Pulsations in short gamma ray bursts from black hole-neutron star mergers

Nicholas Stone, Abraham Loeb, and Edo Berger*

Astronomy Department, Harvard University, 60 Garden Street, Cambridge, Massachusetts 02138, USA(Received 19 September 2012; published 23 April 2013)

The precise of short gamma ray bursts (SGRBs) remains an important open question in relativistic

astrophysics. Increasingly, observational evidence suggests the merger of a binary compact object system

as the source for most SGRBs, but it is currently unclear how to distinguish observationally between a

binary neutron star progenitor and a black hole-neutron star progenitor. We suggest the quasiperiodic

signal of jet precession as an observational signature of SGRBs originating in mixed binary systems, and

quantify both the fraction of mixed binaries capable of producing SGRBs and the distributions of

precession amplitudes and periods. The difficulty inherent in disrupting a neutron star outside the horizon

of a stellar mass black hole biases the jet precession signal towards low amplitude and high frequency.

Precession periods of �0:01–0:1 s and disk-black hole spin misalignments �10� are generally expected,

although sufficiently high viscosity may prevent the accumulation of multiple precession periods during

the SGRB. The precessing jet will naturally cover a larger solid angle in the sky than would standard SGRB

jets, enhancing observability for both prompt emission and optical afterglows.

DOI: 10.1103/PhysRevD.87.084053 PACS numbers: 04.30.Tv, 95.30.Sf, 97.60.Jd, 97.60.Lf

I. INTRODUCTION

The origin of the short-hard gamma ray bursts (SGRBs;durations T90 & 2 s; [1]) was largely a matter of specula-tion until the recent discovery of their afterglows andhost galaxies (e.g., Refs. [2–5]). These observations havedemonstrated that SGRBs are cosmological in origin(z * 0:1; [6]); have a beaming-corrected energy scale of�1049–1050 erg [7–9]; lack associated supernovae (e.g.,Refs. [8,10]); occur in a mix of star-forming and ellipticalgalaxies [11]; have a broad spatial distribution around theirhosts [12], with some events offset by tens of kpc [13]; andhave low-density parsec-scale environments [8,12]. Theconfluence of these characteristics provides support tothe popular model of compact object (CO) mergers [14].

In this context, the key open question that motivates ourpaper is the following: if SGRBs originate in CO mergers,what types of compact objects are merging? Specifically, aneutron star (NS) can theoretically be tidally disrupted by,and produce an accretion disk around, either a more com-pact NS or a sufficiently small stellar mass black hole(BH). Unfortunately, it is not clear how to distinguishbetween mergers of neutron star binaries (NS-NS) andmixed binaries (BH-NS). The advent of gravitationalwave (GW) astronomy will facilitate this task, but distin-guishing the waveforms accompanying NS-NS mergersfrom those in BH-NS mergers is a nontrivial task [15],and furthermore the Advanced LIGO era is still half adecade away. It is also possible that GW signals seen byAdvanced LIGO will lack accompanying electromagneticcounterparts if the SGRB beaming angle is too small, or ifthe intrinsic event rate is too low. In this paper we suggest aclear observational tool for distinguishing between NS-NS

and BH-NS progenitors of SGRBs using electromagneticdata only: the precession of the disks and jets associatedwith these events.Jet precession has previously been discussed as a phe-

nomenon relevant for SGRBs, originally with regard tonow-disfavored SGRB models [16,17] but later in thecontext of CO mergers. Early works in the CO mergerparadigm considered disks fed by stable mass transfer froma NS onto a BH so that the precession was forced by tidaltorques [18], but subsequent models considered the morerealistic neutrino-dominated accretion flows (NDAFs)formed after tidal disruption of a NS by a stellar massBH [19]. This last model is similar to the one presented inthis paper, in that it considers a thick disk precessing as asolid body rotator due to general relativistic Lense-Thirring torques. It has been applied to predict differentobservational signatures, such as the light curves of pre-cessing jets [19,20] or even LIGO-band GW signals emit-ted by forced precession of a large amount of disk mass[21,22]. Others have considered an inclined disk whoseinner regions include a Bardeen-Petterson warp; in theirmodel the inner region of a NDAF precesses along with theBH spin about the total angular momentum vector [23].The above papers generally treat disk precession in an

analytic or semianalytic way, due to the high computa-tional expense of even Newtonian simulations. However,the late inspiral, plunge, and merger of a BH-NS binaryhas been repeatedly simulated in full numerical relativity(see Ref. [24] for a review), providing a detailed picture ofthe initial conditions dictating subsequent disk evolution.Shortly after the first numerical relativity simulations of aBH-BH merger [25], fully relativistic BH-NS mergerswere simulated by multiple groups [26–29]. The resultsof these early simulations were later refined, andsubsequent work considered the effects of varied mass*nstone@cfa.harvard.edu

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ratios [30–32], BH spin [32–34], NS compactness andequation of state [35–37], spin-orbit misalignment[37,38], orbital eccentricity [39], and magnetic fields[40,41]. Although disk precession has been seen in mis-aligned BH-NS merger simulations [38], the semianalyticapproach of this paper has two main advantages overnumerical simulation: we both consider the evolution ofthe remnant accretion disk over long time scales and finelysample the broad parameter space of these events.

This paper follows Ref. [19] in focusing on thick disksprecessing as solid body rotators, which are well motivatedfor the supercritical accretion flows and misaligned angularmomentum vectors characteristic of compact object merg-ers (Sec. II). Our work differs from past efforts, however, inour consideration of the viscous spreading of the disk, aswell as our adoption of simplifying assumptions tailored tomatch results from numerical relativity (NR) simulationsof mixed binary mergers (Sec. III). We quantify for the firsttime the distributions of precession periods and anglesgiven physically motivated assumptions about progenitorspins and masses (Sec. IV), using simple analytic formulaswhen appropriate and more complex empirical fits to NRsimulations when necessary. In the process we estimate thefraction of BH-NS mergers that can actually produce ac-cretion disks. We conclude by considering the observableconsequences of jet precession in the context of SGRBs(Sec. V). Unlike in previous work, we discuss both the casewhere the jet is tied to the BH spin vector and the casewhere it aligns with the disk angular momentum vector.

II. DISK PRECESSION

Nonaxisymmetric torques will, initially, induce smallwarps in accretion disks due to differential precessionbetween adjacent mass annuli. The evolution of thesewarps depends on how they are able to propagate throughthe disk. When the disk is sufficiently viscous, warpspropagate diffusively, allowing differential precession toproduce substantial shear viscosity and dissipating largeamounts of orbital energy. In the context of tilted accretiondisks experiencing Lense-Thirring torques from BH spin,diffusive propagation of warps will align the inner regionsof the disk with the black hole equatorial plane; this isknown as the Bardeen-Petterson effect [42–44].

