Time-Dependent Leptohadronic Modeling of Markarian 421

Hai-Bin Hu; Rui Xue; Ze-Rui Wang

REVIEW 4 major objections 4 minor 91 references

Time-Dependent Leptohadronic Modeling of Markarian 421

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A one-zone time-dependent leptohadronic model can reproduce Mrk 421's 2013 hard X-ray excess at sub-Eddington jet power, but not the 2016 excess without super-Eddington power, and it fails to reproduce the 2017 X-ray/very-high-energy flare.

desk verdict A solid, honestly-hedged epoch-by-epoch leptohadronic study of Mrk 421; the 2013 pp scenario needs a VHE comparison before the 'both' claim can stand. read the letter →

arxiv 2507.01265 v1 pith:4NEUVEJD submitted 2025-07-02 astro-ph.HE astro-ph.GA

Rui Xue , Ze-Rui Wang , Hai-Bin Hu This is my paper

classification astro-ph.HEastro-ph.GA

keywords blazarMarkarian421hardX-rayexcessleptohadronicmodelingphotohadronicinteractionshadronucleartime-dependentspectralvery-high-energygammarays

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether protons, not just electrons, can explain three puzzling episodes in the blazar Markarian 421: the hard X-ray excesses seen in 2013 and 2016 and the correlated X-ray and very-high-energy flare of 2017. Using a time-dependent one-zone model that follows electrons, protons, secondary pairs, and photons together, it finds that the 2013 excess can be produced by secondary emission from either photohadronic or hadronuclear interactions without exceeding the Eddington jet power. For 2016, only the photohadronic channel can fit, and only by allowing a super-Eddington jet power; the hadronuclear channel overshoots the GeV data. For 2017, hadronic interactions fail entirely because the secondary $\pi^0$-decay and cascade emission overshoot the GeV band. The takeaway is that hadronic interpretations of Mrk 421 are viable in some epochs and not others, which sharpens what future multiwavelength observations must look for.

What carries the argument

The load-bearing machinery is a time-dependent one-zone leptohadronic code in which three coupled continuity equations -- for primary electrons and protons, for secondary electron-positron pairs, and for soft and high-energy photons -- are solved simultaneously with a fully implicit difference scheme. Cooling of electrons includes synchrotron and Klein-Nishina-corrected inverse Compton; cooling of protons includes synchrotron, photopion ($p\gamma$), Bethe-Heitler (BH), and proton-proton ($pp$) interactions; secondary pairs enter from all these hadronic channels plus internal $\gamma\gamma$ annihilation. The central identities that carry the fits are the synchrotron peak frequencies of secondary pairs: Bethe-Heitler pairs with $\gamma_e\sim3\times10^6$ land near the hard X-ray band for the 2013 parameters, and $\pi^\pm$ decay pairs with $\gamma_e\sim10^6$ do the same in the hadronuclear case. Solving the equations together matters because the hadronic radiation feeds back into the target photon field and therefore changes the cooling of both primary and secondary particles.

What would settle it

A multi-epoch campaign that measures the hard X-ray excess during a 2013-like low state and looks for a roughly 1.6-hour delay relative to the ~28 eV synchrotron flux would settle the origin: a sustained lag at that scale supports the Bethe-Heitler photohadronic interpretation, while a clean absence of delay would falsify it; a 2016-like excess detected with independent evidence that the jet power stays sub-Eddington would falsify the claim that only super-Eddington photohadronic power can explain that epoch.

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Extended reading notes

Core claim

The paper claims that the hard X-ray excess of Mrk 421 is not a single phenomenon. In the low state of 2013, synchrotron emission from secondary electron-positron pairs produced by Bethe-Heitler pair production (photohadronic) or by charged-pion decay in proton-proton collisions (hadronuclear) can each reproduce the excess with a total jet power near the Eddington luminosity; the photohadronic component lives only in the hard X-ray band, while the hadronuclear channel also predicts a hard TeV component from $\pi^0$ decay. In 2016, the excess is brighter relative to the synchrotron bump, and only the Bethe-Heitler secondary emission can fit, at the price of a super-Eddington jet power, whereas proton-proton secondary emission overshoots the GeV data. For the 2017 correlated X-ray and VHE flare, neither hadronic channel works inside the parameter space allowed by the pre-flare one-zone SSC state, because the secondary pairs and $\pi^0$-decay photons inevitably overshoot the GeV band. The paper therefore concludes that at least one emitting zone beyond the standard proton-bearing blob is needed for Mrk 421's 2017 behavior.

