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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. |
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, in Tables 1-2, each of the figures corresponds to a certain combination of a/M and i (Tab. 1 covers the range of radius up to 100M at low resolution while Tab. 2 covers only the inner 50M at high resolution; otherwise the tables arrangement is identical).
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 yellow circle is photon circular orbit, while an outer yellow 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.
TABLE 1 | a/M | |||
i [deg] | 0 | 1/3 | 2/3 | 1 |
0 | Figure | Figure | Figure | Figure |
20 | Figure | Figure | Figure | Figure |
40 | Figure | Figure | Figure | Figure |
60 | Figure | Figure | Figure | Figure |
80 | Figure | Figure | Figure | Figure |
Archive of all these figures |
Data tables are implemented in XSPEC in such a way that the iteration process can use all of them as necessary. 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.
This (outdated) version of data tables covers equatorial plane up to r=100M (in figures, only a smaller region is shown for clarity).
Correct values are given in the relevant range of radii; outside that range, tables are supplemented to the rectangular m x m matrix with the values that are evidently off.
TABLE 2 | a/M | |||
i [deg] | 0 | 1/3 | 2/3 | 1 |
0 | Figure | Figure | Figure | Figure |
20 | Figure | Figure | Figure | Figure |
40 | Figure | Figure | Figure | Figure |
60 | Figure | Figure | Figure | Figure |
80 | Figure | Figure | Figure | Figure |
Archive of all these figures |