In the opposite regime, a thick disk with a short sound-crossing time scale will propagate warps in a wavelikemanner, redistributing torques throughout the disk andinducing near-rigid body precession [45,46]. In particular,rigid body precession is possible if H=r > �, where H isthe disk height and� the dimensionless viscosity parameterat a radius r. This is the regime most relevant for compactobject mergers and is therefore what we will consider forthe remainder of this paper. Approximately rigid bodyprecession has been seen in hydrodynamical simulationsof protoplanetary disks being torqued by a binary compan-ion [47], in general relativistic magnetohydrodynamic

(GRMHD) simulations of tilted accretion disks aroundspinning black holes [48,49], and, notably, in NR simula-tions of BH-NS mergers [38].In the Newtonian limit, a solid body rotator will precess

with a period Tprec ¼ 2� sin c dðJ=N Þ, where c d is the

misalignment angle between the accretion disk and the BHequatorial plane, J is the total angular momentum of thedisk, and N is the Lense-Thirring torque integrated overthe entire disk. Specifically, if the disk possesses a surfacedensity profile�ðrÞ that is nonzero between an inner radiusRi and an outer radius Ro, and the disk elements possessorbital frequency �ðrÞ, then

J ¼ 2�Z Ro

Ri

�ðrÞ�ðrÞr3dr; (1)

and

N ¼ 4�G2M2

BHaBHc3

sin c d

Z Ro

Ri

1

r3�ðrÞ�ðrÞr3dr; (2)

where the BH’s mass and dimensionless spin are MBH andaBH, respectively. Note that in this section MBH and aBHrefer to postmerger BH quantities; from Sec. III onward wewill distinguish these from the mass and spin of the pre-merger BH. Throughout this paper G is the gravitationalconstant and c is the speed of light. If we use the Keplerian

orbital frequency �k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGMBH=r

3p

, then for a densityprofile of the form �ðrÞ ¼ �0ðr=r0Þ�� , the precessiontime scale is

Tprec¼�Rgð1þ2�Þcð5�2�Þ

r5=2��o r1=2þ�

i ð1�ðri=roÞ5=2�� ÞaBHð1�ðri=roÞ1=2þ� Þ : (3)

Here we have normalized ro ¼ Ro=Rg and ri ¼ Ri=Rg by

the gravitational radius Rg ¼ GMBH=c2. Other effects that

would influence Eq. (3) include nutation and relativisticcorrections to the orbital frequency, which we neglect inthis analysis (but see Refs. [18,19]).We stress that Tprec is an instantaneous precession

period, and that for the nonequilibrium disks expected inCO mergers important quantities such as � and ro will betime dependent (although the cancellation of �0 in J=Nmeans that the secular decrease in disk mass will not affectTprec). The time dependence of these variables will cause

any signal to be quasiperiodic rather than periodic.Because the dominant feature of the disk’s dynamicalevolution will be viscous outward spreading [50,51], weexpect Tprec to increase with time.

Although a power law definition of� is appealing for itssimplicity, both analytical models [50] and NR simulations[38] indicate that the true structure of these disks is morecomplex. To better account for realistic disk structure, andalso to quantify the time evolution of the disk as it spreadsoutward, we adopt the SGRB disk model of Ref. [51],which derives exact � solutions for a viscously spreadingring of matter and then couples these solutions to more

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detailed models of disk energetics and composition. Asmatter from the disk accretes onto the BH, the intiallyadvective NDAF will become optically thin to neutrinosand geometrically thinner. At later times (t * 0:1 sec )the disk will become a geometrically thick, radiativelyinefficient accretion flow. In principle, the intermediateneutrino-cooled period could prevent later disk precessionby aligning the disk into the BH midplane through thecreation of a Bardeen-Petterson warp. In practice, itseems that even the neutrino-cooled phase of accretionstill possesses H=r > �, and is therefore unlikely to align(Ref. [50], Fig. 2).

We are primarily interested in the radiatively inefficientaccretion flow stage, both because it has the longest dura-tion, and because for low disk masses (Md & 0:1MNS,which is the case for most BH-NS mergers—see Sec. III)it is the only phase of accretion. For this stage of diskevolution, we can write the surface density as

�ðr; tÞ ¼ Mdisð1� n=2Þ�R2

disxnþ1=4�

exp

��ð1þ x2�nÞ�

�

� I1=j4�2nj�2x1�n=2

�

�: (4)

HereMdis is the initial disk mass, Rdis is the initial radius ofthe spreading mass ring (i.e., the radius where the NS isdisrupted), Im is a modified Bessel function of orderm, x ¼r=Rdis, � ¼ tð12�0ð1� n=2Þ2=R2

disÞ, and we have assumed

viscosity of the form � ¼ �0xn. We calibrate �0 with the

initial relation tvisc;0 ¼ R2dis=� and the equation

tvisc;0 � 0:11��1�1M

�1=28 R3=2

dis;5 ��

H0

0:3Rdis

��2s; (5)

where � is the dimensionless Shakura-Sunyaev viscositycoefficient and H0 is the characteristic disk height. Weuse the normalizations ��1 ¼ �=0:1, M8 ¼ MBH=8M�,and Rdis;5 ¼ Rdis=10

5 m. The value of � is set by the

magneto-rotational instability (MRI) and has been esti-mated to span a wide range of values, from�0:01 in local,shearing box simulations [52] to �1 in global GRMHDsimulations [53]. However, large � values seen in globalsimulations are confined to small radii, and in these simu-lations �� 0:1 at r * 10rg. The importance of � for our

results lies primarily in how viscosity controls the outwardspreading of the disk, so we follow Ref. [54] and considerlarge radii � values of 0.01, 0.03, and 0.1.

H0 will vary both in radius and in time; generally, H0

grows as one moves further out in the disk [54], and also asthe outer edge of the disk viscously spreads, putting alarger fraction of the disk into a purely advective regimewith large height [51]. Our results are fairly sensitiveto both � and H0, but because the former spans a widerrange we vary � and fixH0 ¼ 0:3R. We arrive at this valueby considering the size of the disk at a time t1=2, a charac-teristic, ‘‘halfway,’’ precession time scale. Specifically,

t1=2 ¼ ððt�1=30 þ 1Þ=2Þ�3 is the time at which half the

SGRB’s precession cycles will have occurred if it lastsfrom t0 to t ¼ 1s and viscous spreading of the disk causes

Tprec / t4=3. Typically, t1=2 � 100 ms, which corresponds

to a disk outer edge at ro � 50; at these distances [54] andtimes [50] more detailed modeling of disk structure indi-cates H=R � 0:3.Using Eq. (4), we plot the time evolution of Tprec in

Fig. 1, and find that it increases in rough agreement withanalytic expectations: at late times, Eq. (4) approaches apower law with � ¼ 1=2, and the outer edge of the disk

expands with time: ro / t2=3. The disk mass declines

slowly, with Md / t�1=3, but the mass accretion rate

declines faster, with _Md / t�4=3. More specifically, thelate-time self-similar solutions are

Md ¼ 0:021��1=3�1 M�1=6

8 R1=2dis;5Mdis;�1t

�1=3 M�; (6)

_Md ¼ 0:007��1=3�1 M�1=6

8 R1=2dis;5Mdis;�1t

�4=3 M�s�1; (7)

Ro ¼ 2:3� 106�2=3�1M

1=38 t2=3 m: (8)

Here Mdis;�1 ¼ Mdis=0:1M� and t is in units of seconds.

