Nuclear Reactions in the Crusts of Accreting Neutron Stars
X-ray observations of transiently accreting neutron stars during quiescence provide information about the structure of neutron star crusts and the properties of dense matter. Interpretation of the observational data requires an understanding of the nuclear reactions that heat and cool the crust during accretion, and define its non-equilibrium composition. We identify here in detail the typical nuclear reaction sequences down to a depth in the inner crust where the mass density is using a full nuclear reaction network for a range of initial compositions. The reaction sequences differ substantially from previous work. We find a robust reduction of crust impurity at the transition to the inner crust regardless of initial composition, though shell effects can delay the formation of a pure crust somewhat to densities beyond . This naturally explains the small inner crust impurity inferred from observations of a broad range of systems. The exception are initial compositions with A 102 nuclei, where the inner crust remains impure with an impurity parameter of due to the shell closure. In agreement with previous work we find that nuclear heating is relatively robust and independent of initial composition, while cooling via nuclear Urca cycles in the outer crust depends strongly on initial composition. This work forms a basis for future studies of the sensitivity of crust models to nuclear physics and provides profiles of composition for realistic crust models.
Correspondence to: ]email@example.com
Approximately 190 Galactic X-ray sources are classified as low mass X-ray binaries (Liu et al., 2007), of which about 100 are confirmed to contain a neutron star accreting matter from a low mass () companion star at typical rates of . Continued mass accretion pushes matter deeper into the crust; as the matter is compressed, the rising pressure and density induce nuclear reactions that generate heat, emit neutrinos, and increase neutron richness. Most low-mass X-ray binaries are expected to be older than , old enough for accretion to have replaced the entire crust of the neutron star. The accreted crust is never heated beyond and therefore differs fundamentally from the original crust, and that of isolated neutron stars, which form via annealing from a high temperature equilibrium (Bisnovatyi-Kogan & Chechetkin, 1979; Sato, 1979; Haensel & Zdunik, 1990). Here we present reaction network calculations that delineate up to a density around , the full set of nuclear reactions that determine the composition and thermal profile of the accreted crust for a given set of astrophysical parameters.
The properties of the accreted crust can be probed observationally in quasi-persistent transiently accreting neutron stars. These systems accrete continuously for years to decades, before accretion turns off and the source switches from outburst to quiescence. Despite the 2–5 order of magnitude drop in luminosity, modern X-ray telescopes can detect these systems in quiescence. The observed soft X-ray component is typically interpreted as thermal emission from the crust heated by nuclear reactions during the outburst (Rutledge et al., 2002), though there is some debate about the potential influence of residual accretion at a very low rate (see, for example, Parikh et al., 2017; Bernardini et al., 2013). The time dependence of this thermal emission reflects the thermal profile of the neutron star crust and its thermal transport properties. For seven sources, the thermal emission in quiescence, following an outburst, has now been tracked observationally for many years (see, for example, summaries in Homan et al., 2014; Turlione et al., 2015; Waterhouse et al., 2016). While there are large differences from source to source, in all cases a decrease in thermal emission over time is observed. This decrease in thermal emission is interpreted as the cooling of the heated crust (Rutledge et al., 2002; Cackett et al., 2006; Shternin et al., 2007; Brown & Cumming, 2009).
Constraints on the physics of neutron star crusts and dense matter in general have been derived from these observations through comparison with models that account for all relevant nuclear processes. Examples include the finding of a relatively high thermal conductivity indicating a relatively well-ordered lattice structure of the solid crust (Cackett et al., 2006; Shternin et al., 2007; Brown & Cumming, 2009) and constraints on its impurity (Brown & Cumming, 2009; Page & Reddy, 2013; Turlione et al., 2015; Ootes et al., 2016; Merritt et al., 2016); evidence for neutron superfluidity (Shternin et al., 2007; Brown & Cumming, 2009); search for signatures of nuclear pasta (Horowitz et al., 2015; Deibel et al., 2017); possible signatures of chemical convection (Degenaar et al., 2014; Medin & Cumming, 2014); constraints on surface gravity (Deibel et al., 2015); and evidence for a strong shallow heat source of unknown origin (Brown & Cumming, 2009; Degenaar et al., 2011; Page & Reddy, 2013; Degenaar et al., 2013, 2015; Deibel et al., 2015; Turlione et al., 2015; Waterhouse et al., 2016; Merritt et al., 2016). Heating and cooling from nuclear reactions in the crust also affects other regions of the neutron star. It influences explosive nuclear burning in regular X-ray bursts and rarer superbursts, which occur above the solid crust (Cumming et al., 2006; Keek & in’t Zand, 2008; Altamirano et al., 2012; Deibel et al., 2016; Meisel & Deibel, 2017), and it contributes towards heating of the neutron star core (Brown et al., 1998; Cumming et al., 2017; Brown et al., 2018). The latter effect can be used to constrain core neutrino emissivities and other core physics (Brown et al., 1998; Colpi et al., 2001). Cumming et al. (2017) recently used core heating constraints in connection with the transient light curve of KS1731-260 to place a lower limit on the core specific heat and concluded that the core could not be dominated by a quark color-flavor-locked phase. Observables related to crust nuclear reactions may not be limited to X-rays. Bildsten (1998) and Ushomirsky et al. (2000) showed that density jumps induced by electron capture reactions in the crust, in combination with a temperature anisotropy, can lead to a mass quadrupole moment and significant gravitational wave emission that may balance the spin-up from the accretion torque and explain observed spin distributions (Patruno et al., 2017).
