Short GRBs: Opening Angles, Local Neutron Star Merger Rate, and Off-axis Events for GRB/GW Association

SGRB 150424A was detected by the Burst Alert Telescope (BAT) on board the Swift satellite at 07:42:57 (UT) on 2015 April 24, (Beardmore et al. 2015). The BAT light curve is composed by a bright multi-peaked short episode (start from T − 0.05 to T + 0.5 s) and a very weak extended emission (lasted to about T + 100 s) (Barthelmy et al. 2015). Konus-Wind also detected a multi-peak short burst with a total duration of ∼0.4 s (Golenetskii et al. 2015). At 1.6 hr after the trigger, the Keck telescope observed the field and found the optical afterglow (Perley & McConnell 2015). Marshall & Beardmore (2015) checked the earlier Swift UV Optical Telescope (UVOT) observations and also found the afterglow. The HST visited the burst site for three epochs within a month (PI: Nial Tanvir, HST proposal ID: 13830). Recently, Knust et al. (2017) reported the multi-wavelength observation data of GRB 150424A and interpreted the central engine as a magnetar. In this work, we focus on the late observations (>0.5 day) to search for possible jet break and the macronova signal.

SGRB 160821B was detected by the Swift BAT at 22:29:13 (UT) on 2016 August 21, (Siegel et al. 2016). Its duration is T₉₀ = 0.48 ± 0.07 s. Following the BAT trigger, the X-ray telescope (XRT) and UVOT slew to the burst site immediately and started the observation in 66 and 76 s, respectively. The XRT found the afterglow at R.A. = 18:39:54.71 decl. =62:23:31.3 (J2000) with an uncertainty of 2.5 arcseconds (Siegel et al. 2016). Based on the observation started at 0.548 hr after the burst, the Nordic Optical Telescope (NOT) first reported the detection of the optical afterglow at R.A. = 18:39:54.56 decl. = 62:23:30.5 (J2000) with an uncertainty of 0.2 arcsecond, and a host galaxy candidate that lies at about 55 away (Xu et al. 2016). Later, the William Herschel Telescope spectral observations of the host galaxy claimed a redshift of z = 0.16 based on the Hα, Hβ and O iii 4959/5007 Å lines. The unambiguous classification as SGRB and the plausible low redshift make it an ideal candidate to search for a macronova. The HST visited the burst site for three epochs within a month (PI: Nial Tanvir, HST proposal ID: 14237).

Now the HST data of both GRB 150424A and GRB 160821B are publicly available, and we have downloaded these data from the Barbara A. Mikulski Archive for Space Telescopes (MAST: http://archive.stsci.edu/).

GRB 150424A was initially found near a bright spiral galaxy R ∼ 20 mag (Perley & McConnell 2015) at a redshift of z = 0.30 (Castro-Tirado et al. 2015). The later deep HST observations, however, found that the field of GRB 150424A is very complicated (Tanvir et al. 2015). There is an extended source to the southeast of afterglow (see Figure 1, marked as S1) that is a candidate of the host galaxy of GRB 150424A, too. It has been detected in HST F105W, F125W, and F160W bands, but not in HST F475W, F606W, and F814W bands (see Figure 2). If the break between F105W and F814W is due to the 4000 Å break, then the redshift is z ∼ 1.2. While the absence of strong evidence for a Lyman break in the Swift UVOT afterglow data (Marshall & Beardmore 2015; Knust et al. 2017) indicates a redshift of z ≤ 1.0. Considering these uncertainties, we suggest that z = 0.3 is favored while z ∼ 1 is still possible. The HST data of GRB 150424A have been analyzed by Knust et al. (2017) via a photometry with a 04 aperture. We have carried out an independent analysis of HST data using a slightly different method. We have adopted the image subtraction technique, taking all the observations at t > 20 days as the reference images, to remove the underlying sources (i.e., S1 and S2) from the afterglow emission. The afterglow is very weak in the second HST observation epoch. We have taken a three-pixels aperture for the photometry and then corrected to an infinite aperture. The results are reported in Table 1. Essentially, our results are in agreement with Knust et al. (2017), except that their photometry included the underlying sources S1 and S2, while we have removed them by image subtraction, hence our afterglow emission are weaker than Knust et al. (2017).

