Star-Forming Main Sequence

• Star-Forming Main Sequence (SFMS) is the tight, log-linear correlation linking star formation rate and stellar mass in galaxies.
• Recent studies reveal symmetric residuals in effective radius and stellar density, underscoring a unified structural regulation mechanism.
• The observed SFR dispersion is driven by gas depletion timescales, with compact, high-density systems exhibiting enhanced variability.

The Star-Forming Main Sequence (SFMS) is the empirical, tight correlation between the star formation rate (SFR) and stellar mass ($M_*$) of star-forming galaxies across cosmic time. It encodes key information about the regulation, variability, and physical drivers of galaxy growth, establishing a benchmark for the interpretation of galaxy evolution in both observational and theoretical contexts. Despite its ubiquity, the physical origin of the approximately 0.3–0.4 dex scatter in SFR at fixed stellar mass—one of its most enduring features—has remained unresolved. Recent developments utilizing large galaxy samples have clarified the structural symmetry and physical regulation underlying both the SFMS relation and its dispersion.

1. The SFR–Mass Correlation and Quantitative Definition

The SFMS is characterized by a log-linear relation between SFR and stellar mass: $\log \mathrm{SFR} = 0.88 \times \log M_* - 8.77$ as established from a sample of nearly 500,000 galaxies. This relation is observed to be remarkably tight across several decades in $M_*$, but exhibits a relatively constant intrinsic dispersion of $\sim 0.3–0.4$ dex at fixed mass. The persistence of this scatter across epochs and environments positions the SFMS as a critical diagnostic of star formation physics and galactic growth modes.

2. Structural Symmetry Across the Main Sequence

Analysis of SDSS galaxies reveals a fundamental symmetry in the structural properties of galaxies distributed above and below the SFMS:

• Residual Effective Radius ($\Delta \log R_{\rm e}$): At fixed $M_*$, the deviation of the galaxy’s effective radius from the median ($\Delta \log R_{\rm e}$) is maximum on the SFMS ridge and decreases symmetrically toward both higher and lower SFRs.
• Residual Stellar Surface Density ($\Delta \log (M_*/R_{\rm e}^2)$): The log deviation from median surface density is minimized on the SFMS and rises similarly above and below it.
• Morphological Symmetry (Sérsic index $n$): Morphological concentration, parameterized by $n$, is lowest (disk-like) at the SFMS and increases symmetrically toward quiescent and starburst populations.

These symmetries rigorously demonstrate that, when controlling for $M_*$, galaxies above and below the SFMS share fundamental structural characteristics, with variations dominated by size and density residuals rather than by distinct morphological classes (He et al., 11 May 2025).

3. Structural Regulation and the Origin of SFMS Scatter

A critical advance is the explicit linkage between structural parameters and SFR dispersion:

• Compact galaxies (smaller $R_{\rm e}$) and those with higher stellar surface densities ($M_*/R_{\rm e}^2$) exhibit significantly larger scatter in SFR at fixed mass (up to $\sim0.5$ dex for the densest systems, versus $\sim0.27$ dex at lowest density).
• These trends hold for both central and satellite galaxies, and for morphological sub-classes (early- and late-type), demonstrating universality of the physical processes regulating the scatter.

The gas depletion timescale ($\tau_{\rm dep}$), derived from an extended Schmidt law,

$\tau_{\rm dep} \propto [M_*/R_{\rm e}^2]^{-0.5}$

serves as the structural link. Galaxies with shorter $\tau_{\rm dep}$ (i.e., high surface density) exhibit enhanced responsiveness to accretion fluctuations and thus higher SFR dispersion.

