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# Black hole-neutron star coalescence: Effects of the neutron star spin on jet
launching and dynamical ejecta mass

Milton Ruiz  Department of Physics, University of Illinois at Urbana-
Champaign, Urbana, IL 61801  Vasileios Paschalidis  Departments of Astronomy
and Physics, University of Arizona, Tucson, AZ 85719  Antonios Tsokaros
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL
61801  Stuart L. Shapiro  Department of Physics, University of Illinois at
Urbana-Champaign, Urbana, IL 61801  Department of Astronomy & NCSA, University
of Illinois at Urbana-Champaign, Urbana, IL 61801

###### Abstract

Black hole-neutron star (BHNS) mergers are thought to be sources of
gravitational waves (GWs) with coincident electromagnetic (EM) counterparts.
To further probe whether these systems are viable progenitors of short
gamma–ray bursts (sGRBs) and kilonovae, and how one may use (the lack of) EM
counterparts associated with LIGO/Virgo candidate BHNS GW events to sharpen
parameter estimation, we study the impact of neutron star spin in BHNS
mergers. Using dynamical spacetime magnetohydrodynamic simulations of BHNSs
initially on a quasicircular orbit, we survey configurations that differ in
the BH spin ($a_{\rm BH}/M_{\rm BH}=0$ and $0.75$), the NS spin ($a_{\rm
NS}/M_{\rm NS}=-0.17,\,0,\,0.23$ and $0.33$), and the binary mass ratio
($q\equiv M_{\rm BH}:M_{\rm NS}=3:1$ and $5:1$). The general trend we find is
that increasing the NS prograde spin increases both the rest mass of the
accretion disk onto the remnant black hole, and the rest mass of dynamically
ejected matter. By a time $\Delta t\sim 3500-5500M\sim 88-138(M_{\rm
NS}/1.4M_{\odot})\,\rm ms$ after the peak gravitational wave amplitude, a
magnetically–driven jet is launched only for $q=3:1$ regardless of the initial
NS spin. The lifetime of the jets [$\Delta t\sim 0.5-0.8(M_{\rm
NS}/1.4M_{\odot})\,\rm s$] and their outgoing Poynting luminosity [$L_{\rm
Poyn}\sim 10^{51.5\pm 0.5}\,\rm erg/s$] are consistent with typical sGRBs
luminosities and expectations from the Blandford–Znajek mechanism. By the time
we terminate our simulations, we do not observe either an outflow or a large-
scale magnetic field collimation for the other systems we considered. The mass
range of dynamically ejected matter is $10^{-4.5}-10^{-2}~{}(M_{\rm
NS}/1.4M_{\odot})M_{\odot}$, which can power kilonovae with peak bolometric
luminosities $L_{\rm knova}\sim 10^{40}-10^{41.4}$ erg/s with rise times
$\lesssim 6.5\,\rm h$ and potentially detectable by the LSST.

###### pacs:

04.25.D-, 04.25.dg, 47.75.+f

##  I Introduction

We are in a golden era of gravitational wave (GW) physics where the
sensitivity of ground-based laser interferometers is rapidly increasing.
During the first observing run $O1$ Abbott  _et al._ (2019), only $3$ GW
events from binary black hole (BBH) mergers were detected. The $O2$ run
observed the first GWs (GW170817) from the inspiral of a stellar compact
binary Abbott  _et al._ (2017), in which at least one of the companions was a
neutron star. The progenitor of this event has been officially classified as a
merging binary neutron star (BNS) system, although the possibility of a
merging black hole-neutron star (BHNS) progenitor cannot be excluded (see e.g.
Yang  _et al._ (2018); Hinderer  _et al._ (2018)). In addition, $7$ new BBHs
were detected Abbott  _et al._ (2019). Finally, during $O3$, whose sensitivity
was increased by $\sim 50\%$ compared to $O1$, at least $53$ GW event
candidates have been reported GW alerts .111Recently $26$ of these events have
been officially confirmed, along with 13 new GW events reported for the first
time in Abbott  _et al._ (2020a). These GW events can be classified as
follows: a) $37$ BBHs candidates; b) $7$ BNS candidates. It should be noted
that the progenitor of GW190425 is a BNS system with a total mass of
$3.4^{+0.3}_{-0.1}M_{\odot}$, which is significantly different from the known
population of Galactic BNS systems (see e.g. Tauris  _et al._ (2017)); c) 4
events in the so-called mass gap (compact objects with masses of $3-5\
M_{\odot}$–see e.g. Mandel  _et al._ (2015); Littenberg  _et al._ (2015);
Tsokaros  _et al._ (2020a)); and d) 5 BHNS candidates, of which only one event
has been confirmed (GW190814) and whose inferred individual masses are
$23^{+1}_{-0.9}M_{\odot}$ and $2.59^{+0.08}_{-0.08}M_{\odot}$ Abbott  _et al._
(2020b). It is worth noting that, although this event is listed in GW alerts ;
Abbott  _et al._ (2020a) as a BHNS candidate with $>99\%$ probability, due to
the lack of any EM counterpart or tidal signature, the nature of the lighter
companion is uncertain (see e.g. Tsokaros  _et al._ (2020b); Most  _et al._
(2020); Godzieba  _et al._ (2020); Tews  _et al._ (2020)). If indeed the
lighter binary companion is a NS then this would be the heaviest NS yet
observed Cromartie  _et al._ (2019). If, on the other hand, the binary
companion is a BH then it would be the lightest BH observed to date. Notice
the mass of the lightest stellar-mass BH candidate observed in X-rays is
$3.8^{+0.5}_{-0.5}M_{\odot}$ Orosz  _et al._ (2004).

Intense EM counterpart-observing campaigns preceding/following GW detections
have led to the following EM detections: a) a weak transient EM signal et al.
(2016); Connaughton  _et al._ (2018) (event GW150914-GBM, observed only by
Fermi) that lasted $1$ s and appeared $0.4$ s after the detection of GW150914,
the first event consistent with the inspiral and merger of a BBH; b) a MeV-
scale EM signal lasting for $32$ ms and occurring $0.46\,\rm s$ before
GW170104 (also consistent with a BBH Abbott  _et al._ (2017)), as reported by
the AGILE mission Verrecchia  _et al._ (2017)); c) EM counterparts across the
spectrum reported by several observatories (see e.g. Abbott  _et al._ (2017a,
b, c); Chornock  _et al._ (2017); Cowperthwaite  _et al._ (2017); Kasen  _et
al._ (2017); Nicholl  _et al._ (2017)) following the detection of GW170817.
Its association with the transient GRB170817A Abbott  _et al._ (2017b) and the
kilonova AT 2017gfo/DLT17ck Valenti  _et al._ (2017) provides the best direct
observational evidence so far that at least some sGRBs are indeed powered by
BNS mergers, or by the merger of a stellar compact binary where at least one
of the companions is a NS (or a hybrid star Paschalidis  _et al._ (2018);
Essick  _et al._ (2020)). The BNS-sGRB connection was anticipated in Paczynski
(1986); Eichler  _et al._ (1989); Narayan  _et al._ (1992), and numerically
demonstrated by self-consistent simulations in full general relativistic
magnetohydrodynamics (GRMHD) of merging BNSs Ruiz  _et al._ (2016); Ruiz and
Shapiro (2017); Ruiz  _et al._ (2020) and BHNSs Paschalidis  _et al._ (2015);
Ruiz  _et al._ (2018a). GW170817 and its EM counterpart signals have been used
to impose some constraints on the physical properties of a NS (see e.g.
Margalit and Metzger (2017); Shibata  _et al._ (2017); Ruiz  _et al._ (2018b);
Rezzolla  _et al._ (2018); Most  _et al._ (2018); Abbott  _et al._ (2017,
2018); Radice  _et al._ (2018); Bauswein  _et al._ (2017); Raithel  _et al._
(2018); Raithel and Ozel (2019)), such as the maximum mass of a spherical NS,
as well as its tidal deformability, equation of state, and radius (see Raithel
(2019); Baiotti (2019); Horowitz (2019); Guerra Chaves and Hinderer (2019);
Radice  _et al._ (2020); Chatziioannou (2020) for reviews); d) an EM flare
observed by the Zwicky Transient Facility, consistent with an ejected BBH
merger remnant in the accretion disk of an active galactic nucleus that may be
associated with GW190521 Graham  _et al._ (2020); and e) a weak EM transient
(GRB190425 event) $\sim 0.5$ s after GW190814 as reported in Pozanenko  _et
al._ (2020).

Unlike the EM counterparts associated with GW170817, the other candidate EM
counterparts were not confirmed by other observatories/satellites operating at
the same time. The absence of observable EM counterparts from candidate BHNS
mergers may question their role as progenitors of the central engines that
power sGRBs. Yet, GRMHD simulations in Paschalidis  _et al._ (2015); Ruiz  _et
al._ (2018a) showed that BHNS remnants of $q=3:1$ mergers can potentially
launch magnetically-driven jets. Now early population synthesis studies found
that the distribution of mass ratios $q$ in BHNSs depends on the metallicity,
and peaks at $q=7:1$ Belczynski  _et al._ (2008, 2010), but more recent work
finds that it is generally less than $10:1$, and peaks at $q\approx 5:1$
Giacobbo and Mapelli (2018); Abbott  _et al._ (2020b). As pointed out in
Foucart (2012); Foucart  _et al._ (2018), the larger the mass ratio, the
higher the BH spin required for the NS companion to be tidally disrupted
before reaching the innermost stable circular orbit (ISCO). So far in BBHs
reported by the LIGO/Virgo scientific collaboration, BHs have high mass and/or
low spins (see e.g. Table VI in Abbott  _et al._ (2020a)). If this trend
continues for LIGO/Virgo BHNSs, then it is expected that LIGO/Virgo BHNS
remnants would have negligible accretion disks and ejecta Foucart (2020),
which might disfavor their role as progenitors of sGRBs and kilonovae.
However, the NS spin could have a strong impact on the tidal disruption and
dynamical ejection of matter, affecting both sGRB and potential kilonovae
signatures. It should be noted that the spins of the binary companions are
only weakly constrained by current GW observations.

