2004MNRAS.355..835H

We present an extension of the scale-invariance formalism by Heinz & Sunyaev which includes the effects of radiative cooling on the X-ray synchrotron emission from core-dominated jets and outflows. We derive the scaling relations between the radio luminosity L_r, the synchrotron X-ray luminosity L_x, the mass M, and the accretion rate of an accreting black hole. We argue that the inclusion of synchrotron losses in the scaling relations is required by the data used in Merloni et al. to define the `fundamental plane' correlation between L<sub>r</sub>, L<sub>x</sub>and M in active galactic nuclei and Galactic X-ray binaries. We then fit the new scaling relations to the `fundamental plane' correlation. This allows us to derive statistical constraints on the contribution from jet synchrotron emission to the X-rays from black holes, or, alternatively, on the particle acceleration scheme at work in these jets. We find that, in order for jet synchrotron X-rays to be consistent with the observed L_r-L_x-M correlation, the global particle spectrum must be both steep,, and unbroken down to energies well below the radio synchrotron regime. Such a steep particle distribution would imply steeper X-ray spectra than observed in most sources contributing to the correlation with available X-ray spectral index measurements. We suggest that, unless some assumptions of the scale-invariance hypothesis are broken, another emission mechanism (likely either disc radiation and/or inverse Compton scattering in the jet) contributes significantly to the X-rays of a sizeable fraction of the sources.