Unequal mass binary neutron star mergers and multimessenger signals

Before describing the characteristics of the GW signals, consider first the 2-body, post-Newtonian approximation for a compact binary system (as can be found in, for example, [39]). This approximation describes the interaction via an expansion with respect to the parameter $x\equiv {(M\omega )}^{2/3}$ where M is the total mass and ω is the orbital frequency. At the lowest order, x = M/R with R the separation of the binary.

The amplitude of the gravitational waves produced, h (and consequently the energy radiated and angular momentum loss), from the system are, to leading order, given by the scaling $h\propto {{ \mathcal M }}^{5/3}{\omega }^{2/3}/r$. Here the chirp-mass is defined as ${ \mathcal M }\equiv {({m}_{1}{m}_{2})}^{3/5}/{({m}_{1}+{m}_{2})}^{1/5}$ and r is the distance to the source. Higher order contributions containing both conservative and dissipative pieces, depending on the constituent masses, spin, radiation-reaction, and internal structure, become increasingly important as the frequency (separation) increases (decreases) (see e.g. [39]).

Particularly relevant for our current discussion are tidal effects and the effect of mass ratio. For the latter, given a mass ratio⁹ $q\equiv {m}_{1}/{m}_{2}$, the chirp mass is ${ \mathcal M }={{Mq}}^{3/5}/{(1+q)}^{6/5}$. In terms of $\delta \equiv 1-q$, ${ \mathcal M }={2}^{-6/5}M(1-3{\delta }^{2}/20+{ \mathcal O }({\delta }^{3}))$. Thus the dominant contribution to the gravitational wave amplitude is already sensitive to the mass ratio.

However, for realistic binary neutron star systems, the parameter q is limited to a small range and 3δ²/20 ≃ {0, 3 × 10⁻³, 9 × 10⁻³} for q = {1, 0.85, 0.76} thus mass ratio differences in BNS systems are small.

Tidal effects, on the other hand, can become quite significant at close separations, appearing with a high power of the inverse of the stellar compaction. In particular, even though tidal effects enter the PN approximation at tenth order in x, this sub-leading effect can still be apparent at sufficiently high frequencies as C_i ranges between 0.05 and 0.2 for neutron stars. That is, the leading contribution to the tidal force on a single star is given by

$$\begin{eqnarray}&&{F}_{T}\propto {k}_{2}^{(i)}{C}_{i}^{-5}{x}^{10},\end{eqnarray} \tag{ 6 }$$

where ${k}_{2}^{(i)}$ is the quadrupolar tidal coupling Love number constant and ${C}_{i}={M}_{i}/{R}_{i}$ (for the ith star) [40, 41]. The dependence of tidal effects on the binary's dynamics can be expressed via ${\kappa }_{2}^{T}$ for the binary in terms of the individual ${k}_{2}^{(i)}$, defined as [42, 43]

$$\begin{eqnarray}&&{\kappa }_{2}^{T}=2\left[\displaystyle \frac{{q}^{4}}{{(1+q)}^{5}}\displaystyle \frac{{k}_{2}^{(1)}}{{C}_{1}^{5}}+\displaystyle \frac{q}{{(1+q)}^{5}}\displaystyle \frac{{k}_{2}^{(2)}}{{C}_{2}^{5}}\right].\end{eqnarray} \tag{ 7 }$$

For neutron stars, the Love number constant typically $\in \,({10}^{-2},0.5)$ [44] and their compaction ratios C ≃ 0.05 − 0.2. Thus for sufficiently high frequencies (close separations) tidal effects can become important as ${\kappa }_{2}^{T}{x}^{10}\simeq { \mathcal O }(1)$. Notice that for a fixed given total mass, stiffer EoS (i.e., with larger radii and smaller compactness) lead to larger ${\kappa }_{2}^{T}$ and therefore stronger tidal effects. Similarly, unequal mass cases also lead to slightly larger ${\kappa }_{2}^{T}$ thus enhancing tidal effects. For low frequencies on the other hand, ${\kappa }_{2}^{T}{x}^{10}\ll { \mathcal O }(1)$, and only the mass ratio can introduce departures when comparing binaries with the same total mass. However, these departures are necessarily small for BNS systems.

The consequences of the above observations are clearly illustrated in figures 1 and 2. Figure 1 shows the (coordinate) separation versus time for all EoS and mass ratios studied while figure 2 displays the gravitational wave signals as computed by the Newman–Penrose scalar Ψ₄.

