# Accurate and Model Independent Radius Determination of Single FGK and M
Dwarfs Using Gaia DR3 Data

Rocio Kiman Department of Astronomy, California Institute of Technology,
Pasadena, CA 91125, USA  Kavli Institute for Theoretical Physics, University
of California, Santa Barbara, CA 93106, USA Rocio Kiman<EMAIL_ADDRESS>Timothy D. Brandt Space Telescope Science Institute, 3700 San Martin Dr,
Baltimore, MD 21218, USA  Department of Physics, University of California,
Santa Barbara, Santa Barbara, CA 93106, USA  Jacqueline K. Faherty Department
of Astrophysics, American Museum of Natural History, Central Park West at 79th
St, New York, NY 10024, USA  Mark Popinchalk Department of Astrophysics,
American Museum of Natural History, Central Park West at 79th St, New York, NY
10024, USA  Physics, The Graduate Center, City University of New York, New
York, NY 10016, USA  Department of Physics and Astronomy, Hunter College, City
University of New York, 695 Park Avenue, New York, NY 10065, USA

(Received 22-Jan-2024; Revised 22-Apr-2024; Accepted 20-May-2024)

###### Abstract

Measuring fundamental stellar parameters is key to fully comprehending the
evolution of stars. However, current theoretical models over-predict effective
temperatures, and under-predict radii, compared to observations of K and M
dwarfs (radius inflation problem). In this work, we developed a model
independent method to infer precise radii of single FGK and M dwarfs using
Gaia DR3 parallaxes and photometry, and we used it to study the radius
inflation problem. We calibrated nine surface brightness-color relations for
the three Gaia magnitudes and colors using a sample of stars with angular
diameter measurements. We achieved an accuracy of $4\%$ in our angular
diameter estimations, which Gaia’s parallaxes allow us to convert to a
physical radii. We validated our method by comparing our radius measurements
with literature samples and the Gaia DR3 catalog, which confirmed the accuracy
of our method and revealed systematic offsets in the Gaia measurements.
Moreover, we used a sample with measured ${\rm H}\alpha$ equivalent width
(${\rm H}\alpha\;{\rm EW}$), a magnetic activity indicator, to study the
radius inflation problem. We demonstrated that active stars have larger radii
than inactive stars, showing that radius inflation is correlated with magnetic
activity. We found a correlation between the radius inflation of active stars
and ${\rm H}\alpha\;{\rm EW}$ for the mass bin $0.5<M[{\rm M_{\odot}}]\leq
0.6$, but we found no correlation for lower masses. This could be due to lack
of precision in our radius estimation or a physical reason. Radius
measurements with smaller uncertainties are necessary to distinguish between
the two scenarios.

††journal: Astronomical Journal††software: scipy (Virtanen et al., 2020);
numpy (Harris et al., 2020); matplotlib (Hunter, 2007); astropy (Astropy
Collaboration et al., 2013, 2018, 2022); emcee (Foreman-Mackey et al., 2013)

##  1 Introduction

Measuring fundamental properties of stars, like radius and effective
temperature, is key to understanding stellar evolution and developing precise
theoretical models. Currently there is disagreement between models and
measurements of radius and effective temperatures of low-mass stars. Measured
radii are larger than what models predict for their mass, age and metallicity,
and effective temperatures are lower. This phenomenon is known as radius
inflation, and since the first time it was discovered (Hoxie, 1970, 1973), the
difference between models and measurements for radius and temperature has been
studied extensively. Inflation has been calibrated by several studies using
both single and binary stars with results such as: $5-15\%$ inflation of the
radius (Ribas, 2006), $15-20\%$ (Berger et al., 2006) and $5-10\%$ (Torres,
2013), but without precise agreement. The cause of radius inflation is still
being debated, though the leading explanation is magnetic activity.

Magnetic activity has been shown theoretically to inhibit convective transport
of energy, which generates an increase in the stellar radius, and a decrease
of surface temperature. This effect could be caused by: strong magnetic fields
(Gough & Tayler, 1966; Mullan & MacDonald, 2001; Chabrier et al., 2007;
MacDonald & Mullan, 2012; Feiden & Chaboyer, 2013) or spot coverage (Chabrier
et al., 2007; Somers & Pinsonneault, 2015; Somers et al., 2020). Although in
theory magnetic activity can explain the difference between radius
measurements and models, not all observational studies agree. Several studies
have found an empirical correlation between radius inflation and magnetic
activity by using eclipsing binaries (López-Morales, 2007; Kraus et al., 2011;
Irwin et al., 2011; Feiden & Chaboyer, 2012; Birkby et al., 2012; Torres et
al., 2018, 2019, 2020, 2021b). Detached eclipsing binaries provide an optimal
stage to study radius inflation because they allow for a precise measurement
of the mass and radius of both components (${\sim}3\%$ uncertainty, Torres et
al. 2010). Nonetheless, these studies do not confirm the relation between
magnetic activity and inflation for single stars, given that the influence of
tidal interactions make the stars in eclipsing binaries spin up, as it was
pointed out in Torres et al. (2021b). In this study the authors say that the
rotation period of the binary is more than an order of magnitude faster than
single stars of the same mass in the cluster (Curtis et al., 2020).

Previous studies have shown empirically that there is a correlation between
radius inflation and magnetic activity for single stars (Morales et al., 2008;
Stassun et al., 2012). On the other hand, several studies have failed to find
a correlation between radius inflation and magnetic activity or rotation for
single stars (López-Morales, 2007; Mann et al., 2015; Kesseli et al., 2018;
Morrell & Naylor, 2019), which could be due to activity being weaker in single
stars in comparison to binary stars (Demory et al., 2009; Boyajian et al.,
2012a; Spada et al., 2013). Although the relation between magnetic activity
and radius inflation for single stars has been studied extensively, a clear
empirical correlation using direct estimation of the radius of the stars is
still missing.

As mentioned above, eclipsing binaries have been used extensively in the
literature to measure radius and mass with high precision, although most of
these studies analyse a small number of stars due to the complexity of these
measurements (e.g., Torres & Ribas, 2002; López-Morales & Ribas, 2005; Torres
et al., 2014, 2018; Cruz et al., 2018; Torres et al., 2020, 2021b, 2021a;
Jennings et al., 2023). There have been recent studies with large samples of
eclipsing binaries, but they have not achieved the same precision in radius
and mass measurements as previous studies (Mowlavi et al., 2022; Cruz et al.,
2022). Another way to perform a precise analysis of radius inflation would be
using a large enough sample to average out statistical uncertainty. In order
to apply such a technique, we need a method to estimate radii which can be
applied to a large sample of stars.

There are different ways to estimate a stellar radius for single stars, but
obtaining an angular diameter measurement is the method that has comparable,
or slightly larger, uncertainty to eclipsing binaries (Spada et al., 2013).
The Center for High Angular Resolution Astronomy (CHARA) Array (ten Brummelaar
et al., 2005; McAlister et al., 2005) is key for interferometric measurements
of stellar angular diameters. CHARA is an optical/IR interferometer, with six
$1$ m aperture telescopes in the shape of a Y, two telescopes in each branch,
providing 15 baselines ranging from $34.1$ to $330.7$ m. CHARA allows precise
measurements of angular diameter for stars ($\lesssim 5\%$, e.g., Berger et
al., 2006; Boyajian et al., 2012b, a). These measurements are observationally
challenging and they require bright stars ($H,K^{\prime}<6.5$, $V<11$,
Boyajian et al. 2012a). Luckily, using a set of measurements of angular
diameters, it is possible to estimate each star’s surface brightness and
calibrate the surface brightness-colour relation (SBCR, Wesselink, 1969;
Barnes & Evans, 1976; Barnes et al., 1976; Fouque & Gieren, 1997), which
allows the estimation of the angular diameter from the color and magnitude of
a star. This relation has been calibrated extensively in the literature using
angular diameter measurements (e.g., Kervella et al., 2004; Boyajian et al.,
2014; Challouf et al., 2014; Salsi et al., 2020, 2021) and using eclipsing
binaries (e.g., Graczyk et al., 2017).

With the release of the Gaia DR3 all-sky survey of stellar astrometry and
photometry (Gaia Collaboration et al., 2016, 2022; Babusiaux, C. et al.,
2022), it is now possible to calibrate the SBCR with unprecedented precision.
Moreover, combining the parallax measurements from Gaia with the SBCR, we can
estimate stellar radius for most of the stars in the Gaia catalog. Recently
Graczyk et al. (2021) used Gaia photometry to obtain the SBCR using eclipsing
binaries, however their relation does not account for the M dwarf regime,
which is the main focus of this article. Salsi et al. (2020, 2021) calibrated
the SBCR using a sample of angular diameter measurements from the literature
(Duvert, 2016) for F, G, K and M dwarfs. However, they fit the relation for
FGK and M main sequence stars separately, which causes a systematic error when
estimating the radius of stars, as we will show in this work. In addition,
they calibrated the relation for the $(G-K_{\rm s})$ color, causing the
calculation of the angular diameter to depend on having magnitudes from The
Two Micron All Sky Survey (2MASS, Skrutskie et al., 2006). Therefore, we
decided to calibrate the SBCR for FGK and M dwarfs using only Gaia photometry,
which will provide a more precise radius estimation.

In this study we calibrate the SBCR using only Gaia photometry. We use a
sample of literature measurements of angular diameters, and we take advantage
of the quality cuts of the Gaia data to select the best possible calibration
sample and fit the relation. The description of the calibration sample and the
fit of the Gaia SBCR is in Section 2. In Section 3 we use our Gaia SBCR to
estimate radius for several literature samples with measured radius to analyze
the performance of our method. In Section 4 we study the influence of
metallicity, extinction and variability on the estimation of radius with our
method. Finally, in Section 5 we estimate the radii of a statistically large
sample of M dwarfs from Kiman et al. (2019) with ${\rm H}\alpha$ equivalent
width measurements to study the relation between magnetic activity and radius
inflation for single stars.

