THE CHEMICAL COMPOSITION OF THE SUN FROM HELIOSEISMIC AND SOLAR NEUTRINO DATA

The development of a new generation of stellar atmosphere models resulted in a downward revision of the solar photospheric abundances. This is shown, for example, in Table 1 where we give the abundances of C, N, O, Ne, Mg, Si, S, and Fe, which are the most relevant elements for solar model construction. By AGSS09met, we represent the (new) photospheric abundances from Asplund et al. (2009) for volatile elements (C, N, O, and Ne) combined with their recommended meteoritic abundances for refractory elements (Mg, Si, S, and Fe), where Si is used as the anchor of both scales. The choice of meteoritic abundances for refractories is rooted in their robustness and superior accuracy and precision over photospheric determinations, and it has traditionally been the preferred choice in SSM calculations (see Serenelli et al. 2009 for a discussion on this topic). By GS98, we represent the (old) heavy element admixture of Grevesse & Sauval (1998). The last column shows the fractional differences between the individual abundances (relative to hydrogen) in the two compilations.

The heavy element admixture determines to a large extent the opacity profile of the Sun and is a crucial input for solar model construction. This is seen in the first two columns of Table 2, where we report the results of two recent SSM calculations (Serenelli et al. 2011) that implement the AGSS09met (Asplund et al. 2009) and GS98 (Grevesse & Sauval 1998) surface compositions. The models have been computed with GARSTEC (Weiss & Schlattl 2008) using the input physics described in Serenelli et al. (2009). We utilize the nuclear reaction rates recommended in Adelberger et al. (2011) with weak screening (Salpeter 1954), and we employ the mixing length theory of convection (Cox 1968; Vitense 1953). We use a gray atmosphere surface boundary condition; this has virtually no influence on the global seismic properties and the predicted neutrino fluxes of the solar model and bears no impact on results presented in this work (see Schlattl et al. 1997 for a comparison of solar models with different model atmospheres). Element diffusion in the solar interior is included according to Thoul et al. (1994). The models have been computed by using the radiative opacities from the Opacity Project (OP; Badnell et al. 2005), complemented at low temperatures with the opacities from Ferguson et al. (2005). Finally, we do not include additional mixing of any sort (rotational mixing, overshooting, or mixing by gravity waves) below the convection zone in the models that would be necessary to explain the solar lithium depletion (a factor of ∼160; Asplund et al. 2009). We come back to this point in Section 6.

Table 2. The Predictions of SSMs Implementing GS98 (Grevesse & Sauval 1998) and AGSS09met (Asplund et al. 2009) Admixtures

Q	AGSS09met	GS98	Obs.	GS98rec
Ys	0.2319 (1 ± 0.013)	0.2429 (1 ± 0.013)	0.2485 ± 0.0035	0.243
Rb/R☉	0.7231 (1 ± 0.0033)	0.7124 (1 ± 0.0033)	0.713 ± 0.001	0.710
Φpp	6.03 (1 ± 0.005)	5.98 (1 ± 0.005)	$6.05(1^{+0.003}_{-0.011})$	5.98
ΦBe	4.56 (1 ± 0.06)	5.00 (1 ± 0.06)	$4.82(1^{+0.05}_{-0.04})$	4.98
ΦB	4.59 (1 ± 0.11)	5.58 (1 ± 0.11)	5.00(1 ± 0.03)	5.49
ΦN	2.17 (1 ± 0.08)	2.96 (1 ± 0.08)	⩽6.7	2.89
ΦO	1.56 (1 ± 0.10)	2.23 (1 ± 0.10)	⩽3.2	2.15

Notes. The theoretical uncertainties (1σ) have been calculated as it is described in the text and do not include the contributions due to errors in the surface composition. In the third column, we show the observational values for helioseismic quantities (Basu & Antia 2004, 1997) and solar neutrino fluxes (Bellini et al. 2011). In the last column, we calculate the GS98 solar model predictions starting from the AGSS09met solar model by using the linear expansion given in Equation (22). The neutrino fluxes are given in the following units: 10¹⁰ cm⁻² s⁻¹ (pp); 10⁹ cm⁻² s⁻¹ (Be); 10⁶ cm⁻² s⁻¹ (B); 10⁸ cm⁻² s⁻¹ (N); and 10⁸ cm⁻² s⁻¹ (O).

