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Neutron star-black hole mergers with a nuclear equation of state and neutrino cooling:Dependence in the binary parameters

Francois Foucart,1 M. Brett Deaton,2 Matthew D. Duez,2 Evan O’Connor,1 Christian D. Ott,3

Roland Haas,3 Lawrence E. Kidder,4 Harald P. Pfeiffer,1,5 Mark A. Scheel,3 and Bela Szilagyi3

1Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada2Department of Physics & Astronomy, Washington State University, Pullman, Washington 99164, USA

3Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA4Center for Radiophysics and Space Research, Cornell University, Ithaca, New York, 14853, USA

5Canadian Institute for Advanced Research, 180 Dundas St. West, Toronto, ON M5G 1Z8, Canada

We present a first exploration of the results of neutron star-black hole mergers using black hole masses inthe most likely range of7M⊙–10M⊙, a neutrino leakage scheme, and a modeling of the neutron star materialthrough a finite-temperature nuclear-theory based equation of state. In the range of black hole spins in whichthe neutron star is tidally disrupted (χBH

>∼ 0.7), we show that the merger consistently produces large amounts

of cool (T <∼ 1MeV), unbound, neutron-rich material (Mej ∼ 0.05M⊙–0.20M⊙). A comparable amount of

bound matter is initially divided between a hot disk (Tmax ∼ 15MeV) with typical neutrino luminosityLν ∼1053 erg/s, and a cooler tidal tail. After a short period of rapid protonization of the disk lasting∼ 10ms, theaccretion disk cools down under the combined effects of the fall-back of cool material from the tail, continuedaccretion of the hottest material onto the black hole, and neutrino emission. As the temperature decreases, thedisk progressively becomes more neutron-rich, with dimmerneutrino emission. This cooling process shouldstop once the viscous heating in the disk (not included in oursimulations) balances the cooling. These mergersof neutron star-black hole binaries with black hole massesMBH ∼ 7M⊙–10M⊙ and black hole spins highenough for the neutron star to disrupt provide promising candidates for the production of short gamma-raybursts, of bright infrared post-merger signals due to the radioactive decay of unbound material, and of largeamounts of r-process nuclei.

PACS numbers: 04.25.dg, 04.40.Dg, 26.30.Hj, 98.70.-f

I. INTRODUCTION

The coalescence and merger of black holes and neutronstars in binary systems is one of the main sources of grav-itational waves that is expected to be detected by the nextgeneration of ground-based detectors (Advanced LIGO [1]and VIRGO [2], KAGRA [3]). If at least one of the mem-bers of the binary is a neutron star, bright post-merger elec-tromagnetic signals could also be observable. For example,the formation of hot accretion disks around remnant blackholes provides a promising setup for the generation of shortgamma-ray bursts (SGRB), while the radioactive decay of un-bound neutron-rich material could power an infrared transientdays after the merger (‘kilonova’) [4]. Numerical simulationsof these mergers in a general relativistic framework are re-quired both to model the gravitational wave signal aroundthe time of merger, and to determine which binaries can pro-duce detectable electromagnetic signals and how the proper-ties of these signals are related to the physical parametersofthe source.

These two objectives demand very different types of simu-lations. To model the gravitational wave signal, long and veryaccurate simulations of the last tens of orbits before mergerare required. But during this phase, the important physicaleffects can be recovered using simple models for the neutronstar matter (e.g. Gamma-law, piecewise polytrope) [5, 6]. Onthe other hand, to assess whether a given binary can powerdetectable electromagnetic signals and to predict nucleosyn-thesis yields, shorter inspiral simulations are acceptable but adetailed description of a wider array of physical effects isre-

quired: magnetic fields, neutrino radiation, nuclear reactionsand the composition and temperature dependence of the prop-erties of neutron-rich, high-density material all play an im-portant role in the post-merger evolution of the system. Anyejected material also has to be tracked far from the merger site,thus requiring accurate evolutions of the fluid in a much largerregion than during inspirals. The simulations presented herefocus on the second issue in the context of neutron star-blackhole (NSBH) mergers.

General relativistic simulations have only recently beguntoinclude the effects of neutrino radiation on the post-mergerevolution of the remnant of binary neutron star (BNS) [7–9] and NSBH [10] mergers through the use of ‘leakage’schemes providing a simple prescription for the cooling of thefluid through neutrino emission. These schemes are largelybased on methods already used in the simulation of post-merger remnants by codes with approximate treatments ofgravity [11–14]. More advanced (and computationally ex-pensive) methods based on a moment expansion of the radia-tion fields [15] have also been developed for general relativis-tic simulations of binary mergers [16]. An energy-integratedversion of the moment formalism was recently used for thefirst time to study BNS mergers [17]. Both general rela-tivistic and non-relativistic simulations have shown thatneu-trino cooling can play an important role in the evolution ofthe disk. Emission and absorption of neutrinos can also af-fect the composition of the ejecta, and the mass of heavy el-ements produced as a result of r-process nucleosynthesis inthe unbound material [17–19]. Finally, energy deposition byneutrino-antineutrino annihilation in the baryon-poor region

2

above the disk and near the poles of the black hole could playa role (either positive or negative) in the production of shortgamma-ray bursts [20].

Magnetic fields are also expected to play a critical role inthe evolution of NSBH remnants. Simulations show mag-netic effects to be unimportant before merger for realisticfieldstrengths [21, 22]. However, if the merger produces an accre-tion disk, the disk will be subject to the magneto-rotationalinstability (MRI) [23], which will induce turbulence, leadingto angular momentum transport and energy dissipation thatdrives the subsequent accretion. Most NSBH simulations withmagnetic fields fail to resolve MRI growth, but it is seen if asufficiently strong poloidal seed field is inserted [24].

Finally, the equation of state of the fluid used to modelthe neutron star plays an important role before, during andafter a NSBH merger. Before merger, it determines the re-sponse of the neutron star to the tidal field of the black hole,which, for low mass black holes, could cause measurabledifferences in the gravitational wave signal [6, 25]. Duringmerger, it determines whether the neutron star is disruptedby the black hole (mostly by setting the radius of the neu-tron star), allowing the formation of an accretion disk andthe ejection of unbound material, or whether it just falls di-rectly into the black hole. Different equations of state canalso lead to different qualitative features for the evolutionof the tidally disrupted material. After merger, knowing thetemperature and composition dependence of the equation ofstate is necessary to properly evolve the forming accretiondisk. Until recently, most general relativistic simulations usedGamma-law or piecewise-polytropic equations of state, whichcan only provide us with accurate results up to the disruptionof the neutron star. For post-merger evolution, composition-and temperature-dependent nuclear-theory based equations ofstate are required both to properly model the properties of thefluid and to be able to compute its composition and its interac-tion with neutrinos. Only two general relativistic simulationsof NSBH mergers using such equations of state have beenpresented so far. First, a single low-mass case without neu-trino radiation [26] showed only moderate differences withan otherwise identical simulation using a simpler Gamma-law equation of state. More recently, a first simulation withour neutrino leakage scheme for a relatively low mass, highlyspinning black hole [10] indicated that the effects of neutrinocooling were more important than the details of the equationof state. Simulations using zero-temperature, nuclear-theorybased equations of state were also reported as part of a cou-ple of studies of the radioactive emission coming from un-bound neutron-rich material [27, 28], but only a few generalproperties of the ejecta have been provided at this point. Ac-cordingly, our best estimates for the dependence of the mergerand post-merger evolution of NSBH binaries on the equationof state of nuclear matter come from a set of studies usingsimpler models for the nuclear matter [26, 29, 30], whichneed to be complemented by simulations using temperature-dependent, nuclear-theory based equations of state.

In this paper, we study with the SpEC code [31] themerger and post-merger evolution of NSBH binaries for blackholes of massMBH = 7M⊙–10M⊙ (around the current es-

timates of the peak of the mass distribution of stellar massblack holes [32, 33]). We use a nuclear-theory based equa-tion of state with temperature and composition dependenceof the fluid properties, and a leakage scheme to approximatethe effects of neutrino cooling. Given the computational costof the numerical simulations, we limit ourselves to a singleequation of state (LS220 equation of state from Lattimer &Swesty [34]) and only consider relatively low mass neutronstars (MNS = 1.2M⊙–1.4M⊙). We also use black holeswith spins high enough for the neutron star to be tidally dis-rupted before plunging into the black hole (dimensionlessspinsχBH = 0.7–0.9, as estimated from simulations usingsimpler equations of state [35]). Indeed, for lower spin blackholes, the post-merger evolution is trivial and we do not ex-pect the production of appreciable nucleosynthesis outputordetectable electromagnetic signals. For the same black holemassesMBH = 7M⊙ − 10M⊙ but higher black hole spins,on the other hand, the neutron star is tidally disrupted dur-ing the merger. We show that large amounts of neutron-rich, low entropy material are ejected, which will undergorobust r-process nucleosynthesis. Bound material which es-capes rapid accretion onto the black hole forms an accretiondisk, albeit generally of slightly lower mass than for bina-ries of more symmetric mass ratio and comparable black holespins (we findMdisk ∼ 0.05M⊙–0.15M⊙). The disk is ini-tially hot (Tmax ∼ 15MeV), with a high neutrino luminosity(Lν ∼ 1053erg/s). But about10ms after merger, the com-bined effect of the accretion of hot material by the black hole,the fall-back of cold material from the tidal tail, the emissionof neutrinos, and the expansion of the disk causes a rapid de-cline of both the temperature and the luminosity. In a realisticdisk, in the inner regions where neutrino cooling is efficient,this decline would presumably stop when the neutrino cool-ing and the viscous heating due to MRI turbulence roughlybalance each other. However, our simulations at this point donot include magnetic effects, and would not resolve the MRIeven if they did. And we do not include any parametrizedviscosity either. Accordingly, here the disk continues to cooluntil we end the simulation, about40ms after merger. Withboth a massive, hot accretion disk and a significant amountof ejected unbound material, NSBH mergers in this part ofthe parameter space would thus be prime candidates for theemission of both prompt (e.g. short gamma-ray bursts) anddelayed (e.g. ‘kilonovae’) electromagnetic signals.