##  2 Gaia DR3 Surface Brightness-Colour Relation

We started with the Stefan-Boltzmann law to derive the Gaia SBCR which will
allow the measurement of stellar radii from photometry and parallaxes. To
estimate the bolometric flux that we measure on Earth, we need to take into
account the angular diameter ($\theta$) of the star, which transforms the
Stefan-Boltzmann law to:

$F_{\rm BOL_{Earth}}=\frac{\theta^{2}}{4}\sigma T_{\rm eff}^{4}.$ (1)

where $\sigma$ is the Stefan-Boltzmann constant and $T_{\rm eff}$ is the
effective temperature of the star. Equation (1) is from Casagrande et al.
(2006). The bolometric flux, $F_{\rm BOL_{Earth}}$, can be estimated as the
flux in a photometric band $m_{\lambda}$ plus a bolometric correction which
can be estimated as a polynomial of a color index ($X$)

$F_{\rm BOL_{Earth}}\approx 10^{-0.4m_{\lambda}}p_{1}(X).$ (2)

Using a similar approximation for the effective temperature, $T_{\rm
eff}^{4}$, we get

$T_{\rm eff}^{4}\approx p_{2}(X)$ (3)

where $p_{2}(X)$ is a polynomial of the color. Combining equations (1) and the
approximations (Equations (2) and (3)) we obtain

$\theta^{2}\approx 10^{-0.4m_{\lambda}}p_{3}(X)$ (4)

where $p_{3}(X)=p_{1}(X)/p_{2}(X)$. Finally, for our calibration we define
$S_{\rm m_{\lambda}}$ such that

$\log_{10}S_{\rm m_{\lambda}}=\log_{10}(\theta^{2}10^{0.4m_{\lambda}})=p(X)$
(5)

for a magnitude $m_{\lambda}$ and a color index $X$. Equation (5) is our SBCR.
In the rest of this section we will calibrate Equation (5) for the three Gaia
bands $m_{\lambda}=G$, $G_{\rm RP}$, $G_{\rm BP}$, and the three Gaia colors
$X=(G_{\rm BP}-G_{\rm RP})$, $(G-G_{\rm RP})$, $(G_{\rm BP}-G)$.

###  2.1 Calibration sample

We calibrated the SBCR described in Equation (5) using the catalog JMMC (Jean-
Marie Mariotti Center) Measured Stellar Diameters Catalog (JMDC, Duvert,
2016), which is a compilation of angular diameter measurements for stars of
different masses and evolutionary stages. The catalog has $2013$ rows, with
$1061$ single sources as of January 22 2024. The JMDC catalog provides uniform
disc angular diameter ($\theta_{\rm UD}$) measurements and the limb-darkened
angular diameter ($\theta_{\rm LD}$). In our analysis we used $\theta_{\rm
LD}$.

The precision of the calibration of the SBCR depends strongly on the precision
of the angular diameter measurements, and we therefore defined quality cuts to
keep the best subsample of the JMDC catalog. The JMDC catalog is a compilation
of angular diameter measurements obtained in different studies. These were
estimated using different techniques: optical interferometry, lunar
occultation or intensity interferometry. For consistency when comparing
angular diameter values, we retained only optical interferometry measurements.
In addition, we kept angular diameter measurements with a value over error
higher than $20$. In the catalog, $429$ stars ($67$ main sequence stars, see
below how we selected giant stars) have more than one stellar diameter
measurement. From those, we kept only the stars whose measurements agreed
within $3\sigma$. We performed tests with different thresholds for consistency
and found $3\sigma$ to be best. We compare the effects of choosing a different
rejection threshold in units of $\sigma$ in Section 2.3. In total we discarded
$12$ main sequence stars which had inconsistent measurements to remove the
scatter caused by the combination of incompatible measurement methods.

We cross-matched the JMDC sample with Gaia DR3, and found that $806$ objects
out of $1061$ had Gaia matches. To perform the cross-match we used astroquery
(Ginsburg et al., 2019) to find the Gaia DR3 (or DR2 in case DR3 was not
available) and 2MASS IDs of each star from their Simbad
ID111http://simbad.u-strasbg.fr/simbad/sim-fbasic, given in the JMDC catalog.
Then we used the Gaia archive222https://gea.esac.esa.int/archive/ to retrieve
the information from the Gaia DR3 catalog (proper motion, parallax and $G$,
$G_{\rm RP}$ and $G_{\rm BP}$ magnitudes, etc.) and the 2MASS catalog (JHK
magnitudes). As the low-mass stars in the JMDC catalog are nearby, and
therefore have high proper motions, this technique helped us avoid the
mismatches which often occur when doing a positional cross-match. To
understand why some stars do not have a cross-match in Gaia, we estimated
$G$-mag from 2MASS magnitudes using the transformation given in Riello et al.
(2021). We found that almost all the stars that do not have a Gaia cross-match
have an estimated magnitude of $G\leq 3$, which is the instrument’s brightness
limit given by the collaboration333https://www.cosmos.esa.int/web/gaia/dr3.

Taking advantage of the quality flags in the Gaia catalog, we included extra
quality cuts to the JMDC sample: 1) Flux over error higher than $10$ for the
three bands ($G$, $G_{\rm RP}$ and $G_{\rm BP}$); 2) Parallax over error
higher than $10$; 3) Renormalised Unit Weight Error (RUWE) $<1.4$ to remove
possible binaries and bad fits, and removed stars indicated in the non-single
star flag by Gaia; 4) Removed sources marked as variable in Gaia; 5) Distance
$<100$ pc to minimize the effect of extinction.

Salsi et al. (2020, 2021) showed that the SBCR is different for main-sequence
and giant stars. We confirmed this difference using our calibration sample and
our giant classification. Therefore, for our calibration we kept only main
sequence stars. To separate the giants from the main sequence stars, we
approximately selected main sequence stars and we fit a $7^{\rm th}$-degree
polynomial, $f(x)$ to the main sequence stars in the color-magnitude diagram
($M_{\rm G}$ versus $(G_{\rm BP}-G_{\rm RP})$ in Figure 1). The results of the
fit was

$\begin{split}f(x)=&0.07\times x^{7}-0.92\times x^{6}+4.67\times x^{5}-\\\
&11.14\times x^{4}+12.38\times x^{3}-6.06\times x^{2}+\\\ &6.27\times
x+0.04\end{split}$ (6)

where $x=G_{\rm BP}-G_{\rm RP}$. All the stars that were at least 2 magnitudes
brighter than $f(x)$ were considered giants. In addition, we removed stars
that had a spectral type classification of I, II, III or IV in the Simbad
database. From the $1061$ single stars in the JMDC catalog, in total we
removed $840$ giants.

We limit the fit to the color range $0.42<G_{\rm BP}-G_{\rm RP}<3.2$ where the
JMDC catalog provides sufficient data. This cut removed two late type M
dwarfs, and limits our calibration to main sequence stars of spectral type
F$0$ to M$4$. After all the cuts, we have $70$ main sequence stars, with $85$
angular diameter measurements in total from different studies (Lane et al.,
2001; Ségransan et al., 2003; Kervella et al., 2004, 2008; Berger et al.,
2006; Baines et al., 2008; Boyajian et al., 2008, 2012b, 2012a, 2013; Bazot et
al., 2011a, b; von Braun et al., 2011, 2012; Crepp et al., 2012; Huber et al.,
2012; Perraut et al., 2013, 2015, 2016; White et al., 2013; Maestro et al.,
2013; Howard et al., 2014; Mennesson et al., 2014; Creevey et al., 2015; Jones
et al., 2015; Kane et al., 2015, 2017; Tanner et al., 2015; Fulton et al.,
2016; Ligi et al., 2016, 2019; Bonnefoy et al., 2018; Karovicova et al., 2018,
2020; Schaefer et al., 2018; Borgniet et al., 2019; Rabus et al., 2019;
Romanovskaya et al., 2019; Wood et al., 2019; Salsi et al., 2021).

Metallicity has been shown to have a significant influence on the SBCR (e.g.,
Boyajian et al., 2012b, a). Therefore, it is necessary to include a
metallicity measurement in the calibration of the SBCR to reduce systematic
uncertainties. We compiled the metallicity measurements ([Fe/H]) for our
sample from the studies mentioned above and other literature measurements
(Gáspár et al., 2016; Soubiran et al., 2016), and kept in our calibration
sample only the $65$ stars with ${\rm[Fe/H]}$ measurements. Our calibration
sample has metallicity values in the range $-2.44<[{\rm Fe/H}]<0.39$ dex, and
the metallicity distribution is shown in Figure 1.

In conclusion, from the $1061$ single stars in the JMDC catalog, after
removing all the giant stars ($840$) and applying our quality cuts, our
calibration sample contains $65$ main sequence stars with metallicity
measurements. This sample will be use in the reminder of the analysis.

Figure 1: Color-magnitude diagram of the JMDC catalog after our quality cuts,
color-coded by the literature metallicities [Fe/H]. The distribution of
metallicities for our sample is shown in the small panel at the top right of
the figure. We also included in light-gray the $100$ pc sample from Gaia DR3
for comparison. We obtained the $100$ pc sample from the Gaia archive applying
a cut in parallax, together with a value over error $>50$ quality cut for
parallax and fluxes in the three Gaia bands.

