# Stellar Half-Mass Radii of $0.5<z<2.3$ Galaxies: Comparison with JWST/NIRCam
Half-Light Radii

Arjen van der Wel Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan
281 S9, 9000 Gent, Belgium  Marco Martorano Sterrenkundig Observatorium,
Universiteit Gent, Krijgslaan 281 S9, 9000 Gent, Belgium  Boris Häußler
European Southern Observatory, Alonso de Cordova 3107, Casilla 19001,
Santiago, Chile  Kalina V. Nedkova Department of Physics and Astronomy, Johns
Hopkins University, Baltimore, MD 21218, USA  Tim B. Miller Department of
Astronomy, Yale University, 52 Hillhouse Avenue, New Haven, CT 06511, USA
Gabriel B. Brammer Cosmic Dawn Center (DAWN)  Niels Bohr Institute, University
of Copenhagen, Lyngbyvej 2, 2100 Copenhagen, Denmark  Glenn van de Ven
Department of Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180
Vienna, Austria  Joel Leja Department of Astronomy & Astrophysics, The
Pennsylvania State University, University Park, PA 16802, USA  Institute for
Computational & Data Sciences, The Pennsylvania State University, University
Park, PA 16802, USA  Institute for Gravitation and the Cosmos, The
Pennsylvania State University, University Park, PA 16802, USA  Rachel S.
Bezanson Department of Physics and Astronomy and PITT PACC, University of
Pittsburgh, Pittsburgh, PA 15260, USA  Adam Muzzin Department of Physics and
Astronomy, York University, 4700 Keele Street, Toronto, Ontario, ON MJ3 1P3,
Canada  Danilo Marchesini Department of Physics and Astronomy, Tufts
University, Medford, MA  Anna de Graaff Max-Planck-Institut für Astronomie,
Königstuhl 17, D-69117, Heidelberg, Germany  Mariska Kriek Leiden Observatory,
Leiden University, P.O. Box 9513, 2300 RA, Leiden, The Netherlands  Eric F.
Bell Department of Astronomy, University of Michigan, 1085 South University
Avenue, Ann Arbor, MI 48109-1107, USA  Marijn Franx Leiden Observatory, Leiden
University, P.O. Box 9513, 2300 RA, Leiden, The Netherlands Arjen van der Wel
<EMAIL_ADDRESS>
(Received January 1, 2999; Revised January 2, 2999; Accepted January3, 2999)

###### Abstract

We use CEERS JWST/NIRCam imaging to measure rest-frame near-IR light profiles
of $>$500 $M_{\star}>10^{10}~{}M_{\odot}$ galaxies in the redshift range
$0.5<z<2.3$. We compare the resulting rest-frame 1.5-2$\mu$m half-light radii
($R_{\rm{NIR}}$) with stellar half-mass radii ($R_{\rm{M_{\star}}}$) derived
with multi-color light profiles from CANDELS HST imaging. In general agreement
with previous work, we find that $R_{\rm{NIR}}$ and $R_{\rm{M_{\star}}}$ are
up to 40% smaller than the rest-frame optical half-light radius
$R_{\rm{opt}}$. The agreement between $R_{\rm{NIR}}$ and $R_{\rm{M_{\star}}}$
is excellent, with negligible systematic offset ($<$0.03 dex) up to $z=2$ for
quiescent galaxies and up to $z=1.5$ for star-forming galaxies. We also
deproject the profiles to estimate $R_{\rm{M_{\star},3D}}$, the radius of a
sphere containing 50% of the stellar mass. We present the $R-M_{\star}$
distribution of galaxies at $0.5<z<1.5$, comparing $R_{\rm{opt}}$,
$R_{\rm{M_{\star}}}$ and $R_{\rm{M_{\star},3D}}$. The slope is significantly
flatter for $R_{\rm{M_{\star}}}$ and $R_{\rm{M_{\star},3D}}$ compared to
$R_{\rm{opt}}$, mostly due to downward shifts in size for massive star-forming
galaxies, while $R_{\rm{M_{\star}}}$ and $R_{\rm{M_{\star},3D}}$ do not show
markedly different trends. Finally, we show rapid size evolution
($R\propto(1+z)^{-1.7\pm 0.1}$) for massive ($M_{\star}>10^{11}~{}M_{\odot}$)
quiescent galaxies between $z=0.5$ and $z=2.3$, again comparing
$R_{\rm{opt}}$, $R_{\rm{M_{\star}}}$ and $R_{\rm{M_{\star},3D}}$. We conclude
that the main tenets of the size evolution narrative established over the past
20 years, based on rest-frame optical light profile analysis, still hold in
the era of JWST/NIRCam observations in the rest-frame near-IR.

galaxies: high-redshift – galaxies: kinematics and dynamics – galaxies:
structure

††journal: ApJ

##  1 Introduction

Projected light profiles of galaxies are widely used as proxies for their
3-dimensional stellar mass profiles, both at low and high redshifts. Under
this assumption, great progress has been made in our understanding of the
structure of galaxies and their assembly history. At the same time, we have
known for decades that galaxies show color gradients (Sandage, 1972; Peletier
et al., 1990; de Jong, 1996), implying that, given the correlation between
color and stellar mass-to-light ratio ($M_{\star}/L$ Bell & de Jong, 2001),
$M_{\star}/L$ varies with radius, and generally peaks in the center for
massive galaxies. The gradients arise due to a combination of radial
variations in attenuation and stellar population properties (age, abundances,
IMF). For early-type galaxies the color gradient is generally understood to be
due to a radial variation in metallicity (e.g., Li et al., 2018), while for
star-forming galaxies the stellar population gradients, in both age and
metallicity, are significant but generally mild (e.g., Sánchez-Blázquez et
al., 2014), and centrally concentrated attenuation plays a dominant role,
especially at higher redshifts as is now being revealed by JWST (Miller et
al., 2022).

In order to interpret observations of light profiles in the context of
theoretical models or simulations, these $M_{\star}/L$ gradients must be taken
into account as their effect on the half- mass radius as inferred from
observations can be very substantial (a factor $\sim 2$), even at near-
infrared wavelengths (e.g., Dutton et al., 2011). In addition, projected light
or mass profiles can be difficult to interpret: a direct comparison with
simulations requires a deprojection in three dimensions (e.g., Prugniel &
Simien, 1997; Baes & van Hese, 2011; Cappellari et al., 2013; van de Ven & van
der Wel, 2021), or the creation of mock observations by projecting simulated
galaxies (de Graaff et al., 2022).

The interpretation of the redshift evolution of galaxy light profiles usually
relies on the assumption that the $M_{\star}/L$ gradient does not (strongly)
evolve, as most studies ignore color and $M_{\star}/L$ gradients. Several
authors used observed color gradients in higher-redshift galaxies, first
reported by Hinkley & Im (2001) and McGrath et al. (2008), to address the
impact of $M_{\star}/L$ gradients on galaxy size estimates (McGrath et al.,
2008; Guo et al., 2011; Wuyts et al., 2012; Szomoru et al., 2013; Mosleh et
al., 2017; Suess et al., 2019a, b; Mosleh et al., 2020; Miller et al., 2023).
Generally, the results point to the existence of qualitatively similar color
gradients at low and high redshift, but even relatively small changes can
strongly affect the interpretation of the observed size evolution of galaxies
(Suess et al., 2019b, 2020; Miller et al., 2023).

Likewise, the evolution of galaxy geometry (the intrinsic, three-dimensional
shape) is often overlooked. Chang et al. (2013); van der Wel et al. (2014a);
Zhang et al. (2019) show that geometry strongly evolves with redshift, which
implies that the interpretation of projected light profiles must change with
redshift, even if its impact has not yet been analyzed.

With the arrival of JWST we can access for the first time the rest-frame near-
IR light profiles of intermediate redshift (here, $0.5<z<2.3$) galaxies that
should provide a more direct proxy of the stellar mass profile since
attenuation becomes negligible in most cases and variations in $M_{\star}/L$
as a function of age and metallicity are less strong. In this paper (Sec. 2)
we use the first batch of NIRCam imaging from the CEERS program (Finkelstein
et al., 2023) that covers $\approx 4$% of the full CANDELS dataset to test the
robustness of stellar half-mass radii estimates based on resolved color
profiles from HST imaging (Koekemoer et al., 2011; Grogin et al., 2011). Early
work by Suess et al. (2022) already demonstrated that the rest-frame near-IR
sizes from JWST/NIRCam are somewhat smaller than the rest-frame optical sizes
as measured from HST/WFC3, supporting the previous results that stellar mass-
weighted profiles are more compact than light-weighted profiles. Sec. 2.6
describes the methodology to convert projected sizes into 3D sizes. In Sec. 3
we present size-mass distributions at for the different size proxies (rest-
frame optical, mass-weighted, deprojected) and the average size evolution for
massive quiescent galaxies. In Sec. 4 we summarize the results.

We assume a flat $\Lambda$CDM cosmology with $H_{0}=70$ km s Mpc-1 and
$\Omega_{\rm{m}}=0.3$, and the Chabrier (2003) stellar initial mass function.

##  2 Data and Methodology

###  2.1 A New Approach for Estimating Stellar Half-Mass Radii

A variety of methods has been developed to convert light distributions into
stellar mass maps, which can be divided along two axes. First, some methods
create 2D mass maps (e.g., Abraham et al., 1999; Zibetti et al., 2009; Meidt
et al., 2012; Wuyts et al., 2012; Meidt et al., 2014; Tacchella et al., 2015;
Abdurro’uf & Akiyama, 2018; Suess et al., 2019a; Mosleh et al., 2020;
Abdurro’uf et al., 2023), which have the advantage of retaining all spatial
information, while other methods create symmetrized (1D) profiles (e.g.,
Szomoru et al., 2013; Fang et al., 2013; Tacchella et al., 2015; Suess et al.,
2019a; Miller et al., 2023), which have the advantage that they are more
easily corrected for the PSF and, relevant to the topic at hand, more easily
compared with standard methods to measure galaxy sizes.

Second, the spatially resolved photometric information can be converted into
$M_{\star}/L$ information by SED fitting (e.g., Suess et al., 2019a; Mosleh et
al., 2020), or by the the application of color-$M_{\star}/L$ relations devised
for integrated galaxy light (Bell & de Jong, 2001; van der Wel et al., 2005;
Meidt et al., 2014) and applied to spatially resolved light distributions
(e.g., Zibetti et al., 2009; Meidt et al., 2014; Fang et al., 2013; Szomoru et
al., 2013; Tacchella et al., 2015; Suess et al., 2019a; Miller et al., 2023).
The former has the advantage that all available information is used, but
(rest-frame) near-IR photometry is required to assign unbiased $M_{\star}/L$
values to dusty regions (e.g, Zibetti et al., 2009; Meidt et al., 2014). The
latter has the advantage that color-$M_{\star}/L$ relations can leverage the
knowledge of $M_{\star}$ obtained from broad-band SED fitting across a wide
wavelength range, including the near-IR. With any method, we should always
keep in mind the uncertainties related to choices made to assign a ‘true’
stellar mass, that is, uncertainties in the stellar population synthesis
models and the implementation of absorption and scattering by dust – a
discussion of these issues is beyond the scope of this paper.