In this paper, we survey fully relativistic BHNS configurations on a
quasicircular orbit undergoing merger in which the BH and/or the NS companions
are spinning. We address two questions: a) Can a moderate high-mass ratio BHNS
binary be the progenitor of an engine that powers sGRBs?; b) Can the spin of a
NS companion change the fraction of the dynamical ejection of matter that may
drive potentially detectable kilonovae signatures?

We consider BHNS configurations with mass ratios $q=3:1$ and $q=5:1$. In the
first case the BH spin is $a_{\rm BH}/M_{\rm BH}=0.75$, while in the latter
one the BH is nonspinning. The NS spin has a spin $a_{\rm NS}/M_{\rm
NS}=-0.17,\,0,\,0.23$ or $0.33$. In all cases, the star is threaded by a
dynamically weak poloidal magnetic field that extends from the stellar
interior into the exterior (as in a pulsar), and whose dipole magnetic moment
is aligned with the orbital angular momentum of the binary. For purposes of
comparison with our earlier studies Paschalidis  _et al._ (2015); Ruiz  _et
al._ (2018a), the NS is modeled by a polytropic equation of state (EOS) with
$\Gamma=2$.

We find that the late inspiral and merger phases of the above BHNS binaries
are roughly the same as in Etienne  _et al._ (2008, 2012), where the magnetic
field is confined to the interior of the star. The fraction of the total rest-
mass NS outside the horizon varies from $\lesssim 1\%$ to $\sim 15\%$
depending strongly on the binary mass ratio. The general trend is that
increasing the NS prograde spin increases both the mass of the accretion disk
remnant and the unbound material (ejecta). In addition, NS spin leads to GW
dephasing, with higher prograde spin increasing the number of GW cycles.

Consistent with our previous results in Ruiz  _et al._ (2018a), we find that
by $\Delta t\sim 3500-5500M\approx 88-138(M_{\rm NS}/1.4M_{\odot})\,\rm ms$
following the GW peak emission a magnetically–driven jet emerges from the BH +
disk remnant of BHNSs with mass ratio $q=3:1$ regardless of the initial NS
spin. However, the jet launching time depends strongly on the latter. As the
initial NS prograde spin increases, the effective ISCO decreases and the
separation at which the star is tidally disrupted increases. These two effects
induce long tidal tails of matter that result in more baryon-loaded
environments. Thus, stronger magnetic fields are required to overcome the
baryon ram-pressure, delaying the launch of the jet while the fields amplify.
Notice that jet launching may not be possible for all EOSs if the matter fall-
back timescale is longer than the disk accretion timescale Paschalidis (2017).
The lifetime of the jet [$\Delta t\sim 0.5-0.8(M_{\rm NS}/1.4M_{\odot})\,\rm
s$] and outgoing Poynting luminosity [$L_{\rm Poyn}\sim 10^{51.5\pm 0.5}\,\rm
erg/s$] are consistent with typical sGRB (see e.g. Bhat  _et al._ (2016); Lien
_et al._ (2016); Svinkin  _et al._ (2016); Ajello  _et al._ (2019)), and with
the Blandford–Znajek (BZ) Blandford and Znajek (1977) luminosities Thorne  _et
al._ (1986). These results are also consistent with a simple, “universal”
model for BH + disk remnants proposed in Shapiro (2017).

The characteristic temperature of the disk remnant is $T\sim 10^{11}\,\rm K$
(or $\sim 8.6$ MeV) and hence it may also emit a copious amount of neutrinos
with peak-luminosity of $10^{53}\,\rm erg/s$ Just  _et al._ (2016); Kyutoku
_et al._ (2018). However, as the lifetime of this process might be too small
to explain typical sGRBs Kyutoku  _et al._ (2018), it has been suggested that
BH + disk remnants powering sGRBs may be dominated initially by thermal pair
production followed by the BZ process Dirirsa (2017).

Finally, we find the dynamical ejection of matter is strongly affected by the
initial NS spin. It ranges between $10^{-4.5}$ and $10^{-2}(M_{\rm
NS}/1.4M_{\odot})M_{\odot}$, and may induce kilonovae signatures with peak
bolometric luminosities of $L_{\rm knova}\sim 10^{40}-10^{41.4}$ erg/s and
rise times $\lesssim 6.5\,\rm h$, potentially detectable by the Large Synoptic
Survey Telescope (LSST) survey Alonso  _et al._ (2018) out to $O(200)$ Mpc.
Similar conclusions were reached in eccentric BHNS mergers with spinning NSs
in East  _et al._ (2015). These preliminary results suggest that moderate high
mass ratio BHNSs that undergo merger, where the NS companion has a non-
negligible spin, may give rise to detectable kilonovae signatures even if
magnetically-driven jets are absent.

The remaining sections of the paper are organized as follows: A short summary
of our numerical methods and their implementation, along with our initial data
and the grid structure used to solve the GRMHD equations, are presented in
Sec. II. We present our results in Sec. III and conclusions in Sec. IV.
Geometrized units ($G=c=1$) are adopted throughout the paper except where
stated explicitly.

Table 1: Initial properties of the evolved BHNS configurations. We list the
mass ratio $q\equiv M_{\rm BH}:M_{\rm NS}$, where $M_{\rm BH}$ is the BH mass
at infinite separation and $M_{\rm NS}$ the NS rest-mass (see Taniguchi  _et
al._ (2006) for details), the BH spin $a_{\rm BH}/M_{\rm BH}$, the NS spin
$a_{\rm NS}/M_{\rm NS}$, which is either aligned or antialigned (indicated
with a $-$ sign) with respect to the total angular momentum of the system.
$T/|W|$ and $P$ are the kinetic-to-binding-energy ratio and the rotation
period in units of $(M_{\rm NS}/1.4M_{\odot})\,\rm ms$ of the NS,
respectively. The dimensionless ADM mass $\bar{M}\equiv\kappa^{-1/2}\,M$ (here
$\kappa$ is the polytropic gas constant), the ADM angular momentum $J$ of the
system, and the orbital angular velocity $\Omega_{0}$. The label for each
configuration includes successively: a mass ratio tag ($q=3$ or $q=5$), and a
tag identifying the spin direction (m = antialigned or p = aligned) and its
magnitude. In all configurations, the NS companions have a nondimensional
rest-mass ${\bar{M}_{\rm NS}}=0.15$, and the initial $M\,\Omega_{0}$
corresponds to an orbital separation of about $D_{0}\simeq 8.7M\sim 66(M_{\rm
NS}/1.4M_{\odot})\,\rm km$.

Model $q$ $a_{\rm BH}/M_{\rm BH}$ $a_{\rm NS}/M_{\rm NS}$ $T/|W|$ $P$ (ms)∗
$\bar{M}^{\dagger}$ $J/M^{2}$ $M\,\Omega_{0}$ q3NS0.0‡ 3:1 0.75 0.0 0.0 0.0
0.55 1.09 0.0328 q3NSm0.17 3:1 0.75 $-0.17$ 0.009 3.2 0.55 1.09 0.0328
q3NSp0.23 3:1 0.75 0.23 0.016 2.4 0.55 1.09 0.0328 q5NS0.00‡ 5:1 0.0 0.0 0.0
0.0 0.83 0.52 0.0333 q5NSm0.17 5:1 0.0 $-0.17$ 0.009 3.2 0.83 0.52 0.0333
q5NSp0.23 5:1 0.0 0.23 0.016 2.4 0.83 0.52 0.0333 q5NSp0.33 5:1 0.0 0.33 0.15
1.9 0.83 0.52 0.0333

∗Normalized to $(M_{\rm NS}/1.4M_{\odot})$. Note that the fastest known pulsar
has a period of $1.40\,\rm ms$ Hessels  _et al._ (2006).

${}^{{\dagger}}M\simeq 5.1M_{\odot}(M_{\rm NS}/1.4M_{\odot})$.

‡ Cases treated previously in Ruiz  _et al._ (2018a).

##  II Numerical schemes

The numerical methods used to evolve the BHNS binaries are the same as in Ruiz
_et al._ (2018a). Therefore, in this section we briefly introduce our notation
and summarize our numerical schemes, along with the initial data. We refer the
reader to Ruiz  _et al._ (2018a) for further details.

