# Asteroseismology of 3,642 Kepler Red Giants: Correcting the Scaling
Relations based on Detailed Modeling

Tanda Li(李坦达) School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham B15 2TT, UK  Stellar Astrophysics Centre (SAC),
Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000
Aarhus C, Denmark  Yaguang Li(李亚光) Sydney Institute for Astronomy (SIfA),
School of Physics, University of Sydney, NSW 2006, Australia  Stellar
Astrophysics Centre (SAC), Department of Physics and Astronomy, Aarhus
University, Ny Munkegade 120, 8000 Aarhus C, Denmark  Shaolan Bi(毕少兰)
Department of Astronomy, Beijing Normal University, Beijing, 100875, China
Timothy R. Bedding Sydney Institute for Astronomy (SIfA), School of Physics,
University of Sydney, NSW 2006, Australia  Stellar Astrophysics Centre (SAC),
Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000
Aarhus C, Denmark  Guy Davies School of Physics and Astronomy, University of
Birmingham, Edgbaston, Birmingham B15 2TT, UK  Stellar Astrophysics Centre
(SAC), Department of Physics and Astronomy, Aarhus University, Ny Munkegade
120, 8000 Aarhus C, Denmark  Minghao Du(杜明昊) Department of Astronomy, Beijing
Normal University, Beijing, 100875, China

###### Abstract

The paper presents a correction to the scaling relations for red-giant stars
using model-based masses and radii. We measure radial-mode frequencies from
_Kepler_ observations for 3,642 solar-like oscillators on the red-giant branch
and use them to characterise the stars with the grid-based modeling. We
determine fundamental stellar parameters with good precision: the typical
uncertainty is 4.5% for mass, 16% for age, 0.006 dex for surface gravity, and
1.7% for radius. We also achieve good accuracy for estimated masses and radii,
based on comparing with those determined for eclipsing binaries. We find a
systematic offset of $\sim 15\%$ in mass and $\sim 7\%$ in radius between the
modeling solutions and the scaling relations. Further investigation indicates
that these offsets are mainly caused by a systematic bias in the $\Delta\nu$
scaling relation: the original scaling relation underestimates $\Delta\nu$
value by $\sim 4\%$, on average, and it is important to correct for the
surface term in the calibration. We find no significant offset in the
$\nu_{\rm max}$ scaling relation, although a clear metallicity dependence is
seen and we suggest including a metallicity term in the formulae. Lastly, we
calibrate new scaling relations for red-giant stars based on observed global
seismic parameters, spectroscopic effective temperatures and metallicities,
and modeling-inferred masses and radii.

star: oscillation

††journal: APJS

##  1 Introduction

Asteroseismology of red giants has been undergoing extensive development in
the last decade, thanks to the space missions like _CoRoT_(Baglin et al.,
2002), _Kepler_(Borucki et al., 2009), _K2_ (Howell et al., 2014) and _TESS_
(Ricker et al., 2015). High-quality photometry data for thousands of red
giants are now available for asteroseismology. Automated data reduction tools
for measuring global seismic parameters (e.g. SYD Pipeline; Huber et al.,
2009a), acoustic modes (e.g. Pbjam; Nielsen et al., 2021), and mixed modes
(e.g. FAMED; Corsaro et al., 2020) make it possible to study solar-like
oscillations in large samples of stars. Compared with studying individual
stars, a population study covering a wide parameter range has great
advantages. For instance, the systematic characterisation of solar-like
oscillations and granulation can constrain stellar masses and radii with good
precision (e.g. Yu et al., 2018a). In addition, measuring gravity-mode period
spacings for red giants allows us to precisely monitor their stellar evolution
from the subgiant stage to the end of helium burning phase (Mosser et al.,
2014; Vrard et al., 2016). Finally, core rotation periods, measured from the
splitting in oscillation modes (e.g. Mosser et al., 2012a), show a significant
spinning down during the red-giant phase and provide constraints on the
angular momentum transport between the core and the envelope (Aerts et al.,
2019; Tayar et al., 2019).

Studying a large sample of stars on the red-giant branch (RGB) could advance
the resolution of puzzles at this evolutionary stage. One of these is the
remarkable accuracy of the scaling relations. Scaling relations were
introduced to predict the oscillation frequencies of solar-type stars (Ulrich,
1986; Brown et al., 1991; Kjeldsen & Bedding, 1995):

$\frac{\Delta\nu}{\Delta\nu_{\odot}}\simeq\left(\frac{\rho}{\rho_{\odot}}\right)^{0.5}$
(1)

and

$\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\simeq\left(\frac{g}{g_{\odot}}\right)\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{-0.5}.$ (2)

Here, $\nu_{\rm max}$ is the frequency at the maximum amplitude of stellar
oscillations, and $\Delta\nu$ is the mean frequency separation between
consecutive modes of the same spherical degree. These relations can be
inverted to estimate masses and radii (Stello et al., 2008; Kallinger et al.,
2010a):

$\frac{M}{M_{\odot}}\simeq\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)^{3}\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{-4}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{1.5},$ (3)

and

$\frac{R}{R_{\odot}}\simeq\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{-2}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{0.5}.$ (4)

The solar reference values have been measured in previous studies with slight
differences, depending on the data and the reduction method. In this work, we
adopt $\nu_{\rm max,\odot}$ = 3090$\pm$30$\mu$Hz, $\Delta\nu_{\odot}$ =
135.1$\pm$0.1$\mu$Hz, and $T_{\rm eff,\odot}$ = 5777K (Huber et al., 2011).

The accuracy of scaling relations has been widely discussed (Stello et al.,
2009; White et al., 2011; Huber et al., 2012; Mosser et al., 2013; Li et al.,
2021) and noticeable systematic offsets have been seen in RGB stars. Stars in
eclipsing binary (EB) systems can provide direct tests of the scaling
relations. For instance, Gaulme et al. (2016) studied ten high-luminosity red-
giant stars in eclipsing binaries by comparing masses and radii obtained from
the scaling relations against spectroscopic orbital solutions using dynamical
modeling. They found that the scaling relations overestimate the radii by
about 5% and the masses by about 15%. With the similar approaches, Brogaard et
al. (2018) and Themeßl et al. (2018) also found overestimated masses and radii
derived by the scaling relations. The radius measured based on interferometry
provides another direct test. For instance, Huber et al. (2012) combined
interferometric angular diameters, Hipparcos parallaxes, asteroseismic
densities, bolometric fluxes and high-resolution spectroscopy to derive near
model-independent fundamental properties for one subgiant and four red giant
stars near the base of RGB. Unlike the EB results, their measurements showed
good agreement with the scaling relations.

Determining masses and radii from the detailed modeling is another way to
verify the scaling relations. Pérez Hernández et al. (2016) characterised 19
low-luminosity _Kepler_ red giants based on a minimisation procedure combining
the frequencies of the p-modes, the period spacing of gravity modes, and the
spectroscopic data. They found no obvious differences between inferred masses
and radii from the modeling and the scaling relations. It should be noted that
the apparent conflict between these various studies may relate to the
evolutionary stage rather than differences in the methods. As presented by
Sharma et al. (2016), the systematic offset of the $\Delta\nu$ scaling
relation varies significantly for different evolutionary stages. Global
diagnosis of the scaling relations that covers a wide parameter range gives a
big picture of the systematic uncertainty in the scaling relations. Huber et
al. (2017), Zinn et al. (2019), and Hall et al. (2019) used _Gaia_ parallaxes
to empirically demonstrate that evolution-dependent corrections to the scaling
relations are necessary to improve the accuracy of asteroseismic radii for
thousands of _Kepler_ RGB stars.

Various methods have been implemented to correct the scaling relations. A
simple approach is changing the solar reference values of $\Delta\nu_{\odot}$
and $\nu_{\rm max,\odot}$ (Kallinger et al., 2010b; Chaplin et al., 2011;
Guggenberger et al., 2016; Themeßl et al., 2018), however, the method does not
consider the change for different evolutionary phases. Sharma et al. (2016),
Rodrigues et al. (2017), and Mosser et al. (2013) provided a correction factor
to $\Delta\nu$ ($f_{\Delta\nu}$), which is calculated based on the theoretical
$\Delta\nu$ from fitting radial mode frequencies. This corrects the scaling
relations for wide mass and metallicity ranges and also for different
evolutionary phases, from the ZAMS to the upper RGB. Including these
correction factors, the masses and radii reached general agreement for stars
in the EB systems (e.g. Brogaard et al., 2018). More details about the efforts
on this topic can be seen in the review by Hekker (2020).

In this paper, we use masses and radii determined with the detailed modeling
to correct the scaling relations. Although theoretical models do introduce
potential systematic uncertainties, the advantage of using modeling solutions
is the coverage of a wide range of stellar parameters and evolution stages.
For instance, Bellinger (2019) used masses and radii previously determined
from fits to theoretical stellar models (Silva Aguirre et al., 2015; Davies et
al., 2016a; Lund et al., 2017a; Silva Aguirre et al., 2017) to calibrate the
scaling relations for Kepler dwarfs. We aim to apply this method to _Kepler_
red-giant oscillators. We calculate radial model frequencies to determine
stellar masses and radii and then use inferred values to calibrate the scaling
relations. The rest of this paper is organised as follow. Section 2 summarises
the target selection and the measurement procedure of oscillating radial
modes; Section 3 discusses our theoretical modeling and the fitting method;
Section 4 demonstrates results of estimated stellar parameters; we correct the
scaling relations with model-determined masses and radii in Section 5; and we
close with a summary of outcomes in Section 6.

