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Galactic star formation is limited by how much gas becomes cold, shielded and self-gravitating, and by how long it remains in that state. A useful local estimate isThe star-formation efficiency per free-fall time ${ϵ}_{\mathrm{ff}}$ absorbs the fact that turbulence, magnetic support and stellar feedback prevent all eligible gas from collapsing in one free-fall time of a uniform sphere. Rotation and disk stability influence the supply of dense clouds, while gravitational collapse and gravitational fragmentation operate inside them.

For galaxy-averaged measurements, the Kennicutt–Schmidt law has ${\Sigma }_{\mathrm{SFR}}\propto {\Sigma }_g^N$, with the classic mixed spiral/starburst sample giving $N\simeq 1.4$. This empirical relation was obtained using H-alpha and far-infrared star-formation tracers together with atomic-hydrogen and CO gas measurements.^[ref] Dust attenuation, contamination by older stellar heating, the stellar initial mass function and the CO-to-molecular-gas conversion must be controlled. A disk-averaged total-gas exponent is not automatically a small-scale cloud law.

Young stars then photoionize gas, drive winds and eventually inject supernova energy and momentum. These processes can disrupt clouds, sustain random motions or expel gas; radiative cooling and fresh inflow rebuild the cold supply. Large-scale star formation is a regulated conversion of gas, rather than unimpeded free fall of the entire disk. A long gas-depletion time ${\Sigma }_g/{\Sigma }_{\mathrm{SFR}}$ is compatible with short galactic dynamical times in the dense fraction.

virial pressure-radius curve with uniform-sphere binding energy. Put $c_s^2=k_{B}T/\mu$. Isothermality makes $\int (p/\rho )dM=c_s^2{M}_0$, so the virial theorem givesThe curve crosses zero at $r_z=B/A=G{M}_0/(5c_s^2)$, rises to its maximum atthen decreases toward zero at large radius. Since $p_0^{\prime }=(4B-3A{r}_0)/r_0^5$, a cloud displaced outward at fixed external pressure has a restoring pressure deficit only on the descending branch. Thus the homologous, isothermal radial mode is stable for $r_0>r_c$, unstable for $r_z<r_0<r_c$, and marginal at $r_c$. The negative-pressure branch is not a physical equilibrium with a positive confining pressure. This conclusion uses the stipulated gravitational-energy approximation, not the exact isothermal density profile.

critical mass from a virial pressure turning point. At the turning point, $B=3A{r}_c/4$; henceUsing the cloud's mean density $\bar{\rho }=3M_0/(4\pi r_0^3)$, the condition $r_0>4G{M}_0/(15c_s^2)$ becomesUsing the boundary density instead changes the numerical coefficient, not this scaling. The coefficient should not be identified with the exact Bonnor-Ebert sphere threshold because the assumed potential energy is different.

mass and luminosity weighting of a stellar initial mass function. Impose finite birth-mass cutoffs $m_l,m_u$. The stellar birth mass and main-sequence luminosity scale asFor a broad mass interval, the low-mass end controls the first integral and the high-mass end the second. Per logarithmic mass interval the weights are respectively $m^2\xi \propto m^{-0.35}$ and $mL\xi \propto m^{2.65}$. Finite lifetimes remove massive stars from an aging population, so the luminosity conclusion applies while those stars remain on the main sequence.

exponential closed-box gas consumption. The gross formation rate is $\Psi =ϵM_g$, but the permanent locking rate is $\alpha \Psi$. The instantaneous recycling approximation and mass conservation give ${\dot{M}}_g=-\alpha ϵM_g$, ${\dot{M}}_{\ast }=\alpha ϵM_g$. ThereforeHere $M_{\ast }$ means surviving locked stellar mass in the galactic gas fraction, not the cumulative gross mass that has passed through star formation.

locked-mass and returned-mass yield conventions. Pre-existing metals are locked at rate $\alpha Z\Psi$; the stipulated newly produced metal source is $b(1-\alpha )\Psi$. ThusFor $\alpha >0$ and initially pristine gas,The effective yield $y$ is per unit permanently locked mass; it is not the source parameter $b$. For $\alpha =0$, $M_g=M_0$ and the direct time equation gives $Z=bϵt$. For $\alpha =1$, the explicitly stipulated metal source vanishes, so this particular enrichment model gives $Z=0$. Constant yield is a trace-enrichment approximation; the formal logarithmic divergence as gas is exhausted must not be interpreted as a physical metal fraction exceeding one.

backward extrapolation of exponential closed-box star formation. Denote the present age of the universe by $t_{\mathrm{now}}$. With the last paragraph's $\alpha =1$,Adopting a present age of the universe $t_{\mathrm{now}}=13.8\mathrm{Gyr}$ for the estimate gives about $365M_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$, requiring $7.3\times 10^{11}{M}_{\odot }$ of gas at that epoch in the same model. The boxed expression makes the dependence on the adopted present age explicit. This is an extreme history for a present Milky-Way-like system, although intense high-redshift star formation by itself is not forbidden.

The extrapolation forces a fixed gas reservoir to decline exponentially over many gas-depletion times. Gas accretion, outflows, evolving gas density and star-formation efficiency, delayed recycling, and galaxy assembly break that assumption. The empirical disk-averaged relation is not a universal constant-efficiency law for all gas: the original measurements combined Balmer series H-alpha, atomic gas, carbon monoxide and infrared data, with tracer calibration and dust corrections entering the inference.^[ref] Cosmological calculations explicitly support continuing cold-gas supply to rapidly forming high-redshift galaxies, providing a physical alternative to this closed-box backward extrapolation.^[ref]

Galactic star formation requires gas to collect into cold dense regions that can become self-gravitating. Disk-scale gravity, turbulent motions, shear, pressure and magnetic fields affect this collection; formation then proceeds within molecular clouds. A useful local parametrization iswhere the star-formation efficiency per free-fall time summarizes inefficient conversion rather than requiring every cloud to collapse at once.

Across galactic disks the empirical Kennicutt–Schmidt law relates star formation rate surface density to gas surface density, ${\Sigma }_{\mathrm{SFR}}\propto {\Sigma }_g^N$. A slope near $1.4$ is a familiar integrated total-gas fit, while molecular-gas relations in ordinary disks can be closer to linear; the slope and normalization depend on tracer, averaging scale and environment. The gas-depletion time $t_{\mathrm{dep}}=M_g/\Psi$ is the corresponding fuel-consumption scale, not a prediction that inflow and recycling cease.

Massive stars supply ionizing radiation, winds and supernova energy and momentum. This stellar feedback can disperse clouds, sustain turbulence and drive outflows, regulating rather than simply extinguishing all future star formation. Fresh accretion and recycled gas replenish the reservoir. UV and recombination-line emission trace young stars, while infrared emission traces absorbed and re-radiated light; extinction, stellar populations and the assumed initial mass function affect inferred rates. A galactic rate reflects supply, cloud formation and feedback together, rather than only the free-fall time of its densest gas.^[ref]