Baryon Mass Surface Density

• Baryon mass surface density is defined as the vertical integral of baryonic density—including stars, gas, and remnants—providing a key diagnostic in galaxies.
• Measurement techniques rely on stellar kinematics and resolved photometric methods, with corrections for disk thickness, gas phases, and dark matter contributions.
• Empirical scaling laws link baryon density to star formation rates and galaxy dynamics, supporting models of self-regulated star formation and inside-out growth.

Baryon mass surface density, conventionally denoted $\Sigma_b$, is the measure of the baryonic mass per unit area, usually expressed in units of $M_\odot\,\mathrm{pc}^{-2}$ or $M_\odot\,\mathrm{kpc}^{-2}$. It provides a local or radially-resolved diagnostic of the collection of baryonic components (predominantly stars and interstellar gas, sometimes including stellar remnants and substellar objects) in galaxies and is a fundamental quantity for interpreting galaxy dynamics, star formation processes, and the interplay between baryonic and non-baryonic (dark) matter.

1. Definition and Components

The surface density of baryonic mass is defined as the vertical integral of the baryonic volume density at a given position:

$$\Sigma_b(R) \equiv \int_{-\infty}^{+\infty} \rho_b(R,z)\,dz$$

where $\rho_b(R,z)$ is the total baryon volume density, including stars, stellar remnants, and the interstellar medium (ISM: molecular, atomic, and ionized gas). In practice, $\Sigma_b$ is often partitioned into resolved contributions:

$$\Sigma_b(R) = \Sigma_*(R) + 1.4[\Sigma_{\mathrm{HI}}(R) + \Sigma_{\mathrm{H}_2}(R)] + \Sigma_{\text{rem}}(R)$$

The factor 1.4 corrects hydrogen columns for helium; $\Sigma_{\text{rem}}$ incorporates white dwarfs, neutron stars, black holes, and brown dwarfs where relevant (McKee et al., 2015).

In surveys with limited direct gas information, baryon mass surface density is sometimes operationally defined as the sum of stars and molecular gas inferred via extinction proxies:

$$\Sigma_b \equiv \Sigma_* + \Sigma_{\mathrm{mol,Av}}$$

with $\Sigma_{\mathrm{mol,Av}}$ calibrated against molecular tracers and Balmer decrement extinction (Barrera-Ballesteros et al., 2021).

2. Measurement Methodologies and Corrections

Determination of $\Sigma_b$ in disk galaxies requires careful dynamical and photometric modeling:

• Kinematic Approach: In face-on spirals, stellar vertical velocity dispersions $\sigma_z(R)$ are used, together with the vertical Jeans and Poisson equations, to derive the local dynamical surface density $\Sigma_{\mathrm{dyn}}(R)$ (Hessman, 2017). Corrections must account for contributions from the dark matter halo, the ISM, and effects due to finite disk thickness. For an exponential vertical profile, $\Sigma_{\mathrm{dyn}}$ is given by:

$$\Sigma_{\mathrm{dyn}}(R) = \frac{\overline{\sigma}_z^2(R)}{\pi k G h_z}$$

with $k=3/2$ in this geometry.

• Finite Thickness Correction: The traditional infinite-sheet approximation neglects the gravitational effect of matter at $R' \ne R$. A correction factor $\xi(R; h/H)$, determined via polynomial fits, adjusts $\Sigma_{\mathrm{dyn}}$ to the true local surface density (Hessman, 2017).
• Component Separation: Dynamical $\Sigma_{\mathrm{dyn}}$ is then decomposed by subtracting the dynamical effects of a thin gas disk, dark matter (using e.g., $(16/3)\rho_{\mathrm{DM}}h_*$ term), and, if necessary, the contribution of a thick disk. The presence of a thick-disk component with shorter scale length boosts the inferred $\Sigma_*$ by $\gtrsim30\%$ in the Milky Way if not separated (Hessman, 2017).
• Resolved Photometric Methods: In large IFU surveys such as MaNGA, spatially-resolved stellar population fits yield $\Sigma_*$, and extinction-corrected Balmer line ratios are calibrated against CO-based gas maps to estimate $\Sigma_{\mathrm{mol,Av}}$; these contribute to an empirical $\Sigma_b$ at kpc scales (Barrera-Ballesteros et al., 2021).

3. Empirical Relations and Scaling Laws

A suite of scaling relations tie $\Sigma_b$ to key galactic observables:

• Central Surface Density Coupling: Analysis of the SPARC galaxy sample finds that the dynamical central surface density, $\Sigma_{\mathrm{dyn}}(0)$, tracks the central stellar surface density, $\Sigma_*(0)$, via a double power law:

$$\Sigma_{\mathrm{dyn}}(0) = \Sigma_0 \left[1 + \frac{\Sigma_*(0)}{\Sigma_{\mathrm{crit}}}\right]^{\alpha-\beta}\left[\frac{\Sigma_*(0)}{\Sigma_{\mathrm{crit}}}\right]^\beta$$

Breaks appear at $\Sigma_*(0) \sim 10^3$ $M_\odot$ pc$^{-2}$, with a tight scatter of $\sim 0.2$ dex (Lelli et al., 2016). High-surface-brightness galaxies follow $\Sigma_{\mathrm{dyn}}(0) \approx \Sigma_*(0)$; low-surface-brightness systems systematically deviate, requiring increasing dark matter.

