CONSTRAINTS ON THE OPACITY PROFILE OF THE SUN FROM HELIOSEISMIC OBSERVABLES AND SOLAR NEUTRINO FLUX MEASUREMENTS

In the last few years, a new solar problem has emerged. Recent determinations of the photospheric heavy element abundances (Asplund et al. 2005, 2009; Caffau et al. 2010) indicate that the Sun's metallicity is lower than previously assumed (Grevesse & Sauval 1998). Solar models that incorporate these lower abundances are no more able to reproduce the helioseismic results. As an example, the sound speed predicted by standard solar models (SSMs) implementing the heavy element admixture of Asplund et al. (2005) disagrees at the bottom of the convective envelope by ∼10σ with the value inferred by helioseismic data, see, e.g., Bahcall et al. (2005) and the black dashed line in Figure 1. In addition, the predicted surface helium abundance is lower by ∼6σ and the inner radius of the convective envelope is larger by ∼15σ with respect to the helioseismic results. Detailed studies have been done to resolve this controversy, see, e.g., Basu & Antia (2008). The latest determinations of the solar photospheric composition (Asplund et al. 2009; Caffau et al. 2010) alleviate the discrepancies but a definitive solution of the "solar composition problem" still has to be obtained.

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The main effect of changing the heavy element admixture is to modify the opacity profile of the Sun. It is, thus, evident that the comprehension of the solar composition problem is intimately related to understanding the role of opacity in solar modeling. Several authors have investigated the effects of opacity changes on the solar structure by using different methods and assumptions, see, e.g., Tripathy & Christensen-Dalsgaard (1998), Christensen-Dalsgaard et al. (2009), and Bahcall et al. (2005). Here, we continue their work, completing and extending the analysis of Tripathy & Christensen-Dalsgaard (1998) by using a different and original approach. The final goal is to provide the instruments to analyze in a transparent and efficient way the role of the opacity in the Sun and to perform a critical "step-by-step" discussion of the present constraints on opacity (and composition) provided by observable properties of the Sun.

In order to calculate the effects of arbitrary opacity changes on the Sun, we use the linear solar model (LSM) approach, presented in Villante & Ricci (2010), Villante (2010), and briefly summarized in Appendices A, B, and C. In this approach, the structure equations of the present Sun are linearized and, by estimating the variation of the present solar composition from the variation of the nuclear reaction rates and elemental diffusion efficiency in the present Sun, we obtain a linear system of ordinary differential equations that can be easily solved and that completely determines the physical and chemical properties of the Sun. It was shown in Villante & Ricci (2010) that this kind of approach reproduces with good accuracy the results of nonlinear evolutionary solar models and, thus, can be used to study the role of parameters and assumptions in solar model construction.

By considering localized opacity changes in LSM approach, we determine numerically the kernels that, in a linear approximation, relate an arbitrary opacity variation to the corresponding modification of the solar observable properties. These opacity kernels are useful in several respects. First, they allow us to individuate the region of the Sun whose opacity is probed with maximal sensitivity by each observable quantity. Then, they permit us to show that effects produced by variations of opacity in different regions of the Sun can compensate among each other. Finally, they will be used to discuss how the different pieces of observational information cooperate to determine the present constraints on the opacity profile of the Sun.

The plan of the paper is the following. In Section 2, we discuss the relation between the effects produced by a variation of the heavy element admixture and those produced by a modification of the radiative opacity. We show that the relevant quantity is the variation of the opacity profile δκ(r) defined in Equation (2), which is approximately given by the superposition of the intrinsic opacity change δκ_I(r) and the composition opacity change δκ_Z(r), defined in Equations (3) and (5), respectively. In Section 3, we define the opacity kernels and describe the method adopted to calculate them. In Section 4, we calculate numerically the kernels for the squared isothermal sound speed u(r) ≡ P(r)/ρ(r), the surface helium abundance Y_b, the inner boundary of the convective envelope R_b, and the various neutrino fluxes Φ_ν. Moreover, we discuss the constraints on δκ(r) provided by the present observational data. Finally, in Section 5 we summarize our results. A conclusive view of the constraints on δκ(r) is provided in Figure 8.

We consider a modification of the opacity κ(ρ, T, Y, Z_i) and/or of the heavy element photospheric admixture {z_i}, expressed here in terms of the quantities z_i ≡ Z_i,b/X_b where Z_i,b is the surface abundance of the i-element and X_b is that of hydrogen. If we neglect the role of metals in the equation of state and in the energy generation coefficient, the only effect of these changes is to modify radiative energy transport in the Sun. The relevant parameter, in this respect, is the total variation of the opacity in the shell r of the present Sun, given by

$$\begin{equation} \delta \kappa ^{\rm tot}(r) = \frac{\kappa (\rho (r),T(r),Y(r),Z_{i}(r))}{\overline{\kappa }(\overline{\rho }(r),\overline{T}(r),\overline{Y}(r),\overline{Z}_{i}(r))} -1, \end{equation} \tag{ 1 }$$

where $\overline{\kappa }(\overline{\rho }(r),\overline{T}(r),\overline{Y}(r),\overline{Z}_{i}(r))$ is the opacity profile of the SSM, while κ(ρ(r), T(r), Y(r), Z_i(r)) is the opacity profile of the solar model that implements the modified opacity and photospheric composition.¹ We note that the quantity δκ^tot(r) is not related in a direct way to the performed variations of opacity and composition. In order to calculate it, we have to take into account that the "perturbed" Sun has different density (ρ), temperature (T), and chemical composition profiles (Y and Z_i) with respect to the SSM. These are not known a priori but have to be obtained as a result of numerical solar modeling.

A relevant simplification is obtained in the LSM approach presented in Villante & Ricci (2010), where one assumes that: (1) the performed changes of opacity and heavy element admixture are small; (2) the variation of the chemical composition of the Sun can be estimated from the variation of nuclear reaction rate and diffusion efficiency of the present Sun; (3) the variation of the metal admixture has a negligible direct effect on nuclear production of helium and on diffusion efficiency. In this case, the relation between δκ^tot(r) and the performed variation of opacity and admixture can be worked out explicitly, as it is described in Appendix A. It can be shown, moreover, that the source term δκ(r) that drives the modification of the solar properties² and that can be constrained by observational data can be written as the sum of two contributions:

$$\begin{equation} \delta \kappa (r) = \delta \kappa _{I}(r) + \delta \kappa _{Z}(r). \end{equation} \tag{ 2 }$$

The first term δκ_I(r), which we refer to as intrinsic opacity change, represents the fractional variation of the opacity along the SSM profile and it is given by

$$\begin{equation} \delta \kappa _{I}(r) = \frac{\kappa (\overline{\rho }(r),\overline{T}(r),\overline{Y}(r),\overline{Z}_{i}(r))}{\overline{\kappa }(\overline{\rho }(r),\overline{T}(r),\overline{Y}(r),\overline{Z}_{i}(r))} -1. \end{equation} \tag{ 3 }$$

This contribution is obtained when we revise the opacity function κ(ρ, T, Y, Z_i) and/or we introduce new effects, like, e.g., the accumulation of few GeVs WIMPs in the solar core that mimics a decrease of the opacity at the solar center, see, e.g., Bottino et al. (2002) and references therein.

