# Support Vector Spectrum Approximations: Efficient Calculations of Light
Scattering Lineshapes

Gregory J. Hunt1,∗, Cody R. Ground2, and Robin L. Hunt2
1William & Mary, PO Box 8763 Williamsburg, VA 23187
2Hypersonic Airbreathing Propulsion Branch, NASA Langley Research Center, 12
Langley Blvd. Hampton, VA 23681.
<EMAIL_ADDRESS>
###### Abstract.

Measuring scattered light is central to many laser-based gas diagnostic
techniques, e.g., coherent anti-Stokes Raman spectroscopy (CARS) and filtered
Rayleigh scattering (FRS). To produce quantitative measurements with such
techniques, a computational model of the scattered spectral lineshape is
necessary. While accurate, these models are often quite computationally
demanding and thus cannot be used in situations where computational speed
matters. To overcome this, approximations of these spectral lineshape models
can be used instead. In this paper, we develop a method called support vector
spectrum approximation (SVSA). This method uses machine learning to create
efficient and accurate approximations of any existing spectral lineshape
model. The SVSA framework improves upon existing methods by allowing efficient
approximations of spectral lineshapes to be calculated in arbitrary flow
regimes. We demonstrate the efficacy of SVSA in approximating coherent and
spontaneous Rayleigh-Brillioun spectra. We also show that SVSA reduces the
computational cost of a simulated filtered Rayleigh scattering experiment by a
factor of 300.

##  1\. Introduction

Measuring scattered light is central to many laser-based gas diagnostic
techniques. For example, coherent anti-Stokes Raman spectroscopy (CARS) and
filtered Rayleigh scattering (FRS) are techniques using Raman and Rayleigh-
Brillouin (RB) scattering, respectively, to quantify properties of a gas such
as pressure, velocity, temperature, and number density (Boguszko and Elliott,
2005; Miles et al., 2001; Forkey, 2000). Such laser-based gas diagnostic
techniques are useful in a diverse set of fields. In atmospheric science,
measurements of RB scattered light are used for remote sensing of clouds,
aerosols, winds, ice crystals, etc. (Witschas et al., 2014; Binietoglou et
al., 2016). In the physics and engineering fields, Raman and RB scattering are
used to nonintrusively investigate flames and other complex flows (Doll et
al., 2017; Miles et al., 2001; Ehn et al., 2017).

To produce quantitative measurements using these techniques, one often needs
to know the spectral lineshape of the scattered light. This lineshape conveys
important information about the gas’ properties. Consequently, accurate
computational models linking gas properties to the spectral lineshape have
been developed in the literature. For example, CARSFT can be used to calculate
coherent anti-Stokes Raman spectra and Tenti’s S6 model or Pan’s S7 model
allow calculation of RB spectra (Palmer and E., 1989; Tenti et al., 1974; Pan
et al., 2002). While typically quite accurate, such lineshape models are often
computationally demanding. Thus, they are not ideal for situations where
computational speed matters, for example, in optimization studies or real-time
diagnostic techniques (Yeaton et al., 2012; Binietoglou et al., 2016).

For RB scattering, this problem has motivated methods to efficiently
approximate the RB spectral lineshape. In particular, approximating Tenti’s S6
model has been previously investigated (Witschas et al., 2014; Ma et al.,
2012; Binietoglou et al., 2016). Such methods have had good success in the
regime on which they are designed. However, there does not presently exist a
general approach for rapidly approximating general spectral lineshapes for any
given scattering process over an arbitrary regime. Existing approximation
frameworks are specific to particular types of scattering, like Rayleigh-
Brillouin, and cannot be used to produce approximations of, for example, Raman
spectra. Similarly, methods may only be valid for a small range of parameters
or they may simplify the lineshape model by treating parameters as constant
even though they are not. Consequently, while these methods may be accurate
for, say, atmospheric regimes, they may not transfer well to highly dynamic
conditions like those in supersonic flows.

This paper proposes a method called support vector spectrum approximation
(SVSA), that creates efficient and accurate approximations of any spectral
lineshape given a preexisting computational model. The method applies machine
learning to produce a general lineshape approximator. The framework is
applicable to preexisting lineshape models of any scattering type in an
arbitrary flow regime. We show that SVSA is both fast and accurate by
demonstrating its efficacy approximating coherent and spontaneous RB spectra.
Furthermore, in application on a real problem, we demonstrate how SVSA might
be used in an FRS study of hypersonic vehicle engines to make the computations
feasible.

##  2\. Methods

Given any preexisting lineshape model, we generically denote the set of input
parameters for the model as $\varphi\in\mathbb{R}^{P}$ where $P$ denotes the
number of parameters. For example, $P=5$ for the S6 and S7 models of RB
scattering, which have as parameters y, Euken factor, internal relaxation
number, internal specific heat, and translational specific heat. Given $M$
values of the (normalized) frequency $x_{1},\ldots,x_{M}$, a lineshape model
$r$ takes $\varphi$ and produces an estimate $r(\varphi)\in\mathbb{R}^{M}$ of
the spectrum at $x_{1},\ldots,x_{M}$. Figure 1 illustrates several examples of
spontaneous Rayleigh-Brillouin scattering spectra estimated by Tenti’s S6
model.

Figure 1. Examples of spontaneous Rayleigh-Brillouin spectral lineshapes for
different sets of input parameters calculated using Tenti’s S6 model. Here,
the order of $\varphi$ is y, internal relaxation number, Euken factor,
internal specific heat, and translational specific heat.

The goal of SVSA is to create fast approximations of $r(\varphi)$. SVSA does
this by leveraging machine learning. Specifically, the method combines support
vector regression (SVR) and the singular value decomposition (SVD) to build a
flexible and general approximator.

###  2.1. The SVSA Model

At the heart of SVSA is an assumption that most of the variability among
spectral lineshapes can be explained by a small number of dominant lineshape
modes denoted $f_{1},\ldots,f_{K}$. That is, for any $\varphi$, we assume that

(1) $r(\varphi)\approx\sum_{k=1}^{K}w_{k}(\varphi)f_{k}$

so that a lineshape is approximately the linear combination of some dominant
global modes $f_{1},\ldots,f_{K}$ with parameter-specific weights
$w_{k}(\varphi)$. Using this approximation, SVSA estimates $r(\varphi)$ by
estimating the $\\{f_{k}\\}$ and the $\\{w_{k}(\varphi)\\}$ and combining them
according to Equation (1).

Given training data generated by a preexisting computational lineshape model,
SVSA does this estimation through a two-step approach:

  1. (1)

Decompose the training data with the SVD to estimate $K$ functional modes
$\hat{f}_{1},\ldots,\hat{f}_{K}\in\mathbb{R}^{M}$.

  2. (2)

Learn the relationship between $\varphi$ and $w_{k}(\varphi)$ using SVR. This
produces, for each $k$, a model $\hat{w}_{k}$ where
$\hat{w}_{k}(\varphi)\approx w_{k}(\varphi)$.

Combining the estimates from these two steps, SVSA predicts $r(\varphi)$ as

(2) $\hat{r}(\varphi)=\sum_{k=1}^{K}\hat{w}_{k}(\varphi)\hat{f}_{k}.$

An important feature of SVSA is that its calculation is fast. We show in the
next section that $\hat{r}(\varphi)$ can be succinctly written as

(3) $\hat{r}(\varphi)=\beta_{0}+\beta z(\varphi)$

with

(4) $\displaystyle z(\varphi)$
$\displaystyle=\exp\left(-||\Sigma-\varphi||^{2}\right)$

where $\beta_{0}\in\mathbb{R}^{M}$ and $\beta\in\mathbb{R}^{M\times S}$ are
the model coefficients from the SVR models expressed in the
$\\{\hat{f}_{k}\\}$ basis and $\Sigma\in\mathbb{R}^{S\times P}$ is a matrix
with rows $\\{\Sigma_{s}\\}_{s=1}^{S}$ that are “support vectors”
$\Sigma_{s}\in\mathbb{R}^{P}$ chosen by SVR. The notation “$\Sigma-\varphi$”
in Equation (4) denotes the $S\times 1$ vector where

$\left(\Sigma-\varphi\right)_{s}=\Sigma_{s}-\varphi.$

Notice that we can expand $-||\Sigma-\varphi||^{2}$ as

$-(\Sigma-\varphi)^{\prime}(\Sigma-\varphi)=2\Sigma\varphi-\text{diag}^{-1}\left(\Sigma^{\prime}\Sigma\right)-\varphi^{\prime}\varphi$

where the $\text{diag}^{-1}$ operator forms the vector of diagonal elements of
a matrix. Thus, calculating $\hat{r}(\varphi)$ boils down to matrix
multiplication and exponentiation. This means SVSA can be calculated very
efficiently using standard scientific computing software.

###  2.2. Fitting SVSA

The SVSA method learns its model parameters $\beta_{0},\beta$ and $\Sigma$
using a set of $N$ training examples. Let
$\varphi_{1},\ldots,\varphi_{N}\in\mathbb{R}^{P}$ be a collection of training
parameters that cover the range of the input space over which we want to learn
the approximator and let
$r(\varphi_{1}),\ldots,r(\varphi_{N})\in\mathbb{R}^{M}$ be the corresponding
training lineshapes calculated by our existing lineshape model. The first task
when fitting SVSA is to learn the dominant functional modes
$f_{1},\ldots,f_{K}$. We do this with the SVD. Let $R\in\mathbb{R}^{M\times
N}$ be the lineshape matrix containing the $N$ training spectra so that
$R_{mn}$ is the value of $r(\varphi_{n})$ at $x_{m}$. If $R$ has an SVD of
$R=UDV^{\prime}$, then define $W=DV^{\prime}$ so that $R=UW$. This
factorization allows us to decompose the $n^{th}$ column of $R$ as

(5) $r(\varphi_{n})=\sum_{k=1}^{M}W_{kn}U_{k}$

where $U_{k}$ is the $k^{th}$ column of $U$. These columns of $U$ (the
noncentered principal components of the training lineshapes) capture the
dominant functional modes of the lineshapes in our training data. Figure 2
displays the top four functional modes of some example training data. The
first mode approximately captures the average lineshape of the training data
and subsequent modes capture typical deviations from this average.

Figure 2. Dominant four lineshape modes for example training data.

Since the $\\{U_{k}\\}$ are ordered decreasing according to how much variation
they explain, if we truncate the linear combination in Equation (5) after
$K<M$ terms, then $\sum_{k=1}^{K}W_{kn}U_{k}$ is the best approximation of
$r(\varphi_{n})$ using only $K$ modes. We thus let $\hat{f}_{k}=U_{k}$ for
$k=1,\ldots,K$ and choose $K$ so that these $\\{\hat{f}_{k}\\}$ capture most
of the variation present in the training lineshapes. We can often achieve a
very good approximation using $K\ll M$. We explore the accuracy of this
approximation in Section 3.

The second step to training SVSA is to learn the relationship between
$\varphi$ and $w_{k}(\varphi)$. To do this, we need training examples of the
$\varphi$ and corresponding weights $w_{k}(\varphi)$ from which to learn. We
can reuse, the training data used to estimate the $\\{f_{k}\\}$ by noting that
for each training parameter $\varphi_{n}$ the correpsonding weight
$w_{k}(\varphi_{n})$ is precisely $W_{kn}$. To see this, consider the similar
form of Equations (1), (2) and (5). The $\\{W_{kn}\\}_{k=1}^{K}$ in Equation 5
are essentially the weights $\\{w_{k}(\varphi_{n})\\}_{k=1}^{K}$ used to
approximately reconstruct $r(\varphi_{n})$ from $f_{1},\ldots,f_{K}$ (Equation
(1)). Thus, to learn the $w_{k}$, we use the training parameters $\varphi_{n}$
and corresponding “training weights” $W_{kn}=w_{k}(\varphi_{n})$ to train $K$
different SVR models. That is, for each $k=1,\ldots,K$, we find a fit to
predict $W_{kn}$ from $\varphi_{n}$. We call this fit $\hat{w}_{k}$ so that
$\hat{w}_{k}(\varphi_{n})\approx w_{k}(\varphi_{n})=W_{kn}$.

The SVR approach used for this fit finds a flexible and regression-like
function to predict $w_{k}(\varphi)$ from $\varphi$. Fitting these models
amounts to using the training data to identify an important subset of training
points $\Sigma_{1},\ldots,\Sigma_{S}\in\mathbb{R}^{P}$, called support
vectors, and an associated set of coefficients
$\alpha_{k0},\ldots,\alpha_{kS}\in\mathbb{R}$. The SVR models use these
parameters to predict $w_{k}(\varphi)$ from $\varphi$ as a linear combination
of the $\\{\alpha\\}$ weighted by the similarity between $\varphi$ and
$\Sigma_{s}$. Specifically,

(6)
$\hat{w}_{k}(\varphi)=\alpha_{k0}+\sum_{s=1}^{S}\alpha_{ks}H(\Sigma_{s},\varphi)$

where $H$ is a kernel function measuring the similarity between $\Sigma_{s}$
and $\varphi$. Essentially, the SVR models use the support vectors as a set of
important landmarks in the parameter space of all $\varphi$ and estimates
$w_{k}(\varphi)\approx w_{k}(\Sigma_{s})$ to the extent that
$\varphi\approx\Sigma_{s}$.

Using SVR-based estimators over more traditional approaches like multiple
regression has substantial advantages. First, SVSA uses $\nu$-SVR, a version
of SVR parameterized by $\nu\in[0,1]$. This type of SVR finds an optimal fit
for the data that uses (approximately) $\nu$ percentage of the training points
as support vectors. Thus, $\nu$ directly controls the model complexity and
allows for an intuitive trade-off between accuracy and speed. A low value of
$\nu$ forces the fit to only retain the few most important support vectors. In
this case, predictions will be less computationally demanding by sacrificing
accuracy. Conversely, a high value of $\nu$ means the model can keep track of
many support vectors and build a nuanced and highly accurate model. However,
this prediction accuracy comes at the cost of computational speed.

