THE INFLUENCE OF UNCERTAINTIES IN THE ¹⁵O(α, γ)¹⁹Ne REACTION RATE ON MODELS OF TYPE I X-RAY BURSTS

Type I X-ray bursts (XRBs) are thought to arise from thermonuclear runaways occurring in a thin layer on the surfaces of neutron stars accreting hydrogen- and helium-rich matter from companion stars in close binary systems (Woosley & Taam 1976; Joss 1977). Accreting neutron stars in binary systems stably fuse hydrogen into helium via the temperature-independent hot CNO cycles at high accretion rates. At low accretion rates they exhibit XRBs due to the explosive burning of H or He of typical duration 10–100 s during which approximately 10³⁹–10⁴⁰ erg is liberated (Cumming 2004). If the accreted material is rich in H, the rp-process occurs after the burst ignites, potentially leading to the synthesis of nuclei up to A ≈ 100 (Schatz et al. 2001) or beyond (Koike et al. 2004; Elomaa et al. 2009). The light nuclei that participate in the CNO cycles and the heavier nuclei of the rp-process are linked by the breakout reactions ¹⁵O(α, γ)¹⁹Ne and ¹⁸Ne(α, p)²¹Na (Wallace & Woosley 1981; Wiescher et al. 1999). The latter reaction becomes important at higher temperatures than the former; hence ¹⁵O(α, γ)¹⁹Ne has a larger effect on the hydrogen mass fraction at the time of burst ignition than does ¹⁸Ne(α, p)²¹Na, thereby more strongly influencing the dynamics of the burst. For this reason, the effect of this rate on XRB models has been the subject of several investigations (Fisker et al. 2006; Cooper & Narayan 2006; Fisker et al. 2007).

While the ¹⁵O(α, γ)¹⁹Ne reaction rate has not been measured directly, intense and steady experimental efforts have enabled measurements of the decay properties of the states in ¹⁹Ne relevant to the reaction rate at XRB temperatures. These states are just above the α threshold, which lies at an excitation energy of 3.53 MeV. As they lie below the proton and neutron thresholds, the states of interest decay only by α or γ ray emission. Hence the resonant rate of ¹⁵O(α, γ)¹⁹Ne can be calculated from the radiative and α widths or equivalently the lifetimes and α decay branching ratios of the relevant states.

In this paper we assess the experimental data on the decay properties of states just above the α emission threshold in ¹⁹Ne and use a Monte Carlo technique to evaluate the reaction rate. With the results of this analysis upper and lower limits on the reaction rate at the 99.73% confidence level are derived in addition to the recommended, median value. These three reaction rates are used as input for three separate calculations of Type I XRBs using spherically symmetric, hydrodynamic simulations of an accreting neutron star. In this way the influence of the ¹⁵O(α, γ)¹⁹Ne reaction rate on the peak luminosity, recurrence time, and associated nucleosynthesis of Type I XRBs is studied and compared with previous findings.

The thermonuclear rate of the ¹⁵O(α, γ)¹⁹Ne reaction at temperatures below 2 GK, the relevant range for Type I XRBs, is dominated by the contributions of resonances corresponding to states in ¹⁹Ne with excitation energies between 4 and 5 MeV. These states lie above the α decay threshold but below the thresholds for neutron or proton decay. Hence they decay only by α and γ emission and the total decay width Γ is the sum of these partial decay widths, i.e., Γ = Γ_α + Γ_γ. As these resonances are narrow compared to the spacing between adjacent levels with the same spin and parity, the contributions of the different resonances add incoherently and can be measured separately. Even in the absence of a direct measurement, the strength of a resonance can be inferred from measurements of the radiative and α widths of a state. Equivalently, measurements of the mean lifetime of a state τ and its α decay branching ratio B_α ≡ Γ_α/Γ suffice to determine the strength of a resonance. The resonance strength ωγ of a ¹⁹Ne state with angular momentum J formed by the capture of an α particle by ¹⁵O is given by

$$\begin{eqnarray} \omega \gamma &=& \frac{2J+1}{(2J_\alpha +1)(2J_{^{15}O}+1)}\frac{\Gamma _\alpha \Gamma _\gamma }{\Gamma }= \frac{2J+1}{2}\frac{(B_\alpha \Gamma)(1-B_\alpha)\Gamma }{\Gamma }\nonumber\\ &=& \frac{2J+1}{2}B_\alpha (1-B_\alpha)\Gamma. \end{eqnarray} \tag{ 1 }$$

