The Electromagnetic Signals of Compact Binary Mergers

1.2 MB·English

by  T. Piran

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed8January2013 (MNLATEXstylefilev2.2)
The Electromagnetic Signals of Compact Binary Mergers
Tsvi Piran1, Ehud Nakar2 and Stephan Rosswog3,4,5
1Racah Institute of Physics, The HebrewUniversity,Jerusalem 91904, Israel
2Raymond and Beverly SacklerSchool of Physics & Astronomy, TelAviv University,TelAviv 69978, Israel
3 3School of Engineering and Science, Jacobs UniversityBremen, Germany
1 4TASC, Department of Astronomy and Astrophysics, Universityof California, Santa Cruz, CA 95064
0 5Astronomy and Oskar KleinCentre, Stockholm University,AlbaNova, SE-10691 Stockholm, Sweden
2

n
a 8January2013
J

7
ABSTRACT

] Compact binary mergers are prime sources of gravitational waves (GWs), targeted
E
by current and next generation detectors. The question “what is the observable elec-
H
tromagnetic (EM) signature of a compact binary merger?” is an intriguing one with
. crucial consequences to the quest for gravitational waves. We present a large set of
h
numericalsimulations that focus on the electromagnetic signals that emerge from the
p
dynamically ejected sub-relativistic material. These outflows produce on a time scale
-
o of a day macronovae - short-lived IR to UV signals powered by radioactive decay.
r Likeinregularsupernovaetheinteractionofthisoutflowwiththesurroundingmatter
t
s inevitably leads to a long-lasting remnant. We calculate the expected radio signals
a
of these remnants on time scales longer than a year, when the sub-relativistic ejecta
[
dominate the emission. We discuss their detectability in 1.4 GHz and 150 MHz and

3 compare it with an updated estimate of the detectability of short GRBs’ orphan af-
v terglows (which are produced by a different component of this outflow). We find that
2 mergers with characteristicssimilar to those of the Galactic neutron star binary pop-
4 ulation(similar massesand typicalcircum-mergerGalactic disk density of∼1 cm−3)
2
that take place at the detection horizon of advanced GW detectors (300 Mpc) yield
6 1.4 GHz [150 MHz] signals of ∼ 50 [300] µJy, for several years. The signal on time
.
4 scales of weeks, is dominated by the mildly and/or ultra-relativistic outflow, which is
0 not accounted for by our simulations, and is expected to be even brighter. Upcoming
2
all sky surveys are expected to detect a few dozen, and possibly more, merger rem-
1
nantsatanygiventimetherebyprovidingrobustlowerlimitstothemergersrateeven
:
v before the advancedGW detectors become operational.The macronovaesignals from
i the same distance peak in the IR to UV rangeatanobservedmagnitude thatmay be
X
as bright as 22-23 about 10 hours after the merger but dimmer, redder and longer if
r the opacity is larger.
a
1 INTRODUCTION 2009), reaching a detection horizon of a few hundred Mpc
forns2 mergersandaboutaGpcfornsbhmergers(445/927
Compact binary (neutron star - neutron star, ns2, or Mpc are adopted by the LIGO-Virgo collaboration as
black hole - neutron star, nsbh) mergers are prime canonical values; Abadieet al. 2010).
sources of gravitational radiation. The GW detectors
LIGO (Abbott et al. 2009a), Virgo (Acernese et al. 2008)
and GEO600 (Grote & theLIGO Scientific Collaboration Understanding the observable EM signature of com-
2008) are designed to optimally detect merger signals. pact binary mergers has several observational implications.
These detectors have been operational intermittently First, once the detectors are operational it is likely that
during the last few years reaching their nominal de- the first detection of a GW signal will be around or even
sign sensitivity (Abbottet al. 2009b; Senguptaet al. 2010; below threshold. Detection of an accompanying EM signal
theLIGO ScientificCollaboration & theVirgo Collaboration will confirm the discovery, thereby increasing significantly
2010) with detection horizons of a few dozen Mpc for ns2 the sensitivity of GW detectors (Kochanek & Piran 1993;
and almost a hundred Mpc for nsbh mergers (the LIGO - Hughes& Holz 2003; Dalal et al. 2006; Arun et al. 2009).
Virgocollaboration adoptsanoptimalcanonicaldistanceof Second, the physics that can be learned from observations
33/70Mpc; Abadieet al. 2010). Both LIGO and Virgo are of a merger event through different glasses is much greater
beingupgradednowandbytheendof2015areexpectedto than what we can learn through EM or GW observations
be operational at sensitivities ∼10−15 times greater than alone.Finally,evenbeforethedetectorsareoperational,the
the initial LIGO (Smith & LIGO Scientific Collaboration detection of EM signatures will enable us to determine the
(cid:13)c 0000RAS
2 T. Piran, et al.
mergerrates1,an issueof utmostimportancefor thedesign drive a short-lived supernova-like event often referred to
and the operation policy of theadvanced detectors. as “macronova” (Kulkarni2005). Metzger et al. (2010) find
The electromagnetic signal that is often consid- that if 0.01M⊙ is ejected then the optical emission from a
ered as the most promising counterpart to gravitational mergerat300Mpcpeaksafter∼1dayatm ≈23mag.For
V
waves is that of short Gamma-Ray Bursts (GRBs), arecentdiscussionofthedetectabilityofthevariouselectro-
which are thought to arise from compact binary merger magneticcounterpartsofGWsourcesseeMetzger & Berger
events (Eichler et al. 1989). The estimated rate of short (2012).
