Clumps in a Cocoon: Geometry and Mixing Set the Universal X-ray to Hα Surface Brightness Ratio

Recent observations reveal a nearly universal X-ray to Hα surface-brightness ratio, SBX/SBHα ~ 3, in galactic winds, ram-pressure-stripped tails, and cluster filaments. This is surprising, given the very different temperatures traced by the X-ray and Hα emission. Plane-parallel mixing-layer models do not recover this ratio. We run 3D wind-tunnel simulations in the high density-contrast (χ ~ 103) regime, where the cold phase shatters into small Hα-emitting clumps embedded in a volume-filling X-ray cocoon. After smoothing on the tail-width scale, the simulated ratio converges to the observed value. The hot-gas entrainment rate sets the bolometric luminosity, Lbol ∝ Ṁᵢₙ. Hα carries a nearly fixed fraction of this luminosity, set mainly by atomic emissivity near 104 K. The X-ray-emitting mass is the same entrainment rate multiplied by the residence time in the X-ray-emitting temperature band, and its luminosity is the corresponding thermal energy divided by the cooling time. Hence LX∝ Ṁᵢₙ\ ∝ Lbol, with the X-ray fraction set by the residence-to-cooling-time ratio. An age diagnostic constrains the residence time to be far shorter than the cooling time, making the X-ray and Hα fractions comparable. Their common scaling with Lbol explains the correlation, while the comparable bolometric fractions explain why the observed ratio is order unity.