompared to fully relativistic simulations in previous work(Liebendorfer et al. 2005). These cases are also discussedhere for reasons of comparison and completeness.

In Eq. (4) the hydrodynamic quantities (like rest-massdensity ρ plus extra terms) are integrated over volume. Inthe Newtonian treatment there is no distinction betweencoordinate volume and local proper volume. Performingthe integral of Eq. (4) therefore leads to a mass – usedas the mass which produces the gravitational potentialin Eqs. (2) and (3) – which is larger than the baryonicmass, mb = 4π

∫dr′ r′2ρ. In particular, it is also larger

than the gravitational mass in a consistent relativistictreatment, which is the volume integral of the total en-ergy density and includes the negative gravitational po-tential energy of the compact object. The latter reducesthe gravitational mass relative to the baryonic mass bythe gravitational binding energy of the star (see, e.g.,Shapiro & Teukolsky 1983, page 125 for a correspond-ing discussion). Therefore, the effective relativistic poten-tial introduced by Rampp & Janka (2002) [our Case R,Eqs. (2–4)] cannot properly reproduce the far field limitof the relativistic case and thus overestimates the effectsof gravity. This particularly applies to the infall velocitiesof the stellar gas ahead of the supernova shock, as shownin Liebendorfer et al. (2005).

Introducing an extra factor Γ in the integral of Eq. (4)for the TOV mass is motivated by the following consider-ations (where for reasons of simplicity contributions fromneutrinos, though important, are neglected and sphericalsymmetry is assumed): In the relativistic treatment thetotal (gravitating) mass of the star is given as

mg = 4π

∫ ∞

0

dr′ r′2(ρ+ e), (8)

where dV ′ = 4πdr′ r′2 is the coordinate volume element,whereas the baryonic mass is

mb = 4π

∫ ∞

0

dr′ r′2Γ−1ρ, (9)

with dV ′ = 4πdr′r′2Γ−1 being the local proper volumeelement. The factor Γ−1 > 1 in the integrand of mb thusensures that mb > mg. The integral for mg can also bewritten as

mg = 4π

∫ ∞

0

dr′ r′2Γ

Γ(ρ+ e) =

∫ ∞

0

dV ′ Γ (ρ+ e). (10)

Since in Newtonian hydrodynamics no distinction is madebetween coordinate and proper volumes, one may identifydV ′ ≡ dV ′, consistent with the rest of our Newtoniancode. This leaves the additional factor Γ in the integrandof Eq. (10), leading to a redefined TOV mass used forcomputing the effective relativistic potential in Case A,

mTOV(r) = 4π

∫ r

0

dr′ r′2Γ

(ρ+ e+ E +

vF

Γ

). (11)

The fact that a factor Γ−1 in the volume integral estab-lishes the relation between gravitating mass, Eq. (8), andbaryonic mass, Eq. (9), in the relativistic case suggeststhat the factor Γ < 1 in Eq. (11) might lead to a suitablereduction of the overestimated effective potential that re-sults when the original TOV mass of Eq. (4) is used inEqs. (2) and (5). Indeed, a comparison of the integral ofEq. (11) for large r with the rest-mass energy of a neutronstar reduced by its binding energy at time t (computed

from the emitted neutrino energy,∫ t

0dt′ Lν(t

′) with Lν

being the neutrino luminosity) reveals very good agree-ment.

The arguments given above only provide a heuristicjustification for the manipulation of the TOV potentialproposed in Case A. A deeper theoretical understandingand more rigorous analytical analysis of its consequencesand implications is certainly desirable, but beyond thescope of the present paper. We plan to return to this ques-tion in future work.

3. Numerical methods

In the following we briefly introduce the two numericalcodes used for the simulations presented in this work. Thecodes are based on state-of-the-art numerical methods forhydrodynamics and neutrino transport coupled to grav-ity, and were used to simulate stellar core collapse andsupernova explosions.

3.1. Hydrodynamic code without transport

The purely hydrodynamic simulations are performed withthe code CoCoNuT developed by Dimmelmeier et al.(2002a,b) with a metric solver based on spectral meth-ods as described in Dimmelmeier et al. (2005). The codeoptionally uses either a Newtonian gravitational potential(or an effective relativistic potential, see Sect. 2), or thegeneral relativistic field equations for a curved spacetimein the ADM 3+1-split under the assumption of the con-formal flatness condition (CFC) for the three-metric. The(Newtonian or general relativistic) hydrodynamic equa-tions are consistently formulated in conservation form,and are solved by high-resolution shock-capturing schemesbased upon state-of-the-art Riemann solvers and third-order cell-reconstruction procedures. Neutrino transportis not included in the code. A simple hybrid ideal gas EoSis used that consists of a polytropic contribution describ-ing the degenerate electron pressure and (at supranucleardensities) the pressure due to repulsive nuclear forces, anda thermal contribution which accounts for the heating ofthe matter by shocks:

P = Pp + Pth, (12)

where

Pp = Kργ , Pth = ρǫth(γth − 1), (13)

and ǫth = ǫ−ǫp. The polytropic specific internal energy ǫpis determined from Pp by the ideal gas relation in combi-

A. Marek et al.: An improved effective potential for supernova simulations 5

nation with continuity conditions in the case of a discon-tinuous γ. In that case, the polytropic constant K also hasto be adjusted (for more details, see Dimmelmeier et al.2002a; Janka et al. 1993).

The CoCoNuT code utilizes Eulerian spherical coor-dinates r, θ, ϕ, and thus axially or spherically symmetricconfigurations can be easily simulated. For the core col-lapse simulations discussed in Sects. 4.1 and 5, the finitedifference grid consists of 200 logarithmically spaced ra-dial grid points with a central resolution of 500 m. A smallpart of the grid covers an artificial low-density atmosphereextending beyond the core’s outer boundary. The spectralgrid of the metric solver is split into 6 radial domains with33 collocation points each. In order to be able to accu-rately follow the dynamics, the domain boundaries adap-tively contract towards the centre during the collapse, asdescribed in Dimmelmeier et al. (2005). For the migrationtest (Sect. 4.3) the finite difference grid consists of 250 log-arithmically spaced radial grid points, of which 60 and 190cover the neutron star and the atmosphere, respectively.For the two-dimensional simulations of rotational core col-lapse (Sect. 5) the finite difference and spectral grids areextended by 30 and 17 equidistant angular grid points,respectively.

Even when using spectral methods the calculationof the spacetime metric is computationally expensive.Hence, in the relativistic spherical (rotational) core col-lapse simulations the metric is updated only once ev-ery 10/1/50 (100/10/50) hydrodynamic time steps be-fore/during/after core bounce, and extrapolated in be-tween. In the migration test the update is performed every10th step during the entire evolution. The numerical ad-equacy of this procedure is tested and discussed in detailin Dimmelmeier et al. (2002a).

The quality of the CFC approximation has been testedcomprehensively in the context of supernova core col-lapse and for neutron star models (Cook et al. 1996;Dimmelmeier et al. 2002a; Shibata & Sekiguchi 2004;Dimmelmeier et al. 2005; Cerda-Duran et al. 2004). Forsingle stellar configurations the CFC approximation isvery accurate as long as neither the compactness of thestar is extremely relativistic nor its rotation is too fast anddifferential. Both criteria are well fulfilled during super-nova core collapse. Concerning core collapse simulationsof rapidly and strongly differentially rotating configura-tions at very high densities, we recently found that theCFC approximation yields excellent agreement with for-mulations solving the exact spacetime metric even in thisextreme regime (Ott et al. 2005).

As the three-metric of any spherically symmetricspacetime can be written in a conformally flat way, thespherical CFC simulations presented in Sect. 4.1 and 4.3involve no approximation, i.e., the exact Einstein equa-tions are solved in these cases.

3.2. Hydrodynamic code with neutrino transport

The Vertex neutrino-hydrodynamics code solves theconservation equations of Newtonian hydrodynamics intheir Eulerian and conservative form using the PPM-basedPrometheus code (Fryxell et al. 1990). The neutrinotransport is treated by determining iteratively a solutionof the coupled set of moments equations and Boltzmannequation, achieving closure by a variable Eddington factormethod (Rampp & Janka 2002). See the latter referencealso for details about the coupling between the hydro-dynamics and neutrino transport parts of the Vertex

code. The models presented in this paper are computedwith the equation of state of Lattimer & Swesty (1991),in agreement with the choice of input physics in thework of Liebendorfer et al. (2005), where results obtainedwith the Vertex code were compared with those fromthe Newtonian and fully relativistic calculations with theAgile-BoltzTran code of the Oak Ridge-Basel collab-oration.

