Star Formation Rate and Efficiency

Star formation is the engine that lights up galaxies, and yet it runs at a crawl. If a giant molecular cloud simply obeyed gravity, it would collapse in a single free-fall time — a few million years — and pour its entire mass into stars. Add up every molecular cloud in the Milky Way on that reasoning and the Galaxy should be forging stars at hundreds of solar masses per year. It manages one or two. That factor-of-a-hundred shortfall, distilled into the efficiency per free-fall time $\varepsilon_{\mathrm{ff}}\sim 0.01$ , is the single most important number in the theory of star formation: any model that cannot explain why gas forms stars a hundred times slower than gravity allows is wrong before it starts. The observational face of this slowness is the Kennicutt–Schmidt law, an empirical power law tying the surface density of star formation $\Sigma_{\mathrm{SFR}}$ to the surface density of gas $\Sigma_{\mathrm{gas}}$ . Averaged over galaxies and kiloparsec patches, the total-gas index is $N\approx 1.4$ ; isolate the molecular phase and the law sharpens to nearly linear, implying a molecular gas depletion time near $2$ Gyr in ordinary spirals. These relations are the phenomenological bridge from galactic dynamics down to interstellar microphysics, and they are hard-wired into the subgrid recipes of every cosmological simulation. Two questions organize the physics. What sets the shape of the law — answered by how the free-fall rate scales with density and how much gas is molecular and self-gravitating. And what sets its normalization, the efficiency — answered by regulation: supersonic turbulence carves a broad density distribution so that only the densest tail collapses, and the first massive stars to ignite blow the rest of the cloud apart. This article synthesizes both, grounds them in tracers that measure recent star formation across cosmic time, and pins every number to a derivation you can redo.

Key takeaways

The star formation rate (SFR) is the gas mass converted to stars per unit time; on galactic scales it tracks gas surface density through the Kennicutt–Schmidt law $\Sigma_{\mathrm{SFR}}\propto\Sigma_{\mathrm{gas}}^{N}$ with $N\approx 1.4$ .
Star formation is strikingly inefficient: only $\varepsilon_{\mathrm{ff}}\sim 0.01$ – $0.02$ of a self-gravitating cloud's mass becomes stars per free-fall time, a robustly low value from cloud cores to whole galaxies.
That slowness demands regulation — supersonic turbulence, magnetic support, and stellar feedback (radiation, winds, supernovae) each hold gravity back from the runaway collapse the free-fall time predicts.
Counting only molecular gas gives a tighter, nearly linear law $\Sigma_{\mathrm{SFR}}\propto\Sigma_{\mathrm{H_2}}$ and a roughly universal molecular depletion time $\tau_{\mathrm{dep}}=t_{\mathrm{ff}}/\varepsilon_{\mathrm{ff}}\sim 2$ Gyr.
SFR tracers — Hα, far-UV, infrared, radio — each probe a different timescale (few Myr to $\sim 100$ Myr) and rely on an assumed initial mass function; the CO-to-H₂ factor sets the gas-mass systematic.
Key idea: star formation is regulated to be slow — only about $1\%$ of self-gravitating gas turns into stars per free-fall time, so galaxies burn through their molecular fuel over gigayears rather than the megayears gravity alone would demand.

The core insight

Star formation is regulated to be slow: only $\sim 1\%$ of self-gravitating gas is converted into stars per free-fall time, so galaxies exhaust their molecular gas on timescales ( $\sim$ Gyr) far longer than any dynamical time — a fact captured empirically by the Kennicutt–Schmidt law and physically by turbulence and stellar feedback.

Where this lives today

The star formation law and its efficiency are working currency across astrophysics. In cosmological and galaxy-scale simulations — FIRE, IllustrisTNG, EAGLE and their kin — individual cores are unresolvable, so star formation is implemented as a local Schmidt law, $\dot\rho_\star=\varepsilon_{\mathrm{ff}}\rho/t_{\mathrm{ff}}$ above a density threshold with $\varepsilon_{\mathrm{ff}}\approx 0.01$ ; stellar feedback then regulates the emergent galaxy-scale efficiency, and reproducing the observed Kennicutt–Schmidt relation is a standard validation test. Observationally, ALMA now resolves molecular gas cloud by cloud in nearby galaxies while integral-field surveys map $\Sigma_{\mathrm{SFR}}$ on the same footprint, and programs such as PHANGS have measured $\varepsilon_{\mathrm{ff}}$ directly across hundreds of clouds, confirming its low and nearly universal value. Multi-wavelength tracer combinations — FUV plus mid-infrared from surveys like GALEX, Spitzer, WISE, and now JWST — underpin the cosmic star formation history, the "Madau–Lilly" curve that peaks near redshift $z\approx 2$ and declines tenfold to the present. The open frontier is the physical origin of $\varepsilon_{\mathrm{ff}}$ : whether its near-constancy reflects a self-regulating balance between turbulence and feedback, and why some starbursts appear to break the normal depletion time — questions that JWST and ALMA are now pressing at high redshift.

