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Abstract The classical radio-loudness parameter R ≡ f ν (5 GHz)/ f ν (4400 Å) divides a radio flux density by an optical/UV accretion tracer, but the two terms do not probe the same clock. The radio numerator can blend compact-core emission from the current engine, lobe and relic plasma left by earlier jet episodes, and host-galaxy synchrotron emission. We introduce a time-domain radio-loudness description that keeps these contributions separate. The radio numerator is written as compact-core and extended-lobe terms, with recovered fractions set by observing frequency, angular resolution, and surface-brightness sensitivity. For a single intermittently jetted active galactic nucleus population, a two-state duty cycle filtered by exponential lobe fading gives an exact stationary Beta distribution for the normalized extended radio response. Its mean is f duty , while its variance scales as ( 1 + χ ν ) − 1 , where χ ν ≡ τ ν / t switch . In this reference limit, the familiar GHz valley near the classical radio-loud/radio-quiet boundary can arise from short radio memory alone, without requiring two intrinsic engine classes. Meter-wave surveys that recover diffuse emission and subtract the host contribution should therefore progressively fill the valley. In the ( R ν core , R ν lobe ) plane, a core–lobe mismatch index separates triggering, sustained, and remnant phases, provided orientation-dependent core beaming is modeled or controlled. A complementary two-barrier picture, involving horizon-threading magnetic flux and jet escape through the nuclear medium, separates jet launching from the formation of large-scale radio structure. This view makes radio-loudness a probe of jet duty cycle, radio memory, and escape through the host environment.
Tao An (Tue,) studied this question.