Baryonic Mass-Size Relation in Galaxies

• The work establishes a robust power-law scaling between baryonic mass and galaxy size with low intrinsic scatter (∼0.1 dex) across disk and spheroid systems.
• The analysis employs precise measurements of stellar mass from Spitzer 3.6 μm data and gas mass from HI and CO profiles, enabling accurate scaling comparisons.
• Implications include a unified framework linking angular momentum retention, disk assembly, and evolutionary tracks across diverse galactic environments.

The baryonic mass-size relation describes how the total baryonic mass—comprising both stellar and cold gas components—scales with the characteristic size of galaxies. This relation underpins structural scaling laws in galaxy formation, traces the retained angular momentum, and offers stringent tests for models of disk and spheroid assembly. Recent surveys and analytic frameworks have demonstrated that the baryonic mass-size relation is more fundamental and less scattered than relations based solely on stellar mass, linking together galaxy types across morphological and evolutionary divisions.

1. Definitions and Measurement Frameworks

The baryonic mass-size relation quantifies the connection between a galaxy's total baryonic mass ($M_b = M_\star + M_\mathrm{gas}$, with $M_\mathrm{gas}$ often including H I and H₂ plus helium corrections) and a characteristic size parameter. Size is usually measured as the exponential disk scale length ($R_\ast$ or $R_d$ for late-types), the baryonic half-mass radius ($R_{h,b}$ or $R_{50,\,\rm bar}$ in disk systems), or the effective stellar radius ($r_e$) for spheroids.

Recent studies standardize mass measurements from Spitzer 3.6 μm luminosities with calibrated stellar mass-to-light ratios, while gas masses are derived from H I profiles and empirical molecular-to-atomic gas scaling relations. The baryonic half-mass radius is extracted from face-on baryonic surface density profiles, integrating both gas and stellar components. For massive systems, size proxies include SDSS Petrosian radii ($R_{50}$, $R_{90}$), allowing controlled comparisons of disk-dominated and bulge-dominated morphologies.

2. Empirical Baryonic Mass-Size Laws

Power-Law Relations Across Galactic Populations

For morphologically late-type (S0 and later) galaxies, $R_\ast$ scales with baryonic mass as a single power law over three orders of magnitude in $M_\mathrm{gas}$0:

$M_\mathrm{gas}$1

or equivalently,

$M_\mathrm{gas}$2

The intrinsic scatter about this relation is $M_\mathrm{gas}$3 dex and nearly constant with $M_\mathrm{gas}$4 (Wu, 2017). Analyses using the SPARC database reveal a dichotomy: high-surface-density (HSD), star-dominated disks (Sa–Sc) follow $M_\mathrm{gas}$5, while low-surface-density (LSD), gas-dominated disks (Sd–dI) follow $M_\mathrm{gas}$6 (Hua et al., 20 Oct 2025). The LSD branch implies a near-constant baryonic surface density, while the HSD branch shows increasing compactness at lower mass.

A distinct scaling arises for passive systems: ellipticals, S0 disks, non-nucleated dwarfs (dEs, dSphs, UDGs), and nucleated dEs each follow separate mass-size relations in the stellar plane, with varying slopes. For instance, lenticular (S0) disks have $M_\mathrm{gas}$7 and non-nucleated dwarfs cluster around $M_\mathrm{gas}$8 (Hua et al., 21 Jan 2026).

Square-Law Scaling in Disks

Tight square-law scaling ($M_\mathrm{gas}$9) is observed in the most physically motivated samples of disk galaxies, with little room for dark matter dominance inside the stellar disk (Schulz, 2017):

$R_\ast$0

This scaling implies nearly constant baryonic surface density at the disk edge and arises directly from balancing Newtonian dynamics and a universal edge acceleration. The orthogonal scatter in the $R_\ast$1–$R_\ast$2 plane is $R_\ast$3 in $R_\ast$4, substantially smaller than optical luminosity-diameter relations.

