Proton Synchrotron & Hadronic Models

• Proton synchrotron and hadronic models are theoretical frameworks that describe high-energy emissions in astrophysical sources by using ultra-relativistic protons and complex interaction processes.
• They combine analytical formulations with numerical kinetic simulations, including CNN surrogate models, to quantify synchrotron emission, photopion production, and electromagnetic cascades.
• Stringent energy budgets and parameter constraints challenge these models, yet they offer key multimessenger diagnostics through correlated gamma-ray, neutrino, and polarization observations.

A proton synchrotron is a theoretical and computational framework within high-energy astrophysics and astroparticle physics in which the high-energy photons in nonthermal astrophysical sources are produced by ultra-relativistic protons radiating synchrotron emission in strong magnetic fields. Hadronic models constitute a broad class of physical scenarios where, in addition to or instead of electrons, relativistic ions (typically protons) serve as the main energy carriers and radiators in compact objects such as blazar jets and gamma-ray bursts (GRBs). These models contrast with leptonic frameworks, where the emission is primarily due to electrons and positrons upscattering ambient photons. Integrating proton synchrotron processes with the full suite of hadronic interactions—such as photopion production, Bethe–Heitler pair creation, and the ensuing electromagnetic cascades—enables the prediction of both electromagnetic and high-energy neutrino emission, thereby forging a unified multimessenger paradigm for compact jet sources (Cerruti et al., 2024, Petropoulou et al., 2012).

1. Physical and Mathematical Foundation of Proton Synchrotron

In a strong magnetic field $B$, a relativistic proton ($\gamma_p \gg1$) emits synchrotron radiation with a characteristic power per particle: $$P_{\rm syn}(\gamma_p) = \frac{4}{3}\,\sigma_T\,c\,U_B\,\left( \frac{m_e}{m_p} \right)^2\,\gamma_p^2\,,$$ where $U_B=B^2/(8\pi)$ is the comoving magnetic energy density, and $\sigma_T$ is the Thomson cross-section. The typical photon energy radiated is

$$\varepsilon_{\rm syn}(\gamma_p) = \frac{3\,e\,B}{4\pi\,m_p\,c}\,\gamma_p^2\,.$$

The synchrotron cooling time for these protons is

$$\tau_{p,\rm syn} = \frac{6\pi\,m_p^3\,c}{\sigma_T\,m_e^2\,B^2\,\gamma_p}\,.$$

For the observed high-energy $\gamma$-ray emission in jets, this necessitates either very large $B$ or extremely high $\gamma_p$, often approaching or exceeding $\gamma_p \gg1$0 (Cerruti et al., 2014, Cerruti et al., 2024).

Modeling these processes, particularly in the context of blazar spectral energy distributions (SEDs), further enforces constraints on the allowed parameter space due to requirements such as confinement (Hillas criterion) and avoidance of excessive internal photon-photon ($\gamma_p \gg1$1) opacity at high $\gamma_p \gg1$2 (Petropoulou et al., 2012).

2. Structure and Regimes of Hadronic Models

Hadronic models for high-energy astrophysical sources generally involve:

• Pure proton-synchrotron regime: Protons are the dominant ultrarelativistic species, and their synchrotron emission produces the observed high-energy hump. This typically requires $\gamma_p \gg1$3 in the range $\gamma_p \gg1$4–$\gamma_p \gg1$5 G, $\gamma_p \gg1$6 (Doppler factor) in the range $\gamma_p \gg1$7–$\gamma_p \gg1$8, and maximum proton energies $\gamma_p \gg1$9–$$P_{\rm syn}(\gamma_p) = \frac{4}{3}\,\sigma_T\,c\,U_B\,\left( \frac{m_e}{m_p} \right)^2\,\gamma_p^2\,,$$0 eV, with often super-Eddington total energy requirements (Petropoulou et al., 2012, Cerruti et al., 2014, Liodakis et al., 2020).
• Cascade–plus–SSC regime: At lower $$P_{\rm syn}(\gamma_p) = \frac{4}{3}\,\sigma_T\,c\,U_B\,\left( \frac{m_e}{m_p} \right)^2\,\gamma_p^2\,,$$1 ($$P_{\rm syn}(\gamma_p) = \frac{4}{3}\,\sigma_T\,c\,U_B\,\left( \frac{m_e}{m_p} \right)^2\,\gamma_p^2\,,$$2–$$P_{\rm syn}(\gamma_p) = \frac{4}{3}\,\sigma_T\,c\,U_B\,\left( \frac{m_e}{m_p} \right)^2\,\gamma_p^2\,,$$3 G), proton $$P_{\rm syn}(\gamma_p) = \frac{4}{3}\,\sigma_T\,c\,U_B\,\left( \frac{m_e}{m_p} \right)^2\,\gamma_p^2\,,$$4 and Bethe–Heitler interactions dominate proton energy losses, resulting in secondary $$P_{\rm syn}(\gamma_p) = \frac{4}{3}\,\sigma_T\,c\,U_B\,\left( \frac{m_e}{m_p} \right)^2\,\gamma_p^2\,,$$5 and photon cascades, accompanied by electron SSC emission, collectively producing the GeV–TeV emission. In mixed (hybrid) lepto-hadronic models, both electron and proton populations contribute significantly to the high-energy component (Cerruti et al., 2014, Cao et al., 2019).

Transitions between particle-dominated and magnetic-dominated regimes are set by the ratio $$P_{\rm syn}(\gamma_p) = \frac{4}{3}\,\sigma_T\,c\,U_B\,\left( \frac{m_e}{m_p} \right)^2\,\gamma_p^2\,,$$6, which, along the minimum Doppler-factor curve $$P_{\rm syn}(\gamma_p) = \frac{4}{3}\,\sigma_T\,c\,U_B\,\left( \frac{m_e}{m_p} \right)^2\,\gamma_p^2\,,$$7, can exceed $$P_{\rm syn}(\gamma_p) = \frac{4}{3}\,\sigma_T\,c\,U_B\,\left( \frac{m_e}{m_p} \right)^2\,\gamma_p^2\,,$$8 (particle-dominated) or be $$P_{\rm syn}(\gamma_p) = \frac{4}{3}\,\sigma_T\,c\,U_B\,\left( \frac{m_e}{m_p} \right)^2\,\gamma_p^2\,,$$9 (magnetically dominated) (Petropoulou et al., 2012).

3. Computational Formulation and Code Systematics

Contemporary implementations solve coupled, stationary or time-dependent kinetic equations for the distributions of protons, electrons, photons, and all secondary particles, incorporating injection, radiative losses, escape, and all relevant hadronic interactions. For instance,

$U_B=B^2/(8\pi)$0

with

$U_B=B^2/(8\pi)$1

and analogous terms for electrons and photons (Cerruti et al., 2014, Cerruti et al., 2011, Cerruti et al., 2024).

Comprehensive code comparisons (Cerruti et al., 2024) between numerical schemes (AM³, ATHENA, B13, LeHa-Paris, LeHaMoC) reveal excellent agreement in spectral shapes, with a normalization uncertainty envelope of $U_B=B^2/(8\pi)$2, which should be systematically included in statistical-hadronic SED modeling.

Surrogate modeling via convolutional neural networks (CNNs), trained on extensive parameter grids from kinetic codes such as SOPRANO, is now used for rapid Bayesian sampling and observational fitting, enabling multidimensional parameter exploration on timescales orders of magnitude faster than traditional solvers (Sahakyan et al., 30 Jun 2025).

4. Energetics, Constraints, and Observational Implications

The necessary conditions for matching observed high-energy emissions with proton-synchrotron models are highly restrictive. For blazars with observed $U_B=B^2/(8\pi)$3 MeV–TeV $U_B=B^2/(8\pi)$4-ray luminosities, minimum jet powers $U_B=B^2/(8\pi)$5 calculated for $U_B=B^2/(8\pi)$6 sources are typically $U_B=B^2/(8\pi)$7 times larger than the Eddington luminosity $U_B=B^2/(8\pi)$8, accretion-disk luminosity $U_B=B^2/(8\pi)$9, or spin-extraction (Blandford–Znajek) power $\sigma_T$0 (Liodakis et al., 2020). Achieving this requires either extreme amplification ($\sigma_T$1) of the jet magnetic field or a production site inside the broad-line region, often incompatible with VLBI or variability constraints.