The steady-state compositional profile of the outer layers of the neutron star is mapped out by the compositional changes of an accreted fluid element as it is incorporated deeper and deeper into the neutron star. These compositional changes are the result of a series of nuclear processes that occur with increasing density. Within hours of arrival on the neutron star, at around g/cm, hydrogen and helium burn into a broad range of heavier elements up to . The reaction sequences are the 3 reaction, the hot CNO cycles, the p-process, and the rapid proton capture process (rp-process) (Wallace & Woosley, 1981; Schatz et al., 1998), and proceed either explosively in regular type I X-ray bursts (Schatz & Rehm, 2006; Schatz et al., 2001; Fisker et al., 2008), or in steady state (Schatz et al., 1999). If the ashes contain significant amounts of carbon, explosive carbon burning in the ocean at g/cm may power the rare superbursts and transform the composition into elements around iron (Schatz et al., 2003; Keek & Heger, 2011). The ashes of these processes form the liquid ocean and eventually solidify around g/cm, setting the initial composition for the nuclear reactions in the solid crust.
The nuclear reactions in the crust of accreting neutron stars, and the associated nuclear heating, were first calculated by Bisnovatyi-Kogan & Chechetkin (1979), Sato (1979) and later by Haensel & Zdunik (1990). They used a simplified model that assumed an initial composition of Fe, the presence of only a single species at a given depth, full - and neutron equilibrium, zero temperature, and no shell structure. They found that electron capture reactions in the outer crust transform Fe stepwise into more neutron rich nuclei. Once the chain of nuclear reactions reaches the neutron drip line on the chart of nuclides (nuclei beyond the neutron drip line are neutron unbound with neutron separation energy ), electron captures with neutron emission in the inner crust continue to transform nuclei to lower . The transition from the outer crust to the inner crust at around is marked by the appearance of free neutrons, which coexist with nuclei. This location in the neutron star is commonly referred to as neutron drip. At density-induced (pycnonuclear) fusion reactions begin to fuse Ne (). The resulting heavy nuclei are then again stepwise reduced in by electron captures with neutron emission. This cycle repeats several times with increasing depth. Haensel & Zdunik (2003, 2008) used the same model to investigate the fate of different initial isotopes, including Cd. Gupta et al. (2007) carried out the first reaction network calculation allowing the presence of an arbitrary mix of nuclei and including nuclear shell structure. They only considered electron capture reactions up to neutron drip, and demonstrated that heating can be substantially increased when taking into account electron capture into excited states. Gupta et al. (2008) carried out a similar study including neutron captures and dissociations and following the electron captures just beyond neutron drip. They found that neutron reactions are not always in equilibrium, resulting in their superthreshold electron capture cascades (SEC), where a sequence of electron captures with neutron emission rapidly transform nuclei to lower Z, instead of the stepwise process found in simpler models. Steiner (2012) developed a simple model similar to Haensel & Zdunik (2003) but allowing for a multi-component plasma and a more realistic mass model. Schatz et al. (2014) used a full reaction network including -decays to follow the crust composition in the outer crust, prior to neutron drip. They found a new type of neutron star crust reaction: nuclear Urca cycles with alternating electron captures and -decays that cool the outer crust. This underlines the importance of using a full reaction network and allowing for the simultaneous presence of multiple species of nuclei.
In this work, we carry out the first full reaction network calculation of the compositional changes in accreted neutron star crusts through neutron drip and into the first pycnonuclear fusion reactions. We follow a broad range of individual reactions and also account for nuclear shell structure. This provides the full picture of nuclear transformations governing the transition from the outer crust to the inner crust. There are a number of open questions that we aim to address: (1) What are the nuclear reaction sequences in the neutron star crust for a realistic multicomponent composition when allowing for branchings and competition between different types of rates? (2) Is the crust evolving towards equilibrium, once free neutrons are available for neutron capture reactions to produce heavier elements, as suggested by Jones (2005), or are previous predictions of an evolution towards lighter elements correct? (3) How does the crust impurity as characterized by the breadth of nuclear composition evolve from the outer to the inner crust? Is the inner crust impurity influenced by nuclear burning at the surface, and therefore likely different from system to system? (4) Can nuclear reactions provide more heating than previously assumed, alleviating, at least in some sources, the need for an exotic additional heat source? Horowitz et al. (2008) proposed that heat released by shallower fusion reactions of lighter nuclei may explain some of the additional heating.
The crust model used here is similar to that in Gupta et al. (2007); Schatz et al. (2014). The crust is modeled as a plane-parallel slab in a local Newtonian frame with constant gravity . We follow the compositional changes of an accreted fluid element induced by the increasing pressure , with local accretion rate and time , to determine the steady state composition of the crust. Time is therefore a measure of depth throughout this work. The mass density is calculated using an equation of state with temperature , and nuclear abundances (including the neutron abundance) as described in Gupta et al. (2007). The pressure of the free neutrons is computed using a zero-temperature compressible liquid-drop model (Mackie & Baym, 1977). Fig. 1 shows the resulting column density as a function of mass density . An accreted fluid element takes about 24,000 yr to reach the end of our calculation around .
In order to track the time evolution of the nuclear abundances, , an implicitly solved nuclear reaction network is used that includes electron captures, -decays, neutron capture, neutron dissociation, and pycnonuclear fusion reactions. The nuclear heat deposited in a time step is obtained as with atomic mass excesses , electron chemical potential (without rest mass) , neutron chemical potential (without rest mass) , electron fraction , neutron abundance , neutrino energy losses from electron captures and beta decays , and lattice energy (Chamel & Haensel, 2008).
2.1 Astrophysical Parameters
Unless otherwise stated, we use in the rest frame at the surface, with the local Eddington accretion rate , and . This accretion rate is in the range for mixed H/He bursts powered by the rp-process as well as superbursts and is therefore appropriate for the initial compositions explored in this work. The calculation starts at a density of . The temperature is treated as a free parameter and set to GK throughout the crust. This corresponds closely to the temperature profile used in Gupta et al. (2007). This approach is suitable for identifying the typical nuclear reactions, independent of specific temperature profiles that vary from system to system and with time, and depend on a number of additional parameters outside of our model (see discussion below). Temperature is not expected to dramatically alter reaction sequences as everywhere and everywhere except for a very narrow layer at neutron drip. Pycnonuclear fusion reaction rates are not temperature sensitive either. The one nuclear process that is strongly temperature dependent is the strength of nuclear Urca cooling in the outer crust (Schatz et al., 2014). Choosing a relatively high constant temperature allows us to clearly identify critical Urca cooling pairs with their intrinsic strengths that may play a role in limiting crustal heating.
