Galaxy Size-Stellar Mass Relation

• Galaxy SMR is the empirical scaling linking effective size measures (e.g., Rₑ) and stellar mass, with slopes and curvature varying by galaxy population and redshift.
• Observations show star-forming galaxies follow a near-single power law, while quiescent systems often require broken power laws, indicating different growth mechanisms.
• Measurement methods, including PSF treatment, bandpass choices, and size definitions, significantly influence the inferred slopes, normalization, and scatter of the SMR.

Searching arXiv for the cited SMR literature to ground the article in current papers and verify identifiers. The galaxy size–stellar mass relation (SMR) is the empirical scaling between a galaxy’s characteristic size and its stellar mass, usually written in terms of an effective or half-light radius such as $R_e$, $r_e$, $R_{50}$, or $R_{\rm eff}$. Across the literature, the SMR is not a single invariant law but a family of relations whose slope, normalization, curvature, and scatter depend on galaxy population, redshift, wavelength, size definition, and analysis pipeline. Over $0.2 \le z \le 2$, star-forming galaxies are well represented by a single power law, whereas quiescent systems generally require a broken or double-power-law description; at low redshift, early-type galaxies exhibit steeper SMR slopes than late types; and the interpretation of apparent environmental trends remains contested because some studies report direct size offsets while others attribute the effect almost entirely to changing population mixtures (Nedkova et al., 2021, Kawinwanichakij et al., 2021, Abdullah et al., 6 Jul 2025).

1. Definition, observables, and measurement conventions

The SMR is operationally defined through a size observable and a stellar-mass estimator. Common size choices include the circularized half-light radius $r_e$, the major-axis half-light radius $R_e$, non-parametric half- and 90 percent-light radii $R_{50}$ and $R_{90}$, and projected half-mass radii derived directly from stellar particles in simulations. The measurement method is correspondingly heterogeneous: PSF-convolved single-Sérsic fits with GALFIT or GalfitM, seeing-convolved 2D single-Sérsic fits with 2DPHOT, SExtractor light-fraction radii, and growth-curve analyses all appear in the literature (Mosleh et al., 2012, Roy et al., 2018, Nedkova et al., 2021).

These definitions are not interchangeable. In the GOODS-N analysis, $r_{50}$ is explicitly distinguished from a Sérsic $r_e$0: if a profile extends formally to infinity, $r_e$1 is model-dependent, whereas $r_e$2 is directly measured from the light distribution. The same study adopts

$r_e$3

with $r_e$4 for $r_e$5 and $r_e$6 for $r_e$7, and shows that de Vaucouleurs-like profiles are especially susceptible to size underestimation when low-surface-brightness outskirts fall below the noise (Ichikawa et al., 2012).

Circularization conventions also differ. In high-$r_e$8 Lyman-break galaxies, the effective radius is taken as

$r_e$9

where $R_{50}$0 is the semimajor-axis half-light radius and $R_{50}$1 is the axis ratio. In cluster work based on adaptive-optics imaging, the adopted circularized effective radius is

$R_{50}$2

Other studies retain the major-axis size explicitly, especially when inclined disks are important (Mosleh et al., 2012, Sweet et al., 2016, Liao et al., 20 Nov 2025).

Rest-frame wavelength is a further structural choice. The HFF+CANDELS analysis derives a redshift-independent size at rest-frame $R_{50}$3 Å and reports little difference when using $R_{50}$4 Å or $R_{50}$5 Å. By contrast, the $R_{50}$6 cluster analysis shows that the fitted SMR slope is strongly wavelength dependent: $R_{50}$7, $R_{50}$8, and $R_{50}$9. This indicates that blue-band sizes need not trace the stellar-mass distribution faithfully, particularly when clumpy star formation affects outer isophotes or deblending (Nedkova et al., 2021, Sweet et al., 2016).

Resolution and pipeline choices can dominate the inferred relation. When the $R_{\rm eff}$0-band AO data are degraded to NICMOS-like resolution, the slope remains broadly consistent, $R_{\rm eff}$1, but degradation to FourStar-like seeing produces $R_{\rm eff}$2, a dramatic bias for compact $R_{\rm eff}$3 systems. In simulations of low-mass galaxies, mock-observed sizes and masses differ systematically from particle-based values: $R_{\rm eff}$4 and $R_{\rm eff}$5, so the same galaxies move to lower normalization and larger scatter in the observational plane (Sweet et al., 2016, Klein et al., 2024).

