ASTROPHYSICAL IMPACT OF THE UPDATED ⁹Be(p,α)⁶Li AND ¹⁰B(p,α)⁷Be REACTION RATES AS DEDUCED BY THM

The light elements lithium, beryllium, and boron experience nuclear burning at stellar depths where temperatures of a few 10⁶ K are reached, ranging from $T\approx 2\times {10}^{6}$ K for ⁶Li to $T\;\approx$(4–5) × ${10}^{6}$ K for boron isotopes. Thus, their surface abundances are strongly influenced by the nuclear burnings as well as by the extension of the convective envelope (see, e.g., Deliyannis et al. 2000; Jeffries 2006). Indeed, in convective regions the matter is completely mixed, and if the burning temperature is reached at least at the bottom of the convective envelope, the element abundance decreases on the surface too. In addition, the extension of the convective envelope has an opposite dependence on stellar age, mass, and chemical composition with respect to the one of the stellar temperature. In more detail, while for a fixed point of the stellar interior, the temperature increases with the mass, age, and original helium abundance, for the same variations the external convective region gets shallower; moreover, the temperature decreases, increasing the stellar metallicity while the convective envelope gets deeper (see, e.g., Jeffries et al. 2000; Dotter et al. 2008; Di Criscienzo et al. 2009, and references therein). This leads to a complex dependence of surface abundances on stellar mass, chemical composition, and age.

Pre-main-sequence (pre-MS) stars show deep convective envelopes at the bottom of which light element ignition temperatures can be reached (depending on the mass), while during the following main-sequence (MS) phase convective envelopes are shallower and light element burning is less favorable, although it could still be triggered depending on the element and on the stellar characteristics.

The prediction of surface abundances for lithium, beryllium, and boron in stars still represents an unsolved and challenging task for astrophysics since they strongly depend on the adopted input physics in theoretical models (e.g., nuclear reaction rates, opacity of the stellar matter, equation of state, efficiency of microscopic diffusion, etc., see, e.g., Piau & Turck-Chièze 2002; D'Antona & Montalbán 2003; Montalbán & D'Antona 2006; Tognelli et al. 2012) as well as on the assumed external convection efficiency. The difficulty in calculating these stellar abundances is proven, for example, by the still present discrepancy between theoretical predictions and observational ⁷Li data (the so-called "lithium-problem," see, e.g., Charbonnel et al. 2000; Deliyannis et al. 2000; Pinsonneault et al. 2000; Baraffe & Chabrier 2010; Talon & Charbonnel 2010).

Besides lithium, remarkable efforts have also been made for studying beryllium and boron stellar abundances. Indeed, the comparison between theory and observation for Be and B surface abundances could provide useful additional information. In particular, ⁹Be is burnt at temperatures higher than the ⁷Li ones (T $\approx \;3.5\times {10}^{6}$ K to be compared with $T\approx 2.5\times {10}^{6}$ K for ⁷Li) thus simultaneous observations of ⁷Li and ⁹Be could constrain theoretical models with particular regard to the extension of the convective envelope.

Unluckily, spectroscopic ⁹Be abundance determinations mainly rely on resonance lines located in the near-UV spectral region of cool stars, whose observation from ground-based telescopes is very difficult. The bulk of observations is for metallicities near to the solar one, even if data are available in a much wider range of chemical compositions.

Up to now, ⁹Be can be safely measured only in stars with temperatures higher than about 5000–5200 K (see, e.g., Garcia Lopez et al. 1995; Randich et al. 2007; Smiljanic et al. 2011; Delgado Mena et al. 2012). In agreement with theoretical predictions, these stars do not show any pre-MS ⁹Be surface depletion and thus they are only marginally useful to further constrain convection efficiency during this evolutionary phase (see, e.g., Smiljanic et al. 2011). On the contrary, smaller stars with a deeper and hotter bottom of the convective envelope, are expected to burn ⁹Be in pre-MS, thus representing a more useful benchmark for testing convective models.

The solar photospheric ⁹Be abundance has been largely investigated in the past. In an early analysis (see, e.g., Chmielewski et al. 1975), a relatively large discrepancy between the meteoric (protosolar) and the solar ⁹Be abundance emerged, in which the solar was smaller by a factor 1.8 with respect to the protosolar one. This discrepancy opened a debate about a possible degree of ⁹Be depletion in the Sun, though not predicted by theoretical models, and consequently raised the quest for additional non-standard depletion mechanisms. However, the more recent observations and accurate solar spectra analysis lead to a revised higher solar ⁹Be abundance (Balachandran & Bell 1998; Lodders 2003; Asplund et al. 2005, 2009; Lodders et al. 2009; Lodders 2010), fully consistent with the meteoric one, solving the long-standing problem of the solar ⁹Be.

Concerning the models, theoretical expectations for solar mass stars predict no surface ⁹Be depletion during the pre-MS nor during the MS phase, in agreement with data. For less massive mid-aged stars (with an effective temperature of ${T}_{\mathrm{eff}}\lesssim 5600$ K), observations reveal an increasing ⁹Be depletion while T_eff decreases, in disagreement with standard theoretical calculations (no significant rotation effects, no magnetic fields, etc., see, e.g., Santos et al. 2004; Randich et al. 2007; Delgado Mena et al. 2012). Such a depletion, which has not yet been seen in young clusters (e.g., Smiljanic et al. 2011), seems to indicate a possible cause acting on the MS timescale. The level of discrepancy between predictions and observations, or the difference between surface Be abundance in depleted and undepleted stars is significantly larger than observational uncertainties and even larger than the initial ⁹Be abundances (solar or meteoritic) adopted by different authors. We note that the inclusion in stellar calculations of non-standard processes acting on MS timescales (such as rotation induced mixing, internal waves mixing, internal magnetic fields, etc.) allows us to better reproduce some of the ⁹Be (and ⁷Li) observational features, thus partially alleviating the discrepancy (see, e.g., Montalbán & Schatzman 2000; Talon & Charbonnel 2010, and references therein).

Similarly to the ⁹Be, solar boron abundance showed a large discrepancy between the meteoric and the photospheric determinations. However, as for ⁹Be, more recent analysis of the solar spectrum have restored the agreement with the meteoric value (Asplund et al. 2009; Lodders 2010).

The spectroscopic measurements of surface ${}^{\mathrm{10,11}}{\rm{B}}$ isotopes abundance are even more problematic than the ⁹Be one. Boron can only be measured from transitions that fall mainly in the ultraviolet, out of the Earth's UV transmission limit. Its abundance can be obtained from neutral boron in cool stars and from ionized boron in high mass stars (spectral types A and B). Moreover, for disk metallicity stars, the neutral boron transition region is affected by strong blending problems (see, e.g., Cunha 2010; Kaufer et al. 2010). In the observed stars, the ratio ¹¹B/¹⁰B seems to be of the order of four, in agreement with solar values and meteorite results, even if it is very difficult to spectroscopically discriminate among the boron isotopes (see, e.g., Chaussidon & Robert 1995; Lambert et al. 1998; Rebull et al. 1998; Proffitt & Quigley 1999; Prantzos 2012).

The ${}^{\mathrm{10,11}}$B-burning temperature is the highest (about (4–5) $\times {10}^{6}$ K) among light elements, thus standard stellar models predict that only masses lower than about 0.5 ${M}_{\odot }$ show surface boron depletion. Cool stars with masses near the solar one with no ⁹Be depletion appear boron undepleted too, in agreement with the results for the Sun (Boesgaard et al. 2005; Lodders et al. 2009). To observe ${}^{\mathrm{10,11}}{\rm{B}}$ depletion, lower masses are needed. However, some F and G MS stars with ⁹Be depletion seem to show slight B depletion as well, in disagreement with standard stellar model predictions (see, e.g., Boesgaard et al. 2005), but the observational difficulties prevent any firm conclusions.

