LINEAR SOLAR MODELS

In the last three decades, there has been enormous progress in our understanding of the Sun. The predictions of the standard solar model (SSM), which is the fundamental theoretical tool to investigate the solar interior, have been tested by solar neutrino experiments and by helioseismology.

The deficit of the observed solar neutrino fluxes, reported initially by Homestake (Davis et al. 1968) and then confirmed by GALLEX (GALLEX Coll. 1999) and SAGE (SAGE Coll. 1999) which subsequently merged into GNO (GNO Coll. 2005), Kamiokande (Hirata et al. 1989), and Super-Kamiokande (Super-Kamiokande Coll. 2008), generated the so-called solar neutrino problem which stimulated a deep investigation of the solar structure, see, e.g., Bahcall (1989). The problem was solved in 2002 when the SNO experiment (SNO Coll. 2002) obtained a direct evidence for flavor oscillations of solar neutrinos and, moreover, confirmed the SSM prediction of the ⁸B neutrino flux with an accuracy which, according to the latest data (SNO Coll. 2008), is equal to about ∼6%, see, e.g., Serenelli (2010).³

At the same time, helioseismic observations have allowed us to determine precisely several important properties of the Sun, such as the depth of the convective envelope which is known at the ∼0.2% level, the surface helium abundance which is obtained at the ∼1.5% level, and the sound speed profile which is determined with an accuracy equal to ∼0.1% in a large part of the Sun; see, e.g., Degl'Innocenti et al. (1997), Gough et al. (1996), Basu & Antia (2008), and references therein. As a result of these observations, the solar structure is now very well constrained, so that the Sun can be used as a solid benchmark for stellar evolution and as a "laboratory" for fundamental physics, see e.g., Ricci & Villante (2002), Fiorentini et al. (2001), and Bottino et al. (2002).

The future could be even more interesting. The KamLAND reactor (anti)neutrino experiment (KamLAND Coll. 2008) has confirmed the flavor oscillations hypothesis and has refined the determination of neutrino parameters. We now reliably know the solar neutrino oscillation probability and we can go back to the original program of solar neutrino studies, i.e., to probe nuclear reactions in the solar core. Present and future solar neutrino experiments, such as Borexino (Borexino Coll. 2008) and SNO+ (SNO Coll. 2006), have the potential to provide the first direct measurements of the CNO and pep neutrinos, thus probing dominant and sub-dominant energy generation mechanisms in the Sun.

At the same time, a new solar problem has emerged. Recent determinations of the photospheric heavy element abundances (Asplund et al. 2005, 2009) indicate that the Sun metallicity is lower than previously assumed (Grevesse & Sauval 1998). Solar models (SMs) that incorporate these lower abundances are no longer able to reproduce the helioseismic results. As an example, the sound speed predicted by SSMs at the bottom of the convective envelope disagrees at the ∼1% level with the value inferred by helioseismic data, see e.g., Bahcall et al. (2005). Detailed studies have been done to resolve this controversy but a definitive solution of the "solar composition problem" still has to be obtained; see, e.g., Basu & Antia (2008) and Asplund et al. (2009).

In this framework, it is important to analyze the role of physical inputs and the standard assumptions for SSM calculations. This task is not always possible in simple and clear terms, since SSM construction relies on (time-consuming) numerical integration of a nonlinear system of partial differential equations. Several input parameters are necessary to fully describe the property of the solar plasma and some of them are not single numbers but complicated functions (e.g., the opacity of the solar interior) which, in principle, can be modified in a nontrivial way. Moreover, any modification of the Sun produces a variety of correlated effects which have to be taken into account all together if we want to correctly extract information from the comparison of theoretical predictions with observational data.

In order to overcome these difficulties, we provide a tool which is at the same time simple and accurate enough to describe the effects of a generic (small) modification of the physical inputs. The starting point is the fact that, despite the present disagreement with helioseismic data, the SSM is a rather good approximation of the real Sun. We can thus assume small variations of the physical and chemical properties of the Sun with respect to the SSM predictions and use a linear theory to relate them to the properties of the solar plasma. With the additional assumption that the (variation of) present solar composition can be estimated from the (variation of) nuclear reaction rate and elemental diffusion efficiency in the present Sun, we obtain a linear system of ordinary differential equations, which can be easily integrated.

We believe that the proposed approach can be useful in several respects. First, the construction of these linear solar models (LSM) can complement the traditional methods for SM calculations, allowing us to investigate in a more efficient and transparent way the role of parameters and assumptions. In a separate paper, we will use LSMs to discuss in general terms the role of opacity and metals in the solar interior (F. L. Villante et al. 2010, in preparation). Moreover, it can help to introduce new effects and to understand their relevance, prior to implementation into the more complicated SSM machinery. Finally, this simplified approach could open the field of solar physics beyond the small community of the SSM's builders with the result of making the Sun a more accessible "laboratory." Clearly, LSMs are not intended to be an alternative to SSMs, which remain the fundamental theoretical tool to compare with observations.

The plan of the paper is as follows. In the next section, we briefly review SSM calculations. In Section 3, we expand to linear order the structure equations of the present Sun. In Section 4, we calculate the properties of the Sun and of the solar plasma that are necessary to define LSMs. In Section 5, we give the integration conditions for the linearized equations. In Section 6, we discuss how to estimate the chemical composition of the Sun. In Section 7, we give the equations that define LSMs in their final form. Finally, in Section 8, we compare the results of LSMs with those obtained by using the standard method for SM calculations, showing the validity of our simplified approach.

In the assumption of spherical symmetry, the behavior of the pressure (P), density (ρ), temperature (T), luminosity (l), and mass (m) in the Sun is described by the structure equations (Kippenhan & Weigert 1991):

$$\begin{eqnarray} \nonumber \frac{\partial m}{\partial r} &=& 4\pi r^{2} \rho \\ \nonumber \frac{\partial P}{\partial r} &=& - \frac{G_{\rm N} m}{r^2} \rho \\ \nonumber \frac{\partial l}{\partial r} &=& 4\pi r^{2} \rho \; \epsilon (\rho,T,X_{i})\\ \nonumber \frac{\partial T}{\partial r} &=& -\frac{G_{\rm N} m T \rho }{r^2 P} \nabla \\ P &=& P(\rho,T,X_{i}), \end{eqnarray} \tag{ 1 }$$

where r indicates the distance from the center, (ρ, T, X_i) is the energy produced per unit time and mass,⁴ the function P = P(ρ, T, X_i) describes the equation of state (EOS) of solar matter, and X_i(r) are the mass abundances of the various chemical elements inside the Sun. In the equation which describes the energy transport, the temperature gradient ∇ is defined as ∇ ≡ ∂ln T/∂ln P. If the energy transfer is due to radiative processes (i.e., at the center of the Sun) one has

$$\begin{equation} \nabla = \nabla _{\rm rad} \end{equation} \tag{ 2 }$$

where the radiative gradient ∇_rad is given by

$$\begin{equation} \nabla _{\rm rad}=\frac{3}{16\pi ac \, G_{\rm N}} \frac{\kappa (\rho,T,X_{i}) \,l \, P}{m \, T^4} \end{equation} \tag{ 3 }$$

and κ(ρ, T, X_i) is the opacity of solar matter. In the presence of convective motions (i.e., in the outer layer of the Sun), the value of ∇ has to be calculated by taking into account the convective energy transport which generally provides the dominant contribution. In a large part of the Sun convective envelope, one has

$$\begin{equation} \nabla \simeq \nabla _{\rm ad}\simeq 0.4. \end{equation} \tag{ 4 }$$

This is expected when convection is very efficient and a negligible excess of ∇ over the adiabatic value ∇_ad is sufficient to transport the whole luminosity. In the outermost layers of the Sun, the situation is more complicated. The precise description of convection in this regime is still an unsolved problem. One uses a phenomenological model which predicts the efficiency of convection as a function of the "mixing length parameter" α, which is related to the distances over which a moving unit of gas can be identified before it mixes appreciably; see, e.g., Kippenhan & Weigert (1991). The transition between the internal radiative core and the external convective envelope occurs at the radius R_b where

$$\begin{equation} \nabla _{\rm rad}(R_{\rm b}) = \nabla _{\rm ad} \;, \end{equation} \tag{ 5 }$$

as is prescribed by the Schwarzschild criterion which states that convective motions occur in the region of the star where ∇_rad(r)>∇_ad.

The chemical composition of the present Sun is not known but has to be calculated by coupling Equations (1), which describe the mechanical and thermal structure, with the equations that describe the chemical evolution of the Sun; see, e.g., Kippenhan & Weigert (1991) for details. It is generally assumed that the Sun was born chemically homogeneous and has modified its chemical composition due to nuclear reactions and elemental diffusion. One assumes that relative heavy element abundances in the photosphere have not been modified during the evolution and uses the observationally determined photospheric composition to fix the initial heavy element admixture. We remark that the chemical abundances are uniform in the convective region due to the very efficient mixing induced by convective motions. The values X_i,b, evaluated at the bottom of the convective region are thus representative for the solar surface composition.

If complete information about the initial composition was available and if a complete theory of convection was known, then there would be no free parameter. In practice, one has three free parameters, namely the initial metal abundance Z_ini, the initial helium abundance Y_ini, and the mixing length parameter α. These are tuned in order to reproduce, at the solar age t_☉ = 4.57 Gyr (Bahcall & Pinsonneault 1995), the observed solar luminosity L_☉ = 3.8418 × 10³³ erg s⁻¹ (Bahcall et al. 2004), the observed solar radius R_☉ = 6.9598 × 10¹⁰ cm (Allen 1976), and the surface metal-to-hydrogen ratio (Z/X)_b. In this paper we use the AS05 composition (Asplund et al. 2005) which corresponds to (Z/X)_b = 0.0165.