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We consider the following observable quantities: the surface helium abundance Y_s, the radius of the inner boundary of the convective envelope R_b, the neutrino fluxes Φ_ν, where the index ν = pp, Be, B, N, O refers to the neutrino producing reactions according to the usual convention, and the fractional sound speed difference δc_i ≡ (c_{obs, i} − c(r_i))/c(r_i) between the predicted sound speed c(r) and the values c_{obs, i} inferred from helioseismic data. The latter is shown in Figure 1, where the black line refers the SSM model implementing the AGSS09met surface composition while the red line is obtained by using the GS98 admixture.

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In the following, we take the SSM implementing the AGSS09met composition as a starting point for our analysis, and we use the notation $\overline{Q}$ ($\overline{I}$) to indicate the prediction (assumption) for the generic quantity Q (input I) in this calculation.

The errors quoted in Table 2 and the light-blue band in Figure 1 correspond to 1σ uncertainties in theoretical predictions. They have been calculated by propagating the errors in the following input parameters: the age of the Sun (age); the diffusion coefficients (diffu); the luminosity (lum); the opacity profile (opa) of the Sun; and the astrophysical factors S₁₁, S₃₃, S₃₄, S₁₇, S_e7, and S_{1, 14}. As it is done, e.g., in Bahcall et al. (2006), albeit diffusion rates are computed individually for each elements, we assume the same fractional error for all, and we treat them as fully correlated, i.e., we rescale them by a common multiplicative factor. Differential variations of the diffusion coefficients within the present uncertainties (∼10%) give negligible contributions to theoretical errors. Note that, as we are interested in establishing bounds on the solar chemical composition, we have explicitly omitted its contribution to theoretical uncertainties. Moreover, we have not considered the uncertainty of the solar radius because it provides subdominant contributions to the error budget of the considered observable quantities.⁷ Our test calculations show that the solar radius uncertainty corresponds to a fractional error at the level of 0.01% and 0.03% for ⁷Be and ⁸B neutrino fluxes; 0.01% and 0.004% for R_b and Y_s; 0.025% at most for the sound speed. The assumed value of R_☉ also affects the determination of the sound speed by inversion of helioseismic frequencies. However, based on results by Basu et al. (2000, see their Figure 9), this uncertainty is of the order of 0.035% and plays a minor role in the full analysis (see the next section and the red band in Figure 1).

By following the standard procedure, we have calculated the logarithmic derivatives

$$\begin{equation} B_{Q, I} = \frac{\partial \ln Q}{\partial \ln I}. \end{equation} \tag{ 1 }$$

B_{Q, I} values are available at Serenelli et al. (2013) for all observables except the sound speed c(r). To our knowledge, the logarithmic derivatives B_{c, I}(r) of the sound speed, defined as

$$\begin{equation} B_{c, I}(r) \equiv \partial\ln c(r)/ \partial\ln I \end{equation} \tag{ 2 }$$

have not been given elsewhere in the scientific literature and are shown in the left panel of Figure 2 as a function of the solar radius.

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As mentioned before, the uncertainty δI in each input parameter I produces fully correlated errors on the SSM predictions of observables Q. To emphasize this point, we use the symbol C_{Q, I} to indicate the fractional variation of Q when a fractional correction δI is applied to the input I. The various error contributions C_{Q, I}, shown in Table 3 and in the right panel of Figure 2, are calculated from the relation

$$\begin{equation} C_{Q, I} = B_{Q, I} \; \delta I \end{equation} \tag{ 3 }$$

with the δI values summarized in Serenelli et al. (2013). The only exception is the opacity, which we discuss below.

