11institutetext: Univ Coimbra, CITEUC, Geophysical and Astronomical
Observatory, Coimbra, Portugal
11email<EMAIL_ADDRESS>22institutetext: Univ Coimbra, CITEUC, Department
of Mathematics, Coimbra, Portugal 33institutetext: Univ Coimbra, CITEUC,
Department of Physics, Coimbra, Portugal
33email<EMAIL_ADDRESS>44institutetext: Niels Bohr Institute, Blegdamsvej
17, DK-2100, Copenhagen, Denmark
44email<EMAIL_ADDRESS>
# Revisiting the mass- and radius-luminosity relations for FGK main-sequence
stars††thanks: This paper is dedicated to the memory of Johannes Andersen.

João Fernandes 11 22 Ricardo Gafeira 33 Johannes Andersen Johannes Andersen
passed away on 28 April 2020.44

(Received ; accepted )

###### Abstract

Context. Scaling relations are very useful tools for estimating unknown
stellar quantities. Within this framework, eclipsing binaries are ideal for
this goal because their mass and radius are known with a very good level of
accuracy, leading to improved constraints on the models.

Aims. We aim to provide empirical relations for the mass and radius as
function of luminosity, metallicity, and age. We investigate, in particular,
the impact of metallicity and age on those relations.

Methods. We used a multi-dimensional fit approach based on the data from
$DEBCat$, an updated catalogue of eclipsing binary observations such as mass,
radius, luminosity, effective temperature, gravity, and metallicity. We used
the PARAM web interface for the Bayesian estimation of stellar parameters,
along with the stellar evolutionary code MESA to estimate the binary age,
assuming a coeval hypothesis for both members.

Results. We derived the mass and radius-luminosity-metallicity-age relations
using 56 stars, with metallicity and mass in the range $-0.34<$[Fe/H]$<0.27$
and $0.66<M/M_{\odot}<1.8$. With that, the observed mass and radius are
reproduced with an accuracy of $3.5\%$ and $5.9\%,$ respectively, which is
consistent with the other results in literature.

Conclusions. We conclude that including the age in such relations increases
the quality of the fit, particularly in terms of the mass, as compared to the
radius. On the other hand, as other authors have noted, we observed an higher
dispersion on the mass relation than in that of the radius. We propose that
this is due to a stellar age effect.

###### Key Words.:

stars:binaries: eclipsing; evolution; fundamental parameters; solar-type

##  1 Introduction

Empirical stellar relations establish the dependence between fundamental
proprieties of stars in a simple way. These relations, frequently, rely on
’obvious’ astrophysical arguments. The most well-known case is the mass-
luminosity relation (MLR). The MLR is an expected relation given that stellar
energy production - and, thus, the luminosity - is strongly dependent on mass.
According to , Jakob Karl Ernst Halm was the first to remark on the existence
of the MLR. In the words of G. P. Kuiper, who points out the relevance of such
a relation, remarking on the importance of this relation to progress in the
field, it stands as a ’a central problem of theoretical astrophysics’ (Kuiper
(1938). Nowadays, the improvement of the accuracy of mass in stellar binaries,
down to 3% or better, as per, for example, Torres et al. (2010), allows us to
establish precise MLRs (Moya et al. (2018) and references. Another example is
the age-metallicity relation, which is based on the simple idea that a gradual
chemical enrichment of the interstellar medium over the time is accompanied by
an increase of the metallically in stellar generations Rocha-Pinto et al.
(2006). Many more examples can be pointed out: mass-radius relations, MRR
(Eker et al., 2018), helium-to-metal enrichment (Verma et al., 2019), seismic
scaling relations (Bellinger, 2019), etc. The substantial advantage of these
kinds of relations is clearly to enable predictions of unknown observed
stellar parameters based on observed ones.

For instance, in terms of the MLR, for single nearby solar-type stars, the
luminosity can be obtained observationally (with a good level of accuracy),
but not the mass. A detailed review of stellar-mass determinations can be
found in the recent paper by Serenelli et al. (2020). However, there are
strong limitations with regard to the accuracy of those predictions other than
the observational errors, that is, the scatter on the relations themselves
thanks to different contributions such as evolution, chemical composition,
stellar rotation, and activity, etc. Over the last few years, we can identify
several works, particularly, on the topic of MLR, which include the
contributions of metallicity and the evolutionary effects on the relation. The
inclusion of the metallicity and the age lies on the fundamental proprieties
of a star as the luminosity and the radius (or the effective temperature)
depends not only on mass, but also on age and chemical composition (cf. the
so-called Vogt–Russell theorem). It is for this reason that the helium
abundance could also be considered. Unfortunately, that is not measurable in
the atmospheres of FGK stars and the determinations via theoretical stellar
models still carry uncertainties larger than 10% (Valcarce et al., 2013). This
value can go down to about 5% if seismic information is included (Nsamba et
al., 2021). We make note of the findings Moya et al. (2018) as theirs is the
most comprehensive analysis on the field to date. Based on 934 stars, they
were able to establish 38 revised or new relations of mass or radius as a
function of observations as effective temperature ($T_{\rm eff}$), luminosity
($L$), surface gravity ($g$), density ($\rho$), and metallicity ([Fe/H]).
Finally, they concluded that the global precision and accuracy obtained is
about 10%, which is similar to previous works with the advantage to provide
’at least one relation for estimating the stellar mass and radius’ as a
function of different combinations of those observations. They also present an
analysis based on machine-learning techniques training a random forest model,
concluding that the accuracy increases in relation to standard linear
regression. However, none of the presented relations includes the stellar age.
The authors explain that such a relation is not useful as the age is not known
with good precision or accuracy.

In an earlier paper, (Gafeira et al. (2012) (hereafter, Paper I), we analysed
26 FGK stars with a known mass, luminosity, metallicity, and age for all
members of binary systems, with observational mass errors less than 3%, taken
from Torres et al. (2010), including the Sun. We derived the MLR taking into
account, as well as the metallicity and age, separately. Our results show that
the inclusion of age and metallicity in the MLR, for FGK stars, improves the
individual mass estimation by 5 % to 15 %. The main limitation of that work is
the low number of stars. In the present paper, we enlarged our sample to 56
stars and we revisited the above analysis for MLR, extending it also to the
radius-luminosity relation (RLR). For that purpose, we used the DEBCat survey,
which is a catalogue of the physical properties of well-studied detached
eclipsing binaries (Southworth, 2015), including $\alpha$ Centauri, the best-
known visual binary. We selected the binaries where both components are in the
main sequence with FGK spectral types, and mass and radius are known as better
as 3% to 4%, respectively.