We find that the impact of using the LS220 equation of stateappears, for the lower mass black holes, fairly weak beforemerger. Most of the difference in post-merger evolution com-pared to previous simulations using neutron stars of similarradii but simpler equations of state can be attributed to theeffect of neutrino cooling. For higher mass black holes, onthe other hand, the disruption occurs very close to the blackhole horizon, and the detail of the tidal response of the neutronstar can cause more significant differences. The disruptionoc-curs later, faster, and creates a much narrower tidal tail for theLS220 equation of state than for the commonly used Gamma-law equation of state withΓ = 2. These differences also causethe disruption of the neutron star to be extremely hard to re-solve numerically, and require the use of a much finer grid

3

10 11 12 13 14 15

RS (km)

1

1.5

2

2.5

MA

DM

(M

Ò)

Γ=2 polytropeLS220H-SoftH-InterH-Stiff

FIG. 1: Mass-radius relationship for the LS220 equation of state, the3 equations of state presented in Hebeler et al. [36], and aΓ = 2polytrope of similar radius for a1.4M⊙ star. We consider the ADMmass of isolated neutron stars, and their areal radius. Dashed linesindicate the masses used in this work. The H-Soft and H-Stiffequa-tions of state bracket the ensemble of mass-radius relationships ob-tained for the family of equations of state presented in [36]. Theseequations of state match both neutron star mass constraintsand nu-clear physics constraints from chiral effective field theory. H-Inter isa representative intermediate case.

close to the black hole horizon than what is typically used ingeneral relativistic simulations of NSBH mergers (see Sec.II).

We will begin with a discussion of the numerical methodsused, the chosen initial configurations and an estimate of theerrors in the simulations (Sec. II). We then discuss the generalproperties of the disruption and the evolution of the accre-tion disk (Sec. III), before a more detailed presentation oftheneutrino emission from the disk, and of the evolution of itscomposition (Sec. IV), and a summary of the expected evo-lution of the disk over timescales longer than our simulations(Sec. V). Finally, Sec. VI briefly discusses the gravitationalwave signal.

Unless units are explicitly given, we use the conventionG = c = 1, whereG is the gravitational constant andc thespeed of light.

II. NUMERICAL SETUP

A. Equation of state

As in our first study of NSBH mergers including neutrinocooling [10], we model the nuclear matter using the Lat-timer & Swesty equation of state [34] with nuclear incom-pressibility parameterK0 = 220MeV and symmetry energySν = 29.3MeV (hereafter LS220), using the table availableon http://www.stellarcollapse.org and described in O’Connor& Ott 2010 [37]. This equation of state lies within the al-lowed range of neutron star radii, as determined by Hebeleret al. [36] from nuclear theory constraints and the existenceof neutron stars of mass∼ 2M⊙ [38, 39]. Although it is

0 2 4 6 8 10 12

riso

(km)

0

0.5

1.0

1.5

ρ 0 (10

15 g

cm

-3) Γ=2 polytrope

LS220H-SoftH-InterH-Stiff

FIG. 2: Density profile as a function of radius (in isotropic coordi-nates) for the LS220 equation of state, the 3 equations of state pre-sented in Hebeler et al. [36], and aΓ = 2 polytrope of similar radiusfor a1.4M⊙ star. Note that in isotropic coordinates, the radius of thesurface is not equal to the circumferential radius.

TABLE I: Properties of the neutron stars before tidal effects becomestrong.Mb

NS is the baryonic mass,MNS the ADM mass of the star,if isolated,RNS the circumferential radius, andCNS ≡ MNS/RNS

the compactness.

MNS RNS CNS MbNS

1.20M⊙ 12.8 km 0.139 1.31M⊙

1.40M⊙ 12.7 km 0.163 1.55M⊙

not fully consistent with the most recent constraints from nu-clear experiments (in particular, measurements of the GiantDipole Resonance [40, 41]), the LS220 equation of state pro-duces neutron stars over a range of masses with structuralproperties within the limits set by chiral effective field the-ory (see Figs. 1-2). Using the LS220 equation of state is, atthe very least, a significant step forward from Gamma-lawequations of state, or even from the temperature-dependentnuclear-theory based equation of state first used in NSBH sim-ulations [26, 42], which do not meet these constraints. Tabu-lated equations of state which do satisfy all known constraintshave recently been developed (see e.g. [41]) and are availablein tabulated form on stellarcollapse.org. Those equationsofstate will be the subject of upcoming numerical studies.

Table I summarizes the properties of the two neutron starsthat we use in this work (with massesMNS = 1.2M⊙ andMNS = 1.4M⊙). With radii of ∼ 12.7 km, they lie withinthe range of sizes allowed by chiral effective field theory con-straints (see Fig. 1) [36]. The most recent measurementsof neutron star radii in X-ray binaries [43] and the millisec-ond pulsar PSR J0437–4715 [44] are also consistent withRNS ∼ 12.7 km (but see [45] for predictions of more com-pact neutron stars).

B. Initial configurations

The initial conditions for our simulations are chosen so thatthe merger leads to the disruption of the neutron star, and thus

4

potentially to the formation of an accretion disk and the ejec-tion of unbound material. We additionally require the blackhole mass to be within the range currently favored by ob-servations of galactic black holes [32, 33] and by populationsynthesis models [46, 47],MBH ∼ 7M⊙ − 10M⊙. In thismass range, the neutron star will only be disrupted for rapidlyspinning black holes and large neutron stars. An approximatethreshold for the disruption of the neutron star to occur is in-deed [35]

CNS<∼

(

2 + 2.14q2/3RISCO

6MBH

)−1

, (1)

whereCNS = MNS/RNS is the compactness of the star,q = MBH/MNS is the mass ratio, andRISCO is the radius ofthe innermost stable circular orbit for an isolated black holeof the same mass and spin. For a canonical neutron star ofmass1.4M⊙ described by the LS220 equation of state, this re-quires a dimensionless spinχBH

>∼ 0.75 (resp.χBH>∼ 0.55)

for a black hole of mass10M⊙ (resp.7M⊙). For the LS220equation of state, the critical spin separating disruptingfromnon-disrupting neutron stars increases with the neutron starmass. Accordingly, we will consider black hole spins in therangeχBH = 0.7–0.9 and neutron star masses in the rangeMNS = 1.2M⊙–1.4M⊙.

The simulations presented in this paper are fairly costly:at our standard resolution, a single run might require100, 000–150, 000 CPU-hours (more than 3 months using 48processors). As we are considering a 3-dimensional param-eter space (neutron star and black hole mass, and black holespin magnitude), we can only afford a very coarse coverageof each parameter. In this first parameter space study usingthe LS220 equation of state, we consider 9 simulations cover-ing 2 black hole masses (7M⊙, 10M⊙), 2 neutron star masses(1.2M⊙, 1.4M⊙), and 3 spins (χBH = 0.7, 0.8, 0.9, the lowerspin being only used forMNS = 1.4M⊙, MBH = 7M⊙).The initial parameters for each configuration are summarizedin Table II, while the properties of the neutron star for eachchoice ofMNS are given in Table I. For reference, our firstsimulation using the LS220 equation of state and a leakagescheme for the neutrino radiation used aMNS = 1.4M⊙ neu-tron star, with a less massive (MBH = 5.6M⊙), rapidly rotat-ing (χBH = 0.9) black hole [10]. In the following sections,we will refer to the various simulations by their names listedin Table II, in which the first two numbers refer to the mass ofthe neutron star and the mass of the black hole, and the thirdnumber to the spin of the black hole, e.g. M12-10-S9 corre-spond to a binary withMNS = 1.2M⊙, MBH = 10M⊙ andχBH = 0.9.

We obtain constraint satisfying initial data using the spec-tral elliptic solver Spells [48, 49], at a separation chosentoprovide 5–8 orbits before merger. The initial data use thequasi-circular approximation [49, 50], which causes the bi-nary to be on slightly eccentric orbits (e ∼ 0.03–0.04, seeTable II).

TABLE II: Initial configurations studied. All cases use the LS220equation of state.MBH andχBH are the mass and dimensionless spinof the black hole,MNS the mass of the neutron star,MΩ0

orbit theinitial orbital frequency multiplied by the total massM = MBH +MNS, ande is the initial eccentricity.

Name MBH χBH MNS MΩ0orbit e

M12-7-S8 7M⊙ 0.8 1.2M⊙ 0.0364 0.027M12-7-S9 7M⊙ 0.9 1.2M⊙ 0.0364 0.026M12-10-S8 10M⊙ 0.8 1.2M⊙ 0.0396 0.031M12-10-S9 10M⊙ 0.9 1.2M⊙ 0.0396 0.033M14-7-S7 7M⊙ 0.7 1.4M⊙ 0.0438 0.039M14-7-S8 7M⊙ 0.8 1.4M⊙ 0.0437 0.037M14-7-S9 7M⊙ 0.9 1.4M⊙ 0.0437 0.037M14-10-S8 10M⊙ 0.8 1.4M⊙ 0.0441 0.042M14-10-S9 10M⊙ 0.9 1.4M⊙ 0.0440 0.043

C. Summary of the neutrino leakage scheme

The neutrino leakage scheme used in this work is a firstattempt at including the effects of neutrino radiation on theevolution of the remnant of NSBH mergers and at providingestimates of the properties of the emitted neutrinos (luminos-ity, species, average energy). A leakage scheme is a localprescription for the number and energy of neutrinos emittedfrom a given point in the disk, based on the local propertiesof the fluid and on an estimate of the optical depth. A de-tailed description of our leakage implementation can be foundin Deaton et al. (2013) [10]. It is essentially a general-ization to 3 dimensions of the leakage code implemented inGR1D (O’Connor & Ott 2010 [37]), which is itself inspiredby the work of Ruffert et al. (1996) [11] and Rosswog &Liebendorfer (2003) [12]. Our leakage scheme aims at mod-eling the effects of neutrino cooling, while we do not considerthe effects of neutrino heating.

As a brief summary, the main components of the leakagecode are:

• A local prescription for the free emission of neutrinosas a function of the fluid properties (i.e. emission inthe optically thin regime). We includeβ-capture pro-cesses,e+-e− pair annihilation, plasmon decay [11]and nucleon-nucleon Bremsstrahlung [51]. We com-pute the luminosity and number emission forνe, νeandνx, whereνx stands for all other neutrino species(νµ,τ , νµ,τ ), which are assumed to all have the same in-teractions with the fluid.

• A local prescription to compute the opacity to neutrinosas a function of the fluid properties and of the chemicalpotentials. We take into account scattering on nucleonsand heavy nuclei and absorptions on nucleons [37].

• A prescription for the computation of the optical depthbetween the current point and the boundaries of thedomain. We estimate this by integrating the opaci-ties along the coordinate axes, and along the two mostpromising diagonal directions (i.e. those correspondingto the coordinate axes along which the optical depth is

5

minimum). The minimum optical depth along the se-lected path is taken as the optical depth at the currentpoint (see Sec. II D for an estimate of the influence ofthis choice).

• A local prescription for the rate at which neutrinoscan escape through diffusion (i.e. in the optically thickregime), for which we follow Rosswog & Liebendorfer(2003) [12].

• A choice of an effective emission rate interpolating be-tween the optically thin and optically thick limit. IfQfree is the emission rate in the free emission regimeandQdiff the emission rate in the diffusive regime, weuse

Qeff =QdiffQfree

Qfree +Qdiff

(2)

(for both the energy and number emission rates).