An SSM, according to the definition of Castellani et al. (1997), is a solution of the above problem which reproduces, within uncertainties, the observed properties of the Sun, by adopting physical and chemical inputs chosen within their range of uncertainties. In this paper, we refer to the SSM obtained by using the FRANEC code (Chieffi & Straniero 1989; Ciacio et al. 1997), including Livermore 2006 EOS (Rogers et al. 1996); Livermore radiative opacity tables (OPAL; Iglesias & Rogers 1996) calculated for the AS05 chemical composition; molecular opacities from Ferguson et al. (2005) and conductive opacities from Cassisi et al. (2006), both calculated for AS05 composition; and nuclear reaction rates from the NACRE compilation (Angulo et al. 1999), taking into account the recent revision of the astrophysical factors S_1,14, S_3,4, and S_1,7 (Marta et al. 2008; Costantini et al. 2008; Junghans et al. 2003) and the ⁷Be electron capture rate from Adelberger et al. (1998).

A variation of the input parameters (EOS, opacities, cross sections, etc.) with respect to the standard assumptions produces an SM which deviates from SSM predictions. The physical and chemical properties of the "perturbed" Sun can be described according to

$$\begin{eqnarray} \nonumber h(r)&=&\overline{h}(r)[1+\delta h(r)]\\ \nonumber X_i(r)&=&\overline{X}_i(r)[1+\delta X_i(r)]\\ Y(r)&=&\overline{Y}(r)+\Delta Y(r), \end{eqnarray} \tag{ 6 }$$

where h = l,  m,  T,  P,  ρ and we use, here and in the following, the notation $\overline{Q}$ to indicate the SSM prediction for the generic quantity Q. For convenience, the deviations from SSM are expressed in terms of relative variations δh(r) and δX_i(r) for all quantities except the helium abundance, for which it is more natural to use the absolute variation ΔY(r).

In this paper, we will be mainly concerned with modifications of the radiative opacity, κ(ρ, T, X_i), and of the energy generation coefficient, (ρ, T, X_i). We indicate with δκ(r) and δ(r) the relative variations of these quantities along the SSM profile, i.e.:

$$\begin{eqnarray} I(\overline{T}(r),\overline{\rho }(r),\overline{Y}(r),\overline{X}_{i}(r))&=& \overline{I}(\overline{T}(r),\overline{\rho }(r),\overline{Y}(r),\overline{X}_{i}(r))\;\nonumber\\ &&\times [1+\delta I(r)], \end{eqnarray} \tag{ 7 }$$

where I = κ,  . When we consider the effect of a perturbation δI(r) on the Sun, we have to take into account that the perturbed SM, due to that modification, has a different density, temperature, and chemical composition with respect to the SSM. The total difference δI^tot(r) between the perturbed Sun and the SSM at a given point r is defined by

$$\begin{eqnarray} I(T(r),\rho (r),Y(r),X_{i}(r))&=& \overline{I}(\overline{T}(r),\overline{\rho }(r),\overline{Y}(r),\overline{X}_{i}(r))\nonumber\\ &&\times [1+\delta I^{\rm tot}(r)]. \end{eqnarray} \tag{ 8 }$$

If we consider small perturbations ($\delta I \ll \overline{I}$),⁵ we can expand to first order in δT(r), δρ(r), δX_i(r) and ΔY(r), obtaining

$$\begin{eqnarray} \delta I^{\rm tot}(r)& =& I_T(r)\, \delta T(r) + I_\rho (r)\,\delta \rho (r) + I_Y(r)\, \Delta Y(r)\nonumber\\ && + \sum _i I_i(r)\, \delta X_i(r) + \delta I(r). \end{eqnarray} \tag{ 9 }$$

Here, the quantities I_j describe the dependence of the properties of the stellar plasma on temperature, density, and chemical composition and are given by

$$\begin{eqnarray} \nonumber I_{\rm \rho }(r)&=&\left.\frac{\partial \ln I}{\partial \ln \rho }\right|_{\rm SSM}\\ \nonumber I_{\rm T}(r)&=&\left.\frac{\partial \ln I}{\partial \ln T}\right|_{\rm SSM}\\ \nonumber I_{i}(r)&=&\left.\frac{\partial \ln I}{\partial \ln X_i}\right|_{\rm SSM}\\ I_{\rm Y}(r)&=&\left.\frac{\partial \ln I}{\partial Y}\right|_{\rm SSM} \end{eqnarray} \tag{ 10 }$$

where I = ,  κ,  P. The symbol |_SSM indicates that we calculate the derivatives I_j along the density, temperature, and chemical composition profiles predicted by the SSM.

By using the above notations, we linearize the structure Equations (1), obtaining

$$\begin{eqnarray} \nonumber \frac{\partial \, \delta m}{\partial r} &=& \frac{1}{l_m} \,[\delta \rho - \delta m] \\ \nonumber \frac{\partial \, \delta P}{\partial r} &=& \frac{1}{l_P} \,[\delta m + \delta \rho - \delta P] \\ \nonumber {\delta P} &=& \bigg[P_\rho \,\delta \rho + P_T \,\delta T + P_Y \Delta Y +\sum _i P_i\, \delta X_i\bigg] \\ \nonumber \frac{\partial \, \delta l}{\partial r} &=& \frac{1}{l_l} \,\bigg[(1+\epsilon _\rho)\delta \rho +\epsilon _T \,\delta T +\epsilon _Y \Delta Y + \sum _i\epsilon _i\, \delta X_i - \delta l + \delta \epsilon \bigg] \\ \nonumber \frac{\partial \, \delta T}{\partial r} &=& \frac{1}{l_T} \bigg[\delta l +(\kappa _T-4) \delta T + (\kappa _\rho +1) \delta \rho + \kappa _Y \Delta Y \nonumber\\ &&+\sum _i \kappa _i\, \delta X_i + \delta \kappa \bigg]\;\;\;\;\;\;\;\;{\rm Rad.}\nonumber\\ \frac{\partial \, \delta T}{\partial r} &=& \frac{1}{l_T} \,[\delta m + \delta \rho - \delta P] \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;{\rm Conv.} \end{eqnarray} \tag{ 11 }$$

where $l_h =\left[d\ln (\overline{h})/dr\right]^{-1}$ represents the scale height of the physical parameter h in the SSM. The last two equations correspond to the linear expansion of the energy transport equation in the radiative and in the convective regime, respectively.⁶ In our approach, we use the radiative transport equation for $r \le \overline{R}_{\rm b}$ and the convective transport equation for $r > \overline{R}_{\rm b}$, where $\overline{R}_{\rm b}=0.730 \, R_{\odot }$ is the lower radius of the convective envelope predicted by our SSM. We calculate a posteriori the relative variation δR_b of the extension of the convective region by applying the Schwarzschild criterion to the solutions of Equations (11). By expanding to first order, we obtain

$$\begin{equation} \delta R_{\rm b} = -\frac{\delta \nabla _{\rm rad, b}}{\zeta _{\rm b}} = -\frac{\delta \kappa _{\rm tot, b} + \delta P_{\rm b} + \delta l_{\rm b} - 4 \delta T_{\rm b} -\delta m_{\rm b}}{\zeta _{\rm b}}, \end{equation} \tag{ 12 }$$

where $\zeta _{\rm b} = d \ln \overline{\nabla }_{\rm rad}(\overline{R}_{\rm b})/d\ln r= 11.71$. Here and in the following, the subscript "b" indicates that the various quantities are evaluated at $r=\overline{R}_{\rm b}$, i.e., at the bottom of the SSM's convective envelope.

In order to define the linearized structure equations, we have to calculate the functions l_h(r) and the logarithmic derivatives I_j(r). In Figure 1, we show the inverse scale heights 1/l_h(r) as a function of the solar radius. The plotted results have been obtained numerically and refer to our SSM. We see that the functions 1/l_P(r) and 1/l_T(r) vanish at the center of the Sun while they grow considerably (in modulus) in the outer regions, as a result of the fast decrease of pressure and temperature close to surface. The kink in the temperature scale height at $\overline{R}_{\rm b}= 0.730\,R_{\odot }$ marks the transition from the internal radiative region to the outer convective envelope. The functions 1/l_m(r) and 1/l_l(r) have the opposite behavior; they diverge at the solar center and vanish at large radii, as is expected by considering that most of the mass and the energy generation in the Sun is concentrated close to the solar center. To be more quantitative, the solar luminosity is produced in the inner radiative core (r ⩽ 0.3 R_☉), which contains approximately 60% of the total solar mass. The radiative region ($r \le \overline{R}_{\rm b}$) which covers about 40% of the total volume of the Sun, includes approximately 98% of the total solar mass.

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In Figure 2, we show the logarithmic derivatives I_j(r) calculated numerically along the temperature, density, and chemical composition profiles predicted by our SSM. The left panel refers to the opacity derivatives κ_j(r) and shows that opacity is a decreasing function of temperature and helium abundance, while it is an increasing function of density. The coefficient κ_Z(r) ≡ ∂ln κ/∂ln Z|_SSM quantifies the dependence of opacity on the total metallicity Z. It has been calculated by rescaling all the heavy element abundances by a constant factor, so that the metal admixture remains fixed. Metals provide about ∼40% of the opacity at the center of the Sun, while they account for about 80% of the total opacity at the bottom of the convective region; see also Basu & Antia (2008).

