Spallation-altered Accreted Compositions for X-Ray Bursts: Impact on Ignition Conditions and Burst Ashes

Thermonuclear explosions on the surface of accreting neutron stars in low-mass X-ray binaries give rise to Type I X-ray bursts (Lewin et al. 1993; Schatz & Rehm 2006; Jose 2016; Meisel et al. 2018). These are among the most frequent thermonuclear explosions in nature with recurrence times ranging from hours to days (Strohmayer & Bildsten 2006). X-ray bursts are powered by nuclear reaction sequences that include the triple-α-process, the α p-process, and the rapid proton capture process (rp-process; Wallace & Woosley 1981; Woosley et al. 2004; Schatz & Rehm 2006; Fisker et al. 2008; José et al. 2010; Cyburt et al. 2016). X-ray burst observations can serve as important probes of neutron stars (Cumming et al. 2006; Zamfir et al. 2012; Özel & Freire 2016). However, to interpret observations in terms of neutron star properties requires reliable burst models. These models depend on a range of input parameters, including the composition of the accreted material (Woosley et al. 2004; Heger et al. 2007; José et al. 2010). This composition strongly affects ignition conditions and the nuclear evolution during the burst (Wallace & Woosley 1981; Cumming & Bildsten 2000; Schatz et al. 2001; Woosley et al. 2004; José et al. 2010). Typical choices of accreted composition are based on the presumed composition of the companion star and range from solar-metallicity to metal-deficient (Z ∼ 10⁻³; Cumming & Bildsten 2000; Schatz et al. 2001; Heger et al. 2007). It has been pointed out previously that proton-induced spallation of the accreted material in the neutron star atmosphere may change the accreted composition before it settles into the deeper layers of the neutron star (Bildsten et al. 1992). Due to Coulomb collisions with atmospheric electrons, heavier elements thermalize at shallower depths compared to hydrogen and helium. At these shallower depths incoming protons still have high enough energies to destroy the heavier elements through nuclear spallation reactions. In Bildsten et al. (1992) it was explicitly mentioned that spallation of the thermalized ions by protons leads to nuclear fragments, which can further undergo fragmentation, hence resulting in a cascading destruction process. Due to lack of knowledge of the relevant spallation cross sections at that time, only the isolated destruction of CNO elements was discussed. Here we study for the first time the full cascading production and destruction processes using a full nuclear reaction network.

In Section 2 we introduce our model, in Section 3 we present our results of the final composition as a function of accretion rate, and in Section 4 we discuss the impact on a 1D multi-zone hydrodynamic X-ray burst model.

The kinetic energy of free-falling material onto a 1.4 M_⊙ neutron star of 10 km radius is ∼200 MeV u⁻¹. Due to Coulomb collisions with atmospheric electrons, heavier elements thermalize at shallower depths compared to hydrogen and helium. At these shallower stopping depths, incoming protons still have high energies, exposing the heavy nuclei to a flux of energetic protons. The time for heavy elements to diffuse from their stopping depths to the proton stopping depth defines the duration of exposure to high-energy protons, the exposure time t_exposure.

We follow Bildsten et al. (1992) to obtain an estimate of t_exposure, which they referred to as t_res, or residence time. The loading time needed to replenish nuclei in the zone where heavier elements are exposed to proton irradiation depends on the range difference of protons and heavy nuclei and can be approximated (Equation (1.5) of Bildsten et al. 1992) as

$$\begin{eqnarray}&&{t}_{\mathrm{load}}=\left(1-\displaystyle \frac{A}{{Z}^{2}}\right)\displaystyle \frac{{y}_{s}(p)}{{j}_{p}},\end{eqnarray} \tag{ 1 }$$

where j_p is the proton beam current, A and Z are mass and atomic numbers of a given element, respectively, and y_s(p) is the electron column density needed to stop protons. The average time a nucleus spends in this region is determined by diffusion and differs from t_load by factor R, i.e., t_exposure = R × t_load. Therefore,

