Cococubed.com Abundance variables
Home Astronomy research Software instruments Presentations Illustrations cococubed YouTube Bicycle adventures Public Outreach Education materials      Solar Systems Astronomy      Energy in Everyday Life      Geometry of Art and Nature      Calculus 2023 ASU Solar Systems Astronomy 2023 ASU Energy in Everyday Life AAS Journals AAS YouTube 2023 MESA VI 2023 MESA Marketplace 2023 MESA Classroom 2022 Earendel, A Highly Magnified Star 2022 White Dwarfs & 12C(α,γ)16O 2022 Black Hole Mass Spectrum 2022 MESA in Don't Look Up 2021 Bill Paxton, Tinsley Prize 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}