Reaction Network Origins

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

Other Stuff:
   Bicycle Adventures

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 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.