The second term δκ_Z(r), which we refer to as composition opacity change, describes the effects of a variation of {z_i}. It takes into account that a modification of the photospheric admixture implies a different distribution of metals inside the Sun and, thus, a different opacity profile, even if the function κ(ρ, T, Y, Z_i) is unchanged. The contribution δκ_Z(r) is given by (see Appendices A and B)

$$\begin{equation} \delta \kappa _{Z}(r) = \frac{ \overline{\kappa }(\overline{\rho }(r),\overline{T}(r),\overline{Y}(r),Z_{i}(r))}{\overline{\kappa }(\overline{\rho }(r),\overline{T}(r),\overline{Y}(r),\overline{Z}_{i}(r))} - 1, \end{equation} \tag{ 4 }$$

where $Z_{i}(r) = \overline{Z}_{i}(r) \, (z_{i} / \overline{z}_{i})$ and can be calculated as

$$\begin{equation} \delta \kappa _{Z}(r) \simeq \sum _{i} \left.\frac{\partial \ln \overline{\kappa }}{\partial \ln Z_{i}} \right|_{\rm SSM} \; \delta z_{i,b}, \end{equation} \tag{ 5 }$$

where δz_i represents the fractional variation of z_i and the symbol |_SSM indicates that we calculate the derivatives along the density, temperature, and chemical composition profiles predicted by the SSM.

Equation (2), although being approximate, is quite useful because it makes the connection (and the degeneracy) between the effects produced by a modification of the radiative opacity and of those produced by a modification of the heavy element admixture explicit. In this paper, we take as a reference the Asplund et al. (2005) composition (AGS05) and we refer with "SSM predictions" to the numerical results obtained by using this composition as input for evolutionary solar model calculations.³ Other compilations can be considered, like, e.g., the Grevesse & Sauval (1998, hereafter GS98) or the more recent Asplund et al. (2009, hereafter AGSS09). The red and blue dashed lines in the right panel of Figure 1 correspond to the opacity change δκ_Z(r) that are obtained when we use the GS98 and the AGSS09 admixture in place of the AS05 composition, as calculated by applying relation (5) and by using the logarithmic derivatives ∂ln κ/∂ln Z_i presented in Figure 12 of Basu & Antia (2008). We observe that the variation from AGS05 to GS98 (AGSS09) heavy element admixture corresponds to increasing the opacity by about 5% (1%) at the center of the Sun and by about 20% (5%) at the bottom of the convective region. The effects of these opacity changes on helioseismic and solar neutrino observables, calculated in the LSM approach, are described in Table 1 and in the left panel of Figure 1. They can be compared with the results of full nonlinear evolutionary codes reported, e.g., in Serenelli (2009), obtaining a satisfactory agreement.

In the following, we consider the effects produced by a generic variation of the opacity profile δκ(r), without discussing whether this is due to a change of the function κ(ρ, T, Y, Z_i) or to a change of the admixture {z_i}. As a result of this modification, we obtain a solar model that deviates from SSM predictions. If the opacity variation is sufficiently small (i.e., δκ(r) ≪ 1), the Sun responds linearly. In this case, the fractional variation of a generic quantity Q, defined as

$$\begin{equation} \delta Q = \frac{Q}{\overline{Q}}-1, \end{equation} \tag{ 6 }$$

can be related to δκ(r) by the linear relation:

$$\begin{equation} \delta Q = \int dr \; K_{Q}(r) \; \delta \kappa (r). \end{equation} \tag{ 7 }$$

The kernel K_Q(r) represents the functional derivative with respect to opacity and allows to quantify the sensitivity of Q to opacity variations in different zones of the Sun.

In this paper, we determine numerically the kernels K_Q(r) for helioseismic observables and solar neutrino fluxes, by using the LSM approach presented in Villante & Ricci (2010) and Villante (2010) and briefly summarized in the appendix. In this approach, the structure equations of the present Sun are linearized and, by estimating the variation of the present solar composition from the variation of the nuclear reaction rates and elemental diffusion efficiency in the present Sun, we obtain a linear system of ordinary differential equations that can be easily solved and that completely determines the physical and chemical properties of the Sun. It was shown in Villante & Ricci (2010) that this kind of approach reproduces with good accuracy the results of nonlinear evolutionary solar models and, thus, can be used to study in an efficient and transparent way the role of parameters and assumptions in solar model construction.⁴

The estimate of K_Q(r) at a given point r = r₀ is obtained by performing a localized increase of opacity in the vicinity of r₀. More precisely, we calculate the variation δQ(r₀) produced by a normalized⁵ Gaussian increase of opacity centered in r₀:

$$\begin{equation} \delta \kappa (r) = G(r-r_0)\equiv \frac{1}{\sqrt{2\pi }\delta r}\exp \left[-\frac{(r-r_0)^2}{2\delta r^2}\right] \end{equation} \tag{ 8 }$$

with δr = 0.01 R_☉, and we assume that

$$\begin{equation} K_{Q}(r_{0}) = \delta Q({r_0}). \end{equation} \tag{ 9 }$$

This corresponds to the approximation:

$$\begin{equation} K_{Q}(r_{0})\simeq \int dr \; K_{Q}(r) \; G(r-r_{0}), \end{equation} \tag{ 10 }$$

which is adequate to describe all the situations in which we consider opacity variations δκ(r) which vary on scales larger than δr = 0.01 R_☉.

The kernels K_Q(r) can be used to calculate the effects of arbitrary opacity changes δκ(r) and allow us to discuss the role of opacity in general terms. In order to consider specific situations and to understand what kind of experimental constraints are provided by each observable quantity Q, it is useful, however, to consider the simple parameterizations:

$$\begin{eqnarray} \delta \kappa (x) &=& A_{\rm in} \; \delta \kappa _{\rm in}(x) + A_{\rm out} \; \delta \kappa _{\rm out}(x) \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} \delta \kappa (x) &=& A_{0} \; \delta \kappa _{0}(x) + A_{1} \; \delta \kappa _{1}(x) \end{eqnarray} \tag{ 12 }$$

with A_in, A_out, A₀, and A₁ as free adjustable parameters and

$$\begin{eqnarray} \nonumber \delta \kappa _{\rm in} (r) &\equiv & \frac{1}{\exp {\left[(r-r_{c})/A \right]}+1}\\ \nonumber \delta \kappa _{\rm out} (r) &\equiv & 1 - \frac{1}{\exp {\left[(r-r_{c})/A \right]}+1} \\ \nonumber \delta \kappa _{\rm 0} (r) &\equiv & 1\\ \delta \kappa _{\rm 1} (r) &\equiv & \frac{r}{\overline{R}_{b}}, \end{eqnarray} \tag{ 13 }$$

where r_c = 0.3 R_☉, A = 0.01, while $\overline{R}_{b} = 0.730\, R_{\odot }$ is the radius of the convective envelope predicted by SSM, see Villante & Ricci (2010). The functions δκ_in(r) and δκ_out(r) correspond to a constant increase of the opacity in the energy producing zone (r ⩽ 0.3 R_☉) and in the outer radiative region (r ⩾ 0.3 R_☉), see Figure 2. They have been defined in such a way that δκ_in(r) + δκ_out(r) ≡ 1. The function δκ₀(r) and δκ₁(r) correspond to a global rescaling and to a linear tilt of the opacity profile.

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In linear approximation, the fractional variation δQ produced by the opacity profiles (11) and (12) can be expressed as

$$\begin{eqnarray} \nonumber \delta Q &=& A_{\rm in} \; \delta Q_{\rm in} + A_{\rm out} \; \delta Q_{\rm out}\\ \delta Q &=& A_{0} \; \delta Q_{0} + A_{1} \; \delta Q_{1}, \end{eqnarray} \tag{ 14 }$$

where the coefficients δQ_j are given by

$$\begin{equation} \delta Q_{j} = \int dr \; K_{Q}(r) \; \delta \kappa _{j}(r) \end{equation} \tag{ 15 }$$

with j = in,  out,  0,  1. We report these coefficients in Table 2 for helioseismic observables and solar neutrino fluxes.

