Analytical Framework for Expanding Bubbles in a Hot Circumgalactic Medium

Abstract

We develop an analytic framework for the evolution of feedback-driven bubbles expanding into a hot, volume-filling circumgalactic medium (CGM), where the ambient pressure and sound speed are non-negligible and radiative cooling is often inefficient. The evolution is organized into four stages -- free expansion, Sedov--Taylor expansion, pressure-modified/transonic transition, and post-transonic relaxation -- and we derive self-consistent scalings for the characteristic radii and timescales that delimit these stages. A central result is that, in hot halos, the end of the strong-shock evolution is frequently set by pressure confinement and transonicity rather than by the onset of catastrophic cooling, implying only a modest late-time overshoot beyond the pressure-balance/transonic point. We connect the dynamics to observable outcomes by estimating bubble sizes and lifetimes, order-of-magnitude band-limited X-ray luminosities, and high-ionization ion column densities, and we provide stitched numerical trajectories that contrast our pressure-modified model appropriate for hot CGM conditions with a classical Sedov--Taylor benchmark. We then discuss physically motivated extensions beyond the single-event baseline, including continuous or episodic energy injection relevant for AGN-driven bubbles and nuclear outflows, highlighting the much higher specific energy of AGN feedback compared to supernovae and the resulting dynamical differences in how bubbles are driven. We further outline how multiphase interaction, mass loading, anisotropic dissipation, intermittency, confinement, and non-thermal channels can increase the emergent X-ray radiation efficiency without requiring changes to the intrinsic feedback energy partition at the launching site. This framework provides a transparent bridge between idealized bubble theory and feedback signatures in hot galactic halos.

Figures (10)

• Figure 1: Time evolution of the shock radius $R(t)$ for a bubble expanding into a hot CGM. Stages I (free expansion) and II (Sedov--Taylor) are evaluated analytically and are smoothly joined near the swept-mass equality time $t_{\rm sw}$ (Eq. \ref{['eq:tsw']}) using the plotting blend described in the text. We also mark $t_{\min}$ (the leftmost vertical dashed line), defined by the condition $\langle T\rangle_M(t_{\min})=T_{\rm ej}$ (Eqs. \ref{['eq:Tej_bound']} and \ref{['eq:Tmass_stageI']}), and indicates the earliest time at which the Stage I mean-temperature scaling is physically meaningful. At $t<t_{\min}$, the partition between bulk kinetic and internal energy is still being established, and the plotted curves should be regarded as formal extrapolations rather than literal predictions. Two alternative Stage III continuations are shown: Stage IIIa (blue solid), which begins at the pressure-equilibrium time $t_P$, and Stage IIIb (orange), which begins at the Mach-unity time $t_{\mathcal{M}}$. For clarity, Stage IIIb is plotted as a dotted curve for $t<t_{\mathcal{M}}$ (formal mathematical continuation) and as a solid curve for $t\ge t_{\mathcal{M}}$ (its physical domain). But the dotted curve is blocked in this plot while visible in a few other plots below. For the fiducial parameters adopted here, $t_P<t_{\mathcal{M}}$, so Stage IIIa represents the physically realized evolution, while Stage IIIb is shown for comparison. The stall radius $R_{\rm stall}$ and corresponding time $t_{\rm stall}$ are marked for both branches.
• Figure 2: Time evolution of the expansion velocity $v(t)$ (left axis) and Mach number $\mathcal{M}(t)=v/c_{s,0}$ (right axis). The horizontal dotted line marks $\mathcal{M}=1$, indicating the transonic transition. Line styles and colors follow the conventions in Fig. \ref{['fig:R']}.
• Figure 3: Time evolution of the mean interior number density $\bar{n}(t)$ of the bubble. The calculation includes both ejecta mass and swept-up ambient mass, with a fiducial mass-loading factor $f_m=1$ appropriate for a hot, well-mixed CGM. Line styles and colors follow Fig. \ref{['fig:R']}.
• Figure 4: Time evolution of the mean interior temperature $\bar{T}(t)$ of the bubble. The horizontal dotted line indicates the ejecta characteristic temperature $T_{\rm ej}$ (Eq. \ref{['eq:Tej_bound']}). The leftmost vertical line marks $t_{\min}$ defined by $\bar{T}(t_{\min})=T_{\rm ej}$, which we use as an early-time interpretation bound for the Stage I mean-temperature curve. At $t<t_{\min}$, the plotted increase of $\bar{T}(t)$ reflects a formal extrapolation of the adopted mean-temperature definition/scaling to arbitrarily early times, when a coherent shocked region and a well-defined thermalized bubble interior have not yet formed; thus $\bar{T}(t)$ in this regime should not be interpreted as a literal gas temperature. The other line styles and colors follow Fig. \ref{['fig:R']}.
• Figure 5: Time evolution of the mean interior pressure $\bar{P}(t)$. The horizontal dotted line marks the ambient thermal pressure $P_0$ of the CGM. The departure from the Sedov--Taylor scaling and the subsequent approach toward pressure equilibrium during Stages III and IV are clearly visible. Line styles and colors follow Fig. \ref{['fig:R']}.