Size-Luminosity Relation in Astrophysics

• Size-Luminosity Relation is an empirical scaling that connects an astronomical system's characteristic size to its luminosity using power-law and context-dependent definitions.
• It applies across various systems—including galaxies, star-forming clumps, AGNs, and protoplanetary disks—with slopes influenced by geometry, radiative processes, and selection effects.
• Inference strategies employ Bayesian analysis, completeness corrections, and model-dependent fits, underscoring the impact of methodology on deriving physical insights.

In astronomy, the size-luminosity relation denotes a family of empirical scalings that connect a characteristic size of an emitting or stellar component to a luminosity defined for that same component. It is commonly written as a power law, $R \propto L^\alpha$, or in logarithmic form, $\log y = K + \beta \log x$, but in some nearby-galaxy applications it is explicitly curved in absolute magnitude rather than a pure power law. The relevant “size” can be a circularized half-light radius, an H$\alpha$ radius, a reverberation lag radius, a luminosity-weighted narrow-line radius, an interferometric torus size, or a millimeter effective radius, so the relation is not a single universal law but a recurring structural diagnostic whose slope and scatter depend on geometry, radiative transfer, angular momentum, and selection effects (Yang et al., 2022, Cho et al., 2023, Husemann et al., 2014, Zormpas et al., 2022).

1. Formal definitions and recurring parameterizations

The size variable is context dependent. In high-redshift galaxy work it is usually the circularized effective radius,

$r_e = r_e^{\rm maj}\sqrt{1-e},$

measured from a Sérsic model in the source plane (Yang et al., 2022). In local and distant star-forming clumps it is the radius of the H$\alpha$-emitting region, $r_{H\alpha}$ (Cosens et al., 2018). In AGN reverberation mapping it is a light-travel size, $R_{\rm BLR}=c\tau$, inferred from the lag between a continuum light curve and a line or dust response (Cho et al., 2023). In ENLR studies it can be a luminosity-weighted [O III] radius,

$$r_{\mathrm{e}} \equiv \frac{\sum_i F_i r_i}{\sum_i F_i},$$

constructed after masking H II-like emission (Husemann et al., 2014). In millimeter disk studies it is the radius enclosing 68% of the continuum flux, $r_{\rm eff}$ (Zormpas et al., 2022).

The luminosity variable is equally heterogeneous. It may be a rest-frame UV luminosity or absolute magnitude $M_{\rm UV}$, H$\log y = K + \beta \log x$0 luminosity, optical continuum luminosity at 5100 Å, [O III] luminosity, mid-infrared monochromatic luminosity, or integrated millimeter continuum flux rescaled to a standard distance. Accordingly, the same phrase, “size-luminosity relation,” refers to distinct but mathematically analogous relations.

Three parameterizations recur in the literature. The first is the generic log-linear regression

$\log y = K + \beta \log x$1

used, for example, in H$\log y = K + \beta \log x$2 BLR work (Cho et al., 2023). The second is the effective-radius power law

$\log y = K + \beta \log x$3

widely adopted for lensed high-redshift galaxies (Yang et al., 2022). The third is a curved relation in magnitude, typically quadratic in $\log y = K + \beta \log x$4, used for nearby galaxy populations when a single power law is inadequate (Bernardi et al., 2012).

Context	Size definition	Representative scaling
Star-forming clumps	$\log y = K + \beta \log x$5	$\log y = K + \beta \log x$6 or $\log y = K + \beta \log x$7, depending on $\log y = K + \beta \log x$8 (Cosens et al., 2018)
Lensed galaxies at $\log y = K + \beta \log x$9–9	circularized $\alpha$0	$\alpha$1, with $\alpha$2 at $\alpha$3–7 and $\alpha$4 at $\alpha$5 for the Bradač model (Yang et al., 2022)
H$\alpha$6 BLR	$\alpha$7	$\alpha$8 and $\alpha$9 (Cho et al., 2023)
ENLR	luminosity-weighted [O III] radius	$r_e = r_e^{\rm maj}\sqrt{1-e},$0 (Husemann et al., 2014)
AGN dust torus	MIR interferometric size	$r_e = r_e^{\rm maj}\sqrt{1-e},$1 (Tristram et al., 2011)
Protoplanetary disks	68% continuum radius $r_e = r_e^{\rm maj}\sqrt{1-e},$2	smooth disks: $r_e = r_e^{\rm maj}\sqrt{1-e},$3; strong substructure: $r_e = r_e^{\rm maj}\sqrt{1-e},$4 (Zormpas et al., 2022)

