Constraints of Flat Spectrum Radio Quasars in the hadronic model: the case of 3C 273
We present a method of constraining the properties of the $γ$-ray emitting region in flat spectrum radio quasars (FSRQs) in the one-zone proton synchrotron model, where the $γ$-rays are produced by synchrotron radiation of relativistic protons. We show that for low enough values of the Doppler factor $δ$, the emission from the electromagnetic (EM) cascade which is initiated by the internal absorption of high-energy photons from photohadronic interactions may exceed the observed $\sim$GeV flux. We use that effect to derive an absolute lower limit of $δ$; first, an analytical one, in the asymptotic limit where the external radiation from the broad line region (BLR) is negligible, and then a numerical one in the more general case that includes BLR radiation. As its energy density in the emission region depends on $δ$ and the region's distance from the galactic center, we use the EM cascade to determine a minimum distance for each value of $δ$. We complement the EM cascade constraint with one derived from variability arguments and apply our method to the FSRQ 3C 273. We find that $δ\gtrsim 18-20$ for $B \lesssim 30$ G and $\sim$day timescale variability; the emission region is located outside the BLR, namely at $r \gtrsim 10 R_{\rm BLR} \sim 3$ pc; the model requires at pc-scale distances stronger magnetic fields than those inferred from core shift observations; while the jet power exceeds by at least one order of magnitude the accretion power. In short, our results disfavour the proton synchrotron model for the FSRQ 3C 273.