Urca nuclide production in Type-I X-ray bursts and implications for nuclear physics studies

ABSTRACT

1 INTRODUCTION

Type I X-ray bursts are thermonuclear explosions that occur in the envelopes of accreting neutron stars. Since their first observation in 1974 (Grindlay & Heise 1975a, b), X-ray bursts have proved to be useful probes of accreting neutron stars thanks to their frequent recurrence (Güver & Özel 2013). Consequently, X-ray bursts aid in understanding the physics of extremely dense matter, often by way of model-observation comparisons (Steiner, Lattimer & Brown 2010; Zamfir, Cumming & Galloway 2012; Goodwin et al. 2019; Meisel, Merz & Medvid 2019; Johnston, Heger & Galloway 2020). However, to minimize uncertainties in model-observation comparisons for X-ray bursts and other accreting neutron star observables, the thermal structure of the neutron star outer layers should be accurately known (Meisel et al. 2018).

Recently, it was discovered that urca cooling, a neutrino-emission process long thought to operate in white-dwarf stars (Tsuruta & Cameron 1970; Schwab, Bildsten & Quataert 2017), can significantly impact the temperature of the accreted neutron star envelope (Schatz et al. 2014; Deibel et al. 2015, 2016; Meisel & Deibel 2017). Urca cooling happens when neutrinos are emitted from a cycle of repeated e⁻-captures and β⁻-decays between a pair of nuclei, creating a local heat sink (Gamow & Schoenberg 1940, 1941). For e⁻-capture at finite temperature T to a final state in a daughter nucleus within k_BT of the ground state, where k_B is the Boltzmann constant, there is phase space available for β⁻-decay to occur despite the electron degeneracy of the environment. For such circumstances, and when the subsequent e⁻-capture is not allowed, significant urca cooling may be possible. These conditions are only expected for nuclei with odd nuclear mass number A (Meisel et al. 2015).

X(A) is set by the ashes of nuclear burning occurring in the neutron star envelope. In the ocean and outer neutron star crust, X(A) from surface burning processes such as X-ray bursts are preserved since burial of ashes by subsequent accretion initially results in transmutation by e⁻-capture.¹ While past numerical calculations have reported X(A) from X-ray bursts that are often substantial for odd-A nuclides, (e.g. Woosley et al. 2004; Fisker, Schatz & Thielemann 2008; José et al. 2010; Cyburt et al. 2016; Meisel et al. 2019), the details of odd-A nuclide production have not yet been described. This is potentially problematic, as nuclear reaction rate uncertainties are known to affect X(A) for urca nuclides up to two orders of magnitude (Cyburt et al. 2016; Meisel et al. 2019). Without knowing the relevant environment temperatures, nuclear physicists lack guidance required to adequately plan for studies of these reactions.

We use one-dimensional models of type-I X-ray bursts to demonstrate that urca nuclides with A ≥ 55 are produced in the hydrogen-burning freeze-out near the end of the burst. In Section 2, we describe our X-ray burst model calculations. Section 3 presents calculation results, demonstrating the late-time urca nuclide production. Section 4 demonstrates the sensitivity of urca nuclide production to nuclear reaction rate variations. Section 5 discuss the implications for nuclear physics studies. We summarize our work in Section 6.

2 MODEL CALCULATIONS

The code MESA (Paxton et al. 2011, 2013, 2015) version 9793 was used for one-dimensional model calculations of Type-I X-ray bursts, following the approach described in Meisel (2018), Meisel et al. (2019). The most pertinent details are repeated here. We followed burst nucleosynthesis within ∼1000 zones constructed to resemble an accreting neutron star envelope. General relativistic effects were accounted for using a post-Newtonian modification to the local gravity (Paxton et al. 2011). Convection was approximated using the time-dependent mixing-length theory of Henyey, Vardya & Bodenheimer (1965). Adaptive time and spatial resolution were employed according to the MESA controls varcontrol_target=1d-3 and mesh_delta_coeff=1.0 (Paxton et al. 2013). The inner boundary of the ∼0.01 km thick envelope is treated as if it is on top of a neutron star with M = 1.4 M_⊙ and R = 11.2 km. Nuclear reaction rates from REACLIB (Cyburt et al. 2010) version 2.2 were used within the 304 isotope network developed by Fisker et al. (2008) for analyses of the rapid proton-capture (rp)-process.