$$\begin{eqnarray}&&{t}_{\mathrm{exposure}}=\displaystyle \frac{R}{{j}_{p}}\left(1-\displaystyle \frac{A}{{Z}^{2}}\right){y}_{s}(p),\end{eqnarray} \tag{ 2 }$$

diffusion calculations give R ∼ 5 (Bildsten et al. 1992). As ${j}_{p}\propto \dot{m}$, where $\dot{m}$ is mass accretion rate per unit area, therefore Equation (2) can be re-expressed in terms of $\dot{m}$ as

$$\begin{eqnarray}&&{t}_{\mathrm{exposure}}\approx \displaystyle \frac{0.5(\mathrm{kg}\,{\mathrm{cm}}^{-2})}{\dot{m}(\mathrm{kg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1})}\left(1-\displaystyle \frac{A}{{Z}^{2}}\right).\end{eqnarray} \tag{ 3 }$$

t_exposure for the elements carbon, neon, silicon, and iron over a range of mass accretion rates per unit area are shown in Figure 1. There are two observations worth noting. First, for a given element the exposure time decreases with increasing mass accretion rate per unit area. Second, for a given $\dot{m}$ the exposure time for different elements is nearly the same. We therefore assume for the remainder of this work that for a given mass accretion rate per unit area the exposure time is independent of the element, and use the exposure time of ¹²C for all elements.

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The spallation process is modeled using the NucNet Tools single-zone reaction network.⁹ Along with the proton-induced spallation reactions, β-decays were also included as the fragments produced can be unstable nuclei. The single-zone reaction network includes 486 isotopes from hydrogen to iron coupled by a total of 13,076 reactions. Out of these reactions, 1421 are weak reactions and the rest are spallation reactions. Spallation reactions were incorporated into the reaction network as decay reactions, since the protons inducing the spallation are constantly supplied by accretion. β-decay rates were taken from the Nuclear Wallet Cards (Tuli 2011). The spallation reaction rates were calculated as ${j}_{p}\times \sigma ({E}_{p})$, where σ(E_p) are energy-dependent partial cross sections. The spallation cross sections σ(E_p) were calculated using the open-source subroutines from the work of Silberberg et al. (1998). These cross sections are based on the semi-empirical formulae of Letaw et al. (1983) and scaled to available experimental data (Gallo et al. 2019). Nevertheless, spallation cross section data for unstable nuclei are scarce and, for the many reaction channels for which data are missing, uncertainties can be as large as a factor of two or more. For a given mass accretion rate, the reaction network is evolved for the corresponding exposure time.

Hydrogen and helium have the same stopping depths (same A/Z²); therefore, no He spallation is considered in the present case. Helium spallation can still occur during slow down in the atmosphere, but spallation products quickly reassemble through various reactions restoring the initial amount of helium (Bildsten et al. 1993).

Element abundances after spallation are shown in Figure 2 for three different exposure times corresponding to three different accretion rates. Clearly spallation can significantly reduce the accreted heavy element abundances from solar to subsolar for all elements heavier than boron. Heavier elements are affected the strongest.

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In Bildsten et al. (1992), only the destruction of CNO elements was considered (isolated destruction). However, in the cascading destruction process, due to the transformation of heavier elements into CNO elements, the CNO elements are partially replenished. To check whether accounting for the full cascading process makes any difference compared to isolated destruction, we followed the abundance evolution of ¹²C in an isolated destruction process and in a full cascading process.

The time-integrated net reaction flow is shown in Figure 3 for isotopes from neon to carbon. The time-integrated reaction flow shows how carbon is replenished via the destruction of neon, oxygen, and nitrogen. The major path of ¹²C production is via spallation of ¹⁴N and ¹⁶O. These production channels will be missing in the isolated destruction scenario.

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Figure 4 shows that the difference in carbon abundance is conspicuous for cascading and isolated destruction models for all mass accretion rates. The final carbon abundance in the isolated destruction scenario is several orders of magnitude smaller. These results demonstrate that CNO destruction can be overestimated if no replenishment is considered.