GRBs is indeed comparable to binary pulsar estimates In addition to electromagnetic signals, there will be a
(Guetta & Piran 2006; Nakaret al. 2006; Guetta& Stella strong (∼ 1053 erg) burst of ∼ 10 MeV neutrinos, similar
2009). However, while appealing, the association is not towhatisproducedbyacore-collapsesupernova.Butsince
proven yet (Nakar 2007), and even if there is an associ- compact binarymergers areordersofmagnituderarerthan
ation, short GRBs are observed only if their relativistic supernovaevents,thechancesofdetectingneutrinosfrom a
jets points towards us. If short GRBs are binary merg- cosmological merger event are essentially zero.
ers then their observed rate, ∼ 10 Gpc−3 yr−1, provides In a companion paper (Rosswog et al. 2012; in the fol-
a lower limit to the merger rate. The true rate is higher lowing called “paper I”) we have investigated to which ex-
and it depends on the poorly constrained beaming angle, tent dynamical collisions as they occur, for example, in
which results in an uncertainty of almost two orders of a globular cluster are different from a gravitational wave
magnitude. While a GRB that is observed off-axis is un- drivencompactbinarymerger.Inthispaperweconcentrate
detectable in gamma-rays, it produces a long-lasting radio entirelyonbinarymergersandwesystematicallyexplorethe
“orphan”afterglow,whichmaybedetectable(Rhoads1997; neutron star binary parameter space in a large set of simu-
Waxman et al. 1998; Frail et al. 2000; Levinson et al. 2002; lations.Weusenumericalsimulationsofthemergerprocess
Gal-Yam et al. 2006). A key point in estimating the de- tofindthepropertiesofthedynamicallyejectedoutflowfor
tectability of GRB orphan afterglows is that the well con- different masses of the coalescing compact stars. We then
strained observables are the isotropic equivalent energy of take the resulting ejecta profiles and calculate the electro-
the flow and the rate of bursts that point towards earth. magnetic transients (i.e., radio flares and macronovae) that
However,thedetectabilityoftheorphanafterglowsdepends arerelatedtothedynamicalejectaofns2andnsbhbinaries.
onlyonthetotalenergyandtruerate,namelyonthepoorly Ourstudydoes not account for othertypesof outflows
constraint jet beaming angle. Levinson et al. (2002) have such as neutrino-driven winds (which yield moderate ve-
shown that while narrower beaming increases the true rate locities of ∼ 0.1 c; Dessart et al. 2009) or mildly and/or
it reduces the total energy, and altogether reduces the de- ultra-relativisticoutflowsthatmayemergefromclosetothe
tectability ofradioorphanafterglows. Thiscounterintuitive compact object at the center of the merger. Since only the
result makesthedetectability oflateemission from adecel- sub-relativistic dynamically ejected material is explored we
erating jet, which produced a GRB when it was still ultra- restrict the light curve calculation to time scales of a year
relativistic, less promising. and longer, when the sub-relativistic component dominates
Regardless of the amount of ultra-relativistic out- the emission. On shorter time scales of weeks and months
flow that is launched by compact binary mergers, and themildlyrelativisticcomponentdominates.Sincetheradio
of whether they produce short GRBs or not, mergers luminosity depends strongly on the outflow velocity, emis-
do launch energetic sub-relativistic and mildly-relativistic sion on time scales of weeks and months will be brighter
outflows (e.g., Rosswog et al. 1999; Ruffert & Janka 2001; thantheonethatwefindhereevenifthemildlyrelativistic
Rosswog 2005; Rosswog & Price 2007; Yamamoto et al. component carries a small fraction out of the total outflow
2008; Rezzolla et al. 2010), unless the equation of state energy(seeNakar& Piran2011fordetails).Theradioflares
at supra-nuclear densities is extremely soft (Rosswog et al. depend sensitively on the surrounding ISM density. We fo-
2000a).Recently,Nakar& Piran(2011)showedthatthein- cus here on physical parameters found in known Galactic
teractionoftheseoutflowswiththesurroundingmatterwill ns binaries. Since all known binaries reside in the Galactic
inevitablyproduceradiocounterparts.Anadditionalsource disk,weconsiderauniformdensityof1cm−3 Draine(2011)
of electromagnetic signal was suggested by Li & Paczyn´ski as themost likely circum-merger environment2.
(1998), which argue that the freshly synthesized, radioac- Our new simulations, that focus on the ejecta also en-
tive elements in the ejected debris from the merger will able us to revise the estimates of the macronovae light
curves. These light curves are determined mostly by three
ingredients: (i) the total amount of the ejecta and their ve-
1 The current rate constraints on compact binary mergers are locitystructurethatwecalculatehere;(ii)theenergyinput
rather loose. The lastLIGO and Virgoruns provided only weak from radioactive decay, for which we use the most accurate
upper limits on the merger rates: 1.4×104 Myr−1(1010L⊙)−1 estimates to date by Korobkin et al. (2012) and (iii) the
correspondingto∼2×105 Gpc−3 yr−1 forns2 and3600Myr−1 (poorly known) material opacity, taken from Metzger et al.
(1010L⊙)−1 (∼ 5 × 104 Gpc−3 yr−1) for nsbh (Abbott etal. (2010).Atafinerlevelthelightcurveandthepeakluminos-
2009b). Estimates based on the observed binary pulsars in ity depend also on the velocity distribution as the emission
the Galaxy are highly uncertain, with values ranging from
20−2×104 Gpc−3 yr−1 (Phinney 1991; Narayanetal. 1991;
Kalogeraetal. 2004a,b; Abadieetal. 2010). There are no di-
rect estimates of nsbh mergers, as no such system has ever 2 Ns2 binaries are in random locations in the Galactic disk.
been observed, and here one has to rely only on a parame- Draine(2011)estimatesthat0.4ofthediskvolumehasadensity
ter dependent population synthesis (e.g., Belczynski etal. 2008; ofabout 0.6cm−3,0.1ofthe volumehas amuchhigher density
Mandel&O’Shaughnessy2010). whiletheresthasalowdensity.
(cid:13)c 0000RAS,MNRAS000,000–000
The Electromagnetic Signals of Compact Binary Mergers 3
from a given mass element moving with a specific velocity (EOS) (Shen et al. 1998a,b), an opacity-dependent multi-
peakswhenthiselementbecomesopticallythin.Weusethe flavor neutrino leakage scheme (Rosswog & Liebend¨orfer
simulationresultstoestimatethistimescaleandinthisway 2003) and a time-dependent artificial viscosity treatment,
we obtain macronovae light curves for our different merger see Rosswog et al. (2000b, 2008) for details. In all nsbh
cases. simulations Newtonian gravity was employed and the
We begin (§2) with a brief discussion of our numeri- black hole was vested with an absorbing boundary at the
cal simulations (more details can be found in paper I). We Schwarzschildradius.Allsimulationsusedasimplegravita-
focus in this discussion on the ejecta properties which are tionalwaveemissionbackreactionforce(Davies et al.1994).
critical both for radio flares and for macronovae. In §3 we The performed simulations complement those that have
first provide an analytic calculation of the radio emission been presented in Paper I. For completeness, we have also
resulting from the interaction of single velocity ejecta with performedtwosimulationsofneutronstarblackholebinary
thesurroundinginterstellarmatter(ISM).Then,weexpand systems. Since we found in earlier studies (Rosswog et al.
this solution to an outflow with a distribution of velocities 2004) cases of long-lived, episodic mass transfer, we started
andwepresentasemi-analyticcalculationtofindthesignal the two cases with lower numerical resolution to be able to
resultingfrom theejectaobtainedinthesimulations,which follow themuntiltheneutronstariscompletelydisrupted3.
show a velocity range of 0.1-0.5 c. We discuss the observa- Our results have turned out to be very robust with respect
tional implications for detectability of merger remnants in tothenumericalresolution,thereforethereducednumerical
§4, including a comparison to an updated estimate of short resolution is not a concern for the purpose of our study.