A second advantage to SVR is our use of a radial basis function (RBF) kernel
$H$ in Equation (6) defined as $H(x,y)=\exp(-||x-y||^{2})$ for
$x,y\in\mathbb{R}^{P}$. This allows SVR to fit highly nonlinear relationships
between $\varphi$ and $w_{k}(\varphi)$. While multiple regression would find a
fit that is linear in $\varphi$, RBF kernelized SVR amounts to fitting a
regression-like estimator in an expanded infinite dimensional space of all
possible polynomials of $\varphi$. Thus, SVSA’s approach allows an increased
amount of flexibility in the model fit. It automatically considers all
possible polynomial relationships between $\varphi$ and $w_{k}(\varphi)$ and
finds the best fit among those while avoiding overfitting.

Finally, once fit, SVSA is very fast to compute. If
$\hat{w}(\varphi)=(\hat{w}_{1}(\varphi),\ldots,\hat{w}_{k}(\varphi))$ then we
can succinctly write $\hat{w}(\varphi)=\alpha_{0}+\alpha z(\varphi)$ with $z$
as in Equation (4), where $\alpha_{0}\in\mathbb{R}^{K}$ and
$\alpha\in\mathbb{R}^{K\times S}$ contain the coefficient estimates across all
$K$ of our SVR models. The parameter $\Sigma$ in $z$ (see Equation 3) is the
combined matrix of all support vectors showing up in any of our models. Here
$\Sigma\in\mathbb{R}^{S\times P}$ where $S$ is the total number of unique
support vectors across the models. Letting
$\hat{F}=\left[\hat{f}_{1}|\cdots|\hat{f}_{K}\right]\in\mathbb{R}^{M\times
K}$, we have

$\hat{r}(\varphi)=\hat{F}\hat{w}(\varphi)=\hat{F}\left(\alpha_{0}+\alpha
z(\varphi)\right)=\beta_{0}+\beta z(\varphi)$

where we define $\beta_{0}=\hat{F}\alpha_{0}$ and $\beta=\hat{F}\alpha$. Thus,
the major components of calculating $\hat{r}(\varphi)$ are (1) calculating
$Z(\varphi)$, which mainly involves the exponential and the product
$\Sigma\varphi$, and (2) calculating a product between $\beta$ and
$Z(\varphi)$. More generally, if we have a matrix
$\Phi=\left[\varphi_{1}|\varphi_{2}|\cdots|\varphi_{L}\right]\in\mathbb{R}^{P\times
L}$ consisting of parameter values
$\varphi_{1},\ldots,\varphi_{L}\in\mathbb{R}^{P}$ for $L$ lineshapes, then we
can efficiently approximate all of the lines as
$\hat{R}(\Phi)\in\mathbb{R}^{M\times L}$ letting

$\hat{R}(\Phi)=\beta_{0}+\beta Z(\Phi)$

with

$\displaystyle Z(\Phi)$ $\displaystyle=\exp\left(-||\Sigma-\Phi||^{2}\right).$

Here, $\Sigma-\Phi$ denotes an $S\times L$ matrix where

$(\Sigma-\Phi)_{s\ell}=\Sigma_{s}-\varphi_{\ell},$

the the norm $||\cdot||$ is applied columnwise and the exponential
elementwise. As $Z$ is a Gaussian kernel, it must be bounded for all $\Phi$
and cannot be ill-conditioned. Again, notice that we can write the exponent as

$-||\Sigma-\Phi||^{2}=2\Sigma\Phi-\text{diag}^{-1}(\Sigma^{\prime}\Sigma)-\text{diag}^{-1}(\Phi\Phi^{\prime})$

where the first subtraction is rowwise and the second is columnwise.

All together, after estimating $\beta_{0},\beta$ and $\Sigma$ using the
training data, SVSA boils down to the following three steps codified in
Algorithm 1:

Algorithm 1 Support Vector Spectrum Approximation

Step 1. Calculate  $\displaystyle E$ $\displaystyle=-||\Sigma-\Phi||^{2}$
$\displaystyle=2\Sigma\Phi-\text{diag}^{-1}(\Sigma^{\prime}\Sigma)-\text{diag}^{-1}(\Phi\Phi^{\prime}).$

Step 2. Exponentiate elements of $E$ to get $Z(\Phi)$.

Step 3. Calculate $\hat{R}(\Phi)=\beta_{0}+\beta Z(\Phi)$.

The computationally expensive parts of each step are (1) calculating $E$ in
$\mathcal{O}\left(SPL\right)$ operations, (2) exponentiating the elements of
$E$ using $SL$ exponentiations, and (3) calculating $\beta Z(\Phi)$ with
$\mathcal{O}\left(MSL\right)$ operations. As each of these steps are fast to
compute using modern scientific computing packages, SVSA allows us to very
efficiently compute a highly flexible, nonlinear approximation of $r(\varphi)$
with basically just two matrix products and exponentiation. Code for fitting
and applying our method can be found on github at gjhunt.github.io/svsa.
Currently, the fitting and lineshape approximation algorithms are implemented
in Python. However, once the coefficients have been estimated in Python, they
may also be exported and used to approximate the lineshape using routines we
have written in Matlab and R.

##  3\. Evaluation on Data

To assess the accuracy and speed of SVSA, we generate example lineshape data
on which to evaluate our method. We generate spontaneous Rayleigh-Brillouin
scattering (RBS) spectral lineshapes according to Tenti’s S6 model (Tenti et
al., 1974). Tenti’s model estimates RBS using the parameters: y, Euken factor,
internal relaxation number, internal specific heat and translational specific
heat. Our primary interest in SVSA is application to a supersonic wind tunnel
where a shock wave system is generated to simulate the flow conditions in a
high speed air-breathing engine. Thus, we generated our training lineshapes
using a range of parameters typically seen in these wind tunnel studies. We
set the internal specific heat (c_int) to $1$, the translational specific heat
(c_tr) to $3/2$, and vary $P=3$ parameters: the Euken factor (Eukenf) over a
grid of 5 points from $1.8$ to $2$, the internal relaxation number (rlx_int)
over a grid of 5 points from $1.5$ to $3$, and the $y$ factor (y) over a grid
of 20 points from 0 to 8. In total, this generates a set of $500$ training
lineshapes. Each lineshape is calculated on a grid of $500$ values of the
nondimensionalized frequency (x) ranging from $-3$ to $3$.

In Figure 3, we plot the cumulative percentage of variance captured in the
training data by the first several estimated modes $\hat{f}_{k}$.

Figure 3. Cumulative percentage of variance captured by first several dominant
modes of the training data.

We can see from this figure that almost all of the variability among spectral
lineshapes can be captured using the first several dominant modes. By
approximating the spectral lineshapes using $K\approx 6$ modes, we are able to
recover $>99\%$ of the important differences among lineshapes. The
dimensionality reduction achieved by these modes makes SVSA efficient to
compute since we need only train $6$ SVR models.

To further explore the relationship between model complexity and accuracy, we
generate a collection of test lineshapes covering the same parameter ranges as
the training data but over a finer grid. For the test data, we generate 32,000
example lineshapes. In Figure 4, we plot the maximum error of SVSA (across all
32,000 test lineshapes) against the average time needed to calculate a
lineshape using the method. We do this for three values of $\nu$, ten values
of $K$, and two ways of calculating lineshapes.

Figure 4. Percent error by average computation time for 32,000 test lineshapes
calculated simultaneously and serially.

The $y$-axis in Figure 4 measures the maximum difference between the SVSA
approximated lineshape and the true lineshape as a percentage of the maximum
height of the true line. The $x$-axis in Figure 4 measures the time it takes
SVSA to make the predictions. We display two sets of curves for two ways of
calculating the 32,000 lineshapes. First, the lineshapes are calculated by
predicting all 32,000 lineshapes simultaneously. This is done by feeding SVSA
a matrix $\Phi\in\mathbb{R}^{3\times 32000}$ of all parameters as per
Algorithm 1. We divide the total time it takes to make all predictions by
32,000. Secondly, we calculate the test lineshapes in a serial fashion feeding
one parameter vector $\varphi\in\mathbb{R}^{3\times 1}$ to SVSA at a time. We
then loop over all 32,000 parameters and report the average computation time.
These two ways of calculating the lineshapes cover two common use-cases for
SVSA. The first is the batch setting where all of the parameters are known in
advance of the calculation. For example, this occurs when analyzing data
collected from an experiment or a computational fluid dynamics (CFD)
simulation. The second situation covers doing calculations in a streaming
context where new data arrives over time. In this case, we do not know the
parameter values in advance and thus, can’t calculate them in bulk. We must
approximate each lineshape separately as the data arrives. This is what our
loop over the 32,000 test cases demonstrates. An example of this situation is
real-time laser diagnostics. For both approaches, the displayed time in Figure
4 is the average number of (wall-time) seconds it takes our Python
implementation to make a prediction on a 4.2 GHz processor.

Figure 4 shows a large difference between estimating the lineshapes
simultaneously and estimating them serially. It is almost an order of
magnitude faster to simultaneously calculate the lineshapes. This happens
because approximating the lineshapes simultaneously allows the entire
calculation to be handled extremely efficiently by low-level linear algebra
packages. Alternatively, the overhead when calculating the lineshapes serially
slows down the average approximation speed. Nonetheless, in either case, SVSA
is still extremely fast. Tenti’s model takes about one-tenth of a second to
calculate a single lineshape on our 4GHz processor. Thus, depending on the
parameter setttings, SVSA is between one-thousand and one-hundred-thousand
times faster.

We also notice in Figure 4 that both $K$ and $\nu$ have substantial impacts on
both speed and accuracy. Increasing either $K$ or $\nu$ increase the model
complexity, and thus decrease error at the cost of speed. For $\nu$, this
effect is straight-forward. Increasing $\nu$ increases the number of support
vectors. This leads to more nuanced and accurate models that are slower to
compute. For $K$, larger values indirectly affect model complexity by
increasing the complexity of the SVR models we build. This is because the
weight functions $w_{k}$, corresponding to a mode $f_{k}$, are increasingly
complex as we increase $k$ since higher-order $f_{k}$ pick up more subtle and
complicated modes. Modeling these complicated modes requires more flexible
models and hence more support vectors. Thus, increasing $K$ correspondingly
increases model complexity which decreases error at the expense of speed.
Notice, however, increasing $K$ quickly reaches diminishing returns. Indeed
using more than about $6$ modes seems to only increase computation time. This
is because the higher order modes explain an ever-smaller proportion of the
variability among the lineshapes. For example, from Figure 3, we see that the
$7^{th}$ mode accounts for less than $0.003\%$ of each line’s shape. Thus,
including these higher order terms increases model complexity for relatively
little gain in approximation accuracy. For our application, it seems that we
do well using $K=6$ and $\nu=0.1$. This yields a maximum error in our test set
of about 4% and an average computation time between about $1\times 10^{-5}$
and $8\times 10^{-5}$ seconds per line.

To test our method further, we build SVSA models to approximate coherent and
spontaneous RB spectra using Tenti’s S6 model and Pan’s S7 model. We build
these for models by training them on 500 example lineshapes over the same flow
parameters previously used and setting $K=6$ and $\nu=0.1$. We then evaluate
these approximators on 32,000 test lineshapes generated over the same
parameter space. In Figure 5, we plot several example predicted lineshapes for
our four models. The black lines are the true spectral lineshapes and the red
lines are the estimated lineshapes by SVSA. In all cases, we can see that SVSA
produces highly accurate approximations of the lineshapes.

Figure 5. Examples of actual and predicted lineshapes for RBS. Black line is
modeled line by the S6 or S7 model. The red line is the SVSA approximation.

In Figure 6, we plot boxplots of the percentage error for each of the four
applications of SVSA across the 32,000 test examples. This figure shows that
we have a median error of about $10^{-2}$ percent for the S6 and S7 models for
both spontaneous and coherent scattering. These errors are well below the
accuracy of the S6 and S7 models as compared to the true scattering lineshape.
This demonstrates that SVSA is a versatile estimator that is applicable across
models, like S6 and S7, and across scattering types, like coherent and
spontaneous RBS. SVSA can approximate these different models with relatively
little tuning and produce low error predictions across a large input parameter
space.

Figure 6. Percent Error for the four SVSA applications approximating the S6
and S7 models of coherent and spontaneous RB scattering.

##  4\. Application

To evaluate the efficacy of SVSA in a real application, we apply SVSA to a
computational fluid dynamics (CFD) study of filtered Rayleigh scattering
(FRS). FRS is an experimental laser-diagnostic technique whereby the total
energy $E$ of filtered Rayleigh-scattered light is used to quantify important
properties of a flow like temperature, pressure, velocity, number density,
etc. The data we use are Reynolds-Averaged Navier-Stokes CFD simulations of
the supersonic shock wave dominated flow field in the experiment mentioned
previously (Baurle et al., 2012). The output from these CFD simulations
contains the aforementioned flow properties on a rectangular 3D grid of
approximately 33 million points. Given these flow properties and the
experimental setup, we can calculate the value of $E$ at each point. Doing
this allows us to study the relationship between $E$, the flow properties, and
the experimental setup. A better understanding of these relationships will aid
in experiment optimization, interpretation, and analysis of the measured
energy $E$ in future real FRS experiments.

To calculate $E$ at each of the 33 million points in our simulation, we need
to know the flow parameters at each point (obtained from the CFD), the
experimental setup (fixed in advance), and the RB scattering lineshape of the
light at each point. To obtain the lineshape, one needs a lineshape model.
Typically, Tenti’s S6 model is used. However, because Tenti’s model is
comparatively slow, using Tenti’s model to compute $E$ would take about 38
days to compute across all 33 million points. Alternatively, as our method is
fast, using SVSA, it takes about $3\times 10^{-4}$ seconds to compute $E$ at
each point and thus, less than three hours to compute across all 33 million
points. In Figure 7, we plot a single cross-sectional plane of $E$ calculated
using the CFD data and either Tenti or SVSA. By eye, there is no discernible
difference between Tenti and SVSA. Additionally, we plot the difference
between the Tenti and SVSA calculated values of $E$. We can see from this plot
that in absolute magnitude the difference between the two methods is never
more than $2\times 10^{-3}$ (keeping in mind that the total scale for $E$ has
been normalized to range between $0$ and $1$). Thus, SVSA produces a value of
$E$ that is highly concurrent with Tenti’s model but does so about 300 times
faster. (While calculating the lineshape is 1-100 thousand times faster with
SVSA, other parts of the calculation of $E$ are identical using Tenti and
SVSA. Thus, the full calculation of $E$ doesn’t see as large of an improvement
as the calculation of the lineshape alone.)

