L(24)-L(Pa α) Correlation in Star Formation

Updated 18 August 2025

The paper demonstrates that extinction-corrected L(Pa α) reliably traces star formation rates in dusty regions with a scatter of ~0.27 dex, enhancing calibration accuracy.
The study uses multi-wavelength observations, including ground-based and space-based imaging, to analyze spatial variations in star formation across various galactic environments.
The research emphasizes that matching evolutionary stages and accounting for disk geometry are essential to reducing dispersion in the L(24)-L(Pa α) correlation.

The L(24)–L(Pa α) correlation refers to the empirical and physical relation between the 24 μm luminosity (L(24)), a proxy for dust-reprocessed emission from young stars, and the Paschen-α recombination line luminosity (L(Pa α)), a direct tracer of ionized gas produced by massive star formation. This correlation is critical for calibrating star formation rate (SFR) indicators in environments characterized by substantial extinction, such as luminous infrared galaxies (LIRGs), starburst regions, and dusty galaxy disks. Multi-wavelength studies exploiting this relationship provide deep insights into the physics of star formation, the structure of galactic disks, and the reliability of SFR tracers in various regimes.

1. Physical Basis and Observational Context

L(Pa α) traces the instantaneous rate of ionizing photon production by massive OB stars, and, due to its reduced sensitivity to dust extinction (only about one-sixth that of Hα), enables reliable SFR measurements in regions inaccessible to optical tracers. L(24) instead measures mid-infrared dust emission, predominantly emitted by small grains heated by UV photons from young stars; it is less direct but more sensitive to deeply embedded sources. Both quantities are coupled through the massive young stellar population, yet their correlation can be modulated by dust geometry, extinction, star formation history, and physical structure of the emitting regions.

Ground-based Paα narrow-band imaging and space-based mid-IR observations have enabled precise measurements of these luminosities in galaxy disks, starburst nuclei, and interacting/merging systems. Key studies analyzing this correlation include spatially resolved imaging in LIRGs using TAO/ANIR (Tateuchi et al., 2014), analysis of disk thickness and power spectra in M51 via HST+JWST (Elmegreen et al., 7 Apr 2025), and extinction-corrected SFR calibrators in starburst systems like Taffy I (Komugi et al., 2012).

2. Mathematical Framework and SFR Conversion

The correlation between L(24) and L(Pa α) is frequently expressed through calibrated conversion formulas relating these luminosities to star formation rates. For Paα, under Case B recombination with T = 10⁴ K, the intrinsic flux ratio Pa α/Hα ≈ 0.12, reflecting the line’s relative strength and extinction properties. SFR calibrations used include: $\text{SFR(Pa α)}\ \left[M_\odot\,\text{yr}^{-1}\right] = 6.4 \times 10^{-41} L(\text{Pa α})\ [\text{erg}\,\text{s}^{-1}]$

$\text{SFR(IR)}\ \left[M_\odot\,\text{yr}^{-1}\right] = 2.8 \times 10^{-44} L(\text{IR})\ [\text{erg}\,\text{s}^{-1}]$

Similarly, empirical calibrations relate L(24) to SFR, with non-linearities and corrections for dust geometry and AGN contamination.

The surface densities of IR luminosity and SFR are often normalized by region size: $\Sigma_{L(\text{IR})} = \frac{L(\text{IR})}{\pi r_m^2}$

$\Sigma_{\text{SFR}} = \frac{\text{SFR}}{\pi r_m^2}$

which enables comparative analysis of star formation efficiency and compaction across different galaxy types and environments.

3. Empirical Results and Scatter

Extinction-corrected Pa α-derived SFRs and IR-based SFRs show a good correlation in LIRGs, with scatter around 0.27 dex (Tateuchi et al., 2014). This scatter arises from residual extinction uncertainties, calibration differences, contrasting dust heating contributions (“cirrus” vs. starburst), and variable physical region sizes. The tightest correlations are observed in matched-age starburst regions (Komugi et al., 2012), where SFR and dense gas surface density relationships (Schmidt–Kennicutt law) exhibit unity slope: $\log \Sigma_{\text{SFR}} = (0.99 \pm 0.08) \log \Sigma_{H_2} - (8.13 \pm 0.12)$ These systems, particularly those with spatially resolved measurements, suggest that in environments of uniform evolutionary stage and dense gas fraction, L(24) and L(Pa α) are linearly related with minimal dispersion.

