# X-ray reverberation for a Schwarzschild black hole with an accretion disk

We want to model the spectra and time of arrival of x-ray emitted from a flare occurring along the symmetry axis perpendicular to the accretion disk (lamp post model). There are a number of steps in the calculation which lend themselves to nice animations.

## Ray tracing black hole with an accretion disk

One of the first steps is to calculate the spatial trajectories of the null rays which connect the accretion disk to the observer. This amounts to ray tracing the spacetime. The animation on the left shows the result of ray tracing a Schwarzschild black hole with an accretion disk for various different observing angles. The accretion disk is red on one side and blue on the other and extends from the inner-most stable orbit (ISCO) at \(r_\text{isco}=6M\) to \(r=16M\). No redshift or doppler effects are included at this stage.

The animation on the right shows the effect of varying the black hole mass (or equivalently varying the inner and outer radii of the disk). In this animation the observer is at an angle of \(\pi/25\) radians with respect to the disk. The smallest mass in the animation is \(M=1/20\) where the least distortion occurs. The largest mass has \(M=1\).

## Timing the x-rays

Once we know the spatial trajectories of the rays that connect the disk to the observer we can calculate the Schwarzschild coordinate time between emission and detection. In addition we calculate the time from the lamp post source to the disk, and the time along a null ray connecting the lamp post with the observer. Using these we can work out the time difference \(\Delta t\) between the direct and reflected emission from the system. The following image and animation have the observer at \(\pi/25\) radians above the disk and the lamp post at \(r_l = 3.1M\)

The left image shows a color map for \(\Delta t\). The shortest \(\Delta t\) are colored violet and the longest are colored red with times in between colored with a 'rainbow' color map. The right animation shows how a pulse at the lamp post would appear to the distant observer to move across the disk.