Integrated luminosity plotted against the power-law spectral index, α, for our sample of PSRs (blue symbols, as in Fig. 4) and SNRs (red symbols, from Abadi et al. 2024). The luminosity is computed by integrating the power-law flux density emission between normalised frequencies x_min = 10⁻² and x_max = 10³. Notice that changing the frequency range over which the integration is performed modifies the asymptotic logarithmic slopes, γ, at the low and high ends of α, corresponding to log(x_min) and log(x_max), respectively. The solid blue and red curves are derived using Eq. (8), with pre-factor parameters S_*, ν_*, and d given by the corresponding median values for PSRs and SNRs.