  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 first
order, an analytical expression for the velocity and an approximate trajectory.
The conservation of the relativistic flux for the energy allows deriving, to
first order, an analytical expression for the velocity, and numerically
determining the trajectory. The back-reaction due to the radiative losses for
the trajectory is evaluated both in the classical and the relativistic case.