2.2 Nuclear Physics Input
Nuclear masses are among the most important input parameters. We use the Atomic Mass Evaluation AME12 (Wang et al., 2012) for experimental masses closer to stability. For the majority of nuclei for which masses are experimentally unknown, we employ the FRDM (Möller et al., 1995) mass model. We do not mix experimental and theoretical masses to calculate reaction Q-values of interest here, such as electron capture thresholds or neutron separation energies. Corrections of the masses of isolated nuclei due to interactions with the free neutron gas can be neglected as our calculations reach at most when the edge of our reaction network is reached. For example, at the most extreme conditions at the end of our calculation, the Mackie & Baym (1977) mass model, which includes interactions with free neutrons, but neglects shell structure, shows an average correction of neutron separation energies of 200 keV (maximum 300 keV), well within mass model uncertainties that even near stability reach keV.
Electron capture rates and -decay rates are determined from strength functions calculated in a model based on wave functions in a deformed folded-Yukawa single-particle potential with residual pairing and Gamow-Teller interactions. They are solved for in a quasi-particle random-phase approximation (QRPA). The original theory was based on a deformed oscillator single-particle potential (Krumlinde & Möller, 1984). To obtain greater global predictive power a folded-Yukawa single-particle model has been used later instead (Möller & Randrup, 1990; Möller et al., 1997). That is the model used here and we refer to it as QRPA-fY. Only allowed Gamow-Teller transitions are considered. Parent nuclei are assumed to be in their ground state, which is a reasonable assumption for the low temperatures ( GK) encountered in neutron star crusts. Weak interaction thresholds are corrected for lattice energy changes following Chamel & Haensel (2008). The weak reaction rates are then calculated for each time step using nuclear masses and a fast phase space approximation (Becerril Reyes et al., 2006; Gupta et al., 2007). Neutron emission is determined individually for each transition from the parent ground state to a daughter state with excitation energy . We make the simplifying assumption that the highest number of emitted neutrons that is energetically possible will occur in all cases, similar to the approach of calculating branchings for -delayed neutron emission in Möller et al. (1997). To take into account Pauli blocking due to the positive neutron chemical potential, we impose the additional condition of with neutron separation energy and number of emitted neutrons .
Neutron capture rates were computed with the TALYS statistical model code as part of a systematic effort to create a reaction rate database for nucleosynthesis studies, using the same atomic masses used to calculate the weak interaction rates (Xu et al., 2013). These neutron capture rates and the rates of the reverse reactions were corrected to account for plasma screening of photons and neutron degeneracy following Shternin et al. (2012). Pycnonuclear fusion rates were calculated from the S-factors of Beard et al. (2010); Afanasjev et al. (2012) using the formalism described in Yakovlev et al. (2006) in the uniformly mixed multicomponent plasma approximation. We implement a total of 4844 pycnonuclear fusion rates from Be to Si.
We performed calculations of the compositional evolution in the accreted crust for different initial compositions. The initial composition is determined by the nuclear ashes of thermonuclear burning near the neutron star surface, which is expected to vary from system to system depending on companion star composition, accretion rate, and neutron star mass. We use here four sets of ashes: a pure Fe ash composition to facilitate comparison with previous work by Haensel & Zdunik (1990), predictions for the ashes of extremely hydrogen rich X-ray bursts powered by an extended rp-process (Schatz et al., 2001), predictions for the ashes of a realistic mixed hydrogen and helium burst with a moderate rp-process expected to power GS 1826-238 (Woosley et al., 2004; Cyburt et al., 2016), and predictions for a superburst powered by explosive carbon burning (Keek et al., 2012).
3.1 Reaction sequence for initial Fe composition
We begin by discussing in detail the reaction sequences for an initial composition of pure Fe. Our calculation follows the compositional change in an accreted fluid element as density and therefore electron chemical potential are slowly rising. The evolution of the main composition as a function of depth is shown in Fig. 2 and the major compositional transitions are listed in Tab. 1.
|Transition||aaPressure in dyne/cm||bbMass density in g/cm||ddNeutron abundance|
The initial reaction sequence up to MeV and takes s and is characterized by a series of three, two step, electron capture (EC) reactions Fe(2EC)Cr, Cr(2EC)Ti, and Ti(2EC)Ca described already in Haensel & Zdunik (1990) (see Fig. 3). These electron captures proceed in steps of two because of the odd-even staggering of the electron capture thresholds. During the last sequence, (,n) reactions release small amounts of neutrons that get recaptured but do not appreciably change the reaction flows. For most EC transitions, the inverse process, -decay, is blocked, as the decay feeds primarily excited daughter states, which reduces the energy of the emitted electrons resulting in effective Fermi blocking. The exception is Ti(EC)Sc where -decay of Sc does occur, leading to a Ti - Sc EC/ Urca cycle (Schatz et al., 2014; Meisel et al., 2015a). However, as discussed in Meisel et al. (2015a), the cycle is weak because of the fast Sc(EC)Ca reaction for the nuclear physics inputs used here, and thus does not affect nuclear energy generation.
At MeV and , the destruction of Ca by electron capture occurs. However, this step proceeds entirely differently owing to the rising significance of free neutrons (see Fig. 3). These neutrons are released in the second step of the two step electron capture sequence, which proceeds as Ca(EC)K(EC,2n)Ar. The neutron separation energy of Ar is sufficiently low for most of the EC transitions from K to proceed to neutron-unbound states leading to the emission of neutrons. The released neutrons are recaptured by the most abundant nucleus, which is still Ca, leading to a neutron capture sequence to Ca. The reaction path therefore splits into two branches leading to Ar and Ca, respectively. However, branchings between electron capture and neutron capture at Ca and K divert some of the reaction flow to Ar via Ca(EC)K(EC,n)Ar and K(n,)K(EC,n)Ar, respectively. The result is a three nuclide composition, dominated by Ar, but with admixtures of Ca and Ar at about 0.2% mass fraction each.