Figure 7. A single cross-sectional plane of $E$ calculated by Tenti and SVSA.
The third plot is the difference between $E$ calculated using these two
models. The energy has been normalized so that the maximum energy in the Tenti
plot is one.

##  5\. Conclusion

In this paper, we developed a method called support vector spectrum
approximation (SVSA) that uses machine learning to create efficient and
accurate approximations of any existing spectral lineshape model in arbitrary
flow regimes.The ability to produce approximations of existing lineshape
models very efficiently is paramount for time-sensitive applications. In
addition to our example of greatly speeding up a computational FRS study, we
expect that SVSA will have application to real-time diagnostic techniques
where efficiently calculating the lineshape from a continuous stream of data
is necessary (Yeaton et al., 2012; Binietoglou et al., 2016). Additionally, we
expect that SVSA will have application to problems outside of FRS and the RB
spectra, for example, in approximating Raman spectra as part of CARS. We
expect to explore such extensions in future work.

## Funding

National Aeronautics and Space Administration No.
NNX15AI20H subaward by the Virginia Space Grant Consortium No.
19-264-100527-010.

## Disclosures

The authors declare that there are no conflicts of interest related to this
article.

## References

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Timur<EMAIL_ADDRESS>XXXXXXX/PAR.20XX.00000 1 An Updated Line
List for Spectroscopic Investigation of G Stars- I: Redetermination of the
Abundances in the Solar Photosphere–An Updated Line List for Spectroscopic
Investigation of G Stars- I: Redetermination of the Abundances in the Solar
Photosphere

# An Updated Line List for Spectroscopic Investigation of G Stars- I:
Redetermination of the Abundances in the Solar Photosphere

T. Şahin${}^{1\cc}$0000-0002-0296-233X  M. Marışmak10000-0002-9397-2778  N.
Çınar10000-0002-5155-9280  and S. Bilir20000-0003-3510-1509
1 Akdeniz University  Faculty of Science  Department of Space Sciences and
Technologies 07058  Antalya  Turkey
2 Istanbul University  Faculty of Science  Department of Astronomy and Space
Sciences 34119  Beyazıt  Istanbul  Turkey

(2023; 00.00.0000 00.00.0000 00.00.0000 00.00.0000 00.00.0000 )

###### Abstract

We propose a line list that may be useful for the abundance analysis of G-type
stars in the wavelength range 4080 – 6780 Å. It is expected that the line list
will be useful for surveys/libraries with overlapping spectral regions (e.g.
ELODIE/SOPHIE libraries, UVES-580 setting of Gaia-ESO), and in particular for
the analysis of F- and G-type stars in general. The atomic data are
supplemented by detailed references to the sources. We estimated the Solar
abundances using stellar lines and the high-resolution Kitt Peak National
Observatory (KPNO) spectra of the Sun to determine the uncertainty in the
$\log gf$ values. By undertaking a systematic search that makes use of the
lower excitation potential and $gf$-values and using revised multiplet table
as an initial guide, we identified 363 lines of 24 species that have accurate
$gf$-values and are free of blends in the spectra of the Sun and a Solar
analogue star, HD 218209 (G6V), for which accurate and up-to-date abundances
were obtained from both ELODIE and PolarBASE spectra of the star. For the
common lines with the Gaia-ESO line list v.6 provided by the Gaia-ESO
collaboration, we discovered significant inconsistencies in the $gf$-values
for certain lines of varying species.

###### keywords:

Line: identification - Sun: abundances – Sun: fundamental parameters - Stars:
individual (HD 218209)

††volume: 0

##  1 Introduction

Mid-spectral type main-sequence stars (F and G-type stars) play a significant
role in understanding the Galactic chemical evolution and history of Galactic
structure. These stars formed in the early Milky Way or are currently forming,
have main-sequence lifetimes comparable to the age of the Galaxy. F and G-type
main-sequence stars have an internal structure consisting of a radiative core
surrounded by a large envelope. Such structural configuration prevents heavy
elements produced in the core from mixing into the stellar atmosphere.
Consequently, mid-spectral type stars carry the chemical composition of the
molecular cloud in which they were born. The study of the abundances of heavy
elements detected in the atmospheres of F and G-type stars provides valuable
insights into the Galactic chemical evolution and the history of Galactic
structure, as noted by (Pagel & Patchett, 1975).

Moreover, these stars offer crucial information about the formation of
different populations within the Galaxy, including the halo, thick disc, and
thin disc. By analyzing the kinematics and orbital dynamics of stars that have
been spectroscopically studied, one can distinguish between these different
population groups in the Milky Way. In addition to providing insights into
Galactic chemical evolution and the formation of the Galaxy, considering the
case for pure spectroscopic analysis, for instance, the $\alpha$-element
abundances determined through spectroscopic analysis of stars play a crucial
role in testing the population membership of host galaxies.

The abundance of elements in a stellar spectrum, such as metallicity ([Fe/H]),
can also be used to estimate the age of the star. This is because the youngest
stars have comparatively higher metal abundances and metallicity, while the
oldest stars have lower ones (Placco et al., 2021). Therefore, pure
spectroscopic analysis alone provides crucial information about the nature of
host galaxies, in addition to insights into the age and chemical composition
of individual stars.

The kinematics, orbital dynamics, and chemical properties of stars are not
only key in understanding the chemical structure of our Galaxy and its
accompanying formation scenarios but also in determining the origins of
(metal-poor) young and old G-spectral type stars. In this context, the
author’s group is currently conducting a comprehensive study involving over 90
metal-poor G-type stars in the Solar neighbourhood. This study aims to
investigate the origins of these stars by analyzing their kinematics, orbital
dynamics, and chemical properties. This research will shed light on the
formation processes and evolutionary pathways of metal-poor G spectral-type
stars in our Galaxy.

Furthermore, the author’s group has previously conducted a thorough
investigation of six metal-poor F-type dwarf stars in the Solar neighbourhood,
with [Fe/H] values ranging from $-$2.4 to $-$1 dex. It is worth noting that
the studied metal-poor F-type dwarf stars exhibited significant changes in
model atmosphere parameters (effective temperature, $T_{\rm eff}$; surface
gravity, $\log g$; metallicity, [Fe/H]; microturbulence, $\xi$), as reported
in the literature. This research, as published in Şahin & Bilir (2020),
employed state-of-the-art analysis methods, including classical spectroscopy
with the help of ELODIE spectra. By combining these analysis techniques, the
authors were able to determine the Galactic origin of these F-type dwarf stars
in the Solar neighbourhood.

The determination of accurate metal abundances depends on the accurate
determination of model parameters. Current spectroscopic sky survey programs
for metal-poor late spectral stars, for example, indicate the need to
determine precisely calibrated model parameters for them. These current sky
survey programs include Gaia-ESO Public Spectroscopic Survey (GES; Gilmore et
al., 2012), GALactic Archaeology with HERMES (GALAH; Heijmans et al., 2012; De
Silva et al., 2015; Martell et al., 2017), the Large Sky Area Multi-Object
Fiber Spectroscopic Telescope (LAMOST; Zhao et al., 2012), the Sloan Extension
for Galactic Understanding and Exploration (SEGUE; Yanny et al., 2009), the
Apache Point Observatory Galactic Evolution Experiment (APOGEE; Allende Prieto
et al., 2008), and the RAdial Velocity Experiment (RAVE; Steinmetz et al.,
2006), and demonstrate that precisely determined and calibrated model
atmosphere parameters are required in different regions of the electromagnetic
spectrum due to differences in adopted observational methods and techniques in
these surveys, for which reliable line lists and atomic data are required.

In this study, we present an up-to-date line list for planned spectral
analyses in the framework of the comprehensive study mentioned above. HD
218209 of G6V, the most metal-rich star among 90 G type program stars
($-2.5<{\rm[Fe/H](dex)}<-0.5$), was selected for the preparation of the line
list which is expected to be useful for the surveys/libraries with overlapping
spectral regions (e.g. ELODIE library; Soubiran et al., 2003) and for the
analysis of F and G-type stars in general.

This paper is organized as follows. Section 2 provides information concerning
the observations. Section 3 explains the procedures used to measure and
identify lines, as well as the techniques employed to determine the model
parameters and conduct chemical abundance analyses of both HD 218209 and the
Sun. Section 4 is dedicated to discussing findings and their potential
implications.

##  2 Observations

For the creation of the line list in this study, high resolution ($R\approx$
42 000) and high signal to noise ($S/N$ = 165 at 550 nm) ELODIE spectrum (HJD
2451184.23521; $R\approx$ 42 000; exposure time 1800 s) of HD 218209 was
selected. The ELODIE cross-dispersed echelle spectrograph provided spectral
coverage from 3900 to 6800 Å. The spectrum was continuum normalized and
wavelength calibrated, and the radial velocity (RV) was corrected by the data
reduction pipeline at the telescope. Since some problems were encountered in
the continuum normalization of the spectra from the library, the spectrum was
renormalized.

Since high resolution ($R\approx$ 76 000 ) and high signal to noise ($S/N$ =
140 at 550 nm) PolarBASE111http://polarbase.irap.omp.eu (Petit et al., 2014)
Narval222Narval spectropolarimeter is adapted to the 2m Bernard Lyot telescope
and provides high-resolution spectral and polarimetric data. spectrum (HJD
2456232.48238; exposure time 400 s) of the star was also available, it was
used to test model parameters for the star. Prior to the line measurement
process, the PolarBASE spectrum was also renormalized and corrected for the
radial velocity (RV). For RV correction, we used a Python interface and the
NARVAL atomic line library containing atomic transitions from 4000 to 6800 Å.
For renormalization, we used an in-house developed interactive normalization
code LIME (Şahin, 2017) in Interactive Data Language (IDL) prior to the
abundance analysis.

The Solar spectrum is certainly a primary reference for stellar astrophysics
and for interpreting physical processes in stars (Molaro & Monai, 2012). The
high-resolution ($R\approx$ 400 000) spectrum of the Sun used in this study
was obtained with the Kitt Peak Fourier Transform Spectrometer (FTS; Kurucz et
al., 1984). The character of the spectra of HD 218209 and the Sun is displayed
in Figure 1.

Figure 1: A small region of the KPNO, PolarBASE, and ELODIE spectra of HD
218209 and the Sun. Identified lines are also indicated.

##  3 The Abundance Analysis

We employed ATLAS9 model atmospheres (Castelli & Kurucz, 2003) computed in
local thermodynamic equilibrium (LTE) with NEWODF opacities for abundance
study of HD 218209 and the Solar spectrum. Elemental abundances were computed
by using the LTE line analysis code MOOG (Sneden, 1973)333The MOOG source code
is available at http://www.as.utexas.edu/chris/moog.html. The details of the
abundance analysis and the source of the atomic data are the same as in Şahin
& Lambert (2009), Şahin et al. (2011, 2016) and Şahin & Bilir (2020). The line
list, atomic data, and model parameter derivation are covered in the following
subsections.

###  3.1 Line List: Line Measurement, Identification and Atomic Data

An essential prerequisite for abundance analysis of a star is a set of
reliably identified lines with reliable atomic data. Our line lists were
created by a systematic search for unblended lines (useful for equivalent
width–EW– analysis technique). For line measurement from the ELODIE spectrum
of HD 218209, a systematic search for unblended lines was performed. The line
centre positions were measured in several segments, each containing a portion
of the spectra of 20 Å. For this, the LIME (Şahin, 2017) code was employed.
The code provides a list of possible transitions in the close neighbourhood of
the measured line together with recent atomic data (e.g. Rowland Multiplet
Number-RMT, $\log gf$, and lower Level Excitation Potential-LEP) that are
compiled from the literature (e.g. from NIST database). The MOORE catalogue
(Moore et al., 1966) was configured as one of the reference atomic line
libraries in the LIME code. Following the line identification step, a
multiplet analysis technique was performed to assess whether the detected
lines belong to the candidate element indicated by the code. Our final list
covers 24 species and 363 lines over the spectrum range from about 4100 – 6800
Å. The identified lines have accurate $gf$-values and are free of blends in
the spectra of the Sun and HD 218209. The number of identified lines in the
respective wavelength regions of the KPNO Solar spectrum is presented in
Figure 2. Our selection of iron lines included 132 Fe i lines with excitation
potentials (LEPs) ranging from 0.05 to $\approx$5 eV and 17 Fe ii lines.
Chosen lines of Fe i and Fe ii are exhibited in Table 1. In Table 2, we
provide the list of identified lines other than iron with the atomic data,
their measured equivalent widths (obtained using the LIME code) and computed
logarithmic abundances in the Solar spectrum and in the spectrum of HD 218209.

Figure 2: The KPNO Solar spectrum and the number of identified lines in the
respective wavelength regions (each bar indicates 50 Å region of the
spectrum).

Atomic data forms the basis for line diagnostics. During the calculation of
element abundances, two basic atomic data stand out, namely the lower level
excitation potential (LEP) and the oscillator strength ($\log gf$). Since the
LEP values are measured with great accuracy, the main source of errors in
abundance analysis usually comes from the $\log gf$ measurements. The
sensitivity of the measurements to determine the $\log gf$ values affects the
sensitivity of the spectral analysis performed. This situation may lead to
incorrect calculation of model atmosphere parameters. Astrophysical
$gf$-values were avoided due to the model dependence of such a procedure. We
calculated Solar abundances using stellar lines to assess the uncertainty in
$\log gf$-values and to minimise systematic errors. The lines were measured
from the Kurucz et al. (1984) Solar flux atlas and analysed with the Solar
model atmosphere from the Castelli & Kurucz (2003) grid for $T_{\rm eff}=5790$
K, and $\log g=4.4$ cgs. Our calculations yielded a microturbulent of 0.7 km
s-1 and the Solar abundances shown in Table 3. As can be noticed, the Solar
abundances were successfully replicated. We use our Solar abundances to
compare stellar abundances to Solar abundances. As a result, the analysis in
this study is done differently in relation to the Sun. Preference for a
differential approach to analysis like this will reduce the errors caused by
uncertainty in oscillator strengths, the influence of spectrograph features,
and departures from LTE. Effective temperature errors in the spectroscopic
excitation technique can also arise from systematic errors in oscillator
strength as a function of excitation potential. The same holds true for errors
with equivalent widths.