#### Formulation and numerical scheme:

We solve Einstein’s equations for the gravitational field coupled to the MHD
equations for the matter and magnetic field using the adaptive–mesh–refinement
(AMR) Illinois GRMHD code embedded in the Cactus/Carpet infrastructure Cac ;
Schnetter  _et al._ (2004). This code uses the
Baumgarte–Shapiro–Shibata–Nakamura (BSSN) formulation Shibata and Nakamura
(1995); Baumgarte and Shapiro (1999) to evolve the metric, and employs moving
puncture gauge conditions cast in first order form Hinder  _et al._ (2014).
Additionally, the code solves the MHD equations in a conservative formulation
[see Eqs.(27)-(29) in Etienne  _et al._ (2010)] using high-resolution shock-
capturing methods Duez  _et al._ (2004). We set the damping parameter $\eta$
appearing in the shift condition to $\eta=3.3/M$ for BHNSs with mass ratio
$q=3:1$, and to $\eta=1.2/M$ for those with mass ratio $q=5:1$ (see Table 1).
Here $M$ is the Arnowitt-Deser-Misner (ADM) mass of the system. Following
Etienne  _et al._ (2012); Farris  _et al._ (2012), we use the generalized
Lorenz gauge to avoid the spurious magnetic fields between AMR levels due to
numerical interpolations. We set the damping parameter $\xi\sim 5.5/M$ for
configurations with mass ratio $q=3:1$, and $\xi=6.4/M$ for those with
$q=5:1$. Finally, we adopt a $\Gamma$–law EOS $P=(\Gamma-1)\rho_{0}\,\epsilon$
with $\Gamma=2$, which allows shock heating during the evolution. Here $P$ and
$\rho_{0}$ are the pressure and the rest-mass density, respectively. As in
standard hydrodynamic and MHD simulations, we integrate the ideal GRMHD
equations everywhere, imposing a tenuous constant–density atmosphere
$\rho_{0,\,{\rm atm}}=10^{-10}\,\rho_{0}^{\rm max}(0)$, where $\rho_{0}^{\rm
max}(0)$ is the initial maximum value of the rest-mass density of the NS.

Figure 1: Maximum value of the rest-mass density $\rho_{0}(t)$ normalized to
its initial maximum value $\rho_{0}(0)$ during the early inspiral for the
nonmagnetized cases (see Table 1). The inset shows the normalized $L_{2}$ norm
of the Hamiltonian constraint (see Eqs. (40) and (41) in Etienne  _et al._
(2008)) for the $q=5:1$ cases. Due to the constraint damping used in our
evolutions (see Eq. 19 in Duez  _et al._ (2003)), after about $t\sim 220M$ (or
around one orbit), the Hamiltonian constraint falls roughly to the same low
value for all cases in Table 1 regardless of the NS spin.

Figure 2: Volume rendering of rest-mass density $\rho_{0}$, normalized to its
initial NS maximum value $\rho_{0}=8.92\times 10^{14}\,(1.4M_{\odot}/M_{\rm
NS})^{2}\rm{g/cm}^{3}$ (log scale) at selected times for the nonmagnetized
case q3NSm0.17 (see Table 1). The BH apparent horizon is shown as a black
sphere. Top panels focus on the binary inspiral, while bottom ones focus on
the NS tidal disruption and disk formation. Despite the central density
oscillations, the shape of the NS is practically unaffected by spin, and
remains nearly spherical during the first five of the seven orbits prior to
merger. Here $M=2.5\times 10^{-2}(M_{\rm NS}/1.4M_{\odot}){\rm ms}=7.58(M_{\rm
NS}/1.4M_{\odot})\,\rm km$.

#### Initial data:

The BHNS initial data used in this work have been presented in Taniguchi  _et
al._ (2008). The configurations correspond to BHNS binaries on a quasicircular
orbit undergoing merger with a separation chosen to be outside the tidal
disruption radius Taniguchi  _et al._ (2008). We consider binaries with mass
ratio $q=3:1$, in which the BH companion has an initial spin parameter $a_{\rm
BH}/M_{\rm BH}=0.75$ ($M_{\rm BH}$ is the BH Christodoulou mass Christodoulou
(1970)) aligned with the total orbital angular momentum of the system, and
binaries with mass ratio $q=5:1$ with a nonspinning (irrotational) BH
companion (see Table 1). In all cases considered here, the companion has a
compaction of ${\cal C}={\mathcal{M}}_{\rm NS}/R_{\rm NS}=0.145$, where
$\mathcal{M}_{\rm NS}$ and $R_{\rm NS}$ are the ADM mass and the
circumferential radius of the star in isolation. Note that for a polytropic
EOS with $\Gamma=2$, the maximum mass configuration has a compaction
$\mathcal{C}=0.215$. For comparative purposes, we rescale the rest-mass of the
star as $M_{\rm NS}=1.4M_{\odot}(\kappa/\kappa_{L})^{1/2}$, and hence the
maximum rest-mass density of the star is $\rho_{0}^{\rm max}=8.92\times
10^{14}\,(1.4M_{\odot}/M_{\rm NS})^{2}\,\rm g/cm^{3}$. Here, $\kappa$ is the
polytropic gas constant used to compute the initial data and defined as
$\kappa=P/\rho_{0}^{2}$, and $\kappa_{L}=189.96\,\rm km^{2}$.

Following Ruiz  _et al._ (2014), to induce spin we endow an irrotational NS
with an uniform angular velocity by modifying the fluid velocity as
$v^{i}=v^{i}_{\rm irrot}+{\epsilon^{i}}_{jk}\Omega^{j}\,x^{k}$, where
$v^{i}=u^{i}/u^{0}$ is the coordinate velocity of the fluid, $u^{\mu}$ is the
fluid 4-velocity, $\Omega^{j}$ is an angular velocity of the NS, and
${\epsilon^{i}}_{jk}$ is the Levi-Civita symbol. As shown in Table 1, we endow
the NS with spins ranging between $-0.17$ and $0.23$ for the two BHNS mass
ratios considered here. In addition, to further assess if highly spinning NS
companions can induce potentially observable EM counterparts in BHNSs with
moderate mass ratios, we consider a more extreme case ($a_{\rm NS}/M_{\rm
NS}=0.33$) for $q=5:1$. In order to measure the spin of the NS, we compute its
quasilocal angular momentum $J_{\rm ql}$ Tsokaros  _et al._ (2019, 2018), and
normalize it to the rest-mass $J_{\rm ql}/M_{\rm NS}^{2}=a_{\rm NS}/M_{\rm
NS}$. If instead we normalized by its ADM mass (in isolation) the
dimensionless spin values will increase correspondingly (e.g. our more extreme
case with spin $a_{\rm NS}/M_{\rm NS}\sim 0.33$ becomes $\sim 0.35$). Using
the measured value of $J_{\rm ql}$, and the rest-mass $M_{\rm NS}$, we adopt
the Cook code Cook  _et al._ (1992, 1994a, 1994b) to generate equilibrium
rotating neutron star models in isolation with these values of angular
momentum and rest-mass, and compute the ratio of kinetic-to-gravitational
potential energy $T/|W|$ as well as the spin period $P$ that we list in Table
1). All our cases have an estimated $T/|W|<0.25$, and hence the NS is stable
against the dynamical bar mode instability Shibata  _et al._ (2000); Baumgarte
and Shapiro (1999); Paschalidis and Stergioulas (2017). While our case with
spin $a_{\rm NS}/M_{\rm NS}=0.33$ might be unstable to the secular $m=2$-bar
mode instability New and Shapiro (2001); Paschalidis and Stergioulas (2017),
we point out that the $T/|W|$ values we provide are only estimated values in
isolation.

Table 2: Summary. Case’s name starting with an NM (or an M) denotes a
nonmagnetized (or magnetized) evolution. $a_{\rm BH}/M_{\rm BH}$ denotes the
spin of the BH remnant, and $\Delta E_{\rm GW}$ and $\Delta J_{\rm GW}$ are
the fraction of energy and angular momentum carried away by GWs, respectively.
The kick velocity due to recoil is denoted by $v_{\rm kick}$ in $\rm km/s$,
$M_{\rm disk}$ is the fraction of the rest-mass of the material outside the
horizon near the end of the simulation, $\dot{M}$ is the rest-mass accretion
rate computed via Eq. (A11) in Farris  _et al._ (2010), and $\tau_{\rm
disk}\sim M_{\rm disk}/\dot{M}$ its lifetime in units of $(M_{\rm
NS}/1.4M_{\odot})\,$s, $\alpha_{\rm SS}$ is the Shakura–Sunyaev viscosity
parameter, and $B^{2}/(8\pi\,\rho_{0})$ is the space-averaged value of the
force-free parameter in a cubical region of length $2R_{\rm BH}$ above the BH
poles near the end of the simulation. Here $R_{\rm BH}$ is the radius of the
BH horizon. $B_{\rm rms}$ is the rms value of the magnetic field above the BH
poles in units of $(1.4M_{\odot}/M_{\rm NS})\,$G, $L_{\rm Poyn}$ is the
Poynting luminosity driven by the jet in units of $\rm erg/s$ and time-
averaged over the last $500M\sim 12.5(M_{\rm NS}/1.4M_{\odot})\rm ms$ of the
evolution, $v_{\rm eje}$ and $M_{\rm eje}$ are the mass-averaged velocity and
rest-mass of the unbound material, while $L_{\rm knova}$ and $t_{\rm peak}$
are the peak luminosity and the rise time of the potential kilonova in units
of $\rm erg/s$ and days, respectively. A dash symbol denotes no
corresponding/unavailable value.

