Physical Implications of the Subthreshold GRB GBM-190816 and Its Associated Subthreshold Gravitational-wave Event

We download the corresponding time-tagged-event data from the public data site of Fermi/GBM according to the time of the event reported by LIGO Scientific Collaboration et al. (2019a). Data reduction follows the standard procedure, as discussed in Zhang et al. (2011, 2016, 2018a). The full-energy-range light curves of all 14 GBM detectors are shown in Figure 1. The weak subthreshold GRB is visible in the light curve of the Na i detector n3 and marginally visible in n1. Indeed, using the best-fit location (17823, 3352) of the GW signal, we calculate that Na i detectors n1 and n3 hold the smallest angular separations with respect to the GW source. Thus, those two detectors are selected for further temporal and spectral analysis. No BGO detector is selected as no significant emission has been observed above 800 keV.

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We perform the following analysis on the gamma-ray signal (Zhang et al. 2011, 2016, 2018a) to study the properties of GBM-190816:

(1) Signal confirmation. We analyze the TTE data of the detector n3 using the Bayesian Block (BB) algorithm (Scargle et al. 2013). Searching in the interval from T₀ − 10 s to T₀ + 10 s, we find a significant sharp signal starting from T₀ + 0.038 s to T₀ + 0.056 s. We then try to derive the significance level of the burst. The background is taken from two intervals T₀ − 15 s to T₀ − 5 s and T₀ + 5 s to T₀ + 15 s. By varying the energy band and the bin size (we make sure that there are at least two bins in the burst block), we find the signal-to-noise ratio (S/N) reaching 3.95. Figure 2 shows the light curves in four different energy channels for detector n3. The details of this method can be found in J. S. Wang et al. (2019, in preparation). The false alarm rate (FAR) of detecting such an event is about 1.2 × 10⁻⁴ (LIGO Scientific Collaboration et al. 2019a).

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(2) Burst duration. For simplicity, we estimate T₉₀ of the burst based on the cumulative net count rate. The background is estimated by applying the "baseline" method (Zhang et al. 2018a) to some long time intervals before and after the signal region. By calculating the time interval during which 90% of the total net counts have been detected, we obtained ${T}_{90}={0.112}_{-0.085}^{+0.185}$ s with the starting and ending times ${T}_{\mathrm{90,1}}={0.032}_{-0.065}^{+0.025}$ s and ${T}_{\mathrm{90,2}}={0.143}_{-0.11}^{+0.17}$ s, respectively (Figure 3). The uncertainties are calculated by a Monte Carlo approach, which takes into account the fluctuations of the observed light curve.

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(3) Amplitude parameter of GBM-190816. Lü et al. (2014) defined two "amplitude parameters" to assist burst classifications: the parameter f denotes the ratio between the peak flux and the average background flux, and f_eff denotes the ratio between the peak flux of a pseudo-burst and the average background flux. The pseudo-burst is defined by scaling down the peak flux until the measured duration of a long burst is shorter than two seconds (Lü et al. 2014). For short GRBs, f = f_eff. Statistically, the f_eff parameters of long GRBs are typically smaller than f of short GRBs, providing a criterion to identify contaminated long GRBs in the observed short GRB sample due to the "tip-of-the-iceberg" effect. We perform the f analysis to GBM-190816, and obtain f = 2.58 ± 0.37. Figures 4(a) and (b) show T₉₀ as functions of f and f_eff for both long and short GRBs, where GBM-190816 is highlighted as a star. We find that its amplitude parameter is generally larger than f_eff of typical long GRBs, consistent with being a typical short GRB. Moreover, we calculate the probability of GBM-190816 being a disguised short GRB according to the p − f relation derived by Lü et al. (2014). We find that such a probability is p ∼ 0.03. All these suggest that GBM-190816 is a genuine short GRB. Nonetheless, there is a nonnegligible probability that the observed spike could still be the "tip of the iceberg" of a longer short burst (see more discussion in Section 4.3).