GRBorphanafterglowsdetectability(§4.2).In§5wepresent The system parameters and some key properties of the
ageneraltheoryofamacronovalightcurveforacasewitha ejecta are summarized in Table 1.
velocity distribution within the ejecta. We use in these cal- In all investigated cases we find ∼ 10−2 m⊙ of unbound
culationsnewradioactiveenergydepositionratescalculated material (E +E > 0). A deviation of the mass ratio
kin pot
recently in a companion paper by Korobkin et al. (2012). from unity has the tendency to enhance the amount of
We then present our detailed calculations of the expected ejectedmass,wefindincontrastnocleartendencywiththe
IRtoUVluminosityofmacronovaethatarisefromradioac- total system mass, see column six in Table 1. In all cases
tivedecaywithintheoutflow.Wesummarizeourresultsand the ejecta velocities are below 0.5 c, their mass-averaged
discuss their implications in §6. valuesaregivenincolumnseven,theyaretypicallycloseto
0.13 c.
Figs. 1 and 2 depict the remnant structures in the or-
2 NUMERICAL SIMULATIONS
bital (XY) plane and perpendicular to it (XZ) at the end
Neutron stars (ns) have long been thought to be narrowly of the simulations. Clearly, visible is the sensitivity to
distributed around 1.35 m⊙ (Thorsett & Chakrabarti deviations from a mass ratio of unity: even differences of
1999). With an increasing number of observed systems it 15%inthestellarmassesleadtolargeasymmetries,i.e.one
has,however,turnedoutthatthemassrangethatisrealized pronounced tidal tail rather than a disk resulting from ini-
in nature is substantially broader. For example, there is tially two such tails. The debris matter cannot cool rapidly
now ample support for neutron star masses significantly enough, therefore it is puffed up, see Fig. 2, but at the end
larger than 1.5 m⊙ (see for example the data compilation of the simulations far from spherical symmetry. Although
in Lattimer & Prakash 2010). A broad peak around 1.5-1.7 this may lead to some viewing angle dependencies, we
m⊙ has been found (Kiziltan et al. 2010; Valentim et al. assumeinthefollowing forsimplicitysphericallysymmetric
2011) for neutron stars with white dwarf companions. In outflows.
addition, there may be a low-mass peak of ∼1.25 m⊙ Compact binary mergers inevitably eject mass in various
neutron stars that have been produced by electron capture forms:
supernovae (Podsiadlowski et al. 2004; van den Heuvel
a) If compact binaries indeed power short GRBs, they
2004; Schwab et al. 2010). PSR J1614-2230 with a mass
have to launch ultra-relativistic outflows with Lorentz fac-
of 1.97 ± 0.04 m⊙ (Demorest et al. 2010) is nowadays tors >100-1000. There are several mechanisms by which
consideredasarobustlowerlimitonthemaximumneutron
this could be achieved (e.g., Blandford & Znajek 1977;
star mass, but even neutron stars with considerably larger
Hawley & Krolik 2006; Piran 2004; Lee & Ramirz-Ruiz
masses are not implausible.
2007; Nakar 2007)
b) Oncethemerger hashappened,thehot remnant emits
These findings are more than enough of a motivation afewtimes1053 erg/sin∼20MeVneutrinos(Eichler et al.
for a broad scan of the neutron star binary parameter
1989; Ruffert & Janka 2001; Rosswog & Liebend¨orfer 2003;
space. We explore neutron star masses between 1.0 to
Sekiguchi 2011). The peak of the neutrino emission is de-
2.0 m⊙ in steps of 0.2 m⊙. All our neutron stars have layed with respect to the merger itself by the time it takes
a negligible initial spin, consistent with the results of
Bildsten & Cutler (1992) and Kochanek (1992). Our sim-
ulations make use of the Smooth Particle Hydrodynamics
3 Phases of stable mass transfer are not necessarily restricted
(SPH) method, see Monaghan (2005) and Rosswog (2009)
tosimulationsthatusetheapproximationofNewtoniangravity.
for recent reviews. Our code is an updated version of the
Relativistic binary simulations with small mass ratios and large
one that was used in earlier studies (Rosswog & Davies bh spin parameters may be particularly prone to a stable mass
2002; Rosswog & Liebend¨orfer 2003; Rosswog et al. 2003; transferphase,seeShibata&Taniguchi (2011)forafurtherdis-
Rosswog 2005). It uses the Shen et al. equation of state cussion.
(cid:13)c 0000RAS,MNRAS000,000–000
4 T. Piran, et al.
Figure 1.Densitycuts throughtheorbitalplanes ofallmergerremnants attheendofeach simulation.Each snapshotshowsaregion
of1000kmx1000km,colorcodedisthelogarithmofmassdensityingcm−3,theannotations indicatethestellarmasses(solarunits)
andthetimeofthesnapshot.
Figure2.Densitycuts(XZ-plane)atthelasttimesliceofthesimulationswith2x1.4m⊙ (left),1.6and1.4m⊙ (middle)and1.8and
1.4m⊙ (right).Thechosencasescorrespondtothesecondlineofpanels(counted frombelow)inFig.1.
