#  Fundamental properties of stars from Kepler and Gaia data: parallax offset
and revised scaling relations

M. Yıldız and S. Örtel

Department of Astronomy and Space Sciences, Science Faculty, Ege University,
35100, Bornova, İzmir, Turkey
E-mail<EMAIL_ADDRESS>
(Accepted 2021 April 6. Received 2021 April 6; in original form 2019 November
19)

###### Abstract

Data from the space missions Gaia, Kepler, CoRoT and TESS, make it possible to
compare parallax and asteroseismic distances. From the ratio of two densities
$\rho_{\rm sca}/\rho_{\pi}$, we obtain an empirical relation $f_{\Delta\nu}$
between the asteroseismic large frequency separation and mean density, which
is important for more accurate stellar mass and radius. This expression for
main-sequence (MS) and subgiant stars with $K$-band magnitude is very close to
the one obtained from interior MS models by Yıldız, Çelik & Kayhan. We also
discuss the effects of effective temperature and parallax offset as the source
of the difference between asteroseismic and non-asteroseismic stellar
parameters. We have obtained our best results for about 3500 red giants (RGs)
by using 2MASS data and model values for $f_{\Delta\nu}$ from Sharma et al.
Another unknown scaling parameter $f_{\nu_{\rm max}}$ comes from the
relationship between the frequency of maximum amplitude and gravity. Using
different combinations of $f_{\nu_{\rm max}}$ and the parallax offset, we find
that the parallax offset is generally a function of distance. The situation
where this slope disappears is accepted as the most reasonable solution. By a
very careful comparison of asteroseismic and non-asteroseismic parameters, we
obtain very precise values for the parallax offset and $f_{\nu_{\rm max}}$ for
RGs of $-0.0463\pm 0.0007$ mas and $1.003\pm 0.001$, respectively. Our results
for mass and radius are in perfect agreement with those of APOKASC-2: the mass
and radius of $\sim$3500 RGs are in the range of about 0.8-1.8 M⊙ (96 per
cent) and 3.8-38 R⊙, respectively.

###### keywords:

stars: distance – stars: evolution – stars: fundamental parameters – stars:
interiors – stars: late-type – stars: oscillations.

††pagerange:  Fundamental properties of stars from Kepler and Gaia data:
parallax offset and revised scaling relations – Fundamental properties of
stars from Kepler and Gaia data: parallax offset and revised scaling relations
††pubyear: 2019

##  1 Introduction

The determination of stellar properties is very important for our
understanding of astronomical structures from planets to galaxies. Thanks to
the Kepler (Borucki et al. 2010), CoRoT (Baglin et al. 2006) and TESS
(Sullivan et al. 2015) space telescopes, asteroseismic scaling relations can
be used to estimate fundamental stellar parameters. One of the essential
parameters that we require to determine fundamental stellar properties or to
check the relations between these properties is the distance of stars from us.
We have already compared asteroseismic distances $d_{\rm sca}$ derived from
the so-called asteroseismic scaling relations with the Gaia distances $d_{\rm
DR1}$ from Data Release 1 (DR1; Brown et al. 2016; Yıldız et al. 2017). There
was good agreement between them for the 64 stars, with a difference of less
than 10 per cent. In the present study, we compare $d_{\rm sca}$ and the
recent Gaia distances $d_{\pi}$ from DR2 (Brown et al. 2018) and we try to
improve asteroseismic relations, also using the data of solar-like oscillating
stars in eclipsing binaries (EBs; Gaulme et al. 2016) and first-ascent red
giants (RGs) in APOKASC-2 (Pinsonneault, Elsworth & Tayar 2018). The total
number of stars analysed is about 3600. The red clump (RC) stars in APOKASC-2
are not included in our analysis.

The Gaia parallax zero-point offset has been widely discussed in the
literature (e.g. Lindegren et al. 2018; Leung & Bovy 2019; Zinn et al. 2019a).
We also discuss the parallax zero-point offset with and without improved
scaling relations.

For the computation of the asteroseismic distance of a star, the most
important outcome of asteroseismology is the scaling relation for radius,
$R_{\rm sca}$. This relates the radius to the so-called large frequency
separation $\left<\mbox{$\Delta\nu$}\right>$ (Tassoul 1980, Christensen-
Dalsgaard 1993), the frequency of maximum amplitude $\nu_{\rm max}$(Brown et
al. 1991; Kjeldsen & Bedding 1995) and the effective temperature $T_{\rm
eff}$:

$\frac{R_{\rm sca}}{\rm R_{\odot}}=\frac{\mbox{$\nu_{\rm max}$}/\nu_{\rm
max\odot}}{(\left<\Delta\nu\right>/\left<\Delta\nu_{\odot}\right>)^{2}}\left(\frac{T_{\rm
eff}}{\rm T_{\rm eff\odot}}\right)^{1/2}.$ (1)

Here, ${\rm R_{\odot}}$, ${\rm T_{\rm eff\odot}}$, $\nu_{\rm max\odot}$ and
$\left<\Delta\nu_{\odot}\right>$ are the solar values of radius, effective
temperature, $\nu_{\rm max}$, and $\left<\Delta\nu\right>$, respectively.
Then, using $R_{\rm sca}$ and spectroscopic $T_{\rm eff}$, we can obtain the
luminosity $L_{\rm sca}$ and the bolometric magnitude. By computing a
bolometric correction, $BC$, from the tables, we can utilize the distance
modulus and hence infer the star’s distance, $d_{\rm sca}$. If we have highly
precise parallaxes, the comparison of $d_{\rm sca}$ and $d_{\pi}$ will clarify
how accurate the asteroseismic relation for $R$ is, and how accurate the
spectroscopic measurement of $T_{\rm eff}$ is.

For the computation of stellar mass from $d_{\pi}$, the surface gravity of
stars, $\log g$, should also be very precisely determined. The exact form of
$M_{\pi}$ is given as

$\frac{M_{\pi}}{\rm M_{\odot}}=\frac{g/g_{\odot}}{(\mbox{$T_{\rm
eff}$}/\mbox{$T_{\rm eff}$}_{\odot})^{4}}10^{-[V-A_{V}+5(1-\log
d_{\pi})+BC-{\rm M}_{\rm bol\odot}]/2.5},$ (2)

in solar units, where $V$, $A_{V}$, ${\rm M_{\odot}}$ and ${\rm M}_{\rm
bol\odot}$ are apparent magnitude, interstellar extinction, the solar mass and
the solar bolometric magnitude, respectively, and $d_{\pi}$ is in units of pc.
Equation (2) is a combination of five different equations. Four equations are
the expressions for the absolute magnitude, the bolometric magnitude, the
gravity and the luminosity. The fifth equation is the relation between the
bolometric magnitude and the luminosity. The standard scaling relation for
mass $M_{\rm sca}$, however, is given as

$\frac{M_{\rm sca}}{\rm M_{\odot}}=\frac{(\mbox{$\nu_{\rm max}$}/\nu_{\rm
max\odot})^{3}}{(\left<\Delta\nu\right>/\left<\Delta\nu_{\odot}\right>)^{4}}\left(\frac{T_{\rm
eff}}{\rm T_{\rm eff\odot}}\right)^{3/2}.$ (3)

It is well known that sound speed is a function of compressibility (the first
adiabatic exponent, $\Gamma_{\rm\negthinspace 1}$), in addition to
temperature. In deriving scaling relations between asteroseismic and non-
asteroseismic quantities, compressibility at the stellar surfaces,
$\Gamma_{\rm\negthinspace 1s}$, is taken as a constant. However, Yıldız, Çelik
Orhan & Kayhan (2016) have shown that $\Gamma_{\rm\negthinspace 1s}$ is not
constant at the stellar surfaces and they modified the expressions for
$\nu_{\rm max}$ and $\Delta\nu$ as functions of $\Gamma_{\rm\negthinspace
1s}$. They also found how compressibility depends on $T_{\rm eff}$ for main-
sequence (MS) stars. A similar modification of the scaling relation may be
required as a function of mean molecular weight as well (Viani et al. 2017).

De Ridder et al. (2016) generally find very good agreement (and a significant
parallax offset) between $d_{\rm sca}$ and $d_{\rm DR1}$ for 22 dwarfs and
subgiant solar-like oscillators. They also confirm that $d_{\rm DR1}$ is more
uncertain than $d_{\rm sca}$ for a sample of 938 RG solar-like oscillators
($d_{\rm sca}$ $>$ 300 pc), more distant than the pulsating dwarfs ($d_{\rm
sca}$ $<$ 250 pc). Davies, Lund & Miglio (2017) compare the Gaia DR1 distances
with $d_{\rm sca}$ of RC stars and the distances obtained using luminosity
determined by EBs. They find that $d_{\rm sca}$ is in better agreement with
the RC distance estimators than the DR1 distances. They propose a correction
to $d_{\rm DR1}$ as a function of distances $>$ 500 pc for the Kepler field of
view. Huber et al. (2017) compare parallaxes and radii from asteroseismology
and Gaia DR1 for 2200 Kepler stars spanning from the MS to the red giant
branch. They find a distance offset no larger than 5 per cent for MS stars and
low-luminosity RGs.

Stassun & Torres (2018) used eclipsing binary distances to compare the
distances from the Gaia DR2 parallaxes. They found a mean offset of $-82\pm
33$ $\mu$as, in the sense that the Gaia parallaxes are too small. A similar
offset ($-52.8\pm 2.4$ $\mu$as) is identified by Zinn et al. (2019a) for the
RGs from the APOKASC-2 catalogue. The Gaia team finds that the offset is
dependent on color and magnitude. While the global mean offset is
statistically determined as $-0.029$ mas in Lindegren et al. (2018), the zero-
point offset from 0.09 to $-0.118$ mas is reported by Arenou et al. (2018) for
various sources. Zinn et al. (2019b), Khan et al. (2019), Hall et al. (2019)
and Sahlholdt et al. (2018) try to revise conventional asteroseismic scaling
relations in order to obtain consistency between $d_{\rm sca}$ and $d_{\pi}$.
We show how the parallax zero-point offset and improvements in scaling
relations are inter-related (see Section 5).