Since, in our case, we do not have spatially resolved rest-frame near-IR
photometric information and the goal is to construct stellar half-mass radii
for comparison with light-weighted radii, we choose to analyze 1D profiles and
apply newly developed color-$M_{\star}/L$ relations. Our method consists of
the following steps. First, we use $M_{\star}$ estimates from SED fits over
the full available wavelength range (UV-to-mid-IR) as described in Sec. 2.2 to
construct a (redshift-dependent) relationship between $M_{\star}/L$ and
multiple HST colors in the observed frame (Sec. 2.3). Second, assuming that
the same relationship holds within galaxies, we convert HST light profiles
(Sec. 2.4.1) into stellar mass profiles via the multi-color-$M_{\star}/L$
relation (Sec. 2.4.2).

The advantages of this method are multiple. Long-wavelength information is
leveraged (via the SED-based $M_{\star}$ estimates) in a redshift-dependent
manner; that is, any evolution in the relationship between color and
$M_{\star}/L$, which is significant (Li & Leja, 2022), is automatically
included. Furthermore, no conversion from observed to rest-frame colors is
required, which removes template-related uncertainties. Finally, rather than a
single color we use the shape of the SED for which spatially resolved
information is available to estimate $M_{\star}/L$. At each step we take care
to formally propagate the uncertainties, resulting in robust uncertainties on
the inferred stellar half-mass radii.

###  2.2 Multiwavelength Photometry and SED Fitting

The 3D-HST/CANDELS photometric catalog provided by Skelton et al. (2014) was
used by Leja et al. (2020) to estimate stellar masses, star-formation rates
and other physical parameters with the Prospector-$\alpha$ model (Leja et al.,
2017; Johnson et al., 2021). The model uses the Conroy et al. (2009) stellar
population FSPS, a non-parametric star-formation history, a two-component dust
model, and optionally indcludes an enshrouded AGN.

The Leja et al. catalog serves as the basis of our work and contains 63413
galaxies in the redshift range $0.5<z<3.0$. This is a stellar mass-complete
sample, where the completeness limit increases from
$\log(M_{\star}/M_{\odot})\approx 8.7$ at $z=0.5$ to
$\log(M_{\star}/M_{\odot})\approx 10.1$ at $z=3.0$. We define galaxies as
quiescent when their 100 Myr-averaged star-formation rates are 0.8 dex or more
below the star-forming sequence as defined by Leja et al. (2022).

###  2.3 Derivation of Color-$M/L$ Relations

Our novel approach to derive mass profiles gradients rests on the assumption
that the color-$M/L$ relation within individual galaxies is identical to the
relation among galaxies. We create a (multi-)color-$M/L$ relation based on the
full SED fitting results described above, for a set of colors for which we
have spatially resolved information from HST. As such we leverage photometry
with a much broader dynamic range in wavelength (UV to mid-IR) to infer $M/L$
estimates based on a more limited range for which spatially resolved profiles
are available (0.6 - 1.6 micron in the observed frame).

In order to capture the effects of cosmological redshift and evolution of
stellar populations simultaneously, we fit a relation of the following form:

$\begin{split}\log{(M_{\star}/F_{160})}=a_{0}+a_{1}\log{(1+z)}+a_{2}\log{(F_{606}/F_{160})}\\\
+a_{3}\log{(F_{814}/F_{160})}+a_{4}\log{(F_{125}/F_{160})}\end{split}$ (1)

where $M_{\star}$ is the stellar mass estimate from the full SED fit, $z$ is
the redshift, and the F values are the total flux densities from the Skelton
catalog in units of AB$=25$ magnitude in the respective HST filters (F606W,
F814W, F125W and F160W). The fit minimizes $\chi^{2}$, which is dominated by
the uncertainties in $M_{\star}$ rather than the photometric data.

Table 1:  Coefficients for Eq. 1 | $a_{0}$ | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$
---|---|---|---|---|---
SF | 7.652 | 2.879 | 0.130 | 0.575 | 0.562
Q | 7.857 | 2.959 | 0.168 | 0.204 | 0.536

We fit two separate relations for quiescent and star-forming galaxies with
stellar masses $M_{\star}>10^{10}~{}M_{\odot}$ and over the redshift range
$0.5<z<2.3$, beyond which only one data point redward of the Balmer/4000$\AA$
remains and the uncertainties in $M_{\star}/L$ estimates increase markedly. We
fit these relations to all galaxies with $M_{\star}>10^{10}~{}M_{\odot}$ and
measured flux densities in the four HST filters and give the coefficients in
Table 1. The resulting $\log{(M_{\star}/F_{160})}$ proxy is shown in Figure 1.
The overall scatter is 0.12 dex, increasing from 0.07 at $z<1$ to 0.15 at
$z\sim 2$, which is less than or comparable to the typical uncertainty in SED-
based $M_{\star}$ estimates.

Figure 1:  Correlation between the ground-truth mass-to-light ratio
$\log{(M_{\star}/F_{160})}$ inferred from UV-to-infrared SED fitting) and our
proxy $\log{(M_{\star}/F_{160})}$ based on four HST/ACS$+$WFC3-filter
photometry as written in Eq. 1 and Table 1. A factor $(1+z)^{3}$, is removed
as a trivial component in the cosmological distance dependence in the
$F_{160}$ flux density.  Figure 2:  Comparison of rest-frame $V$-band stellar
mass-to-light ratio as inferred from Prospector (x-axis) and our HST color-
based proxy (y-axis). Across a 1.5 dex range in $M/L$ the HST color-based
$M/L$ estimates agree well with the ground truth as inferred from fits to the
full SED. The color-coding is the star-formation rate relative to the star-
forming sequence defined by Leja et al. (2022).  Figure 3:  Color-$M/L$
relations for two narrow redshift bins, with in black the ground truth as
inferred from full SED fits with Prospector and in red the HST color-based
stellar $M/L$. The single relation in Eq. 1 produces different color-$M/L$
relations at different redshifts.

The functioning of the method is further illustrated in Figures 2 and 3.
Figure 2 shows a intuitively clearer version of Figure 1, displaying
$M_{\star}/L$ values in the rest-frame $V$ band. The tightness of the
correlation across a large dynamic range demonstrates the general precision
and accuracy of our method. Underlying this result is an observed-frame
color-$M/L$ relation that continuously changes with redshift (see Eq. 1).

In Figure 3 we show for two redshift bins the relationship between
$I_{\rm{814}}-H_{\rm{160}}$ and $M/L$. The $I_{814}-H_{160}$ is one of three
colors used to derive the color-based $M_{\star}/L$ and the scatter is due to
additional color information from $V_{606}$ and $J_{125}$. At low redshift the
set of colors contains sufficient information to reproduce the variety in SED-
based $M/L$ values at fixed color whereas at high redshift this is no longer
the case due to a lack of information. If we fit a single relation to the
joint population of star-forming and quiescent galaxies the distribution of
points in Figure 2 becomes somewhat non-linear for quiescent galaxies, perhaps
because those are greatly outnumbered by star-forming galaxies at higher
redshifts.

Despite the apparent robustness of the method there are several caveats we
have to keep in mind. The color-coding in Figure 2 reveals a remaining
systematic effect: for star-forming galaxies there is a stratification with
star-formation activity in the sense that the HST colors overpredict
$M_{\star}/L_{V}$ for galaxies with high star-formation activity, and vice
versa. This does not translate into a stratification with attenuation; a
degeneracy of stellar population properties must exist for a given (set of)
colors. To what extent this issue affects the estimates of the stellar-half
mass radius will be discussed where relevant.

Additional conceptual caveats are the following. First, the Prospector stellar
mass estimates serve as ground truth for our approach, but this ground truth
itself is uncertain. We test for the sensitivity to this particular choice by
comparing with the original 3D-HST stellar mass estimates presented by Skelton
et al. (2014). Even though the fitted parameters and resulting color-$M/L$
relations change, the results after applying them to the observed color
gradients as explained below do not differ significantly. Second, the work by
Zibetti et al. (2009) demonstrated that average $M/L$ estimates from
integrated photometry can be biased due to the outshining effect of young,
unobscured regions. This bias propagates into our color-$M/L$ relations and,
more importantly, the impact on interpreting spatially resolved color
information may differ.

###  2.4 Converting Light Profiles to Stellar Mass Profiles

####  2.4.1 Light Profile Fits and Color Gradients

Nedkova et al. (2021) describe the Sérsic profile fits performed with galfitM
(Häußler et al., 2013) on CANDELS imaging. The fits are performed on all
available HST images in different filters simultaneously, fitting some
parameters (axis ratio, position angle, position) as a constant while allowing
others to vary quadratically as a function of wavelength (magnitude, effective
radius, Sérsic index). Uncertainties on the parameters are usually
underestimated and we increase the uncertainties as prescribed by van der Wel
et al. (2012), who compared independent parameter estimates for the same
objects and derived the ‘true’ uncertainties as a function of $S/N$.

The quadratic functions that describe the variation of the parameters with
wavelength allow us to calculate Sérsic flux density profiles as a function of
radius $r$ at a common rest-frame wavelength of, e.g., 0.5$\mu$m
($S_{0.5}(r)$), with a half-light radius $R_{0.5}$ and a Sérsic index
$n_{0.5}$. The ratios of Sérsic profiles at different wavelengths produce
color profiles and can be used to define a color gradient between $0.5R_{0.5}$
and $2R_{0.5}$:

$\Delta
C=\frac{S_{0.6}(2R_{0.5})~{}/~{}S_{0.4}(2R_{0.5})}{S_{0.6}(0.5R_{0.5})~{}/~{}S_{0.4}(0.5R_{0.5})}$
(2)

The choices to evaluate the Sérsic profiles at 0.4$\mu$m and 0.6$\mu$m, and
between 0.5 and 2 effective radii, are motivated only by pragmatic
considerations: all galaxies in the sample have this wavelength coverage and
at smaller and larger radii the profiles are more uncertain. Through sampling
the uncertainties in the profile fits we infer propagated uncertainties in the
color profiles.

We note that the color gradient $\Delta C$ is not used in our method to derive
$M_{\star}/L$ profiles and only serves to illustrate the strength of color
gradient as a function of various galaxy parameters: Figure 4 shows the
evolution of the color gradient and its dependence on stellar mass and star-
formation activity. Negative color gradients (redder centers; bluer
outerparts) are ubiquitous, at all $z$ and for all galaxy types. At $z\sim 1$
the measurement uncertainties are smaller than the population scatter, while
at $z\sim 2$ they are similar, implying that at lower $z$ we can distinguish
galaxies with different color gradients while at higher $z$ the observed
scatter is dominated by measurement uncertainties. At fixed stellar mass star-
forming galaxies generally have stronger gradients than quiescent galaxies.

Figure 4: Color gradient as defined in Eq. 2 vs. redshift (left-hand panel),
stellar mass (middle panel), and specific SFR (right-hand panel). The lines
are running 16%th, 50%th, and 84%th percentiles. Red and blue points and lines
correspond with quiescent and star-forming galaxies, respectively. The error
bars reflect the running median uncertainties for individual galaxies. The
uncertainties are always smaller than the scatter, implying that we see
intrinsic variations in the color gradient among galaxies.