Opacity is not a single number but a complicated function of the properties of the solar plasma which can be modified in a non-trivial way. To take this into account, we use the opacity kernels derived in Villante (2010) by adopting the linearization procedure proposed in Villante & Ricci (2010). The kernels K_Q(r) represent the functional derivatives of the observable Q with respect to the opacity profile of the Sun and can be used to calculate the effects produced by an arbitrary opacity variation δκ(r) according to

$$\begin{equation} \delta Q = \int dr \; K_{Q}(r)\protect, \protect \; \delta \kappa (r) \end{equation} \tag{ 4 }$$

where

$$\begin{equation} \delta Q \equiv \frac{Q}{\overline{Q}}-1 \;, \end{equation} \tag{ 5 }$$

and

$$\begin{equation} \delta \kappa (r) \equiv \frac{\kappa (\overline{T}(r),\overline{\rho }(r),\overline{Y}(r), \overline{Z}_j(r))}{\overline{\kappa }(\overline{T}(r),\overline{\rho }(r), \overline{Y}(r),\overline{Z}_j(r))} - \protect 1. \protect \protect \protect \end{equation} \tag{ 6 }$$

In the above relation, the functions κ and $\overline{\kappa }$ are calculated along the density, temperature, and chemical composition profiles predicted by SSM.

By using the kernels K_Q(r), we propagate the uncertainty in the opacity profile of the Sun δκ_opa(r) according to

$$\begin{equation} C_{Q, \rm opa} \equiv \int dr \; K_{Q}(r) \; \delta \kappa _{\rm opa}(r)\protect. \protect \end{equation} \tag{ 7 }$$

The standard prescription, e.g., as adopted in Serenelli et al. (2009), is to consider a 2.5% global rescaling factor that corresponds to take δκ_opa(r) ≡ 0.025. However, this is not realistic because, albeit different groups typically provide opacity values that differ by ∼few percent in the solar interior, there is a complicated dependence on the solar radius. Moreover, it was shown in Villante (2010) that the assumption of a global rescaling underestimates the uncertainties for the sound speed and for the depth of the convective envelope. The opacity kernels for these quantities are not positive everywhere, and a constant δκ produces effects in different regions of the Sun that partially compensate each other. For this reason, we take the difference between the OP (Badnell et al. 2005) and OPAL (Iglesias & Rogers 1996) tables as representative of uncertainties in opacity calculations, i.e., we assume

$$\begin{equation} \delta \kappa _{\rm opa}(r) \equiv \frac{\kappa _{\rm OPAL}(\overline{T}(r),\overline{\rho }(r),\overline{Y}(r),\overline{Z}_i(r))}{\kappa _{\rm OP}(\overline{T}(r),\overline{\rho }(r), \overline{Y}(r),\overline{Z}_i(r))} - \protect 1. \protect \protect \protect \end{equation} \tag{ 8 }$$

The function δκ_opa(r) is shown in Figure 3. The sound speed errors obtained with this assumption are much larger that what is obtained by the standard approach, even if |δκ_opa(r)| ⩽ 0.025 almost everywhere.

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Finally, the total theoretical error for each observable Q is calculated by combining in quadrature all the error contributions, i.e.,

$$\begin{equation} \sigma ^2_{Q,\rm theo} = \sum _{I} C^2_{Q,I}\protect, \protect \end{equation} \tag{ 9 }$$

where the sum extends over the 10 parameters listed in the beginning of this section.

In the third column of Table 2, we give the observational values Q_obs for helioseismic and solar neutrino observables. The solar neutrino fluxes are obtained by performing a fit to all available solar neutrino data (Bellini et al. 2011). Among the various components, the ⁸B and ⁷Be neutrino fluxes are essentially determined by SK (Cravens et al. 2008) and SNO (Ahmad et al. 2002) and by Borexino (Arpesella et al. 2008), respectively, i.e., by two independent sets of experimental data. We thus include the two values