In order to compare the results presented here withthose of the calculations of Liebendorfer et al. (2005)we used the same set of neutrino interaction rates aspicked for Model G15 in Liebendorfer et al. (2005), andexactly the same parameters for the numerical setup(e.g., the grids for hydrodynamics and neutrino trans-port). Information about this setup can be found inLiebendorfer et al. (2005). The initial model for our cal-culations is the 15M⊙ progenitor model “s15s7b2” fromWoosley & Weaver (1995).

Since solving the neutrino transport problem is compu-tationally quite expensive we performed calculations onlyfor Cases A, B, and F (as defined in Sect. 2.2) with theVertex code. The quality of these results is then com-pared to the fully relativistic treatment of the Agile-

BoltzTran code.

3.3. Hydrodynamics and implementation of the

effective relativistic potential

The implementation of an effective relativistic gravita-tional potential or of an effective relativistic gravitationalforce as its derivative into existing Newtonian hydrody-namics codes is straightforward and does not differ fromthe use of the Newtonian potential or force.

The equations solved by the two codes used forour simulations are described in much detail in pre-vious publications (see, e.g., Rampp & Janka 2002;Muller & Steinmetz 1995; Dimmelmeier et al. 2002a,2005, and references therein). The implementation of thesource terms of the gravitational potential as discussed inMuller & Steinmetz (1995) is also applied for the handlingof the effective relativistic potentials investigated in thiswork. The only specific feature here is the mutual depen-dence of Γ and mTOV, which is either accounted for by arapidly converging iteration or by taking Γ from the oldtime step in the update ofmTOV, when the changes duringthe time steps are sufficiently small.

6 A. Marek et al.: An improved effective potential for supernova simulations

Since the actual form of the gravitational source termis unspecified in the conservation laws of fluid dynamics,the Newtonian potential can be replaced by the effectiverelativistic potentials investigated in the current work ina technically straightforward way. Solving an equation forthe internal plus kinetic energy (as in our codes) requiresa treatment of the gravity source term in this equationthat is consistent with its implementation in the equationof momentum.

Of course, the effective potential must be investigatedconcerning its consequences for the conservation of mo-mentum and energy. Since a potential constructed accord-ing to Eqs. (2, 4) does not satisfy the Poisson equation, themomentum equation cannot be cast into a conservationform (cf. Shu 1992, Part I, Chapter 4). As a consequence,the total linear momentum is strictly conserved only whencertain assumptions about the symmetry of the matterdistribution are made, for example in the case of sphericalsymmetry or axially symmetric configurations with equa-torial symmetry, or when only one octant is modeled inthe three-dimensional case. In axisymmetric simulationsthe conservation of specific angular momentum is fulfilledas well, when using the effective relativistic potential. Ingeneral, however, a sufficient quality of momentum (andangular momentum) conservation has to be verified by in-specting the numerical results.

The long-range nature of gravity prohibits to havean equation in pure conservation form for the total en-ergy, i.e., for the sum of internal, kinetic, and gravita-tional energy (Shu 1992, Part I, Chapter 4). In contrastto the Newtonian case, however, our effective relativis-tic potential does also not allow one to derive a conser-vation equation for the total energy integrated over allspace. Monitoring global energy conservation in a simula-tion with effective relativistic potential therefore requiresintegration of the gravitational source terms over all cellsin time. If the local effects of relativistic gravity are con-vincingly approximated – as measured by good agreementwith static solutions of the TOV equation and with fullyrelativistic, dynamical simulations – there is confidencethat the integrated action of the employed gravitationalsource term approximates well also the global conversionbetween total kinetic, internal, and gravitational energiesfound in a relativistic simulation.

The recipes for approximating general relativity shouldbe applicable equally well in hydrodynamic codes differ-ent from our (Eulerian) PPM schemes, provided the ef-fects of gravity are consistently treated in the momentumand energy equations. The proposed effective potentialsare intended to yield a good representation of the effectsof relativistic gravity in particular in the context of stellarcore collapse and neutron star formation. For our approx-imation to work well, the fluid flow should be subrelativis-tic. The numerical tests described in the following sectionsshow that velocities up to about 20% of the speed of lightare unproblematic.

4. Simulations in spherical symmetry

4.1. Supernova core collapse with simplified equation

of state

We first test the quality of the TOV potential and itsvarious modifications as an effective relativistic potentialin the moderately relativistic regime by evolving modelsof supernova core collapse with the simple EoS describedin Sect. 3.1. Using the CoCoNuT code we are able toperform a direct comparison with a Newtonian versionof the code (employing either the Newtonian, TOV, ormodified TOV potential) against the relativistic versionin a curved spacetime.

The initial models are polytropes, which mimic an ironcore supported by electron degeneracy pressure, with acentral density ρc i = 1010 g cm−3 and EoS parametersγi = 4/3 and K = 4.897 × 1014 (in cgs units). To initi-ate the collapse, the initial adiabatic index is reduced toγ1 < γi. At densities ρ > ρnuc ≡ 2.0 × 1014 g cm−3 theadiabatic index is increased to γ2 >∼ 2.5 to approximatethe stiffening of the EoS. This leads to a rebound of thecore (“core bounce”). The inner part of the core comes toa halt, while a prompt hydrodynamic shock starts to prop-agate outward. Since the matter model of the CoCoNuT

code does not account for neutrino cooling, the compactcollapse remnant (corresponding to the hot proto-neutronstar) cannot cool and shrink, but for some models onlyslightly grows in mass due to accretion. These stages ofthe evolution can be seen in the top panel of Fig. 1, wherethe time evolution of the central density ρc is plotted fora typical core collapse model.

As it is known from numerical models with sophisti-cated microphysics that the prompt hydrodynamic shockturns into an accretion shock shortly after core bounce,we choose the EoS parameters γ1 = 1.325, γ2 = 3.0, andγth = 1.2 in our accretion shock model AS (Fig. 1) to re-produce such a behaviour. The conversion of the promptshock into an accretion shock at late times is reflected bythe radial velocity profiles plotted in the bottom panel ofFig. 1.

When recalculating Model AS with regular Newtoniangravity (Case N), the central density remains below theone for relativistic gravity (Case GR) during the entireevolution (see top panel of Fig. 2). This behaviour wasalready observed in the rotational core collapse simula-tions of Dimmelmeier et al. (2002a). Replacing the regu-lar Newtonian potential (Case N) by the TOV potential(Case R) yields the opposite effect, i.e., the central den-sity is too high throughout the evolution. However, whenusing modifications A or B of the TOV potential, the be-haviour of ρc agrees very well with that observed in therelativistic simulation (Case GR), indicating that the dy-namics of the core collapse with this effective relativisticpotential is similar to that of the relativistic case.

When applying the same methods to a model witha prompt shock which continues to propagate, we ob-serve the same qualitative behaviour (see bottom panel

A. Marek et al.: An improved effective potential for supernova simulations 7

1 10 100 1000

r [km]

-0.3

-0.2

-0.1

0.0

v [c

]

80 90 100 110 120 130 140

t [ms]

0

1

2

3

4ρ c [1

014 g

cm

-3]

90.1 ms90.5 ms 140.0 ms

80.0 ms

90.5 ms

90.1 ms

80.0 ms

140.0 ms

ρnuc

Fig. 1. Time evolution of the central density ρc (toppanel) and radial profiles of the velocity v (bottom panel)for the accretion shock model AS in relativistic gravity(Case GR). The velocity profiles are snapshots long be-fore bounce, during bounce shortly before and after max-imum density, and long after bounce, respectively. Att = 140.0 ms the prompt shock has turned into an ac-cretion shock.

of Fig. 2). For this Model PS the EoS parameters areset to γ1 = 1.300, γ2 = 2.0, and γth = 1.5, respectively.Both during core bounce and at late times (see magnify-ing inset) the Newtonian simulations with modifications Aand B of the TOV potential also yield results which agreenoticeably better with the relativistic one (Case GR) thanwith the TOV potential (Case R), and obviously also muchbetter than with the regular potential (Case N).