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(4) Spectral analysis. We extract the time-integrated spectra of GBM-198016 between T_90,1 and T_90,2. Only two GBM detectors, n1 and n3, are selected due to the reasons mentioned in Section 2.1. Background spectra are obtained by empirically modeling the source-free time intervals around the burst. The detector response matrices (DRMs), which are needed in the spectral fitting is generated using the response generator provided by the Fermi Science Tools.¹⁰ Spectral fitting is performed using McSpecfit (Zhang et al. 2018a). A handful of spectral models, such as simple power law (PL), cutoff power law (CPL), Band function (Band), Blackbody (BB), and the combinations of any two or three models, are considered to fit the observed spectra. We then compare the goodness of the fits and find that the CPL is the best one that adequately describes the observed data according to the Bayesian information criteria (BIC). The CPL model fit (Figure 5) gives a peak energy of ${94.84}_{-17.94}^{+114.64}$ keV and a lower energy spectral index of $-{0.92}_{-0.58}^{+0.32}$, both being typical for GRB spectral parameters. The best-fit parameters of CPL fits are listed in Table 1. No further time-resolved spectral fitting is performed due to the low number of photon counts.

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Table 1.  Spectral Properties of GBM-190816 Using the Fit of Cutoff Power-law Model

Time Interval				CPL
t1	t2	Γph	Ep	logNorm	PGSTAT/dof
0.032	0.143	$-{0.92}_{-0.58}^{+0.32}$	${94.84}_{-17.94}^{+114.64}$	${0.53}_{-0.41}^{+0.72}$	130.1/227

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(5) Burst energy. Using the best-fit parameters of the CPL model, we find that the average flux within T₉₀ is ${6.65}_{-2.26}^{+5.72}\times {10}^{-7}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ between 1 and 10,000 keV. The total fluence in the same energy range is ${7.38}_{-2.51}^{+6.35}\,\times {10}^{-8}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}$. Taking into account the burst distance ∼428 Mpc, we further calculate the corresponding isotropic luminosity and energy as ${L}_{\gamma ,\mathrm{iso}}={1.47}_{-1.04}^{+3.40}\times {10}^{49}$ erg s⁻¹ and ${E}_{\gamma ,\mathrm{iso}}={1.65}_{-1.16}^{+3.81}\times {10}^{48}$ erg, respectively.

(6) Amati relation. In order to check if GBM-190816 is an unusual event, we overplot GBM-190816 in the E_p-E_γ,iso correlation of all GRBs with known redshifts (Amati et al. 2002; Zhang et al. 2009). As shown in Figure 6, unlike GRB 170817A, which is an outlier of the short GRB track, GBM-190816 is located well within the 1σ region of the short GRB population, suggesting that it is consistent with typical short GRBs in terms of its spectral peak and total energy.

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A summary of the observed properties of GBM-190816 is listed in Table 2. The observed facts point toward the possibility that GBM-190816 is a short GRB with a sharp peak and typical temporal and spectral properties. The unusually short duration leads to a low fluence, which causes it to be a subthreshold event below the Fermi/GBM triggering threshold.

Table 2.  Observational Properties and Derived Constraints of GBM-190816

Observed Properties
T90 (s)	${0.112}_{-0.085}^{+0.185}$
Peak energy Ep (keV)	${94.84}_{-17.94}^{+114.64}$
Total fluence(erg cm−2)	${7.38}_{-2.51}^{+6.35}\times {10}^{-8}$
Distance (Mpc)	428+/−143
Isotropic energy Eγ,iso (erg)	${1.65}_{-1.16}^{+3.81}\times {10}^{48}$
Luminosity Lγ,iso (erg s−1)	${1.47}_{-1.04}^{+3.40}\times {10}^{49}$
f parameter	2.58+/−0.37
Assumed Parameters
Jet core angle θc,j	assumed 5° (16°)
Viewing angle θv	10°–19° (18°–24°)
Γc	assumed 100
m2 (M⊙)	assumed 1.4 (for NS–BH system)
	assumed 2.8 (for BH–BH system)
Derived Constrains
q from GRB	varies
q from GW	${2.26}_{-0.12}^{+2.75}$
m1 (M⊙)	varies
GW–GRB Time Delay (s)	1.57
Charge of BH (e.s.u.)	${2.162}_{-0.002}^{+0.302}\times {10}^{26}$ (for NS–BH system)
	${2.114}_{-0.042}^{+1.131}\times {10}^{26}$ (for BH–BH system)