$$\begin{eqnarray} \nonumber \Phi _{\rm Be, obs} &=& 4.82\, (1^{+0.05}_{-0.04}) \times 10^{9} \,{\rm cm}^{-2}\, {\rm s}^{-1} \\ \Phi _{\rm B, obs} &=& 5.00\, (1\pm 0.03) \times 10^{6} \,{\rm cm}^{-2}\, {\rm s}^{-1}\protect, \protect \protect \protect \end{eqnarray} \tag{ 10 }$$

and assume their errors are uncorrelated. Generically denoting with U_Q an uncorrelated fractional error in the observable Q, we adopt

$$\begin{eqnarray} \nonumber U_{\rm Be} &=& 0.045\\ U_{\rm B} &=& 0.03 \end{eqnarray} \tag{ 11 }$$

for Φ_Be and Φ_B, respectively. We note that the observational errors are smaller than the uncertainties in theoretical predictions.

The surface helium abundance and the inner radius of the convective envelope are obtained by inversion of helioseismic frequencies. We adopt the values

$$\begin{eqnarray} \nonumber Y_{s, {\rm obs}} &=& 0.2485 \pm 0.0035 \\ R_{b, {\rm obs}} &=& 0.713 \pm 0.001, \end{eqnarray} \tag{ 12 }$$

which are obtained in Basu & Antia (2004) and Basu & Antia (1997), respectively, and we indicate with

$$\begin{eqnarray} \nonumber U_{\rm Y} &=& 0.015 \\ U_{\rm R} &=& 0.0014 \end{eqnarray} \tag{ 13 }$$

the fractional errors that we assume, as it is usually done (see, e.g., Delahaye & Pinsonneault 2006), not being significantly correlated.

The sound speed data points δc_i reported in Figure 1 have been obtained in Basu et al. (2009) with the set of frequencies of solar low-degree p-modes from the BiSON network by using the subtractive optimally localized averages (SOLA) inversion technique. The various points are localized at the target radii r_i of the corresponding averaging kernels. The displayed error bars U_{i, exp} are calculated by propagating the observational uncertainties of the measured frequencies. It is well known, however, that larger errors arise from the choice of the parameters in the inversion procedure and from the assumed starting model for the inversion. An extensive investigation of uncertainties in helioseismic determinations of sound speed has been performed in Degl'Innocenti et al. (1997).⁸ The red band in Figure 1 corresponds to the so-called "statistical" uncertainty U_stat(r) that is obtained in Degl'Innocenti et al. (1997) by combining in quadrature all the relevant error contributions. In our analysis, we define total (fractional) observational errors for the sound speed by combining in quadrature experimental and "statistical" errors according to

$$\begin{equation} U_{c, i} = \sqrt{U_{i,\rm exp}^2 + U^2_{\rm stat}(r_i) }\,, \end{equation} \tag{ 14 }$$

where the index i indicates the considered data point. Clearly, sound speed determinations at different radii are expected to have a certain degree of correlation because uncertainties are mainly related to the inversion procedure. Unfortunately, the information provided in the scientific literature does not allow us to quantify these correlations. For this reason, we include the sound speed errors as being uncorrelated. The correct quantification of helioseismic errors is an important ingredient, and it would be desirable that the complete information were provided in future investigations.

As it is well known, solar models implementing the AGSS09met admixture do not correctly reproduce the helioseismic constraints. By combining in quadrature theoretical and observational errors, we see that the predictions for Y_s and R_b deviate from observational data at ∼3.6σ and ∼3.9σ, respectively.⁹ The sound speed at r ∼ 0.65 R_☉ differs from the results of helioseismic inversion by ∼4.6σ. Helioseismic data are much better fitted by solar models implementing the GS98 surface composition, as can be seen from Table 2 and Figure 1. A reasonable agreement exists between predicted and reconstructed solar neutrino fluxes both for the AGSS09met and the GS98 solar models.