The goals of the present paper are: (i) to provide relations for mass and
radius as a function of luminosity, age, and [Fe/H]; (ii) to quantify the
effects of metallicity and age on the MLR and RLR; and (iii) to discuss the
accuracy of mass and radius estimation from those relations. The paper is
organised as follows. In Sect. 2, we present the observational data for the
sample of binaries. We compute the stellar ages and masses for each star by
means of theoretical stellar models, assuming both binary members are coeval.
In Sect. 3, we derive the MLR and RLR, including the age and metallicity
contributions and we discuss the results. In Sect. 4, we summarise our main
conclusions.

##  2 Eclipsing binaries data and model: Binary age determination

Our stellar sample is taken from
DEBCat111https://www.astro.keele.ac.uk/jkt/debcat/, assuming certain criteria
that were both observational and theoretical. On the observational side, we
chose the stars for which the gravity, effective temperature, luminosity, and
metallicity, as well as the respective uncertainties are available for both
components. For the visual binary system $\alpha\leavevmode\nobreak\
$Centauri, we took the data from Kervella et al. (2017). On the other hand, we
chose the binaries where both components are main-sequence FGK stars and the
metallicities are typical for Population I (which means [Fe/H]$>-0.34$ for
this sample). When the luminosity class is not explicitly written in DEBCat,
we considered a candidate for a main-sequence star if
$3.4<\log\leavevmode\nobreak\ g<4.6$ from Angelov (1996). The reason for this
choice lies in the stellar models used to compute the mass and age (see
below). In spite of the independent age determinations for each binary system
(see respective source papers in DEBCat), for the sake of consistency, we
chose to recompute the age by means of theoretical stellar evolutionary
models. These models are scaled-solar and, thus, $[\alpha/Fe]=0$, which
indicates metal-rich stars (Recio-Blanco et al. (2014)). Moreover, stellar
models on that range are based on the input physics to realistically describe
the internal structure and the evolution of the Sun, and thus, they are
suitable for reproducing FGK stars, as their observations are well-recovered
by means of those theoretical models, such as Lebreton et al. (2008). Within
those criteria, we select 33 binaries, including the visual binary
$\alpha\leavevmode\nobreak\ Cen$. Those binaries are listed in Table 1, except
for five of them (see more information below). The age and the corresponding
error for each binary is also presented (error is computed as the average of
the half width of the $68\%$ confidence interval).

The stellar age values are presented as function of the corresponding solar
value $age_{\odot}=4.6\leavevmode\nobreak\ Gyr$ (Bonanno & Fröhlich, 2015). In
order to compute stellar ages and masses, we used the MESA isochrones from
Rodrigues et al. (2017) and Paxton et al. (2015), with scaled-solar
composition and following the chemical enrichment law, $Y=0.2485+1.007\times
Z,$ assuming the solar metal content is $Z_{\odot}$=0.01756. To achieve this,
we used the public input form PARAM 1.4222http://stev.oapd.inaf.it/cgi-
bin/param_1.4. Ages were determined using a Bayesian estimation, assuming each
star is a single one, taking into account the above observational data for
each star ($\\{T_{\rm eff},\log\leavevmode\nobreak\ g,L,$[Fe/H]$\\}$) and the
respective errors. The age of the system is estimated by the average of the
age of individual stars. We also present the values for the Sun. Using the
same procedure, the solar observations (Prša et al., 2016) were recovered for
a model with $age=4.2904\pm 0.0621\leavevmode\nobreak\ Gyr$ and
$M/M_{\odot}=1.0027\pm 0.0021$. Following the age and mass determinations, we
excluded five binaries for two reasons: there was a ’no solution’ from the
PARAM platform or the computed solution was ’astrophysically inconsistent’.
The label of ’no solution’ indicates that the PARAM solver did not converge
for the input observational data. This is the case for the following binaries:

  * •

ASAS J065134-2211,5: Hełminiak et al. (2019b) presents the orbital and
physical parameters of this binary. Concerning the analysis by means of
stellar models, they use the PARSEC code from Marigo et al. (2017) and they
fit the primary star in $1\sigma$ for an isochrone of $6.3\leavevmode\nobreak\
Gyr$ in the observational plan $\\{(M\leavevmode\nobreak\
vs\leavevmode\nobreak\ T_{\rm eff}),(M\leavevmode\nobreak\
vs\leavevmode\nobreak\ L),(M\leavevmode\nobreak\ vs\leavevmode\nobreak\
R)\\}$. However, the authors conclude that for the secondary, the radius is
larger and $T_{\rm eff}$ is clearly lower than the models. They claim that
this problem could be related to the chromospheric activity of the system.

  * •

ASAS J073507-0905,7: Hełminiak et al. (2019b) also studied this binary. The
analysis was preformed in the same observational plans. Both the proprieties
of primary and secondary are only fitted within $3\sigma$ for a very young age
(10 to 25 Myr) that could indicate that both stars are still in PMS phase.

  * •

NP Per: Lacy et al. (2016) deemed it as a PMS eclipsing binary. On the other
hand, using the MESA models, the authors claim that the observations of both
stars were not fitted with a single isochrone.