The time plotted in Fig. 2 is t − tb, where tb is thetime of bounce for each simulation. We point out thatthe time coordinate t in the relativistic spacetime is thetime of an observer being at rest at infinity, and thus wecan directly relate it to the time used in the Newtoniansimulation (with regular or effective relativistic potential).Moreover, at core bounce the differences between the rel-ativistic coordinate time t and the relativistic proper timetp c =

∫αc dt of the fluid element at the center, where α

is the ADM lapse function, are negligible for the gravita-tional fields encountered in our core collapse models. Inaddition, shortly after core bounce the compact collapseremnant reaches an approximate hydrostatic equilibriumwith nearly constant ρc. Therefore, for the models con-sidered here the time evolution of the central density in

0 5 10

t − tb [ms]

3

4

5

6

7

8

9

ρ c [1014

g c

m-3

]

2

3

4

5

ρ c [1014

g c

m-3

]

AS

PS

Fig. 2. Time evolution of the central density ρc for theaccretion shock model AS (top panel) and prompt shockmodel PS (bottom panel). The regular Newtonian simula-tion (Case N; dashed line) yields a smaller ρc compared tothe relativistic one (Case GR; solid line). If a TOV poten-tial is used (Case R; dashed-dotted line), ρc is too high,while modifications A (dotted line) and B (long-dashedline) of the TOV potential give results close to the rela-tivistic one.

proper time tp c would look similar to the one in coordinatetime t.

The discussion up to now has only been concerned withthe local quantity ρc. While this is a good measure forthe global collapse dynamics, a direct comparison of, e.g.,radial profiles at certain evolution times is more powerful.

In Fig. 3 radial profiles of the density ρ for the core col-lapse model AS are compared at “late” times (t = 150 ms,corresponding to about 60 ms after core bounce). As thismodel exhibits no post-bounce mass accretion, shortly af-ter core bounce the compact remnant has acquired a newequilibrium state with essentially vanishing radial velocity.Its density profile can thus be very well described by thatof a TOV solution constructed with equal central densityρc and an “effective” one-parameter EoS, P = P (ρ), ex-tracted from the two-parameter EoS, P = P (ρ, ǫ), definedby Eqs. (12, 13). Details of this procedure are given inAppendix A. The top panel of Fig. 3 shows that this be-haviour is indeed confirmed by our simulations. The den-sity profile obtained from a dynamic simulation with rel-ativistic gravity (Case GR) and the one obtained fromsolving the TOV structure equations agree extremely wellboth at supranuclear and subnuclear densities. Only ahead

8 A. Marek et al.: An improved effective potential for supernova simulations

0 5 10 15

r [km]

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ρ [1

014 g

cm

-3]

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014 g

cm

-3]

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4ρ

[1014

g c

m-3

]

Newtonian with TOV potential (Case R)

Newtonian with regular potential (Case N)

Relativistic (Case GR)

ρnuc

ρnuc

ρnuc

Fig. 3. Comparison of radial profiles of the density ρ fromthe hydrodynamic evolution of Model AS at late times(solid line) and from a solution of the TOV structure equa-tions with identical central density (dashed line). In boththe relativistic simulation (Case GR; top panel) and theNewtonian simulation with TOV potential (Case R; bot-tom panel) the two curves are almost indistinguishable,while the regular Newtonian simulation (Case N; centrepanel) yields a large mismatch. For comparison, the rela-tivistic profile (solid line in the top panel) is also plottedin dashed-dotted line style in the lower two panels.

of the outward propagating shock (off scale in Fig. 3)where matter is still falling inward, i.e., where the radialvelocity is not negligible, a disagreement can be observed.

When using regular Newtonian gravity (Case N; seecentre panel of Fig. 3) the situation is different in twoaspects. First, due to the shallower Newtonian gravita-tional potential, the density is significantly smaller in thecentral parts of the compact remnant compared to rela-tivistic gravity (Case GR). However, beyond r ∼ 10 kmthe density in the Newtonian model exceeds that of therelativistic simulation, which is the density crossing effectobserved and discussed in detail by Dimmelmeier et al.(2002b). The weaker regular Newtonian potential thusyields a remnant which is less compact and dense in thecentre, but relatively denser in the outer parts.

Second, when comparing the density profile resultingfrom the hydrodynamic evolution with that obtained by

0 5 10 15

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∋ [c

2 ]

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∋ [c

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∋ [c

2 ]

Relativistic (Case GR)

Newtonian with regular potential (Case N)

Newtonian with TOV potential (Case R)

Fig. 4. Same as Fig. 3, but for radial profiles of the specificinternal energy ǫ. The step-like shape of the profile arisesfrom contributions of the thermal energy ǫth in variousstrengths.

solving the TOV structure equations (for the same ρc), aclear disagreement can be noticed. As the core reaches afinal compactness 2Mcore/Rcore ∼ 0.2 as measured by theratio of core mass to core radius, a hydrodynamic simula-tion with regular Newtonian potential cannot be expectedto yield an equilibrium result which is identical to a solu-tion of the (relativistic) TOV structure equations.

If on the other hand the Newtonian hydrodynamic sim-ulation is performed with the TOV potential (Case R; seebottom panel of Fig. 3), the final equilibrium state is (asin Case GR) a solution of the TOV structure equations.However, after core bounce the density profile of Case Rlies well above that obtained in the relativistic simulation(Case GR). Thus Case GR and Case R both exhibit post-bounce density stratifications which are consistent with arelativistic equilibrium state but otherwise differ notice-ably from each other.

Radial profiles of the specific internal energy (seeFig. 4) exhibit a similar behaviour as the density pro-files presented in Fig. 3. As expected, both the relativisticsimulation (Case GR) and the Newtonian simulation withTOV potential (Case R) yield final equilibrium profileswhich closely coincide with the corresponding solution ofthe TOV structure equations (see top and bottom panel,

A. Marek et al.: An improved effective potential for supernova simulations 9

0

1

2

3

4ρ

[1014

g c

m-3

]

0 5 10 15

r [km]

0

1

2

3

4

ρ [1

014 g

cm

-3]

Newtonian with modified TOV potential A

Newtonian with modified TOV potential B

ρnuc

ρnuc

Fig. 5. Comparison of radial profiles of the density ρ fromthe hydrodynamic evolution of Model AS at late times(solid line) and from a solution of the TOV structure equa-tions with identical central density (dashed line). In theNewtonian simulations with either modification A (toppanel) or B (bottom panel) of the TOV potential, the twocurves match well and also agree with the result from therelativistic simulation (Case GR; dashed-dotted line).

respectively). Again, analogous to the density, in Case Rthe internal energy in the compact remnant is significantlyhigher as compared to Case GR. In the Newtonian sim-ulation with regular potential (Case N; centre panel) theprofile deviates clearly from both the corresponding solu-tion of the TOV equation with equal central value and theprofile computed with relativistic gravity (Case GR), anddoes not even show the distinctive steps of that profile.

The above findings imply that the final equilibriumstate of a Newtonian core collapse simulation performedwith the TOV potential (Case R) is consistent with aTOV solution of identical central density and specific in-ternal energy. However, if relativistic effects are important,the density stratification and energy distribution (whichin turn determines the “effective” EoS as described inAppendix A) will differ from that of a consistently rel-ativistic simulation (Case GR). Presumably the combi-nation of the relativistic TOV potential (which dependsnonlinearly on the pressure and the internal energy den-sity) with Newtonian kinematics in Case R is responsiblefor this difference in the stratification of density and spe-cific internal energy in the compact remnant compared toCase GR (as seen in the bottom panels of Figs. 3 and 4).Currently we cannot provide a more detailed analysis forunderstanding the mechanism by which these discrepan-cies are caused. We plan to conduct further research onthis issue.

0.08

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∋ [c

2 ]

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r [km]

0.08

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∋ [c

2 ]

Newtonian with modified TOV potential A

Newtonian with modified TOV potential B

Fig. 6. Same as Fig. 5, but for radial profiles of the specificinternal energy ǫ. The step-like shape of the profile arisesfrom contributions of the thermal energy ǫth in variousstrengths.

As discussed in Sect. 2.2, a modification of aNewtonian simulation incorporating the effects of rela-tivistic gravity must not only yield equilibrium stateswhich (approximately) reproduce a solution of the TOVequations, particularly for compact matter configurations.More importantly, the density and energy profiles mustalso agree as closely as possible with the profiles obtainedfrom the evolution with a relativistic code (which im-plies the previous requirement). Both these requirementsare fulfilled for Newtonian simulations utilizing modifica-tion A and B of the TOV potential, as shown for pro-files of the density and specific internal energy in the finalequilibrium state of the core collapse model AS in Figs. 5and 6, respectively. In these cases the profiles agree wellwith the relativistic result (Case GR) not only near thecentre (as already observed in the time evolution of ρc; seetop panel of Fig. 2) but also throughout the entire core,with the agreement being much better compared to theTOV potential (Case R). Accordingly, the solution of theTOV structure equations for identical central values yieldssimilar profiles as well.

