Constraining the shear acceleration model for the X-ray emission of large-scale extragalactic jets

The nature of the intense X-ray emission from powerful extragalactic jets at large ($>1$ kpc) scale is still debated. The scenario that invokes the inverse Compton scattering of the CMB by electrons is challenged by the lack of gamma-ray emission in the GeV band. An alternative assumes synchrotron emission by a distinct population of ultra-high energy electrons. Here we present a concrete attempt to apply this scenario, exploring the specific model in which the ultra-high energy electrons are accelerated in a shear layer surrounding the jet. We limit the study to non-relativistic flows and particle acceleration is treated by a Fokker-Planck equation. The observed relation between low energy (radio, optical) and X-ray emission prompts us to assume that the required population of pre-accelerated particles is provided by a shock responsible for the acceleration of the electrons emitting at low frequencies. We apply the model to the emission of the principal knots of the jets of PKS 0637-752 and PKS 1136-135, two of the best studied objects. For the set of fiducial parameters adopted, the condition that the jet power does not exceeds a limiting value of $10^{48}$ erg s$^{-1}$ constrains the magnetic field above $10$ $\mu$G and indicates moderate beaming ($\delta\simeq 2$) for PKS 0637-752. For both sources, the requirement that acceleration of the electrons proceeds faster than radiative cooling can be met if the magnetic turbulence in the shear layer follows a Kolmogorov spectrum, $I(k)\propto k^{-q}$ with $q=5/3$, but cannot satisfied in the Bohm-like case ($q=1$).