With regard to the label of a solution that was ’astrophysically inconsistent’
indicates that, cumulatively, the ages of both stars are not compatible with a
coeval hypothesis (within $1\leavevmode\nobreak\ \sigma$) and the masses of
both binary components recovered by the PARAM solver are not compatible with
the corresponding observed values, within $1\leavevmode\nobreak\ \sigma$.
Using these criteria, two more binaries are excluded:

  * •

EF Aqr: Our solutions point to ages for the primary around $2.5\pm
0.4\leavevmode\nobreak\ Gyr$ and $14\pm 1\leavevmode\nobreak\ Gyr$ for the
secondary, which is incompatible with the coeval hypothesis. On the other
hand, both observed masses are outside $1\sigma$ of the masses given by the
models: A - $1.2440\pm 0.0080\leavevmode\nobreak\ vs\leavevmode\nobreak\
1.19\pm 0.03$ and B - $0.9460\pm 0.0061\leavevmode\nobreak\
vs\leavevmode\nobreak\ 0.86\pm 0.02$. The difficulties involved in modelling
EF Aqr are confirmed by Vos et al. (2012), as they conclude that different set
of stellar evolutionary models, scaled on the Sun, are unable to recover the
observations of this binary, mainly $T_{\rm eff},R,M,$ and $L$. However, an
agreement between the models and observations can be obtained if the mixing
length parameter of the models is considerably reduced in relation to the
solar value ($\alpha_{\odot}=1.68$ for GRANADA models - Claret (2004)): 1.3
for the primary and 1.05 for the secondary. We recall that the models in our
paper assume the solar-mixing length parameter. The proposed mixing length
parameter reduction can be explained by the stellar high rotational velocity
and its impact in the convective energy transport (Clausen et al., 2009).

  * •

V636 Cen: Our solutions propose ages for the primary of around $3.5\pm
1\leavevmode\nobreak\ Gyr$ and $14\pm 1\leavevmode\nobreak\ Gyr$ for the
secondary. So, again, they are incompatible with the coeval hypothesis. On the
other hand, both observed masses are outside $1\sigma$ of the masses given by
the models: A - $1.051\pm 0.0051\leavevmode\nobreak\ vs\leavevmode\nobreak\
0.984\pm 0.03$ and B - $0.8540\pm 0.0030\leavevmode\nobreak\
vs\leavevmode\nobreak\ 0.768\pm 0.03$. Clausen et al. (2009) analysed this
binary, drawing a conclusion on the difficulty in modelling the stars using
the same kind of arguments as for EF Aqr: the mixing length parameter of both
stars must be considerably reduced in order to achieve an agreement between
models and observations.

In Fig. 1, we plot our age determinations for the remaining binaries in
comparison to those presented in the corresponding source papers of each
binary and corresponding error bars. For half of the binaries, the error on
age was not published. The mean of the absolute difference between two samples
- the previously published one and ours - is about $1.3\leavevmode\nobreak\
Gyr$. Sahlholdt et al. (2019) discuss the age for Gaia benchmark stars. They
consider as Rank A stars those ’for which the age range is well-defined and
within an interval of a few Gyr’. According to their Table 4, we can infer
that for FG dwarfs ranked A, $1.5\leavevmode\nobreak\ Gyr$ error seems
realistic and can be compared to the above differences. Assuming this value,
all binaries, but one, lies on $1\leavevmode\nobreak\ to\leavevmode\nobreak\
2\sigma$ of mutual error bars. The outlier is IM Vir. The age determined by
Morales et al. (2009) is $2.4\leavevmode\nobreak\ Gyr$ in contrast with our of
about $12\pm 2\leavevmode\nobreak\ Gyr$. The main reason for that is the fact
that Morales et al. (2009) introduced corrections to $T_{\rm eff}$ and to the
mixing-length parameter for both stars to mimic the physical effects of the
stellar chromospheric activity. Naturally, these corrections were not
implemented in our age determinations.

