Molecular Gas Surface Density (Σₕ₂)

• Molecular Gas Surface Density (Σₕ₂) is defined as the molecular hydrogen mass per unit area, typically expressed in M⊙ pc⁻².
• Indirect tracers—such as CO emissions and dust continuum—are used to infer Σₕ₂ while accounting for variations in metallicity and radiation fields.
• Σₕ₂ is crucial for understanding star formation, as it tightly correlates with the star formation rate through the Kennicutt-Schmidt relation.

Molecular gas mass surface density, denoted as Σₕ₂, quantifies the mass of molecular hydrogen (H₂) per unit area, typically expressed in M_⊙ pc⁻². Σₕ₂ is a fundamental parameter for characterizing the cold interstellar medium (ISM) in galaxies, as molecular gas is the immediate precursor to star formation. The distribution and scaling properties of Σₕ₂ underlie key relationships that link galactic gas reservoirs, star formation activity, and broader evolutionary processes.

1. Definition, Measurement, and Tracers of Σₕ₂

Σₕ₂ is defined as: Σₕ₂ ≡ Mₕ₂ / A where Mₕ₂ is the molecular hydrogen mass contained within an area A, usually evaluated on spatial scales ranging from sub-pc (within molecular clouds) to kpc (across galaxy disks).

H₂ is not directly observable in cold ISM conditions due to its lack of a permanent dipole moment and resulting absence of detectable low-excitation transitions. Σₕ₂ is thus typically inferred via indirect tracers:

• CO (Carbon Monoxide) Emission: The rotational transitions of ¹²CO, especially the J=1–0 line, are utilized as proxies for H₂, with corresponding CO intensities converted to Σₕ₂ through an adopted conversion factor, X_CO: Σₕ₂ = X_CO × I_CO × cos i where I_CO is the integrated CO intensity and i is the inclination (Caldu-Primo et al., 2013).
• Dust Continuum and Extinction: FIR dust emission or extinction mapping provides an alternative approach, with the total gas column inferred from dust properties, and the atomic component (Σ_HI) subtracted to yield Σₕ₂ (Lee et al., 2011, Lee et al., 2015).

The accuracy of Σₕ₂ measurements depends critically on the adopted X_CO value, which can vary with metallicity, radiation field, and gas column density. In low-metallicity systems, enhanced X_CO and substantial reservoirs of "CO-dark" H₂ must be considered (McQuinn et al., 2012, Lee et al., 2011).

2. Σₕ₂, Star Formation, and the Kennicutt-Schmidt Relation

The empirical correlation between Σₕ₂ and the star formation rate surface density (Σ_SFR) is a central pillar in star formation theory. The Kennicutt-Schmidt (K–S) law is commonly parameterized as: Σ_SFR ∝ Σₕ₂ⁿ where n ≈ 1 in the molecular regime (Bigiel et al., 2010).

Key findings:

• On kpc scales and in the star-forming disks of spiral galaxies, Σₕ₂ and Σ_SFR are tightly correlated, with a nearly constant molecular gas depletion time τ_dep = Σₕ₂/Σ_SFR ≈ 2 Gyr (Bigiel et al., 2010).
• At high spatial resolution (tens to hundreds of pc), typical values of Σₕ₂ for regions with active star formation are 10–40 M_⊙ pc⁻², but with Σ_SFR exhibiting several orders of magnitude of scatter due to local factors (Onodera et al., 2010).
• Averaging over larger scales mitigates local scatter, revealing the underlying quasi-linear relationship (Onodera et al., 2010, Bigiel et al., 2010).

Theoretical and simulation work supports a direct, often linear dependence of SFR on Σₕ₂ at ensemble-averaged scales but also highlights how stochasticity of star formation and environmental dependencies (metallicity, UV fields) alter both the normalization and slope of the relation (Feldmann et al., 2010, Feldmann et al., 2012).

3. Environmental and Scale Dependence

The behavior of Σₕ₂ and its correlations with star formation and other ISM properties show significant environmental and spatial scale dependence:

• Metallicity and Radiation Field: At fixed Σₕ₂, low-metallicity and/or high-UV field environments exhibit higher Σ_SFR and larger scatter. The Σ_SFR–Σₕ₂ relation steepens in these regimes, with slopes up to ~1.4 and increased normalization (Feldmann et al., 2010).
• Spatial Scale: The tightness of the Σ_SFR–Σₕ₂ correlation degrades on sub-kpc scales due to the discrete, stochastic nature of star formation and the variety of cloud evolutionary states. Scatter increases substantially below ~500 pc, reaching 0.4–0.6 dex (Feldmann et al., 2010, Onodera et al., 2010, Feldmann et al., 2012).
• Atomic-to-Molecular Transition: Σₕ₂ is regulated by the presence of an HI shielding layer. A minimum atomic gas surface density (typically ~6–10 M_⊙ pc⁻²) is required to protect H₂ from dissociation; above this threshold, any additional accumulated gas is converted into molecular form (Lee et al., 2011, Lee et al., 2015, Bialy et al., 2015).