Assuming that ri remains fixed (and ignoring lower-order

t s

Ncycles

10 3

10 2

10 1

100

101

Tpr

ecs

a

10 2 10 1 100

t s

10 1

100

101

Ncy

cles

c

10 1 100

10 1

100

101

b

FIG. 1 (color online). (a) Time evolution of Tprec assuming aviscously spreading disk structure given by Eq. (4). Black dottedcurves represent � ¼ 0:1, dashed magenta curves � ¼ 0:03, andsolid blue curves � ¼ 0:01. Thick curves are for nearly equato-rial disruptions with aBH ¼ 0:9, while thin curves are for aBH ¼0:9 and initial spin-orbit misalignment of 70� or equivalently anearly aligned disruption with a � 0:5. The dash-dotted red lineis / t4=3, the rough time evolution of Tprec. (b) and (c) show

Ncycles, the accumulated number of cycles for 0:1 s< t < 1 s and

0:01 s< t < 1 s, respectively.

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contributions from ro), Eq. (3) then implies Tprec / t4=3. In

Fig. 1 we also plot Ncycles, the total number of precession

cycles undergone during the GRB. For � * 0:1, a vis-cously spreading SGRB disk will generally experienceNcycles & 1.

In the above discussion we have assumed that the angu-lar momentum lost by inspiralling disk matter is redistrib-uted outward solely by internal viscous torques. Analternate possibility is specific angular momentum lossthrough a disk wind, which if magnetized can furthertorque the disk [51]. Adopting the simple ‘‘ADIOS’’ modelfor disk wind losses [55], i.e., _MdðrÞ / rp and _J ¼�C _Mout

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiGMRo

p, we can use the similarity solutions of

Ref. [51] for the evolution of an advective disk. In particu-lar, if the wind is unmagnetized and only specific angular

momentum is lost, then C ¼ 2p=ð2pþ 1Þ and ro / t2=3,just as in the wind free case. The disk mass and accretionrate will decline more rapidly in the presence of a wind,

with Md / t�1 and _Md / t�8=3.On the other hand, if the outflow exerts a significant

magnetic torque on the disk (C � 1), then ro / t1=ðpþ3=2Þ

and Tprec / t2=ðpþ3=2Þ. Both analytic [56] and numerical

works [57] have suggested that a value of p ¼ 1 is roughlyappropriate for radiatively inefficient flows. This yields

ro / t2=5, Tprec / t4=5, and a rapidly declining disk mass,

with Md / exp ð�1:15ðt=tvisc;0Þ2=5. If this is the case,

Ncycles * 10 for all realistic � values, but the rapid loss

of disk mass may make precession more difficult toobserve (as discussed in Sec. V).

For the remainder of this paper, however, we conserva-tively calculate fiducial precession time scales usingEq. (4) and n ¼ 1=2. This may underestimate Ncycles, but

due to the number of additional free parameters, incorpo-rating a detailed outflow model is beyond the scope of thispaper. Our results are generally insensitive to Rdis andMdis

(although we will later use the initial disk mass Mdis as acriterion for whether or not a SGRB can form—seeSec. III). Our results are sensitive to �, with large � valuesincreasing the precession period and decreasing the num-ber of precession cycles that can fit in the duration of anSGRB. In all cases we self-consistently calculate the innerdisk edge ri using the formalism in Perez-Giz 2013 [58] forfinding the innermost stable spherical orbit (ISSO), thetilted analogue to the innermost stable circular orbit(ISCO). Details of the ISSO calculation are in Appendix A.

The precession of the SGRB disk in isolation is unlikelyto be observable and is mainly interesting as a source of jetprecession. The observational signatures of jet precessionwill hinge on two uncertain astrophysical questions: theopening angles of SGRB jets and the alignment directionof a jet in a tilted accretion flow. The first of these questionshas recently become amenable to observational constraint;observations of jet breaks in SGRBs suggest openingangles of �10� [8]. Observational evidence for the second

question is limited, and ambiguous. Observations of arelativistic outflow following tidal disruption of a star bya supermassive BH (Swift J164449:3þ 573451) suggestedthat in that case, the jet aligned with the BH spin axis [59];on the other hand, observations of the microquasar LSIþ61303 have been interpreted as evidence of a precessingjet, aligned with the angular momentum axis of a precess-ing disk [60]. There may not be a universal answer to thisquestion, as different hypothetical jet launching mecha-nisms might each tie the jet axis to a different preferreddirection.However, for the two leading jet launching mechanism

candidates in SGRBs—� �� pair annihilation [61,62], andthe Blandford-Znajek (BZ) mechanism [63,64]—there aretheoretical reasons to believe that the jet will align withthe disk angular momentum vector. The � �� annihilationscenario is independent of BH spin and depends only ondisk properties. Alignment of a BZ-powered jet is moreambiguous, but recent works that have considered jetprecession in SGRBs assumed that a jet powered by theBZ mechanism will align with the disk angular momen-tum vector [19,20] because the magnetic field is anchoredin the disk. This assumption has been further supportedby NR simulations of force-free electromagnetic fieldsaround spinning BHs [65], which found that the directionof Poynting flux from the BZ mechanism is governed bylarger-scale magnetic fields and not the BH spin vector,although we note that GRMHD simulations of tiltedaccretion flows with matter have until very recentlybeen unable to resolve jets [66]. While this paper wasunder peer review, the first GRMHD results on tilted jetsappeared in the literature [67]; in these simulations acollimated jet aligned with the BH spin vector out to adistance of �100Rg before bending to align with the

disk angular momentum vector. Thus radiation fromBZ-powered jets may contain precessing and nonprecess-ing components, depending on the distance from the BHwhere the radiation originates.A final self-consistency check on our model involves

the expected duration of significant jet luminosity fromthese events. Although the details are uncertain, asimple semianalytic model for jet luminosity [68] foundthat accretion rates above _Md * 0:003–0:01M� s�1 arerequired to sustain a luminous GRB (this model alsofound that the BZ mechanism typically dominates � ��annihilation).From this criterion and Eq. (7), we see that our fiducial

SGRB duration of � 1 s, originally chosen on observatio-nal grounds [1,69], is well motivated internally as well.This criterion for jet duration implies that the event typi-cally lasts between one and several viscous time scales,which are largely controlled by �. For � ¼ 0:01 ð0:1Þ, theinitial viscous time scale for a 8M� BH is tvisc;0 �0:7 s ð0:07 sÞ. This time scale grows as the disk expands,so that an SGRB that stays active for 1 s lasts for& 1:5 ð15Þ

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viscous time scales. As only the lower viscosities weconsider are likely to produce Ncycles * 1, a typical

BH-NS SGRB accompanied by an observably precessingjet will last for a few viscous times. When we calculateNcycles at later points in this paper, we integrate between

times of 20 ms and 1 s. The lower bound is the approximatetime by which simulations of misaligned BH-NS mergerdisks reach a steady state and begin exhibiting solid bodyprecession [38].

