Semi-analytic derivation of the threshold mass for prompt collapse in binary neutron star mergers

TL;DR

This paper presents a semi-analytic derivation of the threshold mass for prompt collapse in binary neutron star mergers, showing its robustness and insensitivity to various physical effects, aiding gravitational-wave based EOS constraints.

Contribution

The authors derive a semi-analytic relation for the collapse threshold, demonstrating its insensitivity to thermal effects, deviations from axisymmetry, and rotation law variations.

Findings

The empirical collapse threshold relation is accurately reproduced.

The relation is robust against thermal and rotational law variations.

Implications for gravitational-wave constraints on neutron star EOS.

Abstract

The threshold mass for prompt collapse in binary neutron star mergers was empirically found to depend on the stellar properties of the maximum-mass non-rotating neutron star model. Here we present a semi-analytic derivation of this empirical relation which suggests that it is rather insensitive to thermal effects, to deviations from axisymmetry and to the exact rotation law in merger remnants. We utilize axisymmetric, cold equilibrium models with differential rotation and determine the threshold mass for collapse from the comparison between an empirical relation that describes the angular momentum in the remnant for a given total binary mass and the sequence of rotating equilibrium models at the threshold to collapse (the latter assumed to be near the turning point of fixed-angular-momentum sequences). In spite of the various simplifying assumptions, the empirical relation for prompt collapse is reproduced with good accuracy, which demonstrates its robustness. We discuss implications of our methodology and results for understanding other empirical relations satisfied by neutron-star merger remnants that have been discovered by numerical simulations and that play a key role in constraining the high-density equation of state through gravitational-wave observations.

Tables

Table 1. Table 1: EoSs with references used in this study. M max subscript 𝑀 max M_{\mathrm{max}} and R max subscript 𝑅 max R_{\mathrm{max}} refer to mass and radius of the maximum-mass configuration of non-rotating NSs. M thres eq superscript subscript 𝑀 thres eq M_{\mathrm{thres}}^{\mathrm{eq}} is the threshold mass to prompt collapse derived from equilibrium models (see text). M thres sim superscript subscript 𝑀 thres sim M_{\mathrm{thres}}^{\mathrm{sim}} denotes the threshold binary mass for prompt collapse determined by hydrodynamical simulations for fully temperature-dependent EoSs (data taken from  Bauswein et al. ( 2013a ) ). For zero temperature EoSs the entry for M thres sim superscript subscript 𝑀 thres sim M_{\mathrm{thres}}^{\mathrm{sim}} is empty because these EoS models were not included in  Bauswein et al. ( 2013a ) . R thres eq superscript subscript 𝑅 thres eq R_{\mathrm{thres}}^{\mathrm{eq}} is the corresponding equatorial radius of the equilibrium model at the threshold to collapse with mass M thres eq superscript subscript 𝑀 thres eq M_{\mathrm{thres}}^{\mathrm{eq}} .

