11institutetext: Sternberg Astronomical Institute, M.V. Lomonosov Moscow State
University, 13, Universitetskij pr., 119234, Moscow, Russia 11email:
<EMAIL_ADDRESS>22institutetext: Department of Physics and Astronomy,
University of Southern California, Los Angeles, CA 90089, USA 33institutetext:
Institute of Problems of Chemical Physics RAS, Chernogolovka, Russia
44institutetext: Joint Institute for High Temperatures RAS, Moscow, Russia
55institutetext: Moscow Institute of Physics and Technology, Dolgoprudnyi,
Russia

# Ionization of heavy elements and the adiabatic exponent in the solar plasma

V.A. Baturin Ionization of heavy elements and the adiabatic exponent in the
solar plasmaIonization of heavy elements and the adiabatic exponent in the
solar plasma A.V. Oreshina Ionization of heavy elements and the adiabatic
exponent in the solar plasmaIonization of heavy elements and the adiabatic
exponent in the solar plasma W. Däppen Ionization of heavy elements and the
adiabatic exponent in the solar plasmaIonization of heavy elements and the
adiabatic exponent in the solar plasma S.V. Ayukov Ionization of heavy
elements and the adiabatic exponent in the solar plasmaIonization of heavy
elements and the adiabatic exponent in the solar plasma A.B. Gorshkov
Ionization of heavy elements and the adiabatic exponent in the solar
plasmaIonization of heavy elements and the adiabatic exponent in the solar
plasma V.K. Gryaznov Ionization of heavy elements and the adiabatic exponent
in the solar plasmaIonization of heavy elements and the adiabatic exponent in
the solar plasma I.L. Iosilevskiy Ionization of heavy elements and the
adiabatic exponent in the solar plasmaIonization of heavy elements and the
adiabatic exponent in the solar plasmaIonization of heavy elements and the
adiabatic exponent in the solar plasmaIonization of heavy elements and the
adiabatic exponent in the solar plasma

(Received 26 July 2021 / Accepted 31 January 2022)

###### Abstract

Aims.

Methods.

Results.

###### Abstract

Context. The adiabatic exponent $\Gamma_{1}$ is studied as a thermodynamic
quantity in the partially ionized plasma of the solar convection zone.

Aims. The aim of this study is to understand the impact of heavy elements on
the $\Gamma_{1}$ profile. We calculated $\Gamma_{1}$ with the SAHA-S equation
of state for different chemical compositions of plasma, and we analyzed
contributions of individual elements to $\Gamma_{1}$. We attempted to
determine the mass fractions of the heavy elements using our analysis of the
$\Gamma_{1}$ profile.

Methods. We studied the decrease in $\Gamma_{1}$ due to the ionization of
heavy elements in comparison with the value obtained for a pure hydrogen-
helium plasma. These types of differences are denoted as “$Z$ contributions”,
and we analyzed them for eight elements (C, N, O, Ne, Mg, S, Si, and Fe) as
well as for a mixture of elements corresponding to the solar chemical
composition. The contributions of the heavy elements are studied on an adiabat
in the lower part of the convection zone, where the influence of hydrogen and
helium to the $Z$ contribution is minimal. The $Z$-contribution profiles are
unique for each chemical element. We compared linear combinations of
individual $Z$ contributions with the exact $Z$ contribution. Applying a
least-squares technique to the decomposition of the full $Z$ contribution to a
basis of individual-element contributions, we obtained the mass fractions of
the heavy elements.

Results. The $Z$ contribution of heavy elements can be described by a linear
combination of individual-element $Z$ contributions with a high level of
accuracy of $5\times 10^{-6}$ . The inverse problem of estimating the mass
fractions of heavy elements from a given $\Gamma_{1}$ profile was considered
for the example of solar-type mixtures. In ideal numerical simulations, the
mass fractions of the most abundant elements could be determined with a
relative accuracy better than a few tenths of a percent. In the presence of
random or systematic errors in the $\Gamma_{1}$ profile, abundance estimations
become remarkably less accurate, especially due to unknown features of the
equations of state. If the amplitude of the errors does not exceed $10^{-4}$,
we can expect a determination of at least the oxygen abundance with a relative
error of about 10%. Otherwise, the results of the method would not be
reliable.

###### Key Words.:

Equation of state – Methods: numerical – Sun: abundances – Sun: interior

##  1 Introduction

Properties of the adiabatic exponent $\Gamma_{1}$ are studied for conditions
of the solar interior. The thermodynamic quantity $\Gamma_{1}$ describes the
variation of pressure $P$ under adiabatic compression. It is determined by the
formula $\Gamma_{1}={\left.{\partial\ln P}/{\partial\ln\rho}\;\right|}_{S}$,
where $\rho$ is the density and $S$ is the entropy.

It is commonly known that in a mono-atomic ideal gas
${{\Gamma}_{1}}={5}/{3}\;$. In the present paper, we are mostly interested in
the deviation from this value due to the ionization of chemical elements. It
is well known that a region of ionization generally contributes to a lowering
of $\Gamma_{1}$ from its mono-atomic value. There are several analytical
expressions that illustrate the effect of the hydrogen ionization on
$\Gamma_{1}$ (Cox & Giuli 1968; Hansen et al. 2004). Also, further examples of
analytic expressions are given in Appendix A. The typical result for a one-
stage ionization is a bell-shaped decrease in $\Gamma_{1}$ as a function of
the ionization degree.

Interpretation of this decrease in $\Gamma_{1}$ is analogous to inferences
from the decrease in intensity in an absorption spectral line. While in the
latter the line position indicates the transition energy and the line depth
the mass fraction of the chemical element, in the case of $\Gamma_{1}$ the
position of the decrease indicates an ionization zone of a given chemical
element and the depth of the decrease is proportional to the abundance of that
element.

In the deeper layers of the solar convection zone (CZ), the stratification is
nearly adiabatic. In that part of the Sun, the $\Gamma_{1}$ profile can be
observationally inferred with high accuracy, since uncertainties from
convection matter little. Thus, the observational $\Gamma_{1}$ can be used to
study theoretical models for the ionization of various elements in a wide
range of temperatures and densities.

The computation of thermodynamic values for a mixture of partially ionized
elements requires an advanced equation of state, which – besides ionization –
takes into account complex physical effects: charged particle interaction,
electron degeneration, and relativistic effects. These elaborated methods lead
to significant computational costs, which is why interpolations in pre-
computed tabulated data are often used. Some of the problems associated with
the computation and interpolation of $\Gamma_{1}$ in modern equations of state
are described in (Hansen et al. 2004; Baturin et al. 2017, 2019) and in
Appendix A. In the present paper, we used the data of the SAHA-S equation of
state (Gryaznov et al. 2006, 2013) (see also the SAHA-S web-
site111http://crydee.sai.msu.ru/SAHA-SaEOS). This equation of state was chosen
on the matter of high internal precision and consistency of thermodynamic
functions. This is a requirement for an analysis of rather small effects
associated with heavy elements.

Investigations of the effect of ionization on adiabatic exponents started in
the 1980s. Däppen & Gough (1984) showed that the sound speed and $\Gamma_{1}$
significantly decrease in the second helium-ionization region of the solar
convection zone. This ionization of helium has captured attention, because its
effect on $\Gamma_{1}$ is strong and can be used for a helioseismological
estimation of the helium abundance $Y$. Kosovichev et al. (1992) noticed that
such an estimation of $Y$ can be influenced by the $\Gamma_{1}$ variation due
to heavy elements. The authors examined several different theories of the
ionization of heavy elements and concluded that helioseismological inversions
are sensitive to this description. Däppen et al. (1993) also studied the
influence of heavy elements on $\Gamma_{1},$ with an emphasis on the impact of
excited states in compound species.

Since the stellar plasma contains a mixture of heavy elements, the relative
abundances of the elements in the mixture are of interest. Elliott (1996)
studied the possibility of estimating both total and individual contributions
based on the $\Gamma_{1}$ profile. He used derivatives
$\partial{{\Gamma}_{1}}/\partial{{f}_{i}}$ with respect to parameters
${{f}_{i}}$ that describe the heavy-element mass fractions ${{Z}_{i}}$. The
derivatives are a linear combination of all element contributions. From a
physical viewpoint, such derivatives are similar to the contributions to
$\Gamma_{1}$ from element mass fractions used in our work.

Element contributions were analyzed in the paper by Gong et al. (2001).
Individual contributions of heavy elements were computed by subtracting
$\Gamma_{1}$ profiles obtained in different approximate solar models. The
models were computed for special mixtures, in which the 2% of heavy-element
mass was assigned to only one element (C, N, O, or Ne). Such models were
compared to models without heavy elements. The element contributions were
computed using several different equations of state (OPAL and MHD, among
others). The authors showed that $\Gamma_{1}$ decreases in the lower part of
the convection zone in the lg T=[5.5, 6.5] temperature range, where electrons
are detached from the inner K envelope of ions. At that time, the authors
found that the decrease is due entirely to the heavy-element contribution and
is almost independent of the chosen equation of state. In our work, we
compared $\Gamma_{1}$ for different chemical compositions at fixed
$({{T}_{j}},{{\rho}_{j}})$ points while keeping the same equation of state.
This allowed us to focus solely on the thermodynamics of the $\Gamma_{1}$
decrease.

A helioseismological determination of the heavy element mass fraction $Z$ in
the convection zone has been attempted by several authors. The paper by
Baturin et al. (2000) examined the thermodynamic effect of heavy elements on
the $\Gamma_{1}$ profile. The possibility of a helioseismological estimation
of $Z$ based on $\Gamma_{1}$ was considered in the works by Baturin et al.
(2000) and Lin & Däppen (2005). Antia & Basu (2006) and Lin et al. (2007)
obtained $Z$ = 0.0172. However, Vorontsov et al. (2013, 2014) estimated Z =
0.008–0.013. The low $Z$ value is confirmed by helioseismic inversions in
Buldgen et al. (2017). Their results are close to recent spectroscopic data
(Asplund et al. 2005, 2009, 2021). Inversions by Vorontsov et al. (2013, 2014)
indicated a slightly higher $\Gamma_{1}$ on the Sun compared to the standard
model. This result can be matched by a modern equation of state with a low
$Z$.