2. Empirical forms of the relation across galaxy populations

The SMR is frequently modeled as a power law,

$R_{\rm eff}$6

but many datasets show curvature or broken-power-law behavior. A deep $R_{\rm eff}$7-selected sample in GOODS-N reports a shallow, nearly redshift-invariant slope $R_{\rm eff}$8 over $R_{\rm eff}$9, similar for star-forming and quiescent galaxies. In the resolved-galaxy fit over the full redshift range, the measured slopes are $0.2 \le z \le 2$0 and $0.2 \le z \le 2$1 for all galaxies; star-forming galaxies give $0.2 \le z \le 2$2 and $0.2 \le z \le 2$3; quiescent galaxies are somewhat steeper, with $0.2 \le z \le 2$4 and $0.2 \le z \le 2$5 (Ichikawa et al., 2012).

At lower redshift and with much larger samples, the relation is more evidently non-single-slope. Using DESI Legacy Survey sizes, the full $0.2 \le z \le 2$6–$0.2 \le z \le 2$7 relation has best-fit slopes $0.2 \le z \le 2$8 for the spectroscopic sample and $0.2 \le z \le 2$9 for the photometric-redshift sample, while splitting at $r_e$0 yields steeper high-mass slopes: $r_e$1 and $r_e$2, respectively. The same study notes hints of a transition around $r_e$3, broadly consistent with dwarf-galaxy relations in the literature (Liao et al., 20 Nov 2025).

Population dependence is strong. In the Hyper Suprime-Cam Subaru Strategic Program sample of $r_e$4 million galaxies at $r_e$5, star-forming galaxies are on average larger than quiescent galaxies at fixed redshift and stellar mass over $r_e$6. Both populations follow broken power laws with a pivot stellar mass $r_e$7. For quiescent galaxies, below $r_e$8, the relation follows $r_e$9; above $R_e$0, it steepens to $R_e$1. For star-forming galaxies, below $R_e$2, the relation follows $R_e$3; above $R_e$4, it evolves with redshift and follows $R_e$5 (Kawinwanichakij et al., 2021).

A related low-redshift result appears in the SDSS cluster–void comparison. When the SMR is written as

$R_e$6

early-type classes are consistently steeper than late types. For example, the sSFR-based fits give $R_e$7 for cluster early types and $R_e$8 for cluster late types; the void values are $R_e$9 and $R_{50}$0, respectively. The late types also have higher normalization, meaning they are typically larger at fixed stellar mass (Abdullah et al., 6 Jul 2025).

3. Redshift evolution and the buildup of the relation

A persistent result is that galaxies of fixed stellar mass are smaller at higher redshift. For star-forming Lyman-break galaxies, the stellar mass–size relation persists at least to $R_{50}$1, with slopes $R_{50}$2 at $R_{50}$3 and $R_{50}$4 at $R_{50}$5. At fixed mass, sizes evolve as $R_{50}$6 for $R_{50}$7 and $R_{50}$8 for $R_{50}$9. This near-equality of the two exponents indicates very similar redshift dependence across the two mass bins (Mosleh et al., 2012).

Over $R_{90}$0, the HFF+CANDELS analysis refines this picture by separating star-forming and quiescent systems down to $R_{90}$1. Star-forming galaxies are well described by a single power law

$R_{90}$2

with $R_{90}$3: $R_{90}$4, $R_{90}$5, $R_{90}$6, and $R_{90}$7 from the lowest to highest redshift bins. Quiescent galaxies require a double power law; for $R_{90}$8, the single-power-law slopes are $R_{90}$9, $r_{50}$0, $r_{50}$1, and $r_{50}$2 over the same redshift sequence. The same study finds that star-forming galaxies with $r_{50}$3 and quiescent galaxies with $r_{50}$4 have undergone significant size growth since $r_{50}$5, whereas low-mass galaxies have not (Nedkova et al., 2021).

The quiescent branch shows especially clear curvature. In the full-mass fits, the low-mass slope $r_{50}$6 is flat or slightly negative, while the high-mass slope $r_{50}$7 is steep, about $r_{50}$8–$r_{50}$9, and the flattening is associated with sizes of roughly $r_e$00 kpc at $r_e$01. This is consistent with the HSC result that quiescent galaxies below the pivot mass have a very shallow slope and with the interpretation that large low-mass quiescent galaxies have sizes similar to their star-forming counterparts (Nedkova et al., 2021, Kawinwanichakij et al., 2021).

Intermediate-redshift ground-based surveys recover the same broad differential evolution. In KiDS, the SMR is curved rather than a single power law. Spheroids show significant size evolution above $r_e$02, with $r_e$03 and $r_e$04 at $r_e$05; for disc-dominated galaxies the average slope is much flatter, about $r_e$06, or $r_e$07 if the most uncertain high-mass point is excluded (Roy et al., 2018).