A plausible implication is that, where relativistic hadrons are present, they can only contribute a radiatively subdominant high-energy (steady-state) component rather than accounting for the dominant $\sigma_T$2-ray luminosity of powerful blazars (Liodakis et al., 2020).

The situation is somewhat more favorable for:

• Ultra-high-frequency-peaked BL Lacs (UHBLs), where sub-Eddington jet powers are achievable for $\sigma_T$3–$\sigma_T$4 G, $\sigma_T$5, and $\sigma_T$6 eV (Cerruti et al., 2014).
• Gamma-ray burst afterglows, where proton-synchrotron can explain GeV–TeV afterglow emission if the allowed parameter space (large initial Lorentz factor, $\sigma_T$7–$\sigma_T$8 G, high $\sigma_T$9) is realized (Razzaque, 2010).

5. Polarization as a Discriminant and Multimessenger Diagnostics

Polarization predictions provide an incisive test of proton-synchrotron models. Proton-synchrotron emission in a perfectly ordered field produces a frequency-independent maximum linear polarization,

$$\varepsilon_{\rm syn}(\gamma_p) = \frac{3\,e\,B}{4\pi\,m_p\,c}\,\gamma_p^2\,.$$0

reaching $$\varepsilon_{\rm syn}(\gamma_p) = \frac{3\,e\,B}{4\pi\,m_p\,c}\,\gamma_p^2\,.$$1–$$\varepsilon_{\rm syn}(\gamma_p) = \frac{3\,e\,B}{4\pi\,m_p\,c}\,\gamma_p^2\,.$$2 for typical proton indices (Zhang et al., 2013). Hadronic models predict substantially higher maximal X-ray and $$\varepsilon_{\rm syn}(\gamma_p) = \frac{3\,e\,B}{4\pi\,m_p\,c}\,\gamma_p^2\,.$$3-ray polarization than leptonic SSC or external Compton scenarios, especially in low- and intermediate-synchrotron-peaked blazars. High values of observed high-energy polarization ($$\varepsilon_{\rm syn}(\gamma_p) = \frac{3\,e\,B}{4\pi\,m_p\,c}\,\gamma_p^2\,.$$4 with $$\varepsilon_{\rm syn}(\gamma_p) = \frac{3\,e\,B}{4\pi\,m_p\,c}\,\gamma_p^2\,.$$5 the field-order correction) strongly favor hadronic emission.

Multimessenger diagnostics further leverage the correlated prediction of high-energy neutrinos. Pure proton-synchrotron blazar fits predict neutrino SED peaks at $$\varepsilon_{\rm syn}(\gamma_p) = \frac{3\,e\,B}{4\pi\,m_p\,c}\,\gamma_p^2\,.$$6–$$\varepsilon_{\rm syn}(\gamma_p) = \frac{3\,e\,B}{4\pi\,m_p\,c}\,\gamma_p^2\,.$$7 EeV, with peak all-flavor fluxes $$\varepsilon_{\rm syn}(\gamma_p) = \frac{3\,e\,B}{4\pi\,m_p\,c}\,\gamma_p^2\,.$$8 of the $$\varepsilon_{\rm syn}(\gamma_p) = \frac{3\,e\,B}{4\pi\,m_p\,c}\,\gamma_p^2\,.$$9-ray flux. In the case of TXS 0506+056, only hybrid (SSC + hadronic cascade) scenarios reproduce the observed $$\tau_{p,\rm syn} = \frac{6\pi\,m_p^3\,c}{\sigma_T\,m_e^2\,B^2\,\gamma_p}\,.$$0–$$\tau_{p,\rm syn} = \frac{6\pi\,m_p^3\,c}{\sigma_T\,m_e^2\,B^2\,\gamma_p}\,.$$1 PeV neutrino coincident with the $$\tau_{p,\rm syn} = \frac{6\pi\,m_p^3\,c}{\sigma_T\,m_e^2\,B^2\,\gamma_p}\,.$$2-ray flare (Cao et al., 2019, Cerruti et al., 2018, Sahakyan et al., 30 Jun 2025).

