Gravitational lensing in the Schwarzschild spacetime: Photon rings in vacuum and in the presence of a plasma

Astrophysical black holes are usually surrounded by an accretion disk. At least parts of these accretion disks consist of a plasma in which light rays with different energies are dispersed. However, we usually do not know the exact configurations of these plasmas. In this paper we will now use the example of a Schwarzschild black hole embedded in an inhomogeneous pressureless and nonmagnetised plasma to investigate how the structural changes of the photon rings can help us to determine the properties of a plasma surrounding a black hole using multifrequency observations. For this purpose we will use a simple analytic model which describes a plasma whose electron density increases towards the equatorial plane when we approach the event horizon. For the chosen model we will first derive and then analytically solve the equations of motion. Then we will place an observer in the domain of outer communication and introduce an orthonormal tetrad to relate the constants of motion to latitude-longitude coordinates on the observer's celestial sphere. In the next step we will use the analytic solutions to investigate the geometric structures of the direct image as well as the photon rings of first and second order on the observer's celestial sphere. We will write down a lens equation and calculate the redshift and the travel time. We will compare the obtained results to results for photon rings in vacuum and in the presence of a homogeneous plasma. Finally, we will discuss which of these quantities can be used to extract information about the properties of the plasma.