HYDRODYNAMIC MODELS OF TYPE I X-RAY BURSTS: METALLICITY EFFECTS

Jordi José; Fermín Moreno; Anuj Parikh; Christian Iliadis

doi:10.1088/0067-0049/189/1/204

1. INTRODUCTION

Type I X-ray bursts (hereafter, XRBs) are cataclysmic stellar events. They are powered by thermonuclear runaways (TNRs) in the H/He-rich envelopes accreted onto neutron stars in close binary systems (see reviews by Bildsten 1998; Lewin et al. 1993, 1995; Psaltis 2006; Schatz & Rehm 2006; Strohmayer & Bildsten 2006). These events constitute the most frequent type of thermonuclear stellar explosion in the Galaxy (the third, in terms of total energy output after supernovae and classical novae), in part because of their short recurrence period (hours to days). About 90 Galactic low-mass X-ray binaries exhibiting such bursting behavior (with burst durations of τ_burst∼ 10–100 s) have been found since the discovery of XRBs by Grindlay et al. (1976), and independently, by Belian et al. (1976). Type I XRBs and their associated nucleosynthesis have been extensively modeled by different groups (see pioneering work by Woosley & Taam 1976; Maraschi & Cavaliere 1977; Joss 1977), reflecting the astrophysical interest in determining the nuclear processes that power the explosion, the light curve, as well as in providing reliable estimates for the chemical composition of the neutron star surface (see Schatz et al. 1999; Parikh et al. 2008, and references therein).

With a neutron star hosting the explosion, temperatures, and densities in the accreted envelope reach high values: T_peak > 10⁹ K, and ρ ∼ 10⁶ g cm⁻³. As a result, detailed nucleosynthesis studies require the use of hundreds of isotopes, linked by thousands of nuclear interactions, extending all the way up to the SnSbTe-mass region (Schatz et al. 2001) or beyond (the extent of the nuclear activity⁶ in the XRB nucleosynthesis study of Koike et al. 2004 reaches ¹²⁶Xe). Indeed, the extent of the rp-process in XRBs is still not clear: recent experimental work now shows that it will be more difficult to reach the SnSbTe-mass region (Elomaa et al. 2009). Because of computational constraints, XRB nucleosynthesis studies have been traditionally performed using limited nuclear reaction networks, truncated near Ni (Woosley & Weaver 1984; Taam et al. 1993, 1996; all using a 19-isotope network), Kr (Hanawa et al. 1983—274-isotope network; Koike et al. 1999—463 nuclides), Cd (Wallace & Woosley 1984; 16-isotope network), or Y (Wallace & Woosley 1981—250-isotope network). On the other hand, Schatz et al. (1999, 2001) have carried out very detailed nucleosynthesis calculations with a network containing more than 600 isotopes (up to Xe in Schatz et al. 2001), but using a one-zone approach. Koike et al. (2004) have also performed detailed one-zone nucleosynthesis calculations, with temperature and density profiles obtained from a spherically symmetric evolutionary code, linked to a 1270-isotope network extending up to ¹⁹⁸Bi.

Until recently, it has not been possible to couple hydrodynamic stellar calculations (in one dimension) and detailed networks. Recent efforts include Fisker et al. (2004, 2006, 2007, 2008), and Tan et al. (2007; ∼300 isotopes, up to ¹⁰⁷Te), José & Moreno (2006; 2640 reactions and 478 isotopes, up to Te), or Woosley et al. (2004) and Heger et al. (2007; up to 1300 isotopes with an adaptive network). This has prompted a detailed analysis of the nuclear activity powering the bursts. The most detailed work to date is that of Fisker et al. (2008), in the context of the one-dimensional general relativistic hydrodynamic code AGILE (Liebendörfer et al. 2002), linked to a nuclear reaction network containing 304 isotopes: a thorough analysis of the main nuclear activity in one characteristic burst is reported (although details for a sequence of five consecutive, "representative" bursts are also outlined). However, because of the specific choice of metallicity (Z = 10⁻³, for the accreted matter) and mass-accretion rate ( $\dot{M} \sim 10^{17}$ g s⁻¹) adopted, the nuclear activity does not extend much beyond mass A ∼ 65, as a result of compositional inertia effects, that quench further extension of the nuclear path. Hence, the flow does not reach the SnSbTe-mass region, which was suggested as a natural endpoint in XRB nucleosynthesis studies (see Schatz et al. 1999, 2001).

Clearly, the identification of the most relevant reactions in the A ∼ 65–100 mass region remains to be addressed in detail in the framework of hydrodynamic simulations. This is particularly relevant since, as first pointed out by Hanawa et al. (1983), proton captures on heavy nuclei (i.e., the rp-process) have a dramatic effect on the shape of XRB light curves. To this end, a new set of type I XRBs have been computed with SHIVA, a one-dimensional, spherically symmetric, hydrodynamic, implicit, Lagrangian code, used extensively in the modeling of classical nova outbursts (see José & Hernanz 1998). The code has been linked to a fully updated nuclear reaction network containing 324 nuclides and 1392 nuclear processes, a subset of that used in Parikh et al. (2008), and includes the most relevant charged-particle-induced reactions occurring between ¹H and ¹⁰⁷Te, as well as their corresponding reverse processes. It is worth noting that the size of this network is similar (though slightly larger) to that adopted by Fisker et al. (2008). In order to set up the reaction rate library for our study, we started by adopting the proton drip line predicted by Audi et al. (2003a, 2003b). Experimental rates are available for a small subset of reactions (adopted from Angulo et al. 1999; Iliadis et al. 2001, and some recent updates for selected reactions). For all other reactions for which experimental rates are not available, we used the rates from the Hauser–Feshbach codes MOST (Goriely 1998; Arnould & Goriely 2006) and NON-SMOKER (Rauscher & Thielemann 2000; for details see Parikh et al. 2008). Neutron captures are disregarded since our early test calculations revealed that they play a minor role in XRB nucleosynthesis. All reaction rates incorporate the effects of thermal excitations in the target nuclei (Rauscher & Thielemann 2000). Screening factors are taken from Graboske et al. (1973) and DeWitt et al. (1973). For the weak interactions, β-delayed nucleon emission and laboratory decay rates (Audi et al. 2003a) have been adopted. For a discussion of employing stellar versus laboratory decay rates, see Woosley et al. (2004). It is worth noting, however, that many computed stellar decay rates (Fuller et al. 1982a, 1982b; Langanke & Martinez-Pinedo 2000) do not converge to their laboratory values at lower temperatures and densities, calling into question the model used for these calculations. Studies employing properly converging stellar decay rates for all isotopes relevant to XRB nucleosynthesis have not been performed by any group yet, and would certainly be interesting, although the results presented in this work would not be dramatically affected by their inclusion.

SHIVA uses a time-dependent formalism for convective transport whenever the characteristic convective timescale becomes larger than the integration time step. Partial mixing between adjacent convective shells is treated by means of a diffusion equation (Prialnik et al. 1979). No additional semiconvection or thermohaline mixing is considered. Models make use of Iben's (1975) opacity fits, better suited than the OPAL opacities for astrophysical environments that exhibit strong variations in metallicity, as in XRB nucleosynthesis. However, plans to incorporate these more realistic opacities are currently underway. The adopted equation of state includes contributions from the electron gas (with different degrees of degeneracy; Blinnikov et al. 1996), a multicomponent ion plasma, and radiation; Coulomb corrections to the electronic pressure are also taken into account.

Accretion is computed by redistributing material through a constant number of envelope shells (see Kutter & Sparks 1980 for details). To handle this, a tiny envelope, containing 1.1 × 10¹⁸ g of material (less than 1 permil of the total envelope mass accreted during the first bursting episode), distributed through all the envelope shells, is put initially in place (the influence of the number of envelope shells on burst properties will be discussed in Section 3). The model is then relaxed using a few, very large time steps, to guarantee hydrostatic equilibrium. The temperature at the bottom of the envelope barely reaches 2.7 × 10⁷ K, whereas the density is just 1.4 × 10³ g cm⁻³ (corresponding to a pressure of 5.7 × 10¹⁸ dyn cm⁻²). Mass accretion and nuclear reactions are then initiated.

Special emphasis is placed on the effect of the initial metallicity of the accreted matter on the main nuclear path, which in turn, will affect the final post-burst envelope composition and the shape of the light curves.

The structure of the manuscript is as follows. In Section 2, we analyze the main features (nuclear path, nucleosynthesis, light curves, etc.) of a series of four bursts computed in a model with solar-like accreted material. The effect of the resolution adopted in this model is discussed in Section 3. A detailed analysis of the impact of the metallicity of the accreted material on burst properties is given in Section 4. Finally, a comparison with previous work, together with a thorough analysis of the corrections posed by general relativity, are discussed in Section 5.

2. MODEL 1

We summarize the gross properties of a series of thermonuclear bursts driven by mass accretion onto a 1.4 M_☉ neutron star (L_ini = 1.6 × 10³⁴ erg s⁻¹ = 4.14 L_☉), at a rate ${\dot{M}}_{\rm acc} = 1.75 \times 10^{-9}\;M_\odot$ yr⁻¹ (corresponding to 0.08 ${\dot{M}}_{{\rm Edd}}$ ). The composition of the accreted material (see Table 1) is assumed to be solar-like (X = 0.7048, Y = 0.2752, Z = 0.02). All metals are initially assumed to be in the form of ¹⁴N, following the rapid rearrangement of CNO isotopes that naturally occurs early in the burst (see Woosley et al. 2004). This model is qualitatively similar to model ZM, computed by Woosley et al. (2004) in the framework of the one-dimensional, hydrodynamic, implicit code KEPLER. This choice is made intentionally to compare with previous hydrodynamic studies. Note, however, that Woosley et al. assume a value of 10 km for the neutron star radius. In contrast, our model yields a value of 13.1 km, following the integration of the neutron star structure⁷ from the core to its surface, in hydrostatic equilibrium. Differences in the neutron star size (and in turn, in surface gravity) may effect the strength of the explosion (mass accreted, peak temperature, nucleosynthesis, etc.).

Table 1. Summary of the Models Computed in this Work^a

Model	M_NS (M_☉)	Metallicity	Envelope Shells	Bursts Computed
1	1.4	0.02	60	4
2	1.4	0.02	200	2
3	1.4	1 × 10⁻³	60	5

Note. ^aA mass-accretion rate of 1.75 × 10⁻⁹ M_☉ yr⁻¹ has been adopted for all models.

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2.1. First Burst

The piling-up of solar-like material on top of the neutron star during the accretion stage progressively compresses and heats the envelope (consisting of 60 shells). Indeed, only 145 s since the beginning of accretion, the temperature at the base of the envelope reaches T_base = 5 × 10⁷ K (with ρ_base exceeding 10⁴ g cm⁻³).

The early nuclear activity is fully dominated by H-burning through hot CNO-cycle reactions, initiated by proton captures on ¹⁴N nuclei. At this stage (t = 2327 s), the envelope achieves T_base ∼ 10⁸ K (ρ_base ∼ 6.5 × 10⁴ g cm⁻³), with an energy generation rate of epsilon _nuc ∼ 1.2 × 10¹⁴ erg g⁻¹ s⁻¹. The main reaction path (see Figure 1) is led by ¹⁵N(p, α)¹²C, which powers ¹²C(p, γ)¹³N(p, γ)¹⁴O(β⁺)¹⁴N(p, γ)¹⁵O(β⁺)¹⁵N. This suite of nuclear processes competes with ¹³N(β⁺)¹³C(p, γ)¹⁴N, and to a lesser extent, with ¹⁵N(p, γ)¹⁶O(p, γ)¹⁷F(β⁺)¹⁷O(p, α)¹⁴N. Besides H (X = 0.689) and ⁴He (Y = 0.290), the next most abundant species in the envelope is now ¹⁵O (1.2 × 10⁻²). The amount of unburned ¹⁴N has dropped to 2.4 × 10⁻³ (in the following, when discussing the nucleosynthesis, we will refer to abundances by mass, i.e., mass fractions). Other CNO-group nuclei, such as ¹²C (5.4 × 10⁻⁵), ¹³N (4.2 × 10⁻⁴), ¹⁴O (6.1 × 10⁻³), or ¹⁶O (1.1 × 10⁻⁵), have already achieved an abundance ⩾10⁻⁵.

**Figure 1.** Main nuclear activity at the innermost envelope shell for model 1 (M_NS = 1.4 M_☉, ${\dot{M}}_{\rm acc} = 1.75 \times 10^{-9}\;M_\odot$ yr⁻¹, Z = 0.02), at the early stages of accretion (T_base = 9.9 × 10⁷ K). Upper panel: mass fractions of the most abundant species (X >10⁻⁵). Lower panel: main reaction fluxes (F ⩾10⁻⁹ reactions s⁻¹ cm⁻³).
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**Figure 1.** Main nuclear activity at the innermost envelope shell for model 1 (M_NS = 1.4 M_☉, ${\dot{M}}_{\rm acc} = 1.75 \times 10^{-9}\;M_\odot$ yr⁻¹, Z = 0.02), at the early stages of accretion (T_base = 9.9 × 10⁷ K). Upper panel: mass fractions of the most abundant species (X >10⁻⁵). Lower panel: main reaction fluxes (F ⩾10⁻⁹ reactions s⁻¹ cm⁻³).
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T_base reaches 2.1 × 10⁸ K 4.49 hr (16,163 s) after the beginning of accretion. Mass accretion in highly degenerate conditions has compressed the envelope base to a density of ρ_base = 2.7 × 10⁵ g cm⁻³ (P_base = 9.1 × 10²¹ dyn cm⁻²). The total luminosity of the star has now increased to a value of 2.5 × 10³⁵ erg s⁻¹. The main nuclear activity⁸ (Figure 2) is still dominated by proton captures and β⁺-decays, characteristics of the hot CNO cycle, now supplemented by ¹⁵N(p, γ)¹⁶O(p, γ)¹⁷F(p, γ)¹⁸Ne(β⁺)¹⁸F(p, α)¹⁵O, and by the 3α reaction. The numerous p-captures have reduced the hydrogen content to 0.408. In turn, ⁴He has increased to 0.570, becoming now the most abundant species at the base of the envelope (followed by the short-lived species ¹⁴O [7.7 × 10⁻³] and ¹⁵O [1.4 × 10⁻²]), while most of the CNO nuclei have been reduced to 10⁻⁷–10⁻⁹, by mass). The extension of the main nuclear activity reaches ⁴⁰Ca. Indeed, ³²S and ⁴⁰Ca are the only species in the Ne–Ca mass region with abundances exceeding 10⁻⁹.

**Figure 2.** Same as Figure 1, but for T_base = 2.1 × 10⁸ K.
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Convection sets in erratically, at ∼1 m above the core–envelope interface (the overall envelope size, Δz, is ∼14 m), when T_base reaches 3.9 × 10⁸ K, and progressively extends throughout the whole envelope.

Shortly after, at t = 5.88 hr (21,181 s), T_base reaches 4 × 10⁸ K (with ρ_base = 2.9 × 10⁵ g cm⁻³ and P_base = 1.2 × 10²² dyn cm⁻²). The hydrogen content has dropped to 0.209, whereas ⁴He achieves 0.625. In turn, the rate of nuclear energy generation has increased to a value of 2.8 × 10¹⁶ erg g⁻¹ s⁻¹. The metallicity of this innermost envelope shell has increased from an initial value of 0.02 to 0.17, due to leakage from the CNO cycle (mainly powered by ¹⁵O(α, γ)). As before, the next most abundant species are ¹⁴O (6.9 × 10⁻²) and ¹⁵O (6.5 × 10⁻²), but the number of isotopes with moderately large abundances has now increased. Indeed, ⁴⁰Ca, ²²Mg, ¹⁸Ne, ³⁴Ar, ⁴⁸Cr, and ^42,44Ti have achieved mass fractions of the order of 10⁻³. The nuclear activity extends as far as ⁵³Co. The largest reaction fluxes (defined as the number of reactions per unit time and volume) correspond to the equilibrium processes ²¹Mg(p, γ)²²Al(γ, p)²¹Mg, ³⁰S(p, γ)³¹Cl(γ, p)³⁰S, and ²⁵Si(p, γ)²⁶P(γ, p)²⁵Si. Additional activity is powered by 3α → ¹²C(p, γ)¹³N(p, γ)¹⁴O, followed by ¹⁴O(α, p)¹⁷F(p, γ)¹⁸Ne. The suite of secondary nuclear paths is rich and complex (see Figure 3), and is mainly dominated by p-capture reactions and β⁺-decays, as well as by the CNO-breakout reaction ¹⁵O(α, γ)¹⁹Ne. It is worth noting that the main nuclear path above Ca begins to move away from the valley of stability, toward the proton-drip line (see Figure 3, lower panel).

