FRB 171019: an event of binary neutron star merger?

From the observations of many non-repeating FRBs, one can estimate the luminosity

$$\begin{eqnarray}\begin{array}{lll}L & = & 4\pi {D}^{2}\displaystyle \frac{\delta \Omega }{4\pi }{S}_{\nu }\Delta \nu \\ & \simeq & {10}^{43}{D}_{{\rm{Gpc}}}^{2}\left(\displaystyle \frac{\delta \Omega }{4\pi }\right)\left(\displaystyle \frac{{S}_{\nu }}{10{\rm{Jy}}}\right)\Delta {\nu }_{{\rm{GHz}}}{{\rm{erg\,\,s}}}^{-1},\end{array}\end{eqnarray} \tag{ 1 }$$

where D_Gpc is the source distance in Gpc, δΩ is the beam angle, S_ν is the flux density and Δν_GHz is the frequency bandwidth in GHz. Such a bright emission is generally considered to be coherent radiation.

Totani (2013), Wang et al. (2016) and Murase et al. (2017) concluded that FRB emission can occur during the precursor phase of compact star mergers. In this model, we propose that the one-off FRBs are generated by NSNS or SS-SS merger. Our model is based on the unipolar inductor model, which was first proposed for the Jupiter-Io system (Goldreich & Lynden-Bell 1969). The event rate of the merger bursts should be equivalent to the NS-NS merger rate, viz. ∼ 10⁻¹ − 10³Gpc⁻³ yr⁻¹ (Chruslinska et al. 2018; Paul 2018; Pol et al. 2019).

The orbital angular velocity can be calculated as Ω = (GM(1 + q)/a³)^0.5 ∼ 3.7 × 10³ rad s⁻¹, where q = 1 is adopted as the mass ratio of the two NSs, a = 30 km is adopted as the distance between two NSs and M = 1.4M_⊙ adopted as the mass of NS. In the late inspiral of binary coalescence, the dipolar configuration of the magnetic field between two NSs should be severely distorted due to magnetic interaction, while the far field should remain dipolar with a total magnetic moment μ_tot = μ₁ + μ₂. Inside the region between the two NSs, charged particles are able to move across the orbital plane in the distorted magnetic field, therefore the magnetosphere around the orbital plane can be treated as a conducting plate. Under the assumption of magnetic moment conservation and a parallel configuration, the surface magnetic field can be estimated as

$$\begin{eqnarray}\begin{array}{lll}B & \sim & \displaystyle \frac{{\mu }_{{\rm{tot}}}}{\pi {a}^{2}{R}_{* }}\sim \displaystyle \frac{8}{3}{\left(\displaystyle \frac{{R}_{* }}{a}\right)}^{2}{B}_{* }\\ & = & 3.0\times {10}^{11}{B}_{*,12}{R}_{*,6}^{2}{a}_{6.5}^{-2}{\rm{G}},\end{array}\end{eqnarray} \tag{ 2 }$$

where a ∼ 30 km, R_* ∼ 10 km is the typical pulsar radius, B_* ∼ 10¹² G is the typical surface magnetic field on pulsars, and the convention Q_n = Q/10ⁿ in cgs units is adopted.

The orbital evolution of the binary is dominated by gravitational-wave radiation. Therefore, one can estimate the distance evolution as (Peters 1964)

$$\begin{eqnarray}\begin{array}{lll}a & = & {a}_{0}{\left[1-\displaystyle \frac{256{G}^{3}{M}^{3}q(1+q)}{5{a}_{0}^{4}{c}^{5}}t\right]}^{\displaystyle \frac{1}{4}}\\ & = & 20{(1-1695t)}^{\displaystyle \frac{1}{4}}{\rm{km}}.\end{array}\end{eqnarray} \tag{ 3 }$$

We set a₀ = 20 km for the case when the surfaces of the two NSs touch with each other. One can then estimate the timescale of ∼ 2 ms for the process (from a = 30 km to a₀ = 20 km for q = 1), which is consistent with the FRB duration.

On the conducting plate, the open field line region can be estimated by the proportion of its magnetic flux against the total magnetic flux,

$$\begin{eqnarray}\begin{array}{lll}{r}_{cap} & = & (a+{R}_{* })\sqrt{\displaystyle \frac{{\Phi }_{{\rm{open}}}}{{\Phi }_{{\rm{tot}}}}}\\ & = & (a+{R}_{* }){\left(\displaystyle \frac{\displaystyle {\int }_{{R}_{{\rm{lc}}}}^{+\infty }\displaystyle \frac{r}{{r}^{3}}{\rm{d}}r}{\displaystyle {\int }_{a+{R}_{* }}^{+\infty }\displaystyle \frac{r}{{r}^{3}}{\rm{d}}r}\right)}^{\displaystyle \frac{1}{2}}\\ & = & {(a+{R}_{* })}^{\displaystyle \frac{3}{2}}{R}_{{\rm{lc}}}^{-\displaystyle \frac{1}{2}},\end{array}\end{eqnarray} \tag{ 4 }$$

where R_lc = c/Ω is the radius of light cylinder.