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GRB 160821B is located near a bright (R ∼ 19.2 mag) spiral galaxy (Xu et al. 2016), at a redshift of z = 0.16 (Levan et al. 2016). Deep HST observations found no secure detection of any underlying sources near the afterglow (see Figure 3). We then adopt z = 0.16 for this burst. For GRB 160821B, we have adopted a similar analysis method as GRB 150424A, including image subtraction, although there is no underlying source detected. The results are summarized in Table 2.

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In the GRB jet model (see Table 1 of Zhang & Mészáros (2004) for a comprehensive summary), the afterglow light curve declines as t^−p (or slightly shallower if the sideways expansion of the ejecta is ignorable), as long as the ejecta has been decelerated to a bulk Lorentz factor ≤1/θ_j, no matter whether the observer's frequency ν_obs is between ν_m and ν_c (the spectrum is $\propto {\nu }^{-(p-1)/2}$) or above both of them (the spectrum is ∝ν^−p/2). Therefore, the quick decline t^−α at late time, together with a reliable spectral index that is close to (α − 1)/2 or α/2 for α ≳ 2, are widely taken as strong evidence for the post-jet-break decline behavior.

For GRB 150424A, the F606W, F110W, and F160W data are consistent with the post-jet-break afterglow model for p ∼ 2.5, for which the spectrum should be ∝ν^−0.75 (for ν_m < ν_obs < ν_c) and the decay should be ∝t^−2.5 (see Figures 4 and 5). Converting all bands to i' band with a spectrum of ν^−0.75, then fitting⁵ the GROND and HST data with a broken power-law decay, we find a break at t = 3.73 ± 0.16 days, and the decay indices are 1.49 ± 0.04 and 2.61 ± 0.12, respectively. This fit yields a total ${\chi }^{2}/\mathrm{dof}=27.8/22$, indicating a reasonable fit for both spectral and temporal behavior of the afterglow. Note that the pre-break decline behavior is also well consistent with the theoretical model (for p = 2.5, it is $(3p-2)/4\sim 1.4$). The Swift XRT PC mode spectrum, although taken earlier than HST observations, gives a photon index of ${2.00}_{-0.18}^{+0.19}$ (Melandri et al. 2015), which is consistent with p ∼ 2.5, supposing the X-ray band is above both ν_m and ν_c (i.e., the spectrum should be $\propto {\nu }^{-p/2}$). The data strongly prefer a broken power-law model to a single power-law model, by which the fit gives a ${\chi }^{2}/\mathrm{dof}=97.5/24$. Knust et al. (2017) identified an optical plateau lasting about ∼8 hr and attributed such a shallow decline to the energy injection from a central millisecond magnetar. This is not at odds with our jet-break interpretation because at late times (i.e., when the magnetar spined down and the energy injection rate drops with time as t⁻²) the energy injection was not efficient any longer and the forward shock emission should be rather similar to the normal scenario (i.e., without energy injection; as shown for example in Zhang et al. 2004).

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For GRB 160821B, the publicly available data are rather limited. The HST took two epochs of observations at 3.6 days and 10.4 days after the burst, when the afterglow was still visible. Fitting F606W, F110W, and F160W bands with a same power-law decay yields a flux decline ∝t^{−1.83±0.13} (χ²/d.o.f =1.65/2) (see Figure 6). Fitting the F606W, F110W, and F160W band SED with a power-law spectrum, we find the power-law indices are α = 1.35 ± 0.07 (χ²/d.o.f = 13/1) at t = 3.6 days and α = 1.12 ± 0.39 (χ²/d.o.f = 3.36/1) at t = 10.4 days, respectively. Alternatively, if we fit the data with a thermal spectrum, the resulting temperature is of 4488 ± 81[(1 + z)/1.16] K (χ²/d.o.f = 15/1) at t = 3.6 days and = 4750 ± 473 [(1 + z)/1.16] K (χ²/d.o.f = 0.30/1) at t = 10.4 days, respectively. Though a thermal spectrum can well reproduce the SED at t = 10.4 days, the temperature is much higher than ∼2500 K, the value inferred for the macronova signals of GRB 060614 and GW170817/GRB 170817A in a similar epoch (see Jin et al. 2015; Drout et al. 2017, respectively). Instead, the overall decline behavior as well as the power-law spectrum at t = 10.4 days are largely consistent with the jet model for p ∼ 2 and ${\nu }_{\mathrm{obs}}\gt \max \{{\nu }_{{\rm{m}}},{\nu }_{{\rm{c}}}\}$ (the spectrum is $\propto {\nu }^{-p/2}$). Moreover, the Swift XRT spectrum measured in the time interval of t ∼ 4873–47180 s has a power-law photon index ${2.0}_{-0.6}^{+0.7}$ (http://www.swift.ac.uk/xrt-spectra/00709357/; Evans et al. 2009), which is consistent with p ∼ 2, too. Supposing the sideways expansion of the ejecta is unimportant, at early times the flux declines as t^(2−3p)/4 while in the post-jet-break phase the decline is steepened by a factor of t^−3/4, as shown in Zhang & Mészáros (2004). Therefore, we suggest that the HST data of GRB 160821B were dominated by a power-law afterglow component though at t = 3.6 days an underlying weak macronova component is possible (please see Section 2.3 for further discussion). The X-ray afterglow emission in the time interval of ∼0.05–2 day after the burst also strongly indicate the presence of a jet break at t ≥ 0.3 day. The analytical approach yields a t_jet ∼ 0.7 day (see Figure 6), which is comparable to though a bit larger than the X-ray data based estimate by Lü et al. (2017).