Calculations employed the baseline accretion conditions of Meisel et al. (2019), which were found by Meisel (2018)² to best reproduce observed features for the textbook X-ray burst source GS 1826-24 (Bildsten 2000; Galloway et al. 2008). These conditions include accretion rate |$\dot{M} = 0.17~\dot{M_{\rm E}}$| with the Eddington accretion rate |$\dot{M_{\rm E}}=1.75\times 10^{-8}~\rm M_\odot$| yr⁻¹ from Schatz et al. (1999), accretion-based heating at the base of the envelope Q_b = 0.1 MeV u⁻¹, accreted hydrogen mass fraction X_H = 0.70, accreted metal mass fraction Z = 0.02 with the solar metal distribution of Grevesse & Sauval (1998), and accreted helium mass fraction Y = 1 − X_H − Z. To mitigate the effects of compositional inertia (Woosley et al. 2004), we calculated a sequence of 20 X-ray bursts, where the results of the twentieth burst are shown here.

3 RESULTS AND DISCUSSION

Figs 1 and 2 show X(A) for hydrogen and most suspected urca nuclides (Schatz et al. 2014; Meisel & Deibel 2017) throughout the neutron star envelope over the X-ray burst evolution, where the column depth |$y\equiv \int _{r}^{\infty } \rho \, {\rm d}r^{\prime }$|⁠, ρ is the mass density, and r is the radial coordinate. Of urca nuclides listed in Schatz et al. (2014) and Meisel & Deibel (2017), we only include odd-A nuclides, as it is unlikely that even-A nuclides can satisfy the conditions for urca cycling (Meisel et al. 2015). Additionally, A = 103 and 105 are not investigated as they are not produced in appreciable quantities in the model calculations presented here (see Meisel et al. 2019) and A = 61 is included based on the results of Ong (2018).

Figure 1.

Depth profile of X(A) over the burst evolution, where the light curve is shown for reference.

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Figure 2.

Same as Fig. 1, for A = 59 and hydrogen.

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The evolution of X(A) over the burst reveals two groups. Low-A (29, 31, 33) urca nuclide production primarily occurs in deep burning layers (y ≈ 10^7.5−10^8.5 g cm⁻²) during the rise of the burst light curve (t ≈ 0 − 10 s), while high-A (55, 57, 59, 61, 63, 65) production primarily occurs in shallower layers (y ≈ 10^6.5−10^7.5 g cm⁻²) during the burst light curve decay.³ Juxtaposing X(A) for the high-A group with the hydrogen mass fraction X(H) (see Fig. 2), we see that the increase in urca nuclide X(A) at late times (t ≈ 60 − 120 s) corresponds to a decrease in hydrogen, demonstrating that hydrogen-burning freeze-out⁴ is responsible for producing these nuclides. The low-A group, on the other hand, is produced in a hotter region where all hydrogen is rapidly consumed. Fig. 3 presents the same evidence from an alternative view, where Fig. 3(a) contains X(A) averaged over the entire envelope (above the iron substrate in the model) while the average in Fig. 3(b) is restricted to the shallow region where hydrogen-burning is present at late times.

Figure 3.

Sparklines (Tufte 2001) comparing the time evolution of select X(A) for a weighted average over (panel a) the entire neutron star envelope above the iron substrate and (panel b) only the region with hydrogen burning at late times, where the X-ray flux is shown for reference. Blue (red) circles indicate the minimum (maximum) for a quantity, with corresponding values indicated in the rightward legend. Note that the slight increase in X(H) late in the burst evolution is due to replenishment from accretion.

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We note that Fisker et al. (2008) present a detailed description of rp-process nucleosynthesis for several regions in the neutron star envelope over the evolution of an X-ray burst. By examining their “above ignition” region, one can see that they also find the synthesis of high-A nuclides in shallow depths at late times in the burst. However, Fisker et al. (2008) did not discuss the reaction sequence responsible for high-A production at late times, nor did they discuss the production of odd-A nuclides, as urca cooling was not yet suspected to operate in the accreted neutron star crust. As such, our findings here are not new, they are simply an elaboration on past results.

Figure 4.

Depth profile ofρ and T over the burst evolution, where the light curve is shown for reference.

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For most of the odd-A nuclides in Table 1, production outpaces destruction. For A = 31,  63, and 65, this is at least in part because the destruction reaction involves a higher-Z nuclide and therefore a larger Coulomb barrier. For A = 31,  57,  61, and 65, the destruction reaction involves transmuting an even-Z odd-N nucleus into an odd-Z odd-N nucleus, which is energetically unfavourable due to nuclear pairing. For A = 29,  33,  55, and 59, neither of these conditions are fulfilled and so the production rate is less favoured relative to destruction, and is even smaller than the destruction rate for A = 33.