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In the right panel of Figure 2, we present the logarithmic derivatives _j(r) of the energy generation coefficient. These have been calculated by assuming that the abundances of the secondary elements for PP-chain and CN-cycle, namely ³He, ¹²C, and ¹⁴N, can be estimated as it is described in Appendix A. The bumps at r ≃ 0.28 R_☉ are related to out-of-equilibrium behavior of the ³He abundance. The presence of these features has, however, a negligible influence on the solutions of Equations (11). The function 1/l_l(r), which multiplies all the _j(r) coefficients, drops, in fact, rapidly to zero at r ≃ 0.25 R_☉.⁷

In the more internal region (where ³He assumes the equilibrium value), the displayed results can be understood by considering that energy is produced by the PP-chain (∼99%) with a small contribution (∼1%) by the CN-cycle. The slight increase of _T(r) at r ⩽ 0.15 R_☉ reflects the fact that the CN-cycle, which strongly depends on temperature, gives a non-negligible contribution only at the center of the Sun. The coefficient _ρ(r) is approximately equal to 1, as it is expected by considering that the rate of two-body reactions in the unit mass is proportional to the density. Finally, the behavior of _Y(r) and _Z(r) is understood by considering that the pp-reaction rate is proportional to the hydrogen abundance squared, i.e., to X² = (1 − Y − Z)².

We have also evaluated numerically the logarithmic derivative of the pressure P_j(r) by using the Livermore 2006 EOS (Rogers et al. 1996). The obtained results can be approximated with good accuracy (about 1%) by using the perfect gas EOS: P = (ρk_BT)/(μ m_u), where m_u is the atomic mass unit, k_B is the Boltzmann constant and μ = (2 − 5/4 Y − 3/2 Z)⁻¹ is the mean molecular weight. One obtains

$$\begin{eqnarray} \nonumber P_{\rho }(r)&=&1\\ \nonumber P_{\rm T}(r)&=&1\\ P_{Y}(r) &=& -\frac{\partial \ln \mu }{\partial Y} =-\frac{5}{8-5Y(r)-6Z(r)}. \end{eqnarray} \tag{ 13 }$$

The derivative with respect to the metal abundance P_Z(r) is of the order of few percent and will be neglected in the following. By using the above relations, we obtain the following equation:

$$\begin{equation} \delta \rho (r) = \delta P(r) -\delta T(r) - P_{Y} \Delta Y(r) \end{equation} \tag{ 14 }$$

that can be used to eliminate the quantity δρ from the linear structure Equations (11). We arrive at

$$\begin{eqnarray} \nonumber \frac{d\delta m}{dr} &=& \frac{1}{l_m} \,\left[ \delta P - \, \delta T - \delta m - P_Y \, \Delta Y \right]\\ \nonumber \frac{d\delta P}{dr}&=& \frac{1}{l_P}\, \left[ - \delta T + \delta m - P_Y \, \Delta Y \right]\\ \nonumber \frac{d\delta l}{dr}& = & \frac{1}{l_l}\, \left[ \beta _P \,\delta P + \beta _T \,\delta T - \delta l + \beta _Y\,\Delta Y +\beta _Z\,\delta Z +\delta \epsilon \right]\\ \nonumber \frac{d\delta T}{dr}&=& \frac{1}{l_T}\, \left[ \alpha _P \,\delta P + \alpha _T \,\delta T + \delta l + \alpha _Y\,\Delta Y + \alpha _Z \delta Z +\delta \kappa \right]\;\;\;{\rm Rad.}\\ \frac{d\delta T}{dr}&=& \frac{1}{l_T}\, \left[ - \delta T + \delta m - P_Y \, \Delta Y \right] \;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;{\rm Conv.} \end{eqnarray} \tag{ 15 }$$

where the factors α_h and β_h are given by

$$\begin{eqnarray} \nonumber \alpha _P &=& \kappa _\rho + 1 \;\;\;\;\;\;\;\;\; \alpha _T = \kappa _T - \kappa _\rho - 5 \;\;\;\;\;\;\;\;\; \nonumber\\ \alpha _Y &=& - (\kappa _\rho +1)P_ Y + \kappa _Y \;\;\;\;\;\;\;\;\; \alpha _Z = \kappa _Z \nonumber\\ \beta _P &=& \epsilon _\rho + 1 \;\;\;\;\;\;\;\;\;\; \beta _T = \epsilon _T - \epsilon _\rho - 1 \;\;\;\;\;\;\;\;\;\; \nonumber\\ \beta _Y &=& -(\epsilon _\rho + 1)P_Y + \epsilon _Y \;\;\;\;\;\;\;\;\; \beta _Z = \epsilon _Z. \end{eqnarray} \tag{ 16 }$$

In the above equations, we indicate with δZ the relative variation of the total metallicity, defined by

$$\begin{equation} Z(r)=\overline{Z}(r)\;[1+\delta Z(r)] \end{equation} \tag{ 17 }$$

and we implicitly assume that the heavy element admixture is unchanged.⁸

In order to solve the linear structure equations, one has to specify the integration conditions. This can be done quite easily at the center of the Sun. From the definitions of mass and luminosity, we know that m(0) = 0 and l(0) = 0. This implies that

$$\begin{equation} 1+\delta h(0)=\lim _{r\rightarrow 0} \frac{h(r)}{\overline{h}(r)}= \frac{\frac{dh(0)}{dr}}{\frac{d\overline{h}(0)}{dr}} \end{equation} \tag{ 18 }$$

for h = m,  l, so that one obtains

$$\begin{eqnarray} \nonumber \delta m(0) & = & \delta P_{0} - \delta T_{0} - P_{Y,0} \, \Delta Y_{0} \\ \nonumber \delta P(0) & = & \delta P_{0} \\ \nonumber \delta T(0) &=& \delta T_{0} \\ \delta l(0) &=& \beta _{P,0} \, \delta P_{0} + \beta _{T,0} \,\delta T_{0} + \beta _{Y,0} \, \Delta Y_{0} + \beta _{Z,0} \, \delta Z_0 +\delta \epsilon _{0},\qquad \end{eqnarray} \tag{ 19 }$$

where the subscript "0" indicates that the various quantities are evaluated at the center of the Sun.

To implement surface conditions, we exploit the fact that the Sun has a convective envelope where the solution of the linearized structure equations can be explicitly calculated, as it is explained in the following.

First, we take advantage of the fact that there are no energy producing processes and that a negligible fraction of the solar mass is contained in the convective region. This implies that 1/l_l(r) = 0 and 1/l_m(r) ≃ 0, so that we can integrate the energy generation and the continuity equation. We obtain δl(r) ≡ 0 and δm(r) ≃ 0 where we clearly considered that only solutions that reproduce the observed solar luminosity and solar mass are acceptable.

Then, we consider that the chemical abundances are constant due to the very efficient convective mixing, so that ΔY(r) ≡ ΔY_b and P_Y(r)ΔY_b ≡ −δμ_b, where δμ_b is the relative variation of the mean molecular weight at the bottom of the convective region. By taking this into account, we can rewrite the transport equation in the form

$$\begin{equation} \frac{\partial \, \delta u}{\partial r} = - \frac{\delta u}{l_T} \end{equation} \tag{ 20 }$$

where δu is the fractional variation of squared isothermal sound speed u = P/ρ = k_BT/μm_u. The general solution of this equation is $\delta u(r) = \delta u_{\rm b} \,[\overline{T}_{\rm b}/\overline{T}(r)]$ which shows that, in order to avoid δu(r) to "explode" at the surface of the Sun, it necessarily holds δu(r) = δT(r) − δμ_b ≃ 0 in the internal layers of the convective envelope.

By using this result in the hydrostatic equilibrium equation, we obtain δP(r) = δρ(r) ≃ δC where δC is an arbitrary constant. Finally, by considering that δρ(r) is approximately constant in the internal layers of the convective region (where most of the mass of the convective envelope is contained), we use the continuity equation to improve the condition δm(r) ≃ 0, obtaining $\delta m_{\rm b} = - \overline{m}_{\rm conv} \, \delta C$, where $\overline{m}_{\rm conv}=\overline{M}_{\rm conv}/M_{\odot }=0.0192$ is the fraction of solar mass contained in the convective region, as resulting from our SSM.

In conclusion, we have

$$\begin{eqnarray} \nonumber \delta m(\overline{R}_{\rm b}) &=& - \overline{m}_{\rm conv} \, \delta C \\ \nonumber \delta P(\overline{R}_{\rm b})&=& \delta C \\ \nonumber \delta T(\overline{R}_{\rm b})&=& \delta \mu _{\rm b} = - P_{Y,\rm b} \, \Delta Y_{\rm b} \\ \delta l(\overline{R}_{\rm b}) &=& 0. \end{eqnarray} \tag{ 21 }$$

The factor δC is a free parameter that cannot be fixed from first principles, since we cannot model exactly convection in the outermost super-adiabatic region. It has basically the same role as mixing length parameter in SSM construction.

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. However, this would lead to several complications in our simplified approach. We prefer to use a simple approximate procedure that allows us to estimate with sufficient accuracy the helium and metal abundances of the modified Sun, without requiring us to follow explicitly its time evolution.