2. Star-forming regions and galaxies

In star-forming clumps, the relation is explicitly multiregime rather than universal. A Bayesian analysis of local and high-redshift clumps spanning $r_e = r_e^{\rm maj}\sqrt{1-e},$5 and $r_e = r_e^{\rm maj}\sqrt{1-e},$6 found no evidence for redshift evolution of the H$r_e = r_e^{\rm maj}\sqrt{1-e},$7 size-luminosity relation and no difference in slope between lensed and unlensed samples, but it did find a break with star-formation-rate surface density: low-$r_e = r_e^{\rm maj}\sqrt{1-e},$8 clumps follow $r_e = r_e^{\rm maj}\sqrt{1-e},$9, whereas high-$\alpha$0 clumps follow $\alpha$1 (Cosens et al., 2018). The accompanying interpretation is geometric: a Strömgren-like ionized region gives $\alpha$2 when the radius is smaller than the disk scale height and $\alpha$3 when it exceeds the scale height.

For galaxies during the epoch of reionization, the relation has been measured most aggressively in lensed samples. In the Hubble Frontier Fields, source-plane Sérsic modeling and completeness-corrected Bayesian inference yielded $\alpha$4 with $\alpha$5 at $\alpha$6–7 and $\alpha$7 at $\alpha$8 for the Bradač lens model, while tests across five independent lens models showed that the slopes are consistent within the statistical errors (Yang et al., 2022). A related simultaneous fit of the UV luminosity function and size-luminosity relation in the complete Frontier Fields found $\alpha$9 at $r_{H\alpha}$0–7 and a positively covariant faint-end luminosity-function slope $r_{H\alpha}$1, showing that faint-end LF inference is inseparable from the assumed or fitted size distribution (Kawamata et al., 2017).

The steep faint-end behavior contrasts with earlier blank-field extrapolations. At $r_{H\alpha}$2, CANDELS plus HUDF data already indicated a well-defined relation close to $r_{H\alpha}$3, with galaxies fainter than $r_{H\alpha}$4 having size distributions peaked below $r_{H\alpha}$5, or about $r_{H\alpha}$6 kpc proper (Grazian et al., 2012). JWST later added the first rest-frame optical relation at $r_{H\alpha}$7: in GLASS-JWST-ERS, the median size at $r_{H\alpha}$8 decreases only modestly from rest-frame optical, $r_{H\alpha}$9 pc, to far-UV, $R_{\rm BLR}=c\tau$0 pc, while free-slope fits are consistent with $R_{\rm BLR}=c\tau$1 in most bands but reach $R_{\rm BLR}=c\tau$2 in F150W, where the smallest and faintest galaxies drive a steeper UV trend (Yang et al., 2022). This suggests that the relation can be wavelength dependent, especially once compact UV clumps are resolved.

In the nearby universe, the relation is strongly morphology dependent and often curved. A large SDSS analysis using Bayesian morphological weights found that early types and late types each define distinct non-power-law relations, with S0 galaxies about 15% smaller than ellipticals at fixed luminosity and faint Sbs more than 25% smaller than Scds (Bernardi et al., 2012). Bulge-disk decompositions in EFIGI extended this picture: ellipticals and dwarf ellipticals have steeper and shallower linear size-luminosity relations, respectively, while lenticular and spiral bulges occupy a curved continuation; disks and irregulars also follow a continuous curved relation, such that they first brighten while growing and then approach nearly constant surface brightness (Quilley et al., 2023). A further complication comes from ultra-diffuse galaxies: in a size-limited Coma sample, large red galaxies showed a broad, nearly flat luminosity distribution rather than a peak at the canonical magnitude implied by classical size-luminosity fits, and large faint galaxies outnumbered large galaxies near the canonical relation by a factor of about nine in the $R_{\rm BLR}=c\tau$3 kpc bin (Danieli et al., 2018). A common misconception is therefore that the inverse statement of the classical relation holds; luminous galaxies are often large, but large galaxies are not necessarily luminous.

3. Active galactic nuclei: BLR, ENLR, iron region, and torus

In AGN reverberation work, the size-luminosity relation is central because a lag-derived size can be promoted into a single-epoch mass estimator. For H$R_{\rm BLR}=c\tau$4 broad-line reverberation, the Seoul National University AGN Monitoring Project combined six-year monitoring of six luminous AGNs with literature data to calibrate the H$R_{\rm BLR}=c\tau$5 BLR against both broad H$R_{\rm BLR}=c\tau$6 luminosity and 5100 Å continuum luminosity. The resulting relations are

$R_{\rm BLR}=c\tau$7

and

$R_{\rm BLR}=c\tau$8

with intrinsic scatters of $R_{\rm BLR}=c\tau$9 and $$r_{\mathrm{e}} \equiv \frac{\sum_i F_i r_i}{\sum_i F_i},$$0 dex, respectively (Cho et al., 2023). The same study found $$r_{\mathrm{e}} \equiv \frac{\sum_i F_i r_i}{\sum_i F_i},$$1, confirming stratification between Balmer-line emitting zones.