In the present work, due to the discussed observational problems and the difficulty to consistently reproduce the MS observational data, we decided to restrict our analysis of ⁹Be and ¹⁰B abundances to the pre-MS evolution of low-mass stars (i.e., M $\lesssim \;1.0$ ${M}_{\odot }$).

For a better understanding of this complex scenario, stellar models need to be computed by using input physics that are as accurate as possible. Indeed, it has been shown in several works that pre-MS evolutionary models and, in particular, the surface abundance of light elements (i.e., Li, Be, B) are quite sensitive to both the input physics (i.e., outer boundary conditions, convection, equation of state, reaction rates, opacity, etc.) and/or chemical element abundances (i.e., initial abundance of deuterium, helium, metals, etc.) adopted in stellar models (see, e.g., Burrows et al. 2001; Piau & Turck-Chièze 2002; Burke et al. 2004; Tognelli et al. 2012, 2015b). In this paper, we only focus on the burning nuclear reaction cross sections for ⁹Be and ¹⁰B, taking advantage of the recent measurements performed via the indirect Trojan Horse Method (${\mathtt{THM}}$; Spitaleri et al. 2011; Tribble et al. 2014). We assumed that the other input physics along with the chemical composition are fixed, thus we have performed a differential analysis of the impact of the updated ⁹Be and ¹⁰B reaction rates on the ⁹Be and ¹⁰B surface abundances in pre-MS stars for different masses.

The nuclear burning reaction rates, in fact, constitute key ingredients for light element surface abundance predictions, and they require particular effort in measuring their values in terrestrial laboratories. In the case of charged-particle induced reactions and because, for quiescent burning, the corresponding Gamow peak usually lies in the keV's-regime, direct measurements in laboratories need to be performed at such energies or close to these, as much as possible. However, the presence of the Coulomb barrier among the interacting nuclei cause an exponential drop of the cross-section to nano-or-picobarn values in correspondence of the astrophysical relevant energies, thus often making their measurements impossible and leaving extrapolations as the most common means of determining their values. Usually, extrapolation procedures are performed on the astrophysical S(E)-factor, defined as

$$\begin{eqnarray}&&\ S(E)=E\times \sigma (E)\times \mathrm{exp}(2\pi \eta )\end{eqnarray} \tag{ 1 }$$

for which a more smooth variation with the energy is expected for non-resonant reactions (Rolfs & Rodney 1988). However, it has been shown that experimental S(E)-factor determinations suffer, at astrophysical energies, the presence of electron screening effects for which the "bare-nucleus" cross section is altered by an empirical enhancing factor f_enh given in the laboratory by (Assenbaum et al. 1987; Strieder et al. 2001)

$$\begin{eqnarray}&&{f}_{\mathrm{enh}}=\displaystyle \frac{S{(E)}_{\mathrm{sh}}}{S{(E)}_{{\rm{b}}}}=\displaystyle \frac{\sigma {(E)}_{\mathrm{sh}}}{\sigma {(E)}_{{\rm{b}}}}\approx \mathrm{exp}\left(\pi \eta \displaystyle \frac{{U}_{{\rm{e}}}}{E}\right),\end{eqnarray} \tag{ 2 }$$

where ${\sigma }_{\mathrm{sh}}$ is the shielded nuclear cross section measured in the laboratory, ${\sigma }_{{\rm{b}}}$ is the bare-nucleus cross section, and U_e is the electron screening potential measured in the laboratory.

The direct measurement of the ⁹Be(p,α)⁶Li is discussed in Sierk & Tombrello (1973), where the authors report the measurement of both (p,α) and (p,d) channel from 400 keV down to ∼100 keV, giving a total value of S(0) = 35${}_{-15}^{+45}$ MeV b. From their extrapolation at lower energies, one can obtain a value of S(0) ∼ 17 MeV b for the ⁹Be(p,α)⁶Li reaction. Additionally, in the work of Zahnow et al. (1997), the authors explored the low-energy region down to ∼16 keV extracting the value of U_e = 900 ± 50 eV for the electron screening potential.

In the case of the ¹⁰B(p,α)⁷Be reaction, the low-energy region (i.e., below ∼100 keV), is dominated by the 8.699 MeV ¹¹C excited level intervening as an s-wave resonance at about 10 keV in the ¹⁰B–p center of mass system, as discussed in detail in Angulo et al. (1993). By describing their experimental data, assuming the same enhancing electron screening potential of 430 ± 50 eV measured by the ¹¹B–p interaction, the authors extrapolated a value of S(10 keV) = 2870 ± 500 MeV b.

In order to complement the already available S(E)-factor direct measurements, and to bypass extrapolation procedures at low-energies (i.e., below ∼100 keV's), the previous reactions have additionally been studied by means of the Trojan Horse Method (THM). The method, for which reviews can be found in Spitaleri et al. (2011) and Tribble et al. (2014), allows the experimentalist to bypass the typical difficulties of direct approaches, such as the presence of the Coulomb barrier in the entrance channel or the presence of the electron screening phenomena. Due to its theoretical formalism, ${\mathtt{THM}}$ data need to be normalized to high-energy direct data in which either Coulomb penetrability or electron screening effects are negligible. Thus, ${\mathtt{THM}}$ acts as a complementary experimental technique for nuclear astrophysics. Thanks to its development in the last ∼25 years, ${\mathtt{THM}}$ have been used in investigating the above mentioned reactions of interest here.

The ⁹Be(p,α)⁶Li reaction has been studied in the works of Romano et al. (2006) and of Wen et al. (2008) by properly selecting the quasi-free (QF) contribution of the ²H(⁹Be, ${\alpha }^{6}$ Li)n reaction in two different experiments. The ${\mathtt{THM}}$ measurements lead to a zero-energy S(E)-factor of S(0) = 21.0 ± 0.8 (MeV b) and an electron screening potential of U_e = 676 ± 86 eV.

The ¹⁰B(p,α)⁷Be has been studied in the work of Lamia et al. (2007) and recently in Spitaleri et al. (2014), in order to measure the corresponding S(E)-factor value by means of ${\mathtt{THM}}$ applied to the QF reaction ²H(¹⁰B, ${\alpha }^{7}$ Be)n. The investigation allowed us to measure the S(E)-factor in correspondence with the Gamow energy region in which the 8.701 MeV level of ¹¹C intervenes as an l = 0 resonance at ∼10 keV dominating the whole excitation function from ∼100 keV's down to zero. The S(E)-factor values measured are S(10 keV) = 3127 ± 583 (MeV b) and U_e = 240 ± 200 eV, with this last value being strongly affected by the still present uncertainties on direct measurements at which ${\mathtt{THM}}$ data have been normalized (see the discussion in Spitaleri et al. 2014).

In this paper, the ⁹Be and ¹⁰B (p,α) burning reaction rates have been evaluated by means of the ${\mathtt{THM}}$ cross-section measurements listed above. These were then compared with the widely used NACRE reaction rates (Angulo et al. 1999) and with the more recent NACREII compilation (Xu et al. 2013). The impact of the updated ⁹Be and ¹⁰B burning reaction rates on the ⁹Be and ¹⁰B surface abundances in pre-MS stars is also discussed.

