Eccentric binary neutron star mergers

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by  Roman Gold

Eccentric binary neutron star mergers
Roman Gold,1,2 Sebastiano Bernuzzi,1 Marcus Thierfelder,1 Bernd Bru¨gmann,1 and Frans Pretorius2
1Theoretisch-Physikalisches Institut, Friedrich Schiller Universita¨t Jena, Max-Wien-Platz 1, 07743 Jena, Germany
2Department of Physics, Princeton University, Princeton, NJ 08544, USA.
(Dated: September 26, 2011)
Neutronstarbinariesofferarichphenomenologyintermsofgravitationalwavesandmergerrem-
nants. However, most general relativistic studies have been performed for nearly circular binaries,
with the exception of head-on collisions. We present the first numerical relativity investigation of
mergers of eccentric equal-mass neutron-star binaries that probes the regime between head-on and
circular. In addition to gravitational waves generated by the orbital motion, we find that the sig-
nal also contains a strong component due to stellar oscillations (f-modes) induced by tidal forces,
extendingaclassicalresultforNewtonianbinaries. Themergercanleadtorathermassivediskson
the order of 10% of the total initial mass.
1
1 Introduction. Binaryneutronstar(NSNS)mergersare far poorly understood, even though there is the possibil-
0 amongthemostpromisingsourcesofgravitationalwaves ity of massive disk production, M ∼ 10% of the total
d
2 (GWs)forground-basedinterferometers,aswellasplau- initialbinarymass. Recentresults[16]forBHNSsystems

p sible candidates for the central enginge of short-gamma- indicate a strong variability in properties of the merger
e ray-burst (sGRB). One of the main tools to study NSNS remnant as a function of the eccentricity. The gravita-
S mergers is numerical relativity, which experienced a dra- tional waveforms significantly differ from the chirp sig-

3 matic development in the last 10 years achieving impor- nalofquasi-circularinspiralsthatfeatureslowlyincreas-
2 tant results in NSNS [1–3] and recently also for black ing amplitude. During eccentric orbits, each periastron
hole-neutron star (BHNS) simulations [4, 5] passage leads to a burst of radiation, first studied us-

] In numerical relativity NSNS mergers have been al- ing Newtonian orbits together with leading order rela-
c
q mostexclusivelystudiedforcircularizedinitialdata,with tivistic expressions for radiation and evolution of orbital
- theexceptionofhead-oncollisions[6,7](andsee[8]fora parameters [17, 18], and more recently with numerical
r
g nearly head-on Newtonian simulation with radiation re- simulations of BHBH and BHNS systems in full general
[ action). Somewhat surprisingly, there are no numerical relativity [16, 19–21]. An important Newtonian result

studies of orbits with eccentricity (discounting the resid- concerningNSNS(orBHNS)isthateccentricityleadsto
1
ual eccentricity [9] of quasi-circular data). One reason tidalinteractionsthatcanexciteoscillationsofthestars,
v
8 for this is an expectation that the dominant population which in turn generate their own characteristic GW sig-
2 of NSNS mergers in the universe results from evolution nal[22](seealso[23,24]). InsomecasestheGWscanbe
1 of primordial stellar binaries. For such systems, resid- dominated by these non-orbital contributions [22]. Nei-
5 ual eccentricity at the time of merger is expected to be thertheorbitalnorstellarGWshavebeenstudiedsofar
.
9 low. For example, a recent investigation suggests that for eccentric NSNS orbits in general relativity.
0 only between 0.2% and 2% of this class of mergers de- In this letter we present the first numerical relativity
1 tectablebyAdvanceLIGO/VIRGOwillhaveeccentricity investigation of (highly) eccentric NSNS mergers. We
1 e > 0.01 (and e (cid:46) 0.05) [10]. With such small residual consider an equal-mass binary at fixed initial separation
:
v eccentricity quasi-circular templates will be able to de- and vary the initial eccentricity. Three models are dis-
i tectthemajorityoftheseevents[11],andmoreovertem- cussedthatarerepresentativeofthedifferentorbitalphe-
X
plates have been proposed for weakly eccentric binaries nomenologyobserved: directplunge,mergerafteraclose
r (e (cid:46) 0.10) [12]. However, recent studies have suggested encounter, multiple encounters. We analyze the emitted
a
there may be a significant population of compact object GW, the dynamics in the matter and characterize the
binaries formed via dynamical capture in dense stellar basic properties of the merger remnant. Dimensionless
environments [13, 14]. In [14], it is estimated that the units c=G=M =1 are used if not stated otherwise.
