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We consider light propagation in a non-magnetized pressureless plasma arounda Kerr black hole. We find the necessary and sufficient condition the plasmaelectron density has to satisfy to guarantee that the Hamilton-Jacobi equationfor the light rays is separable, i.e., that a generalized Carter constantexists. For all cases where this condition is satisfied we determine the photonregion, i.e., the region in the spacetime where spherical light rays exist. Aspherical light ray is a light ray that stays on a sphere $r =\mathrm{constant}$ (in Boyer-Lindquist coordinates). Based on these results, wecalculate the shadow of a Kerr black hole under the influence of a plasma thatsatisfies the separability condition. More precisely, we derive an analyticalformula for the boundary curve of the shadow on the sky of an observer that islocated anywhere in the domain of outer communication. Several examples areworked out.

Light propagation in a plasma on Kerr spacetime: Separation of the Hamilton-Jacobi equation and calculation of the shadow