Influence of star cluster mass, age, and galaxy star formation rate on star cluster radii

Star clusters are key components of galaxies, and the relationship between cluster radius and mass encodes information about cluster formation and evolution. Theoretical models predict that age and specific star formation rate (sSFR) should influence cluster size through stellar mass loss and gas dynamics during formation. We hypothesized that if these theoretical predictions hold, multivariate models including age and sSFR should predict cluster radius better than models using mass alone. To test this, we used regression analysis on 5,105 star clusters from the LEGUS survey, comparing a full multivariate model against a mass-only baseline. We found that mass dominated the radius-mass relation: log(Mass) showed a strong correlation with radius (coefficient = 0.131 +/- 0.008, p < 0.001), while log(sSFR) and log(Age) contributed negligibly (0.0002 +/- 0.015 and 0.038 +/- 0.006, respectively). Cross-validation revealed that the mass-only model generalized better (CV R^2 = 0.028 vs -0.017), with the negative value for the multivariate model indicating overfitting. Contrary to our hypothesis, adding age and sSFR did not improve predictive performance. The low R^2 (0.115) indicated that most variance in cluster radius remained unexplained by these variables, suggesting other factors may play important roles. Among the variables tested, our findings were consistent with virial equilibrium predictions, with mass serving as a more fundamental parameter than evolutionary age or galaxy star formation rate.

💡 Research Summary

The paper investigates how star‑cluster mass, age, and the host galaxy’s specific star‑formation rate (sSFR) influence the effective radii of star clusters. Using the LEGUS survey, the authors assembled a clean sample of 5,105 clusters (masses 10³–10⁷ M⊙, ages 1 Myr–12 Gyr) with reliable radius measurements. All variables were log‑transformed to linearize relationships and stabilize variance.

Two regression models were built: (1) a full multivariate linear model including log M, log sSFR, and log Age as predictors of log R_eff, and (2) a baseline model using only log M. The full model achieved an in‑sample R² of 0.122 (adjusted R² = 0.121) and explained only ~12 % of the variance in cluster radii. The coefficients were β_M = 0.131 ± 0.008 (p < 0.001), β_sSFR = 0.0002 ± 0.015 (p ≈ 0.99), and β_Age = 0.038 ± 0.006 (p < 0.001). Thus, mass is the dominant predictor; sSFR contributes essentially nothing, while age has a statistically significant but modest effect (≈ 3 % of the mass effect).

Multicollinearity diagnostics (VIF ≤ 1.56) indicated no severe collinearity, although mass and age show a moderate correlation (r ≈ 0.59), likely reflecting survivorship bias (massive clusters live longer).

The mass‑only model yielded log R_eff = −0.247 + 0.160·log M, with R² = 0.115 and RMSE = 0.272 dex, capturing 94 % of the explanatory power of the full model.

Model comparison employed several metrics. Five‑fold cross‑validation revealed that the simple model generalizes far better (CV R² = 0.028) than the full model (CV R² = −0.017), the latter performing worse than a naïve mean predictor and indicating over‑fitting. Information‑theoretic criteria (AIC, BIC) favored the full model (ΔAIC = −37.2, ΔBIC = −24.2), but this conflicts with the cross‑validation results, highlighting the limitation of AIC/BIC when sample size is large and model complexity modest. A partial F‑test confirmed that adding age and sSFR yields a statistically significant improvement (F = 20.69, p < 0.001), yet the practical gain in predictive power is negligible.

The authors interpret these findings as strong empirical support for the virial‑equilibrium picture: cluster mass fundamentally sets the size scale, while evolutionary effects (mass loss, expansion) and environmental pressure (proxied by sSFR) play at most a secondary role in the LEGUS sample. The weak age effect suggests that either stellar‑evolution‑driven expansion is compensated by other processes (e.g., tidal stripping, early gas expulsion) or that observational selection masks the trend. The near‑zero sSFR coefficient implies that galaxy‑wide star‑formation intensity does not translate into systematic differences in cluster compactness, contrary to some theoretical expectations.

Importantly, the low overall R² (≈ 0.12) indicates that ~88 % of the variance in cluster radii remains unexplained by the three examined variables. Potential missing factors include initial gas pressure, metallicity, ambient density, feedback strength, and dynamical interactions with the host galaxy. The authors recommend future work that expands the sample to a broader range of galactic environments, employs non‑linear or hierarchical Bayesian models to capture hidden dependencies, and integrates high‑resolution simulations to test physical mechanisms.

In summary, the study provides a rigorous statistical test of theoretical predictions concerning cluster size determinants. It confirms that mass is the primary driver of cluster radius, while age and sSFR add little predictive value and can even degrade out‑of‑sample performance when included indiscriminately. This result refines our understanding of cluster formation and evolution and underscores the need to explore additional physical parameters beyond the simple mass‑age‑sSFR framework.