Constraining neutron star radii in black hole-neutron star mergers from their electromagnetic counterparts
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 5 March 2021 (MN L A TEX style file v2.2)
Constraining neutron star radii in black hole-neutron starmergers from their electromagnetic counterparts
Giacomo Fragione , (cid:63) , Abraham Loeb Department of Physics & Astronomy, Northwestern University, Evanston, IL 60202, USA Center for Interdisciplinary Exploration & Research in Astrophysics (CIERA), Evanston, IL 60202, USA Astronomy Department, Harvard University, 60 Garden St., Cambridge, MA 02138, USA
ABSTRACT
Mergers of black hole (BH) and neutron star (NS) binaries are of interest since theemission of gravitational waves (GWs) can be followed by an electromagnetic (EM)counterpart, which could power short gamma-ray bursts. Until now, LIGO/Virgo hasonly observed a candidate BH-NS event, GW190426 152155, which was not followedby any EM counterpart. We discuss how the presence (absence) of a remnant disk,which powers the EM counterpart, can be used along with spin measurements byLIGO/Virgo to derive a lower (upper) limit on the radius of the NS. For the case ofGW190426 152155, large measurement errors on the spin and mass ratio prevent fromplacing an upper limit on the NS radius. Our proposed method works best when thealigned component of the BH spin (with respect to the orbital angular momentum)is the largest, and can be used to complement the information that can be extractedfrom the GW signal to derive valuable information on the NS equation of state.
Key words: stars: black holes – galaxies: kinematics and dynamics – stars: neutron– transients: black hole - neutron star mergers
Together with GWTC-1, from the first two observationalruns (Abbott et al. 2019), the new candidate events pre-sented in Abbott et al. (2020), from the first half of the thirdobservational run, comprise GWTC-2. Among its events,there are both black holes (BHs) and neutron stars (NSs)merging in binary systems. Thanks to the growing numberof detected events, compact objects can be constrained withunparalleled precision and gravitational wave (GW) eventsprovide an unprecedented opportunity to probe fundamen-tal physics (Abbott et al. 2020a,c).The origin of binary mergers is still highly debated. Sev-eral possible scenarios could potentially account for most ofthe observed events (e.g., Antonini & Perets 2012; Belczyn-ski et al. 2016; Askar et al. 2017; Bartos et al. 2017; Liu& Lai 2018; Banerjee 2018; Fragione & Kocsis 2018; Ro-driguez et al. 2018; Fragione et al. 2019; Fragione & Kocsis2019; Hamers & Samsing 2019; Kremer et al. 2019; Rasska-zov & Kocsis 2019). Since several models account for roughlythe same rate, first analyses of the LIGO/Virgo data haveshown that the observed population is likely composed ofmergers originated in more than one scenario (e.g., Wong (cid:63)
E-mail: [email protected] et al. 2020; Zevin et al. 2020). The contribution of differentastrophysical channels will be hopefully disentangled usinga combination of the mass, spin, redshift, and eccentricitydistributions in the upcoming years.Despite the growing number of events, there is onlya candidate BH-NS merger, namely GW190426 152155.Therefore, LIGO has only set a 90% upper limit of ∼ − yr − on the rate of BH-NS mergers. As for BH-BHand NS-NS systems, the origin of BH-NS binaries is stillhighly uncertain. While BH-NS binaries can be producedin isolation as a result of binary evolution (e.g., de Mink& Mandel 2016; Kruckow