Absorption variability as a probe of the multiphase interstellar media surrounding active galaxies

We examine a model for the variable free-free and neutral hydrogen absorption inferred towards the cores of some compact radio galaxies in which a spatially fluctuating medium drifts in front of the source. We relate the absorption-induced intensity fluctuations to the statistics of the underlying opacity fluctuations. We investigate models in which the absorbing medium consists of either discrete clouds or a power-law spectrum of opacity fluctuations. We examine the variability characteristics of a medium comprised of Gaussian-shaped clouds in which the neutral and ionized matter are co-located, and in which the clouds comprise spherical constant-density neutral cores enveloped by ionized sheaths. The cross-power spectrum indicates the spatial relationship between neutral and ionized matter, and distinguishes the two models, with power in the Gaussian model declining as a featureless power-law, but that in the ionized sheath model oscillating between positive and negative values. We show how comparison of the HI and free-free power spectra reveals information on the ionization and neutral fractions of the medium. The background source acts as a low-pass filter of the underlying opacity power spectrum, which limits temporal fluctuations to frequencies $\omega < \dot{\theta}_v / \theta_{\rm src}$, where $\dot{\theta}_v$ is the angular drift speed of the matter in front of the source, and it quenches the observability of opacity structures on scales smaller than the source size $\theta_{\rm src}$. For drift speeds of $\sim 10^3\,$km s$^{-1}$ and source brightness temperatures $\sim 10^{12}\,$K, this limitation confines temporal opacity fluctuations to timescales of order several months to decades.