# Rotating Solar Models in Agreement with Helioseismic Results and Updated
Neutrino Fluxes

Wuming Yang Department of Astronomy, Beijing Normal University, Beijing
100875, China<EMAIL_ADDRESS>
###### Abstract

Standard solar models (SSMs) constructed in accordance with old solar
abundances are in reasonable agreement with seismically inferred results, but
SSMs with new low-metal abundances disagree with the seismically inferred
results. The constraints of neutrino fluxes on solar models exist in parallel
with those of helioseismic results. The solar neutrino fluxes were updated by
Borexino Collaboration. We constructed rotating solar models with new low-
metal abundances where the effects of enhanced diffusion and convection
overshoot were included. A rotating model using OPAL opacities and the Caffau
abundance scale has better sound-speed and density profiles than the SSM with
the old solar abundances and reproduces the observed $p$-mode frequency ratios
$r_{02}$ and $r_{13}$. The depth and helium abundance of the convection zone
of the model agree with the seismically inferred ones at the level of
$1\sigma$. The updated neutrino fluxes are also reproduced by the model at the
level of $1\sigma$. The effects of rotation and enhanced diffusion not only
improve the model’s sound-speed and density profiles but bring the neutrino
fluxes predicted by the model into agreement with the detected ones. Moreover,
the calculations show that OP may underestimate opacities for the regions of
the Sun with $T\gtrsim 5\times 10^{6}$ K by around $1.5\%$, while OPAL may
underestimate opacities for the regions of the Sun with $2\times 10^{6}$ K
$\lesssim T\lesssim 5\times 10^{6}$ K by about $1-2\%$.

Solar abundances — Helioseismology — Solar interior — Solar neutrino fluxes —
Solar rotation

##  1 Introduction

The heavy-element abundance $Z_{s}$ of the Sun, derived by Grevesse & Sauval
(1998, hereafter GS98) from photospheric spectroscopy, is $0.017$, whose
uncertainty is of the order of $10$ percent. The ratio of the heavy-element
abundance to hydrogen abundance is $0.0231$. Since Lodders (2003) and Asplund
et al. (2005) reassessed the value of the $Z_{s}$, it has been revised several
times (Lodders et al., 2009; Asplund et al., 2009; Caffau et al., 2010, 2011;
Lodders, 2020; Asplund et al., 2021; Amarsi et al., 2021). The well-known
values of the new $Z_{s}$ are $0.0133$ (Lodders, 2003), $0.0122$ (Asplund et
al., 2005), $0.0141$ (Lodders et al., 2009), or $0.0134$ (Asplund et al.,
2009, hereafter AGSS09). These revised values are obviously lower than the old
one.

The helium abundance, $Y_{s}$, in the solar convection zone (CZ) and thus
photosphere cannot be inferred directly from spectroscopy, but can be
determined by helioseismology. The values of seismically inferred $Y_{s}$ and
$Z_{s}$ are $0.2485\pm 0.0035$ (Basu & Antia, 2004; Serenelli & Basu, 2010)
and $0.0172\pm 0.002$ (Antia & Basu, 2006), respectively. However, the values
given by Vorontsov et al. (2013) or Vorontsov et al. (2014) are in the range
of $Y_{s}=0.240–0.255$ and $Z_{s}=0.008–0.013$ or $Y_{s}=0.245–0.260$ and
$Z_{s}=0.006–0.011$. Moreover, the value of $Z_{s}$ inferred by Buldgen et al.
(2017) is in the range of $0.008–0.014$. The radius of the base of the CZ
(BCZ) also can be determined by helioseismology. The inferred radius of the
BCZ is $0.713\pm 0.003$ $R_{\odot}$ (Christensen-Dalsgaard et al., 1991) or
$0.713\pm 0.001$ $R_{\odot}$ (Basu & Antia, 1997).

The standard solar models (SSMs) constructed in accordance with the high metal
abundances (old solar abundances, e.g. GS98) are considered to be in good
agreement with the seismically inferred sound-speed and density profiles,
depth and helium abundance of the CZ, but the SSMs constructed in accordance
with the low metal abundances (revised solar abundances) do not completely
agree with the seismically inferred results (Bahcall et al., 2004; Basu &
Antia, 2004; Yang & Bi, 2007; Basu et al., 2009; Serenelli et al., 2009, 2011;
Zhang & Li, 2012) and the neutrino flux constraints (Bahcall & Pinsonneault,
2004; Turck-Chièze et al., 2010, 2011; Turck-Chièze & Couvidat, 2011; Yang,
2016), which is known as solar modeling problem or solar abundance problem
(Basu et al., 2015; Christensen-Dalsgaard, 2021; Salmon et al., 2021; Amarsi
et al., 2021).

In order to reconcile the low-Z models with helioseismology, many physical
effects have been studied. For example, increased opacity at the base of the
CZ was studied by Bahcall et al. (2004), Serenelli et al. (2009), and Buldgen
et al. (2019); enhanced neon abundance was suggested by Bahcall et al. (2005);
mass accretion of metal-rich/poor material or helium-poor material was
investigated by Castro et al. (2007), Guzik & Mussack (2010), Serenelli et al.
(2011), and Zhang et al. (2019); overshooting below the CZ was used to recover
the CZ depth (Montalbán et al., 2006; Castro et al., 2007; Yang, 2019; Zhang
et al., 2019).

In order to match the seismically inferred sound-speed and density profiles
(Basu et al., 2000, 2009), the gravitational settling that reduces the surface
helium abundance by about $11\%$ ($\sim 0.03$ by mass fraction) below its
initial value is required in SSMs. Macroscopic turbulent mixing can reduce the
amount of surface helium settling by around $40\%$ (Proffitt & Michaud, 1991).
The effects of the turbulent mixing on the diffusion and settling of helium
and heavy elements were not considered in the diffusion coefficients of Thoul
et al. (1994). Asplund et al. (2004) suggested that enhanced diffusion and
settling of helium and heavy elements might be able to reconcile the low-Z
models with helioseismology. The increased diffusion can significantly improve
sound-speed and density profiles, but leaves the CZ helium abundance too low
(Basu & Antia, 2004; Montalbán et al., 2004; Guzik et al., 2005; Yang, 2019).

Rotational mixing can transport helium outward. It thus can counteract the
effect of enhanced diffusion on the surface helium abundance (Yang, 2019),
i.e., it can improve the prediction of the surface helium abundance. The
effects of rotation on the low-Z models were studied by Yang & Bi (2007),
Turck-Chièze et al. (2010, 2011), and Yang (2016, 2019). However, the rotating
models with AGSS09 mixtures (Yang, 2019) disagree with the detected neutrino
fluxes of Borexino Collaboration (2018, 2020).

Bahcall & Pinsonneault (2004) and Serenelli et al. (2009) found that an
$11\%-20\%$ increase in OPAL opacities at the BCZ can reconcile the low-Z
models with helioseismology. Badnell et al. (2005) showed that in the region
OP opacity is slightly larger than OPAL opacity but no more than $2.5$
percent. Buldgen et al. (2019) concluded that the solar modeling problem
likely occurs from multiple small contributors. Opacity could be one of the
contributors.

The production of solar neutrinos is sensitive to the central properties of
the Sun. The 8B neutrino flux is strongly dependent on the central temperature
of the Sun (Bahcall & Ulrich, 1988; Turck-Chièze & Couvidat, 2011). The
determinations of solar neutrino fluxes complement helioseismology in
diagnosing the core of the Sun. The constraints of neutrino fluxes on solar
models exist in parallel with those of helioseismology and are studied by many
authors (Bahcall & Pinsonneault, 2004; Turck-Chièze et al., 2010, 2011; Turck-
Chièze & Couvidat, 2011; Yang, 2016, 2019; Zhang et al., 2019). Borexino
Collaboration (2018, 2020) updated solar neutrino fluxes and determined the
total fluxes of 13N, 15O, and 17F neutrinos. Their 8B neutrino flux is higher
than that determined by Bergström et al. (2016). The low-Z model of Zhang et
al. (2019) with helium-poor accretion and solar-wind mass loss agrees with
helioseismic results but disagrees with the neutrino fluxes detected by
Borexino Collaboration (2018, 2020). Moreover, Salmon et al. (2021) showed
that the updated solar neutrino fluxes prefer high-metallicity solar models.
The solar modeling problem has persisted for almost 20 years.

Caffau et al. (2010, 2011) independently analysed carbon, nitrogen, and oxygen
abundances in the solar photosphere. They found that the heavy-element
abundance of the solar surface is $Z_{s}=0.0154/0.0153$, supplemented with
data from Lodders et al. (2009). The value of $Z_{s}/X_{s}$ advocated by
Caffau et al. (2011) is $0.0209$. These values are larger than those of
Lodders et al. (2009) and AGSS09. Hope et al. (2020) studied the possible
origin of the solar abundance problem. Their result favors the solar $Z_{s}$
reported by Caffau et al. (2011) rather than the measurement by AGSS09.

Lodders (2020) reanalysed the solar photospheric abundances and recommended
$Z_{s}=0.0149$ and $Z_{s}/X_{s}=0.0201$. Asplund et al. (2021, hereafter
AAG21) also reassessed the solar abundances with updated atomic data and a 3D
radiative-hydrodynamical model of the solar photosphere. They advocated
$Z_{s}=0.0139\pm 0.0006$, $Y_{s}=0.2423\pm 0.0054$, and $Z_{s}/X_{s}=0.0187\pm
0.0009$. Moreover, using the 3D radiative-hydrodynamical model of solar
photosphere, Amarsi et al. (2021) analysed $408$ molecular lines of C, N, and
O and obtained solar C, N, and O abundances, which are slightly larger than
their previous results in AAG21. With these molecular abundances, the value of
the heavy-element abundance of the solar surface increases from $Z_{s}=0.0139$
to $Z_{s}=0.0142$ (Amarsi et al., 2021). This increase indicates that “it may
be worthwhile to continue improving the atomic and molecular data as well as
the model atmospheres and line formation methods” (Amarsi et al., 2021).

In this work, we mainly focus on whether the solar models constructed in
accordance with AAG21 mixtures and Caffau et al. (2011) mixtures agree with
the seismically inferred results and updated neutrino fluxes. The paper is
organized as follows. Input physics in models are introduced in Section 2,
calculation results are presented in Section 3, and discussion and summary are
given in Section 4.

##  2 Input Physics in Models

All solar models were calculated by using the Yale Rotating Stellar Evolution
Code (Pinsonneault et al., 1989; Yang & Bi, 2007; Demarque et al., 2008) in
its rotation and non-rotation configurations. The frequencies of $p$-modes of
models were computed by using the Guenther (1994) pulsation code. The OPAL
equation-of-state (EOS2005) tables (Rogers & Nayfonov, 2002), OPAL (Iglesias &
Rogers, 1996) and OP opacity tables (Seaton, 1987; The Opacity Project Team,
1995; Badnell et al., 2005; Delahaye et al., 2016) were used, supplemented by
the Ferguson et al. (2005) opacity tables at low temperature. The opacity
tables were reconstructed with the GS98, Caffau et al. (2011), and AAG21
mixtures (see Appendix A). The nuclear reaction rates were calculated with the
subroutine of Bahcall & Pinsonneault (1992) and Bahcall et al. (1995, 2001)
(see Appendix B). Convection was determined by the Schwarzschild criterion and
treated according to the standard mixing-length theory (Böhm-Vitense, 1958;
Kippenhahn et al., 2012). The overshoot region below the BCZ was assumed to be
both fully mixed and adiabatically stratified (Christensen-Dalsgaard et al.,
1991). The depth of the overshoot region is determined by $\delta_{\rm
ov}H_{p}$ (Demarque et al., 2008), where $\delta_{\rm ov}$ is a free parameter
and $H_{p}$ is the local pressure scale height. A convection overshoot of
$\delta_{\rm ov}\approx 0.1$ is required to recover the seismically inferred
depth of the CZ in our rotating models. The diffusion and settling of both
helium and heavy elements were computed using the formulas of Thoul et al.
(1994). In the atmosphere, Krishna Swamy (1966) $T-\tau$ relation was adopted.

We treated the transport of angular momentum and material mixing as a
diffusion process (Endal & Sofia, 1978), i.e.