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For NS–BH mergers, whether or not there is matter left outside the post-merger BH event horizon is determined by a comparison between the tidal disruption radius d_tidal and the radius of the innermost stable circular orbit R_ISCO (Shibata et al. 2009). In general, the total mass M_out of the matter left outside the BH event horizon after tidal disruption of the NS can be divided into two components: the disk mass M_disk and the dynamical ejecta mass M_dyn. Numerical simulations suggest that M_out depends on the mass (M_BH) and the dimensionless spin (χ_BH) of the BH, the baryonic mass of the NS (${M}_{\mathrm{NS}}^{{\rm{b}}}$), as well as the tidal deformability (Λ_NS) of the NS, i.e. (Foucart et al. 2018),

$$\begin{eqnarray}&&{M}_{\mathrm{out}}={M}_{\mathrm{NS}}^{{\rm{b}}}{\left[\max \left(\alpha \displaystyle \frac{1-2\rho }{{\eta }^{1/3}}-\beta {\tilde{R}}_{\mathrm{ISCO}}\displaystyle \frac{\rho }{\eta }+\gamma ,0\right)\right]}^{\delta },\end{eqnarray} \tag{ 11 }$$

where η = q/(1 + q)², $\rho ={\left(15{{\rm{\Lambda }}}_{\mathrm{NS},1.4}\right)}^{-1/5}$ (Λ_NS,1.4 represents Λ_NS for the NS mass at 1.4M_⊙), and the dimensionless ISCO radius follows

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{R}}_{\mathrm{ISCO}} & = & {R}_{\mathrm{ISCO}}{c}^{2}/{{GM}}_{\mathrm{BH}}=3+{Z}_{2}\\ & & -\ \mathrm{sgn}\left({\chi }_{\mathrm{BH}}\right)\sqrt{\left(3-{Z}_{1}\right)\left(3+{Z}_{1}+2{Z}_{2}\right)},\end{array}\end{eqnarray} \tag{ 12 }$$

with ${Z}_{1}=1+{\left(1-{\chi }_{\mathrm{BH}}^{2}\right)}^{1/3}\left[{\left(1+{\chi }_{\mathrm{BH}}\right)}^{1/3}+{\left(1-{\chi }_{\mathrm{BH}}\right)}^{1/3}\right]$ and ${Z}_{2}=\sqrt{3{\chi }_{\mathrm{BH}}^{2}+{Z}_{1}^{2}}$ (Bardeen et al. 1972). The empirical parameters yield to α = 0.308, β = 0.124, γ = 0.283, and δ = 1.536.

The dynamical ejecta mass M_dyn depends on M_BH, the NS gravitational mass M_NS, ${M}_{\mathrm{NS}}^{{\rm{b}}}$, ${\chi }_{\mathrm{BH}}$, the NS compactness ${C}_{\mathrm{NS}}={\sum }_{k=0}^{2}{a}_{k}^{c}{\left(\mathrm{ln}{{\rm{\Lambda }}}_{\mathrm{NS},1.4}\right)}^{k}$ ("C-Love" relation, Yagi & Yunes 2017), and the angle between the BH spin and the binary total angular momentum ι_tilt:

$$\begin{eqnarray}&&\begin{array}{l}{M}_{\mathrm{dyn}}={M}_{\mathrm{NS}}^{{\rm{b}}}\{\max \left[{a}_{1}{q}^{{n}_{1}}\left(1-2{C}_{\mathrm{NS}}\right)/{C}_{\mathrm{NS}}\right.\\ \ -\ \left.{a}_{2}{q}^{{n}_{2}}{\tilde{R}}_{\mathrm{ISCO}}\left({\chi }_{\mathrm{eff}}\right)\left.\,+{a}_{3}\left(1-{M}_{\mathrm{NS}}/{M}_{\mathrm{NS}}^{{\rm{b}}}\right)+{a}_{4},0\right]\right\},\end{array}\end{eqnarray} \tag{ 13 }$$

