Theory of Quasi‐Equilibrium Nucleosynthesis and Applications to Matter Expanding from High Temperature and Density

Abstract

Our first purpose is construction of a formal theory of quasi-equilibrium. We define quasi-equilibrium, in its simplest form, as statistical equilibrium in the face of an extra constraint on the nuclear populations. We show that the extra constraint introduces a uniform translation of the chemical potentials for the heavy nuclei and derive the abundances in terms of it. We then generalize this theory to accommodate any number of constraints. For nucleosynthesis, the most important constraint occurs when the total number of heavy nuclei Yh within a system of nuclei differs from the number that would exist in nuclear statistical equilibrium (NSE) under the same conditions of density and temperature. Three situations of high relevance are (1) silicon burning, wherein the total number of nuclei exceeds but asymptotically approaches the NSE number; (2) alpha-rich freezeout expansions of high entropy, wherein Yh is less than the NSE number; and (3) expansions from high temperature of low-entropy matter, in which Yh exceeds the NSE number. These are of importance, respectively, within (1) supernova shells, (2) Type II supernova cores modestly outside the mass cut, and (3) Type Ia supernova cores in near-Chandrasekhar-mass events. Our next goal is the detailed analysis of situation (2), the high-entropy alpha-rich neutron-rich freezeout. We employ a nuclear reaction network, which we integrate, to compare the actual abundances with those obtained at the same thermal conditions by the quasi-equilibrium (QSE) theory and by the NSE theory. For this detailed comparison, we choose a high-entropy photon-to-nucleon ratio = 6.8, for which we conduct expansions at initial bulk neutron excess η0 = 0.10. We demonstrate that the abundance populations, as they begin expansion and cooling from temperature 10 × 109 K, are characterized by three distinct phases: (1) NSE, (2) QSE having Yh smaller than the NSE value, and (3) final reaction rate-dependent freezeout modifications of the QSE. We demonstrate that the true final abundances are well approximated by the QSE distribution near the freezeout temperature T9f = 4.0. During the expansion, the QSE distribution changes shape continuously in ways that are independent of the reaction cross sections of the heavy nuclei with free light particles. It is this changing shape, rather than nuclear flows, that establish the abundance pattern. The abundance pattern is actually determined by the parameter Yh and the degree to which it differs from the NSE value owing to the slowness with which light particles can be assembled into heavy nuclei (A ≥ 12). We also detail the nature and magnitude of the freezeout corrections to the QSE distribution. The entire distribution depends less upon the values of heavy-element cross sections than has been heretofore thought. Our third goal is to survey the alpha-rich freezeout. We do this by less complete analysis of nine different expansions determined by the matrix of three distinct entropies ( = 1.7, 6.8, and 17) and three distinct initial neutron excesses (η0 = 0.003, 0.10, and 0.1667). The trends are easily comprehended in terms of the concept of quasi-equilibrium, whereas they are not understandable in terms of either NSE or in terms of reaction rates. This secures for the QSE concept a major diagnostic capability within nucleosynthesis theory. We delineate the key trends and also remark on the ways that order arises from disorder in this complex system. We conclude with a discussion of how such systems assemble heavy nuclei.