We conclude that the modifications A and B of theTOV potential combined with Newtonian kinematics pro-vide a very good approximation of the results obtainedwith relativistic gravity (Case GR) in the strongly dy-namic phases during core bounce as well as at post-bouncetimes when the compact remnant acquires equilibrium.The same findings hold for the modifications C and D ofthe TOV potential, although we do not present results forthese cases here. On the other hand, compared to Cases Ato D, the modifications E to G produce less accurate re-

10 A. Marek et al.: An improved effective potential for supernova simulations

sults. Thus, we consider them inappropriate for an effec-tive relativistic potential.

In passing we note that gauge dependences may playan important role when non-invariant quantities (like, e.g.,density profiles) from simulations with a relativistic codeare compared against results from either a relativistic codeusing a different formulation of the spacetime metric orfrom a Newtonian code. For instance, the radial coordinater used in the profile plots in Figs. 3 to 6 is the isotropiccoordinate radius of the ADM-CFC metric used in therelativistic version of the CoCoNuT code (for details,see Dimmelmeier et al. 2002a). In contrast to this, theTOV equations (A.1, A.2) use the standard Schwarzschild-like coordinate radius, which is equivalent to the circum-ferential radius. Thus, a coordinate transformation fromthe standard Schwarzschild-like radial coordinate to theisotropic radial coordinate (as specified in Appendix B)is applied to all profiles of quantities obtained with theTOV solution or Newtonian simulations in those figures.Otherwise, the difference in the two radial coordinates ofup to ∼ 20% would lead to a noticeable distortion of theradial profiles. We also point out that we neglect any in-fluence of the difference between Newtonian and relativis-tic coordinate time (or proper time) when comparing theprofiles at the same coordinate times, as the compact rem-nants are in equilibrium at “late” times (about 60 ms aftercore bounce).

4.2. Supernova core collapse with microphysical

equation of state and neutrino transport

Next we explore the moderately relativistic regime ofstellar core collapse with the microphysical EoS ofLattimer & Swesty (1991) and neutrino transport. Wesimulated the collapse and the post-bounce evolution ofthe progenitor model s15s7b2 with the Vertex code asdetailed in Sect. 3.2. The calculations were performedusing the TOV potential given in Eq. (2) (Model V-R,which is identical with the Vertex calculation of ModelG15 in Liebendorfer et al. 2005), and we also tested themodifications A, B, and F of the TOV potential (ModelsV-A, V-B, and V-F, respectively). For comparison, werefer in the following discussion also to the calculationof Liebendorfer et al. (2005) with the fully relativisticAgile-BoltzTran code (Case GR, Model AB-GR).

Fig. 7 shows the central density as a function of timefor the collapse (left panel) and for the subsequent post-bounce phase (right panel). The “central” density is thedensity value at the centre of the innermost grid zone ofthe AB-GR simulation. Because of a different numericalresolution it was necessary to interpolate the Vertex re-sults to this radial position. During the collapse only mi-nor differences between the relativistic calculation (boldsolid line) and the calculations with the Vertex code arevisible. Note that the trajectories from the Vertex code,with the modifications A, B, and F of the TOV potential

-200 -100 08

10

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14

16

0 100 200 300

3

4

5

6

PSfrag replacements

t − tb [ms]

log(

ρc[g/c

m3 ])

ρc

[101

4g/

cm3 ]V-R

V-AV-BV-F

AB-GR

Fig. 7. Time evolution of the central density ρc for ModelAB-GR (bold solid line), V-A (solid line), V-B (dashed-dotted line), V-F (dashed line), and V-R (dotted line).The left panel shows the collapse phase (note that here allmodels with the Vertex code lie on top of each other),while the right panel shows the post-bounce evolution.Note the different axis scales in both panels.

as well as with the TOV potential (case R), lie on top ofeach other.

We can thus infer that the differences between themodifications A, B, and F of the TOV potential are unim-portant during the collapse phase. Furthermore, we canconclude from the good agreement of the general relativis-tic calculation and theVertex calculations that the TOVpotential works well during the collapse phase. However,after core bounce this potential overestimates the com-pactness of the forming neutron star (just as in the purelyhydrodynamic simulations in Sect. 4.1), and therefore thedensity trajectories of Model AB-GR and Model V-R di-verge (see Fig. 7). At 250 ms after the shock formation thecentral density in Model V-R is about 20% higher than theone in the relativistic calculation. At this time the mod-ifications A and B of the TOV potential give a centraldensity only about 2% higher than Model AB-GR, andthe absolute difference stays practically constant duringthe entire post-bounce evolution. This implies that bothmodifications yield very good quantitative agreement withthe general relativistic treatment. In contrast, in ModelV-F the central density after bounce is lower than the rel-ativistic result of Model AB-GR. This indicates a strongunderestimation of the depth of the gravitational potentialin Case F, where Γ = 1 in the integrand of Eq. (2).

Since the central densities suggest that differencesbetween a fully relativistic calculation and Newtoniansimulations with effective relativistic potential be-come significant only after shock formation (see alsoLiebendorfer et al. 2005), we discuss the implications ofour potential modifications in the following only duringthe post-bounce evolution.

A. Marek et al.: An improved effective potential for supernova simulations 11

0 50 100 150 200 250 3000

50

100

150

200

PSfrag replacements

t − tb [ms]

r s[k

m]

V-RV-AV-BV-F

AB-GR

Fig. 8. Time evolution of the shock position rs for simu-lations with the Vertex code using various modificationsof the TOV potential compared to the general relativisticresult from the Agile-BoltzTran code (Model AB-GR,thick solid line).

Fig. 8 shows the shock positions as functions of time.Both Case A (thin solid line) and Case B (dashed-dottedline) reveal the desirable trend of a closer match with thegeneral relativistic calculation (thick solid line) than seenfor Model V-R, which gives a shock radius that is toosmall, and Model V-F, where the shock is too far out atall times. In particular, Model V-A reveals excellent agree-ment with Model AB-GR. The only major difference isvisible between 170 ms and about 230 ms after shock for-mation when the shock transiently expands in theVertex

calculation. This behaviour is generic for the Vertex re-sults and independent of the choice of the gravitational po-tential. In the Agile-BoltzTran run the transient shockexpansion is much less pronounced and also a bit delayedrelative to the Vertex feature (it is visible as a deceler-ation of the shock retraction between about 200 ms and250 ms). This difference, however, is not caused by gen-eral relativistic effects but is a consequence of a differentnumerical tracking of the time evolution of interfaces be-tween composition layers in the collapsing stellar core (formore details about the numerics and a discussion of theinvolved physics, see Liebendorfer et al. 2005). It is there-fore irrelevant for our present comparison of approxima-tions to general relativity. A good choice for the effectiverelativistic potential (like Case A) should just ensure thatthe corresponding shock trajectory converges again withthe relativistic result (Case GR) after the transient periodof shock expansion.

The time evolution of the central density or the shockposition, however, is not the only relevant criterion for as-sessing the quality of approximations to general relativity,as already discussed in Sect. 4.1. A good approximationdoes not only require good agreement for particular time-dependent quantities (like, e.g., the central density), butalso requires that the radial structure of the models re-

produces the relativistic case as well as possible at anytime.

In the left panels of Fig. 9 we show such profiles ofthe density and velocity (top panel) and of the entropyand electron fraction Ye, i.e., the electron-to-baryon ratio(bottom panel), for Models AB-GR and V-A at a timeof 250 ms after bounce, when the discrepancy betweenrelativistic and approximative treatment was found to belargest in Liebendorfer et al. (2005). Fig. 9 can be directlycompared with Fig. 12 in the latter reference. ObviouslyModel V-A fits the density profile of the general relativis-tic calculation (Model AB-GR) extremely well at all radii.Furthermore, both the velocity ahead of the shock frontand behind it are in extremely good agreement betweenthe two models (the differences at the shock jump haveprobably a numerical reason associated with the differ-ent handling of shock discontinuities in both codes). It isan important result that with this modification A of theTOV potential one is able to approximate the kinematicsof the relativistic run with astonishingly good quality in aNewtonian calculation (at least in supernova simulationswhen the velocities do not become highly relativistic). Incontrast, in Vertex runs with the original TOV potential(Case R) the pre-shock velocities were found to be signif-icantly too large (Liebendorfer et al. 2005), which is dueto the overestimated strength of gravity in the far fieldlimit as compared to the relativistic calculations (see thediscussion in Sect. 2.3).