Compared to Deaton et al. 2013 [10], we have implementedtwo modifications to the leakage scheme. First, we computethe neutrino chemical potentials used in the determinationofthe opacities

µν = µeqν (1 − e−〈τν〉) , (3)

whereµeqν is theβ-equilibrium value of the potential and〈τν〉

the energy-averaged optical depth, by iteratively solvingfor(µν ,τν). This is required becauseτν is itself a function of theopacities. In [10], we instead used an analytical approxima-tion for 〈τν〉 as a function of the local properties of the fluidwhen computingµν . Second, instead of treating the blackhole as an optically thick region, we let neutrinos freely es-cape through the excision boundary. In [10], the black holewas treated as an optically thick region to avoid spuriouslyin-cluding the emission from hot points at the inner edge of thedisk in the total neutrino luminosity. In this paper, when com-puting the optical depth along a direction crossing the excisionboundary, we terminate the integration at that boundary. Butpoints for which the direction of smallest optical depth is to-wards the black hole are excluded when computing the totalneutrino luminosity of the disk. The cooling and composi-tion evolution of the fluid are thus properly computed at theinner edge of the disk, but without affecting estimates of theneutrino luminosity. These changes neither appear to signifi-cantly modify the results of the simulations at late times, northe properties or evolution of the post-merger accretion disk.

D. Error estimates

To estimate the errors in the various observables discussedin this paper for simulations withMBH = 7M⊙, we performa convergence test on simulation M14-7-S9, as well as a seriesof tests checking that the treatment of the low-density regions,the boundary condition on the region excised from the numer-ical domain inside of the apparent horizon of the black hole,and the method used to compute the neutrino optical depth do

not affect the results within the expected numerical errors. Forthe simulations withMBH = 10M⊙, which are significantlyharder to resolve, we also simulate the late disruption phaseusing a fixed mesh refinement and a much finer grid and findthat the standard grid choices used for the lower mass ratiocases are no longer appropriate in this regime. Issues specificto the high mass ratio cases are discussed at the end of thissection.

For the convergence test withMBH = 7M⊙, we performadditional simulations at a lower and a higher resolutions.TheSpEC code [31] uses two separate numerical grids [52]: afinite difference grid on which we evolve the general rela-tivistic fluid equations, and a pseudospectral grid on whichwe evolve Einstein’s equations. The finite difference grid isregularly regenerated so that it covers the region in whichmatter is present, but ignores the rest of the volume cov-ered by the pseudospectral grid. The coupling between thetwo sets of equations then requires interpolation from onegrid to the other, which we perform at the end of each timestep. During the plunge and merger, the three resolutionshaveNFD = (1203, 1403, 1603) points on the finite differ-ence grid, which only covers the region in which matter ispresent. This leads to variation of the physical grid spac-ing over time, from∆x ∼ 200m at the beginning of theplunge to∆x ∼ 2 km − 3 km while we follow the ejectedmaterial as it escapes the grid, to finally∆x ∼ 1 km at theend of the simulation. The resolution on the spectral grid onwhich we evolve Einstein’s equations is chosen adaptively,so that the relative truncation error of the spectral expan-sion of the metric and of its spatial derivatives is kept belowǫsp = (1.9 × 10−4, 1 × 10−4, 5.9 × 10−5) at all times (asmeasured from the amplitude of the coefficients of the spec-tral expansion within each subdomain of the numerical grid).During the short inspiral, the finite difference grid coversamuch smaller region (a box of size2RNS ∼ 26 km) and hasslightly less points (NFD = (1003, 1203, 1403)). The trunca-tion error on the spectral grid is kept at about the same level,except close to the black hole and neutron star where we re-quire the truncation error to be a factor of 10 smaller. At theseresolutions, we find that for many quantities the convergenceis faster than second order, which is the expected convergencerate in the high resolution limit when finite difference errorsdominate the error budget. This is presumably due to contri-butions to the error of terms with a higher convergence rate,such as grid-to-grid interpolation (third order), the evolutionof Einstein’s equations (exponentially convergent everywherebut in regions in which a discontinuity is present), or the time-stepping algorithm (third order). When we observe a con-vergence rate faster than second-order, we assume second or-der convergence at the two highest resolutions to get an upperbound on the error. In previous simulations with the SpECcode, we have generally found this error estimate (which isabout 4 times the difference between the results of the highand medium resolutions) to be much higher than the actual er-ror in the results. Using this estimate, we find the followingupper bounds for the errors in our medium resolution runs:

• ∼ 25% relative error in the final disk mass

6

• ∼ 60% relative error in the mass and kinetic energy ofthe ejected material;

• 1% relative error in the mass and spin of the black holeat a given time;

• ∼ 50% relative error in the neutrino luminosity ob-tained from the leakage scheme – smaller than the er-ror due to the use of the leakage algorithm itself, whichmight be a factor of a few (see e.g. [12]).

In the leakage scheme, one potential source of error is thechoice of specific directions over which the optical depth iscomputed (in our case, the coordinate axes and a subset of thediagonals). To assess the effect of that choice, we tried twoalternative methods. In the first test, we computed the opti-cal depth along every coordinate diagonal. In the second, wecomputed the optical depth by finding the path of smallest op-tical depth among all paths leading from the current point toadomain boundary. We do this by computing the optical depthof a pointτp from the optical depth of its neighborsτi andthe optical depth between the point and its neighborsτp−i, i.e.τ = mini(τi + τp−i), and then iterating untilτ converges atall points – which occurs rapidly ifτp is initialized to its valueat the previous time step. This second method is inspired bythe recent work of Perego et al. [53] and Neilsen et al. [9], andclosely follows the algorithm presented in [9]. Both tests leadto changes of∼ 20% in the neutrino luminosities, thus show-ing that the exact method used to compute the optical depth isa fairly minor source of error compared to the systematic errordue to the use of a leakage scheme.

As a last test of our code for simulations withMBH =7M⊙, we consider the effect of the treatment of the low-density region. This region is dynamically unimportant, butcould conceivably affect the neutrino luminosities if it devel-ops a very hot atmosphere. We also study the impact of theblack hole excision by using linear extrapolation of the den-sity and velocities to the faces of our grid located at the bor-der of the excised region instead of just copying the valueof the fields from the nearest cell center, as well as by lim-iting the temperature of the fluid on the excision boundaryto T = 10MeV. The resulting changes in the post-mergerproperties of the system are well below our estimates of thenumerical error due to the finite resolution of our grid.

We now turn our attention to the higher mass ratio config-urations. These simulations are significantly more difficult toresolve due to a number of issues:

• The disruption of the neutron star occurs very close tothe black hole: except for simulation M12-10-S9, thetidal tail begins to form as high-density material has al-ready begun to cross the apparent horizon of the blackhole. But the accuracy of our code is lower close to theapparent horizon, due to the use of one-sided stencilsclose to the excised region inside of the horizon. Addi-tionally, stricter control of the velocities and tempera-ture within the low-density regions is necessary for sta-bility in that excised region. Hence, properly resolvingthe disruption of the neutron star requires higher reso-lution in such cases.

• Compared to previous simulations using a Gamma-law(Γ = 2) equation of state [54], simulations using theLS220 equation of state show important differences inthe qualitative features of the tidal disruption. Initially,the neutron star remains more compact with the LS220equation of state, and most of the expansion of the neu-tron star occurs later in the plunge (see Fig. 3), but morerapidly. This makes the disruption of the neutron starmore difficult to resolve with the nuclear-theory basedequation of state. This is ultimately a consequence ofthe large difference between the stiffness of the LS220equation of state at high and low densities (with respectto nuclear saturation density).

• Once a tidal tail forms, tidal compression is very effi-cient and forces the tail to be much thinner than for aΓ = 2 equation of state (see Fig. 3). This is, again, aconsequence of the properties of the equation of stateat low densities. Because disruption occurs as the neu-tron star is already plunging into the black hole, the ef-fects of tidal compression can be significant. We findthat the tail has a typical coordinate width (and verti-cal height) of only a few kilometers (note that Fig. 3uses a logarithmic scale for the density). This should becompared with the radius of the apparent horizon of theblack hole, which in our gauge is∼ 30 km. Accord-ingly, the typically resolutions used in the simulationof NSBH mergers are insufficient in this case. Indeed,simulations using fixed mesh refinement typically use20–30 grid points across the radius of the black hole(see e.g. [29]), while our finite difference grid generallyhas a higher resolution at the beginning of the disruption(∼ RNS/50 ∼ 0.2 km), but rapidly becomes coarser aswe progressively expand the grid in order to follow theexpansion of the tidal tail. Additionally, our simulationssuffer from lower accuracy in the immediate neighbor-hood of the excised region, as opposed to codes whichdo not excise the inside of the black hole and can usefinite difference stencils crossing the horizon.

Accordingly, as soon as the resolution of our finite differ-ence grid drops below∆x ≈ 0.5 km–1 km, simulations withMBH = 10M⊙ and the LS220 equation of state suffer greatlosses of accuracy. This typically takes the form of a rapidspurious acceleration of the tail material within the unresolvedregion, which then becomes unbound with large asymptoticvelocities (v/c ∼ 0.6–0.9). To resolve the tidal tail duringthis phase requires resolution that our code cannot maintainwhile following the unbound ejecta and bound tidal tail faraway from the black hole. One solution is to use mesh re-finement for the finite difference grid. Although we have im-plemented a fixed mesh refinement algorithm for our finitedifference grid, the cost of interpolating between the differ-ent evolution grids currently makes it impractical to use morethan 2-3 levels of refinement, and thus to resolve accuratelythe material close to the black hole while following the ejectafor a few milliseconds, especially for high mass ratio simu-lations which require a small grid spacing on both our finitedifference and pseudospectral grids. Instead. we choose to

7

FIG. 3: Comparison of the distribution of matter during disruption for the LS220 equation of state used in this work (left, simulation M14-10-S8), and aΓ = 2 equation of state (right, simulation Q7S9-R12i0 from [54]). Snapshots taken in the equatorial plane at a time at which0.25M⊙ of material remains outside of the black hole. Both simulations lead to∼ 0.1M⊙ of material remaining outside of the black hole atlate times - yet the distribution of that material between bound and unbound material and the properties of the tidal tailare very different. Ineach case, the thick black line shows the location of the apparent horizon of the black hole.

continue simulations M12-10-S8 and M12-10-S9 through dis-ruption while maintaining a resolution in the region close tothe black hole of∆x < 0.7 km (and vertically∆x = 0.2 km)for M12-10-S9, with 2 levels of refinement, and∆x < 0.4 km(and vertically,∆x = 0.1 km) for the more challenging M12-10-S8 , with 3 level of refinements. Each refinement level is abox using1403 grid points, and the grid spacing is multipliedby a factor of 2 between levels. Both of those simulationsconfirmed that the rapid acceleration observed at lower res-olution is indeed a numerical artifact. The higher resolutionsimulations eject∼ 0.1M⊙–0.15M⊙ of material at velocitiesv/c <∼ 0.5 . Using a fixed mesh refinement with a resolu-tion similar to the one used in simulation M12-10-S9 provedto be insufficient to resolve the configurations for which thedisruption occurs as the neutron star crosses the apparent hori-zon of the black hole (M12-10-S8, M14-10-S8, M14-10-S9).Clearly, with these grids covering such a small region aroundthe black hole (up to∼ 5MBH ∼ 75 km), we can only mea-sure the ejected mass and the mass remaining outside of theblack hole (and the latter will only be a rough estimate). Wecannot follow the evolution of the disk as it expands and isimpacted by fall-back from the tidal tail. Hence, for the highmass ratio simulations with fixed mesh refinement (M12-10-S8, M12-10-S9), we stop the evolution when an accretion diskforms. Similarly, for the high mass ratio simulations withoutmesh refinement (M14-10-S8, M14-10-S9), we only obtainestimates of the ejected mass and of the mass remaining out-side of the black hole: we cannot reliably continue the evolu-tion once the resolution drops below∆x ≈ 0.5 km–1 km.