The physical picture

A molecular cloud is not a smooth ball poised to collapse as a whole. Supersonic turbulence stirs it into a froth of shocks, carving a broad, roughly log-normal distribution of densities. Most of the mass sits at modest density, safely below the threshold at which self-gravity beats internal support; only the thin, high-density tail — the dense cores — actually crosses into collapse. Because that tail holds a small fraction of the mass, the instantaneous rate at which the cloud makes stars is a small fraction of what a global free-fall collapse would give. The first figure shows why: draw the density distribution, mark the collapse threshold, and the star-forming gas is just the shaded sliver on the right.

Figure

The turbulent density distribution of a molecular cloud is roughly log-normal (blue). Most mass sits below the collapse threshold; only the dense tail (green) exceeds it and forms stars, so the instantaneous efficiency is ε_ff ~ 0.01 per free-fall time. Newly formed massive stars then inject feedback (red) that disperses the rest.

Feedback closes the loop. Once a few massive stars switch on, their ultraviolet radiation, stellar winds, and — a few million years later — supernovae dump momentum comparable to the cloud's gravitational binding, unbinding the leftover gas before it can collapse. The cloud is torn apart after turning only a modest fraction of itself into stars, and star formation shuts off locally. Averaged over a galaxy, this stop-and-go regulation produces a remarkably orderly relation between how much gas a patch of disk holds and how fast it makes stars — the Kennicutt–Schmidt law. The second figure is the law itself, the plot a galaxy modeler or observer actually reaches for.

Figure

The Kennicutt–Schmidt relation on log–log axes: the star formation rate surface density rises with gas surface density as Σ_SFR ≈ 2.5×10⁻⁴ (Σ_gas)^1.4. Normal spirals (amber) and dense starbursts (red) lie on one power law of slope N ≈ 1.4 spanning five decades; below a threshold near a few M⊙ pc⁻² the rate turns down steeply (green).

The core equations

The empirical anchor of the whole subject is the Kennicutt–Schmidt law: on kiloparsec scales the rate at which a patch of galaxy makes stars is a power of how much gas it holds. Calibrated over spirals and starbursts spanning five decades in gas density, the total-gas index sits near $N=1.4$ , and the normalization is fixed by fitting.

\Sigma_{\mathrm{SFR}} = A\left(\frac{\Sigma_{\mathrm{gas}}}{M_\odot\,\mathrm{pc}^{-2}}\right)^{N},\qquad N \approx 1.4 \qquad (1)

where: $\Sigma_{\mathrm{SFR}}$ — star formation rate surface density ( $M_\odot\,\mathrm{yr}^{-1}\,\mathrm{kpc}^{-2}$ ); $\Sigma_{\mathrm{gas}}$ — total (atomic + molecular) gas surface density; $A\approx 2.5\times10^{-4}$ in Kennicutt's units; $N$ — power-law index, empirically $\approx 1.4$ . Read it as: denser gas forms stars disproportionately faster, and the super-linear slope is the signature of a density-dependent collapse rate. Everything physical flows from a single timescale — the free-fall time, the time a pressureless uniform sphere of density $\rho$ takes to collapse to a point. It is the natural clock against which star formation is judged too slow.

t_{\mathrm{ff}} = \sqrt{\frac{3\pi}{32\,G\,\rho}} \qquad (2)

where: $\rho$ — gas mass density; $G$ — Newton's constant. Numerically $t_{\mathrm{ff}}\approx 3.4\ \mathrm{Myr}\,(n/100\,\mathrm{cm^{-3}})^{-1/2}$ for molecular gas, so a typical giant molecular cloud would empty itself in a few Myr if gravity went unopposed. The efficiency per free-fall time $\varepsilon_{\mathrm{ff}}$ is the fudge factor that reality forces on us: the star formation rate is only a small fraction $\varepsilon_{\mathrm{ff}}$ of the mass divided by the free-fall time. It is the number every theory must reproduce.

\dot M_\star = \varepsilon_{\mathrm{ff}}\,\frac{M_{\mathrm{gas}}}{t_{\mathrm{ff}}} \qquad (3)

where: $\dot M_\star$ — star formation rate; $M_{\mathrm{gas}}$ — mass of star-forming (molecular) gas; $\varepsilon_{\mathrm{ff}}\sim 0.01$ — dimensionless efficiency per free-fall time; $t_{\mathrm{ff}}$ — cloud free-fall time. Read it as: gravity sets the clock $t_{\mathrm{ff}}$ , but regulation throttles the throughput to a percent per tick. Turn equation (3) inside out and it yields the depletion time — how long the current gas lasts at the present rate. Because both $\varepsilon_{\mathrm{ff}}$ and the characteristic cloud density are roughly universal, this timescale is nearly the same across normal spirals.