As a first application, we consider a scenario in which the neon-to-oxygen ratio is fixed to the value prescribed by the AGSS09met compilation, i.e., we further constrain the possible variations of the heavy element admixture by assuming δz_CNO = δz_Ne. In this hypothesis, the χ² is defined in terms of two independent parameters (δz_CNO , δz_met) that are varied to fit helioseismic and solar neutrino constraints. A similar exercise was performed in Delahaye & Pinsonneault (2006), where, however, only the determinations of the surface helium abundance and of the convective radius were considered. Here, we include the information provided by the sound speed profile and the neutrino fluxes in a global quantitative analysis. The redundancy of the different pieces of experimental information allows us to obtain more solid constraints on the solar composition and, even more relevant, to verify that a coherent picture emerges from the data and/or to highlight tensions in SSM assumptions.

Our results are presented in Figure 6, where we use the astronomical scale for logarithmic abundances ε_j in order to facilitate the comparison with observational data. The conversion from δz_j to ε_j is obtained by using the relation

$$\begin{equation} \varepsilon _{j} = \overline{\varepsilon }_{j}+\log {(1+\delta z_{j})} \end{equation} \tag{ 27 }$$

with the AGSS09met abundances $\overline{\varepsilon }_{j}$ given in Table 1. The colored lines are obtained by cutting at $\Delta \chi ^2\equiv \chi ^2-\chi ^2_{\rm min} = 2.3,\, 6.2,\, 11.8$ that correspond to 1, 2, 3 σ confidence levels for a χ² variable with 2 d.o.f. The data points show the observational values and errors for the oxygen and iron abundances in the AGSS09met and GS98 compilations. In order to show how different observational information combine in determining the optimal composition, we present separately the bounds obtained by using the following: the helioseismic constraints on the surface helium abundance and the convective radius (upper left panel); the ⁷Be and ⁸B neutrino flux determinations (upper right panel); the 30 sound speed data points c_{i, obs} from Basu et al. (2009) that are localized at r ⩽ 0.8 R_☉ (lower left panel); and all the observational data simultaneously (lower right panel). The main conclusions of our analysis are discussed in the following.

• 1.  
  The SSM implementing AGSS09met composition is excluded at a high confidence being χ²/d.o.f. = 176.7/32 when all the available observational constraints are considered. This result essentially arises from helioseismic observables that are in severe disagreement with AGSS09met predictions. The ⁷Be and ⁸B solar neutrino flux determinations do not discriminate among different compositions with the sufficient level of accuracy. This is mainly due to theoretical uncertainties which are dominated by the contributions from S₃₄ and S₁₇, respectively, but also it is due to a very mild dependence on CNO abundances.
• 2.  
  There is a reasonable agreement between the information provided by the various observational constraints, as it can be seen by comparing the different panels of Figure 6. The best fit to the observational data is obtained for

  $$\begin{eqnarray} &&\nonumber \delta z_{\rm CNO} = \delta z_{\rm Ne} = 0.45 \pm 0.04 \\ &&\delta z_{\rm met} = 0.19 \pm 0.03, \end{eqnarray} \tag{ 28 }$$

  which corresponds to ε_O = 8.85 ± 0.01 and ε_Fe = 7.52 ± 0.01. These values are consistent at ∼1σ level with those quoted in the GS98 compilation. They are close to the results of Delahaye & Pinsonneault (2006) but have considerably smaller uncertainties. The quality of the fit is quite good being the $\chi _{\rm min}^2/{\rm d.o.f.}= 39.6 / 32$ when all the observational constraints are considered. The errors on the inferred abundances ε_O and ε_Fe are smaller than what is obtained by observational determinations. One caveat is, however, that we are considering a simplified scenario in which different elemental abundances are grouped together and forced to vary by the same multiplicative factors.
• 3.  
  The observational and systematic contribution to $\chi ^2_{\rm min}$ are given by $\chi ^2_{\rm obs} = 35.1$ and $\chi ^2_{\rm syst} = 4.5$, respectively, with the distribution of systematic pulls $\tilde{\xi }_I$ at the best-fit point reported in Table 5. The effects of systematic pulls (that correspond to correlated error sources) are relevant and cannot be neglected. This is seen, e.g., in the upper left panel of Figure 6 where we see that the error ellipse axes do not coincide with the lines Y_s = Y_{s, obs} and R_b = R_{b, obs}, as it would be expected if error correlations were negligible. It is also evident from the red dots and the red line in Figure 7 that show the predictions Q of solar models implementing the best-fit composition calculated by using the linear expansion (22). These predictions deviate from the observational constraints by amounts that are larger than the uncorrelated observational errors. However, this does not imply that the quality of the fit is bad since one has the possibility to change the SSM inputs within their range of uncertainty as it is described in Equation (25). The blue dots and the blue line in Figure 7 show the quantities