$\frac{\partial\Omega}{\partial t}=f_{\Omega}\frac{1}{\rho
r^{4}}\frac{\partial}{\partial r}(\rho r^{4}D\frac{\partial\Omega}{\partial
r})\,$ (1)

for the transport of angular momentum and

$\begin{array}[]{lll}\frac{\partial X_{i}}{\partial
t}&=&f_{c}f_{\Omega}\frac{1}{\rho r^{2}}\frac{\partial}{\partial r}(\rho
r^{2}D\frac{\partial X_{i}}{\partial r})\\\ &&+(\frac{\partial X_{i}}{\partial
t})_{\rm nuc}-\frac{1}{\rho r^{2}}\frac{\partial}{\partial r}(f_{0}\rho
r^{2}X_{i}V_{i})\end{array}$ (2)

for the change in the mass fraction $X_{i}$ of chemical species $i$, where $D$
is the diffusion coefficient caused by rotational instabilities including the
Eddington circulation, the Goldreich-Schubert-Fricke instability (Pinsonneault
et al., 1989), and the secular shear instability of Zahn (1993). The default
values of $f_{\Omega}$ and $f_{c}$ are $1$ and $0.03$ (Yang, 2019),
respectively. We applied a straight multiplier $f_{0}$ to the diffusion
velocity $V_{i}$ to enhance the rates of diffusion and settling, as Basu &
Antia (2004), Montalbán et al. (2004), Guzik et al. (2005), and Yang (2019)
have done, despite the fact that there is no obvious physical justification
for such a multiplier. The value of $f_{0}$ is $1$ for standard cases but
larger than $1$ for an enhanced diffusion model. The angular-momentum loss
from the CZ due to magnetic braking was calculated with Kawaler’s relation
(Kawaler, 1988; Chaboyer et al, 1995). More details of calculation for
rotation were described in Pinsonneault et al. (1989) and Yang (2019).

All models were evolved from a homogeneous zero-age main-sequence model to the
present solar age $4.57$ Gyr, luminosity $3.844\times 10^{33}$ erg
$\mathrm{s}^{-1}$, radius $6.9598\times 10^{10}$ cm, and mass $1.9891\times
10^{33}$ g (Bahcall et al., 1995). The initial metallicity $Z_{0}$, hydrogen
abundance $X_{0}$, and mixing-length parameter $\alpha_{\rm{MLT}}$ are free
parameters. They were adjusted to match the constraints of luminosity and
radius within around $10^{-5}$ and observed $Z_{s}/X_{s}$. The initial helium
abundance is determined by $Y_{0}=1-X_{0}-Z_{0}$. The value of
$\alpha_{\rm{MLT}}$ changes with input physics, i.e. a new solar calibration
of $\alpha_{\rm{MLT}}$ is performed each time the input physics change. The
initial rotation rate, $\Omega_{i}$ of rotating models was adjusted to
reproduce the solar equatorial velocity of about $2.0$ km s-1. The values of
these parameters are shown in Table 1.

##  3 Calculation Results

###  3.1 Solar Models with High Metal Abundances

####  3.1.1 The Models Constructed with OPAL Opacity Tables

Using the OPAL opacity tables constructed with GS98 mixtures, we computed SSM
GS98M and rotating model GS98Mr. The central temperature and density, surface
helium and heavy-element abundances, radius of the BCZ, and other parameters
of the models are listed in Table 1. The fluxes of $pp$, $pep$, $hep$, 7Be,
8B, 13N, 15O, and 17F neutrinos calculated from the models are given in Table
2. Table 3 lists some of the physical configurations of each model.

The surface heavy-element abundance of $0.0174$ of GS98M is in agreement with
that determined by GS98. The surface helium abundance and $r_{cz}$ of GS98M
are also consistent with the seismically inferred ones (see Table 1). We
compared the sound speed and density of models with those inferred by Basu et
al. (2009) using the data from the Birmingham Solar-Oscillations Network
(Chaplin et al., 1996) and the Michelson Doppler Imager (Schou et al., 1998).
The values of relative sound-speed difference, $\delta c_{s}/c_{s}$, and
density difference, $\delta\rho/\rho$, between the Sun and GS98M are less than
$0.0043$ and $0.028$, respectively (see Figure 1). Moreover, the ratios of
small to large frequency separations, $r_{02}$ and $r_{13}$ (Roxburgh &
Vorontsov, 2003), of the model agree with those calculated from observed
frequencies of Chaplin et al. (1999) or García et al. (2011) (see Figure 2).
Table 4 gives the values of $\chi^{2}_{c_{s}+\rho}$ and $\chi^{2}_{d_{02+13}}$
of the models.

Figure 1: Relative sound-speed and density differences, in the sense (Sun-
Model)/Model, between the Sun and models. The solar sound speed and density
are given in Basu et al. (2009).  Figure 2: Distributions of observed and
predicted ratios $r_{02}$ and $r_{13}$ as a function of frequency. The circles
and triangles show the ratios calculated from the frequencies observed by GOLF
& VIRGO (García et al., 2011) and BiSON (Chaplin et al., 1999), respectively.

We compared the neutrino fluxes computed from models with those determined by
different authors (Bellini et al., 2011, 2012; Ahmed et al., 2004; Bergström
et al., 2016; Borexino Collaboration, 2018, 2020) and ones predicted by the
models BP04 (Bahcall & Pinsonneault, 2004) and SSeM (Turck-Chièze & Couvidat,
2011) in Table 2. The neutrino fluxes of GS98M are comparable with those of
BP04. Some differences between the nuclear cross-section factors $S_{0}$
(Bahcall, 1989) used in BP04 and those used in GS98M are listed in Table 6 of
Appendix B. The total fluxes of 13N, 15O, and 17F neutrinos calculated from
GS98M are $\Phi(\rm CNO)=10.2\times 10^{8}$ cm-2 s-1, slightly larger than
$7^{+3}_{-2}\times 10^{8}$ cm-2 s-1 detected by Borexino Collaboration (2020).
The $pp$, $pep$, $hep$, 7Be, and 8B neutrino fluxes of GS98M are in agreement
with those determined by Borexino Collaboration (2018). However, the fluxes of
7Be and 8B neutrinos are larger than those determined by Bergström et al.
(2016) (see Table 2).

Figure 1 shows that the effects of rotation can significantly improve the
sound-speed and density profiles. The value of the relative sound-speed
difference, $\delta c_{s}/c_{s}$, below the CZ can be decreased by about
$50\%$. Moreover, the centrifugal effect leads to a decrease in the central
temperature. The fluxes of 7Be, 8B, 13N, 15O, and 17F neutrinos are sensitive
to the central temperature. Thus the fluxes calculated from rotating models
are generally lower than those computed from non-rotating models (see Table
2). The total fluxes of 13N, 15O, and 17F neutrinos of GS98Mr are $\Phi(\rm
CNO)=9.5\times 10^{8}$ cm-2 s-1, which are in agreement with the detected
value of $7^{+3}_{-2}\times 10^{8}$ cm-2 s-1. The 8B neutrino flux of
$5.29\times 10^{6}$ cm-2 s-1 for GS98Mr is consistent with
$5.16^{+0.13}_{-0.09}\times 10^{6}$ cm-2 s-1 determined by Bergström et al.
(2016) and $5.68^{+0.39}_{-0.41}\times 10^{6}$ cm-2 s-1 detected by Borexino
Collaboration (2018). However, the surface helium abundance of 0.2534 of
GS98Mr is higher than the inferred value of $0.2485\pm 0.0035$. Thus the
high-Z models constructed with OPAL opacities do not completely agree with
helioseismic results and updated neutrino fluxes.

####  3.1.2 The Models Constructed with OP Opacity Tables

In order to study the effects of opacities, we constructed SSM GS98op and
rotating model GS98opr by using OP opacity tables. Tables 1 and 2 list the
fundamental parameters and neutrino fluxes of the models, respectively. For
almost the same metallicity, OP opacities are larger by about $0-2\%$ than
OPAL opacities in the region of the Sun with $0.5$ $R_{\odot}$$\lesssim
r\lesssim 0.8$ $R_{\odot}$ but smaller by about $1-2\%$ in the region with
$r\lesssim 0.4$ $R_{\odot}$ (see Figure 3). The sound-speed and density
profiles and frequency separation ratios of the models are shown in Figures 1
and 2, respectively, which show that OP has almost no improvement in the
reproduction of the frequency separation ratios in comparison to OPAL (see
Figure 2), but obviously improves density profile and the sound-speed profile
below the CZ where OP opacities are mainly larger than OPAL opacities. OP
slightly worsens the sound-speed profile in the inner layers of radiative
region where OP opacities are lower than OPAL opacities (see Figures 1 and 3).

Figure 3: Left panel: comparison of Rosseland mean opacity of different models
relative to that of a reference model constructed with OPAL opacity tables and
$Z_{s}=0.0170$. Right panel: distributions of Rosseland mean opacity of the
models as a function of radius.

In order to obtain the same surface heavy-element abundance and the solar
luminosity and radius at the age of $4.57$ Gyr, the decrease in opacities
requires decreasing the initial helium abundance and changing
$\alpha_{\rm{MLT}}$. As a consequence, the surface helium abundances of the
models constructed with OP opacity tables are lower than those of the models
constructed with OPAL opacity tables. The surface helium abundance of $0.2414$
for GS98op is smaller than the inferred value of $0.2485\pm 0.0035$.
Therefore, GS98op is not a good SSM.

Figure 4: Distributions of hydrogen and metal abundances of models as a
function of radius.

Rotational mixing brings hydrogen into inner layers from outer layers and
transports helium outward, i.e. decreases the hydrogen abundance in the CZ,
but increases the hydrogen abundance in the region with $0.5$ $R_{\odot}$
$\lesssim r\lesssim 0.7$ $R_{\odot}$ (see Figure 4), which changes the
distribution of the mean molecular weight. As a consequence, the density and
sound-speed profiles are significantly improved by the effects of rotation.
The value of the relative sound-speed difference below the CZ is decreased by
about $80\%$ (see Figure 1). Although OP improves the sound-speed profile
below the CZ, the effects of rotation play a more important role in improving
the sound-speed profile. Moreover, the amount of the CZ helium settling is
reduced by about $34\%$. The surface helium abundance of $0.2511$ for GS98opr
is in agreement with the seismically inferred value of $0.2485\pm 0.0035$, and
increased by about $0.01$ compared to that of the non-rotating model.

The central temperatures of models constructed with OP opacity tables are
lower than those of models constructed with OPAL opacity tables. Therefore,
the fluxes of 7Be, 8B, 13N, 15O, and 17F neutrinos computed from GS98op and
GS98opr are smaller than those calculated from GS98M and GS98Mr. The total
fluxes $\Phi(\rm CNO)$ of GS98op and GS98opr are in agreement with that
detected by Borexino Collaboration (2020). The 8B neutrino fluxes of the
models are consistent with that determined by Bergström et al. (2016) but
lower than one detected by Borexino Collaboration (2018). Thus the high-Z
models constructed with OP opacities also do not completely agree with
helioseismic results and updated neutrino fluxes.

We chose GS98M as the best SSM and GS98opr as the best rotating model with
high metal abundances and went on to construct the models with low metal
abundances and compare them with these two models.

###  3.2 Solar Models with Low Metal Abundances

####  3.2.1 Standard and Rotating Models

Figure 5: Relative sound-speed and density differences, in the sense (Sun-
Model)/Model, between the Sun and models. The solar sound speed and density
are given in Basu et al. (2009).

Using OP opacity tables reconstructed with AAG21 mixtures, we computed SSM
Aop01 and rotating model Aop01r with the surface heavy-element abundance
determined by Asplund et al. (2021). Although OP opacities and the effects of
rotation can significantly improve sound-speed and density profiles, the
sound-speed and density profiles of Aop01 and Aop01r are not as good as those
of GS98M (see Figure 5). The models also can not reproduce the observed
frequency separation ratios $r_{02}$ and $r_{13}$ (see Figure 6 or Table 4)
and inferred helium abundance. The total fluxes $\Phi(\rm CNO)$ of Aop01 and
Aop01r are in agreement with the detection of Borexino Collaboration (2020),
but their 7Be and 8B neutrino fluxes are too low (see Table 2).

Figure 6: Distributions of observed and predicted ratios $r_{02}$ and $r_{13}$
as a function of frequency. The circles and triangles show the ratios
calculated from the frequencies observed by GOLF & VIRGO (García et al., 2011)
and BiSON (Chaplin et al., 1999), respectively.

