Fitting the SED of 3C 273

The observed high energy hump of 3C 273 and many quasars peaks near 1 MeV where the spectrum breaks from $\sim E^{-1.7}$ to $\sim E^{-2.5}$ and extends up to a few GeV. The cascade spectrum of escaping gamma-rays is shown in Fig 2(a) and has a slope which is reasonable for the region below the peak, but extends unbroken up to a few GeV. The (linear) cascading that resulted in Fig 2(a) included only external target photons and synchrotron target photons from directly accelerated electrons. However, the photon density of the cascade photons below 1 MeV provides significant pair-production optical depth to cascade photons above 1 MeV if the emission region radius is sufficiently small. The chain curve in Fig 2(b) shows the optical depth for the case of normalizing the chain curve in Fig 2(a) to the observed SED at X-ray energies and using an emission region radius of 0.001 pc and a Doppler factor of 1.3. As can be seen, the optical depth is $\tau_{\gamma\gamma}=1$ at $\sim 0.5$ MeV. In a rigorous calculation, one would treat the non-linear problem by solving the non-linear problem where the whole of the SED can provide target photons for the cascade. Instead, here we make the approximation that the spectrum after such a (non-linear) cascade can be obtained simply by multiplying the result from the linear cascade by $[1-\exp(-\tau_{\gamma\gamma})]/\tau_{\gamma\gamma}$. Further absorption by external photons, e.g. from the accretion disk (dotted in Fig 2b) are negligible, as in this case is absorption outside the emission region by emission region photons. The resulting spectrum is plotted as the solid curve in Fig. 3.