Introduction

Thermonuclear (Type I) X-ray bursts (XRBs) originate in the high-density-temperature degenerate envelope of a neutron star in a close low-mass X-ray binary during thermonuclear runaways (Woosley & Taam 1976;Joss 1977). The envelope consists of stellar material accreted from the low-mass companion star. Every XRB episode encapsulates abundant information on the hydrodynamics and thermal states of the evolution of the degenerate envelope (Woosley et al. 2004), the structure of the accreting neutron star (Steiner et al. 2010), the rapid-proton capture (rp-) process path of synthesized nuclei (Van Wormer et al. 1994;Schatz et al. 1998), and the burst ashes that become compositional inertia for the succeeding bursts before sinking into the neutron-star crust (Keek & Heger 2011;Meisel et al. 2018).

XRBs are driven by the triple-α reaction (Joss 1978), αpprocess (Woosley & Weaver 1984), and rp-process (Wallace & Woosley 1981;Wiescher et al. 1987) and are constrained by βdecay and the proton drip line. After breaking out from the hot CNO cycle, the nuclear reaction flows enter the sd-shell nuclei region via αp-processes; this is also the region in which the αpprocesses are dominant. Then, the reaction flows continue to the pf-shell nuclei region, first going through a few important cycles at the light pf-shell nuclei, e.g., the CaSc cycle, and then reaching the medium pf-shell nuclei where the NiCu and ZnGa cycles reside (Van Wormer et al. 1994). After breaking out from the ZnGa cycles and the GeAs cycle, which may transiently and weakly exist, and passing through Ge and Se isotopes, the reaction flows surge through the heavier proton-rich nuclei where rp-processes actively burn the remaining hydrogen accreted from the companion star; eventually, the reaction flows stop at the SnSbTe cycles (Schatz et al. 2001). This rp-process path is indicated in the pioneering GS 1826-24 clocked burster model (Woosley et al. 2004;Heger et al. 2007).

The 57 Cu(p,γ) 58 Zn reaction that draws material from the 56 Ni waiting point via the 56 Ni(p,γ) 57 Cu branch is located in the NiCu I cycle (Figure 1). The influence of this reaction on the XRB light curve and on burst ash abundances was studied by Cyburt et al. (2016), and they concluded that the 57 Cu(p,γ) 58 Zn reaction is the fifth most influential (p,γ) reaction that affects the light curve of the GS 1826-24 clocked burster (Makino 1988;Tanaka 1989;Ubertini et al. 1999). Forstner et al. (2001) constructed the 57 Cu(p,γ) 58 Zn reaction rate based on shell-model calculations and predicted the properties of important resonances. Later, Langer et al. (2014) experimentally confirmed some low-lying energy levels of 58 Zn, which are dominant resonances contributing to the 57 Cu(p,γ) 58 Zn reaction rate at temperature range 0.3  T(GK)  2.0. With the high-precision measurement of these energy levels, Langer et al. largely reduced the rate uncertainty by up to three orders of magnitude compared to the Forstner et al. reaction rate. Nevertheless, the order of the 1 1 + -and 2 3 + -dominating resonance states was unconfirmed, and the 1 2 + resonance state, which is one of the dominant resonances at the XRB temperature range, 0.8  T(GK)  2, was not detected in their experiment.

The 55 Ni(p,γ) 56 Cu reaction rate was recently determined by Valverde et al. (2019) and Ma et al. (2019) with the highly precisely measured 56 Cu mass (Valverde et al. 2018) and the precisely measured excited states of 56 Cu (Ong et al. 2017). In fact, Ma et al. (2019) found that the 55 Ni (p,γ) 56 Cu reaction rate was underestimated by up to one order of magnitude by Valverde et al. (2018) due to the incorrect penetrability scaling factor, causing a set of wrongly determined burst ash abundances of nuclei A = 55-60. Figure 2 presents the comparison of the 55 Ni(p,γ) 56 Cu reaction rates deduced by Valverde et al. (2019), Ma et al. (2019), and Fisker et al. (2001). The reaction rate was then corrected by Valverde et al. (2019) and used in their updated one-zone XRB model indicating that the reaction flow bypassing the important 56 Ni waiting point could be established. Based on the updated zerodimensional one-zone hydrodynamic XRB model, the extent of the impact the newly corrected 55 Ni (p,γ) 56 Cu reaction induces on the bypassing reaction flow, however, causes merely up to 5% difference in the productions of nuclei A = 55-65 (Valverde et al. 2019). Moreover, due to the zero-dimensional feature of the one-zone XRB model, the distribution of synthesized nuclei along the mass coordinate in the accreted envelope is unknown, and importantly, the one-zone hydrodynamic XRB model does not match with any observation.