With the heavy-element abundance determined by Lodders (2020), we constructed
SSM Aop02 and rotating model Aop02r. We also calculated SSM Cop01 and rotating
model Cop01r by using the OP opacity tables reconstructed with Caffau et al.
(2011) mixtures. The SSMs Aop02 and Cop01 obviously disagree with seismically
inferred results and detected 8B neutrino flux. For rotating models Aop02r and
Cop01r, the overshoot of convection brings the depth of the CZ into agreement
with the seismically inferred one. The surface helium abundances of Aop02r and
Cop01r are $0.2453$ and $0.2467$, respectively, consistent with the inferred
value of $0.2485\pm 0.0035$. The total fluxes $\Phi(\rm CNO)$ predicted by
Aop02r and Cop01r are $7.4$ and $7.6\times 10^{8}$ cm-2 s-1, respectively,
which are in good agreement with that detected by Borexino Collaboration
(2020). The sound-speed profiles of Aop02r and Cop01r are comparable with that
of GS98M (see Figure 5). The observed frequency separation ratios $r_{02}$ and
$r_{13}$ are almost reproduced by the models (see Figure 6). However, the 8B
neutrino fluxes of Aop02r and Cop01r are obviously lower than those determined
by Bergström et al. (2016) and Borexino Collaboration (2018) (see Table 2).
Moreover, their density profiles are not as good as that of GS98M. These
indicate that the effects of OP and rotation can improve the solar model but
can not completely solve the solar modeling problem.

####  3.2.2 Rotating and Enhanced Diffusion Models Constructed in Accordance
with AAG21 Mixtures

The gravitational settling and diffusion reduce the surface helium abundance
of SSMs by around $11\%$ ($\sim 0.03$ by mass fraction) below its initial
value (see Table 1), which plays an important role in shaping the sound-speed
and density profiles of the models. Rotational mixing reduces the amount of
the surface helium settling by about $33\%-36\%$. In order to counteract the
effects of rotational mixing on the settling of elements, we increased the
rates of element diffusion and settling by $36\%$, and then constructed
rotating models Aop12r and Aopal12r by using OP and OPAL opacity tables. The
two models had the surface heavy-element abundance determined by Lodders
(2020). Tables 1 and 2 list their fundamental parameters and neutrino fluxes,
respectively.

Figure 7: Top panels: relative sound-speed and density differences, in the
sense (Sun-Model)/Model, between the Sun and models. The solar sound speed and
density are given in Basu et al. (2009). Bottom panels: distributions of
observed and predicted ratios $r_{02}$ and $r_{13}$ as a function of
frequency. The circles and triangles show the ratios calculated from the
frequencies observed by GOLF & VIRGO (García et al., 2011) and BiSON (Chaplin
et al., 1999), respectively.

The sound-speed and density profiles of Aopal12r are almost as good as those
of GS98M, but those of Aop12r are obviously better than those of GS98M (see
Figure 7). The inferred CZ depth and observed frequency separation ratios
$r_{02}$ and $r_{13}$ are reproduced well by the two models (see Table 1 and
Figure 7). The total fluxes of 13N, 15O, and 17F neutrinos are $\Phi(\rm
CNO)=8.37\times 10^{8}$ cm-2 s-1 for Aop12r and $\Phi(\rm CNO)=8.53\times
10^{8}$ cm-2 s-1 for Aopal12r, which are consistent with the detected value of
$7^{+3}_{-2}\times 10^{8}$ cm-2 s-1 (Borexino Collaboration, 2020). However,
the 8B neutrino fluxes computed from Aop12r and Aopal12r are lower than that
detected by Borexino Collaboration (2018) (see Table 2). The surface helium
abundances of Aop12r and Aopal12r are $0.2413$ and $0.2424$, respectively,
which are consistent with $0.2423\pm 0.0054$ advocated by Asplund et al.
(2021) but lower than $0.2485\pm 0.0035$ inferred by Basu & Antia (2004). Thus
these models also do not completely agree with helioseismic results and
updated neutrino fluxes.

####  3.2.3 Rotating and Enhanced Diffusion Models Constructed in Accordance
with Caffau Mixtures

By using OPAL and OP opacity tables reconstructed with Caffau et al. (2011)
mixtures, we computed rotating models Copal11r and Cop11r. In order to
counteract the effects of rotational mixing on the settling of elements, same
as the cases in models Aopal12r and Aop12r, the rates of element diffusion and
settling were also increased by $36\%$ in models Copal11r and Cop11r. The
surface heavy-element abundance of $0.01548$ for Copal11r is consistent with
that determined by Caffau et al. (2010). Figure 8 shows that Copal11r has
better sound-speed and density profiles (smaller $\chi^{2}_{c_{s}+\rho}$) than
GS98M and reproduces the observed frequency separation ratios $r_{02}$ and
$r_{13}$. The relative differences $\delta c_{s}/c_{s}$ and $\delta\rho/\rho$
between the Sun and Copal11r are smaller than $0.0021$ and $0.025$,
respectively. It also reproduces the seismically inferred surface helium
abundance and radius $r_{cz}$ at the level of $1\sigma$ (see Table 1).

Figure 8: Top panels: relative sound-speed and density differences, in the
sense (Sun-Model)/Model, between the Sun and models. The solar sound speed and
density are given in Basu et al. (2009). Bottom panels: distributions of
observed and predicted ratios $r_{02}$ and $r_{13}$ as a function of
frequency. The circles and triangles show the ratios calculated from the
frequencies observed by GOLF & VIRGO (García et al., 2011) and BiSON (Chaplin
et al., 1999), respectively.

The fluxes of $pp$, $pep$, $hep$, 7Be, and 8B neutrinos and the total fluxes
of 13N, 15O, and 17F neutrinos calculated from Copal11r are in agreement with
those detected by Borexino Collaboration (2018, 2020) at the level of
$1\sigma$. The 8B neutrino flux of $5.41\times 10^{6}$ cm-2 s-1 is also in
good agreement with $(5.21\pm 0.27)\times 10^{6}$ cm-2 s-1 (Ahmed et al.,
2004) but larger than that determined by Bergström et al. (2016) (see Table
2). Copal11r not only is in agreement with updated neutrino fluxes but has
better sound-speed and density profiles than GS98M (see Figure 8 or Table 4).
It is thus better than GS98M.

The surface heavy-element abundance of Cop11r is also $0.01548$. Cop11r has
better sound-speed and density profiles (smaller $\chi^{2}_{c_{s}+\rho}$) than
Copal11r (see Figure 8 and Table 4). It also reproduces the observed $r_{02}$
and $r_{13}$, inferred radius $r_{cz}$, and updated neutrino fluxes except the
8B neutrino flux. The 8B neutrino flux of Cop11r is $5.16\times 10^{6}$ cm-2
s-1, which is lower than $5.68^{+0.39}_{-0.41}\times 10^{6}$ cm-2 s-1
(Borexino Collaboration, 2018) but in good agreement with that determined by
Bergström et al. (2016). The surface helium abundance of $0.2431$ for Cop11r
is lower than $0.2485\pm 0.0035$ inferred by Basu & Antia (2004) but
consistent with $0.2423\pm 0.0054$ advocated by Asplund et al. (2021). OP
significantly improves the sound-speed and density profiles, but leads to the
fact that the 8B neutrino flux and surface helium abundance are lower than the
inferred values. However, if $\Phi(^{8}\rm B)=5.16^{+0.13}_{-0.09}\times
10^{6}$ cm-2 s-1 (Bergström et al., 2016) and $Y_{s}=0.2423\pm 0.0054$
(Asplund et al., 2021) are adopted, the $\Phi(^{8}\rm B)$ and $Y_{s}$ of
GS98op are more consistent with these values than those of GS98M. In this
case, GS98op is the best SSM with high metal abundances rather than GS98M; and
Cop11r is better than GS98op.

Models Cop11r and Copal11r have the same input physics except opacity tables.
The models constructed with OP opacity tables have better sound-speed and
density profiles but lower surface helium abundance and 8B neutrino flux than
those constructed with OPAL opacity tables. The differences in the sound-speed
and density profiles, surface helium abundances, and 8B neutrino fluxes caused
by the discrepancies in the opacities are marked. The results indicate that a
small discrepancy in opacities can obviously affect the solar model.

###  3.3 Possible Problems in Opacities

In order to understand the effects of the differences in opacities on models,
we constructed model Aop12ri with OP opacity tables and Copal11ri with OPAL
opacity tables. The OP opacities in the regions of Aop12ri with $T\gtrsim
4.5\times 10^{6}$ K ($r\lesssim 0.45$ $R_{\odot}$) were increased by about
$1.5\%$ ($k_{0}\leq 1.015$, see Appendix A) to approximately match OPAL
opacities in the same regions. The OPAL opacities for the regions of Copal11ri
with $2\times 10^{6}$ K $\lesssim T\lesssim 5\times 10^{6}$ K were increased
by about $1-2.3\%$ ($k_{0}\leq 1.023$) to roughly match OP opacities (see
Figure 9). The fundamental parameters and neutrino fluxes of these two models
are also listed in Tables 1 and 2, respectively.

Figure 9: Left panel: comparison of Rosseland mean opacity of different models
relative to that of Aopal12r. Right panel: distributions of Rosseland mean
opacity of the models as a function of radius.

Aop12ri is in good agreement with helioseismic results and updated neutrino
fluxes except the surface helium abundance and 8B neutrino flux. The surface
helium abundance of Aop12ri is $0.2436$, which is lower than the seismically
inferred value of $0.2485\pm 0.035$. Moreover, the 8B neutrino flux of Aop12ri
is $5.19\times 10^{6}$ cm-2 s-1, which is lower than that detected by Borexino
Collaboration (2018). But they are higher than those of Aop12r. The increase
in OP opacities mainly improves the predictions of surface helium abundance
and 8B neutrino flux but slightly worsens the sound-speed and density profiles
in comparison to those of Aop12r (see Table 4 or Figures 8 and 10). However,
the sound-speed and density profiles of Aop12ri are still better than those of
GS98M.

Figure 10: Top panels: relative sound-speed and density differences, in the
sense (Sun-Model)/Model, between the Sun and models. The solar sound speed and
density are given in Basu et al. (2009). Bottom panels: distributions of
observed and predicted ratios $r_{02}$ and $r_{13}$ as a function of
frequency. The circles and triangles show the ratios calculated from the
frequencies observed by GOLF & VIRGO (García et al., 2011) and BiSON (Chaplin
et al., 1999), respectively.

The increase in OPAL opacities has almost no effect on surface helium
abundance and neutrino fluxes (see Table 2) but significantly improves the
sound-speed and density profiles (see Figure 10). The relative differences
$\delta c_{s}/c_{s}$ and $\delta\rho/\rho$ between Sun and Copal11ri are
smaller than $0.00117$ and $0.0178$ near the BCZ, decreased by about $46\%$
and $29\%$ in comparison to those of Copal11r, respectively. Copal11ri
reproduces the observed ratios $r_{02}$ and $r_{13}$ (see Figure 10) and the
inferred helium abundance and $r_{cz}$ at the level of $1\sigma$. The neutrino
fluxes calculated from Copal11ri also agree with those detected by Borexino
Collaboration (2018, 2020) at the level of $1\sigma$. Copal11ri is the best
rotating model for the heavy-element abundance determined by Caffau et al.
(2010). It is better than Copal11r and in good agreement with helioseismic
results and updated neutrino fluxes.

The increase in OPAL opacities works well. The changes in OP opacities would
have the same effect. In order to test this case, we constructed model
Cop11ri. The OP opacities for the regions of Cop11ri with $T\gtrsim 4.5\times
10^{6}$ K were increased by about $1.5\%$ to approximately match OPAL
opacities in the same regions. The calculations show that models Cop11ri and
Copal11ri have almost the same sound-speed and density profiles (see Figure
10), surface helium and heavy-element abundances (see Table 1), and neutrino
fluxes (see Table 2). They also have almost the same $Z_{0}$, $Y_{0}$, and
$\alpha_{\rm{MLT}}$. The modified OP and OPAL opacities produce almost the
same rotating models. These imply that the differences between Cop11r and
Copal11r result from discrepancies in opacities, and that OPAL might
underestimate opacities for the regions of the Sun with $2\times 10^{6}$ K
$\lesssim T\lesssim 5\times 10^{6}$ K by about $1-2\%$, or that OP might
underestimate opacities in the regions of the Sun with $T\gtrsim 4.5\times
10^{6}$ K by around $1.5\%$. The possible underestimate in OP leads to the
fact that the models constructed with OP opacity tables have a lower surface
helium abundance and 8B neutrino flux than those constructed with OPAL opacity
tables, while the possible underestimate in OPAL results in the fact that the
sound-speed and density profiles of the models constructed with OPAL opacity
tables are not as good as those of the models constructed with OP opacity
tables.

With the modified opacities, only the models with the metal abundance
determined by Lodders (2020) agree with the 8B neutrino flux determined by
Bergström et al. (2016) and the helium abundance advocated by Asplund et al.
(2021), but the models with the metal abundance determined by Caffau et al.
(2010) are in agreement with the 8B neutrino flux detected by Borexino
Collaboration (2018) and the seismically inferred helium abundance. Thus
precisely determining 8B neutrino flux aids in solving the solar abundance
problem.