(cid:12)
rate of dynamical capture NSNS mergers may be large Numerical Method. We performed numerical simula-
enough to account for a significant fraction of sGRBs, tions in 3+1 numerical relativity solving the Einstein
assuming NSNS mergers are their progenitors. Observa- equations coupled to a perfect fluid matter model (no
tions of sGRBs have also suggested there are different magnetic fields). We employed the BAM code [25, 26],
progenitors for sGRBs that exhibit so called extended which we recently extended to general relativistic hydro-
emission and those that do not [15]; an obvious specula- dynamics[27]. Referringto[26,27]fordetailsandfurther
tion of the cause of the bimodality is primordial vs. dy- references, wesolvetheBSSNformulationinthemoving
namical capture NSNS mergers. A large fraction of this puncture gauge coupled to matter in flux-conservative
latter class of mergers will occur with high eccentricity. form. Metric and matter fields are discretized in space
Eccentricmergerscanleadtoarichphenomenologyin on 3D Cartesian meshes refined with the technique of
the GW and the merger remnant. The properties of the moving boxes. Time integration is performed with the
accretion disk formed as a product of the merger are so method-of-lines using a 3rd order Runge-Kutta scheme.
2
Derivatives of metric fields are approximated by fourth-
order finite differences, while a high-resolution-shock- 25
capturing scheme based on the local-Lax-Friedrich cen- Model 1
20 Model 2
tralschemeandtheconvex-essentially-non-oscillatoryre- Model 3
construction is adopted for the matter. Neutron star 15
matter is modeled with a polytropic equation of state 10
with adiabatic index Γ = 2. Gravitational waveforms
are extracted at finite coordinate radius, r ∼ 90M, by M 5
Y/
using the Newman-Penrose curvature scalar, Ψ4, where 0
M =2.8M(cid:12) is the gravitational mass of the system. 5
AlthoughtheproblemofposinginitialdatatotheEin-
10
stein equations by solving the constraints is in principle
15
well understood, there is no ready-made prescription for
eccentric NSNS that are of interest here. We postpone 20
30 20 10 0 10 20
thesolutionoftheconstraintequationsandinsteadwork X/M
with the following approximation. Initial data are set
by superposing two boosted non-rotating stars with the FIG. 1: Star tracks for Model 1, 2, and 3. The tracks of the
same gravitational mass 1.4M at apoastron. The two second star can be inferred from symmetry.
(cid:12)
starsarelocatedatcoordinates(x ,y,z)=(±25M,0,0)
±
and boosted with parameters ξ = (0,±ξ ,0), respec-
± y
tively. The initial data are asymptotically flat but not
conformally flat, and approximately irrotational. The
maximum constraint violation (at the center of the star)
decays as 1/d as expected for the relatively large initial
proper separation d ∼ 54.7M. The constraint violation
is at the level of the truncation error of the evolution
scheme (see also the discussion in [16]).
Orbital dynamics. We performed a series of simula-
tionsfordifferentinitialboosts(i.e.Newtonianapoastron FIG. 2: Model 2 at t = 523M, shortly after the stars have
velocities), ξy ∈ [0.01,0.05]. Newtonian circular orbits touched and separated again. The rest-mass density (log-
are obtained for ξ (cid:39) 0.07. The grid configuration con- scale) and the three-velocity in the orbital plane are shown.
y
sisted of five refinement levels, maximum spatial resolu-
tionh∼0.1−0.2M andaCourant-Friedrichs-Levyfac-
(cid:12)
tor of 0.25. For ξy (cid:46)0.020, within 1000M a merger and and mass MBH ∼ 2.2M(cid:12). In contrast to Model 1, there
formation of a black hole occured. Selected models were is a massive disk with M ∼ 0.08M ∼ 0.15M , mea-
d 0 (cid:12)
continued for ∼10ms after the merger in order to follow sured 100M after the formation of an apparent horizon;
the accretion process onto the final black hole. During meanwhile accretion has increased the mass and spin to
evolutions the largest violation of the rest-mass conser- M ∼2.64M anda ∼0.81. Attheonsetofmerger,
BH (cid:12) BH
vation occurs before collapse ∆M0/M0 ∼0.01. Similarly the separation remains nearly constant ∼5M. This sug-
theADMmasswasconservedupto1%. Theconsistency gestsatransitionthroughawhirlregimeaswasobserved
of the results was assessed by convergence tests. inBBHsystems[16,19–21,28]. Westress,however,that
We begin by reporting the orbital dynamics in our the binary is in contact at this point and that the mat-
three models. See Fig. 1 for the star tracks as computed ter interaction may lead to additional effects. After the
from the minimum of the lapse function [27]. mergerweobservequasi-periodicoscillationsintheaccre-
InModel1(ξy =0.010)thestarsmovealmostdirectly tion flow at a frequency νQPO ∼ 0.5kHz in the vicinity
towardseachother,collide,thenundergopromptcollapse of the black hole.