In the present work, we reanalyze the nuclear structure information and perform simulations with the aim to constrain the reaction flows in the NiCu cycles and to analyze their impact on the clocked bursts of the GS 1824-26 burster. In Section 2, we present the formalism for the reaction rate calculation and introduce the isobaric-multiplet-mass equation (IMME) that we use to cross-check the order of the 1 1 + and 2 3 + states in 58 Zn, the dominating resonances for the 57 Cu(p,γ) 58 Zn reaction, and to estimate the energy of the 1 2 + resonance state.

The deduced 57 Cu(p,γ) 58 Zn reaction rate is discussed in detail in Section 3. Using the one-dimensional multizone hydrodynamic KEPLER code (Weaver et al. 1978;Woosley et al. 2004;Heger et al. 2007), we model a set of XRB episodes matched with the GS 1826-24 burster with the newly deduced

Reaction Rate Calculations

The total thermonuclear proton-capture reaction rate is expressed as the sum of resonant-(res) and direct-capture (DC) on the ground state and thermally excited states in the target nucleus, and each capture with given initial and final states is weighted by its individual population factor (Fowler & Hoyle 1964;Rolfs & Rodney 1988),

where J are the angular momenta of the initial states of the target nucleus and E are the energies of these initial states.

Resonant rate. The resonant reaction rate for proton capture on a target nucleus in its initial state, i, N v i A res s á ñ , is a sum over all respective compound nucleus states j above the proton separation energy (Rolfs & Rodney 1988;Iliadis 2007). The resonant rate can be expressed as (Fowler et al. 1967;Schatz et al. 2005)

, where the resonance energy in the center-of-mass system,

, is the energy difference between the compound nucleus E j

x state and the sum of the excitation energies of the initial state E i and the respective proton threshold, S p . For the capture on the ground state, E i = 0. μ is the reduced mass of the entrance channel in atomic mass units (μ = A T /(1 + A T ), with A T the target mass number), and T 9 is the temperature in gigakelvin (GK). The resonance energy and strength in Equation (2) are given in units of MeV. The resonance strength, ωγ ij , taken in MeV in Equation (2), reads

where J i is the target spin and J j , ij p G , j G g , and j total G are a spin, proton-decay width, γ-decay width, and total width of the compound nucleus state j, respectively. Assuming that other decay channels are closed (Audi et al. 2017) in the considered excitation energy range of the compound nuclei, the total width becomes

Within the shell-model formalism that we use here, the proton width can be expressed as

where Γ sp is a single-particle width for the capture of a proton with respect to a given (nlj) quantum orbital in a spherically symmetric mean-field potential, while C 2 S(nlj) denotes a corresponding spectroscopic factor containing information of the structure of the initial and final states. The Γ sp can either be estimated from proton-scattering cross sections in a Woods-Saxon potential with the adjusted potential depth to reproduce known proton energies (WSPOT code);foot_0 or alternatively, it can also be obtained from the potential barrier penetrability calculation as (Van Wormer et al. 1994;Herndl et al. 1995)

fm (with r 0 = 1.25 fm) is the nuclear channel radius, and the Coulomb barrier penetration factor P l is

where k E 2m =

 and E is the proton energy in the center- of-mass system; F l and G l are the regular and irregular Coulomb functions, respectively. In the present work, we follow the same procedure as was used by Lam et al. (2016) to get the proton widths of the important 57 Cu(p,γ) 58 Zn resonances up to the Gamow window. The maximum difference between the Γ sp described by the two methods above is below 40% for the present work.

Gamma-decay widths are obtained from electromagnetic reduced transition probabilities B(ΩL; J i → J f ) (Ω stands for electric or magnetic), which contain the nuclear structure information of the resonance states and the final bound states. The corresponding gamma-decay widths for the most contributed transitions (M1 and E2) can be expressed as (Brussaard & Glaudemans 1977 s n =and g 1 l p = , g 0 l n = , whereas the B(E2) values have been obtained from standard effective charges, e p = 1.5e and e n = 0.5e (Honma et al. 2004). We use experimental energies, E γ , when available. The total electromagnetic decay width is obtained from the summation of all partial decay widths for a given initial state.

