Observational determination of the collapsing mass and its systematics: testing the collapse-dominated interpretation of the Kennicutt–Schmidt star formation relations

Ballesteros-Paredes et al. (2024) showed that the family of Kennicutt–Schmidt star formation relations — from the intra-cloud correlation between young stellar objects and dense gas mass to the extragalactic relations between star formation rate and CO, HCN, or CO+HI tracers — can be derived from a single 'fundamental law', SFR = $\epsilon_{\rm ff}\,M_{\rm collapse}/\tau_{\rm ff}$, provided the gas under consideration is in gravitational collapse rather than turbulence-supported. This collapse-dominated interpretation stands on quantities that current observations cannot cleanly deliver: no tracer directly measures the actually-collapsing mass $M_{\rm collapse}$; star formation rates inferred from counts of young stellar objects assume stationary protostellar lifetimes; extragalactic free-fall times assume a fixed line-of-sight depth inside kiloparsec-scale beams; and simulations actively disagree on whether stellar feedback regulates the star formation efficiency per free-fall time itself. The open problem is to determine $M_{\rm collapse}$ and its density distribution in real molecular clouds, to quantify the biases that currently obscure it, and thereby to decide whether the observed constancy of $\epsilon_{\rm ff}$ — interpreted as the ratio of the star formation rate to the gas infall rate — is a genuine signature of collapse-dominated star formation or a methodological artifact. The question remains open and is actively contested on both sides.

Background

The Kennicutt–Schmidt (KS) relation is the empirical correlation between the surface density of the star formation rate (SFR, the rate at which gas is converted into stars) and the surface density of gas, measured either within individual molecular clouds or averaged over whole galaxies (Schmidt 1959; Kennicutt 1998). Its normalization is conventionally expressed through the 'efficiency per free-fall time' $\epsilon_{\rm ff}$: writing $\tau_{\rm ff}=(3\pi/32G\rho)^{1/2}$ for the free-fall time (the time for a gas cloud of density $\rho$ to collapse under its own gravity, with $G$ the gravitational constant), the relation SFR = $\epsilon_{\rm ff}\,M_{\rm gas}/\tau_{\rm ff}$ holds by definition with $\epsilon_{\rm ff} = \tau_{\rm ff}/\tau_{\rm dep}$, where $\tau_{\rm dep} = M_{\rm gas}/{\rm SFR}$ is the depletion time. Observed values cluster near $\epsilon_{\rm ff}$ ~ 0.01–0.03 with roughly one order of magnitude of scatter. Two competing paradigms interpret this. In the turbulence-support picture, molecular clouds last many free-fall times, with supersonic turbulence and stellar feedback keeping the SFR low; in the global hierarchical collapse (GHC) picture, molecular clouds and their substructures are part of a continuous, gravitationally driven flow in which each structure accretes from its parent, and $\epsilon_{\rm ff}$ is better read as the ratio of the SFR to the gas infall rate (Vázquez-Semadeni et al. 2019). Ballesteros-Paredes et al. (2024, 'Gravity or turbulence? VI') sharpened the collapse-side argument: starting from the pure definition of the SFR and basic algebra they derived what they call the fundamental law of star formation, $\langle{\rm SFR}\rangle_{\tau_{\rm YSO}} = \epsilon_{\rm ff}\,M_{\rm collapse}/\tau_{\rm ff}$, where $\tau_{\rm YSO}$ is the lifetime of the counted young stellar objects (YSOs, i.e. protostars and pre-main-sequence stars, observationally classified into evolutionary Classes 0, I, and II) and $M_{\rm collapse}$ is the mass that is actually collapsing. They argued that the constancy of $\epsilon_{\rm ff}$ follows if — and only if — the gas is undergoing Jeans fragmentation (collapse, i.e. self-gravity overwhelming pressure support so the gas breaks into successively smaller fragments), and that the observed variety of KS relations then follows: the intra-cloud $\Sigma_{\rm SFR}$–$\Sigma_{\rm gas}/\tau_{\rm ff}$ correlation (Pokhrel et al. 2021), the linear $N_{\rm YSO}$–$M_{\rm dense}$ correlation (Lada et al. 2010), the resolved and unresolved extragalactic relations traced by CO (carbon monoxide line emission, the standard tracer of molecular gas), HCN (hydrogen cyanide line emission, a tracer of denser gas), and CO+HI (molecular plus atomic gas), and the apparent pressure-regulated correlation with the midplane dynamical equilibrium pressure (the pressure required to support the weight of the galactic gas disk vertically; Sun et al. 2023). They further argued that the sublinear extragalactic slope in $\Sigma_{\rm SFR}$–$\Sigma_{\rm mol}/\tau_{\rm ff}$ found by Sun et al. (2023, who assumed a common line-of-sight depth of 150 pc when converting 1.5 kpc-resolution column densities into volume densities) is a beam-filling artifact: molecular clouds are fractal objects with a small filling factor (the fraction of the beam volume actually occupied by dense gas) inside kiloparsec-scale beams, so the assumed depth biases $\tau_{\rm ff}$ and stretches the correlation. However, the authors explicitly acknowledge that the interpretation rests on quantities that observations do not currently deliver: (i) no tracer measures $M_{\rm collapse}$ directly — tracers such as CO, HCN, or dust extinction select characteristic density regimes, not collapse state; (ii) $\tau_{\rm YSO}$ values used to infer SFRs are estimated under a stationarity assumption (the ratio of protostars in two classes equals the ratio of their lifetimes), while real star formation may be non-stationary and environment-dependent; (iii) the beam-filling and line-of-sight-depth bias in extragalactic $\tau_{\rm ff}$ estimates has not been modelled and calibrated across galaxy types and resolutions; (iv) the relation between measured virial parameters ($\alpha_{\rm vir}$, the ratio of twice the kinetic energy to the gravitational binding energy; overvirial means $\alpha_{\rm vir} > 2$) and the fraction of mass actually participating in collapse is unsettled, since converging motions in a globally collapsing cloud can coexist with overvirial estimates; and (v) the time-dependent coupling between stellar feedback, the evolving density structure, and the instantaneous ratio SFR/infall-rate remains unquantified. The virial parameter, free-fall time, and YSO class system are standard tools of the field; the novel quantity the collapse interpretation needs — and the literature currently lacks — is a direct, calibrated measurement of $M_{\rm collapse}$.