##  2 Data Reduction

###  2.1 Target Selection

We started with _Kepler_ RGB stars classified by Hon et al. (2017), who used a
machine-learning tool to distinguish between RGB stars and core-helium-burning
stars, based primarily on the gravity-mode spacings (Bedding et al., 2011;
Mosser et al., 2012b). According to Hon et al. (2017), their classification
achieved an accuracy of up to 99%. These stars were studied by Yu et al.
(2018b), who measured their global seismic parameters ($\Delta\nu$ and
$\nu_{\rm max}$) using the SYD pipeline (Huber et al., 2009b). We also
included six oscillating red giants in EB systems, which were previously
studied by Li et al. (2018), as the benchmark stars in this study. We cross-
matched _Kepler_ RGB sample with spectroscopically measured atmospheric
parameters ($T_{\text{eff}}$ and [M/H]) from two large-scale spectroscopic
surveys: _APOGEE_ DR16 (Ahumada et al., 2020) and _LAMOST_ DR5 (Xiang et al.,
2019). We only selected stars within a metallicity range of [M/H] = -0.3–+0.3
to fall within the parameter range of our model grid (-0.5–+0.5). Note that
the [M/H] values are not given in _LAMOST_ DR5, and we calculated them from
[Fe/H] and [$\alpha$/Fe] values using the following formula, given by Salaris
& Cassisi (2005):

${\rm[M/H]}={\rm[Fe/H]}+\log(10^{\rm[\alpha/Fe]}0.694+0.396).$ (5)

The cross-match resulted in 4,127 _Kepler_ RGB stars, comprising 2,626
_APOGEE_ and 3,326 _LAMOST_ targets, with 1,825 common sources. We show the
sample on the $T_{\text{eff}}$-$\nu_{\rm max}$ diagram in Figure 1. The sample
covers the lower to the upper RGB, with a $\nu_{\rm max}$ range from 10 to 270
$\mu$Hz. We also inspected the systematic differences in effective
temperatures and metallicities between the two surveys. As shown on the right
of Figure 1, there is an approximately uniform offset in $T_{\text{eff}}$,
which can be described as $T_{\rm eff,APOGEE}$ = $T_{\rm eff,LAMOST}$ \+ 41K.
The systematic offset in the [M/H] values follows a linear function. We fitted
the data and obtained the relation as [M/H]APOGEE = 0.20[M/H]LAMOST \+ 0.023
dex. We show in Section 5 that this systematic offset has an impact on the
correction of $\nu_{\rm max}$ scaling relation.

Figure 1: Left: The _Kepler_ RGB stars on the $T_{\rm eff}$–$\nu_{\rm max}$
diagram. Grey dots are 50,000 MESA models randomly selected from our grid.
Blue and red dots represent the 2,626 APOGEE and 3,326 LAMOST tragets.
Observed $\nu_{\rm max}$ values are from Yu et al. (2018b). Right: Differences
in effective temperature (upper) and metallicity (bottom) of the 1,825 common
stars. Red dashes indicate the systematic offsets, which is constant ($T_{\rm
eff,APOGEE}$ = $T_{\rm eff,LAMOST}$ \+ 41 K) for effective temperature and a
linear function ([M/H]APOGEE = 0.20[M/H]LAMOST \+ 0.023 dex) for metallicty.

###  2.2 Measurement of Radial Oscillation Mode Frequencies

We first identified the radial modes for the aforementioned stars. The
asymptotic relation for p modes can be used to predict the mode frequencies on
the basis that modes of the same $l$-degree and sequential $n$ are separated
by about $\Delta\nu$ (Tassoul, 1980; Gough, 1986):

$\nu(n,l)=\Delta\nu\left(n+\frac{l}{2}+\epsilon\right)-\delta\nu_{0l},$ (6)

where $\epsilon$ defines an offset and $\delta\nu_{0l}$ is the small spacing
between modes of different $l$-degrees. We made use of this regularity to
determine approximate locations of the radial modes, which means determining
$\epsilon$ for each star. We used the background-corrected power spectra
produced by Yu et al. (2018b) and selected data around $\nu_{\rm max}$ in a
range of 5 times the FWHM (full width at half maximum) of the power excess. We
then obtained the so-called collapsed power spectrum where the power was added
from consecutive slices with width of $\Delta\nu$. This collapsed power
spectrum was cross-correlated with a template comprising three peaks,
constructed according to the asymptotic equation, corresponding to the ridges
of $l=0$, 1 and 2 modes. We adopted a value for $\delta\nu_{0l}$ that was a
fixed fraction of $\Delta\nu$, set by a fit to real stars (Huber et al.,
2010). The value of $\epsilon$ was then obtained by maximising the correlation
coefficient. The locations of radial modes were restricted to be in the region
$(\nu_{n,l=0}/\Delta\nu\ {\rm mod}\ 1)\in[\epsilon-0.06,\epsilon+0.12]$. Fig.
2 shows this identification procedure for an example star, KIC 8410637.

Having located the radial modes, we fitted Lorentzian profiles to these
regions to extract the radial mode frequencies:

$M(\nu)=B(\nu)\cdot\eta^{2}(\nu)\cdot\sum_{n}L_{n}(\nu)=B(\nu)\cdot\frac{\sin^{2}(\pi\nu\Delta
t)}{(\pi\nu\Delta
t)^{2}}\cdot\sum_{n}\frac{A_{n}^{2}/(\Gamma_{n})}{1+4(\nu-\nu_{n})^{2}/\Gamma_{n}^{2}},$
(7)

using the fitting method as described by Li et al. (2020b). We used an H1
(odds ratio) approach to evaluate the quality of each fit (Appourchaux et al.,
2012; Corsaro & De Ridder, 2014; Davies et al., 2016b; Lund et al., 2017b). We
retained those modes with a Bayes factor larger than $3$, equivalent to strong
detection (Kass & Raftery, 1995). As an example, we listed model frequencies,
uncertainties of KIC 8410637 in Table 1. This star is the red-giant component
in a classified EB system. Its radial mode frequencies were previously
measured by direct fitting (Li et al., 2018). The comparison shows that the
model frequencies from our pipeline well agree with previous results and we
find one extra mode at low frequency. We also found good consistencies in
extracted modes for five other red-giants components studied by Li et al.
(2018).

The number of identified radial modes is 6 on average, reaching up to 10 for
low-luminosity RGB stars and decreasing with the evolution. For high-
luminosity red giants with $\nu_{\rm max}$ below 20$\mu$Hz, most stars have
only 3–4 modes. We note that a few of the stars may have incorrect results
since they were returned from an automatic pipeline. They do not affect
population studies because of their small number. However, to use frequencies
for a particular star, they should carefully vetted. The source code for the
mode measurement and lists of identified frequencies for the all stars are
available on https://github.com/litanda/SolarlikePeakbagging.

Figure 2: Measurement of radial modes for KIC 8410637. a) The power spectrum in échelle format. b) The collapsed power spectrum. c) The template spectrum. d) The cross-correlation spectrum between b) and c). The red cross shows the maximum in the cross-correlation spectrum, which gives the value of $\epsilon$ and identifies the ridge of radial modes. We used this to define the frequency ranges for the radial modes (shown as black dashed lines) and then fitted the strongest peaks within these ranges. The fitted frequencies are marked by squares.  Table 1: Extracted radial modes of an example star KIC 10513837. $\nu$ ($\mu$Hz) |  $\sigma_{\nu}$ ($\mu$Hz) | Ref.a
---|---|---
37.19 | 0.02 | -
41.62 | 0.02 | $41.63\pm 0.03$
46.28 | 0.02 | $46.29\pm 0.03$
50.85 | 0.02 | $50.86\pm 0.02$
55.55 | 0.06 | $55.53\pm 0.05$
60.22 | 0.05 | $60.37\pm 0.03$
a Mode frequencies from Li et al. (2018).

##  3 Theoretical Modeling

###  3.1 Model Grid and Input Physics

We constructed a stellar model grid that ranges from 0.76 to 2.20$M_{\odot}$
with a mass step of 0.02$M_{\odot}$. The grid includes four independent model
inputs: stellar mass ($M$), initial helium fraction ($Y_{\rm init}$), initial
metallicity ([M/H]), and the mixing-length parameter ($\alpha_{\rm MLT}$).
Ranges and grid steps of these four model inputs are given in Table 2. We
started computation of each stellar evolutionary track from the Hayashi line
and terminated on the upper RGB when $\log g$ $\geq$ 1.5 dex. Note that we did
not include models undergoing core helium burning.

