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Neutrino Emission and Annihilation Near Tori Around Black Holes

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Neutrino Emission and Annihilation Near Tori Around Black Holes ACTA ASTRONOMICA Vol. 43 (1993) pp. 183±191 Neutrino Emission and Annihilation Near Tori around Black Holes by Michaø J a r o s z y nski Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland Received September 20, 1993 ABSTRACT Neutrino radiation ®elds from stationary, axisymmetric, non-selfgravitating tori around stellar mass Kerr black holes are calculated. Such objects may form as a result of a merger between two neutron stars, a neutron star and a stellar mass black hole, or a "failed supernova" collapse of a single rapidly rotating star. The neutrino annihilation rates near the axis of rotation are found with fully general relativistic methods. Our calculations show that about 80% of the energy deposited by annihilating neutrinos is swallowed by the black hole, lowering the energy which can be transported to in®nity. The importance of this effect for the modeling of the gamma ray burst sources is discussed. Key words: Gamma-rays: bursts ± Radiative transfer ± supernovae: general 1. Introduction The aim of this paper is to investigate further the possible observational con- sequences of the presence of hot and dense tori around stellar mass black holes. Such objects may form as a result of a collision ± merger of two neutron stars or a neutron star and a stellar mass black hole (Paczynski 1991, Narayan, Paczynski and Piran 1992, Mochkovitch et al. 1993, and references therein), or by a di- rect collapse of a massive rotating star ± a "failed supernova" (Woosley 1993a,b, Paczynski 1993), and are among the likely candidates for gamma-ray bursters at cosmological distances. The simpli®ed models of such con®gurations have been described by Witt et al. (1994, hereafter WJHPW). According to WJHPW it is possible to obtain selfconsistent models of tori built of neutron matter. Toroidal con®gurations of barotropic ¯uid orbiting Kerr black holes were studied long time ago (Abramowicz, Jaroszynski and Sikora 1978 ± hereafter AJS, and references therein). We use WJHPW models of matter distribution around a black hole. Due to the simplifying assumptions these models are stationary and axially symmet- ric, their self-gravity is neglected, they are barotropic (iso-entropic) and they have constant speci®c angular momentum throughout. 184 A. A. We are speci®cally interested in the neutrino emission and annihilation near a source possesing toroidal geometry. The possible importance of such processes for the initialisation of a gamma-ray burst was pointed out by Meszaros and Rees (1992). Mochkovitch et al. (1993) calculated the rate of energy deposition in the region near the rotation axis by the annihilating neutrino ± antineutrino pairs. Their calculations neglect the effects of general relativity which we are going to include. Similar calculations in spherical geometry were performed in the past(cf. Haensel, Paczynski and Amsterdamski 1991, Woosley and Baron 1992, and references therein). In spherical case the neutrinos radiated from the neutron star drive a slow baryonic wind. The region of possible annihilation of neutrinos becomes polluted by baryons and the radiation of gamma rays or relativistic out¯ow of pair plasma produced by annihilation becomes impossible. In the case of toroidal geometry there is a region near rotation axis where the baryons can not get because of the centrifugal forces barier. One can expect pair plasma generated there to form a relativistic jet, which may radiate gamma rays in the direction of its ¯ow. We investigate the energetics of such a scenario. 2. Model 2.1. Neutrino Optics We are going to describe the neutrino ®eld around the torus. We anticipate, that the neutrino ± neutrino interactions are rare and their in¯uence can be described in the higher order approximation. In the ®rst approximation the neutrinos propagate freely outside the matter. Inside the matter one has to take into account the interactions with matter. The light travel time through the torus is much shorter than any other characteristic scale for the matter con®guration (WJHPW), so one can calculate the distribution of neutrinos in the torus vicinity as if they were emitted by a stationary distribution of sources. We adopt this quasi-stationarity assumption for use in our calculations. One must remember that the results are applicable to the situation at a given time: after the matter con®guration evolves substantially one has to repeat the calculations. The method can be used to calculate the neutrino distribution at a point close to the torus or far away from it. (t x E n) We are interested in the neutrino distribution function f giving the n density of particles with energy E propagating in the direction measured at the t x) time and position ( . In the diffusion approximation the kinetic equation