Problem Statement

Determine observationally the actually-collapsing mass $M_{\rm collapse}$ — its magnitude and its density distribution — in real molecular clouds, together with the quantified systematics needed to interpret it, in order to test the collapse-dominated interpretation of the Kennicutt–Schmidt relations, i.e. to decide whether the quantity $\epsilon_{\rm ff}$ = SFR/($M_{\rm collapse}$/$\tau_{\rm ff}$) is genuinely constant across environments (as required if the star formation law is set by gravitational collapse and $\epsilon_{\rm ff}$ is the ratio of the star formation rate to the gas infall rate) or whether its apparent constancy is a methodological artifact. Here $\epsilon_{\rm ff}$ denotes the collapse-normalized efficiency ${\rm SFR}/(M_{\rm collapse}/\tau_{\rm ff})$, which equals the $M_{\rm gas}$-normalized efficiency $\tau_{\rm ff}/\tau_{\rm dep}$ of the Background only if the gas under consideration is entirely collapsing ($M_{\rm collapse} = M_{\rm gas}$); keeping the two normalizations distinct is part of the task. The linked measurement-systematics components of this single objective are: (1) the mapping from observational tracers (CO, HCN, dust extinction/emission column densities) to the collapsing mass and its density distribution, which no current tracer provides directly — a determination of $M_{\rm collapse}$ must therefore state and justify the operational criterion by which collapsing gas is identified (at minimum a mass or density selection together with a collapse or kinematic test) and quantify how the inferred mass depends on that choice; (2) the impact of non-stationary, environment-dependent protostellar lifetimes $\tau_{\rm YSO}$ on star formation rates and $\epsilon_{\rm ff}$ inferred from young-stellar-object counts, beyond the Monte-Carlo bias quantification already available (Dib et al. 2025); (3) the calibration of the beam-filling and fixed-line-of-sight-depth bias in kiloparsec-scale extragalactic estimates of $\tau_{\rm ff}$, across galaxy types and resolutions; (4) the relation between measured virial parameters (including overvirial estimates) and the fraction of cloud mass actually participating in collapse, including the statistical distribution of converging versus diverging motions; and (5) the time-dependent coupling between stellar feedback, the evolving density structure, and the instantaneous ratio of star formation rate to infall rate. The task is resolved when a determination of $M_{\rm collapse}$ and its density distribution yields an explicit verdict on whether $\epsilon_{\rm ff}$ is environment-independent; the five components are the evidence that verdict must rest on, each required at the depth needed to make the verdict adjudicable rather than as a complete solution of the corresponding research programme.