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The times in figure 2 have been relabeled in terms of a merger time. For each EoS, all mass ratios use the same time. In particular, the merger time is the time at which, for the q = 1 case, the two locations of density maxima (the centers of the two stars) decreases below a threshold distance (generally the sum of the two stellar radii), corresponding to the stars making contact. Because the signals in each panel of figure 2 share the same time, we have included a vertical line to denote the time when such contact occurs for the unequal mass cases. As can be clearly appreciated in both figures, the early behavior of all mass ratios within a given EoS (<7 ms) agree.

However, as the orbit tightens tidal deformations increase and affect the dynamics of the system. Their effects manifest first for binaries described by the NL3, followed by the DD2, and finally the SFHo equations of state. As expected from our discussion above, this succession is tied to the decreasing radii characterizing the stars of each binary. For instance, figure 1 indicates that a more rapid plunge takes place for stars with larger radii, and, interestingly, within the same EoS the merger takes place earlier as the mass ratio is decreased. This behavior is apparent in the gravitational wave signatures in which binaries with large stellar radii demonstrate differences earlier than those binaries with smaller radii. Furthermore, comparison within the same EoS reveals that, as the mass ratio decreases, an earlier departure from a simpler (chirping) sinusoidal wave arises as tidal effects become more important. This pre-merger behavior is especially clear for the NL3 case but is essentially negligible for the softer EoS considered.

Additionally $q\ne 1$ cases also display a stronger l = 2, m = 1 component in the gravitational waves during the late pre-merger and afterwards than the q = 1 case. However, such modes are still at least a couple of orders of magnitude weaker than the main l = 2, m = 2 signal displayed in figure 2. Nevertheless, they can strongly seed a possible m = 1 instability after the merger whose associated frequency would be lower than the one corresponding to m = 2 modes [45, 46].

The post-merger behavior is governed by the properties of the resulting hot, differentially rotating MNS (or hypermassive NS) which loses energy and angular momentum via emission of neutrinos and gravitational waves. Depending on the EoS and the total mass, the remnant MNS may eventually collapse to a black hole or continue as a rotating star. In cases that avoid prompt collapse to a black hole¹⁰ , gravitational waves in this stage are characterized by a few dominant modes with frequencies intimately tied to the EoS. These dominant modes arise are determined by the natural oscillation frequencies of the star, by the details of the merger process itself, and by the possible development of an m = 1 instability if a rather long-lived MNS is possible. Naturally, these modes are a potentially rich source of information about the system (see for instance [47] and references within).

We scrutinize these modes more closely by computing the Fourier transform of the primary Ψ₄ gravitational wave mode (l = 2 = m) in a time window of duration roughly 10 ms after the merger. These spectra, shown in figure 3, demonstrate several characteristic peaks, and these peak frequencies are presented in table 2. The most dominant of these modes, using the terminology of [48], f_peak, is associated with the rotational behavior and quadrupolar structure of the newly formed MNS/HMNS. Secondary modes, at lower frequencies, can be associated with different mechanisms [45, 48–50]. However as noted in [48], in low total mass binaries [51] and for the case of a stiff EoS as in [52], (the majority of) these secondary modes degrade quickly with time in just a few milliseconds and are unlikely to build enough SNR for detection. The primary mode on the other hand, can last for long times and provide the best opportunity for detection. Consequently it has been the focus of increased scrutiny.

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Recently, a fit for these modes has been suggested for the case of equal mass binaries [48], and we include in table 2 the frequencies predicted from this fit together with our measured values. As discussed in our previous work [22], our results for q = 1 are within 10% of the predicted frequencies.

A similar fit for unequal mass binaries described by tabulated EoS is so far unavailable because, contrary to the equal mass case, studies with $q\ne 1$ are still in their infancy within full General Relativity. (For Newtonian or approximate GR, studies have been presented in [53, 54].) The majority of currently available results for unequal mass binaries employ piece-wise polytropes (chosen to be reasonable approximations to tabulated, realistic EoS.). A comparison of results obtained with the so-called MS1, H4, SLy cases, studied in [55], which have similar mass ratios and masses to those used here for the NL3, DD2, SFHo EoS, reveals that the peak frequencies agree to better than 5%. As well, [52] found that, for the polytrope dubbed SLyPP (which yields a NS similar to the one described by the NL3 EoS considered here), the resulting peak frequency, f_peak, has a small dependence on the mass ratio, with a slightly higher frequency for the equal mass case. This finding is also consistent with our results. Last, relatively small differences in the peak frequency with respect to mass ratio have also been reported in other recent works [43, 48, 56]. As can be appreciated in table 2, we also find that the peak frequencies for $q\ne 1$ lie within ≈10% of the corresponding q = 1 case and are typically lower—except for q = 0.85 DD2.