EoS, references	$M_{\max }$	$R_{\max }$	$M_{\mathrm{thres}}^{\mathrm{eq}}$	$M_{\mathrm{thres}}^{\mathrm{sim}}$	$R_{\mathrm{thres}}^{\mathrm{eq}}$
	$\left[M_{\odot }\right]$	[km]	$\left[M_{\odot }\right]$	$\left[M_{\odot }\right]$	[km]
DD2, Typel et al. (2010); Hempel & Schaffner-Bielich (2010)	2.42	11.87	3.24	$\sim$3.35	15.91
LS220, Lattimer & Douglas Swesty (1991)	2.04	10.61	2.94	$\sim$3.05	14.53
LS375, Lattimer & Douglas Swesty (1991)	2.71	12.30	3.39	$\sim$3.65	16.53
NL3, Lalazissis et al. (1997); Hempel & Schaffner-Bielich (2010)	2.79	13.39	3.58	$\sim$3.85	16.68
SFHO, Steiner et al. (2013)	2.06	10.30	2.86	$\sim$2.95	14.08
SFHX, Steiner et al. (2013)	2.13	10.78	2.95	$\sim$3.05	14.32
TM1, Sugahara & Toki (1994); Hempel et al. (2012)	2.21	12.50	3.25	$\sim$3.45	16.05
TMA, Toki et al. (1995); Hempel et al. (2012)	2.02	12.11	3.08	$\sim$3.25	15.73
APR, Akmal et al. (1998)	2.19	9.90	2.77	-	13.92
SLy4, Douchin & Haensel (2001)	2.05	9.97	2.81	-	13.97
ppAPR3, Akmal et al. (1998); Read et al. (2009)	2.38	10.73	3.00	-	14.60
ppENG, Engvik et al. (1996); Read et al. (2009)	2.25	10.40	2.93	-	14.32
ppH4, Lackey et al. (2006); Read et al. (2009)	2.02	11.72	3.08	-	15.48
ppMPA1, Müther et al. (1987); Read et al. (2009)	2.47	11.34	3.15	-	14.89
ppMS1, Müller & Serot (1996); Read et al. (2009)	2.77	13.37	3.59	-	16.85
ppMS1b, Müller & Serot (1996); Read et al. (2009)	2.76	13.28	3.55	-	16.95
ppEoSa, Read et al. (2009), this work	2.05	12.43	3.17	-	16.40
ppEoSb, Read et al. (2009), this work	2.35	12.57	3.32	-	16.23

Equations

$$d{s}^2=-e^{2\nu }d{t}^2+e^{2\psi }(d\varphi -\omega dt{)}^2+e^{2\mu }(d{r}^2+r^{2}d{\theta }^2),$$

$$T^{\alpha \beta }=(e+P)u^{\alpha }{u}^{\beta }+P{g}^{\alpha \beta },$$

$$F(\Omega )=\{\begin{array}{ll}{A}_1^2({\Omega }_c-\Omega )+(A_2^2-A_1^2)(1-\beta ){\Omega }_c,& \Omega \le \beta {\Omega }_c,\\ A_2^2({\Omega }_c-\Omega ),& \beta {\Omega }_c\le \Omega \le {\Omega }_c,\end{array}$$

$$J_{\mathrm{merger}}\simeq a{M}_{\mathrm{tot}}-b,$$

$$b\to b+\frac{R_{1.5}-R_{1.5}^{DD2}}{2km}\ast \mathrm{0.2.}$$

$$k^{\mathrm{sim}}:=M_{\mathrm{thres}}^{\mathrm{sim}}/M_{\max }=-3.38C_{\max }+2.43,$$

$$k^{\mathrm{eq}}:=M_{\mathrm{thres}}^{\mathrm{eq}}/M_{\max }=-3.49C_{\max }+2.40,$$

$$M_{\mathrm{stab}}(J)=c_{1}J+c_2,$$

$$c_1=-0.165(M_{\max }^2/R_{\max })+\mathrm{0.2412.}$$

$$c_2=0.913M_{\max }+\mathrm{0.0133.}$$

$$M_{\mathrm{thres}}=\frac{-c_{1}b+c_2}{1-c_{1}a}.$$

$$k=-3.61C_{\max }+2.44$$

Full text

††pubyear: 2017††pagerange: Semi-analytic derivation of the threshold mass for prompt collapse in binary neutron star mergers–4

1 Introduction

Merging neutron stars (NSs) are the next type of source, which is expected to be detected with the current generation of gravitational-wave detectors. The outcome of a NS merger could be a black hole surrounded by an accretion torus (prompt collapse) or a massive rotating NS merger remnant. In the latter case the remnant may undergo a gravitational collapse at a later time, as a result of angular momentum redistribution and additional losses by mass ejection, neutrino emission and gravitational waves (see e.g. Faber & Rasio (2012) for a review).