Cluster samples show analogous zero-point evolution with little or no strong slope evolution. In SPT-CL J0546$r_e$085345 at $r_e$09, the rest-frame near-IR relation

$r_e$10

has $r_e$11 and is offset from the local relation by $r_e$12 dex in size, corresponding to $r_e$13. In IllustrisTNG100 cluster ETGs, the best-fit relation at $r_e$14 has $r_e$15, $r_e$16, and $r_e$17, while at $r_e$18 it has $r_e$19, $r_e$20, and $r_e$21, reproducing the observed shift toward smaller sizes at higher redshift (Sweet et al., 2016, Matteuzzi et al., 2022).

4. Environmental dependence, progenitor bias, and the status of the controversy

Environmental effects on the SMR remain one of the most disputed aspects of the subject. Several studies report direct size differences. In STAGES, elliptical, lenticular, and high-mass spiral galaxies show no environmental dependence, but intermediate/low-mass spirals with $r_e$22 are about $r_e$23–$r_e$24 larger in the field than in the cluster, attributed to a population of large-$r_e$25 field spirals with extended stellar disks absent from cluster environments (0910.5665).

A local SDSS-based analysis of $r_e$26 galaxies similarly finds that galaxies tend to be larger in the field than in high-density regions. The effect is more pronounced for late-type morphologies, about $r_e$27, and weaker but still present for early types, about $r_e$28. The same work reports that the scatter is larger in low-density regions than in high-density regions, by about $r_e$29 for early-type and $r_e$30 for late-type galaxies on average, with late-type galaxies below $r_e$31 showing the largest scatter differences (Cebrián et al., 2014).

Isolation-selected samples strengthen the late-type signal. In the AMIGA study, there is no difference in the stellar mass–size relation for very isolated and less isolated early-type galaxies, but late-type isolated galaxies are $r_e$32 times larger than less isolated objects with similar mass. The size excess is strongest for massive spirals in the bins $r_e$33, and becomes larger when fully isolated galaxies are compared with galaxies having two or more satellites within $r_e$34 kpc and within $r_e$35 magnitudes (Lorenzo et al., 2013).

More recent large-sample analyses, however, argue that these apparent environmental trends are mostly population-mixing effects. In DESI Legacy Survey data, the full-sample SMR varies with cluster-centric distance, but once galaxies are separated by color and Sérsic index, the environmental offsets largely disappear. The systematic ordering at fixed mass is robust—blue disks are largest, then blue ellipticals, then red disks, and red ellipticals are smallest—and the claim is that the overall size difference with environment is explained by a changing mixture of subpopulations rather than direct size transformation (Liao et al., 20 Nov 2025).

An even more explicit null result appears in the SDSS-DR13 cluster–void comparison. Across cluster versus void, across cluster-mass bins, and across cluster-centric radii, the fitted slopes, normalizations, and intrinsic scatters are consistent within uncertainties once stellar mass and galaxy type are fixed. The paper therefore finds no measurable environmental effect on the SMR at fixed stellar mass and galaxy type, even across the extreme contrast between dense clusters and cosmic voids (Abdullah et al., 6 Jul 2025).

Cluster evolution introduces a distinct but related mechanism: progenitor bias. In IllustrisTNG100, the growth of the cluster ETG SMR from $r_e$36 to $r_e$37 is driven mainly by the acquisition of larger galaxies (“newcomers”) and the transformation of member galaxies at large clustercentric distance (“suburbanites”), whereas the long-lived survivors show almost no evolution in their own $r_e$38–$r_e$39 relation. This suggests that some observed environmental evolution of the SMR can arise from changes in sample membership rather than strong structural transformation of the same galaxies (Matteuzzi et al., 2022).

5. Physical interpretations and theoretical frameworks

The SMR is commonly interpreted as a structural record of the baryon cycle, merger history, and halo assembly. One observationally motivated picture is that the pivot mass marks a change in growth mode. In the HSC analysis, the pivot stellar mass coincides with the mass at which half of the galaxy population is quiescent, implying that the pivot mass represents the transition of galaxy growth from being dominated by in-situ star formation to being dominated by dry mergers (Kawinwanichakij et al., 2021).