(cid:13)c 0000RAS,MNRAS000,000–000
The Electromagnetic Signals of Compact Binary Mergers 5
Table 1.Overviewovertheperformedsimulations
Run m1 [m⊙] m2 [m⊙] NSPH[106] tend [ms] mej [m⊙] hvesci Ekin[1050erg]
1 1.0 1.0 1.0 15.3 7.64×10−3 0.10 1.0
2 1.2 1.0 1.0 15.3 2.50×10−2 0.11 3.4
3 1.4 1.0 1.0 16.5 2.91×10−2 0.13 4.8
4 1.6 1.0 1.0 31.3 3.06×10−2 0.13 5.3
5 1.8 1.0 1.0 30.4* >1.64×10−2 0.13 3.2
6 2.0 1.0 0.6 18.8* >2.39×10−2 0.16 6.0
7 1.2 1.2 1.0 15.4 1.68×10−2 0.11 2.3
8 1.4 1.2 1.0 13.9 2.12×10−2 0.12 3.2
9 1.6 1.2 1.0 14.8 3.33×10−2 0.13 6.2
10 1.8 1.2 1.0 21.4 3.44×10−2 0.14 7.0
11 2.0 1.2 0.6 15.1* >2.95×10−2 0.14 6.0
12 1.4 1.4 1.0 13.4 1.28×10−2 0.10 1.6
13 1.6 1.4 1.0 12.2 2.36×10−2 0.12 4.0
14 1.8 1.4 1.0 13.1 3.84×10−2 0.14 7.6
15 2.0 1.4 0.6 15.0 3.89×10−2 0.15 8.7
16 1.6 1.6 1.0 13.2 1.97×10−2 0.11 2.9
17 1.8 1.6 1.0 13.0 1.67×10−2 0.12 2.7
18 2.0 1.6 0.6 12.4 3.79×10−2 0.14 7.6
19 1.8 1.8 1.0 14.0 1.50×10−2 0.12 2.8
20 2.0 1.8 0.6 11.0 1.99×10−2 0.13 3.7
21 2.0 2.0 0.2 21.4 1.15×10−2 0.11 1.8
22 5+ 1.4 0.2 138.7 2.38×10−2 0.15 6.0
23 10+ 1.4 0.2 139.3 4.93×10−2 0.18 18.2
∗Thesecondaryisstillorbitingattheendofthecomputations.
+Theprimaryisablackhole.
to form a hot accretion torus, about 10 ms. At these huge canbereliablycalculated4.Theirpropertiesareentirelyset
luminosities the neutrinos drive a strong baryonic wind of during the first contact and by the end of the simulation
M˙ ∼10−3 m⊙/s and v∼0.1c (Dessart et al. 2009), prefer- theunboundmaterialismovingballistically,withfastmov-
entially in thedirection of therotation axis. ing ejecta ahead of slower one. The ballistic motion ends
c) It is very likely that the outflows of process a) and b) only when the dynamical ejecta start being decelerated by
interact near the rotation axis/the inner disk. Such an in- theambientmaterialmonths-yearsafterthemerger(seebe-
teraction plausibly produces moderately relativistic matter low). The dynamically ejected debris is expected to be the
outflows near thejet-wind interface. mostenergeticofalltheoutflowcomponents.Asaresultits
velocity profile is not expected to be significantly affected
d) As the accretion disk evolves it spreads viscously un-
by interaction with other outflow components and its pro-
til dissipational heating and/or the recombination of nu-
file at the end of the numerical simulation provides a good
cleons into nuclei unbind a large fraction of the late-time
approximation of the initial conditions for the electromag-
disk (Chen & Beloborodov 2007; Lee & Ramirz-Ruiz 2007;
neticsignal calculations.
Metzger, Piro & Quataert 2008; Beloborodov 2008).
e) Gravitationaltorquesdynamicallyejectmatterdirectly
at first contact with velocities >0.1 c, see Tab. 1.
3 THE RADIO SIGNAL FROM
OUTFLOW-ISM INTERACTION
While all of the above mass loss processes undoubtedly oc-
Webeginbycalculatinganalytically theradiosignalresult-
cur,thequantitativecalculationofprocessesa)tod)istech-
ingfromasinglevelocityoutflowandfromanoutflowwitha
nically very demanding and the relevant physical processes
are not included in the presented simulations. We therefore
do not consider here their contribution to the electromag- 4 Note, that the presented ejecta amounts are numerically fully
netic signature (which will be most important for the radio converged.Theycan,however,dependontheincludedphysicsin-
emission at early times -as discussed later). Instead,wefo- gredientssuchastheequationofstateorthetreatmentofgravity
cus entirely on the signature of the dynamic ejecta, which (Newtonianvs.GR).
(cid:13)c 0000RAS,MNRAS000,000–000
6 T. Piran, et al.
ifthemergersproduceshortGRBs,theemissionduringthe
10−1 relativistic phase will be suppressed by relativistic beam-
ing. Observable emission is produced only once the exter-
nalshockdeceleratestomildlyrelativisticvelocitiesandthe
blast-wave becomes quasi spherical. This takes place when
Γ≈2 namely at R (β =1). From this radius thehydro-
dec 0
10−2 dynamics and the radiation become comparable to that of
a spherical outflow with an initial Lorentz factor Γ ≈ 2.
0
This behavior is the source of the late radio GRB orphan
]n 1.4−1.4 afterglows (Rhoads 1997; Levinson et al. 2002). Ourtheory
Msu 11..02−−11..02 is therefore applicable for the detectability of mildly and
) [ 10−3 11..68−−11..68 non-relativistic outflows as well as for radio orphan GRB
>v 2.0−2.0 afterglows.
M( 11..24−−11..00 EmissionfromNewtonianandmildlyrelativisticshocks
1.4−1.2 is observed in radio SNe and late phases of GRB after-
1.6−1.0
1.6−1.2 glows. These observations are well explained by a theoreti-
10−4 1.6−1.4 cal model involving synchrotron emission of shock acceler-
1.8−1.0
1.8−1.4 ated electrons in an amplified magnetic field. The success
2.0−1.0
of this model in explaining the detailed observations of ra-
2.0−1.4
2.0−1.8 dio Ib/c SNe (e.g., Chevalier 1998; Soderberg et al. 2005;
NS1.4−BH5
NS1.4−BH10 Chevalier & Fransson 2006) allows us to employ the same
10−1 100 microphysicshere.Energyconsiderationsshowthatboththe
v [c] electronsandthemagneticfieldcarrysignificantfractionsof
the total internal energy of the shocked gas, ǫ ≈ ǫ ∼ 0.1
e B
. These values are consistent with those inferred from late
Figure 3. Ejected mass with energy above a given velocity for
radioafterglows oflongGRBs(e.g.,Frail et al.2000,2005).
thedifferentmergercases.