6.1. Notations

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

$$\begin{eqnarray} \nonumber Y(r)&=&Y_{\rm ini}\,\left[ 1+D_{Y}(r) \right] +Y_{\rm nuc}(r)\\ Z(r)&=&Z_{\rm ini}\,\left[ 1+D_{Z}(r) \right]. \end{eqnarray} \tag{ 22 }$$

Here, Y_ini and Z_ini are the initial values which, as explained previously, are free parameters in SM construction and are adjusted in order to reproduce the observed solar properties. In our SSM, we have $\overline{Y}_{\rm ini}= 0.2611$ and $\overline{Z}_{\rm ini}=0.0140$. The terms D_Y(r) and D_Z(r) describe the effects of elemental diffusion. Finally, Y_nuc(r) represents the total amount of helium produced in the shell r by nuclear processes and can be calculated by integrating the rates of the helium-producing reactions during the Sun history.

In Figure 3, we show the quantities $\overline{Y}_{\rm nuc}(r)$, $\overline{D}_{Y}(r),$ and $\overline{D}_{\rm Z}(r)$ estimated from our SSM. Nuclear helium production is relevant in the internal radiative core (r ⩽ 0.3 R_☉) where it is responsible for an enhancement of the helium abundance which can be as large as $\overline{Y}_{\rm nuc,0}\simeq 0.35$ at the center of the Sun. Elemental diffusion accounts for a ∼10% increase of helium and metals in the central regions with respect to external layers of the radiative region. The convective envelope, due to the very efficient convective mixing, is chemically homogeneous and can be fully described in terms of two numbers: the surface helium and metal abundances, indicated with Y_b and Z_b, respectively.⁹ These are related to initial values by

$$\begin{eqnarray} Y_{\rm b}&=&Y_{\rm ini}\,[ 1+D_{Y,\rm b} ]\\ \nonumber Z_{\rm b}&=&Z_{\rm ini}\,[ 1+D_{Z,\rm b} ], \end{eqnarray} \tag{ 23 }$$

where $D_{Y,\rm b}$ and $D_{Z,\rm b}$ parameterize the effects of elemental diffusion. In our SSM, we have $\overline{Y}_{\rm b}=0.229$, $\overline{Z}_{\rm b} = 0.0125$, $\overline{D}_{Y,\rm b}=-0.121$, and $\overline{D}_{Z,\rm b}=-0.105$.

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We are interested in describing how the chemical composition is modified when we perturb the SSM. In the radiative core ($r\le \overline{R}_{\rm b}$), we neglect the possible variations of the diffusion terms¹⁰ and we write

$$\begin{eqnarray} \nonumber \Delta Y(r) &=& \Delta Y_{\rm ini} \,[ 1+\overline{D}_{Y}(r) ] +\Delta Y_{\rm nuc}(r) \\ \delta Z(r) &=& \delta Z_{\rm ini}, \end{eqnarray} \tag{ 24 }$$

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

$$\begin{eqnarray} \nonumber \Delta Y_{\rm b}&=& (1+\overline{D}_{Y, \rm b})\, \Delta Y_{\rm ini} + \overline{Y}_{\rm ini}\, \overline{D}_{Y,\rm b} \, \delta D_{Y,{\rm b}}\\ \delta Z_{\rm b}&=&\delta Z_{\rm ini} +\frac{\overline{D}_{Z, \rm b}}{1+\overline{D}_{Z, \rm b}}\, \delta D_{Z,_{\rm b}}, \end{eqnarray} \tag{ 25 }$$

where $\delta D_{Y,\rm b}$ and $\delta D_{Z, \rm b}$ are the fractional variations of the diffusion terms.

It is important to remark that ΔY_b and δZ_b are related among each other, since the metals-to-hydrogen ratio at the surface of the Sun is observationally fixed. By imposing δ(Z/X)_b = 0, we obtain

$$\begin{equation} \delta Z_{\rm b}=-\frac{1}{1-\overline{Y}_{\rm b}}\Delta Y_{\rm b} \end{equation} \tag{ 26 }$$

where we considered that X_b ≃ 1 − Y_b. This relation can be rewritten in terms of the initial helium and metal abundances, obtaining

$$\begin{equation} \delta Z_{\rm ini} = Q_0 \, \Delta Y_{\rm ini} +Q_1 \, \delta D_{Y,{\rm c}} +Q_2 \, \delta D_{Z,_{\rm c}} \end{equation} \tag{ 27 }$$

where the coefficients Q_i are given by

$$\begin{eqnarray} \nonumber Q_0 &=& -\frac{1+\overline{D}_{Y, \rm b}}{1-\overline{Y}_{\rm b}} = - 1.141 \\ \nonumber Q_1 &=& -\frac{\overline{Y}_{\rm ini}\, \overline{D}_{Y,\rm b}}{1-\overline{Y}_{\rm b}} = +0.041 \\ Q_2 &=& -\frac{\overline{D}_{Z, \rm b}}{1+\overline{D}_{Z, \rm b}} = +0.118 \,. \end{eqnarray} \tag{ 28 }$$

6.2. Production of Helium by Nuclear Reactions

In order to predict the helium abundance in the radiative region, one has to estimate the variation of nuclear production of helium ΔY_nuc(r), see Equation (24). We note that the quantity ${\overline{Y}}_{\rm nuc}(r)$ varies proportionally to $\overline{\epsilon }(r)$ in the SSM, as is seen from Figure 3. This is natural since the helium production rate is directly proportional to the energy generation rate and, moreover, the energy generation profile $\overline{\epsilon }(r)$ is nearly constant during the past history of the Sun. We assume that the observed proportionality holds true also in modified SMs, obtaining as a consequence the following relation:

$$\begin{equation} \Delta Y_{\rm nuc}(r) = \overline{Y}_{\rm nuc}(r) \delta \epsilon ^{\rm tot}(r). \end{equation} \tag{ 29 }$$

If we expand δ^tot(r) according to relation (9) and solve with respect to ΔY(r), we can recast Equation (24) in the form¹¹:

$$\begin{equation} \Delta Y(r)=\xi _Y(r)\,\Delta Y_{\rm ini}+\xi _P(r)\,\delta P(r)+\xi _T(r)\,\delta T(r)+\xi _{\epsilon }(r)\,\delta \epsilon (r), \end{equation} \tag{ 30 }$$

where

$$\begin{eqnarray} \nonumber \xi _Y(r) &=&\frac{1+\overline{D}_{Y}(r)}{1-\overline{Y}_{\rm nuc}(r) [\epsilon _Y(r) -P_{Y}(r)\, \epsilon _{\rho }(r) ]}\\ \nonumber \xi _P(r) &=&\frac{\overline{Y}_{\rm nuc}(r)\,\epsilon _\rho (r)}{1-\overline{Y}_{\rm nuc}(r) [\epsilon _Y(r) -P_{Y}(r)\, \epsilon _{\rho }(r) ]}\\ \nonumber \xi _T(r) &=&\frac{\overline{Y}_{\rm nuc}(r)[ \epsilon _T(r) - \epsilon _\rho (r)]}{1-\overline{Y}_{\rm nuc}(r) [\epsilon _Y(r) -P_{Y}(r)\, \epsilon _{\rho }(r) ]}\\ \xi _\epsilon (r) &=&\frac{\overline{Y}_{\rm nuc}(r)}{1-\overline{Y}_{\rm nuc}(r) [\epsilon _Y(r) -P_{Y}(r)\, \epsilon _{\rho }(r) ]}\;. \end{eqnarray} \tag{ 31 }$$

The factors ξ_j(r) are shown in Figure 4.

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6.3. Elemental Diffusion in the Convective Region

The total mass M_conv,i of the ith element contained in convective envelope evolves due to elemental diffusion according to relation (Aller & Chapman 1960; Thoul et al. 1994):

$$\begin{equation} \frac{1}{M_{{\rm conv}, i}}\frac{d M_{{\rm conv}, i}}{dt}=\frac{\omega _{i}}{H}. \end{equation} \tag{ 32 }$$

Here, the parameter H is the "effective thickness" of the convective region defined by the relation 4πR²_b H ρ_b = M_conv. The quantity ω_i is the diffusion velocity of the ith element at the bottom of the convective region that can be expressed as

$$\begin{equation} \omega _{i}=\frac{T_{\rm b}^{5/2}}{\rho _{\rm b}} \left[(A_{P,i}+\nabla _{\rm ad} \, A_{T,i})\frac{\partial \ln P(R_{\rm b})}{\partial r} \right], \end{equation} \tag{ 33 }$$

where A_P,i, A_T,i are the diffusion coefficients, see Thoul et al. (1994) for details. In the above relation, we neglected, for simplicity, the term proportional to the hydrogen concentration gradient¹² and we took advantage of the fact that ∂ln T/∂r = ∇_ad ∂ln P/∂r at the bottom the convective region, as it is prescribed by the Schwarzschild criterion.

We have to estimate the effect of a generic modification of the SSM on elemental diffusion. In most cases, a good description is obtained by assuming that $D_{i,\rm b}$ varies proportionally to the efficiency of diffusion in the present Sun, i.e., to the right-hand side of Equation (32) evaluated at the present time. This implies that

$$\begin{equation} \delta D_{i,\rm b} = \delta \omega _{i} - \delta H, \end{equation} \tag{ 34 }$$

where δH is the relative variation of the convective envelope effective thickness and δω_i is the relative variation of the diffusion velocity. By following the calculations described in Appendix B, one is able to show that

$$\begin{eqnarray} \nonumber \delta D_{Y,\rm b} &=& \Gamma _{Y} \; \delta T_{\rm b} + \Gamma _{P} \, \delta P_{\rm b} \\ \delta D_{Z,\rm b} &=& \Gamma _{Z} \; \delta T_{\rm b} + \Gamma _{P} \, \delta P_{\rm b}, \end{eqnarray} \tag{ 35 }$$

where Γ_Y = 2.05, Γ_Z = 2.73, and Γ_P = −1.10 and we assumed that all metals have the same diffusion velocity as iron.