Table 2: Input Ranges and Grid Step of the Model Grid. Input Parameter | Range | Increment
---|---|---
| From | To |
$M/$M⊙ |  $-$0.76 | 2.20 | 0.02
$\rm{[Fe/H]}$ |  $-0.5$0 | 0.50 | 0.10
$Y_{\rm init}$ |  $-$0.24 | 0.32 | 0.02
$\alpha_{\rm{MLT}}$ |  $-$1.70 | 2.50 | 0.20

We used the stellar evolution code mesa (Modules for Experiments in Stellar
Astrophysics, version 12115, Paxton et al. 2011, 2013, 2015, 2018) and the
oscillation code gyre (version 5.1, Townsend & Teitler 2013) to compute the
model grid. We adopted the solar chemical mixture [$(Z/X)_{\odot}$ = 0.0181]
provided by Asplund et al. (2009). The initial chemical composition was
calculated by:

$\log(Z_{\rm{init}}/X_{\rm{init}})=\log(Z/X)_{\odot}+\rm{[M/H]_{init}}.\\\ $
(8)

We used the mesa $\rho$–$T$ tables based on the 2005 update of the OPAL
equation of state tables (Rogers & Nayfonov, 2002) and OPAL opacities
supplemented by the low-temperature opacities from Ferguson et al. (2005). The
Eddington photosphere was used for the set of boundary conditions for modeling
the atmosphere. The mixing-length theory of convection was applied, where
$\alpha_{\rm MLT}=\ell_{\rm MLT}/H_{\rm p}$ is the mixing-length parameter. We
considered convective overshooting at the core, the H-burning shell, and the
envelope. The exponential scheme by Herwig (2000) was applied and the
diffusion coefficient in the overshoot region given as

$\displaystyle{D_{\rm{OV}}}={D_{\rm{conv,0}}}\exp\left(-\frac{{2(r-r_{0})}}{{(f_{0}+f_{\rm{ov}}){H_{p}}}}\right).$
(9)

Here, $D_{\rm{conv,0}}$ is the diffusion coefficient from the mixing-length
theory at a user-defined location near the Schwarzschild boundary. The switch
from convection to overshooting was set to occur at $r_{0}$. To consider the
step taken inside the convective region, $(f_{0}+f_{\rm{ov}}){H_{\rm p}}$ was
used, where $f_{\rm 0}$ equals 0.5$f_{\rm{ov}}$. Following Magic et al.
(2010), the overshooting parameter is mass-dependent following a relation as
$f_{\rm{ov}}=(0.13M-0.098)/9.0$, and a fixed $f_{\rm{ov}}$ of 0.018 was
adopted for models above $M=2.0$ M⊙. For a smooth convective boundary, we also
applied the mesa predictive mixing scheme. The mass-loss rate on the red-giant
branch with Reimers’ law was set as $\eta=0.2$, which is constrained by the
seismic targets in old open clusters NGC 6791 and NGC 6819 (Miglio et al.,
2012). Atomic diffusion was only considered for models below 1.1M⊙ at the
main-sequence phase (it is turned off when central hydrogen fraction goes
below 0.01). As discussed by Mowlavi et al. (2012), the mass value of 1.1M⊙ is
a proper switching point, above which the surface helium and heavy elements
would be rapidly drained out of the envelope if atomic diffusion were applied.
The switch of atomic diffusion causes a discontinuity in the surface
metallicity as well as other global parameters during the main-sequence and
early subgiant stages, but most of the effects are not significant at the RGB
phase due to the redistribution of star interiors (Michaud et al., 2010;
Dotter et al., 2017). However, element diffusion also changes the main-
sequence lifetime and hence a discontinuity in star age is expected for the
RGB models at 1.1 M⊙ (e.g. Fig. 11 Mowlavi et al., 2012). To investigate the
systematic uncertainty, we compared stellar ages at given mean density on RGB
for evolutionary tracks with and without atomic diffusion. The comparisons
shows an 8% difference for 1.1M⊙ evolutionary tracks. The age difference is
strongly mass-dependent, going down to 1% at 1.2M⊙ and is approximately zero
at 1.3M⊙. We hence expect a systematic uncertainty of a few per cent in
inferred ages in the mass range of 1.1–1.2 M⊙. Compared with the typical
uncertainty (16%), our estimated ages for stars about 1.1M⊙ could be biased to
some extent for the choice of atomic diffusion, but this systematic difference
is negligible above 1.2 M⊙.

###  3.2 Fitting Method

We used the Maximum Likelihood Estimate (MLE) method for fitting the models to
the observations. In Figure 3, we demonstrate our fitting procedure with the
example star. We first searched for models in the range 0.8–1.2 times
$\Delta\nu$. Note that the $\Delta\nu$ scaling relation was used in this step
and the $\pm$20% range was large enough to cover its systematic uncertainty.
We fitted the observed $T_{\rm eff}$ and [M/H] to models and calculated the
non-seismic likelihood ($L_{\rm non-seismo}$) with the following equation:

$L_{\rm non-seismo}={\rm exp}\left[-\frac{(T_{\rm eff,obs}-T_{\rm
eff,mod})^{2}}{2\sigma^{2}_{T_{\rm eff,obs}}}-\frac{([M/H]_{\rm
obs}-[M/H]_{\rm mod})^{2}}{2\sigma^{2}_{[M/H]_{\rm obs}}}\right],\\\ $ (10)

where the subscripts ‘obs’ and ‘mod’ stand for the observed and model values.
Note that, because our grid resolution for [M/H] is 0.1 dex, an uncertainty of
$\pm$0.1 for [M/H] was used when a star’s observed uncertainty is smaller than
this value. We selected models with $L_{\rm non-seismo}>0.0001$ for the
subsequent seismic analysis.

We corrected the surface term in oscillation frequencies of each selected
model using the two-term formula given by Ball & Gizon (2014). While
inspecting the fits of frequencies, we noticed that some bad models whose
frequencies deviated strongly from the observations could be made to fit well
after an implausibly large surface correction. To avoid this, we set up a
threshold for the surface term when selecting models for the seismic fitting.
Although the surface term of a certain model is unknown before fitting, we can
learn from previous studies to roughly estimate its range. According to
results on _Kepler_ main-sequence and RGB stars (Compton et al., 2018; Li et
al., 2018, 2021), relative surface corrections at $\nu_{\rm max}$
($\delta\nu_{\rm corr}$($\nu_{\rm max}$)/$\nu_{\rm max}$) for most stars are
in a range from $-0.005$ to $-0.01$. Considering the potential systematic
differences between theoretical models, we used a larger range as the
threshold, namely 0 to 0.02. After correcting the surface term, we checked the
relative surface corrections at $\nu_{\rm max}$ and only kept models fitting
this threshold criterion.

We next fitted individual radial mode frequencies and calculated a seismic
likelihood of each model for as

$L_{\rm seismo}=\prod{\rm
exp}\left[-\frac{(\nu_{i\rm,obs}-\nu_{i\rm,mod})^{2}}{2(\sigma^{2}_{\nu_{i\rm,obs}}+\sigma^{2}_{\nu_{\rm
sys}})}\right].\\\ $ (11)

The subscript $i$ refers to the mode frequencies. As discussed by Li et al.
(2020a), the systematic uncertainty in model frequencies cannot be neglected
in the detailed modeling because it could be comparable to or larger than the
observed uncertainty. For this reason, the fit was done in two steps. In the
first step, we only considered the observed frequency uncertainty
($\sigma_{\nu,obs}$) in the likelihood function and found the best-fitting
model. We used the median value of frequency offsets between the observations
and this best-fitting model as the systematic uncertainty ($\sigma_{\rm
sys}$). In the second step, we fitted to individual mode frequencies again but
included the $\sigma_{\rm sys}$ term in the likelihood function. Multiplying
the non-seismic and seismic likelihoods gave us the final likelihood as
$L=L_{\rm non-seismo}\cdot L_{\rm seismo}$. We used the marginal likelihood
distribution to measure the 16%, 50% and 84% cumulative values to determine
global parameters (i.e., mass, age, radius, surface gravity, and luminosity)
and their uncertainties.

Figure 3: The fitting procedure for the example star (KIC 8410637). Top left:
Fitting models to spectroscopic constraints ($T_{\text{eff}}$ and [M/H]) in
the range 0.8–1.2 $\Delta\nu$ and selecting those with $L_{\rm non-
seismo}>0.0001$ for the subsequent seismic analysis. Grey dots represent the
all models, colour-coded dots are selected models and colour indicates the
non-seismic likelihood ($L_{\rm non-seismo}$). Top right: Fitting observed
mode frequencies. We demonstrate the fits on the échelle diagram. Filled black
squares are observed mode frequencies, open blue and red circles represent
theoretical mode frequencies before and after the surface correction. The
échelle diagram is plotted twice to fully present the radial mode pattern, and
the blue dashed line indicates the large separation. Middle and bottom rows:
probability distributions for mass, age, surface gravity, and radius. The
central red dashed line indicates the median of each distribution, and two
black dashed lines mark the 16% and the 84% cumulative values (1-$\sigma$
uncertainty).