Information on nuclear structure.The essential information needed to estimate the resonant rate contribution of 57 Cu(p,γ) 58 Zn consists of the resonance energies of the compound nucleus 57 Cu+p, one-proton transfer spectroscopic factors, and proton-and gamma-decay widths. The properties of resonances sensitive to the 57 Cu(p,γ) 58 Zn reaction rate in the XRB temperature range are provided by Langer et al. (2014). Nevertheless, the order of the In the present study, we use the IMME to constrain the energies of experimentally unknown, but important resonance state, in 58 Zn, i.e., the order of the 1 1 + and 2 3 + states of 58 Zn and the energy of the 1 2 + state. A similar method was exploited earlier by Richter et al. (2011) and Richter & Brown (2012, 2013) to provide the missing experimental information of the nuclear level schemes. Also, the same method was used by Schatz & Ong (2017) to estimate the unknown nuclear masses important for the reverse (p,γ) rates. Assuming that the isospin-symmetry-breaking forces are two-body operators of the isovector and isotensor character, the mass excesses of the members of an isobaric multiplet (I = 1, I z = -I, -I + 1,K, I) show at most a quadratic dependence on I z , as expressed by the IMME (Wigner 1957),

is the mass excess of a quantum state of isospin (I, I z ), and α = (A, J π , N exc ,...) are the nuclear mass number A, excited state number N exc , and all other quantum numbers labeling the quantum state. The a, b, and c coefficients reflect contributions from the isoscalar, isovector, and isotensor parts of the effective nucleon-nucleon interaction, respectively (see Ormand & Brown (1989) or Lam et al. (2013aLam et al. ( , 2013b) ) for details). For an isobaric-triplet state (I = 1, I z = -1, 0, +1), we can form from Equation (8) a system of three linear equations and, therefore, express the IMME c coefficient in terms of three mass excesses as

, 1 2 , 1 2. 9

In turn, if we know the mass excesses of the I z = 0 and I z = 1 isobaric-multiplet members and a theoretical c coefficient, the mass excess of a proton-rich member (I z = -1) can be found via a simple relation:

This equation defines the method that we use in the present paper.

We first obtain a set of theoretical IMME c coefficients for the lowest and excited A = 58 triplets, including those that involve the dominant resonances. To this end, we perform large-scale shell-model calculations in the full pf shell-model space using the NUSHELLX@MSU shell-model code (Brown & Rae 2014) with the charge-dependent Hamiltonian, which is constructed from the modern isospin-conserving Hamiltonian (GXPF1a; Honma et al. 2004Honma et al. , 2005)), the two-body Coulomb interaction, strong charge-symmetry-breaking and chargeindependence-breaking terms (Ormand & Brown 1989), and the pf shell-model space isovector single-particle energies (Ormand & Brown 1995). The Hamiltonian is referred to as "cdGX1A" and was used by Smirnova et al. (2016Smirnova et al. ( , 2017) ) (Seth et al. 1986) dominates the experimental IMME c coefficient uncertainties and propagates to the proton separation energy of 58 Zn, S p ( 58 Zn) = 2.280 ±0.050 MeV (AME2016). In general, theoretical c coefficients are seen to be in robust agreement with the respective experimental values. The comparison yields the rms deviation of about 22 keV, which we assign to the calculated values as the theoretical uncertainty; see Table 1.

According to the recent compilation of IMME c coefficients of isobaric multiplets with A = 6-58 (Lam et al. 2013b), the IMME c coefficients exhibit a gradually decreasing trend as a function of A with values ranging between about 400 and 150 keV. As is well known from the data, the c coefficients of triplets show a prominent staggering effect, being split into two families: the values of c coefficients inherent to isobars with A = 4n + 2 appear to be systematically higher than those for their A = 4n neighbors, with n being a positive integer. These average values decrease with increasing A approximately as A -1/3 as suggested by a uniformly charged liquid drop model. It has also been noticed that the amplitude of staggering decreases with increasing excitation energy manifesting the weakening of the pairing effects in higher excited states (Lam et al. 2013a). In the present study, we extend the compilation of Lam et al. (2013b) and tentatively propose excited isobaric multiplets in the A = 58 triplet. Although the dependence of c coefficients on excitation energy is less known, from theoretical studies in the sd-shell nuclei, the amplitude of staggering in b Presently compiled from the evaluated nuclear masses (AME2016) and experimentally measured levels (Jongsma et al. 1972;Rudolph & McGrath 1973;Honkanen et al. 1981;Rudolph et al. 1998Rudolph et al. , 2000Rudolph et al. , 2002;;Johansson et al. 2009;Langer et al. 2014) according to the procedure implemented by Lam et al. (2013b).