III. PROGENITOR BINARIES

Because large amplitude precession requires large am-plitude misalignment of the postmerger BH and its accre-tion disk, we must consider which COmergers can actuallyproduce misaligned disks. Despite their larger premergerspin-orbit misalignment, NS-NS mergers are unlikely toproduce significantly misaligned disks. If one or bothmembers of the NS-NS binary were millisecond pulsars,disk precession could be feasible: spin angular momentumJMSP � 2

5MNSR2NS�NS � 1� 1042 kgm2=s, and the orbi-

tal angular momentum at the disruption radius LNS-NS �2MNSR

2NS

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2GMNS=R

3NS

q� 1� 1043 kgm2=s. Thus, the

misalignment angle c d between the postmerger accretiondisks (which we assume to lie in the initial orbital plane)and the BH spin axis will be �5� in NS-NS mergersinvolving one star with a ms spin period, if we assumeinitially orthogonal spin and orbital angular momentumvectors. Standard population synthesis channels indicate,however, that the one recycled component of NS binariestypically has a minimum spin period of � 4 ms [70],which would imply c d & 1�, a value that is likely toosmall to carry significant observational consequences.Although the current sample of NS-NS binary spin mea-surements is limited, the fastest rotator discovered so farhas a 22 ms spin period [71], far too slow to produce largeamplitude disk precession.

In BH-NS mergers, however, the BH may possess alarger natal reservoir of spin angular momentum,allowing for greater misalignment between the post-merger BH and the disk formed from NS debris (whichwe have assumed to lie in the initial orbital plane). Natalspin is the most relevant quantity, although subsequentmass transfer onto the BH may produce modest changesin aBH [72]. For a BH-NS system, the relevant numbersare JBH ¼ aBHGMBH=c and LBH-NS � ð1þ qÞMBHR

2NS�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1þ qÞGMBH=R3NS

q, where aBH is the dimensionless

black hole spin, and we define the mass ratio q ¼MNS=MBH. For MBH ¼ 5M�ð10M�Þ and aBH ¼ 0:9, thedisk misalignment angle c d & 20�ð30�Þ, which is largeenough to be observationally interesting. Here we havealso assumed initially orthogonal spin and orbital angularmomentum vectors.

These simple Newtonian estimates motivate an inves-tigation of BH-disk misalignment in BH-NS mergers butare insufficient for accurately estimating either postmergerBH spin a0BH or the misalignment angle c d betweena0BH and LBH-NS. This is because they do not accountfor the fully dynamical, strong field GR effects thataccompany compact object mergers. Recently, empiricalpost-Newtonian (PN) formulas were derived to calculatethese quantities in the case of BH-BH mergers [73]. Theseformulas have a large number of free parameters that werecalibrated based on a suite of NR simulations. Becausedisruptions of NSs by BHs are so marginal (i.e., occur soclose to the ISSO), it is reasonable to expect these formulasto have some utility in making predictions for BH-NSmergers; we show in Appendix B that they are in fact quiteaccurate when tested against NR simulations of BH-NScoalescence. For this reason, we use these PN formulasto calculate the final spin a0BH for a BH formed by themerger of a BH-NS with mass ratio q, premergerBH spin aBH, and a premerger spin-orbit misalignmentangle c . We approximate the premerger NS spinmagnitude as aNS ¼ 0. Finally, we calculate cos c d ¼a0BH �LBH-NS=ða0BHLBH-NSÞ.Even though the PN fomulas in Appendix B reproduce

a0BH with reasonable accuracy, our assumption that thepostmerger disk tilt c d is described by the angle betweenthe initial orbital plane and a0BH may overestimate c d.When we compare our calculations to the late-time(40 ms) results of Ref. [38], we appear to overestimatec d by a factor �3, although our approach provides asignificantly more accurate estimate of early-time disktilt. Similar evolution of disk tilt is not seen in steady-stateGRMHDdisk simulations [48] but is likely at least partiallyphysical for the early stages of BH-NS mergers, as the diskadjusts to an equilibrium configuration. However, numeri-cal viscosity may also play a role in reducing the disk tilt inmisaligned NR simulations [74]. For simplicity, we do notmodel the time evolution of c d, but this probably results inoverestimates of � 2, which we note again in Sec. V.The final component in our calculation is a criterion for

SGRB production in a BH-NS merger. The tidal radius,

defined in the Newtonian limit as rt ¼ RNSq�1=3, is appeal-

ing for this purpose: only a fraction fGRB of all BH-NSmergers will produce an accretion disk and jet, because ifrt < rISSO, the NS is swallowed whole. However, while thetidal radius is cleanly defined in other contexts [75], in thecase of relativistic, comparable mass mergers it is notobvious that the Newtonian definition is applicable [76].Furthermore, the gravitational radius rg � RNS. In light of

these complications, we adopt a fitting formula for theinitial remnant disk mass Mdis, calibrated from NR simu-lations of aligned BH-NS mergers [77]:

Mdis

MNS¼ 0:415q�1=3

�1� 2

GMNS

c2RNS

�� 0:148

rISCORNS

: (9)

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Although this fitting formula was calibrated from NR dataon aligned mergers, we generalize it to misaligned mergersby substituting rISSO for rISCO. This appears to reproduceNR simulations of misaligned postmerger disks reasonablywell (Appendix B). Overall, the fitting formula’s accuracyis comparable to a more complex theoretical model forMdis using the ‘‘affine ellipsoids’’ approximation [78].

The limiting value of Mdis=MNS required to produce aSGRB is highly uncertain, but past theoretical work assum-ing jets are powered by � �� annihilation has suggested thatSGRBs are viable for Mdis=MNS > 0:01 [62]; likewise, arecent attempt to observationally infer disk masses assum-ing � �� annhilation [79] found 0:01<Mdis=MNS < 0:1. Asmentioned in Sec. II, more recent work examining the BZmechanism suggested that initial accretion rates of _Md *0:003–0:01M� s�1 are necessary to power a SGRB, whichin combination with Eq. (7) implies that a SGRB 1 s induration needsMdis * 0:02–0:05M�. In this work, we takeMdis=MNS > 0:05 as the cutoff for SGRB production, butwe discuss the effects of stricter and weaker criteria inSecs. IV and V.

IV. DISTRIBUTIONS

We now integrate the above analytic criteria over dis-tributions of progenitor masses and spins to find distribu-tions of fGRB, c d, Ncycles, and Tprecðt1=2Þ. Because the

distributions of progenitor quantities are not at presentprecisely constrained by observation or population syn-thesis, we consider a wide range of possibilities to bracketthe available parameter space.

For our fiducial case, we take the parametric BH massfunction from Ref. [80] (hereafter the ‘‘OPNM mass func-tion’’), given by

POPNMðMBHÞ ¼ eMc

Mscale

Mscale

8<: e

� MBHMscale ; MBH >Mc

0; MBH � Mc;(10)

where the best-fit values were found to be Mscale � 1:7M�and Mc � 6:2M� [80]. An important qualitative feature ofthe OPNM mass function is the large mass gap betweenNSs and the lowest-mass BH. Although the mass gap hasbeen known from observations for some time [81], it is notfully understood theoretically [82]. Recent populationsynthesis efforts have had some success in reproducing it[83] provided strong assumptions are made about thegrowth of instabilities in supernova explosion mechanisms.Motivated by recent observations [84] suggesting that themass gap may be less distinct than in Ref. [80], we consideras an alternate case a Gaussian mass function where thebest-fit values for mean mass�BH and dispersion�BH werefound to be 7:35M� and 1:25M�, respectively [85].

Recent observations have measured spins for 7 stellarmass BHs and placed upper or lower limits on spins for 3more [86–88]. Although observations of more systems areneeded, the current spin distribution is noticeably bimodal.

