We analytically calculate the influence of a plasma on the shadow of a black hole (or of another compact object). We restrict to spherically symmetric and static situations, where the shadow is circular. The plasma is assumed to be non-magnetized and pressure-less. We derive the general formulas for a spherically symmetric plasma density on an unspecified spherically symmetric and static spacetime. Our main result is an analytical formula for the angular size of the shadow. As a plasma is a dispersive medium, the radius of the shadow depends on the photon frequency. The effect of the plasma is significant only in the radio regime. The formalism applies not only to black holes but also, e.g., to wormholes. As examples for the underlying spacetime model, we consider the Schwarzschild spacetime and the Ellis wormhole. In particular, we treat the case that the plasma is in radial free fall from infinity onto a Schwarzschild black hole. We find that for an observer far away from a Schwarzschild black hole the plasma has a decreasing effect on the size of the shadow. The perspectives of actually observing the influence of a plasma on the shadows of supermassive black holes are discussed. PACS numbers: 04.20.

We consider light propagation in a non-magnetized pressureless plasma around a Kerr black hole. We find the necessary and sufficient condition the plasma electron density has to satisfy to guarantee that the Hamilton-Jacobi equation for the light rays is separable, i.e., that a generalized Carter constant exists. For all cases where this condition is satisfied we determine the photon region, i.e., the region in the spacetime where spherical light rays exist. A spherical light ray is a light ray that stays on a sphere r = constant (in Boyer-Lindquist coordinates). Based on these results, we calculate the shadow of a Kerr black hole under the influence of a plasma that satisfies the separability condition. More precisely, we derive an analytical formula for the boundary curve of the shadow on the sky of an observer that is located anywhere in the domain of outer communication. Several examples are worked out.PACS numbers: 04.20.

We develop a model of gravitational lensing in a non-uniform plasma. When a gravitating body is surrounded by a plasma, the lensing angle depends on the frequency of the electromagnetic wave, due to dispersion properties of plasma, in presence of a plasma inhomogeneity, and of a gravity. The second effect leads, even in a uniform plasma, to a difference of the gravitational photon deflection angle from the vacuum case, and to its dependence on the photon frequency. We take into account both effects, and derive the expression for the lensing angle in the case of a strongly nonuniform plasma in presence of the gravitation. Dependence of the lensing angle on the photon frequency in a homogeneous plasma resembles the properties of a refractive prism spectrometer, which strongest action is for very long radiowaves. We discuss the observational appearances of this effect for the gravitational lens with a Schwarzschild metric, surrounded by a uniform plasma. We obtain formulae for the lensing angle and the magnification factors in this case and discuss a possibility of observation of this effect by the planned VLBI space project Radioastron. We also consider models with a nonuniform plasma distribution. For different gravitational lens models we compare the corrections to the vacuum lensing due to the gravitational effect in plasma, and due to the plasma inhomogeneity. We have shown that the gravitational effect could be detected in the case of a hot gas in the gravitational field of a galaxy cluster.Comment: 12 pages, 3 figure