**Figure 3.** Same as Figure 1, but for T_base = 4 × 10⁸ K.
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Just 2.3 s later (t = 21,183 s), T_base achieves 5 × 10⁸ K. ρ_base has slightly decreased to 2.3 × 10⁵ g cm⁻³ because of a mild envelope expansion (Δz ∼ 15.5 m). Note, however, that P_base = 1.2 × 10²² dyn cm⁻². Hence, the TNR is taking place nearly at constant pressure. A time-dependent, convective mixing with adjacent shells, with a characteristic timescale of τ_conv ∼ 10⁻⁴ s (v_conv ∼ 10³–10⁵ cm s⁻¹), causes a slight increase in the H abundance at the base of the envelope. Indeed, the H abundance is now 0.288, by mass, whereas the ⁴He content has slightly decreased to 0.563 (due to the high temperatures, favoring α-captures). The next most abundant species is now ¹⁸Ne (4.4 × 10⁻²), together with ^14,15O (4.2 × 10⁻² and 1.8 × 10⁻², respectively). Several isotopes, such as ^21,22Mg, ^29,30S, ^50,52Fe, ²⁷P, ^24,25Si, ^49,50,51Mn, and ³⁴Ar, have achieved abundances an order of magnitude lower (∼10⁻³). The nuclear activity extends up to ⁵⁷Cu now, powering an energy generation rate of 1.2 × 10¹⁷ erg g⁻¹ s⁻¹, and an overall luminosity of 1.3 × 10³⁶ erg s⁻¹. The largest fluxes still correspond to the forward and reverse reactions ³⁰S(p, γ)³¹Cl(γ, p)³⁰S, ²¹Mg(p, γ)²²Al(γ, p)²¹Mg, and ²⁵Si(p, γ)²⁶P(γ, p)²⁵Si, together with ¹⁴O(α, p)¹⁷F(p, γ)¹⁸Ne. ¹⁴O+α becomes the most important α-capture reaction, overcoming ¹⁵O(α, γ), or the 3α. Additional activity is driven by ¹⁸Ne(β⁺)¹⁸F(p, α)¹⁵O, ¹⁹Ne(p, γ)²⁰Na(p, γ)²¹Mg, ²¹Na(p, γ)²²Mg(p, γ)²³Al, ¹²C(p, γ)¹³N(p, γ)¹⁴O, and ²⁶Si(p, γ)²⁷P.

A qualitatively similar picture is found when T_base achieves 7 × 10⁸ K (t = 21,185 s), with the most abundant species at the envelope base being H (0.308), ⁴He (0.507), ¹⁸Ne (4.8 × 10⁻²), ²²Mg (2.5 × 10⁻²), ^29,30S (1.1 × 10⁻², and 2.2 × 10⁻², respectively), and ^24,25Si (1.1 × 10⁻² and 1.7 × 10⁻², respectively). The number of species with abundances of the order of 10⁻³ includes now ^54,55,56Ni, ¹⁵O, ²⁸S, ⁵²Fe, ²⁷P, ³⁸Ca, and ^33,34Ar, with the main nuclear activity (see Figure 4) extending all the way up to ⁶⁰Zn. The largest reaction fluxes are achieved by the equilibrium processes described before, supplemented now by ²⁶Si(p, γ)²⁷P(γ, p)²⁶Si, ²²Mg(p, γ)²³Al(γ, p)²²Mg, ²⁹S(p, γ)³⁰Cl(γ, p)²⁹S, and ¹⁶O(p, γ)¹⁷F(γ, p)¹⁶O, followed by p-capture reactions (and β⁺-decays) on Ne–Mg nuclei, such as ¹⁹Ne(p, γ)²⁰Na(p, γ)²¹Mg(β⁺)²¹Na(p, γ)²²Mg, or ¹⁷F(p, γ)¹⁸Ne(β⁺)¹⁸F(p, α)¹⁵O, and by the α-capture reactions ¹⁵O(α, γ)¹⁹Ne, ¹⁴O(α, p)¹⁷F, and the 3α.

**Figure 4.** Same as Figure 1, but for T_base = 7 × 10⁸ K.
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One second later (t = 21,186 s), T_base achieves 9 × 10⁸ K. The hectic nuclear activity, which at this stage releases epsilon _nuc ∼ 3.7 × 10¹⁷ erg g⁻¹ s⁻¹, has reduced the H and ⁴He abundances down to 0.262 and 0.457, respectively. The next most abundant species are now ²²Mg, ²⁵Si, ^28,29,30S, ^33,34Ar, and ⁶⁰Zn, all with mass fractions ∼10⁻². The main nuclear activity has extended up to ⁶⁸Se. Aside from equilibrium (p, γ) and (γ, p) pairs (that involve ¹⁶O–¹⁷F and a handful of species in the mass range Mg–Zn), the largest reaction fluxes correspond to a suite of p-captures and β⁺-decays (see Figure 5, lower panel), mainly ²⁵Al(p, γ)²⁶Si and ^27,28,29P(p, γ)^28,29,30S. Moreover, the most important α-capture reactions, ²²Mg(α, p)²⁵Al, the 3α, ¹⁸Ne(α, p)²¹Na, and ¹⁴O(α, p)¹⁷F, have fluxes of the order of log F ∼ −3 (note the moderate extension of α-captures toward heavier species as a result of the higher temperatures). A very limited nuclear activity in the A = 65–100 mass region is, at this stage, driven by ⁶⁵Ge(p, γ)⁶⁶As(p, γ)⁶⁷Se(β⁺)⁶⁷As(p, γ)⁶⁸Se (with log F ∼ −8), and ⁶⁶As(β⁺)⁶⁶Ge(p, γ)⁶⁷ As (log F ∼ −9).

**Figure 5.** Same as Figure 1, but for T_base = 9 × 10⁸ K.
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At t = 21,188 s, when T_base achieves 1 × 10⁹ K, the energy generation rate by nuclear reactions reaches its maximum value: epsilon _nuc,max ∼ 4.1 × 10¹⁷ erg g⁻¹ s⁻¹. Two seconds later, the envelope will attain maximum expansion, with a size Δz_max ∼ 44 m.

Proton and α-captures continue to reduce the overall H and He abundances at the envelope base (0.191 and 0.400, respectively). The next most abundant species is now ³⁰S (0.103)—a waiting point for the main nuclear path—followed by ^33,34Ar, ^37,38Ca, ⁴²Ti, ⁴⁶Cr, ⁵⁰Fe, ⁵⁶Ni, and ⁶⁰Zn (all with X_i ∼ 10⁻²). The nuclear activity has reached ⁷⁶Sr (see Figure 6). Time evolution of density, temperature, pressure, and rate of nuclear energy generation, at the innermost envelope shell, as well as of the overall neutron star luminosity and envelope size, are shown in Figures 7 and 8. In the A = 65–100 mass region, in particular, the nuclear activity is now dominated by the chains ⁶⁵Ge(p, γ)⁶⁶As(p, γ)⁶⁷Se(β⁺)⁶⁷As(p, γ)⁶⁸Se (with log F ∼ −7), ⁶⁶As(β⁺)⁶⁶Ge(p, γ)⁶⁷As (log F ∼ −8), ⁶⁵As(p, γ)⁶⁶Se(β⁺)⁶⁶As, and ⁶⁸Se(β⁺)⁶⁸As(p, γ)⁶⁹Se(p, γ)⁷⁰Br(p, γ)⁷¹Kr(β⁺)⁷¹Br(p, γ)⁷²Kr (log F ∼ −9). In terms of energy production, the most important contributions come from ³⁹Ca(p, γ)⁴⁰Sc (log F ∼ −2), multiple (p, γ)-reactions involving species in the mass range A ∼ 20–60, and a handful of α-capture reactions, such as ²²Mg(α, p)²⁵Al, ¹⁸Ne(α, p)²¹Na, the 3α, and ¹⁴O(α, p)¹⁷F (log F ∼ −3).

**Figure 6.** Same as Figure 1, but for T_base = 10⁹ K.
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**Figure 7.** Time evolution of density (upper left panel), temperature (upper right), pressure (lower left), and nuclear energy generation rate (lower right), at the innermost envelope shell for model 1 (M_NS = 1.4 M_☉, ${\dot{M}}_{\rm acc} = 1.75 \times 10^{-9}\;M_\odot$ yr⁻¹, Z = 0.02), along the first bursting episode. The origin of the time coordinate is arbitrarily chosen as 21,150 s, for which T_base ∼ 3 × 10⁸ K.
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**Figure 8.** Same as Figure 7, but for the overall neutron star luminosity (left panel), and envelope size (right panel), as measured from the core–envelope interface.
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Four seconds later, the envelope base achieves a maximum temperature of T_peak ∼ 1.06 × 10⁹ K (similar values are reported in the simulations by Fisker et al. 2008). Besides H (0.220) and ⁴He (0.370), the next most abundant isotope is now ⁶⁰Zn (0.159)—another waiting point for the nuclear flow—followed by ³⁰S (3.3 × 10⁻²), ³⁴Ar (3.2 × 10⁻²), ³⁸Ca (2.4 × 10⁻²), ⁴⁶Cr (1.8 × 10⁻²), ⁵⁰Fe (1.3 × 10⁻²), ^55,56Ni (1.8 × 10⁻² and 2.3 × 10⁻², respectively), and ⁵⁹Zn (1.8 × 10⁻²). As shown in Figure 9, the extension of the main nuclear path reaches ⁸⁰Zr. The nuclear activity in the A = 65–100 mass region is now dominated by ⁶⁵Ge(p, γ)⁶⁶As(p, γ)⁶⁷Se(β⁺)⁶⁷As(p, γ)⁶⁸Se (log F ∼ −6), ⁶⁵As(p, γ)⁶⁶Se(β⁺)⁶⁶As (log F ∼ −7), and ⁶⁶As(β⁺)⁶⁶Ge(p, γ)⁶⁷As, ⁶⁸Se(β⁺)⁶⁸As(p, γ)⁶⁹Se(p, γ)⁷⁰Br(p, γ)⁷¹Kr(β⁺)⁷¹Br(p, γ)⁷²Kr (log F ∼ −8). Energy production is due to dozens of (p, γ)-reactions involving species in the mass range A ∼ 20–60, plus some α-capture reactions, such as the 3α, ¹⁴O(α, p)¹⁷F, ²²Mg(α, p)²⁵Al, and ¹⁸Ne(α, p)²¹Na (log F ∼ −3). Less than a second later (t = 21,192.3 s), the neutron star reaches maximum luminosity, L_max = 3.8 × 10³⁸ erg s⁻¹ (9.8 × 10⁴ L_☉).

**Figure 9.** Same as Figure 1, but for the time when temperature at the envelope base reaches a peak value of T_peak = 1.06 × 10⁹ K.
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The numerous proton captures on many species during the decline from T_peak reduce dramatically the H content in the innermost shell. Indeed, when T_base achieves 9.3 × 10⁸ K (t = 21,200 s), the H abundance drops below 0.1, while X(⁴He) = 0.283. Actually, the most abundant species in this shell is now ⁶⁰Zn (0.43 by mass), followed by ³⁰S (2.7 × 10⁻²), ³⁴Ar (2.1 × 10⁻²), ³⁸Ca (1.3 × 10⁻²), ⁵⁶Ni (1.8 × 10⁻²), and ⁶⁴Ge (3.3 × 10⁻²). The nuclear activity reaches ⁹⁰Ru.

Five seconds later (t = 21,205 s), when T_base drops to 9.0 × 10⁸ K, ⁶⁰Zn achieves a maximum abundance of 0.519, by mass. H has been reduced to 1.3 × 10⁻² (X(⁴He) = 0.226). The next most abundant species are now ²⁶Si, ³⁰S, ³⁴Ar, ³⁸Ca, ⁵⁶Ni, and ⁶⁴Ge (all with mass fractions ∼10⁻²). The nuclear activity has not progressed beyond ⁹⁰Ru. The largest reaction fluxes correspond to proton captures and reverse photodisintegration reactions at equilibrium. Many other nuclear processes (β⁺-decays and α-induced reactions like the triple-α, ¹⁴O(α, p)¹⁷F, or ²²Mg(α, p)) contribute to the overall nuclear activity (see Figure 10). In the A = 65–100 mass region, this is driven by ⁶⁵Ge(p, γ)⁶⁶As (followed either by ⁶⁶As(p, γ)⁶⁷Se(β⁺)⁶⁷As or by ⁶⁶As(β⁺)⁶⁶Ge(p, γ)⁶⁷As), ⁶⁷As(p, γ)⁶⁸Se(β⁺)⁶⁸As(p, γ)⁶⁹Se (log F ∼ −5), ⁶⁹Se(p, γ)⁷⁰Br(β⁺)⁷⁰Se(p, γ)⁷¹Br(β⁺)⁷¹Se, ⁷⁴Rb(β⁺)⁷⁴Kr (log F ∼ −6), and ⁷⁰Br(p, γ)⁷¹Kr(β⁺)⁷¹Br, ⁷²Kr(β⁺)⁷²Br(p, γ)⁷³Kr(p, γ)⁷⁴Rb, ⁷⁴Kr(p, γ)⁷⁵Rb(p, γ)⁷⁶Sr (log F ∼ −7). Energy production is due to (p, γ)-reactions involving species in the mass range A ∼ 20–55, and also due to two α-capture reactions, ¹⁴O(α, p)¹⁷F and ²²Mg(α, p)²⁵Al (log F ∼ −3).

**Figure 10.** Same as Figure 1, but for T_base = 9 × 10⁸ K.
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Following the fast decline in temperature, when T_base = 8.0 × 10⁸ K (t = 21,212 s), the ⁶⁰Zn abundance has dropped to 0.509, due to β⁺-decays. H has been heavily depleted (5 × 10⁻¹¹), whereas ⁴He has been slightly reduced to an abundance of 0.190. The next most abundant species are ¹²C, ³⁰P, ³⁹K, ⁵⁶Ni, ⁶⁰Cu, and ⁶⁴Ge. The extent of the nuclear activity (Figure 11) is still restricted to ⁹⁰Ru; it will not proceed beyond this endpoint, first because of the heavy H depletion, and second, because the temperature is already too low to allow proton- or α-captures on heavier species due to their large Coulomb barriers. At this stage, the single, most important reaction, in terms of reaction fluxes, is the triple-α, followed by a suite of β⁺-decay reactions, such as ²⁶Si(β⁺)^26mAl(β⁺)²⁶Mg, ³⁴Cl(β⁺)³⁴S, ⁶⁰Zn(β⁺)⁶⁰Cu, or ²⁷Si(β⁺)²⁷Al. Several α-captures follow the triple-α reaction as a chain: ¹²C(α, γ)¹⁶O(α, γ)²⁰Ne(α, γ)²⁴Mg(α, γ)²⁸Si(α, γ)³²S, or through alternative paths, proceeding close to the valley of stability, up to ∼ Ar, such as ¹³N(α, p)¹⁶O, ^25,27Al(α, p)^28,30Si, ²²Mg(α, p)²⁵Al, ²²Na(α, p)²⁵Mg, or ^26,27Si(α, p)^29,30P, to quote some representative cases (see Figure 11). epsilon _nuc has already declined to a value of ∼3.8 × 10¹⁵ erg g⁻¹ s⁻¹.

**Figure 11.** Same as Figure 1, but for T_base = 8 × 10⁸ K.
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When T_base drops to 4.3 × 10⁸ K (t = 21,254 s), the nuclear energy generation rate has declined to a value of epsilon _nuc ∼ 3 × 10¹⁴ erg g⁻¹ s⁻¹. As shown in Figure 12, the nuclear activity is dominated by ⁶⁰Zn(β⁺)⁶⁰Cu, because of its very large abundance (X(⁶⁰Zn) = 0.416), and is followed by the triple-α reaction (X(⁴He) = 0.137), and by a suite of β⁺-decays of very abundant isotopes, such as ⁶⁴Ge, ³⁰P, ⁶⁴Ga, or ⁶⁰Cu (all with mass fractions ∼10⁻², except ⁶⁰Cu [0.104]), followed by those of ³⁸K, ^26mAl, ⁶⁸Se, ²⁵Al, ⁶⁸As, ⁶³Ga, ⁵⁹Cu, and ⁶¹Zn. Other species, such as ¹²C, ²⁶Mg, ³⁴S, ³⁹K, or ⁵⁶Ni, have achieved an abundance of 10⁻² by mass at this stage. The envelope has already shrunk to a size Δz ∼ 13 m, whereas the overall luminosity of the star has decreased to L_NS = 7.7 × 10³⁶ erg s⁻¹.

When T_base reaches 2 × 10⁸ K (t = 21,618 s), ⁶⁰Zn has remarkably decayed into ⁶⁰Cu, which now constitutes the most abundant species (with 0.393) at the base of the envelope. Because of the relatively low temperatures, the ⁴He abundance is kept constant, at about 0.136. The next most abundant species are now ¹²C, ²⁶Mg, ³⁰Si, ³⁴S, ³⁹K, ⁵⁶Ni, ⁶⁰Ni, ⁶⁴Ga, and ⁶⁴Zn. With the exception of the limited contribution of the triple-α reaction, the main nuclear path (Figure 13) is fully dominated by a suite of β⁺-decays on numerous species, all the way up to ⁷²Br.