Case $a_{\rm BH}/M_{\rm BH}$ $\Delta E_{\rm GW}/M$ $\Delta J_{\rm GW}/J$
$v_{\rm kick}$ $M_{\rm disk}/{M_{\rm NS}}$ $\dot{M}(M_{\odot}/s)$ $\tau_{\rm
disk}$ $\alpha_{\rm SS}$ $B^{2}/(8\pi\,\rho_{0})$ $B_{\rm rms}$ $L_{\rm Poyn}$
$v_{\rm eje}$ $M_{\rm eje}$ $L_{\rm knova}$ $t_{\rm peak}$ NMq3NS0.0† 0.89
$0.97\%$ $14.89\%$ $54.20$ $10.90\%$ $0.39$ $0.4$ $-$ $-$ $-$ $-$ $-$ $-$ $-$
$-$ NMq3NSm0.17 0.89 $1.14\%$ $13.10\%$ $44.62$ $10.84\%$ $0.46$ $0.3$ $-$ $-$
$-$ $-$ 0.24 $10^{-2.4}$ $10^{41.2}$ 0.18 NMq3NSp0.23 0.90 $1.10\%$ $14.48\%$
$63.96$ $14.56\%$ $0.28$ $0.7$ $-$ $-$ $-$ $-$ 0.29 $10^{-2.3}$ $10^{41.2}$
0.18 Mq3NS0.0† 0.85 $1.11\%$ $14.91\%$ $54.20$ $10.00\%$ $0.25$ $0.5$
$0.01-0.03$ $10^{2.0}$ $10^{15.2}$ $10^{51.2}$ $-$ $-$ $-$ $-$ Mq3NSm0.17 0.85
$1.20\%$ $12.90\%$ $46.40$ $8.78\%$ $0.29$ $0.5$ $0.01-0.03$ $10^{2.0}$
$10^{15.1}$ $10^{51.6}$ 0.25 $10^{-2.1}$ $10^{41.3}$ 0.25 Mq3NSp0.23 0.87
$1.12\%$ $14.60\%$ $64.81$ $14.17\%$ $0.23$ $0.8$ $0.01-0.03$ $10^{2.1}$
$10^{15.2}$ $10^{52.1}$ 0.27 $10^{-2.0}$ $10^{41.4}$ 0.27 NMq5NS0.0† 0.42
$1.07\%$ $20.07\%$ $67.96$ $0.29\%$ $0.02$ $0.2$ $-$ $-$ $-$ $-$ $-$ $-$ $-$
NMq5NSm0.17 0.42 $1.00\%$ $19.56\%$ $34.74$ $0.28\%$ $0.02$ $0.2$ $-$ $-$ $-$
$-$ 0.26 $10^{-4.7}$ $10^{40.0}$ 0.01 NMq5NSp0.23 0.42 $1.08\%$ $20.29\%$
$62.88$ $0.65\%$ $0.12$ $0.1$ $-$ $-$ $-$ $-$ 0.27 $10^{-3.7}$ $10^{40.4}$
0.02 NMq5NSp0.33 0.43 $1.07\%$ $20.35\%$ $91.10$ $1.23\%$ $0.15$ $0.1$ $-$ $-$
$-$ $-$ 0.33 $10^{-3.5}$ $10^{40.6}$ 0.04 Mq5NS0.0† 0.42 $1.05\%$ $19.63\%$
$69.96$ $0.53\%$ $0.04$ $0.2$ $-$ $10^{-3.0}$ $10^{12.3}$ $-$ $-$ $-$ $-$ $-$
Mq5NSm0.17 0.42 $1.00\%$ $19.43\%$ $36.82$ $0.53\%$ $0.02$ $0.3$ $-$
$10^{-3.2}$ $10^{12.4}$ $-$ 0.25 $10^{-4.7}$ $10^{40.0}$ 0.01 Mq5NSp0.23 0.42
$1.06\%$ $20.14\%$ $70.31$ $1.04\%$ $0.10$ $0.1$ $-$ $10^{-3.1}$ $10^{12.1}$
$-$ 0.27 $10^{-3.7}$ $10^{40.4}$ 0.03 Mq5NSp0.33 0.43 $1.06\%$ $20.20\%$
$93.80$ $1.42\%$ $0.12$ $0.2$ $-$ $10^{-2.9}$ $10^{12.7}$ $-$ 0.35 $10^{-3.5}$
$10^{40.6}$ 0.04

† Cases treated previously in Ruiz  _et al._ (2018a).

As the new fluid velocity field no longer satisfies the hydrostatic equations,
the NS undergoes small radial oscillations. Fig. 1 shows the relative changes
in the central rest-mass density during the early inspiral for all cases in
Table 1. In contrast to the quasiequilibrium (irrotational) case, where the
amplitude of the oscillations stays below $\sim 1\%$, the amplitude of the
oscillations in cases $a_{\rm NS}/M_{\rm NS}=-0.17$ and $0.23$ (see Table 1)
is larger but remains below $\sim 8\%$, and below $\sim 16\%$ in our extreme
case ($a_{\rm NS}/M_{\rm NS}=0.33$). The inset shows the $L_{2}$ norm of the
Hamiltonian constraint for cases with mass ratio $q=5:1$ during the early
inspiral. We observe that due to the constraint damping (see Eq. 19 in Duez
_et al._ (2003)) used in our numerical evolutions, the constraint violation
induced by our ad-hoc prescription of the new fluid velocity is damped and
propagated away after roughly one orbit (or $t\lesssim 220M$). Similar results
are observed on the other cases in Table 1.

Next, we evolve the above configurations until about two orbits before tidal
disruption. At that point, the NS is threaded by a dynamically weak, dipolar
magnetic field induced by a vector potential generated by a current loop
inside the star (see Eq. 2 in Paschalidis  _et al._ (2013)). As in Paschalidis
_et al._ (2015); Ruiz  _et al._ (2018a), we choose the current $I_{0}$ and the
radius of the loop $r_{0}$ such that the magnetic-to-gas-pressure ratio at the
center of the NS is $P_{\rm mag}/P_{\rm gas}=10^{-2.5}$ (see Fig. 2 in Ruiz
and Shapiro (2017)). The resulting magnetic field at the pole of the star
turns out to be $B_{\rm pole}\sim 6\times 10^{15}(1.4M_{\odot}/M_{\rm
NS})\,\,$G. As pointed out in Paschalidis  _et al._ (2015), although we choose
an astrophysically large magnetic field, it is dynamically unimportant in the
stellar interior and does not affect the late inspiral or the merger phases.
We do expect that the outcome of our numerical results will apply to other
dynamically weak field choices because the magnetic field amplification
following merger will be mainly triggered by magnetic winding and the magneto-
rotational-instability (MRI) Kiuchi  _et al._ (2015).

On the other hand, to capture one of the properties of the force-free
conditions that likely characterize the NS exterior (magnetic-pressure
dominance), we set a variable and low-density magnetosphere outside the star
such that the magnetic-to-gas pressure ratio is $\beta^{-1}=P_{\rm gas}/P_{\rm
mag}=0.01$ everywhere Paschalidis  _et al._ (2015). This one-time reset of the
low-density magnetosphere increases the total rest-mass on the entire grid by
less than $1\%$.

#### Grid structure:

The grid hierarchies used to evolve the BHNS binaries with mass ratio $q=3:1$
and $q=5:1$ are the same as those used to evolve models Tilq3sp0.75 and
Aliq5sp0.0 (see Table II of Ruiz  _et al._ (2018a)), respectively. They
consist of two sets of nested refinement levels centered on both the BH and
the NS. In all cases, the NS is covered by eight refinement levels, while the
BH companion is covered by nine refinement levels for $q=3:1$ cases, or by
eight levels for $q=5:1$ cases. The finest resolution box has a half length of
$\sim 1.5\,R_{\rm BH}$ around the BH and $\sim 1.2\,R_{\rm NS}$ around the NS.
These choices resolve the initial equatorial radius of the BH apparent horizon
by around $40$ grid points, and the initial NS equatorial radius by around
$42$ grid points. In all cases we impose reflection symmetry across the
orbital plane ($z=0$).

##  III Results

The basic dynamics and outcomes of irrotational NS cases in Table 2 have been
previously described in Etienne  _et al._ (2009); Paschalidis  _et al._
(2015); Ruiz  _et al._ (2018a). There it was found that, in contrast to the
high mass ratio cases where the NS basically plunges into the BH, the NS in
$q=3:1$ cases is tidally disrupted before reaching the ISCO. This resulted in
long tidal tails of matter that eventually settle down, forming a significant
accretion disk around the BH regardless of the magnetic field content (see
Figs. 2 and 3).

Figure 3: Volume rendering of rest-mass density $\rho_{0}$ normalized to its
initial NS maximum value $\rho_{0}=8.92\times 10^{14}\,(1.4M_{\odot}/M_{\rm
NS})^{2}\rm{g/cm}^{3}$ (log scale) at selected times for Mq3NSp0.23 (see Table
1). Bottom panels highlight the emergence of the magnetically-driven jet.
White lines denote the magnetic field, arrows denote the fluid velocity, while
the BH apparent horizon is shown as a black sphere. Here $M=2.5\times
10^{-2}(M_{\rm NS}/1.4M_{\odot})\,{\rm ms}=7.58(M_{\rm NS}/1.4M_{\odot})\,\rm
km$.

In the magnetized cases, matter wrapping around the BH drags the frozen-in
magnetic field into a predominantly toroidal configuration. However, initially
the magnetic field lines connect the star with the poles of the BH, and later
the low-density debris ejected during the tidal disruption remains connected
to the accretion disk via these field lines (see top panels in Figs. 3 and 4),
as a result, the external magnetic field maintains a strong poloidal
component. Magnetic winding and the MRI then amplify the magnetic field above
the BH poles from $\sim 10^{13}(1.4M_{\odot}/M_{\rm NS})\,$G to $\sim
10^{15}(1.4M_{\odot}/M_{\rm NS})\,$G, when the accretion disk settles down.
This amplification induces high magnetic pressure gradients above the BH poles
that, when the regions above the BH poles approach force-free values
($B^{2}/8\,\pi\rho_{0}\gg 1$), lead to the launching of a mildly relativistic
outflow with a Lorentz factor $\Gamma_{L}\gtrsim 1.2$ confined inside a
tightly wound, helical magnetic field funnel –an incipient jet (see bottom
panels in Figs. 3 and 4). Here $B$ and $\rho_{0}$ are the strength of the
magnetic field and the rest-mass density, respectively. The lifetime of the
jet and its associated luminosity are consistent with typical sGRBs Bhat  _et
al._ (2016); Lien  _et al._ (2016); Svinkin  _et al._ (2016); Ajello  _et al._
(2019), as well as with the BZ mechanism Blandford and Znajek (1977); Shapiro
(2017).