By using the above relations and taking into account Equations (25) and (26) one obtains the surface abundances variations ΔY_b and δZ_b as a function of the free parameters ΔY_ini and δC. Namely, we obtain

$$\begin{eqnarray} \nonumber \Delta Y_{\rm b} &=& A_{\rm Y} \; \Delta Y_{\rm ini} + A_{\rm C} \; \delta C \\ \Delta Z_{\rm b} &=& B_{\rm Y} \; \Delta Y_{\rm ini} + B_{\rm C} \; \delta C, \end{eqnarray} \tag{ 36 }$$

where

$$\begin{eqnarray} \nonumber A_{\rm Y} & = & \frac{1+\overline{D}_{Y, \rm b}}{ 1 + P_{Y,\rm b} \, \overline{Y}_{\rm ini} \, \overline{D}_{Y, \rm b} \, \Gamma _{Y}} =0.838\\ A_{\rm C} & = & \frac{\overline{Y}_{\rm ini} \, \overline{D}_{Y, \rm b} \, \Gamma _{P}}{ 1 + P_{Y,\rm b} \, \overline{Y}_{\rm ini} \, \overline{D}_{Y, \rm b} \,\Gamma _{Y}} =0.033 \end{eqnarray} \tag{ 37 }$$

and

$$\begin{eqnarray} \nonumber B_{\rm Y} & = & - \frac{A_{\rm Y}}{ 1 - \overline{Y}_{\rm b} } = -1.088 \\ B_{\rm C} & = & - \frac{A_{\rm C}}{ 1 -\overline{Y}_{\rm b} } = -0.043. \end{eqnarray} \tag{ 38 }$$

Moreover, we can use relations (27) and (21) to calculate the variation of the initial metal abundance δZ_ini which, in our approach, coincides with the variation of metal abundance in the radiative region δZ(r). We obtain

$$\begin{equation} \delta Z(r) = \delta Z_{\rm ini} = Q_{\rm Y} \, \Delta Y_{\rm ini} + \, Q_{\rm C} \, \delta C, \end{equation} \tag{ 39 }$$

where

$$\begin{eqnarray} \nonumber Q_{\rm Y} &=& Q_0 -P_{Y,\rm b} \, A_{\rm Y}\, \left(Q_1 \, \Gamma _{Y} + Q_2 \, \Gamma _{Z} \right) = - 0.887\\ Q_{\rm C} &\,{=}\,& -P_{Y,\rm b}\, A_{\rm C} \, \left(Q_1 \,\Gamma _{Y} \,{+}\, Q_2\,\Gamma _{Z}\right) \,{+}\, \Gamma _{P} \, \left(Q_1 + Q_2 \right)\nonumber\\ & =& -0.164. \end{eqnarray} \tag{ 40 }$$

If we had neglected the effect of diffusion, we would have obtained A_Y = 0.879 and A_C = 0, B_Y = −1.141 and B_C = 0, and Q_Y = Q₀ = −1.14 and Q_C = 0.

We have now all the ingredients to formally define LSMs. The equations obtained in Section 6, namely Equations (30), (36), and (39), relate the chemical composition of the modified Sun to the present values of the structural parameters δP(r) and δT(r), to the energy generation coefficient δ(r) and to the free parameters ΔY_ini and δC. These relations can be inserted in Equations (15) and in the integration conditions, Equations (19) and (21), obtaining a linear system of ordinary differential equations that completely determines the physical and chemical properties of the modified Sun. In this section, we give the equations in their final form.

The properties of the Sun in the radiative region ($r\le \overline{R}_{\rm b}$) are described by

$$\begin{eqnarray} \nonumber \frac{d\delta m}{dr} &=& \frac{1}{l_m} \,\left[ \gamma _{P} \, \delta P + \gamma _T \, \delta T - \delta m + \gamma _Y \, \Delta Y_{\rm ini} + \gamma _\epsilon \; \delta \epsilon \right]\\ \nonumber \frac{d\delta P}{dr}&=& \frac{1}{l_P}\, \left[(\gamma _{P} - 1) \, \delta P + \gamma _T \, \delta T + \delta m + \gamma _Y \, \Delta Y_{\rm ini} + \gamma _{\epsilon } \; \delta \epsilon \right]\\ \nonumber \frac{d\delta l}{dr}& = & \frac{1}{l_l}\, [ \beta ^{\prime }_P \,\delta P + \beta ^{\prime }_T \,\delta T - \delta l + \beta ^{\prime }_Y \,\Delta Y_{\rm ini} + \beta ^{\prime }_C \, \delta C + \beta ^{\prime }_\epsilon \,\delta \epsilon ]\\ \frac{d\delta T}{dr}&=& \frac{1}{l_T}\, [ \alpha ^{\prime }_P \,\delta P + \alpha ^{\prime }_T \,\delta T + \delta l + \alpha ^{\prime }_Y \, \Delta Y_{\rm ini}\nonumber\\ &&+ \alpha ^{\prime }_C \, \delta C + \delta \kappa + \alpha ^{\prime }_\epsilon \, \delta \epsilon ]. \end{eqnarray} \tag{ 41 }$$

The coefficients γ_h which define the continuity and the hydrostatic equilibrium equation are given by

$$\begin{eqnarray} \nonumber \gamma _P = 1- P_Y \xi _P \;\;\;\;\;\; \gamma _T &=& - 1 + P_Y \xi _T \;\;\;\;\;\; \gamma _Y = - P_Y \xi _Y\, \;\;\;\;\;\; \nonumber\\ \gamma _\epsilon &=& -P_Y\xi _\epsilon \end{eqnarray} \tag{ 42 }$$

and are shown in Figure 5 as a function of the solar radius. The coefficients α'_h and β'_h which define the transport and the energy generation equation are given by

$$\begin{eqnarray} \nonumber &\beta ^{\prime }_P& = \beta _P + \beta _Y\xi _P \;\;\;\; \beta ^{\prime }_T = \beta _T + \beta _Y\xi _T \;\;\;\;\beta ^{\prime }_Y = \beta _Y\xi _Y +\beta _Z Q_{\rm Y} \;\;\;\;\nonumber\\ & \beta ^{\prime }_C& = \beta _Z Q_{\rm C} \;\;\;\; \beta ^{\prime }_\epsilon = 1+ \beta _Y \xi _{\epsilon } \nonumber\\ &\alpha ^{\prime }_P& = \alpha _P + \alpha _Y\xi _P \;\;\;\; \alpha ^{\prime }_T = \alpha _T + \alpha _Y\xi _T \;\;\;\; \alpha ^{\prime }_Y = \alpha _Y\xi _Y +\alpha _Z Q_{Y} \;\;\;\;\nonumber\\ & \alpha ^{\prime }_C& = \alpha _Z Q_{c} \;\;\;\; \alpha ^{\prime }_\epsilon = \alpha _\epsilon \xi _{\epsilon } \end{eqnarray} \tag{ 43 }$$

and are shown in Figure 6.

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The integration conditions at the center of the Sun (r = 0) are given by

$$\begin{eqnarray} \nonumber \delta m &=& \gamma _{P,0}\, \delta P_{0} + \gamma _{T,0} \, \delta T_{0} + \gamma _{Y,0} \, \Delta Y_{\rm ini} + \gamma _{\epsilon,0} \; \delta \epsilon _{0} \\ \nonumber \delta P &=& \delta P_{0}\\ \nonumber \delta T &=& \delta T_{0}\\ \delta l &=& \beta ^{\prime }_{P,0} \,\delta P_{0} + \beta ^{\prime }_{T,0} \,\delta T_{0} + \beta ^{\prime }_{Y,0} \,\Delta Y_{\rm ini} + \beta ^{\prime }_{C,0} \, \delta C + \beta ^{\prime }_{\epsilon,0} \,\delta \epsilon _{0},\nonumber\\ \end{eqnarray} \tag{ 44 }$$

where γ_P,0 = 1.167, γ_T,0 = −0.254, γ_Y,0 = 0.503, γ_,0 = 0.161 and β'_P,0 = 1.647, β'_T,0 = 1.884, β'_C,0 = 0.006, β'_Y,0 = −1.142, β'_,0 = 0.624. By imposing these conditions, the solution is obtained as a linear function of the four parameters ΔY_ini, δC, δT₀, δP₀ which are fixed by requiring that

$$\begin{eqnarray} \nonumber \delta m &=& - \overline{m}_{\rm conv} \, \delta C\\ \nonumber \delta P &=& \delta C\\ \nonumber \delta T &=& A^{\prime }_Y \, \Delta Y_{\rm ini} + A^{\prime }_{C} \, \delta C\\ \delta l &=& 0 \end{eqnarray} \tag{ 45 }$$

at the convective boundary ($r=\overline{R}_{\rm b}$), where $A^{\prime }_{\rm Y} = -P_{Y,\rm b} \, A_{Y} = 0.626$, $A^{\prime }_{\rm C}=-P_{Y,\rm b} \, A_{\rm C} = 0.025$ and $\overline{m}_{\rm conv} =0.0192$. The solution is, thus, univocally determined.

The chemical composition of the radiative region can be calculated by using relations (30) and (39). The variation of the density profile δρ(r) in the radiative region can be obtained by the relation:

$$\begin{equation} \delta \rho (r) = \gamma _P(r) \delta P(r) + \gamma _T(r) \delta T(r) + \gamma _Y(r) \Delta Y_{\rm ini} + \gamma _\epsilon (r) \delta \epsilon (r) \,. \end{equation} \tag{ 46 }$$

In the lower layers of the convective envelope the quantities δP(r), δT(r), δρ(r) are nearly constant (see Section 5), while the quantity δm(r) gradually goes to zero, as is expected being the total mass of the Sun observationally determined. The surface chemical composition can be determined by using relation (36). We recall that the variations of surface helium and metal abundances are related among each other, in such a way that the hydrogen-to-metal surface ratio is unchanged with respect to the SSM.