where χ_eff = χ_BH cosι_tilt is the effective BH spin, with empirical parameters being a₁ = 4.464 × 10⁻², a₂ = 2.269 × 10⁻³, a₃ = 2.431, a₄ = −0.4159, n₁ = 0.2497, and n₂ = 1.352, respectively (Kawaguchi et al. 2016). For simplicity, we adopt cosι_tilt = 1 in this paper. The NS baryonic mass ${M}_{\mathrm{NS}}^{{\rm{b}}}$ is related to its gravitational mass M_NS by ${M}_{\mathrm{NS}}^{{\rm{b}}}={M}_{\mathrm{NS}}\left(1+\tfrac{0.6{C}_{\mathrm{NS}}}{1-0.5{C}_{\mathrm{NS}}}\right)$ (Lattimer & Prakash 2001; see also Gao et al. 2020). As pointed out in Barbieri et al. (2020), the maximal dynamical ejecta mass M_dyn,max cannot exceed 0.5M_out. We thus assume M_dyn,max = 0.3M_out, which is consistent with the result from numerical simulations of NS–BH mergers in the near-equal-mass regime (Foucart et al. 2019). The disk mass M_disk is obtained by combing Equations (11) and (13):

$$\begin{eqnarray}&&{M}_{\mathrm{disc}}={M}_{\mathrm{out}}-{M}_{\mathrm{dyn}}.\end{eqnarray} \tag{ 14 }$$

We consider a relativistic jet launched from the central engine through the BZ mechanism. The kinetic energy of the jet may be calculated by¹⁷

$$\begin{eqnarray}&&{E}_{{\rm{K}},\mathrm{jet}}=\epsilon \left(1-{\xi }_{{\rm{w}}}\right){M}_{\mathrm{disc}}{c}^{2}{{\rm{\Omega }}}_{{\rm{H}}}^{2}f\left({{\rm{\Omega }}}_{{\rm{H}}}\right),\end{eqnarray} \tag{ 15 }$$

where is a dimensionless constant that depends on the ratio of the magnetic energy density to disk pressure at saturation (Hawley et al. 2015), ξ_w is the fraction of energy that goes to the disk wind (rather than the jet), which is related to the kilonova power, ${{\rm{\Omega }}}_{{\rm{H}}}=\tfrac{{\chi }_{\mathrm{BH},{\rm{f}}}}{2(1+\sqrt{1-{\chi }_{\mathrm{BH},{\rm{f}}}^{2}})}$ is the dimensionless angular velocity evaluated at the BH horizon, and $f\left({{\rm{\Omega }}}_{{\rm{H}}}\right)\,=1+1.38{{\rm{\Omega }}}_{{\rm{H}}}^{2}-9.2{{\rm{\Omega }}}_{{\rm{H}}}^{4}$ is a correction factor for high-spin values.

The dimensionless spin of the final BH remnant, χ_BH,f, is related to the initial BH spin χ_BH in the NS–BH binary through (Buonanno et al. 2008; Pannarale 2013)

$$\begin{eqnarray}&&{\chi }_{\mathrm{BH},{\rm{f}}}=\displaystyle \frac{{\chi }_{\mathrm{BH}}{M}_{\mathrm{BH}}^{2}+{l}_{z}\left({\bar{r}}_{\mathrm{ISCO}},{\chi }_{\mathrm{BH},{\rm{f}}}\right){M}_{\mathrm{BH}}{M}_{\mathrm{NS}}}{{M}^{2}},\end{eqnarray} \tag{ 16 }$$

where M = M_BH + M_NS, and

$$\begin{eqnarray}&&\begin{array}{l}{l}_{z}\left({\bar{r}}_{\mathrm{ISCO}},{\chi }_{\mathrm{BH},{\rm{f}}}\right)=\mathrm{sgn}\left({\chi }_{\mathrm{BH},{\rm{f}}}\right)\\ \ \times \ \displaystyle \frac{{\bar{r}}_{\mathrm{ISCO}}^{2}-2\mathrm{sgn}\left({\chi }_{\mathrm{BH},{\rm{f}}}\right){\chi }_{\mathrm{BH},{\rm{f}}}\sqrt{{\bar{r}}_{\mathrm{ISCO}}}+{\chi }_{\mathrm{BH},{\rm{f}}}^{2}}{\sqrt{{\bar{r}}_{\mathrm{ISCO}}}{\left({\bar{r}}_{\mathrm{ISCO}}^{2}-3{\bar{r}}_{\mathrm{ISCO}}+2\mathrm{sgn}\left({\chi }_{\mathrm{BH},{\rm{f}}}\right){\chi }_{\mathrm{BH},{\rm{f}}}\sqrt{{\bar{r}}_{\mathrm{ISCO}}}\right)}^{1/2}}\end{array}\end{eqnarray} \tag{ 17 }$$

is the orbital angular momentum per unit mass of a test particle orbiting the BH remnant at the ISCO; ${\bar{r}}_{\mathrm{ISCO}}$ is similar to ${\tilde{R}}_{\mathrm{ISCO}}$ but replaces χ_BH by χ_BH,f. Equation (16) in geometric units is the same as that in the normalized units. For simplicity, the rotation of the NS, the mass, and the angular momentum of the tidal material, as well as the GW radiation was not taken into account in Equation (16).