Table 1: Fe i and Fe ii lines. The abundances are obtained for a model of $T_{\rm eff}=5790$ K, $\log g=4.4$ cgs, and $\xi=$ 0.66 km s-1. |  |  |  | Sun | HD 218209 |  |  |  |  |  | Sun | HD 218209 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Spec. | $\lambda$ | LEP | $\log(gf)$ | EW |  $\log\epsilon$(X) | EW |  $\log\epsilon$(X) | RMT | Spec. | $\lambda$ | LEP | $\log(gf)$ | EW |  $\log\epsilon$(X) | EW |  $\log\epsilon$(X) | RMT
| (Å) | (eV) | (dex) | (mÅ) | (dex) | (mÅ) | (dex) |  |  | (Å) | (eV) | (dex) | (mÅ) | (dex) | (mÅ) | (dex) |
Fe i | 4080.22 | 3.28 | -1.23 | 80.9 | 7.34 | 85.1 | 7.16 | 558 |  Fe i | 5618.64 | 4.21 | -1.28 | 49.3 | 7.47 | 34.3 | 7.04 | 1107
Fe i | 4082.11 | 3.42 | -1.51 | 72.7 | 7.60 | 69.8 | 7.34 | 698 |  Fe i | 5624.03 | 4.39 |  -1.20∗ | 49.2 | 7.55 | 38.9 | 7.22 | 1160
Fe i | 4088.56 | 3.64 | -1.50 | 52.4 | 7.43 | 43.9 | 7.09 | 906 |  Fe i | 5633.95 | 4.99 | -0.32 | 67.3 | 7.62 | 53.2 | 7.23 | 1314
Fe i | 4090.96 | 3.37 | -1.73 | 55.5 | 7.39 | 49.9 | 7.10 | 700 |  Fe i | 5636.71 | 3.64 | -2.56 | 19.6 | 7.51 | 13.0 | 7.18 | 868
Fe i | 4204.00 | 2.84 | -1.01 | 125.1 | 7.52 | 115.9 | 7.14 | 355 |  Fe i | 5638.27 | 4.22 | -0.84 | 75.4 | 7.54 | 61.1 | 7.12 | 1087
Fe i | 4207.13 | 2.83 | -1.41 | 83.5 | 7.44 | 88.1 | 7.31 | 352 |  Fe i | 5641.45 | 4.26 | -1.15 | 66.0 | 7.71 | 45.4 | 7.18 | 1087
Fe i | 4220.35 | 3.07 | -1.31 | 91.4 | 7.63 | 70.9 | 7.01 | 482 |  Fe i | 5662.52 | 4.18 | -0.57 | 92.4 | 7.61 | 81.1 | 7.24 | 1087
Fe i | 4291.47 | 0.05 | -4.08 | 92.3 | 7.51 | 87.1 | 7.12 | 3 |  Fe i | 5701.56 | 2.56 | -2.22 | 87.1 | 7.68 | 70.6 | 7.17 | 209
Fe i | 4365.90 | 2.99 | -2.25 | 49.2 | 7.44 | 38.1 | 7.03 | 415 |  Fe i | 5705.47 | 4.30 | -1.36 | 37.5 | 7.38 | 25.8 | 7.01 | 1087
Fe i | 4389.25 | 0.05 | -4.58 | 75.4 | 7.65 | 63.9 | 7.12 | 2 |  Fe i | 5717.84 | 4.28 | -1.10 | 63.3 | 7.62 | 53.8 | 7.29 | 1107
Fe i | 4432.58 | 3.57 | -1.56 | 51.4 | 7.38 | 40.2 | 6.97 | 797 |  Fe i | 5741.86 | 4.26 | -1.67 | 31.5 | 7.51 | 21.8 | 7.18 | 1086
Fe i | 4439.89 | 2.28 | -3.00 | 48.4 | 7.49 | 34.4 | 6.99 | 116 |  Fe i | 5778.46 | 2.59 | -3.43 | 21.5 | 7.41 | 13.6 | 7.03 | 209
Fe i | 4442.35 | 2.20 | -1.25 | 187.7 | 7.54 | 167.6 | 7.09 | 68 |  Fe i | 5806.73 | 4.61 | -1.03 | 56.4 | 7.69 | 39.7 | 7.25 | 1180
Fe i | 4447.14 | 2.20 | -2.73 | 66.2 | 7.66 | 64.1 | 7.45 | 69 |  Fe i | 5916.26 | 2.45 | -2.99 | 54.5 | 7.62 | 40.7 | 7.17 | 170
Fe i | 4447.73 | 2.22 | -1.34 | 171.0 | 7.54 | 155.7 | 7.11 | 68 |  Fe i | 5929.68 | 4.55 | -1.38 | 39.7 | 7.66 | 28.6 | 7.32 | 1176
Fe i | 4502.60 | 3.57 | -2.31 | 28.9 | 7.53 | 29.1 | 7.44 | 796 |  Fe i | 5934.67 | 3.93 | -1.12 | 76.6 | 7.47 | 61.7 | 7.05 | 982
Fe i | 4556.93 | 3.25 | -2.66 | 25.9 | 7.48 | 12.7 | 6.95 | 638 |  Fe i | 5952.73 | 3.98 | -1.39 | 59.6 | 7.50 | 53.0 | 7.22 | 959
Fe i | 4593.53 | 3.94 | -2.03 | 28.3 | 7.54 | 16.1 | 7.10 | 971 |  Fe i | 5956.71 | 0.86 | -4.61 | 52.8 | 7.62 | – | – | 14
Fe i | 4602.01 | 1.61 | -3.15 | 72.0 | 7.58 | 64.8 | 7.21 | 39 |  Fe i | 6027.06 | 4.07 | -1.09 | 62.7 | 7.48 | 48.8 | 7.05 | 1018
Fe i | 4602.95 | 1.48 | -2.22 | 122.5 | 7.51 | 116.3 | 7.13 | 39 |  Fe i | 6065.49 | 2.61 | -1.53 | 118.5 | 7.45 | 102.9 | 7.00 | 207
Fe i | 4619.30 | 3.60 | -1.08 | 84.1 | 7.45 | 73.7 | 7.06 | 821 |  Fe i | 6079.02 | 4.65 | -1.10 | 45.6 | 7.59 | 34.8 | 7.26 | 1176
Fe i | 4630.13 | 2.28 | -2.59 | 72.7 | 7.63 | 61.9 | 7.19 | 115 |  Fe i | 6082.72 | 2.22 | -3.57 | 35.2 | 7.52 | 28.1 | 7.21 | 64
Fe i | 4635.85 | 2.84 | -2.36 | 54.9 | 7.54 | 41.2 | 7.06 | 349 |  Fe i | 6096.67 | 3.98 | -1.88 | 37.5 | 7.55 | 27.2 | 7.22 | 959
Fe i | 4678.85 | 3.60 | -0.83 | 102.5 | 7.47 | 95.2 | 7.13 | 821 |  Fe i | 6127.91 | 4.14 | -1.40 | 48.3 | 7.52 | 37.3 | 7.17 | 1017
Fe i | 4704.95 | 3.69 | -1.53 | 61.5 | 7.55 | 48.6 | 7.12 | 821 |  Fe i | 6137.70 | 2.59 | -1.40 | 135.6 | 7.48 | 122.8 | 7.07 | 207
Fe i | 4728.55 | 3.65 | -1.17 | 81.3 | 7.65 | 69.3 | 7.23 | 822 |  Fe i | 6157.73 | 4.07 | -1.22 | 61.0 | 7.56 | 53.2 | 7.26 | 1015
Fe i | 4733.60 | 1.48 | -2.99 | 82.9 | 7.58 | 84.8 | 7.40 | 38 |  Fe i | 6165.36 | 4.14 | -1.47 | 44.6 | 7.51 | 30.7 | 7.09 | 1018
Fe i | 4735.85 | 4.07 | -1.32 | 64.1 | 7.79 | 57.6 | 7.50 | 1042 |  Fe i | 6173.34 | 2.22 | -2.88 | 68.9 | 7.61 | 58.6 | 7.22 | 62
Fe i | 4741.53 | 2.83 | -1.76 | 72.6 | 7.41 | 63.5 | 7.03 | 346 |  Fe i | 6180.21 | 2.73 | -2.65 | 53.4 | 7.52 | 47.6 | 7.25 | 269
Fe i | 4745.81 | 3.65 | -1.27 | 78.2 | 7.69 | 66.2 | 7.27 | 821 |  Fe i | 6200.32 | 2.61 | -2.44 | 71.5 | 7.59 | 62.6 | 7.23 | 207
Fe i | 4788.77 | 3.24 | -1.76 | 65.6 | 7.61 | 55.0 | 7.20 | 588 |  Fe i | 6213.44 | 2.22 | -2.48 | 82.5 | 7.50 | 75.6 | 7.18 | 62
Fe i | 4802.89 | 3.64 | -1.51 | 60.0 | 7.52 | 44.3 | 7.02 | 888 |  Fe i | 6219.29 | 2.20 | -2.43 | 87.9 | 7.54 | 79.1 | 7.18 | 62
Fe i | 4839.55 | 3.27 | -1.82 | 62.2 | 7.60 | 58.3 | 7.37 | 588 |  Fe i | 6232.65 | 3.65 | -1.22 | 81.5 | 7.60 | 71.9 | 7.25 | 816
Fe i | 4875.88 | 3.33 | -1.97 | 61.0 | 7.60 | 49.2 | 7.20 | 687 |  Fe i | 6240.65 | 2.22 | -3.17 | 47.6 | 7.40 | 39.1 | 7.06 | 64
Fe i | 4917.23 | 4.19 | -1.16 | 62.7 | 7.61 | 49.4 | 7.21 | 1066 |  Fe i | 6252.56 | 2.40 | -1.69 | 119.6 | 7.42 | 103.2 | 6.96 | 169
Fe i | 4918.02 | 4.23 | -1.34 | 53.0 | 7.63 | 45.5 | 7.35 | 1070 |  Fe i | 6265.14 | 2.18 | -2.55 | 86.2 | 7.60 | 82.5 | 7.33 | 62
Fe i | 4924.78 | 2.28 | -2.11 | 92.8 | 7.52 | 85.2 | 7.15 | 114 |  Fe i | 6270.23 | 2.86 | -2.61 | 53.0 | 7.58 | 43.1 | 7.23 | 342
Fe i | 4939.69 | 0.86 | -3.34 | 99.7 | 7.58 | 95.8 | 7.24 | 16 |  Fe i | 6297.80 | 2.22 | -2.74 | 73.6 | 7.56 | 67.1 | 7.26 | 62
Fe i | 4961.92 | 3.63 | -2.25 | 26.5 | 7.42 | 16.4 | 7.03 | 845 |  Fe i | 6301.51 | 3.65 | -0.72 | 112.7 | 7.57 | 106.2 | 7.26 | 816
Fe i | 4962.58 | 4.18 | -1.18 | 54.0 | 7.51 | 40.3 | 7.09 | 66 |  Fe i | 6315.81 | 4.07 | -1.66 | 39.8 | 7.51 | – | – | 1014
Fe i | 4973.10 | 3.96 | -0.92 | 93.8 | 7.72 | 82.4 | 7.34 | 173 |  Fe i | 6322.69 | 2.59 | -2.43 | 78.6 | 7.69 | 61 | 7.16 | 207
Fe i | 5002.80 | 3.40 | -1.53 | 78.8 | 7.54 | 81.2 | 7.35 | 687 |  Fe i | 6335.34 | 2.20 | -2.18 | 96.3 | 7.43 | 88.6 | 7.08 | 62
Fe i | 5022.24 | 3.98 | -0.56 | 99.5 | 7.45 | 93.9 | 7.14 | 965 |  Fe i | 6336.83 | 3.69 | -0.86 | 102.4 | 7.34 | 94.8 | 7.01 | 816
Fe i | 5029.62 | 3.41 | -2.00 | 48.7 | 7.54 | 40.0 | 7.21 | 718 |  Fe i | 6344.15 | 2.43 | -2.92 | 59.2 | 7.61 | 56.4 | 7.40 | 169
Fe i | 5074.75 | 4.22 | -0.23 | 118.8 | 7.50 | 105.8 | 7.12 | 1094 |  Fe i | 6393.61 | 2.43 | -1.58 | 134.6 | 7.48 | 113.4 | 6.99 | 168
Fe i | 5083.35 | 0.96 | -2.96 | 109.9 | 7.43 | 101.8 | 7.02 | 16 |  Fe i | 6408.03 | 3.69 | -1.02 | 97.3 | 7.69 | 88.1 | 7.35 | 816
Fe i | 5088.16 | 4.15 | -1.75 | 34.8 | 7.59 | 23.5 | 7.22 | 1066 |  Fe i | 6419.96 | 4.73 | -0.27 | 86.8 | 7.51 | 71.3 | 7.11 | 1258
Fe i | 5141.75 | 2.42 | -2.24 | 89.3 | 7.68 | 76.5 | 7.23 | 114 |  Fe i | 6430.86 | 2.18 | -2.01 | 115.3 | 7.51 | 97.9 | 7.03 | 62
Fe i | 5145.10 | 2.20 |  -3.08∗ | 52.5 | 7.49 | 49.0 | 7.26 | 66 |  Fe i | 6469.19 | 4.83 | -0.81 | 58.6 | 7.68 | 44.5 | 7.31 | 1258
Fe i | 5198.72 | 2.22 | -2.13 | 94.9 | 7.48 | 91.1 | 7.17 | 66 |  Fe i | 6481.88 | 2.28 | -2.98 | 64.7 | 7.63 | 62.9 | 7.44 | 109
Fe i | 5217.40 | 3.21 | -1.16 | 121.7 | 7.52 | 93.8 | 6.94 | 553 |  Fe i | 6498.94 | 0.96 | -4.69 | 45.7 | 7.59 | 40.9 | 7.32 | 13
Fe i | 5228.38 | 4.22 | -1.26 | 60.0 | 7.68 | 53.5 | 7.41 | 1091 |  Fe i | 6518.37 | 2.83 |  -2.46† | 56.0 | 7.46 | – | – | 342
Fe i | 5242.50 | 3.63 | -0.97 | 85.7 | 7.50 | 73.2 | 7.08 | 843 |  Fe i | 6593.88 | 2.43 | -2.42 | 83.7 | 7.60 | 75.0 | 7.25 | 168
Fe i | 5243.78 | 4.26 | -1.12 | 63.0 | 7.64 | 46.6 | 7.18 | 1089 |  Fe i | 6609.12 | 2.56 | -2.69 | 65.6 | 7.62 | 55.1 | 7.25 | 206
Fe i | 5247.06 | 0.09 | -4.95 | 65.3 | 7.58 | 61.3 | 7.30 | 1 |  Fe i | 6678.00 | 2.69 | -1.42 | 133.5 | 7.54 | 112.1 | 7.05 | 268
Fe i | 5250.22 | 0.12 | -4.94 | 65.9 | 7.62 | 61.5 | 7.32 | 1 |  Fe i | 6703.58 | 2.76 | -3.06 | 37.2 | 7.55 | 31.7 | 7.3 | 268
Fe i | 5250.65 | 2.20 | -2.18 | 100.6 | 7.60 | 95.7 | 7.27 | 66 |  Fe i | 6750.16 | 2.42 | -2.62 | 73.3 | 7.56 | 61.8 | 7.16 | 111
Fe i | 5253.47 | 3.28 | -1.57 | 77.9 | 7.42 | 62.2 | 6.94 | 553 |  Fe ii | 4178.86 | 2.58 |  -2.51∗ | 89.7 | 7.50 | 77.2 | 7.19 | 28
Fe i | 5288.53 | 3.69 | -1.51 | 58.6 | 7.52 | 48.4 | 7.15 | 929 |  Fe ii | 4508.29 | 2.85 |  -2.44∗ | 82.1 | 7.46 | 76.6 | 7.30 | 38
Fe i | 5298.78 | 3.64 | -2.02 | 42.2 | 7.57 | 30.3 | 7.18 | 875 |  Fe ii | 4576.34 | 2.84 | -2.92 | 65.3 | 7.52 | 54.8 | 7.25 | 38
Fe i | 5307.37 | 1.61 | -2.99 | 87.6 | 7.63 | 80.3 | 7.27 | 36 |  Fe ii | 4582.83 | 2.84 | -3.06 | 58.6 | 7.48 | 42.8 | 7.06 | 37
Fe i | 5322.05 | 2.28 | -2.80 | 59.4 | 7.44 | 48.9 | 7.04 | 112 |  Fe ii | 4620.52 | 2.83 | -3.19 | 55.3 | 7.50 | 39.8 | 7.10 | 38
Fe i | 5365.41 | 3.57 |  -1.22∗ | 74.4 | 7.45 | 66.3 | 7.12 | 786 |  Fe ii | 5132.67 | 2.81 | -4.09 | 25.1 | 7.53 | 13.6 | 7.17 | 35
Fe i | 5373.71 | 4.47 | -0.84 | 61.1 | 7.47 | 49.5 | 7.12 | 1166 |  Fe ii | 5197.58 | 3.23 |  -2.22∗ | 81.6 | 7.51 | 68.3 | 7.20 | 49
Fe i | 5379.58 | 3.69 | -1.51 | 59.7 | 7.53 | 46.1 | 7.09 | 928 |  Fe ii | 5234.63 | 3.22 | -2.21 | 83.8 | 7.53 | 68.2 | 7.18 | 49
Fe i | 5398.29 | 4.44 | -0.71 | 71.1 | 7.50 | 60.3 | 7.15 | 553 |  Fe ii | 5264.81 | 3.33 |  -3.13∗ | 45.2 | 7.61 | 29.0 | 7.21 | 48
Fe i | 5473.91 | 4.15 | -0.79 | 78.2 | 7.49 | 65.1 | 7.08 | 1062 |  Fe ii | 5284.11 | 2.89 |  -3.11∗ | 61.1 | 7.58 | – | – | 41
Fe i | 5483.11 | 4.15 | -1.41 | 46.2 | 7.49 | – | – | 1061 |  Fe ii | 5414.07 | 3.22 |  -3.58∗ | 28.6 | 7.50 | 15.5 | 7.12 | 48
Fe i | 5487.15 | 4.41 | -1.51 | 36.5 | 7.61 | 26.3 | 7.28 | 1143 |  Fe ii | 5425.26 | 3.20 |  -3.22∗ | 42.5 | 7.50 | 26.2 | 7.09 | 49
Fe i | 5501.48 | 0.96 | -3.05 | 116.6 | 7.54 | 109.8 | 7.16 | 15 |  Fe ii | 5534.85 | 3.24 |  -2.75∗ | 58.3 | 7.48 | 41.7 | 7.08 | 55
Fe i | 5506.79 | 0.99 | -2.80 | 121.6 | 7.38 | 115.1 | 7.00 | 15 |  Fe ii | 6247.56 | 3.89 |  -2.30∗ | 54.1 | 7.51 | 40.6 | 7.21 | 74
Fe i | 5525.55 | 4.23 | -1.08 | 53.3 | 7.37 | 38.1 | 6.94 | 1062 |  Fe ii | 6432.68 | 2.89 |  -3.57∗ | 40.1 | 7.46 | 27.5 | 7.15 | 40
Fe i | 5543.94 | 4.22 | -1.11 | 62.2 | 7.57 | 48.9 | 7.17 | 1062 |  Fe ii | 6456.39 | 3.90 |  -2.05∗ | 64.6 | 7.50 | 43.9 | 7.04 | 74
Fe i | 5546.51 | 4.37 | -1.28 | 49.7 | 7.62 | 35.8 | 7.22 | 1145 |  Fe ii | 6516.08 | 2.89 |  -3.31∗ | 54.9 | 7.56 | – | – | 40
Fe i | 5560.22 | 4.43 | -1.16 | 50.2 | 7.56 | 35.4 | 7.14 | 1164 |  |  |  |  |  |  |  |  |