Structural Bin	SFR Scatter ($\sigma_{\rm SFR}$, dex)
Large $R_{\rm e}$ / Low Density	$\sim$0.27–0.29
Intermediate	$\sim$0.40–0.41
Small $R_{\rm e}$ / High Density	$\sim$0.49–0.50

4. Accretion Variability, Gas Depletion, and SFR Fluctuations

The underlying physical model conceptualizes SFR as a time-averaged response to fluctuations in cosmic gas accretion ($\Phi$), smoothed by the gas depletion timescale: $$\mathrm{SFR}(t) = \mathrm{SFE} \cdot M_{\rm gas}(t)$$

$$\frac{dM_{\rm gas}}{dt} = \Phi(t) - \mathrm{SFE}(1+\lambda)M_{\rm gas}(t)$$

with $\lambda$ the mass-loading factor from outflows. The SFR variance induced by accretion fluctuations is given by: $$\sigma_{\log \mathrm{SFR}}^\Phi = \frac{\sigma_{\log \Phi}}{\sqrt{1 + (2\pi \tau_{\rm dep}/T_P)^2}}$$ where $T_P$ is the characteristic timescale of accretion fluctuations. Shorter $\tau_{\rm dep}$ admits greater SFR variability, explaining the observed structural dependence of SFMS scatter.

Empirically, the best-fit $T_P \sim 4$ Gyr and $\sigma_\Phi \sim 0.4$–$0.5$ dex, with the fluctuation period scaling as the dynamical free-fall time ($T_{\rm ff} \propto (R_{\rm e}^3/M_*)^{1/2}$), directly tying environmental response to global galaxy structure.

5. Morphological and Environmental Universality

The symmetry and structural regulation of SFMS scatter hold across variations in galaxy environment (central/satellite) with minor quantitative differences in fluctuation timescale, and across morphological type (early/late), irrespective of disk or spheroid dominance at fixed mass. Early-types are more compact at given $M_*$, but the physical scaling of scatter with $M_*/R_{\rm e}^2$ is generic. This universality points to a regulation mechanism that is fundamentally intrinsic and not strongly modulated by large-scale environment.

6. Implications for Galaxy Evolution and Theoretical Modelling

The findings imply the SFMS is not purely a manifestation of deterministic, steady-state galaxy growth but reflects the interaction of stochastic cosmic accretion ("cosmic weather") with galaxy-intrinsic gas processing timescales, as set by structure. The dispersion in SFR is not dominated by measurement errors or observational limitations but is an intrinsic feature of the baryon cycle in galaxies.

• Structural density acts as a regulator of SFR variability: Denser systems exhibit more rapid but variable SFR evolution, leading to more pronounced excursions above and below the SFMS at fixed mass.
• Observational dispersion in the SFMS arises from the response to stochastic accretion regulated by $\tau_{\rm dep}$, with a secondary role played by Poisson noise, significant primarily at low $M_*$ or SFR.
• The symmetry of residuals indicates a fundamental connectedness of galaxy populations above and below the SFMS, persistently shaped by the same physical processes rather than by fundamentally distinct formation histories.

7. Schematic Summary and Key Formulae

Aspect	Observational/Physical Law
SFMS Best Fit	$\log \mathrm{SFR} = 0.88\log M_* - 8.77$
Stellar Surface Density	$\Sigma_* = M_*/(\pi R_{\rm e}^2)$
Depletion Timescale Scaling	$\tau_{\rm dep} \propto [M_*/R_{\rm e}^2]^{-0.5}$
SFR Scatter vs Accretion Fluct.	$$\sigma_{\log \mathrm{SFR}}^\Phi = \frac{\sigma_{\log \Phi}}{\sqrt{1 + (2\pi \tau_{\rm dep}/T_P)^2}}$$

Conclusion: The SFMS is best conceptualized as a dynamic equilibrium reflecting the interplay of stochastic baryonic accretion and the regulating influence of galaxy structure—especially effective radius and surface density. The observed universal scatter and structural symmetries provide stringent constraints for models of star formation and galactic evolution (He et al., 11 May 2025). The inherent variability calls for a shift in theoretical modeling toward frameworks that incorporate stochastic accretion histories and their regulation by internal galactic structure.