References

1. Ballesteros-Paredes, J., Zamora-Avilés, M., Román-Zúñiga, C., Palau, A., Cervantes-Sodi, B., Gutiérrez-Dávila, K., Camacho, V., Jiménez-Andrade, E., Gazol, A. (2024). Gravity or turbulence? – VI. The physics behind the Kennicutt–Schmidt relation. Monthly Notices of the Royal Astronomical Society, 534, 1043–1059. DOI: 10.1093/mnras/stae2036 (arXiv:2408.13636). https://doi.org/10.1093/mnras/stae2036
2. Zamora-Avilés, M., Camacho, V., Ballesteros-Paredes, J., Vázquez-Semadeni, E., Palau, A., Román-Zúñiga, C., Hernández-Cruz, A., Gómez, G. C., Quesada-Zúñiga, F., Naranjo-Romero, R. (2025). Gravity or turbulence? – VII. The Schmidt–Kennicutt law, the star formation efficiency, and the mass density of clusters from gravitational collapse rather than turbulent support. Monthly Notices of the Royal Astronomical Society, 541, 3880–3893. DOI: 10.1093/mnras/staf1123 (arXiv:2409.11588). https://doi.org/10.1093/mnras/staf1123
3. Vázquez-Semadeni, E., Palau, A., Gómez, G. C., Arroyo-Chávez, G., Alig, C., Ballesteros-Paredes, J., Camacho, V., Traficante, A., González-Samaniego, A., Zamora-Avilés, M., Burkert, A. (2025). The turbulent support and global hierarchical collapse models for molecular clouds compared: differences, convergence, and myths. Monthly Notices of the Royal Astronomical Society, 547, article staf2059. DOI: 10.1093/mnras/staf2059 (arXiv:2408.10406). https://doi.org/10.1093/mnras/staf2059
4. Vázquez-Semadeni, E., Palau, A., Ballesteros-Paredes, J., Gómez, G. C., Zamora-Avilés, M. (2019). Global hierarchical collapse in molecular clouds. Towards a comprehensive scenario. Monthly Notices of the Royal Astronomical Society, 490, 3061–3097. DOI: 10.1093/mnras/stz2736 (arXiv:1903.11247). https://doi.org/10.1093/mnras/stz2736
5. Quesada-Zúñiga, F., Zamora-Avilés, M., Vázquez-Semadeni, E., Gómez, G. C., Palau, A., Ballesteros-Paredes, J. (2026). The efficiency per free-fall time as a ratio of the Star Formation Rate to the gas-infall rate in collapsing cores: dependence on the core definition, accretion, and radial structure. arXiv:2606.21878. https://arxiv.org/abs/2606.21878
6. Román-Zúñiga, C. G., Palau, A., Ballesteros-Paredes, J., Zamora-Avilés, M., Peltonen, J., Gutiérrez-Davila, K. (2026). Comparing the $M_{\rm gas}$–$N_{\rm yso}$ relation inside a Giant Molecular Cloud. arXiv:2603.12212. https://arxiv.org/abs/2603.12212
7. Whitworth, D., Vázquez-Semadeni, E., Ballesteros-Paredes, J., Gómez, G. O. (2026). Does supernova feedback regulate the star formation rate in dwarf galaxies? arXiv:2603.12821. https://arxiv.org/abs/2603.12821
8. Dib, S., Zhou, J.-W., Comerón, S., Garduño, L. E., Kravtsov, V. V., Clark, P. C., Li, G.-X., Lara-López, M. A., Liu, T., Shadmehri, M., Doughty, J. R. (2025). Assessing the accuracy of star formation rate measurements by direct star count in molecular clouds. Astronomy & Astrophysics, 693, A51. DOI: 10.1051/0004-6361/202450434 (arXiv:2405.00095). https://doi.org/10.1051/0004-6361/202450434
