A test on external Compton models for $\gamma$-ray active galactic nuclei

There is clear evidence that the $\gamma$-ray emission from active galactic nuclei (AGNs) is attributed to the inverse Compton scatterings in the relativistic blobs near the massive black holes. If the soft seed photons are from the regions outside the blobs, a linear relation between $(\nu F_{\nu,\gamma}/\nu F_{\nu, \rm syn} u^{*})^{1/(1+\alpha)}$ and Doppler factor $\delta$ is expected, where $\nu F_{\nu,\gamma}$ and $\nu F_{\nu, \rm syn}$ are monochromatic $\gamma$-ray and synchrotron fluxes, respectively, and $u^{*}$ is the energy density of soft seed photons \citep{D97}. We estimate the soft photon energy density in the relativistic blobs contributed by the broad line region (BLRs) in these $\gamma$-ray AGNs using their broad-line emission data. The Doppler factors $\delta$ are derived from their radio core and X-ray emission data, based on the assumption that the X-ray emission is produced through synchrotron self-Compton (SSC) scatterings. We find two nearly linear correlations: $(\nu F_{\nu,\gamma}/\nu F_{\rm opt} u^{*})^{1/(1+\alpha)} \propto \delta^{1.09}$, and $(\nu F_{\nu,\gamma}/\nu F_{\rm IR} u^{*})^{1/(1+\alpha)} \propto \delta^{1.20}$, which are roughly consistent with the linear correlation predicted by the theoretical model for external Compton scatterings. Our results imply that the soft seed photons are dominantly from the BLRs in these $\gamma$-ray AGNs.