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If stellar feedback supplies energy $e_{\ast }$ per unit formed stellar mass, an energy-driven outflow requires $\nu v_{\mathrm{esc}}^2/2\lesssim e_{\ast }$. Fixed energy coupling therefore suggests mass-loading factor $\nu \propto v_{\mathrm{esc}}^{-2}$. A fixed momentum supply $p_{\ast }$ instead gives $\nu v_{\mathrm{esc}}\lesssim p_{\ast }$ and momentum-driven outflow scaling $\nu \propto v_{\mathrm{esc}}^{-1}$. These are launch and escape estimates, not universal wind laws; subsequent entrainment, cooling and fallback change the escaping mass.

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 $e_{\ast }$, escape requires $\nu v_{\mathrm{esc}}^2/2\lesssim e_{\ast }$; fixed coupling suggests $\nu \propto v_{\mathrm{esc}}^{-2}$ for an energy-driven outflow. Fixed specific momentum $p_{\ast }$ instead gives $\nu v_{\mathrm{esc}}\lesssim p_{\ast }$, or $\nu \propto v_{\mathrm{esc}}^{-1}$ for a momentum-driven outflow.

Since escape velocity has $v_{\mathrm{esc}}^2\sim GM/R$, 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 $5\times 10^{15}\mathrm{c}{\mathrm{m}}^2{\mathrm{s}}^{-2}$ before coupling losses. At ten percent coupling the energetic upper estimate $2e_{\ast }/v_{\mathrm{esc}}^2$ is about $110$ for $v_{\mathrm{esc}}=30\mathrm{km}{\mathrm{s}}^{-1}$, but about $0.4$ for $500\mathrm{km}{\mathrm{s}}^{-1}$. 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.

Moving radially outward from the quasar, the standard active-galactic-nucleus wind bubble contains: the freely expanding fast wind; a reverse shock that thermalizes it; a hot shocked-wind bubble; a contact discontinuity; a dense swept-up interstellar shell behind a forward shock; and finally the undisturbed interstellar medium.

Let $M_s(R)$ be the swept-up shell mass. If inverse-Compton and atomic radiative cooling remove the shocked wind's thermal energy faster than the bubble expands, the reverse shock is momentum driven andIf cooling is slow, the shocked wind remains hot and the bubble is energy driven:supplemented by the bubble energy equationThe hot bubble stores wind energy and performs work for much longer than the direct photon momentum-crossing time. Equating wind power with shell kinetic power gives the characteristic momentum boostwhen the shell is much slower than the nuclear wind.