for the distribution function has the form (Lifshitz and Pitaevskii 1981): 1 (f f rf = ) ( ) n 0 1 p f d where we neglect the partial time derivative (quasi-stationarity). The term p t d vanishes automatically, since particles move along geodesics. is the neutrino Vol. 43 185 f free path. Outside the matter , so the RHS of the equation vanishes and remains constant along the trajectory. We assume the neutrino distribution inside the torus to be close to the thermal equlibrium distribution, given by the Fermi ± Dirac statistics: 1 1 = ( ) f0 3 2 (E k T ) + h exp 1 k T where h is the Planck constant, the Boltzman constant and ± the temperature. We neglect possible asymmetries between particles and antiparticles and introduce no chemicalpotential. We also neglectthe differences between the different ¯avors. It is convenient to introduce a new variable , which is the neutrino optical depth. Since the cross-section for neutrino interaction with matter depends strongly on its energy, we have to include this dependence in the de®nition: Z ds (E ) = ) (3 0 (E ) al ong r ay a E p v where a is the neutrinoenergy as measured by anobservermovingwith the a v p s four-velocity and a is the particle four-momentum. d is the proper distance 0 a E p u differential measured in the matter frame and a ± neutrino energy in the a same frame, where u is the velocity of matter. With the help of the new variable it is easy to solve the kinetic equation: Z = f ( ) e ( ) f 0 d 4 al ong r ay This equation is reminiscent of the formal solution to the radiative transfer equation. Togetthedistribution functionatagivenpointonehasto perform the above integral many times, for trajectories coming from differrent directions and particles with severalvaluesofenergies. Foreachdirectiononecanpropagate back a null geodesic and calculate the integrals for different energies simultaneously. As a result one obtains an array giving the distribution function on a grid. It is conceptually = equivalent to propagate back geodesics, ®nd the 1 surface for each energy and assume that this surface emits blackbody neutrino radiation at given energy. Then the resulting radiation could be described as a superposition of different shape black bodies with different distribution of temperatures. We think that the solution of the kinetic equation is technically simpler. 2.2. The Torus The methods describedaboveare applicableto any quasi-stationary distribution of hot and dense matter, which can emit neutrinos. We adopt WJHPW models of the matter distribution around the black hole. They are constructed by the methods described by AJS. Below we repeat some information, which may be found in AJS and WJHPW papers for completness and to de®ne parameters of our models. We 186 A. A. choose the simplest models with constant speci®c angular momentum of matter, l u u = u t const, where are the matter four-velocity components. The interesting con®gurations, which may consist of neutron matter must be close to the black hole horizon and have low speci®c angular momentum value, in general l l l ms mb , where 'ms' and 'mb' stand for marginally stable and marginally bound orbits, respectively (Bardeen 1973) For constant angular momentum tori the effective potential W describing gravitational and centrifugal forces is simply W = l n(u ) related to the energy of a particle on a circular orbit, t . Equipotential surfaces have toroidal topology for some range of values of W and the region encompassedby them can be ®lled with matter. There is a maximal (maximally ex- tended) con®guration characterised by the maximal difference between the surface W = W W max cntr and central values of the potential sur f . The actual con®g- W W uration can have max . This parameter is also related to the equation of state: P max Z P P d ( W = ) 5 + P 0 where P is the pressure and ± the energy density. Any torus, orbiting a Kerr a M black hole of the given angular momentum and mass BH , consisting of matter C (l with a known equation of state, can be de®ned by two parameters: l C W W l )(l l ) W max ms ms mb de®ning the angular momentum value and giving the potential difference or central pressure. The calculations of neutrino radiation from neutron stars (Duncan et al. 1986, Woosley and Baron 1992) show that a slow baryonic wind arises from the surface due to the interaction of neutrinos with the outer layers of matter. The energy of annihilating neutrinos deposited in a region polluted by baryonic matter can not be used for the possible gamma-ray burst. One should expect winds to be present near the torus as well, but there is a region near the rotation axis, which is not accessible to matter with non-vanishing angular momentum due to the barier of centrifugal forces. This region is given by the inequality: 2 l g + l g + g ( ) tt 2 t 0 6 g where are the metric components in the standard Boyer-Lindquist coordinates ( + + +) t r and we use the convention for the metric.
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