Our present paper continues the work by Baturin (2010) and Baturin et al.
(2013), where a method for the individual $Z$ contributions was developed
together with an application and improvement of the SAHA-S equation of state.
Baturin (2010) used a SAHA-S version with four heavy elements (C, N, O, and
Ne). Later computations by Baturin et al. (2013) were based on a version with
an extended chemical composition (8 elements, including Fe, Si, Mg, and S) and
a detailed physics of the excited states of all compound species.

The main purpose of the paper is to study the physical and mathematical
foundations that enter the $\Gamma_{1}$ profile in the relevant region of the
Sun, which lies in the solar convection zone. The role of the $\Gamma_{1}$
profile for seismic abundance determinations is indisputable and well
established.

The physical principles studied in the paper are based on the sequential
ionization of each species of a mixture of heavy elements as well as the
computation of the sum of all individual effects. It turns out that there are
two main results concerning the additivity of the effects in the resulting
profile of $\Gamma_{1}$. First, we have additivity of the contributions from
different heavy elements, and second, we have additivity of contributions from
the sequential ionization stages of a particular element.

Mathematically, the function of $\Gamma_{1}$ localized in a piece of the solar
convection zone is decomposed with respect to a set of basis vectors. The
basis vectors, belonging to the effect of each heavy element on $\Gamma_{1}$,
span a linear space. The idea is that a given $\Gamma_{1}$ profile will reveal
the abundances of the heavy elements through the coefficients of the basis
vectors. We want to emphasize that the basis functions are determined by
thermodynamics, in particular the atomic structure of chemical elements. They
depend of course on the details of the underlying equation of state. However,
our analysis is dedicated to the process of adding the various effects.
Surely, any error in the basic atomic and thermodynamic theory would lead to
an overall uncertainty of the final result. However, the analysis of such
errors will have to be done in the future, when the method is applied to real
helioseismic data. In the present work, we deliberately restricted ourselves
to the mechanism of addition of the various effects.

The element-induced dips of $\Gamma_{1}$ in ionization regions are well known
and have been extensively studied before. To isolate the relevant effects, in
our paper we introduce so-called $Z$ contributions instead of the $\Gamma_{1}$
profile itself. Using $Z$ contributions improves the sensitivity to
abundances, but it reduces the domain of applicability to a ”space” around the
exact ideal $\Gamma_{1}$ profile. We computed this ideal solution and
demonstrated that there is a neighborhood around it where the method works.

In this work, we did not touch on the question of determining the heavy-
element abundance in the Sun. The reason is that there are too many factors
precluding such an application at the moment: (1) the absence of a good
$\Gamma_{1}$ from inversions, (2) the unknown entropy in the convection zone,
(3) the presently used equations of state might still not be sufficiently
close to reality, and so on. Therefore, the present paper is somewhat
preliminary. We chose solar conditions merely in order to make our study as
realistic as possible, not to imply that the method can be applied to the Sun
immediately.

The aim of our work is (i) to estimate the effect of the individual heavy
elements on the $\Gamma_{1}$ profile and (ii) to find a solution to the
inverse problem; that is, in determining the mass fractions ${{Z}_{i}}$ out of
the overall contribution of the heavy elements to $\Gamma_{1}$.

We give results of numerical computations as well as analytical formulae for a
better qualitative understanding of the underlying physics. We show that on
the background of an almost completely ionized hydrogen and helium, the
decrease in $\Gamma_{1}$ can be represented as a linear combination of the
individual contributions of heavy elements. We also consider an inverse
problem; that is, the possibility, in principle, of determining the mass
fractions of the individual heavy elements from the $\Gamma_{1}$ profile. The
inverse problem is studied in a preliminary way with a numerical experiment,
because the real task needs more accurate determinations of $\Gamma_{1}$ from
observations, which are currently not available and have to await future
developments.

The plan of the paper is as follows. In Sect. 2 we analyze the profile of the
adiabatic exponent $\Gamma_{1}$. The temperature range for this is selected in
Sect. 2.1, and the sets of heavy-element mass fractions are described in Sect.
2.2. $\Gamma_{1}$ profiles are considered in Sect. 2.3, and the total $Z$
contribution to $\Gamma_{1}$ and the components of the individual
contributions of the elements are defined in Sects. 2.4 and 2.5. In Sect. 2.6,
we demonstrate that the total $Z$ contribution can be presented as a linear
combination of a set of basis functions. In Sect. 2.7, we consider the inverse
problem, that is, an estimation of the heavy-element mass fractions based on
the $\Gamma_{1}$ profile. The influence of the hydrogen content on the
solution is studied in Sect. 2.8. In Sect. 3, we consider the limited space
resolution and pseudo-random errors of $\Gamma_{1}$ profile. The main results
of the work are presented in Sect. 4. Appendix A describes the physics of the
decrease in $\Gamma_{1}$ in regions of ionization analytically. In Appendix B,
we consider some examples of systematic and random errors.

##  2 Analysis of the adiabatic-exponent profile

###  2.1 Domain of analysis

The $\Gamma_{1}$ profile is computed for the points of temperature $T$ and
density $\rho$ of the solar model from (Ayukov & Baturin 2017), labeled
771-0001. In that paper, all model details are described. The main point is
that it is a standard solar model for a high heavy-element abundance. Its
depth of the convection zone is typical for high-$Z$ models
($R_{\mathrm{cz}}/R_{\odot}=0.712$). In Appendix B, we also use the standard
low-$Z$ model 771-0002, which has a shallow convection zone.

In all calculations, we use the fixed set of points $(T,\rho)$ when
calculating all theoretical $\Gamma_{1}$ (for $Z=0$ and for the element
contributions). It is a distinctive feature of our technique. Our method does
not depend on the specific choice of a set of points $T$ and $\rho$. A set can
be chosen from a model or from proper thermodynamic calculations of the
adiabat. Deviations from the adiabat are also possible.

In the framework of an ideal experiment, $\Gamma_{1}$ for different mixtures
$Z$ was calculated on the same set of points. When a different, but fixed,
profile $(T,\rho)$ is used, the result of least-square decomposition (Sect.
2.7) will be the same.

In the general case, however, the function studied could correspond to
$\Gamma_{1}$ calculated at an arbitrary set of points $(T,\rho)$. Within the
framework of our approach, the variations of points $(T,\rho)$ are represented
by perturbations of $\Gamma_{1}$. We performed experiments with $(T,\rho)$
sets that were taken from the two standard models (Appendix B.2).

We use, on the one hand, equation-of-state tables with selected mixtures of
heavy elements and, on the other hand, the tables that contain only one heavy
element at fixed mass fraction $Z=0.01$. The results are compared with those
from tables computed for a pure hydrogen-helium mixture, that is $Z=0$.

The $\left(T,\rho\right)$ points from the solar model are shown by the red
line in Fig. 1. Boundaries of the solar convection zone (CZ) are marked by
filled green points. The convection zone becomes virtually adiabatic at $\lg
T>4.3$, so entropy is constant. However, entropy varies in the colder outer
layers, where convection is inefficient. To illustrate this fact, two limit
adiabats are shown in Fig. 1 by the blue curves, with all points of the
convection zone lying between them.

The black curves in Fig. 1 show the points where the 50% ionization of the
innermost electron occurs. For hydrogen and helium, this is in the outer
layers of the CZ. However, the final ionization of carbon, nitrogen, and
oxygen happens in the lower part of the CZ. Neon ionizes below the CZ.

Figure 1: Temperature and density in the standard solar model (red curve).
Filled green points mark the boundaries of the convection zone. Blue curves
are adiabats. Black curves show points where 50% ionization of the last
electron occurs. The top axis shows the corresponding radius in the solar
model.

All $\Gamma_{1}$ profiles in our work have been analyzed for an interval of
temperatures between $\lg T=5.4$ and 6.4. Hydrogen and helium are near fully
ionized under these conditions. Furthermore, the chosen temperature interval
almost lies in the adiabatic part of the CZ.

There are no exact natural limits for the boundaries of the interval. Of
course, such a choice is always a compromise. That the interval contains
points inside the CZ is not a strict requirement for the method, though it is
desirable, as deviations from the adiabat can introduce some errors. In the
used standard high-$Z$ model, the boundary of the CZ is located at $\lg
T=6.344$. The temperature range has been slightly expanded to $6.4$ to include
the neon ionization zone. Neon would be too important to be ignored.

The same arguments are applied to the low-temperature boundary. It is not
possible to remove the H and He ionization zone completely. A shrinking of the
interval would cause a deterioration of the basis due to an increase of linear
dependence of the basic functions. We also left the extended range to include
an example of neon L-shell ionization.

Undoubtedly, narrowing the interval reduces the contribution of incomplete
ionization of hydrogen and helium, but our choice of boundaries is a
compromise between the stability of the basis when calculating the $Z$
contribution and the linear independence of the basis. In other words, by
narrowing the interval, we strengthen the linear dependence of the basis
functions (see Sect. 2.4).

###  2.2 Mixtures of heavy elements

We analyze the $\Gamma_{1}$ profile for mass fractions of hydrogen $X=0.7$,
helium $Y=0.28$, and heavy elements $Z=0.02$. Obviously, the abundances of the
individual heavy elements satisfy the relation
$\sum\nolimits_{i}{{{Z}_{i}}}=Z$. The SAHA-S equation of state takes into
account all ionization stages of all eight heavy elements considered (C, N, O,
Ne, Mg, Si, S, and Fe). The relative mass fractions ${{A}_{i}}={{Z}_{i}}/Z$ of
the four mixtures studied are presented in Table 1. The mass fractions
${{Z}_{i}}$ are also shown, since they are needed to analyze the results of
the computations.

