Figure 5.12. Left: a delay map ^(r) for an edge-on circular Keplerian orbit, as described by Equation (5.21), at a distance of R = 10light days from a black hole of mass 1O8M0. Right: line profile for an edge-on circular Keplerian orbit, as described by Equation (5.25), at a distance of R = 10 light days from a black hole of mass 1O8M0.
and after some algebra we can write the line profile as
which is plotted in Figure 5.12.
A useful way to characterize the line width is the line dispersion, defined as
By symmetry, (Vlos)
^line = (VOs) - (Vlos)2) 0. The first term is m Vl2os*(VlosMVl
Thus, aline = vorb/21/2. Inspection of Figure 5.12 shows that FWHM = 2vorb, so the ratio of these two line-width measures is FWHM/aline = 2^2.
Generalization to a disk from a ring is obvious, though more parameters are required, specifically, inner and outer radii, the radial responsivity dependence, which is usually parameterized as a power law e(r) x ra, and the inclination. A example of a velocity-delay map for an inclined Keplerian disk is shown in Figure 5.13.
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