##  4 Results

###  4.1 Five Red Giants in Eclipsing Binaries

The accuracy of estimated masses and radii is critical because we use them to
calibrate the scaling relations. We hence examined the accuracy of modeling
solution with five red giants in EB systems, whose masses and radii have been
previously determined with dynamical modeling. We found _APOGEE_ spectra
available for all five red giants and _LAMOST_ data for three of them. The
comparison is demonstrated in Figure 4. We used the dynamical results
determined by Frandsen et al. (2013), Gaulme et al. (2016) and Brogaard et al.
(2018), and the average value was taken when there was more than one estimate
for the same star. With the APOGEE spectra, our model-based masses and radii
agree within $1\sigma$ for three out of the five stars, $1.5\sigma$ for KIC
9970396, and $2\sigma$ for KIC 4663623. The reason for the large offset on KIC
4663623 could be the very different metallicities adopted in our fitting
(${\rm[M/H]}=0.08$) and the dynamical modeling (${\rm[M/H]}=-0.13$). Estimates
for the three stars with the _LAMOST_ data are slightly more discrepant but
still consistent with dynamical determinations. In Table 3, we summarise
estimated masses and radii given by the detailed modeling, the scaling
relations, and dynamical analysis. Compared with the scaling relation, the
fits to radial mode frequencies significantly improve the accuracy of mass and
radius estimates.

Figure 4: Masses and radii determined by fitting radial modes versus those
from dynamical modeling, for five red giants in EB systems. The black dashed
lines show the 1:1 relation.  Table 3: Masses and radii of five red giants in
EB systems derived from our modeling, the scaling relations, and dynamical
modeling.

KIC | $M_{\rm Fit}$ | $R_{\rm Fit}$ | $M_{\rm SR}$ | $R_{\rm SR}$ | $M_{\rm Dyn}$ | $R_{\rm Dyn}$ | Ref.
---|---|---|---|---|---|---|---
| (M⊙) | (R⊙) | (M⊙) | (R⊙) | (M⊙) | (R⊙) |
4663623 | $1.56^{+0.10}_{-0.12}$ | $9.94^{+0.20}_{-0.25}$ | 1.87$\pm$0.13 | 10.78$\pm$0.28 | 1.36$\pm$0.09 | 9.7$\pm$0.2 | G16
7037405 | $1.16^{+0.08}_{-0.08}$ | $13.55^{+0.37}_{-0.31}$ | $1.38\pm 0.10$ | $14.76\pm 0.41$ | $1.25\pm 0.04$ | $14.1\pm 0.2$ | G16
|  |  |  |  | $1.170\pm 0.020$ | $14.000\pm 0.093$ | B18
8410637 | $1.54^{+0.14}_{-0.14}$ | $10.70^{+0.33}_{-0.37}$ | $1.78\pm 0.19$ | $11.45\pm 0.56$ | $1.56\pm 0.03$ | $10.7\pm 0.1$ | F13
9540226 | $1.32^{+0.08}_{-0.08}$ | $12.89^{+0.25}_{-0.28}$ | $1.58\pm 0.11$ | $14.05\pm 0.37$ | $1.33\pm 0.05$ | $12.8\pm 0.1$ | G16
|  |  |  |  | $1.378\pm 0.038$ | $13.060\pm 0.160$ | B18
9970396 | $1.08^{+0.06}_{-0.06}$ | $7.69^{+0.15}_{-0.16}$ | $1.43\pm 0.08$ | $8.68\pm 0.17$ | $1.14\pm 0.03$ | $8.0\pm 0.2$ | G16
|  |  |  |  | $1.178\pm 0.015$ | $8.035\pm 0.074$ | B18

References: F13 – Frandsen et al. (2013), G16 – Gaulme et al. (2016), B18 –
Brogaard et al. (2018)

###  4.2 Estimated Stellar Parameters

We used the fitting method described in Section 3.2 to estimate stellar
parameters (mass, age, surface gravity, and radius) for the 4,127 Kepler RGB
stars. We first removed three types of stars from final results: i) stars with
three or fewer identified radial modes, because the fits will largely affected
by the uncertainty in surface correction; ii) stars with estimated masses
below 0.85 or above 2.10 M⊙, because those values are close to the edge of the
grid; iii) stars with fewer than 100 models for the seismic analysis, because
their observed constraints are either close to the edge or outside the
parameter ranges of the model grid. We then examined age determinations to
identify those targets with unrealistically large estimates. These targets may
bias the calibration for the scaling relation, given that age correlates
strongly with mass for RGB stars. We found 61 out of 5,350 estimated ages that
are larger than the universe age (13.8 Gyr) by more than 2-$\sigma$. We
inspected those targets and found some of them do not have reliable
$T_{\text{eff}}$ measurements and others could be core-helium-burning stars
according to their power spectra. The fraction ($\sim 1.1\%$) of poor age
determinations generally agrees with the accuracy rate of classifications
reported by Hon et al. (2017). We then removed the targets with poor age
estimates from the sample. We finally obtained 5,289 modeling solutions for
3,642 stars, including 2,498 _APOGEE_ and 2,791 _LAMOST_ targets with 1,647
common sources. We list observed constraints and estimated parameters in Table
4 (the full table is included in the supplement).

Table 4: Observed constraints and model-inferred stellar parameters (full table available on-line). | Observed Constraints | Estimates
---|---|---
KIC | Source | $T_{\rm eff}$ | [M/H] | $\Delta\nu$ | $\nu_{\rm max}$ | $M$ | $\tau$ | $R$ | $\log g$
|  | (K) | (dex) | ($\mu$Hz) | ($\mu$Hz) | ($M_{\odot}$) | (Gyr) | ($R_{\odot}$) | (dex)
8161920 | _APOGEE_ | 4551 | -0.046 | 3.65 | 31.9 | $1.14^{+0.08}_{-0.08}$ | $7.7^{+3.3}_{-1.5}$ | $11.26^{+0.22}_{-0.33}$ | $2.393^{+0.007}_{-0.011}$
5358184 | _APOGEE_ | 4632 | 0.104 | 6.23 | 62.4 | $1.14^{+0.02}_{-0.03}$ | $9.7^{+1.2}_{-1.8}$ | $7.87^{+0.08}_{-0.09}$ | $2.701^{+0.004}_{-0.004}$
10793872 | _APOGEE_ | 4780 | -0.229 | 14.86 | 181.7 | $0.94^{+0.03}_{-0.04}$ | $14.6^{+2.9}_{-1.0}$ | $4.19^{+0.04}_{-0.09}$ | $3.17^{+0.003}_{-0.008}$
8815290 | _APOGEE_ | 4821 | -0.112 | 4.46 | 45.5 | $1.72^{+0.10}_{-0.16}$ | $1.9^{+0.5}_{-0.3}$ | $11.38^{+0.25}_{-0.37}$ | $2.56^{+0.008}_{-0.012}$
7749249 | _APOGEE_ | 4703 | 0.099 | 6.66 | 71.8 | $1.22^{+0.08}_{-0.04}$ | $5.1^{+1.2}_{-0.7}$ | $7.72^{+0.17}_{-0.11}$ | $2.751^{+0.007}_{-0.007}$
8161920 | _LAMOST_ | 4560 | 0.249 | 3.65 | 31.9 | $1.14^{+0.08}_{-0.04}$ | $8.6^{+2.5}_{-1.5}$ | $11.25^{+0.23}_{-0.19}$ | $2.394^{+0.006}_{-0.007}$
5358184 | _LAMOST_ | 4573 | -0.018 | 6.23 | 62.4 | $1.14^{+0.02}_{-0.02}$ | $9.08^{+1.9}_{-0.9}$ | $7.87^{+0.06}_{-0.05}$ | $2.702^{+0.001}_{-0.003}$
10793872 | _LAMOST_ | 4925 | 0.110 | 14.86 | 181.7 | $0.94^{+0.03}_{-0.04}$ | $16.2^{+1.4}_{-2.6}$ | $4.16^{+0.07}_{-0.06}$ | $2.17^{+0.005}_{-0.006}$
8815290 | _LAMOST_ | 4749 | -0.195 | 4.46 | 45.5 | $1.68^{+0.12}_{-0.14}$ | $2.4^{+0.4}_{-0.5}$ | $11.26^{+0.31}_{-0.33}$ | $2.558^{+0.010}_{-0.009}$
7749249 | _LAMOST_ | 4681 | 0.117 | 6.66 | 71.8 | $1.24^{+0.08}_{-0.06}$ | $4.8^{+1.2}_{-0.5}$ | $7.76^{+0.17}_{-0.14}$ | $2.752^{+0.007}_{-0.007}$

To inspect the precision of inferred parameters, we show histograms of the
uncertainties in Figure 5. We obtained similar average precision for the
_LAMOST_ and _APOGEE_ samples, but the _LAMOST_ sample spreads over a larger
range. The typical precision (the median of the distribution) for both samples
is 4.5% for mass, 16% for age, 0.006 dex for $\log g$, and 1.7% for radius.