Beyond comparing with particular fits, it is instructive to take a step back and consider estimating the peak frequency based on simple arguments which can shed light on the underlying physics of the system. A simple approach is to consider the contact frequencies, f_c, of the binary, based on the radii of the two stars. This approach can only provide a lower bound but it may potentially lead to improved estimates in the future. The contact frequency¹¹ is given in terms of the binary parameters by (in M⁻¹ units)

$$\begin{eqnarray}&&{f}_{{\rm{c}}}=\displaystyle \frac{1}{\pi {M}_{g}}{\left(\displaystyle \frac{{m}_{g}^{(1)}}{{M}_{g}{C}_{1}}+\displaystyle \frac{{m}_{g}^{(2)}}{{M}_{g}{C}_{2}}\right)}^{-3/2},\end{eqnarray} \tag{ 8 }$$

where ${M}_{g}={m}_{g}^{(1)}+{m}_{g}^{(2)}$, (see equation (36) of [57]). We include with our measured frequencies the estimated contact frequencies in table 2. While they are clearly lower than the observed frequencies as anticipated, they do provide a consistent trend.

This trend is more clearly seen in figure 4 which illustrates the measured peak frequency versus the estimated contact frequency for all cases studied here (which corresponds to total mass M = 2.7M_⊙) together with data from [48] (for equal mass binaries with M = 2.4M_⊙). The observed trend can be expressed quantitatively through a linear regression fit to our data

$$\begin{eqnarray}&&{f}_{\mathrm{peak}}(\mathrm{kHz})=-1.61+2.96{f}_{{\rm{c}}}\left[\displaystyle \frac{2.7{M}_{\odot }}{M}\right](\mathrm{kHz}).\end{eqnarray} \tag{ 9 }$$

This correlation is characterized by a correlation coefficient = 0.96 and has been obtained with total mass M = 2.7M_⊙. For systems with a different total mass, equation (9) effectively rescales the contact frequency.

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For instance, [51] presented peak frequencies from evolutions of equal mass binaries with total mass M = 2.4 M_⊙ concentrating on three EoS: the same DD2 and SFHo EoS studied here along with LS220 [58] which yields radii in between the other two for the same mass. The corresponding peak frequencies computed for DD2, LS220, and SFHo EoS are 2.35 kHz, 2.56 kHz, and 2.96 kHz, respectively. Expression (9) estimates the peak frequencies to be f_peak = 2.32 kHz, 2.49 kHz, and 2.91 kHz, respectively, for these cases. The agreement is quite good. It would be interesting to explore if this simple approach works as well in the case of spinning binaries (e.g. [45, 59]) and to contrast it to alternative fits depending explicitly on tidal deformability parameters [60].

Finally, beyond the specific dynamics and waveform characteristics, it is instructive to explore the extent to which the obtained waveforms could be distinguished with respect to different configurations of aLIGO. To do so, we first obtain a longer wavetrain by employing at earlier times a PN (Taylor T4) expansion augmented to include leading order tidal effects as described in [40, 61]. We match this augmented PN signal at early times to our numerical signal before merger to form a hybrid waveform. (Further improvements in the early signal can be achieved, see e.g. [60, 62], though such improvements would not affect the main conclusions drawn here.) The matching is accomplished by choosing a cycle devoid of the initial numerical transients but still well before merger. Within this cycle, a weighted average of the two waveforms is calculated using a tanh function that transitions from 0 to 1.

Figure 5 illustrates the strain of the waveforms for the different cases considered here with respect to two possible noise curves from aLIGO for binaries at 100 Mpc. As can be appreciated in the figure, essentially no differences exist that could be discerned with the 'no SRM' noise curve with a single event at such a distance. The more aggressive noise curve corresponding to the 'zero detuned, high power' configuration would, however, be sensitive to differences between the stiffest and softest EoS considered.

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As a figure of merit, it is instructive to contrast the distinguishability of the obtained waveforms. Since we start all our simulations at the same separation (orbital frequency) we compute the 'distinguishability' between the different waveforms (as described in e.g. [63], see also [64–66]) using the 'zero detuned, high power' and the 'no SRM' noise curves in aLIGO [67]. Table 3 lists the values obtained for $\delta h\equiv \sqrt{({h}_{i}-{h}_{j},{h}_{i}-{h}_{j})}$ (without allowing for frequency shifts and without normalizing our waveforms). Values above 1 imply such waveforms could be distinguished, indicating therefore that DD2 waveforms for q = 0.76 could be distinguished from all the others when using the more aggressive noise curve. The other cases are not discernible, even when using the zero detuned, high power noise curve.