When the production rate exceeds the destruction rate, an enhancement of X(A) is possible. However, when the mean lifetime for an odd-A nucleus τ = 1/λ₋ is sufficiently short, a substantial fraction of that X(A) will flow into higher-A nuclides. At later times in the burst evolution, τ < 1 s for the low-A group while τ > 100 s for the high-A group. Therefore, hydrogen-burning powering the tail of the burst light curve, when T and X(H) are reduced, results in material being transmuted from low-A nuclides toward higher A and subsequently an enhancement of X(A) in the high-A group. This is reminiscent of the late-time formation of the rare-earth peak in r-process nucleosynthesis (Surman et al. 1997), except here the Coulomb barrier is responsible for funnelling rather than nuclear structure related contours of the nuclear mass surface.

4 IMPLICATIONS FOR CALCULATIONS OF URCA NUCLIDE PRODUCTION

Before addressing whether our findings have implications for nuclear physics studies, we first demonstrate that high-A urca nuclide production is sensitive to plausible variations in nuclear reaction rates, and thereby calculation results will be sensitive to input nuclear physics. We focus on example reactions from Table 1 that are potential targets for future nuclear physics experiments, as discussed in Section 5. Namely, these are ⁵⁵Co(p, γ)⁵⁶Ni, ⁵⁷Ni(p, γ)⁵⁸Cu, ⁶²Zn(p, γ)⁶³Ga, and ⁶⁴Ga(p, γ)⁶⁵Ge.

4.1 Results for typical extreme rate variation factors

In prior works investigating the sensitivity of X-ray burst nucleosynthesis to nuclear reaction rate variations, a factor of 100 rate enhancement or reduction factor has sometimes been adopted (e.g. Cyburt et al. 2016; Meisel & Deibel 2017). The factor of 100 is intentionally generous, but sometimes strains plausibility when doing a realistic assessment of the reaction rate uncertainty. In this spirit, we performed X-ray burst model calculations for the four aforementioned reactions, varying each reaction individually by ×100 or /100, applying the same factor for the associated reverse reaction. Burst ashes were determined from the envelope composition after a sequence of 20 bursts, as described in Meisel et al. (2019). The ratio of X(A) for the two calculations is provided in Section 4.3.

We also performed calculations with temperature-dependent rate modifications. This isolates the impact of the reaction rate variation at low and high temperatures and demonstrates that high-A urca production is sensitive to the low-temperature reaction rate behaviour. We constructed unphysical reaction rates that exhibit a factor of 100 enhancement or reduction near T ∼ 0.5 or 1 GK, as shown in Fig. 5. The ratio between burst ashes obtained when using the rate enhancement and reduction at high or low temperature is provided in Section 4.3. It is apparent that the low-temperature reaction rate variations drive the change in X(A) in the high-mass region, further demonstrating that high-A urca nuclide production occurs in relatively low-T conditions.

Figure 5.

Ratio to the baseline ⁵⁵Co(p, γ) reaction rate for reaction rates constructed to have a rate enhancement or reduction at low or high temperature. Similarly, modified rates were constructed for ⁵⁷Ni(p, γ), ⁶²Zn(p, γ), and ⁶⁴Ga(p, γ).

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4.2 An aside on astrophysical reaction rates

When multiple resonances are present in the astrophysical window, the reaction rate will be characterized by a sum of the narrow resonance rates, still referred to as a resonant reaction rate. However, when roughly 10 or more resonances are present within the astrophysical window, then a statistical treatment for the reaction is adequate (Rauscher, Thielemann & Kratz 1997). This reaction regime is modelled using the Hauser–Feshbach approach, where average resonance properties are used to determine the reaction rate (Wolfenstein 1951; Hauser & Feshbach 1952). In this case, the cross-section to use in equation (5) is |$\sigma _{\rm HF}\propto \lambda ^{2}(\langle \mathcal {T}_{i}\rangle \langle \mathcal {T}_{k}\rangle)/\mathcal {T}_{\rm tot}$|⁠, where λ is the de Broglie wavelength for the entrance channel, |$\langle \mathcal {T}_{i}\rangle$| and |$\langle \mathcal {T}_{j}\rangle$| are the average transmission coefficients for the entrance and exit channels, respectively, and |$\mathcal {T}_{\rm tot}$| is the sum over average transmission coefficients for all channels by which c can decay. Average transmission coefficients are related to average partial widths by |$\langle \mathcal {T}\rangle =2\pi \langle \rho \rangle \langle \Gamma \rangle$|⁠, where 〈ρ〉 is the average level density for the nucleus that the reaction channel populates.

