The effect of the recent 17 O ( p , α ) 14 N and 18 O ( p , α ) 15 N fusion cross section measurements in the nucleosynthesis of AGB stars

The Trojan Horse Method (THM) has been used to investigate the low-energy cross sections of the 17O(p, α)14N and 18O(p, α)15N fusion reactions and to extract the strengths of the resonances that more contribute to the reaction rates at astrophysical energies. Moreover, the strength of the 65 keV resonance in the 17O(p, α)14N reaction, measured by means of the THM, has been used to renormalize the corresponding resonance strength in the 17O + p radiative capture channel. Since, proton-induced fusion reactions on 17O and 18O belong to the CNO cycle network for H-burning in stars, the new estimates of the cross sections have been introduced into calculations of Asymptotic giant branch (AGB) star nucleosynthesis to determine their impact on astrophysical environments. Results of nucleosynthesis calculations have been compared with geochemical analysis of ”presolar” grains. These solids form in the cold and dusty envelopes that surround AGB stars and once that have been ejected by stellar winds, come to us as inclusions in meteorites providing invaluable benchmarks and constraints for our knowledge of fusion reactions in astrophysical environments.


Introduction
Asymptotic Giant Branch phase (AGB) is the last stage of evolution for low mass stars (M ≤ 6M ).During this phase H-burning is the main source of energy and it radiatively takes place in a thin shell placed below an extended and cold convective envelope.In this environment products of stellar nucleosynthesis might condensate in grains.Part of these solids, which came to us as inclusions in meteorites felt on the Earth, provide very precise hints of the nucleosynthesis of stars where they formed.Indeed, geochemical analysis can determine the isotopic composition of these solids with an extremely high precision, which is not allowed to stellar spectroscopy.
Oxide grains (Al 2 O 3 ) of group 1 and 2 have been suggested to condensate, respectively, in the envelopes of RGB 1 and AGB stars, when the C/O ratio is smaller than 1 [1].The oxygen isotopic mix in these grains provide stringent constraints to the nucleosynthesis of these stars, because of the fragility of 18 O and the sensitivity of 17 O abundance to temperature.In particular, group 2 grains show 18 O/ 16 O ratios lower than expected and 17 O/ 16 O ratios larger than accounted for by first dredge-up (FDU, [2,3]).The convective mixing episode that occurs when a star is approaching the RGB and that is supposed to fix the surface oxygen isotopic abundances in the evolved stages This work was supported by the Italian Ministry of University MIUR under the grant 'LNS Astrofisica Nucleare (fondi premiali)'.
a e-mail: palmerini@lns.infn.it 1 The Red Giant Branch phase is an evolutionary stage foregoing the AGB, during which the outermost stellar structure already consists in a H-burning shell surrounded by a convective envelope of low mass stars.Furthermore, low 12 C/ 13 C and C/N ratios are observed in the spectra of RGB and AGB stars, which are at odds with the predictions of standard stellar evolution models, where only purely convective mixing episodes are considered ( [4,5] and reference therein).
As an explanation for the reported anomalies of C and O isotopic ratios in stellar spectra and presolar grains, [2,6] suggested a non convective transport mechanism called 'Cool Bottom Process' (CBP) linking the stellar convective envelope to deep layers where H-burning takes place.The common findings of these authors and of the further updates by [7] and [8] are that 18 O is destroyed through the 18 O(p, α) 15 N reaction, by mixing phenomena, while the maximum temperature experienced by the circulating material determines the 17 O/ 16 O isotopic ratio.In this note we shall concentrate on nucleosynthesis in CBP episodes affecting 17 O and 18 O abundances in the envelope of low-mass AGB stars (M≤1.5M ) using the estimates for the 18 O(p,α) 15 N , 17 O(p,α) 14 N, and 17 O(p,γ) 18 F reaction rate presented by [9][10][11].