et al. 2018), more challenging isthe process of forming BH-NS binaries through dynamicalassembly in star clusters. A number of authors showed thatNSs are generally prevented from forming NS-NS and BH-NS binaries in a star cluster as a result of the strong heatingdue to BH in the cluster core (Fragione et al. 2018; Ye et al.2019; Fragione & Banerjee 2020; Ye et al. 2020), althoughsome authors have claimed higher rates (Rastello et al. 2020;Santoliquido et al. 2020). Only after most of the BHs havebeen ejected from the cluster core, NSs can efficiently seg-regate in the innermost regions and possibly form binaries,that eventually merge. Recently, Fragione & Loeb (2019a,b)have shown that BH-NS mergers can be a natural outcomeof the dynamical evolution of massive triple stellar systems. © a r X i v : . [ a s t r o - ph . H E ] M a r G. Fragione & A. Loeb
What makes BH-NS mergers interesting is the possibil-ity that they can produce an electromagnetic (EM) counter-part after merger. The lifetime of merging BH-NS binariesspans three phases (for a recent short review see Foucart2020), which include the inspiral ( (cid:38) yr) due to GW emis-sion, the merger phase ( ∼ ∼ To compute whether a BH-NS merger produces an EM sig-nature, we compute the remnant baryon mass ( M rem ) out-side the BH after merger. If M rem >
0, a disk is formedand there is EM emission after merger (Foucart 2012; Fou-cart et al. 2013). We use the remnant mass as calibratedby Foucart et al. (2018) using numerical relativity calcula-tions ,ˆ M rem = (cid:20) max (cid:18) α − C NS η / − β ˆ R ISCO C NS η + γ, (cid:19)(cid:21) δ , (1)where C NS = GM NS / ( R NS c ) is the NS compaction, whichdepends on the equation of state, η = Q (1 + Q ) , (2)with Q = M BH /M NS , is the symmetric mass ratio, and (inunits G = c = 1),ˆ R ISCO = R ISCO M BH = 3+ Z − sgn( χ BH ) (cid:112) (3 − Z )(3 + Z + 3 Z ) , (3)is the innermost stable circular orbit (ISCO) radius(Bardeen et al. 1972) with Z = 1 + (1 − χ ) / [(1 + χ BH ) / + (1 − χ BH ) / ] (4) Z = (cid:113) χ + Z , (5) In units of the initial mass of the NS. Assuming circular binaries. For examples of general relativisticsimulations of eccentric BH-NS interactions, see Stephens et al.(2011). B H I = 0 ( M B H / M N S ) m a x B H I = 90 ( M B H / M N S ) m a x R NS (km)0.80.60.40.20.00.20.40.60.8 B H I = 180 ( M B H / M N S ) m a x Figure 1.
Maximum value of the mass ratio M BH /M NS for whicha BH-NS system will disrupt as a function of the NS radius ( R NS )and BH spin ( χ BH ), assuming M NS = 1 . (cid:12) , for different incli-nations I between the BH spin and the orbital angular momen-tum. Top: I = 0 ◦ ; center: I = 90 ◦ ; bottom: I = 180 ◦ . as a solution of Z ( r ) = ( r ( r − − a (2 r (3 r + 14) − a ) = 0 . (6)In Eq. 1, ( α, β, γ, δ ) = (0 . , . , . , . ©000
Maximum value of the mass ratio M BH /M NS for whicha BH-NS system will disrupt as a function of the NS radius ( R NS )and BH spin ( χ BH ), assuming M NS = 1 . (cid:12) , for different incli-nations I between the BH spin and the orbital angular momen-tum. Top: I = 0 ◦ ; center: I = 90 ◦ ; bottom: I = 180 ◦ . as a solution of Z ( r ) = ( r ( r − − a (2 r (3 r + 14) − a ) = 0 . (6)In Eq. 1, ( α, β, γ, δ ) = (0 . , . , . , . ©000 , 000–000 onstraining NS radii in BH-NS star mergers R NS (km)0.80.60.40.20.00.20.40.60.8 B H , ( M BH / M NS ) max = 2.5 BH, [0, 0.1] R NS No tidal disruption R NS (km)0.80.60.40.20.00.20.40.60.8 ( M BH / M NS ) max = 2.5 BH, [0, 0.1] R NS Tidal disruption ( M B H / M N S ) m a x Figure 2.
