We present a method of constraining the properties of the \gamma -ray emitting region in flat spectrum radio quasars ( FSRQs ) in the one-zone proton synchrotron model , where the \gamma -rays are produced by synchrotron radiation of relativistic protons .
We show that for low enough values of the Doppler factor \delta , 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 \delta ; 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 \delta and the region ’ s distance from the galactic center , we use the EM cascade to determine a minimum distance for each value of \delta .
We complement the EM cascade constraint with one derived from variability arguments and apply our method to the FSRQ 3C 273 .
We find that \delta \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 _ { 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 .