4.3 Results for physically motivated rate variation factors

In practice, resonant reaction rates often have large uncertainties because they sensitively depend on Γ and E_r for one or a few states in c, which are often poorly constrained. Meanwhile, averaging over the properties of many states present when in the statistical regime tends to wash-out such uncertainties, resulting in an overall lower uncertainty for a statistical nuclear reaction rate.

For physically motivated uncertainties of statistical reaction rates, we adopt the factors from Rauscher et al. (2016) that were based on comparisons to data, where (p, γ) reactions are subject to a factor of 2 enhancement and factor of 3 reduction with respect to a baseline Hauser–Feshbach estimate.

For resonant rates, one approach is to determine a Monte Carlo uncertainty by calculating the resonant reaction rate many times while sampling the resonance properties from probability distributions (e.g. Longland, Iliadis & Karakas 2012). While the Monte Carlo approach enables statistical rigor, it can underestimate the uncertainty if missing resonances are not accounted for or if the probability distributions for resonance properties are not sufficiently accurate (Cavanna et al. 2015; Sun et al. 2020).

Spectroscopic constraints are inadequate for the compound nuclides produced in the reactions ⁵⁵Co(p, γ)⁵⁶Ni, ⁵⁷Ni(p, γ)⁵⁸Cu, ⁶²Zn(p, γ)⁶³Ga, and ⁶⁴Ga(p, γ)⁶⁵Ge. Instead, we use the theoretical results of Fisker et al. (2001), who used shell model calculations to determine excited state properties in the compound nuclides of interest here. According to those calculations, the heuristic for statistical model validity of 10 levels or more per astrophysical window is fulfilled for ⁶²Zn(p, γ)⁶³Ga and ⁶⁴Ga(p, γ)⁶⁵Ge, but not ⁵⁵Co(p, γ)⁵⁶Ni and ⁵⁷Ni(p, γ)⁵⁸Cu. Therefore, we adopt the × 2 rate increase and × 1/3 rate reduction as plausible uncertainties for these reactions. For ⁵⁵Co(p, γ)⁵⁶Ni and ⁵⁷Ni(p, γ)⁵⁸Cu, we take a resonance that starts out with a large contribution to the reaction rate and enhance ωγ to the Wigner limit. The corresponding reaction rates are shown in Fig. 6, with individual contributions to the total reaction rate, as well as the total rate with the aforementioned resonance enhancement.

Figure 6.

Astrophysical reaction rates calculated using equation (6). The thick solid black line shows the total reaction rate resulting from the sum of the thin lines, which use excited state properties from (Fisker et al. 2001), where the legend indicates E_x in the compound nucleus. The thick grey line shows the total rate when the indicated excited state has ωγ enhanced to the Wigner limit. Thin vertical grey lines help locate 0.5 and 1.0 GK.

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The impact on X(A) for these physically motivated reaction rate uncertainties is shown in Fig. 7. Here, we consider the ratio of X(A) for the reaction rate enhancement to X(A) for the reaction rate reduction (or baseline, in the case of the physically motivated reaction rate uncertainty for ⁵⁵Co(p, γ) and ⁵⁷Ni(p, γ)⁶), which we will denote as |$\mathcal {X}(A)\equiv \frac{X_{\rm enhancement}(A)}{X_{\rm decrease/baseline}(A)}$|⁠. The impact of varying ⁵⁵Co(p, γ) is relatively modest for the physically motivated rate variation, where |$\mathcal {X}(55)=0.7$|⁠, as compared to |$\mathcal {X}(55)=0.02$| for a reaction rate variation factor of 100. The rate variation factor of 100 also leads to |$\mathcal {X}(57)=3.8$| and |$\mathcal {X}(59)=2.1$|⁠. The physically motivated rate variation for ⁵⁷Ni(p, γ) results in |$\mathcal {X}(57)=0.5$|⁠, whereas |$\mathcal {X}(57)=0.01$| for the rate variation factor of 100. The rate variation factor of 100 also leads to a |$\mathcal {X}(59)=2.4$|⁠. For ⁶²Zn(p, γ), |$\mathcal {X}(63)=1.3$| for the physically motivated reaction rate variation, whereas a factor of 100 rate variation leads to a |$\mathcal {X}(63)=4.4$|⁠. For ⁶⁴Ga(p, γ), the physically motivated rate enhancement results in |$\mathcal {X}(65)=1.4$| and |$\mathcal {X}(67)=1.2$|⁠. For the usual factor of 100 rate variation factor, |$\mathcal {X}$| is 3.8, 1.8, and 1.3 for A = 65, 67, and 69, respectively.

