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Time-Dependent Modeling of the Sub-Hour Spectral Evolution During the 2013 Outburst of Mrk 421

MAGIC Collaboration, K. Abe, S. Abe, J. Abhir, A. Abhishek, A. Aguasca-Cabot, I. Agudo, T. Aniello, S. Ansoldi, L. A. Antonelli, A. Arbet Engels, C. Arcaro, T. T. H. Arnesen, A. Babić, C. Bakshi, U. Barres de Almeida, J. A. Barrio, L. Barrios-Jiménez, I. Batković, J. Baxter, J. Becerra González, W. Bednarek, E. Bernardini, J. Bernete, A. Berti, C. Bigongiari, A. Biland, O. Blanch, G. Bonnoli, Ž Bošnjak, E. Bronzini, I. Burelli, A. Campoy-Ordaz, A. Carosi, R. Carosi, M. Carretero-Castrillo, A. J. Castro-Tirado, D. Cerasole, G. Ceribella, Y. Chai, A. Cifuentes, J. L. Contreras, J. Cortina, S. Covino, F. D'Ammando, P. Da Vela, F. Dazzi, A. De Angelis, B. De Lotto, R. de Menezes, J. Delgado, C. Delgado Mendez, F. Di Pierro, R. Di Tria, L. Di Venere, A. Dinesh, D. Dominis Prester, A. Donini, D. Dorner, M. Doro, L. Eisenberger, D. Elsaesser, J. Escudero, L. Fariña, L. Foffano, L. Font, S. Fröse, Y. Fukazawa, R. J. García López, S. García Soto, M. Garczarczyk, S. Gasparyan, J. G. Giesbrecht Paiva, N. Giglietto, F. Giordano, P. Gliwny, T. Gradetzke, R. Grau, D. Green, J. G. Green, P. Günther, A. Hahn, T. Hassan, L. Heckmann, J. Herrera Llorente, D. Hrupec, D. Israyelyan, J. Jahanvi, I. Jiménez Martínez, J. Jiménez Quiles, J. Jormanainen, S. Kankkunen, T. Kayanoki, J. Konrad, P. M. Kouch, G. Koziol, H. Kubo, J. Kushida, M. Laínez, A. Lamastra, E. Lindfors, S. Lombardi, F. Longo, M. López-Moya, A. López-Oramas, S. Loporchio, L. Lulić, E. Lyard, P. Majumdar, M. Makariev, M. Mallamaci, G. Maneva, M. Manganaro, S. Mangano, K. Mannheim, S. Marchesi, M. Mariotti, M. Martínez, P. Maruševec, S. Menchiari, J. Méndez Gallego, S. Menon, D. Miceli, J. M. Miranda, R. Mirzoyan, M. Molero González, E. Molina, H. A. Mondal, A. Moralejo, C. Nanci, A. Negro, V. Neustroev, L. Nickel, M. Nievas Rosillo, C. Nigro, L. Nikolić, S. Nozaki, A. Okumura, J. Otero-Santos, S. Paiano, D. Paneque, R. Paoletti, J. M. Paredes, M. Peresano, M. Persic, M. Pihet, G. Pirola, F. Podobnik, P. G. Prada Moroni, E. Prandini, W. Rhode, M. Ribó, J. Rico, A. Roy, N. Sahakyan, F. G. Saturni, K. Schmitz, F. Schmuckermaier, T. Schweizer, A. Sciaccaluga, G. Silvestri, A. Simongini, J. Sitarek, V. Sliusar, D. Sobczynska, A. Stamerra, J. Strišković, D. Strom, M. Strzys, Y. Suda, H. Tajima, R. Takeishi, F. Tavecchio, T. Terzić, M. Teshima, A. Tutone, S. Ubach, J. van Scherpenberg, M. Vazquez Acosta, S. Ventura, G. Verna, I. Viale, A. Vigliano, C. F. Vigorito, E. Visentin, V. Vitale, I. Vovk, R. Walter, F. Wersig, M. Will, T. Yamamoto, P. K. H. Yeung, M. Petropoulou, M. Polkas, A. Mastichiadis