The observed spectra indicate that the distribution of the
accelerated electrons’ Lorentz factors, γ, is a power-law
power-lawvelocitydistribution.Webegininsection§3.1by dN/dγ ∝ γ−p at γ > γ where p ≈ 2.1−2.5 in mildly
m
repeating (for completeness) and extending the calculation relativistic shocks (e.g., theradio emission from GRB asso-
discussed in the supplementary material of Nakar& Piran ciated SNe and late GRB afterglows) and p ≈ 2.5−3 in
(2011). We then use semi-analytical calculations to gener- Newtonian shocks (as seen in typical radio SNe; Chevalier
alize the result to ejecta with an arbitrary distribution of 1998, and references therein). The value of γ is not ob-
m
velocities. served directly but it can be calculated based on the total
energy of theaccelerated electrons, γ = p−2mpǫ β2.
m p−1me e
The radio spectrum generated by the shock is deter-
3.1 A single velocity outflow
mined by two characteristic frequencies5. Oneis
ConsiderasphericaloutflowwithanenergyEandaninitial ν ≈1 GHzn1/2ǫ1/2 ǫ2 β5, (4)
Lorentz factor Γ , with a corresponding velocity cβ , that m B,−1 e,−1
0 0
propagatesintoaconstantdensity,n,medium.Iftheoutflow thetypicalsynchrotronfrequencyofelectronswiththetyp-
is not ultra-relativistic, i.e., Γ0 −1 ∼< 1 it propagates at a ical (also minimal) Lorentz factor γm. The other is νa, the
constantvelocityuntil,attdec,itreachesradiusRdec,where synchrotron self-absorption frequency. We show below that
it collects a comparable mass to its own (Nakar& Piran since we are interested in the maximal flux at a given ob-
2011): servedfrequency,ν mayplayaroleonlyifitislarger than
a
ν . Its valuein that case is
1/3 m
3E
Rdec =(cid:18)4πnmpc2β02(cid:19) ≈1017 cm E419/3n−1/3β0−2/3, (1) νa(>νm)≈1 GHz R1p7+24n2(6p++p4)ǫB2(2,p−++1p4)ǫe2,(p−p+−141)β(5pp+−42). (5)
and
Fig.4illustratesthetwopossiblespectra,dependingon
t = Rdec ≈30 day E1/3n−1/3β−5/3, (2) the order of νa and νm. The flux at any frequency can be
dec cβ0 49 0 found using these spectra and the unabsorbed synchrotron
where we approximate Γ −1 ≈ β2 and ignore relativistic fluxat νm:
0 0
edffenecottse.sHtehreevaanldueinotfhqe/f1o0lxlowinincg.,g.usn.luesnsitsst.aAtetdaotrhaedriwusiseR,q>x Fm ≈0.5 mJy R137n3/2ǫ1B/,2−1βd−272, (6)
R the flow decelerates assuming the Sedov-Taylor self- where d is the distance to the source (we neglect any cos-
dec
similarsolution,sotheoutflowvelocitycanbeapproximated mological effects).Note that thisis thereal fluxat νm only
as: if νa <νm (see Fig. 4).
Aslongastheshockismovingwithaconstantvelocity
1 R6R ,
β=β0(cid:26) (R/Rdec)−3/2 R>Rddeecc . (3) i.e.,att<tdec,thefluxacrossthewholespectrumincreases
If the outflow is collimated, highly relativistic and
pointsawayfromagenericobserver,aswilltypicallyhappen 5 Thecoolingfrequencyisirrelevantintheradio.
(cid:13)c 0000RAS,MNRAS000,000–000
The Electromagnetic Signals of Compact Binary Mergers 7
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νa νm νm νa ! ) %***’
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Figure4.Asketchofthetwopossiblespectraandtheevolution
of the characteristic flux, Fm, and frequencies, νa and νm. The
arrows show the temporal evolution of the characteristic values.
Figure 5. A sketch of the time evolution of νa and νm in two
ToaafbnhdotevhFete/emmator,prmotohwraeasrlkrwedigdhehpitloeennotldfyhetenhincteeetmahbrpereofolowrerafsetl.tdsNdpeeopectceetnirstduhnemanot,cteetidshaerbfteeeelvleroovwtaldun/etttcioofinostrhonebfooltνeetmfdht Tamcarhaaseretkesr,deadνnbagiy,sedshetchoore<fizνvoνoanmblust,aedaletocdfwa(νsthheoiqepc.dh)Tlaeihnanecdehsvνaeoarnf,tddtiechcmael>acdraνakssmeehds,ediodesncloi(tbnbhseeoetrmrtvioaegmdrhkt)is..sNtAsdeolepstcoe-.
spectra.Theevolutionofνa,markedonlyintherightspectrum, that inthe bottom panel νm andνa arenot crossingeach other
itshactorsrpeeccttrounmly.when νm < νa and is therefore relevant only in att>tdec andonlytwotypes oflightcurves,cases (i)and(iii),
canbeobserved.
(seeequations4-6).Thefluxevolutionatlatertimesdepends
on thespectrum at t , namely on
dec is also the time of the peak. The reason is that while F
m
νm,dec ≡νm(tdec)≈1 GHzn1/2ǫ1B/,2−1ǫ2e,−1β05, (7) increases, νm decreases fast enough so that Fνobs decreases
after t . Note that in that case ν plays no role since it
dec a
and if νa,dec >νm,dec then possibly on decreases after deceleration. Overall, in this case the flux
2 14+3p 2+p 2(p−1) 15p−10 peaksat tdec and Fνobs,peak =Fm,dec(νobs/νm,dec)−(p−1)/2.
