## 3. Computation in the Kerr geometryThe 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 ): with are constants of motion: ## 3.1. The gravitational shiftsFirst, we need the expressions of the gravitational shifts of Eq. (18). Since the hot source A is at rest, one gets: The shift between a point rotating with the disk and the infinity, is given by Cunningham (1975 ) (with since the geodesic crosses the hot source): where is the velocity of the disk in the locally non-rotating frame , which can be express as a function of the coordinate angular velocity of the disk (Cunningham & Bardeen 1973 ): We thus obtain the following expression for the gravitational shift
between The shift between ## 3.2. Computation ofThe disk surface element contained between and is calculated in Appendix A: Thus, we obtain: The derivative is computed numerically by integrating the equation of motion between the hot source and the disk, for a grid of initial values of . The equation of motion has been obtained by Carter (1968 ) taking full advantage of the separation of variables: The signs of and are
always the same as the signs of This gives: Equation (34) is then solved with respect to , for a given . Once all the coefficients are computed, the linear system (2.3) can be solved, by making its determinant vanish. One can extract the values of and and compute the radial effective temperature distribution by means of Eqs. (11), (12) and (15). ## 3.3. Disk emission spectrumThe 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 , measured in the rest frame of the emitter, we obtain: where 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 is simply the Planck function . We spot the apparent position of a point P of the disk by its
coordinates in the plane of the sky. A couple
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
and
Here, is the inclination angle of the accretion disk. The radius of the emitting point of the accretion disk is then calculated by solving the new equation of motion: 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 ): The total spectrum is computed by integrating Eq. (39) over the disk surface. The grid in is obtained from a elliptic polar coordinates sampling (Fig. 2). The polar angles are regularly spaced whereas we use a logarithmic sampling of polar radius . The Cartesian coordinates () are then deduced by the following formulae: © European Southern Observatory (ESO) 1997 Online publication: April 20, 1998 |