**Figure 13.** Same as Figure 1, but for T_base = 2 × 10⁸ K.
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When t = 28,250 s, T_base reaches a minimum value of 1.67 × 10⁸ K, which we consider to mark the end of the first burst in our simulations. ⁶⁰Cu has decayed already into ⁶⁰Ni, now the most abundant species at the envelope base with a mass fraction of 0.504, followed by ⁴He (0.136, not fully consumed during the TNR), and by ¹²C, ²⁶Mg, ³⁰Si, ³⁴S, ³⁹K, ⁵⁶Ni, ⁶⁰Cu, and ⁶⁴Zn (see Figure 14). The marginal nuclear activity played by a suite of β-decays powers a rate of nuclear energy generation of epsilon _nuc ∼ 8.8 × 10¹¹ erg g⁻¹ s⁻¹. At this stage, the size of the envelope has shrunk to Δz ∼ 8 m, whereas the overall luminosity of the star has decreased to L_NS = 8.3 × 10³⁴ erg s⁻¹.

**Figure 14.** Same as Figure 1, but for the time when temperature at the envelope base achieves a minimum value of T_min = 1.67 × 10⁸ K.
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Profiles of density, temperature, rate of energy generation, pressure, and size, along the accreted envelope, for different snapshots during the first bursting episode computed in this model, are shown in Figures 15 and 16.

**Figure 15.** Profiles of density (upper left panel), temperature (upper right), nuclear energy generation rate (lower left), and envelope size (measured from the core–envelope interface; lower right panel), for model 1 (M_NS = 1.4 M_☉, ${\dot{M}}_{\rm acc} = 1.75 \times 10^{-9}\;M_\odot$ yr⁻¹, Z = 0.02), along the first bursting episode. Labels indicate different moments during the TNR, for which T_base reaches a value of (1) 9.9 × 10⁷ K, (2) 2.1 × 10⁸ K, (3) 4 × 10⁸ K, (4) 6 × 10⁸ K, (5) 8 × 10⁸ K, (6) 1.06 × 10⁹ K (T_peak), (7) 9.3 × 10⁸ K, (8) 7 × 10⁸ K, (9) 4.3 × 10⁸ K, and (10) 1.7 × 10⁸ K (T_min). Note the dramatic decrease in ε_nuc in the innermost layers of the envelope (profiles 8 to 10) as H is depleted.
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**Figure 16.** Left panel: same as Figure 15, but for pressure. Right panel: time evolution of the total pressure at the outermost envelope shell.
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It is worth noting that the main nucleosynthetic activity takes place at the innermost, hottest envelope shell. Because of their lower temperatures and densities, all layers above the ignition shell show a similar nucleosynthetic pattern but somewhat diluted, limiting the extent of the nuclear activity to lower masses. Even though the specific reaction sequences have a clear dependence with depth (see Fisker et al. 2008, for the corresponding analysis in one bursting episode), it is clear that the identification of the main nuclear processes responsible for the nucleosynthesis in XRBs can rely on an accurate account of the activity at the ignition shell, at different stages of the TNR, as performed in this paper. However, we would like to outline schematically how depth influences the extent of the nuclear activity throughout the envelope: while in the innermost shells of our computational domain (encompassing 6.8 × 10²⁰ g) the nuclear activity reaches ⁹⁰Ru, at 1.7 × 10²¹ g above the core–envelope interface, the activity stops around ⁷⁸Sr (the final mass fraction of ⁹⁰Ru barely reaches 10⁻¹⁵, by mass), while close to the surface (2.3 × 10²¹ g), the nuclear activity does not extend beyond ⁷²Se (X(⁹⁰Ru) ∼ 10⁻¹⁸). Moreover, the purely nucleosynthetic imprint in these shells is difficult to assess since it is partially poisoned by changes in the chemical composition driven by convective mixing.

All in all, the mean, mass-averaged chemical composition of the envelope at the end of this first burst is mainly dominated by the presence of intermediate-mass elements (far below the SnSbTe-mass region). This includes ⁶⁰Ni (0.32), ⁴He (0.31), ¹H (0.17), ⁶⁴Zn (0.03), ¹²C (0.02), ⁵²Fe (0.02), or ⁵⁶Fe (0.02) (see Table 2, for the mean composition of all species—stable or with a half-life >1 hr—which achieve X_i > 10⁻⁹), with a nucleosynthesis endpoint (defined by the heaviest isotope with X_i > 10⁻⁹) around ⁸⁹Nb (in agreement with the results reported by Fisker et al. 2008). In terms of overproduction factors, f (ratio of the mass-averaged composition of a given isotope over its solar abundance; see also Figures 17 and 19), ⁴³Ca, ⁴⁵Sc, ⁴⁹Ti, ⁵¹V, ^60,61Ni, ^63,65Cu, ^64,67,68Zn, ⁶⁹Ga, ⁷⁴Se, and ⁷⁸Kr achieve a value of f ∼ 10⁴.

**Figure 17.** Left panels: mass fractions of the 10 most abundant, stable (or τ>1 hr) isotopes for model 1, at the end of the first (upper panel) and fourth bursts (lower panel), respectively. Right panels: same as left panels, but for overproduction factors relative to solar (for f > 10⁻⁵).
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Table 2. Mean Composition of the Envelope (X_i > 10⁻⁹) at the End of Each Burst, for Model 1

Nucleus	Burst 1	Burst 2	Burst 3	Burst 4
¹H	1.7 × 10⁻¹	8.8 × 10⁻²	6.3 × 10⁻²	3.9 × 10⁻²
⁴He	3.1 × 10⁻¹	2.0 × 10⁻¹	1.4 × 10⁻¹	1.0 × 10⁻¹
¹²C	1.7 × 10⁻²	2.5 × 10⁻²	2.0 × 10⁻²	1.9 × 10⁻²
¹³C	8.4 × 10⁻⁵	1.4 × 10⁻⁴	1.3 × 10⁻⁴	1.1 × 10⁻⁴
¹⁴N	2.1 × 10⁻³	1.5 × 10⁻³	1.1 × 10⁻³	7.6 × 10⁻⁴
¹⁵N	2.6 × 10⁻³	2.0 × 10⁻³	1.4 × 10⁻³	9.6 × 10⁻⁴
¹⁶O	5.7 × 10⁻⁴	5.2 × 10⁻⁴	3.3 × 10⁻⁴	2.9 × 10⁻⁴
¹⁷O	3.7 × 10⁻⁶	6.3 × 10⁻⁶	6.8 × 10⁻⁶	1.2 × 10⁻⁵
¹⁸O	6.7 × 10⁻⁵	2.3 × 10⁻⁵	1.7 × 10⁻⁵	6.1 × 10⁻⁶
¹⁸F	6.2 × 10⁻⁵	5.4 × 10⁻⁵	3.1 × 10⁻⁵	2.5 × 10⁻⁵
¹⁹F	2.1 × 10⁻⁴	1.8 × 10⁻⁴	1.1 × 10⁻⁴	7.3 × 10⁻⁵
²⁰Ne	6.3 × 10⁻⁴	5.4 × 10⁻⁴	3.5 × 10⁻⁴	4.0 × 10⁻⁴
²¹Ne	2.0 × 10⁻⁵	9.5 × 10⁻⁶	6.7 × 10⁻⁶	6.2 × 10⁻⁶
²²Ne	9.9 × 10⁻⁵	4.2 × 10⁻⁵	3.2 × 10⁻⁵	2.9 × 10⁻⁵
²²Na	3.3 × 10⁻³	1.9 × 10⁻³	1.2 × 10⁻³	9.2 × 10⁻⁴
²³Na	3.7 × 10⁻⁴	2.1 × 10⁻⁴	1.2 × 10⁻⁴	1.3 × 10⁻⁴
²⁴Mg	1.7 × 10⁻³	1.4 × 10⁻³	9.1 × 10⁻⁴	1.3 × 10⁻³
²⁵Mg	2.6 × 10⁻³	1.7 × 10⁻³	1.2 × 10⁻³	1.5 × 10⁻³
²⁶Mg	1.8 × 10⁻³	2.5 × 10⁻³	1.7 × 10⁻³	1.1 × 10⁻³
²⁶Al^g	2.6 × 10⁻⁴	9.8 × 10⁻⁵	5.7 × 10⁻⁵	1.1 × 10⁻⁴
²⁷Al	1.8 × 10⁻³	1.5 × 10⁻³	8.8 × 10⁻⁴	6.7 × 10⁻⁴
²⁸Si	1.2 × 10⁻³	5.3 × 10⁻³	7.0 × 10⁻³	1.3 × 10⁻²
²⁹Si	3.1 × 10⁻⁴	8.6 × 10⁻⁴	4.6 × 10⁻⁴	6.9 × 10⁻⁴
³⁰Si	3.5 × 10⁻³	5.6 × 10⁻³	3.8 × 10⁻³	3.0 × 10⁻³
³¹P	5.1 × 10⁻⁴	1.4 × 10⁻³	1.7 × 10⁻³	1.8 × 10⁻³
³²S	3.8 × 10⁻⁴	2.9 × 10⁻²	5.8 × 10⁻²	9.0 × 10⁻²
³³S	2.9 × 10⁻⁴	3.1 × 10⁻³	4.3 × 10⁻³	5.1 × 10⁻³
³⁴S	3.2 × 10⁻³	1.1 × 10⁻²	1.6 × 10⁻²	1.8 × 10⁻²
³⁵Cl	1.0 × 10⁻³	4.8 × 10⁻³	1.0 × 10⁻²	1.0 × 10⁻²
³⁶Ar	4.2 × 10⁻⁴	4.0 × 10⁻³	9.2 × 10⁻³	1.0 × 10⁻²
³⁷Cl	3.6 × 10⁻⁷	6.4 × 10⁻⁷	1.8 × 10⁻⁶	4.1 × 10⁻⁶
³⁷Ar	2.3 × 10⁻⁴	5.7 × 10⁻⁴	1.0 × 10⁻³	1.1 × 10⁻³
³⁸Ar	1.9 × 10⁻³	4.9 × 10⁻³	8.5 × 10⁻³	8.9 × 10⁻³
³⁹K	3.8 × 10⁻³	7.8 × 10⁻³	1.2 × 10⁻²	1.3 × 10⁻²
⁴⁰Ca	3.2 × 10⁻³	4.3 × 10⁻³	5.6 × 10⁻³	5.3 × 10⁻³
⁴¹K	...	9.3 × 10⁻⁹	4.8 × 10⁻⁸	7.6 × 10⁻⁸
⁴¹Ca	6.5 × 10⁻⁵	6.9 × 10⁻⁵	6.0 × 10⁻⁵	7.2 × 10⁻⁵
⁴²Ca	5.5 × 10⁻⁴	1.2 × 10⁻³	1.7 × 10⁻³	1.9 × 10⁻³
⁴³Ca	1.8 × 10⁻⁴	3.6 × 10⁻⁴	7.7 × 10⁻⁴	1.2 × 10⁻³
⁴³Sc	4.3 × 10⁻⁴	9.5 × 10⁻⁴	1.2 × 10⁻³	9.7 × 10⁻⁴
⁴⁴Ca	4.3 × 10⁻⁸	3.7 × 10⁻⁶	1.8 × 10⁻⁵	3.2 × 10⁻⁵
⁴⁴Sc	2.5 × 10⁻⁷	1.9 × 10⁻⁵	3.6 × 10⁻⁵	2.9 × 10⁻⁵
⁴⁴Ti	5.8 × 10⁻⁴	7.2 × 10⁻⁴	7.3 × 10⁻⁴	7.5 × 10⁻⁴
⁴⁵Sc	8.2 × 10⁻⁵	7.9 × 10⁻⁵	9.3 × 10⁻⁵	1.2 × 10⁻⁴
⁴⁵Ti	1.5 × 10⁻⁴	1.7 × 10⁻⁴	1.1 × 10⁻⁴	1.0 × 10⁻⁴
⁴⁶Ti	9.5 × 10⁻⁴	1.9 × 10⁻³	2.4 × 10⁻³	2.7 × 10⁻³
⁴⁷Ti	6.6 × 10⁻⁴	8.7 × 10⁻⁴	9.2 × 10⁻⁴	9.5 × 10⁻⁴
⁴⁸Ti	2.1 × 10⁻⁷	5.6 × 10⁻⁷	1.6 × 10⁻⁶	3.4 × 10⁻⁶
⁴⁸V	1.2 × 10⁻⁴	1.4 × 10⁻⁴	2.4 × 10⁻⁴	3.7 × 10⁻⁴
⁴⁸Cr	1.8 × 10⁻³	2.1 × 10⁻³	2.0 × 10⁻³	2.0 × 10⁻³
⁴⁹Ti	1.4 × 10⁻⁷	4.4 × 10⁻⁷	6.9 × 10⁻⁷	8.9 × 10⁻⁷
⁴⁹V	1.5 × 10⁻³	1.5 × 10⁻³	1.2 × 10⁻³	1.4 × 10⁻³
⁵⁰Cr	1.5 × 10⁻³	2.3 × 10⁻³	2.7 × 10⁻³	3.0 × 10⁻³
⁵¹V	3.4 × 10⁻⁶	6.4 × 10⁻⁶	1.6 × 10⁻⁵	3.0 × 10⁻⁵
⁵¹Cr	3.2 × 10⁻³	4.5 × 10⁻³	5.1 × 10⁻³	5.8 × 10⁻³
⁵²Cr	1.5 × 10⁻⁵	2.2 × 10⁻⁵	4.9 × 10⁻⁵	9.4 × 10⁻⁵
⁵²Mn	3.4 × 10⁻³	2.3 × 10⁻³	2.7 × 10⁻³	4.0 × 10⁻³
⁵²Fe	2.0 × 10⁻²	1.5 × 10⁻²	9.9 × 10⁻³	1.2 × 10⁻²
⁵³Mn	1.2 × 10⁻³	1.1 × 10⁻³	8.6 × 10⁻⁴	1.1 × 10⁻³
⁵⁴Fe	1.0 × 10⁻³	1.4 × 10⁻³	1.6 × 10⁻³	1.8 × 10⁻³
⁵⁵Mn	6.6 × 10⁻⁹	2.3 × 10⁻⁸	7.1 × 10⁻⁸	1.6 × 10⁻⁷
⁵⁵Fe	2.3 × 10⁻⁴	3.1 × 10⁻⁴	6.4 × 10⁻⁴	1.0 × 10⁻³
⁵⁵Co	2.9 × 10⁻³	3.8 × 10⁻³	4.0 × 10⁻³	4.3 × 10⁻³
⁵⁶Fe	8.0 × 10⁻⁸	5.4 × 10⁻⁷	1.7 × 10⁻⁶	3.5 × 10⁻⁶
⁵⁶Co	2.2 × 10⁻⁴	5.3 × 10⁻⁴	1.1 × 10⁻³	1.8 × 10⁻³
⁵⁶Ni	2.4 × 10⁻²	4.5 × 10⁻²	5.2 × 10⁻²	5.9 × 10⁻²
⁵⁷Fe	2.6 × 10⁻⁸	1.6 × 10⁻⁷	2.0 × 10⁻⁷	2.5 × 10⁻⁷
⁵⁷Co	2.5 × 10⁻⁴	1.2 × 10⁻⁴	1.4 × 10⁻⁴	2.0 × 10⁻⁴
⁵⁷Ni	6.5 × 10⁻³	3.9 × 10⁻³	2.7 × 10⁻³	2.9 × 10⁻³
⁵⁸Ni	4.7 × 10⁻³	2.7 × 10⁻³	1.9 × 10⁻³	2.2 × 10⁻³
⁵⁹Ni	7.3 × 10⁻³	4.3 × 10⁻³	3.3 × 10⁻³	3.8 × 10⁻³
⁶⁰Ni	3.2 × 10⁻¹	3.3 × 10⁻¹	3.1 × 10⁻¹	3.1 × 10⁻¹
⁶¹Ni	3.9 × 10⁻³	1.7 × 10⁻³	1.9 × 10⁻³	2.4 × 10⁻³
⁶¹Cu	8.2 × 10⁻³	6.0 × 10⁻³	3.5 × 10⁻³	3.8 × 10⁻³
⁶²Ni	5.0 × 10⁻⁴	3.1 × 10⁻⁴	4.4 × 10⁻⁴	5.8 × 10⁻⁴
⁶²Zn	3.2 × 10⁻³	2.5 × 10⁻³	1.4 × 10⁻³	1.5 × 10⁻³
⁶³Cu	4.1 × 10⁻³	4.4 × 10⁻³	3.1 × 10⁻³	3.3 × 10⁻³
⁶⁴Zn	3.4 × 10⁻²	1.0 × 10⁻¹	1.3 × 10⁻¹	1.3 × 10⁻¹
⁶⁵Cu	2.5 × 10⁻⁷	2.9 × 10⁻⁷	5.1 × 10⁻⁷	9.2 × 10⁻⁷
⁶⁵Zn	1.4 × 10⁻³	2.5 × 10⁻³	1.9 × 10⁻³	2.2 × 10⁻³
⁶⁶Zn	3.2 × 10⁻⁵	5.0 × 10⁻⁵	1.3 × 10⁻⁴	2.4 × 10⁻⁴
⁶⁶Ga	4.0 × 10⁻⁴	5.1 × 10⁻⁴	5.5 × 10⁻⁴	6.3 × 10⁻⁴
⁶⁶Ge	5.4 × 10⁻⁴	1.3 × 10⁻³	7.5 × 10⁻⁴	8.7 × 10⁻⁴
⁶⁷Ga	4.7 × 10⁻⁴	1.0 × 10⁻³	9.7 × 10⁻⁴	1.1 × 10⁻³
⁶⁸Ge	2.9 × 10⁻³	2.6 × 10⁻²	4.4 × 10⁻²	4.5 × 10⁻²
⁶⁹Ge	2.2 × 10⁻⁴	1.3 × 10⁻³	1.5 × 10⁻³	1.8 × 10⁻³
⁷⁰Ge	7.3 × 10⁻⁵	4.1 × 10⁻⁴	4.6 × 10⁻⁴	6.0 × 10⁻⁴
⁷¹As	7.7 × 10⁻⁵	4.7 × 10⁻⁴	6.0 × 10⁻⁴	7.7 × 10⁻⁴
⁷²Se	2.5 × 10⁻⁴	5.7 × 10⁻³	1.4 × 10⁻²	1.6 × 10⁻²
⁷³Se	3.0 × 10⁻⁵	4.2 × 10⁻⁴	7.6 × 10⁻⁴	1.0 × 10⁻³
⁷⁴Se	1.3 × 10⁻⁵	1.8 × 10⁻⁴	3.3 × 10⁻⁴	4.8 × 10⁻⁴
⁷⁵Br	1.2 × 10⁻⁵	1.8 × 10⁻⁴	3.7 × 10⁻⁴	5.3 × 10⁻⁴
⁷⁶Kr	3.0 × 10⁻⁵	1.4 × 10⁻³	4.5 × 10⁻³	5.6 × 10⁻³
⁷⁷Kr	4.5 × 10⁻⁶	1.5 × 10⁻⁴	4.1 × 10⁻⁴	6.1 × 10⁻⁴
⁷⁸Kr	2.7 × 10⁻⁶	9.2 × 10⁻⁵	2.5 × 10⁻⁴	4.1 × 10⁻⁴
⁷⁹Kr	1.6 × 10⁻⁶	5.9 × 10⁻⁵	1.7 × 10⁻⁴	2.7 × 10⁻⁴
⁸⁰Sr	3.9 × 10⁻⁶	4.0 × 10⁻⁴	1.6 × 10⁻³	2.2 × 10⁻³
⁸¹Rb	8.4 × 10⁻⁷	6.8 × 10⁻⁵	2.7 × 10⁻⁴	4.3 × 10⁻⁴
⁸²Sr	7.9 × 10⁻⁷	8.2 × 10⁻⁵	3.2 × 10⁻⁴	5.6 × 10⁻⁴
⁸³Sr	3.0 × 10⁻⁷	4.3 × 10⁻⁵	1.9 × 10⁻⁴	3.2 × 10⁻⁴
⁸⁴Sr	1.9 × 10⁻⁷	3.8 × 10⁻⁵	1.4 × 10⁻⁴	2.6 × 10⁻⁴
⁸⁵Y	1.1 × 10⁻⁷	4.0 × 10⁻⁵	2.1 × 10⁻⁴	3.5 × 10⁻⁴
⁸⁶Zr	7.1 × 10⁻⁸	4.4 × 10⁻⁵	2.3 × 10⁻⁴	4.6 × 10⁻⁴
⁸⁷Zr	2.7 × 10⁻⁸	3.6 × 10⁻⁵	1.8 × 10⁻⁴	3.9 × 10⁻⁴
⁸⁸Zr	5.6 × 10⁻⁹	1.8 × 10⁻⁵	1.4 × 10⁻⁴	2.6 × 10⁻⁴
⁸⁹Nb	4.0 × 10⁻⁹	3.1 × 10⁻⁵	2.9 × 10⁻⁴	5.8 × 10⁻⁴
⁹⁰Mo	...	1.1 × 10⁻⁵	1.1 × 10⁻⁴	2.2 × 10⁻⁴
⁹¹Nb	...	5.4 × 10⁻⁶	5.3 × 10⁻⁵	1.1 × 10⁻⁴
⁹²Mo	...	2.1 × 10⁻⁶	1.9 × 10⁻⁵	4.5 × 10⁻⁵
⁹³Tc	...	1.3 × 10⁻⁶	1.3 × 10⁻⁵	4.5 × 10⁻⁵
⁹⁴Tc	...	6.9 × 10⁻⁷	7.6 × 10⁻⁶	3.8 × 10⁻⁵
⁹⁵Ru	...	1.1 × 10⁻⁷	1.1 × 10⁻⁶	4.7 × 10⁻⁶
⁹⁶Ru	...	9.4 × 10⁻⁹	1.1 × 10⁻⁷	7.6 × 10⁻⁷
⁹⁷Ru	...	3.9 × 10⁻⁹	5.1 × 10⁻⁸	3.9 × 10⁻⁷
⁹⁸Ru	...	...	1.3 × 10⁻⁸	9.2 × 10⁻⁸
⁹⁹Rh	...	...	2.4 × 10⁻⁹	1.5 × 10⁻⁸
¹⁰⁰Pd	...	...	...	2.2 × 10⁻⁹