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4.1. Impact on the Burst Ignition Conditions

The spallation calculations in the present work show that the resulting ${Z}_{\mathrm{CNO}}$ can be as high as 10⁻⁵ (cascading model). In order to understand the impact of Z_CNO on X-ray burst ignition conditions, we have used the code settle, which computes ignition conditions for type I X-ray bursts. settle uses a multi-zone model of the accreting layer and a one-zone ignition criterion (Cumming & Bildsten 2000). A simple hydrostatic model of the atmosphere during fuel accumulation and immediately prior to X-ray burst ignition is calculated. The model relies on the fact that above a certain mass accretion rate, the accumulating hydrogen is thermally stable and burns via the hot CNO cycles, i.e., the hydrogen-burning rate is constant for a given Z_CNO value. The change in hydrogen mass fraction X_H with column depth y is given by ${{dX}}_{{\rm{H}}}/{dy}=-{\epsilon }_{{\rm{H}}}/\dot{m}{E}_{{\rm{H}}}$, where _H is the energy production rate via hot CNO cycles (which depends on Z_CNO), and E_H is the energy release per gram from burning hydrogen to helium. Neutrino energy losses are neglected. Integrating this equation provides the hydrogen abundance as a function of depth. If helium ignites at a column depth y < y_d (the depth at which hydrogen runs out), a mixed hydrogen–helium-burning flash occurs; otherwise, a pure-helium layer accumulates. In order to obtain a thermal profile of the accumulating layer from the heat equation, the flux from the hot CNO cycles, and the heat released by electron captures, as well as pycnonuclear reactions in the deeper crust (base flux) are considered. Here we assume a base flux of 0.15 MeV/u. The resulting hydrogen mass fractions present at the time of ignition as a function of mass accretion rate per unit area are shown in Figure 5 for different metallicities. Figure 5 shows that for solar Z_CNO, the amount of hydrogen present at the time of helium ignition decreases as the mass accretion rate decreases, and after a certain point no hydrogen is left at the time of helium ignition leading to pure-helium bursts. For Z_CNO = 10⁻¹⁰ (isolated destruction), a negligible amount of hydrogen is burned before the burst and hydrogen is present at helium ignition even at low accretion rates. At these low accretion rates we find a substantial change of X_H between cascading/replenished and isolated destruction models. However, for $\dot{m}\sim 5\,\mathrm{kg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ and below, sedimentation affects the distribution of isotopes and the ignition of H and He in the envelope of an accreting neutron star (Peng et al. 2007). Further studies are required to determine if this counteracts the effect of lower Z. Despite the differences, our results uphold the general conclusion of Bildsten et al. (1992) that spallation leads to burst ignition in a hydrogen–helium mixture for a broad range of accretion rates, even when a full cascading process is considered.

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4.2. Impact of Spallation-altered Accreted Composition on X-Ray Bursts Ashes

We performed 1D multi-zone hydrodynamic X-ray burst calculations with the code MESA (Paxton et al. 2010, 2013, 2015, 2018) to investigate the impact of our new spallation results on the predicted burst ashes. Calculations were performed following the methods of Meisel (2018) and Meisel et al. (2019). Briefly, these consisted of discretizing an 0.01 km atmosphere of a 1.4 M_⊙ 11.2 km neutron star into ∼1000 zones and following the nuclear burning and hydrodynamic evolution induced by accretion. The 304 isotope network of Fisker et al. (2008) and REACLIB (Cyburt et al. 2010) v2.2 reaction rates were used. Hydrodynamic corrections included a post-Newtonian modification of the local gravity to emulate general relativistic effects, and convection was approximated using a time-dependent mixing length theory (Henyey et al. 1965; Paxton et al. 2010). MESA v9793 was used with the time resolution and spatial resolution adapting in time according to the MESA controls varcontrol_target=1d-3 and mesh_delta_coeff=1.0 (Paxton et al. 2013).

Accretion is achieved by adding a small amount of mass to the model's outer layers and readjusting the stellar structure (Paxton et al. 2010). An accretion rate of 10 kg cm⁻² s⁻¹ was used along with a base heating of 0.15 MeV per accreted nucleon.