This admixture is, however, short lived, as at MeV and , Ca and Ar are converted into Ar (Fig. 4). The destruction of Ca proceeds via Ca(EC,1n)K(EC,1n,2n,3n), resulting in a range of Ar isotopes, which, together with the already existing Ar, are quickly transformed into Ar by neutron capture. At this point, the crust is rather pure and mainly composed of Ar.
At =31.6 MeV and , Ar is destroyed by the first previously termed superthreshold electron capture cascade (SEC) (Gupta et al., 2008) (see Fig. 5). This reaction sequence occurs when the neutron emission following an electron capture leads to a nucleus with , which therefore immediately captures electrons again and so on. In this particular case, an SEC leading from Ar all the way to Mg is established. The detailed reaction sequence is shown in Fig. 5 and is characterized by electron captures with the emission of mostly 4–5 neutrons. The released neutrons are recaptured by Ar, which is still the most abundant nuclide. This leads again to a split of the reaction path into the SEC from Ar to Mg and a sequence of neutron captures from Ar to Ar. In the initial phase of the SEC, there is a significant abundance buildup of S produced by neutron capture from the SEC path, and to a lesser extent of Si. However, with only a slight rise of , electron capture quickly destroys these isotopes and they are converted into Mg as well. The end result is a layer that consists primarily of Ar (80% mass fraction) and Mg (20% mass fraction). There is a small admixture of Cl (10 mass fraction). The free neutron abundance is significantly increased to 4.810.
At =33.5 MeV and Ar is destroyed by an SEC and converted into Si and Mg (Fig. 6). The initial reaction sequence proceeds via Ar(EC,3n)Cl(2n,)Cl(EC,11n)S (Fig. 6). At S, four destruction paths carry significant flow: (EC,3n), (EC,4n)P(n,), (EC,5n)P(2n,), and (4n,)S(EC,7n) all leading to P, which then undergoes a (EC,5n) reaction leading to Si (). At this point the neutron abundance has reached (Fig. 7) and this considerable neutron density results in a significant neutron capture branch that drives reaction flow out of into Si. However, Si(EC,n) is not negligible and results in an additional build up of Mg via Si(EC,n)Al(EC,3n and 4n)Mg(2-3n,)Mg. As a consequence, Mg increases significantly in abundance. After the destruction of Ar is complete, the layer consists of Si (44% mass fraction), Mg (32% mass fraction) and neutrons (23% mass fraction).
The destruction of Ar coincides with the onset of a weak reaction flow through the first pycnonuclear fusion reaction, MgMgCr (Fig. 6). The fusion reaction is immediately followed by a rapid SEC sequence leading back to Mg and establishing a pycnonuclear fusion-SEC cycle (Fig. 6). The net effect of the cycle is a MgMgMgn reaction resulting in the conversion of Mg into neutrons with increasing depth. Si is the only significant bottle neck in the cycle besides Mg and maintains a significant, roughly constant abundance while reactions produce and destroy the isotope.
At and , Si begins to be depleted significantly. However, because of its location in the fusion-SEC cycle, its abundance is initially not dropping to zero but is merely reduced to about . At this depth, Mg electron capture begins to initiate a SEC sequence towards lighter nuclei (Fig. 8). This SEC sequence ends at N where the pycnonuclear fusion reaction NMgK (not visible in Fig. 6 because the flow is too weak) dominates over further electron capture. The resulting K is immediately destroyed by a reaction sequence that merges into the SEC of the Mg+Mg main pycnonuclear fusion-SEC cycle. An additional branching occurs at O, where EC is comparable to OMgCa fusion. Again, the resulting Ca does not accumulate but merges into the main SEC. EC on Mg therefore effectively leads to a branching of the reaction flow into a two pycnonuclear fusion sub-cycles. The effect of all these cycles is the same MgMgMgn net reaction converting one Mg nucleus into neutrons on each full loop. The first significant depletion of Mg (Fig. 2) marks therefore the onset of significant pycnonuclear fusion.
At a slightly larger depth, at =37.2 MeV, g/cm, dyne/cm and g/cm abundance builds up at the edge of our network and we stop the calculation. The neutron mass fraction has reached 46%. The calculations indicate, though, that the next step is the conversion of Mg into Mg as a consequence of the increasing neutron density. Deeper fusion-SEC cycles develop then starting on Mg. While Mg is not magic, it is the preferred nucleus at these higher neutron densities, because of the jump in neutron binding from Na to Mg predicted by the FRDM mass model leading to an extension of the neutron drip line by 8 isotopes (see, for example, Fig. 8).
3.2 Reaction sequence for extreme rp-process ashes
For X-ray bursters that do not exhibit superburst burning, the ashes of the rp-process is the appropriate initial composition for the crust processes. We use here the composition calculated by the X-ray burst model of Schatz et al. (2001), which has ignition conditions that correspond to systems with high accretion rate and low metallicity, resulting in a relatively large amount of hydrogen (mass fraction ) at ignition (see Fig. 9). While such bursts would be rare in nature, the model serves as a useful tool to explore the consequences of a maximally extended rp-process that reaches the Sn-Sb-Te cycle. We ignore elements lighter than neon assuming they are destroyed by residual helium burning and other thermonuclear fusion processes near the surface.