The paper is organized in the following way. In Section 2, we briefly recall the main characteristics of the ${\mathtt{THM}}$ method, then we discuss the results of the ⁹Be and ¹⁰B cross-section measurements in Section 3. In Section 4, we derive an analytical expression of the quoted reaction rates to be directly incorporated in the stellar evolutionary code. In Section 5, we present the stellar models and discuss the effects on stellar surface abundances of the adoption of new ⁹Be and ¹⁰B ${\mathtt{THM}}$ reaction rates. We summarize the main results in Section 6.

To by-pass extrapolation procedure and systematic uncertainties due to electron screening effects, the indirect method of the Trojan Horse (Baur 1986; Spitaleri 1990; Cherubini et al. 1996; Spitaleri et al. 1999, 2011; Tribble et al. 2014) has been developed and largely used in the past to shed light on different open issues concerning both pure nuclear physics and nuclear astrophysics.

Indeed, ${\mathtt{THM}}$ allows experimentalists to measure the astrophysically relevant cross sections in correspondence, or very close, to the so-called Gamow peak without experiencing the lowering of the signal-to-noise ratio due to the presence of the Coulomb barrier between the interacting particles.

By referring to the pole diagram of Figure 1, ${\mathtt{THM}}$ selects the QF contribution of the $a+A\to c+C+s$ reaction, where the Trojan nucleus A is chosen because of its large amplitude for the $A=x\oplus s$ cluster configuration. The $a+A$ interaction occurs at energies well above the Coulomb barrier, for extracting the bare nucleus cross section of the astrophysically relevant reaction $a+x\to c+C$ at low-energies, without the action of both Coulomb suppression or electron screening effects (see Spitaleri et al. 2011; Tribble et al. 2014, for more details). In the pole diagram of Figure 1, particle a will then interact only with the cluster x of the TH nucleus A, while s will act as spectator to the $A(x,c)C$ virtual reaction. By invoking the more simple plane wave impulse approximation, the cross section of the $A(a,{cC})s$ reaction can be factorized into two terms corresponding to the poles of Figure 1 via the formula (Spitaleri et al. 2011; Tribble et al. 2014)

$$\begin{eqnarray}&&{\displaystyle \frac{{d}^{3}\sigma }{{{dE}}_{c}d{{\rm{\Omega }}}_{c}d{{\rm{\Omega }}}_{C}}\propto \mathrm{KF}\cdot {| {\rm{\Phi }}({{\rm{p}}}_{\mathrm{sx}})| }^{2}\cdot \displaystyle \frac{d\sigma }{d{\rm{\Omega }}}|}_{\mathrm{cm}}^{\mathrm{HOES}},\end{eqnarray} \tag{ 3 }$$

where

• 1.  
  KF represents the kinematical factor, depending on masses, momenta, and angles of the outgoing particles, that takes into account the final state phase space factor;
• 2.  
  $| {\rm{\Phi }}({{\rm{p}}}_{\mathrm{xs}}){| }^{2}$ is given by the Fourier transform of the radial wave function describing the $x-s$ inter-cluster motion, usually in terms of Hänkel, Eckart, or Hulthén functions depending on the $x-s$ system;
• 3.  
  $d\sigma /d{\rm{\Omega }}{| }_{\mathrm{cm}}^{\mathrm{HOES}}$ is the half-off-energy-shell (HOES) differential cross section for the two-body reaction at the center of mass energy ${E}_{\mathrm{cm}}$ = ${E}_{{cC}}$–$Q$, where Q represents the Q-value of the virtual $A(x,c)C$ reaction, while ${{\rm{E}}}_{{cC}}$ represents the relative $c-C$ energy measured in the laboratory.

The introduction of the penetration factor through the Coulomb barrier, described in terms of the regular and irregular Coulomb functions, and the normalization to the high-energy direct measurements, make the extraction of the bare-nucleus S(E)-factor possible by following Equation (1), where the ${\mathtt{THM}}$ cross section represents the bare nucleus one.

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In recent years, THM data allowed for a better understanding of different astrophysical problems, such as the study of light element burning reactions (Pizzone et al. 2005, 2014; Lamia et al. 2008, 2012b, 2012c, 2013; Tumino et al. 2011a, 2011b, 2014; Grineviciute et al. 2015), CNO reactions (see La Cognata et al. 2010, 2011; Sergi et al. 2010; Palmerini et al. 2013), and removing/producing neutron reactions (Lamia et al. 2008; Gulino et al. 2010, 2013; La Cognata et al. 2012).

The light elements lithium, beryllium, and boron (p,α) burning reactions have been largely investigated by ${\mathtt{THM}}$ in order to access the Gamow energy, known by direct measurements only via extrapolation procedures. Here, the main results obtained are briefly reported.

3.1. The ⁹Be(p,α)⁶Li Reaction

The first ${\mathtt{THM}}$ measurement of the ⁹Be(p,α)⁶Li S(E)-factor has been performed in Romano et al. (2006), by properly selecting the QF-contribution of the three-body reaction ²H(⁹Be, ${\alpha }^{6}$Li)n with a devoted experiment performed at INFN-LNS of Catania, in which a 22 MeV ⁹Be beam hit a 190 μg cm⁻² thick CD₂ target. In such a framework, deuteron ²H has been used as the "TH-nucleus" because of its obvious p–n structure and the relative p–n motion, mainly occurring in s-wave (Lamia et al. 2012a). The transferred proton p and neutron n represent the participant and the spectator, respectively, in agreement with the sketch of Figure 1. The detection setup consisted of a standard ΔE–E telescope, with a position sensitive silicon detector (PSD) as E-stadium, working in logic coincidence with a further PSD detector, placed on the opposite side with respect to the beam axis, as discussed in Romano et al. (2006). The measurement of Romano et al. (2006) allowed for the first time the extraction of the angular distributions at different energies and the investigation of the low-lying resonance at ∼250 keV. Thanks to both experimental and theoretical improvements concerning the method, a second experiment has been performed at the China Institute of Atomic Energy, Beijing, China, and the results reported in Wen et al. (2008), of the detailed analysis of the QF-mechanism selection and background discrimination are discussed. The experimental ${\mathtt{THM}}$ data have then been normalized to the direct ones available in the NACRE compilation, thus allowing for a polynomial fit from ∼100 keV's down to zero as (see Wen et al. 2008)

$$\begin{eqnarray}S(E) & = & 21.0-92.4\times E+4669\\ & & \times {E}^{2}-4.4\times {10}^{4}\times {E}^{3}+\\ & & +2.2\times {10}^{5}\times {E}^{4}-3.8\times {10}^{5}\times {E}^{5},\end{eqnarray} \tag{ 4 }$$

where E represents the center-of-mass energy in the ⁹Be–p system, expressed in MeV. Equation (4) leads to the values of S(0) = 21.0 ± 0.8 (MeV b) and U_e = 676 ± 86 eV, respectively, for the S(0) and U_e values. Since the measured ${\mathtt{THM}}$ zero-energy S(E)-factor deviates from the low-energy extrapolation of NACRE by a factor of ∼1.23, the corresponding reaction rate evaluation needs to be performed to study its impact on astrophysical scenarios where beryllium is destroyed, such as the pre-MS evolution of low-mass stars.