Fusion reaction rates determination through the Trojan Horse Method
In RGB and AGB stars the relevant temperatures for the 17 O and 18 O nucleosynthesis are in the ranges T 9 = 0.01 − 0.1 2 .Thus the cross sections of the 3 fusion reactions of our interest have to be precisely known in the center-of-mass energy lower than E c.m. = 100 keV.At these energies resonance reactions play a decisive role because the astrophysical S(E)-factor might be dramatically enhanced by the presence of a resonance, whose measurement is then crucial to pin down the astrophysical scenario.However, the presence of the Coulomb barrier, exponentially hampering the cross section at astrophysical energies, and of atomic electrons, shielding the nuclear charges (at least partially), makes the direct measurement of lowenergy resonances not accurate enough or even impossible.Indeed the cross section for fusion reactions among charged particles drops below 10 −12 barn, thus making statistical accuracy and signal-to-noise ratio very poor and the recourse to extrapolation from higher energy mandatory [12,13].As a consequence, large uncertainties can be introduced into the astrophysical models because of an incorrect estimate of the relevant cross sections.The THM ([14-16] and references therein) allows one to access the low-energy cross section of an A(x,c)C reaction by extracting the quasi-free (QF) contribution to a suitable A(a,cC)s reaction, having three particles in the exit channel.Particle a, characterized by a prominent x s cluster structure, is referred to as Trojan horse nucleus as it is used to transfer the participant cluster x.Indeed, if the beam energy is chosen larger than the Coulomb barrier for the A+a interacting system, the breakup of the Trojan horse nucleus takes place inside A nuclear field.The transferred particle x is used to feed the excited states of B, later decaying into c+C.In QF kinematics, the other constituent cluster s is emitted without interacting with the system B, thus behaving as a spectator to the A(x,c)C sub-process.
Because the A(a,cC)s reaction is performed at high energies (several tens of MeV), the cross section of the A(x,c)C process is not hindered by the Coulomb interaction of the target-projectile system, while no electron screening enhancement is spoiling the nuclear information [17].
In the energy reagion of our interest the 17 O(p, α) 14 N reaction cross section is dominated by two resonances: one at about 65 keV above the 18 F proton threshold, and the other at 183 keV.In the last years, the E c.m. = 183keV resonance has been measured by several authors [18][19][20].By contrast, only the direct measurement of the 65 keV resonance performed by [21] was available before of the work by [10], which has determined a resonance strength ωγ 1 = (3.66+0.76 −0,64 ) × 10 −9 eV by applying the THM to the quasi-free 2 H( 17 O, 14 Nα)n reaction and by normalizing experimental data to the weighted average of the three values for the 183 keV resonance strength ωγ 2 = (1.66 ± 0.10) × 10 −3 eV, reported in the literature [18,20,22].This result has been used to calculate the contribution of the 65 keV resonance to the total reaction rate adopting the narrow resonance approximation, whose conditions are satisfied for the resonance under investigation [12,13].Panel a) of figure 1 shows the ratio (red middle line) between the reaction rate R extracted including the 65 keV resonance strength measured by THM, and the reaction rate R Cha f a by [18], also reported in the compilation by [23].The other red lines mark the position of the upper and lower limits as deduced in [10].The blue band represents the Chafa reaction-rate [18] range allowed for by the experimental uncertainties.A small difference (∼ 20%) can be seen in the range T 9 = 0.02−0.1,while no significant differences are present for T 9 0.2, where the contribution of the 65keV resonance to the reaction rate is negligible.