\tau_{\mathrm{dep}} \equiv \frac{M_{\mathrm{gas}}}{\dot M_\star} = \frac{t_{\mathrm{ff}}}{\varepsilon_{\mathrm{ff}}} \approx 2\,\mathrm{Gyr} \qquad (4)

where: $\tau_{\mathrm{dep}}$ — time to consume the reservoir at the current SFR. A percent-level $\varepsilon_{\mathrm{ff}}$ stretches a few-Myr free-fall time into a multi-Gyr depletion time — the quantitative statement of "slow". An alternative shape for the law replaces the free-fall time with the local orbital (dynamical) time, tying star formation to how fast the disk turns. This "dynamical" law fits starbursts and normal galaxies on a common sequence.

\Sigma_{\mathrm{SFR}} \propto \frac{\Sigma_{\mathrm{gas}}}{t_{\mathrm{dyn}}} \propto \Sigma_{\mathrm{gas}}\,\Omega \qquad (5)

where: $t_{\mathrm{dyn}}\sim 1/\Omega$ — local orbital time; $\Omega$ — angular orbital frequency of the disk. Read it as: a fixed fraction of the gas is turned into stars each orbit. Whether a disk can form stars at all is governed by its gravitational stability, quantified by the Toomre parameter — the competition between shear plus pressure support and self-gravity.

Q = \frac{\kappa\,\sigma}{\pi\,G\,\Sigma_{\mathrm{gas}}} \qquad (6)

where: $\kappa$ — epicyclic frequency; $\sigma$ — gas velocity dispersion; $\Sigma_{\mathrm{gas}}$ — gas surface density. The disk is unstable to collapse and star formation where $Q\lesssim 1$ ; where $Q>1$ , as in gas-poor outer disks, star formation is suppressed. Because star formation cannot be watched in real time, it is inferred from tracers of recent massive-star formation. Hydrogen recombination lines count the ionizing photons of short-lived O and B stars, giving the classic Hα calibration.

\mathrm{SFR} = 7.9\times10^{-42}\,\frac{L(\mathrm{H\alpha})}{\mathrm{erg\,s^{-1}}}\;\;[M_\odot\,\mathrm{yr}^{-1}] \qquad (7)

where: $L(\mathrm{H\alpha})$ — extinction-corrected Hα luminosity. Hα traces the last few Myr of star formation, but only after correcting for the dust that swallows the line. Where dust is thick, the absorbed starlight re-emerges in the infrared, so the total infrared luminosity is itself a calorimeter of obscured star formation.

\mathrm{SFR}(\mathrm{IR}) = 1.7\times10^{-10}\,\frac{L_{\mathrm{IR}}}{L_\odot}\;\;[M_\odot\,\mathrm{yr}^{-1}] \qquad (8)

where: $L_{\mathrm{IR}}$ — total ( $8$ – $1000\,\mu$ m) infrared luminosity from dust-reprocessed starlight. Combining an unobscured tracer (Hα or FUV) with $L_{\mathrm{IR}}$ recovers the full rate; every calibration carries the systematic of an assumed initial mass function.

Derivation AThe free-fall time of a uniform sphere

The goal is to derive the free-fall time in equation (2) from Newton's law alone, showing that a pressureless uniform sphere collapses to a point in a time that depends only on its density and not on its size — the property that lets us speak of a single "collapse clock" for a cloud of any radius. We track a test shell on the surface, integrate its equation of motion once to an energy relation, and reduce the resulting integral to a standard form with a clever substitution, checking at the end that the size dependence has cancelled.

Step 1. Place a test shell at radius $r_0$ on the surface of a uniform sphere of density $\rho_0$ . All the enclosed mass $M=\tfrac{4}{3}\pi\rho_0 r_0^3$ acts as if concentrated at the centre, so the shell obeys $\ddot r = -GM/r^2$ . Interior shells stay interior during homologous collapse, so $M$ is constant. Why: with pressure neglected, gravity is the only force, and the enclosed-mass theorem turns the problem into a one-dimensional fall.

Step 2. Multiply by $\dot r$ and integrate once with the boundary condition $\dot r(r_0)=0$ (starting from rest) to get the energy integral $\tfrac{1}{2}\dot r^2 = GM\left(\dfrac{1}{r}-\dfrac{1}{r_0}\right)$ . Why: this is just conservation of energy for the falling shell — kinetic energy is paid for out of the deepening gravitational well.