Also the entropy and Ye profiles (bottom left panelof Fig. 9) reveal a similarly excellent agreement betweenModels V-A and AB-GR. The minor entropy differencesahead of the shock are associated with a slightly differ-ent description of the microphysics (nuclear burning andequation of state) in the infall region (for details we referto Liebendorfer et al. 2005).

Not only the radial structure of the forming neutronstar in all relevant quantities is well reproduced, but alsothe neutrino transport results of the relativistic calcula-tion and of the approximative description of Case A arein nearly perfect agreement. Corresponding radial profilesof the luminosities and root mean square energies – bothas defined in Sect. 4 of Liebendorfer et al. (2005) – for elec-tron neutrinos, νe, electron antineutrinos, νe, and heavy-lepton neutrinos1, νx, are displayed in the right panels ofFig. 9. The results for Models V-A and AB-GR for allneutrino flavours share their characteristic features, andin particular agree in the radial positions where the differ-ent luminosities start to rise. While the νx luminosities arenearly indistinguishable below the shock, the jump at theshock is slightly higher for the Vertex run and reflectsthe larger effects due to observer motion, e.g., Dopplerblueshift and angular aberration, for an observer comovingwith the rapidly infalling stellar fluid ahead of the shock.The offset between results of Models V-A and AB-GR de-

1 Since the transport of muon and tau neutrinos and antineu-trinos differs only in minor details we treat all heavy-leptonneutrinos identically in the Vertex simulations.

12 A. Marek et al.: An improved effective potential for supernova simulations

0 50 100 150 2000

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1012

1014

PSfrag replacements

r [km]

s[k

B/b

aryo

n]v

[109

cm/s

]t = 250 ms

ρ

v

ρ[g/c

m3 ]

Ye

s

Ye

V-A

AB-GR

0 50 100 150 200

10

100

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10

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30

40

50

60

PSfrag replacements

r [km]

〈ǫν〉 R

MS

[MeV

]L

[105

1er

g/s]

t = 250 msνe

νe

νe

νe

νx

νx

V-A

AB-GR

Fig. 9. Left: Radial profiles of the velocity v (dashed lines) and density ρ (solid lines) for Model V-A (thin) and therelativistic Model AB-GR (bold) at a time of 250 ms after shock formation (top panel), as well as radial profiles of theentropy s (dashed lines) and of the electron fraction Ye (solid lines) for the same models and the same time (bottompanel). As in Liebendorfer et al. (2005), r is the circumferential radius in case of the relativistic results. Note thedifferent vertical axes on both sides of the two panels. Right: Radial profiles of the luminosities L of electron neutrinos(solid lines), electron anti-neutrinos (dotted lines), and heavy-lepton neutrinos (dashed lines) for Models V-A (thin)and AB-GR (bold) at a time of 250 ms after shock formation (top panel), as well as radial profiles of the root meansquare energies 〈ǫν〉RMS for the number densities of νe, νe, and heavy-lepton neutrinos for Models V-A and AB-GR(bottom panel). The labeling is the same as in the panel above, and all neutrino quantities are given for a comovingobserver.

creases at larger radii where the infall velocities are lower.This discrepancy was not discovered by Liebendorfer et al.(2005), because there the agreement of the radial struc-ture for both investigated models was generally found tobe poorer than in the present work.

General relativistic effects are unlikely as an explana-tion, because they are very small around the shock (seeFig. 13 in Liebendorfer et al. 2005). A detailed analysisreveals that both codes produce internally consistent re-sults, conserving to good precision the luminosity throughthe shock for an observer at rest and showing the ex-pected and physically correct behavior in the limit of

large radii. Most of the observed difference (which hasno mentionable significance for supernova modeling) couldbe traced back to the fact that Vertex achieves only or-der (v/c) accuracy, whereas Agile-BoltzTran producesthe full relativistic result including higher orders in (v/c).Corresponding effects become noticeable when v/c >∼ 0.1.The mean neutrino energies are hardly affected by thisdifference (Fig. 9, bottom right panel). In case of the νeand νe luminosities the Vertex run yields roughly 10%lower values outside of the corresponding neutrino spheres(i.e., between about 50 km and 90 km), but values muchcloser to those from the Agile-BoltzTran calculation

A. Marek et al.: An improved effective potential for supernova simulations 13

0 5 1015 0

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1015202530

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t − tb [ms]

L[1

051 e

rg/s

]L

[105

1 erg/s

]L

[105

1 erg/s

]

L[1

053 e

rg/s

]L

[105

1 erg/s

]L

[105

1 erg/s

] νx

νe

νe

V-AV-R

V-B

AB-GR

Fig. 10. Luminosities as functions of post-bounce timefor different cases computed with the Vertex code andfor Model AB-GR. The top panel shows the results forheavy-lepton neutrinos, the centre panel those for the elec-tron antineutrinos, and the bottom panel the results forelectron neutrinos. The panels on the left magnify theearly post-bounce phase. All luminosities are given for anobserver comoving with the stellar fluid at a radius of500 km. Note the different scales of the vertical axes.

ahead of the shock. Since the neutrinospheric emissionof νe and νe is strongly affected by the mass accretionrate of the nascent neutron star and the correspondingaccretion luminosity (which both seem to have the ten-dency of being slightly higher in Model AB-GR), we re-frain from ascribing the different magnitude of the νe andνe luminosities only to the treatment of relativistic effects.Although such a connection cannot be excluded, the lu-minosity differences might (partly) also be a consequenceof the different accretion histories in Models AB-GR andV-A, which manifest themselves in the shock trajectories(Fig. 8) and are attributable to the different handling of

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s[k

B/b

aryo

n]v

[109

cm/s

]

t = 250 ms

ρ

v

ρ[g/c

m3 ]

s

Lνe

L[1

051er

g/s]

V-AV-R

V-B

Fig. 11. Radial profiles of the velocity v (dashed lines)and density ρ (solid lines) for Models V-A (bold lines),V-B (thin), and V-R (medium) at a time of 250 ms aftershock formation (top panel). Radial profiles of the entropys (dashed lines) and of the electron neutrino luminosityLνe(solid lines) for the same models and the same time(bottom panel). The luminosity is given for an observercomoving with the stellar fluid.

the microphysics and computational grid in the infall layer(see above and Liebendorfer et al. 2005). This interpreta-tion seems to be supported by the time evolution of theneutrino luminosities plotted in Fig. 10. The accretionbump in the νe and νe luminosities which follows afterthe prompt νe burst is stretched in time in case of ModelAB-GR, indicating the delay of mass infall at higher ratesrelative to allVertex simulations. Note that the neutrinoemission reacts with a time lag of some 10 ms (correspond-ing to the cooling timescale of the accretion layer on theneutron star) to variations of the mass accretion rate.

Moreover, Fig. 10 shows that our variations of the ef-fective relativistic potential in the Vertex models have

14 A. Marek et al.: An improved effective potential for supernova simulations

little influence on the prompt burst of νe at shock break-out. But subsequently the overestimated compactness ofthe proto-neutron star in Model V-R, which causes thefaster contraction of the stalled shock after maximal ex-pansion (Fig. 8), also leads to higher neutrino luminosi-ties during the accretion phase. Consistent with the shocktrajectories, Model V-A yields the closest match with thegeneral relativistic run of Model AB-GR also for the neu-trino luminosities. It is very satisfactory that the results(shock radius rs as well as the neutrino luminosities L)from both simulations reveal convergence at later timeswhen the period of massive post-bounce accretion comesto an end.

In Fig. 11 we present the radial structure at 250 msafter bounce for the Vertex simulations with the modi-fications A and B of the TOV potential, compared to theresults with the TOV potential (Case R) which was al-ready discussed in Liebendorfer et al. (2005). Note thatbecause of the excellent agreement seen in Fig. 9, ModelV-A (Case A) can also be considered as a representationof the fully relativistic run of Model AB-GR. Models V-Aand V-B show results of similar quality. The little offsetof the shock position (which is causally linked to the dif-ferences in all profiles) might suggest that Model V-B isslightly inferior to Model V-A in approximating relativ-ity. This conclusion could also be drawn from the post-bounce luminosities in Fig. 10. However, caution seems tobe advisable with such an interpretation, being aware ofthe uncertainties in the accretion phase and infall layerdiscussed above, and in view of the fact that the centraldensities (Fig. 7) and radial density profiles (Fig. 11) agreewell. Moreover, the quality of the agreement at “very late”times cannot be judged, because no information is avail-able for the behaviour of Model AB-GR after 250 ms postbounce, a time when the settling of the shock radius andluminosities to their post-accretion levels seems not yetover in this model (Figs. 8, 10). The TOV potential ofCase R clearly produces too large infall velocities aheadof the shock (and therefore does not agree well with thekinematics of the relativistic calculation), overestimatesthe compactness of the forming neutron star, and thusproduces too high neutrino luminosities during the sim-ulated period of evolution (for a detailed discussion, seeLiebendorfer et al. 2005). Cases A and B clearly performbetter and must be considered as significant improvementsfor use in Newtonian simulations with an effective rel-ativistic potential as approximations to fully relativisticcalculations.

