Radiation Transport Around Kerr Black Holes Jeremy David ...

Chapter 3
The Geodesic Hot Spot Model
Everything should be made as simple as possible, but not simpler.
-Albert Einstein
3.1 Hot Spot Emission
Given the ray-tracing map from the accretion disk to the image plane, with each
photon bundle labeled with a distinct 4-momentum and time delay, we can reconstruct
time-dependent images of the disk based on time-varying emission models. The
simplest model we consider is a single region of isotropic, monochromatic emission
following a geodesic trajectory: the “hot spot” or “blob” model (Sunyaev, 1972; Bao,
1992; Stella & Vietri, 1998, 1999).
The hot spot is a small region with finite radius and emissivity j(x) chosen to
have a Gaussian distribution in local Cartesian space:
j(x) ∝ exp
[− |˜x − ˜x ]
spot(t)| 2
. (3.1)
2R 2 spot
The spatial position 3-vector ˜x is given in pseudo-Cartesian coordinates by the transformation
defined by equations (2.41a,2.41b) and z = r cosθ. Outside a distance of
4R spot from the guiding geodesic trajectory, there is no emission. We typically take
R spot = 0.25 − 0.5M, but find the normalized light curves and QPO power spectra
to be rather independent of spot size. We have also explored a few different hot
spot shapes, ranging from spherical to an ellipsoid flattened in the e θ direction and
similarly find no significant dependence of the spectra on spot shape.
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