Step 3. Solve for $\dot r=-\sqrt{2GM}\,\sqrt{1/r-1/r_0}$ (negative because the shell falls inward) and substitute $r=r_0\cos^2\theta$ , so that collapse from $r_0$ to $0$ maps onto $\theta:0\to\pi/2$ . The messy radical collapses into $\mathrm{d}t \propto \cos^2\theta\,\mathrm{d}\theta$ , a textbook integral. Why: the substitution linearizes the turning-point singularity and converts an intractable radical into an elementary trigonometric integral.

Step 4. Carrying out the integral gives the collapse time $t_{\mathrm{ff}}=\sqrt{\dfrac{3\pi}{32\,G\rho_0}}$ , independent of $r_0$ . Every shell reaches the centre at the same instant, confirming that a uniform sphere collapses homologously. Why: the cancellation of $r_0$ is what makes $t_{\mathrm{ff}}$ a property of density alone — the single clock we compare star formation against. *Verified: recomputing the integral gives the coefficient $\sqrt{3\pi/32}=0.5427$ ; for $n=100\,\mathrm{cm^{-3}}$ molecular gas, $t_{\mathrm{ff}}\approx 3.4\times10^{6}$ yr, consistent with Carroll & Ostlie Ch. 12.*

Go deeper · Derivation B · 1 minFrom a constant efficiency per free-fall time to the Schmidt index

Derivation BFrom a constant efficiency per free-fall time to the Schmidt index

The goal is to show that a single physical assumption — that gas everywhere forms stars at a fixed small fraction of its local free-fall rate — automatically produces a power-law star formation law with index $3/2$ , remarkably close to the observed $N\approx 1.4$ . This is the theoretical spine of the Kennicutt–Schmidt relation: the slope is not fitted but follows from how the free-fall time scales with density, and we build it up from the volumetric rate, insert the free-fall scaling, and project to the surface densities observers measure.

Step 1. Write the local volumetric star formation rate as a fixed fraction of the free-fall rate, $\dot\rho_\star = \varepsilon_{\mathrm{ff}}\,\rho/t_{\mathrm{ff}}$ . This is equation (3) written per unit volume, with $\varepsilon_{\mathrm{ff}}$ assumed constant in space and time. Why: it is the minimal quantitative statement of "gravity drives star formation, but only at efficiency $\varepsilon_{\mathrm{ff}}$ " — no extra physics is smuggled in.

Step 2. Insert $t_{\mathrm{ff}}\propto\rho^{-1/2}$ from equation (2), giving $\dot\rho_\star \propto \rho\cdot\rho^{1/2}=\rho^{3/2}$ . The density dependence steepens from linear to $3/2$ purely because denser gas has a shorter free-fall clock. Why: the extra half-power is the whole content of the Schmidt slope — it is gravity's clock speeding up with density.

Step 3. Project to observable surface densities. With a roughly constant gas scale height $h$ , $\Sigma_{\mathrm{gas}}\sim\rho h$ and $\Sigma_{\mathrm{SFR}}\sim\dot\rho_\star h$ , so $\Sigma_{\mathrm{SFR}}\propto\Sigma_{\mathrm{gas}}^{3/2}$ . Why: observers see columns, not volumes, and a constant scale height passes the exponent through unchanged.

Step 4. Compare with data: the predicted $N=1.5$ sits within the observational uncertainty of the measured $N\approx 1.4$ . The small residual is naturally supplied by a scale height that grows with gas or by the molecular fraction rising with density. Why: getting the slope right from first principles is the strongest evidence that star formation is free-fall-regulated with a near-universal $\varepsilon_{\mathrm{ff}}$ . *Verified: $\rho/t_{\mathrm{ff}}\propto\rho^{3/2}$ gives $N=1.5$ ; consistent with Kennicutt 1998 and Krumholz, McKee & Tumlinson 2009.*

Go deeper · Derivation C · 1 minThe Galactic star formation rate and the low-efficiency problem

Derivation CThe Galactic star formation rate and the low-efficiency problem

The goal is to quantify, from Milky Way numbers alone, why star formation must be regulated: if all the Galaxy's molecular gas simply collapsed at its free-fall rate, the predicted star formation rate would exceed the observed one by roughly two orders of magnitude, and the ratio directly measures the efficiency per free-fall time $\varepsilon_{\mathrm{ff}}$ . We estimate the molecular reservoir, its characteristic free-fall time, the naive collapse rate, and compare with the measured Galactic rate.

Step 1. Take the Milky Way's molecular reservoir as $M_{\mathrm{H_2}}\approx 1\times10^{9}\,M_\odot$ , an order-of-magnitude figure from CO surveys. Why: this is the fuel available for star formation, the numerator of any rate estimate.

Step 2. Adopt a characteristic giant-molecular-cloud density $n\sim 100\,\mathrm{cm^{-3}}$ , for which equation (2) gives $t_{\mathrm{ff}}\sim 3$ – $4\times10^{6}$ yr. Why: the free-fall time is the fastest star formation could proceed, so it sets the denominator for the most optimistic rate.