Taking account of the fact that the mean lifetime and total decay width are related by Γ = ℏ/τ, this implies that

$$\begin{equation} \omega \gamma =\frac{(2J+1)B_\alpha (1-B_\alpha)\hbar }{2\tau }. \end{equation} \tag{ 2 }$$

The results of a number of B_α measurements of states in ¹⁹Ne have been published over the last 20 years (Magnus et al. 1990; Laird et al. 2002; Davids et al. 2003a; Davids et al. 2003b; Rehm et al. 2003; Visser et al. 2004; Tan et al. 2007; Tan et al. 2009). In addition, the lifetimes of these states have been measured by two independent groups over the past five years (Tan et al. 2005; Kanungo et al. 2006; Mythili et al. 2008). Here we critically evaluate the published measurements.

2.1. Mean Lifetime Data

2.2. Spin and Parity Assignments

In the relevant excitation energy range, the spins and parities of most of the states are well known. The exceptions are two of the negative parity states, the 4144 keV state with a tentatively assigned spin of 9/2 and the 4200 keV state tentatively assigned to be spin 7/2. Garrett et al. (1972) initially discussed the 4144 keV state as a 7/2⁻ state and the 4200 keV state as a 9/2⁻ state in the context of the known rotational bands in ¹⁹F and ¹⁹Ne. Based on their distorted-wave Born approximation analyses Garrett et al. (1972) suggested swapping the spin assignments of these two states, a suggestion tentatively adopted by evaluators (Tilley et al. 1995).

We compare the spins of the ¹⁹Ne states with the spins of the presumed isobaric analog states in ¹⁹F by examining the reduced transition probabilities and the branching ratios of putative analog states. The isoscalar component dominates strong E2 transitions. The 4033 keV state in ¹⁹F undergoes a strong E2 transition. Depending on the spin, either the 4144 keV state in ¹⁹Ne or the 4200 keV state undergoes an E2 transition. Both the 3999 and 4033 keV states in ¹⁹F decay to the 5/2⁻ state at 1346 keV. We calculate the reduced transition probabilities of the ¹⁹Ne states using the lifetimes of Mythili et al. (2008) and transition energies of Tilley et al. (1995) and Tan et al. (2005). In ¹⁹F, the 3999 keV state decays to the 1346 keV state via an M1 transition while the 4033 keV state undergoes an E2 transition. These transition strengths are tabulated in the first row of Table 2. The 4144 keV and 4200 keV states in ¹⁹Ne decay to the 5/2⁻ state at 1508 keV. If we accept the tentative spin assignments of 9/2⁻ and 7/2⁻, respectively, the 4200 keV state undergoes an M1 transition and the 4144 keV state an E2 transition. The resulting reduced transition probabilities appear in the second row of Table 2. The reduced transition probabilities calculated with the alternate spin assignments are shown in the third row of Table 2, where we see improved agreement with the ¹⁹F B(E2) value.

Table 2. Reduced M1 and E2 Transition Probabilities for Two Isobaric Analog States in ¹⁹F and ¹⁹Ne

Nucleus	Jπ	Energy Level	B(M1)	Nucleus	Jπ	Energy Level	B(E2)
		(keV)	(MeV fm3)			(keV)	(MeV fm5)
19F	$\frac{7}{2}^{-}$	3999	0.0017+0.0010− 0.0005	19F	$\frac{9}{2}^{-}$	4033	90 ± 20
19Ne	$(\frac{7}{2})^{-}$	4200	0.0008+0.0003− 0.0003	19Ne	$(\frac{9}{2})^{-}$	4144	460+180− 100
19Ne	$(\frac{7}{2})^{-}$	4144	0.0024+0.0010− 0.0009	19Ne	$(\frac{9}{2})^{-}$	4200	150+60− 50

Notes. The first row shows the transition strengths for two states in ¹⁹F. The second row contains the ¹⁹Ne transition strengths with the tentatively adopted spin assignments of Tilley et al. (1995) for the ¹⁹Ne analog states and the third row gives the transition strengths with the alternate assignment of the ¹⁹Ne spins.