Table 1: Relative mass fractions ${{A}_{i}}$ and absolute mass fractions ${{Z}_{i}}=Z\cdot{{A}_{i}}$ for the heavy elements used in our mixtures with $Z=0.02$. | AGSS09 | GN93 | OPAL | SAS5
---|---|---|---|---
Elem. | $A_{i}$ | $Z_{i},\times 10^{-2}$ | $A_{i}$ | $Z_{i},\times 10^{-2}$ | $A_{i}$ | $Z_{i},\times 10^{-2}$ | $A_{i}$ | $Z_{i},\times 10^{-2}$
C | 0.1816 | 0.3633 | 0.1772 | 0.3544 | 0.1907 | 0.3813 | 0.2729 | 0.5459
N | 0.0532 | 0.1064 | 0.0543 | 0.1087 | 0.0558 | 0.1117 | 0.0686 | 0.1371
O | 0.4403 | 0.8805 | 0.4932 | 0.9864 | 0.5430 | 1.0860 | 0.4967 | 0.9934
Ne | 0.0965 | 0.1930 | 0.0986 | 0.1972 | 0.2105 | 0.4210 | 0.0432 | 0.0863
Mg | 0.0544 | 0.1087 | 0.0384 | 0.0768 | 0 | 0 | 0.0404 | 0.0807
Si | 0.0511 | 0.1021 | 0.0414 | 0.0829 | 0 | 0 | 0.0328 | 0.0656
S | 0.0237 | 0.0475 | 0.0216 | 0.0432 | 0 | 0 | 0.0134 | 0.0267
Fe | 0.0992 | 0.1984 | 0.0751 | 0.1503 | 0 | 0 | 0.0321 | 0.0641

Mixture AGSS09 is obtained by proportionally scaling the contents of the eight
heavy elements from the mixture by Asplund et al. (2009). Mixture GN93 is
obtained similarly by using the contents of Grevesse & Noels (1993). The
SAHA-S equation of state also includes a table computed specially for a
comparison with the widely known, often used, OPAL equation of state (Rogers
et al. 1996; Rogers & Nayfonov 2002). The OPAL mixture reproduces the mass
fractions used in the OPAL equation of state. It contains only four heavy
elements (C, N, O, and Ne). SAS5 is an artificial mixture with an increased
carbon content. All mixtures differ in the relative contents of the heavy
elements, whereas the total mass fraction is the same for all: $Z=0.02$ .
Therefore, the mixtures differ in that the abundances of individual heavy
elements are being changed, but the total mass of all heavy elements
considered remains unchanged. Oxygen is the most abundant heavy element in the
solar plasma. In AGSS09 and GN93, carbon takes the second place and neon the
third. In SAS5, carbon and nitrogen are in the second and third places,
respectively. In OPAL, all elements heavier than neon are treated as neon,
which thus has a fictitious abundance. That way, in OPAL, neon is in the
second, and carbon in the third place.

###  2.3 Profile of $\Gamma_{1}$ in the solar model

The theoretical profile of $\Gamma_{1}$ for the points $\left(T,\rho\right)$
of the solar model is represented by the red curve in Fig. 2a. The adiabatic
exponent is equal to 5/3 in a perfect gas. It decreases in regions of
ionization of atoms and ions. The maximum decrease occurs at temperatures $\lg
T=4-5$ due to the ionization of hydrogen and helium. The blue and green curves
illustrate the effect of ionization of the light elements. The blue curve is
computed for pure hydrogen, and the green curve is for a plasma in which the
helium content is twice that of the Sun.

Figure 2: Adiabatic exponent $\Gamma_{1}$ for points $(T,\rho)$ of the
standard solar model for different chemical composition of plasma. (a) Effect
of helium. (b) Effect of different mixtures of heavy elements.

We focused on fine, wave-like perturbations with an amplitude of about
$3\times{{10}^{-3}}$ at temperatures between $\lg T=5.4$ and 6.4. This range
is presented at a larger scale in Fig. 2b. The light blue dashed curve
describes $\Gamma_{1}$ for a hydrogen-helium plasma. The excess of
$\Gamma_{1}$ over $5/3$ arises due to the nonideal effect from Debye
screening, which can lead to an increase in the elasticity of the plasma. The
physical mechanism of this phenomena in the solar CZ is described in Baturin
et al. (2000).

The two-percent addition of heavy elements to the mixture decreases
$\Gamma_{1}$. This decrease is different for the four different mixtures
(AGSS09, GN93, OPAL, and SAS5), which allows us to use the profile of
$\Gamma_{1}$ to determine the detailed chemical composition of the plasma.

###  2.4 Definition of $Z$ contribution to $\Gamma_{1}$

To understand the nature of these perturbations, let us consider the
difference ${{\delta}_{Z}}{{\Gamma}_{1}}$ of $\Gamma_{1}$ computed for a
plasma with a mixture of heavy elements ($X=0.70,Y=0.28,Z=0.02$) and
$\Gamma_{1}$ for a pure hydrogen-helium plasma ($X=0.70,Y=0.30,Z=0$):

${{\delta}_{Z}}{{\Gamma}_{1}}={{\Gamma}_{1}}-\Gamma_{1}^{\mathrm{HHe}}.$ (1)

In the following, we use the term $Z$ contribution to denote this difference.
It is our tool to disentangle the contribution of the heavy elements (Fig. 3).
The $Z$ contribution has a negative sign in the temperature range considered.
The magnitude of ${{\delta}_{Z}}{{\Gamma}_{1}}$ is few fractions of a percent
at $Z=0.02$. The important feature of $Z$ contribution is dependence on the
chemical composition of plasma. Figure 3 shows the $Z$ contributions for
different mixtures.

Figure 3: $Z$ contribution ${{\delta}_{Z}}{{\Gamma}_{1}}$ of 2% admixtures
AGSS09, GN93, OPAL, and SAS5.

Analyzing the $Z$ contribution ${{\delta}_{Z}}{{\Gamma}_{1}}$ instead of the
full $\Gamma_{1}$ plays a fundamental role. Nonideal effects, interpolation
errors, and some effects of hydrogen and helium ionization are subtracted in
this case. The $Z$ contribution is related to the ionization of a small
admixture on the background of the constant ionization of hydrogen and helium.
By “small admixture” we mean that the number of electrons added by the element
under consideration is small compared to all electrons in the plasma. Thus, we
study their ionization over the background of almost fully ionized hydrogen
and helium, which provides the bulk of the electrons.

The effect of the equation of state generally consists of two parts. One is
caused by ideal ionization, the other by nonideal effects, such as the Coulomb
correction and others. By maintaining the equation of state constant, the $Z$
contributions allow us to subtract most of the Coulomb correction. Thus, we
amplify the effect of the heavy-element ionization. If in the same study one
had also used another equation of state for the $\Gamma_{1}$ contribution, the
nonideal part could not have been subtracted. However, since we succeed in
doing that, in this sense we can claim independence of the equation of state.

In the current version of the method, a theoretical background profile of
$\Gamma_{1}^{\mathrm{HHe}}$ is computed for known $X$, fixed points
$(T,\rho)$, in frames of one equation of state. Certainly, if the nature of
$\Gamma_{1}$ is unknown, the true background profile differs from the
suggested theoretical profile, and then additional errors will enter our
method. These errors are an unavoidable price for the increasing sensitivity
to contributions of heavy elements. We give some examples of a possible
variation of $\Gamma_{1}^{\mathrm{HHe}}$ in Appendix B.1. We provide an
example for an alternative equation of state (OPAL) as well (see Appendix
B.3).

###  2.5 Basis of the $Z$ contributions of elements

We now consider individual $Z$ contributions
$\delta_{Z}^{i}{{\Gamma}_{1}}(X,Y,{{Z}_{i}}^{\mathrm{b}})$ of eight heavy
elements available in the SAHA-S equation of state (Fig. 4). Here, the index
“b” in variable $Z_{i}^{\mathrm{b}}$ means “basis”, and $i=1...8$ labels a
heavy element (C, N, O, etc.). The contributions are obtained using special
computations for plasma, in which one percent of mass is represented by only
one heavy element (${{Z}_{i}}^{\mathrm{b}}=0.01$), and the rest of the masses
are hydrogen $(X=0.70)$ and helium ($Y=0.29$). So, we obtain adiabatic
exponent $\Gamma_{1}^{i}$ and then subtract a background that is
$\Gamma_{1}^{\rm HHe}$ for hydrogen-helium plasma ($X=0.70,Y=0.30,Z=0$):

$\delta_{Z}^{i}{{\Gamma}_{1}}=\Gamma_{1}^{i}-{{\Gamma}_{1}}^{\rm HHe}.$ (2)

Figure 4: Basis $\delta_{Z}^{i}{{\Gamma}_{1}}$ of individual contributions to
${{\delta}_{Z}}{{\Gamma}_{1}}$.

The calculation of the $Z$ contribution can be performed either for a fixed
$X$ or for a fixed $Y$. If $Y$ is fixed, the error of the linear synthesis
(see the following section) increases by about two times in comparison with
fixed $X$. It may be explained by the simultaneous incomplete ionization of
hydrogen and helium at the low-temperature edge of the interval. Since the
number of helium particles is about ten times less than the number of hydrogen
particles, the effect of the incomplete hydrogen ionization is stronger. The
$\Gamma_{1}$ profile is explained qualitatively by ionizations (see Appendix
A).

For the most abundant elements (C, N, O, and Ne) in the adiabatic part of the
convection zone, the main role is played by ionization corresponding to
detachment of the last two electrons from the inner, so-called K, shell. For
neon and heavier elements, a contribution of L-shell ionization falls into the
range being investigated.

Every curve has a clear minimum at a certain temperature. That is why these
functions are convenient to use as a basis of decomposition.

###  2.6 Linear synthesis of the $Z$ contribution

Straight modeling of the $Z$ contribution ${{\delta}_{Z}}{{\Gamma}_{1}}$
consists of the computation of the following linear combination (LC
hereafter):

$\delta_{Z}{{\Gamma}_{1}}^{\mathrm{LC}}=\sum\limits_{i}{\left({{{Z}_{i}}}/{\Delta{{Z}^{\mathrm{b}}}}\;\right)\cdot\text{
}}\delta_{Z}^{i}{{\Gamma}_{1}},$ (3)

using known mass fractions ${{Z}_{i}}$, basis functions
$\delta_{Z}^{i}{{\Gamma}_{1}}$ (Eq. (2), and $\Delta{{Z}^{\mathrm{b}}}=0.01$.
Fig. 5a shows all eight terms of the sum (3) and the linear combination
${{\delta}_{Z}}{{\Gamma}_{1}}^{\mathrm{LC}}$(thick red line) for the mixture
AGSS09. The most abundant elements (oxygen, carbon, neon, and nitrogen) give
the main contributions.

An analysis of the contributions in Fig. 5a allows us to interpret the shape
of the resultant profile. Oxygen and neon create the local minimum at $\lg
T=5.4$. Carbon produces the minimum at $\lg T=5.84,$ and oxygen causes the
decrease at $\lg T=6.1$.