###  III.1 Mass ratio $q=3:1$

Figure 2 shows snapshots of the rest-mass density at selected times for
NMq3NSm0.17 (see Table 2 for an explanation of the augmented case labels). We
observe that although the star undergoes small radial oscillations due to our
ad hoc prescription for the NS spin (see Fig. 1), the shape of the star is
nearly spherical during the first five of the almost seven inspiral orbits
before the binary merger (see top panels in Fig. 2). Bottom panels focus on
the NS tail deformation, merger, and the subsequent formation of a
quasistationary disk, as matter having larger specific angular momentum wraps
around the BH. Similar behavior is observed in the other cases independent of
the magnetic field (see Table 2). However, we observe that the larger the NS
prograde spin is, the further out the NS disruption take place.

Notice that as the BH spin increases the ISCO decreases, and hence the tidal
disruption effects become more pronounced, resulting in long tidal tails that
eventually wrap around the BH forming the accretion disk. Similarly, as the
prograde NS spin increases, the effective ISCO decreases (see e.g. Barausse
and Buonanno (2010)). Additionally, as the magnitude of the NS spin increases,
the star becomes less bound, and the tidal separation radius $r_{\rm tid}$
(the separation at which tidal disruption of the NS begins) increases,
resulting also in more pronounced disruption effects. This effect can be
easily understood by estimating $r_{\rm tid}$ through a simple Newtonian
argument. Equating the inward gravitational force exerted by the NS on its
fluid elements with the BH’s outgoing tidal force and the outgoing centrifugal
force we find that (see also East  _et al._ (2015) for a similar expression)

$r_{\rm tid}/M_{\rm BH}\simeq
q^{-2/3}\,\mathcal{C}^{-1}\,\left[1-\Omega^{2}\,M_{\rm
NS}^{2}\,\mathcal{C}^{-3}\right]^{-1/3}\,,$ (1)

where $\Omega=a_{\rm NS}M_{\rm NS}/I$. Here $I$ is the moment of inertia of
the star. Therefore, the larger the magnitude of the NS spin, the larger
$r_{\rm tid}$, and hence the more material spreads out to form the disk.
Consistent with the above predictions, we find that the accretion disk in our
extreme cases has a rest mass ranging between $\sim 9\%$ (for Mq3NSm0.17) and
$\sim 14.2\%$ (for Mq3NSp0.23) of the total rest-mass of the NS (see top panel
in Fig. 5). Slightly more massive disks are found in the nonmagnetized cases
(see Table 2).

Figure 4: Same as Fig. 3 but for Mq3NSm0.17.

In all cases, the BH remnant has a mass of $\simeq 4.76M_{\odot}(M_{\rm
NS}/1.4M_{\odot})$, and its spin is $a_{\rm BH}/M_{\rm BH}\simeq 0.9$ for the
nonmagnetized cases, and $a_{\rm BH}/M_{\rm BH}\simeq 0.85$ for the magnetized
cases. These values seem to be unaffected by the initial NS spin. Similar
behavior was reported in spinning BNS mergers Ruiz  _et al._ (2019).

By contrast to the nonmagnetized cases, where the BH + disk remnant settles
down into an almost steady configuration after $\sim 800M\simeq 20(M_{\rm
NS}/1.4M_{\odot})\,\rm ms$ following merger (see bottom panels of Fig. 2), the
magnetized cases launch a mildly relativistic outflow confined in a tightly
wound, helical magnetic field funnel after $\sim 3500-5500M\simeq
88-138(M_{\rm NS}/1.4M_{\odot})\,\rm ms$ (see below) following merger (see
bottom panels in Figs. 3 and 4).

Figure 5: rest-mass fraction outside the BH apparent horizon as a function of
the coordinate time for cases listed in Table 2. The insets focus on pure
nonmagnetized cases. The coordinate time has been shifted to the merger time
$t_{\rm mer}$.

To probe if magnetic turbulence is operating within the accretion disk, we
first verify that the wavelength of the fastest growing MRI mode $\lambda_{\rm
MRI}$ in all our cases is resolved by $\gtrsim 5$ grid points (see Fig. 9 in
Ruiz  _et al._ (2018a)). It is likely that the MRI is at least partially
captured in our simulations Gold  _et al._ (2014). We also compute the
Shakura–Sunyaev viscosity $\alpha_{\rm SS}$ parameter through Eq. 26 in Penna
_et al._ (2010). We find that in the innermost $12{\rm M}\simeq 91(M_{\rm
NS}/1.4M_{\odot})\,\rm km$ of the disk and outside the ISCO, $\alpha_{\rm SS}$
ranges between $0.01$ and $0.03$ (see Table 2), consistent with values
reported in earlier accretion disk studies Krolik and Hawley (2007); Gold  _et
al._ (2014). Therefore, it is expected that magnetic turbulence driven by MRI
is operating to some degree in our simulations and drives the quasisteady
accretion. However, farther studies at higher resolution are required to
confirm these results. We compute the rest-mass accretion rate $\dot{M}$
through Eq. A11 in Farris  _et al._ (2010) and found after $\sim 1500M\simeq
38(M_{\rm NS}/1.4M_{\odot})\,$ms following the merger, the accretion begins to
settle to a quasi-stationary state and decays slowly afterward. In magnetized
cases, we find that $\dot{M}$ is roughly $0.3M_{\odot}/s$ (see Table 2) once
the outflow reaches a height of $\sim 100M\simeq 760(M_{\rm
NS}/1.4M_{\odot})\,\rm km$. At that time, the disk has a mass of $\sim
0.13M_{\odot}(M_{\rm NS}/1.4M_{\odot})$ for Mq3NSm0.17 and Mq3NS0.0, and of
$\sim 0.2M_{\odot}(M_{\rm NS}/1.4M_{\odot})$ for Mq3NSp0.23 (see Table 2).
Hence the disk (jet’s fuel) is expected to be accreted in $\Delta t\sim M_{\rm
disk}/\dot{M}\sim 0.5-0.8(M_{\rm NS}/1.4M_{\odot})\,\rm s$, consistent with
the lifetime of typical sGRBs Berger (2014).

As pointed out in Paschalidis  _et al._ (2015), following tidal disruption,
magnetic winding and the MRI amplify the magnetic field and induce magnetic
pressure gradients above the BH poles that eventually overcome the baryon ram
pressure of the fall-back debris and drive an outflow collimated by the
magnetic field (see bottom panels in Fig. 3 and 4). In Mq3NSm0.17 we find that
an incipient jet is launched after $\sim 3500M\simeq 88(M_{\rm
NS}/1.4M_{\odot})\,\rm ms$ following merger, while in high prograde spin case
Mq3NSp0.23 the jet emerges after $\sim 5500M\simeq 138(M_{\rm
NS}/1.4M_{\odot})\,\rm ms$. This delay time is not unexpected. As the NS spin
increases, the tidal disruption occurs farther out from the ISCO (see Eq. 1)
and so more material has larger specific angular momentum and spreads out
further. In the prograde NS spin case, the effective ISCO is smaller than in
the retrograde case Barausse and Buonanno (2010). Thus, a larger fraction of
this material remains outside BH horizon and induces a more baryon-loaded
environment that survives for a longer time. As a fraction of this material
rains back, the matter density above the BH poles drops and the magnetic-field
pressure gradients are able to overcome this ram pressure and finally launch a
jet. It should be noted that the jet launching may not be possible for all
EOSs if the matter fall-back timescale is longer than the disk accretion
timescale Paschalidis (2017).

We measure the level of the collimation of the jet through the funnel opening
angle $\theta_{\rm jet}$ defined as the polar angle at which the Poynting flux
drops to $50\%$ of its maximum Paschalidis  _et al._ (2015). Based on the
angle distribution of the outgoing flux on the surface of a coordinate sphere
with radius $60M\sim 455(M_{\rm NS}/1.4M_{\odot})\rm km$ (see Fig. 13 in Ruiz
_et al._ (2018a)), we estimate that the opening angle of the jet is $\sim
25^{\circ}-30^{\circ}$.

Following the emergence of the jet, we verify the outgoing material in the
funnel has specific energy $E=-u_{0}-1>0$ (asymptotic region) and hence is
unbound. The characteristic maximum value of the Lorentz factor reached in the
outflow is $\Gamma_{L}\sim 1.2-1.3$. However, as pointed out in Vlahakis and
Königl (2003), fluid elements can be accelerated to $\Gamma_{L}\simeq
b^{2}/2\rho_{0}\simeq 100$ (see Table 2) consistent with sGRBs which require
$\Gamma_{L}\gtrsim 20$ Zou and Piran (2010).