The delicate nature of distinguishing EoS effects in binary neutron star systems has been thoroughly discussed in [68, 69], which stressed the need for tens of events to disentangling EoS differences if employing gravitational waves alone. Importantly, as discussed in, e.g. [22, 23, 70–73] and later in this work, complementary electromagnetic information can help reveal the EoS from the characteristics of the radioactive decay of heavy elements produced by ejected material.

The merger of a binary with the total mass considered here produces a hot, rotating neutron star that is stable for some time against gravitational collapse to a black hole due to the rapid differential rotation and the thermal pressure. The merger that produces such a star is quite violent and, as a by product, a significant amount of material can be ejected from the binary. The characteristics of this ejecta is important as it can produce an associated electromagnetic signal [74–76] that can provide complementary information about the binary (e.g. [22, 51, 77, 78]).

The recent observation of an infrared transient associated associated with the SGRB 130603B [24, 25] has demonstrated that such a tantalizing prospect is within reach. Additionally as discussed in [79], bound matter might also induce observable electromagnetic signals through the properties of winds emitted by the system [79–81].

We thus concentrate here on describing the properties of outflow material and its dependence on both the nuclear EoS and the binary mass ratio. The latter is particularly relevant as this material generally originates via tidal effects and is therefore especially sensitive to the mass ratio. We analyze in detail the various binaries at ∼3 ms after merger. A key distinction is made between matter remaining bound to the system and ejecta, i.e. material having kinetic energy sufficient to unbind it. Our analysis uses the common approach that material is considered unbound if its energy (defined by assuming that the spacetime is stationary) is positive.

We note that an important issue when studying low-density material in simulations adopting conservative 'Eulerian' schemes (as opposed to Lagrangian schemes) is that the results in low density regions could be significantly affected by the artificial atmosphere employed specifically to handle such regions. The simulations presented here benefit from an improvement to our code's treatment of the atmosphere so that the floor value decreases rapidly with distance from the center of coordinates (similar to that used in [19]). Comparisons of the new atmosphere treatment with the original, as well as studies with different atmosphere values, indicates that the properties being presented are nearly-independent of this improvement when the atmosphere density is sufficiently small ≈5 × 10⁵g cm⁻³ ≈ 10⁻¹⁰ ρ_max and when estimates are calculated shortly after merger of ≤4–5 ms.

We begin by considering all three EoS for the same mass ratio, q = 0.85. Figure 6 displays various properties along the equatorial plane (z = 0), all roughly 3 milliseconds after merger. The density variations are similar to that seen in the equal mass case (see figure 9 of [22]); the SFHo remnant is more centrally condensed compared to the remnants of the stiffer EoS that are less compact. Such behavior is expected as the collision takes place deeper in the gravitational potential of the system which, in turn, implies the remnant of the SFHo case is hotter than the other two. Also, while all three snapshots with different EoS show wisps of electron rich material just outside the remnants, the SFHo has the most significant of such regions. This is due to the larger matter temperatures in the SFHo remnant that drives decompression of the hot material to lower densities where positron capture on neutrons can effectively raise the electron fraction. For all the EoS, there is evidence of tidal tails with one spiral arm stronger than the other. This asymmetry arises from the unequal mass ratio of the binary.

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To more cleanly understand the role of mass ratio, we fix the EoS to that with the intermediate stiffness, DD2, and consider three mass ratios. Figure 7 displays the same quantities as in figure 6 but instead for mass ratios q = {1, 0.85, 0.76}.

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The density panels show that a decreased mass ratio results in a similar looking remnant but with more dispersed material outside it due to stronger tidal effects. The temperature looks roughly similar in the star, although slightly higher in the tidal tails of the equal mass cases. The electron fraction is noticeably lower in the decreased mass ratio simulations as is the neutrino emissivity. All the quantities display a strong spiral arm in the case of unequal masses, in contrast to the two spiral arms of the equal mass case. As we discuss later however, these broad properties do not convey what portion of this material is bound/unbound which is important in determining possible kilonova characteristics.

To this end, we integrate the characteristics of both the bound and the unbound matter and plot them as histograms (such calculations are described in [22]). In particular, we display the electron fraction in figure 8, the temperature in figure 9, the velocity in figure 10 and the polar and azimuthal angles in figure 11. Using these histograms, we present some observations and follow these with a general discussion about these features.