Figure 5: Fractional uncertainties of the estimated parameters for stars with
LAMOST and APOGEE spectroscopic constraints.

We compared inferred values determined with the _APOGEE_ and the _LAMOST_
spectra to investigate any systematic differences caused by the spectroscopy.
In Figure 6, we illustrate comparisons for the 1,655 common targets and find
good consistency at the sample level: the average differences for the four
parameter are all around zero, with standard deviations that are similar to
the typical uncertainties. The large offsets on some stars are due to the very
different effective temperatures given by the two surveys. The results
indicate that our fits are mainly determined by seismic mode frequencies,
while the effective temperature and metallicity are relatively weak
constraints.

Figure 6: Comparisons between estimated parameters with LAMOST and APOGEE
spectroscopic constraints. (In the top left panel, some data points present
stripe-like features. These stars are low-luminous red giants, whose mass loss
is negligible, and hence their estimated masses are about the same as the
initial masses, which are uniformly spaced. Other high-luminous stars have
signficant mass loss, and hence their inferred masses distribute more
randomly.)

##  5 The scaling relations for red giants

Our model-based masses and radii of the _Kepler_ RGB star sample can allow
testing of the scaling relations over a wide parameter range. We have shown
that our estimates for stellar masses and radii are accurate, and hence they
are good reference values for correcting the scaling relations. We start by
inspecting the ratio between the modeling determinations and the scaling
relations. As illustrated in Figure 7, we find that the scaling relations
overestimate masses and radii for most of the stars. The density distributions
give median factors of $\sim$85% for the mass and $\sim$93% for the radius.
These values are consistent with those found for EBs (Gaulme et al., 2016;
Brogaard et al., 2018; Themeßl et al., 2018). In Figure 7, we see that the
ratio weakly depends on mass and this may correspond to some systematic offset
caused by the effective temperature or the metallicity. On the other hand, the
ratio clearly varies with stellar radius, indicating the discrepancy depends
on the evolutionary phase.

We wish to correct the scaling relations by combining the modeling solution
and observations. Rather than correcting Eq. 3 and 4 directly, we prefer to
work with the $\nu_{\rm max}$ and the $\Delta\nu$ scaling relations because
they are more fundamental. The scaling relation for $\nu_{\rm max}$ is based
on the assumption that $\nu_{\rm max}$ in solar-like oscillators is a fixed
fraction of $\nu_{ac}$(Brown et al., 1991; Kjeldsen & Bedding, 1995). The
acoustic cutoff frequency ($\nu_{ac}$) is the frequency above which acoustic
waves are not reflected at the surface and it depends on the effective
temperature and the surface gravity as $\nu_{ac}\propto gT_{\rm eff}^{-0.5}$.
Regarding the $\Delta\nu$ scaling relation, the large frequency separation
$\Delta\nu$ in the asymptotic approximation equals the inverse of the sound
travel time through the star, which results in
$\Delta\nu\propto\sqrt{M/R^{3}}$ (Ulrich, 1986). The $\Delta\nu$ value is
hence proportional to the square root of the mean density ($\overline{\rho}$)
of the star.

Figure 7: Ratios between estimated masses and radii from our modeling and from
the scaling relations. Histograms show the stars with spectroscopic
constraints from LAMOST and APOGEE.

###  5.1 Correcting the $\nu_{\rm max}$ Scaling Relation

We calculated the scaling $\nu_{\rm max}$ using model $T_{\rm eff}$ and $\log
g$, and used the same fitting method to constrain it. We refer to this
estimated scaling $\nu_{\rm max}$ as $\nu_{\rm max,SR}$. Note that we did not
use the measured $\nu_{\rm max}$ as an input constraint in the fitting, and
hence the determination of $\nu_{\rm max,SR}$ is independent of the observed
value.

From the comparison between observed $\nu_{\rm max}$ and $\nu_{\rm max,SR}$,
as demonstrated in Figure 8a, we find no significant systematic offset.
However, the scatter (the standard deviation is $\pm$2.5%) is much larger the
typical observed uncertainty, indicating potential dependencies on other
stellar parameters or that $\nu_{\rm max}$ uncertainties are substantially
underestimated. We therefore investigated the correlations with $\log g$,
$T_{\rm eff}$, and [M/H] in Figures 8b, 8c, and 8d. The reason to consider the
metallicity is that it could impact near-surface properties of solar-like
oscillators, as suggested by hydrodynamic simulations (e.g. Trampedach et al.,
2014). We see that the $\nu_{\rm max}$ offset does not obviously depend on
$\log g$, but clearly correlates with $T_{\rm eff}$ and [M/H]. The same trend
is seen in both the _APOGEE_ and _LAMOST_ samples. From the perspective of
physics, the correlations with the effective temperature may be real because
the thermal structures at the near-surface region are very different between
RGB and main-sequence stars, although there is no theory that quantifies the
difference in $T_{\text{eff}}$-$\nu_{\rm max}$ relations between the two types
of stars. However, we suspect that this correlation is caused by the
systematic offsets in the spectroscopic measurements because the dependency on
$T_{\text{eff}}$ is slightly stronger for the _LAMOST_ data than for the
_APOGEE_ data. The same applies for the metallicity, in that the correlation
is likely a combination of the physical and the instrumental effects and we
cannot separate them because of the lack of theoretical predictions. We will
leave this as an open question for future studies.

Figure 8: Dependencies of systematic offsets in $\nu_{\rm max}$ ($\nu_{\rm
max}$/$\nu_{\rm max,SR}$) on $\nu_{\rm max}$, $T_{\rm eff}$, $\log g$,and
[Fe/H]. The _APOGEE_ sample is presented on the top and the _LAMOST_ sample is
shown at the bottom. Filled triangle is the median of binned data and the
error bar indicates the the standard error of the median, which is calculated
as the standard deviation over the number of data point in a bin ($\sigma/n$).

To correct for the systematics discussed above, we added a metallicity term to
the $\nu_{\rm max}$ scaling relation. We calibrated the revised $\nu_{\rm
max}$ scaling relation with model-based $\log g$, observed $\nu_{\rm max}$,
$T_{\text{eff}}$, and [M/H]. We used the Scipy curve_fit module to do the fits
and obtained corrected $\nu_{\rm max}$ scaling relations as follows:

$\frac{\nu_{\rm max}}{\nu_{\rm max,\odot}}=\frac{g_{\rm
Fit}}{g_{\odot}}\left(\frac{T_{\rm eff}}{T_{\rm
eff,\odot}}\right)^{-0.459}\left(10^{\rm[M/H]}\right)^{-0.022}(\emph{APOGEE})\\\
\\\ $ (12)

and

$\frac{\nu_{\rm max}}{\nu_{\rm max,\odot}}=\frac{g_{\rm
Fit}}{g_{\odot}}\left(\frac{T_{\rm eff}}{T_{\rm
eff,\odot}}\right)^{-0.421}\left(10^{\rm[M/H]}\right)^{-0.039}(\emph{LAMOST}).\\\
$ (13)

We obtained the same power law for the $g$ term as the original relation, but
the exponent of $T_{\rm eff}$ is slightly different from the original
($-0.5$). As discussed above, the difference does not necessarily mean that
the RGB stars have a different $T_{\text{eff}}$–$\nu_{\rm max}$ relation from
the dwarfs. For instance, when we considered a global temperature shift of
$-70K$ in the _APOGEE_ sample, as found by Hall et al. (2019), the fit gave a
very similar power law to the original scaling relation. The additional term
of [M/H] is a secondary term in the formulae, which causes a $\sim$4% change
to the $\nu_{\rm max}$ value when [M/H] varies from -0.3 to 0.3 dex. Note that
the _LAMOST_ [M/H] values are based on the [Fe/H] and [$\alpha$/Fe]
measurements. The choice of formula may bring some systematic uncertainties
and we hence tested the fits with another widely-used formula, namely [M/H] =
[Fe/H] + [$\alpha$/Fe]. The [M/H] values from this formula are higher than
those calculated with Eq. 5 by 0.02 dex, on average, which is significantly
smaller than the typical observed uncertainty. Fitting the scaling relation
with these [M/H] values returns very similar results to Eq. 13. The exponent
is $-0.427$ for the $T_{\text{eff}}$ term and 0.040 for the [M/H] term. The
similarity indicates the choice of [M/H] formula does not significantly affect
the new scaling relation.

###  5.2 The $\Delta\nu$ Scaling Relation

We studied the $\Delta\nu$ scaling relation following the same strategy. We
used the mean density of models to calculate the large frequency separation
from the scaling relation (Eq. 1) and constrained it based on model fitting.
We refer to this as $\Delta\nu_{\rm SR}$. We first examined the systematic
difference between observed $\Delta\nu$ and $\Delta\nu_{\rm SR}$. Note that
the observed $\Delta\nu$ is from Yu et al. (2018b) rather than from individual
radial modes. As shown in Figure 9a, there is a $\sim$4% deviation on average,
and the offset varies from $-6$% to $-2$%. We then inspected the dependencies
of $\Delta\nu$/$\Delta\nu_{\rm SR}$ on the mean density, effective
temperature, and metallicity, as demonstrated in Figures 9b, 9c, and 9d.

