Kennicutt–Schmidt Law

The Kennicutt–Schmidt law (also referred to as the Schmidt–Kennicutt relation) is a foundational empirical scaling relation in extragalactic astrophysics that links the rate of star formation per unit area (Σ_SFR) in galaxies to the local or disk-averaged surface density of cold interstellar gas (Σ_gas). First proposed in three-dimensional volume density form by Dutch astronomer Maarten Schmidt in 1959 and calibrated across five orders of magnitude in sky-projected surface density by American astrophysicist Robert C. Kennicutt Jr. in 1989 and 1998, the relation takes the power-law form Σ_SFR = A Σ_gas^N, with a disk-averaged power-law index N ≈ 1.4 ± 0.15. Spanning quiescent low-surface-brightness spiral disks, normal star-forming galaxies like the Milky Way, and merger-driven ultraluminous infrared starbursts (ULIRGs), the Kennicutt–Schmidt law provides the primary sub-grid star-formation prescription in hydrodynamic cosmological simulations and explains how cold gas reservoirs build up the exponential stellar disks parameterized by the n = 1 Sérsic profile.

In his 1959 investigation of the solar neighborhood within the Milky Way, Maarten Schmidt hypothesized that the three-dimensional volumetric star-formation rate ρ_SFR (measured in M_☉ yr⁻¹ pc⁻³) scales as a power law of the local interstellar gas volume density ρ_gas:

ρ_SFR ∝ ρ_gasⁿ

Comparing the vertical scale height of young Population I stars with that of neutral atomic hydrogen (H I) in the Galactic plane, Schmidt estimated n ≈ 2, which matches the scaling expected if star formation is triggered by binary two-body collisions of interstellar gas clouds. Because measuring the three-dimensional gas scale height h across external galaxies is observationally challenging, Robert C. Kennicutt Jr. (1989, 1998) reformulated the relation in terms of directly observable sky-projected surface densities integrated perpendicular to the galactic disk. Using Hα recombination emission and far-infrared (FIR) dust continuum fluxes to trace Σ_SFR, alongside 21-cm H I hyperfine emission and millimeter rotational transitions of carbon monoxide (CO J = 1 → 0) to trace atomic and molecular hydrogen (H₂), Kennicutt established the canonical global Kennicutt–Schmidt law across 61 normal spiral disks and 36 infrared-selected circumnuclear starburst galaxies:

Σ_SFR = (2.5 ± 0.7) × 10⁻⁴ [Σ_gas / (1 M_☉ pc⁻²)]^{1.4 ± 0.15} M_☉ yr⁻¹ kpc⁻²

where Σ_gas = Σ_{H I} + Σ_H2 is the total cold hydrogen gas surface density (assuming a Salpeter initial mass function). When recalibrated to a modern Chabrier or Kroupa initial mass function (which reduces derived stellar masses and star-formation rates by a factor of ~1.59), the normalization constant shifts to A ≈ 1.5 × 10⁻⁴ M_☉ yr⁻¹ kpc⁻² while the slope N ≈ 1.4 remains invariant.

The non-linear exponent N ≈ 1.4 arises naturally from the physics of gravitational collapse in self-gravitating interstellar gas. Consider a gas cloud or galactic disk patch of volume density ρ_gas collapsing on its gravitational free-fall timescale:

τ_ff = √[3π / (32 G ρ_gas)] ∝ ρ_gas^−1/2

If a fixed dimensionless efficiency ε_ff ≈ 0.01 (roughly 1% of the cold gas mass) is converted into stars per free-fall time, the volumetric star-formation rate obeys the Schmidt relation with n = 1.5:

ρ_SFR = ε_ff (ρ_gas / τ_ff) = ε_ff √(32 G / 3π) ρ_gas^1.5

For a galactic disk of roughly constant vertical scale height h (or a disk in vertical hydrostatic equilibrium where scale height varies slowly with surface density), projecting ρ_SFR ∝ ρ_gas^1.5 along the line of sight directly yields the surface density scaling Σ_SFR ∝ Σ_gas^1.5, matching Kennicutt's empirical exponent N = 1.4 ± 0.15 within observational uncertainties.

An alternative dynamical formulation incorporates the differential rotation of the galactic disk (whose outer circular velocity is set by its Navarro–Frenk–White profile halo and obeys the Tully–Fisher relation). In this dynamical orbital model (Silk, 1997; Elmegreen, 1997), large-scale gravitational instabilities grow on a timescale proportional to the local galactic orbital period τ_orb = 2πR / v_c(R) = 2π / Ω(R):

Σ_SFR ≈ ε_orb Σ_gas Ω(R), with ε_orb ≈ 0.017

where roughly 1.7% of the available cold gas is converted into stars every galactic rotation (~10% per orbital period when normalized to the local disk radius).