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With the break time, the jet half-opening angle can be estimated by (Sari et al. 1999; Frail et al. 2001)

$$\begin{eqnarray}{\theta }_{{\rm{j}}} & \approx & 0.076{\left(\displaystyle \frac{{t}_{\mathrm{jet}}}{1\mathrm{day}}\right)}^{3/8}{\left(\displaystyle \frac{1+z}{2}\right)}^{-3/8}{\left(\displaystyle \frac{{E}_{\mathrm{iso}}}{{10}^{51}\mathrm{erg}}\right)}^{-1/8}\\ & & \times {\left(\displaystyle \frac{{\eta }_{\gamma }}{0.2}\right)}^{1/8}{\left(\displaystyle \frac{n}{0.01{\mathrm{cm}}^{-3}}\right)}^{1/8}.\end{eqnarray} \tag{ 1 }$$

For GRB 150424A, Konus-Wind recorded a total gamma-ray (20 keV–10 MeV) fluence of 1.81 ± 0.11 × 10⁻⁵ erg cm⁻² (Golenetskii et al. 2015), corresponding to an isotropic gamma-ray emission energy of E_iso ∼ 4.3 × 10⁵¹ or ∼1.0 × 10⁵³ erg at z = 0.3 or 1.0, respectively, using cosmological parameters of Planck results (Planck Collaboration et al. 2014). The Fermi gamma-ray (10–1000 keV) fluence is 1.68 ± 0.19 ×10⁻⁶ erg cm⁻² for GRB 160821B (Stanbro & Meegan 2016), so we have E_iso ∼ 10⁵⁰ erg for z = 0.16. Following Frail et al. (2001) we take the radiation efficiency η_γ = 0.2. For the ISM number density n = 0.01 cm⁻³ (see Fong et al. 2015 for the evidence of the low number density of the ISM surrounding SGRBs), we get θ_j ≈ 0.12 for GRB 150424A (if we take z = 1.0 then θ_j ∼ 0.07) and θ_j ≈ 0.1 for GRB 160821B.

A reliable jet half-opening angle has only been rarely inferred for SGRBs/lsGRBs (Please see Table 3 for a summary). Interestingly, all low-redshift (z ≤ 0.4) events (except GRB 050502B, of which the optical afterglow emission was never detected) with deep HST follow-up observations are in this sample. These events include GRB 050709, GRB 060614, GRB 130603B, GRB 150424A, and GRB 160821B (note that for GRB 050709, no jet break was directly measured. But the presence of a macronova signal strongly favors a jet break at t ≤ 1.4 days after the burst; see Jin et al. 2016). Such an observational fact likely points toward the presence of jet break in most events and their non-detection may simply due to the lack of deep follow-up observations. The other interesting feature is the "narrow" distribution of these estimated θ_j that peaks at ∼0.1 rad, indicating that many more SGRBs/lsGRBs were off-beam (or off-axis).

For GRB 150424A, both the temporal and spectral behaviors are well consistent with the afterglow model and there is no macronova signal; see Figures 4 and 5.

