Kennicutt-Schmidt Law

The recipe in one line

Galaxies are machines that turn gas into stars, and the Kennicutt-Schmidt law is the recipe written on the machine. It says something almost embarrassingly simple: a patch of galaxy that holds more gas makes more stars. But the precise shape of that statement is what makes it powerful. The star-formation rate does not just track the gas linearly — it climbs faster than the gas, as the gas surface density raised to a power of about 1.4. Double the gas in a region and you get roughly two-and-a-half times as many new stars per year.

That superlinear kink is the whole story in a number. It tells you that dense gas is not just more fuel but better fuel: it ignites star formation more efficiently per unit mass. The reason, as we will see, is gravity. Denser gas collapses on a shorter timescale, so it converts itself into stars at a higher rate. The Kennicutt-Schmidt law packages all of that messy small-scale physics — molecular clouds, turbulence, feedback — into a single, robust, galaxy-averaged scaling that holds from quiet spirals like the Milky Way all the way up to the most violent starburst nuclei in the universe.

The law and its numbers

The disk-averaged form, as calibrated by Robert Kennicutt in his landmark 1998 paper, is a power law in surface densities:

Σ_SFR = A · Σ_gas^N

N    = 1.4 ± 0.15
A    ≈ (2.5 ± 0.7) × 10^-4    (for Σ in M☉ pc^-2 and Σ_SFR in M☉ yr^-1 kpc^-2)

Here Σ_SFR is the star-formation rate per unit area (solar masses of new stars per year per square kiloparsec) and Σ_gas is the total — atomic plus molecular — gas surface density (solar masses per square parsec). Kennicutt built the relation from 61 normal spiral galaxies and 36 infrared-selected starburst galaxies. The remarkable thing is the dynamic range: the data run from a few solar masses per square parsec in the outer parts of normal disks to more than 10⁵ M☉ pc⁻² in starburst cores, and a single straight line of slope 1.4 threads through all of it on a log-log plot.

An equivalent and very intuitive way to write the same law uses the orbital time of the disk instead of an explicit power:

Σ_SFR ≈ 0.017 · Σ_gas · Ω     (Ω = v_rot / R, the angular rotation rate)

In this version, about 10% of the available gas is turned into stars every orbital period (the factor 0.017 multiplied by the 2π in one full orbit). Both forms fit the global data about equally well; they encode the same idea that the rate is set by a dynamical timescale.

Why the index is 1.4: the free-fall argument

The 1.4 power is not arbitrary — it falls out of a simple gravitational argument. Suppose stars form from gas at a rate governed by gravitational collapse. The natural clock for collapse is the free-fall time, the time for a self-gravitating blob of density ρ to collapse under its own weight:

t_ff = √( 3π / (32 G ρ) )   ∝ ρ^(-1/2)

If a fixed small fraction ε of the gas turns into stars in each free-fall time, the volumetric star-formation rate is

ρ_SFR = ε · ρ_gas / t_ff  ∝ ρ_gas · ρ_gas^(1/2)  =  ρ_gas^(3/2)

Now project this onto the sky. In a disk whose vertical thickness does not change much with radius, the volume density is proportional to the surface density, ρ_gas ∝ Σ_gas, and likewise ρ_SFR ∝ Σ_SFR. So

Σ_SFR ∝ Σ_gas^(3/2)  =  Σ_gas^1.5

which is satisfyingly close to the observed 1.4. The small difference (1.4 versus 1.5) is exactly what you would expect once the disk scale height varies with radius and the conversion efficiency ε is not perfectly constant. The headline lesson stands: the superlinear index is a fingerprint of gravity setting the pace. Denser gas falls in faster, so it makes stars faster per unit mass — and the surplus over linear is the square-root-of-density factor from the free-fall time.

