Electron Capture and Beta-Decay Rates for the Collapse of O + Ne + Mg Cores

Yi Hua Lam1,a, Gabriel Martínez-Pinedo1,2, Karlheinz Langanke2,1, Samuel Jones3, Raphael Hirschi3, Remco G. T. Zegers4,5, and B. Alex Brown4,5 1Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany 2GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany 3Astrophysics Group, Lennard Jones Building, Keele University ST5 5BG, United Kingdom 4NSCL, Michigan State University, East Lansing, Michigan 48824-1321, USA Dept. of Phys. and Astro., Michigan State University, East Lansing, Michigan 48824, USA


Introduction
Stars in the mass range of 8 M/M ⊙ 12 evolve to degenerate O+Ne or O+Ne+Mg cores after ~10 6 years [1].In such a very abundant 20 Ne and 24 Mg matter composition, the e − captures on these two nuclear species play the pivotal role in reducing the e − pressure support, which is supplied by the degenerate relativistic e − gas to against the gravitational contraction of the stellar core.Under the situation of loosing e − to capture process, these stars are driven toward the core collapse stage.Besides, the core environment is cooled by the produced neutrinos (ν) which carry away energy and leave the star unhindered as long as the density is less than ~10 11 g/cm 3 .Moreover, the e − capture process transmute the stellar composition to be more neutron rich by changing protons in the nucleus to neutrons.Therefore, stars in this mass range are crucial in nucleosynthesis of certain nuclides [2].
The e − -capture supernovae and late-stage stellar evolution simulation of 8 − 12 M ⊙ stars utilize the e − -capture rates of 20 Ne and 24 Mg, and β-decay rates of their daughters, i.e. 20 F and 24 Na, respectively, from Ref. [3].However, these rates were tabulated in a sparse density-temperature (ρ-T ) grid.This may hide physics of critical ρ-T region from being manifested in simulations [4].Nevertheless, rates of Oda et al. [3] and Takahara et al. [5] had been calculated prior to the charge-exchange experiments of Gamow-Teller (GT) strength distributions for 20 Ne and 24 Mg [6].In present work, we improve the e − -capture and β-decay rates by including recently measured GT transitions and also forbidden transition, which are the main contributions to the rates, and by considering Coulomb screening effects in dense stellar environment.

Formalism
The e − of 8 − 12 M ⊙ stars in the late-stage stellar environment form a degenerate relativistic Fermi gas under temperatures, T = 10 8 − 10 10 K and densities, ρ = 10 8 − 10 10 g/cm 3 .The total rate of e − capture or β − decay is stated as [7], where the superscript α is for either e − capture (ec) or β − -decay (β − ), and the sums in i and j run over states in the parent and daughter nuclei, respectively.The constant K = 6143.6± 1.7 s, is quoted from superallowed Fermi transitions [8], the partition function of the parent nucleus, G(Z, A, T ) and the reduced transition probability of the nuclear transition is [9], Each of the phase space integral for ec and β − in Eq. ( 2) is calculated by respectively, where the w is the total, rest mass and kinetic, energy of the e − , Q i j is the total energy available in β decay, p = √ w 2 − 1 is the momentum, and S e is the Fermi-Dirac distribution of e − .Both rates of ground state (g.s.) transition and excited-state transition can be written as [7] where