Fig. 6 shows the outgoing EM (Poynting) luminosity computed through $L_{\rm
Poyn}\equiv-\int T^{r(EM)}_{t}\,\sqrt{-g}\,d\mathcal{S}$ Ruiz  _et al._ (2019)
across a spherical surface of coordinate radius $r=80{\rm M}\simeq 606(M_{\rm
NS}/1.4M_{\odot})\,\rm km$. The luminosity is $L_{Poyn}\simeq 10^{51.5\pm
0.5}\,\rm ergs/s$ (see Table 2), and hence is consistent with the BZ
luminosity $L_{\rm BZ}\sim 10^{51}\,a^{2}\,B^{2}_{15}\,M_{5}^{2}\,\rm erg/s$
(see Eq. 4.50 in Thorne  _et al._ (1986)), as well as with typical sGRB
(equivalent isotropic) luminosities (see e.g. Shivvers and Berger (2011)).
Here $B_{15}=B/10^{15}\,\rm G$ and $a=a_{\rm BH}/M_{\rm BH}$ and $M_{5}=M_{\rm
BH}/5M_{\odot}$.

To further assess if the BZ mechanism is operating in the BHNS remnants we
compute the ratio of the angular velocity of the magnetic fields to the
angular velocity of the BH $\Omega_{F}$ (see Eq.12 in Ruiz  _et al._ (2018a))
on a meridional plane passing through the BH centroid and along semicircles of
coordinate radii between $r=R_{\rm BH}$ and $2\,R_{\rm BH}$. In all cases
$\Omega_{F}$ ranges between $\sim 0.4-0.45$ at the BH poles and $\sim 0.1$
near the equator, and hence the field lines are differentially rotating. It
should be noted that deviations from the expected $\Omega_{F}=0.5$ value (see
e.g. Komissarov (2001)) can be attributed to the deviations from the gauge in
which $\Omega_{F}$ is computed ($\Omega_{F}$ is defined for stationary and
axisymmetric spacetimes in Killing coordinates), deviation from a split-
monopole magnetic field or lack of resolution Paschalidis  _et al._ (2015).

We measure the dynamical ejection of matter (ejecta) through $M_{\rm
esc}=\int\rho_{*}\,d^{3}x$ outside a coordinate radius $r>r_{0}$, and under
the following conditions: a) $E=-1-u_{0}>0$, and b) positive (outgoing) radial
velocity of the ejected material. Here
$\rho_{*}\equiv-\sqrt{\gamma}\rho_{0}\,n_{\mu}\,u^{\mu}$, where $\gamma$ is
the determinant of the three metric and $n^{\mu}$ the future pointing normal
vector to a $t=$ constant hypersurface. To verify that our results are
independent of $r_{0}$ at large radius, we compute the mass of the ejecta
varying $r_{0}$ between $30M\simeq 230~{}(M_{\rm NS}/1.4M_{\odot})\,\rm km$
and $100M\simeq 760(M_{\rm NS}/1.4M_{\odot})\,\rm km$. As shown in the inset
of Fig. 6, the initial NS spin has a strong effect on the ejecta (especially
in the $q=5:1$ case that we discuss in the next section). The ejecta in
Mq3NSp0.23 is around $35\%$ higher than in Mq3NSm0.17, where it turns out to
be $10^{-2.1}M_{\odot}(M_{\rm NS}/1.4M_{\odot})$. Slightly smaller values of
the ejecta have been recently reported in BHNS mergers where the NS companion
(irrotational) is modeled with softer (H-type) EOSs Hayashi  _et al._ (2020).
Ejecta masses $\gtrsim 10^{-3}M_{\odot}$ are expected to lead to detectable,
transient kilonovae signatures (see e.g. Metzger (2017)) powered by
radioactive decay of unstable elements formed by the neutron-rich material
ejected during BHNS mergers Li and Paczynski (1998); Metzger (2017). In Barnes
and Kasen (2013) it was shown that the opacities in r-process ejecta are
likely dominated by lanthanides, which induce peak bolometric luminosities for
kilonovae of East  _et al._ (2015)

$L_{\rm knova}\approx 10^{41}\left(\frac{M_{\rm
eje}}{10^{-2}M_{\odot}}\right)^{1/2}\,\left(\frac{v_{\rm
eje}}{0.3c}\right)^{1/2}\,\rm erg/s\,,$ (2)

and rise times of East  _et al._ (2015)

$t_{\rm peak}\approx 0.25\,\left(\frac{M_{\rm
eje}}{10^{-2}M_{\odot}}\right)^{1/2}\,\left(\frac{v_{\rm
eje}}{0.3c}\right)^{-1/2}\,\rm days\,.$ (3)

Here $v_{\rm eje}$ and $M_{\rm eje}$ are the mass-averaged velocity and rest-
mass of the ejecta. Using the above equations, we estimate that the bolometric
luminosity of potential kilonovae signals is $L_{\rm knova}=10^{41.3\pm
0.1}\rm erg/s$ with rise times of $0.18-0.27$ days (see Table 2). These
luminosities correspond to an R band magnitude of $\sim 24$ mag at $200$ Mpc
(inside the aLIGO volume Abbott  _et al._ (2013)), and above the LSST survey
sensitivity of $24.5$ mag Barnes and Kasen (2013); East  _et al._ (2015), and
hence may be detectable by the LSST survey.

Finally, we compute the characteristic interior temperature $T_{\rm disk}$ of
the disk remnant assuming that the specific thermal energy density
$\epsilon_{\rm th}$ can be modeled as Etienne  _et al._ (2009)

$\epsilon_{\rm th}=\frac{3\,k_{\rm B}\,T_{\rm disk}}{2\,m_{n}}+f_{\rm
s}\,\frac{a\,T_{\rm disk}^{4}}{\rho_{0}}\,,$ (4)

where $k_{B}$ is the Boltzmann constant, $m_{n}$ is the mass of a nucleon, and
$a=8\pi^{5}k_{\rm B}^{4}/(15\,h_{\rm P}^{3})$ is the radiation constant. Here
$h_{\rm P}$ is the Plank constant. As pointed out in Etienne  _et al._ (2009),
the first term in Eq. 4 is approximately the thermal energy of the nucleons,
while the second represents the thermal energy due to radiation and thermal
relativistic particles. The factor $f_{\rm s}$ accounts for the number of
species of ultrarelativistic particles that contribute to thermal energy. When
$T\ll 10^{10}\,\rm K$ thermal radiation is dominated by photons and $f_{\rm
s}=1$. When $T\gg 10^{10}\,\rm K$, electrons and positrons become
ultrarelativistic and also contribute to radiation, and hence $f_{\rm
s}=1+2\times(7/8)=11/4$. At sufficiently high temperatures ($T\gtrsim
10^{11}\,\rm K$) and densities ($\rho_{0}\gtrsim 10^{12}\,\rm g/cm^{3}$),
thermal neutrinos and antineutrinos are copiously generated and become
trapped. Taking into account three flavors of neutrinos and antineutrinos
$f_{\rm s}=11/4+3\times(7/8)=43/8$.

We measure the thermal energy generated by shocks through the ratio
$K=P/P_{\rm cold}$ (entropy parameter), where $P_{\rm
cold}=\kappa\rho_{0}^{\Gamma}$ is the pressure associated with the cold EOS
used to build our initial configurations. In all cases we find the
characteristic value of the entropy parameter in the disk is $K\sim 200$.
Next, we compute the specific thermal energy as $\epsilon_{\rm
th}=(K-1)\,\epsilon_{\rm cold}$ with $\epsilon_{\rm cold}=\kappa\,\rho_{0}$
for a polytropic EOS with $\Gamma=2$ (see Eq. 12 in Etienne  _et al._ (2009)).
Plugging these values in Eq. 4, we find the characteristic value of the
temperature in the disk is $T_{\rm disk}\sim 10^{11.0}\,\rm K$ (or 8.6 MeV)
for the nonmagnetized cases, where the characteristic densities in the disk
are $\rho_{0}\sim 10^{12}\,\rm gm/cm^{3}$ (see bottom panels in Fig. 2), and
$T\sim 10^{10.6}\,\rm K$ (or 3.4 MeV) for the magnetized cases, where the
characteristic densities are $\rho_{0}\sim 10^{11}\,\rm gm/cm^{3}$ (see bottom
panels in Figs. 3 and 4). Thus, these hot accretion disks may emit a copious
amount of neutrinos with a peak luminosity of $10^{53}\,\rm erg/s$ through
thermal pair production and subsequent electron/positron captures on free
nucleons Kyutoku  _et al._ (2018). However, their lifetimes might be too small
to explain the majority of sGRBs Just  _et al._ (2016). It has been suggested
that BH + disk engines that power typical sGRBs may be dominated initially by
thermal pair production followed by the BZ process, leading to a transition
from a thermally dominated fireball to a Poynting dominated outflow as
observed in some GRBs, such as GRB 160625B Dirirsa (2017).

Figure 7 shows the GW strain $h_{+}$ of the dominant mode $(2,2)$ for these
configurations. Left column displays the nonmagnetized evolutions, while the
right one displays the magnetized evolutions. The corresponding binaries
(nonmagnetized and magnetized) merge roughly at the same time (here the merger
time $t_{\rm mer}$ is defined as the time of peak amplitude of the GWs). This
result is anticipated because the seed magnetic field is dynamically weak and
there is no significant enhancement of its magnitude during the inspiral (the
seed magnetic field is simply advected with the fluid). However, due to the
hang-up effect Campanelli  _et al._ (2006), the prograde NS spin configuration
($a_{\rm NS}/M_{\rm NS}=0.23$) aligned with the total orbital angular momentum
of the system (bottom panel in Fig. 7) undergoes about one or two more orbits
compared to the irrotational and the retrograde NS spin ($a_{\rm NS}/M_{\rm
NS}=-0.17$) cases (top and middle panels), respectively. A similar effect has
been reported in BBHs Campanelli  _et al._ (2006), and BNSs Tsatsin and
Marronetti (2013); Kastaun  _et al._ (2013); Dietrich  _et al._ (2017); Ruiz
_et al._ (2019); Tsokaros  _et al._ (2019); East  _et al._ (2019).

