2025MNRAS.540.3790Z

We apply a Gaussian process method to the extreme $\gamma$-ray flares of 3C 454.3 and 3C 279 to discover the variable patterns and then to investigate the physical origins of the giant flares. The kernels of stochastically driven damped simple harmonic oscillator (SHO), the damped random-walk (DRW), and Mat$\acute{\rm e}$rn$-3/2$ are respectively used to describe the adaptive-binning $\gamma$-ray light curves of the two flares. Our findings show that both the extreme $\gamma$-ray flares of 3C 454.3 and 3C 279 clearly prefer the SHO kernel in the overdamped mode and the Mat$\acute{\rm e}$rn$-3/2$ kernel over the DRW kernel. The resulted SHO and Mat$\acute{\rm e}$rn$-3/2$ power spectral densities (PSDs) are the same for each object, with the index changing from -4 at high frequencies to 0 at low frequencies. The patterns of the two flares are both approaching the critical damping mode with the quality factor $Q\approx 0.4$ (i.e. the damping ratio $\eta \approx 1.25$), but with slightly different damping time-scales. The characteristic time-scale (corresponding to the broken frequency in the PSD) for 3C 454.3 is 2-3 d and 3-5 d for 3C 279. The variable patterns found here suggest that once the system responds to the energy injection disturbance, the release of the energy in the system is finished abruptly. The obtained time-scale provides a constraint on the size of energy dissipation region for each source.