Figure 1: Age of this work versus the referenced age. Dashed line is a one-to-one relation. The horizontal and vertical blue lines represent respectively the age errors obtained form this work and the observational errors. We did not include any error bars on the cases where the observational error is not determined.  Binary | star | $\log M/M_{\odot}$ | error | $\log R/R_{\odot}$ | error | $\log L/L_{\odot}$ | error | $[M/H]$ | error | Ref. | $age/age_{\odot}$ | error
---|---|---|---|---|---|---|---|---|---|---|---|---
AD Boo | A | 0.1504 | 0.0028 | 0.2074 | 0.0038 | 0.640 | 0.030 | 0.10 | 0.15 | (1) | 0.3729 | 0.1351
AD Boo | B | 0.0824 | 0.0022 | 0.0849 | 0.0036 | 0.280 | 0.040 |  |  |  |  |
AL Dor | A | 0.0425 | 0.0002 | 0.0496 | 0.0039 | 0.159 | 0.033 | 0.10 | 0.10 | (2) | 0.6511 | 0.2179
AL Dor | B | 0.0421 | 0.0002 | 0.0484 | 0.0039 | 0.145 | 0.033 |  |  |  |  |
$\alpha$ Cen | A | 0.0434 | 0.0028 | 0.0878 | 0.0011 | 0.190 | 0.008 | 0.24 | 0.04 | (3) | 0.9495 | 0.1532
$\alpha$ Cen | B | -0.0297 | 0.0028 | -0.0640 | 0.0025 | -0.303 | 0.021 |  |  |  |  |
BG Ind | A | 0.1547 | 0.0024 | 0.3598 | 0.0032 | 0.870 | 0.070 | -0.20 | 0.10 | (4) | 0.5938 | 0.1248
BG Ind | B | 0.1116 | 0.0027 | 0.2253 | 0.0098 | 0.680 | 0.080 |  |  |  |  |
BK Peg | A | 0.1504 | 0.0021 | 0.2984 | 0.0017 | 0.740 | 0.020 | -0.12 | 0.07 | (5) | 0.6135 | 0.0694
BK Peg | B | 0.0993 | 0.0017 | 0.1685 | 0.0050 | 0.490 | 0.030 |  |  |  |  |
BW Aqr | A | 0.1377 | 0.0057 | 0.2388 | 0.0022 | 0.669 | 0.027 | -0.07 | 0.11 | (6) | 0.4939 | 0.0577
BW Aqr | B | 0.1670 | 0.0053 | 0.3161 | 0.0008 | 0.796 | 0.028 |  |  |  |  |
CO And | A | 0.1103 | 0.0025 | 0.2373 | 0.0053 | 0.580 | 0.038 | 0.01 | 0.15 | (7) | 0.7117 | 0.1334
CO And | B | 0.1019 | 0.0025 | 0.2289 | 0.0044 | 0.572 | 0.038 |  |  |  |  |
EPIC 210822691 | A | 0.0374 | 0.0040 | 0.0969 | 0.0035 | 0.245 | 0.028 | 0.00 | 0.05 | (8) | 1.3228 | 0.5111
EPIC 210822691 | B | -0.0177 | 0.0090 | -0.0269 | 0.0046 | -0.118 | 0.038 |  |  |  |  |
EPIC 219394517 | A | 0.0327 | 0.0008 | 0.0233 | 0.0045 | 0.091 | 0.031 | 0.07 | 0.03 | (9) | 0.4999 | 0.2257
EPIC 219394517 | B | 0.0278 | 0.0010 | 0.0179 | 0.0050 | 0.066 | 0.031 |  |  |  |  |
EW Ori | A | 0.0693 | 0.0041 | 0.0674 | 0.0019 | 0.220 | 0.030 | 0.05 | 0.09 | (10) | 0.5105 | 0.1806
EW Ori | B | 0.0504 | 0.0035 | 0.0402 | 0.0020 | 0.120 | 0.030 |  |  |  |  |
FL Lyr | A | 0.0829 | 0.0027 | 0.0948 | 0.0080 | 0.330 | 0.095 | -0.07 | 0.09 | (11) | 0.9479 | 0.5833
FL Lyr | B | -0.0217 | 0.0018 | -0.0458 | 0.0116 | -0.180 | 0.080 |  |  |  |  |
GX Gem | A | 0.1726 | 0.0032 | 0.3668 | 0.0022 | 0.860 | 0.030 | -0.12 | 0.10 | (12) | 0.7018 | 0.0788
GX Gem | B | 0.1664 | 0.0030 | 0.3499 | 0.0023 | 0.820 | 0.030 |  |  |  |  |
IM Vir | A | -0.0083 | 0.0053 | 0.0257 | 0.0065 | -0.012 | 0.034 | -0.10 | 0.25 | (13) | 2.5543 | 0.4300
IM Vir | B | -0.1776 | 0.0031 | -0.1669 | 0.0083 | -0.867 | 0.056 |  |  |  |  |
Kepler-34 | A | 0.0203 | 0.0013 | 0.0651 | 0.0011 | 0.173 | 0.033 | -0.07 | 0.15 | (14) | 1.3305 | 0.2720
Kepler-34 | B | 0.0089 | 0.0009 | 0.0385 | 0.0012 | 0.107 | 0.036 |  |  |  |  |
Kepler-35 | A | -0.0517 | 0.0025 | 0.0122 | 0.0008 | -0.027 | 0.043 | -0.34 | 0.20 | (15) | 2.5429 | 0.3827
Kepler-35 | B | -0.0918 | 0.0024 | -0.1045 | 0.0012 | -0.387 | 0.020 |  |  |  |  |
KIC 10031808 | A | 0.2408 | 0.0022 | 0.4133 | 0.0034 | 1.185 | 0.027 | -0.11 | 0.08 | (16) | 0.2593 | 0.0157
KIC 10031808 | B | 0.2548 | 0.0031 | 0.4810 | 0.0020 | 1.254 | 0.027 |  |  |  |  |
KIC 2557430 | A | 0.2279 | 0.0077 | 0.2742 | 0.0046 | 0.899 | 0.038 | -0.11 | 0.03 | (17) | 0.3651 | 0.2134
KIC 2557430 | B | 0.1303 | 0.0064 | 0.0899 | 0.0106 | 0.249 | 0.075 |  |  |  |  |
KIC 3439031 | A | 0.0890 | 0.0010 | 0.1471 | 0.0008 | 0.450 | 0.060 | 0.10 | 0.13 | (18) | 0.4972 | 0.1591
KIC 3439031 | B | 0.0884 | 0.0009 | 0.1484 | 0.0007 | 0.450 | 0.060 |  |  |  |  |
KIC 7177553 | A | 0.0183 | 0.0058 | -0.0269 | 0.0023 | -0.055 | 0.040 | -0.05 | 0.09 | (19) | 0.3702 | 0.2341
KIC 7177553 | B | -0.0061 | 0.0066 | -0.0264 | 0.0023 | -0.071 | 0.041 |  |  |  |  |
KIC 7821010 | A | 0.1062 | 0.0058 | 0.1059 | 0.0037 | 0.470 | 0.040 | 0.10 | 0.08 | (20) | 0.0712 | 0.0652
KIC 7821010 | B | 0.0867 | 0.0057 | 0.0828 | 0.0050 | 0.390 | 0.050 |  |  |  |  |
KX Cnc | A | 0.0561 | 0.0011 | 0.0269 | 0.0008 | 0.091 | 0.002 | 0.07 | 0.10 | (21) | 0.2186 | 0.0946
KX Cnc | B | 0.0535 | 0.0012 | 0.0208 | 0.0008 | 0.061 | 0.002 |  |  |  |  |
LL Aqr | A | 0.0777 | 0.0003 | 0.1209 | 0.0020 | 0.332 | 0.014 | 0.02 | 0.04 | (22) | 0.8486 | 0.1033
LL Aqr | B | 0.0144 | 0.0003 | 0.0009 | 0.0022 | -0.019 | 0.016 |  |  |  |  |
LV Her | A | 0.0766 | 0.0036 | 0.1329 | 0.0038 | 0.349 | 0.044 | 0.08 | 0.21 | (23) | 0.8695 | 0.2541
LV Her | B | 0.0681 | 0.0030 | 0.1183 | 0.0036 | 0.311 | 0.044 |  |  |  |  |
UX Men | A | 0.0874 | 0.0005 | 0.1294 | 0.0042 | 0.380 | 0.030 | 0.04 | 0.10 | (24) | 0.5995 | 0.1356
UX Men | B | 0.0747 | 0.0005 | 0.1052 | 0.0044 | 0.320 | 0.030 |  |  |  |  |
V1130 Tau | A | 0.1159 | 0.0027 | 0.1729 | 0.0029 | 0.590 | 0.020 | -0.25 | 0.10 | (25) | 0.4551 | 0.0493
V1130 Tau | B | 0.1436 | 0.0025 | 0.2509 | 0.0027 | 0.740 | 0.020 |  |  |  |  |
V785 Cep | A | 0.0426 | 0.0028 | 0.1535 | 0.0058 | 0.340 | 0.030 | -0.06 | 0.06 | (26) | 1.6414 | 0.2150
V785 Cep | B | 0.0338 | 0.0028 | 0.1377 | 0.0060 | 0.300 | 0.030 |  |  |  |  |
VZ Hya | A | 0.1041 | 0.0021 | 0.1186 | 0.0017 | 0.480 | 0.040 | -0.20 | 0.12 | (27) | 0.2798 | 0.1483
VZ Hya | B | 0.0592 | 0.0027 | 0.0461 | 0.0027 | 0.240 | 0.040 |  |  |  |  |
WZ Oph | A | 0.0888 | 0.0025 | 0.1464 | 0.0037 | 0.410 | 0.030 | -0.27 | 0.07 | (28) | 1.0450 | 0.1542
WZ Oph | B | 0.0864 | 0.0021 | 0.1520 | 0.0037 | 0.400 | 0.030 |  |  |  |  |
Sun |  | 0 | 4.4E-05 | 0 | 8.7E-05 | 0 | 0.00062 | 0 | 0.005 | see text | 0.9327 | 0.0621
Table 1: Eclipsing binaries + $\alpha\leavevmode\nobreak\ Cen$ \+ Sun: the
obtained age and errors of the solution. References: (1), (27) and (28)
Clausen et al. (2008a), Clausen et al. (2008b); (2) Gallenne et al. (2019),
Graczyk et al. (2019); (3) Kervella et al. (2017); (4) Rozyczka et al. (2011);
(5) Clausen et al. (2010b); (6) Maxted (2018), Clausen et al. (2010b); (7)
Sandberg Lacy et al. (2010); (8) Hoyman & Çakırlı (2020); (9) Torres et al.
(2018); (10) Clausen et al. (2010a), Popper et al. (1986); (11) Hełminiak et
al. (2019a), Popper et al. (1986); (12) Lacy et al. (2008); (13) Morales et
al. (2009); (14) and (15) Welsh et al. (2012); (16), (18) e (20) Hełminiak et
al. (2019a); (17) Hoyman et al. (2020); (19) Lehmann et al. (2016); (21)
Southworth (2020), Sowell et al. (2012); (22) Graczyk et al. (2016),
Southworth (2013); (23) Torres et al. (2009); (24) Andersen et al. (1989),
Hełminiak et al. (2009); (25) Clausen et al. (2010c); (26) Meibom et al.
(2009)