E. Tracer particles

A major motivation for the simulations presented here isto characterize the unbound outflow: its mass, asymptotic ve-locity distribution, composition, and entropy. In a stationaryspacetime and in the absence of pressure forces, a fluid ele-ment with 4-velocityuα is unbound if it has positive specificenergyE = −ut − 1 (under those conditions,E is constant).In our simulations, we consider a fluid element as unbound ifE > 0 as it leaves the grid – even though it has a finite pres-sure and the metric is not yet completely stationary. Thus, an-other potential source of error in our conclusions comes fromthe finite outer boundary of our fluid evolution grid.

To test that the outflow reaches its asymptotic state beforeexiting the simulation, we evolve one hundred Lagrangiantracer particles in the weakly bound and unbound flow. Thatis, the position of tracer particle A is evolved according tothelocal 3-velocity: dxi

A/dt = vi(xiA), and fluid quantities at

tracer positions are monitored by interpolating from the fluidevolution grid.

In Fig. 4, we plot some fluid quantities for four representa-tive randomly chosen unbound particles in the ejecta of caseM12-7-S9. We see that the energies stabilize after only abouttwo milliseconds of post-merger evolution. The electron frac-tion Ye becomes stationary the soonest because the tail is toocold for strong charged-current interactions that would changeYe. Entropy also levels off, albeit more slowly. Over the4msof evolution, the density decreases by about three orders ofmagnitude.

We are thus confident that near-merger gravitational fluc-tuations, pressure forces, and neutrino emission do not sig-

8

0.01

1

100ρ(

1012

g cm

-3)

0

4

8

S (

k B)

0.06

0.08

Ye

1 2 3 4

t-tmerge

(ms)

-0.05

0

0.05

E

FIG. 4: The evolution of four representative tracer particles repre-senting unbound fluid elements. We plot the baryon densityρ, thespecific entropyS, the electron fractionYe, and the specific energyE ≡ −ut − 1. The small glitches in the entropy evolution are arti-facts of stencil changes with the low-order interpolator used for La-grangian output. Note, the tracer particle denoted by the solid blackline begins closest to the black hole.

nificantly contaminate our predictions about the outflows. Itis important to note that the entropy and energy may changesignificantly outside the evolved region because of recom-bination and nucleosynthesis in the decompressing material.Charged current heating from the background neutrino field,which we do not include in our simulations, can also changethe composition of the ejecta, particularly for material un-bound by disk winds (as opposed to the material ejected dur-ing the disruption of the neutron star). The late-time, large-distance evolution of the ejecta is a matter of ongoing study.

III. DISRUPTION, DISK FORMATION AND OUTFLOWS

A. Disruption of the neutron star

Let us now consider the results of these simulations, start-ing with the disruption and merger dynamics. At the time oftidal disruption, the results are largely unaffected by neutrinos(which act over timescales∼ 10ms, while the merger takes∼ 1ms). Instead, the main components required to properlymodel the system are the size and structure of the neutron star,and general relativity.

We choose the initial parameters of these simulations sothat the neutron star is disrupted, thus allowing the formationof an accretion disk and/or the ejection of material. This isindeed what we observe. The mass remaining outside of theblack hole after disruption (including unbound material) and

the final black hole mass and spin are in agreement with previ-ous numerical studies using simpler equations of state, eitherin the same range of parameters [54], or for slightly lowerblack hole spins [29]. They are also largely consistent withsemi-analytical models mostly based on lower mass simula-tions with simpler equations of state [35, 55]. However, goingbeyond these global properties of the post-merger remnant,we observe differences in the qualitative features of the dis-ruption phase, modifying the distribution of the material re-maining outside of the black hole between unbound material,bound tail and accretion disk (summarized in Table III).

The systems with the more symmetric mass ratio (q = 5,with MNS = 1.4M⊙ andMBH = 7M⊙) provide the mosttraditional results: between0.15M⊙ and0.35M⊙ of mate-rial remains outside of the black hole5ms after merger. Thehigher black hole spins naturally lead to more massive rem-nants, as expected from the fact that the innermost stable cir-cular orbit is at a lower radius for prograde orbits around amore rapidly spinning black hole. We find that the systemejects0.04M⊙–0.07M⊙ of material, which appears compati-ble with the range of ejected mass listed for NSBH binariesat χBH = 0.75 by Hotokezaka et al. [27]: although [27]does not list values for individual systems, they find a rangeMej ∼ 0.02M⊙–0.04M⊙ for the neutron star closest in sizeto the1.4M⊙ neutron star used in this work and mass ratiosq = 3–7. The higher masses found here can easily be at-tributed to larger black hole spins. The average velocity oftheejecta is〈v〉 ∼ 0.2c, where the average〈X〉 of a quantityXon a spatial sliceΣ is here defined as

〈X〉 =∫

ΣρXW

√gdV

∫

ΣρW

√gdV

, (4)

ρ is the baryon density of the fluid,W its Lorentz factor,and g is the determinant of the 3-metric on sliceΣ. Thisis slightly lower than in [27] (〈v〉 ∼ 0.25c–0.29c). As forthe bound material,20ms after merger0.04M⊙–0.14M⊙

has formed a hot accretion disk cooled by neutrinos (dis-cussed in Sections III C-IV),0.04M⊙ − 0.05M⊙ has left thegrid on highly eccentric bound trajectories, while the rest(0.02M⊙–0.10M⊙) has accreted onto the black hole. Ex-trapolating the remnant mass to lower spins indicate that theneutron star will disrupt forχBH

>∼ 0.55, as predicted fromsimulations with simpler equations of state [35].

At higher mass ratios, we generally find that a largeramount of material is ejected. Our two simulations withMNS = 1.2M⊙ and MBH = 7M⊙ find about the sameamount of material in the disk and bound tail as for theq = 5 cases (Mdisk ∼ 0.1M⊙, Mtail ∼ 0.05M⊙), but about0.15M⊙ is ejected at speeds〈v〉 ∼ 0.25c (see Fig. 5). Theclosest results to compare to are again those of Hotokezaka etal. [27], for the H4 equation of state, which has a compactnessCNS = 0.147 for a massMNS = 1.35. Hotokezaka et al. [27]findMej = 0.04M⊙–0.05M⊙ for q = 3−7 andχBH = 0.75,and similar average velocities. Even considering the differ-ent spins used and the expected error bars, the more massiveejecta found here appear to be an indication of a dependenceof the ejected mass on the internal structure of the neutronstar: from our results, we would predictMej ∼ 0.13M⊙ for

9

TABLE III: Properties of the final remnant.MfBH is the mass of the black hole at the end of the simulation,χf

BH is its dimensionless spin,vkick,GW is the recoil velocity due to gravitational wave emission (i.e. not taking into account the larger recoil from the asymmetric ejectionof material), andM5ms

out is the baryon mass outside of the black hole5ms after merger (including unbound material).M20msdisk , M20ms

tail,bound

andMej are the baryon mass of the disk, bound tail material, and unbound material20ms after merger. By conservation of baryon number,M5ms

out −M20msdisk −M20ms

tail,bound−Mej is the mass accreted onto the black hole in the15ms between the two sets of measurements.Eej and〈vej〉are the kinetic energy and average asymptotic velocity of the unbound material. Fluid quantities for the higher mass ratio simulations (secondhalf of the table) generally have larger uncertainties, as discussed in Sec. II D, and are only provided when they can be reliably extracted fromthe simulation.

Name MfBH χf

BH vkick,GW (km/s) M5msout M20ms

disk M20mstail,bound Mej Eej(10

51erg) 〈vej〉/cM12-7-S8 7.79M⊙ 0.85 60 0.31M⊙ 0.09M⊙ 0.04M⊙ 0.14M⊙ 8.5 0.24M12-7-S9 7.74M⊙ 0.91 69 0.37M⊙ 0.13M⊙ 0.03M⊙ 0.16M⊙ 11 0.25M14-7-S7 8.07M⊙ 0.81 79 0.15M⊙ 0.04M⊙ 0.05M⊙ 0.04M⊙ 1.7 0.20M14-7-S8 8.00M⊙ 0.87 76 0.25M⊙ 0.08M⊙ 0.05M⊙ 0.06M⊙ 2.0 0.18M14-7-S9 7.95M⊙ 0.92 94 0.35M⊙ 0.14M⊙ 0.04M⊙ 0.07M⊙ 2.5 0.18M12-10-S8 10.75M⊙ 0.83 45 ∼ 0.25M⊙ – – ∼ 0.10M⊙–0.15M⊙ – –M12-10-S9 10.58M⊙ 0.89 49 ∼ 0.40M⊙ – – ∼ 0.10M⊙–0.15M⊙ ∼ 10–20 ∼ 0.3 −−0.35M14-10-S8 11.00M⊙ 0.85 22 ∼ 0.10M⊙ – – ∼ 0.05M⊙–0.10M⊙ – –M14-10-S9 10.70M⊙ 0.90 54 ∼ 0.30M⊙ – – ∼ 0.10M⊙–0.15M⊙ – –

FIG. 5: Matter distribution during the disruption of the neutron starfor case M12-7-S8. About half of the remnant material is unbound,while a relatively low mass hot disk forms.

q = 5.8 andχBH = 0.75. The difference with the results ofHotokezaka et al. [27] is slightly out of the60% relative errorthat we consider to be a strict upper bound on the error in ourmeasurement ofMej, and which is most likely a significantoverestimate of that error. The most likely reason for an equa-tion of state dependence of the ejected mass at higher massratios is that the disruption of the neutron star then occursasthe neutron star is plunging into the black hole. The mass andproperties of the ejected material then depend not only on theseparation at which disruption occurs, but also on the time-dependent response of the neutron star to the tidal disruption.For the same reason, the properties of the post-merger rem-nant also have a steeper dependence in the mass and spin ofthe black hole for more massive black holes.

