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The peculiarities of the super-sonic expansion of current-carrying plasma in a vacuum arc with and without an axial magnetic field have been studied theoretically. It has been shown that three characteristic modes of plasma expansion into the vacuum ambient may be distinguished according to whether the plasma jet has a conical, parabolic, or cylindrical shape. These modes correspond to the cases of expansion with electron cooling, isothermal expansion, and expansion with electron heating. Numerical calculations were performed for the discharge currents 300-1000 A and the axial magnetic field ranging from zero to 100 mT.