##  3 Mass- and radius-luminosity relations: Impact of metallicity and age

###  3.1 Method

To determine the intended empirical relations that allow us to compute the
mass and the radius, we need to perform a multi-dimensional fit. We follow the
strategy presented in Paper I, where we use inverse problem techniques
associated with the least square method. This method ensures a minimum
deviation between the computed model and the input parameters (Menke, 1989).
The system of equations that represent the model can be written in matrix
form, for example:

$F^{obs}=Gd,$ (1)

where $F^{obs}$ is the matrix containing the logarithm masses in one case and
the radius in the other, and $G$ the matrix with metallicity, age, and the
logarithm of the luminosity. The matrix, $d,$ contains the computed model
parameters that we obtain using inverse problem techniques.
As presented in paper I, we consider the empirical model in the form of:

$F^{obs}=\sum_{i=1}^{N}\sum_{j=1}^{3}a_{i,j}x_{i}^{j}.$ (2)

The index, $i,$ describes the cases when luminosity is considered ($i=1$),
when metallicity is considered ($i=2$), and, finally, when age is used
($i=3$). Depending on the number of physical quantities we used to compute the
model N can have different values. Namely, 1) if just luminosity is used; 2)
when luminosity and metallicity are used; and 3) if all elements are used. The
index, $j,$ is the order for each parameter that goes up to third order.
Rewriting Eq. 1 by taking into account that our system is over-determined and
then taking Eq. 2, we obtain Eq. 3, which we need to solve in order to compute
the desired model:

$d^{est}=(G^{T}G)^{-1}G^{T}\log F^{obs}.$ (3)

With this approach, the errors associated with each measurement and their
influence in the final method can be considered using a Monte Carlo method. We
then performed 20000 fits, where in each case, we used randomly computed
points using the standard deviation associated with each observation. In the
end, we took the average of each parameter, $a_{i,j},$ and their respective
standard deviation. The latter is considered the error of each parameter in
the fitted model.

###  3.2 Results

In solving the Eq. 3, for the values presented in Table 1, we can then compute
the model that better fits the observed mass and radius, for the considered
stars, using the different combinations for the input parameters described in
Sect. 3.1. In plot 2, we can see the results obtained estimating the mass of
the stars for several N’s.

Figure 2: Fitted $\log M/M_{\odot}$ in function of the observed $\log
M/M_{\odot}$. Mass-luminosity relation (blue), mass-luminosity-metallicity
relation (orange), and mass-luminosity-metallicity-age relation (green).

Tables LABEL:lum,LABEL:lummet, and LABEL:lummetage contain the obtained
parameters for N=1, 2, and 3, respectively, for the empirical mass model. The
correlation Pearson factor for each distribution is, respectively, 0.9663,
0.9689, and 0.9787. In plot LABEL:radi_all, we can see the results obtained
estimating the radius of the stars for several Ns. The tables 2,3, and 4
contain the obtained parameters for N=1, 2, and 3, respectively, for the
empirical mass model. The Pearson correlation factor for each distribution is,
respectively, 0.9825, 0.9826, and 0.9869.

Table 2: Parameters obtained used luminosity, to fit the radius using equation 2. $R^{2}=0.9825$ and $\sigma=0.0241$. Parameter | values
---|---
$a_{1,1}$ |  0.3060 $\pm$ 0.0074
$a_{1,2}$ |  0.1036 $\pm$ 0.0067
$a_{1,3}$ |  -0.0386 $\pm$ 0.0092
Table 3: Parameters obtained used luminosity and metallicity to fit the radius using equation 2. $R^{2}=0.9826$ and $\sigma=0.0240$. Parameter | values
---|---
$a_{1,1}$ |  0.3033 $\pm$ 0.0090
$a_{1,2}$ |  0.1008 $\pm$ 0.0148
$a_{1,3}$ |  -0.0359 $\pm$ 0.0138
$a_{2,1}$ |  -0.0176 $\pm$ 0.0358
$a_{2,2}$ |  0.0644 $\pm$ 0.1014
$a_{2,3}$ |  0.2550 $\pm$ 0.4598
Table 4: Parameters obtained used luminosity, metallicity and age to fit the radius using equation 2. $R^{2}=0.9869$ and $\sigma=0.0214$. Parameter | values
---|---
$a_{1,1}$ |  0.2908 $\pm$ 0.0133
$a_{1,2}$ |  0.0896 $\pm$ 0.0255
$a_{1,3}$ |  -0.0170 $\pm$ 0.0195
$a_{2,1}$ |  -0.0112 $\pm$ 0.0364
$a_{2,2}$ |  -0.0603 $\pm$ 0.1123
$a_{2,3}$ |  0.1221 $\pm$ 0.4477
$a_{3,1}$ |  0.0084 $\pm$ 0.0196
$a_{3,2}$ |  0.0145 $\pm$ 0.0249
$a_{3,3}$ |  -0.0057 $\pm$ 0.0074

