SOLAR NEUTRINO PHYSICS OSCILLATIONS: SENSITIVITY TO THE ELECTRONIC DENSITY IN THE SUN'S CORE

Neutrinos, once produced in the core of the Sun, reach the Earth in less than eight minutes. On their interplanetary journey, solar neutrinos very rarely interact with other particles. Luckily for physicists, neutrinos with a scattering cross-section varying from some 10⁻⁴⁶ cm² to some 10⁻⁴² cm² (Bahcall 1989) occasionally interact with the baryons present in the solar neutrino detectors. Therefore, the neutrinos produced in the nuclear reactions of proton–proton (PP) chains and carbon–nitrogen–oxygen (CNO) cycle are natural probe-particles of the constitutive plasma of the Sun's core.

The current theory of neutrino physics, following in the footsteps of the fundamental theory of elementary particles, has shown that neutrinos appear in three (at least) flavors (Hernandez 2010): electron-neutrino (ν_e), muon-neutrino (ν_μ), and tau-neutrino (ν_τ). The theory states that by vacuum flavor oscillations, when moving in vacuum, neutrinos have the ability to switch cyclically between different flavors. This mechanism of neutrino oscillations was proposed by Pontecorvo around 1940. In the late seventies, Lincoln Wolfenstein and colleagues suggested another mechanism for neutrinos to change flavor, as neutrinos move through a high density medium (Wolfenstein 1978): in an environment of high density material, the effective mass of the propagating neutrinos is different from the mass of neutrinos that propagates in a vacuum. Since the oscillations between different flavors of neutrinos depend on their mass, the oscillations of neutrinos in dense media are different from neutrino oscillations in vacuum. This process is now known as Mikheyev–Smirnov–Wolfenstein (MSW) or "matter oscillations." These two mechanisms of flavor oscillations are central to the modern theory of neutrinos. Through these mechanisms the electron-neutrinos produced in the Sun's core have a non-zero probability of being detected as a muon-neutrino or tau-neutrino when they reach the Earth. In particular, a small fraction of electron-neutrinos changes flavor in the Sun's interior when they propagate through the high density plasma medium of the Sun's core.

In recent years, significant progress has been achieved in the understanding of the mechanism responsible for the neutrino oscillations (Eguchi et al. 2003; Fukuda et al. 2001; Ahmad et al. 2001). The current theory of neutrino physics successfully explains the neutrino observations. Furthermore, the theory has been used successfully to determine the fundamental properties of neutrinos (Schwetz et al. 2008).

If, for several decades, the solar neutrino fluxes arriving on Earth have been difficult to interpret, this is no longer the case as the flavor oscillations at different energies are well understood. In particular, the production of these neutrinos has been confirmed by helioseismic data. One may consider a success that these two disciplines (solar neutrinos and helioseismology) agree within the error bars (see Turck-Chièze et al. 2010; Turck-Chièze & Couvidat 2011, and references therein). Such an agreement will now allow us to progress a step further. We believe that solar neutrinos will very likely become a powerful tool to probe the core of the Sun because the production of neutrinos is very sensitive to the local properties of the solar plasma.

In this paper, we focus on the sensitivity of solar neutrinos to the plasma in the solar core, and explore the possibilities of probing the physical properties of such plasma by using the solar neutrino flux measurements. This point has been proposed by John Bahcall and Raymond Davis about 50 years ago (Davis 1964). In principle, the structure of the solar core can be studied by means of neutrino spectroscopy in two fundamental ways: (1) diagnosing the temperature profile in the Sun's core, by measuring the total number of electron-neutrinos produced in each nuclear reaction of PP chains and CNO cycle, in particular the boron flux that is the most sensitive to the temperature (Turck-Chièze et al. 2012; Turck-Chièze & Lopes 2012) and (2) measuring the radial electronic density of matter of the Sun by determining the amount of electron-neutrinos that are converted into another flavor. This article is a first attempt to study this second physical information through neutrino probes.

The neutrino flavor oscillations due to the MSW mechanism are particularly significant in the solar interior, where there is a wide radial variation of the plasma density. The standard solar model (Turck-Chièze & Lopes 1993) partly validated by helioseismology predicts that the density inside the Sun varies from about 150 g cm⁻³ in the center of the star to 1 g cm⁻³ at half of the solar radius.