##  4 Discussion and Summary

Although GS98M was chosen as the best SSM with high metal abundances, the
value of $\Phi(\rm CNO)$ of GS98M is slightly larger than the detected value
of $7^{+3}_{-2}\times 10^{8}$ cm-2 s-1 (Borexino Collaboration, 2020). That of
GS98opr is in agreement with the detected one, but its 8B neutrino flux is
lower than that updated by Borexino Collaboration (2018). Thus the updated
neutrino fluxes do not favour the high-Z models. Moreover, the values of
$\Phi(\rm CNO)$ calculated from the models with the heavy-element abundance
determined by Asplund et al. (2021) are consistent with the detected value,
but the 7Be and 8B neutrino fluxes predicted by the models are much lower than
those determined by Bergström et al. (2016) and ones detected by Borexino
Collaboration (2018). Therefore, the updated neutrino fluxes also do not
prefer the models with the heavy-element abundance determined by Asplund et
al. (2021). For the same input physics, the fluxes of 7Be, 8B, 13N, 15O, and
17F neutrinos predicted by models increase with an increase in metallicity.
These imply that the updated neutrino fluxes prefer a heavy-element abundance
between that determined by GS98 and one advocated by Asplund et al. (2021).

Convection was treated according to the standard mixing-length theory (Böhm-
Vitense, 1958) in this work. The treatment of convection is one of the sources
of uncertainty in modeling of stars. Different treatments of convection, such
as Joyce & Chaboyer (2018), Spada et al. (2018, 2019), and Jermyn et al.
(2022), could affect solar models and deserve more detailed study.

There are many $T-\tau$ relations for solar atmosphere (Krishna Swamy, 1966;
Ball, 2022). The temperature, $T$ reaches $T_{\mathrm{eff}}$ at $\tau=2/3$ for
the Eddington approximation but $\tau=0.312156330$ for the Krishna Swamy
(1966) $T-\tau$ relation, i.e. the solar radius R is defined at an outer layer
for the Krishna Swamy (1966) $T-\tau$ relation. As a consequence, the Krishna
Swamy (1966) $T-\tau$ relation requires a larger $\alpha_{\rm{MLT}}$ to
reproduce solar radius than the Eddington approximation (Demarque et al.,
2008; Joyce & Chaboyer, 2018; Spada et al., 2018). In order to study the
effect of the $T-\tau$ relation on our results, we computed the solar models
with the Eddington approximation. The calculations show that the $T-\tau$
relation has hardly any influence on neutrino fluxes, frequency separation
ratios, and sound-speed and density profiles in the radiative region, but
slightly affects the sound-speed and density profiles in the CZ (lead to a
slight increase in $\chi^{2}_{c_{s}+\rho}$). Choosing between the Eddington
approximation or Krishna Swamy (1966) $T-\tau$ relation does not change our
results.

Rotational mixing can more efficiently inhibit the settling of helium than of
heavy-element abundances because the mixing depends on the gradient of
elements (Yang, 2019). It can reduce the amount of the surface helium settling
by about $33-36\%$ in our models, which is consistent with the result of
Proffitt & Michaud (1991), who found that macroscopic turbulent mixing can
reduce the amount of the surface helium settling by around $40\%$. It leads to
the fact that the surface helium abundances of rotating models are obviously
higher than those of non-rotating models. In the enhanced diffusion models,
the velocity of diffusion and settling was increased by $36\%$. However, we
have no obvious physical justification for the multiplier. In non-rotating
models, the enhanced diffusion leaves the surface helium abundance too low.
However, rotational mixing completely counteracts the effect of the enhanced
diffusion on the surface helium abundance in rotating models. Thus the surface
helium abundances of the rotating models with the enhanced diffusion are
higher than those of non-rotating models. The effects of rotation and enhanced
diffusion bring low-Z models into agreement with helioseismic results (Basu &
Antia, 2004; Basu et al., 2009) and updated neutrino fluxes (Borexino
Collaboration, 2018, 2020). However, the calculations show that the same
effects can not bring high-Z models into agreement with the helioseismic
results and the updated neutrino fluxes at the same time.

The effects of rotation and enhanced diffusion were studied by Yang (2019),
where the value of multiplier ($f_{0}$) was larger than or equal to $1.5$ and
OPAL opacity tables constructed in accordance with AGSS09 mixtures were used.
The flux of 7Be neutrino and the total fluxes of 13N, 15O, and 17F neutrinos
predicted by the best model of Yang (2019) are larger than those detected by
Borexino Collaboration (2018, 2020). The 7Be neutrino flux is also higher than
that determined by Bergström et al. (2016). Different from the earlier models
of Yang (2019), Copal11r and Copal11ri, in which the value of multiplier is
$1.36$ and opacity tables are reconstructed in accordance with Caffau et al.
(2011) mixtures, are in good agreement with the detected neutrino fluxes at
the level of $1\sigma$.

The differences between OPAL and OP opacities are small but can obviously
affect the properties of solar models. The 8B neutrino fluxes computed from
models constructed with OP opacity tables are lower than those calculated from
models constructed with OPAL opacity tables, which derives from the fact that
OP underestimates opacities for the regions of the Sun with $T\gtrsim 5\times
10^{6}$ K by about $1.5\%$ compared to OPAL, especially in the core. As a
consequence, the models constructed with OP opacity tables disagree with the
8B neutrino flux detected by Borexino Collaboration (2018) but can agree with
that determined by Bergström et al. (2016). Thus precisely determining the 8B
neutrino flux aid in diagnosing the opacities in the solar core.

Moreover, the sound-speed and density profiles of models constructed with OPAL
opacity tables are obviously not as good as those of models constructed with
OP opacity tables, which results from the fact that OPAL underestimates
opacities for the regions of the Sun with $2\times 10^{6}$ K $\lesssim
T\lesssim 5\times 10^{6}$ K by about $1-2\%$ compared to OP. If OPAL opacities
in the regions are increased by about $1-2\%$ to approximately match OP
opacities for the same regions, the models constructed with the OPAL opacities
will produce better sound-speed and density profiles. If OP opacities for the
regions of the Sun with $T\gtrsim 5\times 10^{6}$ K are increased by about
$1.5\%$, the model constructed with the OP opacities will be almost the same
as that constructed with the modified OPAL opacities. These imply that the
discrepancies between the sound speed and density of the Sun and those of the
models could partly derive from opacity. The small differences between OPAL
and OP opacities can obviously affect sound-speed and density profiles,
surface helium abundance, and neutrino fluxes of models, which do not depend
on mixture patterns. But the effect on neutrino fluxes slightly relies on the
value of $Z_{0}$. In order to improve the solar model, the discrepancies
between OPAL and OP may deserve more studies.

In this work, by using OP and OPAL opacity tables reconstructed with AAG21 and
Caffau’s mixtures, we constructed rotating solar models in which the effects
of convection overshoot and enhanced diffusion were included. We obtained a
rotating model, Copal11r, that is better than the SSM GS98M and the earlier
rotating models of Yang (2016, 2019). The surface heavy-element abundance of
Copal11r is $0.01548$, which is consistent with the value of $0.0154$
determined by Caffau et al. (2010) and that inferred by Basu & Antia (2004).
The surface helium abundance of $0.2450$ and the radius of the BCZ of $0.714$
$R_{\odot}$ are in agreement with the seismically inferred values at the level
of $1\sigma$. The initial helium abundance is $0.2718$, which is consistent
with the value of $0.273\pm 0.006$ inferred by Serenelli & Basu (2010). The
ratios $r_{02}$ and $r_{13}$ of Copal11r agree with those calculated from
observed frequencies. The sound-speed and density profiles of Copal11r are
better than those of GS98M. Moreover, the fluxes of $pp$, $pep$, $hep$, 7Be,
and 8B neutrinos and the total fluxes of 13N, 15O, and 17F neutrinos
calculated from Copal11r agree with those detected by Borexino Collaboration
(2018, 2020) at the level of $1\sigma$. The fluxes of 7Be and 8B neutrinos are
also consistent with those determined by Bellini et al. (2011) and Ahmed et
al. (2004). To recover the seismically inferred depth of the CZ, a convection
overshoot of $\delta_{\rm ov}\approx 0.1$ is required in rotating models.
Rotation or enhanced diffusion alone can improve sound-speed and density
profiles, but the combination of rotation and enhanced diffusion is required
to bring the rotating model into agreement with seismically inferred results
and detected neutrino fluxes. More details of Copal11r are given in Appendix
C.

If the surface helium abundance of $0.2423\pm 0.0054$ advocated by Asplund et
al. (2021) and the 8B neutrino flux determined by Bergström et al. (2016) are
adopted, we can obtain another rotating model, Cop11r, that is better than
GS98op. The models constructed with OP opacity tables have obviously better
sound-speed and density profiles but lower surface helium abundance and 7Be,
8B, 13N, 15O, and 17F neutrino fluxes than those constructed with OPAL opacity
tables. The calculations show that OPAL may underestimate opacities for the
regions of the Sun with $2\times 10^{6}$ K $\lesssim T\lesssim 5\times 10^{6}$
K by about $1-2\%$, and that OP may underestimate opacities in the regions of
the Sun with $T\gtrsim 5\times 10^{6}$ K by around $1.5\%$. If the possible
underestimate of OPAL or OP were corrected, the model better than Copal11r or
Cop11r would be obtained.

The author thanks the anonymous referee for helpful comments that helped the
author improve this work, as well as Dr Xianfei Zhang and Tanda Li for their
help, and acknowledges the support from the NSFC 11773005 and U1631236.

