Funnel Leakage of the Wien Fireball in the Early Nebula after a Binary Neutron Star Merger

The gist: The funnel leakage of the Wien fireball in the early nebula after a binary neutron star merger suggests that quasi-thermal components observed in extended emission of short gamma-ray bursts could originate from this leakage, which is comparable to or exceeds adiabatic loss at early times

Early Nebula Fireball Physics

A newborn neutron star (NS) surviving a binary NS merger can inject rotational energy into the surrounding ejecta while the ejecta are still optically thick The injected energy is redistributed by the pair cascade, and the nebula becomes a radiation-dominated plasma composed of photons and e± pairs Compton scattering then governs the energy exchange while the photon number is conserved in scattering For typical parameters, we find that the Compton parameter y is much larger than unity at early times The nebula therefore remains in saturated Comptonization for 100 s to a few thousand seconds, depending on the magnetic field strength of the post-merger NS The radiation field takes a Wien spectrum with a Compton equilibrium temperature This nebula is then a so-called Wien fireball The Compton temperature decreases from tens of keV to a few keV at early times As long as the ejecta are optically thick, the funnel leakage is comparable to or exceeds the adiabatic loss The funnel emission is quasithermal, and its isotropic-equivalent luminosity is much higher than the spin-down luminosity

Energy Injection and Pair Cascade

The NS wind initially consists of a Poynting flux and e± pairs, which can originate from the Goldreich–Julian current The pairs injected by the wind are relativistic They cool through synchrotron radiation and inverse Compton (IC) scattering and lose their energy to high-energy photons These photons easily exceed the threshold energy for pair creation They then collide with the background photons of the nebula, creating new e± pairs This is a repeating cycle known as the pair cascade The cascade process is characterized by the compactness parameter, l = E σT R / mec2 At early times l >> 1, so the cascade is saturated The pair-creation rate is N˙ +± = Y Lsd / mec2 Pair creation is accompanied by annihilation at the e+e− collision rate, N˙ −± = 3/16 σT c N2± V-1 Equating the creation and annihilation rates gives the equilibrium pair number, N± = r(16) Y LsdV / (3 σT mec3) This large pair population raises the scattering optical depth of the nebula, which is estimated as τn = 4Y LsdσT / (πR mec3)(1/2) The large optical depth, τn >> 1, shows that the early nebula is optically thick and radiation-trapped, consistent with B. D. Metzger & A. L. Piro (2014) Besides creating copious e± pairs, the cascade converts the spin-down power into a power-law injection spectrum, Jν ∝ ν-α with α ≃ 1 The thermal photons of the nebula, Eq. 2, serve as the low-energy bath of the cascade

Fireball Evolution and Temperature Regulation

The competition between adiabatic expansion and injection from the NS maintains the fireball temperature at the keV level At early times, both the ejecta and the nebula fireball are extremely optically thick, so energy injected by the NS wind is almost fully deposited and thermalized The timescale for adiabatic loss is tP dV = E/LP dV = Re/v, which is roughly the dynamical timescale The ratio to the diffusion luminosity at early times is LP dV / Le ≃ (τn + τe) v / c, which is much larger than unity at early times Adiabatic work therefore dominates the energy loss The fireball temperature remains at the keV level until the optical depth declines and diffusion takes over the energy loss at td ∼ R/v The total energy evolves as dE/dt = Lsd − P dV/dt − Le

Funnel Leakage and Observable Signatures

The funnel leakage consists of two parts: one is the diffusion and escape of photons through the nebula along the funnel, and the other is the advective outflow of e±–photon plasma along the funnel The luminosity of funnel diffusion is Lfd = fb E / td,n Along the funnel direction, photons diffuse only through the nebula, so this diffusion-loss timescale is td,n/fb Because only the solid-angle fraction fb is open, the corresponding diffusion-loss timescale is td n/fb Compared with funnel diffusion, funnel outflow plays a more significant role in the energy loss along the funnel direction At early times, τn >> 1, so the funnel outflow therefore dominates the leakage The total funnel leakage is Lfun = Lfd + Lfo In this construction, adiabatic work and diffusion Le act only on the ejecta direction Since Lfo >> Lfd at early times, the leakage is dominated by the funnel outflow An observer near the funnel direction would then receive radiation from the fireball interior

