Time-dependent Modeling of the Subhour Spectral Evolution during the 2013 Outburst of Mrk 421

Abstract In 2013 April, the TeV blazar Markarian 421 underwent one of its most powerful emission outbursts recorded to date. An extensive multi-instrument campaign featuring MAGIC, VERITAS, and NuSTAR provided comprehensive very high-energy (VHE; E > 100 GeV) and X-ray coverage over nine consecutive days. The VHE flux peaked at approximately 15 times that of the Crab Nebula, with rapid variability detected on timescales down to 15 minutes in both X-ray and VHE bands. This rich data set, characterized by its dense temporal coverage and high photon statistics, offers an unparalleled opportunity to probe the broadband emission dynamics in blazars. In this work, we perform a detailed spectral analysis of the X-ray and VHE emissions on subhour timescales throughout the flare. We identify several clockwise spectral hysteresis loops in the X-rays, revealing a spectral evolution more complex than a simple harder-when-brighter trend. The VHE spectrum extends beyond 10 TeV, and its temporal evolution closely mirrors the behavior in the X-rays. Crucially, we report the first evidence of VHE spectral hysteresis occurring simultaneously with the X-ray loops. To interpret these findings, we apply a time-dependent leptonic model to 240 broadband spectral energy distributions (SEDs) binned on a 15 minute scale, allowing us to self-consistently track the particle distribution’s history. Our modeling shows that the majority of the subhour flux and spectral variations are driven by changes in the luminosity and slope of the injected electron distribution. The required variations in the electron slope are difficult to reconcile with magnetic reconnection, but they are consistent with a shock-acceleration scenario where the shock compression ratio evolves by a factor of ∼2. The model also points to a relatively stable magnetic field and emitting region size, favoring a scenario where the emission originates from a stationary feature in the jet, such as a recollimation shock. However, this scenario requires a jet Lorentz factor that significantly exceeds values from VLBI measurements to account for the high minimum electron energy implied by the lack of variability in the optical band.