Abstract We consider the acceleration of particles due to a velocity shear in relativistic astrophysical flows. The basic physical picture and the formation of power law momentum spectra is discussed for a non-relativistic velocity field using a microscopic approach. We identify possible sites for shear acceleration in relativistic astrophysical jets and analyze their associated acceleration timescales. It is shown in particular that for a mean scattering time τ scaling with the gyro-radius, the acceleration timescale for gradual shear scales in the same manner as the synchrotron cooling timescale so that losses may no longer be able to stop the acceleration once it has started to work efficiently. Finally, the possible role of shear acceleration is discussed with reference to the relativistic jet in the quasar 3C 273.
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