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Although the reduced transition probabilities suggest that perhaps the spins should be swapped, the measured γ decay branching ratios support the tentative spin assignments of Tilley et al. (1995) for the 4144 keV and 4200 keV states in ¹⁹Ne. The 9/2⁻ state at 4033 keV in ¹⁹F decays exclusively to the 1346 keV state; the 4144 keV state in ¹⁹Ne decays exclusively to the 1508 keV state, suggesting that the tentative spin assignment is justified. The 4200 keV state in ¹⁹Ne and the 3999 keV state in ¹⁹F have comparable branching ratios of 80% and 70% to the 1508 keV state in ¹⁹Ne and the 1346 keV state in ¹⁹F, respectively. It is necessary that the spins of the 4144 and 4200 keV states be measured experimentally in order to precisely constrain their contributions to the reaction rate, though this is not the largest uncertainty; for now we adopt the tentative spin assignments of Tilley et al. (1995).

2.3. α Decay Branching Ratio Data

We report here the results of a million event Monte Carlo simulation of the thermally averaged rate of the ¹⁵O(α, γ)¹⁹Ne per particle pair reaction as a function of temperature. This reaction rate is the incoherent sum of the nonresonant rate, as calculated by Dufour & Descouvemont (2000), and the contributions due to the ¹⁹Ne states at 4.03, 4.14, 4.20, 4.38, 4.55, 4.60, 4.71, and 5.09 MeV. The nonresonant contribution is negligible above 0.2 GK. While the nonresonant rate is fixed at a central value due to its totally insignificant impact on the error budget, the resonant contributions are calculated by drawing randomly from the likelihood distributions of the resonance parameters. The energies and spins of the levels adopted here are shown in Table 5. The contribution due to each resonance was calculated as follows. The energy and 1σ uncertainty of each state was taken from the compilation of Tilley et al. (1995) or calculated by using a weighted average of this and the value from Tan et al. (2005), including a discrepancy error to account for potentially discrepant data as described in Cyburt et al. (2001). The likelihood distributions for the level energies are taken to be Gaussians characterized by the means and standard deviations given in Table 5.

The lifetime of the 4.03 MeV state is drawn from the likelihood distributions for this quantity reported by Tan et al. (2005), Kanungo et al. (2006), and Mythili et al. (2008), with the weight of each measurement assigned according to the inverse of its variance. The resulting likelihood distribution realized in the one million event Monte Carlo simulation is shown in Figure 1. For B_α of this state we use the likelihood distribution derived from the measurement of Davids et al. (2003b), shown in Figure 2.

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Prior to the measurement of Davids et al. (2003b), the best information on Γ_α for the 4.03 MeV state came from measurements of α transfer to the ¹⁹F analog state (Mao et al. 1995). The uncertainties in using isospin symmetry to deduce reduced alpha widths of states near threshold from their analog states are fairly well quantified (Fortune et al. 2010), but systematic uncertainties due to the reaction mechanism remain. Fortunately, there are reliable experimental data on B_α for the 4.03 MeV state and we need not rely on the analog state. Moreover, the most likely value of Γ_α for the 4.03 MeV state derived from the Monte Carlo simulation is 8 μeV, perfectly consistent with the value of 8.8(14) μeV given (for q = 9) in Mao et al. (1995).

The mean lifetimes of the 4.14 and 4.20 MeV states are drawn from the measurements of Tan et al. (2005) and Mythili et al. (2008) as described above. As we are not convinced of the reliability of the single B_α measurement for these two states reported in the literature, we calculate the α widths of these states according to

$$\begin{equation} \Gamma _\alpha =\frac{3 \hbar ^2}{\mu a^2} P \theta _\alpha ^2, \end{equation} \tag{ 3 }$$

