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Contact: F.X.Timmes
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$ \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 continuity equation for the number density of species $i$ in an Eulerian framework is \begin{equation} \partf{n_i}{t} + \partf{(n_i v_x)}{x} = \sum_{j,k} r_{jk} n_j n_k \hskip 0.2in \rm{cm}^{-3} \ \rm{s}^{-1} \end{equation} where the reaction rate between two species $j$ and $k$ is \begin{equation} r_{jk} = \ <\sigma v>_{jk} \hskip 0.2in \rm{cm}^3 \ \rm{s}^{-1} \end{equation} and $<\sigma v>_{jk}$ is the cross-section $\sigma$ [in cm$^2$] times the relative speed v [in cm s$^{-1}$] between the two isotopes, and the angled brackets indicates an average over a statistical distribution, usually a Maxwell-Boltzmann. $r_{jk}$ is a function of temperature only. The reaction rate implies a lifetime for isotope $j$ of $\tau_j = 1/(n_j r_{jk})$ s. Nuclear reactions, and expansion or contraction of the plasma can produce changes in the number densities $n_i$. To separate the nuclear changes in composition from hydrodynamic effects, substituting equation ($\ref{eq:y}$) gives \begin{align} \partf{(Y_i n_B)}{t} + \partf{(Y_i n_B v_x)}{x} & = \sum_{j,k} r_{j,k} Y_j Y_k n_B^2 \notag \\[8pt] n_B \partf{Y_i}{t} + Y_i \partf{n_B}{t} + n_B \partf{(Y_i v_x)}{x} + Y_i \partf{(n_B v_x)}{x} & = \sum_{j,k} r_{j,k} Y_j Y_k n_B^2 \notag \\[8pt] n_B \left (\partf{Y_i}{t} + \partf{(Y_i v_x)}{x} \right ) + Y_i \left [ \partf{\rho}{t} + \partf{(\rho v_x)}{x} \right ] & = \sum_{j,k} r_{j,k} Y_j Y_k n_B^2 \end{align} The term in square brackets is zero by the mass continuity equation. Thus, \begin{equation} \partf{Y_i}{t} + \partf{(Y_i v_x)}{x} = \sum_{j,k} r_{j,k} n_B Y_j Y_k \hskip 0.2in \rm{s}^{-1} \end{equation} or in a Lagrangian frame \begin{equation} \drvf{Y_i}{t} = \sum_{j,k} r_{j,k} n_B Y_j Y_k = \sum_{j,k} r_{j,k} N_A \rho Y_j Y_k \hskip 0.2in \rm{s}^{-1} \end{equation} In an operator split Eularian hydrocode, the advection term is done seperately, leading to the same ordinary differential equations to solve as in the Lagrangian form.
Common reaction rate compilations list $\lambda = N_A <\sigma v>_{jk}$ [in cm$^3$ g$^{-1}$ s$^{-1}$], so \begin{equation} \drvf{Y_i}{t} = \sum_{j,k} \lambda_{j,k} \rho Y_j Y_k \hskip 0.2in \rm{s}^{-1} \end{equation} Let $R_{j,k} = \lambda_{j,k} \rho$ be the ``reaction rate'' that subsumes all the temperature and density dependences. Then, \begin{equation} \drvf{Y_i}{t} = \sum_{j,k} R_{j,k} Y_j Y_k \hskip 0.2in \rm{s}^{-1} \end{equation} are the equations that constiture a nuclear reaction network.