In order to connect the BH accretion power with the observed GRB power, we assume a Gaussian-shape structured jet (Beniamini et al. 2019) with an angular distribution of the kinetic energy and Lorentz factor Γ following

$$\begin{eqnarray}&&\displaystyle \frac{{dE}}{d{\rm{\Omega }}}(\theta )={E}_{{\rm{c}}}{e}^{-{\left(\theta /{\theta }_{{\rm{c}},{\rm{j}}}\right)}^{2}},\,\,{\rm{\Gamma }}(\theta )=\left({{\rm{\Gamma }}}_{{\rm{c}}}-1\right){e}^{-{\left(\theta /{\theta }_{{\rm{c}},{\rm{j}}}\right)}^{2}}+1,\end{eqnarray} \tag{ 18 }$$

where ${E}_{{\rm{c}}}={E}_{{\rm{K}},\mathrm{jet}}/\pi {\theta }_{{\rm{c}},{\rm{j}}}^{2}$. Such a structure was long proposed as the typical GRB structured jet (Zhang & Mészáros 2002) and has been successfully applied to model GW170817/GRB 170817A (Lazzati et al. 2018; Lyman et al. 2018; Ghirlanda et al. 2019; Troja et al. 2019; Ryan et al. 2020)

At the viewing angle θ_v, the isotropic gamma-ray radiation energy can be estimated as

$$\begin{eqnarray}&&{E}_{\gamma ,\mathrm{iso}}\left({\theta }_{v}\right)\simeq {\eta }_{\gamma }\int \displaystyle \frac{{D}_{{\rm{p}}}^{3}}{{\rm{\Gamma }}}\displaystyle \frac{{dE}}{d{\rm{\Omega }}}d{\rm{\Omega }},\end{eqnarray} \tag{ 19 }$$

where η_γ the efficiency to convert the EM luminosity to the radiation luminosity in the γ-ray band, D_p = 1/[Γ(1 − βcosα)] is the Doppler factor, and $\cos \alpha =\cos {\theta }_{v}\cos \theta \,+\sin {\theta }_{v}\sin \theta \cos \varphi$.

Combining Equations (11)–(19), we can calculate the isotropic radiation energy of the jet E_γ,iso as a function of some parameters (e.g., q and χ_BH) of the BH and the NS under certain assumptions. Owing to the limited information of this event, we have to assume the event possesses some typical characteristics of short GRBs, e.g.,  = 0.015, ξ_w = 0.01, η_γ = 10%, M_NS = 1.4 M_⊙, and Γ_c = 100 (see Barbieri et al. 2020). Since we do not know the NS equation of state, we take Λ_NS,1.4 = 330 (SFHo EoS) and 700 (DD2 EoS) to cover a range of possible cases. Furthermore, we consider two example cases for a narrow jet core with θ_c,j = 5° and a wide jet core with θ_c,j = 16°. The former value is motivated by GW170817/GRB 170817A (Ghirlanda et al. 2019) while the latter is consistent with the claimed opening angle of some observed short GRBs (Fong et al. 2015). Our constraints on q and χ_BH for different cases are presented in Figures 10 and 11. Despite the flexible allowed range of q and χ_BH, our results suggest that the viewing angle should lie in the most possible range in order to achieve observed E_γ,iso, which is 10°–19° (18°–24°) for the narrow (wide) jet core cases, respectively, as shown in Figure 10 (Figure 11). In addition, different Λ_NS,1.4 values (corresponding to different NS EoSs) can visibly influence the green regions in the q-χ_BH plane achieving the observed E_γ,iso, but it does not significantly change the most possible allowed ranges of the viewing angle.

