SBO Model

Both NS–NS and NS–BH mergers can launch relativistic jets. As the relativistic jet propagates through the sub-relativistic expanding merger ejecta, a high-pressure bubble would be generated, which engulfs the jet and affects its propagation. This will lead to the formation of a jet-cocoon system, which is a structured relativistic outflow with a wide-angle. A successful SGRB jet is expected to penetrate through this ejecta and produce gamma-rays by the internal dissipation processes within the jet. It has been proposed that when the shock driven by a mildly relativistic cocoon breaks out of the ejecta, gamma-ray emission would also be produced [53,54]. This process differs significantly from what occurs in ordinary SGRBs.

For a low-power and short-duration jet, it may not penetrate through the ejecta, and the jet is choked. Both successful jet and chocked jet can drive an at least mildly relativistic SBO from the expanding merger ejecta [53]. The SBO of the relativistic jet or the mildly relativistic cocoon from the sub-relativistic expanding merger ejecta could release a tiny fraction, e.g., <sup>∼</sup> <sup>10</sup>−<sup>4</sup> , of the total kinetic energy of the outflow into gamma-ray.

Two key physical parameters, the final Lorentz factor of the breakout layer ΓSBO and the radius of the SBO *R*SBO, determine the main properties of the observed emissions. The SBO occurring in a sub-relativistic expanding ejecta is very different from that occurring in a static stellar envelope. The shock velocity in the lab frame would determine the boost of the emission to the observer, and the shock velocity in the ejecta frame would determine the strength of the shock. The SBO from the breakout layer would have an optical depth *τ* ∼ *c*/*v* 0 sh, where *v* 0 sh is the shock velocity seen in the ejecta frame. The shock quantities seen in the unshocked merger ejecta frame are marked with a prime. The mass of the breakout layer can be estimated to be

$$M\_{\rm SBO} \sim \frac{4\pi R\_{\rm SBO}^2}{\kappa \beta\_{\rm SBO}^{\prime}} = 4 \times 10^{-8} M\_{\odot} \beta\_{\rm SBO}^{\prime -1} \left(\frac{R\_{\rm SBO}}{10^{12} \,\rm cm}\right)^2 \left(\frac{\kappa}{0.16 \,\rm cm^2 \,\rm g^{-1}}\right)^{-1},\tag{7}$$

where the opacity *κ* = 0.16 cm<sup>2</sup> g −1 is expected for fully ionized heavy elements, *β* 0 SBO = *v* 0 sh/*c*.

If the shock is fast enough (e.g., *v* 0 sh > 0.5*c*), the radiation temperature behind the shock reaches ∼50 keV at which pair production becomes important [49]. In this case, the mean photon energy is in the *γ*-ray regime. The observed energy from the breakout layer *E*SBO can be roughly estimated by the internal energy of the shocked breakout layer and boosted to the observer frame as [32,50]

$$E\_{\rm SBO} \sim M\_{\rm SBO} \epsilon^2 \Gamma\_{\rm SBO} \left(\Gamma\_{\rm SBO}^{\prime} - 1\right) \simeq 7 \times 10^{46} \,\mathrm{erg} \left(\frac{R\_{\rm SBO}}{10^{12} \,\mathrm{cm}}\right)^2 \frac{\Gamma\_{\rm SBO} \left(\Gamma\_{\rm SBO}^{\prime} - 1\right)}{\beta\_{\rm SBO}^{\prime}}.\tag{8}$$

In the case of a spherical breakout, the difference between the light travel time of photons emitted along the line of sight determines the duration of the breakout signal

$$\tau\_{\rm SBO} \sim \frac{R\_{\rm SBO}}{2c\Gamma\_{\rm SBO}^2} = 0.67 \,\mathrm{s} \left(\frac{R\_{\rm SBO}}{10^{12} \,\mathrm{cm}}\right) \left(\frac{\Gamma\_{\rm SBO}}{5}\right)^{-2}.\tag{9}$$

The bolometric luminosity of an SBO could then be roughly estimated as [69]

$$L\_{\rm SBO} \sim \frac{E\_{\rm SBO}}{t\_{\rm SBO}} = \zeta E\_{\rm K,iso} t\_{\rm SBO}^{-1}. \tag{10}$$

where *E*K,iso is the total kinetic energy of the outflow, and *ζ* is the fraction of the total explosion energy emitted in *γ*-rays.

In the framework of the SBO scenario, three SBO parameters: the breakout radius *R*SBO, the ejecta Lorentz factor *γ*ej,SBO, and the shock Lorentz factor *γ*SBO, are related with three main observables: the total observed isotropic equivalent energy *E*SBO, the duration *τ*SBO, and the breakout temperature *T*SBO. The SBO temperature *T*SBO is roughly the immediate downstream temperature of the breakout layer, as observed in the observer frame. The rest-frame temperature at the time that the photons are released is ∼50 keV, the observed temperature of SBO can be estimated as

$$T\_{\rm SBO} \sim 50 \Gamma\_{\rm SBO} \,\mathrm{keV}.\tag{11}$$

The three breakout observed quantities, *τ*SBO, *E*SBO, and *T*SBO, satisfy a closure relation [32,49]

$$
\pi\_{\rm SBO} \sim 20 \,\mathrm{s} \left( \frac{E\_{\rm SBO}}{10^{46} \,\mathrm{erg}} \right)^{1/2} \left( \frac{T\_{\rm SBO}}{50 \,\mathrm{keV}} \right)^{-\frac{9+\sqrt{3}}{4}} \,\mathrm{}\,\tag{12}
$$

This closure relation can be used to see if the detected *γ*-ray flash is consistent with a relativistic SBO origin. It is worth noting that this relation is strongly dependent on the breakout temperature *T*SBO, which is difficult to determine precisely because the SBO spectrum could deviate from a blackbody spectrum.

There are three generic properties of a relativistic SBO from a moving ejecta: (1) the light curve is smooth; (2) only a tiny fraction of the total energy would be emitted at the SBO stage; (3) the spectrum shows a hard to soft evolution [49,53]. Thus for precursors produced by SBO, the observed energies could be orders of magnitude lower, but depend on the viewing angle of the jet. Note, interestingly, all of these properties are observed in GRB 170817A. Therefore, a mildly relativistic cocoon shock breaking out from the merger ejecta provides a natural explanation of the observational properties of GRB 170817A.