To verify whether the improvement in the correlation factors obtained from the
different fits is not caused solely by statistical fluctuations, we use a
bootstrap (Efron, 1979) to evaluate the significance of their differences. We
re-run the fitting procedure explained before, using random stars from Table
1, accepting repetition, until we have achieved the total number of stars. We
repeated this process 20000 times, recording for each one, the obtained
correlation factors. In this case, we did not take into account the
observational errors. That ensures that the input observational values used on
the bootstrap are always the same. The results of the bootstrap both for mass
and radius fittings, assuming the three different combinations of input
parameters, are represented in the histograms in Figs. 3 and 4. Analysing the
results of the bootstrap, we can see that the correlation factor distribution
when we use age is significantly different from the other two, showing better
results. For the first two cases, the difference is not so clear. These
results follow the initial estimation differences presented when a Monte Carlo
method was used and is an indication of the significance of using the age in
the fitting for the mass and radius of the starts. Another aspect that we can
see from the analyses of the histograms is the fact that the distributions
obtained for the radius are less dispersed and they all show, on average, a
better correlation than when the mass is estimated.

Figure 3: Distribution of the bootstrap correlation factor obtained by fitting
$\log M/M_{\odot}$. Mass-luminosity relation (blue), mass-luminosity-
metallicity relation (orange), and mass-luminosity-metallicity-age relation
(green). Figure 4: Distribution of the bootstrap correlation factor obtained
by fitting $\log R/R_{\odot}$. Radius-luminosity relation (blue), radius-
luminosity-metallicity relation (orange), and radius-luminosity-metallicity-
age relation (green).

Below, we compare the mass- and radius-luminosity-metallicity-age relation
derived in this paper (Eq. 2 with $i=3$) with the relation derived in Paper I
(for the mass)333We chose to rewrite the relation as in the Paper I there is a
misprint in the last age term:  $\log M=0.219(\pm 0.023)\log
L/L_{\odot}+0.063(\pm 0.060)(\log L/L_{\odot})^{2}-$ $-0.119(\pm 0.112)(\log
L/L_{\odot})^{3}+0.079(\pm 0.031)\rm{[Fe/H]}-$ $-0.122(\pm
0.119)[Fe/H]^{2}-0.145(\pm 0.234)\rm{[Fe/H]}^{3}+$ $+0.144(\pm
0.062)StellarAge/Age_{\odot}-0.224(\pm 0.104)(StellarAge/Age_{\odot})^{2}+$
$+0.076(\pm 0.045)(StellarAge/Age_{\odot})^{3}$ : we apply both relations to
the observations listed in the Table 1, excluding the stars with observational
mass outside the range $[0.82,1.45]M_{\odot}$, (Paper I relation validity). In
Fig. 5, we plot the comparison between two predictions. The relation derived
in this paper seems to give slightly better results: the mean of the average
differences between observations and predictions is $0.049M_{\odot}$ for Paper
I and $0.028M_{\odot}$ in this paper.

Figure 5: Mass derived by empirical relations: paper I in blue and this paper
in orange, in comparison to the observed values. The black line is the one-to-
one relation.

On the other hand, we compare our empirical relation with the one establish by
Torres et al. (2010), both for mass and radius, using the data from the Table
1 stars:

$\log\leavevmode\nobreak\ M=\log\leavevmode\nobreak\ M\leavevmode\nobreak\
(T_{\rm eff},\rm{[Fe/H]},\log\leavevmode\nobreak\ g),$
$\log\leavevmode\nobreak\ R=\log\leavevmode\nobreak\ R\leavevmode\nobreak\
(T_{\rm eff},\rm{[Fe/H]},\log\leavevmode\nobreak\ g).$

In Figs. 6 and 7, we plot the predictions of mass and radius of our selected
stars using the relations from Torres et al. (2010) and ours in comparison
with the observational data. The corresponding linear regression is also
plotted. On the other hand, the mean of the absolute differences between
predictions and observations is $0.048M_{\odot}$ and $0.026R_{\odot}$ for
Torres et al. (2010) and $0.035M_{\odot}$ and $0.059R_{\odot}$ for our own. It
is interesting to note that these two relations are based in different
approaches: Torres et al. (2010) use only fundamental stellar observations, as
$T_{\rm eff}$, $\log\leavevmode\nobreak\ g$ and [Fe/H]; on the other hand, our
relation includes stellar model contributions (by the age determination).
Despite that, the predictions are globally similar.

Within the framework of the utility of our relations for stars with the age
coming from different stellar evolutionary codes, we recompute the masses and
radius using Eq. 2 ($i=3$) and using the age determination from the source
paper instead of our own. The mean of the absolute differences between
predictions and observations has slight differences: $0.039M_{\odot}$ and
$0.060R_{\odot}$. This means that based on an error of $\sim
1.3-1.5\leavevmode\nobreak\ Gyr$ (see Fig. 1), the impact of the heterogeneity
of the age determination is marginal.

Figure 6: Mass derived by empirical relations: Torres et al. (2010) in blue
and ours in orange. Figure 7: Radius derived by empirical relations: Torres et
al. (2010) in blue and ours in orange.