The initial compositional evolution is characterized by sequences of electron capture reactions along chains of constant mass number (Fig. 10). In this shallow region, the original composition as a function of mass number is preserved and simply pushed to more neutron rich nuclei. In some cases, decay is not completely blocked creating a local nuclear Urca cycle (Schatz et al., 2014) where both, EC and decay occur between a pair of nuclei. This can lead to significant neutrino cooling at the depth where the cycle forms, especially at high temperatures. Such Urca cycles do not occur for all EC transitions. They require a strong ground state to ground state transition (or a transition to a very low lying state with excitation energy ) and an effective blocking of the subsequent EC reaction that would otherwise drain the cycle. EC - -decay pairs therefore occur predominantly in odd chains, though there are a few exceptions, for example in the and chains. The complete set of relevant Urca pairs can be identified in Fig. 11 using the flow as an indicator of the Urca cycling strength. Tab. 2 lists the most important Urca cycling pairs.
|Urca pair||(g/cm)||relative flowaaTime integrated reaction flow relative to the strongest Urca pair.|
At around and the first neutrons are created. These neutrons are immediately recaptured by nuclei in other mass chains. The further evolution is therefore characterized by a combination of EC reactions that drive the composition more neutron rich, and neutron capture reactions that deplete some mass chains, and enhance others. Fig. 10 shows the reaction sequences up to the point where neutrons first appear. The first neutrons are created by Rb(EC,n)Kr, which competes with Rb(EC)Kr. The reason that neutron emission can occur relatively close to stability is that EC on Rb is predicted to occur through a relatively high lying excited state in Kr at 6.9 MeV. This is a consequence of the proximity of the neutron shell closure (Rb has ) resulting in spherically shaped nuclei and EC strength distributions that are concentrated in a few states (Schatz et al., 2014). This highly excited state is sufficiently close to to lead to a significant neutron emission branch. The released neutrons are readily recaptured by nuclei in other mass chains, primarily at higher mass numbers where neutron capture rates tend to be higher. In this case, neutron capture is dominated by Zr, with smaller capture branches on Y, Zr, and Mo, which are the most abundant high nuclei present at this time (Fig. 10). Because of the rapid recapture of the released neutrons, the free neutron abundance stays negligibly small. The chief result of the early neutron release is therefore not the appearance of free neutrons, but changes in the composition as function of mass number (see discussion in Section 4.1.2).
The first fusion reactions are initiated relatively early at and (Fig. 11). Previously, O produced by EC processes from the initially abundant Ne, has been partially converted by neutron capture into O. As soon as O undergoes an EC transition to N, NO and NN fusion reactions occur. Slightly deeper at and =18.0 MeV, two EC transitions on the remaining O produce C, triggering CO, CN, and CC fusion reactions. Other fusion reaction combinations, including reactions involving O produced by neutron capture from O, also occur but are an order of magnitude weaker. At somewhat higher and , the pycnonuclear fusion of oxygen becomes possible. O, produced by electron captures from the initially present Mg, with some contribution from O neutron captures, is destroyed via OO. With this reaction, all oxygen is destroyed, leaving neon as the lightest element present in the crust.
At and nuclei in most reaction chains have reached the neutron drip line (Fig. 11). The free neutron abundance is still low with =. This is sufficient however to drive the composition into a (n,)-(,n) equilibrium within each isotopic chain. Note that the neutron Fermi energy at this stage. Overall, neutron capture reactions have significantly altered the composition as a function of mass number. In particular, abundances in most odd chains have been drastically reduced, the only remaining odd nucleus with a significant abundance is Cu (Section 4.1.2).
Beyond and , SEC chains begin to play a role and rapidly convert nuclei along the neutron drip line into lighter species until a particularly strongly bound nucleus with a large EC threshold is reached (Fig. 12). The associated release of neutrons leads to a drastic increase of the free neutron abundance, marking the location of neutron drip.
Neutron captures also drive the abundance in the neon isotopic chain, predominantly originating from initial Si in the burst ashes, into Ne and Ne. At , pycnonuclear fusion reactions set in and destroy Ne and Ne. The most important reactions are NeNe, NeNe, NeO, and NeC. O is produced via Ne(EC,n)Na(EC,8n)O(n,)O. C is produced from O via O(EC,2n)N(n,)N(EC,5n)C(2n,)C.
All these processes are essentially completed at and , at which point the composition is concentrated in a few nuclei at (Se, abundance =), at (Ca =, near (Si, =), and at (Mg, =). The neutron abundance has reached (Fig. 7) and has reached 0.62 MeV, exceeding keV resulting in degenerate neutrons. The abundance accumulated at the three locations where the neutron drip line intersects the neutron numbers , and 82 can be mapped to different mass ranges in the initial composition. is mostly produced from initial nuclei, with some contribution from . The initial abundance is , already larger than the final abundance. The main branch points that govern leakage to lighter nuclei for material are Kr and Se. At Kr, neutron capture moves material towards , while EC feeds Se via a (EC)(,n)(EC,n) sequence. At Se, (EC,n) moves material ultimately to , while neutron capture feeds . The final abundance may be increased by a small contribution from . The key branchings are Se and again Se, where in each case the EC branch moves the abundance towards .
nuclei are mostly converted into nuclei near the region. is the borderline case, and the reaction sequence is similar to the pure Fe case discussed in section 3.1. An isolated exception is the initial Si abundance which in part ends up in the region due to fusion reactions: Si is converted into neon isotopes via ECs. At Ne neutron capture competes with Ne(EC,n). The Ne(EC,n) branch leads to fluorine and then oxygen, which then fuses into nuclei in the sulfur region. However, a significant fraction of the initial Si abundance is processed through the Ne(n,) branch leading to neutron rich neon isotopes, which then fuse into calcium, and ultimately end up in Ca (). Initial elements lighter than silicon fuse into nuclei below calcium and are therefore converted into nuclei. Initial elements heavier than silicon but lighter than iron end up as magnesium or silicon isotopes, which do not fuse until much greater depths, when nuclei are sufficiently neutron rich for SECs to prevent accumulation at .
We continue the simulation beyond and to and , at which point the increasing neutron density drives the composition towards the edge of our reaction network. At and Ca is destroyed by a SEC and converted into nuclei in the region, leaving only the and regions with significant abundance. We also see the onset of significant MgMg fusion, resulting in a similar fusion-SEC cycle as discussed in section 3.1.