What makes this diagnostic particularly powerful is the possibility of getting a direct measurement of the plasma electronic density almost independently of the solar model. The strong dependence of neutrino oscillations with the local electronic density of matter opens the possibility of inferring the matter density and chemical composition in the nuclear region. Since the neutrino oscillations in matter depend strongly on the matter density and weakly on the chemical composition, this is at first a diagnosis of the radial density profile of matter in the solar core. The composition diagnostic can come from precise CNO neutrino detections. Therefore, we can anticipate that with the expected increased accuracy of the solar neutrino detections in the future, it will be possible to extend this analysis to the solar core chemical composition determination.

The theory of neutrino oscillations in vacuum and matter (Hernandez 2010; Schwetz et al. 2008) has successfully addressed the problem of the solar neutrino deficit—the discrepancy between the neutrino flux detected and the theoretical predictions of the solar standard model (Turck-Chièze & Lopes 1993; Fogli et al. 2009; Gonzalez-Garcia & Maltoni 2008). The theory provides a theoretical solution in full agreement with all solar neutrino experiments (Bellini et al. 2010, 2011b; Aharmim et al. 2010) as well as with the data obtained by the KamLAND reactor experiment (Eguchi et al. 2003). Presently, the theoretical model for neutrino flavor oscillations is defined by means of six parameters: the difference of the squared characteristic masses (Δm²₁₂, Δm²₃₁), the mixing angles (θ₁₂, θ₂₃, θ₃₁), and the charge-parity (CP)-violation phase. The mass square differences and mixing angles are known with reasonable accuracy (Fogli et al. 2009): Δm²₃₁ is obtained from the experiments of atmospheric neutrinos and Δm²₁₂ is obtained from solar neutrino experiments.

The mixing angles are not uniformly well defined. θ₁₂ is obtained from solar neutrino experiments with an excellent precision. θ₂₃ is obtained from the atmospheric neutrino experiments; this is the mixing angle with the highest value. θ₁₃ has been first estimated from the Chooz reactor (Apollonio & Baldini 1999); its value is very small and was still very uncertain (Fogli et al. 2009). Nowadays with Daya Bay and Reno, the situation is largely improved (Fogli et al. 2012), as shown in the next paragraph. However, present experiments cannot fix the value of the CP-violation phase (Hernandez 2010).

An overall fit to the data obtained from the different neutrino experiments, solar neutrino detectors, accelerators, atmospheric neutrino detectors, and nuclear reactor experiments suggests that the parameters of neutrino oscillations are the following ones (Gonzalez-Garcia & Maltoni 2008): Δm²₃₁ ∼ 2.46 ± 0.12 × 10⁻³ eV², Δm²₁₂ ∼ 7.59 ± 0.20 × 10⁻⁵ eV², θ₁₂ = (34.4 ± 1)°, θ₂₃ = (42.8^+4.7_−2.9)°, and θ₁₃ = (5.6^+3.0_−2.7)°. The recent progress leads to θ₁₃ = (8.6^+0.44_−0.46)° and Δm²₃₂ ∼ −2.43_−0.06^+0.42 × 10⁻³ eV² (Gonzalez-Garcia et al. 2012).

In the limiting case where the value of the mass differences, Δm²₁₂ or Δm²₃₁, is large, or one of the mixing angles (θ₁₂, θ₂₃, θ₃₁) is small, the theory of three neutrino flavor oscillations reverts to an effective theory of two neutrino flavor oscillations (Hernandez 2010). Balantekin and Yuksel have shown that the survival probability of solar neutrinos calculated in a model with two neutrino flavor oscillations or three neutrino flavor oscillations has very close values (Balantekin & Yuksel 2003).

In the present work, for reasons of convenience and simplicity, we will restrict our study to the theory of two neutrino flavor oscillations. The generalization of the results to a theory of three neutrino flavor oscillations is obvious (Lisi & Montanino 1997; Gando et al. 2011). The survival probability of the electron-neutrino function $P_{\nu _e} (\equiv P (\nu _e \rightarrow \nu _e)$) in a two flavor neutrino theory is given by

$$\begin{eqnarray} &&P_{\nu _e}(E, r) = \frac{1}{2} + F (\gamma) \cos {(2 \theta _v)} \cos {(2 \theta _m),} \end{eqnarray} \tag{ 1 }$$