Abstract

In April 2013, the TeV blazar Markarian~421 underwent one of its most powerful emission outbursts to date. An extensive multi-instrument campaign featuring MAGIC, VERITAS, and \textit{NuSTAR} provided comprehensive very-high-energy (VHE; $E > 100$\,GeV) and X-ray coverage over nine consecutive days. In this work, we perform a detailed spectral analysis of the X-ray and VHE emissions on sub-hour 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. 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 sub-hour 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 are consistent with a shock-acceleration scenario where the shock compression ratio evolves by a factor of $\sim2$. 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.

Time-Dependent Modeling of the Sub-Hour Spectral Evolution During the 2013 Outburst of Mrk 421

Abstract

In April 2013, the TeV blazar Markarian~421 underwent one of its most powerful emission outbursts to date. An extensive multi-instrument campaign featuring MAGIC, VERITAS, and \textit{NuSTAR} provided comprehensive very-high-energy (VHE; \,GeV) and X-ray coverage over nine consecutive days. In this work, we perform a detailed spectral analysis of the X-ray and VHE emissions on sub-hour 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. 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 sub-hour 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 are consistent with a shock-acceleration scenario where the shock compression ratio evolves by a factor of . 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.

Paper Structure

This paper contains 22 sections, 14 equations, 22 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Summary of the April 2013 flare and the observed broadband evolution characteristics
  3. Scope of this work
  4. Observations and data processing
  5. X-ray & VHE spectral behaviors
  6. Daily VHE spectral analysis
  7. Time-dependent modeling of the flare
  8. Model set-up and initial conditions
  9. Stationary States
  10. Time-dependent intraday SED modeling
  11. Multi-band light curves from the theoretical model and comparison to the measurements
  12. Discussion
  13. Spectral hysteresis in the temporal evolution of the X-ray and VHE emissions
  14. Particle dynamics and acceleration processes
  15. Caveats and short comings of the model
  16. ...and 7 more sections

Figures (22)

  • Figure 1: Spectral parameter $\alpha$ versus the 3-7 keV flux as measured by NuSTAR between MJD 56393 and MJD 56399. $\alpha_{\rm X-ray}$ is derived from a log-parabola fit with the curvature parameter $\beta_{\rm X-ray}$ fixed to 0.38 (see text in Sect. \ref{['sec:analysis']} for more details). The data are binned in 15-min time intervals. Each day is plotted with a different color and black arrows indicate the direction of time.
  • Figure 2: Spectral parameter $\alpha$ versus flux for (a) MAGIC and (b) NuSTAR between MJD 56393 and MJD 56399. The NuSTAR fluxes are evaluated in the 3-7 keV bands, while for MAGIC they are computed above 400 GeV. In each panel the data are binned over 30 min, strictly simultaneous intervals. Each day is plotted with a different color and black arrows indicate the direction of time. The MAGIC points plotted with transparent markers are temporal bins that are not accompanied by a strictly simultaneous NuSTAR measurement.
  • Figure 3: Nightly MAGIC spectra from MJD 56395 and MJD 56397, which show the highest VHE flux of the April 2013 flare. The dotted and dashed lines are the best-fit function using a log-parabola and log-parabola with cut-off model, respectively. For both days, the log-parabola model with a cut-off is preferred at a significance level above $3\sigma$. The SED points and spectral model curves are corrected for the EBL absorption using the model of 2011MNRAS.410.2556D.
  • Figure 4: Stationary states fitted to nightly averaged SEDs, from MJD 56393 to 56398. The light blue dashed line represent the emission from the "slow" zone, the dark blue dotted line is the emission from the "fast" zone and the solid line is the sum of the two components. Data are depicted with dark points. The corresponding model parameters are listed in Tab. \ref{['table:steady']}.
  • Figure 5: Same as Fig. \ref{['stationary_states']}, from MJD 56399 to MJD 56401.
  • ...and 17 more figures