where μ is the reduced mass of the ¹⁵O+α system, a the channel radius, P the Coulomb penetrability, and θ²_α the reduced α width (Teichmann & Wigner 1952). Despite some efforts to determine θ²_α for the ¹⁹F analog states (de Oliveira et al. 1996; Mao et al. 1996), in our judgment no reliable experimental determinations of these reduced widths are available in the literature, so we draw them from distributions of our own construction. We conjecture that these reduced widths can be adequately represented by a lognormal distribution having a mode of θ²_α = 0.005 and standard deviation of its logarithm of 1.5. In a survey of θ²_α values of unbound states in the α nuclei ²⁴Mg, ²⁸Si, ³²S, ³⁶Ar, and ⁴⁰Ca, Longland et al. (2010) find a mean value of 0.01. For comparison, the θ²_α value given in de Oliveira et al. (1996) (without any uncertainty) is 0.135 for both states. We chose a channel radius of a = 5.8 fm, corresponding to r₀ = 1.45 fm; we investigated different channel radii of 5.6 and 6.0 fm and found that the resulting penetrabilities changed by ∼50%, which is totally insignificant compared with the uncertainty due to the reduced width. The likelihood distribution for the B_α value of the 4.20 MeV state resulting from this calculation is shown in Figure 3.

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We treat the 4.38 MeV state in the same way as we do the 4.03 MeV state, with the exception being that only two independent measurements of τ exist. The likelihood distribution of B_α for this state is taken from the measurement of Davids et al. (2003b) and is shown in Figure 4.

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The mean lifetimes of the 4.55 and 4.60 MeV states are drawn from the measurements of Tan et al. (2005) and Mythili et al. (2008). The B_α values of the 4.55 and 4.60 MeV states are drawn from the measurements of Magnus et al. (1990), Laird et al. (2002), Davids et al. (2003b), and Tan et al. (2009), with the weight of each measurement proportional to the inverse of its variance as usual.

In the cases of the 4.71 and 5.09 MeV states for which lifetime data are lacking we assume that isospin symmetry is valid and equate the reduced transition probabilities of these states with their analog states in ¹⁹F. Utilizing the lifetime measurements compiled in Tilley et al. (1995), the γ decay branching ratio measurement of Pringle & Vermeer (1989), and the resonance strength measurement of Wilmes et al. (2002), we find that the radiative widths of the 4.71 MeV and 5.09 MeV states are 46⁺¹¹_{− 8} meV and 110⁺¹¹⁰_{− 60} meV, respectively. The B_α values are drawn from the measurements of Magnus et al. (1990), Davids et al. (2003b), and Tan et al. (2009).

Using the resonance parameter likelihood distributions described above, we calculated the thermally averaged rate of the ¹⁵O(α, γ)¹⁹Ne per particle pair reaction as a function of temperature one million times at a range of temperatures between 0.1 and 2 GK. Analyzing the results, we determine upper and lower limits on the reaction rate at the 99.73% confidence level and adopt the median of the calculated reaction rate distribution as our recommended value. The calculated reaction rates are shown in Figure 5 and numerical values are given in Table 6.

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Graphical comparisons of the Monte Carlo reaction rates to the reaction rates estimated in Caughlan & Fowler (1988) and Fisker et al. (2006) are shown in Figures 6 and 7, respectively. In the temperature range from 0.1 to 2 GK the Monte Carlo lower limit on the reaction rate ranges between 2% and 91% of the lower limit of Tan et al. (2009), the Monte Carlo central value lies between 0.71 and 1.04 times that of Tan et al. (2009), and the Monte Carlo upper limit is between 3.2 and 9.4 times that of Tan et al. (2009). Hence, the central value and limits on the reaction rate estimated by Tan et al. (2009) fall well within the 99.73% CL limits on the reaction rate obtained from the Monte Carlo simulation. Comparing to the recent reaction rate estimates of Iliadis et al. (2010), we find that over the same temperature range of 0.1–2 GK our Monte Carlo lower limit on the reaction rate is between 0.03 and 3.6 times that of Iliadis et al. (2010), our central value is between 1.7 and 3.3 times that of Iliadis et al. (2010), and our upper limit is between 7 and 14 times that of Iliadis et al. (2010). However, throughout the entire temperature range characterizing the initial phase of the thermonuclear runaway (below 0.7 GK), our Monte Carlo lower limit on the reaction rate is smaller than that of Iliadis et al. (2010).