## References

  * Adelberger et al. (1998) Adelberger, E. G., et al. 1998, Rev. Mod. Phys., 70, 1265
  * Adelberger et al. (2011) Adelberger E. G., et al. 2011, RMP, 83, 195
  * Ahmed et al. (2004) Ahmed, S. N., Anthony, A. E., Beier, E. W. et al. 2004, PhRvL, 92, 1301
  * Amarsi et al. (2021) Amarsi, A. M., Grevesse, N., Asplund, M., Collet, R. 2021, A&A, 656, 113
  * Antia & Basu (2006) Antia, H. M., & Basu, S. 2006, ApJ, 644, 1292
  * Asplund et al. (2021) Asplund, M., Amarsi, A. M., Grevesse, N. 2021, A&A 653, A141
  * Asplund et al. (2004) Asplund, M., Grevesesse N., Sauval, A. J., Allende Prieto, C., & Kiselman, D. 2004, A&A, 417, 751
  * Asplund et al. (2005) Asplund, M., Grevesesse, N., Sauval, A. J., Allende Prieto, C., & Blomme, R. 2005, A&A, 431, 693
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A., & Scott, P. 2009, ARA&A, 47, 481 (AGSS09)
  * Baby et al. (2003) Baby, L. T., et al. 2003a, Phys. Rev. Lett., 90, 022501
  * Badnell et al. (2005) Badnell, N. R., Bautista, M. A., Butler, K., Delahaye, F., Mendoza, C., Palmeri, P., Zeippen, C. J., Seaton, M. J. 2005, MNRAS, 360, 458
  * Bahcall (1989) Bahcall, J. N. , 1989, Neutrino Astrophysics (Cambridge University, Cambridge, England).
  * Bahcall et al. (2001) Bahcall, J. N., Pinsonneault, M. H., & Basu, S, 2001, ApJ, 555, 990
  * Bahcall & Pinsonneault (1992) Bahcall, J. N., & Pinsonneault, M. H. 1992, Rev. Mod. Phys., 64, 885 (BP92)
  * Bahcall et al. (1995) Bahcall J. N., Pinsonneault M. H., & Wasserburg G. J. 1995, Rev. Mod. Phys., 67, 781
  * Bahcall et al. (2005) Bahcall, J. N., Basu, S., Pinsonneault, M. H., & Serenelli, A. M. 2005, ApJ, 618, 1049
  * Bahcall & Pinsonneault (2004) Bahcall, J. N., & Pinsonneault, M. H. 2004, Phys. Rev. Lett., 92, 121301 (BP04)
  * Bahcall et al. (2004) Bahcall, J. N., Serenelli, A. M., & Pinsonneault, M. H. 2004, ApJ, 614, 464
  * Bahcall & Ulrich (1988) Bahcall, J. N., & Ulrich, R. 1988, RvMP, 60, 297
  * Ball (2022) Ball, W. H. 2021, RNAAS, 5, 7
  * Basu & Antia (1997) Basu, S., & Antia, H. M. 1997, MNRAS, 287, 189
  * Basu & Antia (2004) Basu, S., & Antia, H. M. 2004, ApJL, 606, L85
  * Basu et al. (2009) Basu, S., Chaplin, W. J., Elsworth, Y., New, R., & Serenelli, A. M. 2009, ApJ, 699, 1403
  * Basu et al. (2015) Basu, S., Grevesse, N., Mathis, S., Turck-Chièze, S. 2015, Space Sci Rev., 196, 49
  * Basu et al. (2000) Basu, S., Pinsonneault, M. H., & Bahcall, J. N. 2000, ApJ, 529, 1084
  * Bellini et al. (2011) Bellini, G. Benziger, J. Bick, D. et al. 2011, PhRvL, 107, 141302
  * Bellini et al. (2012) Bellini, G. Benziger, J. Bick, D. et al. 2012, PhRvL, 108, 51302
  * Bergström et al. (2016) Bergström, J., Gonzalez-Garcia, M. C., Maltoni, M., et al. 2016, JHEP, 3, 132
  * Böhm-Vitense (1958) Böhm-Vitense, E. 1958, ZAp, 46, 108
  * Borexino Collaboration (2018) Borexino Collaboration, Agostini, M., Altenmüller, K., et al. 2018, Nature, 562, 505
  * Borexino Collaboration (2020) Borexino Collaboration, Agostini, M., Altenmüller, K., Appel, S., et al. 2020, Nature, 587, 577
  * Buldgen et al. (2017) Buldgen, G.,Salmon, S. J. A. J., Noels, A., Scuflaire, R., Dupret, M. A., Reese, D. R. 2017, MNRAS, 472, 751
  * Buldgen et al. (2019) Buldgen, G., Salmon, S. J. A. J., Noels, A., et al. 2019, A&A, 621, 33
  * Caffau et al. (2010) Caffau, E., Ludwig, H.-G., Bonifacio, P., Faraggiana, R., Steffen, M., Freytag, B., Kamp, I., Ayres, T. R. 2010, A&A, 514, A92
  * Caffau et al. (2011) Caffau, E., Ludwig, H.-G., Steffen, M., Freytag, B., Bonifacio, P. 2011, Solar Phys., 268, 255
  * Castro et al. (2007) Castro, M., Vauclair, S., & Richard, O. 2007, A&A, 463, 755
  * Chaboyer et al (1995) Chaboyer, B., Demarque, P., Pinsonneault, M. H. 1995, ApJ, 441, 865
  * Chaplin et al. (1996) Chaplin, W. J., Elsworth, Y., Howe, R., et al. 1996, Sol. Phys., 168, 1
  * Chaplin et al. (1999) Chaplin, W. J., Elsworth, Y., Isaak, G. R., Miller, B. A., & New, R. 1999b, MNRAS, 308, 424
  * Christensen-Dalsgaard et al. (1991) Christensen-Dalsgaard, J., Gough, D. O., & Thompson, M. J. 1991, ApJ, 378, 413
  * Christensen-Dalsgaard (2021) Christensen-Dalsgaard, J. 2021, Living Reviews in Solar Physics, 18, 2
  * Delahaye et al. (2016) Delahaye, F., Zwölf, C. M., Zeippen, C. J., Mendoza, C. 2016, JQSRT, 171, 66
  * Demarque et al. (2008) Demarque, P., Guenther, D. B., Li, L. H., Mazumdar, A., Straka, C. W. 2008, Ap&SS, 316, 31
  * Endal & Sofia (1978) Endal, A. S., & Sofia, S. 1978, ApJ, 220, 279
  * Ferguson et al. (2005) Ferguson, J. W., Alexander, D. R., Allard, F. et al. 2005, ApJ, 623, 585
  * García et al. (2011) García, R. A., Salabert, D., & Ballot, J. et al. 2011, JPhCS, 271, 2049
  * Grevesse & Sauval (1998) Grevesse, N., & Sauval, A. J. 1998, in Solar Composition and Its Evolution, ed. C. Fröhlich et al. (Dordrecht: Kluwer), 161 (GS98)
  * Guenther (1994) Guenther D. B., 1994, ApJ, 422, 400
  * Guzik et al. (2005) Guzik, J. A., Watson, L. S. & Cox, A. N. 2005, ApJ, 627, 1049
  * Guzik & Mussack (2010) Guzik, J. A., & Mussack, K. 2010, ApJ, 713, 1108
  * Hope et al. (2020) Hope, C., Bergemann, M., Bitsch, B., Serenelli, A. 2020, A&A, 641, A73
  * Iglesias & Rogers (1996) Iglesias, C., Rogers, F. J. 1996, ApJ, 464, 943
  * Jermyn et al. (2022) Jermyn, A. S., Anders, E. H., Lecoanet, D. & Cantiello, M. 2022, arXiv220600011
  * Joyce & Chaboyer (2018) Joyce, M., & Chaboyer, B. 2018, ApJ, 864, 99
  * Junghans et al. (2003) Junghans, A. R., et al. 2003, Phys. Rev. C, 68, 065803
  * Kawaler (1988) Kawaler, S. D. 1988, ApJ, 333, 236
  * Kippenhahn et al. (2012) Kippenhahn, R., Weigert, A., & Weiss, A. 2012, Stellar Structure and Evolution, (Heidelberg: Springer)
  * Krishna Swamy (1966) Krishna Swamy, K. S. 1966, ApJ, 145, 174
  * Le Pennec et al. (2015) Le Pennec, M., Turck-Chièze, S., Salmon, S., Blancard, C., Cossé, P., Faussurier, G., Mondet, G. 2015, ApJL, 813, L42
  * Lodders (2003) Lodders, K. 2003, ApJ, 591, 1220
  * Lodders et al. (2009) Lodders, K., Palme, H., Gail, H-P. 2009, LanB, 4, 712 (LPG09)
  * Lodders (2020) Lodders, K. 2020, Solar Elemental Abundances, in The Oxford Research Encyclopedia of Planetary Science, Oxford University Press, arXiv:1912.00844
  * Marcucci et al. (2000) Marcucci, L. E., Schiavilla, R., Viviani, M., Kievski, A., & Rosati, S. 2000, PhRvL, 84, 5959
  * Montalbán et al. (2004) Montalbán, J., Miglio, A., Noels, A., Grevesse, N., Di Mauro, M. P. 2004, in Helio- and Asteroseismology: Towards a Golden Future, Proc. of the SOHO 14 / GONG 2004 Workshop, ed. D. Danesy (ESA SP-559; Noordwijk: ESA), 574
  * Montalbán et al. (2006) Montalbán, J., Miglio, A., Theado, S., Noels, A., Grevesse, N. 2006, Commun. Asteroseismol., 147, 80
  * Pinsonneault et al. (1989) Pinsonneault, M. H., Kawaler, S. D., Sofia, S., & Demarqure, P. 1989, ApJ, 338, 424
  * Proffitt & Michaud (1991) Proffitt, C. R., & Michaud, G. 1991, ApJ, 380, 238
  * Rogers & Nayfonov (2002) Rogers, F., & Nayfonov, A. 2002, ApJ, 576, 1064
  * Roxburgh & Vorontsov (2003) Roxburgh, I. W., & Vorontsov, S. V. 2003, A&A, 411, 215
  * Salmon et al. (2021) Salmon, S.J.A.J., Buldgen, G., Noels, A., Eggenberger, P., Scuflaire, R., Meynet, G. 2021, A&A, 651, A106
  * Schiavilla et al. (1994) Schiavilla, R., Wiringa, R. B., Pandharipande, V. R., and Carlson, J. 1992, Phys. Rev. C, 45, 2628
  * Schramm & Shi (1994) Schramm, D. N., & Shi, X. 1994, NuPhS, 35, 321
  * Seaton (1987) Seaton, M. J. 1987, JPhB, 20, 6363
  * Serenelli & Basu (2010) Serenelli, A., & Basu, S. 2010, ApJ, 719, 865
  * Serenelli et al. (2009) Serenelli, A. M., Basu, S., Ferguson, J. W., & Asplund, M. 2009, ApJL, 705, L123
  * Serenelli et al. (2011) Serenelli, A., Haxton, W. C., Peña-Garay, C. 2011, ApJ, 743, 24
  * Schou et al. (1998) Schou, J., Antia, H. M., Basu, S. et al. 1998, ApJ, 505, 390
  * Spada et al. (2018) Spada, F., Demarque, P., Basu, S., Tanner, J. D. 2018, ApJ, 869, 135
  * Spada et al. (2019) Spada, F., & Demarque, P. 2019, MNRAS, 489, 4712
  * The Opacity Project Team (1995) The Opacity Project Team, 1995, The Opacity Project Vol. 1, Institute of Physics Publications, Bristol, UK
  * Thoul et al. (1994) Thoul, A. A., Bahcall, J. N., Loeb, A. 1994, ApJ, 421, 828
  * Turck-Chièze & Couvidat (2011) Turck-Chièze, S., & Couvidat, S. 2011, RPPh, 74, 6901
  * Turck-Chièze et al. (2010) Turck-Chièze, S., Palacios, A., Marques, J. P., Nghiem, P. A. P. 2010, ApJ, 715, 1539
  * Turck-Chièze et al. (2011) Turck-Chièze, S., Piau, L., & Couvidat, S. 2011, ApJL, 731, L29
  * Vorontsov et al. (2013) Vorontsov, S. V., Baturin, V. A., Ayukov, S. V., Gryaznov, V. K. 2013, MNRAS, 430, 1636
  * Vorontsov et al. (2014) Vorontsov, S. V., Baturin, V. A., Ayukov, S. V., Gryaznov, V. K. 2014, MNRAS, 441, 3296
  * Wolfs et al. (1989) Wolfs, F. L. H.,; Freedman, S. J., Nelson, J. E., Dewey, M. S., Greene, G. L. 1989, PhRvL, 63, 2721
  * Yang (2016) Yang, W. 2016, ApJ, 829, 68
  * Yang (2019) Yang, W., 2019, ApJ, 873, 18
  * Yang & Bi (2007) Yang, W. M., & Bi, S. L. 2007, ApJL, 658, L67
  * Zhang & Li (2012) Zhang, Q., & Li, Y. 2012, ApJ, 746, 50
  * Zhang et al. (2019) Zhang, Q. S., Li, Y., Christensen-Dalsgaard, J. 2019, ApJ, 881, 103
  * Zahn (1993) Zahn, J. P. 1993, in Astrophysical Fluid Dynamics, Les Houches XLVII, ed. J.-P. Zahn & J. Zinn-Justin (New York: Elsevier), 561

Table 1: Fundamental Parameters of Solar Models Constructed in Accordance with
Different Mixtures.

Model | $Y_{0}$ | $Z_{0}$ | $\alpha_{\rm{MLT}}$ | $\delta_{\rm ov}$ | $f_{0}$ | $\rho_{c}$ | $r_{cz}$ | $Y_{s}$ | $Z_{s}$ | $(Z/X)_{s}$ | $\Delta Y$ | $\Omega_{i}$
---|---|---|---|---|---|---|---|---|---|---|---|---
Opacity Tables Constructed with GS98 mixtures
GS98M | 0.2761 | 0.01940 | 2.1223 | 0 | 1.0 | 154.59 | 0.716 | 0.2455 | 0.0174 | 0.0237 | 0.0306 | 0
GS98Mr | 0.2735 | 0.01896 | 2.0754 | 0.05 | 1.0 | 154.11 | 0.715 | 0.2534 | 0.0170 | 0.0234 | 0.0201 | 10
GS98op | 0.2713 | 0.01894 | 2.1352 | 0 | 1.0 | 154.25 | 0.716 | 0.2414 | 0.0170 | 0.0230 | 0.0299 | 0
GS98opr | 0.2711 | 0.01890 | 2.0979 | 0.05 | 1.0 | 154.16 | 0.714 | 0.2512 | 0.0170 | 0.0232 | 0.0199 | 10
Opacity Tables Constructed with AAG21 mixtures
Aop01 | 0.2583 | 0.01557 | 2.0722 | 0 | 1.0 | 151.98 | 0.725 | 0.2283 | 0.0139 | 0.0184 | 0.0300 | 0
Aop02 | 0.2648 | 0.01661 | 2.0954 | 0 | 1.0 | 152.89 | 0.722 | 0.2347 | 0.0149 | 0.0199 | 0.0301 | 0
Aop01r | 0.2582 | 0.01554 | 2.0383 | 0.15 | 1.0 | 151.86 | 0.716 | 0.2389 | 0.0140 | 0.0187 | 0.0193 | 10
Aop02r | 0.2648 | 0.01659 | 2.0596 | 0.15 | 1.0 | 152.88 | 0.713 | 0.2453 | 0.0149 | 0.0202 | 0.0195 | 10
Aop12r | 0.2675 | 0.01717 | 2.1055 | 0.10 | 1.36 | 154.42 | 0.714 | 0.2413 | 0.0149 | 0.0201 | 0.0262 | 10
Aopal12r | 0.2690 | 0.01720 | 2.0872 | 0.10 | 1.36 | 154.27 | 0.714 | 0.2424 | 0.0149 | 0.0201 | 0.0266 | 10
Aop12ri | 0.2702 | 0.01722 | 2.0918 | 0.10 | 1.36 | 154.34 | 0.715 | 0.2436 | 0.0149 | 0.0202 | 0.0266 | 10
Opacity Tables Constructed with Caffau mixtures
Cop01 | 0.2666 | 0.01719 | 2.0924 | 0 | 1.0 | 152.87 | 0.720 | 0.2365 | 0.0154 | 0.0206 | 0.0301 | 0
Cop01r | 0.2664 | 0.01714 | 2.0582 | 0.12 | 1.0 | 152.76 | 0.714 | 0.2467 | 0.0154 | 0.0209 | 0.0197 | 10
Cop11r | 0.2695 | 0.01780 | 2.1056 | 0.10 | 1.36 | 154.42 | 0.711 | 0.2431 | 0.01548 | 0.0209 | 0.0264 | 10
Cop11ri | 0.2717 | 0.01777 | 2.0883 | 0.10 | 1.36 | 154.29 | 0.714 | 0.2450 | 0.0154 | 0.0209 | 0.0267 | 10
Copal11r | 0.2717 | 0.01784 | 2.0817 | 0.10 | 1.36 | 154.25 | 0.713 | 0.2450 | 0.01548 | 0.0209 | 0.0267 | 10
Copal11ri | 0.2720 | 0.01774 | 2.0865 | 0.10 | 1.36 | 154.46 | 0.712 | 0.2453 | 0.0154 | 0.0209 | 0.0267 | 10