Load-bearing premise

The whole fit assumes that all of Mrk 421's nonthermal emission comes from one homogeneous spherical zone, and if separate emitting regions contribute -- which the paper itself concludes for 2017 -- the hadronic components and jet-power conclusions for 2013 are not uniquely required.

Editorial extensions

If this is right

For the 2013 low state, a sub-Eddington proton jet is enough; the hard X-ray band becomes a direct probe of Bethe-Heitler or proton-proton secondary emission rather than of primary electrons.
The two channels are distinguishable: Bethe-Heitler hard X-rays should lag the $\sim$28 eV synchrotron photons by a cooling timescale of about 1.6 hours in the observer frame, while proton-proton secondary emission should show no such lag.
For 2016, a hadronic explanation would force the jet power above Eddington, so a two-zone leptonic solution with a narrow electron distribution is preferred for that epoch.
For 2017, the GeV upper limits rule out one-zone hadronic origins for the correlated X-ray/VHE flare, reinforcing the case for multiple emitting zones or a separate narrow electron population.
Future simultaneous hard X-ray and VHE flare observations can test whether hard X-ray variability is correlated with other bands, which the photohadronic model says it should not be.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the single-zone assumption is what makes the 2013 sub-Eddington conclusion possible; if the hard X-ray excess in 2013 actually arises in a separate region, the fitted hadronic components need not be unique, and the sub-Eddington conclusion would be an artifact.
Editorial inference: the branching-ratio argument against hadronic 2017 flares generalizes: because $\pi^\pm$ and $\pi^0$ yields are locked together in proton-proton interactions, any pp scenario that makes hard X-rays will also make VHE $\gamma$-rays, so X-ray-dominant flares with weak VHE variability are generically hard for one-zone hadronic models.
Editorial inference: the same time-dependent machinery could be applied to other nearby high-synchrotron-peaked blazars with hard X-ray excesses, such as PKS 2155-304, to see whether the epoch-dependent pattern found here is universal.
Editorial inference: the predicted roughly 1.6-hour lag is a concrete target for a monitoring campaign; measuring or bounding it would discriminate between photohadronic and hadronuclear origins without any change to the model.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Rui Xue et al. have done something genuinely new here: a time-dependent leptohadronic model applied epoch-by-epoch to Mrk 421's hard X-ray excesses (2013, 2016) and the 2017 X-ray/VHE flare, with a fresh Fermi-LAT GeV dataset for 2016. The mechanisms—Bethe-Heitler pairs and pp secondaries—were proposed earlier, but the epoch-resolved fits and the 2017 branching-ratio argument are not in the cited literature. The 2013 result that both photohadronic and hadronuclear channels can produce the excess at sub-Eddington power is a meaningful existence proof, and the 2017 failure of hadronic channels is a clean, honest negative result. The paper is also refreshingly well-hedged: claims are worded as possibilities, not certainties.

The central soft spot is the pp scenario for 2013. The authors admit that the π0-decay component in the VHE band is unavoidable and has not been observed, but they never quantitatively compare that predicted component to the contemporaneous VHE data plotted in the same figure. That comparison, after EBL absorption at z=0.031, could easily falsify the pp interpretation—and since the abstract's "both" rests on it, this is load-bearing. My second concern is the one-zone assumption: the paper itself concludes the 2017 flare requires multiple zones, so the 2013 sub-Eddington conclusion may be partly an artifact of assigning all emission to a single homogeneous blob. Third, the fits have many hand-tuned parameters and no uncertainties or goodness-of-fit metrics; they are existence proofs rather than constraints. The 2016 "super-Eddington" conclusion should be read as a cost of normalization, not an independent constraint.

These are fixable. The paper deserves a serious referee. The authors should add an EBL-corrected VHE comparison for the pp scenario, discuss the one-zone degeneracy explicitly, and provide at least some measure of fit quality or parameter sensitivity. For readers interested in hadronic models of BL Lacs, this is worth engaging with.