Table 2: Lines used in the analysis of the KPNO Solar spectrum and HD 218209. Abundances for individual lines are those obtained for a model of $T_{\rm eff}=5790$ K, $\log g=4.4$ cgs, and $\xi=$ 0.66 km s-1. |  |  |  | Sun | HD 218209 |  |  |  |  |  |  | Sun | HD 218209 |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Spec. | $\lambda$ | LEP | $\log(gf)$ | EW |  $\log\epsilon$(X) | EW |  $\log\epsilon$(X) | RMT | Ref. | Spec. | $\lambda$ | LEP | $\log(gf)$ | EW |  $\log\epsilon$(X) | EW |  $\log\epsilon$(X) | RMT | Ref.
| (Å) | (eV) | (dex) | (mÅ) | (dex) | (mÅ) | (dex) |  |  |  | (Å) | (eV) | (dex) | (mÅ) | (dex) | (mÅ) | (dex) |  |
Na I | 5682.65 | 2.10 | -0.67 | 100.1 | 6.21 | 82.4 | 5.86 | 6 | 1 |  Ti i | 5145.47 | 1.46 | -0.54 | 37.2 | 4.94 | 31.4 | 4.67 | 109 | 7
Na I | 5688.22 | 2.10 | -0.37 | 121.1 | 6.11 | 99.9 | 5.72 | 6 | 1 |  Ti i | 5147.48 | 0.00 | -1.94 | 38.8 | 4.93 | 37.7 | 4.73 | 4 | 7
Mg i | 4571.10 | 0.00 | -5.40 | 109.2 | 7.54 | 115.1 | 7.44 | 1 | 2 |  Ti i | 5152.19 | 0.02 | -1.95 | 36.9 | 4.92 | – | – | 4 | 7
Mg i | 5711.10 | 4.34 | -1.74 | 104.6 | 7.65 | 107.6 | 7.52 | 8 | 2 |  Ti i | 5192.98 | 0.02 | -0.95 | 84.0 | 5.03 | 85.2 | 4.83 | 4 | 7
Si i | 5645.62 | 4.93 | -2.03 | 35.5 | 7.50 | – | – | 10 | 3 |  Ti i | 5210.39 | 0.05 | -0.82 | 89.0 | 5.03 | 98.6 | 4.95 | 4 | 7
Si i | 5665.56 | 4.92 | -1.99 | 39.6 | 7.53 | 31.0 | 7.37 | 10 | 3 |  Ti i | 5219.71 | 0.02 | -2.22 | 29.1 | 5.00 | 29.0 | 4.82 | 4 | 7
Si i | 5684.49 | 4.95 | -1.58 | 60.6 | 7.52 | 54.3 | 7.35 | 11 | 3 |  Ti i | 5490.16 | 1.46 | -0.84 | 22.4 | 4.85 | 25.6 | 4.80 | 3 | 7
Si i | 5708.40 | 4.95 | -1.47 | 74.7 | 7.64 | 62.2 | 7.37 | 10 | 4 |  Ti i | 5866.46 | 1.07 | -0.79 | 46.9 | 4.97 | 44.9 | 4.78 | 72 | 7
Si i | 5772.15 | 5.08 | -1.62 | 52.0 | 7.53 | 40.0 | 7.27 | 17 | 3 |  Ti i | 6126.22 | 1.07 | -1.42 | 21.7 | 5.00 | 23.0 | 4.88 | 69 | 7
Si i | 5793.08 | 4.93 | -1.86 | 42.4 | 7.46 | 31.6 | 7.20 | 9 | 3 |  Ti i | 6258.11 | 1.44 | -0.39 | 51.4 | 5.00 | 48.3 | 4.78 | 104 | 7
Si i | 5948.54 | 5.08 | -1.09 | 83.8 | 7.51 | – | – | 16 | 3 |  Ti i | 6261.11 | 1.43 | -0.53 | 49.9 | 5.10 | 53.2 | 5.02 | 104 | 7
Si i | 6125.03 | 5.61 | -1.53 | 30.9 | 7.51 | 23.1 | 7.31 | 30 | 3 |  Ti i | 6336.11 | 1.44 | -1.69 | 5.60 | 4.92 | – | – | 103 | 7
Si i | 6142.49 | 5.62 | -1.48 | 33.3 | 7.51 | 26.8 | 7.35 | 30 | 3 |  Ti i | 6743.13 | 0.90 | -1.63 | 18.0 | 4.89 | – | – | 48 | 7
Si i | 6145.02 | 5.61 | -1.39 | 37.9 | 7.50 | 27.6 | 7.26 | 29 | 3 |  Ti ii | 4443.81 | 1.08 | -0.71 | 139.7 | 5.04 | 139.9 | 4.86 | 19 | 7
Si i | 6244.48 | 5.61 | -1.29 | 44.0 | 7.51 | 32.9 | 7.27 | 27 | 3 |  Ti ii | 4468.50 | 1.13 | -0.63 | 134.1 | 4.94 | 149.8 | 4.90 | 31 | 7
Si i | 6721.84 | 5.86 | -0.94 | 42.3 | 7.33 | 29.1 | 7.06 | 38* | 4 |  Ti ii | 4493.53 | 1.08 | -2.78 | 34.2 | 4.90 | 40.1 | 5.01 | 18 | 7
Ca i | 4512.27 | 2.52 | -1.90 | 25.0 | 6.33 | 16.3 | 5.99 | 24 | 5 |  Ti ii | 4568.33 | 1.22 | -2.65 | 32.0 | 4.84 | 26.7 | 4.68 | 60 | 7
Ca i | 4578.56 | 2.52 | -0.70 | 85.8 | 6.33 | 89.4 | 6.18 | 23 | 5 |  Ti ii | 4583.41 | 1.16 | -2.84 | 31.7 | 4.96 | 25.6 | 4.77 | 39 | 7
Ca i | 5260.39 | 2.52 | -1.72 | 33.2 | 6.32 | 25.2 | 6.04 | 22 | 5 |  Ti ii | 4708.67 | 1.24 | -2.35 | 51.0 | 5.02 | 44.0 | 4.82 | 49 | 7
Ca i | 5261.71 | 2.52 | -0.58 | 98.7 | 6.49 | 91.7 | 6.19 | 22 | 5 |  Ti ii | 4874.01 | 3.09 | -0.86 | 36.4 | 4.91 | 27.3 | 4.70 | 114 | 7
Ca i | 5512.99 | 2.93 | -0.46 | 83.9 | 6.36 | 82.2 | 6.16 | 48 | 5 |  Ti ii | 4911.20 | 3.12 | -0.64 | 51.4 | 5.11 | 48.7 | 5.06 | 114 | 7
Ca i | 5581.98 | 2.52 | -0.56 | 94.3 | 6.38 | 88.8 | 6.11 | 21 | 5 |  Ti ii | 5005.17 | 1.57 | -2.73 | 23.9 | 5.02 | 16.2 | 4.76 | 71 | 7
Ca i | 5590.13 | 2.52 | -0.57 | 93.0 | 6.37 | 86.6 | 6.09 | 21 | 5 |  Ti ii | 5013.69 | 1.58 | -2.14 | 49.5 | 5.08 | 43.7 | 4.92 | 71 | 7
Ca i | 5601.29 | 2.52 | -0.52 | 102.8 | 6.47 | 99.0 | 6.21 | 21 | 5 |  Ti ii | 5336.79 | 1.58 | -1.60 | 72.3 | 5.09 | 71.0 | 5.03 | 69 | 7
Ca i | 6102.73 | 1.88 | -0.81 | 122.3 | 6.10 | 123.8 | 5.86 | 3 | 5 |  Ti ii | 5418.77 | 1.58 | -2.13 | 48.7 | 5.02 | 41.3 | 4.83 | 69 | 7
Ca i | 6166.44 | 2.52 | -1.14 | 72.3 | 6.37 | 64.2 | 6.08 | 20 | 5 |  V i | 4437.84 | 0.29 | -0.71 | 37.1 | 4.03 | 32.9 | 3.76 | 21 | 8
Ca i | 6169.04 | 2.52 | -0.80 | 93.5 | 6.34 | 90.8 | 6.09 | 20 | 5 |  V i | 4577.18 | 0.00 | -1.08 | 33.6 | 4.01 | 36.5 | 3.91 | 4 | 8
Ca i | 6169.56 | 2.52 | -0.48 | 114.0 | 6.26 | 109.4 | 5.98 | 20 | 5 |  V i | 5727.06 | 1.08 | -0.02 | 38.7 | 4.04 | – | – | 35 | 8
Ca i | 6455.60 | 2.52 | -1.34 | 56.7 | 6.39 | 47.3 | 6.09 | 19 | 5 |  V i | 6119.53 | 1.06 | -0.36 | 22.1 | 3.93 | 21.8 | 3.78 | 34 | 8
Ca i | 6471.67 | 2.52 | -0.69 | 89.3 | 6.35 | 85.2 | 6.12 | 18 | 5 |  V i | 6243.11 | 0.30 | -0.94 | 29.6 | 3.94 | 29.9 | 3.78 | 19 | 8
Ca i | 6493.79 | 2.52 | -0.11 | 124.4 | 6.26 | 116.6 | 5.95 | 18 | 5 |  Cr i | 4545.96 | 0.94 | -1.38 | 81.6 | 5.68 | 80.4 | 5.43 | 10 | 4
Ca i | 6499.65 | 2.52 | -0.82 | 82.5 | 6.37 | 81.2 | 6.19 | 18 | 5 |  Cr i | 4616.13 | 0.98 | -1.18 | 90.3 | 5.71 | 83.3 | 5.34 | 21 | 4
Ca i | 6572.79 | 0.00 | -4.32 | 32.1 | 6.37 | 33.4 | 6.24 | 1 | 5 |  Cr i | 4626.18 | 0.97 | -1.32 | 82.2 | 5.66 | 72.7 | 5.24 | 21 | 4
Ca i | 6717.69 | 2.71 | -0.52 | 99.5 | 6.36 | 98.7 | 6.04 | 32 | 5 |  Cr i | 4646.17 | 1.03 | -0.71 | 105.5 | 5.55 | 94.3 | 5.11 | 21 | 4
Sc i | 4023.69 | 0.02 | 0.38 | 57.5 | 3.12 | 51.3 | 2.78 | 7 | 6 |  Cr i | 4651.29 | 0.98 | -1.46 | 82.9 | 5.82 | 66.1 | 5.23 | 21 | 4
Sc ii | 4246.84 | 0.31 | 0.24 | 156.6 | 3.17 | 153.8 | 2.94 | 7 | 6 |  Cr i | 4652.17 | 1.00 | -1.03 | 100.8 | 5.77 | 86.3 | 5.26 | 21 | 4
Sc ii | 5239.82 | 1.45 | -0.76 | 52.0 | 3.22 | 48.7 | 3.12 | 26 | 6 |  Cr i | 4708.02 | 3.17 | 0.11 | 57.5 | 5.58 | 41.5 | 5.11 | 186 | 4
Sc ii | 5526.82 | 1.77 | -0.01 | 75.1 | 3.32 | 65.4 | 3.06 | 31 | 6 |  Cr i | 4718.42 | 3.19 | 0.10 | 64.1 | 5.74 | 55.9 | 5.42 | 186 | 4
Sc ii | 5640.99 | 1.50 | -0.99 | 39.7 | 3.18 | 34.7 | 3.04 | 29 | 6 |  Cr i | 4730.72 | 3.08 | -0.19 | 48.3 | 5.67 | 30.5 | 5.12 | 145 | 4
Sc ii | 5657.88 | 1.51 | -0.54 | 66.1 | 3.36 | 55.6 | 3.09 | 29 | 6 |  Cr i | 4737.35 | 3.07 | -0.10 | 55.7 | 5.68 | 43.0 | 5.26 | 145 | 4
Sc ii | 5667.15 | 1.50 | -1.21 | 32.1 | 3.22 | 26.1 | 3.05 | 29 | 6 |  Cr i | 4756.12 | 3.10 | 0.09 | 62.9 | 5.76 | 52.3 | 5.39 | 145 | 4
Sc ii | 5669.04 | 1.50 | -1.10 | 34.1 | 3.16 | 29.7 | 3.03 | 29 | 6 |  Cr i | 4936.34 | 3.11 | -0.34 | 46.2 | 5.77 | 37.6 | 5.46 | 166 | 4
Ti i | 4060.27 | 1.05 | -0.69 | 37.8 | 4.82 | 44.5 | 4.83 | 80 | 7 |  Cr i | 4964.93 | 0.94 | -2.53 | 37.7 | 5.65 | 33.3 | 5.40 | 9 | 4
Ti i | 4186.13 | 1.50 | -0.24 | 42.7 | 4.91 | 43.7 | 4.79 | 129 | 7 |  Cr i | 5247.57 | 0.96 | -1.63 | 82.0 | 5.78 | 74.5 | 5.42 | 18 | 4
Ti i | 4287.41 | 0.84 | -0.37 | 75.3 | 5.20 | 76.7 | 5.01 | 44 | 7 |  Cr i | 5296.70 | 0.98 | -1.41 | 91.9 | 5.77 | 80.7 | 5.33 | 18 | 4
Ti i | 4453.32 | 1.43 | -0.03 | 64.5 | 5.16 | – | – | 113 | 7 |  Cr i | 5300.75 | 0.98 | -2.13 | 58.6 | 5.75 | 44.0 | 5.25 | 18 | 4
Ti i | 4465.81 | 1.74 | -0.13 | 40.9 | 4.95 | 40.7 | 4.81 | 146 | 7 |  Cr i | 5345.81 | 1.00 | -0.98 | 116.8 | 5.73 | 111.0 | 5.37 | 18 | 4
Ti i | 4512.74 | 0.84 | -0.40 | 67.5 | 4.99 | 68.4 | 4.82 | 42 | 7 |  Cr i | 5348.33 | 1.00 | -1.29 | 99.5 | 5.79 | 86.5 | 5.32 | 18 | 4
Ti i | 4518.03 | 0.83 | -0.25 | 73.0 | 4.97 | 74.6 | 4.80 | 42 | 7 |  Cr i | 5787.93 | 3.32 | -0.08 | 45.7 | 5.62 | 32.3 | 5.22 | 188 | 4
Ti i | 4534.79 | 0.84 | 0.35 | 95.9 | 4.86 | 96.5 | 4.60 | 42 | 7 |  Cr ii | 4588.20 | 4.07 | -0.65 | 71.8 | 5.69 | 59.4 | 5.40 | 44 | 9
Ti i | 4548.77 | 0.83 | -0.28 | 70.9 | 4.95 | 82.8 | 4.99 | 42 | 7 |  Cr ii | 5237.32 | 4.07 | -1.17 | 53.0 | 5.75 | 39.2 | 5.42 | 43 | 9
Ti i | 4555.49 | 0.85 | -0.40 | 63.1 | 4.89 | 64.5 | 4.74 | 42 | 7 |  Cr ii | 5305.87 | 3.83 | -1.91 | 24.6 | 5.48 | 13.2 | 5.13 | 24 | 9
Ti i | 4617.28 | 1.75 | 0.44 | 63.7 | 4.94 | 61.3 | 4.72 | 145 | 7 |  Mn i | 4055.55 | 2.14 | -0.08 | 125.5 | 5.76 | 107.2 | 5.26 | 5 | 10
Ti i | 4623.10 | 1.74 | 0.16 | 56.6 | 5.03 | 52.7 | 4.79 | 145 | 7 |  Mn i | 4082.94 | 2.18 | -0.36 | 91.3 | 5.60 | 76.6 | 5.07 | 5 | 10
Ti i | 4639.36 | 1.74 | -0.05 | 43.2 | 4.90 | 49.4 | 4.91 | 145 | 7 |  Mn i | 4451.59 | 2.89 | 0.28 | 95.1 | 5.55 | 80.0 | 5.06 | 22 | 10
Ti i | 4639.66 | 1.75 | -0.14 | 43.5 | 5.01 | 44.4 | 4.89 | 145 | 7 |  Mn i | 4470.14 | 2.94 | -0.44 | 56.9 | 5.46 | 36.7 | 4.82 | 22 | 10
Ti i | 4656.47 | 0.00 | -1.28 | 68.8 | 5.07 | 76.2 | 5.05 | 6 | 7 |  Mn i | 4502.22 | 2.92 | -0.34 | 59.6 | 5.41 | 43.1 | 4.86 | 22 | 10
Ti i | 4722.61 | 1.05 | -1.47 | 18.3 | 5.02 | 18.7 | 4.88 | 75 | 7 |  Mn i | 4709.72 | 2.89 | -0.49 | 68.4 | 5.73 | 44.9 | 5.01 | 21 | 10
Ti i | 4742.80 | 2.24 | 0.21 | 31.6 | 4.85 | 32.5 | 4.74 | 233 | 7 |  Mn i | 4739.11 | 2.94 | -0.61 | 59.3 | 5.66 | 38.7 | 5.01 | 21 | 10
Ti i | 4758.12 | 2.25 | 0.51 | 44.9 | 4.87 | 39.8 | 4.63 | 233 | 7 |  Mn i | 4765.86 | 2.94 | -0.09 | 77.4 | 5.57 | 66.0 | 5.15 | 21 | 10
Ti i | 4759.28 | 2.25 | 0.59 | 44.8 | 4.79 | 43.9 | 4.64 | 233 | 7 |  Mn i | 4766.42 | 2.92 | 0.10 | 93.8 | 5.68 | 76.3 | 5.16 | 21 | 10
Ti i | 4820.41 | 1.50 | -0.38 | 42.3 | 4.96 | 39.9 | 4.77 | 126 | 7 |  Mn i | 4783.42 | 2.30 | 0.03 | 140.1 | 5.75 | 121.1 | 5.27 | 16 | 10
Ti i | 4885.09 | 1.89 | 0.41 | 62.6 | 5.04 | 70.3 | 5.05 | 231 | 7 |  Mn i | 5117.94 | 3.13 | -1.20 | 24.0 | 5.51 | – | – | 32 | 10
Ti i | 4913.62 | 1.87 | 0.22 | 50.6 | 4.91 | 48.1 | 4.71 | 157 | 7 |  Mn i | 5432.55 | 0.00 | -3.79 | 49.6 | 5.61 | 25.9 | 4.86 | 1 | 10
Ti i | 4981.74 | 0.85 | 0.57 | 119.3 | 4.92 | 120.6 | 4.64 | 38 | 7 |  Mn i | 6021.80 | 3.07 | -0.05 | 96.9 | 5.85 | 69.4 | 5.17 | 27 | 10
Ti i | 4999.51 | 0.83 | 0.32 | 102.5 | 4.91 | 104.8 | 4.67 | 38 | 7 |  Co i | 4121.33 | 0.92 | -0.33 | 124.0 | 5.06 | 113.1 | 4.61 | 28 | 11
Ti i | 5009.65 | 0.02 | -2.20 | 24.2 | 4.88 | 34.5 | 4.95 | 5 | 7 |  Co i | 4813.48 | 3.21 | 0.12 | 45.4 | 5.02 | 31.1 | 4.56 | 158 | 11
Ti i | 5016.17 | 0.85 | -0.48 | 64.8 | 4.94 | 64.6 | 4.76 | 38 | 7 |  Co i | 5301.05 | 1.71 | -1.90 | 20.6 | 4.90 | – | – | 39 | 11
Ti i | 5020.03 | 0.83 | -0.33 | 77.6 | 5.07 | 79.9 | 4.91 | 38 | 7 |  Co i | 5342.71 | 4.02 | 0.74 | 30.5 | 4.77 | 21.2 | 4.43 | 190 | 11
Ti i | 5022.87 | 0.83 | -0.33 | 73.2 | 4.97 | 73.6 | 4.78 | 38 | 7 |  Co i | 5352.05 | 3.58 | 0.06 | 24.2 | 4.85 | 16.6 | 4.52 | 172 | 11
Ti i | 5039.96 | 0.02 | -1.08 | 73.8 | 4.95 | 79.9 | 4.87 | 5 | 7 |  Co i | 5483.36 | 1.71 | -1.50 | 49.8 | 5.23 | 39.6 | 4.84 | 39 | 11
Ti i | 5064.65 | 0.05 | -0.94 | 84.8 | 5.09 | 87.7 | 4.92 | 5 | 7 |  Co i | 6093.15 | 1.74 | -2.40 | 9.2 | 4.95 | 8.1 | 4.75 | 37 | 11