Notably, at L(IR) ≈ 8×10¹⁰ L_⊙, a transition occurs from extended star-forming regimes (normal galaxies) to compact, high surface brightness starbursts (U/LIRGs); above this, both L(24) and L(Pa α) are expected to correlate more tightly in compact regions with elevated SFR per area.

4. Disk Structure, Radial Dependence, and Power Spectra

In disk galaxies like M51, detailed analyses of power spectra (PS) from high-resolution Hα, Paα, and mid-IR imaging reveal changes in slope between large and small spatial scales (Elmegreen et al., 7 Apr 2025). These “breaks” occur at characteristic scales, interpreted as disk thickness: ~120 pc in Hα, ~170 pc in Paα, and more gradually in mid-IR bands associated with L(24). As disk thickness varies radially (e.g., thinner in central regions, thicker at large radii), the relative contribution of Paα and L(24) emission per star formation event can shift—affecting the observed correlation. Break scales must be considered when calibrating L(24)-L(Pa α) relations for integrated or resolved regions.

A representative emission measure formula is: $I = 2.0 \times 10^{-18} T_4^{-0.92} EM$ with $EM = n_e^2 L$ , where $n_e$ is electron density and $L$ is line-of-sight thickness.

5. Influence of Evolutionary Stage and Environment

Dispersion in the L(24)-L(Pa α) relation is reduced when regions of matched age and uniform dense gas fraction are analyzed (Komugi et al., 2012). Systems at a similar evolutionary stage—such as immediately post-merger starbursts—exhibit unity slope and low scatter, supporting the hypothesis that evolutionary heterogeneity in molecular clouds underlies much of the global scatter in SFR tracers.

In contrast, in normal or merging galaxies, scatter is amplified by differences in age, dust geometry, and merging stage (Tateuchi et al., 2014). Observational strategies that isolate coeval regions or account for radial disk structure can therefore significantly improve calibration of the L(24)-L(Pa α) correlation for SFR measurement.

6. Statistical and Time Series Analysis Perspectives

For datasets characterized by temporal or spatial nonstationarity in star formation tracers, classical global measures of correlation can be misleading. Local partial autocorrelation estimation (Killick et al., 2020) enables the computation of time-dependent (or locally-varying) correlation functions, revealing shifts in the L(24)-L(Pa α) relationship across environments or timescales. This approach refines calibration schemes and enhances diagnostics of star formation variability, especially for datasets with evolving physical conditions.

7. Theoretical and Future Directions

The calibration and interpretation of the L(24)-L(Pa α) correlation are central to improving SFR estimates in dusty, complex environments. Future research directions include:

High-resolution dense gas tracer studies (e.g., HCN, high-J CO) to directly test if the linear correlation (unity slope) persists as suggested by starburst regions (Komugi et al., 2012).
Expansion to other systems (mergers, quiescent disks, local U/LIRGs) to test whether matched-age selection and thickness corrections further tighten correlations.
Theoretical modeling of star-forming region evolution post-interaction, with attention to turbulent disk structure and implications for luminosity ratios.
Application of local partial autocorrelation techniques for mapping correlation variability in resolved galaxies and time-variable star formation events.

A plausible implication is that the utility of L(24) as a proxy for instantaneous SFR depends sensitively on matched evolutionary stage, local disk geometry, and extinction properties. Accurate calibration thus benefits from multi-wavelength imaging, region-specific selection, and statistical tools adapted to nonstationary phenomena.

In summary, the L(24)-L(Pa α) correlation is underpinned by the physical co-location and coupling of dust-heated and ionized-gas star formation indicators in young stellar environments. Its reliability and dispersion are modulated by region evolution, dust geometry, disk structure, and observation methods. Enhanced understanding and calibration depend on detailed spatially resolved studies, robust extinction corrections, and advanced statistical methodologies. The relationship remains a critical tool for charting star formation in the most dust-obscured and dynamically complex regions of galaxies.