Cococubed.com Nuclear Statistical Equilibrium
Home Astronomy Research    2025 Neutrinos From De-excitation    Radiative Opacity    2024 Neutrino Emission from Stars    2023 White Dwarfs & 12C(α,γ)16O    2023 MESA VI    2022 Earendel, A Highly Magnified Star    2022 Black Hole Mass Spectrum    2021 Skye Equation of State    2021 White Dwarf Pulsations & 22Ne    Software Instruments      Stellar equation of states      EOS with ionization      EOS for supernovae      Chemical potentials      Stellar atmospheres      Voigt Function      Jeans escape      Polytropic stars      Cold white dwarfs      Adiabatic white dwarfs      Cold neutron stars      Stellar opacities      Neutrino energy loss rates      Ephemeris routines      Fermi-Dirac functions      Polyhedra volume      Plane - cube intersection      Coating an ellipsoid      Nuclear reaction networks      Nuclear statistical equilibrium      Laminar deflagrations      CJ detonations      ZND detonations      Fitting to conic sections      Unusual linear algebra      Derivatives on uneven grids      Pentadiagonal solver      Quadratics, Cubics, Quartics      Supernova light curves      Exact Riemann solutions      1D PPM hydrodynamics      Hydrodynamic test cases      Galactic chemical evolution      Universal two-body problem      Circular and elliptical 3 body      The pendulum      Phyllotaxis      MESA      MESA-Web      FLASH      Zingale's software      Brown's dStar      GR1D code      Iliadis' STARLIB database      Herwig's NuGRID      Meyer's NetNuc AAS Journals    2025 AAS YouTube    2025 AAS Peer Review Workshops 2025 ASU Energy in Everyday Life 2025 MESA Classroom Other Stuff:    Bicycle Adventures    Illustrations    Presentations Contact: F.X.Timmes my one page vitae, full vitae, research statement, and teaching statement.	Below $\simeq 10^6$ K it is not energetic enough for nuclear reactions. Up to $\simeq 5 \times10^9$ K one uses a nuclear reaction network to follow abundance evolutions. Above $\simeq 5 \times10^9$ K it is energetic enough for forward and reverse reactions to be balanced, and abundances are in a state of nuclear statistical equilibrium (NSE). For Maxwell-Boltzmann statistics, the mass fractions $X_i$ of any isotope $i$ in NSE is \begin{equation} X_i(A_i,Z_i,T,\rho) = {A \over N_A \rho} \omega(T) \left ( 2\pi kT M(A_i,Z_i) \over h^2 \right )^{3/2} \exp \left [ { \mu(A_i,Z_i) + B(A_i,Z_i) \over kT } \right ] \ , \label{eq1} \tag{1} \end{equation} where $A_i$ is the atomic number (number of neutrons + protons on the nulceus), $Z_i$ is the charge (number of protons), $T$ is the temperature, $\rho$ is the mass density, $N_A$ is the Avogardo number, $\omega(T)$ is the temperature dependent partition function, $M(A_i,Z_i)$ is the mass of the nucleus, $B(A_i,Z_i)$ is the binding energy of the nucleus, and $\mu(A_i,Z_i)$, in the simplest case, is the chemical potential of the isotope \begin{equation} \mu(A_i,Z_i) = Z_i\mu_p + N_i\mu_n = Z_i\mu_p + (A_i-Z_i) \mu_n \ , \label{eq2} \tag{2} \end{equation} where $\mu_p$ is the chemical potential of the protons, $\mu_n$ is the chemical potential of the neutrons. The mass fractions of equation $\ref{eq1}$ are subject to two constraints, conservation of mass (baryon number) and charge, which are expressed as \begin{equation} \sum_i X_i= 1 \hskip 1.0in Y_e = \sum_i {Z_j \over A_i} X_i \ . \label{eq3} \tag{3} \end{equation} Given the triplet of input values $(T, \rho, Y_e)$, an NSE solution boils down to a two-dimensional root find for the chemical potentials of the protons $\mu_p$ and neutrons $\mu_n$. Two constraints and two unknowns. The tool in public_nse.tbz puts a 47 isotope netrork into its NSE state. More serious NSE calculations could modify this tool to use more accurate nuclear data (e.g., ground state spins and temperature dependent partition functions), to add more elaborate couplings (e.g., Coulomb corrections), and to increase the number of isotopes. Still, the figures and movies below, which accompany this article, suggest this tool gives reasonable results for the assumptions made. Abundances vs temperature for varying Ye: ρ = 103 g cm-3    d1p0e3_yevary_3302_a_pdf.mp4    ρ = 104 g cm-3    d1p0e4_yevary_3302_a_pdf.mp4    ρ = 105 g cm-3    d1p0e5_yevary_3302_a_pdf.mp4    ρ = 106 g cm-3    d1p0e6_yevary_3302_a_pdf.mp4    ρ = 107 g cm-3    d1p0e7_yevary_3302_a_pdf.mp4    ρ = 108 g cm-3    d1p0e8_yevary_3302_a_pdf.mp4    ρ = 109 g cm-3    d1p0e9_yevary_3302_a_pdf.mp4    Abundances vs Ye for varying temperature : ρ = 103 g cm-3    d1p0e3_tvary_3302_a_pdf.mp4    ρ = 104 g cm-3    d1p0e4_tvary_3302_a_pdf.mp4    ρ = 105 g cm-3    d1p0e5_tvary_3302_a_pdf.mp4    ρ = 106 g cm-3    d1p0e6_tvary_3302_a_pdf.mp4    ρ = 107 g cm-3    d1p0e7_tvary_3302_a_pdf.mp4    ρ = 108 g cm-3    d1p0e8_tvary_3302_a_pdf.mp4    ρ = 109 g cm-3    d1p0e9_tvary_3302_a_pdf.mp4
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