###  3.3 Discussion

The analysis of these results point us to two main conclusions. Firstly, the
correlation factors, both for mass and radius, increases when metallicity and
age are included in the respective luminosity relations. Secondly, the above
increase is stronger in the mass than in the radius. The first conclusion is
in agreement in our previous study, reported in Paper I. However we must
emphasise that in Paper I, the correlation factor increases more dramatically
from $R^{2}=0.830$ to $0.955$ when, respectively, only the luminosity or
luminosity-metallicity-age is considered. Let us recall the metallicity range
is quite different between these two papers, namely: $-0.60<$[Fe/H]$<0.40$ in
Paper I against $-0.34<$[Fe/H]$<0.27$ in this paper. The second conclusion is
based on the fact that the correlation factors for the radius relations are
higher than the corresponding for mass relations. In some way, the radius
diagram is less dispersed than that of the mass. This can also be seen in
Figures 6 and 7. We decide to investigate the causes, other than the obvious
one related to the connection between luminosity and radius through the
Stefan–Boltzmann law. This highest dispersion in mass, rather than in radius,
could also be a consequence of an eventual larger observational error on mass
than the radius. However, this is not the case here.

In Figs. 8 and 9, we present the corresponding histograms for observational
relative errors for mass and radius taken from DEBCat. The mean value is,
respectively, 0.66% and 0.88%. By using, thus, a single sample t-test, we
reject the null hypothesis of a mean mass error higher than mean radius errors
(confidence level of 0.95 and p-value=0.9971). This test is based on the
normal distribution for mass errors (p-value = 0.00006354, Kolmogorov-Smirnov
test). Thus, observational errors do not explain the higher dispersion in the
mass than as compared to the radius. Curiously, this tendency has been
observed in several other studies; see Fig. 15 in the above cited paper (Moya
et al. (2018). In Torres et al. (2010), we can observe similar results,
involving a scatter from these calibrations is $\sigma_{\log M}=0.027$ and
$\sigma_{\log R}=0.014$ (6.4% and 3.2%, respectively) for main-sequence and
evolved stars that is ’above $0.6M_{\odot}$’. However, also in this situation,
the mean observational error on those stars are 1.2% and 1.3% in mass and
radius, respectively. The same tendency can be observed for seismic scaling
relation; see Figs. 3 and 4 from Bellinger (2019) for more. Cross-checking
these results with our above conclusions regarding the impact of age in the
mass- and radius-luminosity-metallicity-age relations, it is reasonable to
support the conclusion that age has a more significant contribution to mass
than to the radius and this can have an impact on the highest dispersion in
mass empirical model relations.

Figure 8: Histogram of observational mass errors Figure 9: Histogram of
observational radius errors

##  4 Conclusions

In this paper, we revisit the mass and radius-luminosity-metallicity-age
relations for FGK stars. Using a multi-dimensional fit, we derive those
relations for a group of 56 stars members of eclipsing binaries, including the
best-known visual binary, $\alpha\leavevmode\nobreak\ Centauri$. From the
$DEBCat$ catalogue, the mass and radius of those stars are known, at best, as
$3\%$ to $4\%$. The luminosity for both components and metallicity (typical
for Population I) for the system is also known. The age of each binary is
determined by means of theoretical stellar models. The derived relations
reproduce the observed values within a mean accuracy of $3.5\%$ for the mass
and $5.9\%$ for the radius. We compare our results to ones published in our
first paper Gafeira et al. (2012) and with Torres et al. (2010) and find a
similar degree of quality of predictions. Our main conclusions are: (i) the
inclusion of the age improves the quality of the fit and, consequently, the
predictions. This result is more clear with regard to mass than on radius;
(ii) the mass-luminosity-metallicity-age relation is more dispersed than the
one corresponding to the radius, as noted in several other studies (that aim
to fit the observed mass or radius with such empirical relations). We propose
that the explicit inclusion of the age tends to reduce the dispersion and,
thus, to improve the quality of these relations to predict observed values.
With regard to the utility of mass- and radius-luminosity-metallicity-age
relations, we propose to apply it to a coeval and homogeneous (in terms of
chemical composition) group of stars, such as as open clusters (e.g. in the
determination of the IMF). We plan to carry out such a study in a forthcoming
paper.

###### Acknowledgements.

We would like to thank the referee for the valuable comments which helped to
improve the manuscript. The authors are grateful to the PARAM team and to J.
Southworth for having made publicly available, respectively, the web interface
for the Bayesian estimation of stellar parameters and DEBCat. CITEUC is funded
by National Funds through FCT – Foundation for Science and Technology project
_UID/Multi/00611/2019_. J. Fernandes acknowledges funding from the POCH and
Portuguese FCT –Foundation for Science and Technology _ref:
SFRH/BSAB/143060/2018_ and visiting facilities at Niels Bohr Institute
(University of Copenhagen). This research has made use of NASA’s Astrophysics
Data System Bibliographic Services and the open archive ”Astrophysics -
arXiv”.

