Astron. Astrophys. Suppl. Ser. 138, 511-512

On evolution of the pair-electromagnetic pulse of a charged black hole

R. Ruffini¹ - J.D. Salmonson² - J.R. Wilson² - S.-S. Xue¹

1 - I.C.R.A.-International Center for Relativistic Astrophysics and Physics Department, University of Rome "La Sapienza",
I-00185 Rome, Italy
e-mail: ruffini@icra.it
2 - Lawrence Livemore National Laboratory, University of California, Livermore, California, U.S.A.

Received January 21; accepted March 10, 1999

Abstract:

Using hydrodynamic computer codes, we study the possible patterns of relativistic expansion of an enormous pair-electromagnetic-pulse (P.E.M. pulse); a hot, high density plasma composed of photons, electron-positron pairs and baryons deposited near a charged black hole (EMBH). On the bases of baryon-loading and energy conservation, we study the bulk Lorentz factor of expansion of the P.E.M. pulse by both numerical and analytical methods.

Key words: black hole physics -- gamma-ray bursts, theory, observations

In the paper by Preparata et al. (1998), the "dyadosphere" is defined as the region outside the horizon of a EMBH where the electric field exceeds the critical value for ${\rm e}^+{\rm e}^-$pair production. In Reissner-Nordstrom EMBHs, the horizon radius is expressed as  

$$r_{+}={GM\over c^2}\left[1+\sqrt{1-{Q^2\over GM^2}}\right].$$ (1)

The outer limit of the dyadosphere is defined as the radius $r_{\rm ds}$ at which the electric field of the EMBH equals this critical field  

$$r_{\rm ds}= \sqrt{{\hbar e Q \over m^2 c^3}}.$$ (2)

The total energy of pairs, converted from the static electric energy, deposited within a dyadosphere is then  

$$E^{\rm tot}_{{\rm e}^+{\rm e}^-}={1\over2}{Q^2\over r_+} \le... ...ight) \left( 1- \left( {r_+\over r_{\rm ds}}\right)^2 \right) .$$ (3)

In Wilson (1975, 1977) a black hole charge of the order $10\%$ was formed. Thus, we henceforth assume a black hole charge $Q = 0.1 Q_{\max},Q_{\max}=\sqrt{G}M$ for our detailed numerical calculations. The range of energy is of interest as a possible gamma-ray burst source.

In order to model the radially resolved evolution of the energy deposited within the ${\rm e}^+{\rm e}^-$-pair and photon plasma fluid created in the dyadosphere of EMBH, we need to discuss the relativistic hydrodynamic equations describing such evolution.

The metric for a Reissner-Nordstrom black hole is  

$$d^2s=-g_{tt}(r)d^2t+g_{rr}(r)d^2r+r^2d^2\theta +r^2\sin^2\theta d^2\phi ~,$$ (4)

where $g_{tt}(r)=-g^{-1}_{rr}(r)= - \left[1-{2GM\over c^2r}+{Q^2G\over c^4r^2}\right]$.
We assume the plasma fluid of ${\rm e}^+{\rm e}^-$-pairs, photons and baryons to be a simple perfect fluid in the curved spacetime (Eq. (4)). The stress-energy tensor describing such a fluid is given by (Misner et al. 1975)  

$$T^{\mu\nu}=pg^{\mu\nu}+(p+\rho)U^\mu U^\nu$$ (5)

where $\rho$ and p are respectively the total proper energy density and pressure in the comoving frame. The $U^\mu$ is the four-velocity of the plasma fluid. The baryon-number and energy-momentum conservation laws are

$$\begin{eqnarray} (n_{\rm B} U^\mu)_{;\mu}&=&(n_{\rm B}U^t)_{,t}+{1\over r^2}(r^2n_{\rm B}U^r)_{,r}= 0, \\ (T_\mu^\sigma)_{;\sigma}&=&0,\end{eqnarray}$$ (6) (7)

where $n_{\rm B}$ is the baryon-number density. The radial component of Eq. (7) reduces to

$$\begin{eqnarray} &&{\partial p\over\partial r}+{\partial \over\partial t}\left((... ...l r}(U^t)^2+{\partial g_{rr} \over\partial r}(U^r)^2\right] =0 ~.\end{eqnarray}$$ (8)

The component of the energy-momentum conservation Eq. (7) along a flow line is

$$\begin{eqnarray} &&U_\mu(T^{\mu\nu})_{;\nu}=\nonumber\\ &&(\rho U^t)_{,t}+{1\ove... ...[(U^t)_{,t}+{1\over r^2}(r^2U^r)_{,r}\right]=0. \nonumber\\ [-4pt]\end{eqnarray}$$ (9)

Equations (6) and (9) give rise to the relativistic hydrodynamic equations.