• Star Formation Relation: At kpc scales, the star formation rate surface density $\Sigma_{\mathrm{SFR}}$ correlates tightly with the total baryonic surface density $\Sigma_b$:

$$\Sigma_{\mathrm{SFR}} = a\,\Sigma_b^2 + b\,\Sigma_b$$

with $a \approx 4.3 \times 10^{-21}$ (kpc$^2$ yr$^{-1}$ $M_\odot^{-1}$) and $b \approx 6.2 \times 10^{-11}$ (yr$^{-1}$). This bivariate relation outperforms single-component (stellar or molecular gas only) correlations in both strength (Pearson $r \gtrsim 0.77$) and scatter ($\sigma \lesssim 0.28$ dex) (Barrera-Ballesteros et al., 2021).

• Radial Gradients: After correcting for disk thickness, dark halo, and gas, the stellar mass-to-light ratio $M/L_*(R)$ falls with radius proportional to local color gradients, and the baryonic mass scale length $H_*$ is $\sim$80% of the photometric scale length $H_p$ (Hessman, 2017).

4. Baryonic Mass Surface Density in the Solar Neighborhood

A detailed baryonic census at the solar Galactic radius utilizes star counts, gas surveys, and stellar kinematics (McKee et al., 2015). Principal local (solar circle) values are:

Component	Surface Density ($M_\odot\,\mathrm{pc}^{-2}$)	Effective Scale Height (pc)
Main Sequence	27.0 ± 2.7	220–800
M Dwarfs	17.3 ± 2.3	400
Giants	0.4 ± 0.06	400
White Dwarfs	4.9 ± 0.6	430
Brown Dwarfs	1.2 ± 0.36	400
Neutron Stars	0.8 ± 0.2	—
Black Holes	0.1	—
H$_2$ Gas	1.0 ± 0.3	105
HI Gas	10.9 ± 1.6	127–403
HII Gas	1.8 ± 0.1	1,590

Summing these yields a local total baryonic surface density of $47.1 \pm 3.4\,M_\odot\,\mathrm{pc}^{-2}$ (McKee et al., 2015). This value is robust against previous estimates, establishing a benchmark for local mass modeling and constraints on non-baryonic matter.

5. Physical Interpretation and Theoretical Implications

Empirical $\Sigma_b$ scaling relations reveal that in high surface-brightness regions, the baryonic content fully accounts for dynamical measurements—justifying the maximum-disk hypothesis. In low surface-brightness discs, dark matter provides an increasingly dominant gravitational contribution but does so in a manner that preserves a tight scaling with baryons, an outcome challenging to reproduce in unmodified $\Lambda$CDM unless stellar feedback creates large constant-density cores (Lelli et al., 2016, Hessman, 2017).

The observed radial decrease in $M/L$ and shorter baryonic mass scale length compared to the photometric length is consistent with inside-out galaxy formation and radially varying star formation efficiency (Hessman, 2017).

At kpc scales, the tightness of the star formation–$\Sigma_b$ relation supports models in which star formation is self-regulated by the balance of mid-plane pressure (set by $\Sigma_b$) and feedback from young stars. The quadratic term in the empirical star formation law suggests a non-linear increase in pressure and/or feedback efficiency at high $\Sigma_b$ (Barrera-Ballesteros et al., 2021).

6. Limitations, Caveats, and Ongoing Challenges

Significant systematic uncertainties remain:

• Gas Mass Estimation: Corrections for optically thick HI (with observed $\zeta_{\mathrm{HI}} \sim 1.3-2$) and CO-dark ${\rm H}_2$ are essential for accurate $\Sigma_{\rm gas}$ (Hessman, 2017).
• Component Decomposition: Proper accounting for thick-disk stars is required, as blending with the thin disk can inflate $\Sigma_*$ estimates (Hessman, 2017).
• ISM Phases and Scale Heights: The assumption of exponential or $\operatorname{sech}^2$ vertical profiles is an approximation and may mischaracterize the vertical distribution in the Galactic disk (McKee et al., 2015).
• Calibration of Molecular Gas via Extinction: The $A_V \rightarrow \Sigma_{\mathrm{mol,Av}}$ calibration is applicable primarily for low-inclination, gas-rich systems and at surface densities above the CO sensitivity limit (Barrera-Ballesteros et al., 2021).
• Star Formation Tracers: H$\alpha$ SFR tracers can be contaminated by diffuse ionized gas or evolved stellar populations at low $\Sigma_{\rm SFR}$, leading to flattening in scaling relations (Barrera-Ballesteros et al., 2021).

A plausible implication is that future high-resolution molecular gas mapping and more sophisticated vertical dynamical models, especially in the inner galaxy and for thick disk populations, will be critical for further constraining $\Sigma_b$ and its role in disk evolution.

7. Applications and Future Prospects

Baryonic mass surface density serves as a pivotal input for dynamical mass modeling, dark matter searches, and the calibration of star-formation laws. It is the keystone for decomposing mass in external galaxies, constraining the local dark matter density in the Milky Way, and evaluating galaxy formation theories.

The convergence of independent methodologies (stellar kinematics, star counts, and empirical star formation relations) towards consistent local and global baryonic surface densities bolsters confidence in these measurements (McKee et al., 2015, Hessman, 2017, Barrera-Ballesteros et al., 2021). However, the fine-tuned coupling between baryons and the total dynamical mass, especially in low surface brightness systems, remains a critical puzzle for both galaxy formation and fundamental physics (Lelli et al., 2016). Ongoing large-scale IFU surveys, deeper CO and HI imaging, and theoretical advances in feedback and baryon–DM coupling are expected to further refine $\Sigma_b$ determinations and their role in galactic dynamics and star formation.