c Presently calculated with the cdGX1A Hamiltonian based on the full pf shellmodel space. The 1 1 + , 2 3 + , 3 1 + , 1 2 + , and 2 5 + triplets are not taken into the comparison yielding the rms. d An alternative order of the 1 1 + and 2 3 + states according to IMME dominance to the previous order proposed by Langer et al. (2014).

isobaric triplets is expected to gradually diminish in the pf-shell nuclei. Recently, more precise nuclear mass measurements confirmed the persistence of these trends in the pf-shell nuclei (Zhang et al. 2018;Surbrook et al. 2019;Fu et al. 2020). We find that the values of the c coefficients provide a very stringent test for isobaric multiplets as we will see below.

The order of the 1 1 + and 2 3 + states. As was mentioned before, the order of the 1 1 + and 2 3 + states stays undetermined in the work by Langer et al. (2014) with two plausible energies, 2861 keV and 2904 keV. The character of the electromagnetic decay of those states weakly supports the assignment proposed in that work, that the lower state is a 2 3 + state and the higher one is the 1 1 + , I = 1 state. Indeed, we can get the ratio of the partial electromagnetic widths for the decay of these states to 0 g.s.

+ and 2 1 + to be more in reasonable agreement with that assignment as seen from Table 2. Alternatively, a certain constraint can also be imposed by the IMME. Although the 1 1 + and 2 3 + , I = 1 states of 58 Cu are not assigned (Nesaraja et al. 2010), from the existing data we find that the best candidate for 2 3 + could be a state at 3230 ± 20 keV as measured by Rudolph & McGrath (1973) 1). Based on this indication, we suggest here that the 2904 keV state could be tentatively assigned as 2 3 + .

For the 1 1 + , I = 1 state in 58 Cu, only an interval of energies can be proposed. Indeed, no low-lying 1 + , I = 1 states have been observed by Fujita et al. (2002Fujita et al. ( , 2007)). To understand this fact, we have calculated a Gamow-Teller (GT) strength distribution from the 58 Ni ground state to the 1 + states in 58 Cu using the GXPF1a Hamiltonian. The results are summarized in Table 3. First, we remark that there is a relatively good agreement with the data found in Fujita et al. (2002Fujita et al. ( , 2007)). For example, the B(GT) values of 1 + , I = 0 at low energies are comparable. In particular, we also find strong transitions populating the two lowest states. Moreover, our calculation reproduces a relatively large strength fragment at 3.4 MeV, which may be split between two states in the experiment. Second, it can be noticed that the two lowest 1

Note.

a The theoretical B(GT) is quenched with the standard quenching factor of 0.77 (Horoi et al. 2007).

information of the 1 1 + , 1 2 + , and 2 5 + isobaric analog states, and the 1 2 + and 2 5 + states of 58 Zn.

Properties of resonances-With the information on nuclear structure described above, we deduce a set of resonance properties of 58 Zn to construct the new 57 Cu(p,γ) 58 Zn resonant reaction rate within the typical XRB temperature range, e.g., the GS 1826-24 burster. We only consider proton capture on the 3 2 g.s.

-ground state (g.s.) of 57 Cu as the contribution from proton resonant captures on thermally excited states of 57 This is mainly because the main contributions for the 2 3 + and 1 1 + states are the p 1/2 and f 5/2 particle captures, respectively. For higher values of the orbital angular momentum l of the captured proton, the corresponding width becomes more sensitive to the proton energy because barrier penetrability varies faster. Once the 1 + state, governed by the f capture, is assigned at a lower excitation energy, its contribution to the resonant rate becomes drastically reduced.

