Clumps in a Cocoon: Geometry and Mixing Set the Universal X-ray to H$\alpha$ Surface Brightness Ratio

Abstract

Recent observations reveal a universal X-ray to H$\alpha$ surface-brightness ratio, ${\rm SB}_{\rm X}/{\rm SB}_{\rm H\alpha}\sim 3$, in galactic winds, ram-pressure stripped tails, and cluster filaments. This is surprising because H$\alpha$ traces cold ($\sim 10^4$ K) gas while X-rays trace much hotter ($\sim 10^{6}$--$10^{7}$ K) gas. Plane-parallel mixing-layer models do not recover this ratio, and can be off by orders of magnitude. Motivated by recent work showing that geometry controls the temperature PDF of multiphase gas (Chen & Oh 2026), we run 3D wind-tunnel simulations in the high density contrast ($\chi\sim 10^3$) regime. In this limit, the cold phase shatters into many small H$\alpha$-emitting clumps, while X-ray-emitting gas forms a volume-filling cocoon around them. After smoothing on the tail-width scale, the measured surface-brightness ratio converges to the observed value, which can be understood theoretically. The H$\alpha$ luminosity fraction is set by atomic physics, whereas the X-ray luminosity fraction is set by the residence time of gas in the X-ray-emitting band. This residence time is much shorter than the cooling time at X-ray temperatures, but scales roughly inversely with pressure, suggesting that it is tied to the cooling time at a lower-temperature outlet of the mixing cascade. This framework naturally explains why the observed ratio is order unity, and robust to changes in gas pressure.