On kiloparsec scales, the ENLR relation uses a different size metric and a different physical proxy for luminosity. Integral-field spectroscopy of nearby type-1 QSOs measured an effective [O III] radius after QSO-host deblending and BPT masking, then found

$$r_{\mathrm{e}} \equiv \frac{\sum_i F_i r_i}{\sum_i F_i},$$2

with a tighter correlation to continuum luminosity than to integrated [O III] luminosity (Husemann et al., 2014). The slope near $$r_{\mathrm{e}} \equiv \frac{\sum_i F_i r_i}{\sum_i F_i},$$3 is interpreted as consistent with an approximately constant ionization parameter, directly analogous to the square-root intuition often invoked for BLR relations.

The same reverberation logic has been extended to less standard line-emitting regions. A tentative size-luminosity relation for the optical iron-emission region in quasars,

$$r_{\mathrm{e}} \equiv \frac{\sum_i F_i r_i}{\sum_i F_i},$$4

implies Fe-emitting radii comparable to, and at most about twice, the H$$r_{\mathrm{e}} \equiv \frac{\sum_i F_i r_i}{\sum_i F_i},$$5 BLR size (Chelouche et al., 2014). This places the optical Fe blends within the broader BLR scaling framework rather than in a wholly distinct zone.

For the dusty torus, the literature contains both square-root and sub-square-root results. Mid-infrared interferometry at 12 $$r_{\mathrm{e}} \equiv \frac{\sum_i F_i r_i}{\sum_i F_i},$$6m found a characteristic torus size consistent with

$$r_{\mathrm{e}} \equiv \frac{\sum_i F_i r_i}{\sum_i F_i},$$7

and a corresponding radius-luminosity scaling $$r_{\mathrm{e}} \equiv \frac{\sum_i F_i r_i}{\sum_i F_i},$$8, with the mid-IR emitting radius about 30 times the near-IR inner-rim radius (Tristram et al., 2011). More recent uniform WISE reverberation analysis, however, found flatter AD-corrected slopes,

$$r_{\mathrm{e}} \equiv \frac{\sum_i F_i r_i}{\sum_i F_i},$$9

and a stratified wavelength dependence $r_{\rm eff}$0 at $r_{\rm eff}$1, equivalent to $r_{\rm eff}$2 (Mandal et al., 2024). An independent structure-function approach based on ZTF optical variability and WISE mid-IR variability recovered

$r_{\rm eff}$3

showing that ensemble variability statistics can reproduce a torus size-luminosity relation consistent with reverberation mapping (Kim et al., 2024). The current AGN-torus literature therefore does not support a single immutable exponent; rather, it supports wavelength stratification and possible accretion-state dependence.

4. Protoplanetary disks

In protoplanetary disks the size-luminosity relation is defined with respect to continuum emission rather than starlight or line emission. The characteristic size is the 68% effective radius,

$r_{\rm eff}$4

and the luminosity is the submillimeter continuum luminosity $r_{\rm eff}$5, operationally the integrated flux rescaled to 140 pc (Zormpas et al., 2022). The observed empirical relation summarized by Andrews et al. and used as the benchmark in the modeling is approximately $r_{\rm eff}$6.

A population study based on $r_{\rm eff}$7 dust-evolution simulations showed that smooth disks in the radial-drift regime are compatible with this $r_{\rm eff}$8 relation, confirming earlier drift-based interpretations (Zormpas et al., 2022). The same model grid demonstrated that strong substructure changes the slope: planet-induced pressure traps yield

$r_{\rm eff}$9

because optically thick rings produce a different scaling of emitting area and temperature with radius. The study therefore argued that the observed millimeter diagram is plausibly populated by a mixture of smooth and substructured disks rather than a single structural class.

The inferred constraints are correspondingly physical rather than purely geometric. To remain compatible with the observed relation, models require initially high disk masses, $M_{\rm UV}$0, and low turbulence parameters, $M_{\rm UV}$1 (Zormpas et al., 2022). Grain composition and porosity strongly affect where disks lie in the size-luminosity plane, with relatively compact grains including amorphous carbon favored, and a uniformly optically thick disk with high albedo cannot be produced by the evolutionary procedure explored in that work. In this domain, the relation is therefore a probe of dust drift, trapping, opacity, and substructure rather than of a direct equilibrium radius.

5. Inference strategies, completeness corrections, and model dependence

A persistent theme across the literature is that the size-luminosity relation is inseparable from the measurement protocol. In star-forming clumps, the H$M_{\rm UV}$2 scaling was fit with Bayesian inference in Stan using MCMC sampling, explicitly comparing subsamples in redshift, lensing state, and selection (Cosens et al., 2018). In H$M_{\rm UV}$3 BLR work, the lag itself is not unique until one specifies ICCF, zDCF, or JAVELIN/DRW modeling, the centroid threshold, and the FR/RSS uncertainty prescription; the final regression then uses LINMIX_ERR to account for errors in both axes and intrinsic scatter (Cho et al., 2023). ENLR analyses add another layer: censored data from upper limits require ASURV and Buckley-James regression, and the size definition is chosen to avoid the depth dependence of a simple maximal radius (Husemann et al., 2014).