A potential drop produced in the unipolar model is

$$\begin{eqnarray}&&U\simeq \displaystyle \frac{B\Omega {r}_{{\rm{cap}}}^{2}}{2c}.\end{eqnarray} \tag{ 5 }$$

This potential can trigger pair-production avalanches that creates charged bunches. The bunches stream outward along open magnetic field lines, generating coherent radio emissions. The charge density in the magnetosphere of a pulsar is given by (Goldreich & Julian 1969)

$$\begin{eqnarray}\begin{array}{lll}{\rho }_{e} & \simeq & {\rho }_{{\rm{GJ}}}\simeq \displaystyle \frac{\Omega B}{2\pi c}\\ & = & 5.8\times {10}^{3}{M}_{33.44}^{\displaystyle \frac{1}{2}}{R}_{*,6}^{2}{B}_{*,12}{a}_{6.5}^{-\displaystyle \frac{7}{2}}{\left(\displaystyle \frac{1+q}{2}\right)}^{\displaystyle \frac{1}{2}}{\rm{esu}},\end{array}\end{eqnarray} \tag{ 6 }$$

where ρ_GJ is the Goldreich–Julian density, B is the magnetic field. The energy releasing rate of the magnetosphere during merger is given by

$$\begin{eqnarray}\begin{array}{lll}\dot{E} & \simeq & U\pi {r}_{{\rm{cap}}}^{2}{\rho }_{e}c\\ & \simeq & \displaystyle \frac{16}{9{c}^{3}}{B}_{* }^{2}{G}^{2}{M}^{2}{(1+q)}^{2}{\left(\displaystyle \frac{{R}_{* }}{a}\right)}^{4}{\left(1+\displaystyle \frac{{R}_{* }}{a}\right)}^{6}\\ & = & 6.3\times {10}^{44}{B}_{*,12}^{2}{M}_{33.44}^{2}\\ & & \times {\left(\displaystyle \frac{1+q}{2}\right)}^{2}{\left(\displaystyle \frac{{R}_{*,6}}{{a}_{6.5}}\right)}^{4}{\left(1+\displaystyle \frac{{R}_{*,6}}{{a}_{6.5}}\right)}^{6}{{\rm{ergs}}}^{-1}.\end{array}\end{eqnarray} \tag{ 7 }$$

If radio efficiency is ∼ 10⁻³ and the beaming factor is 10², then the isotropic luminosity matches the typical luminosity of non-repeating FRBs, L_iso ∼ 10⁻³ × 10² × Ė ∼ 10⁴³ erg s⁻¹.

This calculation assumed the magnetic moments of the two merger stars are both parallel to the normal direction of the orbital plane (usually denoted as the "U/U case"), which should result in the largest Ė. The above result is consistent with the Ė of the U/U case in Palenzuela et al. 2013; Ponce et al. 2014; Wang et al. 2018a. For oblique configurations, the total magnetic moment |μ_tot| = |μ₁ + μ₂| < |μ₁| + |μ₂|. The antiparallel configuration results in the smallest |μ_tot| = |μ₁ − μ₂|, which can be smaller than μ₁ and μ₂ by several orders of magnitudes.

The compact star mergers may also produce other observable effects. The detection of GRB 170817A corresponding to GW170817 (Abbott et al. 2017) indicates that short γ-ray bursts can be generated by compact star mergers. Therefore, our model infers that a short GRB may be generated as the counterpart of a one-off FRB, which is consistent with the γ-ray counterpart to FRB 131104 (DeLaunay et al. 2016).