Conclusion

The early nebula at t ∼ 100 s is a trapped e±–photon plasma injected by the NS wind It is radiation-dominated, extremely optically thick (τn >> 1), and in saturated Comptonization (y >> 1) The funnel emission reflects the state of the nebula fireball, in particular the Compton temperature Quasi-thermal components with kT ∼ 10–100 keV and durations of tens to hundreds of seconds are observed in the early-phase emission of a fraction of short GRBs We suggest that these quasi-thermal components observed in the extended emission of short GRBs could come directly from the funnel emission of the Wien fireball The nebula fireball can survive for more than 100 s depending on the magnetic field strength of the NS, and then the nebula converts to the normal nebula phase at y = 1 The funnel ultimately changes where the fireball energy goes, and an observer within the beam infers an isotropic-equivalent luminosity much larger than the spin-down luminosity of the NS

How it works

1. A long-lived massive NS can survive after a BNS (or NS–WD) merger if the total mass of the remnant is below the maximum NS mass MTOV Such a surviving post-merger NS initially rotates near the Keplerian period With a high field strength, magnetic dipole spin-down releases rotational energy at a luminosity Lsd = L0 / (1 + tsd-2) The injected energy is almost fully deposited as nonthermal or thermal radiation energy inside the nebula when the ejecta remain extremely optically thick

2. The pair cascade converts the spin-down power into a power-law injection spectrum, Jν ∝ ν-α with α ≃ 1 The injection spectrum extends from the thermal bath to the pair-creation threshold, E˙sd,ν ≃ Lsd / Λ (3kTbb) The mean injected photon energy, ⟨hν⟩ ≈ Λ (3kTbb), lies well below the pair-creation threshold 2mec2

3. The spectrum of the nebula at these early times is characterized by the Compton y parameter For y >> 1 photons escape with negligible energy loss and the spectrum is nonthermal, whereas for y >> 1 they lose most of their energy before escaping, so the Comptonization saturates The fully thermalized early nebula develops a Wien spectrum with µγ < 0 rather than a blackbody spectrum with µγ = 0

4. The funnel leakage dominates the early energy loss because Lfo >> Lfd at early times, and the leakage is dominated by the funnel outflow The escaping photons carry away on average an energy ∼ E/(Nγ + N±) ≈ 3kTC For y >> 1, the escaping photons carry a quasi-thermal spectrum; for y << 1 photons escape before exchanging energy and the spectrum keeps the nonthermal injection shape

5. The funnel shifts the time at which y reaches unity to the right by at most about 30% compared to the no-funnel case of Fig. 5 This is because a larger fb allows more energy to leak through the funnel The parameter y remains above unity at early times for the three magnetic fields and falls below unity on different timescales

1. textbfTheoretically Informed Spectral Modeling of Extended Emission: The improved system can model extended emission spectra by distinguishing between a Wien spectrum and a blackbody spectrum, as noted in Section 5: The nebula fireball is in Compton equilibrium, so the thermal component should be fitted by a Wien spectrum rather than a blackbody. This allows for more accurate characterization of the plasma state, specifically identifying the signature of "µγ < 0 which is self-consistent with the Wien limit."

2. textbfReal-time Funnel Leakage Quantification: The system can predict the evolution of observable luminosity by solving the modified energy equation: dEdt = Lsd − (1 − fb)PdV/dt + Le − Lfun, and specifically calculating the leakage term Lfun using the relationship that dominates at early times, where Lfo ≃ ζ c td,n R ≃ ζ τn ≫ 1.

3. textbfParameter Sensitivity Mapping for Observational Constraints: The system can map how observational features correlate with physical parameters by utilizing the derived relationships, such as how The funnel shifts the time at which y reaches unity to the right by at most about 30% compared to the no-funnel case of Fig. 5. This allows researchers to constrain NS magnetic field strengths or ejecta masses based on observed spectral evolution.

A newborn neutron star (NS) that survives after a binary NS merger can inject its rotational energy into the surrounding ejecta while the ejecta are still optically thick. The injected energy is redistributed by the pair cascade, and the nebula becomes a radiation-dominated plasma composed of photons and e+- pairs. Compton scattering then governs the energy exchange while the photon number is conserved in scattering. This nebula is evolved with a one-zone model that includes pair creation, annihilation, photon injection, diffusion, and in particular the leakage through the jet-induced polar funnel. For typical parameters, we find that the Compton parameter y is much larger than unity at early times. The nebula therefore remains in saturated Comptonization for 100 s to a few thousand seconds, depending on the magnetic field strength of the post-merger NS. The radiation field takes a Wien spectrum with a Compton equilibrium temperature. The nebula is then a so-called Wien fireball. The Compton temperature decreases from tens of keV to a few keV at early times. As long as the ejecta are optically thick, the funnel leakage is comparable to or exceeds the adiabatic loss. The funnel emission is quasi-thermal, and its isotropic-equivalent luminosity is much higher than the spin-down luminosity. We suggest that the quasi-thermal components observed in the extended emission of short gamma-ray bursts could come from the funnel leakage of the nebula fireball.