For X_H = 0.70 and Y = 1 − X_H − Z, calculations were performed with Z = 0.02 (solar), 10⁻⁵ (cascading destruction/replenishment model), 10⁻¹⁴ (isolated destruction model), and 0, each assuming a scaled solar metal distribution (Grevesse & Sauval 1998). Results are shown in Figure 6. For lower Z, the resulting abundance distribution is shifted to higher mass numbers. This is expected, as reduced CNO abundances reduce hydrogen burning during accretion, leading to more hydrogen-rich conditions at burst ignition and therefore to extended hydrogen burning (Heger et al. 2007; José et al. 2010; Meisel et al. 2019). Between Z = 10⁻⁵ and 10⁻¹⁴, the mass fraction of isotopes summed by mass number (X(A)) can vary by more than a factor of two (Figure 6, lower panel). The lower-Z calculations show particularly large differences in the oscillating abundance pattern associated with A = 4n nuclei (Meisel et al. 2019) due to less helium-rich ignition conditions. These changes are significant relative to other changes in astrophysical conditions and nuclear reaction rates (Meisel et al. 2019). Most of the changes in X(A) are in the A ∼ 30–60 region, which potentially affects the urca cooling neutrino luminosity in the crust (Meisel & Deibel 2017). These changes may also alter the thermal conductivity in the inner crust of the neutron star, where it is governed by electron–ion impurity scattering. Nuclear reactions in the crust funnel X-ray burst ashes in the A = 29–55 range into the N = 28 shell closure (Lau et al. 2018) by the time the material reaches the inner crust. Changes in X(A) in this mass region (Figure 6, lower panel) therefore directly affect the inner crust impurity parameter.

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A higher probability of ¹²C survival raises the question of whether this survival leads to enough carbon in the neutron star ocean to ignite the occasionally observed X-ray superbursts (Cumming & Bildsten 2001). However, the resulting carbon abundances remain many orders of magnitude below the mass fractions of the order of 20% estimated to be required to power superbursts (Cumming et al. 2006). Superburst models therefore continue to require significant carbon production by hydrogen- and helium-burning processes either in X-ray bursts or in steady state (Stevens et al. 2014). Our higher initial CNO abundances do not lead to increased carbon production in X-ray bursts (Figure 6); in fact, they rather lead to a reduction due to the lower helium abundance at burst ignition.

We provide new calculations of the spallation-modified composition of material accreted onto a neutron star. These are the first calculations that use a full nuclear reaction network. The resulting compositions can serve as initial parameters for X-ray burst models and differ significantly from the frequently used initial compositions based on the companion star. Our results also differ significantly from estimates of spallation models using isolated destruction processes. We find that when a full cascading destruction model and a full reaction network is used final CNO abundances are significantly larger, though still below the accreted abundances. These larger CNO abundances, especially at lower accretion rates, alter the amount of hydrogen present at the time of burst ignition. We show that this has a significant effect on X-ray burst model predictions of the composition of the burst ashes. Our results must therefore be taken into account in X-ray burst model calculations used to predict the thermal and compositional structure of accreted neutron star crusts (Lau et al. 2018; Meisel et al. 2019).

This work is mainly the product of discussions at the 3rd Astrophysical Reaction Network School supported by the National Science Foundation under grant No. PHY-1430152 (JINA Center for the Evolution of the Elements). J.S.R and R.K. gratefully acknowledge the support from NSERC Canada for this work. Z.M. was supported in part by the U.S. Department of Energy under grant Nos. DE-FG02-88ER40387 and DESC0019042. S.A.G. acknowledges support from the U.S. Department of Energy under Award Number DOE-DE-NA0002847 (NNSA, the Stewardship Science Academy Alliances program). H.S. acknowledges support from NSF under PHY-1102511. B.S.M. was supported by NASA under grant No. NNX17AE32G.

Software: NucNet tools, settle (Cumming & Bildsten 2000), MESA (Paxton et al. 2010, 2013, 2015, 2018).