4.3. Neutron stars in equilibrium – Migration test

The supernova core collapse simulations presented inSect. 4.1 and 4.2 are limited to a maximum core com-pactness 2Mcore/Rcore ∼ 0.2. In order to extend the as-sessment of the quality of a Newtonian simulation withan effective relativistic potential to a more strongly rela-tivistic regime, we construct spherical equilibrium models

of neutron stars using a simple polytropic EoS, P = Kργ ,with γ = 2 and K = 1.455× 105 (in cgs units). The TOVequations are solved on a very fine grid with 105 equidis-tantly spaced radial grid points using a second-order ac-curate Runge–Kutta integration scheme. By varying thevalue of the central density from ρc = 1.0 × 1014 g cm−3

to 100.0 × 1014 g cm−3, we cover the entire range fromweak to strong relativistic gravity. The density varia-tion corresponds to compactness parameters ranging from2Mg/R ∼ 0.02 to 0.5, where Mg is the total gravitationalmass of the neutron star and R its radius. We again pointout that in spherical symmetry the assumption of CFCconstitutes no approximation.

The mass-density curve of the corresponding TOVneutron star models (Fig. 12, top panel) exhibits the well-known form consisting of a stable branch at low densi-ties and an unstable branch at high densities, where thegravitational mass Mg increases or decreases with ρc, re-spectively. The two branches meet at the maximum mass.Configurations on the unstable branch are unstable tosmall perturbations and either collapse to a black holeor migrate to a configuration on the stable branch withthe same gravitational mass but a lower central densityρc f < ρc i (as indicated by the arrow for an exemplaryconfiguration in the top panel of Fig. 12).

While the gravitational mass Mg and the rest massMr are limited in relativistic gravity (Case GR), Mg ≃Mr ≃ 1.6M⊙, regular Newtonian gravity (Case N) allowsfor solutions with arbitrarily high rest mass Mr, which areall stable against (small) perturbations. If an (unphysi-cal) equivalent to the relativistic gravitational mass is in-troduced in Newtonian gravity, Mg ≡ Mr − Mb, whereMb ≡ |Eb| is the mass associated with the binding en-ergy of the self-gravitating object, we find negative valuesat large densities, where the binding energy becomes solarge that Mb > Mr.

On the other hand, when using the modifications Aand B of the TOV potential in the TOV equation for theneutron star model sequence, the corresponding curves forthe TOV gravitational mass Mg in Fig. 12 agree very wellwith the relativistic curve (Case GR) even in the high den-sity regime, and they show the same qualitative behaviourwith an increasing and a decreasing branch. The respec-tive mass maxima are located close to the relativistic onefor both modifications of the TOV potential.

Note that for constructing equilibrium configurationsusing the TOV potential (Case R) is identical to solv-ing the fully relativistic TOV equations (Case GR), andthus the results are identical. Therefore, this test is use-less for predicting the quality of the TOV potential in thecontext of a dynamic evolution. This is different when aneutron star model from the unstable branch is allowed toevolve in time using a dynamic code. Then the (small butnonzero) finite difference discretisation errors of the nu-merical scheme act as perturbations of the unstable equi-librium and excite small oscillations. Depending on the nu-merical algorithms used in the code and the resolution ofthe grid, the neutron star either contracts to higher densi-

A. Marek et al.: An improved effective potential for supernova simulations 15

0 1 2 3 4

t [ms]

0

10

20

30

40

50

ρ c [1014

g c

m-3

]

ρc f

ρc i

1 10 100

ρc [g cm

-3]

0

1

2

3

4M

g [M

O· ]

••ρ

c iρ

c f

Mr

Fig. 12. Mass-density diagram for spherical equilibriumneutron star models (top panel) and time evolution of thecentral density for the migration test from the unstableto the stable branch (bottom panel). In both graphs thebehaviour of the Newtonian models computed with themodification A (dotted line) and B (long-dashed line) ofthe TOV potential is close to that in the relativistic case(Case GR; solid line). The Newtonian equivalent of thegravitational mass Mg (dashed line) as well as the restmass Mr (dashed-dotted line) calculated using a regularNewtonian potential (Case N) deviate strongly from therelativistic solution (top panel). While ρc oscillates aroundits initial value ρc i for Case N (dashed line), in Case R(dashed-dotted line) the neutron star disperses to a statewith ρc ≪ ρc f (bottom panel).

ties (and ultimately to a black hole for a simple polytropicEoS) or approaches a new equilibrium state on the sta-ble branch after a strongly dynamical, nonlinear evolutionphase. This neutron star migration scenario is a standardtest of the relativistic regime with strongly dynamical evo-lution for numerical hydrodynamic codes with relativisticgravity (see, e.g., Font et al. 2002; Baiotti et al. 2005).

The time evolution of the central density is depicted inthe bottom panel of Fig. 12 for such a migration to the sta-ble branch, where the central density ρc drops from ρc i =49.3 × 1014 g cm−3 to ρc f = 8.8 × 1014 g cm−3. Shownis the evolution for a relativistic simulation (Case GR)as well as for Newtonian simulations utilizing a regu-lar Newtonian potential (Case N), the TOV potential(Case R), and its modifications A and B. For these sim-ulations we again use the CoCoNuT code, whose imple-mentation of the hybrid EoS, Eqs. (12, 13) allows one tosuppress the contribution of the thermal pressure Pth, inwhich case the hybrid EoS reduces to a polytropic one.

As expected and as observed in simulations with simi-lar relativistic codes, in relativistic gravity (Case GR) theneutron star initially expands rapidly and its central den-sity decreases strongly. It undershoots the stable equilib-rium at ρc f and experiences several ring-down oscillationsof decreasing amplitude until ρc approaches ρc f at latetimes. The modifications A and B of the TOV potentialboth have initial (final) states on the unstable (stable)branch of the mass-density curve which are very close tothe relativistic one (Case GR). Therefore the evolution ofρc (starting again with the same central density ρc i) issimilar to the relativistic case, as shown in Fig. 12. Thedisagreement in oscillation amplitudes and periods can beattributed to the differences in the Newtonian and rela-tivistic kinematics. When comparing the results of migra-tion tests using various relativistic hydrodynamic codes(Stergioulas et al. 2003), we observe that even small vari-ations in the numerical approach, e.g., a different coordi-nate choice or grid setup, or differences in the treatmentof the artificial atmosphere, can have a strong impact onthe detailed evolution of ρc from the initial to the finalequilibrium state. Consequently, we consider the resultsobtained with the modifications A or B of the TOV po-tential in an otherwise Newtonian formulation to be a verygood approximation of the relativistic one (Case GR).

Using regular Newtonian gravity (Case N), however,yields a totally different behaviour, because there existno unstable configurations in that case. Thus, discreti-sation errors cannot trigger a migration, and the starsimply oscillates around its initial equilibrium state withρc(t) ≃ ρc i. On the other hand, using the TOV potential(Case R) leads to a rapid dispersion (without ring-downoscillations) of the neutron star with a final central densityρc ≪ ρc f , which is in strong qualitative disagreement withthe behaviour in relativistic gravity (Case GR). We thusconclude that in the demanding test case of a migratingspherical neutron star model, both the regular Newtonianpotential (Case N) and the TOV potential (Case R) fail,while the modifications A and B of the TOV potential re-produce the relativistic result very well, both qualitativelyand also quantitatively.

5. Multi-dimensional simulations with rotation

For the multi-dimensional calculations we only use theCoCoNuT code, since parameter studies with MudBath

16 A. Marek et al.: An improved effective potential for supernova simulations

(the two-dimensional version of Vertex) are compu-tationally too expensive when neutrino transport is in-cluded. Moreover, currently there exists no other fully rel-ativistic two-dimensional code including neutrino trans-port to compare our results with. The following discussionhence focuses on purely hydrodynamic simulations usingthe simple matter model described in Sect. 3.1.