Direct-capture rate-Comparing the direct-capture rate deduced by Fisker et al. (2001) (or by Forstner et al. 2001) with the presently deduced resonant capture rate, we notice that the contribution of direct capture is exponentially lower than the contribution of the dominating resonances throughout XRB related temperature range from 0.3 to 2 GK. Hence, the contribution of the direct-capture rate is negligible for the 57 Cu(p,γ) 58 Zn reaction rate; see Figure 3 These parameters, i.e., a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , and a 6 , are listed in Table 6. The running index i is up to 6 for the Present rate for the temperature region, 0.1-2 GK. The parameterized Present rate is evaluated according to an accuracy quantity proposed by Rauscher & Thielemann (2000),

where n is the number of data points, r m are the original Present rate calculated for each respective temperature, and f m are the fitted rate at that temperature. With n = 297, ζ is 4.45 × 10 -3 , and the fitting error is 5.90% for the temperature range from 0.01 to 3 GK. The parameterized rate is obtained with aid from the Computational Infrastructure for Nuclear Astrophysics (CINA). 13 For rates above 3 GK, one may refer to statistical model calculations to match with the Present rate, which is only valid within the mentioned temperature range and fitting errors, see NACRE (Angulo et al. 1999).

We 4. The Hauser-Feshbach statistical model rates, i.e., rath 14 , thra 15 , and ths8 16 are very close to one another from 0.1 to 2.0 GK, and they are lower than the Present rate up to an order of magnitude at temperature T  0.9 GK. Due to the reduction of the contribution from the 1 2 + resonance state, the Present rate is up to a factor of 2 lower than the Langer et al. rate from 0.8 to 2 GK covering the typical maximum temperature of the GS 1826-24 burster, and up to a factor of 4 lower than the wien2 rate (Forstner et al. 2001) recommended by JINA REACLIB v2.2; see the comparison in the respective ratio in the bottom panel of Figure 4. By taking into account the uncertainty of S p ( 58 Zn), we estimate and list the uncertainty of Present 57 Cu(p,γ) 58 Zn reaction rate as upper and lower limits in Table 5. Both upper and lower limits are shown by the red zone in Figure 4, whereas the uncertainty of the Langer et al. rate is indicated by the blue zone. Even if the uncertainty due to the order of the 1 1 + and 2 3 + states would have been removed, the uncertainty of S p ( 58 Zn) propagated from the measured 58 Zn mass (Seth et al. 1986) is still dominant and persistent. Note that this is the first 57 Cu(p,γ) 58 Zn reaction rate constructed from important experimental information supplemented with the full pf-shell space shell-model calculation that yields converged resonance energies, Γ γ , and spectroscopic factors; the uncertainty is 13 http://nucastrodata.org/infrastructure.html 14 Produced by Rauscher & Thielemann (2000) using the NON-SMOKER code with FRDM mass input (Möller et al. 1995). 15 Produced by Rauscher & Thielemann (2000) using the NON-SMOKER code with ETFSI-Q mass input (Pearson et al. 1996). 16 Produced by T. Rauscher using the NON-SMOKER code as part of JINA REACLIB since the v1.0 release (Cyburt et al. 2010).

clearly identified, whereas the Hauser-Feshbach statistical model rates may include unknown systematic errors because of their limited capability in estimating level densities of nuclei near to the proton drip line.

Implication for Multizone X-Ray Burst Models

We explore the influence of the Present 57 Cu(p,γ) 58 Zn reaction rate in characterizing the XRB light curves of the GS 1826-24 X-ray source (Makino 1988;Tanaka 1989) and burst ash composition after an episode of XRBs based on onedimensional multizone hydrodynamic XRB models. The theoretical XRB models matched with the GS 1826-24 clocked burster (Ubertini et al. 1999) are instantiated by the KEPLER code (Weaver et al. 1978;Woosley et al. 2004;Heger et al. 2007) and were used by Heger et al. (2007) to perform the first quantitative comparison with the observed GS 1826-24 light curve. Later, the GS 1826-24 XRB models were used by Cyburt et al. (2016) and by Jacobs et al. (2018) to study the sensitivity of (α, γ), (α, p), (p, γ), and (p, α) nuclear reactions. The GS 1826-24 XRB models are continuously updated and were recently used by Goodwin et al. (2019) and by Johnston et al. (2020) to study the high-density properties of accreted envelopes of the GS 1826-24 clocked burster. The XRB models are fully self-consistent, which take into account the correspondence between the evolution in astrophysical conditions and the feedback of nuclear energy generation in substrates of the accreted envelope. Throughout an episode of outbursts, which may consist of a series of bursts with either an almost consistent or progressively increasing recurrence time, the models are capable to keep updating the evolution of chemical inertia and thermal configurations that drive the nucleosynthesis in the accreted envelope of an accreting neutron star.