This paper is organized as follows. In Section 2, we present DR2 distances and
other observational data for about 3600 stars, and we compare DR1 and DR2
distances of 74 stars (Y17 sample) studied in Yıldız et al. (2017). In Section
3, we briefly explain the method for the computation of asteroseismic
distance. Section 4 is devoted to the results and their comparison. The
parallax offset issue is considered in Section 5. Finally, in Section 6, we
draw our conclusions.

##  2  GAIA DISTANCES: COMPARISON OF DR1 AND DR2 DATA

We have studied 89 solar-like oscillating stars in Yıldız et al. (2016, 2019).
In Yıldız et al. (2016), it is shown that scaling relation between the mean
large separation and mean density depends on $\Gamma_{\rm\negthinspace 1s}$
and the reference frequencies ($\nu_{\rm min}$0, $\nu_{\rm min}$1 and
$\nu_{\rm min}$2) derived from glitches due to the He II ionization zone have
very strong diagnostic potential for the determination of $T_{\rm eff}$. In
Yıldız et al. (2019), the same stars are analysed and their mass, radius and
age are computed from different scaling relations including relations based on
$\nu_{\rm min}$0, $\nu_{\rm min}$1 and $\nu_{\rm min}$2. Parallaxes of the Y17
sample are available in DR1. Their $d_{\rm sca}$ and $d_{\rm DR1}$ are already
compared in Yıldız et al. (2017). Their distances range 21-433 pc.

Figure 1: The figure plots $\log d_{\pi}$ of the Y17 stars with respect to
$\log d_{\rm DR1}$ in units of pc.

In Fig. 1, $\log d_{\pi}$ of the Y17 stars is plotted with respect to $\log
d_{\rm DR1}$, with their uncertainties. The difference between $d_{\pi}$ and
$d_{\rm DR1}$ is less than 4 per cent for 54 stars and greater than 10 per
cent for seven stars. The mean difference is about 2.4 per cent. The greatest
differences occur for KIC 8379927/HIP 97321 and KIC 1435467. The differences
between their $d_{\rm DR1}$ and $d_{\pi}$ are 27 and 30 per cent,
respectively. For five stars, namely, KIC 4914923/HIP 94734, KIC 6933899, KIC
12317678/HIP 97316, KIC 12508433 and HD 181907/HIP 95222, the difference
between the distances is about 9-13 per cent.

We notice that $d_{\pi}$ is much more precise than $d_{\rm DR1}$. In Fig. 2,
uncertainty in $d_{\pi}$ ($\Delta d_{\pi}$) is plotted with respect to
uncertainty in $d_{\rm DR1}$ ($\Delta d_{\rm DR1}$) in the log-log plot. The
solid line is the fitted line. The implication is that $\Delta d_{\pi}$ is
about one-tenth of $\Delta d_{\rm DR1}$. However, it is reported in Brown et
al. (2018) that the astrometric uncertainties may be underestimated (see also
Arenou et al. 2018).

Figure 2: The figure plots $\log\Delta d_{\rm DR1}$ of the Y17 stars with
respect to $\log\Delta d_{\pi}$ in units of pc. The solid line is the fitted
line of $\log\Delta d_{\rm DR1}=1.052\log\Delta d_{\pi}+1.007$. This implies
that $d_{\pi}$ is about 10 times more precise than $d_{\rm DR1}$.

In addition to the Y17 stars, the parallaxes of five more solar-like
oscillating stars studied in Yıldız et al. (2016) are available in DR2. These
stars are KIC 8219268, KIC 11026764, HD 43587/HIP 29860, HD 146233/HIP 79672
and HD 203608/HIP 105858. The latter three stars are relatively very close to
the Sun, the closest stars among the 79 stars (Y16 DR2 sample). Their
distances are 19.30, 14.13 and 9.27 pc, respectively. Their solar-like
oscillations were detected from CoRoT light curves. KIC 8219268 and KIC
11026764 were among the Kepler targets. $d_{\pi}$ of KIC 11026764 is 156.89
pc, while KIC 8219268 is 1345 pc away and the farthest of the Y16 DR2 sample.
It is a red giant and its radius is about 6.7 R⊙. The references for $T_{\rm
eff}$, [Fe/H] and asteroseismic data for Y16 DR2 are given in Yıldız et al.
(2019).

The motivation behind this study is that the parallaxes given in DR2 are much
more precise than in DR1. This may allow us to reach more conclusive results
for the Y16 DR2 sample. The analysis of this sample leads us to extend this
study to include RGs. In addition to the solar-like oscillating RGs in
eclipsing binaries, we also consider 909 RGs from APOKASC-2 for which the Gaia
DR2 parallax and $V$ are available (VRG sample). The spectroscopic parameters
in APOKASC-2 (Nidever et al. 2015; Holtzman et al. 2015, 2018; García Pérez et
al. 2016; Wilson et al. 2019) are taken from the 14th data release of the
Sloan Digital Sky Survey (Gunn et al. 2006; Abolfathi et al. 2018). In our
analysis, we use effective temperatures $T_{\rm A}$ given in APOKASC-2. The
distances of the stars are in the range 200-11 250 pc. While most of the VRG
stars have a $d_{\pi}$ less than 4000 pc, there are only 10 giants with
$d_{\pi}>$ 6000 pc. $V$ and $K$ of these RGs are taken from the SIMBAD data
base, compiled from different sources. For more uniform data, we also use $G$
of the RGs from the Gaia data base (GRG sample). The number of stars for which
the interstellar extinction $A_{g}$ is available, which are also present in
APOKASC-2, is 1323.

In addition to $V$ and $G$ magnitudes, we also employ $J$, $H$ and $K$
magnitudes (JHKRG sample) from the 2MASS (Skrutskie et al. 2006). This sample
consists of 3524 RGs and has the largest number of stars among the samples.
There are very important advantages to using this data set, perhaps the most
important being that interstellar extinction for $J$, $H$ and $K$ is
significantly lower than extinction for $V$ and $G$.

Using the distance of stars and $V$, we can compute absolute magnitude and
then their luminosity $L_{\pi}$. For the computation of luminosity, $BC$ is
derived by interpolating between the colour-$T_{\rm eff}$ tables of Lejeune,
Cuisinier & Buser (1998) or the MESA isochrones and stellar tracks (MIST;
Paxton et al. 2011, 2013; Choi et al. 2016). As we have the $T_{\rm eff}$ of
stars from their spectroscopic observations, the radius of stars $R_{\pi}$ can
be obtained from $R_{\pi}=[L_{\pi}/(4\pi\sigma T_{\rm eff}^{4})]^{0.5}$, where
$\sigma$ is the Stefan-Boltzmann constant. Using either spectroscopic $\log
g_{\rm S}$ or asteroseismic $\log g_{\rm sca}$ gravity, stellar mass can also
be derived. For a reliable stellar mass, in addition to distance with a small
uncertainty, a very precise $\log g$ value is required.

The four samples of stars we analyse (i.e. the Y16 DR2, VRG, GRG and JHKRG
samples) are plotted in Fig. 3. There are few RGs in Y16 DR2, half of the rest
are MS stars and the other half are in subgiant (SG) phase. For Y16 DR2, both
$V$\- and $K$-band magnitudes are used.

Figure 3: The figure plots $\log L/{\rm L}_{\odot}$ of four samples with
respect to $\log\mbox{$T_{\rm eff}$}$. The filled circles, squares, crosses
and open circles denote Y16 DR2, JHKRG, VRG and GRG, respectively.

Interstellar extinction for the $V$-band, namely $A_{V}$, is computed from
$E(B-V)$ from Schlegel, Finkbeiner & Davis (1998), as recalibrated by Schlafly
& Finkbeiner (2011): $A_{V}=3.1E(B-V)$. $A_{G}$ for the $G$-band magnitude is
already given in the Gaia data base for the GRG sample stars. The extinction
coefficient for the $K$ magnitude, $A_{K}$, is obtained as $A_{K}=0.287E(B-V)$
by Fulton & Petigura (2018). We take the extinction coefficients for $H$ and
$J$ magnitudes as $A_{H}=1.74A_{K}$ (Nishiyama et al. 2008) and
$A_{J}=2.7A_{K}$ (Martin & Whittet 1990), respectively.

##  3 ASTEROSEISMIC DISTANCES, MASSES AND RADII

We compute $d_{\rm sca}$ and the distance uncertainty $\Delta d_{\rm sca}$ of
the solar-like oscillating stars as in Yıldız et al. (2017). In short, in
order to compute the distance modulus from the absolute magnitude, we must
first obtain luminosity. The luminosity of these stars is computed using
$T_{\rm eff}$ and $R_{\rm sca}$. In addition to these quantities, visual
magnitude $V$ (or $G$), metallicity ([Fe/H]) and bolometric correction $BC$
are required to compute $d_{\rm sca}$ of a star. We use both the conventional
(equations 1 and 3) and modified (see below) asteroseismic scaling relations
to obtain the mass and radius of the stars.

The scaling relations for $M$ and $R$ are based on two expressions for
$\Delta\nu$ and $\nu_{\rm max}$. The expression for $\Delta\nu$ is

$\frac{\left<\mbox{$\Delta\nu$}\right>}{\left<\mbox{$\Delta\nu$}_{\odot}\right>}=f_{\Delta\nu}\frac{\left<\rho\right>^{1/2}}{\left<\rho_{\odot}\right>^{1/2}},$
(4)

where $f_{\Delta\nu}$ (White et al. 2011; Sharma et al. 2016; Yıldız et al.
2016) encapsulates errors in the assumption of homology and also may capture
issues with the surface correction. In equation (4), $\left<\rho\right>$ and
$\left<\rho_{\odot}\right>$ are the mean stellar and solar density,
respectively. Similarly,

$\frac{\mbox{$\nu_{\rm max}$}}{\mbox{$\nu_{\rm max}$}_{\odot}}=f_{\nu_{\rm
max}}\frac{g/g_{\odot}}{(\mbox{$T_{\rm eff}$}/\mbox{$T_{\rm
eff}$}_{\odot})^{1/2}},$ (5)

where $g_{\odot}$ in equation (5) is the solar surface gravity. The parameter
$f_{\nu_{\rm max}}$, as $f_{\Delta\nu}$, also encapsulates errors in the
assumption of homology and also may capture issues with the surface
correction.