The ratio clearly correlates with mean density and also with the effective
temperature, and slightly depends on the metallicity. It is not clear which
parameter has the true effect because these three parameters are correlated on
RGB. As for its physical significance, we note that $\Delta\nu$ is determined
by the mean density rather than by any surface properties. As pointed out by
Miglio et al. (2012), the differences in internal temperature (hence sound
speed) distributions could cause a noticeable difference in $\Delta\nu$
between different type of stars, despite them having the same mean density.
Their comparison between hydrogen-shell-burning and helium-core-burning stars
with similar mean densities in Kepler open clusters presented a $\sim$3.3%
difference in average $\Delta\nu$. As a consequence, the difference in
$\Delta\nu$–$\rho$ scaling relation between RGB and main-sequence stars is
expected because of their different internal structures. The correlations with
effective temperate and metallicity arise because these two surface properties
relate to the mean density: for stars on RGB, the effective temperature
strongly correlates with mass, and hence with mean density, at given radius.
Moreover, high-resolution spectroscopic surveys have shown a clear
$T_{\text{eff}}$–[M/H] relation on the RGB (e.g. Fig.2 Tayar et al., 2017).
For this reason, we retained the functional form of $\Delta\nu$ scaling
relation and did not include the effective temperature or the metallicity in
the new scaling relation.

We calibrated the $\Delta\nu$–$\rho$ scaling relation using observed
$\Delta\nu$ and modeling-inferred mean density ($\rho_{\rm Fit}$), obtaining
the following results:

$\frac{\Delta\nu}{\Delta\nu_{\odot}}=\left(\frac{\overline{\rho}_{\rm
Fit}}{\overline{\rho}_{\odot}}\right)^{0.507}.\\\ $ (14)

Note that this function is for both the _APOGEE_ and _LAMOST_ samples because
there is no systematic differences in inferred mean densities between the two
at the sample level. Compared with the scaling for $\nu_{\rm max}$, the
scaling for $\Delta\nu$ has a significantly larger systematic offset and the
corrections are especially important for RGB stars. The correction to the
$\Delta\nu$ scaling relation also matches the evolution-dependent systematic
offsets found in previous studies (e.g. Sharma et al., 2016). We see that the
power law exponent relating the large separation and the mean density for RGB
stars is larger than that for dwarfs. This causes $\Delta\nu$ offsets (in
fractional terms) to grow exponentially with decreasing $\Delta\nu$, and hence
leads to larger systematic offsets for more evolved stars. We also note that a
departure from the 0.5 exponent in the $\Delta\nu$ scaling relation has also
been seen in $\delta$ Scuti stars (e.g., García Hernández et al., 2015;
Bedding et al., 2020).

Figure 9: Same as Figure 8 but for $\Delta\nu$.

Our correction to the $\Delta\nu$ scaling relation is similar to previous
studies by White et al. (2011) and Sharma et al. (2016), who checked whether
the $\Delta\nu$ from fitting the model frequencies matched the value from the
scaling the model density. This work is an improvement on that earlier work in
that our corrections include the surface correction to model frequencies. It
would be useful to investigate how much the surface term affect the
corrections $\Delta\nu$ scaling relation. To do this, we followed the method
described by White et al. (2011) and Sharma et al. (2016) to calculate the
correction factor, $f_{\Delta\nu}$. In line with their notation, we defined
our $\Delta\nu$ corrections as $f\,^{\prime}_{\Delta\nu}$ =
$\Delta\nu$/$\Delta\nu$SR. With this notation, we can re-write Eq. 14 as

$\frac{\Delta\nu}{\Delta\nu_{\odot}}=f\,^{\prime}_{\Delta\nu}\left(\frac{\overline{\rho}_{\rm
Fit}}{\overline{\rho}_{\odot}}\right)^{0.5}.\\\ $ (15)

Here, $f\,^{\prime}_{\Delta\nu}$ can be taken as an improved version of
$f_{\Delta\nu}$, with an additional correction for the surface term. Figure 10
shows the comparison between $f_{\Delta\nu}$ and $f\,^{\prime}_{\Delta\nu}$ of
the best-fitting models for all stars. The differences are roughly uniform and
the average differences are 1.6% and 1.4% for the _APOGEE_ and _LAMOST_
samples, respectively. If we adopt a 1.5% difference in the $\Delta\nu$
correction factor and transfer this difference to the mass and radius
determinations, the surface term will cause a systematic offset of $\sim$6%
for the mass and $\sim$3% for the radius, which are highly significant. We
hence conclude that the surface term is worthy of attention in the corrections
to the $\Delta\nu$ scaling relation.

The surface term depends on properties of near-surface layers and we hence
inspected its correlations with the mean density, effective temperature, and
metallicity. As shown in Figure 11, the difference between the two factors
decreases with $\Delta\nu$, implying that the surface term is smaller for more
evolved stars. This trend is consistent with 3D convection simulations of the
surface effect on the RGB (e.g. Trampedach et al., 2017). The effective
temperature also has an effect on the surface term according to those
simulation results, but it is not obvious in our sample because the effective
temperature range on RGB is fairly narrow. We also notice a rough correlation
with the metallicity. This is expected because the metallicity can also change
the convection properties at near surface Magic et al. (e.g. 2015).

Figure 10: The comparison between the two correction factors to the
$\Delta\nu$ scaling relation ($f_{\Delta\nu}$ and $f\,^{\prime}_{\Delta\nu}$)
for the _APOGEE_ sample. Here, $f\,^{\prime}_{\Delta\nu}$ can be taken as the
an improved version of $f_{\Delta\nu}$ due to an additional correction for the
surface term.  Figure 11: The difference in the two correction factors to the
$\Delta\nu$ scaling relation ($f\,^{\prime}_{\Delta\nu}$ \- $f_{\Delta\nu}$)
as a function of four stellar parameters for the _APOGEE_ sample. Green
squares indicates the rolling median of the data. Results for the _LAMOST_
sample (not shown) are similar.

###  5.3 Corrected Scaling Relations for Red Giants

With corrected $\nu_{\rm max}$ and $\Delta\nu$ scaling relations (Eq. 12, 13
and 14), we derived new scaling relations for determining stellar masses and
radii. For the APOGEE sample they are:

$\frac{M_{\rm Fit}}{M_{\odot}}=\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)^{3.0}\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{-3.944}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{1.377}\left(10^{\rm[M/H]}\right)^{0.066},$
(16)

and

$\frac{R_{\rm Fit}}{R_{\odot}}=\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{-1.972}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{0.459}\left(10^{\rm[M/H]}\right)^{0.022}.$
(17)

For the LAMOST sample they are:

$\frac{M_{\rm Fit}}{M_{\odot}}=\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)^{3.0}\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{-3.944}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{1.263}\left(10^{\rm[M/H]}\right)^{0.117},$
(18)

and

$\frac{R_{\rm Fit}}{R_{\odot}}=\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{-1.972}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{0.421}\left(10^{\rm[M/H]}\right)^{0.039}.$
(19)

It worth noting that the above relations are only valid for RGB stars within a
parameter range of $T_{\rm eff}=4400$ to 5200 K, $\nu_{\rm max}=10$ to 270
$\mu$Hz, and ${\rm[M/H]}=-0.3$ to 0.3 dex. We suggest caution when applying
them outside these parameter ranges.

As a test, we computed masses and radii with the new scaling relations and
compared them with model-determined values. We illustrate the distributions of
the residuals in Figure 12. The distributions center at zero with a Half-
Width-at-Half-Maximum of $\sim$5% for the mass and of $\sim$2% for the radius.
These are consistent with the typical estimated uncertainties, which are 4.5%
and 1.7% for mass and radius.

Because the new scaling relations are based on modeling solutions, some
systematic uncertainties are expected. We considered reasonable ranges for the
helium fraction and the mixing-length parameter, so the systematic effects
from these two inputs are covered in the estimated uncertainty. However, the
choices of other input physics could bias the results to some extent. These
include the solar composition, the opacity, the atmosphere boundary condition,
mass-loss rate, and atomic diffusion. The systematic offset, according to
comparisons between seven different pipelines for _Kepler_ main-sequence stars
(Silva Aguirre et al., 2017), is about $\sim$2% for the mass and $\sim$1% for
the radius on average. Systematic offsets related to input physics in our work
could be different because internal structures of RGB stars are very different
from those of dwarfs. Future studies are required to properly estimate the
model systematic uncertainties.

Figure 12: Differences between modeling-inferred results and corrected scaling
relations (represented by the subscript ‘CSR’). Both density distributions
center around zero with a standard deviation of $\sim 5\%$ for mass and $\sim
2\%$ for radius.