####  2.4.2 $M/L$ Gradients and Mass Profiles

The multi-color profiles are converted to $M/L$ profiles using the color-$M/L$
relations described in Section 2.3. At each radius (within an elliptical
annulus) we have a measured value of $F_{160}$, which is multiplied by the
right-hand side of Eq. 1, with $z$ the redshift of the galaxies and where the
flux ratios are given by the ratios of the four Sérsic profiles in same
annulus. The result is a direct conversion of the F160W light profile into an
$M_{\star}$ profile. Since the inferred mass is based on a linear combination
of different, inter-dependent Sérsic profiles, it matches a Sérsic profile
itself (usually to within 0.1%); we refit a Sérsic profile to the mass profile
out to 2$\times$ the effective radius in F160W, propagating – with a Monte
Carlo simulation – the uncertainties on the individual light profile estimates
and the color-$M/L$ relation.

In Figure 5 we show how the stellar half-mass radius $R_{\rm{M_{\star}}}$
compares with the optical half-light radius $R_{0.5\mu\rm{m}}$. There is a
generally tight correlation, with a scatter that increases from $\sim 0.10$
dex at $z<1$ to up to $\sim 0.2$ dex at $z>2$. Since the scatter is generally
similar to the formal uncertainties, we conclude that uncertainties dominate
over intrinsic variations at the level of the precision that we achieve. The
exception is the set of star-forming galaxies at $z<1.5$, for which the
scatter (0.1-0.15 dex) is somewhat larger than the uncertainties (0.08-0.09
dex); this implies that we recover intrinsic variations in
$R_{\rm{M_{\star}}}$$/$$R_{0.5\mu\rm{m}}$from galaxy to galaxy (that are not
explained by uncertainties). As expected based on the color gradients,
$R_{\rm{M_{\star}}}$ is generally smaller (by 0.1-0.15 dex) than
$R_{0.5\mu\rm{m}}$, qualitatively consistent with previous work (Szomoru et
al., 2013; Suess et al., 2019a, b; Mosleh et al., 2020; Miller et al., 2023).
For quiescent galaxies we see an offset that is approximately constant with
redshift ($-0.10$ to $-0.14$ dex across the entire redshift range), while for
star-forming galaxies the difference decreases somewhat, from -0.17 dex at
$z<1$ to -0.05 dex at $z>2$. At $z<1$ the largest galaxies are the most
offset, which is due to the combination of a mild dependence on both
$M_{\star}$ and $R$ (see Sec. 3.2).

It is quite remarkable that, overall the color and $M/L$ gradients are similar
for star-forming and quiescent galaxies, as those in the former are mainly
caused by a radial variation in attenuation (Miller et al., 2022; Matharu et
al., 2023), even though, especially at lower redshifts, stellar population
gradients are also present (Bell & de Jong, 2000), while the latter generally
have little dust and the gradient must be primarily due to stellar population
variations.

The increase in scatter and downturn for galaxies with
$R_{0.5\mu\rm{m}}$$\lesssim 1$ kpc suggests that systematic errors affect the
$R_{\rm{M_{\star}}}$ estimates for the smallest galaxies. This is not
surprising, since the HST/WFC3 PSF has a FWHM of $\sim 1.4$ kpc. While light
profiles can be well constrained at smaller scales, given sufficient S/N and
accurate knowledge of the PSF, combining those from four different filters can
lead to highly non-linear compound uncertainties that are difficult to
propagate formally. Indeed, our formal uncertainties do not increase in line
with the increased scatter. In Section 2.5 we will address the precision and
accuracy of our $R_{\rm{M_{\star}}}$ estimates, including the behavior at
$R_{0.5\mu\rm{m}}$$\lesssim 1$ kpc.

Figure 5:  Stellar half-mass radius vs. rest-frame optical half-light radius
in four redshift bins for quiescent (red) and star-forming (blue) galaxies
with total stellar mass $>10^{10}~{}M_{\odot}$. $\Delta$ is the median offset
in dex, with the 16-84%-ile scatter in parentheses. $\sigma$ is the median
formal uncertainty. Offsets in the range $-0.05$ to $-0.17$ show that stellar
half-mass radii are generally smaller than the optical radius. The scatter is
comparable to the formal uncertainties, which implies that the uncertainties
are certainly not underestimated.

###  2.5 Comparison with NIRCam Effective Radii

Figure 6:  Stellar half-mass radius vs. rest-frame near-infrared half-light
radius derived from JWST/NIRCam imaging in four redshift bins for quiescent
(red) and star-forming (blue) galaxies with total stellar mass
$>10^{10}~{}M_{\odot}$. $\Delta$ is the median offset in dex, with in
parentheses the 16-84%-ile scatter. $\sigma$ is the median formal uncertainty.
The lack of offsets and the similarity between scatter and formal uncertainty
suggest that our $R_{\rm{M_{\star}}}$ estimates do not suffer from biases and
have reliable uncertainties.

435 galaxies in our $M_{\star}>10^{10}~{}M_{\odot}$ sample ($\approx 4$%) fall
within the footprint of JWST/NIRCam imaging from CEERS (Finkelstein et al.,
2023). Martorano et al. (submittted) describe the galfitM (Häußler et al.,
2013) fitting procedure to the 7 short- and long-wavelength filters images and
the estimation of rest-frame near-IR Sérsic profiles and associated half-light
radii. After fitting the individual images a Chebychev polynomal fit is used
to calculate the interpolated half-light radii at the desired rest-frame
wavelength (0.5, 1.5, or 2.0$\mu$m). Relevant for the current paper is that we
find no offset between the $R_{0.5\mu\rm{m}}$ estimates from CANDELS and CEERS
($<0.01$ dex) and small scatter ($\sim$0.05 dex).

In Figure 6 we compare the stellar half-mass radii described in Section 2.4.1
with the NIRCam-based near-infrared half-light radii (rest-frame 2.0$\mu$m for
$z<1.5$; rest-frame 1.5$\mu$m for $z>1.5$). We verified that this sub-sample
is representative of the full CANDELS sample in terms of its
$R_{0.5\mu\rm{m}}$ distribution: the statistical properties of the sub-sample
are not significantly different from those shown in Fig. 5.

Up to $z=1.5$ we see negligible offsets ($\leq 0.03$ dex), implying an absence
of significant systematic biases in our stellar half-mass radii. The typical
formal uncertainty on the stellar half-mass estimates ($\leq 0.10$ dex) is
very similar to the scatter, implying robust uncertainties and a typical level
of precision of 25% or better. For quiescent galaxies at $1.5<z<2$ the
performance is still good, without a systematic offset, and uncertainties and
scatter that are in agreement, while at $z>2$ the uncertainties increase and
we are hampered by the small sample size as well. Reversing the question, we
can also conclude that rest-frame near-IR light profiles represent stellar
mass profiles well. That this is true is not immediately obvious, as age
gradients would still cause $M/L$ gradients even in the near-IR. Either these
effects are small or in the case of massive, high-metallicity galaxies this
trend may be countered by a anti-correlation between metallicity and near-IR
$M/L$.

For star-forming galaxies we see a small bias ($+0.05-0.07$ dex) that is
comparable in magnitude but opposite in sign to the offset with half-light
radius (Fig. 5). This may imply that the stellar half-mass radii for star-
forming galaxies are perhaps overestimated at $z>1.5$. Such issues are
understandable, as it is challenging to obtain accurate $M/L$ estimates with
limited photometric information redward of the 4000$\rm{\AA}$ break. The small
bias in the color-$M_{\star}/L$ relation for star-forming galaxies identified
in Section 2.3 (particularly, Fig. 2) may explain this: if galaxies have a
positive gradient in sSFR (e.g., Tacchella et al., 2017), then their outer
parts will have overesimed $M_{\star}/L$ with our method, leading to
overestimated $R_{\rm{M_{\star}}}$. Clearly, NIRCam-based photometry can
alleviate these concerns, but modeling of NIRCam photometry this is beyond the
scope of this paper.

Unfortunately, the current CEERS NIRCam sample is too small to assess the
robustness of $R_{\rm{M_{\star}}}$ estimates at $<$1 kpc. In the $1<z<1.5$
panel there is a hint that those are indeed somewhat underestimated as
suggested by Fig. 5, if perhaps not by the same amount. The model NIRCam PSF
is known to be inaccurate at some level and further progress in our
understanding of the true NIRCam PSF is required for accurate size estimates
of the smallest galaxies. We note that both the rest-frame optical sizes from
HST and rest-frame near-IR sizes from JWST are based on imaging data with
similar resolution ($\approx 0.15$ arcsec). Our statements regarding the
precision and accuracy of our $R_{\rm{M_{\star}}}$estimates are limited to the
$>$1 kpc regime.

###  2.6 Converting 2D Profiles to 3D Profiles

The methodology developed by van de Ven & van der Wel (2021) allows us to
convert our two-dimensional (projected) Sérsic light and mass profiles into
three-dimensional profiles. The procedure builds on our (statistical)
knowledge of the intrinsic shape distribution of galaxies as described by
Chang et al. (2013); Zhang et al. (2019): these authors constructed models for
the projected shape distributions of galaxies of different types, masses and
at different redshifts. These models assume a Gaussian distribution for the
intrinsic axis ratios of a triaxial ellipsoid $c/a$ (short-to-long axis ratio)
and $b/a$ (middle-to-long axis ratio) and/or a Gaussian distribution for the
triaxiality parameter $T=(a^{2}-b^{2})/(a^{2}-c^{2})$. Depending on the type
of galaxy the model consists of a single oblate population ($a\equiv b$), a
single triaxial population, or a mixed model of two components (one oblate $+$
one triaxial).

Given these models, the redshift, stellar mass and star-formation activity of
a galaxy produce an a priori probability distribution for its intrinsic shape,
parameterized as truncated Gaussian distributions for ellipticity ($E$) and
triaxilaty ($T$) (as published in Table 3 of Chang et al. (2013) and Table 1
of Zhang et al. (2019)), and its projected shape
$q^{\prime}=b^{\prime}/a^{\prime}$ (assuming random viewing angles for the
intrinsic shape distribution). Then, given the measured projected shape
$q^{\prime}$ from the CANDELS imaging (see Sec. 2.4.1), an a posteriori
probability distribution for the intrinsic shape distribution is constructed
(Eq. 9 from van de Ven & van der Wel, 2021): for a given $q$ exists a set of
combinations of intrinsic shape and viewing angle. Instead of calculating this
a posteriori probability distribution for each galaxy we construct a library
of solutions because its calculation requires an inversion of a demanding
numerical integral and we wish to sample from the measurement uncertainty in
$q$ and $R$ in order to propagate this into the inferred constraint on the
intrinsic shape and 3D size. Using these libraries we infer posterior
probability distributions for the 3D profile and the associated parameters
$R_{3D,M*}$, the radius of a sphere that contains 50% of the triaxial stellar
mass distribution.

