The Regulated NiCu Cycles with the new $^{57}$Cu(p,$\gamma$)$^{58}$Zn reaction rate and the Influence on Type-I X-Ray Bursts: GS 1826$-$24 Clocked Burster

During the X-ray bursts of GS 1826$-$24, "clocked burster", the nuclear reaction flow that surges through the rapid-proton capture process path has to pass through the NiCu cycles before reaching the ZnGa cycles that moderate the further extent of hydrogen burning in the region above germanium and selenium isotopes. The $^{57}$Cu(p,$\gamma$)$^{58}$Zn reaction located in the NiCu cycles plays an important role in influencing the burst light curves as found by Cyburt et al. (2016). We deduce the $^{57}$Cu(p,$\gamma$)$^{58}$Zn reaction rate based on the experimentally determined important nuclear structure information, isobaric-multiplet-mass equation, and large-scale shell model calculations. With the isobaric-multiplet-mass equation, we propose a possible order of $1^+_1$ and $2^+_3$ dominant resonance states that the $2^+_3$ resonance state is higher than the $1^+_1$ state, and estimate the resonance energy of $1^+_2$ contributing resonance state. The new rate is up to a factor of five lower than the Forstner et al. (2001) rate recommended by JINA REACLIB v2.2. Using the one-dimensional implicit hydrodynamic code, KEPLER, to model the thermonuclear X-ray bursts of GS 1826$-$24 clocked burster, we find that the new $^{57}$Cu(p,$\gamma$)$^{58}$Zn reaction redistributes the reaction flow in the NiCu cycles and reduces the production of $^{58}$Zn, whereas the $^{59}$Cu(p,$\alpha$)$^{56}$Ni and $^{59}$Cu(p,$\gamma$)$^{60}$Zn reactions suppress the influence of the $^{57}$Cu(p,$\gamma$)$^{58}$Zn reaction and strongly diminish the impact of nuclear reaction flow that by-passes the important $^{56}$Ni waiting point induced by the $^{55}$Ni(p,$\gamma$)$^{56}$Cu reaction on burst light curve. The influence of the newly deduced $^{56}$Ni(p,$\gamma$)$^{57}$Cu is also discussed.