to a black hole at t−r =423M with dimensionless spin In Model 3 (ξ = 0.022) multiple close passages with
y
parameter a ∼ 0.58 and mass M ∼ 2.8M . The no matter exchange are observed. The two bodies do
BH BH (cid:12)
resulting disk has negligible rest-mass, M (cid:46) 10−7M , not merge during a simulation time of t = 5400M. The
d 0
which is at the level of the artificial atmosphere used for orbitsshowasignificantperiastronprecession,leadingto
the numerical treatment of vacuum for the matter. an overall rotation of the quasi-elliptical orbit of almost
In Model 2 (ξ =0.020) the stars come in contact but 90 degrees per encounter. However, zoom-whirl orbits
y
survive the first encounter. During contact matter is ex- areunstable,whichsuggeststhatbyt=5400M Model3
changed between the outer layers of the stars, see Fig. 2, has not yet quite evolved to this regime.
which in turn gain rotation; the density weighted New- Waveforms. The orbital motion of each model results
tonian vorticity is 10 times larger than an irrotational in a characteristic gravitational wave signal. Figs. 3 and
NSNS binary. During a second encounter they merge 4 display the (cid:96) = m = 2 mode, Ψ4 , as well as the cor-
22
formingablackholeatt=975M withinitiala ∼0.79 responding instantaneous gravitational wave frequency,
BH
3
101e 4 discrepancy). The waveforms from Model 3 exhibit sev-
art) eral bursts corresponding to the periastron passages.
alp 5 Thekeyfeatureisthatbetweenthebursts,(seee.g.for
4rΨ(re 22 0 Mfreoqdueeln2cyatstig−narls∼in[55s0ev−er7a5l0GMW]).-mOondeesc,anwhobicsheravreehaigbh-
51e 2 500 600 700 800 sent in the black hole case, but qualitatively similar to
art) 00..46 BHNS eccentric mergers [16]. This signature is progres-
4Ψ(realp 22 0000....0422 sniovteloybsseurpvperdesfsoerdξyat=lo0w.0e5r. eRcecfeenrtrriincgittieosF(ilga.rg4e,rthξey)(2a,n2d)
r 0.6 GWfrequencyisdominatedbyseveralplateauscompati-
0.6 Model 1 blewiththef-modefrequencyofthenon-rotatingstarin
ω2200..45 Model 2 isolation [30–32], Mω ∼ 0.137 (ν ∼ 1.586kHz), inter-
M0.3 f-mode f f
0.2 ruptedbrieflybythecloseencounterbursts(theinterup-
0.1
0.0 tions at t ∼ 1000M,2000M disappear for larger extrac-
0 100 200 300 400 500 600 700 800 900 1000 1100
(t−r)/M tion radii). In Model 2, the signal is mainly in the (2,2)
multipole; the (2,0) multipole (not shown) also contains
FIG.3: RealpartoftherΨ422 waveforms(centralpanel)and the signature (weaker in amplitude by a factor five). In
instantaneous gravitational wave frequency (bottom panel),
contrast, in Model 3 the amplitude of the (2,2) multi-
Mω ,asafunctionofretardedtime,(t−r)/M,forModels1
22 pole is smaller in amplitude by a factor of three than
and 2. The top panel shows a zoom in on the shaded region
the (2,0) multipole, which is the main emission channel
in the central panel for Model 2. The horizontal line in the
between the bursts (encounters).
bottom panel marks the perturbative f-mode frequency.
The natural interpretation is that these GWs are pro-
duced by oscillations of the individual stars, which are
1e 3 tidallyinducedbythecompanion. Physically,theexpec-
Model 3
tation is that the approximately head-on, zoom-part of
0.3
the orbit is responsible for exciting axisymmetric m=0
0.2
art) modes,whileinparticularm=2modesshouldarisefrom
alp 0.1 theapproximatelycircular, partialwhirlnearapoastron.
4rΨ(re 22 00..01 Ttivheenreeslsatoifvethaemdpifflieturednetopfhtahseesmoofdtehseinbdinicaarytesintthereaecfftieocn-
to excite certain modes. Indeed the Newtonian analysis
0.2 ofTurner[22]indicatesthatasufficientlycloseperiastron
passage (i.e. e (cid:46) 1 for fixed apoastron) exerts a pulse-
0.14
0.12 like tidal perturbation which excites the axisymmetric
ω2200..1008
M0.06 f-modes of the star, leading to a GW waveform domi-
0.04
0.02 natedbythestaroscillationsratherthenbytheemission
0.00
0 1000 2000 3000 4000 5000 due to the orbital motion. Note that the phenomenon is
(t r)/M
−
differentfromaresonanttidalexcitation,seee.g.[23,33],
which instead refer to the circular motion case.