  $$\begin{equation} \tilde{Q} = Q \left[ 1 + \sum _{I} C_{Q,I} \,\tilde{\xi }_I \right] \end{equation} \tag{ 29 }$$

  that include the effects due to the pulls of systematic errors. The quantities $\tilde{Q}$ differ quite substantially from the corresponding Q and agree quite well with the observational constraints. The dominant contributions to systematic shifts are shown by the black arrows in Figure 7 and are provided by the following: opacity for the sound speed c(r) (see discussion in the next paragraph); diffusion coefficients which are decreased by ∼11% in order to improve consistency between Y_s and R_b and the sound speed profile c(r); and the astrophysical factors S₃₄ and S₁₇ that are decreased by ∼5% and ∼9%, respectively in order to improve agreement with Φ_B and Φ_Be measurements.
• 4.  
  The large systematic shift of the sound speed due to opacities, $\tilde{\xi }_{\rm opa}=1.07$, indicates that there is tension between observational data and SSMs implementing OP opacity tables. Indeed, if we restrict our analysis to OP opacities, i.e., we choose ξ_opa ≡ 0 in our approach, the quality of the fit is considerably decreased. The best fit is obtained for δz_CNO = 0.33 ± 0.03 and δz_met = 0.19 ± 0.03 with $\chi ^2_{\rm min}/{\rm d.o.f.} = 66.9 / 32$ when all observational constraints are considered. Solar models implementing OP opacities are disfavored because they provide a less satisfactory fit of the sound speed in the region 0.3 < r/R_☉ < 0.6, as it can be seen from Figure 8. It is interesting to note that, when ξ_opa is allowed to vary, the best fit is obtained with $\tilde{\xi }_{\rm opa}\sim 1$ which means that observational data are better described when using OPAL opacity tables.¹¹ The statistical significance of this indication relies on the correct evaluation of the sound speed error in the outer radiative region of the Sun and may be weakened (or strengthened) by the possibility of correlations in the inferences of the solar sound speed at different target radii (not considered here due to lack of the necessary information in the scientific literature).
• 5.  
  The CNO neutrino fluxes are expected to be ∼50% larger than those predicted by SSMs implementing AGSS09met composition. Indeed, solar models providing a good fit to the observational data give Φ_N ≃ 3.4 × 10⁸ cm⁻² s⁻¹ and Φ_O ≃ 2.5 × 10⁸ cm⁻² s⁻¹ as a combined effect of the changes in composition and, to a minor extent, of the systematic shifts in the input parameters. These values are even larger than predictions obtained by assuming GS98 surface composition. However, this result depends on the assumed heavy element grouping. The CNO neutrino fluxes, in fact, are essentially determined by the carbon abundance, see Table 4, while the observational data included in our analysis are basically sensitive to the oxygen content of the Sun, since this element provides a large contribution to the solar opacity.

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Table 5. The Pulls of Systematics $\tilde{\xi }_{I}$ at the Best-fit Point and the Fractional Variations $\tilde{\xi }_I\delta I$ of the Corresponding Input Parameters

	Opa	Age	Lum	Diffu	S11	S33	S34	Se7	S17	S1, 14
$\tilde{\xi }_I$	1.07	0.03	−0.41	−0.74	∼0	0.46	−0.97	0.32	−1.20	∼0
$\tilde{\xi }_I\delta I$	1.07 (κOPAL/κOP − 1)	∼0	−0.0016	−0.11	∼0	0.024	−0.05	0.007	−0.09	∼0

Notes. All the available observational information are simultaneously fitted. The entries with ∼0 are smaller (in magnitude) than 10⁻² in the first line and 10⁻³ in the second line.