$f_{\nu_{\rm max}}$ is related to $f_{\nu}$ introduced by Yıldız et al. (2016)
as

$f_{\nu_{\rm max}}=f_{\nu}\left(\frac{\mu_{\rm s}\Gamma_{\rm\negthinspace
1s}}{\mu_{\rm s\odot}\Gamma_{\rm\negthinspace 1s{\odot}}}\right)^{1/2}$ (6)

where $\Gamma_{\rm\negthinspace 1s}$ and $\mu_{\rm s}$ are the compressibility
and the mean molecular weight at the stellar surface, respectively, and
$\Gamma_{\rm\negthinspace 1s{\odot}}$ and $\mu_{\rm s\odot}$ are
compressibility and the mean molecular weight at the solar surface,
respectively. The compressibility is mainly a function of $T_{\rm eff}$ and is
obtained as

$\frac{1}{\Gamma_{\rm\negthinspace 1s}}=1.6\left(\frac{\mbox{$T_{\rm
eff}$}}{\mbox{$T_{\rm eff}$}_{\odot}}-0.96\right)^{2}+0.607$

for the interior models of MS stars in Yıldız et al. (2016). The solar values
of $\Gamma_{\rm\negthinspace 1s{\odot}}$,
$\left<\mbox{$\Delta\nu$}_{\odot}\right>$ are taken as $1.639$ and $136.0$ in
Yıldız et al. (2016). This value of $\left<\mbox{$\Delta\nu$}_{\odot}\right>$
was preferred because it makes $f_{\Delta\nu}=1$ for the Sun, as it should be.
In this study, we take solar values of $\Delta\nu$ and $\nu_{\rm max}$ as
135.1 and 3090 $\mu$Hz (Sharma et al. 2016), respectively. The solar density
and gravity are computed from the solar data (Sackmann, Boothroyd & Kraemer
1993) as 1.4086 g cm-3 and 27 403 g cm-2, respectively.

The conventional scaling relations assume unity for $f_{\nu_{\rm max}}$ and
$f_{\Delta\nu}$. For the solar-like oscillating MS stars, this can be a good
approach. However, the structure of evolved stars is significantly different
from the solar structure. For the non-standard scaling relation for radius
$R^{\prime}_{\rm sca}$, $R_{\rm sca}$ must be multiplied by
$f_{\Delta\nu}^{2}/f_{\nu_{\rm max}}$:

$\frac{R^{\prime}_{\rm sca}}{\rm R_{\odot}}=\frac{\mbox{$\nu_{\rm
max}$}/\nu_{\rm
max\odot}}{(\left<\Delta\nu\right>/\left<\Delta\nu_{\odot}\right>)^{2}}\left(\frac{T_{\rm
eff}}{\rm T_{\rm eff\odot}}\right)^{1/2}\frac{f_{\Delta\nu}^{2}}{f_{\nu_{\rm
max}}}.$ (7)

For the non-standard scaling relation for mass $M^{\prime}_{\rm sca}$, the
factor is $f_{\Delta\nu}^{4}/f_{\nu_{\rm max}}^{3}$:

$\frac{M^{\prime}_{\rm sca}}{\rm M_{\odot}}=\frac{(\mbox{$\nu_{\rm
max}$}/\nu_{\rm
max\odot})^{3}}{(\left<\Delta\nu\right>/\left<\Delta\nu_{\odot}\right>)^{4}}\left(\frac{T_{\rm
eff}}{\rm T_{\rm eff\odot}}\right)^{3/2}\frac{f_{\Delta\nu}^{4}}{f_{\nu_{\rm
max}}^{3}}.$ (8)

Most of the stars in the APOKASC-2 catalogue are evolved RG stars. Their high
brightness allows detection of their solar-like oscillations by the Kepler and
CoRoT telescopes, despite their large distances. Therefore, any improvement in
scaling relations for these stars will be a substantial advancement not only
in stellar astrophysics but also in our understanding of chemical evolution
and dynamics of the Galactic disc.

##  4 Results and Discussions

###  4.1 Comparison of distances of the Y16 DR2 stars

In Fig. 4, we plot $\log d_{\rm sca}$ with respect to $\log d_{\pi}$ for the
Y16 DR2 sample with (circles) and without (filled circles) interstellar
extinction. The parallax of the five stars is not found in DR1, but is in DR2.
The observational $T_{\rm eff}$ and metallicities of these stars are compiled
from the literature (see table B1 in Yıldız et al. 2019 for the references).
There is, in general, very good agreement between $d_{\rm sca}$ and $d_{\pi}$
of the Y16 DR2 sample without extinction. The most striking feature of Fig. 4
is to see how successful both asteroseismic and astrometric methods are in
determining the distance of a star at $1345$ pc (KIC 8219268). However, there
is a significant difference between $\log d_{\rm sca}$ and $\log d_{\pi}$ if
the extinction is included. This implies that the extinction is very large, in
particular for stars with $d_{\pi}<$ 200 pc.

If we use the K-band magnitude in place of the V-band magnitude, the results
we obtain are very similar to those given in Fig. 4 without extinction. The
mean difference between $d_{\pi}$ and $d_{\rm sca}$ is about 2.8 per cent for
the Y16 DR2 sample. In Yıldız et al. (2017), the difference between $d_{\rm
DR1}$ and $d_{\rm sca}$ was about 5 per cent. In Fig. 5, we plot the histogram
of the normalized difference between Gaia and asteroseismic distances for the
Y16 DR2 stars with K\- and V-band magnitudes. For the K-band magnitude, the
distribution is very similar to a Gaussian distribution. The results for the
V-band magnitude are complicated and do not show a regular Gaussian curve.

Figure 4: The figure plots $\log d_{\rm sca}$ of the Y16 DR2 sample with
(circles) and without (filled circles) interstellar extinction, with respect
to $\log d_{\pi}$ in units of pc.  Figure 5:  Histogram of the normalized
difference between Gaia and asteroseismic distances for the Y16 DR2 sample.
The dotted and solid thick lines are for $K$\- and $V$-band magnitudes,
respectively. The vertical line shows the median and the thin line shows the
fitted Gaussian curves for the $K$-band magnitude.

For 51 stars of the Y16 DR2 sample, the difference between $d_{\pi}$ and
$d_{\rm sca}$ ($\delta d=d_{\pi}-d_{\rm sca}$) is less than 6 per cent. We
notice that distance of most of the stars with $\delta d/d~{}<$ 0.06 is less
than 200 pc. The largest fractional differences ($\delta d/d$) between
$d_{\pi}$ and $d_{\rm sca}$ are 0.28 and 0.22 for KIC 9025370 and KIC 8379927,
respectively. The difference between the DR1 and DR2 distances of KIC 9025370
is about 6 per cent. KIC 9025370 has the most uncertain $\nu_{\rm max}$
($\Delta\mbox{$\nu_{\rm max}$}=215$ $\mu$Hz) among the Y16 DR2 sample. This
might be the reason for the difference between the distances.

If we use the modified scaling relation for the radius of stars with
$\Gamma_{\rm 1s}$ (equation 10 in Yıldız et al. 2016 with $f_{\nu_{\rm
max}}=1$), then the mean difference between $d_{\rm sca}$ and $d_{\pi}$ is
about 1.8 per cent. Although there is a slight difference between distances
computed from conventional and modified scaling relations, the situation
changes if we compare $R_{\rm sca}$ and $R_{\pi}$ (see below).

The $\log g_{\rm S}$ of stars is perhaps the most uncertain parameter derived
from spectroscopic observations. We prepare another sample of stars for which
the difference between $\log g_{\rm S}$ and $\log g_{\rm sca}$ is less than 5
per cent. 32 stars are included in this sample. For this sample, the mean
difference between $d_{\rm sca}$ and $d_{\pi}$ is about 0.4 per cent, much
smaller than that of the Y16 DR2 sample. If $|\log g_{\rm S}-\log g_{\rm
sca}|<0.07$, then the difference in the distance of 42 stars is reduced to 0.1
per cent. Except for seven stars, the difference in the distance for all of
the stars is less than 6 per cent. The largest difference occurs for KIC
9025370: $d_{\pi}$=82.0 and $d_{\rm sca}$=$59.0$ pc.

###  4.2 Comparison of radii of the Y16 DR2 sample stars

Figure 6: The figure plots $\log R_{\pi}$ of the Y16 DR2 sample with respect
to $\log R_{\rm sca}$ in solar units. The solid line shows the fitting line
$\log R_{\pi}=(0.938\pm 0.015)\log R_{\rm sca}+(0.031\pm 0.004)$, and the
dotted line shows $\log R_{\pi}=\log R_{\rm sca}$.

The logarithm of $R_{\pi}$ of the Y16 DR2 sample with and without extinction
is plotted with respect to $\log R_{\rm sca}$ in Fig. 6. $R_{\pi}$ without
extinction is in much better agreement with $R_{\rm sca}$ than $R_{\pi}$ with
extinction. The radii of solar-like oscillating stars are in the range
$0.93-11.4$ R⊙. The largest star is HD 181907. It is a red giant and
relatively close to us. Its distance is $105.6$ pc and its solar-like
oscillations were detected by the CoRoT telescope. For small radii, $R_{\pi}$
and $R_{\rm sca}$ are in good agreement. For the large values of radii,
however, the data deviate from the $\log R_{\pi}=\log R_{\rm sca}$ line. The
fitting line is found to be $\log R_{\pi}=(0.938\pm 0.015)\log R_{\rm
sca}+(0.031\pm 0.004)$; hereafter, radii and masses in similar relationships
are written in solar units. This expression is equivalent to
$R_{\pi}=1.074R_{\rm sca}^{0.938}$. There are clear reasons for having doubts
about whether this deviation is real, as there are only two stars with a
radius larger than 3.5 R⊙. Also, the $\log R_{\pi}=\log R_{\rm sca}$ line and
the fitted line apparently deviate after 3.5 R⊙. Therefore, we obtain another
line by excluding the two largest stars. The new line is found to be
$(0.919\pm 0.024)\log R_{\rm sca}+(0.024\pm 0.006)$. The two fitted lines are
very similar to each other. However, to be sure, the data of the Kepler RG
stars should be analysed (see below).