As for GRB 160821B, the situation is less clear. The HST data at t ∼ 3.6 days are hard to be interpreted as either a power-law or a thermal spectrum (see Section 2.2). Interestingly, they can be interpreted as the superposition of a power-law afterglow component (with a spectrum f_ν ∝ ν^−1.1) and a thermal-like component with a temperature of ∼3100 K (due to the sparse data points, this temperature is fixed to be that of AT2017gfo in the same epoch⁶ ) and the fit yields χ²/d.o.f = 1.36/1, as shown in Figure 7. Such a result is likely to support the speculation of Troja et al. (2016b) that there might be a weak macronova signal in the afterglow of GRB 160821B. Though intriguing, the current limited publicly available HST data alone are insufficient to draw a final conclusion (T. Piran 2017, private communication).

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The SGRB data-based neutron star merger rate was widely estimated in the literature (e.g., Guetta & Piran 2005; Nakar 2007; Coward et al. 2012; Fong & Berger 2013; Petrillo et al. 2013; Fong et al. 2015). However, all these previous approaches were mainly based on jet opening angles measured at relatively high redshifts (though GRB 050709, GRB 060614, and GRB 061201 were included "individually" in some estimates). The reasonable estimate of θ_j for GRB 050709, GRB 060614, GRB 061201, and GRB 160821B, four GRBs at redshifts of ≤0.2, provides the first opportunity to directly though conservatively estimate the neutron star merger rate in the local universe that can be directly tested by (compared with) the ongoing advanced LIGO/Virgo observations.

GRB 050709 was detected by HETE-II with a field of view (FoV) ≈ 3 sr (Villasenor et al. 2005), while GRB 060614, GRB 061201, and GRB 160821B were recorded by the Swift satellite with a FoV ≈ 1.4–2.4 sr (Gehrels et al. 2006; Siegel et al. 2016). Note that HETE-II has a much lower detection rate of SGRBs in comparison with Swift due to the relatively small effective area of the on board detector. Moreover, no GRBs were reported by HETE-II any longer since 2006 March (http://space.mit.edu/HETE/Bursts/). The joint analysis of HETE-II and Swift SGRBs are thus very challenging. In this work, for simplicity we exclude GRB 050709 in the following investigation.

Now we evaluate the "apparent" (i.e., without the jet half-opening angle correction) rate, ${{ \mathcal R }}_{\mathrm{nsm},\mathrm{app}}$, for local SGRBs. The apparent rate is related to the observational number of events as well as the sensitivity and FoV of the instrument. For a given redshift, the sensitivity plays a role in determining the weakest SGRB that can be detected by the instrument while the F.o.V reflects the search portion of the full sky. The BAT on board Swift is a coded mask telescope, and its F.o.V and sensitivity change as a function of the partial coding fraction, which is related to the burst's incident angle (Barthelmy et al. 2005); meanwhile, BAT has a complex trigger algorithm, making detailed analysis on the intrinsic rate through Swift data very complicated (Lien et al. 2014). In this work, we take into account these effects as the following: Lien et al. (2016) showed that the sensitivity of BAT decreases with the burst's duration (which roughly reflects the exposure time) with ${\mathrm{flux}}_{\mathrm{limit}}\propto 1/\sqrt{{T}_{90}};$ they also showed that the sensitivity to 1s flux is ∼3 × 10⁻⁸ erg cm⁻² s⁻¹ for fully coded region. If a burst's incident angle is large, and the detector's plane is partially coded, the effect of partial coding fraction p_f can be expressed with the "effective on-axis exposure time" T_eff = p_fT₉₀ (Baumgartner et al. 2013). With these relations, we collect the 1s peak fluxes of the three bursts from the Swift/BAT Gamma-Ray Burst Catalog and calculate the corresponding smallest partial coding at different distances by

$$\begin{eqnarray}&&{p}_{{\rm{f}}}={\left[\displaystyle \frac{3\times {10}^{-8}\mathrm{erg}/{\mathrm{cm}}^{2}/{\rm{s}}}{{\left(\tfrac{{D}_{{\rm{L}}0}}{{D}_{{\rm{L}}}}\right)}^{2}\left(\tfrac{1+z}{1+{z}_{0}}\right){f}_{0}}\right]}^{2}/{T}_{90},\end{eqnarray} \tag{ 2 }$$

where z₀, D_L0, and f₀ are the observed redshift, luminosity distance, and 1s peak flux, respectively, of a given burst. The yielded p_f is then used to calculate the corresponding BAT FoV, and their relation are inferred from Barthelmy et al. (2005) (we fit their simulated curve adjusted for off-axis projection effects with polynome and obtain a analytical function F.o.V(p_f)). The FoV is then adopted to calculate the spacetime volume of the search for a given burst:

