Uniform Heating Acts at the Wrong Radius: Dissolution Heating and the Precipitation Threshold in a Galactic Halo

Galactic haloes regulate themselves, in the precipitation picture, around a threshold: where the ratio of cooling time to free-fall time falls below roughly ten, hot gas condenses, feeds the central black hole, and the resulting outburst reheats the halo. This paper asks whether a density-independent heating term fed by the surviving vacuum density, Q = f u_L / t_H with u_L = 5.25e-09 erg per cubic centimetre, could play any part in that regulation. For an adopted Milky-Way-mass halo (virial mass 1e12 solar masses, hot gas at 1.2e6 K with central density 4.7e-03 per cubic centimetre falling as a beta profile), the unheated minimum ratio is 1.14 at 14 kpc. The uniform term leaves that minimum essentially untouched — it is still 1.14 at the forest-bounded coupling f = 5.6e-11 and moves only to 1.15 even at f = 1e-03 — because the minimum sits at high density where volumetric heating is negligible against n-squared cooling. To multiply the cooling time at the minimum by the factor of order fourteen that separates the adopted profile from the precipitation band would require f = 1.46e-02, some nine orders of magnitude above the forest bound. Where the uniform term does act is the outer halo: at f = 1e-05 it abolishes net cooling beyond 138 kpc, and at f = 1e-03 beyond 42 kpc. A uniform dissolution term cannot regulate precipitation; its lever is attached to the wrong end of the halo. Computed from stated assumptions; not a measurement.