We predict a new-born pulsar-like compact star left after a fraction of NS-NS or SS-SS merger. We suggest that the remnant object should live for at least several years to generate the subsequent repeating FRBs. NS-NS or SS-SS mergers do not always deliver long-lived compact stars. If the remnant mass is larger than the maximum mass for NS or SS, the merger product should be a blackhole. With uniform rotation, the maximum mass of NS or SS can be larger than the Tolman-Oppenheimer-Volkoff (TOV) maximum mass, hence supermassive. A supermassive remnant NS or SS, collapsing years after merger due to the braking of magnetic dipolar radiation or gravitational wave radiation, may also fit in our model. In addition, the geometric factor of the anisotropy further reduce the event rate of such sources. When taking the theoretical approach, the volumetric density of repeating FRB sources is affected by the star formation history, the mass function of NS binaries, the merger dynamics and maximum mass of NS, some of which are still highly uncertain in theories. On the other hand, taking the observational approach, James (2019) limits the volumetric density of repeating FRBs to be < 70 Gpc⁻³. Adopting the burst rate of repeating FRB 121102, ${5.7}_{-2.0}^{+3.0}{{\rm{day}}}^{{\rm{-1}}}$ (Oppermann et al. 2018), as the burst rate for all repeating FRBs, the observed repeating bursts from such NS-NS remnants should ≲ 10⁵ Gpc⁻³yr⁻¹ for themore sensitive telescopes like GBT and CHIME. In this scenario, it is rather difficult for ASKAP-like small dishes to detect repeating events, which is consistent with the prediction by Yamasaki et al. (2018).

When the initial proto-compact star is formed, gravitational energy would be stored in the compact star as a form of initial thermal energy. Consequently, the temperature of the inner core will be ∼ (30 − 50) MeV. In the first following stage, neutrino emissions make the star becoming cooling very rapidly. However, different equation of states can lead to different cooling behavior.

On the one hand, in the regime of normal neutron star, for a proto-neutron star, it shrinks into ∼ 10 km within several tens of seconds because of the powerful neutrinoinduced cooling down (Pons et al. 1999). The crust has a relatively lower neutrino emissivity than the core. This makes the crust cool more slowly than the core and the surface temperature decreases slowly during the first ten to hundred years (Chamel & Haensel 2008). After that, when the cooling wave from the core reaches the surface, the surface temperature will drop sharply.

On the other hand, the cooling process for a new-born SS consists of three stages (Dai et al. 2011). The first stage is a rapidly cooling process caused by the neutrino and photons emitting at the very beginning. This process spends several tens seconds but in principle faster than the first stage of an NS cooling (Yuan et al. 2017). The SS enters then the second stage, at which the surface temperature remains constant roughly and the liquid SS begins to be solidified, when the temperature drops to the melting point temperature T_p ∼ 0.1 MeV. The time scale of the liquid-solid phase transition can be estimated as

$$\begin{eqnarray}&&{t}_{{\rm{solid}}}=\displaystyle \frac{{E}_{{\rm{in}}}}{4\pi {R}^{2}\sigma {T}_{p}^{4}}=7.8\times {10}^{6}{E}_{{\rm{in}},52}{R}_{6}^{-4}{\rm{s}},\end{eqnarray} \tag{ 8 }$$

where E_in is energy release during the phase transition, σ is the Stefan-Boltzman constant and R is the stellar radius. After the solidification, the newly-born SS will rapidly release its residual inner energy because of its low thermal capacity (Yu & Xu 2011).

Basically, whether it is an NS or an SS, the magnetic field lines are anchored to the stellar surface and their geometry is determined by the motion of the footpoints. For a new-born compact star, there are several kinds of instability which may be driven by gravitational, magnetic or rotational energy. If the instability can grow very fast in the stellar crust, making the pressure exceed a threshold stress, crust quake would happen. Seismic waves, created by the sudden release of energy, diffuse in the crust, which can help some local instabilities growing. The characteristics of self-organized criticality observed in earthquakes, are very expected to be seen in some compact star activities (e.g., Cheng et al. 1996; Duncan 1998; GöǧüŞ et al. 1999). The compact star, which is suggested to be a dead pulsar (i.e., beyond the pulsar death line), can be then excited due to the solid crust quake. A similar but different story of strangeon star was presented in Lin et al. (2015).

In the regime of normal NS, crust shear can trigger the footpoint motion, therefore the magnetic curl or twist are ejected into the magnetosphere from the crust in τ ≃ R/v_A ∼ 1 ms (Thompson et al. 2002), where v_A is the Alvén speed. During this process, charged particles in the magnetosphere, are suddenly accelerated to be ultra-relativistic by the quake-induced magnetic reconnection, and form charged bunches. In general, the cooling timescale for curvature radiation in the observer's rest frame is much smaller than FRB's typical duration. Thus, it requires a strong electric field parallel E_∂ to the B-field that can accelerate electrons, supplying the kinetic energy to balance radiation loss, which is given by (Kumar et al. 2017)

$$\begin{eqnarray}&&{E}_{\parallel }=\displaystyle \frac{{\gamma }_{e}mc}{e{t}_{{\rm{cool}}}}=3\times {10}^{8}{\nu }_{9}{N}_{e,24}{\gamma }_{2}^{-2}{\rm{esu}},\end{eqnarray} \tag{ 9 }$$

where γ_e is the Lorentz factor of electrons, ν is the emission frequency of curvature radiation, and N_e is the electron number. The E_∂ to the B-field formed by the magnetic reconnection is given by

$$\begin{eqnarray}&&{E}_{\parallel }\simeq \displaystyle \frac{2\pi {\sigma }_{s}{v}_{{\rm{A}}}B}{c}=2.1\times {10}^{9}{\sigma }_{s,-3}{v}_{{\rm{A}},8}{B}_{14}{\rm{esu}},\end{eqnarray} \tag{ 10 }$$

where σ_s = ξ/λ is the strain, in which ξ is the amplitude of oscillations and λ is the characteristic wavelength of oscillations.

