4.1.23. Laor: relativistic line broadening model

This multiplicative model broadens an arbitrary additive component with a relativistic line profile. The relativistic line profile is interpolated from tables produced by Laor (1991). In his model, the line transfer function is determined for emission from the equatorial plane around a Kerr black hole, assuming an emissivity law I(\cos\theta) \sim 1. + 2.06\cos\theta. The transfer function is calculated at a grid of 35 radii (r_n = 1600 / (n+1)^2 for n=1,\ldots,35, in units of GM/c^2), 31 inclinations uniformly spaced in \cos i, and 300 frequencies, logarithmically spaced between 0.1 and 5 times the emission frequency, respectively.

Using these numbers, a radial integration is done using an emissivity law

I(r) \sim 1 / (r^2 + h^2)^{q/2},

where h is a characteristic scale height and q an asymptotic power law index (for large r, I(r)\sim r^{-q}). The integration is done between an inner radius r_1 and an outer radius r_2. Given the radial grid that is provided in Laor’s data, the outer radius extends out to at most 400GM/c^2.

Warning

Of course, any line or continuum emission component can be convolved with the this broadening model; for continuum components the computational time may be very large, however, due to the convolution involved.

Warning

The outer radius cannot be larger than 400GM/c^2.

The parameters of the model are:

r1 : Inner radius of the disk, in units of GM/c^2. The minimum allowed value is 1.234 (for a maximally rotating Kerr black hole). For a Schwarzschild black hole, one should take r_i = 6. Default value: 1.234.
r2 : Outer radius of the disk, in units of GM/c^2. Keep this radius less than 400 (default value)
q : Emissivity slope q as described above. Default value: 2.
h : Emissivity scale height. Default value: 0.
i : Inclination angle (in degrees) of the disk (angle between line of sight and the rotation axis of the disk). Default value: 45 degrees.

Recommended citation: Laor (1991).