Table 2 continued
Sun HD 218209 Sun HD 218209 Spec. $\lambda$ LEP $\log(gf)$ EW
$\log\epsilon$(X) EW $\log\epsilon$(X) RMT Ref. Spec. $\lambda$ LEP $\log(gf)$
EW $\log\epsilon$(X) EW $\log\epsilon$(X) RMT Ref. (Å) (eV) (dex) (mÅ) (dex)
(mÅ) (dex) (Å) (eV) (dex) (mÅ) (dex) (mÅ) (dex) Ni i 4410.52 3.31 -1.08 57.8
6.39 43.1 5.94 88 4 Ni i 5748.36 1.68 -3.26 28.8 6.26 19.3 5.87 45 4 Ni i
4470.48 3.40 -0.40 79.5 6.24 76.2 5.98 86 4 Ni i 5805.23 4.17 -0.64 40.5 6.30
28.2 5.93 234 4 Ni i 4606.23 3.60 -1.02 48.0 6.37 38.8 6.05 100 4 Ni i 6007.32
1.68 -3.34 25.4 6.24 22.5 6.04 42 4 Ni i 4686.22 3.60 -0.64 62.3 6.30 58.9
6.08 98 4 Ni i 6086.29 4.26 -0.51 43.5 6.30 28.2 5.88 249 4 Ni i 4731.80 3.83
-0.85 45.4 6.35 26.6 5.82 163 4 Ni i 6108.12 1.68 -2.44 65.6 6.30 55.3 5.92 45
4 Ni i 4732.47 4.10 -0.55 44.3 6.29 30.6 5.89 235 4 Ni i 6128.98 1.68 -3.32
25.3 6.21 20.1 5.94 42 4 Ni i 4752.43 3.66 -0.69 59.1 6.33 43.1 5.86 132 4 Ni
i 6130.14 4.26 -0.96 21.6 6.24 11.5 5.82 248 4 Ni i 4756.52 3.48 -0.34 77.5
6.19 76.1 5.96 98 4 Ni i 6175.37 4.09 -0.54 50.7 6.32 36.4 5.92 217 4 Ni i
4806.99 3.68 -0.64 61.4 6.35 54.7 6.07 163 4 Ni i 6176.82 4.09 -0.53 63.0 6.54
48.0 6.13 228 4 Ni i 4829.03 3.54 -0.33 80.1 6.24 68.7 5.85 131 4 Ni i 6204.61
4.09 -1.14 21.7 6.26 18.2 6.08 226 4 Ni i 4852.56 3.54 -1.07 44.7 6.28 42.3
6.10 130 4 Ni i 6322.17 4.15 -1.17 18.3 6.24 – – 249 4 Ni i 4904.42 3.54 -0.17
87.1 6.19 78.1 5.84 129 4 Ni i 6327.60 1.68 -3.15 37.9 6.34 27.8 5.97 44 4 Ni
i 4913.98 3.74 -0.62 55.3 6.24 42.3 5.84 132 4 Ni i 6378.26 4.15 -0.90 31.5
6.32 23.5 6.04 247 4 Ni i 4935.83 3.94 -0.36 62.9 6.30 48.5 5.88 177 4 Ni i
6414.59 4.15 -1.21 16.8 6.23 – – 244 4 Ni i 4946.03 3.80 -1.29 28.0 6.35 17.0
5.96 148 4 Ni i 6482.81 1.93 -2.63 40.8 6.12 31.8 5.79 66 4 Ni i 4953.21 3.74
-0.66 56.2 6.30 40.9 5.85 111 4 Ni i 6598.61 4.23 -0.98 24.7 6.30 – – 249 4 Ni
i 4998.23 3.61 -0.78 57.5 6.33 43.2 5.90 111 4 Ni i 6635.14 4.42 -0.83 24.0
6.31 15.2 5.97 264 4 Ni i 5010.94 3.63 -0.87 48.4 6.23 33.4 5.79 144 4 Ni i
6767.78 1.83 -2.17 79.6 6.47 67.4 6.06 57 4 Ni i 5032.73 3.90 -1.27 23.8 6.31
16.2 6.00 207 4 Ni i 6772.32 3.66 -0.99 48.4 6.27 35.3 5.89 127 4 Ni i 5035.37
3.63 0.29 107.0 6.05 91.5 5.63 143 4 Zn i 4722.16 4.03 -0.39 66.1 4.70 65.0
4.58 2 12 Ni i 5042.19 3.64 -0.57 61.8 6.21 52.6 5.88 131 4 Zn i 4810.54 4.08
-0.17 72.3 4.66 73.6 4.58 2 12 Ni i 5048.85 3.85 -0.37 66.1 6.29 51.5 5.86 195
4 Sr i 4607.34 0.00 0.28 45.4 2.91 32.0 2.28 2 13 Ni i 5082.35 3.66 -0.54 62.5
6.22 55.2 5.92 130 4 Y ii 4883.69 1.08 0.07 57.6 2.33 53.4 2.15 22 14 Ni i
5084.10 3.68 0.03 91.1 6.14 78.2 5.75 162 4 Y ii 5087.43 1.08 -0.17 48.3 2.25
31.1 1.69 20 14 Ni i 5088.54 3.85 -0.91 31.9 6.12 23.9 5.83 190 4 Zr ii
4208.98 0.71 -0.46 44.7 2.68 39.3 2.45 41 15 Ni i 5102.97 1.68 -2.62 48.1 6.16
36.7 5.74 49 4 Ba ii 4554.04 0.00 0.14 174.6 2.17 154.7 1.78 1 16 Ni i 5115.40
3.83 -0.11 75.2 6.19 68.9 5.90 177 4 Ba ii 5853.69 0.60 -0.91 62.8 2.28 51.2
1.87 2 16 Ni i 5155.13 3.90 -0.66 49.2 6.28 34.5 5.86 206 4 Ba ii 6141.71 0.70
-0.03 112.2 2.23 98.9 1.87 2 16 Ni i 5435.87 1.99 -2.60 52.0 6.52 40.0 6.09 70
4 Ba ii 6496.91 0.60 -0.41 97.5 2.30 90.6 2.01 2 16 Ni i 5587.87 1.93 -2.14
55.8 6.08 39.7 5.56 70 4 Ce ii 4562.37 0.48 0.21 21.9 1.65 18.6 1.47 1 17 Ni i
5593.75 3.90 -0.84 45.4 6.35 28.3 5.88 206 4 Ce ii 4628.16 0.52 0.14 18.4 1.63
12.9 1.34 1 17 Ni i 5625.33 4.09 -0.70 38.1 6.24 28.8 5.94 221 4 Nd ii 4446.40
0.20 -0.35 12.7 1.41 8.0 1.08 49 18 Ni i 5637.12 4.09 -0.80 33.3 6.23 – – 218
4 Nd ii 5092.80 0.38 -0.61 7.0 1.48 – – 48 18 Ni i 5641.89 4.10 -1.08 24.6
6.31 13.1 5.87 234 4 Nd ii 5293.17 0.82 0.10 10.3 1.39 – – 75 18 Ni i 5682.21
4.10 -0.47 51.9 6.30 33.9 5.83 232 4 Sm ii 4577.69 0.25 -0.65 5.2 0.96 – – 23
19