where θ_v is the angle of oscillation in vacuum and θ_m is the angle of oscillation in matter. The function F(γ) is a first-order correction to the adiabatic approximation of the propagation of neutrinos in matter, where γ is the adiabatic parameter which depends of the local properties of the plasma (Landau & Rosenkewitsch 1932). The adiabaticity occurs when the electronic density of the propagating medium is a slowly varying function over the neutrino travel path (or equivalently the solar radius). This approximation is valid in the solar interior, in the case where γ ≫ 1, a condition that is verified in much of the solar core and the radiative region (Landau & Rosenkewitsch 1932; Zener 1932). As F(γ) = 0.5 − P₁₂, where P₁₂ corresponds to the transition between two distinct neutrino flavors, at first order it is reasonable to consider that P₁₂ = 0. The numerical value of F(γ) is approximately equal to 0.5 (Bilenky 2010). The angle θ_v depends on the distance from the Sun to the Earth. In this model we assume θ_v to be equal to θ₁₂. The phase θ_m is the most important term in this analysis, since it depends on the properties of the plasma in the Sun's core, specifically, the local electron density. The neutrino mixing angle θ_m is given by

$$\begin{eqnarray} &&\sin {2\theta _m}= \frac{\sin {2\theta _v}}{\sqrt{(V/\Delta m^2-\cos {(2\theta _v)})^2+ \sin ^2{(2\theta _v)}} }, \end{eqnarray} \tag{ 2 }$$

where Δm² is the difference of the squared masses we consider to be equal to Δm²₁₂. The function of the solar plasma V is given by

$$\begin{eqnarray} &&V (E,r) = 2 \sqrt{2} G_f \; n_e (r) E, \end{eqnarray} \tag{ 3 }$$

where G_f is the Fermi constant, n_e(r) is the electron density of plasma, and E the energy of the neutrino. The electron density n_e(r) = N_o ρ(r)/μ_e(r), where μ_e is the mean molecular weight per electron, ρ(r) is the density of matter, and N_o is the Avogadro's number.

The neutrino fluxes produced in the various nuclear reactions of the PP chains and CNO cycle have been computed for an updated version of the solar standard model (Turck-Chièze et al. 2004, 2010; Bahcall et al. 2005a; Guzik & Mussack 2010; Turck-Chièze & Couvidat 2011). Figure 1 shows the location of the different neutrino emission regions of the nuclear reactions. In the Sun's core, the neutrino emission regions occur in a sequence of shells, following closely the location of nuclear reactions, orderly arranged in a sequence dependent on their temperature.

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The first reaction of the PP chains, the pp nuclear reaction, has the largest neutrino emission shell. This region extends from the center of the Sun up to 30% of the solar radius. The pep reaction has a neutrino emission shell identical to the pp reaction, although only up to 25% of the solar radius. These two key nuclear reactions are strongly dependent on the total luminosity of the star. This is the reason why different solar models with the same total luminosity produced the same pp and pep neutrino emission shells. The neutrino emission shells of ⁸B-ν and ⁷Be-ν extend up to 15% and 22% of the solar radius. Finally, it is worth noting that the maximum emission of neutrinos for the PP chain nuclear reactions follows an ordered sequence (see Figure 1): ⁸B-ν, ⁷Be-ν, pep-ν, and pp-ν with the maximum emission located at 5%, 6%, 8%, and 10% of the solar radius, respectively.

The CNO cycle nuclear reactions produce the following electron-neutrinos sub-species: ¹⁵O-ν, ¹⁷F-ν, and ¹³N-ν released in emission shells identical to the ⁸B-ν. The ¹³N-ν have a second emission shell located between 12% and 25% of the Sun's radius. The emission of neutrinos for ¹⁵O-ν, ¹⁷F-ν, and ¹³N-ν is maximal at 4%–5% of the solar radius. The ¹³N-ν neutrinos have a second emission maximum which is located at 16% of the solar radius.

The neutrino emission reactions of the PP chains and the CNO cycle are produced at high temperatures in distinct layers in the Sun's core. Similarly, the neutrino flavor oscillations occur in the same regions. The average survival probability of electron-neutrinos in each nuclear reaction region is given by

$$\begin{eqnarray} &&\langle P_{\nu _e } (E)\rangle _j = A_j^{-1} \int P_{\nu _e} (E,r)\Phi _j (r) 4\pi \rho (r) r^2 dr, \end{eqnarray} \tag{ 4 }$$

where A_j is a normalization constant given by A_j = ∫Φ_j(r)4πρ(r)r² dr and Φ_j(r) is the electron-neutrino emission function for the j nuclear reaction. j corresponds to the following electron-neutrino reaction subspecies: pp, pep, ⁸B, ⁷Be, ¹³N, ¹⁵O, and ¹⁷F.

