 Solution

Direct evidence for stellar feedback includes expanding ionized shells and superbubbles around young associations, hot X-ray-emitting gas in supernova remnants, broad or split emission lines, and blueshifted absorption showing cool and warm outflows. P-Cygni profiles reveal massive-star winds, while extraplanar filaments and metal-enriched gas demonstrate transport away from star-forming disks. Indirect evidence includes the galaxy mass--metallicity relation, low baryon fractions and suppressed star formation in dwarf galaxies, chemically enriched circumgalactic gas, and correlations of outflow speed and mass loading with the star formation rate.

Rapid gas removal changes the gravitational potential before stellar and dark-matter orbits can respond adiabatically. Positions and velocities are initially unchanged, but orbital binding energies rise; orbits expand and become more eccentric, and some particles escape. Repeated burst--outflow--reaccretion cycles can irreversibly transfer energy to collisionless matter and turn a central dark-matter cusp into a core. This matters because dwarf-galaxy rotation curves are used to test dark-matter microphysics: a feedback-made core can mimic a non-cold or self-interacting dark-matter signature.

Write the initial potential energy as $W=-aG{M}^2/R$. The virial theorem gives $T=-W/2$. If a well-mixed fraction $ϵ$ remains after instantaneous mass loss, the immediate kinetic and potential energies are $T_a=ϵT$ and $W_a={ϵ}^{2}W$, soAfter revirialization at $R^{\prime }$, $E_f=W_f/2=-aG{ϵ}^2{M}^2/(2R^{\prime })$. Equating energies gives the impulsive mass-loss expansion lawThe remnant is bound only for $ϵ>1/2$; loss of half or more of the gravitating mass disrupts this idealized system.

For many infinitesimal, individually revirialized losses, put $ϵ=1+dM/M$ in the impulsive result. To first order, $dR/R=-dM/M$. Integration yields the adiabatic mass-loss expansion lawSlow loss causes finite expansion for every positive remaining mass fraction and has no sharp disruption threshold.

Finally consider an initially circular orbit of radius $R$ around a point mass $M$. Its speed and specific angular momentum obey $v^2=GM/R$ and $h^2=GMR$. Immediately after $M\to ϵM$, these remain unchanged, while the new specific energy isUsing the orbital eccentricity relation $e^2=1+2E^{\prime }{h}^2/(G^2{ϵ}^2{M}^2)$ givesIt is an ellipse for $ϵ>1/2$, parabolic at $ϵ=1/2$, and unbound for smaller $ϵ$, in agreement with the virial argument.