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Given the large uncertainties in the B_α values of the low lying states, it is impossible to precisely specify the fraction of the total reaction rate accounted for by the resonant contribution of any particular state. But in the Monte Carlo approach taken here it is possible to find the likelihood of a given state accounting for any definite fraction of the total reaction rate. Figure 8 shows the likelihood distributions of the fraction of the total reaction rate at 0.5 GK accounted for by the resonant contributions of the 4.03 and 4.14 MeV states. It is possible to say that at this temperature and indeed below 0.6 GK, the 4.03 MeV state likely dominates the reaction rate.

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In order to assess the impact of the current uncertainties in the ¹⁵O(α, γ) rate on Type I XRB models, we have performed a new set of simulations with SHIVA, a one-dimensional (spherically symmetric), hydrodynamic, implicit, Lagrangian code used extensively in the modeling of stellar explosions such as classical novae (José & Hernanz 1998) and XRBs (José et al. 2010). To this end, three different models of thermonuclear bursts, with identical input physics except for the specific choice of the ¹⁵O(α, γ)¹⁹Ne rate have been computed: the lower limit reaction rate dubbed Model A, the recommended rate Model B, and the upper limit Model C. In all cases the accretion of solar-like matter with hydrogen mass fraction X = 0.7048, helium mass fraction Y = 0.2752, and metallicity Z = 0.02 onto a 1.4 M_☉ neutron star (with initial luminosity L_i = 1.6 × 10³⁴ erg s⁻¹ = 4.14 L_☉), at a mass accretion rate $\dot{M}_{{\rm acc}} = 1.75 \times 10^{-9}$ M_☉ yr⁻¹ (corresponding to 0.08 $\dot{M}_{\rm Edd}$), has been adopted. The Eddington accretion rate $\dot{M}_{\rm Edd}$ is the accretion rate that produces a luminosity at which the force due to radiation pressure equals the local gravitational force. All metals are initially assumed to be in the form of ¹⁴N, which is justified considering the rapidity with which the CNO nuclei are converted to ¹⁴N at the high temperatures reached early in the burst and the negligible energy this releases compared with accretion (Woosley et al. 2004). The model adopted in this work is qualitatively similar to model zM computed by Woosley et al. (2004) in the framework of the one-dimensional, hydrodynamic, implicit code KEPLER, as well as to the model computed by Fisker et al. (2006). We note however that whereas Woosley et al. (2004) assume a value of 10 km for the (Newtonian) neutron star radius, our model yields a value of 13.1 km following the integration of the neutron star structure. In turn, the calculations reported by Fisker et al. (2006), in a general relativistic framework, relied on a radius of 11 km for the same neutron star mass. Differences in the neutron star size and thereby surface gravity may affect the strength of the explosion through the accreted mass, peak temperature, nucleosynthesis, etc. to some extent.

The code has been linked to a fully updated nuclear reaction network containing 324 nuclides from ¹H to ¹⁰⁷Te and 1392 nuclear processes, and includes the most important charged particle induced reactions occurring among the nuclides between ¹H and ¹⁰⁷Te as well as their corresponding inverse processes (see Parikh et al. 2008 and José et al. 2010 for details). Neutron captures are not taken into account since they play a very minor role in XRB nucleosynthesis. SHIVA uses a time-dependent formalism for convective transport whenever the characteristic convective timescale becomes larger than the integration time step. Partial mixing between adjacent convective shells is treated by means of a diffusion equation (Prialnik et al. 1979). Accretion is computed by redistributing material through a constant number of envelope shells (Kutter & Sparks 1980): a thin envelope, containing 1.1 × 10¹⁸ g of material (less than 1/1000 of the total envelope mass accreted during the first bursting episode), equally distributed through all the envelope shells, is chosen as an initial condition. The model is then relaxed using a few very large time steps to guarantee hydrostatic equilibrium. More details on the adopted input physics can be found in José et al. (2010).

A sequence of four consecutive bursts has been computed for all three models reported in this work. Summaries of the gross properties of the different sequences are given in Figure 9 and Tables 7–9. For each model, the mean mass fractions of all nuclides i that are stable or have a half-life >1 hr and are present with a mass fraction X_i > 10⁻⁹ in the envelope at the end of the fourth burst are given in Table 10.

