Analytic solution of a magnetized tori with magnetic polarization around Kerr black holes

We present the first family of magnetically polarized equilibrium tori around a Kerr black hole. The models were obtained in the test fluid approximation by assuming that the tori is a linear media, making it is possible to characterize the magnetic polarization of the fluid through the magnetic susceptibility $\chi_{m}$. The magnetohydrodynamic (MHD) structure of the models was solved by following the Komissarov approach, but with the aim of including the magnetic polarization of the fluid, the integrability condition for the magnetic counterpart was modified. We build two kinds of magnetized tori depending on whether the magnetic susceptibility is constant in space or not. In the models with constant $\chi_{m}$, we find that the paramagnetic tori ($\chi_{m}>0$) are more dense and less magnetized than the diamagnetic ones ($\chi_{m} r_{c}$. Now, in the models with non-constant $\chi_{m}$, the tori become more magnetized than the Komissarov solution in the region where $\partial\chi_{m}/\partial r 0$. Nevertheless, it is worth mentioning that in all solutions presented in this paper the magnetic pressure is greater than the hydrodynamic pressure. These new equilibrium tori can be useful for studying the accretion of a magnetic media onto a rotating black hole.