We calculate the opacity kernels for following observable quantities: the squared isothermal sound speed u(r) ≡ P(r)/ρ(r), the surface helium abundance Y_b, the depth R_b of the convective envelope, and the solar neutrino fluxes Φ_ν, where the index ν = pp, Be, B, N, O refers to the neutrino producing reactions.⁶ Our results are presented in the following subsections, together with a discussion of the present observational constraints on δκ(r).

4.1. The Sound Speed

In Figure 3, we discuss the properties of the sound speed kernel K_u(r, r') defined by

$$\begin{equation} \delta u(r) = \int dr^{\prime } \; K_{u}(r, r^{\prime }) \; \delta \kappa (r^{\prime }), \end{equation} \tag{ 16 }$$

where δu(r) is the fractional variation of the squared isothermal sound speed u(r) ≡ P(r)/ρ(r). In the left panel of Figure 3, we show the functions $f_{r^{\prime }}(r) \equiv K_u(r,r^{\prime })\, R_{\odot }$, calculated for the selected values r' = 0.1, 0.2, ..., 0.7 R_☉. In our approach, the functions $f_{r^{\prime }}(r)$ correspond to the sound speed variations produced (in a linear theory) by the localized opacity increases G(r − r'). One has large effects close to r' which are due to the variation of the temperature profile of the Sun, as can be understood by considering that δu = δP − δρ ≃ δT − δμ, where μ is the mean molecular weight of the solar plasma. From Equation (C2), we see that δT(r) is expected to have a sharp decrease close to r', by an amount approximately equal to 1/l_t(r') where $l_t =[d\ln (\overline{T})/dr]^{-1}$, while δμ remains approximately constant, being related to the chemical composition of the solar plasma.

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In the right panel of Figure 3, we show the functions g_r(r') ≡ K_u(r, r') R_☉, calculated for the selected values r = 0.1, ..., 0.7 R_☉, that quantify the sensitivity of the sound speed at a given r to the opacity in the shell r' of the Sun. They clearly indicates that u(r) is maximally sensitive to the value of the opacity at r' ≃ r. However, the displayed functions are different from zero everywhere and have negative values in large parts of the solar radiative region. This suggests that compensating effects can occur, especially when one considers opacity modifications that extend over a broad region of the Sun.

In this respect, it is important to note that the sound speed is practically insensitive to a global rescaling of opacity. This can be appreciated by looking at the black dotted line in the left panel of Figure 3, which is defined by

$$\begin{equation} \delta u_{0}(r) = \int dr^{\prime } K_u(r,r^{\prime }) \end{equation} \tag{ 17 }$$

and corresponds to the sound speed variation produced (in a linear theory) by a constant rescaling δκ₀(r) ≡ 1. For visualization purposes, it is useful to note that the values of δu₀(r) at r = 0.1, ..., 0.7 R_☉ correspond to the integral in r'/R_☉ of the functions g_r(r') displayed in the right panel of Figure 3. We see that δu₀(r) is very small, as a result of an almost perfect compensation between the positive contribution from the region r' ≃ r and the negative contribution from the other regions of the Sun.

A qualitative argument to explain the stability of the sound speed is the following. The virial theorem connects the gravitational energy, E_g = −∫dm Gm/r, and the thermal energy content of a given star, E_i = 3/2∫dm(P/ρ) = 3/2∫dm u, being E_g = −2E_i. For a generic star, a global rescaling of the opacity reflects into a global rescaling of the radial profile and, thus, taking into account the virial theorem, also into a global rescaling of the sound speed. This is not the case for the Sun because the solar radius is observationally determined. We are forced to re-adjust the free parameters in the model in order to keep the solar radius fixed, with the effect of stabilizing the radial profile and, thus, the sound speed profile u(r).

The above result has relevant implications. In particular, the statement that the "sound speed problem" requires an increase of the opacity at the bottom of the convective region is, strictly speaking, not correct because other solutions are possible. We can consider, e.g., the sound speed profiles produced by the opacity variations parameterized by Equations (11) and (12). In linear approximation, we have that

$$\begin{eqnarray} \nonumber \delta u(r) &=& A_{\rm in} \, \delta u_{\rm in}(r) + A_{\rm out} \, \delta u_{\rm out}(r)\\ \delta u(r) &=& A_{0} \, \delta u_{0}(r) + A_{1} \, \delta u_{1}(r) \,. \end{eqnarray} \tag{ 18 }$$

The functions δu_in(r), δu_out(r), δu₀(r), and δu₁(r) are shown in the left panel of Figure 4 and are reported in Table 2 for the selected values r = 0.1, 0.2, 0.4, 0.65 R_☉. We see that δu₀(r) ≪ δu₁(r), indicating that the "tilt" and not the scale of the opacity is fixed by the sound speed. Moreover, we have δu_in(r) ≃ −δu_out(r) that shows that the effect of an enhancement of the opacity in the external radiative region can be equally produced by a decrease of the opacity at the solar center.

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This is confirmed in the right panel of Figure 4 where we show the sound speed profiles obtained by implementing four different opacity modifications, described by:

Model A: 15% decrease of opacity in the energy producing region (A_in = −0.15 and A_out = 0);

Model B: 15% increase of the opacity in the outer radiative region (A_in = 0 and A_out = 0.15);

Model C: a linear tilt of opacity corresponding to a 20% increase at the bottom of the convective envelope (A₀ = 0 and A₁ = 0.2); and

Model D: a linear tilt plus a global rescaling of opacity corresponding to a 20% decrease at the solar center (A₀ = −0.2 and A₁ = 0.2).

The sound speed profiles obtained in all the considered cases reproduce equally well the result inferred by helioseismic data, showing that the helioseismic determination of u(r) translates into a bound on the differential increase A_out − A_in ≃ 0.15 and on the tilt A₁ ≃ 0.2, with no relevant constraint on A₀ (or, equivalently, A_in + A_out) that fix the global scale of opacity.

In light of this observation, it is intriguing the possibility that non-standard effects that mimic a decrease of the opacity at the solar center, like, e.g., the accumulation of few GeVs WIMPs in the solar core (see, e.g., Bottino et al. 2002), could have a role in the solution of the solar composition puzzle. We will see, in the following, that this possibility is disfavored by the determination of the surface helium abundance and by the measurement of the ⁷Be and ⁸B neutrino fluxes.⁷

4.2. The Surface Helium Abundance

In the left panel of Figure 5, we show with solid line the functional derivative K_Y(r) of the surface helium abundance Y_b, defined according to equation

$$\begin{equation} \Delta Y_{b} = \int dr\; K_{Y}(r) \; \delta \kappa (r), \end{equation} \tag{ 19 }$$

where we considered the absolute variation ΔY_b to conform with the notations adopted in Villante & Ricci (2010).⁸

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The surface helium abundance depends on the initial chemical composition and on the effects of elemental diffusion. In Villante & Ricci (2010), by assuming that the variation of the time-integrated effect of diffusion can be estimated from the variation of diffusion efficiency in the present Sun, we have obtained the following relation:

$$\begin{equation} \Delta Y_{b} = A_{Y} \; \Delta Y_{\rm ini} + A_{C} \; \delta C \end{equation} \tag{ 20 }$$

with A_Y = 0.838 and A_C = 0.033, which gives ΔY_b as a function of the absolute variation of the initial helium abundance, ΔY_ini, and of the fractional variation of pressure at the bottom of the convective region, δC = δP_b. The above equation allows us to obtain the functional derivative K_Y(r) as the sum of two contributions, one related to the term A_Y ΔY_ini and the other to A_C δC. These are shown in Figure 5 with red and blue dashed lines, respectively.

