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1973Apj. . .180. .5315 the Astrophysical Journal, 180:531-546 .5315 The Astrophysical Journal, 180:531-546, 1973 March 1 .180. © 1973. The American Astronomical Society. All rights reserved. Printed in U.S.A. 1973ApJ. ACCRETION ONTO BLACK HOLES: THE EMERGENT RADIATION SPECTRUM Stuart L. Shapiro Princeton University Observatory, Princeton, New Jersey Received 1972 August 21 ABSTRACT The luminosity and frequency spectrum of radiation resulting from interstellar gas accreting onto a black hole are calculated. The model is that of spherically symmetric, steady-state accretion onto a nonrotating black hole at rest in the interstellar medium. A fully relativistic treatment (in- cluding special- and general-relativity effects) of both the fluid mechanics and radiation processes has been used. The principal radiation mechanisms are electron-proton and electron-electron bremsstrahlung. The results for a black hole of mass M = 1 M© depend on whether the interstellar -3 medium is an H i or an H n region. For an H i region with «0 = 1 cm and T0 = 100° K, the total luminosity L = 3 x 1025 ergss_1. The spectrum has a bremsstrahlung shape with T ~ 9 0 -3 21 -1 10 K. An H ii region with n0 = 1cm and T0 = 10,000° K produces L = 2 x 10 ergs s . The spectrum is again bremsstrahlung in appearance with T ~ 1011 ° K. In both cases only a small fraction of the available thermal energy is radiated. For different values of «0, To, and M the 3 3 2 spectral distribution remains the same while the total luminosity L cc M To~ n0 , provided 312 2 2 6 (MlMo)(T01100° K)- n0 « 10 for the H i region and (M/Mo)(r0/10,000° K)- ti0 « 10 for the H ii region. The possibility of detecting this radiation and thereby identifying black holes is discussed. Subject headings: black holes — interstellar matter — radiative transfer — relativity I. INTRODUCTION One of the more promising ways of identifying a collapsed star or black hole in space is by observing the electromagnetic radiation spectrum emitted by interstellar gas accreting onto the object. Knowing from theoretical considerations the charac- teristic frequency spectrum of the emitted radiation, one can presumably determine from observations whether or not a black hole has indeed been located. Spherically symmetric, steady-state accretion onto stars for simple polytropic gases has been examined by Bondi (1952) in the nonrelativistic limit. Michel (1972) con- sidered the general-relativistic version of the same problem and applied his analysis to the accretion of matter onto condensed objects. Electromagnetic emission from the gas was not considered in these investigations. The possibility that gas accreting onto a black hole might be an important source of radiant energy was first suggested by Zel’dovich (1964) and Salpeter (1964). Shvartsman (1971) employed nonrelativistic approximations for both the fluid dynamics and radiative processes to estimate the total energy radiated by a fully ionized gas accreting onto a black hole. The purpose of this investigation is to provide a relativistic calculation of the total luminosity and frequency spectrum that emerges from interstellar gas accreting onto a black hole. We consider the case of spherically symmetric, steady-state accretion onto a nonrotating black hole at rest in the interstellar medium. The medium is assumed to consist of pure hydrogen gas of uniform density and temperature far from the black hole. We analyze two cases, one in which the medium is an H i region at 100° K and another in which the medium is an H ii region at 10,000° K. Internal heat loss and ionization of the gas are considered. The treatment of both the fluid 531 © American Astronomical Society • Provided by the NASA Astrophysics Data System 532 STUART L. SHAPIRO Vol. 180 mechanics and radiation processes is fully relativistic, incorporating both special and general relativity. The basic assumptions and equations used in this paper are set forth in § II. In §111 the results of the numerical integrations are presented and analyzed. The pos- sibility of detecting black holes in space by observing the radiation emitted by accreting gas is discussed in § IV. II. BASIC EQUATIONS a) Assumptions In the model, hydrogen gas surrounding a black hole flows radially inward toward the hole under the influence of the gravitational field. In falling toward the hole the gas is heated and compressed as gravitational energy is converted into internal plus kinetic energy. A portion of this internal energy ultimately escapes in the form of radiation which is emitted when the gas particles collide inelastically with each other. We adopt in this paper the field equations of general relativity in a Schwarzschild metric and a fluid description of the accreting gas. The adoption of a continuum model to describe the inflowing gas may seem inappropriate since one can show