3.3 Reaction sequence for initial KEPLER X-ray burst ashes
A more typical estimate of the final composition of mixed H/He bursts is provided by calculations using the 1D multi-zone code KEPLER. We use the model described in more detail in Cyburt et al. (2016), which was shown to reproduce the observed light curve features of GS1826-24 reasonably well (Heger et al., 2007). The final composition entering the crust is calculated by averaging over the deeper layers in the accreted material after a sequence of about 14 bursts, excluding the bottom layers that are produced by the atypical first burst (see Cyburt et al. (2016) for details). Fig. 15 shows the initial composition as a function of mass number. The main difference to the extreme rp-process discussed in section 3.2 is the reduced amount of heavier nuclei beyond . This is due to increased CNO hydrogen burning in between bursts that leads to a lower hydrogen abundance at ignition, and a lower ignition depth that leads to lower peak temperature and a less extended p-process. Both of these effects result in a lower hydrogen to seed ratio and a shorter rp-process (see Eq. 13 in Schatz et al. (1999)).
The evolution of the composition with increasing depth is overall very similar to the extreme rp-process ashes case, though the different mass chains have different relative abundances due to the different initial composition (Fig. 16). The main global difference is a shift of the appearance of significant amounts of free neutrons towards higher densities by about 50%, which is more in line with the calculation for pure Fe ashes (Fig. 7). This is due to the much reduced abundance of heavy () nuclei that tend to reach the neutron drip line at shallower depth and lead to an early release of neutrons in the case of the extreme rp-process ashes. The deeper onset of neutron drip has some consequences for the further evolution of the composition. In particular, when the heavier mass chains reach the neutron drip line, the neutron abundance is lower, and the equilibrium nucleus is therefore closer to stability where electron capture thresholds are lower. SEC cascades therefore set in earlier, compared to the case of extreme rp-process ashes where nuclei are pushed to more neutron rich isotopes that require higher to capture electrons (Fig. 17). This, together with the much lower initial abundance of nuclei leads to a negligible production of nuclei. The final composition after SECs have converted the composition into nuclei at or near closed shells or with large single particle energy level gaps is Ca (, ), Si (, ), and Mg (, ) with a neutron abundance of .
Again we can map these final abundances to the distribution by mass number of the initial composition. As in the case of the extreme rp-process ashes, is the approximate dividing line between material ending up in and near . However, because of the lower free neutron abundance nuclei tend to be less neutron rich and have lower EC thresholds. Therefore, in the case of the KEPLER X-ray burst ashes, there is some leakage from the mass chains towards the region, primarily due to branch points where neutron capture competes with EC such as Ca(EC),Ca(EC,n), and Ca(EC,3n). While the abundance that remains in the Ca isotopic chain is ultimately converted into Ca, any leakage to lighter elements feeds the region. As in the case of the extreme rp-process ashes, the initial Si abundance is fed into the Ca isotopic chain and converted to Ca. Indeed, the sum of the initial abundances of Si and is mol/g exceed the produced abundance slightly, indicating that most of the other nuclei end up near .
At depths beyond () the evolution is essentially the same as in the case of the extreme rp-process ashes (see section 3.2). The nuclear energy release is overall similar (Fig. 14), though there is somewhat stronger nuclear Urca cooling, and a slightly higher nuclear energy generation. The dominant Urca pairs are listed in Tab. 3. Compared to the extreme rp-process ashes there are two additional important Urca pairs in lighter mass chains, Al-Mg and Fe-Mn. The lack of shallower cooling from heavier nuclei is more than offset by the very strong cooling from Mg-Na at around (Fig. 14). This is due to the much larger initial abundance of nuclei ( vs ).
|Urca pair||(g/cm)||relative flowaaTime integrated reaction flow relative to strongest Urca pair.|
3.4 Reaction sequence for initial superburst ashes
In some X-ray bursting systems, rare superbursts may further modify the composition at a depth around (Cumming et al., 2006). For systems that regularly exhibit superbursts, the ashes of superburst burning is the appropriate initial composition for nuclear processes in the crust. We use the final composition produced in a superburst model calculated with the KEPLER code (Keek & Heger, 2011). This composition is shown in Fig. 18. The sequence of EC and neutron captures leading to the neutron drip line is shown in Fig. 19 and, for the mass chains with significant abundance, is very similar to the result with the KEPLER X-ray burst ashes. Fig. 20 shows the reaction sequences and final composition when the composition is consolidated to a few nuclei that are particularly strongly bound due to shell effects. One difference to the calculation with the KEPLER X-ray burst ashes is that this point is reached at a slightly higher density ( instead of ). The reason is that is smaller due to the higher neutron abundance and therefore a higher density is required to achieve , needed to destroy Ar.
The final composition shown in Fig. 20 has been determined at the same as Fig. 17 and demonstrates that indeed the same nuclei are populated. Because this composition is reached at a greater depth, it is much closer to the onset of pycnonuclear fusion of Mg. However, the relative population of Mg (=), Si (=), and Ca (=) is different because of the different initial composition. Most of the abundance is concentrated around with only a smaller contribution from Ca. Because the initial composition has only a small amount of nuclei (= mol/g), the only contribution to comes from parts of the initial and abundances (see discussion in section 3.2). Again the competition of neutron capture and EC at Ca and Ne, respectively, is critical in determining the relative distribution of and nuclei in the inner crust.
In crusts with initial superburst ashes, Urca cooling is comparable to the case of extreme rp-process ashes (Fig. 14), however, cooling comes almost exclusively from the Sc-Ti pair. All other Urca cooling pairs are at least a factor of 50 weaker. Heating is significantly higher than for either of the X-ray burst ashes cases.
4.1 Reaction Sequences
The reaction sequences obtained here with a full reaction network differ substantially from previous work, especially when compared to models that consider only a single species at a given time such as Haensel & Zdunik (1990, 2008). One important difference are the EC/ Urca pairs already discussed in Schatz et al. (2014), which can occur when both the parent and the daughter nucleus of an EC are present, and the decay is not fully blocked.