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Quite apparent in figure 8 is that bound material is generally quite neutron rich and independent of the mass ratio. However, the temperature distribution of this bound material does show some dependence on the mass ratio in figure 9. For the NL3 case, the dependence is minimal with the temperature distribution remaining generally the same among the mass ratios. The SFHo case shows the largest variation with mass ratio with the smaller mass ratio yielding a generally cooler merger remnant.

The ejecta, on the other hand, becomes increasingly neutron rich as the mass ratio departs from unity. Also apparent (in the insets) is that the amount of unbound material generally increases when the mass ratio decreases. We quantitatively summarize the amount of ejecta mass in table 1. The NL3 ejecta for the equal mass case is minuscule but significant for q = 0.85. The DD2 ejecta increases much less dramatically, but the SFHo breaks with this trend, albeit only moderately. The temperature of the ejecta is quite cool (<5 MeV) for every EoS and mass ratio, as displayed in figure 9. The observed trend indicates that for lower values of q the ejected material is cooler, being dominated by neutron-rich, tidal tail material.

Another interesting observation comes from the angular distribution of the ejected material. Figure 11 shows the azimuthal and polar angle distributions of unbound material. The histograms of polar angle do not show significant qualitative differences between the different mass ratio cases other than one observed trend. For low mass ratios the ejecta is more confined towards the equatorial region, giving quantitatively steeper distributions. The azimuthal distribution is more isotropic in the equal mass case (two spiral arms) but more anisotropic in the unequal mass cases (one spiral arm). As expected, the anisotropy grows as the mass ratio decreases from unity.

These observations suggest that a smaller mass ratio results in a less violent collision because the less massive star disrupts earlier and further out of the gravitational well than in the equal mass case. Only the NL3 bound material has essentially the same temperature for the different mass ratios. This EoS, however, has the largest stars generally, and so its equal mass case is already fairly cool. In contrast, the SFHo case has noticeably cooler bound and ejected material.

In general terms, the unbound matter (or ejecta) can be sourced either by tidal disruption during the merger or by thermal pressure and other interactions (e.g. shock heating) (On longer timescales—not studied here—an extra source is provided by winds or outflows from an accretion disk [80]). Our work here concentrates mainly on the material produced by tidal interaction, and we observe that, as the mass ratio decreases, the tidal disruption occurs earlier and further out. Consequently, there is more tidal-ejecta mass for small mass ratios. Furthermore, the temperature of this ejecta is lower, which inhibits positron production and capture on neutrons. Thus, the electron fraction is much closer to that of the original neutron star material. Only the SFHo retains essentially the same amount of ejected material with changes to the mass ratio because of a presumed balance of a competing effect; namely that the equal mass case was quite violent resulting in significant interaction ejecta (with higher values of Y_e and T) from shock heating. In the unequal mass case, the violence decreases (decreasing the interaction ejecta), but the early disruption increases the tidal ejecta. This explanation is consistent with the temperature and Y_e distributions of ejecta mass discussed above and shown in figures 8 and 9.

The current work is so far the only one of which we are aware studying unequal binaries, employing realistic microphysical EoS, with which to compare the amount of ejected materia. Sudies of gravitational wave features of unequal binary neutron star systems with realistic EoS have been recently initiated [82]. As we did for the case of gravitational wave characteristics, we can compare our results to those obtained employing piecewise polytropic EoS. To do so we focus on cases studied using the so-called MS1, H4, SLy piece-wise polytropes from [43] which have similar mass ratios and masses to those used here for the NL3, DD2, SFHo EoS. The results are in reasonable agreement with some of the cases, with typical ejected masses of 10⁻⁴–10⁻³M_⊙ (although they find even 10⁻²M_⊙ for their softest EoS [SLy]). It is not clear if these differences come only from the EoS choice. Nevertheless the overall trend is the same, namely that (a) softer EoS have larger ejecta and (b) unequal mass cases, for a given EoS, have larger ejecta masses for stiff EoS (DD2 and NL3 in our case, MS1 and H4 in the case of [43]) and lower ejecta masses for soft EoS (SFHo in our case, SLy in [43]).