Let us consider the pair production forming charged bunches which emit coherent curvature radiation. Basically, several authors have discussed curvature radiation from charged bunches from pulsar magnetospheres to explain FRBs (e.g., Katz 2014; Kumar et al. 2017; Yang & Zhang 2018). If the charge density prompted by the stellar rotation ρ_GJ and local twist is insufficient to screen E_∥, sparks would occurs at the polar gap regions which creates emitting charged bunches (Wadiasingh & Timokhin 2019). The number density due to the footpoint motion can be estimated as

$$\begin{eqnarray}&&{n}_{e}\simeq \displaystyle \frac{{E}_{\parallel }}{4\pi e\lambda }=3.5\times {10}^{12}{\sigma }_{s,-3}{\Omega }_{{\rm{osc}},3}{B}_{14}{{\rm{cm}}}^{-3},\end{eqnarray} \tag{ 11 }$$

where Ω_osc is the oscillation frequency. Here we assume that the seismic wave is dominated by the base mode which Ω_osc ∼ (π/R)(μ_s/ρ_n)^0.5 ∼ 10³ Hz, where μ_s is the shear modulus and ρ_n the neutron drip density. To generate coherent emissions, charged particles in the bunch would emit with approximately the same phase. Therefore, a comoving volume of the bunch can be estimated as η(γ_e c/ν)³. Only fluctuating electron can contribute to the coherent radiation, the number of which is given by

$$\begin{eqnarray}\begin{array}{lll}{N}_{e} & = & \mu {n}_{e}{\left(\displaystyle \frac{{\gamma }_{e}c}{\nu }\right)}^{3}\\ & = & 9.4\times {10}^{23}\mu {\eta }_{1}{\sigma }_{s,-3}{\Omega }_{{\rm{osc}},3}{B}_{14}{\gamma }_{e,2}^{3}{\nu }_{9}^{-3},\end{array}\end{eqnarray} \tag{ 12 }$$

where μ is the fraction of electrons fluctuation and η is the multiplication factors due to the frame transform (Kumar et al. 2017). The total luminosity of the curvature radiation from charged bunches can be described as ${L}_{{\rm{iso}}}={N}_{{\rm{pat}}}({N}_{e}^{2}\delta {L}_{{\rm{iso}}})$, where $\delta {L}_{{\rm{iso}}}=2{e}^{2}{\gamma }_{e}^{8}c/(3{\rho }^{2})$ is the isotropic luminosity in the observer frame, and N_pat is the patch number. Thus, we have

$$\begin{eqnarray}\begin{array}{lll}{L}_{{\rm{iso}}} & = & 8.3\times {10}^{39}{N}_{{\rm{pat}}}{\mu }_{-1}^{2}{\eta }_{1}^{2}{\sigma }_{s,-3}^{2}{\Omega }_{{\rm{osc}},3}^{2}\\ & & \times {B}_{14}^{2}{\gamma }_{e,2}^{8}{\nu }_{9}^{-4}{{\rm{ergs}}}^{-1}.\end{array}\end{eqnarray} \tag{ 13 }$$

The calculated luminosity is consistent with observations of most repeaters.

The energy release rate during the magnetic reconnection and oscillation-driven activity can both be estimated as

$$\begin{eqnarray}\begin{array}{lll}{\dot{E}}_{{\rm{R}}} & \simeq & 4\pi {R}_{p}^{2}\delta R\displaystyle \frac{{\sigma }_{s}{B}^{2}}{8\pi \tau }\\ & = & 3.3\times {10}^{42}{P}_{-3}^{-1}\delta {R}_{4}{\sigma }_{s,-3}{B}_{14}^{2}{\tau }_{-3}^{-1}{{\rm{ergs}}}^{-1},\end{array}\end{eqnarray} \tag{ 14 }$$

where R_p is the polar cap radius of the remnant star and δR is the height of the crust. The energy release rate for the following starquakes is much smaller than that of the merger stage. Therefore, one expects that there is no any burst detected in the follow-up observation to be as bright as the one triggered by NS-NS merger.