In the late 2000s, sub-kiloparsec spatially resolved surveys of nearby spiral galaxies—most notably The H I Nearby Galaxy Survey (THINGS) coupled with HERACLES CO mapping and Spitzer/GALEX star-formation tracers (Bigiel et al., 2008; Leroy et al., 2008)—dissected the Kennicutt–Schmidt relation on 750-parsec scales corresponding to individual giant molecular cloud complexes. These resolved observations revealed that the relation splits into three distinct physical regimes governed by the phase transition between atomic hydrogen (H I) and molecular hydrogen (H₂):

1. The H I Saturation Ceiling (~10 M_☉ pc⁻²): At low surface densities (Σ_gas < 10 M_☉ pc⁻² at solar metallicity), interstellar hydrogen exists predominantly as diffuse atomic H I, and Σ_SFR drops precipitously below the N = 1.4 extrapolation where the Toomre stability parameter Q = κ c_s / (π G Σ_gas) exceeds unity. Above Σ_gas ≈ 10 M_☉ pc⁻², H I surface density saturates because dust grains and H₂ self-shielding block dissociating Lyman–Werner ultraviolet photons, converting all additional hydrogen into molecular H₂.
2. The Linear Molecular Relation (N_mol ≈ 1.0): In the range 10 ≤ Σ_H2 ≤ 100 M_☉ pc⁻² typical of normal spiral disks, Σ_SFR scales linearly with the molecular gas surface density Σ_H2 (N_mol = 1.0 ± 0.15), corresponding to a nearly constant molecular gas depletion timescale τ_dep,H2 = Σ_H2 / Σ_SFR ≈ 2.0 Gyr. Because giant molecular clouds in normal disks maintain a uniform internal surface density (~100 M_☉ pc⁻²) and internal free-fall time, increasing Σ_H2 on kiloparsec scales simply increases the beam-filling number of similar clouds rather than altering their internal density.
3. The High-Density Starburst Sequence: Above Σ_gas ≈ 100–200 M_☉ pc⁻² in galaxy centers and merging ULIRGs, the interstellar medium becomes a continuous, high-pressure molecular disk whose mean density exceeds that of individual giant molecular clouds, causing the free-fall time τ_ff to shorten and the scaling slope to steepen back toward N ≈ 1.4–1.5.

What is the gas depletion timescale in the Kennicutt–Schmidt law?

The gas depletion timescale τ_dep ≡ Σ_gas / Σ_SFR represents the time required for a galaxy to consume its current cold gas reservoir at its present star-formation rate without cosmological gas infall or stellar mass recycling. For normal spiral galaxies at z = 0, the molecular depletion time is approximately 2.0 billion years (2.0 Gyr), whereas in ultraluminous infrared starbursts (ULIRGs) τ_dep drops to 50–200 million years.

Why does atomic hydrogen (H I) stop increasing above 10 solar masses per square parsec?

Molecular hydrogen (H₂) is destroyed by interstellar ultraviolet photons in the Lyman–Werner bands (912–1108 Å). At solar metallicity, a gas column density of roughly 10 M_☉ pc⁻² (corresponding to visual dust extinction A_V ≈ 0.5 mag) provides sufficient dust shielding and H₂ line self-shielding to block UV dissociation, so any gas accumulated above this column threshold transitions into cold molecular H₂.

How does the CO-to-H2 conversion factor (alpha_CO) affect the Kennicutt–Schmidt law?

Because symmetric H₂ molecules lack a permanent electric dipole moment and do not emit at cold temperatures (10–20 K), molecular gas mass is inferred from rotational CO emission via the conversion factor α_CO = M_H2 / L_CO. In warm, turbulent starburst nuclei, α_CO is roughly 4 to 5 times lower than in Milky Way disk clouds (~0.8 vs. 4.3 M_☉ (K km s⁻¹ pc²)⁻¹), giving rise to a bimodal star-formation sequence between disk galaxies and mergers (Daddi et al., 2010).

1. Schmidt, M. (1959). "The rate of star formation." The Astrophysical Journal, 129, 243–258. https://doi.org/10.1086/146614
2. Kennicutt, R. C., Jr. (1998). "The global Schmidt law in star-forming galaxies." The Astrophysical Journal, 498(2), 541–552. https://doi.org/10.1086/305588
3. Bigiel, F., Leroy, A., Walter, F., Brinks, E., de Blok, W. J. G., Madore, B., & Thornley, M. D. (2008). "The star formation law in nearby galaxies on sub-kpc scales." The Astronomical Journal, 136(6), 2846–2871. https://doi.org/10.1088/0004-6256/136/6/2846
4. Krumholz, M. R., McKee, C. F., & Tumlinson, J. (2009). "The star formation law in atomic and molecular gas." The Astrophysical Journal, 699(1), 850–856. https://doi.org/10.1088/0004-637X/699/1/850
5. Kennicutt, R. C., & Evans, N. J. (2012). "Star formation in the Milky Way and nearby galaxies." Annual Review of Astronomy and Astrophysics, 50, 531–608. https://doi.org/10.1146/annurev-astro-081811-125610