The definition of the resonance strength [12,13] entails that the ωγ parameter of the E 1 = 65keV resonance in the 17 O(p, α) 14 N reaction is proportional to the proton partial width Γ p , the exit channel partial width essentially coinciding with the total width, through the statistical factor.Therefore, the 65 keV resonance strength measured by the THM [10] requires a rescaling of the partial width Γ p and, as a consequence, of the strength of the 65 keV resonance in the 17 O(p, γ) 18 F channel, being proportional to Γ p as well: The TH-scaled resonance strength of the lowest energy resonance is then ωγ T HM p,γ = (1.27+0.26 −0,22 ) × 10 −11 eV to be compared with (1.64 ± 0.28) × 10 −11 eV as given in the recent reviews [18,23,24].Thanks to the THM measurement of the ωγ p,α resonance strength, only the contribution to the reaction rate due to resonance radiative capture can be updated.Furthermore, the resonance THM approach is sensitive to the area subtended by the resonance peak and not to its shape, allowing for the extraction of the strength parameter but not of the S-factor at the Gamow energy, needed to evaluate the tail contribution [12,13].However, inside the temperature range 0.006 ≥ T 9 ≥ 0.06 ( of interest for AGB nucleosynthesis) the contribution of the 65 keV resonance tail to the 17 O(p, γ) 18 F reaction rate is dominant.The modified 17 O(p, γ) 18 F reaction rate including the THM-scaled strength of the 65 keV resonance has been obtained by inserting its contribution, in the place of the corresponding one given by [18], into their recommended reaction rate [9].The THM-modified reaction rate is displayed in Figure 1b as its ratio to the Chafa rate.Figure 1b clearly demonstrates a 20% reduction of the reaction rate at 0.02 ≥ T 9 ≥ 0.06.
At temperatures typical of H-burning in AGB stars, the energy interval where the 18 O(p, α) 15 N is most effective ranges from about 20 to 70 keV.Though nine resonances show up in the 18 O(p, α) 15 N cross section inside the 0-1MeV energy interval, only the 20, 144, and the broad 656 keV resonances are relevant to astrophysics as they determine the reaction rate [25].Despite several direct experimental investigations [26][27][28] and many spectroscopic studies [29][30][31][32], the reaction rate for this process has considerable uncertainty [25].Indeed, only the contribution of the 144 keV resonance has been soundly established by [27].With regard to the 20 keV resonance, its strength was known only from spectroscopic measurements [30] and the direct capture reaction 18 O(p, γ) 19 F [31].The values of the resonance strengths in the literature was then affected by large and not-well-defined uncertainties because they are strongly dependent on the optical model potentials adopted in the data analysis.Since the 20 keV  15 N fusion reaction rates R extracted including resonance strengths measured by the THM, and the reaction rates reported in literature (blue lines [18,25]).The upper and lower limits account for the uncertainty on the THM-scaled resonance strength.In the same way, a blue band is used to show the uncertainties in the reaction rates by [18,23] and [25] resonance is very narrow, according to the measurements in the literature, the narrow-resonance formalism of THM has been employed to obtain its strength [33][34][35].By normalizing to the well-known resonance at 144 keV the TH measurement results in ωγ = (8.3+3.8 −2,6 ) × 10 −19 eV, which is in good agreement with ωγ = (6 +17 −5 9 × 10 −19 eV , reported by [25] but 10 times more accurate.Indeed, the NACRE-recommended value is based on various kinds of estimates while the THM result is obtained from experimental data, thus the accuracy of the resonance strength has been greatly enhanced.As a cross check, the strength of the 90 keV resonance was extracted as well, leading to ωγ = (1.76 ± 0.33) × 10 −7 eV , in good agreement with the strength given by [25], ωγ = (1.6 ± 0.5) × 10 −7 eV. Figure 1c shows the ratio of the reaction rate evaluated by means of the THM data to the one in [25].Clearly, the THM reaction rate shows a much narrower band than the NACRE one over the whole temperature range, especially at low temperatures, thanks to the enhanced precision of the strength of the 20 keV resonance as measured be means of the THM.

Effects of the TH reaction rates on RGB and AGB nucleosynthesis
The THM rates of the 17 O(p, α) 14 N, 17 O(p, γ) 18 F and 18 O(p, α) 15 N fusion reactions have been introduced into the models for proton-capture nucleosynthesis coupled with CBP episodes presented by [8].To safely analyze the consequences on stellar nucleosynthesis of the new nuclear physics inputs and to avoid uncertainties due to the hypothesis about physical cause of CBP (which are still subject of debate), the studied rates have been introduced into the parametric model presented by [7][8][9]36].