###  2.2 Fit and results of the Gaia DR3 SBCR

Using our clean subset of the JMDC catalog cross-matched with Gaia DR3
(Section 2.1), we calibrated the Gaia DR3 SBCR in Equation (5) for main
sequence stars. We used the three Gaia magnitudes ($m_{\lambda}=G$, $G_{\rm
RP}$ and $G_{\rm BP}$), the three colors ($X=G_{\rm BP}-G_{\rm RP}$, $G-G_{\rm
RP}$ and $G_{\rm BP}-G$) and a second degree polynomial for the $p(X)$
function in Equation (5). To fit this relation we used a Gaussian mixture
model. The likelihood for each star is the sum of a Gaussian distribution with
the measurement uncertainty and a second, broader Gaussian to account for
outliers. We modeled the distribution of outliers with standard deviation
$\sigma_{\rm out}=0.3$, which we chose to be larger than the typical
measurement uncertainty, and we assumed that the prior probability of a star
being an outlier is $1-g$, with $g=0.9$. We tried different values for
$\sigma_{\rm out}$ and $g$ and found that the ones mentioned above were the
best suited for our analysis. This test is discussed in Section 2.3. The
likelihood of our model is defined as

$\displaystyle\mathcal{L}=\prod_{i}\bigg{(}$ $\displaystyle
A_{i}\exp\left(\frac{-(\log_{10}S_{i}-\log_{10}S_{\rm
m_{\lambda,i}})^{2}}{2\sigma_{i}^{2}}\right)$
$\displaystyle+B_{i}\exp\left(\frac{-(\log_{10}S_{i}-\log_{10}S_{\rm
m_{\lambda,i}})^{2}}{2(\sigma_{i}^{2}+\sigma_{\rm out}^{2})}\right)\bigg{)}$
(7)

with

$\displaystyle A_{i}$ $\displaystyle=\frac{g}{\sqrt{2\pi\sigma_{i}^{2}}}$ (8)
$\displaystyle B_{i}$
$\displaystyle=\frac{1-g}{\sqrt{2\pi(\sigma_{i}^{2}+\sigma_{\rm out}^{2})}}$
(9)

where $\log_{10}S_{\rm m_{\lambda,i}}$ is the quantity defined in Equation (4)
which we calculated from the angular diameter measurements and Gaia
magnitudes, and $\log_{10}S_{i}$ is calculated from the proposed $p(X)$ such
as

$\log_{10}S_{i}=(a\times X^{2}+b\times X+c)\times(1+d\times{\rm[Fe/H]})$ (10)

where $X$ is the color index, and $a$, $b$, $c$ and $d$ are the parameters of
the models that we are fitting. We compared this functional form with a simple
polynomial, but found that the one in Equation (10) is the most suited for our
problem as suggested by Mann et al. (2019). We discuss the comparison of the
functional forms in Section 2.3. We performed an MCMC-fit using emcee
(Foreman-Mackey et al., 2013) to sample the posterior composed of the
likelihood in Equation (10) and flat priors in all four parameters. We ensured
the convergence of the MCMC by calculating the integrated autocorrelation time
($\tau_{f}$) and performing $100\tau_{f}$ steps, which corresponds to
${\sim}100$ independent samples, following the suggestions in the
documentation of
emcee444https://emcee.readthedocs.io/en/stable/tutorials/autocorr/. We
performed this fit to obtain a posterior distribution for the parameters $a$,
$b$, $c$ and $d$, for the three Gaia magnitudes and the three colors. One
example of the posterior distribution is shown in the Figure 2. As the
distributions of the parameters are nearly Gaussian, we report the
uncertainties as the standard deviation in the remainder of the paper.

Figure 2: Grid plot of the posterior distributions for the four parameters of
the likelihood in Equation (7) ($a$, $b$, $c$ and $d$) obtained with a MCMC-
fit using emcee for the case $S_{\rm G}$ and $(G_{\rm BP}-G_{\rm RP})$.

The results of the fit for each combination of Gaia magnitudes and colors are
shown in Figure 3, and the values of the coefficients are in Table 1. We
included $100$ random samples of the fit in Figure 3 for three cases of
metallicity: [Fe/H]$=\\{-0.5,0,0.1\\}$ dex. We show in Section 4.1 that the
metallicity correction helps reduce systematic uncertainties when estimating
stellar radii. More than $80\%$ of the sample is within $2\sigma$ from the fit
for all the relations. We also found that the largest offsets and residual
scatters are for the lowest-mass stars for the cases where the $(G_{\rm
BP}-G)$ color is included. This is likely due to the low SNR in the flux on
the BP band for the low-mass stars in comparison to the high-mass stars (even
though all stars have high SNR $>300$ in this band), combined with the small
number of low-mass stars to perform the fit. Therefore we suggest choosing
between these nine combinations according to the data in each case. For
example, using the fit with $S_{\rm G}$ and $(G-G_{\rm RP})$ to calculate
angular diameters of faint M dwarfs would provide better results in the case
where the $G_{\rm BP}$ magnitude has low SNR.

Figure 3: Results for our calibration of the Gaia SBCR. We show all the
possible combinations between the fluxes (Equation (5)) for the three Gaia
magnitudes and the three Gaia colors. We color-coded the calibration sample
with metallicity, which covers a range from $-2.44<{\rm[Fe/H]<0.39}$ dex. We
included $100$ random samples of the fit for three metallicities color-coded
using the same scale as shown in the color-bar: ${\rm[Fe/H]}={-0.5,0,0.1}$.
Finally we included the fitted parameters for each case, that are also shown
in Table 1. Table 1: Results of the fit for the parameters of the model shown
in the likelihood in Equation (5).

Magnitude | Color | Color range [mag] | $a$ | $b$ | $c$ | $d$
---|---|---|---|---|---|---
$S_{\rm G}$ | $G_{\rm BP}-G_{\rm RP}$ | $0.44-2.83$ | $-0.143\pm 0.004$ | $1.22\pm 0.01$ | $0.92\pm 0.01$ | $-0.017\pm 0.002$
$S_{\rm G}$ | $G-G_{\rm RP}$ | $0.28-1.24$ | $-0.09\pm 0.03$ | $2.03\pm 0.05$ | $0.85\pm 0.02$ | $-0.011\pm 0.002$
$S_{\rm G}$ | $G_{\rm BP}-G$ | $0.17-1.61$ | $-0.56\pm 0.01$ | $2.18\pm 0.02$ | $1.15\pm 0.01$ | $-0.027\pm 0.002$
$S_{\rm G_{BP}}$ | $G_{\rm BP}-G_{\rm RP}$ | $0.44-2.83$ | $-0.116\pm 0.004$ | $1.37\pm 0.01$ | $0.91\pm 0.01$ | $-0.015\pm 0.002$
$S_{\rm G_{BP}}$ | $G-G_{\rm RP}$ | $0.28-1.24$ | $0.27\pm 0.05$ | $2.03\pm 0.07$ | $0.9\pm 0.03$ | $-0.007\pm 0.002$
$S_{\rm G_{BP}}$ | $G_{\rm BP}-G$ | $0.17-1.61$ | $-0.56\pm 0.01$ | $2.58\pm 0.02$ | $1.15\pm 0.01$ | $-0.025\pm 0.002$
$S_{\rm G_{RP}}$ | $G_{\rm BP}-G_{\rm RP}$ | $0.44-2.83$ | $-0.115\pm 0.004$ | $0.97\pm 0.01$ | $0.91\pm 0.01$ | $-0.019\pm 0.002$
$S_{\rm G_{RP}}$ | $G-G_{\rm RP}$ | $0.28-1.24$ | $-0.09\pm 0.03$ | $1.63\pm 0.05$ | $0.85\pm 0.02$ | $-0.012\pm 0.002$
$S_{\rm G_{RP}}$ | $G_{\rm BP}-G$ | $0.17-1.61$ | $-0.45\pm 0.02$ | $1.73\pm 0.03$ | $1.11\pm 0.01$ | $-0.024\pm 0.003$

Note. — This fit is valid for the metallicity range $-2.44<[{\rm Fe/H}]<0.39$
dex.

Note. — A Python code and the posterior distribution for each of the
parameters of the fit are provided in Zenodo which can be used to estimate
radii applying our calibration: https://zenodo.org/records/11401588.

###  2.3 Test fit choices

The fit of the Gaia SBCR to the calibration sample in Section 2.2 required
several important decisions. Here we discuss each one and the alternatives
that we studied before choosing the final fit.

Our calibration sample was described in Section 2.1, and we discussed a
$3\sigma$ cut to select consistent measurements of angular diameters between
the stars that had more than one measurement. After the cross-match with Gaia
DR3 and removing the giant stars, we found that $67$ stars had more than one
measurement of angular diameter. Out of these stars, $12$ stars had
inconsistent measurements, meaning they did not agree within $3\sigma$ ($18\%$
of the stars with repeated measurements). We tested this same cut using
thresholds of $2\sigma$, $4\sigma$ and $5\sigma$. As expected, when the
threshold increases, the amount of inconsistent measurements decreases (with
cuts of $2\sigma$, $4\sigma$ and $5\sigma$ we found $30\%$, $9\%$ and $7.5\%$
of the stars with repeated measurements were considered inconsistent,
respectively). However we noticed that for the $2\sigma$ sample, the lowest
metallicity stars (${\rm[Fe/H]}<-0.5$) were removed. We applied our new method
to estimate radii for a sample of stars with radius measurements from the
literature (Mann et al. 2015, see Section 3 for a description of the catalog),
and found that when using the fit resulting from the $2\sigma$ sample, our
radius estimations had a significant systematic uncertainty compared to the
literature generated by metallicity. This systematic uncertainty was removed
when using the fit to the $3\sigma$ sample. Finally, the fit to the $a$, $b$,
$c$ and $d$ parameters was similar for the $3\sigma$, $4\sigma$ and $5\sigma$
thresholds, and the MCMC fit with a $3\sigma$ cut took fewer steps to converge
than the others. We took the more conservative approach to select the best and
compatible measurements and chose the $3\sigma$ sample.