Figure 7 shows that the deprojection for galaxies in this mass range has a
small effect on the inferred half-mass radii, as was already demonstrated for
specific cases by van de Ven & van der Wel (2021). The sub-optimal
visualisation of the results is chosen deliberately to highlight the lesser
importance of deprojection compared to the $M_{\star}/L$ gradient correction
shown in Figure 5.

For quiescent galaxies, which show a larger variety in intrinsic shapes than
star-forming galaxies (at least, for $M_{\star}>10^{10}~{}M_{\odot}$), the
scatter in $R_{3D,M_{\star}}~{}/~{}R_{M_{\star}}$ is larger than for star-
forming galaxies, but even for those the full range is no more than 30%, with
a systematic offset of $+$0.05 dex. For star-forming galaxies the scatter is
smaller, and the systematic offset almost zero. In other words, the projected
effective radius, measured as the long axis on the ellipse that encloses 50%
of the light or mass, generally serves as accurate and precise proxy for the
median radius, the radius of a sphere that contains 50% of the light or mass
distribution in 3D. For low-mass star-forming galaxies, as shapes become more
irregular, the deprojection will have a larger effect. Symmetrized
uncertainties are defined as half of the 16th-84th percentile ranges of the
posterior distributions, which produces a typical combined uncertainty in $R$
due to the deprojection of 0.03 dex ($\sigma$ in Fig. 7).

Figure 7:  3D (deprojected) stellar half-mass radius vs. 2D (projected)
stellar half-mass radius for quiescent (red) and star-forming (blue) galaxies
with total stellar mass $>10^{10}~{}M_{\odot}$. $\Delta$ is the median offset
in dex, with in parentheses the 16-84%-ile scatter. $\sigma$ is the median
formal uncertainty.

One caveat is that we have ignored the wavelength dependence on the shape. But
given the lack of sensitivity to changes in intrinsic shape, we can be
confident that this approximation does not affect the results. Another caveat
is related to the finding by Zhang et al. (2019) that a correlation exists
between (intrinsic) shape and size for star-forming galaxies. This implies
that size should, in principle, be included in our construction of the a
posteriori probability distribution for the 3D profile. We test for the
necessity of this additional step by varying the models, shifting the Gaussian
means by 2$\sigma$ up and down. The differences are negligible (on the 1%
level), which implies that the current setup – where we ignore the covariance
between size and shape – is sufficient for our purposes.111The corollary
implication is that differences in shape as a function of, e.g., redshift and
mass are not relevant for the conversion from 2D to 3D profile in the first
place, but this was not a foregone conclusion. Finally, the stellar masses,
star-formation rates and definition of quiescence used in this paper are not
the same as those used by Chang et al. (2013) and Zhang et al. (2019), but
given the minor effects of the deprojection these differences have no impact
on the overall result.

##  3 The Size-Mass Distribution and its Evolution

###  3.1 Comparing Size Proxies

Figure 8:  Size-stellar mass distributions for two redshift bins ($0.5<z<1$ at
the top; $1<z<1.5$ at the bottom) and three different size proxies: rest-frame
optical half-light radius $R_{\rm{opt}}$ (left; projected stellar half-mass
radius $R_{\rm{M_{\star},2D}}$ (middle); deprojected stellar half-mass radius
$R_{\rm{M_{\star},3D}}$ (bottom). The solid lines indicate the 16th, 50th and
84th percentiles of the size distribution in 0.2 dex wide bins of $M_{\star}$
(Table 3). The dotted lines in the middle and right-hand panels repeat, for
reference, the solid lines in the left-hand panels. The color-coding is the
star-formation rate relative to the star-forming sequence defined by Leja et
al. (2022).

The significant but approximately constant offset between the rest-frame
optical half-light radii and stellar half-mass radii, along with a lack of
strong projection effects, imply that the view of the size-mass distribution
of galaxies will not strongly depend on the choice of size proxy. In Figure 8
we show size-mass distributions for the redshift range $0.5<z<1.5$, for light-
weighted radii and both projected (2D) and deprojected (3D) stellar half-mass
radii. Regardless of size proxy we see the same characteristic size
distribution, with a steep slope for quiescent galaxies and a shallower slope
for star-forming galaxies. The combined median trend bends downward at
$M_{\star}\approx 2\times 10^{10}~{}M_{\odot}$, a transitional point in the
structural properties of present-day galaxies first identified by Kauffmann et
al. (2003). The downward trend is particularly noticeable at $z>1$.

Similarities aside, there are a number of small but interesting differences
between the size proxies. Switching from light- to mass-weighted sizes
strengthens the flattening/bending trend with mass further, an immediate
result of stronger color and $M/L$ gradients seen for more massive galaxies.
Projection effects play a relatively small role in the demographics of the
size-mass distribution. But note that the dashed and solid median lines in the
bottom-right panel of Fig. 8 are nearly perfectly parallel. This implies that
the joint effect of $M_{\star}/L$-gradient correction and deprojection leads
to a constant shift in median galaxy sizes across more than 2 orders of
magnitude in $M_{\star}$. The scatter in sizes decreases somewhat when
performing these corrections: rather than increasing the overall error budget,
the corrections get us closer to what can be considered true, physical sizes,
here defined as 3D stellar half-mass radii.

Increased uncertainties in the $R_{M_{\star}}$ estimates prevent us from
presenting a similarly reliable view of the size-mass distribution at higher
redshifts, in particular for star-forming galaxies. We should also keep in
mind that the color-$M_{\star}/L$ relations devised in this paper are
constructed on the basis of galaxies more massive than
$M_{\star}>10^{10}~{}M_{\odot}$, therefore the $R_{\rm{M_{\star}}}$ estimates
of low-mass galaxies may be biased, but this issue is likely not important as
a general absence of strong color gradients implies a general absence of
strong $M_{\star}/L$ gradients. The other caveat is that we concluded in
Section 2.4.1 that $R_{\rm{M_{\star}}}$$\approx 1$ kpc size estimates are
suspect if color gradients are present. The small-size tail of quiescent
galaxies may therefore suffer from currently unknown systematic effects and
increased random uncertainties. That said, the average sizes general exceed 2
kpc so that the trends shown in Figure 8 are robust.

Table 2: Stellar Half-Mass Radii

Field | ID | R.A. | Dec. | z | $\log M_{\star}$ | Quiescent Flag | $R_{M_{\star}}$ | $R_{M_{\star},3D}$
---|---|---|---|---|---|---|---|---
|  | deg. | deg. |  | $M_{\odot}$ |  | kpc | kpc
1 | 19 | 215.299759 | 53.051308 | 1.076 | $9.510^{+0.076}_{-0.079}$ | 0 | 2.578$\pm$0.371 | 2.547$\pm$0.551
1 | 28 | 215.264175 | 53.027222 | 0.763 | $8.893^{+0.167}_{-0.142}$ | 0 | 1.840$\pm$0.234 | 1.746$\pm$0.324
1 | 46 | 215.293457 | 53.048298 | 1.217 | $10.650^{+0.031}_{-0.027}$ | 1 | 0.744$\pm$0.120 | 0.862$\pm$0.154
1 | 55 | 215.298920 | 53.052399 | 0.697 | $9.045^{+0.076}_{-0.088}$ | 0 | 2.557$\pm$0.294 | 2.564$\pm$0.496
1 | 63 | 215.296555 | 53.050770 | 1.219 | $9.931^{+0.049}_{-0.052}$ | 0 | 2.847$\pm$0.317 | 2.997$\pm$0.534
1 | 83 | 215.302460 | 53.055332 | 0.725 | $9.808^{+0.037}_{-0.043}$ | 0 | 2.422$\pm$0.449 | 2.665$\pm$0.633
1 | 110 | 215.297195 | 53.051907 | 0.694 | $10.294^{+0.049}_{-0.036}$ | 1 | 1.528$\pm$0.290 | 1.935$\pm$0.395
1 | 145 | 215.281769 | 53.042686 | 0.784 | $9.360^{+0.055}_{-0.160}$ | 0 | 4.058$\pm$0.504 | 4.034$\pm$0.796
1 | 148 | 215.248138 | 53.019707 | 0.933 | $8.947^{+0.086}_{-0.116}$ | 0 | 1.760$\pm$0.419 | 1.794$\pm$0.511
1 | 151 | 215.252319 | 53.022339 | 0.679 | $9.738^{+0.052}_{-0.062}$ | 0 | 2.300$\pm$0.224 | 2.159$\pm$0.332
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮

Note. — Coefficients from Eq. 1 that describe the fitted relationship between
HST flux density ratios (colors) and $M_{\star}/F_{160}$ across the redshift
range $0.5<z<2.3$ and stellar masses $M_{\star}>10^{10}~{}M_{\odot}$.

Note. — (1): Field (1: EGS; 2: COSMOS; 3: GOODS-N; 4: GOODS-S; 5: UDS); (2):
ID from Skelton et al. (2014); (3): R.A. from Skelton et al. (2014); (4): Dec.
.from Skelton et al. (2014); (5) Stellar mass from Leja et al. (2020), with
16th- and 84th-perceintile uncertainty range; (6) Star-forming (0) or
quiescent (1) (Sec. 2.2); (7) Projected (2D) stellar half-mass radius (Sec.
2.4), defined as semi-major axis of half-mass ellipse; Deprojected (3D)
stellar half-mass radius (Sec. 2.6), defined as radius of sphere containing
50% of the stellar mass.

###  3.2 Separating Star-Forming and Quiescent Galaxies

The clear pattern with star-formation activity in the size-mass distribution
(Fig. 8) invites a closer look at the size distributions for star-forming and
quiescent galaxies separately. Figure 9 shows the star-forming galaxies and
the most eye-catching result is the stellar-half mass radius depends less
strongly on stellar mass than the half-light radius. Galaxies near the knee of
the stellar mass function have $R_{\rm{M_{\star}}}$ just $\approx 2$ times
larger than galaxies 100$\times$ less massive, a slope of 0.15 dex. Also note
the correlation between size and projected axis ratio, particularly at
$M_{\star}<10^{10}~{}M_{\odot}$: as shown by Zhang et al. (2019), galaxies
that are flat in projection are more likely to have prolate/elongated 3D
shapes and have larger (projected) sizes than galaxies with an oblate (disk-
like) shape. When deprojecting the mass distribution this effect is lessened,
but without a meaningful change in median size or scatter. This implies that
for individual galaxies a shape-dependent deprojection correction can improve
the accuracy of the size estimate, but that such a correction is not necessary
to correctly infer the ensemble size distribution.

For quiescent galaxies (Fig. 10) the size-mass distribution is strikingly
different from that of star-forming galaxies, but rather similar for the
different size proxies. The $M_{\star}/L$ gradient correction shifts the sizes
downward, partially countered by an upward shift when correcting for
projection effects. The distribution is flat up until $\approx 2-3\times
10^{10}~{}M_{\odot}$ (also see Nedkova et al., 2021), followed by a steep
increase toward larger masses (seen by many authors). Correlations with
projected axis ratio are less obvious compared to those seen for star-forming
galaxies, as most galaxies are oblate or round/triaxial, but the smallest
galaxies in the mass range $10^{10.5-11}~{}M_{\odot}$ are flatter (and
therefore diskier) than average.