If we use the $K$-band magnitude, we find that $\log R_{\pi}=(0.942\pm
0.018)\log R_{\rm sca}+(0.032\pm 0.005)$. This result is in very good
agreement with the fitting line for $V$-band magnitude without extinction.
This result makes us think that either $A_{V}$ is excessive for these
relatively close stars or the extinction effect is already taken into account
in the $V$ data (see below).

In computation of $R_{\rm sca}$ and $R_{\pi}$, we use the same $T_{\rm eff}$,
so $\delta d/d$ is equal to $\delta R/R=(R_{\pi}-R_{\rm sca})/R_{\rm sca}$.
The explanation is as following. For a given $V$ of a star, we essentially
know the ratio of $L_{\pi}$ to $d^{2}$. For precise determination of $d$, we
must find luminosity. For computation of $L_{\rm sca}$, we have
spectroscopically or photometrically derived $T_{\rm eff}$ and $R_{\rm sca}$
obtained from asteroseismic scaling relations. Because we use the same $T_{\rm
eff}$ in the computation of $R_{\rm sca}$ and $R_{\pi}$, the difference
between $L_{\rm sca}$ and $L_{\pi}$ is due to the difference between the radii
$R_{\rm sca}$ and $R_{\pi}$. In that case, $L/d^{2}\propto R^{2}/d^{2}$. As
this ratio is constant for a given $V$, then $\delta d/d$ is equal to $\delta
R/R$.

There is a substantial difference between $R_{\pi}$ and $R_{\rm sca}$. We can
correct the asteroseismic radius, $R_{\rm cor}$, according to the fitted line
from Fig. 6: $R_{\rm cor}=1.074R_{\rm sca}^{0.938}$, where radii are in solar
units. If we use $R_{\rm cor}$ in place of $R_{\rm sca}$ and recalculate
$d_{\rm sca}$, we see that $d_{\rm sca}$ and $d_{\pi}$ are in better
agreement.

Although the correction in $R_{\rm sca}$ improves the agreement between
$d_{\rm sca}$ and $d_{\pi}$, $R_{\rm sca}$ deviates from $R_{\pi}$ for only a
few stars with large radii. This deviation can be tested using the RG stars in
the APOKASC-2 catalogue and solar-like oscillating RGs in eclipsing binaries.

###  4.3 Solar-like oscillating RGs in EBs and implications for the scaling
relations

For the 10 EBs, the mass $M_{\rm rv}$ and radius $R_{\rm rv}$ of the component
stars are obtained by analysing their light curves and radial velocities
(Gaulme et al. 2016). The RG components in these binaries are solar-like
oscillating stars. Their $\nu_{\rm max}$ and $\left<\mbox{$\Delta\nu$}\right>$
are derived from the Kepler light curves. These stars are a good benchmark for
testing the asteroseismic scaling relations. We can compare the results of our
computations in the previous sections with the asteroseismic and non-
asteroseismic properties of solar-like oscillating RGs in EBs (OEB sample).

Gaulme et al. (2016) have already confirmed discrepancies between quantities
($M$ and $R$) derived from asteroseismic and non-asteroseismic methods. We
first compare $\nu_{\rm max}$ and $\left<\mbox{$\Delta\nu$}\right>$. In Fig.
7, the observed $\nu_{\rm max}$ is plotted with respect to $\nu_{\rm maxsca}$
where $\nu_{\rm maxsca}$ (in units of $\mbox{$\nu_{\rm max}$}_{\odot}=3050$
$\mu$Hz) is computed using equation (5) assuming $f_{\nu_{\rm max}}=1$, and
$g$ is computed from $M_{\rm rv}$ and $R_{\rm rv}$. There is a small but
significant difference between $\nu_{\rm max}$ and $\nu_{\rm maxsca}$. The
fitting line is $\mbox{$\nu_{\rm max}$}=1.038\nu_{\rm maxsca}$. This fixes
that $f_{\nu_{\rm max}}=1.038$ for the OEB sample.

Figure 7: The figure plots $\nu_{\rm max}$ of the 10 RGs in EBs (OEB sample)
with respect to $\nu_{\rm maxsca}$. The solid line is for the fitting line
$\mbox{$\nu_{\rm max}$}=1.038\nu_{\rm maxsca}$, while the dotted line
represents $\mbox{$\nu_{\rm max}$}=\nu_{\rm maxsca}$.

Kallinger et al. (2018) consider OEBs in Gaulme et al. (2016), and RGs and RCs
in the two open clusters, NGC 6791 and 6819. They derive non-linear scaling
relations, and find that the ratio $g/\sqrt{\mbox{$T_{\rm eff}$}}$ is
proportional to $\mbox{$\nu_{\rm max}$}^{1.0075}$. The gravities from
$\mbox{$\nu_{\rm max}$}=1.038\nu_{\rm maxsca}$ and from equation (16) of
Kallinger et al. (2018) (i.e. $g_{\rm sca}$ and $g_{\rm K}$, respectively) are
in very good agreement. The mean difference between $g_{\rm sca}$ and $g_{\rm
K}$ is about 1.1 per cent. While $g_{\rm sca}$ is about 0.6 per cent greater
than the gravity ($g_{\rm rv}$) derived from binary dynamics, $g_{\rm K}$ is
about 0.5 per cent less than $g_{\rm rv}$. The differences between the results
of the two studies are mainly a result of the use of different solar data. For
example, they take $\mbox{$\nu_{\rm max}$}_{\odot}=3140$ $\mu$Hz.

Figure 8: The figure plots $\left<\mbox{$\Delta\nu$}\right>$ of the OEB sample
with respect to $\left<\mbox{$\Delta\nu$}_{\rm sca}\right>$. The solid line
shows the fitting line
$\left<\mbox{$\Delta\nu$}\right>=0.986\left<\mbox{$\Delta\nu$}_{\rm
sca}\right>$, while the dotted line represents
$\left<\mbox{$\Delta\nu$}\right>=\left<\mbox{$\Delta\nu$}_{\rm sca}\right>$.

In Fig. 8, $\left<\mbox{$\Delta\nu$}\right>$ is plotted with respect to
$\left<\mbox{$\Delta\nu$}_{\rm sca}\right>$ found from the binary masses and
radii, in solar unit ($\left<\mbox{$\Delta\nu$}_{\odot}\right>=135.1$
$\mu$Hz). They seem in good agreement. Nevertheless, the slope of the fitting
line is 0.986. The difference, as much as 1.4 per cent, might be considered as
very small, although the influence of such a difference on $R$ and $M$ is
about 3 and 6 per cent, respectively. Therefore, we take into account this
discrepancy in the scaling relations.

The mean densities $\rho_{\rm sca}$ from the scaling relation with
$f_{\Delta\nu}=0.986$ and from equation (17) ($\rho_{\rm K}$, for RGs) of
Kallinger et al. (2018) are in very good agreement. The mean difference
between $\rho_{\rm sca}$ and $\rho_{\rm K}$ is about $-1$ per cent.

If we plot $R_{\pi}$ from the Y16 DR2 sample (with the BC from MIST and
$A_{V}=0$) and $R_{\rm rv}$ of the 10 RGs together with respect to $R_{\rm
sca}$, we obtain

$R=(1.067\pm 0.013)R_{\rm sca}^{0.940\pm 0.005}.$

Here, $R$ is $R_{\pi}$ for the Y16 DR2 sample and $R_{\rm rv}$ for OEB sample.
For some of the stars in the Y16 DR2 sample, $A_{V}$ is very large. This
prevents the two samples (Y16 DR2 and OEB) from being compatible with each
other.

Brogaard et al. (2018) also found the mass and radius of the stars from the
binary dynamics for the three systems among the 10 OEBs. We only used the data
of Gaulme et al. (2016) to ensure regularity in the data in our analysis.

###  4.4 Comparison of distances and radii of RGs in APOKASC-2

####  4.4.1 Comparison of distances of RGs using $V$

We compute the asteroseismic and non-asteroseismic parameters of the VRG stars
using the $BC$ tables of MIST. The $T_{\rm eff}$ and metallicity of the RGs
are taken from APOKASC-2. The mean difference between $d_{\pi}$ and $d_{\rm
sca}$ is about 12.5 per cent, much greater than the corresponding difference
for the Y16 DR2 sample. In Fig. 9, $\log R_{\pi}$ of the VRG stars (with
extinction) is plotted with respect to $\log R_{\rm sca}$. The solid line
shows the fitted line, $\log R_{\pi}=(0.949\pm 0.012)\log R_{\rm
sca}+{0.083\pm 0.012}$. This equation can also be written as
$R_{\pi}=(1.211\pm 0.032)R_{\rm sca}^{0.949\pm 0.012}$. This relationship
between the radii for this group of stars is very similar to that for the Y16
DR2 sample (see Section 4.2 and Fig. 6).

Figure 9: The figure plots $\log R_{\pi}$ of the VRG stars with and without
extinction with respect to $\log R_{\rm sca}$ in units of solar radius. The
solid line shows the fitted line $\log R_{\pi}=(0.949\pm 0.012)\log R_{\rm
sca}+{0.083\pm 0.012}$, while the dotted line denotes $\log R_{\pi}=\log
R_{\rm sca}$.