The distinction between the prompt collapse scenario and the formation of a NS remnant is crucial for several observational aspects of NS mergers. These include the character and strength of the postmerger gravitational-wave emission, the amount of ejecta relevant for heavy element nucleosynthesis (Lattimer et al., 1977; Eichler et al., 1989; Freiburghaus et al., 1999) and nuclear powered electromagnetic emission (Li & Paczyński, 1998; Kulkarni, 2005; Metzger et al., 2010), and the conditions for the launch of a relativistic jet producing a short gamma-ray burst (GRB) (Paczynski, 1986; Eichler et al., 1989).

The outcome of NS mergers (prompt collapse vs. NS remnant or delayed collapse) depends on the binary masses and the equation of state (EoS) of NS matter (e.g. Shibata, 2005; Baiotti et al., 2008; Hotokezaka et al., 2011; Bauswein et al., 2013a). There are still significant uncertainties regarding the EoS of NS matter and various theoretical models are available (e.g. Lattimer & Prakash, 2016; Oertel et al., 2017). The prompt collapse to a black hole occurs for high total binary masses $M_{\mathrm{tot}}$, whereas less massive systems lead to the formation of an at least transiently stable merger remnant. For a given EoS, one can thus introduce a threshold binary mass $M_{\mathrm{thres}}$ that distinguishes the two different scenarios. For $M_{\mathrm{tot}}>M_{\mathrm{thres}}$ prompt collapse occurs, while $M_{\mathrm{tot}}<M_{\mathrm{thres}}$ results in a massive NS remnant that is stable for at least some number of dynamical timescales.

In previous work and within a systematic study of several representative EoSs we found that the threshold mass depends in a particular way on the EoS (Bauswein et al., 2013a). The threshold binary mass can be described as a fraction $k$ of the maximum mass $M_{\mathrm{max}}$ of non-rotating NSs: $M_{\mathrm{thres}}=k\,M_{\mathrm{max}}$ with $k=k(C_{\rm max})$ scaling tightly with the maximum compactness $C_{\mathrm{max}}$ of non-rotating NSs. The maximum compactness is defined by $C_{\mathrm{max}}:=\frac{GM_{\mathrm{max}}}{c^{2}R_{\mathrm{max}}}$ with $R_{\mathrm{max}}$ being the radius of the maximum-mass configuration of non-rotating NSs, whereas $G$ is the gravitational constant and $c$ is the speed of light. Based on results from hydrodynamical simulations, $k(C_{\rm max})$ can be fitted by $k=-3.38\,C_{\mathrm{max}}+2.43$ to good accuracy, which represents a purely empirical finding.

The unique relation between the threshold mass and properties of non-rotating NSs ($M_{\mathrm{max}}$ and $R_{\mathrm{max}}$) is important because it offers the opportunity to infer these quantities (which are directly related to the EoS) from observations. The threshold mass $M_{\mathrm{thres}}$ could be observationally constrained by measuring the total binary mass from the gravitational-wave inspiral signal of a NS merger and by testing for the presence or absence of postmerger gravitational-wave emission originating from a NS remnant (assuming that the detector would have the required sensitivity to detect a postmerger signal if there was one (Clark et al., 2014)). Alternatively, if future theoretical models clarify the exact conditions leading to a short GRB, the observed electromagnetic emission may reveal whether or not in a given event a prompt collapse of the merger remnant occurred. In combination with a simultaneous gravitational-wave observation providing the binary masses, the threshold mass $M_{\mathrm{thres}}$ to black hole formation can be estimated. Similarly, observing a radioactively powered electromagnetic counterpart of a gravitational-wave detection may reveal the occurrence of a prompt collapse, since for equal-mass binaries direct black hole formation leads to smaller ejecta masses and thus different properties of the electromagnetic emission (Hotokezaka et al., 2013; Bauswein et al., 2013b).

The radius $R_{\mathrm{max}}$ can be determined by measuring the dominant oscillation frequency of the postmerger phase for systems with binary masses slightly below $M_{\mathrm{thres}}$ (Bauswein et al., 2013a, 2014). $R_{\mathrm{max}}$ can also be obtained by an extrapolation of the measured postmerger gravitational-wave frequencies of low-mass binary systems (see Bauswein et al., 2014; Bauswein et al., 2015; Bauswein et al., 2016).