The XRB models simulate a grid of Lagrangian zones (Weaver et al. 1978;Woosley et al. 2004;Heger et al. 2007), and each zone independently contains its own isotopic composition and thermal properties. We implement the timedependent mixing length theory (Heger et al. 2000) to describe the convection transferring heat and nuclei between these Lagrangian zones. KEPLER uses an adaptive thermonuclear reaction network that automatically includes or discards the respective reactions out of the more than 6000 isotopes provided by JINA REACLIB v2.2 (Cyburt et al. 2010).

We adopt the XRB model from Jacobs et al. (2018) to compare with the observed burst light curves of the GS 1826-24 clocked burster. The model had been used by Jacobs et al. (2018) -state of 57 Cu in the temperature region of XRB interest are indicated as dashed color lines with the respective resonance energies. Bottom panel: The updated main contributing resonances with the full pf shell-model space calculation for the Γ γ widths and spectroscopic factors, and with the resonance energy of the 1 2 + state using IMME formalism. See details in the text and Table 4. 57 Cu reaction rates are denotedas the Present † , Present ‡ , Present ♠ , Present ♡ , and Present § models, respectively. The Present ♡ and Present § models implement a factor of 0.120 of M Edd  for the accretion rate in order to obtain a modeled recurrence time close to the observation, proposing that either the Present or Langer et al. 57 Cu(p,γ) 58 Zn reaction rate, which is lower than the wien2 rate, shortens the recurrence time by up to 5%.

We then simulate a series of 40 consecutive XRBs for the baseline, Present † , Present ‡ , Present ♠ , Present ♡ , and Present § models; and only the last 30 bursts are summed up with respect to the time resolution and then averaged to yield a burst lightcurve profile. The first 10 bursts simulated from each model are excluded because these bursts undergo a transition from a chemically fresh envelope with unstable burning to an enriched envelope with chemically burned-in burst ashes and stable burning. Throughout the transition, the enriched burst ashes are recycled in the succeeding burst heating, which gradually stabilizes the following bursts. The averaging procedure applied on the modeled light curves is similar to the method performed by Galloway et al. (2017) to produce an averaged light-curve profile from the observed data set of Epoch Jun 1998. The epoch was recorded by the Rossi X-ray Timing Explorer (RXTE) Proportional Counter Array (Galloway et al. 2004(Galloway et al. , 2008(Galloway et al. , 2020) ) and were compiled into the Multi-Instrument Burst Archivefoot_2 by Galloway et al. (2020).

The burst luminosity, L x , obtained from each model is transformed and related to the observed flux, F x , via the relation (Johnston et al. 2020)

where d is the distance; ξ b takes into account of the possible deviation of the observed flux from an isotropic burster luminosity due to the scattering and blocking of the emitted electromagnetic wave by the accretion disk (Fujimoto 1988;He & Keek 2016), and the redshift, z, rescales the light curve when transforming into an observer's frame. d and ξ b are combined to form the modified distance d b x by assuming that the anisotropy factors of the burst and persistent emissions are degenerate with distance. We include the entire burst time span of an averaged observational data to fit our modeled burst light curves of each model to the observed light curve. The best-fit d b

x and (1 + z) factors of the baseline, Present † , Present ‡ , Present ♠ , Present ♡ , and Present § modeled light curves to the averaged-observed light curve and the recurrence time of Epoch June 1998 are 7.28 kpc and 1.29, 7.32 kpc and 1.29, 7.32 kpc and 1.29, 7.32 kpc and 1.28, 7.30 kpc and 1.29, and 7.62 kpc and 1.29, respectively. Using these redshift factors, we obtain a set of modeled recurrence times that are close to the observation. The recurrence times of baseline, Present † , Present ‡ , Present ♠ , Present ♡ , and Present § are 4. 85, 4.91, 4.91, 4.88, 4.96, and 4.95 hr, respectively. Though further reducing the accretion rate for each model improves the matching between modeled and observed recurrence time, all modeled burst light curves remain similar. For instance, the recurrence time of the Present § model Δt rec = 4.95 hr is produced with a defining accretion rate as 0.120 M Edd  , and the produced burst light curve is similar to other modeled light curves in the present work.