Note. — The central density $\rho_{c}$, CZ radius $r_{cz}$, and initial
angular velocity $\Omega_{i}$ are in units of g cm-3, $R_{\odot}$, and
$10^{-6}$ rad s-1, respectively. The quantity $\Delta Y=Y_{0}-Y_{s}$ is the
amount of surface helium settling.

Table 2: Predicted and Measured Solar Neutrino Fluxes.

model | $T_{c}$ | $pp$ | $pep$ | $hep$ |  7Be |  8B |  13N |  15O |  17F
---|---|---|---|---|---|---|---|---|---
BP04 | … | 5.94 | 1.40 | 7.88 | 4.86 | 5.79 | 5.71 | 5.03 | 5.91
SSeM | … | 5.92 | 1.39 | …. | 4.85 | 4.98 | 5.77 | 4.97 | 3.08
Measured | … | 6.06${}^{+0.02a}_{-0.06}$ | 1.6$\pm$0.3b | … | 4.84$\pm$0.24a | 5.21$\pm$0.27c | … | … | …
B16d | … | 5.97${}^{+0.04}_{-0.03}$ | 1.448$\pm$0.013 | 19${}^{+12}_{-9}$ | 4.80${}^{+0.24}_{-0.22}$ | 5.16${}^{+0.13}_{-0.09}$ |  $\leq$13.7 |  $\leq$2.8 |  $\leq$85
Borexinoe | … | 6.1$\pm$0.5 | 1.39$\pm$0.19 |  $<$220 | 4.99$\pm$0.11 | 5.68${}^{+0.39}_{-0.41}$ | 7${}^{+3}_{-2}$ |
GS98M | 15.777 | 5.95 | 1.40 | 9.69 | 5.09 | 5.59 | 5.40 | 4.77 | 5.52
GS98Mr | 15.733 | 5.96 | 1.41 | 9.73 | 4.95 | 5.29 | 5.08 | 4.45 | 5.14
GS98op | 15.706 | 5.98 | 1.42 | 9.81 | 4.92 | 5.18 | 4.97 | 4.35 | 5.02
GS98opr | 15.693 | 5.97 | 1.41 | 9.78 | 4.86 | 5.07 | 4.90 | 4.28 | 4.93
Aop01 | 15.513 | 6.05 | 1.45 | 10.19 | 4.42 | 4.12 | 3.43 | 2.91 | 3.32
Aop02 | 15.604 | 6.02 | 1.43 | 9.79 | 4.66 | 4.60 | 3.97 | 3.43 | 3.93
Aop01r | 15.498 | 6.03 | 1.44 | 10.15 | 4.36 | 4.01 | 3.37 | 2.85 | 3.25
Aop02r | 15.598 | 6.01 | 1.43 | 9.99 | 4.63 | 4.55 | 3.94 | 3.40 | 3.89
Aop12r | 15.680 | 5.97 | 1.42 | 9.84 | 4.82 | 4.98 | 4.45 | 3.88 | 4.47
Aopal12r | 15.706 | 5.97 | 1.42 | 9.81 | 4.87 | 5.09 | 4.54 | 3.96 | 4.57
Aop12ri | 15.715 | 5.96 | 1.41 | 9.77 | 4.91 | 5.19 | 4.61 | 4.04 | 4.66
Cop01 | 15.645 | 6.00 | 1.42 | 9.95 | 4.75 | 4.82 | 4.25 | 3.69 | 4.24
Cop01r | 15.623 | 5.98 | 1.42 | 9.92 | 4.66 | 4.65 | 4.16 | 3.59 | 4.12
Cop11r | 15.716 | 5.95 | 1.41 | 9.77 | 4.89 | 5.16 | 4.73 | 4.15 | 4.79
Cop11ri | 15.751 | 5.94 | 1.40 | 9.70 | 4.99 | 5.38 | 4.90 | 4.32 | 4.99
Copal11r | 15.761 | 5.95 | 1.41 | 9.71 | 4.99 | 5.41 | 4.93 | 4.34 | 5.03
Copal11ri | 15.764 | 5.95 | 1.41 | 9.72 | 5.00 | 5.44 | 4.92 | 4.34 | 5.02

Note. — The table shows the predicted fluxes, in units of $10^{10}(pp)$,
$10^{9}(^{7}\mathrm{Be})$, $10^{8}(pep,^{13}\mathrm{N},^{15}\mathrm{O})$,
$10^{6}(^{8}\mathrm{B},^{17}\mathrm{F})$, and $10^{3}(hep)$ $\mathrm{cm}^{2}$
$\mathrm{s}^{-1}$. The central temperature $T_{c}$ of models is in unit of
$10^{6}$ K. The BP04 is the best model of Bahcall & Pinsonneault (2004) and
has the GS98 mixtures. The SSeM is the best seismic model with high metal
abundances that can reproduce inferred sound speed (Turck-Chièze & Couvidat,
2011). aafootnotetext: Bellini et al. (2011). bbfootnotetext: Bellini et al.
(2012). ccfootnotetext: Ahmed et al. (2004). ddfootnotetext: Bergström et al.
(2016). eefootnotetext: The total fluxes produced by CNO cycle are given in
Borexino Collaboration (2020), other neutrino fluxes are given in Borexino
Collaboration (2018).

Table 3: Physical Configurations of Models.

Model Name | Mixtures | Opacity | Enhanced Diffusion |  $Z_{s}$ value | Rotation | Increased Opacity ($k_{0}$)a
---|---|---|---|---|---|---
GS98M | GS98 | OPAL | No | GS98 | No | No
GS98Mr | GS98 | OPAL | No | GS98 | Yes | No
GS98op | GS98 | OP | No | GS98 | No | No
GS98opr | GS98 | OP | No | GS98 | Yes | No
Aop01 | AAG21b | OP | No | AAG21 | No | No
Aop02 | AAG21 | OP | No | Lodders 2020 | No | No
Aop01r | AAG21 | OP | No | AAG21 | Yes | No
Aop02r | AAG21 | OP | No | Lodders 2020 | Yes | No
Aop12r | AAG21 | OP | Yes | Lodders 2020 | Yes | No
Aop12ri | AAG21 | OP | Yes | Lodders 2020 | Yes | Yes($1\leq k_{0}\leq 1.015$)
Aopal12r | AAG21 | OPAL | Yes | Lodders 2020 | Yes | No
Cop01 | Caffauc | OP | No | Caffau | No | No
Cop01r | Caffau | OP | No | Caffau | Yes | No
Cop11r | Caffau | OP | Yes | Caffau | Yes | No
Cop11ri | Caffau | OP | Yes | Caffau | Yes | Yes($1\leq k_{0}\leq 1.015$)
Copal11r | Caffau | OPAL | Yes | Caffau | Yes | No
Copal11ri | Caffau | OPAL | Yes | Caffau | Yes | Yes($1\leq k_{0}\leq 1.023$)

Note. — a See the Appendix A. bSee the Table 5 in Appendix A and Asplund et
al. (2021). cSee the Table 5 and Caffau et al. (2010, 2011).

Table 4: The Values of $\chi^{2}$ of Different Models.

Model | $\chi^{2\ a}_{c_{s}+\rho}$ | $\chi^{2\ b}_{d_{02+13}}$ | $\chi^{2\ c}_{Borexino-neutrino}$ | $\chi^{2\ d}_{\rm helium}$
---|---|---|---|---
GS98M | 882.4 | 1.2 | 0.53 | 0.7
GS98Mr | 799.2 | 1.4 | 0.49 | 2.0
GS98op | 578.2 | 1.8 | 0.61 | 4.1
GS98opr | 369.6 | 1.5 | 0.88 | 0.6
Aop01 | 6244.7 | 19.0 | 7.08 | 33.3
Aop02 | 3144.1 | 16.2 | 2.85 | 15.5
Aop01r | 4424.0 | 9.0 | 8.43 | 7.5
Aop02r | 1983.0 | 7.0 | 3.23 | 0.8
Aop12r | 389.1 | 1.7 | 1.11 | 4.2
Aopal12r | 917.4 | 1.6 | 0.78 | 3.0
Aop12ri | 524.3 | 2.0 | 0.57 | 2.0
Cop01 | 2440.7 | 8.3 | 1.72 | 11.8
Cop01r | 1402.2 | 6.5 | 2.75 | 0.3
Cop11r | 323.7 | 1.6 | 0.67 | 2.4
Cop11ri | 369.7 | 1.8 | 0.38 | 1.0
Copal11r | 590.6 | 1.5 | 0.37 | 1.0
Copal11ri | 342.6 | 1.6 | 0.35 | 0.8

Note. — The function $\chi^{2}$ defined as
$\chi^{2}=\frac{1}{N}\sum^{N}_{i=1}\frac{(q_{\rm ob,i}-q_{\rm
th,i})^{2}}{\sigma^{2}_{i}}$, where $q_{\rm ob,i}$ and $q_{\rm th,i}$ are the
observed/inferred and theoretical values of quantities $q_{\rm i}$,
respectively; $\sigma_{i}$ are the errors associated to the corresponding
observed/inferred quantities; $N$ is the number of the quantities.
aafootnotetext: The inferred $c_{s}$ and $\rho$ are given in Basu et al.
(2009). bbfootnotetext: The observed $d_{02}$ and $d_{13}$ are calculated from
the frequencies given by Chaplin et al. (1999). ccfootnotetext: In order to
calculate the value of $\chi^{2}_{\rm Borexino-neutrino}$, the
$\Phi(hep)<220\times 10^{3}$ $\mathrm{cm}^{2}$ $\mathrm{s}^{-1}$ of Borexino
Collaboration (2018) was replaced by $\Phi(hep)=19^{+12}_{-9}\times 10^{3}$
$\mathrm{cm}^{2}$ $\mathrm{s}^{-1}$ of Bergström et al. (2016).
ddfootnotetext: The inferred helium is $0.2485\pm 0.0035$ (Basu & Antia,
2004).

##  Appendix A Opacity Tables

The low-temperature opacity tables with the mixtures listed in Table 5 were
downloaded from
https://www.wichita.edu/academics/fairmount_college_of_liberal_arts_and_sciences/physics/Research/opacity.php.
The OPAL and OP opacity tables with the same mixtures were computed from
https://opalopacity.llnl.gov/type1inp.html and http://op-
opacity.obspm.fr/opacity/, respectively. The OP opacities did not consider the
contributions of P, Cl, K, and Ti compared to OPAL opacities. The
contributions of the missing elements (P, Cl, K, and Ti) were automatically
transferred to other elements (see the note of the website of OP). The opacity
tables, used in this work, are also available at the author’s github
repository at https://github.com/yangwuming/SUN.

The low-temperature opacity tables were used in the low-temperature region
with $\lg T<4.0$, and OPAL or OP tables were used in the high-temperature
region with $\lg T>4.1$. In the region with $4.0\leq\lg T\leq 4.1$, the
opacity was linearly interpolated between the high-temperature and low-
temperature opacities. Moreover, we corrected the wrong CALL YALO2D in the
subroutine YALO3D2 of YREC, which mainly affects the value of mixing-length
parameter $\alpha_{\rm{MLT}}$ (lead to a small decrease in
$\alpha_{\rm{MLT}}$). For models Aop12ri, Copal11ri, and Cop11ri, we applied a
straight multiplier $k_{0}$ to opacity $\kappa$ to enhance the opacity in a
given region (see Section 3.3).