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In Figure 6, we also include for reference the solar oxygen and iron abundances determined by Caffau et al. (2011) shown by the blue points labeled "CO⁵BOLD." Unfortunately, Caffau et al. (2011) have not determined the solar photospheric Si abundance, so a direct comparison of their iron abundance with AGSS09met or GS98 cannot be made because it is not possible to construct a meteoritic solar abundance scale. It is, however, interesting to point out that, albeit based on 3D solar atmosphere models with very similar structure (Beeck et al. 2012), the solar abundances they derive are closer to our best-fit values than AGSS09met, particularly for oxygen.

It is important to discuss how the above results change when the neon-to-oxygen ratio is allowed to vary, since Ne lacks photospheric features and the Ne/O ratio has to be inferred indirectly from solar wind measurements. In this assumption, the χ² is described as a function of three parameters (δz_CNO, δz_Ne, δz_met) that can be adjusted independently to reproduce helioseismic and solar neutrino constraints. In order to prevent unphysical results, we add a penalty function to the χ² given by

$$\begin{equation} \chi ^2_{\rm pen} = \left[\frac{\delta z_{\rm Ne}- \delta z_{\rm CNO}}{\Delta \, (1 + \delta z_{\rm CNO}) }\right]^2, \end{equation} \tag{ 30 }$$

where Δ = 0.3, that forces the neon-to-oxygen ratio to the value prescribed by AGSS09met compilation with a 1σ accuracy equal to 30%, as has been observed by Bahcall et al. (2005). The bounds obtained by considering all the available observational constraints are shown in Figure 9. The best-fit composition is

$$\begin{eqnarray} \nonumber \delta z_{\rm CNO} &=& 0.37 \pm 0.07 \\ \nonumber \delta z_{\rm Ne} &=& 0.80 \pm 0.26 \\ \delta z_{\rm met} &=& 0.13 \pm 0.05, \end{eqnarray} \tag{ 31 }$$

which corresponds to ε_O = 8.83 ± 0.02, ε_Ne = 8.19 ± 0.06, and ε_Fe = 7.50 ± 0.02. These values are still consistent at ∼1σ with those obtained in the GS98 compilation. However, the errors in the inferred abundances are larger than before. We note, in particular, that the neon abundance is bounded at the level of accuracy prescribed by the function (30) indicating that the observational data are not effective in constraining it. The neon-to-oxygen ratio is increased by about ∼30% with respect to the AGSS09met value. The quality of the fit, however, is not significantly improved being $\chi ^2_{\rm min}/{\rm d.o.f.} = 37.8/31$ and the assumption 1 + δz_Ne = 1 + δz_CNO is allowed at 1σ.

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The consequence of leaving neon as a free parameter is to introduce degeneracies between the various δz_j, as it is understood from inspection of Figure 9 and, in particular, by comparison of the left panel in Figure 9 and the lower left panel of Figure 6. It exists, in fact, a combination of δz_CNO, δz_Ne, and δz_met that, taking also into account the effects of systematic pulls ξ_I, leaves substantially unchanged the observational properties of the Sun. An increase of the neon-to-oxygen ratio can be compensated for by a slight reduction of CNO and/or Heavy elements according to simple approximate formula

$$\begin{eqnarray} \nonumber \delta z_{\rm CNO} &=& 0.45 - 0.19 \, \Delta ^{\rm Ne}_{\rm CNO} \\ \delta z_{\rm met} &=& 0.19 - 0.14 \, \Delta ^{\rm Ne}_{\rm CNO}, \end{eqnarray} \tag{ 32 }$$

where $\Delta ^{\rm Ne}_{\rm CNO} = \delta z_{\rm Ne} - \delta z_{\rm CNO}$, together with a re-adjustment of the systematic pulls:¹²