As discussed in Sec. II D, for the simulations withMBH =

10M⊙ we can only resolve the disruption of the neutron starand rapid accretion onto the black hole if we use a very highresolution grid covering only a small area around the blackhole. This is due largely to qualitative differences in the dis-ruption process: the neutron star disrupts as high-densityma-terial has already begun to cross the apparent horizon, andthe entire disruption and tail formation process occurs withina distance of about3MBH ∼ 45 km of the black hole center.All material surviving the merger is on highly eccentric orbits,and the tidal tail experiences strong tidal compression in thedirections in which the tidal field of the black hole causes tra-jectories to converge. The tidal tail is reduced to a thin streamof matter only a few kilometers wide. Not surprisingly, in thisregime the result of the merger is very sensitive to the param-eters of the binary: for example, a small change in the spin ofthe black hole (e.g.δχBH ∼ 0.1, as in our simulations) dras-tically changes the amount of material which remains outsideof the black hole after disruption. Changes in the stiffnessofthe equation of state, which affect the distribution of matterduring disruption, also become even more important than inthe previously discussed configurations (see Fig. 3 for a com-parison with a similar simulation with aΓ = 2 equation ofstate). Surprisingly, even in this case, the total amount ofma-terial surviving disruption remains similar for both theΓ = 2and LS220 equations of state. We can for example comparesimulation M14-10-S9 of this work, for which about0.3M⊙

remains outside the black hole after merger, and simulationR13i0 of Foucart et al. 2013 [54], with the same black holeparameters and a similar neutron star radiusRNS = 13.3 km,in which 0.31M⊙ of material remains. But more materialis unbound during merger for the LS220 equation of state:Mej

>∼ 0.1M⊙ here, whileMej ∼ 0.05M⊙ in Foucart et al.2013 [54]. And the disk is initially less massive here: half ofthe material promptly formed a disk in [54], nearly all of thematerial is on highly eccentric orbits here. The long term evo-lution of the disk will thus be more significantly affected bythe fall-back of tail material than expected from the simula-tions with aΓ = 2 equation of state. Compared to the analyti-cal prediction of [35], there is an excess of material surviving

10

0 10 20 30 40

t-tmerge

(ms)

0.01

0.1

1

Mb (M

Ò)

M12-7-S8M12-7-S9M14-7-S7M14-7-S8M14-7-S9

Ejecta leaves grid

Late accretion

FIG. 6: Baryon mass left on the numerical grid as a function oftime,for all simulations withMBH = 7M⊙.

the disruption, which is what we usually find for high spinblack holes and high mass ratio. But the estimate that dis-ruption will only occur forχBH

>∼ 0.65 (resp.χBH>∼ 0.75)

for MNS = 1.4M⊙ (resp.1.2M⊙) appears to be accurate –although it is of course dangerous to draw such a conclusionby extrapolating from only 2 simulations with inaccurate esti-mates of the remnant mass.

All of the simulations presented here thus appear to havepost-merger remnants with large amounts of both bound andunbound material, providing promising setups for potentialelectromagnetic signals. The typical timescales for the mergercan be observed on Fig. 6, which shows the baryon massremaining on the grid at any given time for all simulationswith MBH = 7M⊙. The disruption of the neutron star andrapid accretion of material onto the black hole occur over∼ 1ms. Then accretion slows down and the disk evolvesover timescales of∼ 0.1s. The rapid variation in the masspresent on the grid around5ms–10ms after merger is due tothe unbound material leaving the grid (and discrete jumps aredue to modifications of the location of the outer boundary ofthe grid as the algorithm in charge of deciding which regionsof the spacetime to cover abandons low-density regions con-taining unbound or marginally bound material). Not shownare the simulations forMBH = 10M⊙, for which most of thematerial escaping early accretion onto the black hole rapidlyleaves the grid. In the following sections, we will now lookin more detail into the properties of the unbound material andthe evolution of the disk.

B. Properties of the outflows

Given the large mass of material ejected by these merg-ers, NSBH binaries withMBH ∼ 7M⊙–10M⊙ appear tobe prime candidates to produce post-merger electromagneticcounterparts, and large quantities of r-process elements (seeTable III). Although the mass and velocity of these outflowsvary from case to case, other properties of the ejecta are more

0 0.1 0.2 0.3 0.4 0.5

v/c

0

10

20

30

Mas

s P

erce

ntag

e

M14-7-S9

FIG. 7: Distribution of asymptotic velocities for the unbound mate-rial in simulation M14-7-S9. Other simulations withMBH = 7M⊙

have similar distributions, with a slightly higher peak value whenMNS = 1.2M⊙.

consistent. Since the material is ejected before neutrino emis-sion has had a chance to modify its composition, it has a lowelectron fraction (Ye < 0.1). The neutron star matter initiallyhas a low entropy per baryon, and is not significantly heatedduring the tidal compression of the tail and the ejection of ma-terial: the entropy per baryon of the ejecta for the simulationswith MBH = 7M⊙ is 〈S〉/kB ∼ 4− 5, and it is only slightlyhigher for simulations withMBH = 10M⊙ (〈S〉/kB ∼ 5−7).The outflow is expected to robustly undergo r-process nucle-osynthesis, with the production of material withA > 120insensitive to variations in the properties of the outflow withinthe range of values observed here [19]. Accordingly, all ofthe simulations considered here are promising setups for theproduction of infrared transients [4, 19] and heavy r-processelements.

The asymptotic velocity distribution of the ejecta (its ex-pected velocity at infinite distance from the black hole) isshown in Fig. 7 for simulation M14-7-S9. For all configu-rations withMBH = 7M⊙, we find similar distributions span-ning 0 < v < 0.5c, with slight variations in the location ofthe peak. A more asymmetric mass ratio generally results inhigher asymptotic velocities for the ejecta. This additionalacceleration is presumably due to the fact that, since the dis-ruption occurs on an eccentric orbit passing close to the blackhole, the fluid can be efficiently accelerated by the rapid com-pression and decompression of the tidal tail under the influ-ence of the tidal field of the black hole (see also Sec. III A).This effect is specific to high mass ratio binaries, for whichthe disruption occurs as the neutron star plunges into the blackhole and the properties of the disrupted material are sensitiveto the equation of state (see Fig. 3).

As pointed out in Kyutoku et al. 2013 [56], the ejectionof unbound material is not isotropic. In our simulations, theejecta is confined within∼ 20 of the equatorial plane, andcovers an azimuthal angle∆φ ∼ π (see Fig. 5). An importantconsequence of this anisotropy is that a kick is imparted ontothe black hole withvkick,ej ∼ Mejvej/MBH. For simulationM12-7-S9, for which we follow tracer particles, we measure

11

0 10 20 30 40

t-tmerge

(ms)

0

5

10

S (

k B)

M12-7-S8M12-7-S9M14-7-S7M14-7-S8M14-7-S9

Disk Formation / Shock Heating

Late accretion / Neutrino cooling

FIG. 8: Average entropy per baryon on the computational gridas afunction of time, for all simulations withMBH = 7M⊙.

the linear momentum carried away by the ejecta and find

vkick,ej ∼ 0.5Mej

MBH

〈v〉ej ∼ 770 km/s . (5)

The factor of0.5 comes from the fact that not all of the ejectais ejected in the same direction. We estimate it from the di-rection ofvi as the tracer particles cross spheres of constantradius. Even considering the uncertainties in our measure-ments ofMej and 〈v〉ej, this is clearly larger than the kickvelocity due to gravitational wave emission (see Table III):our simulations predictvkick,ej ∼ 150 km/s–800 km/s butvkick,GW ∼ 20 km/s–100 km/s. These velocities play an im-portant role when assessing whether globular clusters can re-tain a black hole after it merges with a neutron star, and in thedetermination of the rate of NSBH mergers occurring in thoseclusters [57]. At the high end of this velocity range, the kickcould even be above the escape velocity of small galaxies.

C. Disk evolution

Constraining the formation and the long-term evolution ofthe accretion disks resulting from NSBH mergers is an impor-tant step towards assessing their potential as short gamma-rayburst progenitors. Those disks are also likely to power out-flows (e.g. via neutrino-driven or magnetically-driven winds,or recombination ofα-particles), causing optical and radiosignals.

In the simulations withMBH = 7M⊙, we follow the post-merger evolution of the system. We define the time of mergertmerge as the time at which50% of the neutron star has beenaccreted onto the black hole, and the disk massMdisk as themass remaining in the accretion disk20ms after tmerge. Weevolve the disks for about40ms after tmerge, switching to afixed metric evolution after about15ms. We find a range ofdisk massesMdisk = 0.04M⊙–0.14M⊙. At late times, theaccretion timescale in these disks istacc ∼ 150ms for theMNS = 1.2M⊙ simulations andtacc ∼ 75ms for theMNS =1.4M⊙ simulations.

0 100 200 300 400

r (km)

0

0.01

0.02

0.03

0.04

M (

MÒ) 5ms

10ms40ms

t - tmerge

FIG. 9: Distribution of matter around the black hole measured 5ms,10ms and40ms after merger, for simulation M14-7-S8.

1

10

<T

> (

MeV

)

5ms10ms40ms

0 100 200 300 400

r (km)

0

0.1

0.2

0.3

0.4

<Y

e>

t - tmerge

FIG. 10: Distribution of the average temperature and electron frac-tion around the remnant black hole for simulation M14-7-S8,at thesame times as in Fig. 9. The average is taken after subdividing allmaterial into radial bins identical to those shown in Fig. 9.

The evolution of the disk can be subdivided into two mainperiods. For the first5ms–10ms, shocks in the forming diskheat the material and raise the entropy of the fluid (see Fig. 8).The resulting disk is compact and hot (see below). However,average temperatures rapidly decrease through a combinationof effects: the expansion of the hot compact disk, the emis-sion of neutrinos, and the fall-back of cool material from thetail while hot material is accreted onto the black hole. Anexample of this evolution is shown in Figs. 9 and 10, wherewe look in more detail at simulation M14-7-S8.5ms aftermerger, about a third of the remaining material is in a compacthot disk, with most of the mass atr ∼ 5MBH ∼ 60 km andtemperatureT ∼ 5MeV–15MeV. The rest is in a cool tidaltail, part of which is unbound.5ms later, the disk has spreadand slightly cooled down (T ∼ 3MeV–10MeV), while thefall-back rate from the tidal tail begins to decrease. At latertimes, and given that we do not include the effects of viscos-ity, we no longer have any significant source of heating in thedisk. As hotter material accretes onto the black hole, cooler

12

material falls onto the disk, and neutrinos extract energy fromthe system, the disk cools down. At the end of the simulation,most of the disk is atT ∼ 2MeV (except for the small amountof hotter material at the inner edge of the disk). It is also con-fined to a small radial extent of50 km < r < 100 km. As dis-cussed in Deaton et al. 2013 [10], this is a direct consequenceof the loss of energy due to emitted neutrinos. Visualizationsof the disk for the same simulation are shown in Fig. 11, atthe same times. The evolution of the composition of the fluidwill be discussed in the next section, as it is tightly linkedtothe properties of the neutrino emission.