Proof-of-concept case on how to constrain the NS radius by using the information on the effective spin and the maximumvalue of the mass ratio M BH /M NS for which a BH-NS system disrupts the NS. We consider M NS = 1 . (cid:12) and assume M BH /M NS = 2 . χ BH , (cid:107) ∈ [0 , . spin is aligned to the orbital angular momentum. In the caseit is inclined by an angle I , the same fitting formulae can beused by replacing the ISCO radius by the radius of the in-nermost stable spherical orbit (ISSO), which is the solutionof (Stone et al. 2013) S ( r ) = r Z ( r )+ a (1 − C )( a (1 − C ) Y ( r ) − r X ( r )) , (7)where C = cos I , and X ( r ) = a ( a (3 a + 4 r (2 r − r (15 r ( r −
4) + 28)) − r ( r − Y ( r ) = a ( a + r (7 r (3 r −
4) + 36)) + 6 r ( r −
2) (9) × ( a + 2 r ( a (3 r + 2) + 3 r ( r − . Alternatively, Eq. 1 can be used considering only the alignedcomponent of the BH spin χ BH , (cid:107) = χ BH cos θ BH . (10)Figure 1 shows the maximum value of the mass ratio M BH /M NS for which a BH-NS system will disrupt as a func-tion of the NS radius ( R NS ) and BH spin ( χ BH ), assuming M NS = 1 . (cid:12) , for different inclinations I between the BHspin and the orbital angular momentum. For I = 0 ◦ , themaximum mass ratio for disrupting systems is as high as ≈
22 for highly-spinning BHs. In the case I = 90 ◦ , themaximum mass ratio reduces to ≈
6. Obviously, I = 180 ◦ is symmetric with respect to the the case I = 0 ◦ . This isbecause the case I = 0 ◦ can be used for any inclination I with the substitution χ BH −→ χ BH , (cid:107) . Thus, whether or nota BH-NS merger is followed by an EM counterpart is es-sentially determined by the symmetric mass ratio, the NScompaction, and the aligned component of the BH spin. To constrain the NS radius, information on the BH spin isneeded. GW measurements are especially sensitive to theeffective spin, which is the following combination of the BHand NS spins (Abbott et al. 2016; Vitale et al. 2017), χ eff = M BH χ BH cos θ BH + M NS χ NS cos θ NS M BH + M NS , (11)where χ i = cS i GM i (12)is the dimensionless Kerr parameter and S i = | S i | is themagnitude of the intrinsic spin. Approximating the NS as arotating sphere, it can be shown that χ NS ≈ (cid:16) ν − (cid:17) , (13)where ν is the rotational frequency of the NS. Therefore, theNS spin will be typically small, except in the case the NS is a(rare) millisecond pulsar. In the more typical case the NS isnot a millisecond pulsar, and considering that M BH > M NS ,inverting Eq. 11 yields χ BH , (cid:107) ≈ χ eff M BH + M NS M BH . (14)This implies that the aligned component of the BH spincan be approximated by Eq. 14 using the values inferred byLIGO/Virgo analysis pipeline whenever the NS is not a fastrotator. This information, along with the presence or not ofan EM counterpart, can be used to constrain the NS radius.Figure 2 illustrates a proof-of-concept case on how toconstrain the NS radius by using the information on theeffective spin and the maximum value of the mass ratio M BH /M NS for which a BH-NS system disrupts. In our exam-ple, we consider M NS = 1 . (cid:12) and assume that the massratio between the merging BH and NS is M BH /M NS = 2 . © , 000–000 G. Fragione & A. Loeb of the not-negligible uncertainty in the component masses,which would cause a thickening of the M BH /M NS curve onthe plot. Moreover, we assume that the aligned componentof the BH spin is measured in the range χ BH , (cid:107) ∈ [0 , . M BH /M NS ) max = 2 . M BH /M NS ) max = 2 .
5, translating into a lowerlimit on the NS radius. For reference, recent measurementsfrom LIGO/Virgo and NICER constrain the NS radius to ∼ . . GW190426 152155 is the candidate event with the highestfalse alarm rate (1 . − ) among the LIGO/Virgo eventsin GWTC-2. Assuming it is a signal of astrophysical origin,the LIGO/Virgo collaboration has estimated its componentmasses to be 5 . +4 . − . M (cid:12) and 1 . +0 . − . M (cid:12) , rendering it thefirst candidate BH-NS merger. The data are uninformativeabout potential tidal effects, parametrized by (Flanagan &Hinderer 2008)˜Λ = 3239 M ( M NS + 12 M BH )( M NS + M BH ) k C NS , (15)where k is the second Love number. The effective spin ofthe systems has been measured to − . +0 . − . .Figure 3 illustrates the case our method is appliedto GW190426 152155 (Abbott et al. 2020). We both show χ BH , (cid:107) , derived from the measured effective spin − . +0 . − . ,and the BH-to-NS mass ratio 3 . +2 . − . , as derived fromLIGO/Virgo analysis. Note that effective spin and mass ra-tio are strongly correlated. For this event, no EM counter-part has been observed (Abbott et al. 2020b), implying aplunging of the NS onto the BH. Measurement errors on theeffective spin and mass ratio are too large to place an upperlimit on the NS radius for this candidate event. Therefore,GW190426 152155 is not a good candidate to get informa-tion on the NS equation of state from current observations. Ifan EM counterpart had been observed, we would have beenable to place a lower limit on the NS radius, R NS (cid:38)
10 km.
