= Solution
The <escape-speed scaling of wind mass loading> makes the proposed trend dynamically plausible: the shallower <Newtonian gravitational potential> of a <dwarf galaxy> lets <stellar feedback> expel more gas per unit stellar <mass>. If coupled feedback supplies specific <energy> $e_*$, escape requires $\nu v_{\rm esc}^2/2\lesssim e_*$; fixed coupling suggests $\nu\propto v_{\rm esc}^{-2}$ for an <energy-driven outflow>. Fixed specific momentum $p_*$ instead gives $\nu v_{\rm esc}\lesssim p_*$, or $\nu\propto v_{\rm esc}^{-1}$ for a <momentum-driven outflow>.
Since <escape velocity> has $v_{\rm esc}^2\sim GM/R$, galaxies whose characteristic radius grows more slowly than their gravitating <mass> have increasing binding energy per unit <mass> with increasing <mass>. For $R\propto M^a$ with $a<1$, the <energy-driven outflow> estimate gives $\nu\propto M^{a-1}$. Equivalently, greater <velocity dispersion> in massive spheroids implies greater escape energy and more effective retention of metals.
For scale, one <core-collapse supernova> of $10^{51}\,\mathrm{erg}$ per $100M_\odot$ formed supplies about $5\times10^{15}\,\mathrm{cm^2\,s^{-2}}$ before coupling losses. At ten percent coupling the energetic upper estimate $2e_*/v_{\rm esc}^2$ is about $110$ for $v_{\rm esc}=30\,\mathrm{km\,s^{-1}}$, but about $0.4$ for $500\,\mathrm{km\,s^{-1}}$. \b[Efficient loss from shallow potentials and retention in deep potentials can span the required range.] These are illustrative energetics rather than a unique prediction: cooling, entrainment, fallback and the actual <initial mass function> determine the escaping <mass-loading factor>.
Back to article page