The ejecta values obtained here allow us to estimate the lag-times (with respect to merger) and luminosities corresponding to each case. An important property of ejecta with such a low value of Y_e is its ability to produce elements with high atomic mass number (A ≥ 120) [83–85]. As argued in [86], due to the increased opacity as a result of lanthanides contributions, the expected emission from thermal decay of r-process elements would be dimmer/redder and delayed with respect to typical supernova expectations. Estimates in [86] (slightly re-arranged as in [45]) argue that the rise time of the corresponding kilonova emission ${t}_{\mathrm{peak}}^{k}$ (from the time of the merger) is given by

$$\begin{eqnarray}&&{t}_{\mathrm{peak}}^{k}\approx 0.25\,\mathrm{days}{\left[\displaystyle \frac{{M}_{\mathrm{eject}}}{{10}^{-2}{M}_{\odot }}\right]}^{1/2}{\left[\displaystyle \frac{{\rm{v}}}{0.3c}\right]}^{-1/2}\end{eqnarray} \tag{ 10 }$$

with a peak luminosity of

$$\begin{eqnarray}&&L\approx 2\times {10}^{41}\mathrm{erg}\,{{\rm{s}}}^{-1}{\left[\displaystyle \frac{{M}_{\mathrm{eject}}}{{10}^{-2}{M}_{\odot }}\right]}^{1/2}{\left[\displaystyle \frac{{\rm{v}}}{0.3c}\right]}^{1/2}.\end{eqnarray} \tag{ 11 }$$

The values obtained for the cases studied are summarized in table 4.

Table 4.  Estimated properties of electromagnetic (EM) signals after the merger. The rise-time ${t}_{\mathrm{peak}}^{k}$ and luminosity L of EM radiation from a potential kilonova are calculated from equations (10) and (11). We used the middle value of the bin corresponding to the peak velocity (see figure 10), v/c, in calculating these estimates. The lag time t_peak and flux density F, estimated from the collision of the ejecta with the interstellar medium, are calculated from equations (12) and (13).

EoS	q	L (1040erg s−1)	${t}_{\mathrm{peak}}^{k}$ (days)	Meject (10−3 M⊙)	v/c	Ekin(1050 ergs]	tpeak (yr)	F(1 GHz) (mJy)
NL3	1.0	0.9	0.008	0.015	0.45	0.01	0.31	1.8 × 10−3
NL3	0.85	8.8	0.13	2.3	0.25	1.22	4.0	4.4 × 10−2
DD2	1.0	4.1	0.05	0.43	0.3	0.31	1.9	1.9 × 10−2
DD2	0.85	4.1	0.05	0.42	0.3	0.29	1.8	1.7 × 10−2
DD2	0.76	7.2	0.09	1.3	0.3	0.76	2.5	4.6 × 10−2
SFHo	1.0	10.6	0.16	3.4	0.25	1.8	4.6	6.5 × 10−2
SFHo	0.85	8.6	0.13	2.2	0.25	1.8	4.6	6.5 × 10−2

In addition to the electromagnetic signals produced by the decay of radioactive elements, another transient has been proposed as a result from the collision of the ejected material with the interstellar medium [87]. This interaction would produce a radio emission that evolves more slowly. Indeed, the estimated timescales and brightness are expected to obey [87] (employing the re-arranged version of [45])

$$\begin{eqnarray}&&{t}_{\mathrm{peak}}\approx 6\,\mathrm{yr}{\left[\displaystyle \frac{{E}_{\mathrm{kin}}}{{10}^{51}\mathrm{erg}}\right]}^{1/3}{\left[\displaystyle \frac{{n}_{0}}{0.1{\mathrm{cm}}^{-3}}\right]}^{-1/3}{\left[\displaystyle \frac{{\rm{v}}}{0.3c}\right]}^{-5/3.}\end{eqnarray} \tag{ 12 }$$

with a flux density

$$\begin{eqnarray}F({\nu }_{\mathrm{obs}}) & \approx & 0.6\,\mathrm{mJy}\left[\displaystyle \frac{{E}_{\mathrm{kin}}}{{10}^{51}\mathrm{erg}}\right]{\left[\displaystyle \frac{{n}_{0}}{0.1{\mathrm{cm}}^{-3}}\right]}^{7/8}{\left[\displaystyle \frac{{\rm{v}}}{0.3c}\right]}^{11/4}\\ & & {\left[\displaystyle \frac{{\nu }_{\mathrm{obs}}}{1\mathrm{GHz}}\right]}^{-3/4}{\left[\displaystyle \frac{d}{100\mathrm{Mpc}}\right]}^{-2}.\end{eqnarray} \tag{ 13 }$$

Table 4 illustrates the values obtained for t_peak and F(ν_obs) (for a representative value of ν_obs = 1 GHz) assuming a density of n₀ = 0.1 cm⁻³ and a distance of d = 100 Mpc. The values of E_kin have been calculated for the unbound material as the total non-gravitational energy minus the rest mass and the internal energies [19].