Figure 6: Outgoing EM (Poynting) luminosity following jet launching, computed
on a coordinate sphere of radius $r=80{\rm M}\simeq 606(M_{\rm
NS}/1.4M_{\odot})\,\rm km$ for the magnetized cases listed in Table 2. The
inset shows the rest-mass fraction of escaping matter following the peak
amplitude of GWs.  Figure 7: Mode $(l,m)=(2,2)$ of the GW strain $h_{+}$ as
function of the retarded time extracted at a coordinate sphere of radius
$r=80M\sim 606(M_{\rm NS}/1.4M_{\odot})\,$km for nonmagnetized (left column)
and magnetized (right column) cases with mass ratio $q=3:1$ (see Table 2).

###  III.2 Mass ratio $q=5:1$

Figure 8: Same as Fig. 3 but for Mq5NSp0.33.

Fig. 8 summarizes the evolution of Mq5NSp0.33 (our extreme case). All
configurations with mass ratio $q=5:1$ in Table 2 have basically the same fate
independent of the magnitude of the magnetic field or the initial NS spin: The
tidal disruption occurs closer to the ISCO, resulting in short tidal tails
(see middle panel) that leave stellar debris outside the BH horizon with mass
$\lesssim 1.4\%$ of the rest-mass of the NS (bottom panel of Fig. 5). See
Table 2 for other cases.

Figure 9: Same as Fig. 7 but for BHNS configuration with mass ratio $q=5:1$
(see Table 2).

We do not find evidence for an outflow or large-scale magnetic field
collimation (see bottom panel in Fig. 8).

Bottom panel of Fig. 5 shows the rest-mass fraction outside the BH horizon. We
observe that in Mq5NSp0.33 the mass outside the BH horizon is around three
times larger than in Mq5NSm0.17. So, the higher the prograde NS spin, the
larger the fraction of matter ejected. The inset of Fig. 6 shows the dynamical
ejection of matter for the magnetized cases (similar values are found for the
nonmagnetized cases). In the extreme cases, the ejecta ranges between
$10^{-4.5}M_{\odot}(M_{\rm NS}/1.4M_{\odot})$ (for Mq5NSm0.17) and
$10^{-3.5}M_{\odot}(M_{\rm NS}/1.4M_{\odot})$ (for Mq5NSp0.33). The latter is
near the threshold value of ejecta required to give rise to a detectable
kilonova Metzger (2017). Using Eqs. 2 and 3 we estimate peak bolometric
luminosities from potential kilonovae of $L_{\rm knova}\sim
10^{40.0}-10^{40.6}$ erg/s and rise times $\lesssim 1~{}$h (see Table 2). Such
a kilonova is potentially detectable by the LSST survey, although it would
require rapid response and high cadence EM follow-up observations. These
results suggest that even in the absence of a jet, the GWs from BHNS mergers
with moderate mass ratio and/or moderate BH spin may be accompanied by
detectable kilonovae signatures if the companion is a highly spinning NS,
which agrees with East  _et al._ (2015).

The GW strain $h_{+}$ of the dominant mode $(l,m)=(2,2)$ for the nonmagnetized
(left column) and magnetized (right column) evolutions is shown in Fig. 9. We
observe that the more extreme configurations ($a_{\rm NS}/M_{\rm NS}=0.23$ and
$a_{\rm NS}/M_{\rm NS}=0.33$) displayed on the two bottom rows undergo about
half an orbit ($a_{\rm NS}/M_{\rm NS}=0.23$) and a full orbit ($a_{\rm
NS}/M_{\rm NS}=0.33$) more compared to the nonspinning and retrograde NS spin
cases (first two rows). Thus, NS spin can lead to dephasing and should be
accounted for in BHNS waveform templates.

###  III.3 Distinguishability of the gravitational waves

Figure 10: Gravitational-wave power spectrum of the dominant mode
$(l,m)=(2,2)$ at a source distance of $100\rm Mpc$ for our extreme cases with
mass ratio $q=3:1$ (top panel) and mass ratio $q=5:1$ (bottom panel), along
with the aLIGO noise curve. This curve corresponds to the
ZERO$\\_$DET$\\_$HIGH$\\_$P configuration Shoemaker . Solid curves display the
hybrid waveform found by appending the TaylorT1 PN waveform to the raw
numerical signal (dotted curves).

As mentioned in the previous section, as the initial NS spin increases, the
binary inspiral lasts longer, resulting in more gravitational wave cycles (see
Figs. 7 and 9). This enhancement induces a dephasing with respect to the non-
spinning cases, and a change in the amplitude of the GWs of $\lesssim 5\%$
between the respective waveforms, which is reflected in slight changes in the
energy $\Delta E_{\rm GW}$ and angular momentum $\Delta J_{\rm GW}$ carried
away by GWs (see Table 2), as well as in the kick velocity $v_{\rm kick}$ due
to recoil (see Eqs. 3.7 and 3.20 in Ruiz  _et al._ (2008)). In this section,
we probe if these spin-driven and magnetic effects can be distinguished by
aLIGO.

We start by extending the GW spectra in the frequency domain, creating a
hybrid waveform. Following Etienne  _et al._ (2012), we append a TaylorT1
post-Newtonian waveform Ajith  _et al._ (2007) to that of our numerical
relativity simulations. The hybrid waveform is then obtained by minimizing

$\bigintsss_{t_{\rm i}}^{t_{\rm f}}dt\,\left[(h_{+}^{\rm NR}-h_{+}^{\rm
PN})^{2}+(h_{x}^{\rm NR}-h_{x}^{\rm PN})^{2}\right]^{1/2}\,,$ (5)

via the Nelder-Mead algorithm Nelder and Mead (1965), using as free parameters
the initial PN phase, amplitude, and orbital angular frequency. In cases with
mass ratio $q=3:1$ we integrate the above expression between $t_{\rm i}=150M$
and $t_{\rm f}=450M$, while in cases with mass ratio $q=5:1$ the integration
range is between $t_{\rm i}=150M$ and $t_{\rm f}=250M$.

Fig. 10 shows the GW spectrum of the dominant mode $(l,m)=(2,2)$ at a source
distance of $100\rm Mpc$ for our extreme cases listed in Table 2, along with
the aLIGO noise curve of the ZERO$\\_$DET$\\_$HIGH$\\_$P configuration
Shoemaker . Solid (dotted) lines display the hybrid (raw numerical) signals.
We observe that the GW spectrum for cases with mass ratio $q=3:1$ (top panel)
rapidly decays as the star becomes disrupted ($f\sim 900~{}(M_{\rm
NS}/1.4M_{\odot})^{-1}\,\rm Hz$). By contrast, in cases with mass ratio
$q=5:1$ (bottom panel), the signal slowly decays until merger, where the star
basically plunges into the BH ($f\sim 1700~{}(M_{\rm
NS}/1.4M_{\odot})^{-1}\,\rm Hz$), and then the GW power drops significantly.
Unlike the NS spin imprints on the GWs evident at high frequencies, the
magnetic field imprints are not significant. In the lowest mass ratio cases,
the differences in the GW spectrum are marginally observable in the aLIGO band
(top panel), while on those with mass ratio $q=5:1$ are not evident at all
even outside the aLIGO band (bottom panel).

A more precise way to assess the distinguishability is through the match
function $\mathcal{M}_{\text{\tiny{GW}}}$ defined as Allen  _et al._ (2012)

$\mathcal{M}_{\text{\tiny{GW}}}=\underset{(\phi_{c},t_{c})}{{\rm
max}}\frac{\left<{h}_{1}|{h}_{2}(\phi_{c},t_{c})\right>}{\sqrt{\left<{h}_{1}|{h}_{1}\right>\left<{h}_{2}|{h}_{2}\right>}}\,,$
(6)

between two given waveforms. The maximization is taken over a large set of
phase shifts $\phi_{c}$ and time shifts $t_{c}$. Here
$\left<{h}_{1}|{h}_{2}\right>$ denotes the noise-weighted inner product Allen
_et al._ (2012)

$\left<h_{1}|h_{2}\right>=4\,{\rm
Re}\int_{0}^{\infty}\frac{\tilde{h}_{1}(f)\,\tilde{h}^{*}_{2}(f)}{S_{h}(f)}\,df\,,$
(7)

were $h=h_{+}-i\,h_{\times}$, $\tilde{h}$ is the Fourier transform of the
strain amplitude $\sqrt{\tilde{h}_{+}(f)^{2}+\tilde{h}_{\times}(f)^{2}}$ of
the dominant mode $(l,m)=(2,2)$, and $S_{h}(f)$ is the power spectral density
of the aLIGO noise Shoemaker . In our extreme cases, we find that
$\mathcal{M}_{\text{\tiny{GW}}}=0.9980$ between the waveforms of the q3NSm0.17
cases, and $\mathcal{M}_{\text{\tiny{GW}}}=0.9982$ between those of the
q3NSp0.23. Similarly, $\mathcal{M}_{\text{\tiny{GW}}}=0.9998$ for all of our
extreme cases with mass ratio $q=5:1$.