In the case of the initial Fe ashes, we agree with Haensel & Zdunik (1990) that the composition reaches the neutron drip line with the destruction of Ar at around . However, taking into account the finite time needed for the transition and the change in neutron density during the transition, we find that the reaction flow branches into an EC sequence and a neutron capture sequence unlike Haensel & Zdunik (1990). This leads to the appearance of more than one species in the composition. Also, unlike Haensel & Zdunik (1990), we confirm that beyond neutron drip, nuclei are converted rapidly via the superthreshold electron capture cascades (SECs) found in Gupta et al. (2008) into much lighter nuclei. For example, Ar is converted into Mg in a single step so that Mg is already produced at the time of Ar destruction at and =31.6 MeV. This is in contrast to Haensel & Zdunik (1990), where, after Ar destruction at , several EC reactions at stepwise increasing have to occur before Mg is produced at . In addition, we find that the reaction sequence branches at Si leading to the additional production of Si via neutron captures, resulting in a two component composition of Mg and Si.
The onset of pycnonuclear fusion also differs. In Haensel & Zdunik (1990), the first fusion is NeNe, triggered by EC on Mg at and MeV. We find that, due to our mass model that includes shell effects, the threshold for Mg(EC) is higher so that Mg(EC) occurs deeper at . As our calculation allows for the presence of multiple nuclear species, we find that the lighter nuclides produced by the ensuing SEC chains preferably fuse with the still abundant Mg, rather with themselves, leading to the occurrence of fusion between unlike nuclides. Furthermore, the SEC chains on Mg are faster, leading to lighter nuclides before fusion sets in. A branching at O, where EC and fusion competes, leads to the creation of two major species undergoing fusion, resulting in two major fusion reactions, NMgK and OMgCa. In addition, at the larger depth where these reactions are triggered, MgMgCr becomes significant as well. In summary, instead of a single fusion reaction between like species, three fusion reactions occur simultaneously, two of them between very different species.
A fundamental difference here is that we do not find a large abundance buildup of the fusion reaction products as found in Haensel & Zdunik (1990, 2008). Instead, the reaction product is immediately recycled via an SEC. Fusion reactions therefore lead to fusion-SEC cycles. In the case of Mg, for example, the resulting net reaction of all fusion-SEC cycles is MgMgMg+40n. The fusion-SEC cycles slowly convert Mg into neutrons, until the increasing neutron density shifts the composition to more neutron rich nuclei. A single fusion reaction effectively only destroys a single Mg nucleus instead of two, allowing for more fusion reactions at a shallower depth. A fusion induced cycle for the destruction of Mg has been described in Steiner (2012), who used a Quasi Statistical Equilibrium model that allows for the presence of multiple species. He found a cycle that starts at Mg with a SEC to C, followed by C+CMg(,4n)Mg. However, taking into account the finite speed of the nuclear reactions, we find that Mg and the lighter nuclides produced by a SEC coexist leading to asymmetric fusion reactions. Our model also tracks individual reaction channels and can therefore resolve branchings between competing reactions. This also broadens the range of fusion reactions.
The reaction sequences starting with broader initial composition distributions are of similar type, characterized by 4 phases - EC chains without neutrons, EC chains with neutron-induced reactions, SECs at neutron drip, and pycnonuclear fusion. As soon as neutrons are released, the evolution in a given isobaric EC chain starts to depend on what happens in other chains. Initial abundances of lighter species such as Ne, Mg, or Si are transformed into even lighter nuclei, which then undergo pycnonuclear fusion prior to neutron drip. This has already been suggested by Horowitz et al. (2008). We confirm that these reactions occur at a depth around but the types of fusion reactions differ significantly from previous predictions. The most dominant fusion reactions around this depth are O, NN, CO, CN, CC, and OO, but many weaker reactions occur. Ne fusion sets in at slightly higher via NeNe, NeNe, NeO, and NeC. At greater depths we find the pycnonuclear fusion-SEC cycles:
Here the fusion reaction rates determine how rapidly nuclei are converted into free neutrons.
4.1.1 Urca Cooling
Nuclear Urca cooling has been discussed in detail in Schatz et al. (2014); Deibel et al. (2016). With our model temperature of 0.5 GK, we can identify the strongest Urca pairs for the different initial compositions investigated here. For superburst ashes, is the only strong Urca cooling pair, owing to the limited mass range of the nuclei in the burst ashes. The strongest Urca pair identified in Schatz et al. (2014) for superburst ashes, has been shown to be ineffective as a consequence of newly measured masses and newly calculated transition strengths from shell model calculations (Meisel et al., 2015a).
The situation for the pair is less clear. One of the key prerequisites for a strong Urca cycle in an odd chain is a strong allowed ground state to ground state (or within a few 10 keV of the ground state) EC and transition. Experimental studies of the ground state of Ti indicate a spin and parity of 1/2 (Maierbeck et al., 2009). The ground state of Sc is expected to be 7/2, based on systematics. Such a large spin difference would preclude a fast ground state to ground state transition. Crawford et al. (2010) therefore assume that the significant missing strength that they observed in a study of the Sc decay is not due to a ground state transition, but due to a sizable -delayed neutron emission branch. As low lying excited states are not expected in these isotopes, one would have to conclude that, in contrast to the QRPA-fY predictions used here, the Urca pair is not effective, and that therefore there no strong Urca cooling pair exists in crusts composed of superburst ashes. Nevertheless, an experimental confirmation of the absence of a strong ground state to ground state transition in the decay of Sc, or direct evidence of a ground state to ground state -delayed neutron emission branch, would be desirable to clarify whether Urca cooling can play a role in accreting neutron stars with superbursts.