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It is important to stress that the presence of unburned H and ⁴He in the envelope, at the end of the first burst, will have consequences for the subsequent eruptions (see Section 2.2). Note, however (Figure 17), that since the innermost envelope is devoid of H, the next burst will likely initiate well above the core–envelope interface. Moreover, the presence of unburned ¹²C, particularly in the inner envelope layers, has important implications for studies of the physical mechanism that powers superbursts (see Section 2.2).

2.2. Second, Third, and Fourth Bursts

For conciseness, we will focus here on the main differences between the first and successive bursts computed for model 1. A first, remarkable difference is due to the so-called compositional inertia (Taam 1980; Woosley et al. 2004), which accounts for differences in the gross properties of the bursts driven by changes in the chemical content of the envelope. Indeed, after the first burst, the accreted matter will pile up on top of a metal-enriched envelope (the initial metallicity, Z_ini ∼ 0.02, has risen to a mass-averaged value of Z ∼ 0.52, at the end of the first burst) that is devoid of H at its innermost layers. This will cause a shift in the location of the ignition region, progressively moving away from the core–envelope interface in successive flashes (see Figure 18, right panel).

This is schematically shown as well in Figure 18 (left panel), which depicts the mass above the neutron star core, as well as the extension of the convective regions throughout the envelope, for the four bursting episodes computed in this model: the peaks of the explosions (∼10–100 s) correspond to the flat regions of the diagram, whereas the stages of steady accretion (∼5 hr) are indicated by the steep slopes. Note that, in agreement with previous work (Woosley et al. 2004; Fisker et al. 2008), convection mainly develops around the peak of the bursts (during most of the explosion, energy transport is carried by radiation only). Note also that because of fuel consumption (H, He), the location of the ignition shell (and the extent of the convective regions) moves progressively away in mass from the neutron star core.

From the nucleosynthesis viewpoint (Table 2 and Figures 17 and 19), the nuclear activity extends progressively toward heavier species, reaching endpoints (X_i > 10⁻⁹) around ⁸⁹Nb (1st burst), ⁹⁷Ru (2nd burst), ⁹⁹Rh (3rd burst), and ¹⁰⁰Pd (4th burst). The overall mean metallicity of the envelope at the end of each burst is 0.52 (1st burst), 0.71 (2nd burst), 0.80 (3rd burst), and 0.86 (4th burst). This increase in Z reflects both the nuclear activity during each individual burst and the accumulated ashes from previous bursts. A similar mass-averaged ¹²C yield of ∼0.02 is systematically obtained at the end of each of the four bursts computed. This is not enough to power a superburst, which requires X(¹²C)_min ⩾ 0.1, at the envelope base (see Cumming & Bildsten 2001; Strohmayer & Brown 2002; Brown 2004; Cooper & Narayan 2004, 2005; Cumming 2005, or Cooper et al. 2006). With respect to overproduction factors, the increase in nuclear activity reported for successive bursts translates also into larger values, as high as f ∼ 10⁶, for ⁷⁶Se, ^78,80Kr, and ⁸⁴Sr, or f ∼ 10⁵, for species such as ^64,68Zn, ^72,73Ge, ^74,77Se, ⁸²Kr, ^86,87Sr, ⁸⁹Y, and ⁹⁴Mo, for the 4th bursting episode (see Figures 17 and 19).

**Figure 19.** Mean post-burst composition in the envelope for each of the bursting episodes computed for model 1 (left) and model 2 (right).
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A summary of the gross properties of the four bursts computed in this model is given in Table 3. Peak temperatures and luminosities amount to T_peak ∼ (1.1–1.3) × 10⁹ K and L_peak ∼ (1–2) × 10⁵ L_☉, respectively. Recurrence times between bursts of τ_rec ∼ 5–6.5 hr and ratios between persistent⁹ and burst luminosities of α ∼ 35–40 (except for the first burst) have been obtained. These values are in agreement with those inferred from some observed XRB sources (see Galloway et al. 2008) such as the textbook burster GS 1826-24 [τ_rec = 5.74 ± 0.13 hr, α = 41.7 ± 1.6], 4U 1323-62 [τ_rec = 5.3 hr, α = 38 ± 4], or 4U 1608-52 [τ_rec = 4.14–7.5 hr, α = 41–54]. As reported by Woosley et al. (2004), there is also some trend toward stabilization of these values with increasing burst number.

**Figure 20.** Light curves corresponding to the second (upper left panel), third (upper right), and fourth bursts (lower left), and for the overall computed time (lower right), for model 1.
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Table 3. Summary of Burst Properties for Model 1

Burst	T_peak (K)	t(T_peak) (s)	τ_rec (hr)	L_peak (L_☉)	τ_0.01 (s)	α
1	1.06 × 10⁹	21192	5.9	9.7 × 10⁴	75.8	60
2	1.15 × 10⁹	44342	6.4	1.7 × 10⁵	62.3	40
3	1.26 × 10⁹	62137	4.9	2.1 × 10⁵	55.4	34
4	1.12 × 10⁹	80568	5.1	1.2 × 10⁵	75.7	36

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Figure 20 depicts the corresponding light curves from the second to the fourth burst. A quite interesting feature, observed in some XRBs such as 4U 1608-52 (Penninx et al. 1989), 4U 17+2 (Kuulkers et al. 2002), or 4U 1709-267 (Jonker et al. 2004), is the appearance of a double-peaked burst in Figure 20 (lower left panel). Double- (or triple-) peaked bursts can be classified in two categories (Watts & Maurer 2007): the first one corresponds to the so-called photospheric radius expansion bursts, which exhibit multi-peaked bursts in the X-ray band but not in the bolometric luminosity. The second type of multi-peaked events is also visible in the bolometric light curves, and has been attributed to different causes, such as a stepped release of thermonuclear energy caused either by mixing induced by hydrodynamic instabilities (Fujimoto et al. 1988) or driven by a nuclear waiting-point impedance in the thermonuclear reaction flow (Fisker et al. 2004). A preliminary analysis of the 4th burst reported for model 1 suggests a likely nuclear physics origin (waiting-point impedance) for this double-peaked feature (see J. José & F. Moreno 2010, in preparation). The main nuclear activity and the dominant reaction fluxes at peak temperature, for the 4th burst computed in this model, are shown in Figure 21.

**Figure 21.** Upper panel: main nuclear activity at the ignition shell (∼5.6 m above the core–envelope interface), when temperature reaches a peak value of T_peak = 1.16 × 10⁹ K, during the 4th burst computed for model 1. Lower panel: main reaction fluxes (F ⩾10⁻⁹ reactions s⁻¹ cm⁻³).
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3. MODEL 2

In the previous section, we have reported results from a sequence of type I XRBs computed with a coarse resolution, in which the accreted envelope was discretized in 60 shells. We have checked the influence of the adopted number of envelope shells on the gross properties of the bursts by performing another simulation, identical to model 1, but computed with a finer resolution: 200 shells (hereafter, model 2).

A summary of the main properties of the two bursts computed for model 2 is given in Table 4: the recurrence times obtained are in the same range as those reported for model 1 (4–6 hr). The same applies to the ratios between persistent and burst luminosities, as well as to peak temperatures and luminosities. Similar light curves have also been obtained.

Table 4. Summary of Burst Properties for Model 2

Burst	T_peak (K)	t(T_peak) (s)	τ_rec (hr)	L_peak (L_☉)	τ_0.01 (s)	α
1	1.05 × 10⁹	21189	5.9	9.0 × 10⁴	59.2	62
2	1.20 × 10⁹	37783	4.6	1.5 × 10⁵	73.9	31

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There is also good agreement from the nucleosynthesis viewpoint, with only minor differences in the final, mass-averaged abundances, as shown in Table 5 (particularly, for the heavier species of the network, since their low abundances are very sensitive to the specific thermal history of the explosion; see also Figure 19). It is worth noting that both models reach almost identical nucleosynthesis endpoints (X_i > 10⁻⁹): ⁸⁹Nb, for the first burst computed in both models, and ⁹⁷Ru (model 1) and ⁹⁹Rh (model 2), for the second burst. Furthermore, the amounts of unburned H, ⁴He, and ¹²C are very similar in both models. As expected from the above-mentioned similarities, there is also good agreement in terms of overproduction factors, dominated by ⁶⁴Zn and ⁶⁰Ni (with f ∼ 10⁴) in the first burst and by ⁶⁴Zn, ⁷²Ge, ^74,76Se, ^78,80Kr, and ⁸⁴Sr (f ∼ 10⁵), in the second, for both models.

Table 5. Mean Composition of the Envelope (X_i > 10⁻⁹) at the End of Each Burst, for Model 2