Referee Report

4 major / 4 minor

Summary. This paper presents a time-dependent, one-zone leptohadronic model for Mrk 421. The model solves coupled continuity equations for primary electrons and protons, secondary electron-positron pairs, and photons, including synchrotron, SSC/IC with Klein-Nishina corrections, p-gamma, Bethe-Heitler, pp interactions, and gamma-gamma absorption. The authors apply it to the 2013 hard X-ray excess (MJD 56302), the 2016 hard X-ray excess (MJD 57422-57429), and the 2017 correlated X-ray/VHE flare (MJD 57788). Their principal result is epoch-dependent viability: in 2013 both photohadronic (BH-dominated) and hadronuclear (pp-dominated) secondary emissions can reproduce the excess with sub-Eddington proton power; in 2016 only the photohadronic interpretation works and requires super-Eddington proton power; in 2017 hadronic interactions cannot explain the correlated flare. They also present a two-zone leptonic alternative for the 2016 state and discuss acceleration scenarios.

Significance. The paper is a useful addition to the hadronic-modeling literature because it treats the cascade time-dependence self-consistently and gives concrete, falsifiable predictions: a delayed hard X-ray component relative to ~28 eV photons for the BH scenario, and a hard TeV pi0-decay component for the pp scenario. The numerical scheme is described in enough detail to be reproducible, and the authors use public tools (NAIMA, easyFermi, iminuit). If the claims survive the missing quantitative checks identified below, the paper would establish that hadronic contributions are a viable, epoch-dependent explanation for Mrk 421's hard X-ray excesses, with important implications for jet power constraints.

major comments (4)

[Sec. 3.1, Fig. 2 (lower right), Abstract] The statement that both photohadronic and pp interactions can explain the 2013 hard X-ray excess without super-Eddington power rests on the pp fit, whose unavoidable pi0-decay VHE component is admitted to be 'not yet observed.' The manuscript never computes the EBL-attenuated pi0-decay flux at z = 0.031 and compares it quantitatively with the contemporaneous VHE points or upper limits already shown in Fig. 2. This comparison is essential: if the predicted EBL-corrected pi0 flux lies above the observed VHE data in any energy bin, the pp scenario is ruled out and the 'both' conclusion in the Abstract reduces to photohadronic only. Please either add this comparison or weaken the pp claim accordingly.
[Sec. 3.1, Table 1] The sub-Eddington result for 2013 is partly built into the model by construction: for the p-gamma/BH fit the authors set Gamma^2 L_p,inj ≈ L_Edd, and for the pp fit they set Gamma^2 L_p,inj ≈ P_p,cold ≈ 0.5 L_Edd before fitting. Consequently the phrase 'without introducing a super-Eddington jet power' describes an input assumption, not a fit outcome. The text should be rephrased to make this explicit, and ideally the authors should report how the fits would change if L_p,inj were a free parameter, so the reader can see how much headroom exists.
[Sec. 2 first paragraph; Sec. 3.2; Sec. 5] The model assumes that all emission comes from one homogeneous spherical zone, yet the paper itself concludes that the 2017 flare requires multiple emitting zones (Sec. 5) and explores a two-zone leptonic model for 2016 (Sec. 3.2). If the hard X-ray excess in 2013 originates in a separate leptonic region rather than in the same zone as the primary synchrotron/SSC component, the fitted hadronic components are not uniquely required and the Eddington-based conclusions are weakened. The authors should explicitly state this degeneracy and discuss whether the available variability information (e.g., the delay test proposed in Sec. 3.1) can distinguish co-spatial hadronic emission from a separate leptonic zone.
[Table 1; Figs. 2–4] No goodness-of-fit measures or parameter uncertainties are reported for any of the SED fits. With roughly a dozen free parameters (delta_D, B, R, L_e,inj, spectral indices, break energies, L_p,inj, n_H, gamma_p,max), the visual agreement in Figs. 2–4 does not by itself establish that one scenario is preferred over another. Please report residuals or a chi-square/TS value per scenario, and demonstrate, at least for the 2013 case, that the conclusion is stable under the expected parameter degeneracies.

minor comments (4)