$$\begin{eqnarray} \nonumber \tilde{\xi }_{\rm opa} &=& 1.07 + 0.76 \, \Delta ^{\rm Ne}_{\rm CNO}\\ \nonumber \tilde{\xi }_{\rm diffu} &=& -0.74 - 0.41 \, \Delta ^{\rm Ne}_{\rm CNO}\\ \nonumber \tilde{\xi }_{\rm s_{33}} &=& 0.46 - 0.28 \, \Delta ^{\rm Ne}_{\rm CNO}\\ \nonumber \tilde{\xi }_{\rm s_{34}} &=& -0.97 + 0.58 \, \Delta ^{\rm Ne}_{\rm CNO}\\ \tilde{\xi }_{\rm s_{17}} &=& -1.20 + 0.62 \, \Delta ^{\rm Ne}_{\rm CNO}. \end{eqnarray} \tag{ 33 }$$

From the above relations, we see that a 30% uncertainty in the neon-to-oxygen ratio roughly corresponds to ∼6% and ∼4% errors in the inferred CNO and heavy element abundances.

This degeneracy can be discussed at a more basic level by considering that the main effect produced by a change of the solar composition is the modification of the opacity profile of the Sun. The source term δκ(r) that drives the modification of the solar properties and that is probed by observational data can be written as the sum of two contributions (Villante 2010):

$$\begin{equation} \delta \kappa (r) = \delta \kappa _{\rm I}(r) + \delta \kappa _{\rm Z}(r)\protect. \protect \end{equation} \tag{ 34 }$$

The intrinsic opacity change, δκ_I(r), represents the fractional variation of the opacity along the SSM profile and it is given, in our approach, by $\delta \kappa _{\rm I}(r) = \tilde{\xi }_{\rm opa}\, \delta \kappa _{\rm opa}(r)$. The composition opacity change δκ_Z(r) can be approximately calculated as

$$\begin{equation} \delta \kappa _{\rm Z}(r) \simeq \sum _{j} \frac{\partial \ln \kappa (r)}{\partial \ln Z_{j}} \; \delta z_{j} \end{equation} \tag{ 35 }$$

by using the logarithmic derivatives ∂ln κ/∂ln Z_j that are presented in the left panel of Figure 10. Taking advantage of relation (34), we calculate the effective opacity change δκ(r) that corresponds to the models that provide a good fit to observational data. We see that δκ(r) is well constrained by the available observational information. Opacity should be increased by ∼few percent at the center of the Sun and by ∼25% at the bottom of the convective envelope, as it was calculated by Villante (2010). The moderate increase at the solar center improves the agreement with Y_{s, obs} without affecting the solar neutrino fluxes. The increasing trend of δκ(r) is required to fit the convective radius R_b and sound speed profile δc_i (see Villante 2010). The wavy behavior at intermediate radii improves consistency with inferred sound speed values in the region 0.3 < r/R_☉ < 0.6. The general features of δκ(r) are essentially independent on the assumptions about the opacity uncertainty. In this respect, the increase of the CNO and/or Ne content is interpreted as providing the "tilt" to δκ(r) and is a solid conclusion of our analysis.

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Figure 10, right panel, also compares the effective variations of opacity δκ(r) obtained in the two and three parameter analysis. In particular, the black dashed line corresponds to the solar model with the composition given by Equation (31) and the value $\tilde{\xi }_{\rm opa} = 1.40$ calculated from Equation (33). The red dotted line represents the effective opacity variation obtained with parameters given by Equation (28) and $\tilde{\xi }_{\rm opa} = 1.07$. We see that the two lines coincide at the 2% level or better. From this, we infer that the reconstructed opacity profile does not depend on the assumed heavy element grouping. Moreover, we understand that the compositions (28) and (31) cannot be discriminated by the adopted observational constraints. More generally, they cannot be distinguished by any conceivable observational test that is dominated by the opacity profile in the radiative region of the Sun: the 2% difference is indeed smaller than the accuracy to which the opacity of the solar plasma is known. In summary, the neon-to-oxygen ratio cannot be effectively constrained with current data.