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3.2. The ¹⁰B(p,α)⁷Be reaction

The study of the ¹⁰B(p,α)⁷Be reaction is of importance in nuclear astrophysics because of the difficulty in measuring the corresponding S(E)-factor at Gamow energies and the astrophysical community's interest in the production/destroying processes of the unstable ⁷Be isotope (see, for instance, Simonucci et al. 2013). Indeed, the 8.701 MeV excited level of ¹¹C dominates the S(E)-factor trend at low-energies, being this an s-wave resonance in the ¹⁰B–p system at 10 keV. However, due to the action of both Coulomb barrier and electron screening effects, the S(10 keV)-factor was only extrapolated from the high energy measurement (Angulo et al. 1999). To by-pass extrapolations, a first ${\mathtt{THM}}$ measurement has been discussed in Lamia et al. (2007), where the dominance of the QF reaction mechanism intervening in the ²H(¹⁰B, α⁷Be)n has been constrained via the study of the experimental momentum distribution. The work reports on the ²H(¹⁰B, α⁷Be)n experiment performed at USP (University of Sao Paulo, Brazil) by means of a 24 MeV ¹⁰B beam hitting a 190 μg cm⁻² thick CD₂ target. The adopted detection setup allowed for ⁷Be identification and alpha-particle detection, as described in Lamia et al. (2007). The experiment has made it possible to detect the population of the ∼10 keV resonance intervening in the ¹⁰B–p center-of-mass system, although the limited energy resolution (of about ∼60 keV) did not allow us to get any definitive results. A further experimental run has been performed at INFN-LNS (INFN-Laboratori Nazionali del Sud, Catania, Italy) with the aim of enhancing the energy resolution and constraining the S(10 keV)-factor. Thanks to the available CAMERA2000 scattering chamber, a very extreme angular resolution has been obtained (i.e., of about 01), which is a key requirement for ${\mathtt{THM}}$ purposes as deeply discussed in the paper of Spitaleri et al. (2011). The INFN-LNS measurement allowed us, at the end, to reach an overall energy resolution of ∼16 keV. The complete analysis of the experiment, together with a detailed DWBA calculation for the momentum distribution made with the FRESCO code and the proposed R-matrix calculation, are deeply discussed in Spitaleri et al. (2014). The ${\mathtt{THM}}$ experimental data have been folded for the experimental resolution and normalized to the available direct data of Angulo et al. (1993) in the center of mass energy region ranging from 60 keV up to 100 keV. Thus, following the procedure described in Spitaleri et al. (2014), the experimental data have been fitted by means of a standard Breit–Wigner function centered at 10 keV and with a total width of Γ = 15 keV (as reported in Angulo et al. 1993) superimposed on a not-resonant contribution, thus leading to the value of S(10 keV) = 3127 ± 583 (MeV b) in correspondence with the resonance energy. The result of such an analysis is summarized in Figure 3, where the empty triangles are the direct data of Angulo et al. (1993), while the full black line represents the ${\mathtt{THM}}$ bare-nucleus S(E)-factor at "infinite resolution." Its uncertainties are represented by the blue area. By using this measured ${\mathtt{THM}}$ S(E)-factor, the low-energy direct data (i.e., ${E}_{\mathrm{cm}}\lt 40$ keV) have been fitted by means of the enhancing factor given in Equation 2 with an electron screening potential of U_e = 240 ± 200 eV. The result of such a fit is represented by the red-line of Figure 3 while the gray area marks the corresponding uncertainties.

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Starting from the ${\mathtt{THM}}$ experiment discussed above, the reaction rate at astrophysical energies has been deduced via the standard formula given in Rolfs & Rodney (1988)

$$\begin{eqnarray}&&{N}_{A}\langle \sigma v\rangle ={\left(\displaystyle \frac{8}{\pi \mu }\right)}^{\frac{1}{2}}\displaystyle \frac{{N}_{A}}{{{kT}}_{9}^{\frac{3}{2}}}{\displaystyle \int }_{0}^{\infty }{S}_{{\rm{b}}}(E){e}^{-2\pi \eta -\frac{E}{{kT}}}{dE}\\ &&({\mathrm{cm}}^{3}\;{\mathrm{mol}}^{-1}\;{{\rm{s}}}^{-1}),\end{eqnarray} \tag{ 5 }$$

where the temperature T₉ is expressed in units of 10⁹ K and the center of mass energy E in MeV. In Equation (5), the bare-nucleus S(E)-factor, S_b(E), is the one measured at the Gamow energies via the ${\mathtt{THM}}$ and discussed in the previous sections. The integration has been performed through the energy intervals covered by each experiment, i.e., from ∼200 keV down to about ∼10 keV, depending on the involved reaction. Thus, the reaction rate has been fitted via the following formula:

$$\begin{eqnarray}{N}_{A}\langle \sigma v\rangle & = & \mathrm{exp}\left[{a}_{1}+\displaystyle \frac{{a}_{2}}{{T}_{9}}+\displaystyle \frac{{a}_{3}}{{T}_{9}^{1/3}}+{a}_{4}\times {T}_{9}^{1/3}\right.\\ & & \left.+{a}_{5}\times {T}_{9}+{a}_{6}\times {T}_{9}^{5/3}+{a}_{7}\times \mathrm{ln}{T}_{9}\right],\end{eqnarray} \tag{ 6 }$$

where the a_i coefficients have been left as free parameters for the two ⁹Be(p,α)⁶Li and ¹⁰B(p,α)⁷Be reactions. In Equation (6), the temperature T₉ is expressed in units of 10⁹ K and the final reaction rate is given in (cm³ mol⁻¹ s⁻¹). The resulting a_i coefficients are listed in Table 1. Figure 4 reports the discrepancy in percentage between the calculated ${\mathtt{THM}}$ reaction rate and its parametrization via Equation (6) for the ⁹Be(p,α)⁶Li case, leading to a maximum variation of ±0.6% at temperatures ${T}_{9}\lt 0.2$ thus confirming the goodness of the adopted procedure.

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The deduced ${\mathtt{THM}}$ reaction rates, together with the corresponding uncertainties, have been compared to the ones given in the literature with the aim of evaluating the deviation caused by the present ${\mathtt{THM}}$ investigation. In particular, we compared our results with the NACRE compilation, largely used for astrophysical purposes, and with the more recent NACREII compilation by Xu et al. (2013), in which a comprehensive Distorted Wave Born Approximation (DWBA) calculation has been performed by including all of the measurements "post-NACRE." Here, we are interested in evaluating the impact of nuclear inputs in the astrophysical scenario by only using the ${\mathtt{THM}}$ bare-nucleus measurements at astrophysical energies.

Figure 5 reports the THM-to-NACRE ratio, i.e.,

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{A}{\langle \sigma v\rangle }_{{\mathtt{THM}}}}{{N}_{A}{\langle \sigma v\rangle }_{{\mathtt{NACRE}}}}\end{eqnarray} \tag{ 7 }$$

for the ⁹Be(p,α)⁶Li case. The ${\mathtt{THM}}$ reaction rate has been evaluated by considering the S(E)_b-factor given by Equation (4) while the Angulo et al. (1999) compilation adopts the low-energy extrapolation leading to S(0) = 17${}_{-7}^{+25}$ MeV b. In Figure 5, the blue line is the ratio between the adopted ${\mathtt{THM}}$ and NACRE reaction rates while the filled red area refers to the range values allowed by the experimental errors on S(E)-measurements given in Wen et al. (2008) and shown in Figure 2. From the present reaction rate determination, a strong reduction of the uncertainties on the reaction rate is clearly visible. In particular, at temperatures lower than 10⁸ K, the ${\mathtt{THM}}$ allows us to reduced the reaction rate uncertainties to about ∼20%, while the NACRE one is given with an uncertainty of ∼70%–90% at the same temperatures. By also comparing the ${\mathtt{THM}}$ reaction rate extracted here with that given in the NACREII compilation, a reduction of the uncertainty region is clearly visible from Figure 6 while the ratio between the adopted values does not show any significant deviation. This is not unexpected since in the comprehensive fit of Xu et al. (2013) and the ${\mathtt{THM}}$ data of Wen et al. (2008) have been also included, thus dominating the low-energies S(E)-factor data set.