BH-NS mergers are interesting since they could produce anEM counterpart in the form of a short gamma-ray burst(GRB), which can provide crucial information on theirorigin and NS structure. Despite the growing number ofevents, there is only one candidate BH-NS merger, namelyGW190426 152155.We have shown how to use information on the EM R NS (km)0.80.60.40.20.00.20.40.60.8 B H , GW190426_152155Mass ratioMeasured
BH, ( M B H / M N S ) m a x Figure 3.
The case of the candidate BH-NS mergerGW190426 152155 from GWTC-2 (Abbott et al. 2020), for whichno EM counterpart has been observed (Abbott et al. 2020b). Mea-surement errors on the effective spin and mass ratio are too largeto place an upper limit on the NS radius for this candidate event. counterpart in BH-NS mergers can be used to place con-straints on the NS radius. In particular, we have illustratedhow to derive a lower (upper) limit based on the presence(absence) of an accretion disk that powers the EM coun-terpart (Foucart et al. 2018). We have concluded that ourmethod works best when the aligned component of the BHspin is the largest. Ascenzi et al. (2019) presented a methodbased on a similar idea in Pannarale & Ohme (2014) toexploit joint GW-short GRB observations of BH-NS coales-cences. Hinderer et al. (2019) performed a similar analysison GW170817, working under the assumption that the eventoriginated from a BH-NS coalescence, and exploited the EMconstraints from the kilonova light curve. We have also ap-plied our method to the case of the candidate LIGO/Virgoevent GW190426 152155, for which no EM counterpart hasbeen observed (Abbott et al. 2020b). We have found thatlarge measurement errors on the spin and mass ratio preventfrom placing an upper limit on the NS radius for this can-didate event. Li et al. (2020) reanalyzed GW190426 152155using several waveforms with different characteristics andfound that the results are influenced by the priors of massratio, showing that the chance of observing an EM counter-part is rather unpromising for GW190426 152155. Cough-lin et al. (2020) employed three different kilonova modelsand derived upper bounds on the ejecta mass for some ofthe LIGO candidate events, which they used to put con-straints on the mass-ratio, spin, and NS compactness forGW190426 152155 (for recent candidates for more recentcandidates see Anand et al. 2021).We finally note that we have assumed an EM counter-part will be seen in the case of tidal disruption. However, thismight not be the case due to observational strategies andlimitations. The current search methods in the LIGO/Virgopipeline include both modelled searches for a GW signal(compatible with the inspiral of a NS-NS or BH-NS bi-nary) within 6 s of data associated with an observed shortGRB and unmodelled search for generic transients, consis-tent with the sky localization and time window for each © , 000–000 onstraining NS radii in BH-NS star mergers GRB, that begins 600 s before the GRB trigger time andends 60 s after it (see Abbott et al. 2020b, and referencestherein). A GRB not pointed at us would not be detectable.Disrupting BH-NS binaries could also have a lot of dynami-cal ejecta, which may power a kilonova. However, follow-upobservations of potential BH-NS mergers are so far only cov-ering a small portion of the sky consistent with the detectedGW signals and are typically not deep enough to observeall kilonovae. Thus, the lack of an observed EM signal fol-lowing a BH-NS merger event is significantly less constrain-ing in our model. Finally, existing theoretical models remainlimited by the lack of understanding of post-merger outflowsand by nuclear physics and radiation transport uncertainties(e.g., Barnes et al. 2016; Barbieri et al. 2020).While obtaining precise models for the observable EMsignals powered by BH-NS binaries can be challenging, thedependence of these signals on the properties of BH-NS bi-naries is essential for extracting valuable information (e.g.,Ascenzi et al. 2019; Hinderer et al. 2019; Sridhar et al. 2021).This can complement the information that can be obtainedfrom the GW signal, either from the potential tidal de-phrasing or the cut-off frequency when disruption occurs( ∼ ACKNOWLEDGEMENTS
GF acknowledges support from CIERA at NorthwesternUniversity. This work was supported in part by Harvard’sBlack Hole Initiative, which is funded by grants from JFTand GBMF.
Data Availability
The data underlying this article will be shared onreasonable request to the corresponding author.
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