6. Generalizations and Applications across Source Classes

The proton-synchrotron and broader hadronic framework is applied across multiple source classes:

• Blazars: SED modeling for FSRQs, BL Lacs, and UHBLs, with code implementations (SOPRANO, AM³, etc.) allowing parameter studies over $$\tau_{p,\rm syn} = \frac{6\pi\,m_p^3\,c}{\sigma_T\,m_e^2\,B^2\,\gamma_p}\,.$$3, $$\tau_{p,\rm syn} = \frac{6\pi\,m_p^3\,c}{\sigma_T\,m_e^2\,B^2\,\gamma_p}\,.$$4, $$\tau_{p,\rm syn} = \frac{6\pi\,m_p^3\,c}{\sigma_T\,m_e^2\,B^2\,\gamma_p}\,.$$5, and particle injection rates (Cerruti et al., 2024, Sahakyan et al., 30 Jun 2025).
• Gamma-Ray Bursts: Hadronic models for the prompt and afterglow emission of GRBs invoke proton-synchrotron and photohadronic cascades to account for high-energy ($$\tau_{p,\rm syn} = \frac{6\pi\,m_p^3\,c}{\sigma_T\,m_e^2\,B^2\,\gamma_p}\,.$$6100 MeV) LAT data, with strong constraints from energy budget and neutrino upper limits (Crumley et al., 2012, Florou et al., 2021, Razzaque, 2010).
• Jet Simulation and Microphysics: Integration of hadronic energy losses and proton-synchrotron cooling into particle-in-cell and general-relativistic MHD schemes is achieved using relativistic Boris-pusher and guiding-center algorithms, incorporating continuous and stochastic hadronic collision terms, validated against analytic benchmarks (Zou et al., 2024).

These methodologies enable not only electromagnetic SED fitting, but increasingly detailed predictions for neutrino event rates and polarimetric signatures, with systematic uncertainties in SED normalization ($$\tau_{p,\rm syn} = \frac{6\pi\,m_p^3\,c}{\sigma_T\,m_e^2\,B^2\,\gamma_p}\,.$$7) now quantified for robust inference (Cerruti et al., 2024, Sahakyan et al., 30 Jun 2025).

7. Limitations, Systematics, and Outlook

The proton-synchrotron model, while conceptually attractive for linking $$\tau_{p,\rm syn} = \frac{6\pi\,m_p^3\,c}{\sigma_T\,m_e^2\,B^2\,\gamma_p}\,.$$8-ray and neutrino astrophysics, faces severe constraints:

• Energetic requirements for steady blazar emission are typically orders of magnitude above the Eddington or disk luminosity (Liodakis et al., 2020).
• Viable parameter regimes require high $$\tau_{p,\rm syn} = \frac{6\pi\,m_p^3\,c}{\sigma_T\,m_e^2\,B^2\,\gamma_p}\,.$$9, large $\gamma$0, and ultrarelativistic proton populations that challenge standard acceleration scenarios and jet energetics.
• Secondary $\gamma$1 production via $\gamma$2 and $\gamma$3 interactions is generally subdominant in the TeV band for high $\gamma$4, but can reshape the spectrum below the peak unless $\gamma$5-ray emission zones are highly compact and magnetized (Florou et al., 2021).

These constraints suggest a radiatively subdominant role for protons in most steady blazar jets, though episodic or highly-magnetized environments may admit plausible hadronic dominance, particularly in orphan flares or narrow SED intervals (Petropoulou et al., 2012, Cerruti et al., 2014, Cerruti et al., 2024). The integration of deep learning surrogates and code cross-comparison establishes a reproducible, systematics-aware foundation for future multimessenger studies (Sahakyan et al., 30 Jun 2025).