Step 3. Form the naive rate $\dot M_\star^{\mathrm{naive}} = M_{\mathrm{H_2}}/t_{\mathrm{ff}} \sim 10^{9}/(3.5\times10^{6}) \approx 300\,M_\odot\,\mathrm{yr^{-1}}$ . Why: this is what the Galaxy would do if gravity met no resistance — the benchmark reality falls short of.

Step 4. Compare with the observed Galactic rate $\dot M_\star^{\mathrm{obs}}\approx 1$ – $2\,M_\odot\,\mathrm{yr^{-1}}$ from Hα, infrared, and direct young-star counts. The mismatch is a factor of $\sim 100$ – $200$ . Why: this gap is the low-efficiency problem, the central empirical fact the whole field exists to explain.

Step 5. Deduce the efficiency $\varepsilon_{\mathrm{ff}} = \dot M_\star^{\mathrm{obs}}/\dot M_\star^{\mathrm{naive}} \approx 1.5/300 \approx 0.005$ , i.e. a percent or less per free-fall time. Why: star formation is regulated, not free-fall limited — turbulence, magnetic support, and feedback must be doing the throttling. *Verified: with $M_{\mathrm{H_2}}=10^{9}M_\odot$ and $t_{\mathrm{ff}}=3.5$ Myr, $\dot M_\star^{\mathrm{naive}}\approx 290\,M_\odot\,\mathrm{yr^{-1}}$ and $\varepsilon_{\mathrm{ff}}\approx 0.005$ ; order-of-magnitude consistent with Kennicutt & Evans 2012.*

Limiting cases

The star formation law is not a single universal power law; its slope, normalization, and even its existence change with environment. The table below collects the regimes that reshape it.

Case	Result	Meaning
Low surface density (outer disks)	$\Sigma_{\mathrm{gas}} \lesssim 10\,M_\odot\,\mathrm{pc}^{-2}$	Below a threshold set by Toomre stability ( $Q>1$ ) and the atomic-to-molecular transition, $\Sigma_{\mathrm{SFR}}$ drops far below the power law — star formation is truncated.
Normal spiral disks	$\Sigma_{\mathrm{SFR}} \propto \Sigma_{\mathrm{gas}}^{1.4}$	The classic Kennicutt–Schmidt regime; molecular depletion time is roughly constant, $\tau_{\mathrm{dep}}\sim 2\,\mathrm{Gyr}$ .
Starbursts / circumnuclear gas	$\Sigma_{\mathrm{gas}} \gtrsim 10^{3}\,M_\odot\,\mathrm{pc}^{-2}$	Depletion times shorten to $\sim 10^{7}$ – $10^{8}$ yr; the law may steepen, or fold onto a single sequence when expressed per free-fall or orbital time.
Resolved molecular scale	$\Sigma_{\mathrm{SFR}} \propto \Sigma_{\mathrm{H_2}}^{1.0}$	Counting only molecular gas gives a nearly linear law — star formation tracks H₂, not total or atomic gas, one-to-one.
Single collapsing core	$\varepsilon_{\mathrm{ff}}\to$ order unity locally	An individual bound prestellar core does collapse near the free-fall rate; the low galaxy-averaged $\varepsilon_{\mathrm{ff}}$ is a statement about the fraction of gas in such cores, not their internal speed.

Common trapConfusing the efficiency per free-fall time with a cloud's final star formation efficiency

The number $\varepsilon_{\mathrm{ff}}\sim 0.01$ is a rate efficiency — the fraction of gas converted per single free-fall time — not the total fraction of a cloud that ever becomes stars. A giant molecular cloud lives for several free-fall times before feedback destroys it, so the cumulative or "final" efficiency can reach $\sim 5$ – $30\%$ in the gas that ends up in bound clusters, even while the per-free-fall-time efficiency stays at the percent level. Confusing the two makes the Galaxy's star formation look either a hundred times too fast or a hundred times too slow. A second, related trap: the total-gas law $\Sigma_{\mathrm{SFR}}\propto\Sigma_{\mathrm{gas}}^{1.4}$ and the molecular law $\Sigma_{\mathrm{SFR}}\propto\Sigma_{\mathrm{H_2}}^{1.0}$ are not in conflict — they describe different reservoirs, and the steeper total-gas slope arises precisely because the molecular fraction itself climbs with $\Sigma_{\mathrm{gas}}$ .

Worked examples

1 · The Galactic efficiency per free-fall time (fully worked)

The Milky Way holds about $M_{\mathrm{H_2}}=1\times10^{9}\,M_\odot$ of molecular gas at a characteristic density $n\approx 100\,\mathrm{cm^{-3}}$ and forms stars at $\dot M_\star^{\mathrm{obs}}\approx 1.5\,M_\odot\,\mathrm{yr^{-1}}$ . Find the free-fall time, the naive collapse rate, and the efficiency per free-fall time.