We used a mixture model to fit our SBCR, as discussed in Section 2.2. For this
fit we assumed the outlier distribution to have a width of $\sigma_{\rm out}$,
a larger dispersion than the typical uncertainty value, and each point had a
probability of being an outlier of $(1-g)$. We performed the fit for the
different combinations of Gaia colors and magnitudes for a grid of values for
$g=\\{0.8,0.85,0.9,0.95\\}$ and $\sigma_{\rm
out}=\\{0.2,0.25,0.3,0.35,0.4\\}$, and compared the results for the four free
parameters $a$, $b$, $c$ and $d$. We found that the best-fit values and
confidence intervals on $a$, $b$, $c$, and $d$ parameters do not depend on
$\sigma_{\rm out}$ for values $>0.2$, so we decided to fix $\sigma_{\rm
out}=0.3$. In addition, we found that the parameters do not change
significantly with $g$. For a fixed $\sigma_{\rm out}=0.3$, there was a
maximum difference in the parameters of less than $1\%$ for the set of values
of $g$. Thus we decided to fix $g=0.9$, given that different values did not
modify the results.

Last we discuss the choice of functional form. We tried two different
functional forms to include $[{\rm Fe/H}]$ in the model: the one indicated at
Equation (10) and $a\times X^{2}+b\times X+c+d\times{\rm[Fe/H]}$. We found no
significant difference between the two in the metallicity range of our
calibration sample, however the MCMC-fit for the second case took longer to
converge, and in the case of $\log_{10}(S_{\rm G_{RP}})$ with the $(G_{\rm
BP}-G)$ color, the fit did not converge according to the criterion described
in Section 2.2, even after increasing the total amount of steps. Therefore we
concluded that Equation (10) (as confirmed by Mann et al. 2019) is more suited
to include metallicity in the model.

###  2.4 Radius estimation with the Gaia SBCR

Combining the calibrated Gaia SBCR described in the previous section and the
Gaia parallaxes, we can estimate stellar radius. For any given star in the
valid color and metallicity range, using a Gaia magnitude and a color with the
Gaia SBCR we can estimate an angular diameter. Then with the star’s distance
measurement ($D$) from Gaia, we can estimate the stellar radius ($R$) using
the following relation where $\theta_{\rm LD}$ is the limb darkened angular
diameter:

$R=\frac{D\theta_{\rm LD}}{2}$ (11)

for $\theta_{\rm LD}$ small, and where $\theta_{\rm LD}$ is in radians555The
$\theta_{\rm LD}$ measurements in Duvert (2016) are published in mas,
therefore we converted the units using the equation $\theta_{\rm
LD_{rad}}=\theta_{\rm LD_{mas}}\times\pi/(1000\times 180\times 3600)$.

To study the accuracy of our method, we compared the stellar radii from the
JMDC sample calculated using their angular diameters and the parallaxes from
Gaia, with the stellar radii calculated using our Gaia SBCR. We found that for
$68\%$ of the sample, meaning between $16^{\rm th}$ and $84^{\rm th}$
percentiles, the difference between the radii estimated with our method and
the original radii is $4\%$ or less. The rest of the sample, has a difference
of $10\%$ or less. This result holds for all the relations shown in Figure 3,
meaning it is not dependent on the Gaia band used. We note that the JMDC
sample contains only bright stars with excellent photometry in the three Gaia
bands. For fainter star with larger flux uncertainties, the radius estimation
will have larger uncertainties as well.

###  2.5 Comparison with SBCR from the literature

To facilitate the comparison of our method to literature SBCRs, we used
different relations to estimate radii for the clean sample of stars from the
JMDC catalog, meaning the stars left after our quality cuts, described in
Section 2.1. Using the most recent SBCRs from Graczyk et al. (2021) and Salsi
et al. (2021), we estimated angular diameters using the Gaia $G$ magnitude and
the $(G-K_{\rm s})$ color. We then estimated radii from the angular diameter
measurements combined with the Gaia parallax, as described in Section 2.4. The
comparison of the radii estimated using our method to the other two SBCRs is
in Figure 4.

Using the Graczyk et al. (2021) SBCR, we found good agreement with our radius
estimations. The largest differences corresponds to the objects with large
uncertainty in the $K_{\rm s}$ magnitude, which propagates into a large radius
uncertainty. For the rest of the stars, the radius measurements agree within
$\approx 3\%$. Finally, we note that we cannot use the Graczyk et al. (2021)
SBCR to estimate the radii of M dwarfs, given that the relation was not
calibrated for such small masses.

We also found good agreement between our radii and the ones estimated using
the SBCR from Salsi et al. (2021). We found that the difference between radii
is within $\approx 4\%$, except for the stars that have high uncertainty in
the $K_{\rm s}$ magnitude, which propagates into a large radius uncertainty.
In addition, for the low-mass stars ($<0.6~{}{\rm M_{\odot}}$), there is a
significant systematic where the fractional difference seems to have a
negative slope with radius. In Section 3 we compare our radius estimations to
two samples from the literature with measured radii (Mann et al. (2015) and
the radii from the Exoplanet Archive). Using those same samples, we calculated
radii using the SBCR from Salsi et al. (2021) and we compared them to the
literature values. We found the same systematic described above in the two
samples, therefore we conclude that the SBCR from Salsi et al. (2021) is
causing the issue, and thus is not appropriate to estimate radius of low-mass
stars. We also compared the radius estimation using the two literature SBCR,
for higher mass stars, and found good agreement with a difference between
radii within $1\%$. The main difference in this comparison is that the radius
for the stars with high $K_{\rm s}$ uncertainty agree between the two
estimations.

When comparing the uncertainties in the radius estimation we note that our
radii can be significantly more precise than in these two studies when the
uncertainty of the $K_{\rm s}$ magnitude is large. This shows the advantage of
our method to estimate stellar radii: it depends only on Gaia photometry which
has higher precision than previous surveys, which translate into smaller
radius uncertainties.

Figure 4: Comparison of the results of estimating radii for the JMDC sample
using our Gaia SBCR against the results using two recent SBCRs from Graczyk et
al. (2021) and Salsi et al. (2021) for the $(G-K_{\rm s})$ color and $S_{\rm
G}$. We color-coded the sample with literature metallicities. In the bottom
panels we show the fractional radius difference with three horizontal blue
dashed lines that show the median, $16^{\rm th}$ and $84^{\rm th}$ percentile,
which are also included as blue numbers in each panel.

##  3 Comparison to previous radius measurements

To study the accuracy of our method to estimate stellar radii, we applied our
calibration of the Gaia SBCR on several samples from the literature which had
radius and metallicity measurements. The samples we compared to are a set of M
dwarfs from Mann et al. (2015) and a sample of FGK and M dwarfs from the
Planetary Systems Composite Parameters Table from the NASA Exoplanet Archive
(NASA Exoplanet Archive, 2019; NASA Exoplanet Science Institute,
2020)666https://exoplanetarchive.ipac.caltech.edu/. The result of the
comparisons are shown in Figure 5. In addition we compared our radii
estimations to the radii from the $50$ pc sample of main sequence stars from
Gaia DR3, and the results are shown in Figure 6. We obtained the $50$ pc
sample from the Gaia archive applying a cut in parallax. We applied similar
quality cuts for the two literature samples as for our calibration sample,
described in Section 2.1. We required stars to have a distance $<100$ pc
(except for the Gaia DR3 sample which has a $50$ pc limit), RUWE $<1.4$,
parallax SNR $>10$ and photometric flux in the $G$, $G_{\rm BP}$ and $G_{\rm
RP}$ bands with SNR $>10$. In addition, we only estimated a radius for stars
in the valid metallicity (${\rm[Fe/H]}$) and color range for our calibration
described in Section 2.1. To estimate radii using our method we used the fit
from the Table 1 corresponding to $S_{\rm G}$ and the color $(G_{\rm
BP}-G_{\rm RP})$.

###  3.1 Comparison to radii from Mann et al. (2015)

Mann et al. (2015) measured spectra for a sample of M dwarfs to estimate
bolometric flux ($F_{\rm BOL}$) and effective temperature ($T_{\rm eff}$).
They calculated $F_{\rm BOL}$ by integrating under the radiative flux density
and $T_{\rm eff}$ by comparing the spectra to the CFIST suite of the BT-SETTL
version of the PHOENIX atmosphere models (Allard et al., 2013). Mann et al.
(2015) used their measured $F_{\rm BOL}$ and $T_{\rm eff}$ combined with
parallaxes from the literature to estimate the radius of each star using the
Stefan-Boltzmann law. They also estimated metallicities using empirical
relations between equivalent widths of atomic features like Na and Ca and
metallicity, that were calibrated for M dwarfs using wide binaries with an FGK
primary and M dwarf secondary (Mann et al., 2013, 2014). We cross-matched this
sample with Gaia DR3 to obtain parallaxes and photometry in the Gaia bands,
using the same method described in Section 2.1. We used the Gaia photometry
and parallaxes, together with the $[{\rm Fe/H}]$ measurements from Mann et al.
(2015) to estimate radii for the sample with our Gaia SBCR. In total we
compared the radii estimation from Mann et al. (2015) for $100$ stars with our
calculations, and the results are in the left panel of Figure 5. We also
included a panel with the fractional radius difference as a function of
radius. We find that most of our radius measurements agree within $1\sigma$
with the radii from Mann et al. (2015), with a radius fractional difference
smaller than $3.6\%$, and there is no significant offset between the
measurements. In addition, we do not find a correlation with metallicity.
Therefore we conclude that there is no systematic added by our method. The
agreement is expected given that Mann et al. (2015) used the angular diameter
measurements from the JDMC catalog to calibrate their method, which we also
used to calibrate our SBCR.