Because of the small number of data points we do notattempt to fit a parametrized spin function, and insteadsimply take a flat prior on aBH, sampling it uniformly inthe ranges (0, 0.3) and (0.7, 1) for our fiducial, ‘‘bimodal’’case. For nonfiducial cases, we also consider three alternatespin functions. The ‘‘flat,’’ ‘‘slow,’’ and ‘‘fast’’ cases uni-formly sample aBH along the intervals (0, 1), (0, 0.5), and(0.5, 1), respectively. We explore a variety of possible spindistributions because theoretical expectations are quiteuncertain; in particular, the spin distribution at time ofmerger depends strongly on both the (unknown) birthspins, and on poorly constrained details of a hypercriticalcommon envelope phase [72,89].Kicks resulting from asymmetric supernova explosions

are expected to produce spin-orbit misalignment in BH-NSbinaries. Past research has constrained the allowed pre-merger misalignment angle c as a function of progenitormasses and separation and kick velocity distributions [90].More recent population synthesis of BH-NS binaries hasfound a wide spread in premerger spin-orbit misalignmentc , but with �50% of systems possessing c < 45� [72].Our fiducial, ‘‘prograde’’ case samples the premerger spin-orbit misalignment uniformly in c between 0� and 90�,but we also consider an alternate, ‘‘isotropic’’ case wherec is sampled uniformly from 0� to 180�; physically thiswould represent larger supernova kicks, or formation ofbinaries through nonstandard processes such as dynamicalcapture.Finally, we sample NS masses from a Gaussian distri-

bution peaked at a mean �NS ¼ 1:35M� with standarddeviation �NS ¼ 0:13M�. These values, taken from thedouble NS binaries examined in Ref. [91], are in goodqualitative agreement with other studies of the NS massfunction [92]. Because most NS equations of state that arenot in conflict with observations of � 2M� NSs [93] areroughly constant radius in the relevant mass range, we takea fiducial radius of 13.5 km but as an alternate case con-sider a NS radius of 11 km. For reasons described earlier,we neglect NS spin.With these distributions defined, we are now ready to

populate a large Monte Carlo sample of BH-NS mergers.Our precise procedure is as follows, for any desired set ofdistributions:(1) Generate 2� 105 BHmassesMBH, spin magnitudes

aBH, initial misalignment angles c , and NS massesMNS.

(2) Compute the premerger rISSO from Eq. (A3).(3) Calculate the postmerger BH mass M0

BH, BH

spin a0BH, and spin-disk misalignment c d using

Eqs. (B2) and (B3).(4) Flag the disruption as GRB producing if

Mdis=MNS > 0:05.(5) Compute Ncycles, Tprecðt1=2Þ, and fGRB (using the

postmerger r0ISSO). In these calculations, set

Rdis ¼ r0ISSO.

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In Table I, we summarize the scenarios considered inthis work, along with averaged results. Typical precessionperiods are hTprecðt1=2Þi � 20 ms–200 ms, with mean disk-

BH misalignment typically hc di � 20�. These results aregenerally insensitive to variation of assumptions about theprogenitor population. The fraction of BH-NS mergers thatproduce SGRBs (fGRB) varies more with the properties ofprogenitor binaries, and, in particular, depends strongly onthe value of premerger BH spin aBH. Values of aBH * 0:5are generally necessary to produce a postmerger disk.

Scenario A, our fiducial case, results in GRBs for � 30%of BH-NS mergers. Results for scenario A are quite similarto those in scenarios B (Gaussian BH mass function) and D(flat prior on the BH spin distribution). The probability ofdisruption of the NS falls by a factor � 2 in scenario C(softer NS equation of state) and becomes almost negli-gible in scenario E (bottom-heavy spin distribution). Thefraction of BH-NS mergers capable of producing SGRBs ismaximized in scenario F (fast spins), reduced somewhat ifwe switch to an isotropic c distribution (scenario G), and

TABLE I. Choices of black hole mass, black hole spin, and spin-orbit misalignment distributions we have surveyed in this paper. Wealso summarize key results here: the fraction of BH-NS mergers that can produce a disk and GRB (fGRB), a typical precession period inmilliseconds hTprecðt1=2Þi, and the mean postmerger misalignment angle between the disk and the BH equatorial plane hc di. The firstnumber printed in the hTpreci and hNcyclesi columns is the fiducial value for � ¼ 0:03, while the numbers in parentheses represent

� ¼ 0:01 and � ¼ 0:1 cases.

Scenario BH masses BH spins c Mdis=MNS RNS fGRB hNcyclesi hTprecðt1=2Þi hc diA OPNM bimodal prograde 0.05 13.5 km 0.303 6.0 (16.7, 1.6) 50 (21, 180) ms 20.9�B Gaussian bimodal prograde 0.05 13.5 km 0.333 5.7 (16.1, 1.5) 53 (22, 188) ms 20.9�C OPNM bimodal prograde 0.05 11 km 0.131 7.0 (20.3, 1.8) 40 (15, 143) ms 15.9�D OPNM flat prograde 0.05 13.5 km 0.229 5.3 (14.9, 1.4) 57 (24, 198) ms 19.0�E OPNM slow prograde 0.05 13.5 km 0.010 2.4 (6.3, 0.7) 116 (53, 376) ms 9.3�F OPNM fast prograde 0.05 13.5 km 0.447 5.4 (15.2, 1.5) 56 (23, 194) ms 19.3�G OPNM bimodal isotropic 0.05 13.5 km 0.248 4.5 (12.5, 1.2) 108 (58, 364) ms 32.0�H OPNM bimodal isotropic 0.05 11 km 0.099 6.4 (18.3, 1.7) 56 (34, 239) ms 23.2�I OPNM bimodal prograde 0.01 13.5 km 0.348 5.7 (15.9, 1.5) 54 (23, 190) ms 22.6�J OPNM bimodal prograde 0.2 13.5 km 0.102 7.4 (21.5, 1.9) 37 (14, 136) ms 15.3�

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Pd

(a) (c) (e) (g)

(b) (d) (f) (h)

0 20 40 60 80d

0.000

0.005

0.010

0.015

0.020

0.025

Pd

0 20 40 60 80d

0 20 40 60 80d

0 25 50 75 100125150175d

FIG. 2 (color online). The probability distribution of postmerger misalignment angles c d for scenarios A–H (labeled as panels (a)–(h), respectively). The vertical axis is a probability and the horizontal axis is c d. The dark blue curves show the c d distribution for all2� 105 BH-NS mergers in each scenario, while the smaller, light orange curves show only the subset of events where the NS isdisrupted outside the ISSO (i.e., the subset where a disk and jet can form). Generally, fGRB, the ratio of the light orange area to the darkblue area, falls for bottom-heavy spin distributions and smaller NS radii. All distributions of c d cut off fairly sharply above 50�.

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reduced significantly for a stricter lower cutoff onMdis=MNS (scenario J). Our fiducial scenario comes closeto maximizing fGRB; only switching over to top-heavy BHspin distributions (scenario F) or laxer SGRB criteria(scenario I) increase its value. More detail on fGRB anddistributions of c d can be seen in Fig. 2.