Figure 9:  Size-stellar mass distributions of star-forming galaxies for two
redshift bins ($0.5<z<1$ at the top; $1<z<1.5$ at the bottom) and three
different size proxies: rest-frame optical half-light radius $R_{\rm{opt}}$
(left; projected stellar half-mass radius $R_{\rm{M_{\star},2D}}$ (middle);
deprojected stellar half-mass radius $R_{\rm{M_{\star},3D}}$ (bottom). The
solid lines indicate the 16th, 50th and 84th percentiles of the size
distribution in 0.2 dex wide bins of $M_{\star}$ (Table 4). The dotted lines
in the middle and right-hand panels repeat, for reference, the solid lines in
the left-hand panels. The color-coding is the projected axis ratio.

Figure 10:  Size-stellar mass distributions of quiescent galaxies for two
redshift bins ($0.5<z<1$ at the top; $1<z<1.5$ at the bottom) and three
different size proxies: rest-frame optical half-light radius $R_{\rm{opt}}$
(left; projected stellar half-mass radius $R_{\rm{M_{\star},2D}}$ (middle);
deprojected stellar half-mass radius $R_{\rm{M_{\star},3D}}$ (bottom). The
solid lines indicate the 16th, 50th and 84th percentiles of the size
distribution in 0.2 dex wide bins of $M_{\star}$ (Table 5). The dotted lines
in the middle and right-hand panels repeat, for reference, the solid lines in
the left-hand panels. The color-coding is the projected axis ratio.  Table 3:
Median Radii and Percentiles

| $0.5<z<1$ | $1<z<1.5$
---|---|---
| $\log R_{0.5\mu\rm{m}}$ | $\log R_{M_{\star}}$ | $\log R_{M_{\star},3D}$ | $\log R_{0.5\mu\rm{m}}$ | $\log R_{M_{\star}}$ | $\log R_{M_{\star},3D}$
$\log M_{\star}~{}(M_{\odot})$ | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84%
8.8 | 0.01 | 0.26 | 0.50 | 0.03 | 0.25 | 0.47 | 0.03 | 0.25 | 0.46 | -0.04 | 0.17 | 0.41 | -0.03 | 0.17 | 0.37 | -0.02 | 0.16 | 0.35
9.0 | 0.06 | 0.31 | 0.54 | 0.06 | 0.30 | 0.50 | 0.07 | 0.29 | 0.49 | 0.02 | 0.29 | 0.50 | 0.02 | 0.25 | 0.45 | 0.02 | 0.23 | 0.43
9.2 | 0.09 | 0.34 | 0.56 | 0.09 | 0.32 | 0.53 | 0.10 | 0.32 | 0.52 | 0.05 | 0.31 | 0.50 | 0.04 | 0.26 | 0.45 | 0.05 | 0.25 | 0.43
9.4 | 0.13 | 0.41 | 0.62 | 0.12 | 0.37 | 0.58 | 0.14 | 0.38 | 0.57 | 0.10 | 0.35 | 0.56 | 0.10 | 0.31 | 0.50 | 0.10 | 0.31 | 0.48
9.6 | 0.15 | 0.42 | 0.65 | 0.14 | 0.38 | 0.59 | 0.15 | 0.38 | 0.58 | 0.16 | 0.39 | 0.60 | 0.14 | 0.34 | 0.54 | 0.15 | 0.34 | 0.52
9.8 | 0.21 | 0.46 | 0.70 | 0.19 | 0.40 | 0.61 | 0.21 | 0.41 | 0.60 | 0.18 | 0.42 | 0.63 | 0.17 | 0.37 | 0.56 | 0.18 | 0.37 | 0.56
10.0 | 0.25 | 0.49 | 0.71 | 0.19 | 0.42 | 0.63 | 0.20 | 0.42 | 0.63 | 0.23 | 0.47 | 0.68 | 0.17 | 0.42 | 0.59 | 0.19 | 0.42 | 0.57
10.2 | 0.26 | 0.54 | 0.73 | 0.17 | 0.45 | 0.62 | 0.19 | 0.45 | 0.62 | 0.24 | 0.51 | 0.69 | 0.17 | 0.44 | 0.61 | 0.19 | 0.45 | 0.61
10.4 | 0.18 | 0.50 | 0.75 | 0.07 | 0.40 | 0.64 | 0.12 | 0.41 | 0.64 | 0.26 | 0.57 | 0.72 | 0.17 | 0.45 | 0.63 | 0.22 | 0.47 | 0.64
10.6 | 0.18 | 0.49 | 0.75 | 0.05 | 0.37 | 0.61 | 0.11 | 0.39 | 0.62 | 0.04 | 0.47 | 0.71 | -0.05 | 0.36 | 0.61 | 0.00 | 0.39 | 0.62
10.8 | 0.21 | 0.45 | 0.76 | 0.08 | 0.32 | 0.60 | 0.13 | 0.37 | 0.62 | 0.06 | 0.39 | 0.74 | -0.03 | 0.26 | 0.64 | 0.00 | 0.31 | 0.64
11.0 | 0.31 | 0.49 | 0.76 | 0.20 | 0.39 | 0.63 | 0.24 | 0.45 | 0.68 | 0.14 | 0.36 | 0.72 | 0.02 | 0.27 | 0.59 | 0.07 | 0.32 | 0.62
11.2 | 0.41 | 0.59 | 0.81 | 0.29 | 0.51 | 0.70 | 0.35 | 0.57 | 0.75 | 0.29 | 0.53 | 0.74 | 0.21 | 0.41 | 0.62 | 0.25 | 0.44 | 0.65
11.4 | 0.57 | 0.77 | 0.94 | 0.48 | 0.66 | 0.88 | 0.56 | 0.70 | 0.97 | 0.50 | 0.65 | 0.81 | 0.35 | 0.53 | 0.72 | 0.41 | 0.57 | 0.74
11.6 | 0.71 | 0.85 | 0.95 | 0.54 | 0.71 | 0.89 | 0.60 | 0.79 | 0.99 | 0.67 | 0.82 | 0.98 | 0.61 | 0.66 | 0.95 | 0.66 | 0.72 | 0.98

Note. — Percentiles correspond with the lines in Figure 8.

Table 4: Median Radii and Percentiles for Star-Forming Galaxies

| $0.5<z<1$ | $1<z<1.5$
---|---|---
| $\log R_{0.5\mu\rm{m}}$ | $\log R_{M_{\star}}$ | $\log R_{M_{\star},3D}$ | $\log R_{0.5\mu\rm{m}}$ | $\log R_{M_{\star}}$ | $\log R_{M_{\star},3D}$
$\log M_{\star}~{}(M_{\odot})$ | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84%
8.8 | 0.01 | 0.26 | 0.50 | 0.03 | 0.25 | 0.47 | 0.03 | 0.25 | 0.46 | -0.04 | 0.17 | 0.41 | -0.03 | 0.17 | 0.38 | -0.02 | 0.16 | 0.35
9.0 | 0.06 | 0.32 | 0.55 | 0.06 | 0.31 | 0.50 | 0.07 | 0.30 | 0.49 | 0.02 | 0.29 | 0.50 | 0.02 | 0.25 | 0.45 | 0.02 | 0.23 | 0.43
9.2 | 0.09 | 0.35 | 0.57 | 0.10 | 0.33 | 0.54 | 0.10 | 0.33 | 0.52 | 0.05 | 0.31 | 0.50 | 0.05 | 0.26 | 0.45 | 0.05 | 0.25 | 0.44
9.4 | 0.14 | 0.42 | 0.63 | 0.13 | 0.38 | 0.58 | 0.14 | 0.38 | 0.57 | 0.11 | 0.35 | 0.56 | 0.11 | 0.31 | 0.50 | 0.11 | 0.31 | 0.48
9.6 | 0.17 | 0.45 | 0.67 | 0.16 | 0.40 | 0.60 | 0.17 | 0.40 | 0.59 | 0.16 | 0.39 | 0.60 | 0.15 | 0.35 | 0.54 | 0.16 | 0.34 | 0.52
9.8 | 0.24 | 0.48 | 0.71 | 0.22 | 0.42 | 0.62 | 0.23 | 0.42 | 0.61 | 0.19 | 0.43 | 0.63 | 0.18 | 0.38 | 0.57 | 0.20 | 0.37 | 0.56
10.0 | 0.29 | 0.52 | 0.72 | 0.23 | 0.44 | 0.65 | 0.24 | 0.45 | 0.64 | 0.24 | 0.48 | 0.68 | 0.19 | 0.43 | 0.59 | 0.20 | 0.43 | 0.58
10.2 | 0.36 | 0.58 | 0.74 | 0.29 | 0.48 | 0.63 | 0.30 | 0.47 | 0.64 | 0.31 | 0.54 | 0.71 | 0.24 | 0.46 | 0.62 | 0.26 | 0.46 | 0.62
10.4 | 0.39 | 0.61 | 0.80 | 0.29 | 0.50 | 0.67 | 0.31 | 0.51 | 0.66 | 0.39 | 0.61 | 0.74 | 0.33 | 0.50 | 0.65 | 0.35 | 0.51 | 0.66
10.6 | 0.45 | 0.63 | 0.80 | 0.31 | 0.49 | 0.67 | 0.32 | 0.50 | 0.68 | 0.37 | 0.59 | 0.76 | 0.24 | 0.48 | 0.65 | 0.27 | 0.49 | 0.66
10.8 | 0.48 | 0.70 | 0.86 | 0.33 | 0.54 | 0.70 | 0.35 | 0.55 | 0.71 | 0.45 | 0.67 | 0.82 | 0.34 | 0.56 | 0.70 | 0.36 | 0.58 | 0.71
11.0 | 0.56 | 0.74 | 0.90 | 0.42 | 0.59 | 0.74 | 0.45 | 0.60 | 0.75 | 0.43 | 0.65 | 0.83 | 0.35 | 0.52 | 0.69 | 0.38 | 0.54 | 0.72
11.2 | 0.67 | 0.76 | 0.89 | 0.52 | 0.63 | 0.73 | 0.54 | 0.65 | 0.76 | 0.58 | 0.72 | 0.82 | 0.42 | 0.59 | 0.70 | 0.44 | 0.60 | 0.72
11.4 | 0.66 | 0.87 | 1.02 | 0.57 | 0.71 | 0.93 | 0.65 | 0.76 | 1.00 | 0.56 | 0.69 | 0.86 | 0.45 | 0.59 | 0.72 | 0.49 | 0.62 | 0.72

Note. — Percentiles correspond with the lines in Figure 9.