νa,dec ≡νa(tdec)≈1 GHz E439(4+p)n6(4+p)ǫB2(,4−+1p)ǫe,4−+1p β03(4+p).(8) In the two other cases, (ii) and (iii), νobs < νm,dec
and/or ν < ν and the flux keeps rising at t > t
The flux at that time can be found using the unabsorbed obs a,dec dec
until ν = ν (t) or ν = ν (t), whichever comes last.
synchrotron flux at ν : obs m obs a
m,dec To find out which one of the two frequencies is it, we com-
Fm,dec ≈0.5 mJy E49n1/2ǫ1B/,2−1β0−1d−272. (9) pare νobs with νeq. At t > tdec, νm decreases faster than
ν . Therefore in case (ii) where ν < ν , the last fre-
a eq obs
Consider now a given observed frequency νobs. We are quency to cross νobs is νm and the peak flux is observed
interested in the light curve near the peak flux at this fre- when ν = ν (t). In case (iii) where ν < ν , the last
obs m obs eq
quency. There are three possible types of light curves near frequency to cross ν is ν and the peak flux is observed
obs a
the peak corresponding to: (i) νm,dec,νa,dec < νobs, (ii) when νobs = νa(t). Now, it is straight forward to calculate
νweeqd<efiνnoebs < νm,dec < and (iii) νobs < νeq,νa,dec. Where tthe p,efaokr dfliuffxe,reFnνtobfsr,epqeuakenacnieds.tIhteistiamlseotshtaratigithtisfoorbwsaerrdvetdo,
peak
ν =1 GHzE1/7n4/7ǫ2/7 ǫ−1/7 (10) calculate the flux temporal evolution prior and after tdec
eq 49 B,−1 e,−1 using equations 4-6 and the relation t ∝ R which holds at
as the frequency at which6 ν = ν . In Fig. 5 we show a t<t andβ∝t−3/5att>t .Thepeakfluxes,thetimes
m a dec dec
sketch of the time evolution of ν and ν and the corre- ofthepeakandthetemporalevolutionofthethreedifferent
a m
sponding ranges of ν in which each of the cases is ob- cases are summarized in table 2. The overall different light
obs
served. curvesare depicted in Fig. 6
To estimate the time and value of the peak flux we The most sensitive radio facilities are at frequencies of
recall, that at all frequencies the flux increases until t . 1.4 GHz and higher. Equations 7 and 8 imply that in this
dec
In case (i), ν ,ν <ν , the deceleration time, t , frequency range, for most realistic scenarios, it is a case (i)
m,dec a,dec obs dec
light curve, i.e., ν ,ν <ν . Therefore, Newtonian
a,dec m,dec obs
and mildly relativistic outflows as well as relativistic GRB
6 Notethatifνm,dec<νa,dec thisequalitywillnevertakeplace. orphan afterglows peak at tdec with (Nakar& Piran 2011):
In that case νeq is the frequency at which this equality would
havehappenedifΓ0 wouldhavebeenlargeenough(seeFig.5) Fνobs,peak(νa,dec,νm,dec <νobs)≈
(cid:13)c 0000RAS,MNRAS000,000–000
8 T. Piran, et al.
Case Fνobs,peak/Fm,dec tpeak/tdec Fν†obs Fνobs
t<tpeak tpeak <t
(i)νm,dec,νa,dec<νobs νobs/νm,dec −p−21 1 ∝t3 ∝t−15p1−021
(cid:0) (cid:1)
(ii)νeq <νobs<νm,dec νobs/νm,dec −1/5 (νobs/νm,dec)−1/3 ∝t85 ∝t−15p1−021
(cid:0) (cid:1)
(iii)νobs<νeq,νa,dec νmp−2,d1ec νa−,d3e(pc1+0(43)p(−5p2−)7) νo(5b3(s23pp−−427)) (νobs/νa,dec)−34p+−p2 ∝t23 ∝t−15p1−021
Table 2.Theobservedfluxbeforeandaftertpeak inthethreedifferentregimes.
†Thetemporalevolutiononlyduringthelastpower-lawsegmentbeforetpeak.Atearliertimesthetemporal
evolutionmaybedifferent.
To date, the best observed signal from a mildly rel-
ativistic blast wave is the radio emission that follows GRB
associatedSNe.Themaindifferenceisthatinthesecasesthe
circumburstmediumistypicallyawind(i.e.,n∝R−2)and
thereforethedensityatearlytimesismuchlargertheninthe
ISMand self absorption playsthe main role in determining
the light curve. A good example for comparison of equa-
tion 11 with observations is the light curve of SN 1998bw.
This light curve is observed at several frequencies at many
epochs, enabling a detailed modeling that results in tight
constraintsoftheblastwaveandmicrophysicalparameters.
Li & Chevalier (1999) find that at the time of the peak at
1.4 GHz, about 40 daysafter the SN,taking ǫ =ǫ =0.1,
e B
the energy in the blast wave is ∼ 1049 erg, its Lorentz fac-
tor is ∼ 2 and the external density at the shock radius is
n∼1 cm−3.Thepeakisobservedwhenν ,ν 6ν andit
m a obs
dependsonlyontheseparameters(itisonlyweaklysensitive
to the density profile). Therefore, equation 11 is applicable
in that case. Indeed, plugging these numbers into equation
11weobtainafluxof20mJyatthedistanceofSN1998bw
(40Mpc),compared totheobserved fluxof30 mJy.This is
Figure 6. Schematic light curves of the three cases. The rising notsurprising giventhatthemodelweuseisbased onthat
phase,markedindashedlineforeachofthephases,isthatofthe of radio SNe.
lasttemporalpowerlawsegmentbeforethepeak.Afterthepeak
allcasesshowthesamepower-lawdecay.
3.2 An outflow with a power-law velocity profile
The numerical simulations provide profiles of the outflow
0.3 mJy E49np+41ǫBp+4,−11ǫpe,−−11β05p2−7d−272(cid:16)1.4νGobsHz (cid:17)−p(−2111). eanneorguytflaoswawfuitnhctaiosninogflethveelvoecliotcyitwy.eSaimppilraorxlyimtaotethtehecabselaostf
TheregimeofF atlowerradiofrequencies(<1 wave to be spherical. The main difference when a range of
νobs,peak
GHz) depends on the various parameters. If the outflow velocities is considered, is that in addition to the forward
is Newtonian or the density is low or the energy is low shock, which is driven intothe circum burst medium,there
then ν ,ν < 100 MHz and equation 11 is applica- is a continuous reverse shock that is driven into the ejecta.
a,dec m,dec
ble. Otherwise low radio frequencies are in regime (iii), i.e., If most of the outflow energy resides in low velocities then
ν <ν ,ν . The fluxpeaks in this case at this reverse shock drives an increasing amount of energy
obs eq a,dec
into the shocked region, mitigating the deceleration of the
tpeak(νobs<νeq,νa,dec)≈ forwardshock.Inthatcasethemasscollectedbytheforward
200 day E41591n272ǫB292,−1ǫe16,1−1 150νoMbsHz 1131 , (12) schomocpka,rMab(lRe)t,owthheenmitassvselioncitthyeisejβectaandwiittshrvaedlioucsitiys >R,βis.