The function K_Y(r) is positive everywhere, showing that an increase of opacity in an arbitrary shell of the Sun translates into an increase of the helium abundance.⁹ Moreover, the kernel K_Y(r) has a rather broad profile. This implies that the determination of Y_b effectively constrains the opacity scale and breaks the degeneracy between the possible solutions of the sound speed problem presented in the previous section. The SSM that implements the AGS05 heavy element admixture predicts the value $\overline{Y}_{b}=0.229$ (see, e.g., Villante & Ricci 2010), which is about 6σ lower than the helioseismic determination Y_b = 0.2485 ± 0.0034 (Basu & Antia 2008). This discrepancy requires an increase of the opacity, as it can be obtained, e.g., by increasing the metal content of the Sun. Models that accounts for a reduction "effective" opacity at the solar center are expected to decrease Y_b, increasing the disagreement with helioseismic results.

A simple quantitative analysis can be performed by considering the opacity profiles (Equations (11) and (12)) that produce the variations ΔY_b given by

$$\begin{eqnarray} \Delta Y_{b} &=& 0.073 \; A_{\rm in} + 0.069 \; A_{\rm out} \simeq 0.07 \;(A_{\rm in} + A_{\rm out}) \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} \Delta Y_{b} &=& 0.142 \; A_{0} + 0.062 \; A_{1}. \end{eqnarray} \tag{ 22 }$$

By comparing the SSM prediction with the helioseismic result, we obtain A_in + A_out = 0.28 ±  0.05 that, combined with the "orthogonal" constraint A_out − A_in ≃ 0.15 provided by the sound speed, gives A_in = 0.04 → 0.09 and A_out = 0.19 → 0.24. Alternatively, we can use Equation (22) and the information on the tilt A₁ ≃ 0.2 provided by the sound speed, to fix the opacity at the solar center obtaining A₀ = 0.025 → 0.075.

4.3. The Depth of the Convective Region

In the right panel of Figure 5, we show with a solid line the functional derivative K_R(r) of the inner radius of the convective envelope R_b defined according to the relation:

$$\begin{equation} \delta R_{b} = \int dr\; K_{R}(r) \; \delta \kappa (r). \end{equation} \tag{ 23 }$$

We see that the kernel K_R(r) has a very sharp peak at $r \simeq \overline{R}_{b}= 0.730\, R_{\odot }$ that reflects the (well-known) fact that the depth of the convective zone is particularly sensitive to the opacity at the bottom of the convective envelope. The shape of this peak has not a precise physical meaning and depends on the method of calculation. The effect of opacity changes has been, in fact, estimated within the LSM approach, in which the fractional variation δR_b is calculated from

$$\begin{equation} \delta R_{b} = \Gamma _{Y} \; \Delta Y_{\rm ini} + \Gamma _{C} \; \delta C + \Gamma _\kappa \; \delta \kappa _{b}, \end{equation} \tag{ 24 }$$

where Γ_Y = 0.449, Γ_C = −0.117, and Γ_κ = −0.085, while δκ_b is fractional variation of opacity at the bottom of the convective envelope, i.e., $\delta \kappa _{b} = \delta \kappa (\overline{R}_{b})$. The "local" term Γ_κ δκ_b translates into a delta-function contribution $\Gamma _\kappa \, \delta (r-\overline{R}_{b})$ to the functional derivative. Since we evaluate numerically the kernel by applying a localized Gaussian increase of opacity, this is convolved with the function G(r − r₀) given in Equation (8). As a final result, one obtains the contribution $\Gamma _\kappa \, G(r-\overline{R}_{b})$ to the kernel K_R(r) which is shown by the green dashed line in Figure 5, whereas the red and blue dashed lines describe the contributions arising from Γ_Y ΔY_ini and Γ_C δC, respectively. We remark that the area under the peak at $r=\overline{R}_{b}$ is approximately equal to Γ_κ and does not depend on the calculation method. We can, thus, safely use the functional derivative K_R(r) to describe all the situations in which opacity varies on scale larger than δr = 0.01 R_☉.

Equations (20) and (24) can be combined to obtain a direct determination of δκ_b from quantities that are all determined by helioseismic observations. We can, in fact, eliminate ΔY_ini from Equation (24), obtaining

$$\begin{equation} \delta \kappa _{b} = C_{Y}\;\Delta Y_{b} + C_{R} \; \delta R_{b} + C_{\rho } \; \delta \rho _{b}, \end{equation} \tag{ 25 }$$

where

$$\begin{eqnarray} C_{Y} &=& - \frac{\Gamma _{Y}}{A_{Y}\;\Gamma _{\kappa }} = 6.27\\ C_{R} &=& \frac{1}{\Gamma _{\kappa }} = -11.71 \\ C_{\rho } &=& \frac{1}{\Gamma _{\kappa }}\left[\frac{A_{C}\; \Gamma _{Y}}{A_{Y}}-\Gamma _{C}\right] = -1.58. \end{eqnarray} \tag{ 26 }$$

In the derivation of the above result, we have considered that δC = δP_b ≃ δρ_b, since the fractional variation of the sound speed δu(r) is expected to vanish at the bottom of the convective region, i.e., δu_b = δP_b − δρ_b ≃ 0, as it is discussed in Villante & Ricci (2010). The discrepancy between the helioseismic determinations of R_b and Y_b and the predictions of SSMs implementing AS05 admixture is quantified as δR_b = −0.0205 ± 0.0015 and ΔY_b = 0.0195 ± 0.0034. The density at the bottom of the convective region deviates from value inferred by helioseismology by δρ_b = 0.08, as it is discussed, e.g., in Serenelli (2009). We obtain δκ_b ≃ 0.24 ± 0.03, where errors have been combined in quadrature and we have neglected the (sub-dominant) contribution to the total error budget due to the uncertainties in the density determination. We remark that the obtained result is model-independent, since it does not rely on any assumption or parameterization for the function δκ(r).

As a final application, we consider the response of R_b to the opacity changes parameterized by Equations (11) and (12). We obtain the relations

$$\begin{eqnarray} \delta R_{b} &=& 0.12 \; A_{\rm in} - 0.14 \; A_{\rm out} \simeq 0.13 \; (A_{\rm in} - \; A_{\rm out}) \end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray} \delta R_{b} &=& -0.02 \; A_{0} -0.10 \; A_{1} \end{eqnarray} \tag{ 30 }$$

which show that the inner radius of the convective envelope, just like the sound speed, provides bounds on the differential increase A_out − A_in and on tilt A₁, with no relevant constraints on A₀ (or equivalently A_in + A_out) that fix the global scale of opacity. By considering δR_b = −0.0205 ± 0.0015, we obtain A_out − A_in ∼ 0.15 and A₁ ∼ 0.2 in substantial agreement (and complete degeneracy) with the information provided by the sound speed measurement. By combining this information with the constraints provided by the surface helium abundance and performing a simple χ² analysis, we obtain A_in = 0.07 ± 0.04 and A_out = 0.21 ± 0.04 for the parameterization given in Equation (11) and A₀ = 0.056 ±  0.040 and A₁ = 0.187 ± 0.023 for that given in Equation (12). The corresponding bounds on the opacity change δκ(r) are shown in Figure 8 and commented upon in the conclusive section.