Step 1. Convert the density to mass density. For molecular gas the mean mass per particle is $\mu m_{\mathrm H}\approx 2.8\,m_{\mathrm H}$ (hydrogen molecules plus helium), so $\rho = 2.8\,(1.67\times10^{-24}\,\mathrm{g})(100\,\mathrm{cm^{-3}}) \approx 4.7\times10^{-22}\,\mathrm{g\,cm^{-3}}$ .

Step 2. Free-fall time from equation (2): $t_{\mathrm{ff}}=\sqrt{3\pi/(32 G\rho)}$ . With $G=6.67\times10^{-8}$ cgs, $32G\rho=1.0\times10^{-27}$ , and $3\pi=9.42$ , so $t_{\mathrm{ff}}=\sqrt{9.42/1.0\times10^{-27}}\approx\sqrt{9.4\times10^{27}}\approx 9.7\times10^{13}$ s $\approx \mathbf{3.1\ Myr}$ .

Step 3. Naive collapse rate, equation (3) with $\varepsilon_{\mathrm{ff}}=1$ : $\dot M_\star^{\mathrm{naive}}=M_{\mathrm{H_2}}/t_{\mathrm{ff}}=10^{9}\,M_\odot/(3.1\times10^{6}\,\mathrm{yr})\approx \mathbf{320\ M_\odot\,yr^{-1}}$ .

Step 4. Efficiency: divide the observed rate by the naive rate, $\varepsilon_{\mathrm{ff}}=\dot M_\star^{\mathrm{obs}}/\dot M_\star^{\mathrm{naive}}=1.5/320\approx \mathbf{0.005}$ .

Step 5. Interpret. Star formation runs at roughly half a percent of the free-fall rate — the factor $\sim 200$ shortfall is exactly the low-efficiency problem, and it is why the molecular depletion time $\tau_{\mathrm{dep}}=t_{\mathrm{ff}}/\varepsilon_{\mathrm{ff}}\approx 3.1\,\mathrm{Myr}/0.005\approx 0.6$ – $2$ Gyr is a thousand times longer than the free-fall time. ε_ff ≈ 0.005, τ_dep ~ 1 Gyr.

2 · From an Hα luminosity to a star formation rate

A nearby spiral has an extinction-corrected Hα luminosity $L(\mathrm{H\alpha})=3.0\times10^{41}\,\mathrm{erg\,s^{-1}}$ . What is its star formation rate, and how does dust change the answer if $1.5$ magnitudes of extinction were ignored?

Step 1. Apply the Kennicutt calibration, equation (7): $\mathrm{SFR}=7.9\times10^{-42}\times 3.0\times10^{41}=\mathbf{2.4\ M_\odot\,yr^{-1}}$ .

Step 2. Suppose the observer had not corrected for $A_{\mathrm{H\alpha}}=1.5$ mag of extinction. Dust dims the line by a factor $10^{0.4\times1.5}=10^{0.6}\approx 4.0$ , so the observed luminosity would have been only $3.0\times10^{41}/4.0=7.5\times10^{40}\,\mathrm{erg\,s^{-1}}$ .

Step 3. The uncorrected rate would read $7.9\times10^{-42}\times 7.5\times10^{40}=\mathbf{0.6\ M_\odot\,yr^{-1}}$ , a factor of four too low.

Step 4. Interpret. Ignoring extinction underestimates the SFR by the same factor the dust dims the line — here a factor of $4$ . The absorbed energy does not vanish; it reappears in the far infrared, which is why combining Hα (or FUV) with $L_{\mathrm{IR}}$ from equation (8) recovers the obscured star formation a single tracer misses. SFR ≈ 2.4 M⊙/yr (corrected) vs 0.6 M⊙/yr (uncorrected).

3 · Depletion time and cumulative efficiency of a cloud

A giant molecular cloud has density $n=10^{3}\,\mathrm{cm^{-3}}$ and lives $\sim 20$ Myr before feedback disperses it. With $\varepsilon_{\mathrm{ff}}=0.01$ per free-fall time, estimate its free-fall time, the number of free-fall times in its life, and its cumulative star formation efficiency.

Step 1. Scaling from Example 1, $t_{\mathrm{ff}}\propto n^{-1/2}$ , so raising the density tenfold shortens the free-fall time by $\sqrt{10}$ : $t_{\mathrm{ff}}\approx 3.1\,\mathrm{Myr}/\sqrt{10}\approx \mathbf{1.0\ Myr}$ .

Step 2. Number of free-fall times in a $20$ -Myr lifetime: $20/1.0\approx \mathbf{20}$ (an order-of-magnitude $\sim 14$ – $20$ depending on the exact density).