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By following the same procedure, the ${\mathtt{THM}}$ ¹⁰B(p,α)⁷Be reaction rate has been evaluated and the corresponding coefficients reported in the right column of Table 1. Also, in this case, small variations (∼±2%) have been found when comparing the obtained reaction rate with its parametrization of Equation (6) at temperatures of ${T}_{9}\lt 0.2$. Figure 7 reports the ratio (blue line) between the ${\mathtt{THM}}$ and NACRE reaction rate, together with the corresponding uncertainties given by the available data, while Figure 8 reports the comparison between the ${\mathtt{THM}}$ and the NACREII reaction rate. A careful examination of both figures suggests a reduction of uncertainties at lower temperatures, i.e., close to the Gamow peak typical of quiescent boron burning, while at higher temperatures the ${\mathtt{THM}}$ reaction rate is inevitably affected by the experimental uncertainties discussed in Spitaleri et al. (2014). However, besides the small deviations among the adopted values, the NACREII compilation reports the value of S(0.001) = 1.3${}_{-0.9}^{+0.2}\;\times \;$10³ MeV b, while the ${\mathtt{THM}}$ data of Spitaleri et al. (2014) suggest S(0.001) = 1405 ± 450 MeV b.

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The described change in the ⁹Be and ¹⁰B proton burning rates due to the ${\mathtt{THM}}$ cross-section measurements is significant; thus it is worthwhile to evaluate their effects on stellar evolutionary models. The ⁹Be and ¹⁰B burning reactions are negligible regarding the stellar energetics, thus their effects on stellar structures are unimportant; however, a change in the burning rates sensibly affects Be and B surface abundances.

For the reasons discussed in the Introduction, we restrict our analysis on low-mass pre-MS stars; ⁹Be and ¹⁰B surface abundances obtained with the present rates are compared with the ones resulting from the adoption of the still widely used NACRE compilation.

5.1. Stellar Evolution Models

5.2. ⁹Be and ¹⁰B Surface Abundances

As discussed in the Introduction, due to their low burning temperatures, ⁹Be and ¹⁰B are completely destroyed in hot stellar interiors, while their surface abundance depends on the temperature reached at the bottom of the external convective envelope. In order to understand the pre-MS surface abundance behavior, one has to remember that during this phase, the gravitational contraction leads to an increase of the stellar temperature until, at the pre-MS end, nuclear reactions provide the energy necessary to counterbalance the radiative losses at the stellar surface, stabilizing the structure on nuclear timescales.

At the first pre-MS stages stars are fully convective and the surface matter is continuously mixed with the stellar interior one. When the central temperature is high enough, the light elements start to be destroyed, thus changing their surface abundances because of the action of convective phenomena. However, as the central temperature increases, depending on the stellar mass, a radiative core develops while the envelope remains convective. Adopting a solar-like chemical composition (i.e., [Fe/H] = $+0.0$), for stellar masses lower than about 0.3 ${M}_{\odot }$, such a radiative core never forms (or it only temporarily appears, as in the 0.3 ${M}_{\odot }$ model) while, for higher masses, the larger the mass is, the lower the age is at which it is formed. If a radiative core is present, the surface matter can no longer reach the hottest central regions of the star, though it can be dragged down to a depth given by the extension of the convective envelope. Thus, the external matter experiences the maximum temperature (as well as the highest light element burning efficiency) at the bottom of the convective envelope, whose depth depends on both the stellar mass, age, and chemical composition. From this general discussion it emerges that the phase of efficient light element burning is strictly correlated with the formation of a radiative core in the star and on its temporal evolution.

Figure 9 shows the temporal evolution of the temperature at the bottom of the convective envelope (T${}_{{\rm{b}}.{\rm{c}}.{\rm{e}}.}$) for different masses, for stellar evolutionary models with [Fe/H] = $+0.0$, from the early pre-MS evolution up to the beginning of the MS (marked by an open diamond). The minimum mass plotted in the figure (i.e., 0.08 ${M}_{\odot }$) approximately corresponds to the minimum mass (M_min) that reaches a central temperature high enough to burn ⁹Be (${M}_{\mathrm{min}}\approx 0.07$ ${M}_{\odot }$) and ¹⁰B (${M}_{\mathrm{min}}\approx 0.08$ ${M}_{\odot }$). For the selected chemical composition, masses larger than about 0.7 ${M}_{\odot }$ do not deplete ⁹Be (and consequently they do not destroy ¹⁰B) during the pre-MS and thus they are not shown. In Figure 9, we also plotted a rough mean temperature estimate at which the ⁹Be and ¹⁰B begin to be destroyed, useful to make clearer the following discussion. Notice that ⁹Be and ¹⁰B ignition temperatures are different in the upper ($T{(}^{9}\mathrm{Be})\approx 3.2\times {10}^{6}$ K and $T{(}^{10}{\rm{B}})\approx 3.6\times {10}^{6}$ K) and lower ($T{(}^{9}\mathrm{Be})\approx 3.6\times {10}^{6}$ K and $T{(}^{10}{\rm{B}})\approx 4.1\times {10}^{6}$ K) panel. This happens because the rate at which an element is destroyed in stars depends not only on the temperature but also (even if weakly) on the density, which increases as the mass decreases, with a resulting decrease of the burning temperature.

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Referring to the the upper panel of Figure 9, models with $M\lt 0.3$ ${M}_{\odot }$ remain fully convective in pre-MS and MS. In these cases, ${T}_{{\rm{b}}.{\rm{c}}.{\rm{e}}.}$ always corresponds to the central temperature which progressively increases as the star approaches the MS, where its maximum value is reached. Thus, the light element burning gets more and more efficient as the star evolves. The M = 0.3 ${M}_{\odot }$ is, approximately, the transition mass between stars that are always fully convective during the pre-MS and MS, and masses that develop a radiative core that continuously grows (in mass) in the pre-MS. In the 0.3 ${M}_{\odot }$ model, a temporarily radiative core develops in the pre-MS, but it disappears before the star reaches its ZAMS location.¹⁰ Notice that the ${T}_{{\rm{b}}.{\rm{c}}.{\rm{e}}.}$ of the 0.3 ${M}_{\odot }$ model continues to increase even when the radiative core forms, but at a much reduced rate.

