The case of a/M=1, i=60 deg
Figure 1. Content of data tables is illustrated by this plot, showing contour levels of various quantities. Equatorial plane of Kerr black hole is considered (the view along rotation axis). Observer is located towards top of the figure with inclination of 60 deg. See the text for details.

Global parameters of the model and its tabular representation

There are two global parameters: black hole angular momentum, 0<a/M<1 and observer inclination angle, 0<i<90. Non-rotating Schwarzschild black hole has a/M=0, while extremely rotating Kerr black hole has a/M=1; observer located along the rotation axis (pole-on) has inclination i=0, while equatorial (edge-on) observer has i=90 deg (we do not consider strictly edge-on position because it appears unnecessary; in real systems radiation would be obscured by an outer torus). Below, the figures correspondng to a certain combination of a/M and i are shown (use the form to get the figure files).

Figures illustrate the content of these tables for different parameter values. They show a view of equatorial plane along rotation axis. Schwarzschild coordinates are used (units of M). Observer is located on top of the figure at spatial infinity (we do not consider strictly edge-on position of the observer because it appears unnecessary; in real systems radiation would be obscured by an outer torus; maximum inclination of 80 deg appears to be suffucient). Black hole is indicated by a black circle in the centre. An inner circle is photon circular orbit, while an outer circle is the innermost stable orbit (co-rotating). All these radii are identical in the case of extremele rotating hole. Notice that gravitational radius depends on black-hole rotation parameter as Rg=M+(M2-a2)1/2. Other critical orbits also depend on a. We use c=G=1 units.

Figures and tables

Each figure contains four sets of contour lines (corresponding to four data tables):
  1. Levels of redshift function are plotted with solid red curves. Values greater than unity indicate shift of energy and observed flux towards higher values. Doppler effect and gravitational redshift play a role here (the latter one prevails near horizon). The curves would be symmetrical with respect to vertical axis (butterfly-shaped) if relativistic effects were ignored.
  2. Levels of constant time delay are plotted with dashed blue lines. They would be perfectly horizontal if relativistic effects were ignored. Values are in geometric units and scaled with black-hole mass.
  3. Levels of constant emission angle (with respect to normal to the disc plane in a frame corotating with the disc material) are plotted with black dash-dotted curves. Abberation and lensing play a role to determine the values of emission angle. Far from the hole they attain the value equal to observer inclination. Local emission angle should be considered if the emissivity is anisotropic.
  4. Levels of constant magnification (lensing effect) are plotted with solid curves of magenta color. They have been computed by integrating geodesic deviation equation which determines the change of cross-sectional area between neighbouring light rays (the rays are parallel to each other at infinity, therefore the change of area is zero in case of flat space). Even with black-hole geometry, a significant effect occurs only for light originating near behind the hole (i.e. the place of upper conjunction for the observer, r<15M) and when observer is close to equatorial plane (large inclination, i>60 deg). Far from the hole, the values approach cosi asymptotically, which corresponds to simple effect of geometrical projection in the euclidean geometry (without lensing); larger values indicate light enhancement from the corresponding region of the disc. Note: magnification is generally large only in very small areas close to caustics; therefore contours may appear somewhat wigly if they are drawn for very tiny magnification, especially in the case of a small inclination; this should not introduce any significant error in spectra computations.
Content of data tables for different parameter values. In order to get a figure file, select parameters in the following form.
Horizon  [GM/c2] 1.00  1.05  1.10  1.15  1.20  1.25  1.30 
1.35  1.40  1.45  1.50  1.55  1.60  1.65 
1.70  1.75  1.80  1.85  1.90  1.95  2.00 
Inclination  [deg] 0.1  10  15  20  25 
30  35  40  45  50  55  60 
65  70  75  80  85  89 
Function Boyer-
Lindquist
Redshift  Delay  Projected  
Lensing 
Orthogonal
Lensing 
Angle1 Angle2 Polarization
Kerr
ingoing
Redshift  Delay  Projected  
Lensing 
Orthogonal
Lensing 
Angle1 Angle2 Polarization

Local emissivity given, the four data sets (one set for each combination of global parameters) are sufficient to compute observed spectrum of a source orbiting in the disc plane.

Data tables have implemented in XSPEC in such a way that the iteration process can use all of them, as needed. Additional tables can be included, and they can be computed with an increaed resolution, or in a different range of radii. Hence it seems to be a good idea if the routine is prepared for such additions/modifications, although it may be unnecessary to do it in near future. Current version of data tables covers equatorial plane up to r=103GM/c2. (We keep also old tables which were used in previous versions of the code; they have lower resolution than the current ones.)

Implicit assumptions

Here we have assumed a Keplerian disc which is geometrically thin and planar (aligned with equatorial plane), optically thick, and non-selfgravitating. These are standard assumptions but they can be relaxed in future. We did consider more general cases in other papers, e.g. the case of self-gravity. Other authors discussed non-equatorial (twisted) disks or the role of highre order images which are neglected here. For the moment we postpone additional complications which would introduce other free parameters.

Technical notes about data tables

Read more in the paper...


Link to this project main page and to my homepage.