¹⁷O Destruction Rate in Stars

Abstract

In recent years, several laboratory studies of CNO cycle-related nuclear reactions have been carried out. Nevertheless, extant models of stellar nucleosynthesis still adopt CNO reaction rates reported in old compilations, such as NACRE or CF88. In order to update these rates, we performed new calculations based on a Monte Carlo R-Matrix analysis. In more detail, a method was developed that is based on the collection of all the available data, including recent low-energy measurements obtained by the LUNA collaboration in the reduced background environment of the INFN-LNGS underground laboratory, on R-Matrix cross-section calculations with the AZURE2 code and on uncertainty evaluations with a Monte Carlo analysis. As a first scientific benchmark case, the reactions ¹⁷O$(\mathrm{p},\gamma )$¹⁸F and ¹⁷O$(\mathrm{p},\alpha )$¹⁴N were investigated. Among the different stellar scenarios they can influence, the ¹⁶O/¹⁷O abundance ratio in RGB and AGB stars is the one that can be directly confirmed from spectroscopic measurements. The aim is to reduce the nuclear physics uncertainties, thus providing a useful tool to constrain deep mixing processes eventually taking place in these stars. In this work, we present the procedure we followed to calculate the ¹⁷O$(\mathrm{p},\gamma )$¹⁸F and the ¹⁷O$(\mathrm{p},\alpha )$¹⁴N reaction stellar rates and preliminary comparisons with similar rates reported in widely used nuclear physics libraries are discussed.

1. Introduction

In recent years, substantial progress has been made in the investigation of low-energy reaction cross-sections included in the CNO cycle. Despite these huge efforts, the reaction rates eventually provided to the astrophysical community were often non-exhaustive, i.e., not including all the relevant open channels, or obtained by means of too simple approximations, such as the single-level Breit–Wiegner formalism, to describe the resonant contribution. In order to provide more rigorous reaction rates, the LUNA collaboration ([1,2,3,4] and references therein) decided to develop a reaction rate Monte Carlo analysis procedure based on the R-Matrix formalism, which is becoming more common in the nuclear astrophysics field (e.g., [5,6]). Together with ASFIN [7,8], n-TOF [9,10], ERNA [11,12,13,14] and PANDORA [15], LUNA is one of the main contributors to nuclear astrophysics research in Italy.

We applied this method, which will be described in Section 2, to the ¹⁷O$(\mathrm{p},\gamma )$¹⁸F and ¹⁷O$(\mathrm{p},\alpha )$¹⁴N reactions. These reactions are the main destruction channels for the ¹⁷O. They operate in the deepest and hottest layer of the H-rich envelope of RGB and AGB stars, eventually undergoing non-convective deep-mixing processes capable of modifying the composition at the stellar surface. In this context, the atmospheric ¹⁶O/¹⁷O isotopic ratio is an optimal diagnostic tool to understand the real physical nature of the deep mixing. Therefore, measurements of oxygen-isotopic ratio in giant stars and in pre-solar grains, which are supposed to form in the circumstellar envelope of these stars, are of pivotal importance in stellar astrophysics. Obviously, for a successful application of this diagnostic tool, precise evaluations of the relevant nuclear reaction rates are mandatory.

2. Materials and Methods

The goal of this work is to describe the rigorous method developed to produce accurate reaction rates and uncertainty bands to be used in models of stellar nucleosynthesis. We employed the R-Matrix formalism ([16] and references therein) to perform cross-section estimates, which were then used to evaluate stellar rates $N_A\langle \sigma v\rangle$ following the relation [17,18]

$$\begin{array}{c}\hfill N_A\langle \sigma v\rangle ={\displaystyle \frac{3.7318·{10}^{10}}{T_9^{3/2}{\mu }^{1/2}}}{\int }_0^{\infty }{E}_6\sigma \left(E_6\right)e^{-11.605E_6/T_9}d{E}_6\end{array}$$

(1)

where $N_A$ is Avogadro’s number, $\sigma$ is the reaction cross-section, v is the reacting nuclei relative velocity, $\mu$ is their reduced mass in amu, $T_9$ is the temperature in GK and $E_6$ is the center of mass energy in units of MeV. For narrow resonances, whose particle widths are so small as to contribute only at resonant energy [19], one can rewrite Equation (1) as

$$N_A{\langle \sigma v\rangle }_{NR}={\displaystyle \frac{1.5399·{10}^{11}}{T_9^{3/2}{\mu }^{3/2}}}\omega \gamma e^{-11.605E_{R6}/T_9}$$

(2)

where $E_{R6}$ and ωγ are the resonance energy and strength, both in units of MeV. For the cross-section evaluation through R-Matrix, we adopted the input parameter presented in the starlib compilation [20,21], updated with all the most recent works available (see below). In case of missing information, e.g., branching ratios for primary $\gamma$-ray decays not used in starlib, we took them from the latest available nuclear property compilation [22], updated with more recent literature if applicable. Nuclei mass was also gathered from a recent review work by [23], while channel radius required by the R-Matrix formalism was evaluated as in [24]

$$r=r_0(A_1^{1/3}+A_2^{1/3})$$

(3)

where $A_i$ is the atomic number of the species involved in the reaction and $r_0=1.5$ fm. R-Matrix can also include non-resonant components of the cross-section, through the introduction of high-energy poles, and in the case of radiative capture reactions, sub-threshold states. Poles are also used to add the contribution of low-energy tails of broad high-energy resonances. According to [25], sub-threshold state contributions rely on asymptotic normalization coefficients which are mostly evaluated from indirect experiments. Therefore, a preliminary fit is required.