The difference between $d_{\rm sca}$ and $d_{\pi}$ is greater than 50 per cent
for some stars. Such a big difference may have some unusual causes. For KIC
10273236, for example, the unusual difference between $d_{\rm sca}$ and
$d_{\pi}$, about 62 per cent, seems to arise from its $V$. If we plot $V$ of
stars with respect to $K$, there is a strict relationship between them. Some
of the stars do not obey this relation. KIC 10273236 is among these stars and
its $V$ according to its $K$ must be about 12.25 rather than 10.2. If we take
$V$ as 12.25, the difference reduces to 3 per cent. $A_{V}$ of the VRG stars
range from 0.1 to 4.0. The ratio $R_{\pi}/R_{\rm sca}$ depends on $A_{V}$ if
$A_{V}>0.6$. If we exclude the stars with $A_{V}>0.6$, we obtain the
relationship between $R_{\pi}$ and $R_{\rm sca}$ as

$R_{\pi}=(1.131\pm 0.037)R_{\rm sca}^{0.967\pm 0.010}.$

The mean difference between $\log R_{\pi}$ and $\log R_{\rm sca}$ is 0.02 (5
per cent). For the VRG sample without extinction, $\log R_{\pi}-\log R_{\rm
sca}$= $-0.048$ (12 per cent). These results show that the extinction improves
the relationship between $R_{\pi}$ and $R_{\rm sca}$. For the Y16 DR2 sample
with the $V$-band magnitude, however, we see a better agreement when the
extinction effect is excluded. The reason for this result is that the two
samples have similar range for $A_{V}$, but very different distance ranges.
The Y16 DR2 sample is very close to the Sun in comparison to VRG, so $A_{V}$
of the Y16 DR2 sample should be smaller than that of VRG.

####  4.4.2 Comparison of distances and radii of RGs using $G$ magnitude

Figure 10: The figure plots $\log R_{\pi}$ of the GRG stars with interstellar
extinction with respect to $\log R_{\rm sca}$ in solar units. The solid line
shows the fitting line: $\log R_{\pi}=(0.996\pm 0.012)\log R_{\rm
sca}+{0.048\pm 0.013}$. The dotted line denotes $\log R_{\pi}=\log R_{\rm
sca}$.

The $V$ magnitudes of stars in the SIMBAD data base are compiled from
different sources with different $V$ bandwidths, but most of them are from
TYCHO-2 (Høg et al. 2000). This may cause some problems because $BC$ depends
on the bandwidth of measurements. Therefore, we also use the Gaia catalogue
and we use $G$ magnitudes for the computation of $BC$. In Fig. 10, $\log
R_{\pi}$ of the GRG stars is plotted with respect to their $\log R_{\rm sca}$.
For this figure, $T_{\rm eff}$ is taken from APOCASK-2 and interstellar
extinction is taken into account. The fitting curve is derived as

$\log R_{\pi}=(0.996\pm 0.012)\log R_{\rm sca}+{0.048\pm 0.013},$

where $\log R_{\pi}$ is about 12 per cent greater than $\log R_{\rm sca}$.

The $d_{\rm sca}$ and $d_{\pi}$ of RG KIC 5395942/Gaia DR2 2075037891998301312
are very different from each other. Therefore, we neglect this star in our
analysis. The problem seems to be due to its parallax (13.09 mas).

####  4.4.3 Comparison of distances and radii for the JHKRG sample

In Fig. 11, $\log R_{\pi}$ of the JHKRG stars is plotted with respect to $\log
R_{\rm sca}$ for the $J$ magnitude with extinction. The solid line shows the
fitted line, $\log R_{\pi}=(1.0217\pm 0.0029)\log R_{\rm sca}-{0.0040\pm
0.0029}$, which can also be written as $R_{\pi}=0.991R_{\rm sca}^{1.0217}$.
Note that $\log R_{\pi}$ and $\log R_{\rm sca}$ are in such good agreement
that the fitted line and the line for $\log R_{\pi}=\log R_{\rm sca}$ are
almost indistinguishable. A similar situation appears in other figures for $H$
and $K$.

Figure 11: The figure plots $\log R_{\pi}$ of the JHKRG stars for $J$ with
respect to $\log R_{\rm sca}$ in units of solar radius. The solid line shows
the fitted line $\log R_{\pi}=(1.0217\pm 0.0029)\log R_{\rm sca}-{0.0040\pm
0.0029}$, while the dotted line denotes $\log R_{\pi}=\log R_{\rm sca}$.

Most of the JHKRG stars have a distance of less than $4000$ pc. The distance
of the JHKRG stars extends up to $10000$ pc. However, at these distances,
asteroseismic and non-asteroseismic parameters are usually not in agreement.
For most of the cases, the problem seems to be the Gaia DR2 parallax, as in
the case of KIC 5395942.

###  4.5 Comparison of masses

Figure 12: The figure plots $M_{\pi}$ of the GRG stars with extinction with
respect to $M_{\rm sca}$ in solar units. The thick solid line is the fitting
curve for the GRG stars with $T_{\rm A}$: $\log M_{\pi}=(0.558\pm 0.034)\log
M_{\rm sca}+{0.152\pm 0.006}$, $M_{\pi}=(0.930\pm 0.053)M_{\rm sca}^{0.683\pm
0.045}$. The thin solid line shows the fitted line for the GRG without
extinction, $M_{\pi}=(0.930\pm 0.053)M_{\rm sca}^{0.683\pm 0.045}$, while the
dotted line denote $\log M_{\pi}=\log M_{\rm sca}$. The OEB sample (filled
circles) is also shown.

The same gravity $g_{\rm sca}$ and $T_{\rm eff}$ are used in computation of
asteroseismic and non-asteroseismic distance, radius and mass. Therefore,
$\delta g$ is equal to zero. This implies that $\delta M/M=2\delta
R/R=2(R_{\pi}-R_{\rm sca})/R_{\rm sca}$.

In our analysis, we use $\log g_{\rm sca}$ to compute $BC$ and therefore
$R_{\pi}$ is not a purely non-asteroseismic parameter. Nevertheless, the use
of $\log g_{\rm sca}$ has a very limited effect on $R_{\pi}$ because $BC$ is
not a very sensitive function of $\log g$. However, this is not the case for
$M_{\pi}$. Therefore, $M_{\pi}$ and $M_{\rm sca}$ are not independent.

Fig. 12 plots $M_{\pi}$ of the GRG stars without extinction with respect to
$M_{\rm sca}$. The fitted curve is found to be

$M_{\pi}=(0.930\pm 0.053)M_{\rm sca}^{0.683\pm 0.045}.$

The fitting line for the GRG stars with extinction is also plotted in Fig. 12.
For both cases, $M_{\rm sca}$ and $M_{\pi}$ are not in good agreement.

Also shown are the solar-like oscillating RGs in the OEB sample. Their
positions in Fig. 12 share the same area with the most of the GRG stars,
except for the RG component with $M_{\rm sca}$ of about 2.5 M⊙.

For the OEB sample, if the power of $M_{\rm sca}$ is taken as $0.698$, then

$M_{\pi}=(0.926\pm 0.041)M_{\rm sca}^{0.683}.$

This is close to the curve for GRG without extinction.

Fig. 13 plots $\log M_{\pi}$ of the JHKRG stars for $K$ magnitude with respect
to $\log M_{\rm sca}$ in solar units. The fitting line for the JHKRG stars
with $\Delta\pi/\pi<0.02$ is obtained as

$\log M_{\pi}=(0.906\pm 0.030)\log M_{\rm sca}+0.000\pm 0.004,$

which can also be written as $M_{\pi}=M_{\rm sca}^{0.906}$. For JHKRG, $M_{\rm
sca}$ and $M_{\pi}$ are in good agreement (see also Section 5.4 and Fig. 19).

Figure 13: The figure plots $\log M_{\pi}$ of the JHKRG stars (circles) for
the $K$-band magnitude with respect to $\log M_{\rm sca}$ in solar units. The
filled circles represent the stars with $\Delta\pi/\pi<0.02$. The solid line
shows the fitting line for them, $\log M_{\pi}=(0.906\pm 0.030)\log M_{\rm
sca}+0.000\pm 0.004$, and the dashed line denotes $\log M_{\pi}=\log M_{\rm
sca}$.

###  4.6 ${f_{\Delta\nu}}$ and ${f_{\nu_{\rm max}}}$ from comparison of masses
and radii

Figure 14: (a) For the Y16 DR2 sample with (circles) and without (filled
circles) interstellar extinction for the $V$-band magnitude, $f_{\Delta\nu}$
is plotted with respect to $T_{\rm eff}$ in solar units. The thick solid line
shows the fitting curve obtained by excluding three stars (open circles) in
the upper part of the figure, $f_{\Delta\nu}=(-1.280\pm 0.433)(T_{\rm
eff}/{\rm T}_{\rm eff\odot}-0.958)^{2}+1.031\pm 0.005$. The dotted line shows
$f_{\Delta\nu}$ derived from the interior models by Yıldız et al. (2016). If
we take $\nu_{\rm max\odot}$= 3000$\mu$Hz, $f_{\Delta\nu}$ for $A_{V}=0$ is in
very good agreement with that of Yıldız et al. (2016). (b) $f_{\Delta\nu}$ for
the $K$-band magnitude with and without extinction, $f_{\Delta\nu}=(-1.219\pm
0.313)(T_{\rm eff}/{\rm T}_{\rm eff\odot}-0.970)^{2}+1.026\pm 0.004$ and
$f_{\Delta\nu}=(-1.207\pm 0.301)(T_{\rm eff}/{\rm T}_{\rm
eff\odot}-0.970)^{2}+1.018\pm 0.003$, respectively. The fitting lines for
these two cases are very similar to each other.

In principle, we can obtain expressions for $f_{\nu_{\rm max}}$ and
$f_{\Delta\nu}$ from the ratios $M_{\pi}/M^{\prime}_{\rm sca}$ and
$R_{\pi}/R^{\prime}_{\rm sca}$. Assuming $M_{\pi}$ and $R_{\pi}$ as exact
values and using equations (7) and (8), we obtain

$\frac{R_{\rm\pi}}{R_{\rm sca}}=\frac{f_{\Delta\nu}^{2}}{f_{\nu_{\rm max}}},$
(9)

and

$\frac{M_{\rm\pi}}{M_{\rm sca}}=\frac{f_{\Delta\nu}^{4}}{f_{\nu_{\rm
max}}^{3}}.$ (10)

If we take the cube of equation (9) and divide it by equation (10), the right-
hand side becomes $f_{\Delta\nu}^{2}$. Then, we obtain

$f_{\Delta\nu}=\sqrt{(R_{\pi}/R_{\rm sca})^{3}/(M_{\pi}/M_{\rm sca})}.$

Similarly, we derive the expression for $f_{\nu_{\rm max}}$ as
$(R_{\pi}/R_{\rm sca})^{2}/(M_{\pi}/M_{\rm sca})$.