###  3.2 Comparison to radii from the NASA Exoplanet Archive

The NASA Exoplanet Archive contains a compilation of all the known exoplanets
with their characteristics and the properties of the host star. We used the
Planetary Systems Composite Parameters Table (NASA Exoplanet Science
Institute, 2020), which contains a vetted sample of the archive with the best
estimation of parameters. Among those properties are radius and metallicity.
In general, the calculations of radius and metallicities are done similarly to
the work done by Mann et al. (2015), where the fitting of observations is done
on spectra or spectral energy distributions (SEDs). In some cases, the [Fe/H]
was estimated using fitting of theoretical stellar models. We kept only the
host stars with radii over uncertainty larger than $10$ to remove a few clear
outliers with large radius uncertainty. We cross-matched the sample of host
stars with Gaia DR3 using the same method described in Section 2.1. With the
Gaia DR3 photometry and parallax, and metallicity we estimated radii using our
Gaia SBCR. In total we used $434$ stars from the NASA Exoplanet Archive to
compare to our method, and the results are in the right panel of Figure 5. For
most of the stars, our measurements and the values in the Exoplanet Archive
agree within $1\sigma$, with a radius fractional difference smaller than
$3.6\%$, and there is no significant offset. For $6$ stars in the sample we
found that our radii estimation were larger than the literature value by
$\Delta R/R{\sim}0.5$. All of these radii where estimated by the same study
(Demangeon et al., 2021). We found that all $6$ stars had more than one radius
estimation in the Archive, and that the second measurement agreed with our
estimations. Therefore we decided to remove them from the comparison. The
comparison with the Exoplanet Archive sample has several outliers ($18$ stars
with $\Delta R/R\gtrsim 0.1$) which are likely due to the combination of
different methods to estimate radius in the Archive. However, in total the
outliers account for $4\%$ of the sample, and they do not modify significantly
the median difference or the scatter. In addition we found no significant
trend with metallicity. Therefore we conclude that our method does not add
systematic uncertainty to the radius calculations, and it is estimating
accurate radii.

Figure 5: We compared the radius estimations using our method with two samples
from the literature to asses the accuracy of our method. The two samples are a
set of M dwarfs from Mann et al. (2015) (left panel in the figure) and a
sample of FGK and M dwarfs from the Planetary Systems Composite Parameters
Table from the NASA Exoplanet Archive (right panel in the figure). We color-
coded the stars in each panel by the metallicities provided by each source. We
included in the bottom panels the fractional radius difference as a function
of the radius in the literature. In these panels we also show the zero
difference as a horizontal black line, and in a blue dashed line we show the
median, $16^{\rm th}$ and $84^{\rm th}$ percentile, which are also included as
blue numbers in the panel. We found good agreement between the measurements.

###  3.3 Comparison to radii from Gaia DR3.

As part of Gaia DR3, the collaboration released stellar radius measurements.
Therefore we decided to compare our estimation of radii using the Gaia SBCR
with the results from Gaia. We used the Gaia Archive to extract photometry,
parallaxes, radius measurements and metallicities for the $50$ pc sample,
which we selected applying a parallax cut. The Gaia collaboration used two
methods to estimate radius (Andrae et al., 2022). One is the General Stellar
Parameterizer from Photometry (GSP-Phot). This method utilizes a main
algorithm (Aerneas Bailer-Jones 2011) which fits the measured BP/RP spectra
(De Angeli et al., 2022), parallax and G magnitude to estimate stellar age,
mass, metallicity ([M/H]) and extinction using stellar PARSEC iscochrones
(PARSEC 1.2S Colibri S37 models Tang et al. 2014; Chen et al. 2015; Pastorelli
et al. 2020), and four different stellar atmosphere models: MARCS for
$2,500<T_{\rm eff}<8,000$ K, PHOENIX for $3,000<T_{\rm eff}<10,000$ K, A-stars
models for $6,000<T_{\rm eff}<20,000$ K, and OB stars models for
$15,000<T_{\rm eff}<55,000$ K (Creevey et al., 2022). The Gaia collaboration
also uses the General Stellar Parametrizer from Spectroscopy (GSP-Spec Recio-
Blanco et al. 2022) method to estimate stellar atmospheric parameters and
individual chemical abundances ($T_{\rm eff}$, $\log g$, [M/H], [Fe/H], etc)
from the Radial Velocity Spectrometer (RVS) spectra of single stars, with a
resolution of $R{\sim}11500$. The second method used by the Gaia collaboration
to estimate radii is the Final Luminosity Age Mass Estimator (FLAME). FLAME
uses BaSTI stellar evolution models (Hidalgo et al., 2018) assuming solar
metallicity to estimate mass, age and evolutionary stage, and it also produces
radius, luminosity and gravitational redshift. These estimations are done
based on $T_{\rm eff}$, $\log g$ and [M/H] from GSP-Phot or GSP-Spec.

We estimated radii for the $50$ pc sample of main sequence stars from Gaia DR3
using our Gaia SBCR and Gaia photometry, parallaxes, and assuming solar
metallicity. The comparison of our results and the radius estimation using the
GSP-Phot method are shown in Figure 6. We also compared our radii estimates to
the values from GSP-Spec and Flame, but we did not include a figure. In
general we found that the radii estimated with our method are systematically
smaller than the values in the Gaia catalog, but the difference depends on the
models used and the radius. We found that for a radius $>0.6\,{\rm
R_{\odot}}$, our radius estimations are $3-4\%$ smaller than the values in the
Gaia catalog calculated using the MARCS or A-stars models, and a slightly
smaller shift ($2\%$) using the PHOENIX models. In the case where the radii
were estimated with the FLAME method with parameters from GSP-Phot, we found
also a $4\%$ offset for radius $>0.6\,{\rm R_{\odot}}$, and $<2\%$ for FLAME
using parameters from GSP-SPEC. For the M dwarfs (radius $<0.6\,{\rm
R_{\odot}}$) the offset is between $10-25\%$ when the radii from Gaia were
estimated using the GSP-Phot method with the PHOENIX model and $7\%$ for the
MARCS models. We found a difference of $13\%$ when the Gaia radii were
estimated using the FLAME method with parameters from GSP-Phot. The difference
between the MARCS and PHOENIX models was already discussed by Andrae et al.
(2022). We conclude that the radii estimations from Gaia should be used with
care. We found that the best agreement between our calculations and the
results in the Gaia catalog for higher mass stars ($>0.6\,\mbox{${\rm
R}_{\sun}$}$) are for the FLAME method using the GSP-SPEC models and for the
low-mass stars ($<0.6\,\mbox{${\rm R}_{\sun}$}$) the best agreement is using
the GSP-Phot method with the MARCS models. However, for low-mass stars these
radii calculations have an offset ($7\%$) as well, which should be taken into
account.

To study this offset between radius estimations in more detail, we used our
calibration sample of main sequence stars with angular diameter measurements,
described in Section 2.1. We used the angular diameters and the parallaxes
from Gaia DR3 to estimate radii for the sample and we compared these with the
Gaia radius estimations using GSP-Phot. We found the same $3\%$ offset between
the two radius estimations for radius $>0.6\,{\rm R_{\odot}}$, and around
$10\%$ for smaller stars. Therefore we conclude that the offset is not being
generated by our method. We note that Fouesneau et al. (2022) performed a
similar comparison for the Gaia radius estimations using the GSP-Phot method
against angular diameters measurements, but included giants in the analysis
that make the difference less evident.

The difference between our calculation and the Gaia radii for M dwarfs is not
surprising given that evolutionary and atmospheric models have been shown to
still have several discrepancies with observed data (e.g., Baraffe et al.,
2015; Dieterich et al., 2021). Significantly, the models used in the Gaia
catalog for M dwarfs over-predict the radius for all these stars, and not just
the magnetically active.

Figure 6: Comparison of the radius measurements using our Gaia SBCR and the
radii in the Gaia DR3 catalog for the $50$ pc sample. We show in this figure
only the radii calculated by Gaia with the GSP-Phot method, and we color-coded
the stars by the model used in this calculation. In general we found that our
radius estimations are systematically smaller than the values in the Gaia
catalog, and that the shift is different according to the model used and the
radius. We found that for radius $>0.6\,{\rm R_{\odot}}$, the systematic
difference is $3-4\%$ for the MARCS and A-stars models, but the difference is
slightly smaller ($2\%$) using the PHOENIX models. For the M dwarfs (radius
$<0.6\,{\rm R_{\odot}}$) the offset is between $10-25\%$ for GSP-Phot using
PHOENIX, $7\%$ for the MARCS models and $13\%$ for the FLAME method using GSP-
Phot parameters.

##  4 Possible biases

To further test the precision of our method of using the Gaia SBCR to estimate
radius, we study different properties that could affect our measurements,
given that we rely on photometry and parallax. In this section we discuss the
following properties: metallicity, variability and extinction.

###  4.1 Influence of metallicity

Prior studies estimated radii using stellar models and compared measured radii
to models which showed the importance of having precise metallicity values
(Burrows et al., 2011; Irwin et al., 2011; Feiden & Chaboyer, 2012; Spada &
Demarque, 2012; Morrell & Naylor, 2019). Other studies showed that stars with
different metallicities follow a different SBCR (Boyajian et al., 2012b, a).
Therefore we calibrated our Gaia SBCR taking into account the metallicity
([Fe/H]) of our sample. However, it is common in the literature to assume
solar metallicity ([Fe/H] $=0$) when a measurement is not available. In order
to test this assumption applying our method, we used the literature samples
discussed in Section 3 to compare the radius estimations that account for
metallicity with those that assume solar metallicity.