Nucleus	Burst 1	Burst 2
¹H	2.3 × 10⁻¹	1.0 × 10⁻¹
⁴He	3.0 × 10⁻¹	1.9 × 10⁻¹
¹²C	1.8 × 10⁻²	2.2 × 10⁻²
¹³C	6.8 × 10⁻⁵	6.2 × 10⁻⁵
¹⁴N	2.0 × 10⁻³	1.4 × 10⁻³
¹⁵N	2.6 × 10⁻³	2.0 × 10⁻³
¹⁶O	5.3 × 10⁻⁴	3.9 × 10⁻⁴
¹⁷O	4.8 × 10⁻⁶	3.0 × 10⁻⁶
¹⁸O	6.3 × 10⁻⁵	2.8 × 10⁻⁵
¹⁸F	2.1 × 10⁻⁴	5.7 × 10⁻⁵
¹⁹F	3.6 × 10⁻⁴	1.7 × 10⁻⁴
²⁰Ne	5.4 × 10⁻⁴	4.1 × 10⁻⁴
²¹Ne	2.2 × 10⁻⁵	7.3 × 10⁻⁶
²²Ne	7.7 × 10⁻⁵	4.4 × 10⁻⁵
²²Na	3.6 × 10⁻³	1.6 × 10⁻³
²³Na	2.3 × 10⁻⁴	1.7 × 10⁻⁴
²⁴Mg	1.2 × 10⁻³	1.3 × 10⁻³
²⁵Mg	1.8 × 10⁻³	1.4 × 10⁻³
²⁶Mg	3.1 × 10⁻³	1.6 × 10⁻³
²⁶Al^g	1.0 × 10⁻⁴	1.1 × 10⁻⁴
²⁷Al	1.6 × 10⁻³	1.1 × 10⁻³
²⁸Si	1.2 × 10⁻³	7.5 × 10⁻³
²⁹Si	4.6 × 10⁻⁴	6.7 × 10⁻⁴
³⁰Si	4.7 × 10⁻³	4.6 × 10⁻³
³¹P	6.1 × 10⁻⁴	1.4 × 10⁻³
³²S	4.7 × 10⁻⁴	2.7 × 10⁻²
³³S	4.0 × 10⁻⁴	2.4 × 10⁻³
³⁴S	4.3 × 10⁻³	9.6 × 10⁻³
³⁵Cl	1.1 × 10⁻³	3.8 × 10⁻³
³⁶Ar	5.8 × 10⁻⁴	2.9 × 10⁻³
³⁷Cl	1.6 × 10⁻⁷	5.3 × 10⁻⁷
³⁷Ar	2.9 × 10⁻⁴	4.0 × 10⁻⁴
³⁸Ar	2.6 × 10⁻³	5.2 × 10⁻³
³⁹K	4.9 × 10⁻³	8.7 × 10⁻³
⁴⁰Ca	3.1 × 10⁻³	3.4 × 10⁻³
⁴¹K	...	8.7 × 10⁻⁹
⁴¹Ca	6.8 × 10⁻⁵	4.5 × 10⁻⁵
⁴²Ca	7.3 × 10⁻⁴	1.2 × 10⁻³
⁴³Ca	9.6 × 10⁻⁵	4.3 × 10⁻⁴
⁴³Sc	7.2 × 10⁻⁴	1.2 × 10⁻³
⁴⁴Ca	5.8 × 10⁻⁹	2.3 × 10⁻⁶
⁴⁴Sc	9.6 × 10⁻⁸	1.1 × 10⁻⁵
⁴⁴Ti	5.6 × 10⁻⁴	4.8 × 10⁻⁴
⁴⁵Sc	3.5 × 10⁻⁵	4.9 × 10⁻⁵
⁴⁵Ti	2.1 × 10⁻⁴	1.2 × 10⁻⁴
⁴⁶Ti	1.2 × 10⁻³	1.9 × 10⁻³
⁴⁷Ti	6.8 × 10⁻⁴	6.3 × 10⁻⁴
⁴⁸Ti	2.8 × 10⁻⁸	2.5 × 10⁻⁷
⁴⁸V	4.4 × 10⁻⁵	9.4 × 10⁻⁵
⁴⁸Cr	1.9 × 10⁻³	1.7 × 10⁻³
⁴⁹Ti	2.4 × 10⁻⁸	2.9 × 10⁻⁷
⁴⁹V	1.4 × 10⁻³	1.2 × 10⁻³
⁵⁰Cr	1.6 × 10⁻³	2.2 × 10⁻³
⁵¹V	6.4 × 10⁻⁷	5.5 × 10⁻⁶
⁵¹Cr	3.4 × 10⁻³	4.9 × 10⁻³
⁵²Cr	1.6 × 10⁻⁶	8.1 × 10⁻⁶
⁵²Mn	1.0 × 10⁻³	1.5 × 10⁻³
⁵²Fe	1.7 × 10⁻²	1.3 × 10⁻²
⁵³Mn	9.9 × 10⁻⁴	8.4 × 10⁻⁴
⁵⁴Fe	9.2 × 10⁻⁴	1.2 × 10⁻³
⁵⁵Mn	...	1.3 × 10⁻⁸
⁵⁵Fe	7.9 × 10⁻⁵	2.6 × 10⁻⁴
⁵⁵Co	2.8 × 10⁻³	3.7 × 10⁻³
⁵⁶Fe	1.4 × 10⁻⁸	3.6 × 10⁻⁷
⁵⁶Co	1.1 × 10⁻⁴	5.0 × 10⁻⁴
⁵⁶Ni	3.2 × 10⁻²	5.0 × 10⁻²
⁵⁷Fe	2.6 × 10⁻⁹	6.6 × 10⁻⁸
⁵⁷Co	6.9 × 10⁻⁵	7.4 × 10⁻⁵
⁵⁷Ni	5.1 × 10⁻³	3.4 × 10⁻³
⁵⁸Ni	3.4 × 10⁻³	2.3 × 10⁻³
⁵⁹Ni	4.8 × 10⁻³	3.7 × 10⁻³
⁶⁰Ni	2.9 × 10⁻¹	3.2 × 10⁻¹
⁶¹Ni	9.1 × 10⁻⁴	1.4 × 10⁻³
⁶¹Cu	6.3 × 10⁻³	5.8 × 10⁻³
⁶²Ni	9.9 × 10⁻⁵	1.9 × 10⁻⁴
⁶²Zn	1.8 × 10⁻³	2.1 × 10⁻³
⁶³Cu	2.2 × 10⁻³	3.2 × 10⁻³
⁶⁴Zn	2.3 × 10⁻²	1.2 × 10⁻¹
⁶⁵Cu	2.4 × 10⁻⁸	2.7 × 10⁻⁷
⁶⁵Zn	5.4 × 10⁻⁴	3.1 × 10⁻³
⁶⁶Zn	1.7 × 10⁻⁶	2.9 × 10⁻⁵
⁶⁶Ga	6.4 × 10⁻⁵	6.2 × 10⁻⁴
⁶⁶Ge	2.8 × 10⁻⁴	1.7 × 10⁻³
⁶⁷Ga	1.8 × 10⁻⁴	1.2 × 10⁻³
⁶⁸Ge	1.6 × 10⁻³	2.8 × 10⁻²
⁶⁹Ge	7.3 × 10⁻⁵	1.8 × 10⁻³
⁷⁰Ge	2.0 × 10⁻⁵	5.9 × 10⁻⁴
⁷¹As	2.4 × 10⁻⁵	6.2 × 10⁻⁴
⁷²Se	1.2 × 10⁻⁴	5.8 × 10⁻³
⁷³Se	8.0 × 10⁻⁶	6.1 × 10⁻⁴
⁷⁴Se	3.2 × 10⁻⁶	2.7 × 10⁻⁴
⁷⁵Br	3.3 × 10⁻⁶	2.7 × 10⁻⁴
⁷⁶Kr	1.2 × 10⁻⁵	1.6 × 10⁻³
⁷⁷Kr	1.2 × 10⁻⁶	2.3 × 10⁻⁴
⁷⁸Kr	6.7 × 10⁻⁷	1.5 × 10⁻⁴
⁷⁹Kr	4.0 × 10⁻⁷	9.2 × 10⁻⁵
⁸⁰Sr	1.5 × 10⁻⁶	4.9 × 10⁻⁴
⁸¹Rb	2.3 × 10⁻⁷	1.1 × 10⁻⁴
⁸²Sr	2.2 × 10⁻⁷	1.4 × 10⁻⁴
⁸³Sr	9.4 × 10⁻⁸	7.0 × 10⁻⁵
⁸⁴Sr	6.4 × 10⁻⁸	6.3 × 10⁻⁵
⁸⁵Y	4.5 × 10⁻⁸	6.4 × 10⁻⁵
⁸⁶Zr	3.1 × 10⁻⁸	7.8 × 10⁻⁵
⁸⁷Zr	1.4 × 10⁻⁸	6.9 × 10⁻⁵
⁸⁸Zr	3.5 × 10⁻⁹	3.3 × 10⁻⁵
⁸⁹Nb	2.5 × 10⁻⁹	5.7 × 10⁻⁵
⁹⁰Mo	...	2.4 × 10⁻⁵
⁹¹Nb	...	1.3 × 10⁻⁵
⁹²Mo	...	7.3 × 10⁻⁶
⁹³Tc	...	6.9 × 10⁻⁶
⁹⁴Tc	...	5.0 × 10⁻⁶
⁹⁵Ru	...	9.2 × 10⁻⁷
⁹⁶Ru	...	9.1 × 10⁻⁸
⁹⁷Ru	...	3.7 × 10⁻⁸
⁹⁸Ru	...	9.1 × 10⁻⁹
⁹⁹Rh	...	1.8 × 10⁻⁹

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All in all, we conclude that the resolution adopted in model 1 is appropriate for XRB simulations. This is in agreement with the studies performed by Fisker et al. (2004), who concluded that the minimum discretization of the envelope, in one-dimensional hydrodynamic simulations of XRBs, is about 25 shells.

4. MODEL 3

To test the impact of the metallicity of the accreted material (which reflects the surface composition of the companion star) on the overall properties of the bursts, we have computed another series of bursts (hereafter, model 3), driven by accretion of metal-deficient material (Z ∼ Z_☉/20) onto a 1.4 M_☉ neutron star (L_ini = 1.6 × 10³⁴ erg s⁻¹ = 4.14 L_☉), at a rate ${\dot{M}}_{\rm acc} = 1.75 \times 10^{-9}\;M_\odot$ yr⁻¹. The composition of the accreted material is assumed to be X = 0.759, Y = 0.240, and Z = 10⁻³, and as for model 1, all metals are initially added up in the form of ¹⁴N. This model is indeed qualitatively similar to model zM, from Woosley et al. (2004; see also Fisker et al. 2008; Schatz et al. 2001). Both the envelope zoning and the initial relaxation phase are identical to those described for model 1.

4.1. First Burst

The piling-up of matter on top of the neutron star during the accretion stage progressively compresses and heats the envelope.

At t = 4337 s, the envelope achieves T_base ∼ 10⁸ K (ρ_base ∼ 9.8 × 10⁴ g cm⁻³). The nuclear activity is fully dominated by the CNO cycle. In contrast to model 1, the smaller metallicity of this model limits substantially the role of proton captures. Indeed, at this stage, H has only been reduced to 0.757 at the envelope base. The main reaction fluxes are actually an order of magnitude lower than those reported from model 1, for the same temperature, powering an energy generation rate of ε_nuc ∼ 6.2 × 10¹² erg g⁻¹ s⁻¹. Because of the lower metallicity of this model, the time required to achieve T_base ∼ 10⁸ K is about twice the value reported for model 1, resulting in a thicker, more massive envelope which will affect the forthcoming explosion. Besides H and He (0.242), by far the most abundant nuclei in the envelope, the nuclear activity in the CNO region increases the chemical abundances of many species in this mass range, with ¹⁵O (6.3 × 10⁻⁴) being the most abundant CNO-group nucleus at the envelope base.

16.8 hr (60,347 s) after the beginning of accretion, T_base has reached 2 × 10⁸ K (with ρ_base = 5.7 × 10⁵ g cm⁻³ and P_base = 3.4 × 10²² dyn cm⁻²). The total luminosity of the star is only 6.9 × 10³⁴ erg s⁻¹. The main nuclear activity is governed by ¹⁵N(p, α)¹²C and other reactions of the CNO cycle. ⁷Be(p, γ)⁸B is at equilibrium with its reverse photodisintegration reaction ⁸B(γ, p)⁷Be. Because of the limited number of CNO catalysts in this low-metallicity model, some proton–proton chain reactions, such as the pep, ³He(α, γ)⁷Be, or ⁸Be → 2⁴He, are relatively frequent. In terms of chemical abundances, the now frequent p-captures have reduced the hydrogen content down to a value of 0.695 (while X(⁴He) = 0.303). The next most abundant isotopes in the network are the short-lived species ¹⁵O (1.4 × 10⁻³) and ¹⁴O (8 × 10⁻⁴). The nuclear activity (X_i > 10⁻⁹) reaches ⁴⁰Ca at this stage.

18.1 hr (65,081 s) from the onset of accretion, T_base reaches 4 × 10⁸ K. Hydrogen continues to decrease smoothly (X(H) = 0.689), whereas the ⁴He abundance reaches 0.280. The next most abundant nuclei are ¹⁴O (1.4 × 10⁻²) and ¹⁵O, now followed by ⁵²Fe and ¹⁸Ne, with mass fractions of the order of 10⁻³. The nuclear activity extends up to ⁵⁸Cu. The largest reaction fluxes correspond to different processes that operate almost at equilibrium with their inverse photodisintegration reactions, such as ²¹Mg(p, γ)²²Al(γ, p)²¹Mg, ³⁰S(p, γ)³¹Cl(γ, p)³⁰S, ²⁵Si(p, γ)²⁶P(γ, p)²⁵Si, and ⁷Be(p, γ)⁸B(γ, p)⁷Be.

Six seconds later (t = 65,087 s), T_base achieves 5 × 10⁸ K. ρ_base has slightly decreased to 4.3 × 10⁵ g cm⁻³ because of a mild envelope expansion (Δz ∼ 19.7 m). ⁴He has slightly decreased to 0.267, as a result of the frequent α-captures driven by the high temperatures achieved. The next most abundant species are now ¹⁸Ne (10⁻²), together with ^14,15O (3.5 × 10⁻³ and 7.5 × 10⁻³, respectively), ⁵²Fe (2.8 × 10⁻³), and ³⁴Ar (1.1 × 10⁻³). The nuclear activity reaches ⁶¹Ga, powering an energy generation rate of 2.8 × 10¹⁶ erg g⁻¹ s⁻¹. The overall stellar luminosity is now 2.5 × 10³⁵ erg s⁻¹.

A qualitatively similar picture is found when T_base achieves 7 × 10⁸ K (at t = 65,090 s), with the nuclear activity extending all the way up to ⁶⁸Se.

One second later (t = 65,091 s), T_base achieves 10⁹ K. The nuclear activity (with epsilon _nuc ∼ 1.5 × 10¹⁷ erg g⁻¹ s⁻¹; see Figure 22) continues to reduce the H and ⁴He abundances down to 0.680 and 0.224, respectively. The next most abundant species are now ^28,29,30S, ^33,34Ar, ²⁵Si, ⁶⁰Zn, and ³⁸Ca (all with mass fractions ∼10⁻²), with the main nuclear path reaching ⁷²Kr. The largest absolute fluxes are achieved by nuclear interactions between equilibrium (p, γ)–(γ, p) pairs, which do not contribute to the net energy balance. Instead, the most important contributors to the energy production at this stage are ²⁵Al(p, γ)²⁶Si, ^27,29,30P(p, γ)^28,30,31S, ²⁸Si(p, γ)²⁹P, ^32,33Cl(p, γ)^33,34Ar, ^32,33Cl(p, γ)^33,34Ar, ³⁵Ar(p, γ)³⁶K, ^35,36,37K(p, γ)^36,37,38Ca, the chain 3α→ ¹²C(p, γ)¹³N(p, γ)¹⁴O(α, p)¹⁷F(p, γ)¹⁸Ne(α, p)²¹Na(p, γ)²²Mg, and a suite of β⁺-decays, such as ²⁵Si(β⁺)²⁵Al, ^28,29,30S(β⁺)^28,29,30P, and ³³Ar(β⁺)³³Cl. The activity in the A = 65–100 mass region is dominated by the suite of reactions depicted in Figure 22 (lower panel), mainly ⁶⁵Ge(p, γ)⁶⁶As, ⁶⁷Se(β⁺)⁶⁷As(p, γ)⁶⁸Se, ⁶⁵As(p, γ)⁶⁶Se(β⁺)⁶⁶As (with log F ∼ −7), and ⁶⁵Se(β⁺)⁶⁵ As (log F ∼ −8).

In contrast to model 1, which achieved a peak temperature of 1.06 × 10⁹ K, model 3 reaches relatively higher values. Hence, at t = 65,093 s, T_base achieves 1.2 × 10⁹ K. The H and ⁴He abundances have been reduced to 0.648 and 0.205, respectively. The next most abundant nucleus is still ³⁰S (3.1 × 10⁻²), followed by ³⁸Ca (2 × 10⁻²), and by a large number of species with abundances of the order of 10⁻³, such as ^36,37Ca, ^28,29S, ^32,33,34Ar, ^58,59,60Zn, ^62,63,64Ge, ^48,49,50Fe, ^53,54,55Ni, ⁴¹Ti, and ^44,45,46Cr. At this stage, the main nuclear path reaches ⁷⁶Sr.

One second later, at t = 65,094 s, the rate of nuclear energy generation achieves a maximum value of epsilon _nuc,max ∼ 2.1 × 10¹⁷ erg g⁻¹ s⁻¹.

At t = 65,095 s, while T_base = 1.3 × 10⁹ K, the main nuclear path reaches ⁸⁰Zr. Because of the large temperature achieved, the number of proton- and α-captures increases, which in turn efficiently reduces the H (0.621) and ⁴He (0.197) abundances. The next most abundant nucleus is ⁶⁰Zn (2.8 × 10⁻²), followed by ⁶⁴Ge, ³⁸Ca, ³⁰S, ⁵⁵Ni, and ⁵⁹Zn (with X_i ∼ 10⁻², see Figure 23). The nuclear activity in the A = 65–100 mass region is now dominated by ⁶⁵As(p, γ)⁶⁶Se(β⁺)⁶⁶As(p, γ)⁶⁷Se (log F ∼ −4), ⁶⁵Ge(p, γ)⁶⁶As, and ⁶⁷Se(β⁺)⁶⁷As(p, γ)⁶⁸Se (log F ∼ −5; see Figure 23). Energy production is mainly due to suite of (p, γ) reactions and β⁺-decays involving nuclear species in the mass range A = 30–62.

**Figure 23.** Same as Figure 22, but for T_base = 1.3 × 10⁹ K.
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Shortly after, at t = 65,098 s, a peak temperature of T_peak = 1.4 × 10⁹ K is achieved at the envelope base. This is followed, less than a second later, by a maximum expansion of the envelope, Δz_max∼ 73.9 m, and by a maximum luminosity, L_max = 4.0 × 10³⁸ erg s⁻¹ (10⁵ L_☉). The main nuclear path reaches ⁹³Pd (already beyond the nucleosynthesis endpoint achieved in model 1). With respect to the chemical abundances, the envelope base is still dominated by H (0.560) and ⁴He (0.175), with ⁶⁰Zn reaching a mass fraction of 0.111. The next most abundant species are ⁶⁴Ge (6.3 × 10⁻²) and ⁶⁸Se (2 × 10⁻²). Regarding the activity in the A > 65 mass region, at this stage is dominated by ⁶⁶Se(β⁺)⁶⁶As(p, γ)⁶⁷Se(β⁺)⁶⁷As(p, γ)⁶⁸Se (log F ∼ −4), ⁶⁵Ge(p, γ)⁶⁶As, and ⁶⁸Se(β⁺)⁶⁸As(p, γ)⁶⁹Se(p, γ)⁷⁰Br(p, γ)⁷¹Kr(β⁺)⁷¹Br(p, γ)⁷²Kr (log F ∼ −5; see Figure 24, for additional processes down to log F ∼ −8). The most important contributors to the energy production are at this stage ^44,45V(p, γ)^45,46Cr, ⁴⁹Mn(p, γ)⁵⁰Fe(β⁺)⁵⁰Mn(p, γ)⁵¹Fe, ⁵²Co(p, γ)⁵³Ni, ⁵³Co(p, γ)⁵⁴Ni(β⁺)⁵⁴Co(p, γ)⁵⁵Ni(β⁺)⁵⁵Co(p, γ)⁵⁶Ni, and ⁵⁸Zn(β⁺)⁵⁸Cu(p, γ)⁵⁹Zn(β⁺)⁵⁹Cu(p, γ)⁶⁰Zn (log F ∼ −3).

