Example double smoothly broken power law jet apparent structure model and induced L and E_p probability distributions. Panels a.1 and a.2 show the apparent jet structure functions and (solid red and green line, respectively) adopting the parametrisation defined in Eq. (16) and using the parameter values reported in the middle panel. Panel a.0 shows the induced E_p(L) relation due to the structure (purple line). In panels a.1 and a.2, the dotted black lines show the transition angles θ_c and θ_w. The dashed lines demonstrate the analytical trends discussed in the text. Panel b.1 shows the induced luminosity function ϕ(L) = dP/dlnL = L∫P(L, E_p) dE_p (red line), with dashed lines demonstrating approximate analytical trends (significant departures are due to the core dispersion), and with the core luminosity distribution shown with a dashed dark grey line, centred at L_c and with its integral normalised to 1 − cos θ_c. Panel b.2 is the analogue of b.1, but showing the dP/dlnE_p probability distribution (green line).