Astron. Astrophys. 326, 99-107 (1997)

2. Relativistic energy balance equations

The geometry of the system, formed by the hot source and the accretion disk, is sketched in Fig. 1. The energy balance can be solved self-consistently, for a given disk emissivity law. Actually, the emission of the disk is entirely controlled by the hard radiation angular distribution, which is at turn determined by the disk emissivity through the anisotropic IC process. In Paper I, we solved the Newtonian case of this radiative balance by solving a system of 3 equations between the 3 Eddington parameters characterizing the photon field. When relativistic effects are taken into account, the same principle can be used, but with some modifications. First, the photons do not follow any more straight trajectories, but geodesics whose equations must be deduced from the metrics. Second, one must take care of gravitational and Doppler shifts between the hot source, the rotating accretion disk and the observer at infinity.

Fig. 1. The general picture of the model. We have also drawn the trajectory of a beam of photons emitted by the hot source in a solid angle and absorbed by a surface ring on the disk. The letters A and B refer to the indices define in part (2.1).

2.1. The different frames

The Kerr metrics describes the exterior metrics of an stationary axisymmetric gravitational field around a rotating body. It is completely specified by its total mass M and angular momentum per unit mass a. The line element can be written in Boyer-Lindquist coordinates as follows (we use convenient units such that ; then the unit length is , being the Schwarzschild radius):

where

Yet, as shown by Bardeen et al. (1972 ), physics is not simple in the Boyer-Lindquist coordinate frame. First, the dragging of inertial frame becomes so severe as we approach the Kerr black hole, that the line element goes time-like. Second, the metrics is non-diagonal which introduces algebraic complexities. For those reasons, Bardeen introduced the locally non-rotating frames (LNRF, Bardeen et al. 1972 ) to cancel out, as much as possible, the "frame dragging" effects of the hole rotation. They are linked to observers whose world lines are , and . We will used the same method here. We defined the set of frames as these LNRF. For a Schwarzschild black hole, they correspond simply to the curvature coordinates frame . We also define the set of frames which locally rotate with the disk. All terms computed in these frames are labeled with a star.

The quantities expressed at the hot source location () are indexed with an A. They are indexed with a B when they are computed on the surface of the disk (). For example, a differential elementary surface of the disk, expressed in the rotating frame , will be noted (cf. Fig. 1). The gravitational shift between any point P and Q is noted , with , being the emitted wavelength of a photon and the observed wavelength at infinity along the geodesic connecting P and Q.

2.2. Computation of the specific intensity

The radiative balance between the energy radiated by the disk and that radiated by the hot source of relativistic leptons gives the relation:

Here, is the flux emitted in the frame rotating with the disk, by the surface ring which radius is in the range [ , ], and we use the fact that the ratio (where is the radiative power, measured in Q, released by the hot source) is a relativistic invariant. Using, now, the covariance of the space-time quadrivolume between the 2 inertial frames and , we obtain:

where and are the elementary space intervals in the Z direction. Since there is no motion along this direction, and thus , combining Eqs. (8) and (9), one gets:

We suppose the disk to radiate isotropically like a blackbody at the temperature . So, one gets also:

The ratio of the power emitted by the hot source by solid angle unit, is derived in Eq. (48) of Paper I . One gets (with , cf. Fig. 1):

where and are the three Eddington parameters defined by:

The gravitational shifts and and the Jacobian of Eq. (10) will be developed in the next section. We deduce from Eqs. (10), (11) and (13) the general expression of the specific intensity of the radiation emitted by the disk, in the rotating frame :

2.3. Computation of Eddington parameters

Now, we can find a system of 3 equations between the 3 Eddington parameters , and characterizing the radiation field near the hot source. Using Eqs. (14) and (15), and the fact that is a relativistic invariant (Liouville's theorem), one obtains:

If we define, now, the following parameters:

we can rewrite the expression of , and those of the second and third Eddington moment and in the same way, in order to obtain the following linear system:

We will find the values of and by making the determinant of this system vanish, that is by solving a cubic equation in . The only physical constraints on the choice of that we have to respect are and . Note that the Newtonian case of an infinite disk of Paper I can be recovered by setting . In the general case, we need to compute the values of the variables , that is, to express the gravitational shifts and , and the ratio . This will be the subject of Sect. 3 where we use the Kerr metrics to calculate explicitly these coefficients.

© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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