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As a framework for our discussion, we present first a summary of the results for Model B, in which the recommended ¹⁵O(α, γ)¹⁹Ne rate has been adopted. As shown in Table 8, peak temperatures T_peak ∼ 1.1–1.3 GK and peak luminosities L_peak ∼ (1–2) × 10⁵ L_☉ are reached in this model. Recurrence times between bursts around τ_rec ∼ 4.5–6 hr and α ∼ 28–50, have also been obtained. The ratio between persistent and burst luminosities, α, is defined as

$$\begin{equation} \alpha = \frac{\int _t^{t+\tau _{{\rm rec}}} L(t)\,dt}{\int _{t^{\prime }}^{t^{\prime }+\tau _{0.01}} L(t)\,dt}, \end{equation} \tag{ 4 }$$

with the latter integrated over the time during which the burst exceeds 1% of its peak luminosity, τ_0.01. During the interburst period, the accretion luminosity, $L_{{\rm acc}} = {\it GM} \dot{M} / R \sim 1.5 \times 10^{37}$ erg s⁻¹, will hide the thermal emission from the cooling ashes.

The values reported for Model B are qualitatively in agreement with those inferred from several XRB sources, such as the textbook burster GS 1826-24 [τ_rec = 5.74 ± 0.13 hr, α = 41.7 ± 1.6], 4U 1323-62 [τ_rec = 5.3 hr, α = 38 ± 4], or 4U 1608-52 [τ_rec = 4.14–7.5 hr, α = 41–54], to quote a few examples from Galloway et al. (2008). From the nucleosynthesis viewpoint, the main nuclear flow is dominated by the rp-process (rapid proton-captures and β⁺ decays), the 3α-reaction, and the αp-process (a sequence of (α, p) and (p, γ) reactions). The main flow proceeds away from the valley of stability, eventually reaching the proton drip line beyond A = 38. In this context, much of the initial H and ⁴He is transformed into heavier nuclei. However, it is important to stress the presence of some unburned H (X = 0.05) and ⁴He (Y = 0.11) in the envelope at the end of the fourth burst (see Table 10). Indeed, the mean, mass-averaged metallicity at the end of the fourth bursting episode has increased from the original (accreted) metallicity of Z_i ∼ 0.02 to a value of Z ∼ 0.84. The nuclei in the envelope with the largest mass fractions are ⁶⁰Ni (0.29), ⁶⁴Zn (0.14), ⁴He (0.11), ³²S (0.09), and ⁵⁶Ni (0.06). A mass-averaged ¹²C yield of ∼0.02 is obtained at the end of the fourth burst, not enough to power a superburst (which requires X(¹²C)_min ∼ 0.1 at the envelope base; see Cumming & Bildsten 2001; Strohmayer & Brown 2002; Brown 2004; Cooper & Narayan 2005; Cumming 2005; Cooper & Narayan 2006). The corresponding nucleosynthetic endpoint (defined by the heaviest isotope with a mass fraction X_i > 10⁻⁹) is found around ¹⁰⁰Pd.

When the lower limit for the ¹⁵O(α, γ)¹⁹Ne reaction rate is adopted (Model A), peak temperatures and luminosities of T_peak ∼ 1.1–1.3 GK and L_peak ∼ (1–3) × 10⁵ L_☉ are found (see Table 7). Recurrence times between bursts around τ_rec ∼ 4.5–6 hr and ratios between persistent and burst luminosities, α ∼ 29–50, have also been obtained. All in all, these values are very similar to those found with Model B. But more important, the model exhibits quasi-periodic bursting episodes in sharp contrast with the results reported in a similar analysis by Fisker et al. (2006). In that work, the adopted lower limit on the reaction rate, which happens to lie within a factor of two from the lower limit found in the present Monte Carlo simulation, led to steady state burning of the accreted layer without bursting. With respect to nucleosynthesis, similar amounts of unburned H (0.03, by mass) and ⁴He (0.1) are found at the end of the fourth burst (see Table 10). The mean, mass-averaged metallicity at the end of the fourth bursting episode reached a value of Z ∼ 0.87 in this model. The most abundant nuclides in the envelope in this model are ⁶⁰Ni (0.26), ⁶⁴Zn (0.14), ⁵⁶Ni (0.14), ⁴He (0.1), and ³²S (0.09). A mass-averaged ¹²C yield of ∼0.02 is also obtained at the end of the fourth burst, while the nucleosynthetic endpoint corresponds to ¹⁰¹Pd.