Motivated by the promising results in spherical sym-metry, we now apply the modified effective relativistic po-tentials according to Eq. (7) to simulations of rotationalcore collapse in axisymmetry. The initial models of thepre-collapse core are described by the same EoS parame-ters as the ones in Sect. 4.1 with ρc i = 1010 g cm−3, butnow rotate at various rotation rates with different degreesof differential rotation. To construct these rotating equilib-rium configurations, we use Hachisu’s self-consistent fieldmethod (see Komatsu et al. 1989). The initial pressure re-duction to trigger the collapse and the stiffening of the EoSfollows the prescription given in Sect. 4.1.

For these multi-dimensional tests we choose three rota-tional core collapse models as representative cases. Theirevolution is influenced by the effects of rotation at vari-ous degrees (for details about the specifications of thesemodels, see Dimmelmeier et al. 2002a). The core of ModelA1B3G1 is an almost uniform rotator with a moderaterotation rate and collapses nearly spherically, forming acompact remnant immediately after core bounce. ModelA4B1G2 is a very differentially rotating model, which de-velops considerable deviations from spherical symmetryduring its evolution, but otherwise shares the qualitativebounce behavior of the previous model A1B3G1. Finally,in the multiple bounce model A2B4G1 the increase of thecentrifugal forces during the contraction phase causes aseries of subsequent multiple bounces at subnuclear den-sities. The time evolution of the central density for thesethree models computed relativistically (Case GR) is shownwith solid lines in the top, centre, and bottom panels ofFig. 13, respectively.

In a Newtonian simulation with regular Newtoniangravity (Case N) all three models exhibit several consecu-tive bounces (at subnuclear densities for Models A1B3G1and A2B4G1, and just above nuclear matter densities forModel A4B1G2), with the central density being consid-erably lower during the entire evolution (Fig. 13). Sucha qualitative change of the collapse dynamics betweena Newtonian and a general relativistic simulation of thesame model was also observed in the comprehensive pa-rameter study of rotational core collapse performed byDimmelmeier et al. (2002b).

When applying the TOV potential (Case R) or itsmodifications A and B to Model A1B3G1 (top panel ofFig. 13) the outcome is similar to the situation in spher-ical symmetry shown in Fig. 2. Case R yields a centraldensity noticeably higher than the one in the relativisticsimulation (Case GR). For Case A the result agrees verywell with the relativistic one, while for Case B the centraldensity is too small.

-5 0 5 10

t − tb [ms]

0

1

2

3

4

ρ c [1014

g c

m-3

]

-20 0 20 40 60 80

t − tb [ms]

0.0

0.2

0.4

0.6

ρ c [1014

g c

m-3

]

0 10 20 30

t − tb [ms]

0

1

2

3

4

ρ c [1014

g c

m-3

]

ρnuc

ρnuc

A1B3G1

A4B1G2

A2B4G1

Fig. 13. Time evolution of the central density ρc forthe rotating core collapse models A1B3G1 (top panel),A4B1G2 (centre panel), and A2B4G1 (bottom panel), re-spectively. With increasing influence of rotation on the col-lapse dynamics, the results of the Newtonian simulationswith the modifications A (dotted line) and B (long-dashedline) of the TOV potential and eventually also with theTOV potential (Case R; dashed-dotted line) move awayfrom the relativistic result (Case GR; solid line). Thecentral density obtained with regular Newtonian gravity(Case N; dashed line) is lowest for all three models.

As the influence of rotation increases, the ordering ofthe curves for Cases R, A, and B (from higher to lower ρc)

A. Marek et al.: An improved effective potential for supernova simulations 17

remains unaltered. However, all curves are shifted to lowerdensities (Fig. 13). For Model A4B1G2 the result from theNewtonian simulation with the TOV potential (Case R)matches the relativistic result (Case GR) best, which is acoincidence associated with the particular rotational stateof this model. For Model A2B4G1, where the influence ofrotation is stronger, the results obtained for Cases A, B,and R differ considerably from the relativistic results.

A behaviour consistent with that of these three rep-resentative models is observed for other rotational corecollapse models as well. Thus, in rotational core col-lapse, even a Newtonian simulation with the TOV po-tential (Case R) yields lower central densities than thecorresponding relativistic simulation (Case GR) beyond acertain rotation rate (which depends both on the initialmodel and on the EoS parameters).

We therefore conclude that for rotational core collapsethe use of neither the modifications A and B of the TOVpotential nor the (stronger) TOV potential itself (Case R)can closely reproduce the collapse behaviour of a relativis-tic code for all degrees of rotation. Hence, in a Newtoniancode either the effective relativistic potential has to in-clude the effect of rotation, or the collapse kinematics (andthus the hydrodynamic equations) have to be modified.However, in the latter case the advantage of a Newtonianapproach over a fully relativistic formulation in terms ofsimplicity and computational speed may be lost.

On the other hand, we find that the qualitative dynam-ical behaviour of relativistic rotational core collapse mod-els (Case GR), particularly their collapse type (regularcollapse, multiple bounce collapse, or rapid collapse as in-troduced by Zwerger & Muller 1997) is correctly capturedin a Newtonian simulation for a wide range of parameterspace, if the TOV potential (Case R) or its modified ver-sions (Cases A to D) are used. Moreover, the correspond-ing results agree much better with the relativistic resultthan that obtained from a Newtonian simulation with reg-ular Newtonian potential (Case N). Hence, the usage ofthe effective relativistic potentials is an improvement ofNewtonian simulations also in the case of rotational corecollapse.

6. Summary and Conclusions

We investigated different modifications of the TOV poten-tial used as effective relativistic potentials in Newtonianhydrodynamics with the aim to test their quality of ap-proximating relativistic effects in stellar core collapse andsupernova simulations. This work was motivated by a re-cent comparison of the neutrino radiation-hydrodynamicscodes Agile-BoltzTran and Vertex used by the OakRidge-Basel collaboration and by the Garching group, re-spectively (Liebendorfer et al. 2005). While the formercode is a fully relativistic implementation of neutrinotransport and hydrodynamics in spherical symmetry, thelatter code employs an approximative relativistic descrip-tion by treating the self-gravity of the stellar fluid withan effective relativistic potential tailored from a compar-

ison of the Newtonian and relativistic equations of mo-tion. Otherwise Vertex solves the Newtonian equationsof hydrodynamics, ignoring the effects of relativistic kine-matics and space-time curvature (Rampp & Janka 2002).The comparison by Liebendorfer et al. (2005) shows verygood agreement of the results from both codes for thephases of stellar core collapse, bounce and prompt shockpropagation. However, it also revealed a moderate butgradually increasing overestimation of the compactness ofthe nascent neutron star during the subsequent accretionphase, accompanied by an overestimation of the luminosi-ties and mean energies of the emitted neutrinos.

These systematic differences suggested that an im-provement might be possible by changing the effectiverelativistic potential such that the strength of the grav-itational field is reduced relative to the TOV potential.To this end we considered a variety of modifications(Cases A–F; see Sect. 2.2) of the TOV potential (whichdefines our reference Case R), and compared correspond-ing hydrodynamic as well as hydrostatic solutions withregular Newtonian (Case N) and fully relativistic calcula-tions (Case GR). While some modifications did not pro-vide any significant advantage over the TOV potential,several options for the potential were identified which al-low for a much better reproduction of relativistic resultswith Newtonian hydrodynamics. In particular, reducingthe TOV mass, Eq. (4), that appears in the expressionof the TOV potential, turned out to be very successful inthis respect.

For example, in Case A we introduced an additionalmetric factor Γ < 1, Eq. (5), in the integrand of the TOVmass, and in Case B we ignored the internal energy (andall terms depending on neutrinos) in the TOV mass. Inboth cases not only characteristic model parameters likethe central density and shock radius were in excellentagreement with relativistic simulations during all stagesof the evolution including the long-term accretion phaseof the collapsing core after bounce. Also the radial profilesof the hydrostatic central core as obtained in the hydrody-namic simulations were found to be very good approxima-tions of the profiles from the fully relativistic treatment.Consequently, they are nearly consistent with solutionsof the TOV equations for equal central density. Migrationtests of extremely relativistic, ultradense, hydrostatic con-figurations from the unstable to the stable branch of themass-density relation confirmed the good qualitative andquantitative performance of the approximation even forsuch a demanding test problem with a strongly dynamictransition from the initial to the final state. This suggeststhat the use of a modified TOV potential in an other-wise Newtonian hydrodynamics code also ensures a goodreproduction of relativistic kinematics.