The top panel of Figure 5 illustrates the comparison between the best-fit modeled and observed XRB light curves. The evolution time of the light curve is relative to the burst-peak time, t = 0 s. The overall averaged flux deviations between the observed epoch and each of these theoretical models, baseline, Present † , Present ‡ , Present ♠ , Present ♡ , and Present § , in units of 10 -9 erg cm -2 s -1 , are 1. 154, 1.170, 1.172, 1.133, 1.181, and 1.147, respectively. The deviations between the Present § (and baseline) and observed light curve throughout the whole time span of the observed light curve are displayed in the bottom panel of Figure 5.

The observed burst peak is thought to be located in the time regime t = -2.5 to 2.5 s (top-left inset in Figure 5), and at the vicinity of the modeled light-curve peaks of baseline, Present † , Present ‡ , Present ♠ , Present ♡ , and Present § . The modeled light curves of baseline, Present † , Present ‡ , Present ♠ , Present ♡ , and Present § at the near-burst-peak region t = -4.5 to 5.5 s are almost indiscernible.

All modeled light curves are less enhanced than the observed light curve at t = 8-80 s, and the decrement is even augmented around t = 13 and 40 s, increasing the deviation between the modeled and observed light curves (bottom panel in Figure 5).

From the time regime at t = 78 s onward until the burst-tail end, all modeled burst light curves are enhanced. Overall, all modeled light-curve profiles are similar and note that the observed burst tail is reproduced from t = 78 s onward until the burst-tail end.

To investigate the microphysics behind the difference between both modeled burst light curves of the baseline and Present § models, we consider the 39th, the 42nd, and the 41st bursts for the baseline, Present ♡ , and Present § models, respectively. These bursts resemble the respective averaged light-curve profile presented in Figure 5. The reference time of the accreted envelope and nucleosynthesis in the following discussion is also relative to the burst-peak time, t = 0 s.

The moment before and during the onset. After the preceding burst, the synthesized proton-rich nuclei in the accreted envelope go through β + decays and enrich the region around stable nuclei with long half-lives, e.g., 60 Ni, 64 Zn, 68 Ge, and 78 Se, which are the remnants of waiting points. When the accreted envelope evolves to the moment just before the onset of the succeeding XRB, due to the continuing nuclear reactions that occur in unburned hydrogen above the base of the accreted envelope, the temperature of the envelope increases up to a maximum value of about 0.93 GK at the moment t = -10 s for the baseline, Present ♡ , and Present § scenarios; see Figure 6. At the moment just before the onset, some nuclei have already been synthesized and stored in the NiCu cycles, i.e., the NiCu I and II cycles (Van Wormer et al. 1994), and the sub-NiCu II cycle (Figure 1), see the top-left and bottom-right insets of the top-left, bottom-left, and top-right panels of Figure 6. Among the isotopes in the NiCu cycles, the highly synthesized nuclei having mass fractions of more than 2 × 10 -4 are 59 Zn, 58 Zn, 57 Zn, 58 Ni, 59 Cu, 58 Cu, 57 Cu, and 60 Cu isotopes, and the 56 Ni and 60 Zn waiting points, whereas the 56 Co and 60 Cu isotopes having analogous mass-fraction distributions in the envelope are converted to 57 Ni and 61 Zn, respectively (the lower-right insets in the top-left, bottom-left, and top-right panels of Figure 6).