Figure 7.

Ratio between X(A) averaged over the envelope after 20 bursts for reaction rate increase/decrease (or increase/baseline), |$\mathcal {X}(A)$|⁠, using the rate variation factor of 100 (open grey circles), 100 at low temperature (small blue circles), 100 at high temperature (small red circles), and physically motivated (black filled circles). For ⁵⁵Co(p, γ) and ⁵⁷Ni(p, γ), the X(A) ratio with the physically-motivated rate variation factor is increase/baseline, for all others it is increase/decrease. See Sections 4.1 and 4.3 for a description of the temperature-dependent and physically motivated reaction rate variation factors, respectively.

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4.4 Are these X(A) impacts significant?

Prior works have demonstrated some of the astrophysical consequences of urca cooling in the accreted neutron star crust for realistic estimates of L_ν. Deibel et al. (2016) showed that urca cooling can alter the carbon-ignition depth for X-ray superbursts, widening the discrepancy between the ignition depth inferred from observations and model calculations. Meisel & Deibel (2017) showed that urca cooling could lead to observable features in the light curve of a neutron star cooling after sustained accretion ceases.

Whether the impact of a nuclear reaction rate uncertainty on X(A) results in a significant uncertainty contribution to L_ν depends on other influences on X(A) as well as the relative uncertainty contribution from other inputs in equation (1).

X(A) is also sensitive to changes in assumed astrophysical conditions for a given bursting source. Recent works have constrained the astrophysical conditions for the source GS 1826-24 by performing model-observation comparisons of the X-ray burst light curve over several bursting epochs (Meisel 2018; Johnston et al. 2020). Though performed with different codes, the Meisel (2018) (MESA) and Johnston et al. (2020) (KEPLER) results are largely consistent. For instance, the uncertainty in the accretion rate considering the uncertainties in each individual work, along with the discrepancy between the two results, is roughly 10 per cent. For a similar change in accretion rate, Meisel et al. (2019) found X(A) for urca nuclides is changed roughly 20 per cent or less. According to the Meisel (2018) calculations, using the upper limit for X_H from Johnston et al. (2020) would result in modifications of X(A) for urca nuclides of roughly 30–50 per cent. We caution that a rigorous benchmarking of the MESA and KEPLER codes has not been performed to date. Thus comparing the codes’ results provides only a rough guide for the uncertainties that can potentially be achieved in X-ray burst light curve model-observation comparisons.

The uncertain quantities in equation (10) are Q_EC and ft. The majority of nuclear masses required to calculate Q_EC for the relevant nuclei are already well constrained (Meisel et al. 2020), with most contributing on the order of 10 per cent uncertainty to L_ν. It is anticipated that in the near future higher precision constraints will further reduce these mass uncertainties such that the uncertainty contribution of Q_EC to L_ν will be less than 1 per cent (Meisel 2020). Experimental ft constraints are presently more sparse. However, with the significant increase in statistics at near-future radioactive ion beam facilities, it should be possible to apply current techniques (e.g. Baumann et al. 1987) to obtain uncertainties on the tens of percent level.

Therefore, any nuclear reaction rate variations that alter X(A) of urca nuclides, and thereby L_ν, by tens of per cent or more can be considered significant. In the context of the previously discussed impact on X(A) from nuclear reaction rate variations, this would be |$\mathcal {X}(A)\lesssim 0.8$| or |$\mathcal {X}(A)\gtrsim 1.2$|⁠. By this criterion, all of the reaction rate variations described in this section for ⁵⁵Co(p, γ)⁵⁶Ni, ⁵⁷Ni(p, γ)⁵⁸Cu, ⁶²Zn(p, γ)⁶³Ga, and ⁶⁴Ga(p, γ)⁶⁵Ge are significant. We stress that more detailed spectroscopy is needed for ⁵⁶Ni, ⁵⁸Cu, ⁶³Ga, and ⁶⁵Ge so that more rigorous reaction rate uncertainty assessments can be performed.