Minor mergers recur as a specific explanation for late-time size growth, especially in massive systems. In the GOODS-N study, $r_e$40 increases by roughly $r_e$41–$r_e$42 from $r_e$43 to $r_e$44 for massive galaxies, whereas $r_e$45 grows more weakly; the decline of $r_e$46 toward low redshift is interpreted as evidence that minor mergers preferentially build low-surface-brightness outskirts. In the $r_e$47 cluster sample, the absence of slope evolution from $r_e$48 to $r_e$49 is taken to favor processes that increase the size without significant morphological interference, such as minor mergers and/or adiabatic expansion (Ichikawa et al., 2012, Sweet et al., 2016).

A more dynamical interpretation is that the ordinary SMR is a projection of a deeper relation involving binding energy. In a heterogeneous sample of $r_e$50 galaxies spanning nine orders of magnitude in stellar mass, the tightest universal relation is between $r_e$51 and $r_e$52, where $r_e$53 represents the fourth root of the total binding energy within $r_e$54. The reported scatter is $r_e$55 dex in $r_e$56 and $r_e$57 dex in $r_e$58. A plausible implication is that size alone is not the most fundamental structural coordinate (Shi et al., 2021).

In halo-based models, the SMR is coupled to the stellar mass–halo mass relation and halo response. In EAGLE, matching both abundance matching and the Tully–Fisher relation requires galaxy sizes similar to observed spirals and some halo contraction. The authors emphasize that the TFR is degenerate with galaxy formation efficiency and the mass-size relation: simulations that fail to match the galaxy stellar mass function may fit the observed TFR if galaxies follow a different mass-size relation (Ferrero et al., 2016).

Analytic work on dynamically hot galaxies offers a more explicit halo-size homology. In the two-phase model of halo assembly, the bulge size and the fast-assembly halo size obey

$r_e$59

approximately independent of redshift and halo mass. In this framework, the observed size–mass relation of dynamically hot galaxies is the combination of galaxy–halo homology and the non-linear stellar mass–halo mass relation, and this reproduces the low-mass flattening and the high-mass upturn (Chen et al., 2023).

Other approaches place dark matter even more centrally in the interpretation. Clausius’ virial maximum theory treats the ETG mass–size relation as a consequence of baryon–dark matter virial energy sharing, with the scaling

$r_e$60

where $r_e$61. The theory is presented as a route to connect the SMR, the Fundamental Plane, the cusp/core problem, and a possible lower boundary $r_e$62 in the $r_e$63 plane (Bindoni et al., 2011).

6. Systematics, caveats, and current synthesis

Any encyclopedia-level account of the SMR must treat methodology as part of the phenomenon. Size estimates depend on rest-frame bandpass, PSF treatment, Sérsic versus non-parametric measurement, major-axis versus circularized convention, and light-based versus mass-based definitions. The low-mass FIRE-2 analysis is especially explicit: the intrinsic relation has slope $r_e$64, the mock-observed relation has $r_e$65, and the main changes are in normalization and scatter because color–MLR assumptions and positive radial MLR gradients bias both axes downward. If the simulations are accurate, then the true low-mass SMR may be tighter than observationally inferred (Klein et al., 2024).

Completeness and selection limits further shape inferred slopes and curvature. KiDS imposes $r_e$66 and finds structural parameters unbiased to better than $r_e$67 only in that regime; SDSS-based environment analyses often exclude $r_e$68 from fitting because the low-mass regime is incomplete; DESI-LS requires angular-size and surface-brightness cuts that are important for completeness at low size and low mass. This suggests that claims about low-mass breaks or dwarf-galaxy transitions should be read together with the survey selection function (Roy et al., 2018, Abdullah et al., 6 Jul 2025, Liao et al., 20 Nov 2025).

A cautious synthesis of the literature is therefore possible. First, the SMR is unquestionably real across a broad dynamic range in mass and redshift. Second, a single power law is usually adequate for star-forming galaxies, with slope near $r_e$69 in several datasets, whereas quiescent or dynamically hot populations show steeper high-mass branches and frequent low-mass flattening (Nedkova et al., 2021, Kawinwanichakij et al., 2021). Third, fixed-mass sizes generally decrease with redshift, but the strength of evolution depends strongly on population and mass, with high-mass spheroids and quiescent galaxies evolving more strongly than low-mass systems (Mosleh et al., 2012, Roy et al., 2018). Fourth, environmental dependence is not settled at the level of the full population, but the most recent large-sample results argue that once color, morphology, or star-formation class is fixed, the residual dependence is small or absent (Liao et al., 20 Nov 2025, Abdullah et al., 6 Jul 2025).

The remaining ambiguities are methodological as much as physical. This suggests that future progress will depend less on fitting ever larger samples with a single functional form and more on cross-calibrating size definitions, rest-frame wavelengths, MLR prescriptions, and subpopulation labels across surveys and simulations.