Figure 2 shows the energetic neutrino spectra (Bahcall 1989) for an updated version of our standard solar model before and after the flavor conversion. Figure 3(a) shows the survival probability of electron-neutrinos produced in the regions where the different nuclear reactions occur. This survival probability of electron-neutrinos $\langle P_{\nu _e} (E)\rangle _j$ is very similar for low- and high-energy neutrinos but has some differences between them. Unfortunately, as is well known, the emitted neutrino energy spectrum is limited to a specific energy range for each nuclear reaction. Nevertheless, to highlight the sensitivity of neutrino to MSW flavor oscillations, we choose to represent the survival probability of electron-neutrinos in all the available energy range, so that the regions of interest are indicated in color or by a single color square in the case of line fluxes (cf. Figure 2).

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The neutrino flavor change for low-energy neutrinos is due to vacuum oscillations, and for high-energy neutrinos it is caused by a cumulative effect of oscillations in vacuum and matter (MSW effect). The experimental neutrino flux measurements for low-energy neutrinos have been used to determine the value of θ₁₂ (Bellini et al. 2010). The survival probability of the electron-neutrinos $\langle P_{\nu _e } (E)\rangle _j$ with intermediate values of energy, between 1 MeV and 10 MeV, has a strong dependence on the electron density of the plasma. Since the production of electron-neutrinos occurs in various nuclear reactions at different layers in the Sun's interior, a sharp differentiation is observed between the different $\langle P_{\nu _e} (E)\rangle _j$ curves for the neutrinos with intermediate values of energy. Figure 3(b) shows a well-ordered sequence of curves that corresponds to the difference between the survival probability of electron-neutrinos produced by each nuclear reaction j and the survival probability of electron-neutrinos in the center of the Sun. This sequence of curves occurs as a result of the regular decrease of matter density from the center of the Sun. As a consequence, the difference between the survival probability of electron-neutrinos produced by each nuclear reaction j and the survival probability of electron-neutrinos produced in the center of the star increases with the distance of the location of the nuclear reactions from the center (cf. Figure 3(b)). For example, in the case of pp electron-neutrinos this difference is of the order of 0.06 which is three times larger than in the case of ⁸B electron-neutrinos. This is due to the fact that the ⁸B electron-neutrino source is located near the center of the star (cf. Figure 1). Nevertheless, the possibility of observing such effect is somewhat limited, once the neutrino flavor oscillations induced by matter depend on the location of the neutrino source in the Sun's core (cf. Figure 1), as well as on the energy of the emitted electron-neutrinos, i.e., the neutrino emission spectrum (cf. Figure 2). In the case of the pp nuclear reaction, the emitted neutrinos have a maximum energy of 0.42 MeV; consequently, the survival probability difference is smaller than 0.005. This effect is more pronounced in the case of the ⁸B nuclear reaction for which the emitted neutrinos have a maximum energy of 14.02 MeV. In this case, the effect is maximum for neutrinos with a energy of 4 MeV which have a survival probability difference of 0.02. Identical behavior will be observed for the CNO nuclear reactions (namely the nuclear reactions related to the production of ¹³N, ¹⁵O, and ¹⁷F chemical elements) which emitted neutrinos with maximum energies slightly above 1 MeV, for which the survival probability difference is of the order of 0.014. An equally pronounced difference is observed for the spectral line of pep neutrinos. In the case of the ⁷Be neutrino the effect is important for the high-energy neutrino line and very small in the case of the small-energy neutrino line. In particular, it should be possible to check experimentally the modification of the ⁸B energy neutrino spectrum caused by matter flavor oscillations due to the radial dependence of the ⁸B electron-neutrino source.

The solar model has been checked in most of the solar radiative region by means of the high-precision data from the Solar and Heliospheric Observatory (SOHO) helioseismic instruments (Turck-Chièze & Couvidat 2011; Turck-Chièze & Lopes 2012).

As shown in Equations (1)–(3), the radial profile of the electron density of the solar standard model is a fundamental ingredient to test neutrino physics theory. Up to now this quantity has been checked by the detection of the first gravity modes which are really sensitive to the central region of the Sun (Turck-Chièze et al. 2012; Turck-Chièze & Lopes 2012). In the next few years, with the increase of accuracy in the measurements of several solar neutrino experiments, the situation could reverse: neutrino fluxes should start to be used to diagnose and to infer the thermodynamic properties of the Sun's core.