4.5 Are these the most significant X(A) impacts?

Likely not. The reactions we investigate in this work are based on the nuclear reaction network flow for the rp-process. However, a full reaction rate sensitivity study, varying a large number of reaction rates within estimated uncertainties and quantifying the impact on X(A), would be required to produced a ranked list of reaction rate importance. We leave that significant undertaking for a future work. The fact that we find important sensitivities in the present work indicates that such a sensitivity study would be worthwhile.

4.6 Pertinence for sensitivity study results

The different temperature range of interest for urca nuclide production, as opposed to temperatures of interest for X-ray burst light curve impacts, explains why certain reaction rates impact model calculation results for the light curve but not the burst ashes. For instance, the reaction rate sensitivity studies of Cyburt et al. (2016) and Meisel et al. (2019) each found that the competition between the ⁵⁹Cu(p, γ) and ⁵⁹Cu(p, α) reaction rates plays a key role in the decaying phase of the X-ray burst light curve. However, neither study observed an impact on X(59) from changes to the ⁵⁹Cu(p, α) reaction rate.⁷

Fig. 8 shows that when the ⁵⁹Cu(p, α) reaction rate is enhanced by a factor of 100 from the nominal best estimate, as performed by Meisel et al. (2019) and Cyburt et al. (2016), then the (p, γ) and (p, α) rates are comparable for T ≈ 1 GK. At that temperature, of relevance for light curve impacts, the two reactions would compete in terms of reaction flow. However, for the T ≲ 0.6 GK conditions of relevance for urca nuclide production, the enhanced (p, α) rate is an order of magnitude or more lower than the (p, γ) rate, meaning a minor fraction of the overall reaction flow will proceed through this path. Hence, in this example, the details of (p, α) versus (p, γ) rate competition are important for understanding light curve impacts, but not A = 59 urca nuclide production.

Figure 8.

Reaction rates for ⁵⁹Cu + p from REACLIBv2.2 label ths8 (Cyburt et al. 2010), acquired from https://reaclib.jinaweb.org/.

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Note that the lack of influence on urca nuclide production does not preclude the influence of a reaction rate on overall nucleosynthesis. A reaction like ⁵⁹Cu(p, α) that influences nuclear energy generation, and therefore the X-ray burst light curve, will modify the thermodynamic conditions of the burst environment. The modified conditions, e.g. maintaining a higher temperature for a longer time, will subsequently modify the rp-process path, leading to changes in X(A) far from A of the heavy nuclide participating in the influential reaction (Parikh et al. 2008; Cyburt et al. 2016; Meisel et al. 2019).

5 IMPLICATIONS FOR NUCLEAR PHYSICS STUDIES

An important consequence of the previous section is that high-A urca nuclides are primarily produced where T ≲ 0.6 GK. This is much lower temperature than is typically associated with impacts of reactions involving high-A nuclides on the X-ray burst light curve. For instance, fig. 1 of Meisel et al. (2019) shows that ⁵⁹Cu(p, γ)⁶⁰Zn alters the light curve by stalling or enhancing nuclear burning around t ≈ 10 s after burst ignition. Comparing to Figs 2(a) and 4(b), we see that this nuclear burning occurs at y ≈ 10⁸ g cm⁻², where T ≈ 1.0 GK.

Since lower temperatures are of interest for studies constraining high-A urca nuclide production relative to studies constraining light curve impacts, experimental designs may need to be modified relative to studies focused on light curve impacts. We stress that for many cases our findings will not change experimental designs. Below we detail a few examples where our findings may have an impact, to demonstrate that the temperature difference between high-A urca nuclide production and X-ray burst light curve impacts is worth keeping in mind during the planning phase of nuclear physics experiments. These examples require some speculation, as spectroscopy for the relevant nuclides is currently insufficient and some of the experimental devices intended to perform the described experiments are still under development.

For the reactions of relevance here, direct measurements will require inverse kinematics, where a radioactive ion beam impinges on a hydrogen target. Radiative capture reactions on short-lived nuclides tend to have low yields and high γ-backgrounds, which favours the use of recoil separators (Ruiz, Greife & Hager 2014). For the purposes of our estimates below, we will consider hypothetical future measurements using the JENSA gas-jet target (Schmidt et al. 2018) coupled to the SECAR recoil separator at the upcoming Facility for Rare Isotope Beams (FRIB) (Berg et al. 2018), as this will likely be the first set-up where direct measurements of rp-process reactions in the astrophysical energy window will be possible. However, we provide all relevant experimental details such that our estimates could be applied to other facilities as they come online in the future, e.g. CRYRING at the Facility for Antiproton and Ion Research (Lestinsky et al. 2016). We anticipate that the length of a typical SECAR experiment at FRIB will be limited to 10 d of beam-on-target based on the precedent established at current radioactive ion beam facilities.

