The effect of adiabatic losses on spectra of stationary jets and the origin of soft radio spectra of accreting black hole sources

It has been suggested that adiabatic energy losses are not effective in stationary jets, where the jet expansion is not associated with net work. Here, we study jet solutions without them, assuming that adiabatic losses are balanced by electron reacceleration. The absence of effective adiabatic losses makes electron advection along the jet an important process, and we solve the electron kinetic equation including that process. We find analytical solutions for the case of conical jets with advection and synchrotron losses. We show that accounting for adiabatic losses in the case of sources showing soft partially self-absorbed spectra with the spectral index of alpha < 0 in the radio-to-IR regime requires deposition of large amounts of energy at large distances in the jet. On the other hand, such spectra can be accounted for by advection of electrons in the jet. We compare our results to the quiescent spectrum of the blazar Mrk 421. We find its soft radio-IR spectrum can be fitted either by a model without adiabatic losses and advection of electrons or by one with adiabatic losses, but the latter requires injection of a very large power at large distances.