In Section 3 we used the offset and scatter between our radius estimation and
the literature to study the precision of our method. To study the effect of
metallicity, we assumed solar metallicity for all the stars and re-calculated
the offset and scatter. For the samples from Mann et al. (2015) and the
Exoplanet Archive which had metallicity measurements, we did not find a
significant change in the offset or the scatter of the stars. This is not
surprising given that the stars in these samples are close to solar
metallicity (the bulk of the stars are in the range $-0.3<[{\rm Fe/H}]<0.3$
dex).

In addition, when we assumed solar metallicity to estimate radii with our
method, we found that for the Mann et al. (2015) and the Exoplanet Archive
samples we underestimated radii for low-metallicity stars and overestimated
radii for high-metallicity stars. We found that for stars with
$-1.5<{\rm[Fe/H]}<0$ dex assuming solar metallicity underestimates the radius
by about $3\%$, and for stars with $0<{\rm[Fe/H]}<0.4$ dex assuming solar
metallicity overestimates the radius by $2\%$. Outside of this range, assuming
solar metallicity generates a larger loss of precision. In this work we study
radius inflation, which is normally a $7-10\%$ difference in radius. Therefore
we chose to work with stars with metallicities between $-0.3<{\rm[Fe/H]}<0.3$
dex for the rest of our analysis, to minimize the radius offset due to
metallicity.

###  4.2 Influence of variability

Measurement error due to variability could also affect our calibration of the
SBCR and the radius estimation using our method, which depends strongly on
photometry. Magnetic activity can generate short and long term variability in
the photometry of a star (e.g., Irving et al., 2023). It has been shown that
measurements in eclipsing binaries are affected by variability (Morales et
al., 2010; Han et al., 2019) or the presence of large polar spots (Morales et
al., 2010; Kraus et al., 2011; Torres et al., 2019). To test the effect of
photometric variability on the radius estimation, we used the light-curves
provided by Gaia DR3. We used the sample of M dwarfs with known ages cross-
matched with Gaia DR2 from Kiman et al. (2021), and we used the Gaia DR2 id to
obtain the Gaia DR3 id. We selected the stars from this sample that were
indicated to be members of the Pleiades cluster. This young cluster is $120$
Myr old (Dahm, 2015), with ${\rm[Fe/H]}=+0.03\pm 0.05$ dex (Soderblom et al.,
2009) and its members are known to show high variability (e.g., Kiman et al.,
2021). We note that this test is a proof of concept of the effect of high
variability when calculating radii. However, our method to estimate radii
using the Gaia SBCR is not calibrated for pre-main sequence stars.

The acquisition of the Gaia light-curve works as follows: first Gaia takes a
photometry measurement in the G band, then $\approx 27$ minutes later a
measurement in the BP band and $\approx 7.5$ minutes later a measurement in
the RP band. The time between these sequences of three band measurements
varies. However, this time is always larger than the time between band
observations. In addition, most of the stars for which we downloaded a light-
curve, were observed for $900$ days in total, but not continuously. In
conclusion, we assumed that the photometry points in each band were taken at
the same time. Therefore, we can combine the light-curves in each band with
the parallax and estimate a time sequence of radius using our method.

We used this time sequence of radii to estimate the fractional difference
between these radii and the radius calculated with the published Gaia
photometry for each star. The median of the fractional difference for each
object in our sample as a function of the radius estimated from the published
photometry are in Figure 7. These stars are young and variable in their
photometry due to phenomena such as spots and magnetic cycles (Irving et al.,
2023). Our analysis shows that if only one photometry point is taken for these
stars, the change in radius can be as large as $4\%$ for most of the stars and
even $7\%$ and $10\%$ for two stars in the sample. We note that all the stars
in our sample are M dwarfs, therefore these are fast rotating stars. However,
given that Gaia takes several photometry measurements and then averages all of
them for the final photometry, the effect of variability is reduced, as shown
by the median difference in radius being $<1\%$ for all the stars in Figure 7.
Therefore, when using Gaia photometry variability is not a concern for
calculating the radius of stars, or for the calibration of the surface
brightness-color relation.

Figure 7: Fractional radius difference between our radius estimation using the
published Gaia photometry and the median of all the photometric points taken
by Gaia for each star. We included the $1\sigma$ and $2\sigma$ deviations to
show what the radius difference could be if Gaia would take only one
photometry point per star. These stars are from the Pleiades cluster ($120$
Myr) so we expect high photometric variability. Given that Gaia averages
several photometry points to estimate the photometry of the star, photometric
variability it is not an issue when estimating radii using Gaia photometry.

###  4.3 Influence of extinction

In this work we used only nearby stars for most of our analysis ($<100$ pc) to
minimize the effect of extinction on the photometry. In reality, most stars
are not closer than $100$ pc, and extinction will affect their photometry. An
extinction correction is necessary in these cases to use the method described
in this paper. To approximate the effect extinction would have on radius
measurements, we selected a random sample of $20,000$ stars within $500$ pc
from the sample of Anders et al. (2022). In this work, the authors used the
code StarHorse to estimate distances, extinctions (at $\lambda=542$ nm), ages,
masses, effective temperatures, metallicities, and surface gravities for field
stars, from the parallaxes and photometry from Gaia EDR3 (Gaia Collaboration
et al., 2021), combined with photometry from 2MASS (Cutri et al., 2003),
AllWISE (Cutri et al., 2013), Pan-STARRS1 (Chambers et al., 2016), and
SkyMapper (Onken et al., 2019).

To clean the sample we removed giant stars according to their position in the
color-magnitude diagram, using the same method described in Section 2.1. We
estimated radii for the resulting sample using our method with the corrected
and uncorrected magnitudes, and calculated the percentage of radius difference
between these two values. We repeated this process for all nine relations we
derived for the three Gaia colors combined with the three magnitudes, which
are shown in Table 1. In this random subsample from Anders et al. (2022), we
found that the effect of extinction on magnitudes increases with distance and
it is larger on the $(G_{\rm BP}-G_{\rm RP})$ color, as expected. However, the
effect of extinction on the radius estimation is more complex, and it is
described below.

We found that the extinction correction can generate a radius difference of up
to $10\%$, for stars within $500$ pc. In addition, the influence of extinction
on the radius estimation does not depend on which magnitude is used to
estimate radius. As expected, we found that on average the influence of
extinction in the radius estimation increases as a function of the distance,
however the increase of the absolute value of the radius difference as a
function of distance depends on the color used to estimate the radius and the
spectral type of the star. To study this effect we divided the sample in three
bins of spectral type: F and G dwarfs, K dwarfs and M dwarfs. We found that
the influence of extinction for F,G and K dwarfs is similar. For these three
spectral types we found that the radius estimated with any of the three Gaia
color is larger when the magnitudes are not corrected for extinction. The
extinction effect is larger when the $(G-G_{\rm RP})$ color is used to
calculate the radius ($6\%$ average radius difference at $500$ pc), while the
radius estimation is almost not modified ($0.1\%$ at $500$ pc) when the
$(G_{\rm BP}-G)$ color is used. For M dwarfs we found that when using the
$(G-G_{\rm RP})$ color to estimate radii, the radii estimated with magnitudes
without the extinction correction are larger, but using the $(G_{\rm
BP}-G_{\rm RP})$ and $(G_{\rm BP}-G)$ colors, the radii calculated with
magnitudes without extinction correction are smaller. Finally, we found that
using the $(G_{\rm BP}-G)$ to estimate radii, the average radius difference is
of $6\%$ at $500$ pc, the largest between the three Gaia colors. Using
$(G-G_{\rm RP})$ and $(G_{\rm BP}-G_{\rm RP})$ we found an average radius
difference of $2\%$ when the $G$ or $G_{\rm RP}$ magnitudes are used. When the
$G_{\rm BP}$ magnitude is used in combination with the $(G-G_{\rm RP})$, we
found an average radius difference of $5\%$. In conclusion, the radii
estimated using the $(G-G_{\rm RP})$ color are the most affected by extinction
for F, G and K, while for M dwarfs $(G_{\rm BP}-G)$ is the most problematic
color.

##  5 Radius inflation in M dwarfs

We used the Gaia DR3 SBCR to study the relationship between radius and
magnetic activity and mass for low-mass stars. To do this, we measured radius
and mass for the Kiman et al. (2019) M dwarf sample which has magnetic
activity measurements and estimates of metallicity.

###  5.1 The Kiman et al. (2019) sample

Kiman et al. (2019) compiled a sample of M dwarfs which have ${\rm H}\alpha$
equivalent width (${\rm H}\alpha\;{\rm EW}$) measurements from spectra from
the Sloan Digital Sky Survey (SDSS, West et al., 2011; Schmidt et al., 2015),
which is a magnetic activity indicator (e.g., West et al., 2011; Newton et
al., 2017; Kiman et al., 2019, 2021). This sample also has SDSS and 2MASS
photometry. We used the Gaia DR2 source ids provided in the catalog to obtain
the corresponding Gaia DR3 source from the cross-match available in the Gaia
archive. We refer to Kiman et al. (2019) for details on the Gaia cross-match.