The neutrino flux emissions inside the Sun are sensitive to the local values of temperature (specifically some of them), matter density, chemical composition, and electronic density where the nuclear reactions are taking place.

In particular, as the MSW flavor oscillation is strongly dependent on the local electronic density, it is possible to infer this quantity from the survival probability of electron-neutrinos. Furthermore, since electronic density depends on matter density and chemical composition, it will also be possible to obtain some information about these quantities.

5.1. The Sensitivity of Neutrino Flavor Oscillations to the Central Density

Any standard solar model calibrated for the present day total solar luminosity has a radial profile of temperature and density in the core that results from the balance found between the energy produced by the nuclear reactions and the energy transported to the surface. Solar models with slightly different physical assumptions have different radial profiles of temperature and matter density among other quantities. This is the result of the readjustment of the Sun's internal structure caused by the need to obtain the same total luminosity.

Our evolution code is an up-to-date version of the one-dimensional stellar evolution code CESAM (Morel 1997). The code has an up-to-date and very refined microscopic physics (updated equation of state, opacities, nuclear reactions rates, and an accurate treatment of microscopic diffusion of heavy elements), including the solar mixture of Asplund et al. (2005, 2009). The solar models are calibrated to the present solar radius R_☉ = 6.9599 × 10¹⁰ cm, luminosity L_☉ = 3.846 × 10³³ erg s⁻¹, mass M_☉ = 1.989 × 10³³ g, and age t_☉ = 4.54 ± 0.04 Gyr (e.g., Turck-Chièze & Couvidat 2011; Turck-Chièze et al. 2011). The models are required to have a fixed value of the photospheric ratio (Z/X)_☉, where X and Z are the mass fraction of hydrogen and the mass fraction of elements heavier than helium. The value of (Z/X)_☉ is determined according to the solar mixture proposed by Asplund et al. (2005). Our reference model is a solar standard model that shows acoustic seismic diagnostics and solar neutrino fluxes near other solar standard models (Turck-Chièze & Lopes 1993; Turck-Chièze et al. 2004, 2010; Bahcall et al. 2005b; Guzik & Mussack 2010; Serenelli et al. 2009).

In Table 1 we present the central density and the mean electronic molecular weight of this standard solar model and of two other non-standard solar models. These last two models have their radiative energy transport slightly modified in the core in order to obtain a model of the Sun with a higher central density. Figure 4(a) shows the radial density profile in the Sun's core for the three models. The amount of neutrinos converted to the non-electron flavor by the MSW flavor oscillation mechanism is dependent on the central density profile. Figure 4(b) shows the difference between the survival probability of electron-neutrinos for each of the two modified solar models relative to the standard solar model. The difference is more significant for the model with the highest central density. It is clearly illustrated that an increase of the central density leads to an increase in the neutrinos converted by the MSW flavor oscillation in all electron-neutrino subspecies.

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Table 1. Central Parameters of Standard and Modified Solar Models

Solar Modelsa	SSM	Sun A	Sun B
Model Values
Density (g cm−3)	151	161	167
Mean molecular
Weight per electron μe	1.69	1.65	1.64
Neutrino energy (MeV)
Lower cutoff Ercb	2.0	1.83	1.75
Inverted Values
Density (g cm−3)
with μe solar model	151.5 [0.3%]	161 [+0.4%]	168 [+0.6%]
Density (g cm−3)
with μe = μSSM = 1.69	151.5 [0.3%]	165 [+2.8%]	173 [+3.6%]

Notes. ^aThe solar standard model (SSM) and the modified models, Sun A and Sun B, were calibrated in order to keep the total luminosity of the present Sun (Figure 4 shows the corresponding density profile). ^bE_rc is the minimum energy that a neutrino must have to be affected by the MSW flavor oscillation mechanism. The values between [...] show the percentage difference between the model values and the inverted values.

