The extended Schmidt law, ΣSFR∝(Σstar0.5Σgas)1.09\Sigma_{\rm SFR}\propto(\Sigma_{\rm star}^{0.5}\Sigma_{\rm gas})^{1.09}, empirically describes star formation from dwarf-galaxy outer disks and molecular clouds to starburst mergers and high-redshift galaxies, implying that existing stars regulate the efficiency of star formation. What physical mechanism produces the Σstar0.5\Sigma_{\rm star}^{0.5} dependence remains unidentified: mid-plane gas pressure set by stellar gravity is consistent with the data, but stellar radiation, heating, and feedback, or a purely statistical mediation by molecular gas, have not been excluded. Compounding this, the law was established with a uniform CO-to-H2_2 conversion factor, while that factor is known to drop by large factors in starburst nuclei, so the true slope, scatter, and faint-end behavior of the relation are not yet pinned down. The problem is to identify the physical origin of the stellar-mass scaling — causal or statistical — and to quantify how an environment-dependent conversion factor (and faint-end measurement errors) reshape the law.

Background

Star formation in galaxies is commonly described by the Kennicutt–Schmidt law, an empirical correlation between the star formation rate surface density ΣSFR\Sigma_{\rm SFR} and the gas surface density Σgas\Sigma_{\rm gas}; the Silk–Elmegreen variant replaces gas density with the orbital time of the disk. Both laws fail in certain environments: outer disks of dwarf galaxies, low-surface-brightness galaxies, and extremely metal-poor systems fall far below the mean relations. Shi et al. (2011) proposed the extended Schmidt law, in which the star formation efficiency (SFE, the ratio ΣSFR/Σgas\Sigma_{\rm SFR}/\Sigma_{\rm gas}) depends explicitly on the stellar mass surface density Σstar\Sigma_{\rm star}, roughly as SFE ∝Σstar0.5\propto\Sigma_{\rm star}^{0.5}; in a 2018 revisit this was refined to ΣSFR∝(Σstar0.5Σgas)1.09\Sigma_{\rm SFR}\propto(\Sigma_{\rm star}^{0.5}\Sigma_{\rm gas})^{1.09} with about 0.3 dex scatter, holding from ΣSFR∼10−5 M⊙ yr−1 kpc−2\Sigma_{\rm SFR}\sim10^{-5}\,M_\odot\,{\rm yr}^{-1}\,{\rm kpc}^{-2} in dwarf outer disks to starburst mergers and high-redshift galaxies, and bringing most Kennicutt–Schmidt outliers onto the relation. The natural interpretation is gravitational: the weight of the interstellar gas in the galactic potential, and hence its mid-plane pressure, is set largely by the stellar disk, and star formation may be self-regulated so that the SFR maintains the pressure needed to balance turbulent dissipation and cooling (the pressure-regulated, feedback-modulated picture of Ostriker, Kim, and collaborators). Molecular gas masses in these studies are, however, derived from CO line luminosities through the CO-to-H2_2 conversion factor XCOX_{\rm CO} (often written αCO\alpha_{\rm CO}), which converts CO intensity to molecular hydrogen column; Shi et al. (2018) adopted a single Galactic value XCO=2×1020 cm−2/(K km s−1)X_{\rm CO}=2\times10^{20}\,{\rm cm^{-2}/(K\,km\,s^{-1})} everywhere, explicitly noting that it is not understood under what physical conditions starburst regions require a smaller factor or by how much, even though resolved studies show the factor varies strongly in galaxy centers. The faint outer-disk measurements also carry errors up to about 0.5 dex in SFR, gas, and stellar mass, leaving unclear how much of the observed faint-end dispersion is intrinsic. Since 2018, systematic alternatives to a causal stellar-gravity interpretation have also emerged: the volumetric star formation law (relating volume rather than surface densities), the finding that a single three-dimensional relation links Σ∗\Sigma_\ast, ΣSFR\Sigma_{\rm SFR}, and Σmol\Sigma_{\rm mol}, and partial-correlation analyses suggesting the stellar term may be statistically mediated by the molecular gas.

Problem Statement

Establish the physical foundation of the extended Schmidt law ΣSFR∝(Σstar0.5Σgas)1.09\Sigma_{\rm SFR}\propto(\Sigma_{\rm star}^{0.5}\Sigma_{\rm gas})^{1.09}, as posed by Shi et al. (2018), on two linked fronts. (1) Mechanism: determine whether the observed Σstar0.5\Sigma_{\rm star}^{0.5} dependence of the star formation efficiency is causally produced by a physical effect of the existing stellar population — gravity acting through the mid-plane gas pressure it sets, stellar radiation and heating, or feedback-coupled self-regulation — or is instead a statistical by-product of the correlations between molecular gas, stellar mass, and star formation rate, or is set by gas self-gravity alone in the dense regime, and, if it is causal, identify the dominant mechanism. (2) Conversion factor: determine under what physical conditions the CO-to-H2_2 conversion factor should be smaller than the Galactic value in starburst regions, by how much, and how an environment- and spatially-dependent XCOX_{\rm CO} (together with reduced faint-end measurement errors in the outer disks of dwarf galaxies) changes the slope, scatter, and claimed universality of the extended Schmidt law. The common objective is a mechanistic, conversion-factor-controlled account of why existing stars regulate star formation efficiency across environments ranging from ΣSFR∼10−5 M⊙ yr−1 kpc−2\Sigma_{\rm SFR}\sim10^{-5}\,M_\odot\,{\rm yr}^{-1}\,{\rm kpc}^{-2} in dwarf outer disks to starburst mergers and high-redshift galaxies.

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