[Fig. 2 and Fig. 3] Several axis labels and legends contain typographical errors: 'ν (H)' should be 'ν (Hz)', and 'To al', 'ca cade', 'lepto ic', and 'erg −1 cm−2' should be corrected to 'Total', 'cascade', 'leptonic', and 'erg s−1 cm−2'.
[References [66] and [69]] The bibliographic entries for Dermer & Menon (2009) and Press et al. (1989) contain '???' as the publisher information; these should be completed.
[Sec. 3.1] The statement that the target photon ratio n_BH_t(28 eV)/n_pγ_t(81 keV) is 'much larger than 143' should be supported by a quantitative estimate from the fitted low-energy SED, since this ratio is a key input to the conclusion that BH dominates over pγ.
[Sec. 4] The reference to 'the upper panel of FIG. 2' should specify the upper right SED panel, rather than the timescale panel, to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SED fits are tested against independent multiwavelength data and yield testable predictions.

full rationale

The central derivations are not circular. The model solves coupled continuity equations with standard microphysical rates (synchrotron, IC, pgamma, Bethe-Heitler, pp, gamma-gamma), and the parameters are fitted to external simultaneous SEDs from NuSTAR, Fermi-LAT, and MAGIC/VERITAS. The sub-Eddington character of the 2013 solutions is an explicitly declared modeling constraint ('we boldly assume that the jet power is equal to the Eddington luminosity'; 'we set nH = 0 cm-3 and Gamma^2 Lp,inj ~ LEdd'), and the scientific content is the nontrivial existence of hadronic fits at that assumed power, not a derived measurement of the power. The 2016 super-Eddington conclusion follows from fitting Lp,inj to reproduce the observed hard-X-ray flux, which is a physical inference from the fit rather than a renamed input. The pp scenario's hard-TeV pi0-decay prediction and the Bethe-Heitler hard-X-ray delay prediction are independent of the fitted normalizations (fixed by isospin branching and cooling timescales), giving the model falsifiable content. The self-citations [23,24,84] supply parameter priors and an analytical method, but they do not by themselves force the fitted SEDs, which are anchored to external data. A genuine caveat is that the predicted pi0-decay TeV component is not quantitatively compared with contemporaneous VHE data points, but that is an unaddressed falsification risk and a physical-consistency issue, not circularity.

Assumptions & free parameters 13 free parameters · 5 assumptions · 0 invented entities

The central claims rest on a one-zone homogeneous jet geometry, assumed injection spectra, Eddington-normalized proton power for the 2013 fits, standard analytic cross sections, and neglect of external photon fields and EBL absorption. The main free parameters are the Table 1 model parameters, adjusted by hand to match SEDs; the 2016 super-Eddington conclusion follows from normalizing L_p to the excess flux, while the 2013 fits impose a sub-Eddington cap before fitting.

free parameters (13)

Doppler factor delta_D = 13 (2013 pgamma/BH), 40 (2013 pp), 14 (2017)
Chosen to match SED peaks and variability timescale; strongly affects luminosities and times.
Magnetic field B = 0.18 G (2013 pgamma/BH), 0.9 G (2013 pp), 0.4 G (2017)
Set to place secondary-pair synchrotron emission in the hard X-ray band.
Blob radius R = 5e16 cm (2013 pgamma/BH), 1.4e15 cm (2013 pp), 2.3e15 cm (2017)
Controls escape, cooling, and variability timescales; varied per epoch.
Electron injection luminosity L_e,inj = 5e41 erg/s (2013 pgamma/BH), 1.2e40 erg/s (2013 pp), 3e41 erg/s (2016)
Normalized to reproduce the low-energy synchrotron bump.
Electron spectral index alpha_e1 = 1.7 (2013 pgamma/BH), 1.8 (2013 pp)
Sets low-energy synchrotron slope.
Electron spectral index alpha_e2 = 4.5 (2013 pgamma/BH), 6.5 (2013 pp), 6.0 and 4.5 (2016 fits)
Sets high-energy tail of the low-energy bump.
Electron break Lorentz factor gamma_e,b = 7e4 (2013 pgamma/BH), 3e4 (2013 pp), 1e5 and 2e4 (2016)
Places the synchrotron peak frequency.
Proton spectral index alpha_p = 2.0 (2013 pgamma/BH), 1.5 (2013 pp), 1.3 (2016 pp)
Hard spectrum required for pp scenario to yield hard X-ray excess.
Maximum proton Lorentz factor gamma_p,max = 2.5e7 (2013 pgamma/BH), 7.1e4 (2013 pp), 5e6 (2016), 2.5e6 (2017)
Caps proton energy; low values suppress pgamma and BH channels in pp scenarios.
Proton injection luminosity L_p,inj = 1.1e45 erg/s (2013 pgamma/BH), 5.1e43 erg/s (2013 pp), 150 LEdd/delta^2 (2016), 4.5e47 erg/s (2017)
Normalizes secondary emission; 2013 values chosen to satisfy Eddington cap, 2016 value fitted to excess flux.
Cold proton density n_H = 0 (2013 pgamma/BH), 2e5 cm^-3 (2013 pp)
Tuned to switch dominance between photohadronic and hadronuclear interactions.
Electron Lorentz factor bounds gamma_e,min and gamma_e,max = 3e2 and 5e6 (all scenarios)
Fixed boundaries of injected electron distribution; do not affect fitted bands significantly.
Minimum proton Lorentz factor gamma_p,min = 1 (all scenarios)
Conventional lower bound for injected protons.