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The survival probability of electron-neutrinos can be used to infer the matter density in the center of the Sun, once precise neutrino measurements will become available. In the following we describe a simple procedure that highlights the sensitivity of neutrino flavor oscillations to the Sun's core density. Only neutrinos with energy above a given minimum value will have their flavor changed by the MSW flavor oscillation mechanism. Several resonances occur in Equation (2) for neutrinos with energy E_r such that V(E_r, r) = Δm²cos (2θ_v). We call the lowest neutrino energy value E_r that verifies the previous equation E_rc—the cutoff minimum resonance neutrino energy for a given solar model. The value of E_r is minimum for the maximum value of the matter density (Equation (3)). Therefore, each solar model has a unique cutoff minimum neutrino energy E_rc given by

$$\begin{eqnarray} &&E_{\rm rc}= \frac{\Delta m^2 \cos {(2\theta _v)}}{2\sqrt{2} G_f N_o \rho _c /\mu _{{\rm ec}}}, \end{eqnarray} \tag{ 5 }$$

where ρ_c is the central matter density and μ_ec is the central value of the mean molecular weight per electron.

In Table 1, we show the value of E_rc for the three solar models computed from the survival probability electron-neutrino function: if the resonance condition is verified, θ_m = π/4 and the survival probability of electron-neutrinos (Equation (1)) has the specific value of 0.5. The value E_rc is determined for each solar model by identifying the minimum neutrino energy value for which the survival probability of electron-neutrinos is equal to 0.5. The models with the higher central densities have the lowest values of E_rc (see Table 1). The central density can be calculated from Equation (5) once the value of E_rc is known for each solar model and assuming that μ_ec (or the chemical composition) is known. In Table 1 we show the "inferred values" of the central density. The small variation μ_ec between different models allows us to make a reasonable estimation of the central density, assuming that the variation of the resonance energy E_rc is due solely to variations in the central density. The central density is inverted at most with a difference of a few percent higher than the value of the model, even in the case where μ_ec is poorly known.

In fact, the central value of the density in the Sun might be more difficult to obtain since the electron-neutrinos are not produced just in the center, and the different ranges of neutrino energies are limited (cf. Figure 2). An interesting possibility is to probe the solar core by using the neutrinos produced near the center of the star, such as ⁸B-ν, ¹⁵O-ν, and ¹⁷F-ν neutrino fluxes. The simultaneous measurement of two different neutrino fluxes produced in the same layer of the Sun's interior, such as ⁸B-ν and ¹⁵O-ν, could be used to estimate simultaneously the matter density and the average molecular weight per electron in the core of the Sun.

5.2. The Sensitivity of Neutrino Flavour Oscillations to the Radial Density Profile

All the solar models validated by acoustic and gravity mode detections predict that the density of matter decreases by 90% from its central value to the value at 30% R_☉. The neutrino fluxes produced by the different nuclear reactions are sensitive to the local values of the electronic density. The different neutrino fluxes Φ_j(r) can be interpreted as a local average of n_j(r).⁴ In principle taking into account the high sensitivity of solar neutrino fluxes to the temperature changes in the solar core, we should expect important variations in the location of the neutrino source in the Sun's core. However, the effect is actually quite small. The reason is related to the calibration procedure that solar models are subjected to, which require that the Sun in the present age must have the observed solar luminosity. Consequently, the production of neutrinos inside the Sun, which are strongly dependent on the luminosity of the star, namely, the neutrinos produced in the PP chain nuclear reactions, such as ⁸B and ⁷Be, occur sensibly in the same location, i.e., at the same distance from the center of the star. As an example, solar models which experiment with a decrease of 10% in the central density (and a 3% reduction in the central temperature) have a variation of the maximum of any of the neutrino sources (both PP chain nuclear reactions and CNO nuclear reactions) displaced by an amount smaller than 1% of the solar radius, although the neutrino fluxes change significantly due to their high sensitivity to the temperature. Actually, the PP neutrino source has the same location, as this nuclear reaction more than any other depends directly on the total luminosity of the star. For the purpose of this article the neutrino sources of the different solar models are considered to be located within the same fraction of solar radius. Therefore, if the fundamental parameters of neutrino oscillations are known, the probability of survival of electron-neutrinos can be used to measure the radial electronic density profile in the solar core, i.e., the matter density and molecular weight per electron in several layers of the Sun. A simple procedure to obtain the electronic density is now proposed. We first consider an electron-neutrino survival probability $\bar{p}_j$ determined, at a specific neutrino energy E_j, by some neutrino experiment. One assumes that $\bar{p}_j$ is determined fully independently of the solar standard model by a method identical to the one described in Berezinsky & Lissia (2001). Preferentially, E_j is the value chosen within the energy interval for which the MSW neutrino flavor oscillations are significant (cf. Figure 3). For each pair $(E_j, \bar{p}_j)$, the value of the electronic density for a certain layer can be computed from Equations (1)–(3). We estimate $\bar{n}_ {j} \equiv \left\langle n_e (r)\right\rangle _j$ corresponding to the electronic density of several layers in the Sun's core, as $\bar{n}_ {j}= 1/(2 \sqrt{2} G_f) \times \alpha _j \left\langle V\right\rangle _j / E_ {j}$, where 〈V〉_j is computed from p_j using the previous equations, and α_j is a weight-correction parameter unique for each j-nuclear reaction. α_j is estimated for the solar standard model and it takes a value of the order of a unit, increasing slightly when the neutrino source Φ_j(r) moves away from the center of the star. Similarly, by using the experimental value $\bar{p}_j$, which is different from the theoretical prediction p_j, we estimate the value $\bar{n}_j$ as a correction to the theoretical value n_j, from the equation Δn_j/n_j = β_jΔp_j/p_j, where $\Delta n_{j}= \bar{n}_{j}-n_{j}$, $\Delta p_{j}=\bar{p}_{j}-p_{j}$, and β_j is a coefficient computed from the solar standard model. The previous expression is obtained from a perturbation analysis of Equations (1)–(3). In this analysis, $\bar{n}_j$ is the inverted value deduced from the experimental data.