For ashes from regular X-ray bursts that produce a wider range of nuclei, additional strong nuclear Urca pairs can be populated. For KEPLER X-ray burst ashes, the dominant cooling comes from the pair (Tab. 3). This pair had also been identified in Schatz et al. (2014) when using the FRDM mass model (see below). Experimental data indeed indicate a strong ground state to ground state transition, although the experimentally derived indicates a roughly a factor of 4 slower transition than what is used in our model (Klotz et al., 1993; Guillemaud-Mueller et al., 1984). Another very strong Urca pair for KEPLER X-ray burst ashes is . The experimentally derived value of 5.2 is very close to the QRPA-fY prediction (5.0) (Tripathi et al., 2008) and would confirm a very strong Urca pair. However, there is some debate about the experimental interpretation, in particular about the parity of the Al ground state (Yordanov et al., 2010).
For our calculation with the ashes of an extreme X-ray burst, the production of nuclei opens up a number of additional possible Urca cooling pairs (Tab. 2), the strongest of which had already been identified in Schatz et al. (2014). Indeed, while is also important, Urca cooling is largely dominated by the and chains. Not much is known experimentally about the relevant nuclei Sr, Y, Zr, and Nb. Data on ground state to ground state transitions, and the masses of Sr and Y remain to be determined to put the existence and strengths of these Urca cooling pairs on solid experimental footing.
The strongest Urca cooling pairs identified here are located at depths in the range of to . Pairs at shallower depths are considerably weaker, though they may still be important in limiting the strong shallow heating that is indicated by observations of cooling transients and superburst ignition depths (Deibel et al., 2016; Meisel & Deibel, 2017). Urca cooling at greater depths is largely precluded by the onset of neutron emission and capture reactions, that tend to deplete odd mass chains (see below), and prevent the coexistence of parent and daughter nuclides once the drip line has been reached.
4.1.2 Appearance of free neutrons
We find that free neutrons start to play a role long before the composition reaches the neutron drip line, the traditional point where free neutrons appear. This early release of neutrons stems from EC reactions that populate neutron unbound excited states (), which then decay by neutron emission. There are two basic mechanisms for EC to populate high lying excited states. First, the EC threshold of a particular reaction may be increased by the excitation energy of the lowest lying daughter state for an allowed transition. If this state is above the neutron separation energy, neutron emission will occur. Also, once the transition proceeds at threshold, can be higher than the threshold of the subsequent EC reaction, leading again to the population of excited daughter states that may be above the neutron separation energy. These effects can occur in even and odd chains. An example is Rb(EC,n)Kr discussed in section 3.2, where the lowest lying EC transition is predicted to go to a 6.9 MeV state in Kr, close to . This leads to neutron emission relatively close to stability. It will be important to explore how lower lying forbidden transitions not included in the QRPA-fY calculations may reduce this effect.
The second mechanism to release neutrons prior to reaching the neutron drip line is the odd-even staggering of in even chains, . In these chains, an EC reaction on an even-even nucleus is immediately followed by an EC reaction on an odd-odd nucleus (Haensel & Zdunik, 1990), where excited states up to can be populated (Gupta et al., 2007). Neutron emission is possible if . depends strongly on the mass model (Meisel et al., 2015a). For typical values of 3 MeV neutron release would only start closer to the drip line at MeV. However, the FRDM mass model predicts significantly larger in some cases.
The early release of neutrons does not lead to a buildup of a large free neutron abundance. Instead, the released neutrons are recaptured by other nuclei present at the same depth. This is a feature of the multi-component composition of the outer crust. Nuclei with the largest abundance and largest neutron capture cross sections will dominate the neutron absorption. Interestingly this tends to lead to the depletion of odd chains, starting as early as at (see Fig. 21). As the odd mass chains tend to have most of the nuclear Urca pairs, the early release of neutrons strongly limits Urca cooling in the deeper regions of the outer crust.
The required for pre-drip line neutron release varies greatly from mass chain to mass chain. To illustrate this point, we provide a simple estimate for the minimum for neutron release in each mass chain, based on nuclear mass differences () (Fig. 22). EC transitions are assumed to proceed when , with being the daughter excitation energy of the lowest lying EC transition. This simple estimate neglects lattice energy and finite temperature corrections, which depend on overall composition and astrophysical parameters and are included in the full network calculation. Clearly, the depth of early neutron release depends strongly on the mass chain and therefore on the initial composition created by thermonuclear burning on the neutron star surface. While transitions into excited states move the release of neutrons to shallower depths, on average by 6 MeV in (red solid line in Fig. 22), the odd-even staggering of alone leads to significant neutron release (red dashed line in Fig. 22) prior to reaching the neutron drip line (blue line in Fig. 22). All curves in Fig. 22 show a pronounced variation in from mass chain to mass chain of up to about 10 MeV. Therefore, regardless of the detailed transition energies and odd-even staggering, there will be a transition region between the outer and inner crust where some mass chains release neutrons, and others capture them. The characteristics of the compositional evolution in this region will depend sensitively on the composition of the X-ray burst ashes.
4.1.3 Superthreshold Electron Capture Cascades
In agreement with Gupta et al. (2008) we find that EC and neutron emission sequences at the neutron drip line proceed not in single steps but in a rapid sequence spanning many isotopic chains. Once the neutron drip line is reached, the composition is therefore rapidly converted into lower nuclei. We also find that similar rapid sequences of EC and neutron emission drive the products of pycnonuclear fusion instantly back to the originating nucleus, leading to pycnonuclear fusion-SEC cycles (section 4.1).
In the SEC mechanism, EC with neutron emission drives the composition away from the neutron drip line towards lower EC thresholds. EC reactions can then become faster than neutron capture reactions and another EC reaction follows immediately, before neutron capture can restore (n,)-(,n) equilibrium in the isotopic chain. If the subsequent EC reaction again leads to neutron emission, the sequence can repeat many times, greatly accelerating the conversion of heavier elements into lighter ones.
An example is the SEC sequence shown in Fig. 5 for the initial Fe composition. For the neutron density and temperature at the location shown, the dominant (n,)-(,n) equilibrium abundance in the isotopic chains would be