FIG.4: SameasthecentralandbottompanelsofFig.3but
for Model 3. If these waves indeed correspond to stellar f-modes,
they should be detectable as oscillations of the stel-
lar matter. A way to identify the modes is to con-
Mω ,ofthecurvaturescalar,whichrepresentsthemain sider the projections of the rest-mass density of one of
22
emission channel for the binary signal. the stars onto the spherical harmonics, e.g. [31], ρ ≡
(cid:96)m
Waveforms from Model 1 are characterized by a peak (cid:82) Y∗ ρd3x, and perform an analysis of the spectrum.
(cid:96)m
at the merger and the subsequent quasi-normal-mode Results are reported in Fig. 5 for Model 2 and 3, where
(QNM) ringing of the final black hole. The GW fre- the ρ (t) projections and their power spectral density
22
quencyoftheQNMisMω ∼0.48,compatiblewiththe (PSD)areshown. Inbothmodelswecanclearlyidentify
22
fundamentalQNMfrequencyoftheKerrblackholecom- frequencies compatible with the linear f-mode (broad-
puted from the apparent horizon parameters, e.g. [29]. ened by the signal length in Model 2), together with
In fact, we find the value ν ∼5.9kHz, which differs secondary peaks eventually due to nonlinear couplings.
QNM
by6%fromthevalueobtainedfromtheGW.Waveforms Similarresultsareobtainedfortheaxisymmetric(m=0)
fromModel2showaburstatretardedtimet−r ∼500M quadrupolemode,whileinthiscasethesignalisstrongly
related to the orbital dynamics [18], followed by a fea- modulatedbytheorbitalmotion. Overalltheanalysisin-
ture between t − r ∼ [800M − 900M] akin to a tran- dicatesthatbothnon-axisymmetricandaxisymmetricf-
sition through a whirl phase and the final merger sig- modesareexcitedduringtheencounter. Notethat,given
nal around t−r ∼ 1000M. In this case the QNM fre- anuncertaintyonthefrequencies,wecanestimateanup-
quency is higher, Mω ∼ 0.632 (ν ∼ 7.6kHz, 2% perlimitontherotationofthestar. Inmodel3thereare
22 QNM
4
modes are thus not expected, but they may be present
0.01 in the outer layers of Model 2; g-modes are absent here
ρ22 0 by construction.
−0.01 Conclusion. Mergers of eccentric NSNS could be very
0 100 200 300 400 500 600 700 800
t/M interestingsourcesforthirdgenerationGWdetectors, in
x 10−3
2
particulargiventhe(insomerespect)surprisinglystrong
ρ22 0 and clear signal from orbit-induced stellar oscillations
−2 (OISOs). If detected, this would pose constraints on the
0 200 400 600 800 1000 1200 1400 1600 1800
t/M equation of state. The present results are likely to be
x 10−3
highlydependentonthecompactnessofthestarsandon
4
D themassratio,thusthesecasesshouldbestudiedaswell.
PS2
Furthermore,evenlargerdiskmassesareexpectedforthe
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Mω unequal mass case which could be interesting for sGRB
astrophysics. More work is required in these directions
FIG. 5: Mode analysis. The panels display the (cid:96) = m = 2 together with a detailed understanding of the eccentric
spherical-harmonicprojectionoftherest-massdensityofone NSNS distribution in the Universe.
of the stars (Model 2 top, Model 3 center), and its power
Acknowledgments. We gratefully acknowledge helpful
spectral density (PSD, bottom), where ν is marked. The
f
widthofthegreenshadedarea,73Hz,spanstheperturbative discussions with William East, Branson Stephens and
values of the mode frequencies of models A0-A5 (right-left). Kostas Kokkotas. This work was supported in part by
DFG SFB/Transregio 7. RG was supported by DFG
GRK 1523, in particular during a stay at Princeton, FP
more cycles leading to the most restrictive estimate for was supported by NSF grant PHY-0745779 (FP) the Al-
the maximum rotation rate (rotational energy/potential fredP.SloanFoundation. Computationswereperformed
energy) of ∼ 0.05, i.e. at most mildly rotating [30]. r- on JUROPA (Ju¨lich) and at the LRZ (Munich).
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