5.1 Implications for resonance strength measurements

Fig. 6(a) shows that, when assuming resonance properties of Fisker et al. (2001), the ⁵⁵Co(p, γ)⁵⁶Ni reaction rate has the most significant contributions from populating the ⁵⁶Ni excited states at E_x =7.906, 7.938, and 8.268 MeV at 0.5 GK, whereas at 1.0GK, the state at 8.268 MeV provides the dominant contribution. For the anticipated FRIB reaccelerated beam intensity⁸ of ⁵⁵Co, I_b = 1.4 × 10⁹ s⁻¹, the yield from the 8.268 MeV state could be measured to 10 per cent statistics in tens of minutes. However, considerably more time would be required to measure ωγ from the two lower E_x contributions.⁹ For the Fisker et al. (2001) ωγ, it would take the full hypothetical 240-h t_meas in order to achieve 20 per cent precision for both resonances. Hence, based on these theoretical resonance properties and as-yet-unproven radioactive ion beam facility capabilities, the requested measurement time would be vastly different depending on if one is interested in the impacts of ⁵⁵Co(p, γ) on X-ray burst light curves or high-A urca X(A).

Fig. 6(b) highlights the case of ⁵⁷Ni(p, γ)⁵⁸Cu, where the reaction rate at 1 GK primarily comes from the contribution of the hypothetical 3.772 and 3.901 MeV states in ⁵⁸Cu, while at 0.5 GK there most significant contributions come from E_x =3.369, 3.561, and 3.772 MeV. For the predicted I_b = 1.4 × 10⁹ s⁻¹ and Fisker et al. (2001) ωγ, 30 h of on-target time would be required to measure Y_res for the 3.772 and 3.901 MeV states to 10 per cent statistical precision. Alternatively, if the reaction rate at 0.5 GK is of primary interest, then measuring the 3.561 MeV state would require an additional 210 h to obtain ∼20–500 events from this resonance, assuming ωγ falls between the Fisker et al. (2001) calculation result and the Wigner limit.

Of course, for many cases measurement plans would not be affected by considering an environment temperature of 0.5 GK instead of 1.0 GK. An example is ⁵⁴Co(p, γ)⁵⁵Ni, whose reaction rate across this entire temperature range is primarily determined by the resonant reaction into the 5.273 MeV state of ⁵⁵Ni, when using the nuclear structure estimates of Fisker et al. (2001).

5.2 Implications for cross-section measurements

Fig. 9 focuses on ⁶²Zn(p, γ)⁶³Ga and ⁶⁴Ga(p, γ)⁶⁵Ge, using the Hauser-Feshbach σ calculated by Rauscher & Thielemann (2001) and I_b = 1.4 × 10⁹ s⁻¹. For both cases, 10 per cent statistical precision in Y_thin could be achieved within a day for a measurement energy reaching roughly half-way into the 1.0-GK astrophysical window, whereas nearly the entire plausible measurement time would be required to make a single measurement near the upper end of the 0.5 GK window.

Figure 9.

Hauser–Feshbach cross-sections for ⁶²Zn(p, γ) (solid) and ⁶⁴Ga(p, γ) (dashed) from Rauscher & Thielemann (2001) with the 0.5 and 1.0 GK astrophysical windows indicated. The grey horizontal line indicates the energy range in which N_evt could be measured to 10 per cent statistical precision in one day or less.

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While measuring at the higher energies may help constrain σ at lower energies, the constraints may not be particularly stringent. As discussed in the following subsection, the sensitivity of Hauser–Feshbach results to various input physics changes substantially as a function of energy for σ_HF and temperature for N_A〈σv〉. For ⁶²Zn(p, γ) and ⁶⁴Ga(p, γ), σ_HF in the measurable region is significantly impacted by uncertainties in the γ-strength function, whereas in the 0.5 GK astrophysical window, σ_HF is essentially only sensitive to the proton optical model potential (Rauscher 2012).