We find that although the reaction flow induced by the 55 Ni(p,γ) 56 Cu(p,γ) 57 Zn branch noticeably bypasses the 56 Ni waiting point and enriches 57 Zn for the baseline, Present ♡ , and Present § scenarios, it eventually has to go through the 57 Zn(β + ν) 57 Cu(p,γ) 58 Zn branch and combines with the NiCu cycles and then breaks out from the NiCu cycles to the ZnGa cycles; see the upper-left insets in the top-left, bottom-left, and top-right panels of Figure 6. Due to the rather weak 57 Zn(p,γ) 58 Ga reaction, the 55 Ni(p,γ) 56 Cu(p,γ) 57 Zn reactions and the subsequent 57 Zn(β + ν) 57 Cu(p,γ) 58 Zn branch redirect an appreciable amount of material away from the 56 Ni waiting point, but the redirecting branch does not store material. Moreover, the newly corrected 55 Ni(p,γ) 56 Cu reaction rate is lower than the one recommended in JINA REACLIB v2.2 (Figure 2), causing less enrichment of 57 Zn in the Present ♡ and Present § scenarios (bottom-right panel of Figure 6). This explains why neither the newly corrected nor the recommended 55 Ni(p,γ) 56 Cu reaction rate exhibiting significant influence on the light curve of the GS 1826-24 burster and abundances of synthesized heavier nuclei. Also, the corrected 55 Ni(p,γ) 56 Cu reaction rate is not as influential as claimed by Valverde et al. (2018Valverde et al. ( , 2019)). Note that the one-zone models used by Valverde et al. (2018Valverde et al. ( , 2019) ) do not reproduce any burst light curves that are matched with observations. We remark that the baseline model that uses the recommended 55 57 Cu reaction rates. The continuous impact from the correlated influence among these reactions and 59 Cu(p,α) 56 Ni that cycles the reaction flow back to the reaction series in the NiCu cycles since the onset subsequently influences the burst ash composition at the burst-tail end. The mass fraction of 57 Cu in the baseline is lower than the one in the Present ♡ and Present § scenarios because the newly updated 56 Ni(p,γ) 57 Cu by Kahl et al. (2019) implemented in Present ♡ and Present § is about up to a factor of 9 higher than the recommended 56 Ni(p,γ) 57 Cu rate from JINA REACLIB v2.2 used in baseline at a temperature region of around 1 GK. Nevertheless, the mass fraction of 58 Zn in the baseline is about a factor of 1.2 higher than the one in the Present § scenario. This reflects a stronger flow of 57 Cu(p,γ) 58 Zn in the baseline than in the Present § scenario. Such stronger flow is because the recommended wien2 57 Cu(p,γ) 58 Zn reaction rate from JINA REACLIB v2.2 used in baseline is about up to a factor of 4 higher than the Present 57 Cu(p,γ) 58 Zn reaction rates at the temperature region around 1 GK. Meanwhile, the induced 57 Zn(β + ν) 57 surging through nuclei heavier than 68 Se. We find that the GeAs cycle that involves the two-proton sequential capture of 64 Ge consisting of 64 Ge(p,γ) 65 As(p,γ) 66 Se reactions could weakly exist in the middle of onset until the moment after burst peak (Lam et al. 2022); see the nucleosynthesis charts in Figures 6,7, and 8. A new 65 As(p,γ) 66 Se reaction rate based on a more precise 66 Se mass is desired to constrain the transient period, nonetheless, the fact that the transient existence of the weak GeAs cycle is not ruled out for the GS 1826-24 burster.

The ZnGa cycles were recently investigated by Y. H. Lam et al. (2022, in preparation) using the same GS 1826-24 clocked burster model as is used in this work and the full pfmodel space shell-model calculation. They found that the GeAs cycle that follows the ZnGa cycles only weakly exists for a brief period, which could last until t = 21.4-58.6 s after the burst peak (Lam et al. 2022). This causes some reactions relevant to the ZnGa cycles to become decisive in controlling the reaction flow reaching nuclei heavier than the Ge and Se isotopes where extensive H-burning via (p,γ) reactions occur. These influential reactions are 59 Cu(p,γ) and 61 Ga(p,γ), which were identified and marked by Cyburt et al. (2016) as the top four most sensitive reactions on clocked burst light curve. Y. H. Lam et al. (2022, in preparation) found that the 59 Cu(p,γ) and 61 Ga(p,γ) reactions characterize the burst light curve of the GS 1826-24 clocked burster at t ≈ 8-30 s after the burst peak and the burst-tail end. Preliminary results of the investigation of the ZnGa cycles were presented in the supplemental material of Hu et al. (2021) prior to Y. H. Lam et al. (2022, in preparation).

We notice that the balance between the 56 Ni(p,γ) 57 Cu and 57 Cu(p,γ) 58 Zn reactions also redistributes the reaction flow to the NiCu II cycle and then the reaction flow eventually joins with the NiCu I cycle and branches out to the ZnGa cycles at the 60 Zn waiting point or follows the 60 Cu(p,γ) 61 Zn(p,γ) 62 Ga reactions branches out to the ZnGa II cycle. Then, the joint reaction flow surges through the proton-rich region heavier than 64 Ge where (p,γ) reactions actively burn hydrogen and intensify the rise of burst light curve from t = -10 s up to t = 0 s (burst peak).

The moment at the immediate vicinity of the burst peak. As the redistributing and reassembling of reaction flow from the moment of onset until the burst peak regulate a rather similar feature of abundances in the NiCu cycles (the lower-right insets