$$\begin{eqnarray}&&\langle {VT}\rangle =0.9T{\int }_{0}^{0.2}\displaystyle \frac{{\rm{F}}.{\rm{o}}.{\rm{V}}({p}_{{\rm{f}}})}{4\pi }\displaystyle \frac{1}{1+z}\displaystyle \frac{{{dV}}_{c}(z)}{{dz}}{dz},\end{eqnarray} \tag{ 3 }$$

where the factor 0.9 represents the fraction of the time that BAT spends on searching for GRBs. Assuming a negligible evolution of rate in the local universe, and the observed number of event in a given spacetime volume should follow a Poission distribution, we use the Bayesian inference to derive the posterior distribution of ${{ \mathcal R }}_{\mathrm{nsm},\mathrm{app}}$ for each burst by (see Wang et al. 2017, and references therein)

$$\begin{eqnarray}&&P({{ \mathcal R }}_{\mathrm{nsm},\mathrm{app}})\propto {P}_{\mathrm{Poission}}(1| {\rm{\Lambda }})\times P^{\prime} ({{ \mathcal R }}_{\mathrm{nsm},\mathrm{app}}),\end{eqnarray} \tag{ 4 }$$

where ${P}_{\mathrm{Poission}}(1| {\rm{\Lambda }})$ is the likelihood of observing one event from a Poission distribution with a mean number ${\rm{\Lambda }}\,={{ \mathcal R }}_{\mathrm{nsm},\mathrm{app}}\langle {VT}\rangle$, and $P^{\prime} ({{ \mathcal R }}_{\mathrm{nsm},\mathrm{app}})$ is the prior (for which we choose a uniform distribution). The inferred rates from GRB 060614, GRB 061201 and GRB 160821B are ${0.25}_{-0.18}^{+0.37},{0.26}_{-0.19}^{+0.40}$ and ${0.30}_{-0.22}^{+0.45}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$, respectively. In this work, the errors stand for the 68% credible intervals unless specifically noted. These rates are similar, indicating that these three bursts are bright and hence the influence from the change of F.o.V is not significant.

To get the local neutron star merger rate, the geometry correction (including the uncertainties of θ_j) should be addressed. We assume that the probability distribution of the true values of θ_j of GRB 060614 and GRB 061201 follow uniform distribution in the intervals reported in Table 3. For GRB 160821B, we assume its θ_j distributes uniformly within 0.08–0.18 rad since t_jet should be within ∼0.3–3 days after the burst (see the X-ray and optical data in Figure 6). The posterior distribution of local neutron star merger rate derived from each burst, ${{ \mathcal R }}_{\mathrm{nsm}}$, is then calculated by

$$\begin{eqnarray}&&p({{ \mathcal R }}_{\mathrm{nsm}})\propto {\int }_{{\theta }_{{\rm{j}},\min }}^{{\theta }_{{\rm{j}},\max }}p(1| {{ \mathcal R }}_{\mathrm{nsm}},{\theta }_{{\rm{j}}})p^{\prime} ({{ \mathcal R }}_{\mathrm{nsm}})p^{\prime} ({\theta }_{{\rm{j}}})d{\theta }_{{\rm{j}}}\end{eqnarray} \tag{ 5 }$$

in which the priors for θ_j follow the distributions discussed above, and the prior for ${{ \mathcal R }}_{\mathrm{nsm}}$ is assume to be uniform. The likelihood term can be written as

$$\begin{eqnarray}&&p(1| {{ \mathcal R }}_{\mathrm{nsm}},{\theta }_{{\rm{j}}})={p}_{\mathrm{poisson}}\left(1| {{ \mathcal R }}_{\mathrm{nsm}}\displaystyle \frac{{{\theta }_{{\rm{j}}}}^{2}}{2}\langle {VT}\rangle \right).\end{eqnarray} \tag{ 6 }$$

The local neutron star merger rates inferred from GRB 060614, GRB 061201, and GRB 160821B are ${68}_{-50}^{+103},{832}_{-625}^{+1407}$ and ${33}_{-27}^{+84}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$, respectively. The total rate, dominated by GRB 061201, is obtained by convoluting the three distributions together, with which we have

$$\begin{eqnarray}&&{{ \mathcal R }}_{\mathrm{nsm}}={1109}_{-657}^{+1432}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 7 }$$

Correspondingly, the total rate with the 90% credible interval is ${1109}_{-840}^{+2872}{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$.