Finally, the variation of the convective radius is calculated by using relation (12). We obtain

$$\begin{equation} \delta R_{\rm b} = \Gamma _{Y}\, \Delta Y_{\rm ini} + \Gamma _{C} \, \delta C + \Gamma _{\kappa }\, \delta \kappa _{\rm b}, \end{equation} \tag{ 47 }$$

where

$$\begin{eqnarray} \nonumber \Gamma _{Y} & = & -\frac{A_Y}{\zeta _{\rm b}}\left[-P_{Y,\rm b} \, (\kappa _{T,\rm b}-4) + \kappa _{Y,\rm b} -\frac{\kappa _{Z,\rm b}}{1-\overline{Y}_{\rm b}}\right] = 0.449\\ \nonumber \Gamma _{C} & = & -\frac{A_{C}}{\zeta _{\rm b}} \left[-P_{Y,\rm b} \, (\kappa _{T,\rm b}-4) + \kappa _{Y,\rm b} -\frac{\kappa _{Z,\rm b}}{1-\overline{Y}_{\rm b}}\right] -\frac{\kappa _{\rho,\rm b}+1}{\zeta _{\rm b}}\nonumber\\ & =& -0.117\nonumber\\ \Gamma _{\kappa } & = & -\frac{1}{\zeta _{\rm b}} = -0.085. \end{eqnarray} \tag{ 48 }$$

In the derivation of the above relation, we considered that δl_b = 0, δm_b ≃ 0 and we used Equations (21) and (36).

In order to show the validity of the proposed approach, we consider four possible modifications of the standard input and we compare the results obtained by LSMs with those obtained by the full nonlinear SM calculations. Three of the considered cases concern opacity modifications that are generally described as

$$\begin{equation} \kappa (\rho,T,X_{i})=F(T) \,\overline{\kappa }(\rho,T,X_{i}), \end{equation} \tag{ 49 }$$

where $\overline{\kappa }(\rho,T,X_{i})$ represent the standard value and F(T) is a suitable function of the temperature. Namely, we consider

OPA1: overall rescaling of the opacity by a constant factor F(T) ≡ 1.1. This clearly corresponds to introduce the opacity variation:

$$$ \delta \kappa (r)\equiv 0.1 $$$

in Equations (41) for LSM calculations.

OPA2: smooth decrease of the opacity at the center of the Sun described by the function:

$$$ F(T) = 1 + \frac{A}{1+\exp {\big[\frac{T_{\rm c} - T}{fT_{\rm c}}\bi]}} $$$

where A = −0.1, T_c = 9.4 × 10⁶ K, f = 0.1. In our approach this corresponds to a variation δk(r) along the solar profile given by

$$$ \delta \kappa (r) = \frac{A}{1+\exp {\big[\frac{T_{\rm c} - \overline{T}(r)}{fT_{\rm c}}\big]}} $$$

where $\overline{T}(r)$ is the temperature profile predicted by the SSM (see the left panel of Figure 7).

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OPA3: sharp increase of the opacity at the bottom of the convective envelope described by assuming F(T) = 1.1 for T ⩽ 5 × 10⁶ K (and F(T) = 1 otherwise). In our approach, this corresponds to

$$\begin{eqnarray} \nonumber \delta \kappa (r) &=& 0.1 \;\;\;\;\;\;\; {\rm if} \;\;\;\; \overline{T}(r) \le 5\times 10^{6}\, {\rm K} \\ \nonumber \delta \kappa (r) &=& 0 \;\;\;\;\;\;\;\;\;\;\; {\rm otherwise}. \end{eqnarray}$$

The above inequality can be rewritten in terms of a condition on the distance from the center of the Sun obtaining δκ(r) = 0.1 for r ⩾ 0.4 R_☉ (and δκ(r) = 0 otherwise).

The fourth studied case concerns with modification of energy generation in the Sun. More precisely, we consider

Spp: increase of the astrophysical factor S_pp of the p + p → d + e⁺ + ν_e reaction by +10%. In order to introduce this effect in LSM calculations, we consider the following variation of the energy generation profile:

$$\begin{equation} \delta \epsilon (r) =\epsilon _{S_{\rm pp}}(r) \;\delta S_{\rm pp}, \end{equation} \tag{ 50 }$$

where

$$\begin{equation} \epsilon _{S_{\rm pp}}(r) = \frac{\partial \ln \epsilon }{\partial \ln S_{\rm pp}} |_{\rm SSM} \end{equation} \tag{ 51 }$$

and δS_pp = 0.1. The function δ(r) obtained in this way is shown in the right panel of Figure 7. The bump at r ≃ 0.28 R_☉ is due to the out-of-equilibrium behavior of helium-3.

8.1. Physical and Helioseismic Properties of the Sun

In Figure 8, we show with solid lines the physical properties of LSMs, obtained by solving the linear system of Equations (41), and with dotted lines the results obtained by the "standard" nonlinear SM calculations.¹³ The four panels correspond to the input modifications introduced in the previous section. We use different colors to show the variations of pressure (red), temperature (black), mass (blue), and luminosity (green) as a function of the solar radius.

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We see that a very good agreement exists between LSM and nonlinear SM results. To be more quantitative, the response of the Sun to the input modifications is reproduced at the 10% level or better, in all the considered cases. It is important to note that all the relevant features of the δh(r) are obtained, indicating that the major effects are correctly implemented in our approach. The small differences between LSMs and nonlinear SMs, typically more evident just below the convective region and/or at the center of the Sun, basically reflects the accuracy of the assumptions which have been used to estimate the variation of the chemical composition of the present Sun. In the OPA1 case, a disagreement exists in the δl(r) behavior at the center of the Sun. This difference, which has no observable consequences, is mainly due to nonlinear effects. One should note, in this respect, that the constant 10% increase of the radiative opacity induces variations of temperature and pressure which are at the ∼1% level and a much smaller variation of δl(r) (at the 0.1% level). In this situation, cancellations between different first-order competing contributions may occur.

In Table 1, we present the variations of the initial abundances and of the photospheric chemical composition obtained by using relations (39) and (36). Moreover, we show the variation of the convective radius δR_b calculated according to Equation (47). Finally, in Figure 9, we present the variations of the density profile δρ(r) calculated according to Equation (46) and the variation of the squared isothermal sound speed u = P/ρ given by

$$\begin{equation} \delta u(r) = \delta P(r) - \delta \rho (r) \,. \end{equation} \tag{ 52 }$$

We see that LSMs reproduce the results obtained by "standard" calculations with a very good accuracy. In particular, the variations of the helioseismic observables are obtained within 10% or better (unless they are extremely small). All this shows that our approach is sufficiently accurate to use LSMs as a tool to investigate the origin of the present discrepancy with helioseismic data. In a separate paper, we will use LSMs to analyze the role of opacity and metals in the Sun (F. L. Villante et al. 2010, in preparation).

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Table 1. Comparison between the Predictions of LSM and "Standard" Nonlinear SM for the Initial and Surface Chemical Abundances, the Convective Radius, and the Solar Neutrino Fluxes

	OPA1		OPA2		OPA3		Spp
	SM	LSM	SM	LSM	SM	LSM	SM	LSM
ΔYini	0.016	0.017	−0.0056	−0.0058	0.0021	0.0019	0.0015	0.0016
δZini	−0.018	−0.016	0.000	−0.001	−0.012	−0.012	−0.010	−0.011
ΔYb	0.014	0.014	−0.0037	−0.0036	0.0038	0.0038	0.0031	0.0034
δZb	−0.018	−0.018	0.0049	0.0047	−0.0050	−0.0049	−0.0040	−0.0044
δRb	−0.0020	−0.0020	−0.0067	−0.0070	−0.014	−0.015	−0.0058	−0.0064
δΦpp	−0.011	−0.010	0.0045	0.0052	−0.0020	−0.0011	0.0090	0.0092
δΦBe	0.13	0.13	−0.067	−0.064	0.017	0.016	−0.11	−0.11
δΦB	0.27	0.27	−0.17	−0.17	0.029	0.028	−0.27	−0.28
δΦN	0.14	0.14	−0.10	−0.094	0.003	0.004	−0.21	−0.22
δΦO	0.21	0.22	−0.14	−0.14	0.012	0.012	−0.29	−0.31

Notes. Note that the absolute variations are reported for helium, whereas the relative variations are shown for all the other quantities.