(cid:16) (cid:17)
Thus the relation between the radius and the velocity can
with
befound at any time byequating:
Fνobs,peak(νobs <νeq,νa,dec)≈ M(R)(βc)2=E(>β), (14)
50 µJy E4549n15ǫB15,−1ǫe35,−1d−272 150νoMbsHz 56 . (13) where this equality is correct up to a factor of order unity
(cid:16) (cid:17) that depends on the exact ejecta profile. Chevalier (1982)
Inthelast twoequationsweusedp=2.5 (otherpvaluesin calculatesself-similarsolutionsforanoutflowwithapower-
therange2.1-3yieldslightlydifferentnumericalfactorsand law distribution ρ ∝ β−k in which most of the energy is
power laws). at low velocities, i.e., for k > 5 since E(> β) ∝ β−(k−5).
(cid:13)c 0000RAS,MNRAS000,000–000
The Electromagnetic Signals of Compact Binary Mergers 9
Chevalier (1982) provides the exact coefficients of the solu- Inspection of thevarious ns2 merger simulations reveal
tionforseveralvaluesofk,allprovidingcorrectionsoforder relatively large kinetic energies (at least 1050 erg and at
unityto equation 14. times near 1051 erg, see Tab. 1) but with relatively low
In§3.3weapplyequation14totheresultsofthenumer- average velocities, around 0.1c - 0.16c. The corresponding
ical simulations and we calculate the resulting light curves peakfluxesanddurationscorrespond,therefore,tothesub-
from the ejecta profiles. However before doing so, we use relativisticcaseexaminedinNakar& Piran (2011).Wecal-
equation 14 to find R(t) and β(t) for an outflow with a culate, for these simulated mergers, the light curves at two
power-law velocity profile and we find an analytic solution frequencies, 1.4 GHz and 150 MHz, and for two values of
forthefluxforapower-lawvelocityprofile.Consideranout- external densities n = 1 cm−3 and n = 0.1 cm−3. The flux
flowwithapower-lawvelocityprofileandaminimalvelocity normalizationisforeventsatadistanceof1027 cm,roughly
β and a total energy E(>β )=E , that propagates atthedetectionhorizon forns2 mergersbyadvancedLIGO
min min tot
in a constant density medium. Equation 14 implies that its and Virgo.
radius and velocity evolvewith time as: Here we calculate theradio emission only from thedy-
1 namical component of the ejecta (see section 2). This com-
R = 3kk−3(βminc)k−5Etot k tk−k3 ponentislaunchedpreferentiallyalongtheequatorialplane,
(cid:18) 4π(k−3)k−3nmp (cid:19) while faster moving outflows (related to processes a and c
≈ 4·1017 Etot,50 k1 βmin k−k5 tk−k3 cm (15) dAisscausrseesdulitntshecetifoanst2m)aorveinlaguenjcehcteadcaalonngprtohpeargoattaetitoonlaaxrgise.
n 0.2 year
(cid:16) (cid:17) (cid:16) (cid:17) distances, ahead of the dynamical ejecta, and interact with
the ambient medium. The faster moving components are
1
β = 3(k−3)3(βminc)k−5Etot k t−k3 expected to carry less energy than the slow moving ejecta.
(cid:18) 4πk3nmp (cid:19) Yet,thestrongdependenceoftheradiopeakfluxontheout-
1 k−5 flow velocity (equation 11) and the short deceleration time
≈ 0.3 Etot,50 k βmin k t−k3 (16) of the fast moving ejecta (weeks to months) imply that it
n 0.2 year
(cid:16) (cid:17) (cid:16) (cid:17) is expected to dominate radio emission before the dynami-
plugging these values into equations 4-6 we find that for cal ejecta start decelerating (months-years). Thus,our esti-
relevant parameters νa,νm <1.4 GHz and matesin thissection areonly lower limits onthetrueradio
(3+5p)(k−5) emission at early time (up to about a year for n = 1 cm−3
F (ν >ν ,ν ) ≈ 50 µJy e−3.4(p−2.5)E3+2k5p βmin k and about three years for n= 0.1 cm−3). A glimpse of the
ν a m tot,50(cid:16) 0.2 (cid:17) influenceofamildlyrelativisticmaterialcanbeseeninFigs.
n5k−10pk+pk−6ǫBp+4,−11ǫpe,−−11d−272ty3e(2akr−2k3−5p) 1er1s.anOdur12sibmeulolawtiwonhsicfihnddetphicattsatmheilldiglyhtrecluartviveisstoifcnmsbahtemriaerlgis-
= 50 µJy E0.86 βmin 3.44nǫ7/8 ǫ3/2 d−2t5/12 ejectedinthe1.4m⊙ns-5m⊙bhcase.Althoughitcarriesa
tot,50(cid:16) 0.2 (cid:17) B,−1 e,−1 27 year muchlower energy thantheslower movingejecta its strong
(k=9 ; p=2.5). (17) effect is seen a early times. Since we expect the dynamical
ejecta to be the most massive and most energetic of all the
Thisequationisapplicablestartingatt thatcorresponds
dec
outflowcomponents,itisnotexpectedtobeaffectedsignif-
to the maximal velocity7, β , and the energy that is
max
icantlybypossibleinteractionbetweenthedifferentoutflow
carried by the material that moves at the maximal veloc-
ity (i.e., E ≈ E (β /β )−k+5) and up to t components. If this is the case then our radio predictions
βmax tot max min dec aregoodapproximationstothetrueemissionatlatertimes.
that corresponds to β and E . At earlier times then
min tot
If however,the fast outflow is more energetic than theslow
t (β ,E ) the light curve is the one described in
dec max βmax one, then also our late time radio estimates are only lower
table 2 (with β and E ) and at later times then
max βmax limits of thetrue emission.
t (β ,E ) it joins the decaying light curve described
dec min tot
in table 2 (F ∝t(21−15p)/10). TheresultinglightcurvesareshowninFigs.7-12.Figs.