that, as each gas element falls inward, the mean free path of gas particles, /c, with respect to Coulomb collisions becomes larger than the characteristic dimension r of the region. Here r is the radius of the fluid element from the black hole. However, the presence of a very weak magnetic field frozen into the hydrogen plasma serves to couple particles together and effectively provide collisions. Zel’dovich and Novikov (1971) have shown that even for field strengths many orders of magnitude less than the mean interstellar field of 10_ 6 gauss the Larmor radius of protons moving at thermal velocity remains much smaller than r everywhere throughout the fluid. More- over, observations of the solar wind and experience with high-temperature laboratory plasmas provide further evidence for the general validity of fluid picture even when lc » r (Perkins 1972). We henceforth assume the existence of a very weak magnetic field frozen into our plasma to hold particles together. We neglect all other dynamical and radiative effects of the magnetic field, postponing a discussion of these topics for a subsequent report. In the model we suppose the gas to be pure hydrogen. We assume spherical sym- metric, steady state flow onto a nonrotating black hole of mass M at rest in the interstellar medium. Shvartsman (1971) has argued that the assumption of spherical symmetric flow is also appropriate for a black hole moving with an arbitrary velocity V through the medium. If F > cs, where cs is the sound velocity in the gas, a conically shaped shock wave forms behind the black hole. After compression in the shock wave 2 the gas increases in temperature so that kT0 ~ mpV . For any F, gas particles within 2 2 a distance rc ~ GM/(V + cs ) from the black hole will be captured. Gas pressure serves to symmetrize the flow, and for r « rc the motion of the gas can be considered radial. We assume the gas pressure to be isotropic at each point in the fluid. In an H n region the gas is assumed to be completely ionized by ultraviolet photons from an exciting star. In an H i region collisional ionization of neutral hydrogen by electrons is assumed to predominate over other ionization processes, including ionization by cosmic rays and photoionization by the outgoing radiation flux. The principal heating and cooling mechanisms in the gas depend on whether the medium is an H i or H n region and are discussed in detail in § lib. Reabsorption and scattering in the fluid of the outgoing X-ray and y-ray flux, as well as the production of electron-positron pairs, are shown to be unimportant. © American Astronomical Society • Provided by the NASA Astrophysics Data System .5315 No. 2, 1973 ACCRETION ONTO BLACK HOLES 533 .180. b) Relativistic Fluid Equations A black hole of mass M is represented by the Schwarzschild line element, 1973ApJ. dr 2 ds2 = (1 — 2m/r)c2dt2 — ^ 2m¡r) ~ r2(d02 + sin2 6d(f>2), (1) where m = GM/c2. In writing equation (1) we have ignored the influence of the sur- rounding plasma on the static line element; the increase in M due to accretion is shown below to be negligible over any relevant timescale. The fundamental equations of relativistic fluid dynamics are given by Landau and Lifshitz (1965). The equation of continuity is (nU% = 0, (2) where the 4-velocity vector of the fluid U' = dx'/ds and n is the scalar number density of particles (neutral hydrogen atoms plus protons) measured in the frame in which the fluid element is at rest. Since we are assuming spherically symmetric and steady- state flow, equation (2) becomes jrlnUW-g) = 0, (3) or 2 Annur = A = constant, (4) where 11/1! = u¡c. The relativistic generalizations of Euler’s equations are k dp dP_ ojU UUk = 1 - UiU* k (5) dx dx where P is the pressure, ou = e + P is the internal enthalpy per unit proper volume, and e is the proper internal energy density, including rest mass energy and ionization energy. For / = 1, equation (5) reduces to Id/u2\ _ 1 dP iu2 2 dr \c2/ (P + e) dr \c2 + For i = 2, 3, equation (5) reduces to the trivial identity 0 = 0 while for / = 0 it reproduces equation (6) above. Finally, we have the entropy equation which is given by the thermodynamic identity ± _ idn _Pdn = A(P) - F(P) dr n dr n dr u . ' In the above equation the decrease in the entropy of the inflowing gas has been set equal to the energy lost through inelastic collisions, A(P), minus the energy gained by absorption, F(P). Here A(P) and F(P) are both measured in ergs cm-3 s_1. Equations (4), (6), and (7) represent the three basic fluid equations that characterize gas accreting onto a black hole. In addition to these three equations, we require the relations that characterize the internal properties of the gas, namely, the equation of state, the equation for the proper internal energy density, and the equations describing the energy gain and energy loss functions.
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