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The constraints presented in Figures 10 and 11 are by no means definite. This is because from merger parameters to the observed GRB parameters, there are three major steps in modeling, which involve many unknown parameters: First, the uncertainties in the NS–BH merger physics may introduce a large error in the disk mass. Second, jet launching from a BH–disk system in principle involves two possible mechanisms: neutrino-antineutrino annihilation (Popham et al. 1999) or Blandford–Znajek (BZ) mechanism (Blandford & Znajek 1977) and we only considered the latter mechanism. Furthermore, we have adopted a simple analytical formula to denote the BZ power. Third, the radiation efficiency (which depends on the energy dissipation site and mechanism, e.g., photosphere emission, internal shocks, or magnetic dissipation) and geometry (jet structure and viewing angle) are essential to determine how a BZ-powered jet is observed as a short GRB. As explained above, when deriving (q, χ_BH) presented in Figures 10 and 11, we have adopted the typical values for some parameters based on the best known models. Allowing broader distributions of these parameters would further weaken the constraints.

For an NS–BH merger with a relatively large q (e.g., ∼5), the NS would plunge into the BH as a whole. Alternative mechanisms (e.g., McWilliams & Levin 2011; Tsang et al. 2012; D'Orazio et al. 2016; Levin et al. 2018; Dai 2019; Pan & Yang 2019; Zhang 2019a; Zhong et al. 2019) have to be introduced to explain the observed GRB. One group of the mechanisms, which have recently received increasing interest, are the electric and magnetic dipole radiation, magnetic reconnection, and BZ mechanism of the charged objects in the binary system.

The cCBC models involve at least one member of the binary carrying either a constant (Zhang 2016, 2019a) or increasing (Levin et al. 2018; Dai 2019) charge. The high-energy EM emission can be produced either before (Dai 2019; Zhang 2019a) or after (Pan & Yang 2019; Zhong et al. 2019) the merger.

4.2.1. cCBC with a Constant Charge

Here we consider the simplest case in which both objects carry a constant charge, which are denoted as Q₁ and Q₂, respectively. The following derivation applies to both plunging NS–BH and BH–BH scenarios.

Two components can contribute to the EM luminosity of such a constant charge binary. The electric dipole radiation (Deng et al. 2018) component reads (Zhang 2019a)

$$\begin{eqnarray}&&{L}_{{\rm{e}},\mathrm{dip}}=\displaystyle \frac{1}{6}\displaystyle \frac{{c}^{5}}{G}\left({\hat{q}}_{1}^{2}+{\hat{q}}_{2}^{2}\right){\left(\displaystyle \frac{{r}_{s}\left({m}_{1}\right)}{a}\right)}^{2}{\left(\displaystyle \frac{{r}_{s}\left({m}_{2}\right)}{a}\right)}^{2},\end{eqnarray} \tag{ 20 }$$

where a is the semimajor axis with the eccentricity e = 0 assumed, m₁ and m₂ are the masses of compact objects, ${\hat{q}}_{i}\equiv {Q}_{i}/{Q}_{c,i}$ (i = 1, 2) are the dimensionless charges, ${Q}_{c,i}\equiv 2\sqrt{G}{m}_{i}$ are the critical charges (Zhang 2016), and r_s(m_i) are the Schwarzschild radii of the two merging objects.

Following Zhang (2016, 2019a), the magnetic dipole radiation luminosity reads

$$\begin{eqnarray}&&{L}_{{\rm{B}},\mathrm{dip}}=\displaystyle \frac{196}{1875}\displaystyle \frac{{c}^{5}}{G}{\left(\displaystyle \frac{{\hat{q}}_{1}{m}_{1}+{\hat{q}}_{2}{m}_{2}}{M}\right)}^{2}\times {\left(\displaystyle \frac{{r}_{s}\left(\mu \right)}{a}\right)}^{4}{\left(\displaystyle \frac{{r}_{s}(M)}{a}\right)}^{11}.\end{eqnarray} \tag{ 21 }$$

Since at the final moment of the merger, the global open field lines in the binary system cover almost the full sky (Zhang 2016) and since there is no matter outside the BH event horizon to collimate the Poynting flux outflow, the estimated EM luminosity is the isotropic equivalent one:

$$\begin{eqnarray}&&{L}_{\gamma ,\mathrm{iso}}={\eta }_{\gamma }\left({L}_{{\rm{e}},\mathrm{dip}}+{L}_{{\rm{B}},\mathrm{dip}}\right).\end{eqnarray} \tag{ 22 }$$

For the most optimistic cases, we assumed η_γ ∼ 1.

