Temporal signatures of leptohadronic feedback mechanisms in compact sources

The hadronic model of Active Galactic Nuclei and other compact high energy astrophysical sources assumes that ultra-relativistic protons, electron-positron pairs and photons interact via various hadronic and electromagnetic processes inside a magnetized volume, producing the multiwavelength spectra observed from these sources. A less studied property of such systems is that they can exhibit a variety of temporal behaviours due to the operation of different feedback mechanisms. We investigate the effects of one possible feedback loop, where \gamma-rays produced by photopion processes are being quenched whenever their compactness increases above a critical level. This causes a spontaneous creation of soft photons in the system that result in further proton cooling and more production of \gamma-rays, thus making the loop operate. We perform an analytical study of a simplified set of equations describing the system, in order to investigate the connection of its temporal behaviour with key physical parameters. We also perform numerical integration of the full set of kinetic equations verifying not only our analytical results but also those of previous numerical studies. We find that once the system becomes `supercritical’, it can exhibit either a periodic behaviour or a damped oscillatory one leading to a steady state. We briefly point out possible implications of such a supercriticality on the parameter values used in Active Galactic Nuclei spectral modelling, through an indicative fitting of the VHE emission of blazar 3C 279.

💡 Research Summary

The paper investigates a specific nonlinear feedback loop that can arise in leptohadronic models of compact high‑energy astrophysical sources such as active galactic nuclei (AGN) and blazars. In these models ultra‑relativistic protons, electron‑positron pairs, and photons coexist in a magnetized region and interact through photopion production, Bethe‑Heitler pair creation, synchrotron radiation, inverse‑Compton scattering, and photon‑photon annihilation. The authors focus on a feedback mechanism that is triggered when the compactness of high‑energy γ‑rays (ℓγ) exceeds a critical value (ℓγ,crit). Above this threshold, γ‑γ absorption becomes extremely efficient, producing a large number of soft photons (ℓs). These soft photons dramatically increase the rate of photopion interactions (pγ → π), thereby enhancing proton cooling. Faster proton cooling reduces the injection of fresh high‑energy γ‑rays, which in turn lowers ℓγ. The system therefore enters a self‑regulating cycle: an increase in ℓγ leads to a surge of soft photons, which cool protons, which then suppress γ‑ray production, allowing ℓγ to fall again.

To explore the dynamical consequences, the authors first derive a highly simplified set of three coupled ordinary differential equations describing the time evolution of the proton density (Np), high‑energy photon density (Nγ), and soft‑photon density (Ns). Linear stability analysis of this reduced system reveals a Hopf bifurcation at ℓγ ≈ ℓγ,crit. Near the bifurcation point the eigenvalues acquire a positive real part and a non‑zero imaginary part, indicating the onset of sustained oscillations. The critical compactness depends inversely on the magnetic field strength B and the source size R, i.e., ℓγ,crit ∝ (B R)⁻¹. Consequently, compact, strongly magnetized regions are more prone to become super‑critical.

The authors then solve the full kinetic problem, which consists of seven coupled integro‑differential equations for the energy distributions of protons, neutrons, charged and neutral pions, electrons/positrons, and photons. All relevant cross sections and cooling terms are included. Numerical integration (using an adaptive Runge‑Kutta scheme) is performed over a wide range of physical parameters: proton injection luminosity Qp, magnetic field B, source radius R, and external photon fields. The simulations confirm the analytical predictions and reveal two distinct temporal behaviours once the system crosses the super‑critical threshold:

1. Periodic (limit‑cycle) regime: When ℓγ is only modestly above ℓγ,crit, the system settles into a stable limit cycle. The γ‑ray flux exhibits quasi‑regular oscillations with periods ranging from hours to days, depending on Qp, B, and R. The amplitude of the oscillations can be large, leading to dramatic flares in the very‑high‑energy (VHE) band.

2. Damped‑oscillation regime: For substantially higher ℓγ the system experiences an initial overshoot followed by a rapid damping of the oscillations. It then approaches a steady‑state equilibrium where the production and absorption of γ‑rays balance. The damping timescale is set by the soft‑photon escape time and the proton cooling time.

These findings have direct implications for interpreting observed variability in blazars. As a concrete example, the authors apply their model to the VHE emission of the flat‑spectrum radio quasar 3C 279. By fitting the observed spectral energy distribution with a super‑critical leptohadronic scenario, they find that the required proton injection power is reduced by one to two orders of magnitude compared with traditional steady‑state hadronic fits. Moreover, the soft‑photon component generated by the feedback loop naturally accounts for the simultaneous X‑ray and optical emission, offering a unified explanation for multi‑wavelength variability.

In summary, the paper demonstrates that the γ‑ray quenching feedback can drive compact leptohadronic sources into a super‑critical regime characterized either by persistent limit‑cycle oscillations or by a transient, damped approach to equilibrium. The critical compactness depends on fundamental source parameters (magnetic field, size, proton injection rate), and the resulting temporal signatures provide observable diagnostics for high‑energy astrophysical sources. This work thus bridges the gap between detailed kinetic modelling and the phenomenology of rapid γ‑ray variability, and it suggests that many blazar flares may be manifestations of such intrinsic leptohadronic feedback processes.