*
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}