  * References for the adopted $gf$-values: (1) Takeda et al. (2003), (2) Pehlivan Rhodin et al. (2017), (3) Shi et al. (2011), (4) NIST Atomic Spectra Database (http://physics.nist.gov/PhysRefData/ASD), (5) Den Hartog et al. (2021), (6) Lawler et al. (2019), (7) Lawler et al. (2013), (8) Lawler et al. (2014), (9) Lawler et al. (2017), (10) Den Hartog et al. (2011), (11) Lawler et al. (2015), (12) Biemont & Godefroid (1980), (13) Hansen et al. (2013), (14) Hannaford et al. (1982), (15) Biemont et al. (1981), (16) Klose et al. (2002), (17) Lawler et al. (2009), (18) Den Hartog et al. (2003), (19) Lawler et al. (2006), ($\dagger$) Bard & Kock (1994), (*) Meléndez & Barbuy (2009)

Table 3: Solar abundances obtained by employing the Solar model atmosphere
from Castelli & Kurucz (2003) compared to the photospheric abundances from
Asplund et al. (2009). The abundances were calculated using the line EWs.

| HD 218209 |  Sun† | Asplund et al. (2009) |
---|---|---|---|---
Species |  $\log\epsilon$(X)1 |  [X/Fe]1 |  [X/Fe]2 | $N^{\rm 1}$ |  $\log\epsilon_{\rm\odot}$(X) | $N$ |  $\log\epsilon_{\rm\odot}$(X) |  $\Delta\log\epsilon_{\rm\odot}$(X)
| (dex) | (dex) | (dex) |  |  |  | (dex) |
Na i |  5.79$\pm$0.09 |  -0.02$\pm$0.14 |  0.00$\pm$0.18 | 2 |  6.16$\pm$0.07 | 2 |  6.24$\pm$0.04 | 0.08
Mg i |  7.48$\pm$0.06 |  0.23$\pm$0.13 |  0.09$\pm$0.20 | 2 |  7.60$\pm$0.08 | 2 |  7.60$\pm$0.04 | 0.00
Si i |  7.28$\pm$0.09 |  0.13$\pm$0.14 |  0.13$\pm$0.19 | 11 |  7.50$\pm$0.07 | 12 |  7.51$\pm$0.03 | 0.01
Ca i |  6.09$\pm$0.10 |  0.10$\pm$0.15 |  0.10$\pm$0.19 | 18 |  6.34$\pm$0.08 | 18 |  6.34$\pm$0.04 | 0.00
Sc i |  2.78$\pm$0.00 |  0.01$\pm$0.08 |  -0.33$\pm$0.14 | 1 |  3.12$\pm$0.00 | 1 |  3.15$\pm$0.04 | 0.03
Sc ii |  3.05$\pm$0.06 |  0.17$\pm$0.13 |  0.18$\pm$0.20 | 7 |  3.23$\pm$0.08 | 7 |  3.15$\pm$0.04 | 0.08
Ti i |  4.82$\pm$0.12 |  0.21$\pm$0.17 |  0.21$\pm$0.21 | 39 |  4.96$\pm$0.09 | 43 |  4.95$\pm$0.05 | 0.01
Ti ii |  4.86$\pm$0.13 |  0.22$\pm$0.17 |  0.18$\pm$0.19 | 12 |  4.99$\pm$0.08 | 12 |  4.95$\pm$0.05 | 0.04
V i |  3.81$\pm$0.07 |  0.17$\pm$0.12 |  0.05$\pm$0.15 | 4 |  3.99$\pm$0.05 | 5 |  3.93$\pm$0.08 | 0.06
Cr i |  5.30$\pm$0.11 |  -0.06$\pm$0.15 |  -0.01$\pm$0.18 | 19 |  5.71$\pm$0.07 | 19 |  5.64$\pm$0.04 | 0.07
Cr ii |  5.32$\pm$0.16 |  0.03$\pm$0.23 |  -0.01$\pm$0.20 | 3 |  5.64$\pm$0.14 | 3 |  5.64$\pm$0.04 | 0.00
Mn i |  5.06$\pm$0.15 |  -0.21$\pm$0.21 |  -0.23$\pm$0.23 | 12 |  5.62$\pm$0.13 | 13 |  5.43$\pm$0.05 | 0.19
Fe i |  7.17$\pm$0.12 |  -0.02$\pm$0.17 |  -0.01$\pm$0.20 | 128 |  7.54$\pm$0.09 | 132 |  7.50$\pm$0.04 | 0.04
Fe ii |  7.16$\pm$0.07 |  0.00$\pm$0.11 |  0.00$\pm$0.19 | 15 |  7.51$\pm$0.04 | 17 |  7.50$\pm$0.04 | 0.01
Co i |  4.62$\pm$0.15 |  0.00$\pm$0.23 |  0.03$\pm$0.26 | 6 |  4.97$\pm$0.15 | 7 |  4.99$\pm$0.07 | 0.02
Ni i |  5.91$\pm$0.12 |  -0.02$\pm$0.17 |  -0.03$\pm$0.21 | 50 |  6.28$\pm$0.09 | 54 |  6.22$\pm$0.04 | 0.06
Zn i |  4.58$\pm$0.01 |  0.25$\pm$0.09 |  0.26$\pm$0.17 | 2 |  4.68$\pm$0.03 | 2 |  4.56$\pm$0.05 | 0.12
Sr i |  2.28$\pm$0.00 |  -0.28$\pm$0.08 |  -0.16$\pm$0.14 | 1 |  2.91$\pm$0.00 | 1 |  2.87$\pm$0.07 | 0.04
Y ii |  1.92$\pm$0.32 |  -0.02$\pm$0.33 |  -0.15$\pm$0.22 | 2 |  2.29$\pm$0.05 | 2 |  2.21$\pm$0.05 | 0.08
Zr ii |  2.45$\pm$0.00 |  0.12$\pm$0.08 |  0.27$\pm$0.14 | 1 |  2.68$\pm$0.00 | 1 |  2.58$\pm$0.04 | 0.10
Ba ii |  1.88$\pm$0.09 |  -0.01$\pm$0.13 |  0.06$\pm$0.20 | 4 |  2.24$\pm$0.06 | 4 |  2.18$\pm$0.09 | 0.06
Ce ii |  1.40$\pm$0.09 |  0.11$\pm$0.12 |  0.25$\pm$0.21 | 2 |  1.64$\pm$0.02 | 2 |  1.58$\pm$0.04 | 0.06
Nd ii |  1.08$\pm$0.00 |  0.01$\pm$0.09 |  0.19$\pm$0.14 | 1 |  1.42$\pm$0.05 | 3 |  1.42$\pm$0.04 | 0.00
Sm ii | – | – | – | – |  0.96$\pm$0.00 | 1 |  0.96$\pm$0.04 | 0.00

(1) The abundances are obtained using the ELODIE spectrum. (2) The abundances
are obtained using the PolarBASE spectrum. (*) $\Delta{\rm
log}\epsilon_{\rm\odot}{\rm(X)=log}\epsilon_{\rm\odot}{\rm(X)}_{\rm
This\,study}-{\rm log}\epsilon_{\rm\odot}{\rm(X)}_{\rm Asplund}$. ($\dagger$)
The Solar abundances calculated in this study.

Further verification of the $\log gf$ values was performed by comparing the
$\log gf$ values used in this study with those in the Gaia-ESO line list v.6
provided by the GES collaboration (Heiter et al., 2021). It should be noted
that the $gf$-values for the chosen lines of Fe i and Fe ii in this study were
taken from the compilation of Fuhr & Wiese (2006). The GES line list contains
a list of recommended lines and atomic data (hyperfine structure-has-corrected
$gf$-values) for the analysis of FGK stars. It is worth noting that several
lines in the spectra of FGK stars have not yet been identified (Heiter et al.,
2015).

The GES line list (v.6) contains a total of 141 233 lines in the wavelength
range 4200 – 10 000 Å. This number is reduced to 84 735 lines in the
wavelength range 4200 - 6800 Å. For species that match those in the line list
(24 in total) provided in this study in the same wavelength range, the GES
contains 50 326 lines. This wavelength range is the same as the wavelength
range for the ELODIE spectrum used for diagnostics. Comparing the GES line
list with the new line list provided useful information. We used a total of
363 lines from the line list to match the GES lines. Out of 363 lines, 11
lines were found to be out of range ($<$ 4200 Å) in the GES line list and the
two were missing (Y ii lines at 4883.69 Å and 5087.43 Å). The search resulted
in a one-to-one match of 350 lines in terms of wavelength, element, $\log gf$
and low LEP. The difference between the $\log gf$ values444$\log gf_{\rm
This\,study}$ \- $\log gf_{\rm GES}$ for the 128 common Fe i lines in the GES
line list was -0.03$\pm$0.06 dex. For the 16 Fe ii lines, the difference in
$\log gf$ values was -0.002$\pm$0.079 dex. For, Na i, Si i, Ca i, Sc ii, Ti i,
Ti ii, V i, Cr i, Ni i, Zn i, Sr i, Zr ii, and Sm ii, the difference in the
$\log gf$ values was less than 0.05 dex. For 54 common Ni i lines, the
difference was found to be -0.045$\pm$0.114 dex. A formal least square
solution gives a gradient of 0.99 and a zero point difference of -0.02 dex
over 350 common lines. However, there were large discrepancies in the $\log
gf$ values for transitions for some elements common to both line lists.
Details about this discrepancy are included in the following section.