4.4. Neutrino Fluxes

In the left panel of Figure 6, we show the functional derivatives K_ν(r) of the neutrino fluxes Φ_ν defined according to relation:

$$\begin{equation} \delta \Phi _{\nu } = \int dr\; K_{\nu }(r) \; \delta \kappa (r), \end{equation} \tag{ 31 }$$

where the index ν = pp,  Be,  B,  N,  O refers to the neutrino producing reactions. The kernel K_ν(r) have been calculated by using the LSM approach and by taking into account that the fractional variations of the fluxes δΦ_ν are related to physical and chemical properties of the Sun by

$$\begin{eqnarray} \delta \Phi _\nu &=& \int dr [ \phi _{\nu,\rho }(r) \, \delta \rho (r) +\phi _{\nu,T}(r) \, \delta T(r)\nonumber\\ &&+ \phi _{\nu,Y}(r) \, \Delta Y(r) + \phi _{\nu,Z}(r) \, \delta Z(r)]. \end{eqnarray} \tag{ 32 }$$

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The functions ϕ_ν,j(r) have been defined and calculated in Figure 10 of Villante & Ricci (2010).

Our results show that neutrino fluxes probe the opacity of the Sun in the region r ⩽ 0.45 R_☉ which is larger than the neutrino producing zone.¹⁰ This is not surprising. A modification of the opacity translates, indeed, into a different temperature profile (and into a readjustment of the parameters of the model in such a way that the observed solar luminosity is recovered). Thus, even an opacity change localized outside the energy producing region can lead to a different central temperature and to modification of the solar neutrino fluxes.

The kernels K_B(x), K_Be(x), K_N(x), and K_O(x) are positive-valued almost everywhere, while the kernel K_pp(r) is negative. This indicates that an increase of opacity generally translates into an enhancement of ⁸B, ⁷Be, and CNO neutrino fluxes and into a (slight) decrease of the pp-neutrino component. In Table 2, we show the coefficients $\delta \Phi _{\nu,\rm in}$, $\delta \Phi _{\nu,\rm out}$, δΦ_ν,0, and δΦ_ν,1 that allow to describe the effects of opacity changes parameterized by Equations (11) and (12), through the simple relations:

$$\begin{eqnarray} \delta \Phi _\nu &=& A_{\rm in} \; \delta \Phi _{\nu,\rm in} + A_{\rm out} \;\delta \Phi _{\nu,\rm out} \end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray} \delta \Phi _\nu &=& A_{0} \; \delta \Phi _{\nu,0} + A_{1} \;\delta \Phi _{\nu,1}. \end{eqnarray} \tag{ 34 }$$

We see that ⁸B and CNO neutrinos are extremely sensitive to opacity changes, as it is expected since they strongly depend on the temperature of the central regions of the Sun. We also see that $\delta \Phi _{\nu, \rm in}\ge \delta \Phi _{\rm \nu, out}$, as a consequence of the fact that the kernels are peaked in the most internal region of the Sun. We finally note, in the right panel of Figure 6, that the normalized neutrino kernels K_ν(x)/Φ_ν,0 have a common behavior, with two maxima at r ∼ 0.1 R_☉ and r ∼ 0.3 R_☉, and one minimum at r ∼ 0.2 R_☉. This shows that different fluxes basically probe the same quantity, constraining the opacity profile with the maximal sensitivity in the two regions of the Sun r ∼ 0.05 ÷ 0.15 and r ∼ 0.2 ÷ 0.45.

Some insights on the above results can be obtained by recalling that the total neutrino flux is essentially fixed by the solar luminosity constraint. We, thus, expect that $\Delta \Phi _{\rm tot} = \sum _{\nu } \overline{\Phi }_{\nu } \cdot \delta \Phi _{\nu } \simeq 0$, where $\overline{\Phi }_{\nu }$ are the SSM predictions for the various neutrino components. Considering that about 99% of the total flux is provided by pp and Be neutrinos, the luminosity constraint implies $\delta \Phi _{pp} \simeq -(\overline{\Phi }_{\rm Be} /\overline{\Phi }_{pp}) \, \delta \Phi _{\rm Be} = -0.075\, \delta \Phi _{\rm Be}$ which explains the smallness of the pp-neutrino kernel, the ratio between the pp and Be-neutrino coefficients in Table 2 and the equality between the normalized pp and Be-neutrino kernels observed in the right panel of Figure 6. The wavy shape of the kernels K_ν(r) reflects, instead, the response of the central temperature to localized opacity modifications. This was first noted and discussed by Tripathy & Christensen-Dalsgaard (1998) and it is seen in the right panel of Figure 6, where we show with a dashed line the functional derivative of the central temperature T_c, defined by

$$\begin{equation} \delta T_{c} = \int dr \; K_{T}(r) \; \delta \kappa (r). \end{equation} \tag{ 35 }$$

For convenience, we plot the normalized kernel K_T(x)/δT_c,0, where the normalization factor is δT_c,0 = 0.138.

The peculiar behavior with r of the kernel K_T(r) cannot be explained in simple terms. A qualitative comprehension can be obtained from Figure 7, where we show the temperature profiles δT(r) produced by the opacity changes δκ(r) = G(r − r₀) with r₀ = 0.1,  0.2,  0.3 R_☉. We see that the performed opacity modifications translate into a large increase of temperature close to r₀, that necessarily alters nuclear burning rates since r₀ is inside or close to the energy producing region. The free parameters of the solar model are readjusted in such a way that the same luminosity is obtained with a different temperature profile.

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In the two cases r₀ = 0.1 R_☉ (i.e., opacity increase well inside the energy producing region) and r₀ = 0.3 R_☉ (i.e., opacity increase just outside the energy producing region), the entire energy producing zone is affected. A new "equilibrium" situation is achieved in which nuclear burnings occur at higher temperatures with a larger helium abundance, favoring ⁸B, CNO, and ⁷Be neutrinos at the expense of pp neutrinos. For r₀ = 0.2 R_☉, we have a peculiar situation since the maximal effect is produced in a region of the Sun where only pp neutrinos are produced, as it is seen from Figure 10 of Villante & Ricci (2010). This necessarily shifts energy production outward with respect to the SSM case. In order to avoid overproduction of energy and to recover the observed luminosity, the central temperature T_c has to be (slightly) decreased so that the burning rates at the center of the Sun are suppressed.

At present, the best studied components of the solar neutrino flux are the ⁸B neutrino flux which is determined by the SNO neutral current measurement with about 6% accuracy, Φ_B = (5.18 ± 0.29) × 10⁶ cm⁻² s⁻¹ (Ahmad et al. 2002; Aharmim et al. 2005, 2008),¹¹ and the ⁷Be neutrino flux which is measured by Borexino, Φ_Be = (5.18 ± 0.51) × 10⁹ cm⁻² s⁻¹ (Arpesella et al. 2008). These fluxes have to be compared with the results of theoretical calculations. SSMs implementing AGS05 heavy elements admixture predict values for Φ_B and Φ_Be which are about 10% lower than the experimental results. We take, for definiteness, the values $\overline{\Phi }_{\rm B} = 4.66 \times 10^6 \;{\rm cm}^{-2}\; {\rm s}^{-1}$ and $\overline{\Phi }_{\rm Be} = 4.54 \times 10^9 \;{\rm cm}^{-2}\; {\rm s}^{-1}$ obtained in Serenelli (2009) which are affected by ∼9% and ∼5% theoretical uncertainties,¹² respectively. The difference between the theoretical predictions and the experimental data points toward a moderate increase of the central opacity of the Sun. As an example, a 5% increase of the opacity in the region r ⩽ 0.3 R_☉ would produce a 9.7%(4.3%) increase of the ⁸B(⁷Be) neutrino flux. Models that account for a reduction of the "effective" central opacity, due, e.g., to WIMP accumulation, increase the disagreement and are, thus, disfavored by solar neutrino data.