For an NS–BH merger system, at least the NS is charged (Michel 1982; Zhang 2019a). We adopt the following simplest assumptions: (1) only the NS carries a constant charge; (2) the NS mass is 1.4M_⊙; (3) a = a_min = r_s(m_BH) + 2.4r_s(m_NS) (r_NS = 2.4 r_s for neutron star) at the merger time; (4) mass ratio q is $q={2.26}_{-0.12}^{+2.75}$, which is constrained by the GW signal. We can then obtain that ${\hat{q}}_{\mathrm{NS}}$ is ${1.495}_{-0.001}^{+0.210}\times {10}^{-4}$. Consequently, the absolute charge Q_NS is ${2.162}_{-0.002}^{+0.302}\times {10}^{26}$ e.s.u. The dimensionless charge of an NS can be estimated as (Zhang 2019a)

$$\begin{eqnarray}&&{\hat{q}}_{\mathrm{NS}}\simeq \displaystyle \frac{3{\rm{\Omega }}{B}_{p}{R}^{3}}{2c\sqrt{G}M}\cos \alpha =(4.4\times {10}^{-4}){B}_{15}{P}_{-3}^{-1}{R}_{6}^{3}{M}_{1.4}^{-1}\cos \alpha .\end{eqnarray} \tag{ 23 }$$

In order to satisfy the observational constraint, one requires ${B}_{15}/{P}_{-3}\sim {0.340}_{-0.001}^{+0.047}$. This implies that the neutron star has to be a millisecond magnetar before the merger. The condition to form such a magnetar in BNS mergers is contrived, so this scenario is disfavored.

Similarly, for a charged BH–BH merger system we adopt the following two most straightforward assumptions: (1) the lighter BH has a mass of 2.8 M_⊙, which is less than 3 M_⊙ and falls into the BH mass regime; (2) only the lighter BH carries a constant dimensionless charge $\hat{q}$ (for the same absolute charge Q, a lighter BH carries a higher $\hat{q}$ which is more relevant). The mass ratio q of this system should be different from the range constrained above assuming an NS–BH merger, but this ratio does not enter the problem in view of assumption (2) above. We constrain the black hole charge as ${\hat{q}}_{\mathrm{BH}}={7.308}_{-0.147}^{+3.909}\,\times {10}^{-5}$ and the corresponding absolute charge ${Q}_{\mathrm{BH}}={2.114}_{-0.042}^{+1.131}\,\times {10}^{26}$ e.s.u. The demanded dimensionless charge is comparable to the one required to explain the putative γ-ray event (Connaughton et al. 2016) associated with the first BH–BH merger event (Zhang 2016). Contrived conditions are again needed for a BH to carry such a large charge.

4.2.2. cCBC with an Increasing Charge

This scenario involves a plunging BH–NS system in which the BH is immersed in the magnetic field of the NS and gains charge via the Wald mechanism (Wald 1974) in an initial electro-vacuum approximation. Levin et al. (2018) suggested that the BH can be charged stably to carry the Wald's charge quantity Q_W until it could transit from the electro-vacuum state to the force-free state thanks to abundant pair production induced by the strong electric field. At this point, the BH may reach the maximal Wald charge. In this scenario, there are four possible pre-merger mechanisms (first and second magnetic dipole radiation, electric dipole radiation, and magnetic reconnection close to BH's equatorial plane; Dai 2019) and two possible post-merger mechanisms (magnetic reconnection at polar regions and the BZ mechanism; Zhong et al. 2019) to generate γ-ray emission. Following Dai (2019) and Zhong et al. (2019), we calculate (Figure 12) that the subthreshold GRB could be produced by the pre-merger magnetic reconnection or the post-merger BZ mechanism if the NS surface magnetic field satisfies $\mathrm{log}({B}_{{\rm{S}},\mathrm{NS}}/{\rm{G}})\gt 13.5$ or $\mathrm{log}({B}_{{\rm{S}},\mathrm{NS}}/{\rm{G}})\sim 13.5-14.6$, respectively, given the following conditions: the radiative efficiency η_γ = 1, the mass ratio is q = 5, the minimal separation between the BH and the NS is a_min = 2GM_BH/c² + r_NS, the NS mass is M_NS = 1.4 M_⊙ and its radius is r_NS = 12 km. The following two points are worth mentioning in our calculation: (1) We consider that the pre-merger magnetic reconnection in Equation (19) of Dai (2019) should be the BH's magnetic field produced by the Wald charge Q_W rather than that of the NS. This is because the BH's magnetic field should be always lower than that of the NS, as pointed out in Levin et al. (2018). (2) For the post-merger magnetic reconnection and BZ mechanism, the parameters such as the BH's spin and mass and their derived parameters should be relevant to the final BH rather than the pre-merger BH in the binary system. However, they can be linked to those of the pre-merger BH through Equations (16) with M = M_BH + M_NS.