After the first 10 ms, matter in the disks largely settles intocircular orbits, although they remain highly non-axisymmetricthroughout the post-merger evolution. In Fig. 12, we showvertically- and azimuthally-averaged profiles of the density,temperature, entropy, and electron fraction at three times, sep-arated by 8 ms, for the case M12-7-S8. Other systems thatproduce large disks show very similar profiles. The diskevolves under the influence of accretion and neutrino emis-sion, causing the disk to cool and the inner regions to evac-uate. TheS andYe profiles approach the “U” shape seenin our previous simulation [10]. Fig. 13 shows the angularvelocity and angular momentum profiles 31 ms after merger.The inner disk is almost entirely rotation-supported, as can beseen from the agreement between the actual rotation and thecurves for a circular geodesic orbit. Beyond 70 km, the actualrotation is significantly slower than geodesic, a sign that theouter disk has significant pressure support. This is confirmedby the agreement between the actual rotation rate and whatis required for a hydrodynamic equilibrium circular orbit,anagreement that only breaks down in the very outer regionswhere the disk has not equilibrated. It is also observed in thefact that the sound speedcs decreases much more slowly withradius than the angular velocity. The small change incs is aconsequence of the moderate variation in temperature acrossthe disk and the very weak dependence ofcs on density for the

disk’s range of densities and temperatures. The low∂cs∂ρ

∣

∣

∣

T,Ye

comes from passing through a minimum ofcs; at lower densi-ties, the pressure is dominated by relativistic particles,whileat higher densities the degeneracy pressure is stronger. Weobserve that the angular momentum of a freely falling par-ticle on a circular orbit actually has a minimum at around34 km, which is also where the radial epicyclic frequency ofperturbed circular orbits (measured as in [58]) vanishes. Thisis the innermost stable circular orbit (ISCO). Inside the ISCO,the fluid picks up a significant radially ingoing velocity com-ponent, and, as would be expected, the angular momentumprofile is relatively flat. It is worth pointing out the differencein this inner-disk behavior from what was found in our previ-ous paper [10], which involved a less-massive black hole andamore-massive accretion disk. In those simulations, we did notresolve a geodesic ISCO, but we did see a sharp increase in theorbital energy in the inner disk, connected to strong pressureforces. This was associated with an apparent inner disk insta-bility. No such effects are seen for these systems with moremassive black holes and less massive disks (see Sec. IV).

IV. NEUTRINO COOLING AND DISK COMPOSITION

Neutrino emission plays an important role in the post-merger evolution of NSBH remnants. It is indeed the mainsource of energy loss for the resulting accretion disk, and mayalso deposit energy in the low-density regions above the diskthroughνν annihilation, or drive outflows through neutrinoabsorption in the upper regions of the disk. The simple leak-age scheme used in our simulations can only address the cool-ing of the disk, as we do not take into account the effects ofneutrino heating. A first guess at the importance ofνν anni-hilation and neutrino absorptions can however be made fromthe intensity of the neutrino emission coming from the disk.

To understand the evolution of our disks under neutrinoemission, it is helpful to compare our results to previous sim-ulations of accretion tori using a pseudo-Newtonian potentialand with more controlled initial conditions (Setiawan et al.2006 [13]). There are a few important differences betweenthis paper’s simulations and those presented in [13]. The ap-proximate treatment of gravity used in [13] might significantlyaffect the behavior of the inner disk, especially for high-spincases. Also, the initial conditions in [13] were axisymmet-ric (taking the azimuthal average of the final result of an ear-lier NSBH merger simulation performed with a Newtoniancode [59]), were colder than the initial post-merger state ofour simulations (T ∼ 1MeV–2MeV in most of the disk), andused lower black hole spin parameters (as the post-Newtonianpotential performs worse with higher spins). Some of theirsimulations also include an explicit viscosity, which in themost extreme cases (α = 0.1) causes the disk to heat to tem-peratures similar to our initial conditions (T ∼ 10MeV). Thetypical density (ρ0 ∼ 1010–1011g/cm3) and electron fraction(Ye < 0.05) are fairly similar to those found in our disks re-sulting from general relativistic mergers.

For disks withT ∼ 10MeV, Setiawan et al. [13] find to-tal neutrino luminositiesLν ∼ 1053erg/s, dominated by theemission of electron antineutrinos. This is expected becausethe disk is both extremely neutron-rich and at a low enoughdensity that, at this temperature, beta-equilibrium wouldre-quire a more balanced distribution of neutrons and protons.As a consequence, the disk rapidly protonizes, with the elec-tron fraction rising toYe ∼ 0.1–0.3 on a timescale of20ms.This is fairly similar to the conditions observed in our diskatearly times (∼ 10ms after merger), except that our initial con-ditions are asymmetric and that the high temperatures comefrom shock heating during disk formation rather than viscos-ity. Accordingly, the early evolution of the disk and its neu-trino emission follow a similar pattern. In Fig. 14, we showthe total neutrino luminosity for all simulations withMBH =7M⊙. At early times, we findLν ∼ (2–5) × 1053erg/s,which appears consistent with Setiawan et al. [13], given theslightly higher temperatures present in our disks. As in [13],the luminosity also increases with the disk mass – althoughthe high variability of our disks makes it difficult to deriveanactual scaling (it wasLν ∝ M2

disk in [13]). The emission isdominated by electron antineutrinos, withτ andµ neutrinoshaving fairly negligible contributions to the total luminosity(see Fig. 15 for simulation M14-7-S8, and the summary of the

13

t− tmerge = 5ms

t− tmerge = 10ms

t− tmerge = 40ms

FIG. 11: Disk evolution for case M14-7-S8. The 3 snapshots are taken5ms after merger (top, unbound material leaves the grid),10ms aftermerger (middle, end of rapid protonization of the disk), and40ms after merger (bottom, end of the simulation).Left: 3D distribution ofbaryonic matter.Center: Electron fraction in the vertical slice perpendicular to the orbital plane of the disk which passes through the initiallocation of the black hole and neutron star. The white line isthe density contourρ = 1010g/cm3. Right: Temperature in the same verticalslice. In the first two snapshots, the disk is hot and rapidly protonizing. At later times, the disk cools down toT ∼ 2− 3MeV and becomesvery neutron rich again (Ye ∼ 0.05).

TABLE IV: Neutrino luminositiesLν and average energy〈ǫν〉 10ms after merger. All luminosities are in unites of1052erg/s, and energiesare in MeV.Lνx is the luminosity for each of the 4 types of neutrinosνµ,τ , νµ,τ individually.

Name L10msνe L10ms

νe L10msνx 〈ǫ〉10ms

νe 〈ǫ〉10msνe 〈ǫ〉10ms

νx

M12-7-S8 9 13 0.2 11 14 16M12-7-S9 10 14 0.2 10 14 14M14-7-S7 6 11 0.2 12 16 18M14-7-S8 10 14 0.2 11 15 16M14-7-S9 13 23 1.5 11 14 16

14

0

1

2

3ρ

(1011

g cm

-3)

23ms31ms39ms

2

4

T (

MeV

)

51015

S (

k B)

50 100 150

r (km)

0

0.1

0.2

Ye

FIG. 12: The late-time post-merger evolution of the disk formedin case M12-7-S8. We plot density-weighted average values alongcylindrical shells, with the shell radius defined from the proper cir-cumference of the cylinder on the equator. Data at three times areshown: 23, 31, and 39 ms after the merger time.

0

0.2

0.4

0.6

0.8

v/c

ΩrΩ

Geodesicr

ΩEquilibrium

rc

s

50 100 150

r (km)

8

10

12

14

L (1

016cm

2 s-2) L

LGeodesic

LEquilibrium

FIG. 13: The equatorial velocity in theφ-direction,vφ ≡ uφr/ut,and the specific angular momentumL ≡ −uφ/ut in the disk forcase M12-7-S8 measured 31 ms after merger. For comparison, thesound speed,cs, is also included in the top panel, as are thevφ andL values for geodesic circular orbit and hydrodynamic equilibriumcircular orbit for the given metric and pressure profile in both panels.

properties of the neutrino emission for all simulations in Ta-ble IV). The net lepton number emission, shown in Fig. 16, ishighly variable but clearly causes a protonization of the disk– an effect confirmed by the evolution of the average elec-tron fraction in the fluid (Fig. 17). Finally, the energy of theemitted neutrinos is very consistent across all disks (see Ta-ble IV), with the electron neutrinos having the lowest energy(〈ǫνe〉 ∼ 10MeV–12MeV), and the electron antineutrinosbeing slightly more energetic (〈ǫνe〉 ∼ 14MeV–16MeV).The other species of neutrinos have energies comparable to〈ǫνe〉, are more variable, and are emitted at much lower rates.This is slightly lower than the neutrino energies observed forthe hottest disks in [13] (〈ǫ〉 ∼ 20MeV). The large variations

0 10 20 30 40

t-tmerge

(ms)

1052

1053

1054

Lν (

erg

s-1)

M12-7-S8M12-7-S9M14-7-S7M14-7-S8M14-7-S9

FIG. 14: Total neutrino luminosity (all species, at infinity) as a func-tion of time, for all simulations withMBH = 7M⊙.

0 10 20 30 40

t-tmerge

(ms)

1050

1051

1052

1053

Lν (

erg

s-1)

νe

νa

νx

FIG. 15: Luminosity of the electron neutrinos, electron antineutrinos,and for each of the other 4 species (νµ,τ , νµ,τ ), for simulation M14-7-S8.

observed in the neutrino luminosity during the first∼ 10msfollowing the merger (see Fig. 14) are due to the chaotic na-ture of the disk formation process, and are particularly largefor the lower mass disks. For example, simulation M14-7-S7has a period∼ 5ms after merger in which there is very littlematerial in the hot inner disk (most of the material shockedat early times has been accreted by the black hole), and theneutrino luminosity is very low. As the remaining materialfrom the cold tidal tail falls onto the forming disk and heatsup, the luminosity rises again. And once a massive and moresymmetric accretion disk forms, the variability in the neutrinoluminosity decreases. Simulations with more massive post-merger remnants, on the other hand, rapidly form massive, hotinner disks and show less variability during these first10ms.