3. Theoretical Foundations and Physical Interpretations

Virial Equilibrium and Dark Matter Halo Coupling

The Clausius' Virial Maximum Theory (CVMT) models the baryonic mass-size relation as a consequence of baryons virializing in the tidal field of the dark matter halo (Bindoni et al., 2011). By partitioning virial energy between baryons and dark matter (dependent on the inner slope $R_\ast$5 of the halo density profile), the theory predicts a scaling $R_\ast$6, with $R_\ast$7 and $R_\ast$8 computed from halo structural parameters and cosmological mass variance. The observed slope varies between $R_\ast$9 at low mass and $R_d$0 at high mass, mediated by the dark-to-baryon mass ratio $R_d$1 and potential cusp/core structure.

Self-Gravitating DM Flow and Energy Cascade

Analytic models based on self-gravitating collisionless dark matter flow (SG-CFD) attribute the mass-size scaling to a constant energy cascade rate $R_d$2 across halos, yielding distinct scaling laws in incompressible (small, $R_d$3) and irrotational (large, $R_d$4) regimes (Xu, 2022):

$R_d$5

$R_d$6

The critical break occurs near $R_d$7 and $R_d$8 kpc, with a maximal baryonic-to-halo mass ratio $R_d$9 at $R_{h,b}$0. These analytic structures are matched to SPARC data and naturally recover empirical square-law scaling in disks.

Angular Momentum Retention and Evolutionary Paths

The baryonic mass-size relation encodes the retained specific angular momentum of the baryonic component, tightly linking it to evolutionary pathways. Star-forming HSD disks connect structurally to S0 disks; LSD disks, upon gas removal, populate the non-nucleated dwarf and UDG branches. Morphological transformation is thus driven primarily by gas consumption or removal, with minimal change in stellar size or surface density (Hua et al., 21 Jan 2026).

4. Comparison to Other Mass-Size Relations

Stellar mass-size relations show systematically larger scatter and changing slope at low $R_{h,b}$1, due to the failure of stellar mass to trace the full baryonic budget in gas-rich galaxies. Deviations are most stark in ultra-diffuse galaxies (UDGs), which are extreme outliers in the $R_{h,b}$2–$R_{h,b}$3 plane but standard in $R_{h,b}$4–$R_{h,b}$5.

Baryonic Tully-Fisher ($R_{h,b}$6–$R_{h,b}$7) and angular momentum ($R_{h,b}$8–$R_{h,b}$9) relations remain tight across HSD and LSD disk populations, with dichotomies in the mass-size plane manifesting as vertical offsets at fixed $R_{50,\,\rm bar}$0 but not in $R_{50,\,\rm bar}$1 (Hua et al., 20 Oct 2025).

5. Statistical Properties, Scatter, and Structural Sequence Dichotomies

Measured intrinsic scatter in baryonic mass-size relations is consistently low ($R_{50,\,\rm bar}$2–0.3 dex), far surpassing stellar-only relations. The division into HSD and LSD disk sequences is robust, as revealed by clustering analyses in multidimensional space (baryonic mass, size, gas fraction, Hubble type, mean surface density). Passive systems further subdivide into four scaling branches in the stellar mass-size plane, each with characteristic slopes and normalizations.

Offsets between active and passive sequences indicate distinct evolutionary tracks: spirals may quench and slide onto the S0 disk branch with little structural change; gas-dominated LSD disks require external processes (ram pressure, tides) to transition onto the non-nucleated dwarf branch (Hua et al., 21 Jan 2026). UDGs mark the diffuse extension of the dwarf sequence, originating from the largest LSD disks after gas removal.

6. Broader Implications for Galaxy Formation and Evolution

The baryonic mass-size relation is a stringent constraint for models of disk assembly, star formation efficiency, and environmental quenching. Models must reproduce the observed dichotomies (HSD vs LSD, disk vs spheroid) and tight BTFR scaling while accommodating the diverse outcomes in surface density and angular momentum retention.

The scaling laws clarify the effects of feedback processes (insufficient for removing gas in dwarfs), importance of environmental transformation (dominant in gas-deficient systems), and the role of modified gravity in stabilizing low-surface-density disks. The coupling to cosmological parameters (e.g., the inner slope $R_{50,\,\rm bar}$3 of halos, cosmic mass variance) means that observed mass-size relations retain information about initial conditions and dark matter structure.

A plausible implication is that mass-size scaling reflects both local baryonic assembly history and global properties of the host dark matter halo, making it central for understanding and modeling galaxy evolution across varied environments and epochs.