5.3 Implications for indirect measurements

Often times direct measurements of nuclear reaction data in the astrophysical window are not possible due to insufficient yields, in which case indirect techniques must be applied. For statistical reaction rates, this consists of constraining the quantities required to calculate |$\mathcal {T}$|⁠, in particular the nuclear level density, optical model potential for particle channels, and the γ-strength function for photon emission. Generally, indirect experimental approaches are well suited to constraining one or two of these quantities, but seldom all three. For example, elastic scattering can be used to constrain an optical model potential (e.g. Patronis et al. 2012), whereas the β-Oslo technique simultaneously constrains a nucleus’s level density and γ-strength function (Spyrou et al. 2014).

The sensitivity of σ_HF to each of these inputs is energy-dependent (Rauscher 2012). For a (p, γ) reaction near the reaction threshold, the Coulomb barrier is substantial and so P_ij(E) is small. As such, 〈Γ_p〉 < <〈Γ_γ〉 and so |$\mathcal {T}_{\rm tot}\approx \langle \mathcal {T}_{p}\rangle +\langle \mathcal {T}_{\gamma }\rangle \approx \langle \mathcal {T}_{\gamma }\rangle$|⁠. This means |$\sigma _{\rm HF}\propto \lambda ^{2}\langle \mathcal {T}_{p}\rangle$| at low energies, and consequently, at low temperatures the (p, γ) reaction rate is mostly sensitive to the proton optical model potential. At higher temperatures the Coulomb barrier is no longer a significant impediment and the γ-strength function, which determines |$\langle \mathcal {T}_{\gamma }\rangle$|⁠, is the most important input to the statistical reaction rate calculation.

Consider the ⁶¹Zn(p, γ)⁶²Ga reaction rate. Rauscher (2012) found that this rate is almost exclusively sensitive to the proton optical model potential at 0.5 GK, whereas at 1.0 GK there is roughly equal sensitivity to the γ-strength function. Cyburt et al. (2016) found that ⁶¹Zn(p, γ) does not significantly impact the X-ray burst light curve, but does significantly impact X(61) and X(63) in the burst ashes. Having shown that high-A urca nuclide production occurs near 0.5 GK, we conclude that a measurement of the ⁶²Ga γ-strength function is not necessary for X-ray burst applications, but instead future measurements should work to constrain the ⁶¹Zn + p proton optical model potential.

6 CONCLUSION

In summary, we investigated the conditions responsible for urca nuclide production in Type-I X-ray bursts using model calculations performed with the code MESA. Using conditions that reproduce observables from the source GS 1826-24, we follow the nucleosynthesis of a single X-ray burst at the end of a sequence of 20. Our results show that urca nuclides with A = 55, 57, 59, 61, 63, and 65 are produced at late times in the evolution of the X-ray burst light curve in environment temperatures near 0.4–0.6 GK. This is cooler than the roughly 1 GK conditions typically relevant for X-ray burst light curve impacts from nuclear reactions involving high-A nuclides. Our results also show that high-A urca nuclide production is sensitive to plausible variations in nuclear reaction rates, with impacts that could alter model-observation comparisons.

These results could impact plans for nuclear physics studies aimed at reducing the uncertainty in the production of urca nuclides during X-ray bursts. The relatively lower temperature range of interest can shift priorities of direct resonance strength measurements to focus on lower-lying excitation energy regions in the compound nuclei than would be of interest for studies motivated by impacts on the X-ray burst light curve. Similarly, direct cross-section measurements may require much more measurement time to be within the astrophysical window. Or, such measurements may be altogether unfeasible. Plans for indirect measurements may also have to be modified, as statistical reaction rate sensitivities differ between low- and high-temperature conditions. These results also explain why some nuclear reaction rates significantly impact the burst light curve, but not X(A) for the A of the heavy nuclide participating in the reaction.

Ultimately, our findings will help in planning nuclear physics efforts aimed at better constraining urca nuclide production in X-ray bursts and thereby improve constraints on the thermal properties of the accreted neutron star crust.

ACKNOWLEDGEMENTS

We thank the developers of the MESA software and contributors to the MESA Marketplace (http://cococubed.asu.edu/mesa_market/) and the associated forum, who enabled the calculations presented here. This work was supported by the U.S. Department of Energy under grants DE-FG02-88ER40387 and DESC0019042. We benefited from support by the National Science Foundation under grant PHY-1430152 (Joint Institute for Nuclear Astrophysics–Center for the Evolution of the Elements).

DATA AVAILABILITY

MESA data were generated using input files available at https://doi.org/10.5281/zenodo.2598715. The data are available upon request.

Footnotes

REFERENCES

Author notes