Minimalist approach to BL Lacertae: Explaining gamma-ray spectral and temporal variability with a single physical parameter

The eponymous BL Lac object BL Lacertae is one of the most well-monitored active galactic nuclei, frequently observed from radio to gamma rays. Its relatively soft $γ$-ray spectrum peaks near 500~MeV, and since 2020 it has undergone an exceptional series of flaring episodes. The observed emission is well described by synchrotron self-Compton (SSC) models, with negligible contribution from external seed photons. We investigate the physical origin of BL~Lacertae's $γ$-ray temporal and spectral variability using data from the Large Area Telescope (LAT) on board the \textit{Fermi} Gamma-ray Space Telescope, and show that this variability can be explained by a single varying parameter, namely the electrons' peak energy, $γ_p$, under a single-zone SSC scenario with a log-parabolic electron distribution. We use a Markov chain Monte Carlo to estimate the spectral parameters of BL Lacertae over time, selected from an adaptive-binned gamma-ray light curve. We then study the correlation between the inverse Compton peak luminosity, $L_{IC}$, and the position of this peak on the SED energy axis, $E_p$, and compare it with what is expected for a single-zone SSC scenario when only one parameter is free to vary. We find a correlation $L_{IC} = 10^{42.33\pm0.15\pm0.18_{sys}}E_p^{0.98\pm0.05\pm0.06_{sys}}$, consistent, within the errors, with the linear relation $L_{\mathrm{IC}} \propto E_p$, expected when $γ_p$ is the only free parameter in the assumed SSC model. This result supports a minimalist SSC scenario in which changes in $γ_p$ dominate the observed temporal and spectral variability of BL~Lacertae.