**Figure 24.** Same as Figure 22, but for the time when temperature at the envelope base reaches a peak value of T_peak = 1.4 × 10⁹ K.
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At t = 65,110 s, following the decline from peak temperature, the envelope base achieves T_base = 1.3 × 10⁹ K (Figure 25). At this stage, the main nuclear activity has already reached the SnSbTe-mass region (¹⁰⁴Sn, in particular). The chemical abundances at the envelope base are still dominated by H (0.471), now followed by ⁶⁴Ge (0.162) and ⁶⁸Se (0.161), while ⁴He has dropped to 0.131. The next most abundant species shift to ⁶⁰Zn (2.1 × 10⁻²) and ⁷²Kr (1.9 × 10⁻²), with a suite of nuclei reaching ∼10⁻³ (³⁰S, ⁶⁷Se, ^37,38Ca, ⁷⁶Sr, ^62,63Ge, ⁵⁹Zn, ⁵⁵Ni, ³⁴Ar, or ⁵⁰Fe). The nuclear activity in the A > 65 mass region is now powered by ⁶⁵As(p, γ)⁶⁶Se(β⁺)⁶⁶As(p, γ)⁶⁷Se(β⁺)⁶⁷As(p, γ)⁶⁸Se(β⁺)⁶⁸As(p, γ)⁶⁹Se(p, γ)⁷⁰Br(p, γ)⁷¹Kr(β⁺)⁷¹Br(p, γ)⁷²Kr (log F ∼ −4), ⁶⁵Ge(p, γ)⁶⁶As, ⁶⁹Br(p, γ)⁷⁰Kr(β⁺)⁷⁰Br, ⁷²Kr(β⁺)⁷²Br(p, γ)⁷³Kr(p, γ)⁷⁴Rb(p, γ)⁷⁵Sr(β⁺)⁷⁵Rb(p, γ)⁷⁶Sr, ⁷⁸Y(p, γ)⁷⁹Zr, and ⁸⁶Tc(p, γ)⁸⁷Ru (log F ∼ −5; see Figure 25). Energy production is not due to a handful of nuclear processes but to dozens of different reactions (from 3α→ ¹²C all the way to ⁷¹Br(p, γ)⁷²Kr).

Twenty-two seconds later (t = 65,132 s), the temperature at the envelope base has decreased to T_base = 1.2 × 10⁹ K (Figure 26). The H content has been slightly reduced to 0.370, whereas ⁴He reaches 8.74 × 10⁻². Indeed, after H, the most abundant species at the envelope base are now ⁶⁸Se (0.270) and ⁶⁴Ge (0.104), followed by ⁷²Kr (8.7 × 10⁻²), ⁷⁶Sr (2.8 × 10⁻²), and ⁸⁰Zr (1.2 × 10⁻²). At this stage, some of the heaviest species of the network have already achieved an abundance of ∼10⁻³ (such as ^88,89Ru, ^92,93Pd, ⁹⁶Cd, ⁹⁹In, or ^101,102Sn, together with the lighter nuclei ³⁰S, ³⁸Ca, ^59,60Zn, and ⁶⁷Se), with the nuclear activity extending all the way up to ¹⁰⁷Te. The nuclear activity in the A = 65–100 mass region is similar to that described above for T_base = 1.3 × 10⁹ K, and is depicted in Figure 26 (lower panel).

At t = 65,264 s, the envelope base achieves T_base = 10⁹ K (Figure 27). Now, the most abundant element at the envelope base is ¹⁰⁵Sn (0.228). This is followed by a large number of species with abundances ∼10⁻², such as ¹⁰⁴Sn, ⁶⁸Se, ⁷²Kr, ¹⁰⁴In, ⁹⁴Pd, ⁶⁴Ge, ¹⁰³In, ⁷⁶Sr—all more abundant than H (1.8 × 10⁻²) and ⁴He (2.6 × 10⁻²), at this stage—together with ^102,103Sn, ⁹⁵Ag, ¹⁰⁷Te, ^100,101,102In, ⁸⁰Zr, ⁶⁰Zn, and ^98,99Cd. Note that, since the heaviest element included in our network, ¹⁰⁷Te, achieved already an abundance of 2.2 × 10⁻², leakage from the SnSbTe-mass region cannot be discarded. Note, however, that α-emission for ¹⁰⁷Te was not included, which may account for the high abundances reported here. Further studies to explore possible nucleosynthesis beyond the SnSbTe-mass region are underway with a larger network, so detailed abundances in this region should be taken with caution. The set of equilibrium (p, γ)–(γ, p) pairs is now accompanied by a handful of β⁺-decays (such as ⁸⁰Zr(β⁺)⁸⁰Y, ⁷⁶Sr(β⁺)⁷⁶Rb, ⁸⁴Mo(β⁺)⁸⁴Nb, and ⁸²Nb(β⁺)⁸²Zr) as the nuclear processes with largest absolute fluxes, since H depletion and the low temperature limit the extent of charged-particle reactions. Indeed, these weak interactions will become progressively more important during the last stages of the burst. At this stage, the activity in the A = 65–100 mass region is dominated by ⁷⁶Sr(β⁺)⁷⁶Rb(p, γ)⁷⁷Sr(p, γ)⁷⁸Y, ⁷⁹Y(p, γ)⁸⁰Zr(β⁺)⁸⁰Y(p, γ)⁸¹Zr(p, γ)⁸²Nb(β⁺)⁸²Zr(p, γ)⁸³Nb(p, γ)⁸⁴Mo(β⁺)⁸⁴Nb(p, γ)⁸⁵Mo, ⁸⁹Ru(p, γ)⁹⁰Rh (log F ∼ −4), and more than 60 different nuclear processes with log F ∼ −5 (see Figure 27, lower panel), involving nuclei in the mass range A = 65–104. Indeed, energy production is driven by (p, γ) and β⁺ processes involving species in this mass range.

At t = 65,362 s, the temperature at the envelope base has already declined to T_base = 7.6 × 10⁸ K (Figure 28). H is now fully depleted (7.6 × 10⁻¹²), while ⁴He barely reaches 1.8 × 10⁻². As before, the most abundant element at the envelope base is ¹⁰⁵Sn (0.251), followed by ¹⁰⁴In (0.142), and by a large number of species with abundances ∼10⁻². The depletion of H dramatically reduces the fluxes of most of the (p, γ) reactions, which are now overcome by many β⁺-decays (such as those affecting ^68,69Se, ⁶⁸As, ⁶⁴Ge, ^71,72Br, ⁶⁰Zn, ⁶⁰Cu, ^72,73Kr, ¹⁰⁴Sn, ⁶⁴Ga, ⁸²Zr, or ⁷⁶Sr), and by a suite of α-capture reactions, such as 3α → ¹²C(α, γ)¹⁶O(α, γ)²⁰Ne(α, γ)²⁴Mg(α, γ)²⁸Si(α, γ)³²S, or ¹³N(α, p)¹⁶O.

At t = 69,715 s, after a long decline, a minimum temperature is achieved at the envelope base, T_base = 2 × 10⁸ K (Figure 29), which we consider to mark the end of the first bursting episode for this model. At this stage, H is fully depleted (3.2 × 10⁻²³) at the envelope base, while ⁴He has a mass fraction of 1.5 × 10⁻² only. The distribution of the most abundant elements almost follows the one described for T_base = 7.6 × 10⁸ K (Figure 28), and is dominated by ¹⁰⁵Sn (0.251), followed by ¹⁰⁴In (0.147), and by a large number of species with abundances ∼10⁻², such as ⁹⁴Pd, ^{100,101,102,103}In, ⁶⁸Ge, ⁶⁴Zn, ⁷²Se, ⁷⁶Rb, ¹⁰⁷Te, ⁶⁰Ni, ⁹⁹Cd, ^97,98Ag, ^89,90Ru, and ⁸⁰Y. At this stage, the dominant interactions are all β⁺-decays (^60,61Cu(β⁺)^60,61Ni, ^66,67Ge(β⁺)^66,67Ga, ⁶⁵Ga(β⁺)⁶⁵Zn, ⁵¹Mn(β⁺)⁵¹Cr, ⁵²Fe(β⁺)⁵²Mn, ⁶³Zn(β⁺)⁶³Cu, ⁵⁶Ni(β⁺)⁵⁶Co, or ⁴³Sc(β⁺)⁴³Ca), except for the triple-α reaction.

Depth also influences the extent of the nuclear activity throughout the envelope, but in contrast to model 1, the nuclear activity in all shells of our computational domain essentially reaches the SnSbTe-mass region. Indeed, the inner part of the envelope (encompassing 3.4 × 10²¹ g) is, at the end of the burst, dominated by large amounts of ¹⁰⁵Sn and ¹⁰⁴In, the most abundant nuclei with mass fractions ∼ 0.1–0.2; at 5.6 × 10²¹ g above the core–envelope interface, the most abundant isotopes are H (0.26) and ⁴He (0.12), while the most abundant species in the SnSbTe-mass region achieve a mass fraction ∼10⁻³; and close to the surface (7.7 × 10²¹ g), shells are largely dominated by the presence of unburned H (0.75) and ⁴He (0.24), with X(SnSbTe) ∼ 10⁻⁷.

The mean, mass-averaged chemical composition of the whole envelope, at the end of the first bursting episode, is dominated by the presence of unburned H (0.18) and ⁴He (0.084), followed by ¹⁰⁵Ag (0.075), ¹⁰⁴Pd (0.053), ⁶⁴Zn (0.042), ⁹⁵Ru (0.031), ⁶⁸Ge (0.028), ⁹⁴Tc (0.026), and ¹⁰³Ag (0.026), with a nucleosynthesis endpoint around ¹⁰⁷Cd. In contrast, the first burst computed in model 1 yielded, in general, lighter nuclei, ⁶⁰Ni, ⁴He, ¹H, ⁶⁴Zn, ¹²C, and ^52,56Fe, with a more modest nucleosynthesis endpoint around ⁸⁹Nb (Table 2).

In terms of overproduction factors, f (Figure 30), while model 1 showed moderate values (f ∼ 10⁴) for a handful of intermediate-mass elements, such as ⁴³Ca, ⁴⁵Sc, ⁴⁹Ti, ⁵¹V, ^60,61Ni, ^63,65Cu, ^64,67,68Zn, ⁶⁹Ga, ⁷⁴Se, or ⁷⁸Kr, model 3 achieves moderate overproduction factors (⩾10⁴), for all stable species heavier than ⁶⁴Zn, and as high as ∼10⁸ for ⁹⁸Ru, ^102,104Pd, and ¹⁰⁶Cd.

4.2. Second, Third, Fourth, and Fifth Bursts

Tables 6–8 summarize the most relevant properties that characterize the bursting episodes computed for model 3. Recurrence times between bursts of τ_rec ∼ 9 hr (except for the first one, for which τ_rec ∼ 18 hr), ratios between persistent and burst luminosities of α ∼ 20–30, and peak luminosities around L_peak ∼ 10⁵ L_☉ represent the basic observables associated with this model. Indeed, the recurrence times obtained are in agreement with the values reported for the XRB sources (see Galloway et al. 2008) 1A 1905+00 [τ_rec = 8.9 hr], 4U 1254-69 [τ_rec = 9.2 hr], or XTE J1710-281 [τ_rec = 8.9 hr, α = 22–100]. A striking result is the quick stabilization of the recurrence times, that show a regular periodicity after the second burst.

Table 6. Properties of the Last Burst Computed in Models 1 and 3

Property	Model 1	Model 3
ρ_max,base (g cm⁻³)	1.3 × 10⁶	2.6 × 10⁶
P_max,base (dyn cm⁻²)	4.2 × 10²²	1.1 × 10²³
ρ_max,ign (g cm⁻³)	5.4 × 10⁵	1.0 × 10⁶
P_max,ign (dyn cm⁻²)	1.3 × 10²²	2.9 × 10²²
τ_acc (hr)	5.1	8.8
Δm_acc (M_☉)	1.0 × 10⁻¹²	1.8 × 10⁻¹²
T_peak (K)	1.1 × 10⁹	1.3 × 10⁹
L_peak (L_☉)	1.2 × 10⁵	1.0 × 10⁵
Δz_max (m)	40	44
α	36	30

Note. ρ_max,base and P_max,base are the maximum density and pressure achieved at the base of the envelope, whereas ρ_max,ign and P_max,ign correspond to the maximum values attained at the ignition shell (defined as the first shell that reaches T > 4 × 10⁸ K, for this burst).

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Table 7. Summary of Burst Properties for Model 3

Burst	T_peak (K)	t(T_peak) (s)	τ_rec (hr)	L_peak (L_☉)	τ_0.01 (s)	α
1	1.40 × 10⁹	65110	18.1	1.0 × 10⁵	423	34
2	1.39 × 10⁹	98879	9.4	1.1 × 10⁵	296	24
3	1.32 × 10⁹	130816	8.9	9.8 × 10⁴	281	24
4	1.30 × 10⁹	162777	8.9	1.0 × 10⁵	252	27
5	1.26 × 10⁹	194266	8.8	1.0 × 10⁵	250	30

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Table 8. Mean Composition of the Envelope (X_i > 10⁻⁹) at the End of Each Burst, for Model 3