This formulation would be valid if we precisely determined $\log g_{\rm S}$
from stellar spectra. Unfortunately, $\log g_{\rm S}$ is not so accurate and
we have to use $\log g_{\rm sca}$. Then, $M_{\pi}$ and $R_{\pi}$ are not
purely non-asteroseismic quantities. Therefore, $f_{\nu_{\rm max}}$ and
$f_{\Delta\nu}$ are inter-related (see Section 5).

For MS and SG stars, $f_{\Delta\nu}$ is a function of $T_{\rm eff}$ (or
$\Gamma_{\rm 1s}$) as in Yıldız et al. (2016). However, this relation is
derived for the solar composition and $f_{\Delta\nu}$ may also be a function
of metallicity, for example. Fig. 14(a) plots $f_{\Delta\nu}$ of the Y16 DR2
sample with respect to $T_{\rm eff}$ , assuming $f_{\nu_{\rm max}}=1$. $BC$ is
computed from the MIST tables. Three stars (KIC 8379927, KIC 9025370 and KIC
11772920) are taking place in the upper part of Fig. 14(a). The fitting curve
is obtained by neglecting these three stars as

$f_{\Delta\nu}=(-1.280\pm 0.433)(T_{\rm eff}/{\rm T}_{\rm
eff\odot}-0.958)^{2}+1.031\pm 0.005.$

Fig. 14(a) also shows $f_{\Delta\nu}$ obtained from the interior models by
Yıldız et al. (2016) due to a variation of $\Gamma_{\rm 1s}$. The difference
between these two expressions for $f_{\Delta\nu}$ is almost constant and is
about 0.02. Such a small difference may be the result of the modelling of the
outer regions or the uncertainties in any of the observed quantities such as
$T_{\rm eff}$ and $\pi$. Another way of removing the discrepancy is to use
$\nu_{\rm max\odot}$=3000$\mu$Hz. If we compute $BC$ from Lejeune et al.
(1998) rather than MIST and use $V$ magnitude, we obtain
$f_{\Delta\nu}=-1.286(T_{\rm eff}/{\rm T}_{\rm eff\odot}-0.96)^{2}+1.017$.

In Fig. 14(b), $f_{\Delta\nu}$ is plotted for the $K$-band magnitude with and
without extinction. For the extinction case, we obtain

$f_{\Delta\nu}=(-1.219\pm 0.313)(T_{\rm eff}/{\rm T}_{\rm
eff\odot}-0.970)^{2}+1.026\pm 0.004.$ (11)

We can use spectroscopically derived $\log g_{\rm S}$ in the computation of
$M_{\pi}$ of the Y16 DR2 sample, and we try to obtain $f_{\nu_{\rm max}}$.
However, the accuracy of $\log g_{\rm S}$ does not allow us to reach a
definite relationship for $f_{\nu_{\rm max}}$, because, the spectroscopic
$\log g_{\rm S}$ is calibrated on asteroseismic $\log g_{\rm sca}$.

Similarly, we can also derive an expression for $f_{\Delta\nu}$ of RGs as a
function of $T_{\rm eff}$, again assuming $f_{\nu_{\rm max}}=1$. In Fig. 15,
$f_{\Delta\nu}$ is plotted with respect to $\mbox{$T_{\rm eff}$}/\mbox{$T_{\rm
eff}$}_{\odot}$ for the GRG stars. We notice that the most of the stars have
$f_{\Delta\nu}$ significantly less than 1 for the case without extinction. The
fitting curve is obtained as

$f_{\Delta\nu}=(-3.28\pm 1.12)(T_{\rm eff}/{\rm T}_{\rm eff\odot}-0.777\pm
0.007)^{2}+0.964\pm 0.002.$ (12)

Its maximum value is 0.964. This implies that the corrections for $R_{\rm
sca}$ and $M_{\rm sca}$ are about 7 and 14 per cent, respectively. Also shown
in Fig. 15 is the fitting line for the $G$-band magnitude with extinction,
which is very different from equation (12).

Figure 15: The figure plotd $f_{\Delta\nu}$ of the GRG stars with (circles)
and without (filled circles) interstellar extinction with respect to $T_{\rm
eff}$ in solar units. The thick solid line shows the fitting curve for
$A_{G}=0$: $f_{\Delta\nu}=(-3.28\pm 1.12)(T_{\rm eff}/{\rm T}_{\rm
eff\odot}-0.777\pm 0.007)^{2}+0.964\pm 0.002$. The dotted line denotes the
case with $A_{G}$: $f_{\Delta\nu}=(-0.679\pm 0.053)T_{\rm eff}/{\rm T}_{\rm
eff\odot}+1.590\pm 0.042$.

There are two very important uncertainties for the expression for
$f_{\Delta\nu}$ given in equation (12). These uncertainties are due to the
assumption about $f_{\nu_{\rm max}}=1$ and the $G$ magnitude. The use of $J$,
$H$ and $K$ magnitudes may change the situation (see Section 5.3).

Figure 16: The figure plots $M_{\pi}$ of the GRG stars with (circles) and
without extinction (filled circles) with respect to the corrected
asteroseismic mass $M_{\rm cor}$ in solar units using $f_{\Delta\nu}$ derived
from Fig. 15. The line denotes $M_{\pi}=M_{\rm cor}$.

In Fig. 16, $M_{\pi}$ is plotted with respect to the corrected $M_{\rm sca}$
($M_{\rm cor}$) according to $f_{\Delta\nu}$ derived for GRG from Fig. 15
(equation 12). The relationship between $M_{\pi}$ and $M_{\rm cor}$ is linear
and the positions of the most of the stars are close to the $M_{\pi}$=$M_{\rm
cor}$ line. Most of the stars have mass in the range of 0.9-1.5 M⊙ . However,
there is also significant scattering. For 319 of the GRG stars, the difference
between $M_{\pi}$ and $M_{\rm cor}$ is less than 5 per cent. There are some
stars with mass $0.5<M<0.9{\rm M}_{\odot}$. The MS lifetime of these stars is
greater than the age of the Galaxy. Therefore, they cannot be in the RG branch
as long as these masses are their initial masses. However, at least for some
of these stars, both $M_{\rm cor}$ and $M_{\pi}$ are small. This leads us to
rely on the computed masses. Then, these stars might have experienced serious
mass loss during their past evolution.

If we use $f_{\Delta\nu}$ derived from Fig. 15 for the case with extinction,
we obtain different values for the mass of the GRG stars. The mean mass
difference between the cases with and without extinction is about 0.55 M⊙ .

The $f_{\Delta\nu}$ for the VRG stars is completely different from that given
in Fig. 15 (dotted line). It has a positive slope. This conflicting situation
may also be due to extinctions for $V$\- and $G$-band magnitudes.

###  4.7 Effective temperature tests

In this section, we want to test the effect of the uncertainty of $T_{\rm
eff}$ on the differences between asteroseismic ($d_{\rm sca}$, $R_{\rm sca}$
and $M_{\rm sca}$) and non-asteroseismic quantities ($d_{\pi}$, $R_{\pi}$ and
$M_{\pi}$). We try to make the corrected $R_{\rm sca}$ and $R_{\pi}$ equal by
changing only $T_{\rm eff}$. We first clarify how $R_{\rm sca}$ and $R_{\pi}$
are sensitive to changes in $T_{\rm eff}$. For $R_{\rm sca}$, we find from the
scaling relation that

$\frac{\Delta R_{\rm sca}}{R_{\rm sca}}=\frac{1}{2}\frac{\Delta T_{\rm
eff}}{T_{\rm eff}}.$ (13)

For $R_{\pi}$, the situation is slighty more complicated. We first compute
$L_{\pi}$ from the absolute magnitude and $BC$. The change in $T_{\rm eff}$
influences $BC$ and hence $L_{\pi}$. If we neglect the temperature dependence
of $BC$, then the dependence of $R_{\pi}$ is determined by
$R_{\pi}=(L_{\pi}/4\pi\sigma)^{1/2}/\mbox{$T_{\rm eff}$}^{2}$ (for the coolest
stars, this is not the case; see below). For a given value of $L_{\pi}$, we
can show that

$\frac{\Delta R_{{\pi}}}{R_{{\pi}}}={-2}\frac{\Delta T_{\rm eff}}{T_{\rm
eff}}.$ (14)

While $R_{\rm sca}$ is proportional to $\mbox{$T_{\rm eff}$}^{1/2}$, $R_{\pi}$
is inversely proportional to $\mbox{$T_{\rm eff}$}^{2}$. As the $T_{\rm eff}$
dependences of $R_{\rm sca}$ and $R_{\pi}$ are completely different, we
attempt to remove the difference between the asteroseismic and non-
asteroseismic quantities by changing only $T_{\rm eff}$.

The estimated $T_{\rm eff}$ ($T^{\prime}_{\rm eff}$) as a function of $R_{\rm
sca}$ and $R_{\pi}$ is found from equations (13) and (14) as

$T^{\prime}_{\rm eff}=\mbox{$T_{\rm eff}$}+\Delta\mbox{$T_{\rm
eff}$}=\mbox{$T_{\rm eff}$}(1+x_{T})$ (15)

where

$x_{T}=\frac{\Delta\mbox{$T_{\rm eff}$}}{\mbox{$T_{\rm
eff}$}}=\frac{R_{\pi}-R_{\rm sca}}{2R_{{\pi}}+R_{\rm sca}/2}.$ (16)

If $R_{\pi}$ is less than $R_{\rm sca}$, then $T^{\prime}_{\rm eff}$ is lower
than $T_{\rm eff}$; otherwise, $T^{\prime}_{\rm eff}$ is higher than $T_{\rm
eff}$.

