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

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3. Computation in the Kerr geometry

The photons follow null geodesics either between the disk and the hot source, or between the disk/hot source and the observer at infinity. We recall the general expressions of the momentum along a null geodesic (Carter 1968 , Cunningham 1975 ):

[EQUATION]

with

[EQUATION]

[FORMULA] are constants of motion: E is the energy-at-infinity and [FORMULA] and q are closely related to the angular momentum. For geodesics intersecting the Z axis, one has [FORMULA], which is the case for every photon coming from or reaching the hot source.

3.1. The gravitational shifts

First, we need the expressions of the gravitational shifts of Eq. (18). Since the hot source A is at rest, one gets:

[EQUATION]

The shift [FORMULA] between a point [FORMULA] rotating with the disk and the infinity, is given by Cunningham (1975 ) (with [FORMULA] since the geodesic crosses the hot source):

[EQUATION]

where [FORMULA] is the velocity of the disk in the locally non-rotating frame [FORMULA], which can be express as a function of the coordinate angular velocity of the disk [FORMULA] (Cunningham & Bardeen 1973 ):

[EQUATION]

We thus obtain the following expression for the gravitational shift between A and [FORMULA]:

[EQUATION]

The shift between B and [FORMULA] is deduced from Lorentz transformation between the 2 inertial frames [FORMULA] and [FORMULA], that is:

[EQUATION]

3.2. Computation of [FORMULA]

The disk surface element [FORMULA] contained between [FORMULA] and [FORMULA] is calculated in Appendix A:

[EQUATION]

Thus, we obtain:

[EQUATION]

The derivative [FORMULA] is computed numerically by integrating the equation of motion between the hot source and the disk, for a grid of initial values of [FORMULA]. The equation of motion has been obtained by Carter (1968 ) taking full advantage of the separation of variables:

[EQUATION]

The signs of [FORMULA] and [FORMULA] are always the same as the signs of dr and [FORMULA], respectively. In this case, [FORMULA] is always positive (we do not take into account geodesics spinning round the black hole). Only dr can change its sign at a turning point in r. The constant of motion [FORMULA] and q must be taken such that, at the starting point A, one has:

[EQUATION]

This gives:

[EQUATION]

Equation (34) is then solved with respect to [FORMULA], for a given [FORMULA]. Once all the coefficients [FORMULA] are computed, the linear system (2.3) can be solved, by making its determinant vanish. One can extract the values of [FORMULA] and [FORMULA] and compute the radial effective temperature distribution [FORMULA] by means of Eqs. (11), (12) and (15).

3.3. Disk emission spectrum

The power carried to the observer by the photons emitted by a surface element of the disk, will be the product of its observed solid angle and specific intensity. Using again the Liouville's theorem to relate the observed power to the emitted specific intensity [FORMULA], measured in the rest frame of the emitter, we obtain:

[EQUATION]

where [FORMULA] is the redshift between the disk and the observer at infinity. Here again, we are only interested in the "direct" geodesics and do not compute photon trajectories crossing several times the equatorial plane between the black hole and the observer. If we suppose that the disk radiates like a black-body, the specific intensity [FORMULA] is simply the Planck function [FORMULA].

We spot the apparent position of a point P of the disk by its coordinates [FORMULA] in the plane of the sky. A couple [FORMULA] represents the coordinates of the impact parameter of the null geodesic between the disk and the observer. They are measured relative to the direction of the center of the black hole, in the sense of the angular momentum (see Fig. 2). Once again, we use the Carter's formalism to compute the geodesic between the disk and the observer. We can simply expressed [FORMULA] and q as a function of the impact parameters [FORMULA] (Cunningham & Bardeen 1973 ):

[FIGURE] Fig. 2. The impact parameters [FORMULA] in the plane of the sky deduced from the polar coordinates ([FORMULA]). If [FORMULA] is the inclination angle of the disk, one have the relation [FORMULA].

[EQUATION]

Here, [FORMULA] is the inclination angle of the accretion disk. The radius [FORMULA] of the emitting point of the accretion disk is then calculated by solving the new equation of motion:

[EQUATION]

Finally, we can expressed the gravitational redshift between a point of the accretion disk and an observer at infinity, needed in Eq. (39), as follows (Cunningham & Bardeen 1973 ):

[EQUATION]

The total spectrum is computed by integrating Eq. (39) over the disk surface. The grid in [FORMULA] is obtained from a elliptic polar coordinates sampling (Fig. 2). The polar angles [FORMULA] are regularly spaced whereas we use a logarithmic sampling of polar radius [FORMULA]. The Cartesian coordinates ([FORMULA]) are then deduced by the following formulae:

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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