Given $M_{\mathrm{thres}}$ and $R_{\mathrm{max}}$, the maximum mass of non-rotating NSs could then be deduced by inverting the relation $M_{\mathrm{thres}}=k(C_{\mathrm{max}})M_{\mathrm{max}}=M_{\mathrm{thres}}(M_{\mathrm{max}},R_{\mathrm{max}})$ describing the collapse behavior of merger remnants. We remark that the ratio $k$ can be similarly described as function of $M_{\mathrm{max}}/R_{1.6}$ with $R_{1.6}$ being the radius of a non-roating NS with a gravitational mass of 1.6 $M_{\odot}$ (Bauswein et al., 2013a). Compared to $R_{\mathrm{max}}$ the radius $R_{1.6}$ may be easier to measure, e.g. by gravitational-wave detections (Bauswein et al., 2012; Clark et al., 2014, 2016). We also note that the relation between $k$ and $C_{\mathrm{max}}$ (Eq. 6) has been found empirically through NS merger simulations for equal-mass binaries. For some candidate EoSs it has been verified that the same relation holds for slightly asymmetric binaries with mass ratios $q=M_{1}/M_{2}\approx 0.9$ with $M_{1}$ and $M_{2}$ being the masses of the binary compenents (Bauswein et al., 2013a). A future measurement of the inspiral gravitational-wave signal of a NS merger will determine the mass ratio $q$ sufficiently accurate, (e.g. Rodriguez et al., 2014; Farr et al., 2016), to decide if the observed binary should follow the relation established for perfectly symmetric mergers. More general relations for arbitrary mass ratios still have to be determined through simulations although one may expect that the relation for symmetric binaries provides a fairly good estimate.

In this paper we provide a more general view on the stability of NS merger remnants by considering equilibrium models of rotating NSs. Using a simplified setup we corroborate in a more general context the specific dependence of the threshold mass on stellar parameters of non-rotating NSs. In doing this, we do not intend to construct equilibrium models that quantitatively resemble merger remnants to high accuracy. This would require significant fine-tuning and an extensive analysis of available hydrodynamical data. Instead, we aim at reproducing only the qualitative behavior with minimal assumptions. Such an approach is important because it is independent of time-consuming and sophisticated hydrodynamical simulations, whereas it may allow a first qualitative investigation of a large sample of EoS models without employing computationally expensive calculations for many different binary configurations.

Efforts to interpret equilibrium models of differentially rotating NSs in the context of merger remnants have been presented in e.g. Baumgarte et al. (2000); Lyford et al. (2003); Morrison et al. (2004); Galeazzi et al. (2012); Kaplan et al. (2014); Studzińska et al. (2016); Gondek-Rosińska et al. (2017) (see Paschalidis & Stergioulas (2016) for a review). Various studies have also considered rigidly rotating NSs as models for the late-time structure of merger remnants, when uniform rotation is enforced on a viscous or MRI timescale (e.g. Lasky et al., 2014; Fryer et al., 2015; Lawrence et al., 2015; Dall’Osso et al., 2015; Margalit et al., 2015; Gao et al., 2016).

The novelty of our approach lies in the fact that we relate equilibrium models to NS merger remnants by considering the detailed angular momentum budget provided by binary mergers as a function of mass. The relatively small computational demands of stellar equilibrium computations permit the investigation of a large number of different NS EoSs.

Finally, we note that the empirical relation that determines the available angular momentum in the merger remnant for a given total binary mass, should allow the construction of sequences of models resembling remnants of various masses. In turn, this will allow a detailed analysis of the oscillation modes of merger remnants which are relevant for the interpretation of postmerger gravitational-wave emission.

The paper is organized as follows. In Sect. 2 we describe the numerical method to compute equilibrium models and provide details on the employed EoSs as well as basic results from NS merger calculations. In the next section we discuss properties of differentially rotating stars and relate the results to the collapse behavior of NS mergers.

