How do synthetic far-infrared observations explain star formation efficiency?

What the efficiency per free-fall time measures

The star formation efficiency per free-fall time, written ε_ff, asks how much of a cloud's mass turns into stars during one gravitational free-fall time. Formally it is the star formation rate divided by the cloud mass over that free-fall time. A value near 100% would mean a cloud converts its gas into stars about as fast as gravity alone could collapse it; observed clouds instead sit far lower, historically at ε_ff ≤ 1% in early measurements.

The difficulty is that none of the quantities in that ratio — the stellar accretion rate, the total gas mass, the free-fall time, or the gas density — is directly measurable in a real cloud with substructure and blended lines of sight. Observers therefore build ε_ff from proxies: far-infrared dust emission converted into a gas surface density, plus a count of embedded protostars converted into a star formation rate using assumed masses and lifetimes. The paper's question is whether that proxy chain, applied to a simulation that knows the true answer, recovers the same low efficiencies.

Mock observations of one simulated cloud

The study uses the fiducial STARFORGE radiation-magnetohydrodynamics simulation of a giant molecular cloud with mass 2×10⁴ M☉ and radius about 10 pc, giving a mean surface density of 64 M☉ pc⁻². The cloud starts with supersonic turbulence at virial parameter ≈2 and a magnetic field making it magnetically supercritical, and the simulation follows star cluster formation with stellar feedback — irradiation, dust thermal re-emission, protostellar outflows, winds, and supernovae — until feedback disrupts the cloud at about 10 Myr, roughly 2.5 free-fall times.

To make the comparison, the authors post-process the simulation into synthetic dust emission maps at 150, 250, 350, and 500 µm, matching Herschel/SPIRE survey wavelengths, and fit them with a modified blackbody assuming a 1% dust-to-gas ratio. The fitted 500 µm optical depth becomes a mock observed gas surface density. Embedded protostars are represented by sink particles younger than 0.5 Myr, the paper's stand-in for Class 0 young stellar objects. The synthetic maps reach 0.01 pc resolution and reproduce the cloud's morphology and reasonable surface densities, which the authors treat as validation of the pipeline.

Efficiency varies across the cloud and over time

Across surface density thresholds from about 100 to 2500 M☉ pc⁻², the inferred ε_ff ranges from 0.05% to 10%, with most estimates between 1% and 3% — the same range surveys report. The relation is not fixed. In the first 3–5 Myr, ε_ff is very low (about 0.3–0.5%) at low surface density and rises sharply toward higher densities, forming a brief plateau near 5 Myr before declining at the highest densities. Between 6 and 7 Myr the relation flattens; by 8–10 Myr it becomes more stochastic, with local spikes as feedback disperses the cloud.

Tracking three fixed thresholds of 100, 500, and 1000 M☉ pc⁻² shows the same story in time. The low-density threshold rises later, by nearly an order of magnitude over the first 2–4 Myr, while higher thresholds start with higher efficiencies. By about 5 Myr the curves converge and evolve in parallel, with episodic peaks reaching roughly 5%. The authors read this as evidence that efficiency is set by the cloud's evolutionary stage rather than simply by how much gas is enclosed — and therefore that much of the scatter among observed clouds may reflect where each cloud sits in its lifetime, not intrinsic cloud-to-cloud differences.

Coarser resolution lowers the inferred efficiency

The authors repeat the measurement at 0.01 pc, 0.04 pc, and 0.08 pc, the last matching the effective resolution of Herschel/SPIRE observations of nearby clouds. All three agree at very low surface density, near log ε_ff ≈ −1.6, but diverge as density rises. Lower resolution consistently reduces the inferred ε_ff, because beam dilution smooths the compact gas around protostars, lowers peak surface densities, and shrinks dense contours, which redistributes young stellar objects and lengthens the inferred free-fall time.

This is the paper's dominant observational bias. The authors note that future far-infrared instruments with sub-0.1 pc resolution could reduce it, but current instruments cannot, so surveys must account for resolution smoothing or risk attributing an instrumental limit to a physical mechanism.

Where method assumptions flip the bias

The paper separates the observational and simulation definitions of both the star formation rate and the free-fall collapse rate, then mixes them to isolate which assumption drives which discrepancy. The observational star formation rate assumes a fixed mean protostellar mass of about 0.5 M☉ and a Class 0 lifetime of about 0.5 Myr; the simulation rate uses the actual stellar accretion rate averaged over a time window. The observational free-fall rate assumes a uniform sphere, while the simulation sums each gas particle's mass divided by its own free-fall time.

The two approaches cross near log Σ_gas ≈ 2.3 M☉ pc⁻², a transition the authors say depends on resolution. Above that threshold, smoothing dense structure increases both the inferred free-fall time and the enclosed gas, and the gas term dominates, suppressing ε_ff. Below it, the discrepancy is driven mainly by the star formation rate assumptions: a fixed 0.5 M☉ average underestimates the rate in dense regions where massive protostars accrete fast and live short, and becomes statistically unreliable in low-density regions where too few stars exist to populate the mass distribution. The hybrid tests show that the geometrical treatment of the cloud and free-fall collapse, not the star formation rate alone, is the primary driver of the offset.

AGN contamination matters most for nearby clouds

Injecting synthetic active galactic nuclei into the young stellar object catalogs at a surface density of 9 deg⁻² tests a second observational bias: background sources misidentified as protostars. For a snapshot at about 6 Myr, the effect is largest at distances of 0.1–0.4 kpc, where ε_ff is offset by up to about 0.1 dex at log Σ_gas ≲ 2.5 M☉ pc⁻². The spread shrinks with distance and is negligible by 1 kpc.

Compared with resolution smoothing, AGN contamination is a secondary effect confined to nearby, lower-density regimes. The paper's ordering is explicit: resolution is the dominant bias in far-infrared studies of ε_ff, and contamination is smaller but still worth accounting for when connecting measurements to theory.

What the simulated cloud still idealizes

The cloud begins as a uniform sphere with fixed mass, radius, metallicity, and uniform magnetic field, seeded with a turbulent velocity field; it does not capture the poorly constrained assembly history of a real molecular cloud. Turbulence decays freely rather than being continuously driven, unlike the Milky Way where large-scale flows, feedback, and shear replenish it. Cosmic ray heating, grain-surface chemistry, and dust physics are treated simply, and the radiative transfer uses an M1 solver with a reduced speed of light, which can underestimate radiation pressure in complex geometries.

These choices can shift gas temperatures, abundances, and feedback strength, all of which feed into star formation rates and ε_ff. They bound how far the result transfers: the paper's claim is that a feedback-regulated cloud with these standard initial conditions naturally produces observationally inferred efficiencies of 1–3%, not that every real cloud's history is reproduced.

The threshold that should guide comparisons

The most useful takeaway for anyone comparing theory with survey data is the split at log Σ_gas ≈ 2.3 M☉ pc⁻². Below it, mismatches between simulated and observationally inferred ε_ff trace the assumed star formation rate — a fixed 0.5 M☉ protostellar mass and 0.5 Myr Class 0 lifetime — so improving those assumptions is the priority. Above it, the dominant error comes from how the gas geometry and free-fall time are treated, and from resolution smoothing dense structure, so the priority shifts to resolution and to how contours and free-fall times are defined.

That ordering also reframes the scatter seen in surveys: because ε_ff moves strongly over a cloud's roughly 10 Myr life, a sample of clouds at different evolutionary stages will show spread even if the underlying physics is identical. The paper's contribution is not a new efficiency value but a map of which assumption to fix at which surface density — and the demonstration that feedback-regulated star formation reaches observed efficiencies without extreme initial conditions.

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