Figure 5b shows contributions of magnesium, silicon, sulfur, and iron at the
larger scale. The violet curve is their sum. The total contribution of these
elements reaches ${{10}^{-3}}$ , which is comparable to the contribution of
carbon. The contribution of iron is dominant in the group. Its minimum almost
coincides with carbon, which complicates the problem of differential analysis.
The OPAL equation of state does not include these elements.

Figure 5: Weighted contributions ${{Z}_{i}}\cdot\delta_{Z}^{i}{{\Gamma}_{1}}$
of individual elements and their sum
${{\delta}_{Z}}{{\Gamma}_{1}}^{\mathrm{LC}}$ for the AGSS09 mixture. (a)
Contributions of all eight considered elements. (b) Contributions of Mg, Si,
S, and Fe at a larger scale.

Figure 6 illustrates the linear synthesis of $\delta_{Z}\Gamma_{1}$ for three
mixtures: AGSS09, OPAL, SAS5. It shows only contributions of the most abundant
elements (C, N, O, and Ne) and their sum without the contribution of the
heavier group. The figure demonstrates that variations of contents in the C-Ne
group can significantly change the total Z-profile. The figure allows us also
to compare the content of elements in the considered mixtures. For example,
enhanced content of neon in OPAL is the main feature of the mixture. The
direct effect of neon can be noticed at $\lg T=5.4$ and at $\lg T=6.4$; oxygen
contribution is also significant there. The content of oxygen and neon is more
prominent in OPAL than in other mixtures, which results in deeper total $Z$
contribution (blue solid thick curve). Similarly, a large percentage of carbon
in mixture SAS5 produces a relative decrease at $\lg T=5.84$ (green solid
thick curve), which is minor in other mixtures. However, this carbon minimum
is influenced by the heavy group, first of all by iron, and can disappear in
the total curve.

Figure 6: Weighted contributions ${{Z}_{i}}\cdot\delta_{Z}^{i}{{\Gamma}_{1}}$
of main elements (C, N, O, and Ne) in ${{\delta}_{Z}}{{\Gamma}_{1}}$ for
mixtures AGSS09, OPAL, and SAS5.

We now compare the linear combination ${{\delta}_{Z}}{{\Gamma}_{1}}^{\rm LC}$
and the exact $Z$ contribution ${{\delta}_{Z}}{{\Gamma}_{1}}$ to estimate the
accuracy of the linear combination. The difference is shown in Fig. 7. It does
not exceed $5\times{{10}^{-6}}$ for all considered mixtures. The relative
error is about 0.1%, which demonstrates the reliability of the linear
synthesis. The errors of the linear synthesis for the mixtures are generally
similar, and the divergence between them does not exceed
$(2-3)\times{{10}^{-6}}$.

The error profiles have a peculiar character. The error is positive at low
temperatures and negative at high temperatures of the range in Fig. 7. The
physical nature of the error of the linear synthesis is related to ionization
of elements in a mixture. We assume that the electron concentration in a
mixture slightly differs from that upon computation of individual basis
contributions. This leads to somewhat different ionization conditions for key
components. The error of the linear synthesis plays an important role in the
inverse problem discussed in the next section.

Figure 7: Difference between the linear combination
${{\delta}_{Z}}{{\Gamma}_{1}}^{\mathrm{LC}}$ and exact $Z$ contribution
${{\delta}_{Z}}{{\Gamma}_{1}}$ for mixtures AGSS09, GN93, OPAL, and SAS5.

###  2.7 Inverse problem of least-squares decomposition

The inverse problem consists of determining mass fractions of heavy elements
${{Z}_{i}}$ using a given ${{\delta}_{Z}}{{\Gamma}_{1}}$ profile. We solve the
problem using the method of linear, unconstrained least squares (LS).

We analyze ${{\delta}_{Z}}{{\Gamma}_{1}}$ computed in the SAHA-S equation of
state for the chemical composition $X=0.70,$ $Y=0.28,$ $Z=0.02$ and different
mixtures. Then, we estimate mass fractions ${{Z}_{i}}^{\mathrm{LS}}$ using the
LS method. The results are presented in Table 2.

Table 2: Mass fractions of heavy elements ${{Z}_{i}}^{\mathrm{LS}}$ determined
from a given $Z$ contribution ${{\delta}_{Z}}{{\Gamma}_{1}}$ , absolute error
$\Delta{{Z}_{i}}$ and relative error $\Delta{{Z}_{i}}/{{Z}_{i}}$ .

| AGSS09 | GN93 | OPAL | SAS5
---|---|---|---|---
Elem. | $Z_{i}^{\mathrm{LS}},$ | $\Delta Z_{i},$ | ${\Delta Z_{i}}/{Z_{i}},$ | $Z_{i}^{\mathrm{LS}},$ | $\Delta Z_{i},$ | ${\Delta Z_{i}}/{Z_{i}},$ | $Z_{i}^{\mathrm{LS}},$ | $\Delta Z_{i},$ | ${\Delta Z_{i}}/{Z_{i}},$ | $Z_{i}^{\mathrm{LS}},$ | $\Delta Z_{i},$ | ${\Delta Z_{i}}/{Z_{i}},$
| $\times 10^{-2}$ | $\times 10^{-6}$ | % | $\times 10^{-2}$ | $\times 10^{-6}$ | % | $\times 10^{-2}$ | $\times 10^{-6}$ | % | $\times 10^{-2}$ | $\times 10^{-6}$ | %
C | 0.3666 | 33 | 0.9 | 0.3578 | 34 | 1.0 | 0.3830 | 17 | 0.4 | 0.5505 | 48 | 0.9
N | 0.1066 | 2 | 0.2 | 0.1092 | 5 | 0.4 | 0.1136 | 19 | 1.7 | 0.1363 | -8 | -0.6
O | 0.8803 | -2 | -0.02 | 0.9856 | -8 | -0.1 | 1.0841 | -19 | -0.2 | 0.9923 | -11 | -0.1
Ne | 0.1937 | 7 | 0.3 | 0.1980 | 8 | 0.4 | 0.4204 | -6 | -0.1 | 0.0889 | 26 | 3.0
Mg | 0.1154 | 66 | 6.1 | 0.0830 | 61 | 8.0 | 0.0057 | 57 | - | 0.0854 | 47 | 5.8
Si | 0.0970 | -51 | -5.0 | 0.0783 | -45 | -5.5 | -0.0032 | -32 | - | 0.0616 | -40 | -6.1
S | 0.0472 | -3 | -0.6 | 0.0425 | -7 | -1.6 | -0.0012 | -12 | - | 0.0256 | -11 | -4.2
Fe | 0.1961 | -24 | -1.2 | 0.1481 | -21 | -1.4 | -0.0006 | -6 | - | 0.0614 | -27 | -4.2
$Z^{\mathrm{LS}}$ | 0.020028 | 0.020026 | 0.020018 | 0.020024
$\eta$ | $2.320\times 10^{-6}$ | $2.525\times 10^{-6}$ | $2.704\times 10^{-6}$ | $2.996\times 10^{-6}$
$\varepsilon$ | $2.513\times 10^{-7}$ | $2.778\times 10^{-7}$ | $2.280\times 10^{-7}$ | $5.249\times 10^{-7}$
$\sigma$ | $2.307\times 10^{-6}$ | $2.509\times 10^{-6}$ | $2.694\times 10^{-6}$ | $2.949\times 10^{-6}$

222The results are obtained from the decomposition using all eight considered
elements, whereas the OPAL mixture includes only four elements.

Table 2 shows absolute and relative differences between the computed values
${{Z}_{i}}^{\mathrm{LS}}$ and original mass fractions ${{Z}_{i}}$ (see Table
1):

$\Delta{{Z}_{i}}={{Z}_{i}}^{\mathrm{LS}}-{{Z}_{i}},\qquad\frac{\Delta{{Z}_{i}}}{{{Z}_{i}}}=\frac{{{Z}_{i}}^{\mathrm{LS}}-{{Z}_{i}}}{{{Z}_{i}}}\cdot
100\%.$ (4)

The results demonstrate good agreement between the values. For the AGSS09
mixture, discrepancies are smaller or about one percent. The exceptions are Mg
and Si, for which relative differences reach 5-6%. The errors have opposite
signs and compensate each other in the total sum. The LS method estimates the
mass fraction of all heavy elements, ${{Z}^{\mathrm{LS}}}=0.020028,$ which is
close to the original value $Z=0.02$. In the case of GN93, the errors are
similar. The OPAL mixture includes only four elements: carbon, nitrogen,
oxygen, and neon; the accuracy is several tenths of a percent for C, O, and
Ne, and 1.7% for N. The negative coefficients of Si, S, and Fe have no
physical meaning and represent an error in the method. In the SAS5 mixture,
the errors for carbon, nitrogen, and oxygen are fractions of a percent. For
the rest elements, they reach several percent. The total mass fraction of
heavy elements $Z$ is estimated with an accuracy of one tenth of a percent in
all considered cases. Thus, we conclude that the LS method gives an accurate
estimation for mass fractions of heavy elements under the ideal condition that
we know the percentages of hydrogen and helium, which are needed to compute
the background $\Gamma_{1}^{\mathrm{HHe}}$ and basis
$\left\\{\delta_{Z}^{i}{{\Gamma}_{1}}\right\\}$ .

Figure 8 schematically presents a geometrical interpretation of the LS method.
Every point in the figure corresponds to a certain
${{\delta}_{Z}}{{\Gamma}_{1}}\left(T\right)$. Point $E$ corresponds to the
exact $Z$ contribution ${{\delta}_{Z}}{{\Gamma}_{1}}$, computed in the
equation of state for a mixture of elements. Plane $\alpha$ is a subspace of
all linear combinations and its dimension equals the number of basis elements.
Every point from subspace $\alpha$ corresponds to vector of mass fractions
$\left\\{{{{\tilde{Z}}}_{i}}\right\\}$.

Figure 8: Geometrical interpretation of the least-squares method.

