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We consider a turbulent jet which is moving in a Lane--Emden ($n=5$) medium.The conserved quantity is the energy flux, which allows finding, to firstorder, an analytical expression for the velocity and an approximate trajectory.The conservation of the relativistic flux for the energy allows deriving, tofirst order, an analytical expression for the velocity, and numericallydetermining the trajectory. The back-reaction due to the radiative losses forthe trajectory is evaluated both in the classical and the relativistic case.

Classical and relativistic evolution of an extra-galactic jet with back-reaction