Results
We obtained theoretical GT strengths within the shell-model approach in the full sd shell using USDB interaction [10].However, for the relevant ρ-T range, both e − -capture and β-decay rates of 20 Ne⇋ 20 F and 24 Mg⇋ 24 Na are fully determined by the experimental GT strengths [6].Rates shown in Figs.1-4 were calculated without Coulomb screening effect.This medium effect was included in the final rates.
Rates of Urca pairs 20 Ne⇋ 20 F. Comparing to previous calculations of Refs.[3,5], c.f. Fig. 1, we find that the e − capture on 20 Ne is enhanced by several orders of magnitude, in the density range 4 ρ/(10 9 g/cm 3 ) 10 and temperatures below 0.7 GK, due to the contribution of the second forbidden transition from 20 Ne g.s. to 20 F g.s.The forbidden transition also enhances β-decay rates in this density regime, c.f. Fig. 2. The e − capture rate of ( 20 Ne)2 + 1 → ( 20 F)2 + is higher than the rate of 20 Ne g.s. to 20 F g.s.(double-dot-dashed line) at density ρ 9.3 × 10 9 g/cm 3 , c.f. Fig. 1.The reason is, the µ e is much lower than the e − -capture Q-value in the early evolution stage of an O-Ne-Mg core, and the Q i f is decreased with increasing E i .Hence, the decrement of rate caused by the Boltzmann probability, exp [−E i /(kT )], for the thermal population of the excited state with increasing E i in Eq.( 4) is compensated by the exponential increment, i.e. higher value of Fermi integrals in Eq.( 4), of the number of e − contributing to the capture process.Rates of Urca pairs 24 Mg⇋ 24 Na.Overall, due to the recent β-decay data from Nishimura et al. [6], the e − capture on 24 Mg is enhanced by about a factor of two compared to Takahara et al. [5], c.f. Fig. 3.The e − capture on 24 Mg dominates over β decay of 24 Na at ρ 9.3 × 10 9 g/cm 3 , c.f. Fig. 6.Although forbidden transitions do not contribute to the e − capture rate, the β decay rate of 24 Na is enhanced by these transitions at temperature T 0.4 GK.

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Coulomb screening effect.We included this effect according to Ref. [11].Coulomb screening reduces e − capture rate in dense astrophysical environment in two ways.First, the threshold energy in the medium is increased, Q ec, med i f . Second, the chemical potential of e − is decreased, µ med e = µ e − V s .Contrarily, β-decay rate is increased in two ways.First, the threshold energy in the medium is decreased, Q β, med i f = Q i f − ∆Q c (Z + 1).Second, the decrement of µ e causes the reduction of Pauli blocking in the final state, i.e. lower values of [1 − S e (w)] in Eq.(3b).The comparison of rates with screening effect and without screening effect is shown in Figs. 5 and 6.Stellar evolution simulation.Jones et al. showed that stellar evolution simulations using recently updated rates produce lower temperatures at the start of e − -capture phase for super-AGB star progenitors [4].Here, we performed a stellar evolution simulation of 8.8 M ⊙ star, which is failed massive star (see Ref. [4] for definition).The central ρ-T of the late-stage region is shown in Fig. 7.The first dip at the central density, ρ c ≈ 9.1 × 10 9 g/cm 3 is due to 25 Mg⇋ 25 Al, whereas the second dip at ~9.2 × 10 9 g/cm 3 is due to 23 Na⇋ 23 Ne.Both dips correspond to the URCA cooling, when β decay and e − capture proceed on similar timescale and thus provide neutrino cooling.The near-adiabatic contraction is noticeable at 9.2 ρ c /(10 9 g/cm 3 ) 9.5.During the contraction, there are two late Ne-O shell flashes which provide an additional support for the outer layers of the core, hence the center expands (lower down the density) and cools adiabatically each time before contracting again.The heat releasing process of 24 Mg→ 24 Na→ 24 Ne only happens on the input rates of Takahara et al. [5] (in a coarse ρ-T grid) at the region of 9.55 ρ c /(10 9 g/cm 3 ) 9.8.However, it does not happen on the other two sets of fine grid, because consistent resolution of e − capture and β-decay rates are used in the simulation.Interestingly, these two input rates cause a few spikes at ρ c ≈ 9.85 × 10 9 g/cm 3 .It may be the center heats up quickly when the e − capture on 20 Ne starts, but at some point it becomes convectively unstable because there is so much energy being released.This unstable behavior is more profound for the present input rates.We present an improved calculation of e − capture and β-decay rates of 20 Ne⇋ 20 F and 24 Mg⇋ 24 Na.The impact of new rates on latestage stellar evolution of 8.8 M ⊙ model at core density ρ c 9.85×10 9 g/cm 3 is demonstrated.Rates of above Urca pairs and other Urca pairs are available upon request from the author.

Conclusion
a e-mail: LamYiHua@gmail.comDOI: 10.1051/ C Owned by the authors, published by EDP Sciences,