Point $E$ does not belong to subspace $\alpha$ in general; that is, no linear
combination can fully accurately reproduce the $Z$ contribution
${{\delta}_{Z}}{{\Gamma}_{1}}$. There are two particular points in subspace
$\alpha$. Point $L$ corresponds to linear combination
${{\delta}_{z}}{{\Gamma}_{1}}^{\mathrm{LC}}$ with original values ${{Z}_{i}}$
from Table 1. Point $P$ describes linear combination
${{\delta}_{z}}{{\Gamma}_{1}}^{\mathrm{LS}}$ with values
$\left\\{{{Z}_{i}}^{\mathrm{LS}}\right\\}$ obtained using the LS method. Point
$P$ is the nearest point in subspace $\alpha$ to $E$ by definition of the LS
method. Segment $EP$ is orthogonal to $\alpha$, because the LS method finds
orthogonal projection of point $E$ onto subspace $\alpha$. Every segment is a
difference of $Z$ contributions. The segments can be characterized by
Euclidean norms (distance) computed as standard deviations of $Z$ profiles.
For the three segments in Fig. 8, they are

$\eta={{\left\|EL\right\|}_{2}}=\sqrt{\frac{1}{N}\sum\limits_{j=1}^{N}{{{[{{\delta}_{z}}{{\Gamma}_{1}}^{\mathrm{LC}}-{{\delta}_{Z}}{{\Gamma}_{1}}]}^{2}}}},$
(5)
$\varepsilon={{\left\|EP\right\|}_{2}}=\sqrt{\frac{1}{N}\sum\limits_{j=1}^{N}{{{[{{\delta}_{z}}{{\Gamma}_{1}}^{\mathrm{LS}}-{{\delta}_{Z}}{{\Gamma}_{1}}]}^{2}}}},$
(6)
$\sigma={{\left\|PL\right\|}_{2}}=\sqrt{\frac{1}{N}\sum\limits_{j=1}^{N}{{{[{{\delta}_{z}}{{\Gamma}_{1}}^{\mathrm{LC}}-{{\delta}_{Z}}{{\Gamma}_{1}}^{\mathrm{LS}}]}^{2}}}}.$
(7)

Here, summation is performed over all points $({{T}_{j}},{{\rho}_{j}}),\text{
}j=1...N$ of the solar model (Fig. 1) within the $\lg T=5.4-6.4$ range. The
orthogonality of the segments results in the relation
${{\eta}^{2}}={{\varepsilon}^{2}}+{{\sigma}^{2}}.$

Segment $EL$ of length $\eta$ estimates the inaccuracy of linear synthesis
that is caused by physical nonlinearity of ionization in the mixture that is
considered in Sect. 2.6. $\eta\approx(2-3)\times{{10}^{-6}}$(see Table 2).
Segment $\varepsilon=EP$ is a distance from point $E$ to plane $\alpha$. It
describes a part of $Z$ contribution that cannot be represented as a linear
combination of the element contributions. Value $\varepsilon$ is mainly
determined by the completeness of the basis and thus depends on subspace
$\alpha$. Including additional elements expands $\alpha$ and results in a
decrease in $\varepsilon$. However, if a considered temperature range is
narrow, then $\alpha$ cannot be expanded because new components are linearly
dependent on those already included in $\alpha$.

It is interesting to consider LS decomposition in the case of applied
constraints to the space $\alpha$. Constraints may be applied by a limited
number of basic elements or a requirement of nonnegative abundances, or by the
known total Z content. It is important to note that the squeezing of space may
lead only to an increase of least squares $\varepsilon$. These approaches do
not increase the accuracy of the LS solution and could simply help to reject
physically inappropriate solutions.

Value $\sigma={{\left\|PL\right\|}_{2}}$ characterizes an error of estimation
$Z_{i}^{\mathrm{LS}}$ compared to the original values ${{Z}_{i}}$. If
$\varepsilon\ll\sigma,$ then $\sigma\to\eta$. Thus, an estimation error cannot
be eliminated, because a possible decrease in $\varepsilon$ does not decrease
$\sigma$.

In the inverse problem, the LS method estimates
$\left\\{{{Z}_{i}}^{\mathrm{LS}}\right\\}$ and hence $\varepsilon$ (not $\eta$
or $\sigma,$ because we do not know, a priori, the original
$\left\\{{{Z}_{i}}\right\\}$). So, $\varepsilon$ deserves special attention.
In our test experiments, value $\varepsilon$ is six-to-ten times smaller than
$\eta$. Therefore, we conclude that $\varepsilon$ is close to its minimal
value and eight elements are sufficient for space $\alpha$. The obtained mass
fractions and their errors are not likely to be improved by including
additional elements. If a profile contains an additional undecomposed
component, the residuals increase and the LS estimation of
$\left\\{{{Z}_{i}}^{\mathrm{LS}}\right\\}$ becomes unreliable (Sect. 2.8).

###  2.8 Case of unknown hydrogen content

The computation of $Z$ contributions (Eqs. (1) and (2)) uses background
profile ${{\Gamma}_{1}}^{\mathrm{HHe}}$ for hydrogen-helium plasma and
profiles $\Gamma_{1}^{i}$ for different elements. However, hydrogen content
can be unknown in the investigated profile $\Gamma_{1}$ and hence in the
theoretical functions ${{\Gamma}_{1}}^{\mathrm{HHe}}$ and in $\Gamma_{1}^{i}$.
In this section, we estimate the sensitivity of the method to the hydrogen
content in the studied profile ${{\delta}_{Z}}{{\Gamma}_{1}}$ and in the
profiles $\delta_{Z}^{i}{{\Gamma}_{1}}$ used for decomposition.

We use the mixture AGSS09 as an example. The test function $\Gamma_{1}$ is
computed for $X=0.70,Y=0.28,Z=0.02$, but we take $X=0.69$ for the computation
of the background ${{\Gamma}_{1}}^{\mathrm{HHe}}$and basis
$\left\\{\delta_{Z}^{i}{{\Gamma}_{1}}\right\\}$. Thus, the content of hydrogen
in the studied function differs from the background and the basis. The results
of LS decomposition are presented in Table 3.

Table 3: Mass fractions of heavy elements ${{Z}_{i}}^{\mathrm{LS}}$determined from the test profile $\Gamma_{1}$ at $X=0.70$ that differs from $X=0.69$ in the background ${{\Gamma}_{1}}^{\mathrm{HHe}}$and basis $\left\\{\delta_{Z}^{i}{{\Gamma}_{1}}\right\\}$.  Elem. | $Z_{i}^{\mathrm{LS}},\times 10^{-2}$ |  $\Delta Z_{i}/Z_{i}$, %
---|---|---
C | 0.2620 | -28
N | 0.0900 | -15
O | 0.8342 | -5
Ne | 0.1435 | -26
Mg | -0.0777 | -171
Si | 0.2505 | 145
S | 0.0884 | 86
Fe | 0.2931 | 48
$Z^{\mathrm{LS}}$ | 0.018840
$\eta$ | $6.925\times 10^{-5}$
$\varepsilon$ | $1.044\times 10^{-5}$
$\sigma$ | $6.846\times 10^{-5}$

Errors $\Delta{{Z}_{i}}/{{Z}_{i}}$ are significantly increased (see Table 2).
They reach tens of percent in most cases and exceed 100% for magnesium and
silicon. Moreover, a negative mass fraction $Z_{i}$ is obtained for magnesium
that has no physical meaning. The standard deviations $\eta$, $\varepsilon$,
and $\sigma$ are also increased significantly. The results show that hydrogen
content in the background and basis plays a critical role.

The problem of unknown $X$ can be resolved by varying hydrogen content in the
background and basis and subsequently determining minimal $\varepsilon$.
Figure 9a presents the results of the approach. Standard deviation
$\varepsilon$ is minimal at $X=0.70$ in the background and basis, which
corresponds to the original value of hydrogen content in the analyzed profile
$\Gamma_{1}$. Fig. 9b shows the errors of LS estimations at various $X$ in the
background and basis, which are minimal at $X=0.70.$

Figure 9: (a) Standard deviation $\varepsilon$ obtained using LS method for
various assumptions about hydrogen content $X$ . (b) Errors of mass fractions
for carbon, oxygen, and neon at various $X$ in the background and basis.

This is a special experiment in which we assume that there are no errors, and
only the parameter $X$ is unknown. The purpose of this epsilon analysis is not
to obtain an independent $X$ determination, but the best $X$ value, for which
the application of the LS decomposition is optimal. Using an independently
provided $X$ that differs from our optimal value would lead to a deterioration
of the LS estimation of the $Z$ abundances.

##  3 Emulation of limited space resolution and pseudo-random and systematic
errors

We emulate random disturbances of $\Gamma_{1}$ at limited space resolution.
The main reason for this is to obtain a relation between the amplitude of the
disturbances and the level of error in the LS decomposition.

We consider an experiment with a $\Gamma_{1}$ profile that has some error
function added to it. We choose an error function defined at 20 points with a
spacing of $\Delta\lg T\approx 0.05$ in the $\lg T=5.4-6.4$ range. At these
points, random errors were generated according to a normal (Gaussian)
distribution with standard deviation $\sigma_{\Gamma}=10^{-4}$. Of course,
this experiment is a particular one. However, we think that these added errors
are representative for possible errors in helioseismic inversions. The result
is described in Appendix B.4, and it serves to estimate the error tolerance in
the $\Gamma_{1}$ profile.

Obviously, adding errors to the $\Gamma_{1}$ profile deteriorates the results.
However, if the standard deviation of errors $\sigma_{\Gamma}$ is under
$10^{-4}$ , the abundance of oxygen could still be estimated with a relative
accuracy of 10%.

An additional experiment was performed with a set of $\Gamma_{1}$ points taken
from a solar model based on the OPAL equation of state. Twenty points in
radius were used. Details of this experiment are described in Appendix B.5.
This example of the decomposition somewhat differs from other experiments in
the paper. The most represented elements (C, O, and Ne) are determined with a
relative accuracy better than 10%. Despite the satisfactory accuracy, we
cannot draw any further conclusions, since this last experiment stands apart
from the main idea of the paper and its interpretation would require
considerable additional analysis.

##  4 Conclusion

The main result of the paper is that the $Z$ contribution in $\Gamma_{1}$ for
a mixture of elements is a linear combination of the contributions of
individual elements. Each contribution is proportional to the content of the
element and does not depend on the contribution of other elements. The
hypothesis of linearity is not always satisfied. In the following, we list the
factors that are essential for linearity.

