3 Energy and number of GRBs

The SBHs under consideration are remnants of Population I-II and III stars located in the disc or in the halo of galaxies. In the first case their masses are greater than $10\, M_\odot$, while in the second are greater than $10^3\,M_\odot$.In this hypothesis, SBHs are made of baryonic dark matter and have integral mass about $(0.1 - 1)M_{\rm gal}\sim (10^9 - 10^{10})M_\odot$; this means that the number of SBHs is 10⁶ - 10⁸ per galaxy. These estimates come from dynamical, photometric and chemical data [1, (Bahcall 1984;] [8, Carr 1994).] Of course the density of SBHs has to be higher in galaxies with star-forming regions.

We show here that the maximum energy of cosmological GRBs can be accumulated in magnetospheric inhomogeneities near massive () SBHs. Observing isotropic energy of GRBs is 10⁵¹ - 10⁵⁴ ergs [17, (Piran 1998).] There are many reasons to think that their radiation is beamed with a factor $\sigma = {\theta^2}/4\pi\sim 10^{-4} - 10^{-5}$. $\theta$ is either angular size of particles stream or beaming angle of these particles radiation ($1/\gamma$)[17, (Piran 1998).] After taking into account this effect, the energy of GRBs becomes 10⁴⁷ - 10⁵⁰ ergs in a rest frame. It was shown the SBH accretes the plasma from ISM with a rate (for recent review see [9, Chakrabarti 1996)] $\dot M = 1.4\ 10^{13}nM_3^2V_{300}^{-3}\,{\rm g} \,{\rm s}^{-1}$, where n is number density of ISM in units of cm^-3, M₃ is the SBH mass in units of $10^3\ M_\odot$, V₃₀₀ is velocity of SBH in units of 300 km s^-1. The magnetic field frozen in the plasma during accretion is reinforced and conserved near the horizon (e.g. [6, Bisnovatyj-Kogan & Ruzmaikin 1976;] [13, Macdonald et al. 1986;] [11, Kardashev 1995;] [3, Beskin 1997).] There is not a self-consistent model of this process and we will estimate possible parameters for the SBH magnetosphere in a phenomenological frame only.

If the part of the full mass-energy of the accreted plasma $\delta\dot{M}c^2$ is stored in form of magnetosphere energy during the time t, then $(H_{\rm g}/8\pi)(4/3)\pi{r_{\rm g}}^3\sim\delta\dot{M}c^2t$,where $H_{\rm g}$ value of global magnetic field, $r_{\rm g} = 2\,GM/c^2$, and $H_{\rm g} \sim 3 \ 10^{12} \delta_{-1}^{1/2} {M_3}^{-1/2} V_{300}^{-3/2} n^{1/2} t_9^{1/2}$ G, where $\delta_{-1}$ is expressed in units of 0.1, and t in units of 10⁹ years.

For a magnetic field $H_{\rm l}$ in inhomogeneities of size r for constant magnetic flux it is easy to obtain that: $H_{\rm l} \sim H_{\rm g} (r_{\rm g} / r)^2 \sim 3 \ 10^{14} \alpha_{0.1}^{-2} \delta_{-1}^{1/2} M_3 ^ {-1/2} V_{300}^{-3/2} n^{1/2} t_9^{1/2}$ G, where $\alpha = r_{\rm g}/r$ is in units of 0.1.

For the magnetic energy concentrated in the inhomogeneities we obtain $E_{\rm l} \sim (H_{\rm l}^2/8\pi) \cdot (4/3) \pi r^3 \sim 4 \ 10^{50} \alpha_{0.1} ^{-1} M_3^3 \delta_{-1} n V_{300}^{-3} t_9$ ergs. This means that this amount of energy, accumulated during 10⁹ years, can be released as observable GRBs. We estimate that the mass of matter trapped in the inhomogeneity region during the time t is

$m = \dot{M} (r/r_{\rm g})^3 t$ $\sim$ $4.3 \ 10^{26} n M_3^2 V_{300}^{-3} \alpha_{0.1}^3 t_9 \sim 2$ $10^{-7} M_\odot\, n M_3^2 V_{300}^{-3} \alpha_{0.1}^3 t_9$.

Only $2 \ 10^{-7} M_\odot$ is accumulated during 10⁹ years. This means that the baryon contamination is absent. In the results of instabilities development in inhomogeinity its magnetic energy converts to kinetic energy of electrons and protons. If $\beta$ is the efficiency of this conversion, then $\beta n_{\rm e} \gamma_{\rm e} m_{\rm e} c^2 \sim \beta n_{\rm p} \gamma_{\rm p} m_{\rm p} c^2 \sim (H_{\rm l}^2)/8\pi$, and we obtain $\gamma_{\rm e} \sim 10^6 \beta$, $\gamma_{\rm p} \sim 10^3 \beta$. The stream of particles is extremely relativistic as it is necessary to explain the GRBs observational properties.

We postulate a universal form for flares number distribution with energy if they are connected with the magnetosphere in inhomogeneities. This means that the parameter k of the distribution ${\rm d}N(E)/{\rm d}E = k E^{-\beta}$($\beta \sim 1.7$) is the same for solar and UV Ceti flares stars and GRBs, they have different energy ranges: 10²⁷ - 10³² ergs for Sun and UV Ceti stars, 10⁴⁷ - 10⁵⁰ ergs for GRBs. Taking $k \sim 6 \ 10^{24}$ yr^-1 from observations of UV Ceti stars [5, (Beskin et al. 1988)] we obtain for a minimal GRBs energy of 10⁴⁷ ergs (in the rest frame) about 10^-8 flares per SBH per year. Using estimated number of SBH per galaxy, 10⁶ - 10⁸, we have 10^-2 - 10⁰ GRBs per year per galaxy. [17, Piran (1998)] gives for this number 10^-2 (for beaming factor $\sigma \sim 10^{-4}$). Taking into account all of our assumptions, this coincidence is very promising!

Acknowledgements

This investigation was supported by the Russian Fund of Fundamental Research, by the Educational Scientific Centre "Cosmion", by the italian Ministry of Foreign Affairs and by the University of Bologna (Funds for selected research topics).