###  3.2 Model Parameters and Abundances

We used neutral and ionised Fe lines to determine model atmospheric parameters
such as effective temperature, surface gravity, microturbulence, and
metallicity (Table 4). First, the effective temperature was calculated by
requiring that the resulting abundance be independent of the lower LEP.

Table 4: Model atmosphere parameters for HD 218209 and the Sun.

Star | $T_{\rm eff}$ | $\log g$ | [Fe/H] | $\xi$
---|---|---|---|---
| (K) | (cgs) | (dex) | (km s-1)
HD 218209† | 5650${}_{-170}^{+165}$ | 4.55${}_{-0.21}^{+0.18}$ | -0.35 ${}_{-0.07}^{+0.07}$ | 0.60 ${}_{-0.50}^{+0.50}$
HD 218209∗ | 5630${}_{-160}^{+170}$ | 4.43${}_{-0.26}^{+0.22}$ | -0.31 ${}_{-0.13}^{+0.13}$ | 0.35 ${}_{-0.50}^{+0.50}$
Sun | 5790${}_{-125}^{+125}$ | 4.40${}_{-0.23}^{+0.16}$ | 0.00 ${}_{-0.04}^{+0.04}$ | 0.66 ${}_{-0.50}^{+0.50}$

  * ($\dagger$) The ELODIE spectrum was used.

  * (*) The PolarBase spectrum was used.

If all lines have the same LEP and a similar wavelength, the microturbulence
($\xi$) is determined by requiring that the calculated abundance be
independent of the reduced equivalent width (EW). The precision in the
determination of the microturbulent velocity is $\pm$0.5 km s-1. We calculated
surface gravity ($\log g$) by requiring ionisation equilibrium, which means
that the Fe i and Fe ii lines yield the same iron abundance (Figures 3 and 4).
Due to the interdependence of model parameters, an iterative procedure is
required. Between each of the above steps, minor adjustments are made to the
model parameters. We also confirmed that there is no substantial trend in iron
abundances (see Figures 3 and 4 for the Sun and HD 218209, respectively).

Figure 3: An example for the determination of the atmospheric parameters
$T_{\rm eff}$ and $\xi$ using abundance ($\log\epsilon$) as a function of both
lower LEP (panels a and b for Fe and Ti, respectively) and reduced EW (REW;
$\log$ (EW/$\lambda$), panels c for Fe and Ti, respectively) for the Sun. In
all panels, the solid red line represents the least-squares fit to the data.

Figure 4: An example for the determination of the atmospheric parameters
$T_{\rm eff}$ and $\xi$ using abundance ($\log\epsilon$) as a function of both
lower LEP (panels a and b for Fe and Ti, respectively) and reduced EW (REW;
$\log$ (EW/$\lambda$), panels c for Fe and Ti, respectively) for HD 218209 and
the ELODIE spectrum. In all panels, the solid red line represents the least-
squares fit to the data.

The model parameters obtained for the Sun as a result of the Solar analysis
were determined as $T_{\rm eff}=5790$ K, $\log g=4.4$ cgs, [Fe/H]=0 dex and
$\xi=0.66$ km s-1. These values are similar to those recommended by Heiter et
al. (2015) as ($T_{\rm eff}$, $\log g$) = (5771, 4.438). Element abundances
for the Sun were calculated with these model parameters. For HD 218209, the
model parameters were obtained from ELODIE and PolarBASE spectrum of the star.

During the calculation of the model atmosphere, the effect of convection on
the reported abundances is also taken into consideration. In one-dimensional
(1D) atmosphere modeling, it is currently common practice to assume a
universal value of 1.5 for the mixing length parameter, $\alpha$. However, in
order to test the impact of convection on final abundances in this study, the
ATLAS models were produced using two different methods for calculating
$\alpha$. Two well known references in the literature, Ludwig et al. (1999)
and Magic et al. (2015), for the calculation of $\alpha$, yields two different
values (1.59 and 1.98) even when the same set of model parameters is employed.
The logarithmic abundance of ionized iron showed a 0.01 dex increase for the
model atmosphere computed with the $\alpha$ parameter set to 1.59 and the
model parameters from the ELODIE spectrum. For $\alpha=$1.98, it is +0.02 dex.
With regards to the FGK dwarf field stars, it is expected that the influence
of the mixing length parameter, for instance, on the determination of
metallicity would be minimal, amounting to less than 0.02 dex over eight dwarf
stars (Song et al., 2020). Thus, our results for HD 218209 confirm those of
Song et al. (2020). Regarding the model parameters of HD 218209 from the
PolarBASE spectrum, the logarithmic abundance of ionized iron remained the
same for the model atmosphere computed with $\alpha=$1.59. The difference in
logarithmic abundance is 0.02 dex for $\alpha=$1.98.

The resulting stellar model parameters for HD 218209 along with our
determination of model parameters of the Sun are listed in Table 4. Table 5
lists the model parameters reported for the star in the literature. Rebolo et
al. (1988), Abia et al. (1988), and Kim et al. (2016) show the largest
variations compared to our $\log g$. The abundances obtained for the Solar
photosphere as a result of the Solar analysis are given in Table 3. The Solar
abundances from Asplund et al. (2009) are also included for comparison. In
Table 3, we provide a summary of element abundances based on LTE-based model
parameters. $\log\epsilon$ is the logarithm of the abundances. The errors
reported in $\log\epsilon$ abundances are the 1$\sigma$ line-to-line scatter
in abundances. [X/H] is the logarithmic abundance ratio of hydrogen to the
corresponding Solar value, and [X/Fe] is the logarithmic abundance considering
the Fe i abundance. The error in [X/Fe] is the square root of the sum of the
quadratures of the errors in [X/H] and [Fe/H]. Table 3 also presents the
abundances obtained using the PolarBase spectrum of the star as [X/Fe] ratio.

Table 5: Atmospheric parameters of HD 218209 from this study. $T_{\rm eff}$ | $\log g$ | [Fe/H] |  Notes
---|---|---|---
(K) | (cgs) | (dex) |
5650 | 4.55 | -0.37 | This study (ELODIE)
5630 | 4.43 | -0.32 | This study (PolarBASE)
5506 | 4.38 | -0.46 | Takeda (2023)
5636 | 4.56 | -0.47 | Rice & Brewer (2020)
5518 | 4.39 | -0.49 | Aguilera-Gómez et al. (2018)
5623 | 4.46 | -0.40 | Luck (2017)
5555 | 4.26 | -0.49 | Boeche & Grebel (2016)
5565 | 4.29 | -0.49 | Boeche & Grebel (2016)
5669 | 4.24 | -0.39 | Kim et al. (2016)
5575 | 4.11 | -0.46 | Kim et al. (2016)
5607 | 4.07 | -0.58 | Kim et al. (2016)
5536 | 4.37 | -0.46 | Da Silva et al. (2015)
5705 | 4.50 | -0.43 | Mishenina et al. (2015)
5705 | 4.50 | -0.43 | Mishenina et al. (2013)
5592 | 4.25 | -0.51 | Lee et al. (2011)
5592 | 4.33 | -0.61 | Lee et al. (2011)
5592 | 4.31 | -0.65 | Lee et al. (2011)
5705 | 4.50 | -0.43 | Mishenina et al. (2011)
5539 | 4.37 | -0.50 | Da Silva et al. (2011)
5600 | – | – | Casagrande et al. (2011)
5473 | 4.57 | -0.64 | Sozzetti et al. (2009)
5506 | 4.38 | -0.46 | Takeda et al. (2007)
5693 | – | – | Masana et al. (2006)
5585 | 4.60 | -0.46 | Valenti & Fischer (2005)
5684 | – | – | Kovtyukh et al. (2004)
5705 | 4.50 | -0.43 | Mishenina et al. (2004)
5665 | 4.40 | -0.60 | Gehren et al. (2004)
5478 | 4.00 | -0.60 | Rebolo et al. (1988)
5478 | 4.00 | -0.60 | Abia et al. (1988)

##  4 Summary and Conclusion

We present a line list that can be useful for abundance analysis of G-type
stars over 4080 – 6780 Å wavelength region (i.e. for spectra from the
ELODIE/SOPHIE library). The line list is expected to be useful for the
surveys/libraries with overlapping spectral regions (e.g. ELODIE library,
UVES-580 setting of Gaia-ESO, and, especially, for the analysis of F- and
G-type stars in general. We identified 363 lines of 24 species that have
accurate $gf$-values and are free of blends in the spectra of the Sun and a
Solar analogue star, HD 218209. It should be noted that the GES line list does
not contain any atomic transitions below 4200 Å and unlike the GES line list,
the line list created in this study contains additional four Fe i, one Fe ii,
1 Sc i, 2 Ti i and 1 Co i line below this limit. To assess the uncertainty in
$\log gf$-values and to minimise systematic errors, we calculated Solar
abundances using stellar lines and the high-resolution KPNO spectrum of the
Sun. For the creation of the line list in this study, high resolution and high
signal to noise ELODIE spectrum of HD 218209 was used. During the comparison
with the GES line list, discrepancies in the $\log gf$ values of some ions
were detected. For example, the average difference in $\log gf$ values for 5 V
i lines was 0.325 +- 0.328. Similarly, for 11 Mn i lines, the difference was
0.288 +- 0.283 dex. The largest difference detected was 0.585$\pm$0.161 dex
over 6 Co i lines. For the other elements reported in the new line list
presented in this study, the $\log gf$ difference was less than 0.1 dex.

To verify the accuracy of the model parameters obtained for HD 218209 from the
ELODIE spectrum, we analyzed the PolarBASE spectrum of the same star. The
model parameters from both spectra matched quite well, taking into account the
error margins. The element abundances obtained from both spectra also matched
quite well, except for Sci which was represented by a single line at 4023.69 Å
in both spectra. The logarithmic abundance from this line in the ELODIE showed
a 0.3 dex increase. Additionally, three other species (Sr i, Zr ii, and Ce
ii), with the latter two species presented by two transitions in the spectra,
had discrepancies in their logarithmic abundances at around 0.2 dex level.
Although the PolarBASE spectrum had higher resolution than the ELODIE, several
lines in the PolarBASE did not have clear line profiles. Unfortunately, there
was only one spectrum available in the PolarBASE archive, so a more thorough
analysis is recommended to check the abundances of the star from the PolarBASE
spectrum.

The model parameters obtained for HD 218209 clearly indicate a Solar analogue
character for the star. Although no definite classification scheme exists,
Solar analogue stars are defined as the dwarf stars of early G-type which have
properties analogous to the Sun (e.g. differences in $T_{\rm eff}$ and $\log
g$ are $\pm\leq$ 100–200 K and $\pm\leq$ 0.1–0.2 dex). Indeed, seeing the star
listed in a recent study by Takeda (2023) as a Solar analogue was not
surprising for us. The model parameters ($T_{\rm eff}=5506$ K, $\log g=4.38$
cgs, [Fe/H]=-0.46 dex, $\xi=$ 0.74 km s-1) reported by Takeda (2023) are in
good agreement with those obtained in this study. However, we report up-to-
date abundances for the star using 24 species identified in both ELODIE and
PolarBase spectra of the star.

HD 218209 is listed in Gaia DR2 (Gaia Collaboration et al., 2018) with Gaia
DR2 2213415145505277824 designation and a $T_{\rm eff}$ of 5648 K. The surface
gravity and metallicity of the star were not reported. HD 218209 is also
listed in the Gaia DR3 (Gaia Collaboration et al., 2023). Our spectroscopic
measurements of the temperature and surface gravity of the star align very
well with the values reported by the Gaia consortium, which are $T_{\rm
eff}=5528$ K and $\log g=4.40$ cgs. However, there seems to be an error in the
reported metallicity of the star by Gaia, which states [Fe/H] = -0.79 dex.
This discrepancy is likely due to the incorrect selection of the template
spectrum for the star in the Gaia DR3, where the metallicity of the template
star is reported as -0.50 dex.

In the Gaia DR3 (Gaia Collaboration et al., 2023), an important addition is a
collection of 220 million low-resolution BP/RP spectra. The GSP-Phot catalogue
provides consistent estimations of stellar parameters for 471 million sources
with $G<19$ mag, derived from the BP/RP spectra, parallax, and integrated
photometry. It assumes that each source is a single star and that any
intrinsic time variability is lost when using combined BP/RP spectra. However,
it is important to note that the GSP-Phot results from the BP/RP spectra in
the Gaia DR3 may still be affected by systematic effects, particularly in the
[M/H] estimates with large systematic errors. Consequently, it is advised not
to rely on these estimates as they can only provide qualitative information at
best (Beck et al., 2023).

In a study of stellar and sub-stellar companions of nearby stars from the Gaia
DR2 (Gaia Collaboration et al., 2018), Kervella et al. (2019) examined the
status of stars exhibiting anomalies in their proper motions as binary stars.
HD 218209 as a high proper-motion star happens to be one of the common stars.
The study found no evidence of a companion around HD 218209, even at distances
ranging from 1 to 50 astronomical units (au) and down to planetary masses.
Therefore, the authors concluded that HD 218209 is a single star (P. Kervella,
private communication, 2023).

Soubiran & Girard (2005), in their exploration of abundance discrepancies
between the thin disc and thick disc of the Galaxy, identified HD 218209 out
of the 44 stars as being part of the Hercules stream with a likelihood of
0.76. The Hercules stream shares chemical characteristics with the thin disc,
which bolsters the dynamic hypothesis (i.e. the influence of the central bar
of the Galaxy impacting stars) of its origin. However, kinematic and orbital
dynamics analysis for the star is the subject of another study.

## Acknowledgements

This study has partly been supported by the Scientific and Technological
Research Council (TÜBİTAK) MFAG-121F265. We thank Ferhat Güney for his help
with the preparation of Figure 2. We also thank Gizay YOLALAN and Sena A.
ŞENTÜ̈RK for helpful discussion.

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