Coupled Time-Dependent Proton Acceleration and Leptonic-Hadronic Radiation in Turbulent Supermassive Black Hole Coronae

TL;DR

We address the challenge of linking fast proton acceleration in compact SMBH coronae to multi-messenger emissions by introducing a time-dependent framework that couples proton acceleration via a Fokker-Planck equation with leptonic-hadronic radiation cascades. The method is validated on a steady NGC 1068 corona, reproducing the IceCube neutrino spectrum while keeping gamma-ray flux within limits, and extended to transient TDE coronae to reveal delayed, cascade-influenced signatures with peak neutrino energies around $E_\nu\sim 100~\mathrm{TeV}$ and delays of order $\sim$100 days. We find that radiation feedback can significantly alter proton cooling and maximum energies in weaker coronae, while in steady coronal environments the solution rapidly converges to a stable spectrum, largely independent of the injection form. The framework offers a versatile, efficient tool for modeling time-dependent multi-messenger signals from SMBH coronae and related transients, adaptable to alternative acceleration mechanisms and a broad class of sources such as gamma-ray bursts and TDEs.

Abstract

Turbulent coronae of supermassive black holes can accelerate non-thermal particles to high energies and produce observable radiation, but capturing this process is challenging due to comparable timescales of acceleration, cooling, and the development of cascades. We present a time-dependent numerical framework that self-consistently couples proton acceleration--modeled by the Fokker-Planck equation--with leptonic-hadronic radiation. For the neutrino-emitting Seyfert galaxy NGC 1068, we reproduce the neutrino spectrum observed by IceCube, while satisfying gamma-ray constraints. We also consider a transient corona scenario, potentially emerging in tidal disruption events like AT 2019dsg, and show that cascade feedback on proton cooling can impact proton acceleration and radiation processes in weaker coronae, producing delayed optical/ultraviolet, X-ray, and neutrino emissions of $\mathcal O(100~\rm d)$. This flexible tool efficiently models multi-messenger signals from both steady and transient astrophysical sources, providing insights in combining particle acceleration and radiation mechanisms.

Figures (6)

• Figure 1: Test of numerical solutions to FP equations: comparison between time-dependent solutions (solid curves) and steady-state solutions (red dashed curve), normalized to arbitrary units. The injection rates (blue dash-dotted curve) are also shown. The insets display the proton escape, acceleration, and cooling rates. Different injection functions $q\propto p\exp(-p/p_{\rm inj})$ and $q\propto \delta(p-p_{\rm inj})$ with $p_{\rm inj}=10^2m_pc$ are used respectively in the left and right panels.
• Figure 2: NGC 1068: Observed all-flavor neutrino (dashed) and EM cascade (solid) spectra (left panel), and in-source proton density spectra (right panel) at times ranging from $0.1t_{\rm fs}=0.1R_{\rm co}/c$ to $500t_{\rm fs}$. Radio Chang:2019cdu, gamma-ray Fermi-LAT:2019ylaMAGIC:2019fvw, and neutrino IceCube:2022der observations are shown as gray points, blue points, and the red-shaded area. The green and blue curves respectively show the coronal X-ray and OUV spectra. The solid curves in the right panel show the proton spectra from time dependent FP equation, whereas the dashed red curve denotes the steady-state solution from the reduced ordinary differential equation (ODE).
• Figure 3: Left panel: Same as the left panel of Fig. \ref{['fig:NGC1068']}, but for the transient corona of the TDE AT 2019dsg. The time is scaled to the TDE mass fallback time $t_{\rm fb}\simeq45$ d, up to $10t_{\rm fb}$. Right panel: Coronal cascade optical/UV (solid red), X-ray (solid blue), and neutrino (dash-dotted black) light curves of TDE AT 2019dsg. The injected OUV and X-ray light curves are shown as dashed lines. Red and blue points represent the UV (193 nm) Stein:2020xhk and X-ray (0.3–10 keV) Cannizzaro:2020xzc observations, respectively.
• Figure 4: Test of the impact of the EM cascade feedback for a corona with weaker X-ray luminosity (e.g., $L_{X,\rm bol} < L_p$). NGC 1068's corona is used as an example, with parameters identical to those used in §\ref{['subsec:ngc1068']} except for a lower luminosity of $L_{X,\rm bol} = 5\times10^{41}\rm erg~s^{-1}$. Left panel: Proton spectra obtained with EM cascade feedback (solid curves) and without cascade feedback (red solid curve). Right panel: Proton cooling rates at times from $0.1t_{\rm fs} = 0.1R_{\rm co}/c$ to $500t_{\rm fs}$, including the contribution from cascade photons. The acceleration and escape rates are also shown. The green and purple areas represent the regimes where $pp$/BH and $p\gamma$ processes, respectively, dominate proton cooling.
• Figure 5: Left panels: EM cascade spectral components, including contributions from $\gamma\gamma$/BH pairs, leptons from $p\gamma$/$pp$ interactions, and proton synchrotron radiation, are shown for NGC 1068 (upper panel) at $500t_{\rm fs}$ and for the TDE AT 2019dsg (bottom panel) at $t_{\rm fb}$, where $t_{\rm fs}=R_{\rm co}/c$ is the free escaping time and $t_{\rm fb}$ is the TDE mass fallback time. Radio Chang:2019cdu and gamma-ray Fermi-LAT:2019ylaMAGIC:2019fvw observations of NGC 1068 are shown as the gray and blue points. The OUV and coronal X-ray spectra are also shown. Both calculations account for $\gamma\gamma$ attenuation by the extragalactic background light. Right panels: Proton cooling rates at various times are shown, with colors ranging from yellow to black. The horizontal dashed gray curve represents the acceleration rate, while the blue lines depict the proton escape rates. The shaded regions from left to right indicate proton cooling dominated by $pp$/BH interactions, by $p\gamma$ interactions with coronal X-ray photons, and by interactions with OUV and cascade photons.