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8.2. Neutrino Fluxes

As a final application, we calculate the solar neutrino fluxes predicted by LSMs and we compare them with those obtained by using "standard" nonlinear SM calculations. The flux Φ_ν of solar neutrinos can be expressed as

$$\begin{equation} \Phi _\nu = \frac{1}{D^2} \int dr \; r^2 \rho (r) \, n_{\nu }(r), \end{equation} \tag{ 53 }$$

where D is the Sun-earth distance, n_ν(r) is the total number of neutrinos produced per unit time and unit mass in the Sun, and the index ν = pp, B, Be, N, O labels the neutrino producing reaction according to commonly adopted notations. If we expand to first order, the relative variation of the flux δΦ_ν can be expressed as

$$\begin{eqnarray} \nonumber \delta \Phi _\nu &=& \frac{1}{\overline{\Phi }_{\nu }} \int dr \; (r/D)^2 \, \overline{\rho }(r)\, \overline{n}_{\nu }(r) \\ & & \times [ n_{\nu,\rho }(r) \, \delta \rho (r) + n_{\nu, T}(r)\,\delta T(r)\,+\, n_{\nu,Y}(r)\,\Delta Y(r) \nonumber\\ &&+ n_{\nu,Z}(r) \,\delta Z(r) + n_{\nu,\rm Spp}(r)\, \delta S_{\rm pp} ], \end{eqnarray} \tag{ 54 }$$

where $\overline{\Phi }_{\nu }$ is the SSM value and

$$\begin{eqnarray} \nonumber n_{\nu, \rm \rho }(r)&=&\left.\frac{\partial \ln n_\nu }{\partial \ln \rho }\right|_{\rm SSM}+1\\ \nonumber n_{\nu,\rm T}(r)&=&\left.\frac{\partial \ln n_\nu }{\partial \ln T}\right|_{\rm SSM}\\ \nonumber n_{\nu,Z}(r)&=&\left.\frac{\partial \ln n_\nu }{\partial \ln Z}\right|_{\rm SSM}\\ \nonumber n_{\nu,Y}(r)&=&\left.\frac{\partial \ln n_\nu }{\partial Y}\right|_{\rm SSM}\\ n_{\nu,\rm Spp}(r)&=&\left.\frac{\partial \ln n_\nu }{\partial \ln S_{\rm pp}}\right|_{\rm SSM}. \end{eqnarray} \tag{ 55 }$$

The derivatives n_ν,j(r) have been calculated numerically by assuming that the abundances of secondary elements (helium-3, carbon-12, and nitrogen-14) can be estimated as is described in Appendix A. The symbol |_SSM indicates that we calculate the derivatives along the density, temperature, and chemical composition profiles predicted by the SSM.

The term $n_{\nu,\rm Spp}(r) \, \delta S_{\rm pp}$ is introduced to describe the effects of a variation of S_pp on the rates of neutrino producing reactions. It clearly holds $n_{\rm pp,\rm Spp}(r) \equiv 1$, since the pp-neutrinos are produced by p + p → d + e⁺ + ν_e. Boron and beryllium neutrinos are influenced by S_pp because this parameter determines the helium-3 production rate (through deuterium) and, thus, also the rate of ³He + ⁴He → ⁷Be + γ reaction. As a consequence, we have

$$\begin{equation} n_{\rm Be, \rm Spp}(r) = n_{\rm B, \rm Spp}(r) = \left.\frac{\partial \ln X_3}{\partial \ln S_{\rm pp}}\right|_{\rm SSM}, \end{equation} \tag{ 56 }$$

where X₃ is the helium-3 abundance. The CN-cycle efficiency, instead, does not depend on the value of S_pp and, thus, we have $n_{\rm N,\rm Spp}(r) = n_{\rm O,\rm pp}(r) \equiv 0$.

It is useful to recast Equation (54) in the form

$$\begin{eqnarray} \delta \Phi _\nu &=& \int dr \; [ \phi _{\nu,\rho }(r) \, \delta \rho (r) + \phi _{\nu, T}(r) \,\delta T(r) + \phi _{\nu,Y}(r) \,\Delta Y(r) \nonumber\\ &&+ \phi _{\nu,Z}(r) \,\delta Z(r) + \phi _{\nu,\rm Spp}(r) \, \delta S_{\rm pp} ], \end{eqnarray} \tag{ 57 }$$

where

$$\begin{equation} \phi _{\nu,j} (r) = \frac{r^2 \, \overline{\rho }(r) \, \overline{n}_{\nu }(r) \, n_{\nu,j}(r)}{\int dr \; r^2 \, \overline{\rho }(r) \, \overline{n}_{\nu }(r)}. \end{equation} \tag{ 58 }$$

The functions ϕ_ν,j(r) are displayed in Figure 10. They show explicitly the well-known fact that the boron, beryllium, and CNO neutrinos are produced in a more internal region with respect to pp-neutrinos and that they more strongly depend on temperature. The complicated behavior of the functions Φ_N,j(r) at r ≃ 0.15 R_☉ is due to the out-of-equilibrium behavior of carbon-12 abundance. The N-neutrinos originate, in fact, from the decay of ¹³N, which is produced by ¹²C + p → ¹³N + γ reaction. The Sun was born with a relatively large amount of carbon-12, which has been converted by CN-cycle into nitrogen-14 in the more internal regions. The ¹²C abundance is, thus, larger where CN-cycle is less effective and, as a consequence, a non-negligible production of N-neutrinos occurs at relatively large radii, where equilibrium conditions do not hold.

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In Table 1, we compare the LSM results, i.e., the values δΦ_ν obtained by applying Equation (57) to the solutions of Equations (41), with the results obtained from "standard" calculations. We see that an excellent agreement exists, for all the cases considered in this paper.¹⁴ All this shows that our approach can be used as a tool to investigate the dependence of the solar neutrino fluxes on the input parameters adopted in SSM construction.

We have proposed a new approach to studying the properties of the Sun which is based on the following points:

• 1.  
  We have considered small variations of the physical and chemical properties of the Sun with respect to SSM predictions and we have linearized the stellar equilibrium equations to relate them to the properties of the solar plasma (see Sections 3, 4, and 5).
• 2.  
  We have derived simple relations which allow us to estimate the (variation of) present solar composition from the (variation of) nuclear reaction rates and elemental diffusion efficiency in the present Sun (see Section 6).As a final result, we have obtained a linear system of ordinary differential equations (see Section 7) which can be easily solved and that completely determine the physical and chemical properties of the "perturbed" Sun.

In order to show the validity of our approach, we have considered four possible modifications of the input parameters (opacity and energy generation profiles) and we have compared the results of our LSMs with those obtained by "standard" methods for SM calculations (see Section 8). A very good agreement is achieved for all the structural parameters (mass, luminosity, temperature, pressure, etc.) and for all the helioseismic and solar neutrino observables.

We believe that our approach can complement the traditional method for SM calculations, allowing us to investigate in a more efficient and transparent way the role of the different parameters and assumptions. In particular, it could be useful to study the origin of the present discrepancy between SSM results and helioseismic data.

We thank A. Formicola from LUNA Collaboration for providing us useful information about nuclear cross sections and G. Fiorentini for comments and discussions. We are extremely grateful to S. Degl'Innocenti and to the other members of the Astrophysics group of University of Pisa for very useful discussions, for providing us updated opacity tables, for reading the manuscript, and for fruitful collaboration.

Secondary elements are those elements which are both created and destroyed in a reaction chain or in a reaction cycle. The relevant secondary elements to understand energy and neutrino production in the Sun are ³He, ¹²C, and ¹⁴N.

A.1. The ³He Abundance

The evolution of helium-3 abundance, X₃, is described by the equation¹⁵:

$$\begin{equation} \frac{\partial X_{3}}{\partial t} = \frac{3 \rho }{ m_{u}}\left[\frac{X^2}{2}\langle \sigma v \rangle _{\rm pp} - \frac{X_{3}^2}{9}\langle \sigma v \rangle _{33} - \frac{X_{3} Y}{12}\langle \sigma v \rangle _{\rm 34} \right], \end{equation} \tag{ A1 }$$

where 〈σv〉_pp is the reaction rate per particle pair of the p + p → d + e⁺ + ν_e reaction, while 〈σv〉₃₃ and 〈σv〉₃₄ refer to ³He + ³He → ⁴He + 2p and ³He + ⁴He → ⁷Be + γ reactions, respectively. In the most internal region (r ⩽ 0.25 R_☉), the rates of these reactions are fast with respect to the Sun evolutionary times and equilibrium is achieved:

$$\begin{equation} X_{3,\rm eq} = \frac{3 \, Y}{8} \frac{\langle \sigma v \rangle _{34} }{\langle \sigma v \rangle _{33} }\left[-1 + \sqrt{ 1+32 \left(\frac{X}{Y}\right)^2 \frac{\langle \sigma v \rangle _{33}\; \langle \sigma v \rangle _{\rm pp}}{\langle \sigma v \rangle _{\rm 34} ^2}} \right]. \end{equation} \tag{ A2 }$$

In the outer regions (R ⩾ 0.2 R_☉), we can neglect the contribution from ³He + ⁴He → ⁷Be + γ and we obtain

$$\begin{equation} \frac{\partial X_{3}}{\partial t} = {\mathcal C}_{3} - {\mathcal D}_{3} \; X_{3}^2, \end{equation} \tag{ A3 }$$

where ${\mathcal C}_3 = (3\, X^2 \, \rho \, \langle \sigma v \rangle _{\rm pp}) / (2\, m_{u})$ and ${\mathcal D}_3 = (\rho \, \langle \sigma v \rangle _{\rm 33}) /(3\, m_{u})$. If we assume that the factors ${\mathcal D}_3$ and ${\mathcal C}_3$ are approximately constant over the time scale $\tau _3 = {\mathcal C_3}^{-1}$ during which ³He is produced, the above equation can be explicitly solved. We obtain

$$\begin{equation} X_{3} = X_{3,\rm eq} \; \tanh \left[\frac{X_{3,\rm max} }{X_{3,\rm eq} }\right], \end{equation} \tag{ A4 }$$

where $X_{3,\rm max} = {\mathcal C_3} \, t$ represents the helium-3 abundance which would have been produced in the integration time t if we had neglected the ³He destruction processes (and we implicitly assumed that the helium-3 initial abundance is negligible).