In Fig. 3, we explore the sensitivity of fGRB and c d toour uncertain assumption about the minimum normalizeddisk mass, Mdis=MNS, needed to produce a SGRB. In ourstandard scenarios, we take a cutoff value of 0.05, but inscenarios I and J we change this cutoff to 0.01 and 0.2,respectively. While relaxing the cutoff seems to have littleeffect on our overall results, increasing the cutoff aboveMdis=MNS � 0:1 very quickly suppresses fGRB, and biasesthose SGRBs that are produced toward slightly tilted ac-cretion disks.In general, c d is strongly cut off above 40�–50� in all

scenarios except E, and there is little variation betweenthe mean value hc di in each of our progenitor scenarios.Specifically, hc di ranges from 9� to 32�. Interestingly,our two scenarios with isotropic premerger c only mod-estly extend the range of postmerger c d values: stronglymisaligned (c > 90�) BH-NS mergers simply do notproduce massive disks. As mentioned earlier, our calcu-lation of c d likely overestimates its true value by a factor�2, because the tilt angle will decrease as the disk settlesinto equilibrium, so the true range of hc di would moreaccurately be �5� to �15�. For rapidly spinning BHs, itis possible that our c d estimate accurately describes thefirst few precession cycles, before disk tilt has time todecrease.As can be seen in Table I, the key observable hNcyclesi

depends much more sensitively on � than on the parame-ters of BH-NS binary populations. As usual, the outlier isscenario E, which is biased toward few cycles and longprecession time scales, but all other scenarios produce4:5 & hNcyclesi & 7:5 and 30ms & hTprecðt1=2Þi & 100ms

0.0050.0100.0150.0200.0250.030

Pd

a

20 40 60 80d

0.000

0.005

0.010

0.015

0.020

0.025

Pd

b

FIG. 3 (color online). The probability distribution of post-merger misalignment angles c d for scenarios I and J (in panels(a) and (b), respectively), illustrating the dependence of fGRBand c d on the strictness of our requirement for Mdis=MNS.Specifically, in the top panel we have imposed the strict require-ment that Mdis=MNS > 0:2 in order to produce a SGRB, while inthe lower panel we have imposed the much laxer requirement ofMdis=MNS > 0:01. In all other respects both of these cases areidentical to scenario a. Results in the bottom panel are quitesimilar to scenario a, but in the top panel, fGRB has been stronglysuppressed, particularly at higher c d. The axes and curves arethe same as in Fig. 2.

0.0

0.2

0.4

0.6

0.8

1.0

PN

Cyc

les

(a) (c) (e) (g)

(b) (d) (f) (h)

5 10 15 20 25 30NCycles

0.0

0.2

0.4

0.6

0.8

1.0

PN

Cyc

les

5 10 15 20 25 30NCycles

5 10 15 20 25 30NCycles

5 10 15 20 25 30NCycles

FIG. 4 (color online). Probability distributions of Ncycles for scenarios A–H. As in Fig. 1, the dotted black lines represent � ¼ 0:1, thedashed magenta lines � ¼ 0:03, and the solid blue lines � ¼ 0:01. There is relatively little variation between progenitor scenarios,with the notable exception of scenario e (slow spins). The total number of precession cycles accumulated depends strongly on �.

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for our fiducial � ¼ 0:03 case. We plot distributions ofNcycles in Fig. 4, where we see that low � values produce

many more disk precession cycles.In Fig. 5 we plot contours to indicate the relative proba-

bility of BH-NS mergers producing combinations of Tprec

and c d. These two quantities in general appear fairlyuncorrelated, although in some scenarios (A, B, and G)we do see a weak positive correlation, indicating that themost dramatically precessing SGRB disks will typicalprecess with longer periods. In no scenario do we seetypical Tprecðt1=2Þ values above 0.4 s; if we set aside

scenario E, only a small fraction of events haveTprecðt1=2Þ above 0.1 s when � ¼ 0:03. Scenarios C and

H (soft NS equation of state) generally have the shortestprecession periods, as a combination of rapidly spinningBHs and fairly low c are required to disrupt thesemore compact NSs. Generally, Tprecðt1=2Þ * 0:01 s in all

scenarios.

V. DISCUSSION

In this paper, we have explored much of the relevantparameter space for BH-NS mergers and found that quan-tities relevant for disk precession are generally insensitiveto the assumptions we have made about progenitor binary

parameters. The important results of our calculations(hTprecðt1=2Þi, hNcyclesi, hc di, fGRB) typically change by

factors & 2 as we have varied our assumptions about theunderlying populations of premerger BH-NS binaries. Theone exception to this concerns premerger BH spin: in orderfor a significant fraction of BH-NS mergers to produceSGRBs, aBH * 0:5 is required. Because of the difficulty ofincreasing BH spin through premerger mass transfer [72],this is equivalent to requiring modestly large natal aBHvalues. SGRB formation could also be inhibited if thetypical NS radius is significantly below the smallest valuewe consider (11 km). The strongest precession effects (i.e.,large misalignment c d and short period Tprec) arise from

populations with top-heavy BH spin distributions and stiffNS equations of state. Our results are notably insensitive tothe choice of BH mass function, so long as the massfunction peaks near 6M�, as is suggested by observationsof x-ray binaries.On the other hand, our results do depend fairly sensi-

tively on the details of how the disk will viscously spreadoutward. Using the � viscosity parametrization, we haveseen that effective � values * 0:1 will strongly suppressthe number of observable precession cycles, to the pointwhere jet precession will rarely if ever be detectable.Likewise, � * 0:1 could be large enough to induce a

FIG. 5 (color online). Each contour plot in this figure shows the range of ‘‘halfway’’ precession periods (vertical axis) andpostmerger misalignment angles c d (horizontal axis) for every BH-NS merger that produces a disk. Tprec is plotted in seconds, and in

all scenarios is generally � 0:4 s, although we note again that it will grow with time as the disk viscously spreads outward. As inprevious plots, we show scenarios A–H. The logarithmic contour scale is included in panel (e); the range in colors covers a probabilityrange of 2 orders of magnitude. In this plot we use the fiducial value � ¼ 0:03.

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Bardeen-Petterson warp in the thinner, inner regions of thedisk, limiting global precession. We note again, however,that what matters most for the time evolution ofTprec is not the value of � in the inner regions of the disk,

where GR effects enhance the turbulent viscosity producedby the MRI [53], but rather the outer regions of the disk,where local shearing box simulations that find lower(0:01 & � & 0:1) levels of MRI-generated viscosity aremore appropriate [52].

If jets align with the disk angular momentum axis,then they will precess around the total angular momen-tum vector by an angle � c d, because JBH is signifi-cantly larger than Jdisk. In this case, observations ofSGRBs associated with BH-NSs will often be markedby a clear ‘‘lighthouse effect,’’ so long as c d * �jet. This

seems plausible, as observations of jet breaks in SGRBssuggest typical opening angles of �10� [8]. If c d & �jet,

then jet precession would, typically, be encoded moresubtly as a variation in the portion of the jet presented tothe observer. Alternatively, if jets align with the BHspin axis, they will precess by a much smaller amount,since the angle between the BH spin vector and the

total system angular momentum vector is cos c BH ¼ð1þ j cos c dÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2j cos c d þ j2

p, where j ¼ Jd=JBH.