(i) If several elements simultaneously ionize, the interaction between them
occurs through free electrons. The linearity of the contributions will be
fulfilled in the ionization model of small admixtures at the background of
almost completely ionized hydrogen and helium. Each heavy element is
represented by a small mass fraction, and the ionization of hydrogen and
helium practically does not change. Then, the total number of free electrons
is approximately constant, which is a condition for the linearity of the
contributions.

(ii) In addition to the linearity of the contributions of individual elements,
the independence of the contributions from successive ionizations of one
element plays an important role. If the ionization energies of the different
ionization stages are close to each other, the contributions of the successive
ionizations to $\Gamma_{1}$ are superimposed and summed up. This happens
during the ionization of L electrons for example. As a result, the $Z$
contribution of the element becomes wide and poorly localized. A broad dip as
a sum of narrow dips from sequential ionizations contains less information
than several separated dips. This circumstance complicates the use and
interpretation of such contributions as basis functions. In this case,
linearity is preserved, but sensitivity of $\Gamma_{1}$ to this element
becomes weak due to poor localization of the ionization region by temperature.

(iii) On the other hand, the ionization of K electrons for C, N, O, and Ne
plays an important role. The contribution to $\Gamma_{1}$ of K-electron
ionization is relatively isolated from the contributions of other ionizations.
Such contributions cover almost all the investigated adiabatic part of the
convective zone, which allows us to use them as basis functions.

The inverse problem consists of estimating the content of elements based on a
theoretical $\Gamma_{1}$ profile and contributions from individual elements
and $\Gamma_{1}$ for the H-He background. We use the LS method of
decomposition of the $Z$ contribution $\Gamma_{1}$ by $Z$ contributions of
eight elements. The method gives a satisfactory result in an ideal numerical
experiment with a known hydrogen content. Mass fractions of heavy elements are
estimated with relative accuracy up to a few tenths of a percent.

Numerical experiments allow us to make several conclusions regarding LS
estimations. There is a residual irreducible error in the LS method. It is
caused by the nonlinearity of the $Z$ contribution for a mixture of elements.
The error is marked $(\varepsilon)$ in Fig. 8. Our result consists of an
estimation of the minimum value of this error in an ideal experiment.

We now discuss the optimal number of elements in the basis. The contribution
of additional elements in the basis turns out to be linearly dependent on the
$Z$ contributions of the already included elements. This linear independence
can be studied with an SVD decomposition of the array of the basis vectors.
Specifically, the condition number $\kappa=s_{max}/s_{min}$ for eight elements
is $134$, while for four major elements $\kappa=10$. Here, $s_{max}$ and
$s_{min}$ are the largest and smallest singular values. By adding additional
elements to the basis, $s_{max}$ can only increase and $s_{min}$ can only
decrease (Golub & Van Loan 2013). This proves that increasing the number of
basis elements would not give a substantial gain but rather lead to an ill-
posed LS problem. Yet, a too-small number of elements would lead to an
increase of $\varepsilon$, and consequently to large error in the abundance
determination. However, the precise optimal number of basic elements is
unknown and will demand further study.

The standard deviation $\sigma$ (Eq. (7)) characterizes the difference between
mass fractions $Z_{i}^{\mathrm{LS}}$ obtained using LS decomposition and
original ${{Z}_{i}}$. The value of $\sigma$ is six-to-ten times greater than
$\varepsilon$. It depends on the structure of the basis functions of the
individual elements.

The analysis of $Z$ contributions depends on the hydrogen content $X$. The
difference between the true $X$ and that used in the calculated basis
contributions can lead to the appearance of an erroneous component in the
profile ${{\delta}_{z}}{{\Gamma}_{1}}$. However, we can perform computations
for various $X$ and then use the value $X$ that gives minimal $\varepsilon$,
and therefore we obtain an optimal result of the LS decomposition. This
approach is illustrated in Sect. 2.8.

The listed reasons do not cover all possible sources of errors. For example,
there is the possibility of an error in the assumed entropy. The results have
an erroneous component if the basis functions are calculated for points of a
different adiabat. It can be corrected using an approach such as that
described for hydrogen, but a detailed experiment is beyond the scope of this
paper. Errors can also appear due to differences in the equations of state.
Estimations of mass fractions can be unstable if the calculations are
performed using different equations of state.

The basic conclusion is the existence of an ideal solution, with which it is
possible to obtain a reliable estimation of the content of most abundant
elements. The practical meaning of our results lies in finding the conditions
under which the LS decomposition can give relevant results.

The LS estimation can be applied to a helioseismic inversion of $\Gamma_{1}$,
provided that such a profile is obtained with a high accuracy. To successfully
apply the method, the amplitude of error in the solar $\Gamma_{1}$ profile
should be at least below $10^{-4}$. This is a necessary, but not sufficient
condition. Another important condition is that the theoretical equation of
state should be close enough to the true plasma properties in the Sun. The
measure of closeness can be expressed with the help of the epsilon deviation.

###### Acknowledgements.

We owe sincere thanks to Prof. Douglas Gough whose inspiring idea motivated
the present study. We are most grateful to Dr. Gaël Buldgen for helpful
comments and discussions. The addition of the experiments in Section 3
(described in detail in Appendix B) was suggested by the anonymous referee.

## References

  * 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. (2005) Asplund, M., Grevesse, N., & Sauval, A. J. 2005, Astronomical Society of the Pacific Conference Series, Vol. 336, The Solar Chemical Composition, ed. I. Barnes, Thomas G. & F. N. Bash, 25
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481
  * Ayukov & Baturin (2017) Ayukov, S. V. & Baturin, V. A. 2017, Astronomy Reports, 61, 901
  * Baturin (2010) Baturin, V. A. 2010, Ap&SS, 328, 147
  * Baturin et al. (2013) Baturin, V. A., Ayukov, S. V., Gryaznov, V. K., et al. 2013, Astronomical Society of the Pacific Conference Series, Vol. 479, The Current Version of the SAHA-S Equation of State: Improvement and Perspective, ed. H. Shibahashi & A. E. Lynas-Gray, 11
  * Baturin et al. (2000) Baturin, V. A., Däppen, W., Gough, D. O., & Vorontsov, S. V. 2000, MNRAS, 316, 71
  * Baturin et al. (2017) Baturin, V. A., Däppen, W., Morel, P., et al. 2017, A&A, 606, A129
  * Baturin et al. (2019) Baturin, V. A., Däppen, W., Oreshina, A. V., Ayukov, S. V., & Gorshkov, A. B. 2019, A&A, 626, A108
  * Buldgen et al. (2017) Buldgen, G., Salmon, S. J. A. J., Noels, A., et al. 2017, MNRAS, 472, 751
  * Cox & Giuli (1968) Cox, J. P. & Giuli, R. T. 1968, Principles of stellar structure (Gordon and Breach, New York)
  * Däppen & Gough (1984) Däppen, W. & Gough, D. O. 1984, in Liege International Astrophysical Colloquia, Vol. 25, Liege International Astrophysical Colloquia, 264–268
  * Däppen et al. (1993) Däppen, W., Gough, D. O., Kosovichev, A. G., & Rhodes, E. J., J. 1993, Astronomical Society of the Pacific Conference Series, Vol. 40, On the Influence of Treatment of Heavy Elements in the Equation of State on the Resulting Values of the Adiabatic Exponent, ed. W. W. Weiss & A. Baglin, 304
  * Elliott (1996) Elliott, J. R. 1996, MNRAS, 280, 1244
  * Golub & Van Loan (2013) Golub, G. H. & Van Loan, C. F. 2013, Matrix Computations, 4th edn. (Johns Hopkins University Press, Baltimore)
  * Gong et al. (2001) Gong, Z., Däppen, W., & Nayfonov, A. 2001, ApJ, 563, 419
  * Grevesse & Noels (1993) Grevesse, N. & Noels, A. 1993, in Origin and Evolution of the Elements, ed. N. Prantzos, E. Vangioni-Flam, & M. Casse, 15–25
  * Gryaznov et al. (2006) Gryaznov, V. K., Ayukov, S. V., Baturin, V. A., et al. 2006, Journal of Physics A Mathematical General, 39, 4459
  * Gryaznov et al. (2013) Gryaznov, V. K., Iosilevskiy, I. L., Fortov, V. E., et al. 2013, Contributions to Plasma Physics, 53, 392
  * Hansen et al. (2004) Hansen, C. J., Kawaler, S. D., & Trimble, V. 2004, Stellar interiors : physical principles, structure, and evolution (Springer-Verlag, New York)
  * Kosovichev et al. (1992) Kosovichev, A. G., Christensen-Dalsgaard, J., Däppen, W., et al. 1992, MNRAS, 259, 536
  * Lin et al. (2007) Lin, C.-H., Antia, H. M., & Basu, S. 2007, ApJ, 668, 603
  * Lin & Däppen (2005) Lin, C.-H. & Däppen, W. 2005, ApJ, 623, 556
  * Rogers & Nayfonov (2002) Rogers, F. J. & Nayfonov, A. 2002, ApJ, 576, 1064
  * Rogers et al. (1996) Rogers, F. J., Swenson, F. J., & Iglesias, C. A. 1996, ApJ, 456, 902
  * 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

##  Appendix A Relation between ionization of elements and their contributions
to $\delta_{Z}\Gamma_{1}$

This appendix presents a simple picture of the plasma elasticity decrease when
a heavy element is ionized. The definition of the adiabatic exponent
${\Gamma_{1}}={\left.{\left({\partial\ln
P/\partial\ln\rho}\right)}\right|_{S}}$, with the help of the basic
thermodynamic law for adiabatic processes $dU=-PdV,$ can be rewritten as

${\Gamma_{1}}={\left.{\frac{{\partial({P\mathord{\left/{\vphantom{P\rho}}\right.\kern-1.2pt}\rho})}}{{\partial
U}}}\right|_{S}}+1.$ (8)

In an ideal plasma, without the ionization, the internal energy and pressure
are given by

$U=\frac{3}{2}N{k_{B}}T\quad\mathrm{and}\quad P/\rho=N{k_{B}}T,$ (9)

where $N$ is total number of all particles (atoms, ions, and electrons) per
gram, and ${k_{B}}$ is theBoltzmann constant. So in this case
$U=\left(3/2\right)P/\rho$ and from Eq. (8) one has ${\Gamma_{1}}=5/3.$