Besides the issues examined above, some other factors may further increase the uncertainties to our results (see e.g., Coward et al. 2012). (a) The redshift of GRB 061201 is less secure (Stratta et al. 2007) than the other events.⁷ Because our sample number is small, this will increase the uncertainty of the result. In the following, we will exclude GRB 061201 and do the calculation again for comparison. (b) Only ≲1/4 of the SGRBs have measured redshifts (Berger 2014) and some other SGRBs could be nearby, too.⁸ For example, recently Siellez et al. (2016) claimed that GRB 070923 and 090417A were at z < 0.1. Moreover, in our analysis two "local" bursts (GRB 080905A and GRB 150101B) without a reliable jet opening angles are excluded in the rate estimate. It is also probable that only a fraction of neutron star mergers can produce either SGRBs or lsGRBs. Taking into account these factors, the real neutron star merger rate density should be enhanced by a factor of f_c > 1. As a conservative estimate on ${{ \mathcal R }}_{\mathrm{nsm}}$, we do not correct this further. (c) There are arguments that GRBs shorter than 2 s may have collapsar origin, while bursts with duration longer than 2 s may still have non-collapsars origin (Bromberg et al. 2013). This is indeed the case for GRB 060614, which is a long-duration burst, but the identified macronova component in its late afterglow (Jin et al. 2015; Yang et al. 2015) revealed its neutron star merger origin. For the other nearby events (GRB 050709, GRB 061201, and GRB 160821B), no supernova emission were detected down to very stringent limits, which strongly disfavored the collapsar origin. Therefore, for our sample, this correction is likely irrelevant. (d) Due to its limited energy range, the SGRB detection rate of Swift is found to be a factor of R_b/s = 6.7 lower than that of BATSE. Coward et al. (2012) amplified the Swift SGRB rate by such a factor to crudely correct the bias. Though interesting, it is unclear whether this is a reasonable approximation for the local events that are of our interest. Moreover, such a correction likely enhances the merger rate. In summary, despite all these factors playing roles in the rate estimation, only point (a) seems to be able to significantly change our "conservative" estimate. If we exclude GRB 061201 from the sample, the apparent SGRB rate and neutron star merger rate are ${0.81}_{-0.39}^{+0.60}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ and

$$\begin{eqnarray}&&{{ \mathcal R }}_{\mathrm{nsm},{\rm{w}}/{\rm{o}}061201}={162}_{-83}^{+140}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 8 }$$

respectively. The resulting neutron merger rate is significantly smaller than that found in Equation (7), suggesting that our results are sensitively dependent on the narrowly beamed burst GRB 061201, and a larger sample is crucial to get a more reliable estimate on ${{ \mathcal R }}_{\mathrm{nsm}}$.

Intriguingly, the successful detection of GW170817 in the O2 run of advanced LIGO yields a neutron star merger rate of (Abbott et al. 2017)

$$\begin{eqnarray*}&&{{ \mathcal R }}_{\mathrm{nsm},\mathrm{gw}}={1540}_{-1220}^{+3200}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1},\end{eqnarray*}$$

which is in agreement with the local SGRB-based neutron star merger rate reported in Equation (7). For Equation (8), the consistence with ${{ \mathcal R }}_{\mathrm{nsm},\mathrm{gw}}$ is marginal.

The advanced LIGO detectors can detect the gravitational wave radiation from double neutron star mergers within a typical distance D ∼ 220 Mpc at its designed sensitivity (Abadie et al. 2010). The detection rate of neutron star mergers is thus

$$\begin{eqnarray}&&{R}_{\mathrm{gw},\mathrm{nsm}}=\displaystyle \frac{4\pi {D}^{3}}{3}{{ \mathcal R }}_{\mathrm{nsm},\mathrm{gw}}\sim {69}_{-55}^{+143}\,{\mathrm{yr}}^{-1}\,{\left(\displaystyle \frac{D}{220\mathrm{Mpc}}\right)}^{3}.\end{eqnarray} \tag{ 9 }$$

Notice that the above number assumes a perfect GW detector; a realistic one operates with a duty cycle and in reality the expected value would decrease by the coincident duty cycle. Nevertheless, the detection prospect is indeed quite promising and we expect that much more binary neutron star merger events will be detected in the near future.