At later times, interesting changes in the properties of thedisk arise due to the cooling of the disk and the fall-back oftail material. As hotter, more proton-rich material close to theblack hole is accreted, and cooler, more neutron-rich material(Ye < 0.05) from the tidal tail is added to the disk, we ob-serve a decrease in the temperature of the disk (see previoussection), and in its average electron fraction. Additionally, atthe lower temperature observed in the disk after20ms–40ms

15

0 10 20 30 40

t-tmerge

(ms)

-6

-4

-2

0

2R

ν e-Rν e (

1057

s-1)

M12-7-S8M12-7-S9M14-7-S7M14-7-S8M14-7-S9

FIG. 16: Neutrino number emission as a function of time, for all sim-ulations withMBH = 7M⊙. A short period of rapid protonizationoccurs during the first10ms–15ms after merger. Then, the emissionof electron neutrinos and antineutrinos become more balanced.

0 10 20 30 40

t-tmerge

(ms)

0

0.05

0.1

0.15

0.2

<Y

e>

M12-7-S8M12-7-S9M14-7-S7M14-7-S8M14-7-S9

Leptonization

Late accretionMass Loss

FIG. 17: Average electron fraction of the material on the grid, forall simulations withMBH = 7M⊙. The early rise in〈Ye〉 is dueto preferential emission of electron antineutrinos, whilethe late timeevolution is due to the accretion of the higherYe material onto theblack hole and, forMNS = 1.2M⊙, the emission of a slight excessof electron neutrinos.

the matter is mainly optically thin. In this regime, and un-der the assumption of no neutrino heating, the lepton numberemission (and therefore theYe evolution) is determined by theelectron and positron capture rates alone. TheYe where theserates are balanced, denoted asYe,eq, as a function of densityand temperature is shown in Fig. 18. For the conditions of ourdisks at late times (T ∼ 2−3MeV,ρ ∼ 1011 g/cm3), the equi-librium value ofYe is lower than the instantaneous value ofYe ∼ 0.1. Therefore, at that time, the neutrino emission fromthe optically thin parts of the disks is predominantly of posi-tive lepton number and can lead to net deleptonization. Thisis particularly visible in simulations M12-7-S8 and M12-7-S9, with a net excess of electron neutrinos over antineutrinostowards the end of the simulation (see Fig. 16). In general,the neutronization of the disk is due to a combination of theaccretion of neutron rich material and the preferential emis-sion of electron neutrinos. Note that, even at late times, the

1 2 5 10 15 20

T (MeV)

0.1

0.2

0.3

0.4

0.5

Ye,

eq

12

11

10

9

FIG. 18: Equilibrium electron fractionYe,eq in the free streamingregime, defined as the value ofYe at which the rate of positron andelectron captures are equal for optically thin material, i.e. ignoringthe effect of neutrino absorption on the matter. The black (thick)set of curves result from the balancing of the electron and positroncapture rates used in our leakage code. There are corrections to thesecapture rates that make the energy dependence not∝ E2. These can-not be included in our leakage scheme, but would adjust the value ofYe,eq to the red set of curves. These corrections include those duetothe neutron-proton rest mass difference, weak magnetism, and finiteelectron mass. The largest difference between these rates occurs ei-ther at low densities, where neutrino emission is low, or high temper-atures, where the free streaming assumption is not valid. The variouslines of each color showYe,eq(T ) for a range of baryon densities(solid lines, labeled bylog ρ = 9, 10, 11, 12 in g/cm3). The dot-ted lines are separated by∆ log ρ = 1/3. We only show the range0.035 < Ye < 0.55 covered by our equation of state table.

luminosity of the electron antineutrinos remains larger thanthe luminosity of the electron neutrinos, since their averageenergy is larger. Due to the steep dependence on tempera-ture of the charged-current reactionsn + e+ → p + νe andp+ e− → n+ νe, which are the main contributors to the neu-trino luminosity, the disk also becomes significantly dimmeras it cools down.

In nature, this evolution of the disk towards a lower elec-tron fraction will eventually stop. Indeed, a physical diskwillhave a non-zero effective viscosity (presumably provided bythe magneto-rotational instability). When the heating duetoviscous effects compensates the cooling due to neutrino emis-sion, the disk should reach a more steady temperature profile.In the range of black hole masses and spins studied here, wethus expect the neutrino emission to be composed of a shortburst of mostly electron antineutrinos, withLν ∼ 1053erg/s,lasting∼ 10ms and followed by a period of more constantluminosity at a level set by the viscosity of the disk, decay-ing on a timescale comparable to the lifetime of the disk. Theemission should cause rapid protonization of the high-densityregions of the disk in the first10ms, but can then contributeto re-neutronization as the disk cools down.

For such a configuration, the evolution ofYe in the low-density regions above the disk (from which a wind might belaunched) is expected, at early times, to be affected by theasymmetry in the number of electron neutrinos and antineutri-nos emitted, and can slow down the protonization of the wind

16

material resulting from neutrino captures [18]. The effectofthe neutrino emission on a disk wind over longer timescalesis more uncertain, particularly when general relativisticef-fects on the neutrino trajectories are taken into account. Thisis mainly because the antineutrinos can begin free-streamingmuch closer to the black hole than the neutrinos, for whichthe optical depth is generally higher. Surman et al. [60] haverecently shown, based on neutrino fluxes derived from New-tonian simulations, that disk winds could then be proton-rich,with an evolution resulting in the production of large amountsof Ni56 but no heavy elements (A > 120), as opposed tothe material unbound during tidal disruption (see also [61]).Given the sensitivity of the process to the geometry of theneutrino radiation, which we do not directly compute, and thegeometry of the wind, which we could only obtain by includ-ing neutrino heating, further studies are required to determinein which category the winds emitted by the disks produced inour simulations fall.

It is also worth noting some differences with the disk ob-tained for a lower mass black hole in Deaton et al. 2013 [10].In [10], the combination of a high black hole spin (χBH = 0.9)and a low black hole mass (MBH = 5.6M⊙) led to the forma-tion of a massive disk (Mdisk ∼ 0.3M⊙). The disks presentedhere initially have optical depths of a few (τνe ∼ 3–5 > τνe ).Towards the end of the simulations, they haveτνe ∼ 1 and areoptically thin to electron antineutrinos. On the contrary,themore massive disk in [10] remained optically thick until theend of the simulation,40ms after merger (withτνe ∼ 15,τνe ∼ 5). The inner region of that disk was also suscep-tible to a convective instability, due to the negative gradientdL/dr < 0 of the angular momentumL of the disk at radiir <∼ 27 km, which prevented the disk to evolve towards an ax-isymmetric configuration at late times. The less massive disksobserved in this work, on the other hand, satisfy the Solberg-Høiland criteria for convective stability of an axisymmetricequilibrium fluid [62] everywhere outside of their ISCO. Moregenerally, the massive accretion disk in Deaton et al. [10] washotter, denser, and had a higher electron fraction than any ofthe disks observed here. However, unless astrophysical blackholes are less massive than expected, or are very rapidly spin-ning (χBH > 0.9), we expect the less optically thick disksobtained in this paper to be more representative of the post-merger remnants of NSBH binaries.

V. LONG-TERM EVOLUTION OF THE DISK

Since our code does not include the effects of magneticfields, or any ad-hoc prescription for the viscosity in the disk,important physical effects will be missing if we attempt toevolve the post-merger disks over timescales comparable tothe disk lifetime. From existing simulations using approx-imate treatments of gravity [13, 63], we can however ob-tain reasonable estimates of what should happen. As men-tioned in the previous section, the cooling of the disk willstop when neutrino cooling and viscous heating balance eachother. Forα ∼ 0.01–0.1, this is expected to lead to maxi-mum temperatures in the disk ofTmax ∼ 5MeV–10MeV and

Ye ∼ 0.1–0.3 [13]. Over longer timescales, viscous transportof angular momentum will drive mass accretion onto the blackhole, and the viscous spreading of the disk. Over timescalesofa few seconds, nuclear recombination and viscous heating canthen unbind∼ 10% of the matter in the disk [63]. Althoughthis ejecta is more proton-rich than the material ejected duringthe disruption of the neutron star, Fernandez & Metzger [63]estimate that its electron fraction (Ye ∼ 0.2) and specific en-tropy (S ∼ 8kB) are low enough to allow the production of2nd and 3rd peak r-process elements. In NSBH mergers, thisprocess would probably be of less importance than the promptejection of material:10% of the disk mass is much less thanthe mass ejected at the time of merger, and likely difficult todetect if its properties are similar to those of the dynamicalejecta. This is quite different from the results from BNS merg-ers, in which only10−4M⊙–10−2M⊙ of material is promptlyejected, and the ejected material is strongly heated by shocksand significantly affected by the neutrino radiation field [17]:the different components of the outflow can then be of similarimportance, and have very different properties.

The exact mechanism leading to the production of rela-tivistic jets from the post-merger remnants of NSBH bina-ries is more uncertain. Mildly relativistic outflows along thespin-axis of the black hole have been obtained by Etienneet al. [24], but only after seeding a coherent poloidal fieldBθ ∼ 1014G in the disk resulting from a NSBH merger.For black hole spins initially aligned with the orbital angu-lar momentum of the binary (and thus aligned with the post-merger accretion disk), the magnetic flux in the post-mergerremnant of a NSBH binary is likely to be too low to gener-ate the highly efficient relativistic jets which have been ob-served in numerical simulations of magnetically choked ac-cretion flows [64], at least until the accretion rate onto theblack hole decreases by several orders of magnitude, or un-less the MRI can somehow lead to the growth of a coherentpoloidal field. But given the relatively high spins obtainedaf-ter merger (χBH ∼ 0.9), if such a flow can be established itwould have an efficiencyη = Ejet/(Mc2) ∼ 100% [64], ora jet powerEjet ∼ (1053 erg/s) × (M)/(0.1M⊙/s). Evenif relativistic jets can only be launched when the accretionrate drops toM ≪ 1M⊙/s, this remains sufficient to explainthe energy output of short gamma-ray bursts. For black holespins misaligned with the orbital angular momentum of thebinary, the angular momentum of the post-merger accretiondisk can be misaligned by5 − 15 with respect to the finalblack hole spin [65, 66]. Etienne et al. [24] suggest that alarge scale, coherent poloidal field could be obtained from themixing between poloidal and toroidal fields in a tilted accre-tion disk, thus allowing the formation of a jet at earlier times.Unfortunately, the small number of existing general relativis-tic simulations of NSBH binaries with misaligned black holespins [54, 65] do not include the effects of magnetic fields.

Finally, a few percents of the energy radiated in neutrinosis expected to be deposited in the region along the spin axisof the black hole [13], throughνν annihilations. This energydeposition, which at early times in our disks would beEνν ∼1051 erg/s, might also be sufficient to power short gamma-ray bursts. Whether jets can be launched prompty or only

17

after the accretion rate drops, the disks produced by disruptingNSBH mergers forMBH ∼ 7M⊙ − 10M⊙ thus appear to bepromising candidates for the production of short gamma-raybursts.