The cleaner law: it is really about molecular gas

Stars do not form out of just any gas — they form in cold, dense, molecular clouds. So when astronomers split the gas into its atomic (HI, traced by the 21-cm line) and molecular (H₂, traced by CO) components, the picture sharpens dramatically. Spatially resolved surveys of nearby galaxies — THINGS for HI, HERACLES for CO, at sub-kiloparsec resolution — found that:

• The molecular law is nearly linear: Σ_SFR ∝ Σ_H₂^N with N ≈ 1.0 (Bigiel et al. 2008 found N = 1.0 ± 0.2).
• The atomic gas shows almost no correlation with star formation — HI saturates near ~9 M☉ pc⁻² and the extra gas piles up as molecular.
• The molecular gas is consumed at a remarkably constant depletion time of roughly 2 billion years across normal spiral disks.

The resolution of the apparent contradiction (global 1.4 versus molecular 1.0) is the two-step nature of star formation. The 1.4 you see in the disk-averaged total-gas law is partly a statement about where gas is dense enough to go molecular, while the linear molecular law is a statement about how fast molecular gas, once formed, makes stars. Put differently: the hard step is turning diffuse atomic gas into molecular clouds; once you have molecular gas, it forms stars at a near-universal rate of about one part in two billion per year.

By the numbers: normal disks vs. starbursts

The single most instructive way to read the law is to compare regimes. Here is what the relation looks like across the cosmic zoo of star-forming systems:

System	Σ_gas (M☉ pc⁻²)	Σ_SFR (M☉ yr⁻¹ kpc⁻²)	Depletion time	Example
Outer disk (atomic)	~1–5	~10⁻⁴–10⁻³	> 10 Gyr	Outer HI of M33
Normal spiral disk	~10	~3 × 10⁻³	~2 Gyr (molecular)	Milky Way, M51
Galactic centre / bar	~100	~0.1	~1 Gyr	Central Molecular Zone
Nuclear ring	~10³	~1–10	~0.1–0.5 Gyr	NGC 1097 ring
Starburst nucleus	~10⁴	~10²	~50–100 Myr	M82, NGC 253
Ultraluminous IR galaxy	~10⁵	~10³–10⁴	~10–50 Myr	Arp 220

Five orders of magnitude in gas density map onto roughly seven orders of magnitude in star-formation-rate density — which is exactly what a slope of 1.4 (or 1.5) on a log-log plot predicts. Arp 220, a merger remnant ~250 million light-years away, packs the gas of a whole spiral galaxy into its two compact nuclei and is forming stars hundreds of times faster than the entire Milky Way, even though the Milky Way is the larger galaxy. That is the Kennicutt-Schmidt law in action: density, not total mass, sets the rate.

Star-formation efficiency and depletion time

Two derived quantities turn the law into something you can reason about physically. The first is the gas depletion time — how long current star formation would take to exhaust the gas:

t_dep = Σ_gas / Σ_SFR     (the inverse of the star-formation efficiency per unit time)

Because Σ_SFR ∝ Σ_gas^1.4, the depletion time itself depends on density: t_dep ∝ Σ_gas^(-0.4). Denser regions burn their fuel faster. For the molecular phase in normal disks, t_dep,mol ≈ 2 Gyr — long compared with a galaxy's orbital period (~250 Myr for the Sun) but short compared with the age of the universe, which is why galaxies need a continual supply of fresh gas to keep forming stars for a Hubble time.

The second is the per-free-fall-time efficiency, ε_ff, the fraction of a cloud's mass converted into stars in a single free-fall time. Observations across an enormous range of densities — from whole galaxies down to individual Milky Way clouds — converge on a strikingly low and roughly constant value:

ε_ff ≈ 0.01    (about 1% of gas turns into stars per free-fall time)

This is one of the deep puzzles of star formation: gravity wants to collapse molecular gas in a single free-fall time of a few million years, yet the gas is converted at only ~1% of that rate. The brake is feedback — stellar winds, radiation pressure, supernovae, and turbulence — which keeps clouds from running away into stars. The Kennicutt-Schmidt law is, in this light, the macroscopic average of that slow, feedback-regulated burn.

