Cococubed.com Nuclear Energy Generation Expressions
Home Astronomy Research    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    2024 AAS YouTube    2024 AAS Peer Review Workshops 2024 ASU Energy in Everyday Life 2024 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} \label{eq:y} \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 \label{eq:mconserv} \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. The rest-mass energy is \begin{equation} E = - M c^2 \hskip 0.2 in {\rm erg} \, \end{equation} where $M$ is the total baryonic mass [g]. The minus sign indicates that creating mass reduces the energy reservoir of a closed system. For $M$ being composed of $i$ isotopes \begin{equation} E = - \sum_{i=1}^{k} N_i \ m_i \ c^2 \hskip 0.2 in {\rm erg} \ . \end{equation} Multiplying by the constant $N_A / N_B$ and using equation ($\ref{eq:y}$) gives gives the specific nuclear energy of the baryons \begin{equation} \epsilon = - N_A c^2 \sum_{i=1}^{k} Y_i \ m_i \hskip 0.2 in {\rm erg \ g}^{-1} \ . \end{equation} Taking the time derivative yields the specific nuclear energy generation rate \begin{equation} \dot{\epsilon} = - N_A c^2 \sum_{i=1}^{k} \dot{Y_i} \ m_i \hskip 0.2 in {\rm erg \ g}^{-1} \ {\rm s}^{-1} \label{eq:epsnuc} \end{equation} Note one only needs to evaluate, not integrate, the right-hand sides of the $\dot{Y_i}$ ODEs defining the nuclear reaction network to obtain the instantaneous $\dot{\epsilon}$. Nice. The atomic mass $m_i$ in equation ($\ref{eq:epsnuc}$) can be defined as \begin{equation} m_i = {\rm A}_i m_u + \Delta_i \hskip 0.2in {\rm g} \label{eq:mass} \end{equation} where $\Delta_i$ is the mass excess of each species in atomic mass units [g]. This expression neglects the electronic binding energy, and $\Delta_i$ is thus independent of the ionization state of a given species. However, the electron rest masses are included in this definition since the $m_i$ are atomic masses - this is important for accurately tracking weak reactions. The mass excess is related to the binding energy $B_i$, mass excess of the proton $\Delta_p$ and mass excess of the neutron $\Delta_n$ by \begin{equation} \Delta_i = {\rm Z}_i \Delta_p + N_i \Delta_n - \frac{B_i}{c^2} \hskip 0.2in {\rm g} \label{eq:massex} \end{equation} Substituting equations ($\ref{eq:mass}$) and ($\ref{eq:massex}$) into equation ($\ref{eq:epsnuc}$) gives \begin{equation} \dot{\epsilon} = - N_A c^2 \sum_{i=1}^{k} \dot{Y_i} \ \left ( A_i m_u + {\rm Z}_i \Delta_p + N_i \Delta_n - \frac{B_i}{c^2} \right ) \hskip 0.2 in {\rm erg \ g}^{-1} \ {\rm s}^{-1} \end{equation} From equation ($\ref{eq:mconserv}$), $\sum \dot{X_i} = \sum \dot{Y_i} {\rm A}_i = 0$, hence the above reduces to \begin{equation} \dot{\epsilon} = N_A \sum_{i=1}^{k} \dot{Y_i} B_i - \sum_{i=1}^{k} \dot{Y_i} ({\rm Z}_i \Delta_p + N_i \Delta_n) \hskip 0.2 in {\rm erg \ g}^{-1} \ {\rm s}^{-1} \end{equation} This is a handy form as changes due to weak reactions are isolated by the second term on the right-hand side. If weak reactions are not important, \begin{equation} \dot{\epsilon} = N_A \sum_{i=1}^{k} \dot{Y_i} B_i \hskip 0.2 in {\rm erg \ g}^{-1} \ {\rm s}^{-1} \end{equation} Equation ($\ref{eq:epsnuc}$) is the {\it instantaneous} energy generation rate. In this form it can be added as an ODE to the nuclear reaction network equations, or added to the system of PDEs in a fully coupled hydrocode. In an operator-split hydrocode, over a timestep $\Delta t$ one usually uses the finite difference approximation \begin{equation} \dot{Y_i} = \frac{Y_{i,{\rm end}} - Y_{i,{\rm start}}}{\Delta t} \hskip 0.2in {\rm s}^{-1} \end{equation} and equation ($\ref{eq:epsnuc}$) becomes \begin{equation} \left < \dot{\epsilon} \right > = - N_A c^2 \sum_{i=1}^{k} \left [ \frac{Y_{i,{\rm end}} - Y_{i,{\rm start}}}{\Delta t} \right ] \ m_i \hskip 0.2 in {\rm erg \ g}^{-1} \ {\rm s}^{-1} \ , \end{equation} and represents the average energy generation rate over a finite timestep. A note on the mass of isotope $i$. The intuitive mass counting of nucleons minus the binding energy is \begin{equation} m_i = {\rm Z}_i m_p + N_i m_n - \frac{B_i}{c^2} \hskip 0.3in {\rm g} \end{equation} Eliminating the binding energy $B_i$ using equation ($\ref{eq:massex}$) \begin{align} m_i & = {\rm Z}_i m_p + N_i m_n - \Delta_i - {\rm Z}_i \Delta_p - N_i \Delta_m \notag \\ & = {\rm Z}_i (m_p - \Delta_p) + N_i (m_n - \Delta_n) - \Delta_i \end{align} By definition, $m_p = m_u + \Delta_p \simeq (1 + 0.007276) m_u$ and $m_n = m_u + \Delta_n \simeq (1 + .008664) m_u$. Substituting, \begin{align} m_i & = {\rm Z}_i m_u + N_i m_u - \Delta_i \notag \\ & = ({\rm Z}_i + N_i) m_u - \Delta_i \notag \\ & = {\rm A}_i m_u - \Delta_i \end{align} which is equation ($\ref{eq:mass}$) for the atomic mass.