assumptions (5)

domain assumption All jet emission comes from a single homogeneous spherical blob.
Sec. 2 first paragraph; multi-zone alternative is considered only for the 2016 two-zone leptonic fit and is invoked in Sec. 5 for 2017.
ad hoc to paper Eddington-normalized proton power: Gamma^2 L_p,inj = L_Edd for the 2013 photohadronic fit and Gamma^2 L_p,inj approximately P_p,cold approximately 0.5 L_Edd for the 2013 pp fit.
Sec. 3.1; this constraint guarantees the 'no super-Eddington power' result and is not derived from observations.
domain assumption Continuous constant-rate simultaneous injection of electrons and protons, with no acceleration model.
Eqs. (1)-(2) and Sec. 5; injection spectrum stands in for unknown acceleration.
standard math Standard analytic cross sections and secondary spectra for pgamma, Bethe-Heitler, and pp interactions.
Eqs. (7)-(9) rely on Kelner et al. (2006, 2008) and Chodorowski et al. (1992).
domain assumption Photon absorption includes only internal synchrotron self-absorption and gamma-gamma pair production; no EBL absorption for VHE photons.
Eqs. (10)-(12) define absorption; EBL is never mentioned, although it affects the TeV flux predicted in the pp scenario.

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Cite this review

Pith. "Pith review of Time-Dependent Leptohadronic Modeling of Markarian 421." pith.science (2026). https://pith.science/paper/4NEUVEJD

@misc{pith2026250701265,
  author       = {Pith},
  title        = {Pith review of: Time-Dependent Leptohadronic Modeling of Markarian 421},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4NEUVEJD}},
  note         = {Machine review of arXiv:2507.01265}
}

read the original abstract

Due to its proximity, Markarian 421 is one of the most extensively studied jetted active galactic nuclei. Its spectral energy distribution and light curve are widely studied, serving as primary means for understanding jet radiation mechanisms. Numerous intriguing observational results have been discovered, some of which, such as the hard X-ray excess, and the associated variability between X-ray and very-high-energy (VHE) emissions, challenge the commonly adopted one-zone leptonic model. In this work, by establishing a time-dependent leptohadronic model, we explore whether the hard X-ray excess and the associated variability between X-ray and VHE emissions could be interpreted by emission from hadronic interactions. Our modeling finds that for the hard X-ray excess found in 2013, both of the secondary emissions from photohadronic and hadronuclear interactions could be a possible explanation for the hard X-ray excess without introducing a super-Eddington jet power. The emission from the photohadronic interactions contributes only to the hard X-ray band, while the hadronuclear interactions also predict VHE emissions associated with the hard X-rays. While for the hard X-ray excess found in 2016, only the secondary emissions from photohadronic interactions provide an interpretation at the cost of introducing a super-Eddington jet power. For the associated variability between X-ray and VHE emissions in 2017, we find that hadronic interactions fail to provide a possible interpretation.

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