## References

  * Andersen et al. (1989) Andersen, J., Clausen, J. V., & Magain, P. 1989, A&A, 211, 346
  * Angelov (1996) Angelov, T. 1996, Bulletin Astronomique de Belgrade, 154, 13
  * Bellinger (2019) Bellinger, E. P. 2019, MNRAS, 486, 4612
  * Bonanno & Fröhlich (2015) Bonanno, A. & Fröhlich, H. E. 2015, A&A, 580, A130
  * Claret (2004) Claret, A. 2004, A&A, 424, 919
  * Clausen et al. (2009) Clausen, J. V., Bruntt, H., Claret, A., et al. 2009, A&A, 502, 253
  * Clausen et al. (2010a) Clausen, J. V., Bruntt, H., Olsen, E. H., Helt, B. E., & Claret, A. 2010a, A&A, 511, A22
  * Clausen et al. (2010b) Clausen, J. V., Frandsen, S., Bruntt, H., et al. 2010b, A&A, 516, A42
  * Clausen et al. (2010c) Clausen, J. V., Olsen, E. H., Helt, B. E., & Claret, A. 2010c, A&A, 510, A91
  * Clausen et al. (2008a) Clausen, J. V., Torres, G., Bruntt, H., et al. 2008a, A&A, 487, 1095
  * Clausen et al. (2008b) Clausen, J. V., Vaz, L. P. R., García, J. M., et al. 2008b, A&A, 487, 1081
  * Efron (1979) Efron, B. 1979, Ann. Statist., 7, 1
  * Eker et al. (2018) Eker, Z., Bakıs, V., Bilir, S., et al. 2018, MNRAS, 479, 5491
  * Gafeira et al. (2012) Gafeira, R., Patacas, C., & Fernandes, J. 2012, Ap&SS, 341, 405
  * Gallenne et al. (2019) Gallenne, A., Pietrzyński, G., Graczyk, D., et al. 2019, A&A, 632, A31
  * Graczyk et al. (2019) Graczyk, D., Pietrzyński, G., Gieren, W., et al. 2019, ApJ, 872, 85
  * Graczyk et al. (2016) Graczyk, D., Smolec, R., Pavlovski, K., et al. 2016, A&A, 594, A92
  * Hełminiak et al. (2019a) Hełminiak, K. G., Konacki, M., Maehara, H., et al. 2019a, MNRAS, 484, 451
  * Hełminiak et al. (2009) Hełminiak, K. G., Konacki, M., Ratajczak, M., & Muterspaugh, M. W. 2009, MNRAS, 400, 969
  * Hełminiak et al. (2019b) Hełminiak, K. G., Tokovinin, A., Niemczura, E., et al. 2019b, A&A, 622, A114
  * Hoyman & Çakırlı (2020) Hoyman, B. & Çakırlı, Ö. 2020, MNRAS, 493, 2329
  * Hoyman et al. (2020) Hoyman, B., Çakırlı, Ö., & Özdarcan, O. 2020, MNRAS, 491, 5980
  * Kervella et al. (2017) Kervella, P., Bigot, L., Gallenne, A., & Thévenin, F. 2017, A&A, 597, A137
  * Kuiper (1938) Kuiper, G. P. 1938, ApJ, 88, 472
  * Lacy et al. (2016) Lacy, C. H. S., Fekel, F. C., Pavlovski, K., Torres, G., & Muterspaugh, M. W. 2016, AJ, 152, 2
  * Lacy et al. (2008) Lacy, C. H. S., Torres, G., & Claret, A. 2008, AJ, 135, 1757
  * Lebreton et al. (2008) Lebreton, Y., Montalbán, J., Christensen-Dalsgaard, J., Roxburgh, I., & Weiss, A. 2008, Ap&SS, 316, 187
  * Lehmann et al. (2016) Lehmann, H., Borkovits, T., Rappaport, S. A., et al. 2016, ApJ, 819, 33
  * Marigo et al. (2017) Marigo, P., Girardi, L., Bressan, A., et al. 2017, ApJ, 835, 77
  * Maxted (2018) Maxted, P. F. L. 2018, Research Notes of the American Astronomical Society, 2, 39
  * Meibom et al. (2009) Meibom, S., Grundahl, F., Clausen, J. V., et al. 2009, AJ, 137, 5086
  * Menke (1989) Menke, W. 1989, Geophysical Data Analysis: Discrete Inverse Theory (Dmowska, Renata and Holton, James REditors)
  * Morales et al. (2009) Morales, J. C., Torres, G., Marschall, L. A., & Brehm, W. 2009, ApJ, 707, 671
  * Moya et al. (2018) Moya, A., Zuccarino, F., Chaplin, W. J., & Davies, G. R. 2018, ApJS, 237, 21
  * Nsamba et al. (2021) Nsamba, B., Moedas, N., Campante, T. L., et al. 2021, MNRAS, 500, 54
  * Paxton et al. (2015) Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15
  * Popper et al. (1986) Popper, D. M., Lacy, C. H., Frueh, M. L., & Turner, A. E. 1986, AJ, 91, 383
  * Prša et al. (2016) Prša, A., Harmanec, P., Torres, G., et al. 2016, AJ, 152, 41
  * Recio-Blanco et al. (2014) Recio-Blanco, A., de Laverny, P., Kordopatis, G., et al. 2014, A&A, 567, A5
  * Rocha-Pinto et al. (2006) Rocha-Pinto, H. J., Rangel, R. H. O., Porto de Mello, G. F., Bragança, G. A., & Maciel, W. J. 2006, A&A, 453, L9
  * Rodrigues et al. (2017) Rodrigues, T. S., Bossini, D., Miglio, A., et al. 2017, MNRAS, 467, 1433
  * Rozyczka et al. (2011) Rozyczka, M., Kaluzny, J., Pych, W., et al. 2011, MNRAS, 414, 2479
  * Sahlholdt et al. (2019) Sahlholdt, C. L., Feltzing, S., Lindegren, L., & Church, R. P. 2019, MNRAS, 482, 895
  * Sandberg Lacy et al. (2010) Sandberg Lacy, C. H., Torres, G., Claret, A., et al. 2010, AJ, 139, 2347
  * Serenelli et al. (2020) Serenelli, A., Weiss, A., Aerts, C., et al. 2020, arXiv e-prints, arXiv:2006.10868
  * Southworth (2013) Southworth, J. 2013, A&A, 557, A119
  * Southworth (2015) Southworth, J. 2015, in Astronomical Society of the Pacific Conference Series, Vol. 496, Living Together: Planets, Host Stars and Binaries, ed. S. M. Rucinski, G. Torres, & M. Zejda, 164
  * Southworth (2020) Southworth, J. 2020, arXiv e-prints, arXiv:2012.05977
  * Sowell et al. (2012) Sowell, J. R., Henry, G. W., & Fekel, F. C. 2012, AJ, 143, 5
  * Torres et al. (2010) Torres, G., Andersen, J., & Giménez, A. 2010, A&A Rev., 18, 67
  * Torres et al. (2018) Torres, G., Curtis, J. L., Vanderburg, A., Kraus, A. L., & Rizzuto, A. 2018, ApJ, 866, 67
  * Torres et al. (2009) Torres, G., Sandberg Lacy, C. H., & Claret, A. 2009, AJ, 138, 1622
  * Valcarce et al. (2013) Valcarce, A. A. R., Catelan, M., & De Medeiros, J. R. 2013, A&A, 553, A62
  * Verma et al. (2019) Verma, K., Raodeo, K., Basu, S., et al. 2019, MNRAS, 483, 4678
  * Vos et al. (2012) Vos, J., Clausen, J. V., Jørgensen, U. G., et al. 2012, A&A, 540, A64
  * Welsh et al. (2012) Welsh, W. F., Orosz, J. A., Carter, J. A., et al. 2012, Nature, 481, 475