Table 5: Median Radii and Percentiles for Quiescent Galaxies

| $0.5<z<1$ | $1<z<1.5$
---|---|---
| $\log R_{0.5\mu\rm{m}}$ | $\log R_{M_{\star}}$ | $\log R_{M_{\star},3D}$ | $\log R_{0.5\mu\rm{m}}$ | $\log R_{M_{\star}}$ | $\log R_{M_{\star},3D}$
$\log M_{\star}~{}(M_{\odot})$ | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84% | 16% | 50% | 84%
8.8 | -0.06 | 0.13 | 0.33 | -0.12 | 0.11 | 0.29 | -0.05 | 0.17 | 0.29 | 0.13 | 0.13 | 0.16 | 0.05 | 0.14 | 0.17 | 0.09 | 0.14 | 0.19
9.0 | 0.04 | 0.22 | 0.36 | 0.05 | 0.19 | 0.33 | 0.09 | 0.23 | 0.35 | 0.02 | 0.23 | 0.35 | 0.04 | 0.19 | 0.37 | 0.11 | 0.23 | 0.36
9.2 | 0.02 | 0.18 | 0.36 | -0.03 | 0.13 | 0.36 | 0.04 | 0.19 | 0.43 | -0.07 | 0.10 | 0.40 | -0.09 | 0.12 | 0.36 | -0.08 | 0.15 | 0.39
9.4 | 0.05 | 0.20 | 0.36 | 0.01 | 0.18 | 0.34 | 0.06 | 0.22 | 0.40 | -0.01 | 0.16 | 0.41 | -0.03 | 0.16 | 0.40 | -0.00 | 0.20 | 0.40
9.6 | 0.04 | 0.24 | 0.34 | -0.05 | 0.16 | 0.29 | 0.02 | 0.21 | 0.35 | 0.08 | 0.22 | 0.43 | 0.05 | 0.22 | 0.34 | 0.09 | 0.22 | 0.36
9.8 | -0.02 | 0.22 | 0.37 | -0.03 | 0.19 | 0.34 | -0.02 | 0.22 | 0.36 | -0.04 | 0.18 | 0.26 | -0.19 | 0.05 | 0.27 | -0.14 | 0.12 | 0.33
10.0 | -0.04 | 0.19 | 0.40 | -0.11 | 0.09 | 0.32 | -0.08 | 0.15 | 0.35 | 0.07 | 0.31 | 0.47 | 0.05 | 0.21 | 0.46 | 0.11 | 0.23 | 0.50
10.2 | -0.10 | 0.23 | 0.44 | -0.17 | 0.12 | 0.32 | -0.11 | 0.14 | 0.36 | -0.07 | 0.11 | 0.36 | -0.15 | 0.00 | 0.27 | -0.06 | 0.05 | 0.29
10.4 | 0.00 | 0.19 | 0.41 | -0.12 | 0.12 | 0.30 | -0.08 | 0.15 | 0.35 | -0.08 | 0.11 | 0.49 | -0.15 | 0.08 | 0.37 | -0.11 | 0.14 | 0.40
10.6 | 0.06 | 0.24 | 0.50 | -0.07 | 0.13 | 0.41 | -0.03 | 0.19 | 0.44 | -0.06 | 0.08 | 0.41 | -0.19 | -0.02 | 0.29 | -0.12 | 0.04 | 0.35
10.8 | 0.15 | 0.33 | 0.49 | 0.03 | 0.21 | 0.38 | 0.09 | 0.25 | 0.43 | 0.00 | 0.18 | 0.44 | -0.10 | 0.05 | 0.30 | -0.05 | 0.11 | 0.35
11.0 | 0.29 | 0.42 | 0.61 | 0.17 | 0.31 | 0.51 | 0.21 | 0.36 | 0.58 | 0.10 | 0.29 | 0.54 | -0.03 | 0.17 | 0.43 | 0.01 | 0.20 | 0.49
11.2 | 0.40 | 0.56 | 0.75 | 0.28 | 0.48 | 0.66 | 0.34 | 0.54 | 0.73 | 0.25 | 0.38 | 0.63 | 0.15 | 0.29 | 0.50 | 0.17 | 0.34 | 0.56
11.4 | 0.56 | 0.71 | 0.91 | 0.47 | 0.64 | 0.88 | 0.55 | 0.69 | 0.95 | 0.46 | 0.61 | 0.78 | 0.32 | 0.51 | 0.69 | 0.39 | 0.56 | 0.76
11.6 | 0.75 | 0.87 | 0.99 | 0.67 | 0.74 | 0.88 | 0.68 | 0.79 | 0.98 | 0.66 | 0.81 | 0.86 | 0.60 | 0.65 | 0.88 | 0.66 | 0.70 | 0.94

Note. — Percentiles correspond with the lines in Figure 10.

###  3.3 Size Evolution of Massive Quiescent Galaxies

Figure 11:  Stellar half-mass/light radius vs. redshift for massive
($M_{\star}>10^{11}~{}M_{\odot}$) quiescent galaxies. The small data points
are the 3D stellar half-mass radii for the individual objects, the large red
symbols their mean values in bins of redshift. The open green symbols are the
biweight mean values of the 2D stellar half-mass radii; the blue symbols the
2D stellar half-light radii at rest-frame 0.5$\mu$m. The uncertainties on the
mean values are of the same order as the sizes of the symbols. The lines show
that the pace of evolution, determined by simple regression fits, is always
similar and in agreement with previously published results on the evolution of
light-weighted sizes.

For massive ($M_{\star}>10^{11}~{}M_{\odot}$), quiescent galaxies our size
estimates are robust at $z>1.5$ and we can probe the evolution of size-mass
distribution out to $z=2.3$.222The data for this subset of galaxies is
included in Table 2. Figure 11 shows the evolution with redshift of the sizes
of massive, quiescent galaxies, comparing the three definitions: light-
weighted ($R_{0.5\mu\rm{m}}\propto(1+z)^{-1.64\pm 0.09}$), mass-weighted
($R_{M_{\star}}\propto(1+z)^{-1.72\pm 0.15}$), and deprojected mass-weighted
($R_{M_{\star},3D}\propto(1+z)^{-1.79\pm 0.16}$). The key result is that the
size evolution is significant, regardless of the choice of size proxy. At all
redshifts the mass-weighted sizes are smaller than the light-weighted sizes by
similar amounts by 0.1$-$0.15 dex, in line with the early NIRCam-based results
from Suess et al. (2022) that showed that NIRCam/F444W sizes are smaller than
NIRCam/F150W by $\approx 0.15$ dex. The deprojection from 2D to 3D (Sec. 2.6)
shifts the sizes upward by about 0.05 dex. For most purposes, the measured
sizes of massive galaxies do not require a deprojection correction to enable a
meaningful comparison with the sizes of simulated galaxies based on 3D stellar
particle distributions.

The above analysis does not consider the steep slope of the size-mass relation
for massive galaxies, and the effect that evolution in slope or differences in
slope among the three size proxies might have on the inferred evolution. But
the similarity in slope for the three size proxies seen in Figure 10 and a
lack of strong change in slope with redshift suggest that the effects are mild
at most. Indeed, the size evolution result shown in Figure 11 does not depend
on the precise selection in $M_{\star}$: galaxies with
$10^{11}<M_{\star}/M_{\odot}<2\times 10^{11}$ and galaxies with
$M_{\star}>2\times 10^{11}~{}M_{\odot}$ show the same pace of evolution for
all three size proxies, well within the uncertainties.

There are two effects that may introduce a bias the inferred size evolution.
First, differences in slope among the size proxies depend The slopes are very
similar for the three size proxies (see Fig.10), so that any dependence on
slope in the parameterization of size evolution is the same for all three
proxies. A shift in the $M_{\star}$ distribution with redshift would introduce
a bias in the estimated pace of evolution. Repeating the analysis by adopting
a fixed slope of $\Delta\log{R}/\Delta\log{M_{\star}=0.6}$ does not change the
pace of evolution by less than 25% of the uncertainty.

Our result that the half-mass and half-light radii of massive quiescent
galaxies evolve rapidly wiht redshift and in a similar manner (with
$R\propto(1+z)^{\alpha=-1.6\dots-1.8}$) is in tension with previous work.
Suess et al. (2019b); Miller et al. (2023) argue that a correction for
$M_{\star}/L$ gradients removes much of the size evolution at $z>1$ seen in
the rest-frame optical so that the stellar half-mass radius, on average,
evolves much less than the stellar half-light radius or not at all.

This begs the question how the previously published stellar half-mass radii
compare with the rest-frame near-IR sizes from NIRCam. In Appendix A we
provide and extensive and quantitative comparison, the result of which is, in
short, that the $R_{\rm{M_{\star}}}$ estimates constructed in this paper
produce the smallest offset and scatter when compared to NIRCam-based sizes.

In addition, for our half-mass radii the uncertainties are similar to the
scatter in the comparison with the NIRCam sizes (typically, 0.10 dex at $z<2$;
also see Fig. 6), whereas for previously published estimates the formal
uncertainties are smaller ($\lesssim 0.05$ dex) and likely underestimated, as
was already pointed out by Miller et al. (2023). The accurate agreement over
the redshift range $0.5<z<2$ for our half-mass radii argue in favor of our
conclusion that the sizes of massive quiescent galaxies strongly evolve with
cosmic time, in line with previous results based on rest-frame optical size
measurements (e.g., Trujillo et al., 2004, 2006; van der Wel et al., 2008; van
Dokkum et al., 2008; Newman et al., 2012; Carollo et al., 2013; van der Wel et
al., 2014b).

##  4 Summary & Outlook

Our novel method to estimate stellar half-mass radii for a large sample of
galaxies at $0.5<z<2.5$ drawn from CANDELS and 3D-HST rests on leveraging the
integrated UV-to-midIR photometric information that is available for these
galaxies. We derive a relationship between the HST/ACS$+$WFC3 colors and the
$M/L$ estimated from the UV-to-midIR SED (Sec. 2.3) and apply that
relationship to the spatially resolved color profiles (Sec. 2.4.2). The
underlying assumption is that the distribution of physical properties (age,
metallicity, attenuation, etc.) among galaxies is comparable to that within
galaxies. Moreover, we infer 3D sizes based on the deprojection machinery
developed by van de Ven & van der Wel (2021) and our knowledge of the shape
distribution of galaxies and its dependence on stellar mass and redshift (Sec.
2.6). The $R_{\rm{M_{\star}}}$ and $R_{\rm{M_{\star},3D}}$ estimates are made
publicly available online – see Table 2 for the first 10 entries of the
catalog.

An essential test of the reliability of our stellar half-mass radii is
provided by the comparison with size measurements from JWST/NIRCam imaging in
the rest-frame near-IR for a (for now) limited subset galaxies in CEERS. The
agreement is excellent (Sec. 2.5). First, systematic offsets are less than 10%
for quiescent galaxies up to $z=2$ and for star-forming galaxies up to
$z=1.5$. Second, the scatter is small (typically, 25%) and consistent with the
formal error budget. The comparison with NIRCam demonstrates that our stellar
half-mass radii are precise and accurate under the assumption that rest-frame
near-IR sizes are, indeed, a good proxy for stellar-mass weighted sizes. As
briefly discussed in Section 2.5 this is not self-evident. Even in the near-IR
the $M_{\star}/L$ evolves strongly with age. Either our
$R_{\rm{M_{\star}}}$and $R_{\rm{NIR}}$ sizes agree because they both
accurately trace the stellar mass distribution, or they suffer from the same
systematic bias. The latter is a distinct possibility: if our $M_{\star}/L$
are overestimated in regions with high star-formation activity (see Sec. 2.3)
then lower $M_{\star}/L_{\rm{NIR}}$ are to be expected as well.