We calibrated scaling relations for the APOGEE and LAMOST samples separately
because of the systematic differences in their measurements. Given the large
number of APOGEE and LAMOST spectra, the new scaling relations should be
useful for future studies. It is also worth considering whether to apply these
new scaling relations to stars observed by other spectroscopic surveys. To
test this, we applied the _APOGEE_ formulae to the _LAMOST_ data. As
determined from Figure 1, we uniformly increased the _LAMOST_ $T_{\text{eff}}$
by 44K, and calibrated the _LAMOST_ [M/H] using the linear function as
0.20[M/H] + 0.023. We used calibrated _LAMOST_ $T_{\text{eff}}$ and [M/H] in
Eq. 16 and 17 to calculate masses and radii. Comparing with model-based
determinations, we found good consistency: the average difference was zero
with a spread of $\pm$1.3% for the mass and $\pm$0.4% for the radius. This
result indicates that the new scaling relations can be applied with
spectroscopic data from other sources, provided proper calibrations to the
effective temperature and the metallicity are implemented.

##  6 Conclusions

We studied the solar-like oscillations of 3,642 _Kepler_ RGB stars. We
measured radial mode frequencies and used them to characterise stars with
detailed modeling, including correction for the surface effect. We provide
model-based estimates for mass, age, radius, and surface gravity for the full
sample. For five red giants in eclipsing binaries, we found good consistency
with masses and radii determined from dynamical modeling. We also showed that
systematic differences between the _APOGEE_ and the _LAMOST_ measurements of
effective temperature and metallicity do not significantly affect the modeling
solutions at the sample level because the fits are dominated by seismic mode
frequencies.

We used our results to study the systematic offsets in the widely-used
$\nu_{\rm max}$ and $\Delta\nu$ scaling relations. We did not find a
significant systematic offset in the $\nu_{\rm max}$ scaling relation.
However, a clear dependency on observed metallicity was seen, and we hence
added a metallicity term to the scaling relation. We used the observed
$\nu_{\rm max}$, $T_{\text{eff}}$, [M/H], and model-inferred $\log g$ to
calibrate our revised $\nu_{\rm max}$ scaling relation for RGB stars. We noted
that the correction to the $\nu_{\rm max}$ scaling relation depends on the
selection of spectroscopic measurements. For the $\Delta\nu$–density relation,
our results showed that the scaling relation overestimates the $\Delta\nu$
value by $\sim$4% on average. We calibrated a new $\Delta\nu$–$\rho$ relation
for RGB stars based model-inferred $\overline{\rho}$ and observed $\Delta\nu$,
and showed that the surface correction plays a very important part in this
calibration.

Combining our results, we derived revised scaling relations for determining
masses and radii for RGB stars. The new relations can characterise stars with
precision of $\sim$5% in mass and $\sim$2% in radius. We suggest that the new
scaling relations can be applied to stars observed by other spectroscopy
surveys, provided any differences in the effective temperature and metallicity
scales are properly calibrated.

We gratefully acknowledge the Kepler team, whose efforts made these results
possible. This work has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (CartographY GA. 804752). This work is supported by the Joint
Research Fund in Astronomy (U2031203) under cooperative agreement between the
National Natural Science Foundation of China (NSFC) and Chinese Academy of
Sciences (CAS) and also by NSFC grants 12090040 and 12090042. We also
gratefully acknowledge support from the Australian Research Council (grant
DP210103119), and from the Danish National Research Foundation (Grant DNRF106)
through its funding for the Stellar Astrophysics Center (SAC).