We applied similar quality cuts as described in previous sections to estimate
precise radii (see Sections 2.1 and 3). We kept stars with parallax
measurements with SNR $>40$ (not $>10$ as in previous section because these
stars are at a larger distance: $100-200$ pc), photometry in the $G$ and
$G_{\rm RP}$ Gaia bands with flux with SNR $>10$ and RUWE $<1.4$. Following
Kiman et al. (2019), we selected stars that do not need an extinction
correction using the estimation of extinction from the $(r-z)$ color available
in the catalog. We estimated stellar mass using the relation between mass and
absolute 2MASS $K_{\rm s}$ magnitude from Mann et al. (2019). In order to
estimate precise masses, we kept stars with $K_{\rm s}$ uncertainty $<0.07$
mag ($\approx$7% uncertainty in flux). After the quality cuts we were left
with $7,640$ stars. The stars from Kiman et al. (2019) have measurements of
$\zeta_{\rm TiO/CaH}$ calculated by West et al. (2011). $\zeta_{\rm TiO/CaH}$
is a metallicity dependent parameter for M dwarfs defined by Lépine et al.
(2007) as the relation between the CaH and TiO molecular indices such that

$\zeta_{\rm TiO/CaH}=\frac{1-{\rm TiO5}}{1-{\rm[TiO5]_{Z_{\odot}}}}$ (12)

where

$\begin{split}{\rm[TiO5]_{Z_{\odot}}}=&-0.164\times({\rm CaH}2+{\rm
CaH}3)^{3}\\\ &+0.670\times({\rm CaH}2+{\rm CaH}3)^{2}\\\ &-0.118\times({\rm
CaH}2+{\rm CaH}3)-0.050.\end{split}$ (13)

Woolf et al. (2009) calibrated the relation between $\zeta_{\rm TiO/CaH}$ and
[Fe/H] by measuring the two quantities for a sample M dwarfs with high
resolution spectra. They fit a linear function to the data such that
${\rm[Fe/H]}=a+b\times\zeta_{\rm TiO/CaH}$ where $a=-1.685\pm 0.079$ and
$b=1.632\pm 0.096$. This relation is valid for $-2<{\rm[Fe/H]}<0.5\,{\rm dex}$
according to Figure 6 from Woolf et al. (2009). We used this calibration to
estimate approximate [Fe/H] for our sample.

###  5.2 Mass-radius relation for active and inactive stars

We calculated radii for the Kiman et al. (2019) sample using our Gaia SBCR
combining the Gaia photometry and parallaxes with the estimated [Fe/H]. We
used the relation for the $(G-G_{\rm RP})$ color and $S_{\rm G}$ in Table 1.
We decided to use this relation instead of the one corresponding to the
$(G_{\rm BP}-G_{\rm RP})$ color as in previous sections (see for example
Section 3) because this sample is on average fainter than other samples in
this work, and therefore has high uncertainties in the $G_{\rm BP}$ photometry
(Kiman et al., 2019). In addition, we estimated stellar mass using the
relation between mass and absolute 2MASS $K_{\rm s}$ magnitude from Mann et
al. (2019), as we discussed in Section 5.1. The relation between our estimated
radii and masses are shown in Figure 8. We only included the median
uncertainty to simplify the visualization of the data. The relation from Mann
et al. (2019) is valid for masses $<0.7\,{\rm M_{\odot}}$, so we removed all
the stars which had estimated masses outside this range. We divided the sample
into active and inactive stars with the active flag in the Kiman et al. (2019)
sample. Moreover, we color-coded the active stars according to ${\rm H}\alpha$
by fractional ${\rm H}\alpha$ luminosity ($L_{{\rm H}\alpha}/L_{\rm bol}$),
and we show the inactive stars in gray. Analysing Figure 8, we found that
active stars have larger radii on average than inactive stars all along the M
dwarf regime. For this same sample, Kiman et al. (2019) found that active
stars are redder in the Gaia color-magnitude diagram than inactive stars (See
Figure 17 in their work). Therefore we can conclude that the offset between
active and inactive stars is due to an inflated radius rather than
underestimated masses. As ${\rm H}\alpha$ is an indicator of magnetic
activity, this result indicates that stars that are magnetically active have,
on average, larger radii than their inactive counterparts with the same mass,
which indicates that radius inflation is correlated with magnetic activity. In
addition, the difference in radius between active and inactive becomes smaller
for smaller stars, which agrees with previous results (e.g., López-Morales,
2007; Morales et al., 2022).

The result in Figure 8 agrees with the results from Stassun et al. (2012). In
that work the authors estimated radii from $T_{\rm eff}$ and bolometric
luminosity measurements, and they approximated $T_{\rm eff}$ from spectral
type, and bolometric luminosity from 2MASS $K_{\rm s}$ magnitudes and
parallaxes. They used the isochrones from Baraffe et al. (1998) to estimate
masses from the bolometric luminosity. In comparison, our parameters are fully
empirical given that we rely only on measured data, and our radius estimations
were done directly from a SBCR, instead of depending on spectral type. In
addition, we have a larger sample which allowed us to study the difference
between active and inactive stars as a function of mass.

Figure 8: Mass-Radius relation for the sample from Kiman et al. (2019).
Inactive stars according to ${\rm H}\alpha$ are shown in gray and active stars
are color-coded by fractional ${\rm H}\alpha$ luminosity. We only include the
typical uncertainly (median) of the sample. Active stars show larger radii
than inactive stars indicating that radius inflation is correlated with
magnetic activity.

Figure 8 has a clear separation between active and inactive stars, however
most of the active stars have the same locus as some inactive stars. To study
this overlap in more detail, we divided the sample in different metallicity
bins. We fit the mass-radius relation for active and inactive stars for two
different metallicity bins: a low metallicity bin of $0.73\leq\zeta_{\rm
TiO/CaH}<0.94$ which corresponds roughly to $-0.5\leq{\rm[Fe/H]}<-0.15$ dex
and a solar metallicity bin of $0.97\leq\zeta_{\rm TiO/CaH}<1.09$ which is
approximately $-0.1\leq{\rm[Fe/H]}<0.1$ dex. Our results are in Figure 9. We
included the fractional difference between radii of active and inactive stars
for both metallicity bins, and we calculated the median of the difference for
four mass bins. We estimated the uncertainty of this difference by doing a
Monte Carlo propagation of uncertainties. The median values of fractional
difference between radii for active and inactive stars, in order of increasing
mass bin are: for low metallicity ${0.005}_{-0.037}^{+0.041}$,
${0.04}_{-0.04}^{+0.03}$, ${0.05}_{-0.03}^{+0.04}$ and
${0.06}_{-0.04}^{+0.04}$, and for solar metallicity
${0.002}_{-0.044}^{+0.031}$, ${0.02}_{-0.03}^{+0.03}$,
${0.05}_{-0.04}^{+0.04}$ and ${0.09}_{-0.08}^{+0.04}$. These values of radius
inflation are consistent with the measurements from the literature. By
separating in metallicity, the difference between active and inactive stars
becomes more clear, and the overlap between the two types of stars is
significantly reduced, in comparison to Figure 8. Finally, the fractional
difference in radius as a function of mass shows that the difference between
active and inactive is significant and decreases with mass. This difference is
not significant for masses $<0.3\,{\rm M_{\odot}}$ because the uncertainty in
our radii calculation becomes comparable to the expected inflation (e.g.,
$5.8\%$ found by Morales et al. 2022).

Figure 9: Same data as in Figure 8 divided in two bins of metallicity relation
with metallicity: $0.73\leq\zeta_{\rm TiO/CaH}<0.94$ (left panel) which
corresponds roughly to $-0.5\leq{\rm[Fe/H]}<-0.15$ dex and a solar metallicity
bin of $0.97\leq\zeta_{\rm TiO/CaH}<1.09$ (right panel) which is approximately
$-0.1\leq{\rm[Fe/H]}<0.1$ dex. We include in the lower panels the fractional
radius difference between the radii of the active stars and the fit to the
inactive stars for the same mass. In dark blue we show the median and standard
deviation for four bins in mass. When we separate the stars by metallicity,
the overlap between active and inactive stars seen in Figure 8 decreases.

Besides metallicity, age could also affect the scatter in Figures 8 and 9. M
dwarfs do not change their properties significantly once they converge onto
the main sequence. Therefore, a significant difference would only appear for
very young stars ($<1$ Gyr). Most of the stars in this sample are within $200$
pc, which means that if they are young, they belong to a known moving group.
Kiman et al. (2019) showed that these stars needed to be $24$ Myr old to
explain their position in the color-magnitude diagram, which was unlikely. In
addition, Kiman et al. (2021) checked the membership of all these stars and
found that only 46 stars from this sample belong to moving groups, therefore
most of the stars are from the field. From the members of moving groups, most
of the stars belong either to the Praesepe cluster or Coma Berenices. By
comparing these members of young associations with the active stars, we found
that age alone cannot explain the shift in radius between active and inactive
stars.

In our work we compared stars that are not young and have similar
metallicities, which allows as to identify a difference in radius between
active and inactive stars in a large sample. We found that active stars have a
larger radius than inactive stars on average, which indicates that the radius
inflation problem could be caused by activity. In this work we do not compare
to models because the comparison of data to models requires precise ages and
metallicities for each star (Torres et al., 2018, 2019, 2020, 2021b), which we
do not have for this sample.