In agreement with these authors, the transport of materials is described by the mixing rate Ṁ (in units of 10 −6 M /yr) and the temperature T P of the deepest zones affected by the circulation.We shall refer to this temperature through the logarithmic difference ΔT P = log T H − log T P , where T H is the temperature at which the maximum energy of the H-burning shell is released.In Figure 2, we present a comparison between our results and ones obtained by [8] adopting in calculation the reaction rates described in the previous section and those reported in [23] and [25], respectively.In the figure the black squares along the almost horizontal dashed line report the oxygen isotopic abundances left by the FDU in the envelope of RGB stars with different mass (from 1 to 2M as indicated by the labels).Such values, which have been taken as initial ones for our CBP calculations, do not result to be sensitive to the 17 O+p or the 18  (2) where Y(i) means the abundance in number of the ith isotope and any reaction rate R is temperature and density dependent.From the previous equation, the equilibrium value of the 17 O/ 16 O ratio turns out to be One can notice that equilibrium values of the 17 O/ 16 O isotopic ratio do not depend on the initial isotopic abundances, but they are instead determined by the values of the reaction rates.As a consequence, in our CBP calculations the 17 O/ 16 O ratio result larger because of THM cross sections of the 17 O +p reactions are smaller (see Figure 3, 4 and 5 in [9]) than ones determined reported by [23].
Conversely, no significant changes in the AGB nucleosynthesis predictions arise from the investigation of 18 O(p, α) 15 N fusion reaction by means of THM.Indeed CBP calculations adopting 18 O(p, α) 15 N reaction rate from [11] and [25] show differences in the final 18    For both the stellar masses a moderate mixing with Δ = 0.22, Ṁ = 10 −8 M /yr.has been considered at play during the previous RGB phase.The CBP calculations executed using the THM 18 O(p,α) 15 N , 17 O(p,α) 14 N, and 17 O(p,γ) 18 F reaction rates are drawn in red, while the results obtained by using the rates by [23].
smaller than the 0.2%.Because of the 18 O fragility, which is so easily destroyed that even an increase of about 30% in the rate of the most important destruction channel (namely the 18 O(p, α) 15 N reaction) does not produce any appreciable variation.Furthermore, the resulting increase in 15 N production is also negligible, in fact in AGB stars 15 N is mostly synthesized through the 14 N(p, γ) 15 O(β + ) 15 N chain.

Figure 1 .
Figure 1.Ratios between the 17 O(p, α) 14 N, 17 O(p, γ) 18 F and 18 O(p, α)15 N fusion reaction rates R extracted including resonance strengths measured by the THM, and the reaction rates reported in literature (blue lines[18,25]).The upper and lower limits account for the uncertainty on the THM-scaled resonance strength.In the same way, a blue band is used to show the uncertainties in the reaction rates by[18,23] and[25]

Figure 2 .
Figure 2. Comparison between the oxygen isotopic mix measured in a sample of oxide grains (from WUSTL Presolar Database: http://presolar.wustl.edu/?pgd/ ) and CBP calculations performed using the model in the paper by[8] for a 1M and a 1.2M solar-metallicity AGB star.The solid curves show the evolution of the O-isotopic ratios in the envelope of the 1.2 M star calculated for two efficient cases, namely, (a) Δ = 0.1, Ṁ = 10 −6 M /yr and (b) Δ = 0.1, Ṁ = 3 × 10 −6 M /yr.The effects of CBP in the composition of the 1 M (dotted lines) are shown instead just for Δ = 0.1, Ṁ = 3 × 10 −6 M /yr.For both the stellar masses a moderate mixing with Δ = 0.22, Ṁ = 10 −8 M /yr.has been considered at play during the previous RGB phase.The CBP calculations executed using the THM18 O(p,α)15 N ,17 O(p,α)14 N, and 17 O(p,γ)18 F reaction rates are drawn in red, while the results obtained by using the rates by[23].
[23]reaction rates employed in calculations.Instead, it is clearly shown that, for the same CBP cases, models considering TH data are in better agreement with with the 18 O/16O versus17O/16O values in group 2 grains.The temporal evolution of chemical abundances under the effect of CBP (during the RGB and AGB phase) are the downward red and black curves, which deal with calculations employing THM measurements and data from[23], respectively.In more detail, the lowest18O/16O values exhibited by group 2 grains can be explained by the most efficient CBP case applied to a 1.2M AGB model (curve b).