This angle will initially be smaller than c d by a factor�10 if Jd � 0:1JBH [38], and this difference will in-crease slowly in time as angular momentum is accretedfrom the disk onto the BH (or quickly, if angular mo-mentum is lost in outflows). Unless the typical �jet is

quite small, less than a few degrees, the dramatic light-housing effect previously discussed would be unlikely.A quasiperiodic signal will still be present in all thesescenarios, but it will be strongest for tightly collimatedjets aligned with Jdisk. When we correct our results forthe � 2 overestimate in c d, we have found hc di � �jet,

although there are substantial observational uncertaintiesregarding the distribution of jet opening angles.Detection of a precession signal appears observatio-

nally feasible: our study found a floor of �20 ms forthe ‘‘halfway’’ precession period hTprecðt1=2Þi, but preces-sion periods of �50 ms were much more common, and inall cases the precession period grows in time. Both theSwift satellite’s Burst Alert Telescope and the Fermisatellite’s Gamma-ray Burst Monitor have much finerintrinsic timing resolution, so the observability of pureprecession signals will be ultimately limited by photon-counting statistics. Analysis of Fermi SGRB light curvesfinds evidence of structure in the �10–50 ms range [94],while Swift’s Burst Alert Telescope also produces signalswith structure in the tens of ms [95]. For comparison, theBurst and Transient Source Experiment (BATSE) archivaldata sample contains sufficiently high-resolution timingdata to enable a search for �kHz frequency quasiperiodicoscillations [96,97], indicating that the much longerprecession signals should be clearly resolvable.On the other hand, precession is unlikely to be the only

source of variability in the noisy prompt emission lightcurves of SGRBs. Observational constraints on the natureand sources of SGRB variability are limited by the lowsignal-to-noise ratio and small rate of these events (e.g.,SGRBs made up �1=4 of the BATSE sample). However,analysis of BATSE data indicates that SGRB prompt emis-sion is highly variable, with typical variability time scales�50–100 ms, uncorrelated with the total burst duration[98]. In rare cases, variability on millisecond or shortertime scales has been seen [99]. From a theoretical perspec-tive, the observed variability is generally attributed tointernal rather than external shocks [100], althoughrelativistic turbulence [101] and magnetic dissipation inPoynting-dominated outflows [102] are other possibilities.All of these would represent sources of confusion for arealistic precession signal, although a lighthouse effect forhigh-amplitude precession could serve as a distinguishingfeature.The two largest challenges for observability are (1) the

relatively low number of precession cycles, and (2) therapidly growing precession period Tprec. However, we have

TABLE II. Comparison of the full NR results for postmergerBH spin a0BH to the predictions of our PN fitting formula,

Eq. (B3). Here c is the initial spin-orbit misalignment angle,q is the mass ratio, and the ‘‘Ref.’’ column refers to the paperwhose NR data we are comparing the PN results to. Theagreement is quite strong for small q. For q ¼ 1=3, roughlythe upper limit considered in our BH mass functions, the error is&10%, which is still acceptable for our purposes. In the last twocolumns we also demonstrate the reasonable agreement betweenNR estimates for initial postmerger disk mass, MNR

d , and the

prescription of Eq. (9),MFdis. The disk mass fitting formula works

very well for low and moderate values of spin but becomes lessaccurate for the aBH ¼ 0:9 runs. In each simulation in this table,the NS mass is 1:4M�.

Ref. q c aBH RNS a0BH;NR a0BH;PNMNR

dis

MNS

MFdis

MNS

[34] 1=7 0� 0.5 14.4 km 0.67 0.658 � 0:4% 0%

[34] 1=7 0� 0.7 14.4 km 0.80 0.786 6% 7.2%

[34] 1=7 0� 0.9 14.4 km 0.92 0.913 28% 22.9%

[34] 1=5 0� 0.5 14.4 km 0.71 0.681 6% 6.5%

[37] 1=7 0� 0.9 12.2 km 0.923 0.913 10% 11.8%

[37] 1=7 0� 0.9 13.3 km 0.919 0.913 20% 17.4%

[37] 1=7 0� 0.9 14.4 km 0.910 0.913 30% 22.1%

[37] 1=7 20� 0.9 14.4 km 0.909 0.911 28% 20.3%

[37] 1=7 40� 0.9 14.4 km 0.898 0.900 15% 13.8%

[37] 1=7 60� 0.9 14.4 km 0.862 0.870 3% 1.3%

[38] 1=3 0� 0.0 14.6 km 0.56 0.54 5.2% 5.21%

[38] 1=3 0� 0.5 14.6 km 0.77 0.70 15.5% 16.3%

[38] 1=3 0� 0.9 14.6 km 0.93 0.829 38.9% 28.3%

[38] 1=3 20� 0.5 14.6 km 0.76 0.699 14.5% 15.8%

[38] 1=3 40� 0.5 14.6 km 0.74 0.691 11.5% 14.1%

[38] 1=3 60� 0.5 14.6 km 0.71 0.671 8.0% 11.4%

[38] 1=3 80� 0.5 14.6 km 0.66 0.636 6.1% 8.0%

NICHOLAS STONE, ABRAHAM LOEB, AND EDO BERGER PHYSICAL REVIEW D 87, 084053 (2013)

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shown that for most progenitor populations there exists alarge value tail to the Ncycles distribution, from which

events with Ncycles * 5 can be observed, provided �<

0:1. Furthermore, searches for evolving quasiperiodicityare feasible, provided the time axis of time series data canbe rescaled to match appropriate theoretical models. Pastsearches for simple periodicity in SGRBs [97] would nothave been able to detect the more complex time evolutionof a realistic jet precession signal. In this paper we haveemployed a simple analytic model for the viscous spread-ing of the disk and for the first time have found that

precession periods grow as t4=3. We hope that this willprovide a starting point for searches for jet precessionin SGRBs, but detailed hydrodynamical, and perhapsGRMHD, simulations are necessary to validate or refinethis analytic expectation. We also note that both of thesechallenges are mitigated if a significant fraction of diskangular momentum is lost through an outflow, which will

reduce the late-time scaling of Tprec to t4=5—although in

this scenario, only emission aligned with Jd, not JBH, willprecess significantly.

Pulsation of prompt emission may not be the onlyobservable implication of disk precession in SGRBs.A precessing jet will sweep out a larger solid angle in thesky, enabling BH-NS mergers to make up a larger fractionof the observed SGRB rate than would be implied by asimple calculation (i.e., the intrinsic BH-NS merger ratemultiplied by fGRB). This would likewise enhance observ-ability of BH-NS optical afterglows [103].

Unless there exists a large population of NS-NSbinaries with millisecond spin periods (not accountedfor by current observations or population synthesis esti-mates), SGRBs due to BH-NS mergers will be distin-guishable from those due to NS-NS mergers by thepresence of a quasiperiodic signal, with a typical periodof �30–100 ms. We have shown that this signal is robustto a large number of assumptions about the progenitorbinaries but evolves quickly and could become difficultto observe if jets align with the BH spin axis, or if thepostmerger disk viscosity is large. A better understand-ing of both viscous disk spreading, and how GRB jetintensity varies with angle of observation, will aid futuresearches for this discriminant between SGRB progenitorbinaries.