We applied this procedure to the ¹⁷O$(\mathrm{p},\gamma )$¹⁸F and ¹⁷O$(\mathrm{p},\alpha )$¹⁴N reactions as a first case study. The R-Matrix analysis was performed with the AZURE2 code [26], a powerful tool which allows us to perform multi-level and multi-channel calculations and fits. Both channels were treated at the same time.

Resonance parameters of the ¹⁷O destruction process were gathered from well-known compilations [21,22], updated with the most recent experimental results taken from [27,28,29,30,31,32]. An arbitrary energy of 15 MeV was assigned to the poles (see, e.g., [5,29]). Therefore, a fit of the cross-section, as derived from all available experimental data [27,28,33,34,35,36,37], was carried out to evaluate the pole parameters. The ¹⁸F compound nucleus states included in the analysis are reported in

Table 1. Resonance parameters included in the analysis: compound nucleus state energy $E_x$, spin channel $J^{\pi }$ and partial widths of the entrance channel ${\mathsf{\Gamma }}_p$ and the exit ones ${\mathsf{\Gamma }}_{\alpha }$ and ${\mathsf{\Gamma }}_{\gamma }$. For each exit channel, states with 0 partial width were not included in this analysis.

${\mathbf{E}}_{\mathbf{x}}$	${\mathit{J}}^{\mathit{\pi }}$	${\mathsf{\Gamma }}_{\mathit{p}}$	${\mathsf{\Gamma }}_{\mathit{\alpha }}$	${\mathsf{\Gamma }}_{\mathit{\gamma }}$
[keV]		[eV]	[eV]	[eV]
5671.6(2)	1−	3.50(68) $\times {10}^{-8}$	130(5)	0.44(2)
5789.9(3)	2−	4.00(24) $\times {10}^{-3}$	13.3(5.5)	1.1(3) $\times {10}^{-2}$
6096.4(1.1)	4−	138(26)	106(17)	3.07(50) $\times {10}^{-2}$
6108(3)	1+	0.20(2)	33.6(3.3)	0
6163.2(9)	3+	140,074(257)	5.0(6)	0.595(134)
6240.4(8)	3−	58.2(7.0)	133(24)	0
6242(3)	3−	40.8(3.7)	137(35)	0.73(11)
6262.0(2.5)	1+	27(3)	575(120)	0
6283.2(9)	2+	11,121(186)	28.1(5.0)	0.603(29)
6310.5(8)	3+	525(117)	426(82)	0.17(4)
6385.5(1.7)	2+	109(11)	286(87)	0.270(68)
6484.9(1.5)	3+	277(91)	123(25)	0
6567.0(1.5)	5+	1.2(1)	560(132)	2.6(5) $\times {10}^{-2}$
6633(10)	1−	2920(315)	77,090(2000)	0
6643.7(8)	2−	368(61)	231(40)	0
6777.0(1.4)	4+	9000(1000)	150(24)	0.31(8)
6809(5)	2−	16,570(1600)	71,500(2000)	0
6811(7.5)	2+	2750(450)	210(67)	0
6857(10)	3−	5000(1000)	30(7)	0
7201(2)	4+	29,400(1000)	500(58)	0
7247(2)	1+	5000(1000)	55,000(5000)	0
7291(2)	3+	15,820(1426)	44,180(15,000)	0

. All interference patterns have been set to constructive, except for those involving poles, whose values were an output of the fit. As usual, additional contributions from narrow resonances were included directly in the calculation of the reaction rate by means of Equation (2).

A Monte Carlo analysis was used to evaluate the reaction rate uncertainty. Having resonance parameters from the literature and pole parameters from our preliminary fit, we performed 15,000 R-Matrix calculations sampling both resonance and pole parameters assuming normal error distributions. In practice, at each iteration, a set of resonance and pole parameters were extracted, an R-Matrix calculation was performed and the resulting cross-section was used to calculate the reaction rate by means of Equation (1). For the ¹⁷O$(\mathrm{p},\gamma )$¹⁸F channel, the contribution of narrow resonances was added later, according to Equation (2), providing the resonance strengths sampled through a normal error distribution. Those resonance strengths and the relative standard deviations are reported in

Table 2. Center of mass energy $E_{cm}$ and strength ωγ of narrow resonances included in the ¹⁷O$(\mathrm{p},\gamma )$¹⁸F channel.