VI. GRAVITATIONAL WAVES

Given the relatively small number of orbits simulated in thiswork (5 to 8) and the lower phase accuracy of simulations witha tabulated equation of state compared to simulations with aΓ-law equation of state, our simulations are not competitivewith longer, more accurate recent results (e.g. [54, 67]) asfar as the gravitational wave modeling of the inspiral phaseis concerned. There are however two effects which can bestudied even with these lower accuracy simulations: the influ-ence of the tidal disruption of the neutron star on the gravi-tational wave signal, and the kick velocity given to the blackhole as a result of an asymmetric emission of gravitationalwaves around the time of merger.

A. Neutron star disruption

The effect of the disruption of the neutron star on the grav-itational wave signal generally falls into one out of three cat-egories, as shown in Kyutoku et al. 2011 [29] (see also Pan-narale et al. 2013 [68] for a phenomenological model of thewaveform amplitude). When tidal disruption occurs far outof the innermost stable circular orbit (i.e., for low mass blackholes), the spectrum of the gravitational wave signal showsa sharp cutoff at the disruption frequency. At the oppositeend, when the neutron star does not disrupt (that is, for highermass black holes of low to moderate spins), the gravitationalwave spectrum is similar to what is seen during the merger-ringdown of a binary black hole system (i.e. a bump in thespectrum at the ringdown frequency, and an exponential cut-off at higher frequencies). Finally, for neutron stars disrupt-ing close to the innermost stable circular orbit or during theirplunge into the black hole (for higher mass black holes withlarge spin), the spectra show a shallower cutoff of the sig-nal at frequencies above the disruption frequency (in effect, asmaller contribution of the ringdown of the black hole to thesignal).

Most of the simulations presented here fall into the thirdcategory, as should be expected from the dynamics of themerger. A simple approximation for the frequency of tidaldisruption can be obtained by equating the tidal forces actingon the neutron star with the gravitational forces on its surface,i.e. in Newtonian theory

MBH

d3tideRNS ∝ MNS

R2NS

, (6)

wheredtide is the binary separation at which tidal disruptionoccurs. We then obtain a gravitational wave frequency at tidal

100 1000

f × (1.2MÒ/M

NS)1/2

(RNS

/12.7km)3/2

(Hz)

0.1

1

h(f)

f1/

2 / M

chirp

Scaled (1.2 MÒ)

Scaled (1.4 MÒ)

Unscaled (1.2 MÒ)

Unscaled (1.4 MÒ)

FIG. 19: Gravitational wave spectra for all simulations. Solidlines are rescaled in amplitude by the chirp massMchirp =(MNSMBH)

0.6/(MNS+MBH)0.2 and in frequency by the expected

correction to the tidal disruption frequency. Dashed curves show thespectra before rescaling in amplitude and frequency. The amplitudeis chosen arbitrarily (as it scales with the distance between the binaryand the observer). Once rescaled, all spectra have roughly the samecut-off frequency.

disruption which scales as

fGW,tide ∝√

GMNS

R3NS

. (7)

We thus expect very similar high frequency signals for all sim-ulations, with the cutoff occurring at a frequency∼ 9% largerfor simulations using a neutron star of massMNS = 1.4M⊙

than for those using a neutron star of massMNS = 1.2M⊙,and the amplitude scaling with the chirp massMchirp =(MNSMBH)

0.6/(MNS +MBH)0.2. Fig. 19 shows that this is

indeed the case – the mass and spin of the black hole only con-tribute to the disruption signal by determining how sharp thecutoff is, and not where the disruption starts. For the LS220equation of state, we get

fGW,tide ∼ 1.2 kHz

(

MNS

1.2M⊙

)1/2 (12.7 km

RNS

)3/2

. (8)

The shape of the merger-ringdown waveform shows morevariations for the heavier neutron stars (MNS = 1.4M⊙), assimulations withMBH = 10M⊙ are close to the transitionbetween disrupting and non-disrupting systems.

B. Kick velocity

The second type of information that we can extract from themeasured gravitational wave signal is the kick velocity givento the black hole at merger as a result of asymmetric emis-sion of gravitational waves. We find kicks spanning the rangevkick ∼ 25 km/s–95 km/s, with the expected approximate

18

scaling ofvkick ∝ q−2. These kick values are only slightly be-low what we would expect for binary black hole mergers withthe same masses and spins (less than 20% smaller, see [69]for the binary black hole predictions). The kicks are how-ever increasing with the black hole spin, while the oppositetrend is observed in the fitting formula for binary black holeswhen the spins are aligned with the orbital angular momen-tum andq >∼ 4. In contrast, the difference with binary blackhole results was shown to be much larger for lower mass blackholes [65], which showed a stronger relative suppression ofthe gravitational recoil.

For disrupting NSBH binaries of large mass ratio andrapidly spinning black holes, however, those gravitational-wave induced kicks appear to be significantly weaker than therecoil velocities due to the asymmetric ejection of unboundmaterial (see Sec. III B).

VII. CONCLUSIONS

We performed a first numerical study of NSBH mergersin the most likely range of black hole masses,MBH =7M⊙–10M⊙, using a general relativistic code, a leakagescheme to estimate neutrino emission, and a hot nuclear-theory based equation of state (LS220). We consider relativelyhigh black hole spinsχBH = 0.7–0.9 and low mass neutronstars (MNS = 1.2M⊙–1.4M⊙), so that the neutron star is dis-rupted by the tidal field of the black hole before merger. Un-der those conditions, our simulations show that NSBH merg-ers reliably eject large amounts of neutron-rich material,withMej = 0.04M⊙–0.20M⊙. The ejected material has a low en-tropy per baryon〈S〉 < 10kB, and is ejected early enough thatits composition is largely unaffected by neutrino emissionandabsorption. Accordingly, it should consistently produce heavyr-process elements – as opposed to BNS mergers, which showa wider variety of outflow compositions [17].

The mergers produce accretion disks of massesMdisk ∼0.05M⊙–0.15M⊙, similar to the ejected masses. Dueto shock heating, these disks are initially hot (T ∼5MeV–15MeV), and are luminous in neutrinos (Lν ∼1053erg/s). Preferential emission of electron antineutrinoscauses the disks to rapidly protonize during the first10msafter merger, fromYe ∼ 0.06 to Ye ∼ 0.1–0.4. However,at later times, the cooling of the disk combined with the fall-back of cold, neutron rich material from the tidal tail reversesthis process. About30ms after merger, the denser areas ofthe disks are already much colder (T ∼ 2MeV–3MeV) andmore neutron rich (Ye < 0.1) than at early times. The disk isthen expected to evolve mainly under the influence of MRI-driven turbulence, and spread from its initial dense and com-pact state (ρ ∼ 1011g/cm3, r ∼ 5MBH ∼ 50 km) over aviscous timescale oftν ∼ 0.1s. These effects are however notmodeled in our code, and we thus stop the simulations40msafter merger. Outflows from these accretion disks are likelytobe subdominant compared to the material dynamically ejectedduring the disruption of the neutron star. On the other hand,the disks are promising configurations for the production ofshort gamma-ray bursts.

Our simulations confirm that, for disrupting NSBH bina-ries, large recoil velocitiesvkick,ej >∼ 100 km/s can be im-parted to the black hole. These recoil velocities are not duetothe gravitational wave emission, but rather to the asymmetryin the ejection of unbound material, as proposed by Kyutokuet al. [29].

Finally, we showed that the disruption of a neutron star witha nuclear-theory based equation of state has, for the largerblack hole masses considered here, very different propertiesfrom those observed for lower mass black holes or simplerequations of state. The tidal tail is strongly compressed, towidth of only a few kilometers, and the entire disruption pro-cess occurs within a distancer <∼ 3MBH of the black hole (inthe coordinates of our simulation). In such cases, the prop-erties of the post-merger remnant are more sensitive to theinitial parameters of the binary, and to the details of the equa-tion of state. Unfortunately, due to the differences of scalebetween the width of the tail, the radius of the black hole, andthe much larger distance to which we need to follow the ejecta,this also makes these configurations significantly harder tore-solve numerically. In the current version of our code, we canonly evolve these mergers by using a very fine grid close tothe black hole, and letting the disrupted material exit the gridearly. Improvements to the adaptivity of our grid will be re-quired in order for such binaries to be reliably evolved overthe same timescales as lower mass configurations.

This first study involved a fairly small number of cases,so many questions have yet to be addressed. Perhaps mostinteresting is the variation in merger outcomes from differ-ent plausible assumptions about the equation of state. Wehave found that LS220 andΓ = 2 results differ, but it wouldbe much more interesting to know if viably ”realistic” equa-tions of state can be distinguished by their merger outcomes.Also, a more complete understanding of any of these NSBHpost-merger evolutions will require simulations that includethe remaining crucial physical effects. As ejecta continuesto expand far from the merger location, its density and tem-perature drop below the thresholds for maintaining nuclearstatistical equilibrium, and the subsequent compositional andthermal evolution can only be modeled using nuclear reactionnetworks. Additionally, tracking the disk evolution will re-quire including the effects of MRI turbulence, either directlythrough MHD evolutions (the optimal solution), or at least in-directly through some sort of effective viscosity prescription.Finally, using radiation transport instead of a leakage schemeto evolve the neutrino radiation field is needed to capture theproduction of neutrino-driven winds, to estimate the deposi-tion of energy and the production ofe+e− pairs in the low-density region along the black hole spin axis, and to generallyimprove the accuracy of our predictions for the neutrino emis-sion from the post-merger accretion disk.

Acknowledgments

The authors wish to thank Luke Roberts, CurranMuhlberger, Nick Stone, Carlos Palenzuela, Albino Perego,Zach Etienne and Alexander Tchekhovskoy for useful dis-

19

cussions over the course of this project, and the members ofthe SXS collaboration and the participants and organizers ofthe MICRA 2013 workshop for their suggestions and sup-port. F.F. gratefully acknowledges support from the Vincentand Beatrice Tremaine Postdoctoral Fellowship. The authorsat CITA gratefully acknowledge support from the NSERCCanada, from the Canada Research Chairs Program, andfrom the Canadian Institute for Advanced Research. M.D.D.and M.B.D. acknowledge support through NASA Grant No.NNX11AC37G and NSF Grant PHY-1068243. L.K. grate-fully acknowledges support from National Science Founda-tion (NSF) Grants No. PHY-1306125 and No. AST-1333129,

while the authors at Caltech acknowledge support from NSFGrants No. PHY-1068881 and No. AST-1333520 and NSFCAREER Award No. PHY-1151197. Authors at both Cal-tech and Cornell also thank the Sherman Fairchild Founda-tion for their support. Computations were performed on thesupercomputer Briaree from the Universite de Montreal,man-aged by Calcul Quebec and Compute Canada. The operationof this supercomputer is funded by the Canada Foundationfor Innovation (CFI), NanoQuebec, RMGA and the Fonds derecherche du Quebec - Nature et Technologie (FRQ-NT); andon the Zwicky cluster at Caltech, supported by the ShermanFairchild Foundation and by NSF award PHY-0960291.

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