If not noted otherwise we use the term “mass” for the gravitational mass in isolation. If we refer to “binary masses”, we mean the sum of the gravitational masses of the binary components at infinite binary separation. We work in geometrical units $G=c=1$ with the remaining scale set by $M_{\odot}=1$ if units are not explicitly mentioned.

2 Setup and numerical method

2.1 Stellar equilibrium code

We use the RNS code (Stergioulas & Friedman, 1995) to construct axisymmetric equilibrium models of differentially rotating NSs (Stergioulas et al., 2004), assuming a spacetime metric of the form

[TABLE]

where $\nu$, $\psi$, $\omega$ and $\mu$ are four metric functions that depend on the coordinates $r$ and $\theta$ only. Matter is assumed to be a perfect fluid with stress energy tensor

[TABLE]

where $\alpha,\beta$ are spacetime indices, $g_{\alpha\beta}$ is the metric tensor, $u^{\alpha}$ is the four-velocity, $P$ is pressure and $e$ is energy density.

We have extended the RNS code to a new, 3-parameter rotation law, that allows for a different rotational description of the envelope, compared to the core of the star. Specifically, the usual 1-parameter rotation law introduced in Komatsu et al. (1989) and used in a many previous studies (see Friedman & Stergioulas (2013); Paschalidis & Stergioulas (2016) for recent reviews) is extended as

[TABLE]

where $F(\Omega):=u^{t}u_{\phi}$ and $A_{1},A_{2},\beta$ are the three parameters of the rotation law, while $\Omega:=u^{t}/u^{\phi}$ is the angular velocity as measured by an observer at infinity, $\Omega_{c}$ is the angular velocity at the center of the star. The rotation law reduces to the usual 1-parameter law in Komatsu et al. (1989) when setting $\beta=1$. For the current qualitative study, we choose $\hat{A}_{1}:=A_{1}/r_{e}=1,$ $\hat{A}_{2}:=A_{2}/r_{e}=1.225$ and $\beta=0.8$ with the equatorial radius $r_{e}$. Similar qualitative behaviour is obtained for other values of the parameters that are within the range that produces equilibrium models with similar bulk properties as those of the remnants in simulations of binary NS mergers.

We stress that since we are interested in the prompt collapse of remnants, we are not concerned with the detailed rotational profile several dynamical timescales after merging, which has been extracted e.g. in Shibata et al. (2005); Kastaun & Galeazzi (2015); Guilet et al. (2016); Hanauske et al. (2016); Kastaun et al. (2016b, a). Hence, rotational law (3) suffices for a first qualitative investigation such as the one presented here. In fact, our main result is rather insensitive to the details of the rotation law. It is only important to allow for a slower-rotating envelope such that stars can reach high masses (as those typical for remnants) without encountering mass-shedding. Further refinement of our findings can be performed in the future with more sophisticated rotation laws.

2.2 Equations of state

For constructing equilibrium models of rotating NSs, we are neglecting, to a first approximation, thermal effects, since we are only interested in qualitatively reproducing the collapse behavior of merger remnants. Remnants are in fact non-barotropic and constructing corresponding equilibrium models would in any case require an averaging step, to produce pseudo-barotropic equilibria (Friedman & Stergioulas, 2013). However, for typical temperatures of a few ten MeV as expected in merger remnants, the stellar structure is only moderately altered at higher densities: at fixed density the pressure is increased by order of 10 per cent compared to the pressure at zero temperature (see e.g. Fig. 1 in Bauswein et al. (2010)). Hence, the qualitative collapse behaviour is retained, to a first approximation, even when considering zero-temperature EoSs111To asses the quantitative impact of thermal effects we redid hydrodynamical simulations for the DD2 EoS as in Bauswein et al. (2013a). For this EoS we determined the threshold mass in simulations with the full temperature-dependent EoS table, in runs with the EoS at zero temperature and in calculations that employ an appximate treatment of thermal effects choosing different values of $\Gamma_{\mathrm{th}}$, which regulates the strength of the thermal pressure contribution. We find $M_{\mathrm{thres}}=3.35~{}M_{\odot}$ for the full table, $M_{\mathrm{thres}}=3.425~{}M_{\odot}$ for the zero-temperature calculation, and $M_{\mathrm{thres}}=3.35~{}M_{\odot}$ for $\Gamma_{\mathrm{th}}=1.5$, $M_{\mathrm{thres}}=3.425~{}M_{\odot}$ for $\Gamma_{\mathrm{th}}=1.75$ and $M_{\mathrm{thres}}=3.425~{}M_{\odot}$ for $\Gamma_{\mathrm{th}}=2$. We thus conclude that the influence of thermal effects on the collapse behavior is relatively small.. Following the same arguments, we assume neutrino-less beta-equilibrium to compute stellar equilibrium models.