Table 5:  Fractional Abundances of Heavy Elements Used to Construct Opacity
Tables.

Element | Number fraction | Number fraction
---|---|---
| (Caffau’s mixtures) | (AAG21 mixtures)
C | 0.260408 | 0.266509
N | 0.059656 | 0.062476
O | 0.473865 | 0.452597
Ne | 0.096751 | 0.106099
Na | 0.001681 | 0.001534
Mg | 0.029899 | 0.032788
Al | 0.002487 | 0.002487
Si | 0.029218 | 0.029903
P | 0.000237 | 0.000238
S | 0.011632 | 0.012182
Cl | 0.000150 | 0.000189
Ar | 0.002727 | 0.002217
K | 0.000106 | 0.000109
Ca | 0.001760 | 0.001844
Ti | 0.000072 | 0.000086
Cr | 0.000385 | 0.000385
Mn | 0.000266 | 0.000243
Fe | 0.027268 | 0.026651
Ni | 0.001431 | 0.001465

Note. — The number fraction of Caffau’s mixtures were listed by Ferguson in
the WSU low temperature opacities, while those of AAG21 mixtures were
calculated by using the data in Table 2 of Asplund et al. (2021).

##  Appendix B Nuclear reactions

The nuclear reaction rates in conjunction with the corresponding energy
release of three alternative $pp$ branches ($pp1,pp2,pp3$), CNO cycle, and
helium burning are calculated in YREC. The sequence of the reactions is
explicitly listed in Demarque et al. (2008). The standard reaction rates
implemented are identical to the rates published in Bahcall (1989). The
calculated energy generation and neutrino fluxes are dependent on Q-values and
nuclear cross-section factors $S_{0}$, respectively. The default Q-values are
taken from the Table 21 of Bahcall & Ulrich (1988) and not changed. The
default values of nuclear cross-section factors $S_{0}$ are taken from the
Tables 3.2 and 3.4 of Bahcall (1989). Compared with the default and those
adopted by Bahcall & Pinsonneault (1992), we adopted different $S_{0}(hep)$
and $S_{0}(^{7}\rm{Be}+p)$, which are shown in Table 6. The value of $0.0202$
keV barns of $S_{0}(^{7}\rm{Be}+p)$ (Schramm & Shi, 1994) is in agreement with
$0.0205\pm 0.0007$ keV barns of Baby et al. (2003) and $0.0208\pm 0.0007$ keV
barns of Adelberger et al. (2011). Moreover, if $S_{0}(hep)=15.3\times
10^{-20}$ keV barns (Wolfs et al., 1989) is adopted, the value of $hep$ flux
of model Copal11r is $14.4\times 10^{3}$ cm2 s-1, which is in better agreement
with that determined by Bergström et al. (2016). The changes in $S_{0}(hep)$
and $S_{0}(^{7}\rm{Be}+p)$ affect, respectively, only the $hep$ and 8B fluxes
(Bahcall & Pinsonneault, 2004). All models were computed by using the same
Q-values and $S_{0}$.

Table 6: Some of Nuclear Cross-section Factors $S_{0}$ (keV barns).

Reaction | Defaulta | BP92 | BP04 | This work
---|---|---|---|---
1H($p$, e${}^{+}\nu_{e}$ ) 2H | $4.07\times 10^{-22}$ | $4.00^{+0.06}_{-0.04}\times 10^{-22}$ | $3.94(1\pm 0.004)\times 10^{-22}$ | $4.00\times 10^{-22}$
1H($p+e^{-}$, $\nu_{e}$) 2H | Eq. (3.17)a | Eq. (3.17)a | Eq. (3.17)a | Eq. (3.17)a
3He(3He, $2p$) 4He | $5.15\times 10^{3}$ | $5.00(1\pm 0.06)\times 10^{3}$ | $5.4\times 10^{3}$ | $5.00\times 10^{3}$
3He(4He, $\gamma$) 7Be | 0.54 | $0.533(1\pm 0.032)$ | 0.533 | 0.533
7Be($e^{-}$, $\nu_{e}$) 7Li | Eq. (3.18)a | Eq. (3.18)a | Eq. (26)b | Eq. (3.18)a
7Be($p$, $\gamma$) 8B | 0.0243 | $0.0224(1\pm 0.093)$ | 0.0206$\pm 0.0008^{c}$ | 0.0202d
3He($p$, e${}^{+}\nu_{e}$ )4He | $8\times 10^{-20}$ | $1.30\times 10^{-20}$ | $(8.6\pm 1.3)\times 10^{-20}$ | $2.30\times 4.5^{e}\times 10^{-20}$

Note. — aBahcall (1989). bThe equation of Adelberger et al. (1998).
${}^{c}S_{20\ \rm{keV}}(^{7}\rm{Be}+p)$ given by Junghans et al. (2003). dThe
value given by Schramm & Shi (1994). eThe value of $S_{0}(hep)$ given by
Schiavilla et al. (1994) is $2.30\times 10^{-20}$, which should be multiplied
by a factor of $4.5$ (Marcucci et al., 2000).