ACKNOWLEDGMENTS

We would like to thank Francois Foucart, TassosFragos, Vassiliki Kalogera, Raffaella Margutti, BrianMetzger, Cole Miller, and Gabriel Perez-Giz for helpfuldiscussions and suggestions. N. S. and A. L. were sup-ported in part by NSF Grant No. AST-0907890 and NASAGrants No. NNX08AL43G and No. NNA09DB30A. E. B.acknowledges support for this work from the NationalScience Foundation under Grant No. AST-1107973, andby NASA/Swift AO7 Grant No. NNX12AD71G.

APPENDIX A: INNERMOST STABLESPHERICAL ORBITS

We follow the simple formalism of Perez-Giz 2013 tocalculate the ISSO radius for Kerr metric geodesics andpresent it here for completeness (in this appendix we usegeometrized units,G ¼ c ¼ 1, for radial distance r and BHspin a). As mentioned in Sec. II, the ISSO is the innermoststable orbit at constant radius but fixed nonzero inclinationaround a Kerr BH. We define an inclination angle suchthat C ¼ cos , the Carter constant Q ¼ L2sin 2, and con-served vertical angular momentum Lz ¼ LC, i.e., ¼ 0corresponds to equatorial prograde orbits. Calculation ofthe equatorial plane ISCO, rISCO, is well documented in theliterature [104] and consists of finding the roots of thepolynomial

ZðrÞ ¼ ðrðr� 6ÞÞ2 � a2ð2rð3rþ 14Þ � 9a2Þ ¼ 0; (A1)

with one root the prograde and one the retrogradeISCO. The polar ( ¼ �=2) ISSO can be found at theroot of

PðrÞ ¼ r3ðr2ðr� 6Þ þ a2ð3rþ 4ÞÞþ a4ð3rðr� 2Þ þ a2Þ ¼ 0 (A2)

that lies between r ¼ 6 and r ¼ 1þ ffiffiffi3

p þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þ 2

ffiffiffi3

pp.

Finally, the generic ISSO is a root of the polynomial

SðrÞ ¼ r8ZðrÞ þ a2ð1� C2Þða2ð1� C2ÞYðrÞ � 2r4XðrÞÞ;(A3)

with auxiliary functions defined as

XðrÞ ¼ a2ða2ð3a2 þ 4rð2r� 3ÞÞ þ r2ð15rðr� 4Þ þ 28ÞÞ� 6r4ðr2 � 4Þ; (A4)

YðrÞ ¼ a4ða4 þ r2ð7rð3r� 4Þ þ 36ÞÞþ 6rðr� 2Þða6 þ 2r3ða2ð3rþ 2Þ þ 3r2ðr� 2ÞÞÞ:

(A5)

Specifically, rISSO is the root of SðrÞ located between theappropriate rISCO (prograde or retrograde) and the polarISSO.

APPENDIX B: POST-NEWTONIANMERGER TREATMENT

The Ref. [73] formulas discussed in Sec. III arepresented in this appendix, along with a discussion oftheir applicability to BH-NS mergers. These formulasare calibrated to define the outcome of a BH-BH merger,although we apply them more generally to the case ofBH-NS mergers with masses m1 and m2, and mass ratio

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q ¼ m1=m2 � 1. We define the symmetric mass ratio ¼ q=ð1þ qÞ2 and denote the dimensionless CO spin

vectors ai as having components aki and a?i , which are

parallel to and perpendicular to the binary angularmomentum, respectively. We further define � as theangle made between the radial direction and thevector � ¼ ðm1 þm2Þðm2a2 m1a1Þ. Then the post-merger remnant mass is found to decrease by a fraction

�M=M¼ ~EISCOþE22þE3

3þ 2

ð1þqÞ2 ðESðak2þq2ak1Þ

þE�ð1�qÞðak2�qak1ÞþEAja2þqa1j2þEBja?2 þqa?1 j2ðcos2ð�þ��2ÞþECÞþEDja2�qa1j2þEEja?2�qa?1 j2ðcos2ð����3ÞþEFÞÞ: (B1)

Here the energy lost during the inspiral from infinitydown to the plunge is fit as

~EISCO�1�ffiffiffi8

p3þ0:103803þ 1

36ffiffiffi3

p ð1þqÞ2 ðqð1þ2qÞak1

þð2þqÞak2Þ�5

324ffiffiffi2

p ð1þqÞ2 ða22�3ðak2Þ2

�2qða1 �a2�3ak1ak2Þþq2ða21�3ðak1Þ2ÞÞ: (B2)

The final spin vector of the postmerger BH is

a0 ¼�1��M

M

��2�~JISCOþðJ22þJ3

3Þnk

þ 2

ð1þqÞ2�ðJAðak2þq2ak1ÞþJBð1�qÞðak2�qak1ÞÞnk

þ ð1�qÞja?2 �qa?1 jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ� cos ð2ð����4ÞÞþJM

qn?

þja?2 þq2a?1 jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJS cos ð2ð�S��5ÞÞþJMS

qn?

��;

(B3)

where the angular momentum radiated during the inspiralfrom infinity to the plunge is

~JISCO ��2

ffiffiffi3

p � 1:5255862� 1

9ffiffiffi2

p ð1þ qÞ2 ðqð7þ 8qÞak1

þ ð8þ 7qÞak2Þ þ2

9ffiffiffi3

p ð1þ qÞ2 ða22 � 3ðak2Þ2

� 2qða1 � a2 � 3ak1ak2Þ þ q2ða21 � 3ðak1Þ2ÞÞ

�nk

� 1

9ffiffiffi2

p ð1þ qÞ2 ðqð1þ 4qÞa1 þ ð4þ qÞa2Þ

þ 1

a2 þ q2a1ð1þ qÞ2 : (B4)

The other variables in these formulas are empirical fittingconstants, with values found to be E2 ¼ :341,E3 ¼ 0:522,ES ¼ 0:673, E� ¼ �0:3689, EA¼�0:0136, EB ¼ 0:045,EC ¼ 0, ED ¼ 0:2611, EE ¼ 0:0959, EF¼0, J2 ¼ �2:81,J3 ¼ 1:69, JA ¼ �2:9667, JB ¼ �1:7296, and J� ¼JM ¼ JS ¼ JMS ¼ 0. We also set �2 ¼ �3 ¼ �4 ¼�5 ¼ 0 because of the weak dependence of the resultson these parameters. With extremely low probability(� 10�5), Eq. (B3) can give superextremal spin values,of a0 > 1; we discard these cases from our Monte Carlosample when they appear.As mentioned in Sec. III, these formulas, which were

derived, and calibrated, for BH-BH mergers, are found togive surprisingly good agreement with detailed results forBH-NS mergers simulated in full NR. We demonstrate theagreement in Table II, where we also plot the generallygood agreement between Eq. (9) and NR results for theinitial mass of the postmerger accretion disk. The largesterrors in the disk mass fitting formula seem to occur forlarge disk masses (Mdis > 0:2MNS) and for high premergerspin-orbit misalignments (c * 60�). The first of thesecases occurs only for a small subset of our BH-NSmergers;the second occurs for a larger fraction and may result in amodest overestimate of fGRB.

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