tokens: (− z rec for the counter-jet) for which a recollimation shock is formed due to the jet and wind momentum fluxes balancing each other in the y -axis direction (perpendicular to the jet propagation). This condition can be approximately expressed as follows ( Bosch-Ramon Barkov 2016 ): where Ṗ w = Ṁ w v w is the wind thrust. For γ j = 1.2 we obtain z rec = 1.4 × 10 12 cm ≈ 0.47 a , whereas for γ j = 3, z rec = 3.1 × 10 11 cm ≈ 0.10 a . Some combinations of Ṗ w , Ṗ j and γ j yield non-physical (complex) values of z rec in Eq. ( 4 ), which means that for those sets of parameters no recollimation shock is formed, as the momentum flux balance is never reached (see Yoon et al. 2016 , for a non-relativistic simulation where this effect is studied). Electrons are injected at , since the jet displacement in the x and y coordinates is small at z = ± z rec (≲0.005 a in the worst case). The particles are assumed to follow a power-law energy distribution with spectral index −2, typical for acceleration in strong shocks via the Fermi I mechanism (e.g Drury 1983 ), with a minimum energy of MeV. An exponential cutoff is also assumed: where E ′ is the electron energy in the FF. The normalization of Q ′( E ′) is obtained from the condition that the total power available for non-thermal electrons in the FF is . Electrons are convected from along the jet path while they cool down via adiabatic, synchrotron and IC processes. For simplicity, we neglect particle acceleration beyond , even though weak shocks and turbulence may (re)accelerate particles further downstream in the jets. Analytical expressions are used for the energy losses in the FF (see Longair 1981 ; Khangulyan et al. 2014 for synchrotron and IC losses, respectively). The characteristic timescale for adiabatic cooling is , where R j ( l )∝ l is the jet radius. The magnetic pressure at the accelerator (located at ) is characterized in the FF as a fraction η B of the total jet energy density: where and r j, 0 = R j ( z rec ) are the magnetic field and the jet radius at , respectively. For the same , this fraction relates to the ratio of magnetic pressure to stellar photon energy density in the LF as , which is useful to compare the expected radiation outputs of synchrotron and IC processes. This yields for γ j = 1.2, and for γ j = 3. The jet magnetic field is assumed perpendicular to the flow in the FF, since at the scales considered in this work, far from the jet launching point, a dominant toroidal component for the magnetic field is expected (e.g., Pudritz et al. 2012 ). Given the adopted assumptions plus frozen in conditions, the magnetic field at each jet point can be computed as: The cutoff energy is the maximum energy that the electrons can achieve in the accelerator region in the FF, and it is obtained as follows: We compare the acceleration timescale with the cooling timescale , where Ė ′ is the cooling rate accounting for adiabatic, synchrotron and IC losses. In this work we set η acc = 0.1 as a
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