##  Appendix C Numerical Solar Model and Data Availability

Table 7 presents a basic numerical description of model Copal11r. All models
and data used in this work are availabe at the author’s github repository at
https://github.com/yangwuming/SUN. In the old YREC, opacity was calculated by
linearly interpolating between the opacity of a given $Z_{1}$ and that of
another given $Z_{2}$ for heavy-element diffusion. We modified the code to
linearly interpolate between a series of $Z$ values (with $\delta Z=0.001$)
for the heavy-element diffusion. The amended YREC package, including pulsation
code, are available from the author at<EMAIL_ADDRESS>
Table 7: Numerical Description of Model Copal11r.  M/M⊙ | r/R⊙ | L/L⊙ | T (K) |  $\rho$ (g cm-3) | p (dyn cm-2) |  $\kappa$ (cm2 g-1) | X | Y | Z |  $\Omega$ (10-6 rad s-1)
---|---|---|---|---|---|---|---|---|---|---
0.000001 | 0.00190 | 0.00001 | 1.576e+07 | 1.542e+02 | 2.368e+17 | 1.2914e+00 | 0.3364 | 0.6444 | 0.0192 | 1.50e-05
0.000002 | 0.00263 | 0.00002 | 1.576e+07 | 1.542e+02 | 2.367e+17 | 1.2916e+00 | 0.3365 | 0.6443 | 0.0192 | 1.50e-05
0.000003 | 0.00310 | 0.00003 | 1.576e+07 | 1.541e+02 | 2.367e+17 | 1.2917e+00 | 0.3366 | 0.6442 | 0.0192 | 1.50e-05
0.000005 | 0.00365 | 0.00005 | 1.576e+07 | 1.541e+02 | 2.366e+17 | 1.2918e+00 | 0.3367 | 0.6441 | 0.0192 | 1.50e-05
0.000009 | 0.00430 | 0.00008 | 1.576e+07 | 1.540e+02 | 2.366e+17 | 1.2920e+00 | 0.3369 | 0.6439 | 0.0192 | 1.50e-05
0.000014 | 0.00507 | 0.00013 | 1.575e+07 | 1.539e+02 | 2.364e+17 | 1.2922e+00 | 0.3371 | 0.6437 | 0.0192 | 1.50e-05
0.000023 | 0.00597 | 0.00021 | 1.575e+07 | 1.538e+02 | 2.363e+17 | 1.2926e+00 | 0.3375 | 0.6434 | 0.0192 | 1.50e-05
0.000038 | 0.00704 | 0.00034 | 1.574e+07 | 1.537e+02 | 2.361e+17 | 1.2931e+00 | 0.3379 | 0.6430 | 0.0191 | 1.50e-05
0.000062 | 0.00829 | 0.00055 | 1.574e+07 | 1.534e+02 | 2.357e+17 | 1.2938e+00 | 0.3385 | 0.6424 | 0.0191 | 1.50e-05
0.000102 | 0.00977 | 0.00089 | 1.573e+07 | 1.531e+02 | 2.353e+17 | 1.2947e+00 | 0.3393 | 0.6415 | 0.0191 | 1.50e-05
0.000166 | 0.01152 | 0.00146 | 1.571e+07 | 1.527e+02 | 2.347e+17 | 1.2960e+00 | 0.3405 | 0.6403 | 0.0191 | 1.50e-05
0.000272 | 0.01358 | 0.00238 | 1.570e+07 | 1.521e+02 | 2.339e+17 | 1.2979e+00 | 0.3421 | 0.6387 | 0.0191 | 1.50e-05
0.000444 | 0.01601 | 0.00387 | 1.567e+07 | 1.512e+02 | 2.328e+17 | 1.3004e+00 | 0.3444 | 0.6365 | 0.0191 | 1.50e-05
0.000726 | 0.01889 | 0.00629 | 1.564e+07 | 1.501e+02 | 2.312e+17 | 1.3039e+00 | 0.3475 | 0.6334 | 0.0191 | 1.49e-05
0.001187 | 0.02230 | 0.01022 | 1.559e+07 | 1.485e+02 | 2.291e+17 | 1.3087e+00 | 0.3517 | 0.6292 | 0.0191 | 1.49e-05
0.001939 | 0.02634 | 0.01655 | 1.552e+07 | 1.463e+02 | 2.261e+17 | 1.3149e+00 | 0.3576 | 0.6233 | 0.0191 | 1.48e-05
0.003168 | 0.03116 | 0.02669 | 1.543e+07 | 1.434e+02 | 2.221e+17 | 1.3236e+00 | 0.3656 | 0.6153 | 0.0191 | 1.48e-05
0.005179 | 0.03691 | 0.04284 | 1.530e+07 | 1.394e+02 | 2.166e+17 | 1.3357e+00 | 0.3766 | 0.6044 | 0.0191 | 1.47e-05
0.008465 | 0.04382 | 0.06827 | 1.512e+07 | 1.342e+02 | 2.092e+17 | 1.3526e+00 | 0.3914 | 0.5895 | 0.0190 | 1.45e-05
0.013830 | 0.05217 | 0.10770 | 1.488e+07 | 1.274e+02 | 1.993e+17 | 1.3763e+00 | 0.4114 | 0.5696 | 0.0190 | 1.43e-05
0.022608 | 0.06237 | 0.16760 | 1.455e+07 | 1.187e+02 | 1.863e+17 | 1.4098e+00 | 0.4378 | 0.5433 | 0.0189 | 1.41e-05
0.036949 | 0.07497 | 0.25560 | 1.410e+07 | 1.079e+02 | 1.694e+17 | 1.4577e+00 | 0.4720 | 0.5092 | 0.0188 | 1.37e-05
0.056968 | 0.08871 | 0.36217 | 1.358e+07 | 9.657e+01 | 1.509e+17 | 1.5169e+00 | 0.5091 | 0.4722 | 0.0187 | 1.33e-05
0.080715 | 0.10213 | 0.46885 | 1.304e+07 | 8.624e+01 | 1.333e+17 | 1.5799e+00 | 0.5433 | 0.4380 | 0.0186 | 1.30e-05
0.109607 | 0.11617 | 0.57552 | 1.248e+07 | 7.636e+01 | 1.161e+17 | 1.6526e+00 | 0.5758 | 0.4057 | 0.0185 | 1.26e-05
0.146362 | 0.13193 | 0.68219 | 1.185e+07 | 6.642e+01 | 9.836e+16 | 1.7429e+00 | 0.6072 | 0.3744 | 0.0184 | 1.22e-05
0.196727 | 0.15132 | 0.78841 | 1.110e+07 | 5.573e+01 | 7.918e+16 | 1.8647e+00 | 0.6383 | 0.3434 | 0.0183 | 1.18e-05
0.254970 | 0.17201 | 0.87065 | 1.033e+07 | 4.595e+01 | 6.194e+16 | 2.0069e+00 | 0.6631 | 0.3188 | 0.0182 | 1.14e-05
0.312918 | 0.19172 | 0.92300 | 9.650e+06 | 3.798e+01 | 4.845e+16 | 2.1549e+00 | 0.6810 | 0.3010 | 0.0181 | 1.04e-05
0.369383 | 0.21066 | 0.95531 | 9.044e+06 | 3.149e+01 | 3.790e+16 | 2.3107e+00 | 0.6909 | 0.2911 | 0.0180 | 9.57e-06
0.423543 | 0.22902 | 0.97493 | 8.501e+06 | 2.610e+01 | 2.965e+16 | 2.4761e+00 | 0.6975 | 0.2846 | 0.0179 | 9.20e-06
0.474881 | 0.24694 | 0.98679 | 8.011e+06 | 2.160e+01 | 2.319e+16 | 2.6534e+00 | 0.7022 | 0.2800 | 0.0179 | 8.80e-06
0.523093 | 0.26456 | 0.99366 | 7.567e+06 | 1.786e+01 | 1.814e+16 | 2.8464e+00 | 0.7054 | 0.2768 | 0.0178 | 8.40e-06
0.568053 | 0.28198 | 0.99705 | 7.162e+06 | 1.474e+01 | 1.419e+16 | 3.0574e+00 | 0.7078 | 0.2745 | 0.0177 | 8.03e-06
0.609743 | 0.29929 | 0.99852 | 6.789e+06 | 1.215e+01 | 1.110e+16 | 3.2874e+00 | 0.7098 | 0.2725 | 0.0177 | 7.66e-06
0.648210 | 0.31657 | 0.99923 | 6.446e+06 | 1.001e+01 | 8.682e+15 | 3.5379e+00 | 0.7115 | 0.2709 | 0.0177 | 7.31e-06
0.683576 | 0.33387 | 0.99960 | 6.126e+06 | 8.229e+00 | 6.792e+15 | 3.8109e+00 | 0.7129 | 0.2695 | 0.0176 | 6.96e-06
0.715985 | 0.35125 | 0.99980 | 5.829e+06 | 6.762e+00 | 5.312e+15 | 4.1078e+00 | 0.7141 | 0.2683 | 0.0176 | 6.64e-06
0.745591 | 0.36875 | 0.99991 | 5.550e+06 | 5.552e+00 | 4.156e+15 | 4.4296e+00 | 0.7151 | 0.2673 | 0.0176 | 6.32e-06
0.772576 | 0.38642 | 0.99997 | 5.287e+06 | 4.556e+00 | 3.251e+15 | 4.7782e+00 | 0.7161 | 0.2664 | 0.0175 | 6.02e-06
0.797116 | 0.40428 | 1.00000 | 5.040e+06 | 3.737e+00 | 2.543e+15 | 5.1556e+00 | 0.7169 | 0.2656 | 0.0175 | 5.74e-06
0.819381 | 0.42237 | 1.00001 | 4.806e+06 | 3.064e+00 | 1.989e+15 | 5.5644e+00 | 0.7177 | 0.2648 | 0.0175 | 5.46e-06
0.839548 | 0.44069 | 1.00001 | 4.584e+06 | 2.511e+00 | 1.556e+15 | 6.0050e+00 | 0.7186 | 0.2640 | 0.0174 | 5.21e-06
0.857782 | 0.45929 | 1.00001 | 4.373e+06 | 2.058e+00 | 1.217e+15 | 6.4815e+00 | 0.7194 | 0.2632 | 0.0174 | 4.96e-06
0.874233 | 0.47816 | 1.00000 | 4.172e+06 | 1.686e+00 | 9.520e+14 | 7.0002e+00 | 0.7203 | 0.2624 | 0.0174 | 4.73e-06
0.889056 | 0.49732 | 1.00000 | 3.980e+06 | 1.382e+00 | 7.447e+14 | 7.5656e+00 | 0.7212 | 0.2614 | 0.0174 | 4.51e-06
0.902387 | 0.51678 | 0.99999 | 3.795e+06 | 1.132e+00 | 5.825e+14 | 8.1805e+00 | 0.7223 | 0.2604 | 0.0173 | 4.30e-06
0.914354 | 0.53654 | 0.99998 | 3.618e+06 | 9.283e-01 | 4.557e+14 | 8.8490e+00 | 0.7234 | 0.2593 | 0.0173 | 4.11e-06
0.925080 | 0.55659 | 0.99998 | 3.446e+06 | 7.614e-01 | 3.564e+14 | 9.5772e+00 | 0.7247 | 0.2580 | 0.0173 | 3.92e-06
0.934675 | 0.57691 | 0.99997 | 3.280e+06 | 6.251e-01 | 2.788e+14 | 1.0375e+01 | 0.7262 | 0.2565 | 0.0173 | 3.75e-06
0.943239 | 0.59749 | 0.99997 | 3.118e+06 | 5.137e-01 | 2.181e+14 | 1.1257e+01 | 0.7278 | 0.2549 | 0.0173 | 3.58e-06
0.950869 | 0.61828 | 0.99996 | 2.958e+06 | 4.230e-01 | 1.706e+14 | 1.2250e+01 | 0.7296 | 0.2531 | 0.0173 | 3.42e-06
0.957649 | 0.63923 | 0.99996 | 2.799e+06 | 3.491e-01 | 1.334e+14 | 1.3427e+01 | 0.7316 | 0.2510 | 0.0174 | 3.28e-06
0.963657 | 0.66025 | 0.99997 | 2.637e+06 | 2.894e-01 | 1.044e+14 | 1.5065e+01 | 0.7338 | 0.2485 | 0.0177 | 3.14e-06
0.968963 | 0.68119 | 0.99997 | 2.467e+06 | 2.417e-01 | 8.165e+13 | 1.6387e+01 | 0.7360 | 0.2467 | 0.0173 | 3.02e-06
0.973526 | 0.70160 | 0.99998 | 2.295e+06 | 2.041e-01 | 6.424e+13 | 1.7759e+01 | 0.7382 | 0.2454 | 0.0164 | 2.92e-06
0.976002 | 0.71333 | 1.00022 | 2.185e+06 | 1.854e-01 | 5.560e+13 | 1.8426e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.977337 | 0.72008 | 1.00037 | 2.114e+06 | 1.764e-01 | 5.118e+13 | 1.9703e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.980936 | 0.73943 | 1.00030 | 1.918e+06 | 1.522e-01 | 4.004e+13 | 2.3806e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.984045 | 0.75784 | 1.00024 | 1.741e+06 | 1.313e-01 | 3.132e+13 | 2.8373e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.986711 | 0.77528 | 1.00019 | 1.579e+06 | 1.133e-01 | 2.450e+13 | 3.3468e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.988982 | 0.79177 | 1.00016 | 1.433e+06 | 9.775e-02 | 1.916e+13 | 3.9246e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.990904 | 0.80729 | 1.00013 | 1.300e+06 | 8.435e-02 | 1.499e+13 | 4.5830e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.992521 | 0.82188 | 1.00011 | 1.180e+06 | 7.278e-02 | 1.173e+13 | 5.3219e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.993874 | 0.83555 | 1.00009 | 1.070e+06 | 6.280e-02 | 9.173e+12 | 6.1516e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.995001 | 0.84832 | 1.00008 | 9.710e+05 | 5.419e-02 | 7.175e+12 | 7.0972e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.995933 | 0.86022 | 1.00007 | 8.811e+05 | 4.677e-02 | 5.613e+12 | 8.2430e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.996702 | 0.87130 | 1.00006 | 7.995e+05 | 4.036e-02 | 4.390e+12 | 9.6834e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.997334 | 0.88158 | 1.00006 | 7.254e+05 | 3.483e-02 | 3.434e+12 | 1.1441e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.997850 | 0.89110 | 1.00005 | 6.583e+05 | 3.006e-02 | 2.686e+12 | 1.3527e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.998271 | 0.89991 | 1.00005 | 5.975e+05 | 2.594e-02 | 2.101e+12 | 1.5967e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.998613 | 0.90804 | 1.00005 | 5.423e+05 | 2.239e-02 | 1.644e+12 | 1.8885e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.998889 | 0.91554 | 1.00005 | 4.923e+05 | 1.932e-02 | 1.286e+12 | 2.2482e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999113 | 0.92244 | 1.00005 | 4.469e+05 | 1.668e-02 | 1.006e+12 | 2.7065e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999292 | 0.92878 | 1.00005 | 4.058e+05 | 1.439e-02 | 7.867e+11 | 3.3250e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999436 | 0.93461 | 1.00005 | 3.685e+05 | 1.242e-02 | 6.154e+11 | 4.1670e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999552 | 0.93995 | 1.00004 | 3.348e+05 | 1.072e-02 | 4.814e+11 | 5.3263e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999644 | 0.94484 | 1.00004 | 3.042e+05 | 9.247e-03 | 3.765e+11 | 6.9405e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999718 | 0.94933 | 1.00004 | 2.765e+05 | 7.979e-03 | 2.946e+11 | 9.2024e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999777 | 0.95343 | 1.00004 | 2.514e+05 | 6.883e-03 | 2.304e+11 | 1.2436e+03 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999823 | 0.95718 | 1.00004 | 2.288e+05 | 5.936e-03 | 1.802e+11 | 1.7082e+03 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999860 | 0.96060 | 1.00004 | 2.083e+05 | 5.118e-03 | 1.410e+11 | 2.3811e+03 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999890 | 0.96373 | 1.00004 | 1.898e+05 | 4.411e-03 | 1.103e+11 | 3.3552e+03 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999913 | 0.96659 | 1.00004 | 1.731e+05 | 3.798e-03 | 8.626e+10 | 4.7501e+03 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999931 | 0.96921 | 1.00004 | 1.581e+05 | 3.269e-03 | 6.748e+10 | 6.6841e+03 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999946 | 0.97159 | 1.00004 | 1.447e+05 | 2.810e-03 | 5.278e+10 | 9.2032e+03 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999957 | 0.97378 | 1.00004 | 1.326e+05 | 2.412e-03 | 4.129e+10 | 1.2213e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999967 | 0.97578 | 1.00004 | 1.217e+05 | 2.068e-03 | 3.230e+10 | 1.5468e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999974 | 0.97761 | 1.00004 | 1.118e+05 | 1.772e-03 | 2.526e+10 | 1.8710e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999979 | 0.97930 | 1.00004 | 1.028e+05 | 1.516e-03 | 1.976e+10 | 2.1823e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999984 | 0.98084 | 1.00004 | 9.461e+04 | 1.298e-03 | 1.546e+10 | 2.4907e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999987 | 0.98225 | 1.00004 | 8.699e+04 | 1.111e-03 | 1.209e+10 | 2.8321e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999990 | 0.98355 | 1.00004 | 7.994e+04 | 9.521e-04 | 9.458e+09 | 3.2562e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999994 | 0.98581 | 1.00004 | 6.744e+04 | 7.004e-04 | 5.788e+09 | 4.5889e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999996 | 0.98771 | 1.00004 | 5.714e+04 | 5.143e-04 | 3.541e+09 | 6.9908e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999998 | 0.98929 | 1.00004 | 4.894e+04 | 3.749e-04 | 2.167e+09 | 1.0278e+05 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999999 | 0.99064 | 1.00004 | 4.252e+04 | 2.701e-04 | 1.326e+09 | 1.2664e+05 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999999 | 0.99179 | 1.00004 | 3.746e+04 | 1.923e-04 | 8.113e+08 | 1.2530e+05 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999999 | 0.99231 | 1.00004 | 3.531e+04 | 1.616e-04 | 6.346e+08 | 1.1637e+05 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
0.999999 | 0.99279 | 1.00004 | 3.337e+04 | 1.355e-04 | 4.964e+08 | 1.0456e+05 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
1.000000 | 0.99354 | 1.00004 | 3.044e+04 | 9.966e-05 | 3.255e+08 | 8.2105e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
1.000000 | 0.99444 | 1.00004 | 2.716e+04 | 6.534e-05 | 1.845e+08 | 5.6764e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
1.000000 | 0.99594 | 1.00004 | 2.214e+04 | 2.646e-05 | 5.699e+07 | 2.3956e+04 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
1.000000 | 0.99786 | 1.00004 | 1.643e+04 | 4.852e-06 | 6.879e+06 | 4.6756e+03 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
1.000000 | 0.99837 | 1.00004 | 1.495e+04 | 2.667e-06 | 3.292e+06 | 2.5776e+03 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
1.000000 | 0.99881 | 1.00004 | 1.360e+04 | 1.469e-06 | 1.576e+06 | 1.3462e+03 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
1.000000 | 0.99919 | 1.00004 | 1.230e+04 | 8.173e-07 | 7.542e+05 | 6.2276e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
1.000000 | 0.99952 | 1.00004 | 1.090e+04 | 4.663e-07 | 3.610e+05 | 2.0916e+02 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
1.000000 | 0.99979 | 1.00004 | 8.935e+03 | 2.874e-07 | 1.728e+05 | 2.7659e+01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
1.000000 | 0.99998 | 1.00004 | 6.028e+03 | 2.073e-07 | 8.270e+04 | 5.0052e-01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06
1.000000 | 1.00002 | 1.00004 | 5.777e+03 | 1.719e-07 | 6.569e+04 | 3.1819e-01 | 0.7396 | 0.2450 | 0.0155 | 2.87e-06