9. Leroy, A. K., Sun, J., Meidt, S., Agertz, O., Chiang, I.-D., Gensior, J., Glover, S. C. O., Gnedin, O. Y., Hughes, A., Schinnerer, E., Barnes, A. T., Bigiel, F., Bolatto, A. D., Colombo, D., den Brok, J., Chevance, M., Chown, R., Eibensteiner, C., Gleis, D. R., Grasha, K., Henshaw, J. D., Klessen, R. S., Koch, E. W., Oakes, E. K., Pan, H.-A., Querejeta, M., Rosolowsky, E., Saito, T., Sandstrom, K., Sarbadhicary, S. K., Teng, Y.-H., Usero, A., Utomo, D., Williams, T. G. (2025). Cloud-scale Gas Properties, Depletion Times, and Star Formation Efficiency per Freefall Time in PHANGS–ALMA. The Astrophysical Journal, 985, 14. DOI: 10.3847/1538-4357/adbcab (arXiv:2502.04481). https://doi.org/10.3847/1538-4357/adbcab
10. Khullar, S., Matzner, C. D., Murray, N., Grudić, M. Y., Guszejnov, D., Wetzel, A., Hopkins, P. F. (2024). Playing with FIRE: A Galactic Feedback-halting Experiment Challenges Star Formation Rate Theories. The Astrophysical Journal, 973, 40. DOI: 10.3847/1538-4357/ad6311 (arXiv:2406.18526). https://doi.org/10.3847/1538-4357/ad6311
11. Bertram, E. (2026). A Momentum-Regulated Model For Star Formation Efficiency in Giant Molecular Clouds. arXiv:2607.06727. https://arxiv.org/abs/2607.06727
12. Sun, J., Leroy, A. K., Ostriker, E. C., Meidt, S., Rosolowsky, E., Schinnerer, E., Wilson, C. D., Utomo, D., Belfiore, F., Blanc, G. A., Emsellem, E., Faesi, C., Groves, B., Hughes, A., Koch, E. W., Kreckel, K., Liu, D., Pan, H.-A., Pety, J., Querejeta, M., Razza, A., Saito, T., Sardone, A., Usero, A., Williams, T. G., Bigiel, F., Bolatto, A. D., Chevance, M., Dale, D. A., Gensior, J., Glover, S. C. O., Grasha, K., Henshaw, J. D., Jiménez-Donaire, M. J., Klessen, R. S., Kruijssen, J. M. D., Murphy, E. J., Neumann, L., Teng, Y.-H., Thilker, D. A. (2023). Star Formation Laws and Efficiencies across 80 Nearby Galaxies. The Astrophysical Journal Letters, 945, L19. DOI: 10.3847/2041-8213/acbd9c (arXiv:2302.12267). https://doi.org/10.3847/2041-8213/acbd9c
13. Pokhrel, R., Gutermuth, R. A., Krumholz, M. R., Federrath, C., Heyer, M., Khullar, S., Megeath, S. T., Myers, P. C., Offner, S. S. R., Pipher, J. L., Fischer, W. J., Henning, T., Hora, J. L. (2021). The Single-cloud Star Formation Relation. The Astrophysical Journal Letters, 912, L19. DOI: 10.3847/2041-8213/abf564. https://doi.org/10.3847/2041-8213/abf564
14. Lada, C. J., Lombardi, M., Alves, J. F. (2010). On the Star Formation Rates in Molecular Clouds. The Astrophysical Journal, 724, 687–693. DOI: 10.1088/0004-637X/724/1/687. https://doi.org/10.1088/0004-637X/724/1/687
15. Kennicutt, R. C., Jr. (1998). The Global Schmidt Law in Star-forming Galaxies. The Astrophysical Journal, 498, 541–552. DOI: 10.1086/305588. https://doi.org/10.1086/305588
16. Bazzi, Z., Kobayashi, M. I. N., Colombo, D., Bigiel, F., Leroy, A. K., Meidt, S. E., Klessen, R. S., Rosolowsky, E., Chown, R., Dale, D. A., Dlamini, S., Greve, M., Stuber, S. K., Boquien, M., Williams, T. G., Pan, H.-A., Querejeta, M., Ramambason, L., Romanelli, A., Saito, T., Romano, L. E. C., Jiménez-Donaire, M. J., Kim, H., Pathak, D., Koziol, H., Sutter, J., Lee, J. C., and the PHANGS collaboration (2026). The lifetime of 100,000 molecular clouds in the nearby Universe. arXiv:2604.24759. https://arxiv.org/abs/2604.24759