We, finally, discuss the constraints provided by solar neutrinos on opacity changes parameterized by Equations (11) and (12). We note that the coefficients A_in and A_out and/or A₀ and A₁ required to fit helioseismic results (see the previous section) produce enhancements of the ⁸B and ⁷Be neutrino fluxes which are equal to 28% and 14%, respectively. While the ⁷Be component would be consistent with the observational data, the ⁸B neutrino flux is too large with respect to the SNO measurement. When we fit simultaneously helioseismic and solar neutrino data, we obtain a slight reduction of the required opacity change. A simple χ² analysis gives A_in = 0.05 ± 0.03, A_out = 0.19 ± 0.03 for the parameterization (11) and A₀ = 0.038 ± 0.034, A₁ = 0.192 ± 0.023 for the parameterization (12) with the best-fit values corresponding to χ²_min/d.o.f. = 2.1/2 and χ²_min/d.o.f. = 1.7/2, respectively. The corresponding bounds on δκ(r) are displayed in Figure 8.

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Motivated by the solar composition problem and by using the recently developed LSM approach (Villante & Ricci 2010), in this paper we have discussed the effects of arbitrary opacity changes on the Sun. Our main results can be summarized as follows.

• 1.  
  We have discussed the relation between the effects produced by a variation of the heavy element admixture and those produced by a modification of the radiative opacity. We have shown that the relevant quantity is the variation of the opacity profile δκ(r) defined in Equation (2), that is approximately given by the superposition of the intrinsic opacity change δκ_I(r) and the composition opacity change δκ_Z(r), defined in Equations (3) and (5), respectively.
• 2.  
  We have studied the response of the Sun to an arbitrary modification of the opacity δκ(r). Namely, we have calculated numerically the kernels that, in a linear approximation, relate the opacity change δκ(r) to the corresponding modifications of the solar observable properties. We have considered the following observable quantities: the squared isothermal sound speed u(r), the surface helium abundance Y_b, the inner boundary of the convective envelope R_b, and the solar neutrino fluxes Φ_ν.
• 3.  
  We have shown that different observable quantities probe different regions of the Sun. Moreover, effects produced by variations of opacity in distinct zones of the Sun may compensate among each other. In this respect, we noted that the sound speed u(r) and the depth of the convective envelope R_b are practically insensitive to a global rescaling of the opacity.
• 4.  
  As a consequence of the above result, we have seen that the discrepancy between the SSM predictions for u(r) and R_b and the helioseismic inferred values can be equally solved by a ∼15% decrease of the opacity at the center of the Sun or by a ∼15% increase of the opacity in the external radiative region. The degeneracy between these two possible solutions is broken by the "orthogonal" information provided by the measurements of the surface helium abundance and of the boron and beryllium neutrino fluxes, that fix the scale of opacity and indicate that only the second possibility can effectively solve the solar composition problem.
• 5.  
  We have derived a model-independent relation, Equation (25), that allows us to obtain a direct determination of the opacity change required at the bottom of the convective envelope δκ_b from quantities that are determined by helioseismic observations, i.e., the surface helium abundance, the depth of the convective region, and the density at the bottom of the convective envelope. By considering the present observation values, we have obtained δκ_b = 0.24 ± 0.03.

A conclusive view of the present constraints on δκ(r) is contained in Figure 8. The red (blue) dashed lines correspond to the composition opacity changes obtained when we replace the AGS05 composition with the GS98 (AGSS09) admixture. The blue tick at $r=\overline{R}_{b}$ is the value of the opacity at the bottom of the convective region that is obtained by applying the model-independent relation (25) to helioseismic data. The dark and light areas individuate the opacity changes δκ(r) that, for each value of r, are obtained at 68.3%(1σ) and 95.4%(2σ) C.L. by applying a simple χ² analysis to the opacity modification parameterized by Equation (12). The left panel takes into account only helioseismic observables. Namely, it is obtained by using the present observational determinations of the surface helium abundance and of the inner radius of the convective zone. The selected opacity profile δκ(r) is also consistent with the helioseismic determination of the sound speed, as it is has been discussed in Figure 4. The right panel also includes the observational information on boron and beryllium neutrinos. These data move the required opacity change δκ(r) toward slightly smaller values.

In summary, we see that helioseismic and solar neutrino data require a tilt of the opacity profile of the Sun, in such a way that the opacity at the bottom of the convective region is increased by about 25% while a more moderate increase is obtained at the solar center. The required variation is large with respect to uncertainties in opacity calculations which are at the few percent level, as can be estimated by comparing different calculations (see Figure 7 of Badnell et al. 2005), and corresponds well with the composition opacity change that is obtained by using the GS98 heavy element admixture in place of the AGS05 composition. Finally, we note that the obtained results disfavor solutions of the solar composition problem that predict a decrease of the opacity at the solar center, like, e.g., that recently proposed by Frandsen & Sarkar (2010).

It is a pleasure for me to thank B. Ricci for collaboration on the subject presented in this paper, for precious suggestions, and very useful comments.

In order to calculate the total variation of opacity δκ_tot(r) at a given radius r, we have to take into account that the perturbed Sun has different temperature, density, and chemical composition profiles with respect to SSM. We define

$$\begin{equation} \delta \kappa ^{\rm tot}(r) = \frac{\kappa (\rho (r),T(r),Y(r),Z_{i} (r))}{\overline{\kappa }(\overline{\rho }(r),\overline{T}(r),\overline{Y}(r),\overline{Z}_{i} (r)) } -1 \end{equation} \tag{ A1 }$$

and, by expanding the above equation to linear order, we obtain

$$\begin{equation} \delta \kappa ^{\rm tot}(r) = \kappa _{T} \, \delta T(r) + \kappa _{\rm \rho } \, \delta \rho (r) + \kappa _{Y} \, \Delta Y(r) + \sum _{i} \kappa _{i} \, \delta Z_{i} (r) + \delta \kappa _{I}(r), \end{equation} \tag{ A2 }$$

where

$$\begin{eqnarray*} \nonumber \kappa _{T}(r) &=& \left. \frac{\partial \ln \kappa }{\partial \ln T} \right|_{\rm SSM}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \nonumber \kappa _{\rho } (r) = \left. \frac{\partial \ln \kappa }{\partial \ln \rho }\right|_{\rm SSM}\\ \nonumber \kappa _{Y}(r) &=& \left. \frac{\partial \ln \kappa }{\partial Y}\right|_{\rm SSM} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \kappa _{i}(r) = \left. \frac{\partial \ln \kappa }{\partial \ln Z_{i}}\right|_{\rm SSM} \end{eqnarray*}$$

and the symbol |_SSM indicates that we calculate the derivatives κ_j(r) along the density, temperature, and chemical composition profiles predicted by the SSM. The quantity δκ_I(r) represents the intrinsic opacity change and it is given by

$$\begin{equation} \delta \kappa _{I}(r) = \frac{\kappa (\overline{\rho }(r),\overline{T}(r),\overline{Y}(r),\overline{Z}_{i} (r)) }{\overline{\kappa } (\overline{\rho }(r),\overline{T}(r),\overline{Y}(r),\overline{Z}_{i} (r)) } -1. \end{equation} \tag{ A3 }$$

