Trapped fireshell (halo) of photons and pairs around black-hole horizon: source for ultra-high-energy particles

TL;DR

This work proposes a mechanism for producing ultra-high-energy charged particles in a gravitationally bound halo (trapped fireshell) of photons and $e^-e^+$ pairs surrounding a black-hole horizon, as formed in GRB central engines. It analyzes the Compton-rocket acceleration in both optically thin and opaque regimes, showing that Klein–Nishina corrections can trigger avalanche runaway in the opaque fluid, yielding nontrivial fractions of UHE electrons and protons and subsequent very-high-energy photons and neutrinos. The authors compute time-dependent UHE luminosities and halo cooling via UHE emissions and blackbody radiation, predicting two-phase evolution and signatures potentially observable in GRB afterglows or testable in high-intensity laser experiments. The work highlights observational relevance and calls for numerical simulations to quantify UHE/VHE outputs and their signals, offering a framework to connect horizon-halo dynamics with high-energy astrophysical phenomena and early-Universe scenarios.

Abstract

We study the Compton-rocket effect of multi-photon interacting with electrons in an opaque fireball (or fire spot) of photons and $e^-e^+$ pairs at temperature $T_γ\gg m_e$. We find the charged-particle acceleration and the avalanche runaway process, leading to a non-trivial probability of ultra-high-energy (UHE) electrons and protons, which subsequently produce very-high-energy (VHE) photons and neutrinos. We show such peculiar dynamics using the Gamma-Ray Burst central engine fireball, whose inner part inflows and forms a gravitationally trapped fireshell (halo) around a black hole. The halo is a metastable, cooling via UHE particle emissions and blackbody radiation. We calculate the UHE particle luminosity varying in time, and discuss the peculiar features of such produced UHE particles, which lead to VHE particles, in connection with possible numerical simulations, observations and experiments.

Figures (7)

• Figure 1: Figure 2 of Ref. Ruffini2003 shows the world lines of fireball hydrodynamics expanding outflow (dot line) and trapped inflow (solid line) in the Schwarzschild geometry of a collapsing core (dashed line) of the mass $M$, and horizon radius $r_+=2M$. The separatrix radius between out and in flows is $\bar{R} = 4M$, and the trapped inflow energy is about half of the expanding outflow energy. We can approximately obtain the separatrix by estimating the balance between thermal repulsive pressure and gravitational attractive force $4\pi r_+^2\rho_\gamma(r_+)c \sim G m_\gamma M/r_+^2$ on the photon-pair shell of mass energy $m_\gamma=\rho_\gamma(r_+)4\pi r_+^2\bar{R}$. It is in distinct contrast with the processes of pressure-less baryon dust or a test particle free-falling in a gravitational collapse. The gravitational time dilation $\delta t=\left(1-{\frac{2Gm(r)}{r}}\right)^{-1/2}\delta\tau\rightarrow \infty$ for an observer at infinity is evident near the horizon $r\rightarrow r_+$, and $\delta\tau$ is a finite proper time (distance) for a local observer.
• Figure 2: Left: the trapped fireshell (halo) energy density $\rho_\gamma$ is expressed in units of $m_e^4$ and as functions of radius $r$, which is in the unit of the horizon radius $r_+=2GM=2.33\times 10^6$ cm for the black hole mass $M=7.75 M_\odot$ (\ref{['bhcon']}). Right: the trapped fireshell (halo) temperature $T_\gamma$ is expressed in units of the electron mass $m_e\approx 0.5 {\rm MeV}$. The maximal temperature $T^+_\gamma$ near the horizon rapidly decreases to the asymptotic value $\approx 2m_e$ near the boundary $2r_+=4GM$, where the electron-positron pair annihilation takes place. The Compton energy density $m_e^4\approx 4.54\times 10^{21}{\rm erg/cm}^3$ is of the same order as the critical energy density $\rho_c=E_c^2/(8\pi) =5.45 m_e^4$, where $E_c=m_e^2/e$ is the critical electric field for electron and positron pair production in the semi-classical approximation Ruffini2010. We use the $\hbar=c=1$ and Compton unit of the electron mass $m_e$, unless otherwise specified.
• Figure 3: The photon and pair number density $n_\gamma$ (left) and mean-free path $\xi_\gamma=(\sigma_\gamma n_\gamma)^{-1}$ (right) are plotted as functions of radius $r$ in the local observer frame. The Compton number density $m_e^3= 1.74\times 10^{35}/{\rm cm}^3$ and length $\lambda_e=m_e^{-1}=3.86\times 10^{-11}{\rm cm}$.
• Figure 4: We calculate the fraction $N_e/\bar{N}_e$ of accelerated electrons drifting out of the bulk electrons for the Thomson (\ref{['prob2i0']}) and the Klein-Nishina (\ref{['prob2i']}) scattering cases. It shows that the probability of accelerated electrons reaching high energies is non-trivial, implying the runaway behaviour, in the Klein-Nishina scattering case, but is completely negligible in the Thomson scattering case. These figures also present the proton case with the substitution of the subscript $e \rightarrow p$.
• Figure 5: We show the fraction $N_e/\bar{N}_e$ (\ref{['prob2i']}) of accelerated electrons drifting out of bulk electrons at different temperatures $T^+_\gamma\gg m_e$ near the horizon. The $N_e/\bar{N}_e$ value is exponentially sensitive to the horizon temperature $T_\gamma^+$. The probability of accelerated electrons reaching high energies is small for temperatures $T^+_\gamma \lesssim 10 m_e$, and the Compton-rocket effect and runaway dynamics might not be observationally relevant. These figures also present the proton case by substituting the subscript $e\rightarrow p$.