###  5.3 Radius as a function of ${\rm H}\alpha\;{\rm EW}$

The mass-radius relation of active and inactive stars in Figure 8 shows that
active stars have larger radii than inactive stars. In this section we used
the same sample from Kiman et al. (2019) to study the dependence of radius
inflation on ${\rm H}\alpha\;{\rm EW}$ for different mass bins. We calculated
the difference between the radius of active stars and the fit to the mass-
radius relation for inactive stars with close to solar metallicity from Figure
9. We also estimated the uncertainties of the difference by doing a Monte
Carlo propagation. The fractional difference between the active and inactive
radii for each active star as a function of ${\rm H}\alpha\;{\rm EW}$ is on
Figure 10. As the value of ${\rm H}\alpha\;{\rm EW}$ for inactive stars does
not provide extra information besides that the stars are inactive, we did not
include them in the plot. We divided the stars in four mass bins: $0.5<M[{\rm
M_{\odot}}]\leq 0.6$, $0.4<M[{\rm M_{\odot}}]\leq 0.5$, $0.3<M[{\rm
M_{\odot}}]\leq 0.4$ and $0.2<M[{\rm M_{\odot}}]\leq 0.3$. As in the mass-
radius relation in Figure 8, we can see the dependence on mass of the radius
inflation, where active higher mass stars have a larger difference with
inactive stars than lower mass stars. In addition, the top two panels show a
correlation between the fractional radius difference and ${\rm H}\alpha\;{\rm
EW}$. We estimated the Kendall’s $\tau$ coefficient and the p-value to test
the hypothesis of no correlation in the data777To calculate the Kendall’s
$\tau$ coefficient and the p-value we used the python function
scipy.stats.kendalltau. We show the results in the label of each panel of
Figure 8. As was clear by eye, both bottom panels have low Kendall’s $\tau$
coefficients and relatively high p-values, which indicates that there is no
evidence for correlation. We found similar results for the mass bin
$0.4<M[{\rm M_{\odot}}]\leq 0.5$. On the other hand, the panel for the mass
bin $0.5<M[{\rm M_{\odot}}]\leq 0.6$ seems to present a correlation. The
p-value for the data in this mass bin is $3\times 10^{-6}$ which means that
the two parameters appear substantially more correlated than would be expected
by chance. This indicates that for the mass bin $0.5<M[{\rm M_{\odot}}]\leq
0.6$, the radius inflation of active stars over inactive ones is increasing
with increasing magnetic activity indicated by ${\rm H}\alpha\;{\rm EW}$. That
there is no clear correlation between radius inflation and ${\rm
H}\alpha\;{\rm EW}$ on the other three mass bins could be explained by the
radius uncertainty of our method ($4\%$). If our uncertainty is similar to or
larger than the percentage of radius inflation, then that would not allow us
to distinguish a clear trend in the data. The lack of correlation could also
be physical, however more precise radii measurements are required to
distinguish between the two scenarios.

To ensure that we understand the correlation in Figure 10, we analyzed each
axis separately. The ${\rm H}\alpha\;{\rm EW}$ measurements were made from
SDSS spectra which has a resolution R$\sim 2000$. It has been shown that ${\rm
H}\alpha\;{\rm EW}$ increases with decreasing mass for M dwarfs (e.g., West et
al., 2011; Schmidt et al., 2015; Kiman et al., 2019; Popinchalk et al., 2021).
Therefore it is important to divide the sample into mass bins, to remove the
dependence of ${\rm H}\alpha\;{\rm EW}$ with color. The mass bin size selected
in Figure 10 is small enough that there should not be an effect due to the
color dependence. The masses on this plot were estimated from absolute $K_{\rm
s}$ magnitude, meaning that effectively we are selecting a small bin in
absolute $K_{\rm s}$ magnitude. Given that the scatter in the mass-luminosity
relation from Mann et al. (2019) is small, we can consider this narrow bin in
$K_{\rm s}$ magnitude to be narrow in mass as well. In addition, the radius
estimations using our Gaia SBCR are proportional to the color of the stars
(Equation (10)); an increase in radius at fixed absolute ${\rm K_{s}}$ implies
a redder color. This correlation is expected given that when stars become
inflated, their radius increases at constant bolometric luminosity, which
generates a decrease in effective temperature and therefore, a redder color.

The results in Figure 10 partially agree with the results from Stassun et al.
(2012). In that work, the authors studied radius inflation as a function of
$L_{{\rm H}\alpha}/L_{\rm bol}$ combining all the masses between
$0.3-0.7\,M_{\odot}$ in the same analysis. As we mentioned above, the
difference in radius between active and inactive stars is dependent on mass,
so it is necessary to study radius inflation in bins of mass to distinguish
between dependence on mass and on magnetic activity. Stassun et al. (2012)
also studied this relation for eclipsing binaries, assuming that the
dependence on magnetic activity is the same for eclipsing binaries and single
stars. Although we see radius inflation both in single stars and eclipsing
binaries, it has been shown that eclipsing binaries tend to rotate faster than
single stars of the same mass and age (Torres et al., 2021b). Therefore we
only compared our analysis to their results on single stars.

Figure 10: Estimated radii for the Kiman et al. (2019) sample of M dwarfs as a
function of ${\rm H}\alpha\;{\rm EW}$, divided in four mass bins: $0.5<M[{\rm
M_{\odot}}]\leq 0.6$, $0.4<M[{\rm M_{\odot}}]\leq 0.5$, $0.3<M[{\rm
M_{\odot}}]\leq 0.4$ and $0.2<M[{\rm M_{\odot}}]\leq 0.3$. We estimated the
Kendall’s $\tau$ coefficient and the p-value to test the hypothesis of no
correlation in the data. The results are shown in the label of each panel.

##  6 Conclusions

In this work we developed a method to estimate radii of FGK and M dwarfs using
Gaia DR3 photometry and astrometry. This method has the advantage that it
estimates precise radii ($4\%$ average uncertainty) using only photometry and
parallaxes provided by the Gaia catalog. Therefore, it is straightforward to
estimate radii for large samples. We used this method to study radius
inflation for a statistically large sample of M dwarfs and showed that radius
inflation on single stars is correlated with magnetic activity.

We calibrated the surface brightness-color relation (SBCR) using Gaia DR3
magnitudes. We used the sample of angular diameter measurements from Duvert
(2016) as a calibration sample. We cross-matched it with Gaia DR3 to obtain
photometry and parallax measurements for the sample. In addition, we searched
the literature for metallicity measurements. We fit the relation between the
flux $S_{\rm m_{\lambda}}$ defined in Equation (5), color ($X$), and
metallicity ([Fe/H]) for each of the three magnitudes in Gaia
($m_{\lambda}=\\{G,G_{\rm RP},G_{\rm BP}\\}$) and the three colors
($X=\\{(G_{\rm BP}-G_{\rm RP}),(G_{\rm BP}-G),(G-G_{\rm RP})\\}$). We used a
mixture model with the likelihood shown in Equation (7) to fit the relation,
and the results are shown in Table 1. We found that our method has an average
precision of $4\%$.

We found good agreement between our Gaia SBCR and two recent calibrations of
this relation for the $(G-K_{\rm s})$ color and $S_{\rm G}$ from Graczyk et
al. (2021) and Salsi et al. (2021). However, Graczyk et al. (2021) did not
calibrate the relation for masses smaller than $0.6\,\mbox{${\rm M}_{\sun}$}$
and the calibration in Salsi et al. (2021) presents a systematic trend when
estimating radii for M dwarfs which is added by the method. In addition, both
SBCRs have a much larger radius uncertainty in comparison to our results for
several stars when the uncertainty of the $K_{\rm s}$ magnitude is larger than
the uncertainty in Gaia magnitudes. All of these results show the advantage of
using our Gaia SBCR. We also confirmed the accuracy of our method when we
found good agreement between our radius measurements and two sample from the
literature: a set of M dwarfs from Mann et al. (2015) and a sample of FGK and
M dwarfs from the Planetary Systems Composite Parameters Table from the NASA
Exoplanet Archive. We also derived radius with our method for the $50$ pc
sample from Gaia DR3 which contains model derived radius measurements. We
found that the Gaia radius measurements for high mass stars
($>0.6\,\mbox{${\rm R}_{\sun}$}$) are systematically larger by $3-4\%$ in
comparison to our results, and for smaller stars their radii are also larger
than our calculations but the offset is as large as $25\%$. We found that this
offset is generated by the model-dependent method used in the Gaia catalog.

There are several properties of the star that can affect radius measurements.
We studied metallicity, variability and extinction. We found that when using
our method to estimate radii, assuming solar metallicity is approximately
correct for $-0.3<{\rm[Fe/H]}<0.3$ dex. Outside that range it is necessary to
take into account metallicity. We also found that, given that Gaia averages
several photometric measurements, photometric variability is not an issue when
estimating radii with Gaia photometry. Finally, we found that the radii
estimated using the $(G-G_{\rm RP})$ color are the most affected by extinction
for F, G and K, while for M dwarfs $(G_{\rm BP}-G)$ is the most problematic
color.

We studied the influence of magnetic activity on radius measurements –known as
radius inflation– using the sample of M dwarfs from Kiman et al. (2019). By
comparing field stars of similar metallicity we found that active stars
according to ${\rm H}\alpha$ have larger radii than inactive stars. The
percentage of inflation decreases with decreasing mass, with a maximum of
$9\%$ which agrees with literature measurements. Finally, we used the same
sample to study the dependence of radius inflation with ${\rm H}\alpha\;{\rm
EW}$ for different mass bins. We found a clear correlation for the mass bin
$0.5<M[{\rm M_{\odot}}]\leq 0.6$. We found no significant correlation for
lower mass stars, although we cannot distinguish if this is due to lack of
precision in the radius measurements or to a physical reason. More precise
radius measurements are needed to distinguish between the two cases. In future
work we will study radius inflation using rotation period as magnetic activity
indicator. The use of rotation periods will allow us to expand our analysis to
higher mass stars.

##  7 Acknowledgements

The authors would like to thank Dr. Tabetha Boyaijan, Dr. Aurora Kesseli, Dr.
Mercedes López-Morales and Dr. Lars Bildsten for helpful discussions that
contributed to the results of this work. This project was developed in part at
the 2022 NYC Gaia DR3 Fête Workshop at the Center for Computational
Astrophysics of the Flatiron Institute in 2022 June. This research was
supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958 and by the Simons Foundation (668346, JPG). This research has made
use of the NASA Exoplanet Archive, which is operated by the California
Institute of Technology, under contract with the National Aeronautics and
Space Administration under the Exoplanet Exploration Program. This work has
made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

Exoplanet Archive.

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