Previously published half-mass radii do not perform equally well, as described
in Sec. 3.3, with larger systematic offsets, and larger scatter while
reporting smaller formal uncertainties. The main caveat of the present
analysis is that for small objects $\lesssim$ 1 kpc the systematic
uncertainties are not well understood, not only because this is near the
resolution limit of HST in the optical and JWST/NIRCam in the near-IR, but
also because the NIRCam PSF is not sufficiently well understood at the moment.

In Section 3 we show the effects of correcting for $M_{\star}/L$ gradients and
deprojecting the 2D distribution on the size-mass distribution of galaxies at
$0.5<z<1.5$. Compared to the rest-frame optical size distribution, the stellar
half-mass radius - stellar mass relation is less steep (Fig. 8), while
deprojection affects the size-mass distribution only little. A separation
between star-forming and quiescent galaxies (Sec. 3.2) shows that the
flattening of the size-mass relation is driven by massive star-forming
galaxies, which have the largest downward $M_{\star}/L$ correction.

For quiescent galaxies, the deprojection counters the downward $M_{\star}/L$
and the size-mass distribution in the rest-frame optical is very similar to
the 3D half-mass radius distribution, modulo a small ($\approx 0.05$ dex)
downward shift. The medians and percentile values of the various size
distributions shown in Figures 8, 9 and 10 are provided in Tables 3, 4, and 5,
respectively. In Section 3.3 we show that the average $R_{\rm{M_{\star}}}$and
$R_{\rm{M_{\star},3D}}$of massive, quiescent galaxies evolve rapidly from
$z=2.3$ to $z=0.5$, with $R\propto(1+z)^{-1.7\pm 0.1}$, and at the same pace
as the rest-frame optical half-light radii.

Now that NIRCam imaging datasets across larger volumes become available from
COSMOS-Web (Casey et al., 2022) and JADES (Robertson et al., 2022), and the
sample sizes become similar to those drawn from CANDELS, then our stellar mass
profiles will be superseded by rest-frame near-IR profiles and the modeling of
the optical-to-near-IR light profiles as pioneered by Miller et al. (2022);
Abdurro’uf et al. (2023); Ji et al. (2023). But our work provides a simple
conversion from light-to-stellar mass weighted sizes for galaxies without
spatially resolved near-IR imaging, that is, those without NIRCam imaging or
those at high redshift ($z>6$) when even NIRCam only samples the rest-frame
optical. The key point of our work is that spatially resolved optical colors
accurately predict the sizes of galaxies in the rest-frame near-IR, which is
generally considered a robust proxy for the stellar mass distribution.

## Acknowledgments

MM acknowledge the financial support of the Flemish Fund for Scientific
Research (FWO-Vlaanderen), research project G030319N. All the HST and JWST
data used in this paper can be found in MAST:
http://dx.doi.org/10.17909/z7p0-8481 (catalog 10.17909/z7p0-8481),
http://dx.doi.org/10.17909/T94S3X (catalog 10.17909/T94S3X).

## References

  * Abdurro’uf & Akiyama (2018) Abdurro’uf, & Akiyama, M. 2018, MNRAS, 479, 5083, doi: 10.1093/mnras/sty1771
  * Abdurro’uf et al. (2023) Abdurro’uf, Coe, D., Jung, I., et al. 2023, ApJ, 945, 117, doi: 10.3847/1538-4357/acba06
  * Abraham et al. (1999) Abraham, R. G., Ellis, R. S., Fabian, A. C., Tanvir, N. R., & Glazebrook, K. 1999, MNRAS, 303, 641, doi: 10.1046/j.1365-8711.1999.02059.x
  * Baes & van Hese (2011) Baes, M., & van Hese, E. 2011, A&A, 534, A69, doi: 10.1051/0004-6361/201117708
  * Bell & de Jong (2000) Bell, E. F., & de Jong, R. S. 2000, MNRAS, 312, 497, doi: 10.1046/j.1365-8711.2000.03138.x
  * Bell & de Jong (2001) —. 2001, ApJ, 550, 212, doi: 10.1086/319728
  * Cappellari et al. (2013) Cappellari, M., Scott, N., Alatalo, K., et al. 2013, MNRAS, 432, 1709, doi: 10.1093/mnras/stt562
  * Carollo et al. (2013) Carollo, C. M., Bschorr, T. J., Renzini, A., et al. 2013, ApJ, 773, 112, doi: 10.1088/0004-637X/773/2/112
  * Casey et al. (2022) Casey, C. M., Kartaltepe, J. S., Drakos, N. E., et al. 2022, arXiv e-prints, arXiv:2211.07865, doi: 10.48550/arXiv.2211.07865
  * Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763, doi: 10.1086/376392
  * Chang et al. (2013) Chang, Y.-Y., van der Wel, A., Rix, H.-W., et al. 2013, ApJ, 773, 149, doi: 10.1088/0004-637X/773/2/149
  * Conroy et al. (2009) Conroy, C., Gunn, J. E., & White, M. 2009, ApJ, 699, 486, doi: 10.1088/0004-637X/699/1/486
  * de Graaff et al. (2022) de Graaff, A., Trayford, J., Franx, M., et al. 2022, MNRAS, 511, 2544, doi: 10.1093/mnras/stab3510
  * de Jong (1996) de Jong, R. S. 1996, A&A, 313, 377, doi: 10.48550/arXiv.astro-ph/9604010
  * Dutton et al. (2011) Dutton, A. A., van den Bosch, F. C., Faber, S. M., et al. 2011, MNRAS, 410, 1660, doi: 10.1111/j.1365-2966.2010.17555.x
  * Fang et al. (2013) Fang, J. J., Faber, S. M., Koo, D. C., & Dekel, A. 2013, ApJ, 776, 63, doi: 10.1088/0004-637X/776/1/63
  * Finkelstein et al. (2023) Finkelstein, S. L., Bagley, M. B., Ferguson, H. C., et al. 2023, ApJ, 946, L13, doi: 10.3847/2041-8213/acade4
  * Grogin et al. (2011) Grogin, N. A., Kocevski, D. D., Faber, S. M., et al. 2011, ApJS, 197, 35, doi: 10.1088/0067-0049/197/2/35
  * Guo et al. (2011) Guo, Y., Giavalisco, M., Cassata, P., et al. 2011, ApJ, 735, 18, doi: 10.1088/0004-637X/735/1/18
  * Häußler et al. (2013) Häußler, B., Bamford, S. P., Vika, M., et al. 2013, MNRAS, 430, 330, doi: 10.1093/mnras/sts633
  * Hinkley & Im (2001) Hinkley, S., & Im, M. 2001, ApJ, 560, L41, doi: 10.1086/323940
  * Ji et al. (2023) Ji, Z., Williams, C. C., Tacchella, S., et al. 2023, arXiv e-prints, arXiv:2305.18518, doi: 10.48550/arXiv.2305.18518
  * Johnson et al. (2021) Johnson, B. D., Leja, J., Conroy, C., & Speagle, J. S. 2021, ApJS, 254, 22, doi: 10.3847/1538-4365/abef67
  * Kauffmann et al. (2003) Kauffmann, G., Heckman, T. M., White, S. D. M., et al. 2003, MNRAS, 341, 54, doi: 10.1046/j.1365-8711.2003.06292.x
  * Koekemoer et al. (2011) Koekemoer, A. M., Faber, S. M., Ferguson, H. C., et al. 2011, ApJS, 197, 36, doi: 10.1088/0067-0049/197/2/36
  * Leja et al. (2017) Leja, J., Johnson, B. D., Conroy, C., van Dokkum, P. G., & Byler, N. 2017, ApJ, 837, 170, doi: 10.3847/1538-4357/aa5ffe
  * Leja et al. (2020) Leja, J., Speagle, J. S., Johnson, B. D., et al. 2020, ApJ, 893, 111, doi: 10.3847/1538-4357/ab7e27
  * Leja et al. (2022) Leja, J., Speagle, J. S., Ting, Y.-S., et al. 2022, ApJ, 936, 165, doi: 10.3847/1538-4357/ac887d
  * Li et al. (2018) Li, H., Mao, S., Cappellari, M., et al. 2018, MNRAS, 476, 1765, doi: 10.1093/mnras/sty334
  * Li & Leja (2022) Li, Y., & Leja, J. 2022, ApJ, 940, 88, doi: 10.3847/1538-4357/ac97e6
  * Matharu et al. (2023) Matharu, J., Muzzin, A., Sarrouh, G. T. E., et al. 2023, ApJ, 949, L11, doi: 10.3847/2041-8213/acd1db
  * McGrath et al. (2008) McGrath, E. J., Stockton, A., Canalizo, G., Iye, M., & Maihara, T. 2008, ApJ, 682, 303, doi: 10.1086/589631
  * Meidt et al. (2012) Meidt, S. E., Schinnerer, E., Muñoz-Mateos, J.-C., et al. 2012, ApJ, 748, L30, doi: 10.1088/2041-8205/748/2/L30
  * Meidt et al. (2014) Meidt, S. E., Schinnerer, E., van de Ven, G., et al. 2014, ApJ, 788, 144, doi: 10.1088/0004-637X/788/2/144
  * Miller et al. (2023) Miller, T. B., van Dokkum, P., & Mowla, L. 2023, ApJ, 945, 155, doi: 10.3847/1538-4357/acbc74
  * Miller et al. (2022) Miller, T. B., Whitaker, K. E., Nelson, E. J., et al. 2022, ApJ, 941, L37, doi: 10.3847/2041-8213/aca675
  * Mosleh et al. (2020) Mosleh, M., Hosseinnejad, S., Hosseini-ShahiSavandi, S. Z., & Tacchella, S. 2020, ApJ, 905, 170, doi: 10.3847/1538-4357/abc7cc
  * Mosleh et al. (2017) Mosleh, M., Tacchella, S., Renzini, A., et al. 2017, ApJ, 837, 2, doi: 10.3847/1538-4357/aa5f14
  * Nedkova et al. (2021) Nedkova, K. V., Häußler, B., Marchesini, D., et al. 2021, MNRAS, 506, 928, doi: 10.1093/mnras/stab1744
  * Newman et al. (2012) Newman, A. B., Ellis, R. S., Bundy, K., & Treu, T. 2012, ApJ, 746, 162, doi: 10.1088/0004-637X/746/2/162
  * Peletier et al. (1990) Peletier, R. F., Davies, R. L., Illingworth, G. D., Davis, L. E., & Cawson, M. 1990, AJ, 100, 1091, doi: 10.1086/115582
  * Prugniel & Simien (1997) Prugniel, P., & Simien, F. 1997, A&A, 321, 111
  * Robertson et al. (2022) Robertson, B. E., Tacchella, S., Johnson, B. D., et al. 2022, arXiv e-prints, arXiv:2212.04480, doi: 10.48550/arXiv.2212.04480
  * Sánchez-Blázquez et al. (2014) Sánchez-Blázquez, P., Rosales-Ortega, F. F., Méndez-Abreu, J., et al. 2014, A&A, 570, A6, doi: 10.1051/0004-6361/201423635
  * Sandage (1972) Sandage, A. 1972, ApJ, 176, 21, doi: 10.1086/151606
  * Skelton et al. (2014) Skelton, R. E., Whitaker, K. E., Momcheva, I. G., et al. 2014, ApJS, 214, 24, doi: 10.1088/0067-0049/214/2/24
  * Suess et al. (2019a) Suess, K. A., Kriek, M., Price, S. H., & Barro, G. 2019a, ApJ, 877, 103, doi: 10.3847/1538-4357/ab1bda
  * Suess et al. (2019b) —. 2019b, ApJ, 885, L22, doi: 10.3847/2041-8213/ab4db3
  * Suess et al. (2020) —. 2020, ApJ, 899, L26, doi: 10.3847/2041-8213/abacc9
  * Suess et al. (2022) Suess, K. A., Bezanson, R., Nelson, E. J., et al. 2022, ApJ, 937, L33, doi: 10.3847/2041-8213/ac8e06
  * Szomoru et al. (2013) Szomoru, D., Franx, M., van Dokkum, P. G., et al. 2013, ApJ, 763, 73, doi: 10.1088/0004-637X/763/2/73
  * Tacchella et al. (2017) Tacchella, S., Carollo, C. M., Faber, S. M., et al. 2017, ApJ, 844, L1, doi: 10.3847/2041-8213/aa7cfb
  * Tacchella et al. (2015) Tacchella, S., Lang, P., Carollo, C. M., et al. 2015, ApJ, 802, 101, doi: 10.1088/0004-637X/802/2/101
  * Trujillo et al. (2004) Trujillo, I., Rudnick, G., Rix, H.-W., et al. 2004, ApJ, 604, 521, doi: 10.1086/382060
  * Trujillo et al. (2006) Trujillo, I., Förster Schreiber, N. M., Rudnick, G., et al. 2006, ApJ, 650, 18, doi: 10.1086/506464
  * van de Ven & van der Wel (2021) van de Ven, G., & van der Wel, A. 2021, ApJ, 914, 45, doi: 10.3847/1538-4357/abf047
  * van der Wel et al. (2005) van der Wel, A., Franx, M., van Dokkum, P. G., et al. 2005, ApJ, 631, 145, doi: 10.1086/430464
  * van der Wel et al. (2008) van der Wel, A., Holden, B. P., Zirm, A. W., et al. 2008, ApJ, 688, 48, doi: 10.1086/592267
  * van der Wel et al. (2012) van der Wel, A., Bell, E. F., Häussler, B., et al. 2012, ApJS, 203, 24, doi: 10.1088/0067-0049/203/2/24
  * van der Wel et al. (2014a) van der Wel, A., Chang, Y.-Y., Bell, E. F., et al. 2014a, ApJ, 792, L6, doi: 10.1088/2041-8205/792/1/L6
  * van der Wel et al. (2014b) van der Wel, A., Franx, M., van Dokkum, P. G., et al. 2014b, ApJ, 788, 28, doi: 10.1088/0004-637X/788/1/28
  * van Dokkum et al. (2008) van Dokkum, P. G., Franx, M., Kriek, M., et al. 2008, ApJ, 677, L5, doi: 10.1086/587874
  * Wuyts et al. (2012) Wuyts, S., Förster Schreiber, N. M., Genzel, R., et al. 2012, ApJ, 753, 114, doi: 10.1088/0004-637X/753/2/114
  * Zhang et al. (2019) Zhang, H., Primack, J. R., Faber, S. M., et al. 2019, MNRAS, 484, 5170, doi: 10.1093/mnras/stz339
  * Zibetti et al. (2009) Zibetti, S., Charlot, S., & Rix, H.-W. 2009, MNRAS, 400, 1181, doi: 10.1111/j.1365-2966.2009.15528.x

