Ultra-compact spherically symmetric dark matter charged star objects

We study the properties of ultra-compact spherically symmetric dark matter sector star objects, being the solution of Einstein equations with two $U(1)$-gauge fields. One of them is the ordinary Maxwell field, while the auxiliary gauge field pertains to the hidden sector, and mimics the properties of dark matter. The visible and hidden sectors are coupled by a kinetic mixing term with a coupling constant $\alpha$. We also investigate the possibility of condensation of charged scalar field around the reflecting dark matter star object. It happens that dark matter sector, both dark matter charge and coupling constant, cause shrinking of the star radius for which the condensation may occur.