$BC$ is almost constant for about $5000<\mbox{$T_{\rm eff}$}<6000$ K. For
$\mbox{$T_{\rm eff}$}<4100$ K, however, $BC$ is a very sensitive function of
$T_{\rm eff}$. Therefore, we follow an iterative method to find the most
appropriate $T^{\prime}_{\rm eff}$ for a star for which $R_{\pi}=R_{\rm sca}$.
We first compute all the asteroseismic and non-asteroseismic quantities using
the catalogue value of $T_{\rm eff}$ and then we find $T^{\prime}_{\rm eff}$.
We recompute all the quantities using the new effective temperature and we
repeat the procedure until the effective temperature is fixed at a certain
value.

However, for $\mbox{$T_{\rm eff}$}<$ 4100 K, where $\partial
BC/\partial\mbox{$T_{\rm eff}$}$ is not small, we find that the coefficient
($-2$) on the right-hand side of equation (14) becomes $-2-(\partial
BC/\partial\ln\mbox{$T_{\rm eff}$})\ln(10)/5$. For $\mbox{$T_{\rm eff}$}<$
4100 K, we obtain $(\partial BC/\partial\ln\mbox{$T_{\rm eff}$})=3.934$ from
the MIST tables. This method can be applied to the much more precise data for
parallax than Gaia DR2.

The difference between $d_{\rm sca}$ and $d_{\pi}$ can be removed in a variety
of ways, one of which is to change $T_{\rm eff}$. The temperature difference
between $T_{\rm A}$ and effective temperature from LAMOST (Zong et al. 2018)
reaches 120 K. If we decrease $T_{\rm A}$ by about 183 K, there is no mean
difference between $d_{\rm sca}$ and $d_{\pi}$. However, a change of 183 K in
the APOGEE temperature scale would be very large.

##  5 Parallax offset and scaling relations

The difference between asteroseismic and non-asteroseismic distance can arise
either from the difference between true ($\pi_{\rm true}$) and Gaia DR2
parallaxes or from the difference between true radius ($R_{\rm true}$) and
$R_{\rm sca}$. If there is an offset, then $\Delta\pi=\pi-\pi_{\rm true}$ is a
constant. $R_{\rm true}$ can be written as a function of $f_{\nu_{\rm max}}$
and $f_{\Delta\nu}$ : $R_{\rm true}=R_{\rm sca}f_{\Delta\nu}^{2}/f_{\nu_{\rm
max}}$. If $R_{\rm sca}$ is perfect, then $f_{\Delta\nu}=f_{\nu_{\rm max}}=1$,
so the defect is pertaining to the Gaia DR2 data and we can obtain
$\Delta\pi$. If $\pi$ is perfect, then $\Delta\pi=0$ and we can at least find
$f_{\Delta\nu}^{2}/f_{\nu_{\rm max}}$ from the relation between $R_{\rm true}$
and $R_{\rm sca}$ (see below). However, we can do something better than these
two options, if we can obtain a relationship between $f_{\Delta\nu}^{2}$ ,
$f_{\nu_{\rm max}}$ and $\Delta\pi$ (see also Khan et al. 2019; Zinn et al.
2019b).

The radius and distance of a star are related through its luminosity. We
assume that $T_{\rm eff}$, $V$ and $BC$ are given. If we determine the radius
(e.g. from asteroseismic scaling relations), then we can find the distance
from the comparison of the flux at the surface of the star and the flux we
receive. If we know the radius, we can determine distance, or vice versa. If
the difference between ’true’ and estimated radii is, for example, 2 per cent,
then the difference between true and estimated distances is also 2 per cent
(see section 4.2).

This is the case for both sets of quantities based on Gaia DR2 parallax and
asteroseismic scaling relations.

$R_{\rm true}=R_{\pi}\frac{d_{\rm true}}{d_{\pi}}=\frac{R_{\pi}}{(1-\Delta\pi
d_{\pi})}.$

Using the relation between $R_{\rm sca}$ and $R_{\rm true}$,

$R_{\rm true}=R_{\rm sca}\frac{f_{\Delta\nu}^{2}}{f_{\nu_{\rm
max}}}=\frac{R_{\pi}}{(1-\Delta\pi d_{\pi})}.$

Then,

$f_{\nu_{\rm max}}=f_{\Delta\nu}^{2}(1-\Delta\pi d_{\pi})\frac{R_{\rm
sca}}{R_{\pi}}.$ (17)

For MS and RG stars we know $f_{\Delta\nu}$ from models. Then, we have a
relationship between $f_{\nu_{\rm max}}$ and $\Delta\pi$ from equation (17).

###  5.1 Parallax offset for the MS and SG stars

If we take $f_{\Delta\nu}=f_{\nu_{\rm max}}=1$, the parallax offset for the
$K$-band magnitude of the Y16 DR2 sample with and without interstellar
extinction is $-0.189$ and $-0.094$ mas, respectively. We have already derived
an expression for $f_{\Delta\nu}$ for the former case (equation 11). Use of
this expression gives a negligibly small offset, $\Delta\pi=0.005$ mas.

###  5.2 Parallax offset for RG stars for approximate values of
$f_{\Delta\nu}$ and $f_{\nu_{\rm max}}$

We have already derived $f_{\Delta\nu}$ assuming no parallax offset. If we
take $f_{\Delta\nu}=f_{\nu_{\rm max}}=1$, then we can find the value of
$\Delta\pi$. For $J$-, $H$\- and $K$-band magnitudes, we obtain $\Delta\pi$ as
$-0.0232$, $-0.0127$ and $-0.0145$ mas, respectively. Another way to compute
$\Delta\pi$ is to use values of $f_{\Delta\nu}$ and $f_{\nu_{\rm max}}$ from
the OEB sample. In that case, $\Delta\pi$ is equal to $-0.0621$, $-0.0506$ and
$-0.0525$ mas for the $J$-, $H$\- and $K$-band magnitudes, respectively. These
values of $\Delta\pi$ are in good agreement with the values given in the
literature. In some studies on the analysis of Gaia DR2 data, potential
color-, magnitude-, and spatial-dependent terms to $\Delta\pi$ are found (see,
e.g., Leung & Bovy 2019; Zinn et al. 2019a). Similar associations can be made
for our results. However, there are some cases where these dependences
disappear (see below).

###  5.3 Relationship between parallax offset and $f_{\nu_{\rm max}}$ for RG
stars with $f_{\Delta\nu}$ from models

Model values of $f_{\Delta\nu}$ ($f_{\Delta\nu{\rm mod}}$) are available from
Sharma et al. (2016). $f_{\Delta\nu{\rm mod}}$ is computed using the code
given by Sharma et al. (2016), as a function of $T_{\rm eff}$, $\log Z$, $M$
and $\log g$. It is in very good agreement with the expression derived by
Yıldız et al. (2016) for MS models. For RGs, however, $f_{\Delta\nu{\rm mod}}$
is very different from that given in Fig. 15 and its mean value for the JHKRG
stars is about 0.975. This value is in good agreement with the value found
from the OEB sample.

If we use $f_{\Delta\nu{\rm mod}}$, there are now two unknowns in our
analysis: $\Delta\pi$ and $f_{\nu_{\rm max}}$. In this subsection, we first
consider $K$-band magnitude of the JHKRG stars. If we take $f_{\nu_{\rm
max}}=1$, then $\Delta\pi$ becomes about -0.044 mas. For $f_{\nu_{\rm
max}}=1.038$, the value for the OEB sample, $\Delta\pi$ is about -0.067 mas.
In Fig. 17, $f_{\nu_{\rm max}}$ is plotted with respect to $\Delta\pi$. For
each pair of solutions in Fig. 17, the difference ($\delta\pi$) between
$\pi^{\prime}=\pi-\Delta\pi$ and $\pi_{\rm sca}$ shows $\pi$ dependence. We
make a line equation fit to the graph of $\delta\pi$ and $\pi$:
$\delta\pi=a\pi+b$, where $a$ and $b$ are plotted with respect to $\Delta\pi$
in Fig. 18. While the slope $a$ is less than zero for $\Delta\pi=-0.067$ mas,
it is greater than zero for $\Delta\pi=-0.021$, for example. This means that
at an intermediate value of $\Delta\pi$, the slope becomes zero. At
$\Delta\pi=-0.0463$ mas, $f_{\nu_{\rm max}}=1.003$. At this point, $a$ and $b$
intersect each other and they are very close to zero. This is reasonable
because the real mean difference between the two parallaxes is zero where the
slope is zero.

We apply the same method to $J$ and $H$ data. A summary of the results is
provided in Table 1. Values of $\Delta\pi$ for $H$\- and $K$-band magnitudes
are very close to each other. This is the case also for the values of
$f_{\nu_{\rm max}}$.

Figure 17: The figure plots $f_{\nu_{\rm max}}$ with respect to $\Delta\pi$ for JHKRG stars with $f_{\Delta\nu{\rm mod}}$ from Sharma et al. (2016). This figure is for the $K$-band. Very similar figures are obtained for the $J$\- and $H$-band.  Figure 18: $\delta\pi=\pi^{\prime}-\pi_{\rm sca}$ is a function of $\pi$ for given values of $f_{\nu_{\rm max}}$ and $\Delta\pi$ and can be represented by a line equation: $\delta\pi=a\pi+b$. Here, $a$ (plus symbols) and $b$ (circles) for the $K$-band are plotted with respect to $\Delta\pi$. At about $\Delta\pi=-0.0463\pm 0.0007$ ($f_{\nu_{\rm max}}=1.003\pm 0.001$), $a$ and $b$ both vanish and intersect each other. The typical uncertainty of $f_{\nu_{\rm max}}$ is obtained from the uncertainties of $a$ and $b$ at $\Delta\pi=-0.0463\pm 0.0007$: $\Delta f_{\nu_{\rm max}}=\Delta a\Delta\pi+\Delta b$. The typical uncertainty for $\Delta\pi$ is found to be 0.0007 mas from the $\Delta\pi$ difference between the points $a=0$ and $b=0$.  Table 1: Mean values of asteroseismic scaling parameters and parallax offset for $J$-, $H$\- and $K$-band magnitudes for JHKRG. $\overline{f_{\Delta\nu{\rm mod}}}$ is the mean of $f_{\Delta\nu{\rm mod}}$ computed from Sharma et al. (2016). For comparison, the scaling parameters of OEBs are also given. We notice that there is a very good agreement between the scaling parameters for $H$ and OEBs. $\Delta M$ given in the last column is the mean difference between $M^{\prime}_{\rm sca}$ and $M_{\rm AP}$. In the last row, uncertainties in $f_{\nu_{\rm max}}$ and $\overline{f_{\Delta\nu}}$ are given for OEBs.  Data |  $\Delta\pi$(mas) | $f_{\nu_{\rm max}}$ | $\overline{f_{\Delta\nu{\rm mod}}}$ |  $\Delta M$(M⊙ )
---|---|---|---|---
$J$ | $-0.0333$ | 0.966 | 0.975 | 0.148
$H$ | $-0.0424$ | 1.000 | 0.975 | 0.010
$K$ | $-0.0463$ | 1.003 | 0.975 | $-0.001$
OEBs | — | 1.038 | 0.986 | —
$\epsilon$(OEB) | — | 0.010 | 0.006 | —

We also obtain solutions for $J$-, $H$\- and $K$-band magnitudes without
extinction. The results are summarized in Table 2. It is worth noting that the
results for $H$ are very consistent with the data of OEBs.