Nucleus	Burst 1	Burst 2	Burst 3	Burst 4	Burst 5
¹H	1.8 × 10⁻¹	8.6 × 10⁻²	5.5 × 10⁻²	4.5 × 10⁻²	3.7 × 10⁻²
⁴He	8.4 × 10⁻²	5.8 × 10⁻²	4.4 × 10⁻²	3.8 × 10⁻²	3.3 × 10⁻²
¹²C	7.7 × 10⁻⁴	1.7 × 10⁻³	1.8 × 10⁻³	1.7 × 10⁻³	1.5 × 10⁻³
¹³C	4.2 × 10⁻⁶	8.9 × 10⁻⁶	1.4 × 10⁻⁵	2.3 × 10⁻⁵	2.7 × 10⁻⁵
¹⁴N	2.8 × 10⁻⁴	2.1 × 10⁻⁴	1.7 × 10⁻⁴	1.5 × 10⁻⁴	1.4 × 10⁻⁴
¹⁵N	4.9 × 10⁻⁴	3.6 × 10⁻⁴	2.9 × 10⁻⁴	2.4 × 10⁻⁴	2.2 × 10⁻⁴
¹⁶O	1.4 × 10⁻⁵	2.9 × 10⁻⁵	4.1 × 10⁻⁵	3.7 × 10⁻⁵	3.8 × 10⁻⁵
¹⁷O	4.8 × 10⁻⁷	6.8 × 10⁻⁷	2.1 × 10⁻⁶	1.3 × 10⁻⁶	2.0 × 10⁻⁶
¹⁸O	...	3.7 × 10⁻⁹	9.4 × 10⁻⁹	9.5 × 10⁻⁹	8.1 × 10⁻⁹
¹⁸F	6.4 × 10⁻⁹	1.3 × 10⁻⁸	2.8 × 10⁻⁸	2.4 × 10⁻⁸	2.3 × 10⁻⁸
¹⁹F	8.7 × 10⁻⁸	1.6 × 10⁻⁷	1.7 × 10⁻⁷	2.0 × 10⁻⁷	8.0 × 10⁻⁸
²⁰Ne	1.5 × 10⁻⁵	2.9 × 10⁻⁵	3.5 × 10⁻⁵	3.0 × 10⁻⁵	3.1 × 10⁻⁵
²¹Ne	1.9 × 10⁻⁸	4.4 × 10⁻⁸	4.9 × 10⁻⁸	5.2 × 10⁻⁸	4.4 × 10⁻⁸
²²Ne	2.9 × 10⁻⁷	6.6 × 10⁻⁷	4.3 × 10⁻⁷	1.1 × 10⁻⁶	7.5 × 10⁻⁷
²²Na	1.1 × 10⁻⁵	2.4 × 10⁻⁵	2.0 × 10⁻⁵	2.3 × 10⁻⁵	1.9 × 10⁻⁵
²³Na	2.6 × 10⁻⁶	6.5 × 10⁻⁶	6.5 × 10⁻⁶	6.5 × 10⁻⁶	5.7 × 10⁻⁶
²⁴Mg	8.2 × 10⁻⁵	1.4 × 10⁻⁴	1.4 × 10⁻⁴	1.2 × 10⁻⁴	1.1 × 10⁻⁴
²⁵Mg	4.9 × 10⁻⁵	7.3 × 10⁻⁵	7.4 × 10⁻⁵	7.2 × 10⁻⁵	6.6 × 10⁻⁵
²⁶Mg	1.1 × 10⁻⁵	1.9 × 10⁻⁵	1.8 × 10⁻⁵	2.2 × 10⁻⁵	2.0 × 10⁻⁵
²⁶Al^g	1.2 × 10⁻⁵	1.4 × 10⁻⁵	1.3 × 10⁻⁵	1.2 × 10⁻⁵	1.1 × 10⁻⁵
²⁷Al	4.1 × 10⁻⁵	6.3 × 10⁻⁵	6.5 × 10⁻⁵	6.8 × 10⁻⁵	6.2 × 10⁻⁵
²⁸Si	2.6 × 10⁻⁴	6.2 × 10⁻⁴	5.2 × 10⁻⁴	3.9 × 10⁻⁴	3.6 × 10⁻⁴
²⁹Si	8.6 × 10⁻⁶	1.1 × 10⁻⁵	9.2 × 10⁻⁶	1.3 × 10⁻⁵	1.1 × 10⁻⁵
³⁰Si	5.5 × 10⁻⁵	8.8 × 10⁻⁵	8.0 × 10⁻⁵	1.1 × 10⁻⁴	9.9 × 10⁻⁵
³¹P	2.3 × 10⁻⁵	3.6 × 10⁻⁵	2.6 × 10⁻⁵	3.2 × 10⁻⁵	2.6 × 10⁻⁵
³²S	3.7 × 10⁻⁵	2.1 × 10⁻³	4.9 × 10⁻³	5.8 × 10⁻³	7.7 × 10⁻³
³³S	5.7 × 10⁻⁶	4.2 × 10⁻⁵	6.1 × 10⁻⁵	5.6 × 10⁻⁵	9.2 × 10⁻⁵
³⁴S	2.6 × 10⁻⁵	6.7 × 10⁻⁵	7.2 × 10⁻⁵	7.3 × 10⁻⁵	7.5 × 10⁻⁵
³⁵Cl	2.9 × 10⁻⁵	1.5 × 10⁻⁴	4.2 × 10⁻⁴	5.0 × 10⁻⁴	5.9 × 10⁻⁴
³⁶Ar	8.2 × 10⁻⁶	2.9 × 10⁻³	6.8 × 10⁻³	9.9 × 10⁻³	1.2 × 10⁻²
³⁷Cl	3.5 × 10⁻⁹	2.8 × 10⁻⁸	5.4 × 10⁻⁸	1.5 × 10⁻⁷	2.2 × 10⁻⁷
³⁷Ar	3.7 × 10⁻⁶	2.6 × 10⁻⁵	5.1 × 10⁻⁵	1.1 × 10⁻⁴	1.6 × 10⁻⁴
³⁸Ar	1.5 × 10⁻⁵	6.4 × 10⁻⁵	1.1 × 10⁻⁴	1.7 × 10⁻⁴	2.0 × 10⁻⁴
³⁹K	4.7 × 10⁻⁵	9.4 × 10⁻⁴	1.4 × 10⁻³	2.1 × 10⁻³	2.5 × 10⁻³
⁴⁰Ca	1.3 × 10⁻⁴	1.3 × 10⁻³	1.8 × 10⁻³	2.3 × 10⁻³	2.6 × 10⁻³
⁴¹K	...	...	...	...	1.1 × 10⁻⁹
⁴¹Ca	2.5 × 10⁻⁶	6.4 × 10⁻⁶	7.7 × 10⁻⁶	1.4 × 10⁻⁵	1.7 × 10⁻⁵
⁴²Ca	3.7 × 10⁻⁶	1.6 × 10⁻⁵	1.8 × 10⁻⁵	3.3 × 10⁻⁵	3.5 × 10⁻⁵
⁴³Ca	3.3 × 10⁻⁶	1.1 × 10⁻⁵	1.0 × 10⁻⁵	2.0 × 10⁻⁵	2.6 × 10⁻⁵
⁴³Sc	1.4 × 10⁻⁵	4.0 × 10⁻⁵	3.7 × 10⁻⁵	5.3 × 10⁻⁵	5.2 × 10⁻⁵
⁴⁴Ca	...	4.2 × 10⁻⁹	5.7 × 10⁻⁷	1.6 × 10⁻⁶	5.1 × 10⁻⁶
⁴⁴Sc	6.0 × 10⁻⁹	2.8 × 10⁻⁸	2.2 × 10⁻⁶	3.3 × 10⁻⁶	9.2 × 10⁻⁶
⁴⁴Ti	1.5 × 10⁻⁵	4.9 × 10⁻⁵	6.3 × 10⁻⁵	8.0 × 10⁻⁵	9.0 × 10⁻⁵
⁴⁵Sc	1.2 × 10⁻⁶	1.9 × 10⁻⁶	6.0 × 10⁻⁶	1.3 × 10⁻⁵	2.3 × 10⁻⁵
⁴⁵Ti	4.0 × 10⁻⁶	6.0 × 10⁻⁶	1.8 × 10⁻⁵	1.9 × 10⁻⁵	2.3 × 10⁻⁵
⁴⁶Ti	7.0 × 10⁻⁶	1.4 × 10⁻⁵	1.8 × 10⁻⁵	3.9 × 10⁻⁵	5.4 × 10⁻⁵
⁴⁷Ti	1.4 × 10⁻⁵	3.0 × 10⁻⁵	3.9 × 10⁻⁵	5.7 × 10⁻⁵	6.7 × 10⁻⁵
⁴⁸Ti	2.6 × 10⁻⁹	5.8 × 10⁻⁹	4.9 × 10⁻⁸	2.0 × 10⁻⁷	5.4 × 10⁻⁷
⁴⁸V	2.4 × 10⁻⁶	4.8 × 10⁻⁶	1.3 × 10⁻⁵	2.7 × 10⁻⁵	4.7 × 10⁻⁵
⁴⁸Cr	5.2 × 10⁻⁵	1.0 × 10⁻⁴	1.2 × 10⁻⁴	1.3 × 10⁻⁴	1.3 × 10⁻⁴
⁴⁹Ti	1.9 × 10⁻⁹	2.2 × 10⁻⁸	4.8 × 10⁻⁸	8.3 × 10⁻⁸	1.2 × 10⁻⁷
⁴⁹V	4.7 × 10⁻⁵	6.3 × 10⁻⁵	7.9 × 10⁻⁵	8.8 × 10⁻⁵	9.5 × 10⁻⁵
⁵⁰Cr	3.9 × 10⁻⁵	5.7 × 10⁻⁵	6.9 × 10⁻⁵	1.0 × 10⁻⁴	1.3 × 10⁻⁴
⁵¹V	5.3 × 10⁻⁸	1.1 × 10⁻⁷	3.2 × 10⁻⁷	7.9 × 10⁻⁷	1.7 × 10⁻⁶
⁵¹Cr	1.2 × 10⁻⁴	1.9 × 10⁻⁴	2.2 × 10⁻⁴	2.7 × 10⁻⁴	3.1 × 10⁻⁴
⁵²Cr	5.6 × 10⁻⁷	7.4 × 10⁻⁷	4.0 × 10⁻⁶	1.3 × 10⁻⁵	2.8 × 10⁻⁵
⁵²Mn	2.1 × 10⁻⁴	2.5 × 10⁻⁴	4.0 × 10⁻⁴	6.0 × 10⁻⁴	7.7 × 10⁻⁴
⁵²Fe	2.1 × 10⁻³	2.2 × 10⁻³	2.1 × 10⁻³	1.8 × 10⁻³	1.6 × 10⁻³
⁵³Mn	8.8 × 10⁻⁵	2.1 × 10⁻⁴	3.6 × 10⁻⁴	4.0 × 10⁻⁴	4.2 × 10⁻⁴
⁵⁴Fe	7.7 × 10⁻⁵	1.2 × 10⁻⁴	1.5 × 10⁻⁴	1.8 × 10⁻⁴	1.9 × 10⁻⁴
⁵⁵Mn	...	...	5.1 × 10⁻⁹	1.6 × 10⁻⁸	3.4 × 10⁻⁸
⁵⁵Fe	1.0 × 10⁻⁵	1.7 × 10⁻⁵	4.5 × 10⁻⁵	7.7 × 10⁻⁵	1.1 × 10⁻⁴
⁵⁵Co	2.2 × 10⁻⁴	3.1 × 10⁻⁴	3.0 × 10⁻⁴	3.0 × 10⁻⁴	2.9 × 10⁻⁴
⁵⁶Fe	1.0 × 10⁻⁹	1.3 × 10⁻⁸	1.4 × 10⁻⁷	4.7 × 10⁻⁷	9.9 × 10⁻⁷
⁵⁶Co	4.6 × 10⁻⁶	2.2 × 10⁻⁵	8.4 × 10⁻⁵	1.6 × 10⁻⁴	2.5 × 10⁻⁴
⁵⁶Ni	7.8 × 10⁻⁴	2.1 × 10⁻³	2.7 × 10⁻³	3.3 × 10⁻³	3.6 × 10⁻³
⁵⁷Fe	...	3.3 × 10⁻⁸	6.2 × 10⁻⁸	1.0 × 10⁻⁷	1.5 × 10⁻⁷
⁵⁷Co	1.3 × 10⁻⁵	1.5 × 10⁻⁵	2.9 × 10⁻⁵	4.8 × 10⁻⁵	6.6 × 10⁻⁵
⁵⁷Ni	5.6 × 10⁻⁴	5.8 × 10⁻⁴	5.7 × 10⁻⁴	5.1 × 10⁻⁴	4.8 × 10⁻⁴
⁵⁸Ni	4.1 × 10⁻⁴	4.0 × 10⁻⁴	4.0 × 10⁻⁴	3.7 × 10⁻⁴	3.6 × 10⁻⁴
⁵⁹Ni	6.9 × 10⁻⁴	6.4 × 10⁻⁴	6.3 × 10⁻⁴	5.8 × 10⁻⁴	5.7 × 10⁻⁴
⁶⁰Ni	2.4 × 10⁻²	3.7 × 10⁻²	4.6 × 10⁻²	5.2 × 10⁻²	5.8 × 10⁻²
⁶¹Ni	1.5 × 10⁻³	2.1 × 10⁻³	3.8 × 10⁻³	5.5 × 10⁻³	6.3 × 10⁻³
⁶¹Cu	5.7 × 10⁻³	6.9 × 10⁻³	6.5 × 10⁻³	4.7 × 10⁻³	3.7 × 10⁻³
⁶²Ni	3.3 × 10⁻⁴	5.8 × 10⁻⁴	1.4 × 10⁻³	1.8 × 10⁻³	2.0 × 10⁻³
⁶²Zn	3.6 × 10⁻³	4.4 × 10⁻³	3.3 × 10⁻³	2.3 × 10⁻³	1.9 × 10⁻³
⁶³Cu	4.1 × 10⁻³	3.3 × 10⁻³	2.9 × 10⁻³	2.5 × 10⁻³	2.5 × 10⁻³
⁶⁴Zn	4.2 × 10⁻²	5.6 × 10⁻²	7.0 × 10⁻²	8.1 × 10⁻²	9.1 × 10⁻²
⁶⁵Cu	8.1 × 10⁻⁷	1.2 × 10⁻⁶	4.9 × 10⁻⁶	1.1 × 10⁻⁵	1.7 × 10⁻⁵
⁶⁵Zn	8.3 × 10⁻³	9.7 × 10⁻³	1.2 × 10⁻²	1.2 × 10⁻²	1.2 × 10⁻²
⁶⁶Zn	1.3 × 10⁻⁴	2.3 × 10⁻⁴	9.9 × 10⁻⁴	1.8 × 10⁻³	2.3 × 10⁻³
⁶⁶Ga	2.8 × 10⁻³	2.7 × 10⁻³	3.1 × 10⁻³	2.7 × 10⁻³	2.2 × 10⁻³
⁶⁶Ge	6.8 × 10⁻³	4.9 × 10⁻³	3.8 × 10⁻³	2.3 × 10⁻³	1.9 × 10⁻³
⁶⁷Ga	4.1 × 10⁻³	3.4 × 10⁻³	3.6 × 10⁻³	3.2 × 10⁻³	3.0 × 10⁻³
⁶⁸Ge	2.8 × 10⁻²	4.5 × 10⁻²	5.6 × 10⁻²	6.5 × 10⁻²	7.1 × 10⁻²
⁶⁹Ge	1.0 × 10⁻²	1.0 × 10⁻³	1.2 × 10⁻²	1.1 × 10⁻²	1.1 × 10⁻²
⁷⁰Ge	6.5 × 10⁻³	5.2 × 10⁻³	5.3 × 10⁻³	4.6 × 10⁻³	4.2 × 10⁻³
⁷¹As	6.1 × 10⁻³	4.8 × 10⁻³	5.0 × 10⁻³	4.4 × 10⁻³	4.2 × 10⁻³
⁷²Se	1.7 × 10⁻²	2.8 × 10⁻²	3.4 × 10⁻²	3.8 × 10⁻²	4.1 × 10⁻²
⁷³Se	9.1 × 10⁻³	6.5 × 10⁻³	7.0 × 10⁻³	6.4 × 10⁻³	6.3 × 10⁻³
⁷⁴Se	7.2 × 10⁻³	6.5 × 10⁻³	7.7 × 10⁻³	7.4 × 10⁻³	7.2 × 10⁻³
⁷⁵Br	8.4 × 10⁻³	4.5 × 10⁻³	3.3 × 10⁻³	2.2 × 10⁻³	2.0 × 10⁻³
⁷⁶Kr	1.3 × 10⁻²	1.8 × 10⁻²	2.0 × 10⁻²	2.2 × 10⁻²	2.3 × 10⁻²
⁷⁷Kr	7.2 × 10⁻³	6.0 × 10⁻³	6.2 × 10⁻³	5.9 × 10⁻³	5.7 × 10⁻³
⁷⁸Kr	8.7 × 10⁻³	7.3 × 10⁻³	7.5 × 10⁻³	6.9 × 10⁻³	6.5 × 10⁻³
⁷⁹Kr	7.0 × 10⁻³	3.5 × 10⁻³	3.0 × 10⁻³	2.5 × 10⁻³	2.4 × 10⁻³
⁸⁰Sr	1.3 × 10⁻²	1.3 × 10⁻²	1.4 × 10⁻²	1.5 × 10⁻²	1.5 × 10⁻²
⁸¹Rb	8.0 × 10⁻³	6.4 × 10⁻³	6.4 × 10⁻³	6.0 × 10⁻³	5.8 × 10⁻³
⁸²Sr	1.7 × 10⁻²	1.3 × 10⁻²	1.2 × 10⁻²	1.2 × 10⁻²	1.1 × 10⁻²
⁸³Sr	1.3 × 10⁻²	1.2 × 10⁻²	1.4 × 10⁻²	1.3 × 10⁻²	1.3 × 10⁻²
⁸⁴Sr	1.3 × 10⁻²	6.1 × 10⁻³	3.8 × 10⁻³	2.6 × 10⁻³	2.2 × 10⁻³
⁸⁵Y	9.8 × 10⁻³	7.9 × 10⁻³	7.7 × 10⁻³	7.2 × 10⁻³	6.9 × 10⁻³
⁸⁶Zr	1.7 × 10⁻²	1.9 × 10⁻²	2.1 × 10⁻²	2.1 × 10⁻²	2.0 × 10⁻²
⁸⁷Zr	2.1 × 10⁻²	9.1 × 10⁻³	5.1 × 10⁻³	3.8 × 10⁻³	3.0 × 10⁻³
⁸⁸Zr	5.9 × 10⁻³	5.5 × 10⁻³	5.0 × 10⁻³	5.1 × 10⁻³	5.3 × 10⁻³
⁸⁹Nb	1.8 × 10⁻²	2.0 × 10⁻²	1.9 × 10⁻²	1.8 × 10⁻²	1.8 × 10⁻²
⁹⁰Mo	1.2 × 10⁻²	1.1 × 10⁻²	9.9 × 10⁻³	9.6 × 10⁻³	9.2 × 10⁻³
⁹¹Nb	8.2 × 10⁻³	9.4 × 10⁻³	1.0 × 10⁻²	1.0 × 10⁻²	9.9 × 10⁻³
⁹²Mo	7.6 × 10⁻³	4.1 × 10⁻³	2.3 × 10⁻³	1.8 × 10⁻³	1.4 × 10⁻³
⁹³Tc	9.5 × 10⁻³	8.8 × 10⁻³	8.1 × 10⁻³	7.6 × 10⁻³	7.4 × 10⁻³
⁹⁴Tc	2.6 × 10⁻²	4.6 × 10⁻²	4.6 × 10⁻²	4.6 × 10⁻²	4.6 × 10⁻²
⁹⁵Ru	3.1 × 10⁻²	2.4 × 10⁻²	1.4 × 10⁻²	9.3 × 10⁻³	7.6 × 10⁻³
⁹⁶Ru	8.1 × 10⁻³	8.7 × 10⁻³	7.4 × 10⁻³	6.6 × 10⁻³	6.2 × 10⁻³
⁹⁷Ru	1.0 × 10⁻²	1.2 × 10⁻²	1.0 × 10⁻²	9.5 × 10⁻³	9.1 × 10⁻³
⁹⁸Ru	1.2 × 10⁻²	1.4 × 10⁻²	1.3 × 10⁻²	1.3 × 10⁻²	1.2 × 10⁻²
⁹⁹Rh	1.3 × 10⁻²	2.1 × 10⁻²	2.1 × 10⁻²	2.0 × 10⁻²	1.9 × 10⁻²
¹⁰⁰Pd	1.6 × 10⁻²	9.2 × 10⁻³	8.7 × 10⁻³	8.0 × 10⁻³	8.2 × 10⁻³
¹⁰¹Pd	1.7 × 10⁻²	1.8 × 10⁻²	1.8 × 10⁻²	1.8 × 10⁻²	1.7 × 10⁻²
¹⁰²Pd	1.6 × 10⁻²	1.8 × 10⁻²	1.9 × 10⁻²	1.9 × 10⁻²	1.8 × 10⁻²
¹⁰³Ag	2.6 × 10⁻²	3.2 × 10⁻²	3.2 × 10⁻²	3.3 × 10⁻²	3.2 × 10⁻²
¹⁰⁴Pd	5.3 × 10⁻²	7.2 × 10⁻²	7.4 × 10⁻²	7.5 × 10⁻²	7.5 × 10⁻²
¹⁰⁵Ag	7.5 × 10⁻²	1.3 × 10⁻¹	1.4 × 10⁻¹	1.4 × 10⁻¹	1.4 × 10⁻¹
¹⁰⁶Cd	1.2 × 10⁻²	9.8 × 10⁻³	5.3 × 10⁻³	3.2 × 10⁻³	2.5 × 10⁻³
¹⁰⁷Cd	9.5 × 10⁻³	6.4 × 10⁻³	3.1 × 10⁻³	1.9 × 10⁻³	1.5 × 10⁻³

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It is worth noting that both the recurrence periods and the ratios between persistent and burst luminosities are larger than those reported for model 1 (see Table 6, for comparison), showing a clear dependence on the metallicity of the accreted material: the smaller the metal content, the larger the recurrence time (and the smaller the value of α).

