We analyse processes of electron acceleration in the Fermi Bubbles in order to define parameters and restrictions of the models, which are suggested for the origin of these giant radio and gamma-ray structures. In the case of leptonic origin of the nonthermal radiation from the Bubbles, these electrons should be produced somehow in-situ because of relatively short lifetime of high energy electrons, which lose their energy by synchrotron and inverse Compton processes. It has been suggested that electrons in Bubbles may be accelerated by shocks produced by tidal disruption of star accreting onto the central black hole or a process of re-acceleration of electrons ejected by supernova remnants. These processes will be investigated in subsequent papers. In this paper we focus to study in-situ stochastic (Fermi) acceleration by a hydromagnetic/supersonic turbulence, in which electrons can be directly accelerated from the background plasma. We showed that the acceleration from the background plasma is able to explain the observed fluxes of radio and gamma-ray emission from the Bubbles but the range of permitted parameters of the model is strongly restricted.
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