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The delay time between GBM-190816 and the putative GW event is about 1.57 s in the observer frame and is about 1.43 s in the cosmological proper frame. This is similar to the 1.70 s GW–GRB delay observed in GW170817/GRB 170817A (Abbott et al. 2017) which is also 1.68 s in the cosmological proper frame. In the literature, the origin of 1.7 s delay has been extensively discussed. The delay due to the effects of exotic physics is likely small (Wei et al. 2017; Shoemaker & Murase 2018; Burns 2019), and the main contribution is likely due to astrophysical processes (Zhang et al. 2018b; Zhang 2019b).

Following the convention introduced in Zhang (2019b), we discuss the three terms of the astrophysical GW–GRB delay timescale. Since the cCBC scenario is not favored, we limit ourselves to the hyperaccreting NS–BH merger scenario.

(1) Δt_jet: the delay time to launch a clean relativistic jet. In general, such a delay includes three parts for a hyperaccreting BH central engine, namely, the waiting time Δt_wait for a central object (BH) to form, the accretion timescale Δt_acc, and the time Δt_clean for the jet to become clean. In our considered scenario for the event GBM-190816, since at least one BH already exists in the pre-merger system, Δt_wait should be 0. For a black hole engine, Δt_clean ∼ 0 and Δt_acc is typically ∼10 ms. So Δt_jet is ∼0.01 s.

(2) Δt_bo: the delay time for the jet to break out from the surrounding medium. For an NS–BH progenitor, this timescale is typically 10–100 ms.

(3) Δt_GRB: the delay time for the jet to reach the energy dissipation and GRB emission site. Such a delay is directly related to the emission radius, i.e., t_GRB = R/2cΓ², where Γ is the Lorentz factor of the eject and c is the speed of light. In view that the first two terms are negligibly small for NS–BH mergers, the cosmological-proper-frame 1.43 s delay should be mainly defined by this term. The falling timescale of a burst is defined by the angular spreading time, which carries the same expression as t_GRB, one would then expect that the true duration of GBM-190816 would be of the same order of the delay timescale (1.43 s). The observed T₉₀ ∼ 0.1 s is apparently much shorter than this. However, it is possible that the true burst is longer and the observed T₉₀ is simply the tip of the iceberg of the true burst. The fact that the amplitude parameter f is not very large allows such a possibility.

The fact that the 1.43 s-delay in GBM-190816 is similar to the 1.68 s-delay in GW170817/GRB 170817A also sheds light on the origin of the delay in the latter system. Since GW170817/GRB 170817A is an NS-NS merger system, the final merger product is quite uncertain, which depends on the unknown neutron star equation of state (Ai et al. 2020). If the merger product turns into a black hole before the GRB jet is launched, it is possible that there is a significant delay attributable to Δt_jet (e.g., Nakar & Piran 2018). However, this scenario has to introduce chance coincidence to explain the apparent consistency between the delay time and the duration of the burst. Alternatively, if the merger product does not collapse to a black hole before the jet is launched, then there is no immediate reason to suggest the existence of a significant Δt_jet. The fact of a comparable delay time and duration then favors the possibility that Δt_GRB is the dominant contribution to the observed Δt (Zhang et al. 2018b; Zhang 2019b).

Since for NS–BH mergers, the observed time delay should be mostly contributed by Δt_GRB, the fact that the GBM-190816 has a comparable amount of the delay from its GW counterpart suggests that Δt_GRB itself can be this long. This indirectly suggests that the jet in GW170817/GRB 170817A was launched promptly without significant delay (Zhang et al. 2018b; Zhang 2019b). This conclusion is also supported by a recent independent study by Beniamini et al. (2020).