Finally, when the upper limit for the ¹⁵O(α, γ) rate is adopted (Model C), peak temperatures and luminosities of T_peak ∼ 1–1.3 GK and L_peak ∼ (0.8–2) × 10⁵ L_☉ are found (with the lower value obtained for the first burst, interpreted as driven by the initial conditions; see Table 9). Recurrence times between bursts around τ_rec ∼ 4.5–6 hr and ratios between persistent and burst luminosities, α ∼ 27–52, have also been obtained. In summary, the results are very similar to those reported above for Model B. As for the associated nucleosynthesis, similar amounts of unburned H (0.04, by mass) and ⁴He (0.1) are found at the end of the fourth burst (see Table 10). The mean, mass-averaged metallicity at the end of this fourth bursting episode is Z ∼ 0.86 for this model. The most abundant species in the envelope are ⁶⁰Ni (0.29), ⁶⁴Zn (0.17), ⁴He (0.1), ³²S (0.07), and ⁵⁶Ni (0.06). A mass-averaged ¹²C yield of ∼0.02 is also obtained at the end of the fourth burst, while the nucleosynthetic endpoint corresponds now to ¹⁰³Ag, slightly above the endpoints reported for Models A and B. Indeed, as shown in Table 10, there is a trend when comparing the nucleosynthesis associated with Models A, B, and C: a small increase in the final abundances of intermediate-mass and heavy elements, likely driven by the specific choice of ¹⁵O(α, γ)¹⁹Ne rate, since a larger adopted value for the rate will translate into a more effective breakout from the CNO cycles.

In summary, we have evaluated the available experimental data on the states in ¹⁹Ne that serve as resonances in the ¹⁵O(α, γ)¹⁹Ne reaction in Type I XRBs. Using the likelihood distributions inferred from α decay branching ratio and lifetime measurements, we calculated the thermally averaged ¹⁵O(α, γ)¹⁹Ne reaction rate via the Monte Carlo method, determining the median and 99.73% confidence level upper and lower limits on the reaction rate. At this confidence level, the uncertainty in the rate is very large, exceeding a factor of 1000 at some temperatures. Given this large uncertainty, the central value of the rate, which does not differ from the recommended rates of Caughlan & Fowler (1988) or Fisker et al. (2006) by more than a factor of three at these temperatures, is not particularly meaningful and the whole range of reaction rates allowed by experiment must be considered.

Nevertheless, with the exception of the weak nucleosynthetic trend outlined above, we conclude that variation of the ¹⁵O(α, γ)¹⁹Ne reaction rate within the present 99.73% confidence level allowed range has no substantial effect on either the burst properties or the accompanying nucleosynthesis in our models of Type I XRBs. In striking contrast to Fisker et al. (2006), we do not find that a small value of the ¹⁵O(α, γ)¹⁹Ne reaction rate at the lower limit allowed by experiment leads to steady state burning but rather to marginally more energetic bursts than are predicted with the recommended value of the reaction rate. Further astrophysical model calculations are required to resolve the apparent discrepancy between different sensitivity studies concerning the importance of the ¹⁵O(α, γ)¹⁹Ne reaction in models of Type I XRBs.

B.D. acknowledges support from the Natural Sciences and Engineering Research Council of Canada. TRIUMF receives federal funding via a contribution agreement through the National Research Council of Canada. R.H.C. acknowledges helpful discussions with Manoel Couder and Wanpeng Tan. The work of R.H.C. was supported by the U.S. National Science Foundation Grant PHY-08-22648 (JINA). This work has been partially supported by the Spanish MICINN grants AYA2010-15685 and EUI2009-04167, by the E.U. FEDER funds, and by the ESF EUROCORES Program EuroGENESIS.