The models in the bottom panel of Figure 9 (i.e., $M\geqslant 0.4$ ${M}_{\odot }$) form a radiative core during the pre-MS, evidenced by the first plateau in $\mathrm{log}\;{T}_{{\rm{b}}.{\rm{c}}.{\rm{e}}.}$. Even in these cases, during the first total convective phase ${T}_{{\rm{b}}.{\rm{c}}.{\rm{e}}.}$, which coincides with the central temperature, increases following the stellar contraction. However, when the radiative core develops, it pushes the bottom of the convective envelope toward more and more external regions. In a first phase, the decrease of ${T}_{{\rm{b}}.{\rm{c}}.{\rm{e}}.}$ caused by the growth (in mass) of the radiative core is counterbalanced by the temperature increase due to the stellar contraction and ${T}_{{\rm{b}}.{\rm{c}}.{\rm{e}}.}$ almost stabilizes. However, as the star evolves toward the zero-age main sequence (ZAMS), the contraction rate slows down and ${T}_{{\rm{b}}.{\rm{c}}.{\rm{e}}.}$ decreases, reaching its minimum value in ZAMS. Note that increasing the stellar mass the radiative core develops at early ages and reaches more external regions, thus lowering the maximum ${T}_{{\rm{b}}.{\rm{c}}.{\rm{e}}.}$ value reached during the pre-MS evolution (and also ${T}_{{\rm{b}}.{\rm{c}}.{\rm{e}}.}$ in ZAMS) and reducing the light element burning efficiency. For example, while for M = 0.4 ${M}_{\odot }$ in ZAMS the convective envelope contains about 40% of the stellar mass for M = 0.7 ${M}_{\odot }$ it reduces to about 10%.

From Figure 9, it is evident that, depending on the mass, the formation of a radiative core, pushing the convective envelope toward more external regions, might lead to a temperature at the bottom of the convective envelope that is lower than that needed to (efficiently) destroy the considered element. Regarding the masses for which this happens, the larger the mass is, the more rapid the shift of the convective envelope toward the surface is, and the earlier the temperature at the bottom of the convective envelope decreases below the burning threshold. Referring to Figure 9, one can see that while for $M\leqslant 0.4$ ${M}_{\odot }$, ⁹Be is destroyed both in the pre-MS and in the MS, for masses between 0.5 and 0.7 ${M}_{\odot }$ ⁹Be is destroyed only in the pre-MS. Moreover, the temporal duration of the burning phase steeply decreases as the mass increases. For ¹⁰B, the transition mass is much sharper; indeed, while for $M\lesssim 0.4$ ${M}_{\odot },$ ¹⁰B is destroyed both in the pre-MS and the MS, for $M\gt 0.5$, it is never destroyed.

Figure 10 shows the temporal evolution of surface ⁹Be (top panel) and ¹⁰B abundance (bottom panel), normalized to one. The models have been computed adopting both the ${\mathtt{THM}}$ (solid line) and the NACRE (dashed line) reaction rates for the ⁹Be(p,α)⁶Li and ¹⁰B(p,α)⁷Be reactions. As discussed in Section 4, the ${\mathtt{THM}}$ rate for ⁹Be burning is about 25% larger than the NACRE one (at the temperature of interest, see Figure 5), thus leading to a faster ⁹Be destruction in the ${\mathtt{THM}}$ models. In addition, the adoption of a larger reaction rate at a given temperature reduces the age at which ⁹Be depletion becomes efficient. Thus, at the same age, models with the ${\mathtt{THM}}$ rate show a lower ⁹Be surface abundance with respect to models with the NACRE one.

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The differences in the predicted surface abundances between the ${\mathtt{THM}}$ and NACRE models are significant if surface ⁹Be is efficiently destroyed (i.e., for $M\lesssim 0.5$ ${M}_{\odot }$), while at larger masses they are quite negligible. The effect of the update of the ⁹Be burning rate is also significant for $M\leqslant 0.3$ ${M}_{\odot }$ even if not clearly visible in the figure because of the steep decrease in the surface abundance (almost vertical in the plot).

It is worth noticing that in stellar models ⁹Be is destroyed following two channels: (1) ⁹Be(p,α)⁶Li (R₁, the rate analyzed here) and (2) ⁹Be(p, $2\alpha$)²H (R₂). The ratio between the ⁹Be(p,α)⁶Li and ⁹Be(p, $2\alpha$)²H reaction rates in stellar conditions at the temperature of interest is ${R}_{1}/{R}_{2}\approx 1.2$, thus the ⁹Be(p, $2\alpha$)²H contribution to beryllium destruction is not negligible. As a consequence, the reaction rate variation for the first channel affects the final beryllium abundance for a factor that is given by the reaction rate change (about 25%) multiplied by the probability that the ⁹Be burning occurs in that channel, i.e., $25\%\times {R}_{1}/({R}_{1}+{R}_{2})\approx 14\%$.

The predicted temporal evolution of the ¹⁰B surface abundance for the selected stellar masses is shown in the bottom panel of Figure 10. In this case the ${\mathtt{THM}}$ ¹⁰B burning rate is smaller (by about 25%), at the temperatures of interest, with respect to the NACRE one, thus leading to a less pronounced ¹⁰B destruction in the ${\mathtt{THM}}$ models and to a larger surface ¹⁰B abundance at a fixed age. Because of the larger ¹⁰B burning temperature with respect to the ⁹Be one, the effect of changing the reaction rate is relevant only for masses $M\lesssim 0.4$ ${M}_{\odot }$. Also notice that the typical timescale at fixed mass in which ¹⁰B is destroyed is longer than that corresponding to ⁹Be.

We mention that for ages typical of the MS evolution, microscopic diffusion might become a dominant effect in determining the surface abundance of light elements. As an example, such an effect is visible in Figure 10, for $\mathrm{log}\;t\gtrsim 8.5$, in the case of M = 0.5 ${M}_{\odot }$ for ⁹Be and of M = 0.4 ${M}_{\odot }$ for ¹⁰B. However, the efficiency of diffusion is independent of the analyzed burning reaction rates, and being only interested in the pure effect of nuclear burnings, we only mention it without further discussions.

The panels of Figure 10 are useful for understanding the impact of the updated reaction rates on the theoretical expectations. Additionally, for an easier comparison between the theoretical predictions made here and the observational data, we have also reported the predicted ⁹Be and ¹⁰B abundances for several stellar masses as a function of the effective temperature, T_eff, thus defining the A(X) versus the T_eff plane. Since this plot is often used when studying galactic open clusters for which the comparison between observations and models is made at a fixed age (i.e., that of the cluster), three different ages, suitable for ⁹Be and ¹⁰B depletion timescales, have been selected in the figure, namely 50, 100. and 300 Myr for ⁹Be depletion and 100, 200, and 500 Myr for the analysis of ¹⁰B.

The panels in Figure 11 show, for each selected age, the comparison between the surface logarithmic abundances obtained using the ${\mathtt{THM}}$ and NACRE reaction rates as a function of the effective temperature, for [Fe/H] = +0.0. Each curve represents the abundance isochrone, i.e., the locus of models with the same age but different masses, in the range [0.06, 0.8] ${M}_{\odot }$. The surface abundances are shown as a function of T_eff, which does not change with the variation of ⁹Be and ¹⁰B reaction rates. To clearly show the impact of the new reaction rates on the surface abundances, Table 2 lists the effective temperature, the corresponding mass, and the surface depletion levels, namely,

$$\begin{eqnarray*}{\rm{\Delta }}A({}^{9}\mathrm{Be}) & \equiv & A({}^{9}\mathrm{Be})-{A}_{\mathrm{ini}}({}^{9}\mathrm{Be})\equiv \mathrm{log}\displaystyle \frac{N({}^{9}\mathrm{Be})}{{N}_{\mathrm{ini}}({}^{9}\mathrm{Be})}\\ {\rm{\Delta }}A({}^{10}{\rm{B}}) & \equiv & A({}^{10}{\rm{B}})-{A}_{\mathrm{ini}}({}^{10}{\rm{B}})\equiv \mathrm{log}\displaystyle \frac{N({}^{10}{\rm{B}})}{{N}_{\mathrm{ini}}({}^{10}{\rm{B}})}.\end{eqnarray*}$$

These have been evaluated for some of the models plotted in Figure 11 by means of both ${\mathtt{THM}}$ and NACRE reaction rates.