High-redshift galaxy work makes completeness modeling especially consequential. In the Hubble Frontier Fields, sizes were measured in the source plane with Lenstruction built on Lenstronomy, while two-dimensional completeness maps $M_{\rm UV}$4 were generated with GLACiAR2 and folded into a forward Bayesian model of the intrinsic joint distribution (Yang et al., 2022). The simultaneous HFF fit of the UV luminosity function and the size-luminosity relation went further by using glafic lens models, a lognormal size distribution, and a Schechter luminosity function in one likelihood; the resulting positive covariance between $M_{\rm UV}$5 and the LF faint-end slope $M_{\rm UV}$6 is an explicit warning that adopting a fixed size law can bias the inferred abundance of faint galaxies (Kawamata et al., 2017). The steep faint-end slopes inferred in lensed work are thus methodological results as much as observational ones.

Nearby galaxy work shows a different but equally important systematic: structural model mismatch. In SDSS, fitting a single Sérsic profile to galaxies better represented by two-component SerExp models overestimates half-light radii, with the bias increasing for higher fitted Sérsic index and peaking near $M_{\rm UV}$7 (Bernardi et al., 2012). The global size-luminosity relation changes only modestly over most of the population, but the luminous tail curves upward more strongly under single-Sérsic fitting. Morphological definition matters as well: Bayesian-classifier weights and hard morphological cuts do not yield identical early- and late-type relations. A recurring misconception is therefore that a measured slope is an intrinsic physical constant; in practice, it can depend materially on size definition, tracer choice, fit family, completeness correction, and whether luminosity and size are solved simultaneously.

6. Physical interpretation, non-universality, and open questions

The literature supports several distinct physical mechanisms rather than a universal exponent. Slopes near one half emerge naturally in photoionized or radiatively heated systems: the ENLR relation is interpreted in terms of an approximately constant ionization parameter (Husemann et al., 2014), and early torus interferometry interpreted $M_{\rm UV}$8 as the signature of dust in radiative equilibrium at roughly fixed temperature (Tristram et al., 2011). In reverberation-mapped line regions, H$M_{\rm UV}$9 BLR slopes of $\log y = K + \beta \log x$00–$\log y = K + \beta \log x$01 are consistent with the canonical H$\log y = K + \beta \log x$02 relation within the quoted uncertainties (Cho et al., 2023).

Other regimes are explicitly geometry dependent. In star-forming clumps, the transition between $\log y = K + \beta \log x$03 and $\log y = K + \beta \log x$04 was proposed to depend on whether the ionized region is smaller or larger than the disk scale height, which is why the measured slopes bifurcate with $\log y = K + \beta \log x$05 rather than with redshift (Cosens et al., 2018). In protoplanetary disks, $\log y = K + \beta \log x$06 corresponds to smooth drift-dominated evolution, whereas $\log y = K + \beta \log x$07 corresponds to optically thick rings in strongly substructured disks (Zormpas et al., 2022).

Galaxy relations are further shaped by angular momentum and morphology. The simultaneous HFF analysis argued that a steep high-redshift slope can be reproduced by a simple analytical model in which smaller galaxies have lower specific angular momenta (Kawamata et al., 2017). Local SDSS and EFIGI studies, by contrast, emphasize curvature, morphological bifurcation, and special stellar-mass scales near $\log y = K + \beta \log x$08 and $\log y = K + \beta \log x$09, so that even the nearby galaxy relation is better described as a set of connected sequences than as a single law (Bernardi et al., 2012). The Coma ultra-diffuse-galaxy result adds a further caution: at fixed large size, luminosity can vary enormously, so size alone is not a reliable indicator of halo mass in that regime (Danieli et al., 2018).

The AGN torus currently illustrates the main open issue. Interferometric and classical dust-reverberation work are broadly consistent with square-root scaling, but the uniform WISE reverberation analysis found flatter W1 and W2 slopes and a moderate correlation between residual torus size and Eddington ratio, suggesting that accretion state may flatten the global relation (Mandal et al., 2024). A plausible implication is that anisotropic illumination or slim-disk self-shadowing modifies the simple equilibrium picture at high accretion rate, although the paper itself presents this as a suggested influence rather than a closed explanation. More generally, the modern literature treats the size-luminosity relation as an empirical interface between structure and selection: its numerical form is often informative, but only once the emitting component, wavelength, geometry, and inference scheme are specified precisely.