Finally we turn our attention to the neutrinos emitted by the system. We estimate the luminosity via a leakage scheme [88–90] and determine the thermodynamic properties of the neutrinospheres via a ray-tracing formalism [91]. The algorithms we employ to obtain both estimates have been described in our previous papers [20] and [22], respectively.

Figures 6 and 7 illustrate both the spatial extent of the temperature and of the neutrino emissivities in the equatorial plane at ∼3 ms after merger for the q = 0.85 mass ratio simulations across all EoS (NL3, DD2, SFHo), and for the DD2 EoS simulations across the three mass ratios considered (q = 1.0, 0.85, and 0.76), respectively. The first figure indicates that softer EoS will give rise to higher neutrino luminosities which is consistent with observations in the equal mass case [22]. The higher luminosities are natural consequences of both the increased temperature and the increased amount of shock heating for the softer EoS. The higher temperatures also imply that the neutrino average energies will be higher.

These observations and conclusions are also supported by the results of our ray-tracing shown in figure 12. This figure shows the spatial distribution of the temperature of the electron neutrinosphere and electron antineutrinosphere for the q = 0.85 mass ratio simulations using three different EoS at ∼3 ms after merger. They show higher peak matter temperatures at the electron (anti)neutrinosphere for the SFHo EoS, locally as high as ∼16 MeV (∼18 MeV), when compared to the NL3 EoS where the peak temperatures at the (anti)neutrinosphere reach ∼12 MeV (∼14 MeV). Our ray-tracing gives average electron neutrino (antineutrino) energies (as measured at infinity) for these same snapshots of ∼12.6 MeV (∼15.6 MeV), ∼15.1 MeV (∼18.1 MeV), and ∼15.3 MeV (∼17.8 MeV), for the NL3, DD2, and SFHo EoS, respectively. These average energies are also summarized in table 5.

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Table 5.  Neutrino properties at ∼3 ms after the BNS merger for each EoS and mass ratio studied. The average energies $\langle {E}_{{\bar{\nu }}_{e}}\rangle$ and $\langle {E}_{{\nu }_{e}}\rangle$ are computed with (without) the gravitational redshift from our ray-tracing scheme while the electron antineutrino luminosity for the corresponding time is predicted from the leakage scheme. Also included is the estimated instantaneous detection rate in a SuperKamiokande-like water Cherenkov detector for a merger at 10 kpc from Earth, equation (14). The times are given with respect to t_merger, which is defined for each EoS as the time when the q = 1 mass ratio binary reaches first contact.

EoS	q	t	$\langle {E}_{{\bar{\nu }}_{e}}\rangle$	$\langle {E}_{{\nu }_{e}}\rangle$	${L}_{{\bar{\nu }}_{e}}$	Rν
		(ms)	(MeV)	(MeV)	(1053 erg s−1)	(#/ms)
NL3	1.0	3.4	18.5 (22.4)	15.2 (18.3)	0.7	18
NL3	0.85	3.0	15.6 (18.7)	12.6 (15.1)	0.8	18
DD2	1.0	3.3	18.3 (22.1)	14.6 (17.4)	1.1	28
DD2	0.85	2.8	18.1 (21.7)	15.1 (18.0)	1.0	25
DD2	0.76	2.4	19.7 (23.9)	14.8 (17.9)	1.3	36
SFHo	1.0	3.5	24.6 (29.7)	23.5 (28.3)	3.5	121
SFHo	0.85	3.9	17.8 (21.3)	15.3 (17.9)	2.0	50

We next consider the impact of the mass ratio on the properties of the emitted neutrinos. Figure 13 shows the neutrino surfaces ∼3 ms after merger for the three mass ratios considered for the DD2 EoS. The maximum temperatures achieved at the neutrinospheres are ∼14 MeV and correspond to the equal mass case. These maximum temperatures decrease to ∼12 MeV for the smaller mass ratios. As q decreases, the matter distribution becomes more dispersed, a consequence of stronger tidal effects, and this effect is reflected in more dispersed neutrino surfaces that, in addition to reaching smaller maximum temperatures, display a more uniform temperature. This more dispersed, but uniform, behavior can also be seen close to the core in the equatorial slices of figure 7. Our neutrino analysis shows that for the NL3 EoS there is a drop of ∼2–3 MeV in the average neutrino energy emitted from the merger remnants when the mass ratio goes from q = 1 to q = 0.85. For the DD2 EoS, no significant change or trend is seen in the neutrino average energies for these three particular snapshots.