## References

  * Aerts et al. (2019) Aerts, C., Mathis, S., & Rogers, T. M. 2019, ARA&A, 57, 35, doi: 10.1146/annurev-astro-091918-104359
  * Ahumada et al. (2020) Ahumada, R., Prieto, C. A., Almeida, A., et al. 2020, ApJS, 249, 3, doi: 10.3847/1538-4365/ab929e
  * Appourchaux et al. (2012) Appourchaux, T., Chaplin, W. J., García, R. A., et al. 2012, A&A, 543, A54, doi: 10.1051/0004-6361/201218948
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, Annual Review of Astronomy and Astrophysics, 47, 481, doi: 10.1146/annurev.astro.46.060407.145222
  * Baglin et al. (2002) Baglin, A., Auvergne, M., Barge, P., et al. 2002, in ESA Special Publication, Vol. 485, Stellar Structure and Habitable Planet Finding, ed. B. Battrick, F. Favata, I. W. Roxburgh, & D. Galadi, 17–24
  * Ball & Gizon (2014) Ball, W. H., & Gizon, L. 2014, A&A, 568, A123, doi: 10.1051/0004-6361/201424325
  * Bedding et al. (2011) Bedding, T. R., Mosser, B., Huber, D., et al. 2011, Nature, 471, 608, doi: 10.1038/nature09935
  * Bedding et al. (2020) Bedding, T. R., Murphy, S. J., Hey, D. R., et al. 2020, Nature, 581, 147, doi: 10.1038/s41586-020-2226-8
  * Bellinger (2019) Bellinger, E. P. 2019, MNRAS, 486, 4612, doi: 10.1093/mnras/stz714
  * Borucki et al. (2009) Borucki, W., Koch, D., Batalha, N., et al. 2009, in Transiting Planets, ed. F. Pont, D. Sasselov, & M. J. Holman, Vol. 253, 289–299, doi: 10.1017/S1743921308026513
  * Brogaard et al. (2018) Brogaard, K., Hansen, C. J., Miglio, A., et al. 2018, MNRAS, 476, 3729, doi: 10.1093/mnras/sty268
  * Brown et al. (1991) Brown, T. M., Gilliland, R. L., Noyes, R. W., & Ramsey, L. W. 1991, ApJ, 368, 599, doi: 10.1086/169725
  * Chaplin et al. (2011) Chaplin, W. J., Kjeldsen, H., Christensen-Dalsgaard, J., et al. 2011, Science, 332, 213, doi: 10.1126/science.1201827
  * Compton et al. (2018) Compton, D. L., Bedding, T. R., Ball, W. H., et al. 2018, MNRAS, 479, 4416, doi: 10.1093/mnras/sty1632
  * Corsaro & De Ridder (2014) Corsaro, E., & De Ridder, J. 2014, A&A, 571, A71, doi: 10.1051/0004-6361/201424181
  * Corsaro et al. (2020) Corsaro, E., McKeever, J. M., & Kuszlewicz, J. S. 2020, A&A, 640, A130, doi: 10.1051/0004-6361/202037930
  * Davies et al. (2016a) Davies, G. R., Silva Aguirre, V., Bedding, T. R., et al. 2016a, MNRAS, 456, 2183, doi: 10.1093/mnras/stv2593
  * Davies et al. (2016b) —. 2016b, MNRAS, 456, 2183, doi: 10.1093/mnras/stv2593
  * Dotter et al. (2017) Dotter, A., Conroy, C., Cargile, P., & Asplund, M. 2017, ApJ, 840, 99, doi: 10.3847/1538-4357/aa6d10
  * Ferguson et al. (2005) Ferguson, J. W., Alexander, D. R., Allard, F., et al. 2005, ApJ, 623, 585, doi: 10.1086/428642
  * Frandsen et al. (2013) Frandsen, S., Lehmann, H., Hekker, S., et al. 2013, A&A, 556, A138, doi: 10.1051/0004-6361/201321817
  * García Hernández et al. (2015) García Hernández, A., Martín-Ruiz, S., Monteiro, M. J. P. F. G., et al. 2015, ApJ, 811, L29, doi: 10.1088/2041-8205/811/2/L29
  * Gaulme et al. (2016) Gaulme, P., McKeever, J., Jackiewicz, J., et al. 2016, ApJ, 832, 121, doi: 10.3847/0004-637X/832/2/121
  * Gough (1986) Gough, D. O. 1986, in Hydrodynamic and Magnetodynamic Problems in the Sun and Stars, ed. Y. Osaki, 117
  * Guggenberger et al. (2016) Guggenberger, E., Hekker, S., Basu, S., & Bellinger, E. 2016, MNRAS, 460, 4277, doi: 10.1093/mnras/stw1326
  * Hall et al. (2019) Hall, O. J., Davies, G. R., Elsworth, Y. P., et al. 2019, MNRAS, 486, 3569, doi: 10.1093/mnras/stz1092
  * Hekker (2020) Hekker, S. 2020, Frontiers in Astronomy and Space Sciences, 7, 3, doi: 10.3389/fspas.2020.00003
  * Herwig (2000) Herwig, F. 2000, A&A, 360, 952.  https://arxiv.org/abs/astro-ph/0007139
  * Hon et al. (2017) Hon, M., Stello, D., & Yu, J. 2017, MNRAS, 469, 4578, doi: 10.1093/mnras/stx1174
  * Howell et al. (2014) Howell, S. B., Sobeck, C., Haas, M., et al. 2014, PASP, 126, 398, doi: 10.1086/676406
  * Huber et al. (2009a) Huber, D., Stello, D., Bedding, T. R., et al. 2009a, Communications in Asteroseismology, 160, 74.  https://arxiv.org/abs/0910.2764
  * Huber et al. (2009b) —. 2009b, Communications in Asteroseismology, 160, 74.  https://arxiv.org/abs/0910.2764
  * Huber et al. (2010) Huber, D., Bedding, T. R., Stello, D., et al. 2010, ApJ, 723, 1607, doi: 10.1088/0004-637X/723/2/1607
  * Huber et al. (2011) —. 2011, ApJ, 743, doi: 10.1088/0004-637X/743/2/143
  * Huber et al. (2012) Huber, D., Ireland, M. J., Bedding, T. R., et al. 2012, ApJ, 760, 32, doi: 10.1088/0004-637X/760/1/32
  * Huber et al. (2017) Huber, D., Zinn, J., Bojsen-Hansen, M., et al. 2017, ApJ, 844, 102, doi: 10.3847/1538-4357/aa75ca
  * Kallinger et al. (2010a) Kallinger, T., Weiss, W. W., Barban, C., et al. 2010a, A&A, 509, A77, doi: 10.1051/0004-6361/200811437
  * Kallinger et al. (2010b) Kallinger, T., Mosser, B., Hekker, S., et al. 2010b, A&A, 522, A1, doi: 10.1051/0004-6361/201015263
  * Kass & Raftery (1995) Kass, R. E., & Raftery, A. E. 1995, Journal of the American Statistical Association, 90, 773, doi: 10.1080/01621459.1995.10476572
  * Kjeldsen & Bedding (1995) Kjeldsen, H., & Bedding, T. R. 1995, A&A, 293, 87
  * Li et al. (2020a) Li, T., Bedding, T. R., Christensen-Dalsgaard, J., et al. 2020a, MNRAS, 495, 3431, doi: 10.1093/mnras/staa1350
  * Li et al. (2018) Li, T., Bedding, T. R., Huber, D., et al. 2018, MNRAS, 475, 981, doi: 10.1093/mnras/stx3079
  * Li et al. (2020b) Li, Y., Bedding, T. R., Li, T., et al. 2020b, MNRAS, 495, 2363, doi: 10.1093/mnras/staa1335
  * Li et al. (2021) Li, Y., Bedding, T. R., Stello, D., et al. 2021, MNRAS, 501, 3162, doi: 10.1093/mnras/staa3932
  * Lund et al. (2017a) Lund, M. N., Silva Aguirre, V., Davies, G. R., et al. 2017a, ApJ, 835, doi: 10.3847/1538-4357/835/2/172
  * Lund et al. (2017b) —. 2017b, ApJ, 835, 172, doi: 10.3847/1538-4357/835/2/172
  * Magic et al. (2010) Magic, Z., Serenelli, A., Weiss, A., & Chaboyer, B. 2010, ApJ, 718, 1378, doi: 10.1088/0004-637X/718/2/1378
  * Magic et al. (2015) Magic, Z., Weiss, A., & Asplund, M. 2015, A&A, 573, A89, doi: 10.1051/0004-6361/201423760
  * Michaud et al. (2010) Michaud, G., Richer, J., & Richard, O. 2010, A&A, 510, A104, doi: 10.1051/0004-6361/200912328
  * Miglio et al. (2012) Miglio, A., Brogaard, K., Stello, D., et al. 2012, MNRAS, 419, 2077, doi: 10.1111/j.1365-2966.2011.19859.x
  * Mosser et al. (2012a) Mosser, B., Goupil, M. J., Belkacem, K., et al. 2012a, A&A, 548, A10, doi: 10.1051/0004-6361/201220106
  * Mosser et al. (2012b) —. 2012b, A&A, 540, A143, doi: 10.1051/0004-6361/201118519
  * Mosser et al. (2013) Mosser, B., Michel, E., Belkacem, K., et al. 2013, A&A, 550, A126, doi: 10.1051/0004-6361/201220435
  * Mosser et al. (2014) Mosser, B., Benomar, O., Belkacem, K., et al. 2014, A&A, 572, L5, doi: 10.1051/0004-6361/201425039
  * Mowlavi et al. (2012) Mowlavi, N., Eggenberger, P., Meynet, G., et al. 2012, A&A, 541, A41, doi: 10.1051/0004-6361/201117749
  * Nielsen et al. (2021) Nielsen, M. B., Davies, G. R., Ball, W. H., et al. 2021, AJ, 161, 62, doi: 10.3847/1538-3881/abcd39
  * Paxton et al. (2011) Paxton, B., Bildsten, L., Dotter, A., et al. 2011, The Astrophysical Journal Supplement Series, 192, 3, doi: 10.1088/0067-0049/192/1/3
  * Paxton et al. (2013) Paxton, B., Cantiello, M., Arras, P., et al. 2013, The Astrophysical Journal Supplement Series, 208, 4, doi: 10.1088/0067-0049/208/1/4
  * Paxton et al. (2015) Paxton, B., Marchant, P., Schwab, J., et al. 2015, The Astrophysical Journal Supplement Series, 220, 15, doi: 10.1088/0067-0049/220/1/15
  * Paxton et al. (2018) Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, The Astrophysical Journal Supplement Series, 234, 34, doi: 10.3847/1538-4365/aaa5a8
  * Pérez Hernández et al. (2016) Pérez Hernández, F., García, R. A., Corsaro, E., Triana, S. A., & De Ridder, J. 2016, A&A, 591, A99, doi: 10.1051/0004-6361/201628311
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003, doi: 10.1117/1.JATIS.1.1.014003
  * Rodrigues et al. (2017) Rodrigues, T. S., Bossini, D., Miglio, A., et al. 2017, MNRAS, 467, 1433, doi: 10.1093/mnras/stx120
  * Rogers & Nayfonov (2002) Rogers, F. J., & Nayfonov, A. 2002, ApJ, 576, 1064, doi: 10.1086/341894
  * Salaris & Cassisi (2005) Salaris, M., & Cassisi, S. 2005, Evolution of Stars and Stellar Populations
  * Sharma et al. (2016) Sharma, S., Stello, D., Bland-Hawthorn, J., Huber, D., & Bedding, T. R. 2016, ApJ, 822, 15, doi: 10.3847/0004-637X/822/1/15
  * Silva Aguirre et al. (2015) Silva Aguirre, V., Davies, G. R., Basu, S., et al. 2015, MNRAS, 452, 2127, doi: 10.1093/mnras/stv1388
  * Silva Aguirre et al. (2017) Silva Aguirre, V., Lund, M. N., Antia, H. M., et al. 2017, ApJ, 835, 173, doi: 10.3847/1538-4357/835/2/173
  * Stello et al. (2008) Stello, D., Bruntt, H., Preston, H., & Buzasi, D. 2008, ApJ, 674, L53, doi: 10.1086/528936
  * Stello et al. (2009) Stello, D., Chaplin, W. J., Bruntt, H., et al. 2009, ApJ, 700, 1589, doi: 10.1088/0004-637X/700/2/1589
  * Tassoul (1980) Tassoul, M. 1980, ApJS, 43, 469, doi: 10.1086/190678
  * Tayar et al. (2019) Tayar, J., Beck, P. G., Pinsonneault, M. H., García, R. A., & Mathur, S. 2019, ApJ, 887, 203, doi: 10.3847/1538-4357/ab558a
  * Tayar et al. (2017) Tayar, J., Somers, G., Pinsonneault, M. H., et al. 2017, ApJ, 840, 17, doi: 10.3847/1538-4357/aa6a1e
  * Themeßl et al. (2018) Themeßl, N., Hekker, S., Southworth, J., et al. 2018, MNRAS, 478, 4669, doi: 10.1093/mnras/sty1113
  * Townsend & Teitler (2013) Townsend, R. H. D., & Teitler, S. A. 2013, MNRAS, 435, 3406, doi: 10.1093/mnras/stt1533
  * Trampedach et al. (2017) Trampedach, R., Aarslev, M. J., Houdek, G., et al. 2017, MNRAS, 466, L43, doi: 10.1093/mnrasl/slw230
  * Trampedach et al. (2014) Trampedach, R., Stein, R. F., Christensen-Dalsgaard, J., Nordlund, Å., & Asplund, M. 2014, MNRAS, 445, 4366, doi: 10.1093/mnras/stu2084
  * Ulrich (1986) Ulrich, R. K. 1986, ApJ, 306, L37, doi: 10.1086/184700
  * Vrard et al. (2016) Vrard, M., Mosser, B., & Samadi, R. 2016, A&A, 588, A87, doi: 10.1051/0004-6361/201527259
  * White et al. (2011) White, T. R., Bedding, T. R., Stello, D., et al. 2011, ApJ, 743, doi: 10.1088/0004-637X/743/2/161
  * Xiang et al. (2019) Xiang, M., Ting, Y.-S., Rix, H.-W., et al. 2019, ApJS, 245, 34, doi: 10.3847/1538-4365/ab5364
  * Yu et al. (2018a) Yu, J., Huber, D., Bedding, T. R., et al. 2018a, ApJS, 236, 42, doi: 10.3847/1538-4365/aaaf74
  * Yu et al. (2018b) —. 2018b, ApJS, 236, 42, doi: 10.3847/1538-4365/aaaf74
  * Zinn et al. (2019) Zinn, J. C., Pinsonneault, M. H., Huber, D., et al. 2019, ApJ, 885, 166, doi: 10.3847/1538-4357/ab44a9