ν 7and8depicttheradiolightcurvesofthevariousns2merg-
ersfor n=1cm−3.Ourcanonicalcase, amergeroftwo1.4
3.3 Radio Light curves from the simulated m⊙neutronstarsismarkedinbothfigureswithathicksolid
outflows line. The flux of the canonical merger is 0.04mJy (0.2mJy)
at 1.4GHz (150 MHz) 1-4 years after the merger. It shows
Tocalculatetheelectromagneticsignaturesweusetheejecta
almostminimalemissionamongallequalmassmergers(Fig.
velocityprofilesfromthesimulationsandapplytheapprox-
7),theemissionincreasesbyafactorof5(at1.4GHz)fora
imations of §3.2. We use equation 14 to find R(t) and β(t)
and subsequently plug them into equations 4-6 to calculate 1.8-1.8m⊙ pairanditisalmost similarfor1.0-1.0 m⊙.Fig.
8 depicts mergers with combinations of different nsmasses.
thelightcurve.Weapproximatetheejecta-ambientmedium
Inmergerswithalargermassdifferencethesecondaryisdis-
interaction as a spherical blast wave that propagates into a
rupted more completely, producing a more prominent tidal
constant density, n. Behind the shock constant fractions of
tail. Therefore, the ejected mass rises with the mass ratio
the internal energy, ǫ =0.1 and ǫ =0.1, are deposited in
e B (seeTab.1andFig. 1).Asaresult,thedynamicallyejected
relativistic electrons and in magnetic field, where the elec-
outflows from unequal mass mergers, produce in general
trons are accelerated to a power-law with an index p=2.5.
brighter and longer lived radio remnants. Indeed, in most
cases of unequalmass mergers theluminosities, both in the
7 Weassumethat themaximalvelocityejectaisstillmildlyrel- 1.4GHzandat150MHz,arebrighteratlatetimesthanthe
ativistic,i.e.,γmaxβmax∼<1. equalmass case. In some cases they reach 0.3 mJy (2mJy)
(cid:13)c 0000RAS,MNRAS000,000–000
10 T. Piran, et al.
Table3.BasicpropertiesofRadioFlaresfromthesub-relativisticdynamicallyejectedoutflowforselected
cases
Run Masses n=1cm−3 n=0.1cm−3
1.4GHz 150MHz 1.4GHz 150MHz
Fν(peaka) t(peaka) Fν(peaka) t(peaka) Fν(peaka) t(peaka) Fν(peaka) t(peaka)
m⊙ mJy yr mJy yr µJy yr µJy yr
8 1.4-1.2 0.09 4 0.5 4 10 9 50 9
12b 1.4-1.4 0.04 1.5 0.2 2 5 3 30 3
15c 1.4-2.0 0.3 5 2 6 50 10 200 10
23d 1.4-10 1.5 4 4 8 200 10 1000 10
a This is the peak of the sub-relativistic outflow. A mildly relativistic outflow, not calculated here, may
produceastrongerandearlierpeak.
b Thecanonicalns2 case.
c Thisisthemaximalsignalfromourrunsofns2 mergers
d nsbhmerger.
a few years after the merger. Note that even a small mass
differenceoflessthan15%canmakealargedifferenceinthe
observedluminosity.Forexample,a1.4-1.2m⊙ mergerpro-
duces twice as bright and twice as long remnant compared
to a 1.4-1.4 m⊙ merger.
The external density is a critical parameter. Figs. 9
and 10 depict the resulting light curves when the density
isn=0.1 cm−3 fort>3yr.Inthatcasethesignal, at the
twoconsideredfrequencies,isalmost anorderofmagnitude
fainterinall cases,compared with n=1cm−3.At1.4GHz
the signal also evolves slower. Since the self absorption fre-
quency is below 150 MHz in that case, the light curves at
both 1.4 GHz and 150 MHzhavequite similar shapes.
Mergers of a black hole and a neutron star eject even
largeramountsofmass.Theirsignals,showninFigs.11and
12, are stronger reaching 1 mJy at 1.4 GHz and a few mJy
at150MHz.Thetimescalesarealsolongerandreachafew
yearsatdensitiesof1cm−3 andabout10yearat0.1cm−3.
Figure7.Radiolightcurvesgeneratedbyinteractionofthedy-
4 DETECTION AND IDENTIFICATION namically ejected sub-relativistic outflows from equal mass ns2
mergersat150MHzand1.4GHzfor1cm−3 cirum-mergerden-
4.1 Detectability sity,ǫB =ǫe=0.1andp=2.5.Themergerdistanceis1027 cm,
roughlythedetectionhorizonforns2 mergersbyadvancedLIGO
We use the radio light curves of our canonical ns2 merger
and Virgo. The shaded region at t < 1 yr reflect the fact that
(1.4-1.4 m⊙) to estimate their detectability by current and atthis timethe emissionisexpected to be dominated bymildly
future radio facilities. The estimates based on this merger
relativisticoutflows, whicharenotincludedinoursimulations.
case are conservative. First, since other merger cases, and
especially those with even a minor mass difference between
the coalescing stars, are brighter and second since our sim- for this particular population. An additional hypothetical
ulations do not include all outflow sources. We also con- population of ns2 mergers that take place in a much lower
siderherethedetectabilityofnsbhmergerbasedonthelight density environment and might produce weaker signals is
curvescalculatedfora1.4-10m⊙ merger.Table4showsthe not included in our radio counterpart estimates.
detection horizons for these two merger scenarios for differ- A ns2 merger that takes place in n = 1cm−3 environ-
ent radio telescopes and two external densities (n=1cm−3 mentproduces aradio remnant that can beeasily observed
and n=0.1cm−3). by EVLA (1 hour integration) all the way to the advanced
The radio flares depend sensitively on the surrounding LIGO/Virgo detection horizon, with a rise time of ∼1 yr
circum-merger density. Current robust knowledge concern- and a decay time of several years. However, given the rela-
ing ns2 binaries arises from the observed Galactic popula- tive narrow EVLA field of view it requires either a ∼ deg2
tion. These binaries are all observed in the Galactic disk localizationoradedicatedsearchinalargererrorbox.Given
whosetypicaldensityisabout1cm−3 Draine(2011).Hence that the planned localization of advanced GW detectors is
weexpectthatthisisthemostrelevantdensitytoconsider. ∼ 10−100 deg2 for events with a detector network signal
In particular we stress that the estimated rate of ns2, that to noise ratio of 10 (Wen & Chen 2010), a targeted search
are based on these binaries (and used later when we esti- in the error box of a GW signal is certainly feasible. For
matethedetectionrateoftheseradioflares)areallrelevant that purpose other facilities, with lower sensitivity but sig-
(cid:13)c 0000RAS,MNRAS000,000–000