Ionization increases the internal energy through ionization energy
${U_{\mathrm{ioniz}}}$ (Hansen et al. 2004):

$U=\frac{3}{2}N{k_{B}}T+{U_{\mathrm{ioniz}}}.$ (10)

Adiabatic exponent ${\Gamma_{1}}$ changes in the presence of ionization.
Equation (8) can be rewritten in the following form:

${\Gamma_{1}}=\frac{1}{{{{\left.{\frac{{\partial
U}}{{\partial\left({P/\rho}\right)}}}\right|}_{S}}}}+1=\frac{1}{{\frac{3}{2}+{{\left.{\frac{{\partial{U_{\mathrm{ioniz}}}}}{{\partial\left({P/\rho}\right)}}}\right|}_{S}}}}+1=\frac{2}{3}\frac{1}{{1+\frac{2}{3}{{\left.{\frac{{\partial{U_{\mathrm{ioniz}}}}}{{\partial\left({P/\rho}\right)}}}\right|}_{S}}}}+1.$
(11)

The derivative in the denominator is smaller than unity in the case of
ionization of a small abundant element, and expression 11 is expanded into a
Taylor series:

${\Gamma_{1}}\approx\frac{2}{3}\left({1-\frac{2}{3}{{\left.{\frac{{\partial{U_{\mathrm{ioniz}}}}}{{\partial\left({P/\rho}\right)}}}\right|}_{S}}}\right)+1=\frac{5}{3}-\frac{4}{9}{\left.{\frac{{\partial{U_{\mathrm{ioniz}}}}}{{\partial\left({P/\rho}\right)}}}\right|_{S}}.$
(12)

The derivative in the second term is positive in ionization regions, and
therefore ${\Gamma_{1}}$ becomes smaller.

As an example, we apply these expressions to small impurity of carbon at the
background of fully ionized hydrogen and helium. Mass percentage of carbon is
0.01, hydrogen 0.70, helium 0.29. The sequence $(T,\rho)$ is taken from the
model adiabatic curve corresponding to deep part of the convection zone.
Therefore, the analytical description can be applied.

A carbon atom has six electrons: four on the outer L-shell and two on the
inner K-shell. The ionization internal energy is written in two forms by
regrouping the sum:

${U_{\mathrm{ioniz}}}=\sum\limits_{j=1}^{6}{\left({{N_{{C^{j}}}}\sum\limits_{i=1}^{j}{{\chi_{{C^{i}}}}}}\right)=}\sum\limits_{j=1}^{6}{\left({{\chi_{{C^{j}}}}\sum\limits_{i=j}^{6}{{N_{{C^{i}}}}}}\right).}$
(13)

Here, $\chi_{{C^{i}}}$ is the ionization potential of the j-th ion of carbon,
and ${N_{{C^{j}}}}$ represents the concentration of the j-th ion. The
concentration of ions along the selected adiabatic curve is shown in Fig. 10.
Carbon is mainly in the neutral state at low temperatures $\lg T<3.7$, and it
is almost fully ionized at $\lg T>6.7$.

Figure 10: Distribution of carbon atoms and ions along the adiabatic curve.

Derivative
${\left.{\left[{\partial{U_{\mathrm{ioniz}}}/\partial\left({P/\rho}\right)}\right]}\right|_{S}}$
is shown in Fig. 11a by the red dashed curve together with the individual
terms of expression (13). The dotted line has three local maxima. The first
peak, at $\lg T=3.8$, is due to the ionization of neutral atoms. The second
one, at $\lg T=4.6-5$, is associated with the ionization of the L shell. The
third maximum, at $\lg T=5.9$, corresponds to the ionization of the K shell.
We note that individual ionizations are linearly added in total derivatives in
our approximation. Outside of the carbon ionization range, the derivative goes
to zero. The ${\Gamma_{1}}$-lowering is shown in Fig. 11b. At the temperatures
$\lg T=5.4-6.4$, which are analyzed in the article, the
${\Gamma_{1}}$-lowering is mainly due to the ionization of the carbon K-shell
electron.

Figure 11: (a) Derivative
${{\left.\left[\partial{{U}_{\mathrm{ioniz}}}/\partial\left(P/\rho\right)\right]\right|}_{S}}$
for the case of carbon ionization (red dashed curve), and derivatives of terms
${{\chi}_{{{C}^{j}}}}\sum\limits_{i=j}^{6}{{{N}_{{{C}^{i}}}}}$(colored solid
curves). (b) Adiabatic exponent $\Gamma_{1}$ for ideal plasma without
ionization (black horizontal line) and with ionization (green curve).

Simple considerations in this appendix explain the formation of the
${\Gamma_{1}}$ lowering of each element as a sum of individual contributions
of every ionization stage. Position of minima in the ${\Gamma_{1}}$-lowering
profile is directly associated with the ionization energy of corresponding
ions. For computations that are more accurate, the Coulomb interaction,
electron degeneracy, and simultaneous background ionization should be taken
into account. All of them are included in the SAHA-S calculation used in our
work.

##  Appendix B Some examples of error effect

###  B.1 Background adiabatic exponent for hydrogen-helium plasma

We used the adiabatic exponent $\Gamma_{1}^{\mathrm{HHe}}$ to compute the $Z$
contributions. It depends on the hydrogen mass fraction $X$, the
$\left(T,\rho\right)$ profile that is on entropy, and the equation of state.
Figure 12a shows several examples of the $\Gamma_{1}^{\mathrm{HHe}}$ profiles
for different parameters. Mass fractions of hydrogen are $X=0.70$ and
$X=0.71$. $\left(T,\rho\right)$ profiles are taken from the standard solar
models with high and low mass fractions of heavy elements Z (corresponding to
models 771-0001 and 771-0002 from (Ayukov & Baturin 2017)). There are two
examples of ${{\Gamma}_{1}}$ profiles computed with the SAHA-S and OPAL
equations of state. The adiabatic exponents are very similar on the figure
scale. The detailed difference between them is presented in Fig. 12b.

Figure 12: (a) Adiabatic exponents for hydrogen-helium plasma for different
input parameters. (b) Difference between them.

The red curve represents $\Gamma_{1}^{\mathrm{HHe}}$ computed for $X=0.70$, at
points $\left(T,\rho\right)$ from the high-$Z$ model and with the SAHA-S
equation of state. The green curve differs from it only by hydrogen, which is
$X=0.71$. The cyan curve corresponds to $\Gamma_{1}^{\mathrm{HHe}}$ computed
with the same $X$, but on points $\left(T,\rho\right)$ from the low-$Z$ model.
The deviation at high temperatures is due to the shallow convection zone in
the standard low-$Z$ model. The blue curve shows effect of the equation of
state. There, the OPAL is used instead of SAHA-S. The differences are rather
moderate and do not exceed $1.3\times{{10}^{-4}}$. The intrinsic distinction
between equations of state demonstrates different behavior in case $X$ or
adiabat variations. All these curves demonstrate the behavior of the
$\Gamma_{1}^{\mathrm{HHe}}$ profile itself, and these variations should not be
translated to errors in ${{\Gamma}_{1}}$ contribution. On the contrary, this
effect can be used as diagnostic tool.

###  B.2 LS decomposition for different $(T,\rho)$ profiles

In the main part of the paper, we considered cases when points
$\left(T,\rho\right)$ were taken from the high-$Z$ model 771-0001 to compute
the analyzed $\Gamma_{1}$, background $\Gamma_{1}^{\mathrm{HHe}}$, and basis
$\delta_{Z}^{i}{{\Gamma}_{1}}$. Let us suppose that we do not know the points
$\left(T,\rho\right)$ of the analyzed profile ${{\Gamma}_{1}}$. We take the
profile ${\left(T,\rho\right)}_{\mathrm{hz}}$ from the high-$Z$ model 771-0001
to compute $\Gamma_{1}^{\mathrm{HHe}}$ and $\delta_{Z}^{i}{{\Gamma}_{1,}}$
whereas the analyzed ${{\Gamma}_{1}}$ is computed for the
${\left(T,\rho\right)}_{\mathrm{lz}}$ from the low-$Z$ model 771-0002. As in
the main part, $X=0.70$ and $Z=0.02$ for the mixture AGSS09. Figure 13
presents the difference between ${{\Gamma}_{1}}$ profiles computed on points
$\left(T,\rho\right)$ from low-$Z$ and high-$Z$ models. The difference does
not exceed $8\times{{10}^{-5}}$. Thus, we study how this difference influences
the results of LS-decomposition.

Figure 13: Difference between $\Gamma_{1}$ profiles computed at points
$\left(T,\rho\right)$ from low-$Z$ and high-$Z$ models. Computations are
performed with SAHA-S for the AGSS09 mixture, $X=0.70$, $Z=0.02$.

The results are presented in Table 4. The mean deviation is
$\varepsilon=2.849\times{{10}^{-6}}$, which is an order of magnitude larger
than in the ideal case for which the profiles $\left(T,\rho\right)$ coincide
(see Table 2 of the main part). The mass fractions of the most abundant
elements C, N, O, and Ne are determined with an accuracy of a few percent.
Thus, an error in the $\left(T,\rho\right)$ profile leads to increasing
$\varepsilon$. This property can be used for diagnostic purposes.

Table 4: LS decomposition for different $(T,\rho)$ profiles in the analyzed function and background. Elem. | $Z_{i}^{\mathrm{LS}},\times 10^{-2}$ |  $\Delta Z_{i}/Z_{i}$, %
---|---|---
C | 0.3724 | 2.5
N | 0.1026 | -3.6
O | 0.8876 | 0.8
Ne | 0.1959 | 1.5
Mg | 0.0969 | -10.9
Si | 0.1244 | 21.8
S | 0.0361 | -24.0
Fe | 0.1947 | -1.9
$Z^{\mathrm{LS}}$ | 0.020105
$\eta$ | $1.445\times 10^{-5}$
$\varepsilon$ | $2.849\times 10^{-6}$
$\sigma$ | $1.417\times 10^{-5}$

###  B.3 LS decomposition for the OPAL equation of state

Figure 14a shows adiabatic exponents computed in the OPAL and SAHA-S (OPAL
mixture) equations of state at points $(T_{j},\rho_{j})$ from the solar model
771-0001. The curves for a hydrogen-helium plasma (dashed lines) agree well,
whereas curves for $Z=0.02$ (solid lines) differ by about $10^{-3}$ at $\lg
T=5.4-5.5$ because of spurious fluctuations contained in OPAL.