In this work, we need to calculate the helium-3 response to a generic modification of the solar properties. We use the following approach. We calculate the ³He equilibrium value in a generic SM by using Equation (A2). We assume that Equation (A4) is valid at each point of the Sun and we estimate the value $\overline{X}_{3,\rm max}$ in the SSM by inverting it (and by using the SSM-abundance $\overline{X}_3$). Then, we calculate the value of X_3,max in the modified SM by considering that $X_{3,\rm max}$ scales as $X_{3,\rm max}\propto X^2 \, \rho \, S_{\rm pp}$. Finally, we calculate the value of X₃ in the modified Sun by using Equation (A4) with the modified values for $X_{3,\rm eq}$ and $X_{3,\rm max}$. The obtained results reproduce with very good accuracy the variation of the ³He abundances in all the cases considered in this paper.

A.2. The ¹²C and ¹⁴N Abundances

The evolution of carbon-12 and nitrogen-14 abundances, indicated with X₁₂ and X₁₄ respectively, is described by the equations¹⁶:

$$\begin{eqnarray} \nonumber \frac{\partial X_{12}}{\partial t} &=& \frac{12 \, \rho }{ m_{u}}\left[\frac{X\,X_{14} }{14}\langle \sigma v \rangle _{1, 14} - \frac{X\, X_{12}}{12}\langle \sigma v \rangle _{1, 12} \right]\\ \frac{\partial X_{14}}{\partial t} &=& \frac{14 \, \rho }{ m_{u}}\left[\frac{X\,X_{12} }{12}\langle \sigma v \rangle _{1, 12} - \frac{X\, X_{14}}{14}\langle \sigma v \rangle _{1, 14} \right], \end{eqnarray} \tag{ A5 }$$

where 〈σv〉_1,12 and 〈σv〉_1,14 are the reaction rates per particle pair of ¹²C + p → ¹³N + γ and ¹⁴N + p → ¹⁵O + γ, respectively. It is useful to rewrite these equations in terms of the variables η = X₁₄/14 − X₁₂/12 and N = X₁₂/12 + X₁₄/14. We obtain

$$\begin{eqnarray} \nonumber \frac{\partial \eta }{\partial t} &=& {\mathcal C}_\eta \, N- {\mathcal D}_\eta \, \eta \\ \frac{\partial N}{\partial t} &=& 0, \end{eqnarray} \tag{ A6 }$$

where ${\mathcal C}_\eta = \rho \, X (\langle \sigma v \rangle _{1, 12} - \langle \sigma v \rangle _{1, 14}) / m_{u}$ and ${\mathcal D}_\eta = \rho \, X (\langle \sigma v \rangle _{1, 14} + \langle \sigma v \rangle _{1, 12}) / m_{u}$. From the above equations, we see that N is constant and that the equilibrium value for η is given by

$$\begin{equation} \eta _{\rm eq} = N\; \frac{\langle \sigma v \rangle _{1, 12} - \langle \sigma v \rangle _{1, 14} }{\langle \sigma v \rangle _{1, 12} + \langle \sigma v \rangle _{1, 14} }\;. \end{equation} \tag{ A7 }$$

In the assumption that the coefficients ${\mathcal C}_\eta$ and ${\mathcal D}_\eta$ are approximately constant over the time scale $\tau _{\eta } = 1/({\mathcal C}_\eta \, N)$, the solution for η can be explicitly calculated, obtaining

$$\begin{equation} \eta = \eta _{\rm eq} + (\eta _0 - \eta _{\rm eq}) \exp \left(- \frac{\eta _{\rm max}}{\eta _{\rm eq}}\right), \end{equation} \tag{ A8 }$$

where η₀ is the initial value and $\eta _{\rm max} = {\mathcal C}\, N\, t$ represents the η value which would have been obtained in the integration time t if we had neglected the "destruction" term in Equation (A6).

In analogy to what was done for helium-3, we use the following approach to calculate the response of carbon-12 and nitrogen-14 to a generic modification of the solar properties. We calculate the η_eq value in a generic SM by using Equation (A7). We assume that Equation (A8) is valid at each point of the Sun and we estimate the value $\overline{\eta }_{\rm max}$ in the SSM by inverting it (and using the SSM-values $\overline{\eta }$ and $\overline{\eta }_{0}$). Then, we calculate the value of η_max in the modified Sun by considering that it scales as η_max ∝ N ρ X(〈σv〉_1,12 − 〈σv〉_1,14). Finally, we calculate η in the modified SM by using Equation (A8) with the new values for η_eq, η_max, and η₀. The carbon-12 and nitrogen-14 abundances can be calculated from η and N by the simple relations:

$$\begin{eqnarray} \nonumber X_{12} &=& 6 (N - \eta)\\ X_{14} &=& 7 (N + \eta). \end{eqnarray} \tag{ A9 }$$

The obtained results reproduce with very good accuracy the results obtained by full numerical calculations in all the cases considered in this paper.

The effect of elemental diffusion on the convective envelope chemical composition can be estimated from

$$\begin{equation} \delta D_{i,\rm b} = \delta \omega _{i} - \delta H, \end{equation} \tag{ B1 }$$

where δH is the relative variation of the convective envelope effective thickness, while δω_i is the relative variation of the diffusion velocity of the i-element at the bottom of the convective region. One should note that the quantity δω_i is defined as

$$\begin{equation} \delta \omega _{i} = \frac{\omega _i(R_{\rm b}) - \overline{\omega }_i(\overline{R}_{\rm b})}{\overline{\omega }_i(\overline{R}_{\rm b})} \end{equation} \tag{ B2 }$$

and, thus, involves the difference between the diffusion velocities evaluated at two different points. By taking into account the expression for ω_i given in Equation (33) and by considering that ∂ln P/∂r ≃ −(G_NM_☉/R²_b)(ρ/P) at r = R_b, we obtain

$$\begin{eqnarray} \delta \omega _{i} &=& \frac{5}{2}\,\frac{ T(R_{\rm b})-\overline{T}(\overline{R}_{\rm b}) }{\overline{T}(\overline{R}_{\rm b})} -\frac{ P(R_{\rm b}) -\overline{P}(\overline{R}_{\rm b}) }{\overline{P}(\overline{R}_{\rm b})}\nonumber\\ && -2\, \delta R_{\rm b} + \frac{d \ln A_{\rm tot, i} }{d Y}\,\Delta Y_{\rm b} \end{eqnarray} \tag{ B3 }$$

where A_tot,i = A_P,i + ∇_ad A_T,i. This can be rewritten as

$$\begin{eqnarray} \nonumber \delta \omega _{i} &=& \frac{5}{2}\,\delta T_{\rm b} -\delta P_{\rm b} -2\, \delta R_{\rm b} + \frac{d \ln A_{\rm tot, i} }{d Y}\,\Delta Y_{\rm b} \\ & & + \left[\frac{5}{2}\frac{\partial \ln \overline{T}(\overline{R}_{\rm b})}{\partial \ln r} - \frac{\partial \ln \overline{P}(\overline{R}_{\rm b})}{\partial \ln r} \right] \delta R_{\rm b}, \end{eqnarray} \tag{ B4 }$$

where according to the notation convention adopted in this paper, we have indicated with $\delta T_{\rm b} = [ T(\overline{R}_{\rm b})- \overline{T}(\overline{R}_{\rm b}) ] / \overline{T}(\overline{R}_{\rm b})$ and $\delta P_{\rm b} = [ P(\overline{R}_{\rm b})- \overline{P}(\overline{R}_{\rm b}) ] / \overline{P}(\overline{R}_{\rm b})$.¹⁷ We note that the last term in the above equation can be neglected since ∂ln T/∂ln P = ∇_ad ≃ 2/5 at the bottom of the convective region.

The effective thickness of the convective envelope is defined by the condition H = M_conv/(4πR²_bρ_b). The relative variation δH is thus given by

$$\begin{equation} \delta H = \delta M_{\rm conv} - 2 \delta R_{\rm b} - \delta \rho _{\rm b} -\frac{\partial \ln \rho (\overline{R}_{\rm b})}{\partial \ln r} \, \delta R_{\rm b}. \end{equation} \tag{ B5 }$$

If we consider that $M_{\rm conv}= \int _{R_{\rm b}} dr \; 4\pi r^2 \rho (r)$ and that δρ(r) is approximately constant in the convective region, we obtain

$$\begin{equation} \delta M_{\rm conv} = \delta \rho _{\rm b} - \frac{\overline{R}_{\rm b}}{\overline{H}}\,\delta R_{\rm b}. \end{equation} \tag{ B6 }$$

By using this relation and taking into account Equations (B5) and (B4), we arrive at the expression:

$$\begin{equation} \delta D_{i,\rm b} = \frac{5}{2}\,\delta T_{\rm b} -\delta P_{\rm b} + \frac{d \ln A_{\rm tot, i}}{d Y}\, \Delta Y_{\rm b} + \left[\frac{\partial \ln \rho (\overline{R}_{\rm b})}{\partial \ln r} + \frac{\overline{R}_{\rm b}}{\overline{H}} \right] \delta R_{\rm b}. \end{equation} \tag{ B7 }$$

Finally, by using the expression for δR_b given in Equation (12) (with δm_b = 0 and δl_b = 0) and by taking advantage of the integration conditions at the bottom of the convective region expressed by Equations (21), we obtain the final relations:

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

where Γ_Y = 2.05, Γ_Z = 2.73, Γ_P = −1.10, and Γ_κ = −0.06 and we assumed that all metals have the same diffusion velocity as iron. In the calculations presented in this paper, we neglect for simplicity the terms proportional to δκ_b which generally give a very small contribution.