##  Appendix A Comparison with $R_{\rm{M_{\star}}}$ Estimates from the
Literature

Figure 12: Comparison of rest-frame optical half-light sizes and stellar-half
mass radii, equivalent to Figure 5, but now for four different estimates of
the half-mass radii as labeled above the panels. $R_{0.5\mu\rm{m}}$ is taken
from the respective authors. Bottom panels show results for $1<z<2$; top
panels show results for $0.5<z<1$, with the exception of the top-right panel:
Miller et al. (2023) did not provide $R_{\rm{M_{\star}}}$ estimates for $z<1$.
Instead, here we show the comparison of the Miller et al. (2023)
$R_{\rm{M_{\star}}}$ estimates and the $R_{0.5\mu\rm{m}}$ estimates used in
this paper (the bottom panel compares with the Miller et al. (2023)
$R_{0.5\mu\rm{m}}$estimates). Results are shown for quiescent (red) and star-
forming (blue) galaxies with total stellar mass $>10^{10}~{}M_{\odot}$.
$\Delta$ is the median offset in dex, with the 16-84%-ile scatter in
parentheses. $\sigma$ is the median formal uncertainty as reported by the
respective authors.  Figure 13: Comparison of rest-frame near-infrared half-
light sizes from NIRCam and stellar-half mass radii, equivalent to Figure 5,
but now for four different estimates of the half-mass radii as labeled above
the panels. Results are shown for quiescent (red) and star-forming (blue)
galaxies with total stellar mass $>10^{10}~{}M_{\odot}$. $\Delta$ is the
median offset in dex, with the 16-84%-ile scatter in parentheses. $\sigma$ is
the median formal uncertainty as reported by the respective authors. Suess et
al. (2020) does not provide $z>1$ $R_{\rm{M_{\star}}}$ estimates for galaxies
in the CEERS footprint; Miller et al. (2023) does not provide $z<1$
$R_{\rm{M_{\star}}}$ estimates.

Previously published estimates of stellar half-mass radii based on CANDELS
data are shown in Figures 12 and 13 in the same manner as in Figures 5 and 6.

Figure 12 compares, for four different authors, the rest-frame optical half-
light radii $R_{\rm{opt}}$333Except for our own estimates, which are at rest-
frame 0.5$\mu$m, the plotted values are measured from HST/WFC3/F160W CANDELS
imaging with half-mass radii $R_{\rm{M_{\star}}}$, each time comparing the
radii as published by the authors. The one exception is the top-right panel
(see figure caption for details). Even though all $R_{\rm{M_{\star}}}$
estimates are systematically smaller than $R_{\rm{opt}}$, the offsets and
scatter vary from author to author. Our estimates show similar offsets and
scatters as those by Suess et al. (2020), adopting their preferred Method 1
estimates. The main difference is that their uncertainties are several times
smaller than ours. The $R_{\rm{M_{\star}}}$ estimates by Mosleh et al. (2020)
and Miller et al. (2023) show much larger scatters, and very small formal
uncertainties ($\lesssim 0.05$ dex). Taken at face value this means that these
authors find a large (0.2$-$0.3 dex) galaxy-to-galaxy scatter in the
$R_{\rm{M_{\star}}}$$/$$R_{0.5\mu\rm{m}}$ (in fact, larger than the scatter in
$R_{\rm{opt}}$ at fixed mass).

We note that in all cases the $R_{\rm{opt}}$ agree well between the authors,
with small scatter and no systematic offsets. The largest scatter (0.1 dex) is
found when comparing with Miller et al. (2023), who model the light profiles
in a fundamentally different manner (multi-gauss expansion rather than Sérsic
profile fitting). These differences do not explain the different trends and
patterns seen in Figure 12 – rather, those are due to the variety in
techniques to correct for $M_{\star}/L$ gradients (see discussion in Sec.
2.1).

The comparison with NIRCam-based rest-frame near-infrared sizes used in this
paper provides additional insights, as illustrated in Figure 13. As already
demonstrated in Section 2.5 the $R_{\rm{M_{\star}}}$ estimates presented in
this paper compare well with the NIRCam sizes, with a reasonably small scatter
of similar magnitude as the formal uncertainties. The $R_{\rm{M_{\star}}}$
estimates by Mosleh et al. (2020) and Miller et al. (2022) show much larger
scatter, especially for quiescent galaxies (0.2-0.3 dex) suggesting low
precision, especially when compared to the very small formal uncertainties
(0.01-0.06 dex).

Unfortunately, Suess et al. (2020) $R_{\rm{M_{\star}}}$ estimates for the
CEERS NIRCam sample are not available for $z>1$ and the comparison is limited
to $z<1$. For those galaxies the Suess et al. (2020) $R_{\rm{M_{\star}}}$
estimates agree fairly well with the NIRCam sizes, with somewhat larger
scatter than our $R_{\rm{M_{\star}}}$estimates. It should be kept in mind that
light profiles used by Suess et al. (2020) are not identical to our profiles
so that the comparison is not entirely fair and straightforward. The scatter
in $R_{0.5\mu\rm{m}}$between Suess et al. (2020) and ours is $0.05$ dex, which
can explain the difference in scatter for the star-forming galaxies. The
sample of quiescent galaxies is too small to make a reliable statement.

We must now address the question where the tension arises between the
conclusions presented by Suess et al. (2020), who find little or no evolution
in $R_{\rm{M_{\star}}}$for quiescent galaxies at $z>1$ and the results
presented here in Figure 11, with strong evolution in $R_{\rm{M_{\star}}}$for
massive quiescent galaxies up to $z=2.3$. The size comparisons in Figures 12
and 13 do not provide immediate answers.

Figure 14: Redshift evolution of the ratio
$R_{\rm{M_{\star}}}$$/$$R_{0.5\mu\rm{m}}$ of massive quiescent galaxies. Left:
Results from this paper; Right: Results from Suess et al. (2020). Due to the
systematic difference in $M_{\star}$ the stellar mass limit is shifted
accordingly (see text for discussion).

Figure 14 shows the evolution in the light-to-mass weighted size ratio
($R_{\rm{M_{\star}}}$$/R_{\rm{opt}}$), comparing the results presented in this
paper and those from Suess et al. (2020). In both cases the measurements from
the respective authors are used without matching catalogs. Differences in
redshift measurements, stellar mass estimates and size estimates can all
contribute to differences in the comparison. Notably, the stellar mass
estimates used here are systematically larger by 0.2 dex, which is here
accounted for by lowering the stellar mass cut. Up to $z\approx 2$ the
patterns are similar, with $R_{\rm{M_{\star}}}$ estimates that are 0.1-0.2 dex
lower than $R_{0.5\mu\rm{m}}$, both works showing no evidence for a different
pace of evolution in $R_{\rm{M_{\star}}}$ compared to $R_{0.5\mu\rm{m}}$.

The main difference arises at $z>2$, where Suess et al. (2020) find no offset
and we do. For Suess et al. (2020) the scatter in
$R_{\rm{M_{\star}}}$$/$$R_{0.5\mu\rm{m}}$at $z>1.5$ ($>$0.2 dex)is larger than
the scatter in $R_{\rm{opt}}$. For our own estimates this is not the case, but
we see an increased number of outliers at $z>1.5$. These trends suggest that
uncertainties start dominating over the corrective effect of accounting for
$M_{\star}/L$ gradients at $z>2$. A more definitive statement on the evolution
of $R_{\rm{M_{\star}}}$ beyond $z=2$ will have to wait for NIRCam size
measurements for larger samples.