We now turn to the analysis of ${\rm e}^+{\rm e}^-$ pairs initially created in the Dyadosphere. Let $n_{{\rm e}^\pm}$ be the proper densities of electrons and positrons (${\rm e}^\pm$). The rate equation for ${\rm e}^\pm$ is  

$$(n_{{\rm e}^\pm}U^\mu)_{;\mu}=\overline{\sigma v} \left[n_{{\rm e}^-}(T)n_{{\rm e}^+}(T) - n_{{\rm e}^-}n_{{\rm e}^+}\right],$$ (10)

where $\sigma$ is the mean pair annihilation-creation cross-section, v is the thermal velocity of ${\rm e}^\pm$, and $n_{{\rm e}^\pm}(T)$are the proper number-densities of ${\rm e}^\pm$, given by appropriate Fermi integrals with zero chemical potential. The equilibrium temperature T is determined by the thermalization processes occurring in the expanding plasma fluid with a total proper energy-density $\rho$,governed by the hydrodynamical Eqs. (6,9). We have  

$$\rho = \rho_\gamma + \rho_{{\rm e}^+}+\rho_{{\rm e}^-}+\rho^{\rm b}_{{\rm e}^-}+\rho_{\rm B},$$ (11)

where $\rho_\gamma$ is the photon energy-density and $\rho_{{\rm e}^\pm}$ is the proper energy-density of ${\rm e}^\pm$. In Eq. (11), $\rho^{\rm b}_{{\rm e}^-}+\rho_{\rm B}$ are baryon-matter contributions. We can also, analogously, evaluate the total pressure p. We define the total proper internal energy density $\epsilon$ and the baryon mass density $\rho_{\rm B}$ in the comoving frame, and have the equation of state ($\Gamma$ is thermal index)  

$$\epsilon \equiv \rho-\rho_{\rm B},\quad \rho_{\rm B}\equiv n_{\rm B}mc^2,\quad \Gamma = 1 + { p\over \epsilon}\cdot$$ (12)

The calculation is initiated by depositing the total energy (3) between the Reissner-Nordstrom radius r₊ and the dyadosphere radius $r_{\rm ds}$. The calculation is continued as the plasma fluid expands, cools and the ${\rm e}^+{\rm e}^-$ pairs recombine, until it becomes optically thin:  

$$\int_R (n_{\rm pairs}+n^{\rm b}_{\rm e}) \sigma_{\rm T} {\rm d}r = {2 \over 3}$$ (13)

where $\sigma_{\rm T}$ is the Thomson cross-section, $n^{\rm b}_{\rm e}$ is the number-density of ionized electrons and integration is over the radial size of the expanding plasma fluid in the comoving frame. Here, we only present $n^{\rm b}_{\rm e}=0,\rho_{\rm B}=0$ case.

  

Figure 1: Lorentz $\gamma$ as a function of radius. Three models for the expansion pattern of the ${\rm e}^+{\rm e}^-$ pair plasma are compared with the results of the one dimensional hydrodynamic code for a $1000\ M_\odot$ black hole with charge $Q = 0.1 Q_{\max}$. The 1-D code has an expansion pattern that strongly resembles that of a shell with constant coordinate thickness

We use a computer code (Wilson et al. 1997, 1998) to evolve the spherically symmetric hydrodynamic equations for the baryons, ${\rm e}^+{\rm e}^-$-pairs and photons deposited in the Dyadosphere. In addition, we use an analytical model to integrate the spherically symmetric hydrodynamic equations with the following geometries of plasma fluid expansion: (i) spherical model: the radial component of four-velocity $U(r)=U{r\over {\cal R}}$, where U is four-velocity at the surface (${\cal R}$) of the plasma, (ii) slab 1: $U(r)=U_r={\rm const.}$, the constant width of expanding slab ${\cal D}= R_\circ$ in the coordinate frame of the plasma; (iii) slab 2: the constant width of expanding slab $R_2-R_1=R_\circ$ in the comoving frame of the plasma.

We compute the relativistic Lorentz factor $\gamma$ of the expanding ${\rm e}^+{\rm e}^-$ pair and photon plasma. We compare these hydrodynamic calculations with simple models of the expansion. In Fig. 1 we see a comparison of the Lorentz factor of the expanding fluid as a function of radius for all of the models. We can see that the one-dimensional code (only a few significant points are pesented) matches the expansion pattern of a shell of constant coordinate thickness (slab 1).

We have shown that a relativistically expanding P.E.M. pulse can originate from the Dyadosphere of a EMBH. The P.E.M. pulse can produce gamma-ray bursts having the general characteristics of observed bursts. For example, the burst energy for a $1000\ M_\odot$ BH is 3 10⁵⁴ ergs with a spectral peak at 500 keV and a pulse duration of 40 seconds (Ruffini et al. 1999). This oversimplified model is encouraging enough to demand further study of the Dyadosphere created by EMBHs.