The standard choice for the threshold match for distinguishing two signals is
$1-1/(2\rho^{2})$, where $\rho$ here is the signal-to-noise ratio (SNR). For a
SNR of 15 Harry and Hinderer (2018), two signals are distinguishable when
$\mathcal{M}_{\text{\tiny{GW}}}\lesssim 0.9956$. Using Equation (18) in Baird
_et al._ (2013) (with one degree of freedom since we compare configurations
varying one parameter), a match of $1-1/(2\rho^{2})$ corresponds to a 68%
confidence level. Thus, it seems unlikely that aLIGO detectors can detect the
magnetic field imprints on the GWs even at 68% confidence level. By contrast,
the imprints of the NS spin on the waveform are more easily detectable. The
match function between the waveform of Mq3NSp0.23 and Mq3NSm0.17 is $0.9844$,
and $0.9874$ between those of Mq5NSp0.33 and Mq5NSm0.17, and hence smaller
than the threshold match value.

##  IV Conclusions

We previously reported self-consistent MHD numerical simulations in full GR
showing that BHNS binaries undergoing merger and significant NS tidal
disruption outside the ISCO can launch a magnetically-driven jet Paschalidis
_et al._ (2015); Ruiz  _et al._ (2018a). This happens whenever a net poloidal
magnetic flux is accreted onto the BH with $B^{2}/8\pi\rho_{0}\gg 1$ above the
BH poles, and hence these systems serve as a possible progenitor of the
central engine that powers a sGRB. However, population synthesis studies
Belczynski  _et al._ (2008, 2010) along with the reported aLIGO/Virgo GW
detections Abbott  _et al._ (2017); GW alerts ; Abbott  _et al._ (2020a)
suggest that in typical BHNSs we may have $M_{\rm BH}/M_{\rm NS}\gtrsim 5$.
For such high mass ratios the NS companion simply plunges into the BH before
undergoing tidal disruption, leaving a negligible amount of matter outside the
BH horizon ($\lesssim 2\%$ of the NS rest-mass). Thus, BHNS mergers may not be
accompanied by a near simultaneous, observable EM counterpart. However, the NS
spin can have a strong impact on the dynamical ejection of matter, and so may
lead to subsequent kilonovae signatures. It should be noted that the current
GW observations set constraints on the effective spin of the binary but not on
the individual spins.

In this paper, we explored the impact of the NS spin companion on the
dynamical ejection of matter, the mass of the accretion disk, and the jet
launching, from BHNS binaries undergoing merger with moderate mass ratios
($q=3:1$ and $q=5:1$). For comparative purposes, we considered the
quasiequilibrium BHNS initial data used previously in Ruiz  _et al._ (2018a);
Etienne  _et al._ (2009) and endowed the NS companion with an effective spin
Ruiz  _et al._ (2014).

Consistent with our previous results Paschalidis  _et al._ (2015); Ruiz  _et
al._ (2018a), we found that all magnetized cases with mass ratio $q=3:1$ in
Table 2 launch a magnetically-driven jet after $\Delta t\sim 3500M-5500\approx
88-138(M_{\rm NS}/1.4M_{\odot})\,\rm ms$ following the peak GW signal (see
bottom panels in Figs. 3 and 4). At these times the force-free parameter above
the poles of the BH reaches values of $b^{2}/(2\,\rho_{0})\gtrsim 100$ ( see
Table 2). The time delay between GW peak and jet launching depends strongly on
the NS spin. The larger the NS spin, the longer the delay. This result can be
explained by the fact that as the prograde NS spin increases the effective
ISCO Barausse and Buonanno (2010) decreases, while the NS becomes less bound,
so that the onset of NS tidal disruption occurs farther out from the ISCO.
This causes long tidal tails of matter having larger specific angular momentum
that spread out and form a baryon-loaded environment that persists for a
longer time. The lifetime of the jet [$\Delta t\sim 0.5-0.8(M_{\rm
NS}/1.4M_{\odot})\rm s$] and outgoing Poynting luminosity [$L_{\rm Poyn}\sim
10^{51.5\pm 0.5}\,\rm erg/s$] are consistent with typical sGRBs, as well as
with the BZ mechanism Blandford and Znajek (1977). Consistent with our
previous results Paschalidis  _et al._ (2015); Ruiz  _et al._ (2018a), we
estimated that the opening angle of the jet is $\sim 25^{\circ}-30^{\circ}$.
In contrast to the $q=3:1$ cases, we do not find evidence of outflow or large-
scale magnetic field collimation in any of the BHNS cases with mass ratio
$q=5:1$, (see bottom panel in Fig. 8). Persistent fall-back debris in the
atmosphere are observed until the termination of our simulations.

We estimated the characteristic interior temperature of the disk remnant via
Eq. 4, and found that it is $T_{\rm disk}\sim 10^{11}\,\rm K$ (or 8.6 MeV).
Thus, it may emit a copious amount of neutrinos with peak luminosity of
$10^{53}\,\rm erg/s$ Kyutoku  _et al._ (2018). It has been suggested that, as
neutrino annihilation may carry away a significant amount of energy from inner
regions of the disk, it may help the jet development. Thus, a BH + disk
remnant that powers a typical sGRB may be dominated initially by thermal pair
production followed by the BZ process Dirirsa (2017).

We observed that the dynamical ejection of matter is strongly affected by the
initial NS spin. The ejecta ranges between $\sim 10^{-4.5}(M_{\rm
NS}/1.4M_{\odot})M_{\odot}$ and $\sim 10^{-2}(M_{\rm
NS}/1.4M_{\odot})M_{\odot}$, and may induce kilonovae signatures with peak
bolometric luminosities of $L_{\rm knova}\sim 10^{40}-10^{41.4}$ erg/s and
rise times $\lesssim 6.5\,\rm h$, potentially detectable by the LSST survey
Alonso  _et al._ (2018). These preliminary results suggest that moderately
high-mass ratio BHNS binaries undergoing merger, where the NS companion has a
significant spin, may give rise to a detectable kilonovae signatures even if a
magnetically-driven jet is absent.

Furthermore, we probed if magnetic-field and NS spin imprints on the GWs can
be distinguished by aLIGO. To assess this possibility, we stitched a TaylorT1
Post-Newtonian waveform to that of our numerical simulations. Next, we
computed the GW power spectrum and the match function
$\mathcal{M}_{\text{\tiny{GW}}}$ (see Eq. 6). Unlike the NS spin imprints on
the GWs evident at high frequencies (see Fig. 10), we found that in the lowest
mass ratio cases, the GW power spectrum of the corresponding nonmagnetized and
magnetized signals differ marginally inside the aLIGO band, while for those
with mass ratio $q=5:1$, the differences are negligible even outside the aLIGO
band. The match function for our extreme cases is
$\mathcal{M}_{\text{\tiny{GW}}}=0.9980$ between the waveforms of q3NSm0.17
cases, and $\mathcal{M}_{\text{\tiny{GW}}}=0.9982$ between those of q3NSp0.23.
Similarly, $\mathcal{M}_{\text{\tiny{GW}}}=0.9998$ for all of our extreme case
with mass ratio $q=5:1$. It appears unlikely that aLIGO detectors can detect
magnetic-field imprints on the GWs which require
$\mathcal{M}_{\text{\tiny{GW}}}\lesssim 0.9956$ for a signal-to-noise ratio of
15 Harry and Hinderer (2018). By contrast, the imprints of the NS spin on the
waveform are more easily detectable. The match function between the waveform
of Mq3NSp0.23 and Mq3NSm0.17 is $0.9844$, and $0.9874$ between those of
Mq5NSp0.33 and Mq5NSm0.17.

Notice that due to the finite computational resources at our disposal, we
explored only two extreme cases to probe the impact of the NS spin on the
dynamical ejection of matter when the NS is (or is not) tidally disrupted
before merger. Our motivation for considering a non-spinning BH in the $q=5:1$
case was to observe if the NS spin alone can change the amount of mass left
outside the BH horizon. Using the fitting model in Foucart (2012) (which
applies to non-spinning BHs) for a BHNSs with mass ratio $q=5:1$, a BH spin
larger than $0.6$ is required to tidally disrupt the NS before merger and to
leave matter outside the BH. Thus, by considering a worst case scenario for
ejecta and matter outside the BH we can study the NS spin effects alone. We
will explore more general cases in the future.

Finally, some caveats are in order. Despite the fact that our initial data
with NS spin slightly violate the constraints, as the evolution proceeds our
efficient constraint damping scheme decreases the constraint violations down
to the same levels as our constraint-satisfying initial data with irrotational
neutron stars. However, some of the reported results in this work may be
affected by the fact that our method for endowing the NS with spin throws the
NS slightly off equilibrium for the highest spin values we consider. While the
trends with NS spin we reported are robust against this property of our
initial data, given that they are supported by analytic arguments, the precise
values of ejecta and disk masses we reported could be affected. We plan to
address these points in future work.

###### Acknowledgements.

We thank the Illinois Relativity group REU team (K. Nelli, M. N.T Nguyen, and
S. Qunell) for assistance with some of the visualizations. This work was
supported by NSF Grants No. PHY-1662211 and No. PHY-2006066, and NASA Grant
No. 80NSSC17K0070 to the University of Illinois at Urbana-Champaign, and NSF
Grant PHY-1912619 to the University of Arizona. This work made use of the
Extreme Science and Engineering Discovery Environment (XSEDE), which is
supported by National Science Foundation Grant No. TG-MCA99S008. This research
is also part of the Frontera computing project at the Texas Advanced Computing
Center. Frontera is made possible by National Science Foundation award
OAC-1818253. Resources supporting this work were also provided by the NASA
High-End Computing (HEC) Program through the NASA Advanced Supercomputing
(NAS) Division at Ames Research Center.

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