4. Σₕ₂ in Diverse Galactic Environments

Disk Galaxies

• Radial Profiles: In the inner regions (r ≲ 0.2 r₍₂₅₎), molecular gas dominates, with Σₕ₂ reaching several hundred M_⊙ pc⁻² in some galaxies. Beyond this, Σₕ₂ declines exponentially with radius. When combined with the relatively flat atomic component, the total neutral gas profile (Σ_gas = Σ_HI + Σₕ₂) exhibits a universal exponential form once scaled by the optical radius and transition surface density (Σ_HI = Σₕ₂) (Bigiel et al., 2012).
• MGMS (Molecular Gas Main Sequence): Σₕ₂ is tightly correlated with the local stellar mass surface density (Σ⋆), forming the molecular gas main sequence (MGMS): log₁₀ Σₕ₂ = a · log₁₀ Σ⋆ + b, with a ≈ 1.10 ± 0.01 (Lin et al., 2019, Morselli et al., 2020).

Starburst Dwarfs and LSB Galaxies

• In low-mass, low-metallicity starburst dwarfs, inferred Σₕ₂ based on SF history analysis often exceeds that implied by HI observations, indicating substantial molecular reservoirs with high X_CO factors and low CO visibility (up to 40× Galactic X_CO) (McQuinn et al., 2012).
• In giant LSB galaxies such as Malin 1, deep CO observations yield only upper limits: Σₕ₂ < 0.3 M_⊙ pc⁻² and molecular-to-atomic mass ratios <0.13, placing these disks at the extreme low end of the molecular gas regime (Galaz et al., 2022).

Molecular Clouds

• On parsec scales within individual clouds (e.g., Perseus), Σₕ₂ derived from dust continuum is found to increase steeply with total column density once the HI shielding threshold is met. The ratio Σₕ₂/Σ_HI rises rapidly above total columns of (8–14)×10²⁰ cm⁻² (Lee et al., 2011, Lee et al., 2015).

5. Theoretical Models, ISM Structure, and Star Formation Efficiency

H₂ Formation and Shielding

• Analytic models such as those by Krumholz, McKee, Tumlinson (KMT09) and Sternberg et al. capture the physics of atomic-to-molecular transitions. For solar metallicity, a nearly constant HI column density is needed before molecular gas appears in bulk, consistent with observed ΣHI ~ 6–9 M⊙ pc⁻² in regions like Perseus (Lee et al., 2011, Bialy et al., 2015).
• The critical HI shielding layer is set by the dimensionless parameter αG ∝ I_UV/n (FUV field to gas density), which determines the thickness of the atomic envelope needed to protect H₂ from photodissociation (Bialy et al., 2015).

Star Formation Efficiency (SFE) and Gas Depletion Time

• In inner spiral disks, τ_dep ≈ 2 Gyr is remarkably uniform, indicating self-regulation. In HI-dominated regions or outer disks, τ_dep is much longer, indicating inefficiency in forming stars from atomic gas (Bigiel et al., 2010).
• In molecular clouds, star formation efficiency increases super-linearly with Σgas, following Σ⋆ ∝ Σ_gas², consistent with models of thermal fragmentation or turbulent hydrodynamics and implying continuous star formation with no clear column density threshold (Pokhrel et al., 2020).

ISM Structure

• Observations indicate a substantial fraction of diffuse, dynamically hot (σ_CO ≈ σ_HI ≈ 12 km s⁻¹) thick molecular gas disks in galaxies, challenging the notion that the molecular phase is exclusively confined to cold, self-gravitating GMCs (Caldu-Primo et al., 2013).
• "CO-dark" H₂ can constitute ~30% of the molecular mass, especially in low-metallicity or low-extinction regions where CO is under-abundant relative to H₂ (Lee et al., 2011).

6. Scaling Relations, Galaxy Evolution, and Cosmic Context

Σₕ₂ not only shapes local star formation, but also underpins galaxy scaling relations and regulates cosmic star formation history:

• At z ≈ 1–1.5, the cosmic molecular gas density is ~10× higher than at z ≈ 0, in lockstep with the elevated cosmic star formation rate density (Maeda et al., 2016).
• Resolved studies (ALMaQUEST, HERACLES, ALMA–MaNGA) show that Σₕ₂, ΣSFR, and Σ⋆ form a fundamental 3D relation, with the resolved star-forming main sequence (rSFMS) emerging from the combination of the Schmidt-Kennicutt and MGMS relations (Lin et al., 2019, Morselli et al., 2020).
• In disk galaxies, the presence of a universal exponential distribution for the total neutral gas profile beyond the transition radius links Σₕ₂ to the galactic stellar disk size, implying regulated gas inflow and consumption across the galactic disk (Bigiel et al., 2012).

7. Observational and Methodological Considerations

• Accurate estimation of Σₕ₂ requires careful calibration of the CO-to-H₂ conversion factor, accounting for environmental variations in metallicity and radiation field (Feldmann et al., 2010, Feldmann et al., 2012).
• Methods that invert star formation laws, such as the multi-freefall formalism using Hα flux and gas kinematics, provide additional avenues for estimating Σₕ₂ and have demonstrated better predictive accuracy (within 32%) compared to simple inversions of the K–S law (Federrath et al., 2017).
• New IFU and interferometric surveys (SAMI, ALMA, MaNGA) have enabled spatially resolved mapping of Σₕ₂ at sub-kpc scales, directly linking molecular gas to star formation and stellar structure.

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In summary, molecular gas mass surface density (Σₕ₂) is a critical diagnostic of the ISM phase most intimately linked to star formation, exhibiting tight scaling relations with ΣSFR, Σ⋆, and environmental parameters. Its spatial variation, environmental sensitivity, and interplay with atomic gas establish indispensable constraints on models of galactic star formation and evolution, from cloud scales to the cosmic history of galaxies.