We consider a wide range of a total of 18 EoSs. 8 of these EoSs are available with full temperature and composition dependence (DD2, LS220, LS375, NL3, SFHO, SFHX, TM1 and TMA, see Table 2.2 for the definition of the acronyms and references), but are used in the zero-temperature limit for constructing equilibrium models. 8 EoSs (APR, ppAPR3, ppENG, ppH4, ppMPA1, ppMS1, ppMS1b and Sly4) are zero-temperature EoSs and are implemented in their piecewise polytropic form provided in Read et al. (2009) (except for APR and SLy4, which are provided by tables taken from the Lorene package http://www.lorene.obspm.fr). Finally, we include two additional piecewise polytropes, where the parameters were chosen in order to obtain models with properties that are not covered by other EoSs. For these two additional piecewise polytropes we set $\{\log{p_{1}}=34.75,\Gamma_{1}=3.0,\Gamma_{2}=2.0,\Gamma_{3}=2.0\}$ and $\{\log{p_{1}}=34.75,\Gamma_{1}=3.0,\Gamma_{2}=2.6,\Gamma_{3}=2.0\}$ in the terminology of Read et al. (2009).

Table 2.2 lists the mass $M_{\mathrm{max}}$ and radius $R_{\mathrm{max}}$ of the maximum-mass configuration of non-rotating NSs described by these 18 EoSs (obtained by solving the Tolman-Oppenheimer-Volkoff (TOV) equations (Tolman, 1939; Oppenheimer & Volkoff, 1939). All EoSs in our sample are compatible with the lower bound on the maximum mass $M_{\mathrm{max}}$ of non-rotating NSs set by the observation of NSs with a gravitational mass of $\sim 2~{}M_{\odot}$ (Demorest et al., 2010; Antoniadis & et al., 2013). The variety of NS properties within our sample of 18 EoSs is apparent from the mass-radius relations of cold, non-rotating NSs displayed in Fig. 1. The maximum mass $M_{\mathrm{max}}$ ranges between 2.02 $M_{\odot}$ and 2.79 $M_{\odot}$, while the radius of the maximum-mass configuration $R_{\mathrm{max}}$ ranges between 9.90 km and 13.39 km. The compactness $C_{\mathrm{max}}$ of the maximum-mass TOV configuration, which has been found to have a decisive impact on the collapse behavior of NS mergers, varies between 0.243 and 0.328 for the EoS models within our sample.

For the first 8 EoSs which are available with full temperature and composition dependence the threshold binary mass for prompt collapse to a black hole, $M_{\mathrm{thres}}^{\mathrm{sim}}$, was determined by hydrodynamical simulations in Bauswein et al. (2013a) and is listed in Table 2.2. For the other EoSs in Table 2.2, where $M_{\mathrm{thres}}^{\mathrm{sim}}$ is not listed, estimates for $M_{\mathrm{thres}}^{\mathrm{sim}}$ may be found in the literature, e.g. in Hotokezaka et al. (2011); Bauswein et al. (2012). Note, however, that the approximate treatment of thermal effects, which is required for these zero-temperature EoSs, may lead to ambiguities in determining $M_{\mathrm{thres}}^{\mathrm{sim}}$.