Figure 14: (a) Adiabatic exponents and (b) $Z$ contributions computed in the
OPAL and SAHA-S (OPAL mixture) equations of state. $X=0.70$, $Z=0.02$.

Figure 14b shows $Z$ contributions computed in different ways. The peak at low
temperature region $\lg T=5.4-5.5$ is caused by the artefact in $\Gamma_{1}$
at $Z=0.02$ in OPAL.

The $Z$ contribution for OPAL was calculated in two ways. In the first case,
the $\delta_{Z}\Gamma_{1}^{\mathrm{HHe}}$ for the hydrogen-helium plasma was
calculated with the SAHA-S (blue curve), in the second case it was calculated
with OPAL (green dashed curve). The difference between the two values of
$\delta_{Z}\Gamma_{1}$ is about $10^{-4}$. The origin of this difference is
unknown and will require additional research. These examples are given only to
demonstrate the required level of precision of a modern equation of state and
the absolute accuracy with which the solar equation of state has to be known.

Table 5 shows the results of the LS method for the $Z$ contribution from OPAL.
Decomposition over the entire interval $\lg T=5.4-6.4$ gives an accuracy of
14% for the oxygen mass fraction and 81% for carbon. A bad carbon estimation
is caused by a peak in $\delta_{Z}\Gamma_{1}$ at low temperatures. If the
investigated temperature range is reduced to $\lg T=5.5-6.4$ and thereby the
peak is removed, then the estimates are improved: oxygen 8%, carbon 14%.
Switching the profile $\Gamma_{1}^{\mathrm{HHe}}$ from SAHA-S to OPAL is not
essential.

Table 5: LS decomposition for OPAL equation of state calculated at different intervals $\lg T$ and with different background $\Gamma_{1}^{\mathrm{HHe}}$ . |  $\Gamma_{1}^{\mathrm{HHe}}$ from SAHA-S, |  $\Gamma_{1}^{\mathrm{HHe}}$ from SAHA-S, |  $\Gamma_{1}^{\mathrm{HHe}}$ from OPAL,
---|---|---|---
| $\lg T=5.4-6.4$ | $\lg T=5.5-6.4$ | $\lg T=5.5-6.4$
Elem. |  $Z_{i}^{\mathrm{LS}},$ $\times 10^{-2}$ |  ${\Delta Z_{i}}/{Z_{i}},$ % |  $Z_{i}^{\mathrm{LS}},$ $\times 10^{-2}$ |  ${\Delta Z_{i}}/{Z_{i}},$ % |  $Z_{i}^{\mathrm{LS}},$ $\times 10^{-2}$ |  ${\Delta Z_{i}}/{Z_{i}},$ %
C | 0.0738 | -81 | 0.4262 | 12 | 0.4301 | 13
N | 0.0763 | -32 | 0.1017 | -9 | 0.1025 | -8
O | 0.9288 | -14 | 0.9993 | -8 | 1.0187 | -6
Ne | 0.3522 | -16 | 0.5298 | 26 | 0.5380 | 28
Mg | -0.8653 | - | -0.2545 | - | -0.2295 | -
Si | 0.7998 | - | 0.1474 | - | 0.1814 | -
S | -0.0370 | - | -0.0112 | - | -0.0387 | -
Fe | 0.3221 | - | -0.0308 | - | -0.0370 | -
$Z^{\mathrm{LS}}$ | 0.016089 | 0.019079 | 0.019557
$\eta$ | $1.788\times 10^{-4}$ | $8.683\times 10^{-5}$ | $4.326\times 10^{-5}$
$\varepsilon$ | $7.107\times 10^{-5}$ | $1.950\times 10^{-5}$ | $1.824\times 10^{-5}$
$\sigma$ | $1.641\times 10^{-4}$ | $8.461\times 10^{-5}$ | $3.922\times 10^{-5}$

The analysis of the OPAL equation of state thus makes it possible to determine
oxygen, as the most abundant heavy element, with an accuracy of about 10%.
Selecting a background $\Gamma_{1}^{\mathrm{HHe}}$ from SAHA-S or OPAL is not
essential. An accurate equation of state plays an important role in the
reliability of our method.

###  B.4 An example of finite space resolution and random errors

Until now, all computations were performed for 1309 points
$\left(T_{j},\rho_{j}\right)$ in the interval $\lg T=5.4-6.4$, i.e. simulating
a continuous function $\Gamma_{1}$. Generally, such a function can show some
small details that cannot be expected in helioseismic profiles $\Gamma_{1}$.
The expected resolution for the inverted profile $\Gamma_{1}$ is not much
better than $\Delta\lg T\approx 0.05$.

To simulate a profile with a finite resolution, we switched to a discrete
version of LS decomposition with a limited number ND of points. Varying the ND
from 1309 to about 40-20, we did not find significant changes in the LS
decomposition results. In Table 6, we present the results for the discrete
case with ND = 20. It can be compared with Table 2 (AGSS09). Thus, the
discrete LS decomposition is computationally equivalent to its quasi-
continuous counterpart in the case of an ideal experiment.

Table 6: LS decomposition for $\Gamma_{1}$ profile with limited space resolution (20 points instead of 1309). Elem. | $Z_{i}^{\mathrm{LS}},\times 10^{-2}$ |  $\Delta Z_{i}/Z_{i}$, %
---|---|---
C | 0.3662 | 0.8
N | 0.1064 | -0.02
O | 0.8803 | -0.02
Ne | 0.1936 | 0.3
Mg | 0.1137 | 4.6
Si | 0.0988 | -3.2
S | 0.0468 | -1.5
Fe | 0.1965 | -0.9
$Z^{\mathrm{LS}}$ | 0.020024
$\eta$ | $2.232\times 10^{-6}$
$\varepsilon$ | $2.336\times 10^{-7}$
$\sigma$ | $2.219\times 10^{-6}$

The use of the discrete version also turns out to be convenient for simulating
random errors in the investigated $\Gamma_{1}$. We use 20 points in the $\lg
T=5.4-6.4$ range. In these points, normally distributed random errors were
added to the theoretical profile. Their standard deviation is
$\sigma_{\Gamma}=10^{-4}$. The errorless $\delta_{Z}\Gamma_{1}$ from SAHA-S is
presented as a blue curve in Fig. 15. Red points show an example of distorted
data. Only these 20 points were used for the LS decomposition.

Figure 15: (a) Precise Z contribution $\delta_{Z}\Gamma_{1}$ from SAHA-S (blue
line) and an example of random errors (red points). (b) Resulting errors of
oxygen and carbon max fractions (points) in 100 experiments. Dashed lines are
standard deviations normalized to main values.

We performed 100 experiments with different sets of 20 distorted points. The
standard deviation was $\sigma_{O}=0.07\times 10^{-2}$ for oxygen and
$\sigma_{C}=0.17\times 10^{-2}$ for carbon. Fig. 15b shows the relative errors
of the mass fractions in every experiment. Dashed lines in Fig. 15b show the
standard deviations normalized to the theoretical profile. Thus, the expected
error of LS-decomposition is about 10% for oxygen and 50% for carbon.
Estimations for other elements are not reliable.

These experiments with artificial errors show that adding random errors on the
order of $\sigma_{\Gamma}=10^{-4}$ in the function $\Gamma_{1}$ will lead to a
deterioration of the results compared to an error-free experiment. However,
even then, the most abundant heavy element, oxygen, can be determined with a
relative accuracy of about 10 percent.

###  B.5 LS decomposition for $\Gamma_{1}(r)$ profile from standard solar
model with the OPAL equation of state

In the final section of Appendix B, we consider an LS decomposition and
estimation of heavy elements for a special illustrative example of the
$\Gamma_{1}$ profile. This $\Gamma_{1}$ profile is obtained from a standard
solar model calculated with the OPAL equation of state (version 2005). Here,
the $\Gamma_{1}$ profile is considered as a function of radius, not of
temperature as above. This solar model, named 772, was specifically calculated
for this experiment. It was computed and calibrated identically to model
771-0001, except that the OPAL equation of state is used.

Thus, the $\Gamma_{1}\left(r\right)$ profiles in these models differ not only
due to the different physics in the equation of state, but also due to a
discrepancy in the $\left(T,\rho\right)$ adiabats. Fig. 16 shows these model
differences as functions of radius. In the convection zone, the temperature
difference is less than 0.06%, and it does not exceed 0.5% in density. The
density difference is mainly caused by the different adiabates of the
calibrated models.

Figure 16: Relative differences of temperatures (a) and densities (b) in
models 772 and 771-0001.

Following the technique proposed in Appendix B.4, we used 20 points uniformly
distributed along the radius in the selected interval,
$r/R_{\odot}=0.714-0.943$, which corresponds to temperatures $\lg T=5.5-6.34$.
These points are plotted as red circles on Fig. 17. The background
$\Gamma_{1}^{HHe}$ and the basic functions $\delta_{Z}^{i}\Gamma_{1}$ were
computed at the same points $r_{j}$ in the model 771-0001 and the
corresponding $\left(T,\rho\right)$ were used in the SAHA-S calculations.

Figure 17: Adiabatic exponent as a function of radius in model 772. Red points
show data used for LS decomposition.

As a result, the LS decomposition for the four chemical elements used in the
OPAL mixture is obtained (Table 7). The most represented elements (C, O, and
Ne) are determined with a relative accuracy better than 10%. The total errors
are comparable with the results in Table 5.

Table 7: LS decomposition for $\Gamma_{1}(r)$ profile from standard solar model with OPAL equation of state. Elem. | $Z_{i}^{\mathrm{LS}},\times 10^{-2}$ |  $\Delta Z_{i}/Z_{i}$, %
---|---|---
C | 0.3206 | -6
N | 0.1393 | 39
O | 0.9329 | -4
Ne | 0.3465 | -8
$Z^{\mathrm{LS}}$ | 0.01739
$\eta$ | $7.482\times 10^{-5}$
$\varepsilon$ | $3.961\times 10^{-5}$
$\sigma$ | $6.348\times 10^{-5}$

We want to note that this numerical blind-test experiment is completely
illustrative. We do not draw any conclusions in addition to those already made
in the article. A detailed study of individual factors and their combinations
in this experiment is far beyond the scope of this paper.