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The impact of the mass ratio on the neutrino emission properties of the post merger evolution appears to be stronger for the SFHo EoS simulations. Recall the observation in the temperature and Y_e distributions (figures 8 and 9) of the ejected mass and bound mass from the previous section. For the soft SFHo EoS, when comparing the q = 1 to the q = 0.85 mass ratio simulation, we saw a transition from shock heated—or interaction—ejecta to tidal ejecta and overall lower temperatures. Our neutrino analysis suggests that this transition has a direct and significant impact on the neutrino quantities; we observe significantly lower average neutrino energies (∼6–8 MeV; see table 5) and consequently predict a lower rate of neutrino detection for close by mergers. This analysis would indicate that observations of a few systems with different values of q (extracted from gravitational wave measurements of the individual NS masses) could hint at properties of the underlying EoS (see e.g. [92]).

We caution however that these observation are drawn from a detailed analysis at particular moments of time after merger in different simulations. These disks are highly perturbed and are undergoing rapid changes. It is important to explore this possibility more fully in the future by employing actual neutrino transport in order to obtain a better handle on the neutrino energy and luminosity evolution as well as to capture the effects of neutrino absorption.

We end our discussion of the neutrino properties of the merger remnants by exploring the evolution of the neutrino luminosity. Figure 14 shows the total neutrino luminosity as a function of time for all cases considered. Since neutrino production is strongly dependent on the temperatures achieved during and after the merger, a higher emission is expected as the EoS softens. Our simulations follow this trend (note the different scales in figure 14 for the different EoS).

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As discussed throughout this paper, the merger yields a less violent collision for a given EoS as the mass ratio is decreased. This observation is also apparent in the neutrino luminosities. The equal mass cases show various episodes of higher neutrino luminosity (and neutrino average energy; see table 5) tied to the decompression stage that follows both the bounce (i.e. intensity maxima) and the occurrence of periods with higher relative intensity in gravitational waves. Our results suggest that unequal mass cases display a less oscillatory total neutrino luminosity. Although our sampling frequency of the this luminosity is less than it was for the equal mass cases, we see no suggestions of the quite apparent oscillations of the q = 1 cases. Furthermore, physically one would naturally expect that the oscillations would be more apparent in the symmetric, equal mass case as the bar-like remnant oscillated about a centrally condensed state. Once again, we stress the importance of a more detailed computation including neutrino transport to confirm the conclusions reached here.

The very rare observation of neutrinos from a close-by (galactic) neutron star merger would greatly complement the GW signature and reveal important information regarding the nuclear EoS. In combination with the electron antineutrino luminosities from our leakage scheme, we can use the average energies computed from our ray-tracing to estimate detection rates in a SuperKamiokande-like water-Cherenkov detector. These numbers are summarized in table 5. In addition to the three q = 0.85 mass ratio simulations, we also include in the table the results from the q = 1 simulations (with all three EoS) and the q = 0.76 simulation performed with the DD2 EoS. We note that the change in definition of merger time between this work and that of [22] causes slight differences in the value of t for the q = 1 snapshots ¹² . To estimate the neutrino detection rates presented in table 5, we have made use of the following approximation

$$\begin{eqnarray}&&{R}_{\nu }\sim \displaystyle \frac{21.05}{\mathrm{ms}}\left[\displaystyle \frac{{M}_{\mathrm{water}}}{32\,\mathrm{kT}}\right]\left[\displaystyle \frac{{L}_{{\bar{\nu }}_{e}}}{{10}^{53}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right]\left[\displaystyle \frac{\langle {E}_{{\bar{\nu }}_{e}}\rangle }{15\,\mathrm{MeV}}\right]{\left[\displaystyle \frac{10\mathrm{kpc}}{D}\right]}^{2},\end{eqnarray} \tag{ 14 }$$

where ${L}_{{\bar{\nu }}_{e}}$ and $\langle {E}_{{\bar{\nu }}_{e}}\rangle$ are the electron antineutrino luminosity and average energy as listed in table 5. The details of this expression and the tracing technique are discussed in [22]. Our calculations suggest that discriminating the mass ratio based on neutrino detection alone is unlikely. However such a detection, especially if coincident or correlated with GWs, could shed light on the dynamics of stellar material near, and soon after, the point of merger.