Time-Domain Radio Loudness of Active Galactic Nuclei: Intermittency, Memory, and Jet Escape

Abstract

The classical radio-loudness parameter $R \equiv f_ν(5\,\mathrm{GHz})/f_ν(4400\,\textÅ)$ compares a prompt accretion tracer with a radio numerator that mixes rapidly varying compact-core emission, lobe plasma surviving over millions of years, and host-galaxy synchrotron emission. We introduce a time-domain radio-loudness (TDRL) framework that makes this timescale mismatch explicit. The radio numerator is decomposed into compact-core and extended-lobe contributions, each weighted by a recovered fraction that depends on observing frequency, angular resolution, and surface-brightness sensitivity. For a single intermittently jetted AGN population, a two-state jet duty cycle convolved with exponential lobe fading yields an exact stationary Beta distribution for the normalized extended-radio response, whose mean is $f_{\rm duty}$ and whose variance scales as $(1+χ_ν)^{-1}$ with $χ_ν\equiv\taunu/t_{\rm switch}$. This result serves as an analytic reference model, while precise inference will require population models matched to specific survey selections. In this minimal reference model, the familiar GHz valley near the classical radio-loud/quiet boundary can in principle arise from short radio memory alone, without invoking two intrinsic engine classes; metre-wave surveys that recover diffuse emission and model the host galaxy contribution should progressively fill that valley. In the $(R^{core}_ν,R^{lobe}_ν)$ plane a core--lobe mismatch index distinguishes triggering, sustained, and remnant jet phases. A complementary two-barrier phase diagram in event-horizon-threading magnetic flux and jet-escape parameter provides a heuristic organizing scheme for jet launching and propagation through the nuclear medium. The framework offers testable, frequency-dependent predictions for current and future radio surveys.

Figures (4)

• Figure 1: The TDRL diagnostic plane. The vertical axis shows the prompt/core radio loudness $R_{\nu}^{\rm core}$, tracing present-day engine activity; the horizontal axis shows the memory/lobe radio loudness $R_{\nu}^{\rm lobe}$, tracing the time-integrated jet history over $\tau_{\nu}$. The dashed diagonal marks approximate quasi-equilibrium between injection and accumulated lobe power; the perpendicular offset defines the mismatch index $\mathcal{I}_\nu$ (Eq. \ref{['eq:mismatch']}). Four regimes are indicated: triggering or restarting (core bright, lobe faint, $\mathcal{I}_\nu\gg0$), sustained activity (both components bright, $\mathcal{I}_\nu\simeq0$), remnant or fading (lobe bright, core faint, $\mathcal{I}_\nu\ll0$), and quiescent or weak systems (both components faint). The two-dimensional decomposition separates what the engine is doing now from what it has done in the past, a distinction that scalar $R$ cannot make. Orientation-dependent beaming of the core component can shift sources along the vertical axis and must be modelled or controlled for in quantitative applications.
• Figure 2: Time-domain origin of radio loudness in the TDRL framework. Panels show: (a) the underlying engine state $J(t)$ modelled as a stochastic on/off process, (b) the prompt compact-core emission modulated by beaming, (c) the lobe emission smoothed by the frequency-dependent fading function for short-memory (high frequency, green) and long-memory (low frequency, red) observations, and (d) the resulting observed scalar radio loudness $R$. A single intermittent engine produces either a bimodal or a continuous $p(\log R)$ depending on $\chi_\nu=\tau_{\nu}/t_{\rm switch}$ and on whether the survey recovers diffuse emission.
• Figure 3: Heuristic two-barrier phase diagram for AGN jet lifecycles. The horizontal axis shows the dimensionless horizon-threading magnetic flux $\phi_{\rm BH}$ (SANE$\rightarrow$MAD); the vertical axis shows the escape parameter $\Xi_{\rm env}=t_{\rm on}/t_{\rm bo}$, separating confined from escaping jets. Three regimes are schematically indicated: quiescent or weak systems (flux-starved SANE), frustrated or compact sources such as CSO/CSS objects (MAD jets confined by dense environments, $\Xi_{\rm env}\lesssim1$), and classical large-scale radio galaxies (MAD jets that escape and inflate lobes, $\Xi_{\rm env}\gtrsim1$). Engine power and environmental permeability are independent axes: a jet can be powerful yet remain compact if the circumnuclear gas column is sufficiently high. The diagram classifies known phenomenology into physically motivated regions but does not predict population fractions (see Section \ref{['sec:phenomenology']} for discussion).
• Figure 4: Illustrative frequency-dependent radio-loudness distributions from a single intermittent engine population ($f_{\rm duty}\simeq 10\%$). Both panels adopt the illustrative parameters of Section 2.6 ($t_{\rm switch}=5\,$Myr, $B=10\,\mu$G, $z=0.5$) with a log-normal host star-formation floor ($\langle\log R_{\rm SF}\rangle=-0.3$, scatter $0.3\,$dex). (a) At high frequencies ($\nu_{\rm obs}=5\,$GHz, $\chi_\nu\approx 2.4$, short memory) the Beta shape parameter $\alpha=\lambda_\uparrow\tau_\nu<1$, so the stationary distribution of the normalized extended-radio response piles up near $Y\approx 0$; combined with the host star-formation floor this produces a bimodal $p(\log R)$ with a pronounced valley near the classical $\log R=1$ boundary. (b) At low frequencies ($\nu_{\rm obs}=150\,$MHz, $\chi_\nu\approx 14$, long memory) $\alpha>1$ and the Beta distribution becomes unimodal; fading and remnant systems fill the valley, yielding a broad, continuous distribution. The variance of the normalized extended-radio response shrinks as $(1+\chi_\nu)^{-1}$ (Eq. \ref{['eq:varY']}). These distributions are an existence proof, not a fit to data. The adopted parameters are illustrative (Section \ref{['subsec:forward_demo']}). Proper interpretation of observed samples requires subtracting the host contribution and accounting for surface-brightness selection.