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Table 2.  Temperature, Mass, and Depletion Levels of ⁹Be and ¹⁰B Surface Logarithm Abundances for Some of the Models Plotted in Figure 11, Computed with the ${\mathtt{THM}}$ and NACRE Reaction Rates, for [Fe/H] = $+0.0$

Element: 9Be, Aini(9Be) = 1.32, [Fe/H] = $+0.0$
50 Myr				100 Myr				300 Myr
Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$	Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$	Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$
3059	0.10	−0.00	−0.00	2912	0.08	−0.00	−0.00	2695	0.08	−1.55	−1.37
3168	0.14	−0.02	−0.02	3030	0.10	−0.09	−0.08	2914	0.10	$\lt -10.00$	$\lt -10.00$
3255	0.18	−0.32	−0.28	3161	0.14	−5.07	−4.77	3609	0.42	−2.76	−2.46
3326	0.22	−2.35	−2.09	3244	0.18	$\lt -10.00$	$\lt -10.00$	3681	0.46	−0.71	−0.63
3384	0.26	−9.36	−8.28	3547	0.38	−5.74	−5.39	3767	0.50	−0.23	−0.20
3433	0.30	−9.29	−8.07	3613	0.42	−1.85	−1.63	4062	0.60	−0.03	−0.02
3480	0.34	−4.86	−4.47	3694	0.46	−0.57	−0.50	⋯	⋯	⋯	⋯
3526	0.38	−2.30	−2.04	3798	0.50	−0.20	−0.18	⋯	⋯	⋯	⋯
3574	0.42	−0.97	−0.86	4160	0.60	−0.02	−0.02	⋯	⋯	⋯	⋯
3624	0.46	−0.40	−0.36	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
3680	0.50	−0.17	−0.15	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
3893	0.60	−0.02	−0.02	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
Element: 10B, Aini(10B) = 2.20, [Fe/H] = $+0.0$
100 Myr				200 Myr				500 Myr
Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$	Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$	Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$
3161	0.14	−0.02	−0.02	2964	0.10	−0.02	−0.03	2864	0.10	−1.76	−2.30
3244	0.18	−0.36	−0.47	3145	0.14	−3.00	−3.89	3131	0.14	$\lt -10.00$	$\lt -10.00$
3314	0.22	−3.32	−4.26	3239	0.18	$\lt -10.00$	$\lt -10.00$	3480	0.34	−6.01	−7.74
3377	0.26	−7.26	−9.46	3486	0.34	−0.34	−0.44	3538	0.38	−0.16	−0.19
3434	0.30	−0.53	−0.69	3551	0.38	−0.03	−0.03	3603	0.42	−0.02	−0.02
3489	0.34	−0.08	−0.11	3622	0.42	−0.00	−0.01	⋯	⋯	⋯	⋯
3547	0.38	−0.02	−0.02	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
3613	0.42	−0.00	−0.01	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
3694	0.46	−0.00	−0.00	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
3798	0.50	−0.00	−0.00	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
4160	0.60	−0.00	−0.00	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯

Note. The adopted initial ⁹Be and ¹⁰B logarithm abundance A_ini is indicated in each table.

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As expected, the models that do not efficiently destroy ⁹Be (${}^{10}$B) are not affected by the reaction rate variation. The situation is different moving toward the region where ⁹Be (¹⁰B) is destroyed (${T}_{\mathrm{eff}}\lesssim 3600$ K); in this case, the differences between the adoption of the NACRE and ${\mathtt{THM}}$ reaction rates can be as large as about $-1$ dex for ⁹Be and almost 2 dex for ¹⁰B.

Notice that the temperature range where the efficient depletion occurs depends on both the age and the considered element. For a 50 Myr age, the ⁹Be depletion occurs for 3300 ${\rm{K}}\lesssim {T}_{\mathrm{eff}}\lesssim$ 3700 K, while for larger ages the range shifts to [3000, 3800] K at 100 Myr and to [2600, 3800] K at 500 Myr. The ¹⁰B shows, at a fixed age, a thinner range of effective temperature for which burning is efficient, about [3200, 3500] K for 100 Myr, [3000, 3500] K at 200 Myr, and [2800, 3500] K for ages of 500 Myr.

As anticipated in Section 5.1, stellar models with [Fe/H] = $-1.5$ have also been computed and the corresponding results reported in Figure 12. The corresponding T_eff, stellar mass values and surface depletion levels, namely ${\rm{\Delta }}A{(}^{9}\mathrm{Be})$ and ${\rm{\Delta }}A{(}^{10}{\rm{B}})$, for some of the models plotted in Figure 12 are listed in Table 3, similarly to the [Fe/H] = $+0.0$ case of Table 2.

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Table 3.  The Same as in Table 2, but for the [Fe/H] = $-1.5$ Models Plotted in Figure 12

Element: 9Be, Aini(9Be) = 0.00, [Fe/H] = $-1.5$
50 Myr				100 Myr				300 Myr
Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$	Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$	Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$
3591	0.10	−0.01	−0.01	3348	0.08	−0.04	−0.03	3006	0.08	−1.75	−1.58
3727	0.14	−1.18	−1.05	3517	0.10	−1.90	−1.73	3348	0.10	$\lt -10.00$	$\lt -10.00$
3811	0.18	−9.30	−8.08	3703	0.14	$\lt -10.00$	$\lt -10.00$	4188	0.42	−0.30	−0.27
3880	0.22	$\lt -10.00$	$\lt -10.00$	4130	0.38	−1.19	−1.05	4300	0.46	−0.02	−0.02
4059	0.34	−4.52	−4.33	4233	0.42	−0.08	−0.07	4460	0.50	−0.00	−0.00
4122	0.38	−0.64	−0.56	4353	0.46	−0.01	−0.01	⋯	⋯	⋯	⋯
4204	0.42	−0.07	−0.06	4477	0.50	−0.00	−0.00	⋯	⋯	⋯	⋯
4335	0.46	−0.01	−0.01	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
4566	0.50	−0.00	−0.00	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
Element: 10B, Aini(10B) = 0.85, [Fe/H] = $-1.5$
100 Myr				200 Myr				500 Myr
Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$	Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$	Teff(K)	$M/$ ${M}_{\odot }$	ΔA${}_{{\mathtt{THM}}}$	ΔA${}_{{\mathtt{NACRE}}}$
3517	0.10	−0.00	−0.00	3169	0.08	−0.00	−0.00	2746	0.08	−0.00	−0.00
3703	0.14	−0.70	−0.92	3405	0.10	−0.19	−0.24	3304	0.10	−3.15	−4.06
3803	0.18	$\lt -10.00$	$\lt -10.00$	3673	0.14	$\lt -10.89$	$\lt -10.89$	3666	0.14	$\lt -10.89$	$\lt -10.89$
4051	0.34	−0.09	−0.12	4037	0.34	−4.63	−5.92	4102	0.38	−0.14	−0.18
4130	0.38	−0.00	−0.00	4107	0.38	−0.03	−0.04	4184	0.42	−0.00	−0.00
4233	0.42	−0.00	−0.00	4192	0.42	−0.00	−0.00	⋯	⋯	⋯	⋯

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As is well known, by changing the chemical composition, the degree of depletion at a given age and mass changes. This is expected because the adopted chemical composition affects the structure of a star, thus its effective temperature (a reduction of the metallicity produces hotter models), convective envelope extension, age, etc.