By taking advantage of the equation of state of the stellar plasma, we can eliminate the density from Equation (A2). We use the relation

$$\begin{equation} \delta \rho (r) = \delta P(r) - \delta T(r) - P_{Y}(r) \Delta Y(r), \end{equation} \tag{ A4 }$$

where P_Y(r) ≃ −∂ln μ/∂Y ≃ −5/[8-5Y(r)] and μ represents the mean molecular weight, and we obtain

$$\begin{eqnarray} \delta \kappa ^{\rm tot}(r) &=& (\kappa _{\rm T} -\kappa _{\rho })\, \delta T(r) + \kappa _{\rm \rho } \, \delta P(r) + (\kappa _{Y} - P_{Y} \, \kappa _{\rho }) \, \Delta Y(r)\nonumber\\ &&+ \sum _{i} \kappa _{i} \, \delta Z_{i} (r) + \delta \kappa _{I}(r). \end{eqnarray} \tag{ A5 }$$

In order to evaluate Equation (A5), we need to estimate the chemical composition of the perturbed Sun, i.e., the quantities δZ_i(r) and ΔY(r). By using the approximate method introduced in Villante & Ricci (2010) and reviewed in the next section, we can relate the chemical composition profiles to the modification of the photospheric heavy element admixture, to the present values of δT(r) and δP(r), and to the parameters ΔY_ini and δC which represent the absolute variation of the initial helium abundance and the fractional variation of the pressure at the bottom of the convective region, respectively. We obtain

$$\begin{eqnarray} \delta \kappa ^{\rm tot}(r) &=& \kappa ^{\prime }_{T} \, \delta T(r) + \kappa ^{\prime }_{P} \, \delta P(r) + \kappa ^{\prime }_{Y} \, \Delta Y_{\rm ini}\nonumber\\ &&+ \kappa _{C} \, \delta C + \left[ \delta \kappa _{I}(r) +\delta \kappa _{Z}(r) \right], \end{eqnarray} \tag{ A6 }$$

where δκ_Z(r) is the composition opacity change given by

$$\begin{equation} \delta \kappa _{Z} (r) = \sum _{i} \kappa _{i} \, \delta z_{i}, \end{equation} \tag{ A7 }$$

where δz_i is the fractional variation of z_i = Z_i,b/X_b, while

$$\begin{eqnarray*} \nonumber \kappa ^{\prime }_{T} &=& \kappa _{T} -\kappa _{\rho }\left(1+P_{Y} \, \xi _{T}\right) -K_{Y} \,\xi _{T} \\ \nonumber \kappa ^{\prime }_{P} &=& \kappa _{\rho }\left(1 - P_{Y} \, \xi _{P}\right) -K_{Y} \,\xi _{P} \\ \nonumber \kappa _{Y} &=& (\kappa _{Y} - \kappa _\rho \, P_{Y}) \, \xi _{Y} + Q_{Y} \, \kappa _{Z}\\ \nonumber \kappa _{C} &=& Q_{C} \, k_{Z} \end{eqnarray*}$$

with the coefficients Q_h and ξ_h defined in the next section. In the derivation of the above equation, we took into account that ∑_iκ_i = κ_Z where $\kappa _{Z} = \partial \ln \overline{\kappa }/\partial \ln Z$ is the partial derivative of opacity with respect to the total metal abundance (i.e., calculated by rescaling all the heavy element abundances by a constant factor, so that the metal admixture remains fixed).

It is useful to define the opacity change δκ(r) given by

$$\begin{equation} \delta \kappa (r) = \delta \kappa _{I}(r) + \delta \kappa _{Z}(r) \end{equation} \tag{ A8 }$$

which groups together the contributions to δκ^tot(r) which are directly related to the variation of the input parameters. The other terms in Equation (A6) represent derived quantities that have to be determined by solving the structure equations and/or by fitting the observed properties of the Sun.

The chemical composition of the perturbed Sun should be calculated by integrating the perturbed structure and chemical-evolution equations starting from an ad hoc chemical homogeneous ZAMS model. In Villante & Ricci (2010), we proposed a simplified approximate procedure that allows to estimate with sufficient accuracy the helium and metal abundances of the modified Sun, without requiring to follow explicitly its time-evolution. We review this procedure and we extend it to take into account the effect of a variation of the photospheric composition.

In order to quantify the relevance of the different mechanisms determining the present composition of the Sun, we express the helium and metal abundance according to

$$\begin{eqnarray} \nonumber Y(r)&=&Y_{\rm ini}\,\left[ 1+D_{Y}(r) \right] +Y_{\rm nuc}(r)\\ Z_i (r)&=&Z_{i, \rm ini}\,\left[ 1+D_{Z}(r) \right]. \end{eqnarray} \tag{ B1 }$$

Here, Y_ini and $Z_{i, \rm ini}$ are the initial values for the abundances, the terms D_Y(r) and D_Z(r) describe the effects of elemental diffusion, and Y_nuc(r) represents the total amount of helium produced in the shell r by nuclear processes. Note we, implicitly, assumed that heavy elements have all the same diffusion velocity by introducing a common diffusion term D_Z(r) for all metals.

We are interested in describing how the chemical composition is modified when we perturb the SSM. In the radiative core ($r\le \overline{R}_{b}$), we neglect the effects produced by variations of the diffusion terms¹³ and we write

$$\begin{eqnarray} \nonumber \Delta Y(r) &=& \Delta Y_{\rm ini} \,[ 1+\overline{D}_{Y}(r) ] +\Delta Y_{\rm nuc}(r) \\ \delta Z_{i}(r) &=& \delta Z_{i, \rm ini}, \end{eqnarray} \tag{ B2 }$$

where ΔY_nuc(r) is the absolute variation of the amount of helium produced by nuclear reactions. A better accuracy is required in the convective region, because the surface helium abundance Y_b is an observable quantity. We, thus, discuss explicitly the role of diffusion and we write

$$\begin{eqnarray} \nonumber \Delta Y_{b}&=& (1+\overline{D}_{Y, b})\, \Delta Y_{\rm ini} + \overline{Y}_{\rm ini}\, \overline{D}_{Y,b} \, \delta D_{Y,{b}}\\ \delta Z_{i, b}&=&\delta Z_{i, \rm ini} +\frac{\overline{D}_{Z, b}}{1+\overline{D}_{Z, b}}\, \delta D_{Z,_{b}}, \end{eqnarray} \tag{ B3 }$$

where δD_Y,b and δD_Z,b are the fractional variations of the diffusion terms D_Y,b and D_Z,b. The quantities ΔY_b and δZ_b are related among each other, since the metals-to-hydrogen ratios at the surface of the Sun are observationally fixed. If we indicate with z_i = Z_i,b/X_b the surface abundance of the i-element (rescaled to that of hydrogen), we obtain

$$\begin{equation} \delta Z_{i, b}=-\frac{1}{1-\overline{Y}_{b}}\Delta Y_{b} + \delta z_{i}, \end{equation} \tag{ B4 }$$

where we considered that X_b ≃ 1 − Y_b, while δz_i is defined by

$$\begin{equation} \delta z_{i} = \frac{(Z_{i,b}/X_{b}) -(\overline{Z}_{i, b}/\overline{X}_{b})}{(\overline{Z}_{i, b}/\overline{X}_{b})}. \end{equation} \tag{ B5 }$$

The above relation can be rewritten in terms of the initial helium and metal abundances, obtaining:

$$\begin{equation} \delta Z_{i, \rm ini} = Q_0 \, \Delta Y_{\rm ini} +Q_1 \, \delta D_{Y,{b}} +Q_2 \, \delta D_{Z,_{b}} + \delta z_{i}. \end{equation} \tag{ B6 }$$

The coefficients Q_i have been calculated explicitly in Villante & Ricci (2010) and are given by Q₀ = −1.141, Q₁ = 0.041, and Q₂ = +0.118, respectively.