In addition to these purely hydrodynamic simulationswith the CoCoNuT code, we performed runs includingneutrino transport and compared them to relativistic cal-culations with the Agile-BoltzTran code of the OakRidge-Basel collaboration (cf. the simulations for ModelG15 in Liebendorfer et al. 2005). Also in this application,

18 A. Marek et al.: An improved effective potential for supernova simulations

the modifications A and B of the TOV potential pro-duced a major improvement relative to the original TOVpotential (Case R), for which results with the Vertex

code were already presented in Liebendorfer et al. (2005).Good agreement between the relativistic and approxima-tive treatments was seen in the transport quantities aswell. The Vertex code performs well despite the factthat the approximative description takes into account onlyrelativistic redshift in the transport but – for reasons ofconsistency with the Newtonian hydrodynamics code –ignores the difference between proper radius and coordi-nate radius. Remaining (minor) quantitative differences(below about 10% for neutrino data and even smaller forhydrodynamic quantities) are probably only partly asso-ciated with relativistic effects. We suspect that a signifi-cant contribution to these differences originates from small(but unavoidable because linked to the specifics of theemployed numerical methods) discrepancies in the infalllayer ahead of the shock as described by the Vertex andAgile-BoltzTran codes.

The results discussed in this paper therefore demon-strate that a very good, simple, and computationally ef-ficient approximation to a relativistic treatment of stel-lar core collapse and neutron star formation in sphericalsymmetry can be achieved by modeling the self-gravity ofthe stellar plasma with an effective relativistic potentialbased on the TOV potential as suggested in this work.Two-dimensional axisymmetric collapse calculations (us-ing a simple equation of state and ignoring neutrino trans-port) for rotating stellar cores with a wide range of condi-tions showed that even in this case the TOV potential andits modifications reproduce the characteristics of the rel-ativistic collapse dynamics quantitatively well for not toorapid rotation. There is still good qualitative agreementwhen the rotation becomes fast, in contrast to Newtoniansimulations with regular Newtonian potential (Case N),which mostly fail even qualitatively. Multi-dimensionalsimulations with an effective relativistic potential (whichin general does not satisfy the Poisson equation), however,fulfill strict momentum conservation only for special casesof symmetry (for details, see Sect. 3.3).

Although our Cases A and B yield results of similarquality for all the test problems considered here and nei-ther is preferred when a simple equation of state is used,the situation is more in favour of Case A when micro-physics is taken into account in the equation of state.When particles can annihilate or nuclear interactions takeplace, rest mass energy is converted into internal energyand vice versa. In such a situation baryon number andlepton number are conserved, but not the total rest massof the particles, and only the total (“relativistic”) energydensity, ρ + e, is a well defined quantity, but not the in-dividual energies. Case B therefore becomes ambiguous(see Sect. 2.2) while Case A does not suffer from suchproblems. We therefore are tempted to recommend usingCase A as an effective relativistic potential in Newtonianhydrodynamics codes. The extra factor Γ < 1 in the in-tegrand of the TOV mass, Eq. (4), reduces the mass in-

tegral below the mass equivalent of the total (rest massplus internal) energy. Its introduction may be justified byheuristic arguments and consistency considerations whenthe relativistic concept of the gravitating mass is appliedin a Newtonian description of the dynamics, which doesnot distinguish between proper volumes and coordinatevolumes (see Sect. 2.3).

Acknowledgements. We thank M.-A. Aloy and M. Obergau-linger for helpful discussions and M. Rampp for his support.We are particularly grateful to the referee, M. Liebendorfer,for his interest in our work and his knowledgeable comments,which helped us to improve our manuscript. We also thankhim for providing us with data of his general relativistic run.This work was supported by the German Research FoundationDFG (SFB/Transregio 7 “Gravitationswellenastronomie” andSFB 375 “Astroteilchenphysik”). The simulations were per-formed at the Max-Planck-Institut fur Astrophysik and theRZG Rechenzentrum in Garching.

Appendix A: Construction of an “effective”

one-parameter equation of state

The relativistic TOV solution for a spherical, self-gravitating matter distribution in equilibrium can befound by solving the following two coupled ordinary dif-ferential equations for the pressure P and the mass m:

∂P

∂r= −ρ(1 + ǫ) + P

r2(m+ 4πr3P

)(1− 2m

r

)−1

, (A.1)

∂m

∂r= 4πρ(1 + ǫ)r2. (A.2)

with appropriate boundary conditions at the inner andouter boundary at r = 0 and r = R, respectively. Notethat for a TOV solution the massm(R) is equivalent to thetotal gravitational mass Mg of the matter configuration.

The above system is closed by choosing a two-parameter EoS P (ρ, ǫ) and assuming a distribution profilefor the specific internal energy ǫ, which effectively trans-forms the EoS into a one-parameter relation, P = P (ρ) [oralternatively ρ = ρ(P )], from which the specific internalenergy can be determined as ǫ = ǫ(P, ρ). A typical exam-ple for this is to demand a polytropic relation for P andto specify the internal energy ǫ by the ideal gas EoS:

P = Kργ , ρ =

(P

K

)1/γ

,

−→P = ρǫ(γ − 1), ǫ =

P

ρ(γ − 1).

(A.3)

For a two-parameter EoS of the form of Eqs. (12, 13),where a priori no profile for the specific internal energyis known, the following procedure can be applied to con-struct an “effective” one-parameter EoS. For matter inequilibrium, e.g., at late times long after the core bouncephase, both the radial profiles of the pressure P (r) andthe density ρ(r), as well as the specific internal energyǫ(r) can be assumed to be stationary. If the pressure pro-file is monotonic, then for each value of P there exists

A. Marek et al.: An improved effective potential for supernova simulations 19

a unique location r. By inverting the pressure profile,P (r) −→ r(P ), we can thus relate a specific value ρ(r)and ǫ(r) to each value of P . This way a one-parameterEoS ρ = ρ(r(P )) [and ǫ = ǫ(r(P ))] can be constructed.

The excellent matching of the density profiles in thecentral part of a collapsed core in equilibrium obtainedfrom a dynamic evolution with profiles from a TOV solu-tion using equal central density and the EoS transformedas described above (see Figs. 3 to 5) demonstrates theapplicability of this method.

Appendix B: Radial coordinate transformation

To compare a solution of the TOV equations (A.1, A.2),which are formulated in standard Schwarzschild-like coor-dinates, to results from our evolution code using relativis-tic gravity, which is based on radial isotropic coordinates,we must perform a coordinate transformation of the radialcoordinate.

In standard Schwarzschild-like coordinates, the line el-ement ds2st reads

ds2st = −Adt2st +B dr2st + r2st dΩ2st, (B.1)

while the line element ds2iso in radial isotropic coordinatesis given by

ds2iso = −α2 dt2iso + φ4(dr2iso + r2isodΩ

2iso

). (B.2)

In vacuum outside the matter distribution, the metriccomponents become

A = 1− 2Mg

rst, B =

(1− 2Mg

rst

)−1

= A−1 (B.3)

and

α =1− Mg

2riso

1 +Mg

2riso

, φ = 1 +Mg

2riso, (B.4)

respectively.As with both metrics one can describe a spherical mat-

ter distribution in equilibrium, equating the line elements,ds2st = ds2iso, yields conditions for each of the coordinatesand metric components in such a case.

Setting the two coordinate times equal, we obtain acondition for A and α:

Adt2st = α2 dt2iso,

dtst = dtiso,

−→

√A = α. (B.5)

A similar condition can be obtained for the angle element:

r2st dΩ2st = φ4r2isodΩ

2iso,

dΩst = dΩiso,

−→ rst = φ2riso. (B.6)

From the radial elements, we get a relation for the radialdifferentials:√B drst = φ2 driso. (B.7)

Knowing the solution in the isotropic radial coordi-nate, it is straightforward to derive a simple expressionfor rst(riso) from Eq. (B.6).

On the other hand, obtaining the inverse relationriso(rst) is more complicated. For this, we insert Eq. (B.6)into Eq. (B.7) and arrive at

drisodrst

=

√B

φ2=

risorst

√B. (B.8)

This differential equation can be solved by numericalintegration using, e.g., a Runge–Kutta integration schemewith the following boundary conditions:

r0 iso = r0 st = 0, (B.9)

Riso =Rst

2

(1− Mg

Rst

+

√1− 2Mg

Rst

). (B.10)

An alternative method is to rewrite Eq. (B.8) in termsof ln riso,

d ln risodrst

=

√B

rst, (B.11)

which can be directly integrated numerically.

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