A minimalist approach to BL Lacertae: explaining gamma-ray spectral and temporal variability with a single physical parameter

TL;DR

BL Lacertae's gamma-ray variability is addressed within a minimalist one-zone SSC model using a log-parabolic electron distribution, positing that changes in the electron peak energy $\gamma_p$ drive both flux and spectral evolution. The authors analyze nearly 17 years of Fermi-LAT data with adaptive-bin light curves and MCMC fits to a log-parabolic spectrum per bin to test the $L_{IC}$–$E_p$ relation, finding a robust near-linear scaling $L_{IC} \propto E_p$ consistent with a sole varying $\gamma_p$. The result supports a single-parameter variability mechanism in which $\gamma_p$ dominates the observed gamma-ray behavior, with $\beta$ showing little dependence on $E_p$ and hints of a Thomson/Klein-Nishina transition during bright flares. This minimalist interpretation, if extended to other BL Lacs, could provide strong constraints on high-energy emission regions and radiative processes in blazars.

Abstract

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.

Figures (4)

• Figure 1: Temporal (top) and spectral (bottom) evolution of BL Lacertae in gamma rays. In the integrated photon-flux light curve shown in the top panel, the three arrows mark the time bins corresponding to the three spectra shown in the bottom panels. The chosen TS for upper limits is set to 25. In the bottom panels, we can see that the log-parabola (green solid line) maximum likelihood spectral peak (orange vertical line) shifts towards higher energies when the gamma-ray flux increases. The dotted vertical lines represent the $68\%$ containment interval for the MCMC posterior distribution on $E_p$. The spectra presented here are in the observed frame.
• Figure 2: Rest-frame correlations of $L_{IC}$ and $\beta$ with $E_p$. Top: $L_{IC}$ scales nearly linearly with $E_p$, consistent with variability driven mainly by changes in $\gamma_p$. The inset shows the jackknife resampling distribution of the slope, centered at $\Gamma = 0.98$. Orange points mark upper limits on $E_p$ (not included in the fit), defined when $E_p < 300$ MeV or when uncertainties extend below 100 MeV. Dashed lines indicate the correlations expected if variability depends on a single parameter (see text). Bottom: The spectral curvature $\beta$ shows no significant dependence on $E_p$, suggesting BL Lacertae may be transitioning out of the Thomson regime. $N_0$ and $\Gamma$ are described in the text.
• Figure 3: Evolution of BL Lacertae's SED when only the electron peak energy $\gamma_{p}$ (or equivalently $\gamma_{0}$) is free to vary, while all other parameters are fixed to the values in Table \ref{['tab:starting_parameters']}. The black line tracks the IC peak positions, the green line shows the jackknife correlation derived from the data, and the color gradient represents the SSC emission models for different values of $\gamma_{p}$. The results indicate that BL Lacertae remains mostly in the Thomson regime, with hints of entering a Thomson/Klein-Nishina transition during its brightest flares. The red curves correspond to the log-parabolic models fitted to the highest- and lowest-flux bins shown in the light curve of Fig. \ref{['fig:LCs_and_specs']}.
• Figure 4: The best fit one-zone SSC model for the multiwavelength data described in Sect. \ref{['subsec:SED_multiwavelength_data']}. The rounded parameters for this fit are found in Table \ref{['tab:starting_parameters']}. We use these parameters as a starting point in the investigation of the SED spectral evolution when $\gamma_{p}$ (or, equivalently here, $\gamma_0$) is the only free variable. The Fermi-LAT data shown here covers the 17 years of observations.