Table 2: The same as Table 1 but without interstellar extinction.  Data |  $\Delta\pi$(mas) | $f_{\nu_{\rm max}}$ | $\overline{f_{\Delta\nu{\rm mod}}}$ |  $\Delta M$(M⊙ )
---|---|---|---|---
$J$ | $-0.0351$ | 1.019 | 0.975 | 0.048
$H$ | $-0.0432$ | 1.036 | 0.975 | 0.103
$K$ | $-0.0459$ | 1.025 | 0.975 | 0.065
OEBs | — | 1.038 | 0.986 | —
| — | 0.010 | 0.006 | —

###  5.4 Masses of RGs

Figure 19: The figure plots $M^{\prime}_{\pi}$ and $M_{\rm AP}$ with respect
to $M^{\prime}_{\rm sca}$ for the $J$-, $H$\- and $K$-band magnitudes. The
solid line denotes $M^{\prime}_{\pi}=M^{\prime}_{\rm sca}$.

The mean difference between mass from the corrected parallax
$M^{\prime}_{\rm\pi}$ and mass $M_{\rm AP}$ given in APOKASC-2 is given in the
last column of Table 1. The difference is negligibly small for $K$ and $H$
magnitudes, about $-0.001$ and 0.010 M⊙ , respectively.

Fig. 19 shows the first systematic comparison of Gaia and asteroseismic masses
across a wide range in metallicity. Zinn et al. (2019b) compare the mass
scales in the metal-poor regime, but not across all metallicities. In Fig. 19,
$M^{\prime}_{\rm\pi}$ and $M_{\rm AP}$ are plotted with respect to
$M^{\prime}_{\rm sca}$. $M_{\rm AP}$ is calibrated to the dynamical mass scale
from the NGC 6791 and 6819 clusters, and theoretical $\Delta\nu$ corrections
are applied. While there is an agreement between $M^{\prime}_{\rm sca}$ and
$M^{\prime}_{\rm\pi}$ for the $J$ magnitude, a substantial difference is
observed between $M^{\prime}_{\rm sca}$ and $M_{\rm AP}$. For $H$\- and
$K$-band magnitudes, $M^{\prime}_{\rm sca}$ and $M_{\rm AP}$ are in perfect
agreement to the greatest extend. They are slightly different if $M<0.75$ M⊙ .

For some of the stars, fundamental properties are very accurate. For 580 RGs,
the difference between $M^{\prime}_{\rm sca}$ and $M^{\prime}_{\rm\pi}$ is
less than 2 per cent. The number of stars with mass difference less than 3 per
cent is 837. A number of selected stars deserve to be studied in detail.

Fig. 20 exhibits how the masses of RGs are distributed; 96 per cent of the
stars have the mass range 0.8-1.8 M⊙ . The radii of these stars range from 3.8
to 38 R⊙. The number of stars is the maximum at about 1.2 M⊙ .

Figure 20: Mass ($M^{\prime}_{\rm sca}$) distribution of RGs.

##  6 Conclusions

Combining all photometric, spectral, astrometric and asteroseismic data, we
find the distance, radius and even mass of about 3600 stars from two different
methods and compare these findings in order to assess Gaia parallaxes and to
improve the asteroseismic scaling relations for stellar mass and radius. In
this regard, the Gaia, Kepler and TESS space missions open a new era in our
understanding of stellar evolution theory.

We first compare the first two data releases of Gaia. The DR2 parallaxes are
more precise than the DR1 parallaxes. The 5 per cent mean difference between
$d_{\rm sca}$ and $d_{\rm DR1}$ (Yıldız et al. 2017) reduces to 2.8 per cent
for $d_{\pi}$ for the Y16 DR2 (MS + SG + 2RGs) stars with $K$-band magnitude
and without any correction. From the comparison of $R_{\rm sca}$ and
$R_{\pi}$, we find that $R_{\pi}=1.076R_{\rm sca}^{0.942}$. As there are only
two RGs in this sample, we extend our study to the RGs in the APOKASC-2
catalogue. There are three samples of RGs we analyse. The luminosity of the
stars is computed from the $V$ magnitude for one group (VRG) and from the Gaia
$G$ magnitude for the other group (GRG). About 493 RGs are common in both
groups. We also consider $J$-, $H$\- and $K$-band magnitudes of RGs (JHKRG) in
the APOKASC-2 catalogue. Almost for all of the RGs in the catalogue,
$\sim$3500, their $J$-, $H$\- and $K$-band magnitudes are available. From the
comparison of $R_{\rm sca}$ and $R_{\pi}$, we obtain relationships for the
VRG, GRG and JHKRG samples that are very similar to the relationship for the
Y16 DR2 sample. These relationships leads us to show how to improve the
conventional scaling relations. Similar relationships are valid also for OEBs
(solar-like oscillating RGs in eclipsing binaries). We also find the
relationship between $M_{\rm sca}$ and $M_{\pi}$. From the ratio
$\sqrt{(R_{\pi}/R_{\rm sca})^{3}/(M_{\pi}/M_{\rm sca})}$, we find the
expression for $f_{\Delta\nu}$, assuming $f_{\nu_{\rm max}}=1$. This
expression for Y16 DR2 with $K$-band magnitude is almost identical to the one
we have obtained from interior models for MS stars. Interstellar extinction
complicates the situation in most cases. In this regard, we obtained the best
solutions for the $H$\- and $K$-band magnitudes.

Using these expressions for $f_{\Delta\nu}$, we can improve $M_{\rm sca}$ and
$R_{\rm sca}$. Such a correction for $R_{\rm sca}$ and $M_{\rm sca}$ of the
OEB stars means that there is agreement with radius and mass derived from
orbital analysis.

Improvement for $M$ and $R$ can also be done over $T_{\rm eff}$, because the
dependences of asteroseismic ($d_{\rm sca}$, $R_{\rm sca}$ and $M_{\rm sca}$)
and non- asteroseismic quantities ($d_{\pi}$, $R_{\pi}$ and $M_{\pi}$) on
$T_{\rm eff}$ are completely different. While $R_{\rm sca}$, for example, is
proportional to $T_{\rm eff}^{1/2}$, we find $R_{\pi}$ $\propto$ $T_{\rm
eff}^{-2}$ for RGs with $T_{\rm eff}>4100$ K. Therefore, slight modifications
in $T_{\rm eff}$ remove all the differences between asteroseismic and non-
asteroseismic quantities for most of the stars. For the RGs with $T_{\rm
eff}<4100$ K, we also take into account the derivative of $BC$ with respect to
$T_{\rm eff}$.

We also show how $f_{\Delta\nu}$, $f_{\nu_{\rm max}}$ and $\Delta\pi$ are
inter-related (see equation 17). If we employ $f_{\Delta\nu}$ as derived for
Y16 DR2 with $K$-band magnitude (equation 11) and take $f_{\nu_{\rm max}}=1$,
we obtain a parallax offset of about 0.005 mas.

We reach the best solution for RGs by using the model results of Sharma et al.
(2016) for $f_{\Delta\nu}$. Using the data of JHKRG, we show how $f_{\nu_{\rm
max}}$ and $\Delta\pi$ are inter-related and we find that the difference
between $\pi-\Delta\pi$ and $\pi_{\rm sca}$ is $\pi$ dependent. We fit a line
to this trend for $\Delta\pi$ derived using $K$-band magnitudes and we see
that the slope vanishes at about $\Delta\pi=-0.0463$ mas for the $K$-band
magnitude (see Table 1). At this value of $\Delta\pi$, the difference between
$\pi-\Delta\pi$ and $\pi_{\rm sca}$ is very small. In this case, the mean
value is $f_{\nu_{\rm max}}=1.003$. For the $H$ magnitude, we find that
$f_{\nu_{\rm max}}=1.000$ and $\Delta\pi=-0.0424$. The use of these values of
$f_{\nu_{\rm max}}$, $\Delta\pi$ and $f_{\Delta\nu\rm mod}$ from the models
ensures that the asteroseismic and non-asteroseismic quantities are very
compatible with each other (see Table 1), in particular for the $H$\- and
$K$-band magnitudes. For these magnitudes, the asteroseismic mass
$M^{\prime}_{\rm sca}$ is in perfect agreement with the mass given in
APOKASC-2 (see Fig. 19). The mean difference for the $K$-band magnitude, for
example, is 0.001 M⊙ . We find that the mass and radius of the stars range
from 0.8 to 1.8 M⊙ and from 3.8 to 38 R⊙ , respectively. The number of stars
is maximum at about 1.2 M⊙ .

## Acknowledgements

We acknowledge discussions with Daniel Huber, M. Salaris and Antonio Frasca.
We would like to thank Dr. Frederick A. Magill for his professional checking
of the language in the manuscript. This work is supported by the Scientific
and Technological Research Council of Turkey (TÜBİTAK: 118F352).

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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