The comparison between the models in Figures 11 and 12 evidences the shift and the lengthening of the effective temperature range where ⁹Be or ¹⁰B are efficiently destroyed. Furthermore, one can see that the effect of the updated reaction rates in the A(X) versus the T_eff plane at a given age and fixed depletion level from the original ⁹Be, ¹⁰B values is similar when calculating [Fe/H]$\;=\;-1.5$ or [Fe/H] = $+0.0$ models. This is even clearer if one compares, in Tables 2 and 3, the differences between columns ΔA${}_{{\mathtt{THM}}}$ and ΔA${}_{{\mathtt{NACRE}}}$ (taken at approximately the same level of surface depletion, ΔA${}_{{\mathtt{THM}}}$) for the two [Fe/H] values. As an example, for a surface ⁹Be reduction of about 1 dex (i.e., A(⁹Be) $\approx \;0$ for [Fe/H] = $+0.0$ and A(⁹Be) $\approx \;-1$ for [Fe/H] = $-1.5$) the adoption of the ${\mathtt{THM}}$ instead of the NACRE reaction rate leads to a reduction of the surface logarithmic abundance of 0.1–0.2 dex in both [Fe/H] = $+0.0$ and [Fe/H] = $-1.5$ models. This clearly indicates that the effect of the change of the ⁹Be and ¹⁰B reaction rates on the predicted depletion level is weakly dependent on the adopted metallicity.

Another point worth discussing is the effect of the ${}^{10}{\rm{B}}$ reaction rate on the $N{(}^{11}{\rm{B}})/N{(}^{10}{\rm{B}})$ ratio. Figure 13 shows the temporal evolution of the ¹¹B–¹⁰B numerical abundance ratio for several masses, for [Fe/H] = $+0.0$ (top panel) and [Fe/H] = $-1.5$ (bottom panel). Models have been computed by using the ${\mathtt{THM}}$ (full line) and the NACRE (dashed line) ¹⁰B(p,α)⁷Be reaction rate. The points corresponding to a surface ¹⁰B reduction of 80%, 50%, 10%, and 1%, with respect to its initial abundance, are also marked in the plot.

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The initial $N{(}^{11}{\rm{B}})/N{(}^{10}{\rm{B}})$ is 4 for [Fe/H] = $+0.0$ and 4.5 for [Fe/H] = $-1.5$ models (as discussed in Section 5.1), but it significantly changes in time in those models that destroy boron (i.e., $0.1\lesssim M/{M}_{\odot }\lesssim 0.3$). This is expected, because the depletion of ¹⁰B and ¹¹B occurs at different ages, due to the slightly different "ignition" temperatures (about $4\times {10}^{6}$ K for ¹⁰B and $5\times {10}^{6}$ K for ¹¹B), the ¹⁰B burning starting at younger ages. Consequently, the $N{(}^{11}{\rm{B}})/N{(}^{10}{\rm{B}})$ value is expected to increase with time.

As already discussed, the new ¹⁰B ${\mathtt{THM}}$ reaction rate is smaller, at a given temperature, than the NACRE one. This produces a shift of the ¹⁰B depletion at progressively larger ages. This feature is clearly visible in Figure 13; the change of the ¹⁰B reaction rate has a strong impact on the $N{(}^{11}{\rm{B}})/N{(}^{10}{\rm{B}})$ ratio at a fixed age, and the differences increase with the depletion degree. Notice that the effect on the predicted $N{(}^{11}{\rm{B}})/N{(}^{10}{\rm{B}})$ value of updating the ¹⁰B reaction rate is weakly affected by the adopted metallicity, as already discussed.

The light elements lithium, beryllium, and boron offer an important opportunity for a deeper understanding of stellar structures and mixing phenomena. They are destroyed by nuclear reactions at temperatures of a few millions of kelvin and thus their surface abundances depend on the temperature reached at the bottom of the external convective envelope (that is on its depth) in which the matter is completely mixed.

For this reason spectroscopic observations of light element stellar surface abundances allow us to constrain the available theoretical models. By focusing our attention on pre-MS models, we evaluated the impact of the updated ${\mathtt{THM}}$ reaction rates for the two (p,α) destruction channels of the ⁹Be and ¹⁰B isotopes. A variation of the quoted rates is expected to affect light element external abundances only, leaving the stellar structure unchanged. ${\mathtt{THM}}$ allows the experimentalists to get low-energy S(E)-factor measurements without the need for extrapolations, thus strongly reducing the uncertainties typical of direct determinations. In the present work, the ${\mathtt{THM}}$ ⁹Be(p,α)⁶Li S(E)-factor measurement by Wen et al. (2008) and the ¹⁰Be(p,α)⁷Be S(E)-factor evaluation by Spitaleri et al. (2014) have been used to calculate the corresponding reaction rates for which analytical forms have been given with the parameters shown in Table 1. The obtained rates have been compared with the NACRE ones, widely adopted in the astrophysical literature, finding, at temperatures of about (3–5) × ${10}^{6}$ K, variations of ∼25% in both cases.

Then, we calculated stellar evolutionary models in the mass range of [0.06, 0.8] ${M}_{\odot }$ from the pre-MS to the MS phase by varying only the evaluated reaction rates from NACRE to THM. The temporal behavior of the surface ⁹Be and ¹⁰B abundances, for [Fe/H] = $+0.0$, when the NACRE and ${\mathtt{THM}}$ reaction rates are alternatively adopted is reported in Figure 10 showing, in some cases, significant variations. Moreover to reproduce typical observations in galactic open clusters, we also evaluated the ⁹Be and ¹⁰B abundances for different masses (effective temperatures) at fixed ages, for two metallicities typical of galactic and halo clusters, namely [Fe/H] = $+0.0$ and [Fe/H] = $-1.5$. The plots shown in Figures 11 and 12 clearly underline the differences in the results when the reaction rates change from NACRE to THM, especially for models for which an efficient ⁹Be or ¹⁰B burning is expected. An interesting example are models with masses of about $0.08\lesssim M/{M}_{\odot }\lesssim 0.5$ in the temperature range of 2600–3600 K (for [Fe/H] = $+0.0$) or 3000–4000 K (for [Fe/H] = $-1.5$) for which a maximum difference of the ⁹Be and ¹⁰B logarithmic abundances of more than 1 dex, has been found. We emphasize that the effect of the reaction rate update is weakly dependent on the adopted metallicity, if the same level of depletion is considered. We also showed that the change of the ¹⁰B reaction rate update has a significant impact in the predicted $N{(}^{11}{\rm{B}})/N{(}^{10}{\rm{B}})$ temporal evolution.

Although observational ⁹Be and ${}^{10}$B abundances are still not available for the low temperature/masse regimes typical of efficient ⁹Be and/or ¹⁰B burning (i.e., ${T}_{\mathrm{eff}}\lesssim 4000$ K), the present work is an attempt to estimate the role of the improvements in nuclear physics in the computation of realistic and accurate theoretical stellar evolutionary models.

This work has been partially supported by the Italian Ministry of the University under grant RBFR082838 and "LNS-Astrofisica Nucleare (fondi premiali)" by PRIN-MIUR 2010-2011 (Chemical and dynamical evolution of the Milky Way and Local Group galaxies, PI F. Matteucci), by PRIN-INAF 2012 (The M4 Core Project with Hubble Space Telescope, PI L. Bedin), and by INFN (Iniziativa specifica TAsP).