${\mathbf{E}}_{\mathbf{cm}}$	ωγ
[keV]	[eV]
64.48(54)	2.95(60) $\times {10}^{-11}$
182.73(58)	1.66(12) $\times {10}^{-6}$
529.4(6)	0.110(25)
633.3(9)	0.16(26)
877.8(1.6)	1.93(17) $\times {10}^{-2}$
1036.6(9)	0.275(28)
1196.0(1.6)	2.70(92) $\times {10}^{-2}$
1270.3(1.8)	5.0(1.9) $\times {10}^{-2}$

.

3. Results and Discussion

In Figure 1, we compare our preliminary results with the reaction rates reported in the starlib compilation [38]. About the ¹⁷O$(\mathrm{p},\gamma )$¹⁸F rate, we find a slight difference (<10%) for ${\mathrm{T}}_9$ < 0.02 and ${\mathrm{T}}_9$ > 0.6, likely due to the R-Matrix treatment of the non-resonant component. For $0.02<{\mathrm{T}}_9<0.1$, the new value of the $E_{cm}$ = 64.5 keV resonance ωγ from the work of [30] causes an enhancement up to a factor of 2. Minor changes are also due to the revision of the proton separation energy [39]. The difference with the previous rate gradually reduces to 20% in the range $0.1<{\mathrm{T}}_9<0.5$. At these temperatures, the rate is dominated by the $E_{cm}$ = 183 keV resonance, whose ωγ value was updated with the more recent and precise measurement of [27].

Regarding the ¹⁷O$(\mathrm{p},\alpha )$¹⁴N reaction, two scenarios were investigated in our analysis, representing the two opposite interference patterns with the 64.5 keV resonance. They differ by a factor as high as 7, but only up to ∼0.03 GK, being indistinguishable at higher temperatures. Similarly to the radiative capture, the new determination of the strengths of the 64.5 keV [30] and the 183 keV resonances [27] implies a reaction rate enhancement up to a factor of 2.4, in the temperature range $0.03<{\mathrm{T}}_9<0.1$, and about 25%, in the range of $0.1<{\mathrm{T}}_9<0.3$. A gradual reduction in this enhancement to about 5% is obtained at higher temperatures, once again due to the R-matrix poles.

4. Conclusions

The development of more sophisticated technologies for stellar spectroscopy and pre-solar grain analyses, coupled to the construction of bigger and more powerful telescopes, both ground-based and space-borne, provides more precise determinations of the isotopic abundances eventually modified by nucleosynthesis processes occurring in stellar interiors. The correct interpretation of these observations requires more accurate stellar models. Hence, a better understanding of the mixing processes active in stars, coupled to an improvement of the relevant reaction rates controlling the energy generation and the stellar nucleosynthesis, is mandatory. In particular, the CNO cycle is active in the core of main-sequence stars with mass above ∼1.2 M_⊙, as well as in a thin shell located at the base of the H-rich envelope of RGB, AGB and red supergiant stars. However, the latest reaction rate compilations are more than 10 years old, in spite of a number of updates published after their releases.

With this work, we provide a procedure to produce reliable stellar rates, which involves a critical analysis of literature data and both R-Matrix and Monte Carlo methods.

As a first application, we applied our procedure to the ¹⁷O$(\mathrm{p},\gamma )$¹⁸F and ¹⁷O$(\mathrm{p},\alpha )$¹⁴N reactions. Preliminary results show, for the former, a major update on the reaction rate up to a factor of 2, mainly due to the recent measurements of the low-energy resonance strengths [27,30] and, to a lesser extent, to the new proton separation energy [39]. Similarly, our reanalysis of the ¹⁷O$(\mathrm{p},\alpha )$¹⁴N channel shows an increase in the reaction rate up to a factor of 2.4, again due to the new values for the low-energy resonances. In this case, the use of R-Matrix provided non-negligible differences with respect to the previously published reaction rate. In particular, below 30 MK, the possible interference patterns for the 64.5 keV resonance can lead to different estimates which differ by up to a factor of 7 among them. Nevertheless, it is worth noting that those temperatures have a minor impact on stellar model predictions of the ¹⁷O abundance.

For the future, we are planning to apply this procedure to other reactions. A method similar to the one described here has been already used to calculate the rates of the ¹²C$(\mathrm{p},\gamma )$¹³N and ¹³C$(\mathrm{p},\gamma )$¹⁴N reactions [6]. Here, new low-energy non-resonant data show relevant systematic discrepancies with respect to previous experimental studies. Therefore, a fit of resonance parameters has been performed and a Bayesian analysis applied to obtain more accurate reaction rates. The next step will be the reevaluation of the ¹⁸O$(\mathrm{p},\gamma )$¹⁹F and ¹⁸O$(\mathrm{p},\alpha )$¹⁵N reaction rates, which are a breaking point and a recycling mechanism of the CNO cycle, respectively.