Stellar feedback

Stellar feedback is the transfer of energy, momentum, mass, and newly formed elements from stars to surrounding matter through radiation, winds, supernovae, and cosmic rays.

Depositing supernova energy into dense unresolved gas can radiate it away before a numerically resolved expansion develops. Equal injected energy therefore need not imply equal outflow momentum or heating at different resolution. A feedback model should represent the coupling and cooling phase it intends to approximate, and comparisons should include gas recycling and the resulting stellar and circumgalactic properties.

Before local expulsion, a linear star-formation law $\psi =\nu {\rho }_g$ without return gives ${\rho }_g={\rho }_0{e}^{-\nu t}$. If $q=\gamma E_{\mathrm{SN}}$ is the energy supplied per unit stellar mass, accumulated energy is $q{\rho }_0(1-e^{-\nu t})$. A fixed binding energy per unit remaining gas mass $B(r)$ gives the threshold $q(e^{\nu t}-1)=B(r)$. This assumes a specified retained gravitational profile and neglects exchange between volumes.

The local shell-energy prescription sets $B(r)=GM(<r)/r$ rather than the full escape work. It predicts simultaneous quenching for $M(<r)\propto r$, but a centre-to-edge front for $M(<r)\propto r^2$. Exterior-shell contributions explain why this ordering can differ from a true escape-potential calculation.

With test-parcel escape work $B(r)=-\Phi (r)$, outer gas in a spherical isolated attractive potential is less tightly bound and quenches first under identical local specific heating. A truncated inverse-square density gives $r_q=r_0\exp [1-q(e^{\nu t}-1)/(2k)]$ after edge onset. The result is a fixed-potential closure, not a self-consistent potential update after mass loss.

For flux $F$ and effective dust opacity $\kappa$, radiative acceleration is $\kappa F/c$. Comparing it with the gravitational acceleration gives an Eddington luminosity condition appropriate to dusty gas. In an optically thick starburst galaxy, feedback can limit the star formation rate per area and drive a galactic outflow, while the gas supply limits the duration. A surface-density limit is not a universal ceiling on total star formation rate: the star-forming area matters. Radiation-supported starburst disk calculations give characteristic optically thick infrared fluxes near $10^{13}{L}_{\odot }{\mathrm{kpc}}^{-2}$.

If a virialized, well-mixed self-gravitating system instantaneously retains mass fraction $ϵ>1/2$, homologous revirialization gives $R^{\prime }/R=ϵ/(2ϵ-1)$. At or below one half, the idealized remnant is unbound.

Take an initially isolated virialized self-gravitating stellar population with kinetic energy $T=|W|/2$ and Newtonian gravitational potential energy $W<0$. Instantaneous spatially uniform removal of mass, without kicks to surviving material, leaves fraction $f$ with $T^{\prime }=fT$ and $W^{\prime }=f^{2}W$. Hence $E^{\prime }=|W|f(1/2-f)$: the homologous whole-system energy is negative only for $f>1/2$. Positive whole-system energy does not forbid a bound core after selective escape, and slow mass loss follows the adiabatic mass-loss expansion law instead.

For an initial mass function $dN\propto m^{-\alpha }dm$ on $0.1<m/M_{\odot }<100$, complete loss of material initially above $10M_{\odot }$ removes fractionThe impulsive disruption by stellar mass loss threshold is $F=1/2$, with unique finite root $\alpha \simeq 1.79101236$. Decreasing $\alpha$ shifts mass to larger stars and increases $F$, so the idealized system is globally unbound for $\alpha <{\alpha }_{\mathrm{crit}}$. The derivation requires rapid, well-mixed loss and absence of a confining external potential.

If a self-gravitating system loses mass slowly compared with its orbital times, adiabatic invariance gives $MR=\text{constant}$ in the homologous approximation, so retaining fraction $ϵ$ expands the system by $R^{\prime }/R=1/ϵ$.