Figure 5 shows the inverted electronic density values obtained from the survival probability of electron-neutrinos as described previously. There is a good agreement between the values of the electronic density obtained by inversion and the electronic density of the standard solar model. In Figure 5(a) we show the electronic density values that one can deduce from the survival probabilities obtained from the neutrino detector measurements. More specifically, we present the electronic density inverted from the ⁷Be neutrino fluxes using the measurements of Bellini et al. (2010) and the ⁸B neutrino fluxes using the measurements of Aharmim et al. (2007).

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The Borexino experiment measures ⁷Be solar neutrino rates with an accuracy better than 5%. This corresponds to a ⁷Be neutrino flux Φ(⁷Be) = 4.87^+0.24_−0.24 × 10⁹ cm⁻² s⁻¹, under the assumption of the MSW-LMA scenario of solar neutrino oscillations (Bellini et al. 2011a; Arpesella et al. 2008). The estimated survival probability, assuming a high-metallicity solar standard model, was initially estimated to be $P_{\nu _e} (^7{\rm Be})=0.56\pm 0.1$ at the energy 0.862 MeV (Arpesella et al. 2008). Later, the Borexino collaboration (Bellini et al. 2011a) updated their estimate to $P_{\nu _e} (^7{\rm Be})=0.52^{+0.07}_{-0.06}$. Similarly, as proposed by Berezinsky & Lissia (2001), we compute the survival probability for ⁸B as $P_{\nu _e}(^8{\rm B})=\Phi _{\nu _e}(^8{\rm B})/\Phi _{t}(^8{\rm B})$, where Φ_t(⁸B) is the total neutrino flux integrating all neutrino flavors measured by SNO (Aharmim et al. 2010): Φ_t(⁸B) = 5.046^+0.226_−0.275 × 10⁶ cm⁻² s⁻¹ and $\Phi _{\nu _e}(^8{\rm B})$ is the electron-neutrino flux measured by the same detector (Aharmim et al. 2007): $\Phi _{\nu _e}(^8{\rm B})=1.76^{+0.14}_{-0.14} \times 10^6\; {\rm cm^{-2}\, s^{-1}}$. It follows that $P_{\nu _e}(^8{\rm B})=0.35\pm 0.047$ for SNO. It is clear that presently the error bars are still too large to really estimate the small gap with the standard model values. But one may hope for progress in the future as we already have several improved detections for ⁸B neutrinos: the neutrino flux measured by the Kamiokande-III experiment (Abe et al. 2011), $\Phi _{\nu _e}(^8{\rm B})=2.32^{+0.09}_{-0.09} \times 10^6\; {\rm cm^{-2}\, s^{-1}}$, the neutrino flux measured by the Borexino experiment is $\Phi _{\nu _e}(^8{\rm B})=2.4^{+0.5}_{-0.5} \times 10^6\; {\rm cm^{-2}\, s^{-1}}$ (Bellini et al. 2010). Moreover, we can already measure the energy dependence of these fluxes down to 3 MeV. So, to illustrate the potential of this diagnostic and to indicate some research perspectives, we present in Figure 5(b) the electronic density values deduced assuming an error bar on the survival probability of the order of 4%. It follows that the error bar on the electronic density that one can deduce for ⁷Be and ⁸B fluxes is largely reduced. This is due to the high sensitivity of neutrino flavor oscillations to the electronic density. This first study shows the potentiality of such type of neutrino diagnostic. Another possibility is to fix values of the electron-neutrino survival probability curve for neutrinos of low and high energy, as such neutrinos have respectively pure vacuum oscillations and vacuum plus MSW oscillations. In this case neutrinos with energy in the interval 0.5–10 MeV can probe the electronic density of different solar layers.