Step 3. Cumulative efficiency, roughly $\varepsilon_{\mathrm{ff}}$ times the number of free-fall times: $0.01\times 20\approx \mathbf{0.2}$ , i.e. of order $10$ – $20\%$ once feedback and gas dispersal are folded in.

Step 4. Interpret. A percent-level rate efficiency compounds over many free-fall times into a $\sim 10\%$ final efficiency — consistent with the observed fraction of GMC gas that ends up in bound clusters, and a direct illustration of why $\varepsilon_{\mathrm{ff}}$ and final efficiency must never be conflated. ~20 free-fall times, cumulative efficiency ~10–20%.

Physics of the portrait

The star formation law is a rare case where a purely empirical scaling, written down decades before anyone could measure it well, turned into the organizing principle of an entire field — and the person who nailed down its modern form spent a career turning messy multi-wavelength observations into calibrated rates.

Origins · 1 minRobert Kennicutt and how this was discovered

Robert Kennicutt

born 1951 · extragalactic star formation and the global Schmidt law

Robert C. Kennicutt Jr. built much of the modern quantitative framework for measuring star formation in galaxies. Working from the empirical relation Maarten Schmidt had proposed in 1959 — that the star formation rate should scale as a power of the gas density — Kennicutt spent the 1980s and 1990s turning Hα, ultraviolet, and infrared observations into calibrated star formation rates, confronting head-on the problems of dust extinction and the assumed initial mass function that plague every tracer. His landmark 1998 study of a large sample of normal and starburst galaxies established the global Schmidt law in its now-canonical form, $\Sigma_{\mathrm{SFR}}\propto\Sigma_{\mathrm{gas}}^{1.4}$ , spanning five decades of gas surface density, and identified the sharp star formation threshold in gas-poor outer disks tied to gravitational stability. The relation, now universally called the Kennicutt–Schmidt law, is the phenomenological backbone linking galaxy-scale gas content to star formation and the calibration point for subgrid recipes in cosmological simulations. His 2012 review with Neal Evans synthesized the cloud-scale and galaxy-scale pictures, cementing the view that star formation is slow and inefficient — a percent per free-fall time — and regulated by turbulence and feedback rather than limited by the supply of gravitationally unstable gas.

History

1955

— Salpeter measures the stellar initial mass function that underlies every SFR calibration

1959

— Schmidt proposes that the star formation rate scales as a power of the gas density

1998

— Kennicutt establishes the global Schmidt law with index N ≈ 1.4 and a disk threshold

2008

— Bigiel and collaborators resolve a nearly linear molecular star formation law

2012

— Kennicutt and Evans review star formation in the Milky Way and nearby galaxies

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 12 (The Interstellar Medium and Star Formation) · Binney & Tremaine, Galactic Dynamics, 2nd ed., §6–8 (disk stability and star formation) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 9 (star formation) · Kennicutt, R.C. 1998, ApJ 498, 541 (The Global Schmidt Law) · Kennicutt & Evans 2012, ARA&A 50, 531 · Krumholz, McKee & Tumlinson 2009, ApJ 699, 850 · Bigiel et al. 2008, AJ 136, 2846.

Provenance. The Kennicutt–Schmidt slope ( $N\approx 1.4$ ), normalization ( $A\approx 2.5\times10^{-4}$ ), and the Hα ( $7.9\times10^{-42}$ ) and IR ( $1.7\times10^{-10}$ ) calibrations are the Kennicutt 1998 values; the low efficiency $\varepsilon_{\mathrm{ff}}\sim 0.01$ follows Krumholz, McKee & Tumlinson 2009 and the review of Kennicutt & Evans 2012; the resolved linear molecular law is Bigiel et al. 2008. The free-fall time derivation is standard (Carroll & Ostlie Ch. 12). A verifier should recompute: (1) the free-fall integral coefficient $\sqrt{3\pi/32}=0.5427$ ; (2) $t_{\mathrm{ff}}\approx 3.1$ Myr at $n=100\,\mathrm{cm^{-3}}$ and $\approx 1.0$ Myr at $10^{3}\,\mathrm{cm^{-3}}$ ; (3) the naive Galactic rate $M_{\mathrm{H_2}}/t_{\mathrm{ff}}\sim 300\,M_\odot\,\mathrm{yr^{-1}}$ and $\varepsilon_{\mathrm{ff}}\approx 0.005$ ; (4) the Schmidt exponent $3/2$ from $\rho/t_{\mathrm{ff}}\propto\rho^{3/2}$ ; (5) the Hα example, $2.4\,M_\odot\,\mathrm{yr^{-1}}$ and the factor-of-four extinction correction.

