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Von Zeipel's theorem for a magnetized circular flow around a compact object

We analyze a class of physical properties, forming the content of the so-called von Zeipel theorem, which characterizes stationary, axisymmetric, non-selfgravitating perfect fluids in circular motion in the gravitational field of a compact object. We consider the extension of the theorem to the magnetohydrodynamic regime, under the assumption of an infinitely conductive fluid, both in the Newtonian and in the relativistic framework. When the magnetic field is toroidal, the conditions required by the theorem are equivalent to integrability conditions, as it is the case for purely hydrodynamic flows. When the magnetic field is poloidal, the analysis for the relativistic regime is substantially different with respect to the Newtonian case and additional constraints, in the form of PDEs, must be imposed on the magnetic field in order to guarantee that the angular velocity $Ω$ depends only on the specific angular momentum $\ell$. In order to deduce such physical constraints, it is crucial to adopt special coordinates, which are adapted to the $Ω={\rm const}$ surfaces. The physical significance of these results is briefly discussed.