Where the law shows up

• The Milky Way and the solar neighbourhood. At the Sun's galactocentric radius the gas surface density is ~10 M☉ pc⁻² and the star-formation rate density is a few × 10⁻³ M☉ yr⁻¹ kpc⁻² — squarely on the Kennicutt line. The whole Galaxy makes roughly 1–2 M☉ of new stars per year.
• M51, the Whirlpool. The textbook resolved case: CO and Hα maps show the star formation tracking the molecular spiral arms, with the gas piling up at the density-wave-driven shocks where the arms compress the interstellar medium.
• M82, the prototype starburst. A nearby (~12 million light-years) edge-on galaxy whose central kiloparsec is forming stars ~10× faster per unit area than the Milky Way disk, driving a galactic superwind of hot gas out of the plane.
• Arp 220 and ULIRGs. Gas-rich major mergers funnel ~10⁹–10¹⁰ M☉ of molecular gas into the central few hundred parsecs, reaching Σ_gas ~ 10⁵ M☉ pc⁻² and total star-formation rates of hundreds of M☉ yr⁻¹ — the most extreme end of the relation.
• High-redshift disks. Surveys like PHIBSS show that gas-rich galaxies at redshift z ~ 1–2 lie on roughly the same molecular law, but with shorter depletion times (~0.7 Gyr), explaining the peak of cosmic star formation ("cosmic noon") as simply the era when galaxies held the most molecular gas.

Thresholds, breaks, and the outer disk

The law is not a single unbroken line all the way down. Below a critical gas surface density of roughly 5–10 M☉ pc⁻², the disk-averaged relation steepens sharply or breaks: the outer, atomic-dominated parts of galaxies form stars far less efficiently than the slope-1.4 extrapolation would predict. Two physical ingredients are usually invoked:

• Gravitational (Toomre) stability. A gas disk is unstable to collapse only when the Toomre parameter Q = κ c_s / (π G Σ_gas) drops below ~1, where κ is the epicyclic frequency and c_s the gas sound speed. Where the gas is too diffuse — large Q — random motions and shear stabilise it against collapse, and star formation shuts off. This sets a star-formation threshold that roughly coincides with the observed break radius.
• The atomic-to-molecular transition. Diffuse atomic gas must first cool and shield itself to become molecular before it can form stars. In low-density, low-metallicity outer disks, that conversion is inefficient, so even where gas exists it stays atomic and sterile. This is why the relation is so much tighter when you plot only Σ_H₂.

The takeaway is that the Kennicutt-Schmidt law is really two laws stitched together: a tight, nearly linear molecular law that governs star formation once cold clouds exist, and a steeper, threshold-controlled transition that governs whether the gas gets cold and dense enough to count in the first place.

Common misconceptions and edge cases

• "The 1.4 index is fundamental physics." It is an empirical, galaxy-averaged slope that emerges from the free-fall argument plus disk geometry. Resolved, molecular-only measurements give a nearly linear slope; the 1.4 reflects the total-gas, disk-averaged view, not a law of nature.
• "More gas always means more stars." Only above the threshold. In stable, diffuse outer disks the gas just sits there. Density and dynamical state — not raw gas mass — control the rate.
• "Atomic and molecular gas count the same." They do not. HI saturates near ~9 M☉ pc⁻² and barely correlates with star formation; the action is entirely in the molecular phase. Lumping them together is what produces the steeper total-gas slope.
• "Starbursts just have more gas." At a fixed molecular surface density, starbursts and merger nuclei still form stars several times faster than quiet disks — a separate, higher-efficiency sequence with ~10–100 Myr depletion times. The split partly depends on the assumed CO-to-H₂ conversion factor, which is lower in dense starburst environments.
• "It predicts when an individual cloud will form stars." No. The law is a statistical average over many clouds and many free-fall times. A single molecular cloud forms stars erratically; the smooth power law only emerges after averaging over a kiloparsec-sized patch.