Cococubed.com Abundance variables
Home Astronomy Research    2025 Neutrinos From De-excitation    2024 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 AAS Journals    2025 AAS YouTube    2025 AAS Peer Review Workshops 2025 ASU Energy in Everyday Life 2025 MESA Classroom Outreach and Education Materials     Solar Systems Astronomy     Energy in Everyday Life     Geometry of Art and Nature     Calculus Other Stuff:    Bicycle Adventures    Illustrations    Presentations Contact: F.X.Timmes my one page vitae, full vitae, research statement, and teaching statement.	$ \def\drvop#1{{\frac{d}{d{#1}}}} \def\drvf#1#2{{\frac{d{#1}}{d{#2}}}} \def\ddrvf#1#2{{\frac{d^2{#1}}{d{#2}^2}}} \def\partop#1{{\frac{\partial}{\partial {#1}}}} \def\ppartop#1{{\frac{\partial^2}{\partial {#1}^2}}} \def\partf#1#2{{\frac{\partial{#1}}{\partial{#2}}}} \def\ppartf#1#2{{\frac{\partial^2{#1}}{\partial{#2}^2}}} \def\mpartf#1#2#3{{\frac{\partial^2{#1}}{\partial{#2} \ {\partial{#3}}}}} $ A pdf of this note is avaliable. Baryon number is an invariant. Define the abundance of species $Y_i$ by \begin{equation} Y_i = \frac{n_i}{n_B} = \frac{N_i}{N_B} \end{equation} where $N_i$ is the number of particles of isotope $i$, $N_B$ is the number of baryons, $n_i$ is the number density [cm$^{-3}$] of isotope $i$ and $n_B$ is baryon number density [cm$^{-3}$]. The number of baryons in isotope $i$ divided by the total number of baryons is the baryon fraction $X_i$, \begin{equation} X_i = Y_i \ A_i = \frac{n_i \ A_i}{n_B} \end{equation} where $A_i$ is the atomic mass number, the number of baryons in an isotope. Usually the baryon fraction is called the ``mass fraction''. Note \begin{equation} \sum X_i = \frac{n_B}{n_B} = 1 \end{equation} is invariant under nuclear reactions. Define the baryon density, in atomic mass units, as \begin{equation} \rho_B = n_B \ m_u = \frac{n_B}{N_A} \hskip 0.2in {\rm g \ cm}^{-3} \end{equation} where $m_u$ is the atomic mass unit [g] and $N_A$ is the Avogadro number [g$^{-1}]$ in a system of units where the atomic mass unit is {\it defined} as 1/12 mass of an unbound atom of $^{12}$C is at rest and in its ground state. Mean atomic number \begin{equation} \overline{\rm A} = \frac{\sum n_i {\rm A}_i}{\sum n_i} = \frac{n_B}{\sum n_i} = \frac{\sum Y_i {\rm A}_i}{\sum Y_i} = \frac{1}{\sum Y_i} \end{equation} Mean charge \begin{equation} \overline{\rm Z} = \frac{\sum n_i {\rm Z}_i}{\sum n_i} = \frac{\sum Y_i {\rm Z}_i}{\sum Y_i} = \overline{\rm A} \sum Y_i {\rm Z}_i \end{equation} Electron to baryon ratio, where the second equality assumes full ionization \begin{equation} Y_e = \frac{n_e}{n_B} = \frac{\sum n_i Z_i}{n_B} = \sum Y_i Z_i = \frac{\overline{\rm Z}}{\overline{\rm A}} \end{equation} Neutron excess \begin{equation} \eta = \sum ({\rm N}_i - {\rm Z}_i) Y_i = \sum ({\rm A}_i - 2 {\rm Z}_i) Y_i = \sum {\rm A}_i Y_i - 2 Y_e = 1 - 2 Y_e \end{equation} Mean ion molecular weight \begin{equation} \mu_{{\rm ion}} = \overline{\rm A} \end{equation} Mean electron molecular weight \begin{equation} \mu_{{\rm ele}} = \frac{1}{Y_e} = \frac{\overline{\rm A}}{\overline{\rm Z}} \end{equation} Mean molecular weight \begin{equation} \mu = \left [ \frac{1}{\mu_{ion}} + \frac{1}{\mu_{ele}} \right ]^{-1} = \left [ \frac{1}{\overline{\rm A}} + Y_e \right ]^{-1} = \left [ \frac{1}{\overline{\rm A}} + \frac{\overline{\rm Z}}{\overline{\rm A}} \right ]^{-1} = \frac{\overline{\rm A}}{\overline{\rm Z} + 1} = \frac{ n_B}{\sum n_i + n_e} \end{equation}