Doppler factor and ρ_eff as a function of a jet’s apparent half-opening angle θ_app. The semi-logarithmic Doppler factor plot in the upper panel is for a jet with Γ = 10 and ρ = 0.21, and uses Eq. (4) to convert θ_app to θ_ob. The lower panel plot shows ρ_eff as a function of θ_app from Eq. (9) using ρ = 0.21 and f_sh = 0.33, the best fit values found in Sect. 3.2.