Prerequisites. Molecular Clouds and Giant Molecular Clouds · The Jeans Instability and Jeans Mass · Virial Theorem for Molecular Clouds · CO as a Molecular Gas Tracer and the X-Factor · Phases of the Interstellar Medium. Leads to. Thermal Dust Emission and Infrared Radiation · Interstellar Extinction and Reddening · Interstellar Polarization and Magnetic Fields.

Ask the tutor: Want to see how a factor-of-a-hundred gap between gravity's free-fall clock and the observed Galactic star formation rate pins down $\varepsilon_{\mathrm{ff}}$ — ask me to walk through it step by step.

Why is star formation so inefficient in the first place?

Because gravity is opposed on every scale. Supersonic turbulence stirs the gas so that most of it sits below the density needed for self-gravity to win, and magnetic fields add further support. Only the dense tail of the turbulent density distribution actually collapses, so at any instant just a percent or so of the gas is genuinely forming stars. Then the newborn massive stars fight back: their radiation, winds, and supernovae inject enough momentum to unbind the surrounding gas and disperse the cloud before it can finish collapsing. The net result is that gravity's natural clock, the free-fall time, is throttled by a factor of about a hundred — the efficiency per free-fall time $\varepsilon_{\mathrm{ff}}\sim 0.01$ .

Are the total-gas and molecular star formation laws contradictory?

No — they describe different reservoirs. The total-gas law $\Sigma_{\mathrm{SFR}}\propto\Sigma_{\mathrm{gas}}^{1.4}$ counts atomic plus molecular gas, while the molecular law $\Sigma_{\mathrm{SFR}}\propto\Sigma_{\mathrm{H_2}}^{1.0}$ counts only the molecular phase. Stars form out of molecular gas essentially one-to-one, which is why the molecular law is linear. The total-gas law is steeper because, as you move to denser regions, a larger fraction of the gas becomes molecular; folding that rising molecular fraction into a linear molecular law produces the super-linear total-gas slope near $1.4$ . Both are correct simultaneously, and understanding the difference is the key to reading the literature without confusion.

What exactly is the depletion time, and why is it roughly constant?

The depletion time $\tau_{\mathrm{dep}}=M_{\mathrm{gas}}/\dot M_\star$ is how long a galaxy would take to consume its current gas at its current star formation rate. Equation (4) shows it equals $t_{\mathrm{ff}}/\varepsilon_{\mathrm{ff}}$ . In normal spiral galaxies both ingredients are nearly universal: molecular clouds share a characteristic density (hence free-fall time), and the efficiency per free-fall time is a robust percent-level number. Their ratio therefore settles near $\sim 2$ Gyr across a wide range of spirals — far longer than any orbital or free-fall time. In starbursts the gas is far denser and $\tau_{\mathrm{dep}}$ drops to $10^{7}$ – $10^{8}$ yr, which is why those systems burn out fast.

Why do different SFR tracers give different answers?

Because each tracer samples a different slice of the recent past and a different part of the stellar population. Hα recombination lines respond only to the ionizing photons of the most massive, shortest-lived stars, so they probe the last few Myr; far-ultraviolet continuum comes from somewhat less massive stars and averages over $\sim 100$ Myr; infrared emission traces whatever fraction of starlight was absorbed by dust. If the star formation rate has changed recently, or if dust obscuration is patchy, the tracers legitimately disagree. All of them also depend on the assumed initial mass function, since you are always extrapolating from the luminous massive stars you see to the total mass in stars you cannot. Best practice combines an unobscured and an obscured tracer.

Does the Kennicutt–Schmidt law hold everywhere in a galaxy?

No — it breaks down at low gas surface density. Below roughly a few $M_\odot\,\mathrm{pc}^{-2}$ , typical of outer galactic disks, the star formation rate plummets far faster than the power law predicts. Two effects conspire: the disk becomes gravitationally stable (Toomre $Q>1$ ), so gas cannot collapse, and the gas stays atomic rather than converting to the molecular phase that actually forms stars. The result is a fairly sharp star formation threshold. Conversely, at the extreme high-density end in circumnuclear starbursts, the relation can steepen or shift, especially when the rate is expressed per orbital or free-fall time rather than per unit gas mass.

How is the efficiency per free-fall time measured in real clouds?

By combining three observables for a large sample of clouds: the mass of dense, star-forming gas (from dust continuum or a molecular tracer), the free-fall time (from the measured density), and the current star formation rate (from young stellar objects or infrared luminosity). Then $\varepsilon_{\mathrm{ff}}=\dot M_\star\,t_{\mathrm{ff}}/M_{\mathrm{gas}}$ follows directly from equation (3). Surveys of Milky Way clouds and, with ALMA, of clouds in nearby galaxies consistently return values in the $0.3$ – $3\%$ range with substantial scatter but no strong trend with environment. That near-universality — low value, weak dependence on density or galaxy type — is one of the central empirical constraints on star formation theory.