The corresponding light curves (see Figure 31) exhibit, in turn, a clear pattern: as shown in Figure 32 (left panel), where light curves of the third bursting episode computed in models 1 and 3 are compared, explosions in metal-deficient envelopes (such as model 3) are characterized by lower peak luminosities and longer decline times. A similar pattern has been reported by Heger et al. (2007), in the framework of one-dimensional, hydrodynamic models of XRBs performed with the KEPLER code. It is worth noting that no double-peaked bursts have been obtained in model 3.

Larger peak temperatures, around T_peak ∼ (1.3–1.4) × 10⁹ K, have also been obtained in model 3. This, together with the longer exposure times to high temperatures (driven by the slower decline phase) cause a dramatic extension of the main nuclear path toward the SnSbTe-mass region or beyond.

From the nucleosynthesis viewpoint, and as shown in Table 8 and Figures 31 and 32, the nuclear activity already reaches the end of the network (¹⁰⁷Te) at the late stages of the first bursting episode. The overall mean metallicity of the envelope at the end of each burst is now 0.74 (1st burst), 0.85 (2nd burst), 0.90 (3rd burst), 0.92 (4th burst), and 0.93 (5th burst). Note that, although the accreted material is more metal deficient in model 3 than in model 1, the post-burst mean metallicity of the envelope is larger in model 3. This results from the combination of higher temperatures and longer burst durations, which favors the extension of the nuclear activity: for short bursts (like those obtained in metal-rich envelopes), only the fastest p- and/or α-capture reactions can naturally occur (those that proceed with a characteristic time shorter than the overall exposure time to high temperatures); in contrast, for long-duration bursts, the overall number of p- and/or α-capture reactions increases dramatically. This, in particular, affects CNO breakout through ¹⁵O(α, γ)¹⁹Ne and ¹⁴O(α, p)¹⁷F, which are favored in the longer bursts obtained for model 3. It is also worth noting that the abundance pattern obtained after the different bursts is very similar. This is clearly shown in Figure 33, that depicts the main nuclear activity at peak and at the end of the fourth burst computed for model 3, when compared with that corresponding to the first burst—Figures 24 and 29. This fact justifies our emphasis on the reaction sequences that characterize the first burst (see discussion in Section 5.2).

**Figure 33.** Upper panel: main nuclear activity at the ignition shell when temperature reaches a peak value of T_peak = 1.23 × 10⁹ K, during the 4th burst computed for model 3. Lower panel: same as in the upper panel, but at the end of the burst.
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A final ¹²C yield of ∼2 × 10⁻³ is obtained at the end of each bursting episode (except in the first one, for which X(¹²C) = 8 × 10⁻⁴). As reported for model 1, the amount of unburned ¹²C left over turns out to be too small to power a superburst. Finally, huge overproduction factors (see Figures 31 and 32), involving heavy species such as ^102,104,105Pd, ⁹⁸Ru, or ⁹⁴Mo (with f ∼ 10⁸), have been obtained in model 3, in contrast with the somewhat more modest values achieved in model 1, where maximum overproduction factors are about f ∼ 10⁶, and involving lighter species, such as ⁷⁶Se, ^78,80Kr, or ⁸⁴Sr.

5. DISCUSSION

5.1. General Relativity Corrections

The calculations reported here have been performed assuming Newtonian gravity. Since the envelope layers are very thin, it is easy to introduce general relativity corrections to this Newtonian framework (see Ayasli & Joss 1982; Lewin et al. 1993; Taam et al. 1993; Cumming et al. 2002; Woosley et al. 2004). To this end, the surface gravity is rewritten as g = GM_*/R²_*(1 + z), where M_* is the mass, R_* is the stellar radius (defined in such a way that the surface area is 4πR²_*), and z is the gravitational redshift given by 1 + z = (1 − 2GM_*/R_*c²)^−1/2. Our models of M_* = 1.4 M_☉ require R_* = 14.3 km, and a gravitational redshift of z = 0.19.

Following Woosley et al. (2004), once the redshift and radius are determined, it is straightforward to derive the set of correcting factors to the physical magnitudes described above for a suitable observer at infinity. Hence, recurrence times and burst durations should be increased by a factor of 1 + z. The mass-accretion rate as well as the burst luminosity have to take into account both the difference in surface area (compared to the Newtonian framework) and the gravitational redshift term. The energy and rest mass-accretion rate scale as R²_*/(1 + z), while the luminosity ∝ R²_*/(1 + z)². However, when M_* is taken exactly as M_NS (Newtonian framework), the surface area and redshift corrections for energy and mass-accretion rate cancel out, since g ∝ (1 + z)/R²_* = const, and hence, no correction to the observed burst energy or mass-accretion rate is necessary, while the luminosity correction is simply given by 1/(1 + z) = 0.84. In addition, the accretion luminosity for an observer at infinity changes only by a factor of 1.012; that is, the ratio between gravitational energy released per unit mass in general relativity, c².z/(1 + z), and the Newtonian value, GM_NS/R_NS. Finally, the luminosity measured at infinity will be smaller by a factor of (1 + z) = 1.19.

5.2. Comparison with Previous Work

For consistency, the results discussed in this paper have been compared with those reported in previous work (obtained with similar hydrodynamic codes or in the framework of one-zone models).

As emphasized in Section 2, model 1 is qualitatively similar to model ZM of Woosley et al. (2004). The 12 bursts computed by Woosley et al. (2004) in a Newtonian frame were characterized by recurrence times of about ∼2.7 hr, peak luminosities of L_peak ∼ (1.5–2) × 10³⁸ erg s⁻¹, and ratios between persistent and burst luminosities of α ∼ 60–65. Our calculations (model 1, Newtonian frame) yield τ_rec ∼ 5–6.5 hr, L_peak ∼ (3–7) × 10³⁸ erg s⁻¹, and α ∼ 35–40.

The role played by the metallicity of the accreted material (model 3, with Z = Z_☉/20 = 0.001) qualitatively agrees with the pattern reported by Woosley et al. (2004; see also, Heger et al. 2007). Longer recurrence times of ∼9 hr, peak temperatures of about (1.3–1.4) × 10⁹ K, and ratios between persistent and burst luminosities of α ∼ 20–30 (with L_peak ∼ 10³⁸ erg s⁻¹) have been obtained in the five bursts computed in model 3. In turn, the 15 bursts computed by Woosley et al. (2004) for model zM are characterized by recurrence times of about 3–3.5 hr, peak luminosities of L_peak ∼ 10³⁸ erg s⁻¹, and ratios between persistent and burst luminosities of α ∼ 50–60. Results reveal a dependence of burst properties on the metallicity of the accreted material: the smaller the metal content, the larger the recurrence time (and the smaller the α). In turn, explosions in metal-deficient envelopes (i.e., model 3) are characterized by lower peak luminosities and longer decline times, in agreement with the pattern described in Woosley et al. (2004) and Heger et al. (2007). Model 3 bears as well a clear resemblance with the model computed by Fisker et al. (2008). In that work, five representative bursting sequences were analyzed, with τ_rec ∼ 3.5–4 hr, L_peak ∼ (7–8) × 10³⁷ erg s⁻¹, and α ∼ 65–70, as measured at infinity.

Despite the qualitative similarities in the gross properties of the bursts presented in this paper (as well as in the role played by the metallicity of the accreted material) and those reported in previous work, a quantitative comparison reveals some discrepancies that are worth analyzing. In model 1 (with Z = Z_☉), our computations yield systematically larger (by a factor of ∼ 2) recurrence times and peak luminosities (and hence, lower α) than model ZM of Woosley et al. (2004). Similar results are found in the low-metallicity case (model 3, with Z = Z_☉/20) when compared with model zM of Woosley et al. (2004), except for the peak luminosities that turn out to be very similar. It is also worth noting that the values reported by Fisker et al. (2008) show discrepancies with respect to Woosley et al. (2004), in particular, lower peak luminosities (and larger α). A major difference concerns the much larger effect played by the metallicity of the accreted material in this work as compared with Woosley et al. (2004), who explained the moderate effect found as due to compositional inertia washing out the influence of the initial metallicity. Another striking issue concerns the extremely large differences in the gross physical characteristics—nucleosynthesis, energies, or recurrence times—between the first and subsequent bursts, as reported by Woosley et al. (2004). In terms of nucleoynthesis or nuclear activity, Figures 30–32 reveal a similar behavior for the different bursts (although a somewhat lower production of intermediate-mass elements as well as of the heaviest elements is reported for the first burst computed in model 3).

Very limited information on the nucleosynthetic yields obtained in model ZM is given in Woosley et al. (2004). Thus, we will restrict the discussion on the extent of the nuclear activity and on the resulting chemical abundance pattern to model 3, through a brief comparison with the work reported by Schatz et al. (2001), Fisker et al. (2008), and Woosley et al. (2004; for model zM). It is worth noting that both the nucleosynthetic endpoint (located in the SnSbTe-mass region) and the main nuclear path in the A ∼ 50–100 mass region obtained in this work (passing through a suite of different nuclei, such as ⁵⁵Co, ⁶⁰Zn, ⁷⁰Br, ⁷⁵Rb, ⁸⁵Mo, ⁹⁰Rh, or ^100,105Sn) are similar to those reported by Schatz et al. (2001) in the framework of one-zone calculations. Whereas the main nuclear path in the first burst of model zM (Woosley et al. 2004) is very similar to the one reported in this work, compositional inertia causes a more limited extension of the nuclear activity in the successive bursts of Woosley et al.: hence, while the three most abundant nuclei at the bottom of the envelope are ¹⁰⁶Sn and ^104,106In at the end of the first burst, this switches to ⁶⁴Zn, ⁶⁸Se, and ³²S (a similar trend is also reported by Fisker et al. 2008). In this work, the mass-averaged composition at the end of the first burst computed for model 3 (see Table 8) is dominated (aside from some residual H and ⁴He) by the presence of ¹⁰⁴Pd (0.05, by mass) and ¹⁰⁵Ag (0.08), while X(⁶⁴Zn) ∼ 0.04. But the peak at the end of the abundance distribution (see Figure 30) increases with subsequent bursts up to a plateau value, which indicates that these heavy nuclei are still produced in similar quantities. This is very different to the results reported by Woosley et al. (2004). Indeed, at the end of the fifth burst, the abundance pattern shows still a significant presence of heavy species (i.e., X(¹⁰⁵Ag) ∼ 0.1, X(¹⁰⁴Pd) ∼ 0.08, and X(⁹⁴Tc) ∼ 0.05), together with a simultaneous increase in the abundances of intermediate-mass elements, such as ⁶⁰Ni (0.06), ⁶⁴Zn (0.09), ⁶⁸Ge (0.07), or ⁷²Se (0.04). It is finally worth noting that, in agreement with all previous hydrodynamic studies, both models 1 and 3 yield very small post-burst abundances of ¹²C, below the threshold amount required to power superbursts. Even though only a few bursts have been computed for these models, they already show a trend on the amount of ¹²C that may be expected after many more bursts.

Finally, it is also worth mentioning that large differences exist between the hydrodynamic simulations reported here (see also Woosley et al. 2004; Fisker et al. 2008) and those based on one-zone models (i.e., Schatz et al. 1999, 2001) as regards the shape of the light curve accompanying the bursting episodes (the primary difference being the presence of a long-lasting plateau in the latter).

The origin of the discrepancies reported is not totally clear and would require additional hydrodynamic studies. Note, however, that the local surface gravity of our model is somewhat smaller than that adopted in the above-mentioned works: whereas a 10 km radius is assigned to the 1.4 M_☉ neutron star in Woosley et al. (2004; g = 1.86 × 10¹⁴ cm s⁻²), the integration of the neutron star structure from the core to its surface, in hydrostatic equilibrium, yielded 13.1 km (14.3 km, after general relativity corrections are introduced; see Section 5.1), for our 1.4 M_☉ neutron star (corresponding to a surface gravity of g = 1.08 × 10¹⁴ cm s⁻²); in turn, the calculations reported by Fisker et al. (2008), in a general relativity framework, relied on a 11 km (1.4 M_☉) neutron star, for which g = 1.53 × 10¹⁴ cm s⁻². Although XRB properties depend weakly upon the neutron star mass (or surface gravity), part of the differences outlined between the three studies can be attributed to the combined effect of the adopted neutron star size (surface gravity) and to differences in the input physics (i.e., nuclear reaction network, opacities, treatment of convection). In particular, the use of Iben's opacities may have some effect on the peak luminosities achieved since the larger OPAL opacities will likely decrease the amount of energy radiated away from the star. Moreover, the inclusion of semiconvection and thermohaline mixing would have a minor effect in the properties of the explosions, likely affecting the appearance of marginal convective transport between bursts (see Woosley et al. 2004; Fisker et al. 2008). It is however worth noting that the convective pattern shown in Figures 18 and 32 is similar to those reported in previous work: namely that convection sets in as soon as superadiabatic gradients are established in the envelope, following the early stages of the TNR and the corresponding rise in temperature; it reaches the surface and begins to recede before the observed burst properly commences, shutting off thereafter (Woosley et al. 2004; Fisker et al. 2008).

The potential impact of XRB nucleosynthesis on Galactic abundances is still a matter of debate. Matter accreted onto a neutron star of mass M and radius R releases GMm_p/R ∼ 200 MeV nucleon⁻¹, whereas only a few MeV nucleon⁻¹ are released from thermonuclear fusion. Thus ejection from a neutron star is unlikely. However, it has been suggested that radiation-driven winds during photospheric radius expansion may lead to ejection of a tiny fraction of the envelope (containing nuclear processed material; see Weinberg et al. 2006; MacAlpine et al. 2007). Indeed, XRBs have been proposed as a possible source of the light p-nuclei ^92,94Mo and ^96,98Ru (Schatz et al. 1998, 2001). No matter is ejected in any of the models reported in this work, a result fully independent of the adopted resolution and in agreement with all previous hydrodynamic simulations (Woosley et al. 2004; Fisker et al. 2008). Moreover, it is worth noting that, as shown in Figures 17 and 30, the abundances of many species synthesized during the bursts decrease remarkably toward the outer envelope layers, because of inefficient convective transport (see Figures 18 and 32). To assess the possible contribution to the Galactic abundances, one has to rely on the abundances of the outer envelope layers (the only ones that have a chance to be ejected by radiation-driven winds). This shows the limitations posed by one-zone nucleosynthesis calculations, in which the chemical species synthesized in the innermost layers are, by construction, assumed to represent the whole (chemically homogeneous) envelope. The mass fractions of these p-nuclei, obtained in model 3, drop by more than an order of magnitude in the outer envelope layers (as compared with the values achieved at the innermost envelope; see Figure 34); the resulting overproduction factors, f ∼ 10⁶, are several orders of magnitude smaller than those required to account for the origin of these problematic nuclei (see Weinberg et al. 2006; Bazin et al. 2008), in sharp contrast with the results obtained on the basis of one-zone calculations (Schatz et al. 1998, 2001).

**Figure 34.** Distribution of the unstable species ⁹²Pd, ⁹⁴Ag, and ^96,98In (that power the light p-nuclei ^92,94Mo and ^96,98Ru) throughout the accreted envelope, at the end of the fifth bursting episode of model 3.
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The authors are grateful to the anonymous referee for a critical reading of the manuscript and for valuable comments that have improved the presentation of this paper. This work has been partially supported by the Spanish MICINN grants AYA2007-66256 and EUI2009-04167, by the E.U. FEDER funds, by the U.S. Department of Energy under Contract No. DE-FG02-97ER41041, by the DFG cluster of excellence "Origin and Structure of the Universe," and by the ESF EUROCORES Program EuroGENESIS.

HYDRODYNAMIC MODELS OF TYPE I X-RAY BURSTS: METALLICITY EFFECTS

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ABSTRACT

1. INTRODUCTION