Up to this point, the derived relations have a general validity, since the only assumption implied by our analysis is that the heavy elements have all the same diffusion velocity (we take iron as representative for all metals). To complete our calculation, we have to estimate the term ΔY_nuc(r) in Equation (B2) and the quantities δD_Y,b and δD_Z,b in Equations (B3) and (B6). We use the procedure adopted in the LSM approach, where we assumed that the helium produced by nuclear reactions scales proportionally to the energy generation coefficient (and, thus, the helium production rate) in the present Sun, i.e.,

$$\begin{equation} \Delta Y_{\rm nuc} = \overline{Y}_{\rm nuc} (r) \, \delta \epsilon ^{\rm tot}(r), \end{equation} \tag{ B7 }$$

where δ^tot(r) is the fractional variation of the energy generation rate. The effect of elemental diffusion is modeled by assuming that the terms D_i,b vary proportionally to the efficiency of diffusion in the present Sun, obtaining (see Section 6.3 and Appendix C of Villante & Ricci 2010)

$$\begin{eqnarray} \nonumber \delta D_{Y,b} &=& \Pi _Y \, \delta T_{b} + \Pi _{P} \, \delta P_{b}\\ \delta D_{Z,b} &=& \Pi _Z \, \delta T_{b} + \Pi _{P} \, \delta P_{\rm b}, \end{eqnarray} \tag{ B8 }$$

where Π_Y = 2.05, Π_Z = 2.73, and Π_P = −1.10.

By following the calculations described in Section 6.2 of Villante & Ricci (2010), we obtain the following expression for the variation of the helium abundance in the radiative region:

$$\begin{equation} \Delta Y(r) = \xi _{Y}(r) \, \Delta Y_{\rm ini} + \xi _T(r) \, \delta T(r) + \xi _{P}(r) \, \delta P(r). \end{equation} \tag{ B9 }$$

The coefficients ξ_h(R) are defined in Equation (31) of Villante & Ricci (2010) and are shown in their Figure 4.

By taking into account relations (B8) and by considering the conditions that hold at the bottom of the convective region (expressed in Equation (21) of Villante & Ricci 2010), we can estimate the variation of metal abundances in the radiative region, obtaining:

$$\begin{equation} \delta Z_{i} (r) = \delta Z_{i, \rm ini} = Q_{Y} \, \Delta Y_{\rm ini} + Q_{C} \, \delta C + \delta z_{i}, \end{equation} \tag{ B10 }$$

where Q_Y = −0.887, Q_C = −0.164, and δC = δP_b represent the variation of pressure at the bottom of the convective envelope.

Finally, we can calculate the abundances in the convective region obtaining

$$\begin{eqnarray} \delta Y_{b} &=& A_{Y} \, \Delta Y_{\rm ini} + A_{C} \, \delta C \end{eqnarray} \tag{ B11 }$$

$$\begin{eqnarray} \delta Z_{i, b} &=& B_{Y} \, \Delta Y_{\rm ini} + B_{C} \, \delta C + \delta z_{i}, \end{eqnarray} \tag{ B12 }$$

where A_Y = 0.838, A_C = 0.033, B_Y = −1.088, and B_C = −0.043.

We remark that, while the quantities ΔY_ini and δC are parameters which are univocally determined by imposing the appropriate integration conditions (see the next section), the quantities δz_i represent input parameters for solar model calculations.

By expanding to linear order the structure equations of the present Sun close to the SSM solution and by assuming that the variation of the chemical abundances of the Sun can be estimated by the procedure outlined in the previous section, we obtain a linear system of ordinary differential equations that completely determine the physical and chemical properties of the "perturbed" Sun, see Villante & Ricci (2010) for details. Namely, we obtain

$$\begin{eqnarray} \frac{d\delta m}{dr} &=& \frac{1}{l_m} \,\left[ \gamma _{P}\, \delta P + \gamma _T \, \delta T - \delta m + \gamma _Y \, \Delta Y_{\rm ini} \right]\\ \nonumber \frac{d\delta P}{dr}&=& \frac{1}{l_P}\, \left[\left(\gamma _{P}-1\right) \, \delta P + \gamma _T \, \delta T + \delta m + \gamma _Y \, \Delta Y_{\rm ini} \right]\\ \nonumber \frac{d\delta l}{dr}& = & \frac{1}{l_l}\, [ \beta ^{\prime }_P \,\delta P + \beta ^{\prime }_T \,\delta T - \delta l + \beta ^{\prime }_Y \,\Delta Y_{\rm ini} + \beta ^{\prime }_C \, \delta C ]\\ \nonumber \frac{d\delta T}{dr}&=& \frac{1}{l_T}\, [ \alpha ^{\prime }_P \,\delta P + \alpha ^{\prime }_T \,\delta T + \delta l + \alpha ^{\prime }_Y \, \Delta Y_{\rm ini}+ \alpha ^{\prime }_C \, \delta C + \delta \kappa ]. \end{eqnarray} \tag{ C1 }$$

The coefficients γ_h, β'_h, and α'_h and the scale heights $l_{h}\equiv [d\ln (\overline{h})/dr]^{-1}$ have been calculated in Villante & Ricci (2010) and are shown in their Figures 1, 5, and 6. The parameters ΔY_ini and δC represent the absolute variation of the initial helium abundance and the relative variation of pressure at the bottom of the convective envelope and can be univocally determined by imposing the appropriate integration conditions. At the center of the Sun (r = 0), we have

$$\begin{eqnarray*} \nonumber \delta m &=& \gamma _{P,0}\, \delta P_{0} + \gamma _{T,0} \, \delta T_{0} + \gamma _{Y,0} \, \Delta Y_{\rm ini} \;\;\;\;\;\;\;\;\;\;\;\;\;\,\qquad \delta P = \delta P_{0}\\ \nonumber \delta l &=& \beta ^{\prime }_{P,0} \,\delta P_{0} + \beta ^{\prime }_{T,0} \,\delta T_{0} + \beta ^{\prime }_{Y,0} \,\Delta Y_{\rm ini} + \beta ^{\prime }_{C,0} \, \delta C \;\;\;\;\;\;\; \delta T = \delta T_{0}, \end{eqnarray*}$$

where the subscript "0" indicates that a given quantity is evaluated at r = 0. At the bottom of the convective envelope ($r=\overline{R}_{b}$), we have instead

$$\begin{eqnarray*} \nonumber \delta m &=& - \overline{m}_{\rm conv} \, \delta C \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \delta P = \delta C\\ \nonumber \delta l &=& 0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \delta T = A^{\prime }_Y \, \Delta Y_{\rm ini} + A^{\prime }_{C} \, \delta C, \end{eqnarray*}$$

where A'_Y = 0.626 and A'_C = 0.025 and $\overline{m}_{\rm conv}=\overline{M}_{\rm conv}/M_{\odot }=0.0192$ is the fraction of solar mass contained in the convective region.

The term δκ(r) contains the contributions to the modification of the opacity profile of the Sun that are directly related to the variation of the input parameters. It is given by

$$\begin{equation} \delta \kappa (r) = \delta \kappa _{I}(r) + \delta \kappa _{z}(r), \end{equation} \tag{ C2 }$$

where δκ_I(r) and δκ_Z(r) are the intrinsic and composition opacity changes, defined in Equations (3) and (5), respectively (see Appendix A). It represents the source term that drives the modification of the solar properties and that can be bounded by observational data.