The capability to make such a study is real because new neutrino detection experiments are arriving and others are planned for the near future. The diversity of neutrino flux measurements expected at different energy levels will allow us to make reliable and detailed neutrino inversions.

In this first study we have used only two mean survival probabilities, without taking into account the energy dependence of the boron flux and the detailed radial distribution. But the Borexino detector also measures the pep neutrino flux, Φ(pep) = 1.6^+0.3_+0.3 × 10⁸ cm⁻² s⁻¹ (Bellini et al. 2012). This preliminary result is still insufficiently accurate, but it is not in contradiction with the fact that the Sun is hotter than what is suggested by the present standard solar model. The pep neutrino flux is strongly dependent on the luminosity of the star. Therefore, it is an indirect measurement of the total energy (luminosity) produced in the nuclear region. So it represents a powerful probe of the physics of the nuclear region of the Sun, in parallel to the information introduced in the seismic model (Turck-Chièze & Lopes 2012).

Today, there is some uncertainty about the chemical composition in the Sun's core. In parallel, all the neutrino measurements agree with the predicted values of the solar seismic model but not so well with the standard model (Turck-Chièze & Couvidat 2011), showing some capability of understanding the difference with the standard model if one extracts the electronic density profile. The results obtained so far suggest that in the near future we will be able to probe the physics of the Sun's core using neutrinos.

In this work we have proposed a strategy that allows the use of solar neutrino flux measurements to invert the electron density of the solar plasma at specific locations of the Sun's core. Furthermore, in the near future, it will also be possible to put important constraints on the matter density and on the molecular weight per electron of the solar plasma. The inversion is made based upon the assumption that the future Earth neutrino experiments (e.g., reactor experiments, superbeams, beta beams, and neutrino factories) will allow the precise determination of basic neutrino oscillation parameters.

Today, the neutrino fluxes of ⁷Be-ν, ⁸B-ν, and pp-ν are already well measured and obtained separately. The pep-ν neutrino flux is also measured although with much less precision. It will be interesting to verify if such results hold with the increase of accuracy in the observations within the next few years. The present electron-neutrino survival probability curve is fixed at low neutrino energy by pp-ν, sensitive to pure vacuum oscillations, and by ⁸B-ν, sensitive to vacuum plus MSW oscillations. Therefore it will be relevant to verify if future measurements of ⁷Be-ν and pep-ν can give us some information about the electronic density in two distinct layers of the Sun, namely 0.1 R_☉ (pep-ν) and 0.07 R_☉ (⁷Be-ν). There is a real possibility that Borexino or SNO + experiments will be able to measure accurately these different neutrino fluxes, as well as the neutrino fluxes of some of the CNO cycle neutrino emission reactions.

The Low Energy Neutrino Astrophysics solar neutrino detector is expected to start operating within the next few years. This detector will be able to perform very accurate measurements of the different sources of the solar neutrino spectrum, allowing not only very precise measurements of the solar core plasma, but also to identify possible seasonal neutrino variations.

In conclusion, we have shown that in the near future one may hope to use neutrino spectroscopic measurements to infer the electronic density of the plasma in the core of the Sun. This will be an important and totally independent test of the nuclear region of the Sun properties in parallel with the helioseismic inversions of matter density and sound speed. The proposed method, which could be improved, allows us to infer the properties of the plasma in the very central region of the Sun, which until now have only been explored by some gravity mode detections with GOLF/SOHO. We have shown in Table 1 that the inversion of a different density profile can be done with reasonable quality if the neutrino fluxes are obtained with good accuracy. Of course this method supposes that the description of neutrino oscillations is independent of solar model predictions. If that becomes the case, one will be able to extract from the different solar neutrino detections the electronic density, and matter density profiles with some hints of the CNO composition in the core of the Sun.

This work was supported by grants from "Fundação para a Ciência e Tecnologia" and "Fundação Calouste Gulbenkian." We thank the anonymous referee for the helpful comments that much improved the clarity of the paper.