Available Reaction Networks

iso7, aprox13, aprox19, and aprox21

These are alpha-chains (with some other nuclei) from Frank Timmes. These networks share common rates (from Microphysics/rates), plasma neutrino loses (from Microphysics/neutrinos), and electron screening (from Microphysics/screening).

Energy generation.

These networks store the total binding energy of the nucleus in MeV as bion(:). They then compute the mass of each nucleus in grams as:

\[M_k = (A_k - Z_k) m_n + Z_k (m_p + m_e) - B_k\]

where \(m_n\), \(m_p\), and \(m_e\) are the neutron, proton, and electron masses, \(A_k\) and \(Z_k\) are the atomic weight and number, and \(B_k\) is the binding energy of the nucleus (converted to grams). \(M_k\) is stored as mion(:) in the network.

The energy release per gram is converted from the rates as:

\[\edot = -N_A c^2 \sum_k \frac{dY_k}{dt} M_k - \edotnu\]

where \(N_A\) is Avogadro’s number (to convert this to “per gram”) and \(\edotnu\) is the neutrino loss term.

general_null

general_null is a bare interface for a nuclear reaction network – no reactions are enabled. The data in the network is defined at compile type by specifying an inputs file. For example, networks/general_null/triple_alpha_plus_o.net would describe the triple-\(\alpha\) reaction converting helium into carbon, as well as oxygen and iron. This has the form:

# name       short name    aion     zion
helium-4       He4          4.0      2.0
carbon-12      C12         12.0      6.0
oxygen-16      O16         16.0      8.0
iron-56        Fe56        56.0     26.0

The four columns give the long name of the species, the short form that will be used for plotfile variables, and the mass number, \(A\), and proton number, \(Z\).

The name of the inputs file by one of two make variables:

• NETWORK_INPUTS : this is simply the name of the “.net” file, without any path. The build system will look for it in the current directory and then in $(MICROPHYSICS_HOME)/networks/general_null/.

  For the example above, we would set:

  NETWORK_INPUTS := triple_alpha_plus_o.net
  

• GENERAL_NET_INPUTS : this is the full path to the file. For example we could set:

  GENERAL_NET_INPUTS := /path/to/file/triple_alpha_plus_o.net
  

At compile time, the “.net” file is parsed and a network header network_properties.H is written using the python script write_network.py. The make rule for this is contained in Make.package.

CNO_extras

This network replicates the popular [MESA “cno_extras” network](https://docs.mesastar.org/en/latest/net/nets.html) which is meant to study hot-CNO burning and the start of the breakout from CNO burning.

We add \({}^{56}\mathrm{Fe}\) as an inert nucleus to allow this to be used for X-ray burst simulations.

CNO_He_burn

This network is meant to study explosive H and He burning. It combines the CNO_extras network (with the exception of the inert \({}^{56}\mathrm{Fe}\) with the subch_simple network. This allows it to capture hot-CNO and He burning.

ECSN

ECSN is meant to model electron-capture supernovae in O-Ne white dwarfs. It includes various weak rates that are important to this process.

ignition_chamulak

This network was introduced in our paper on convection in white dwarfs as a model of Type Ia supernovae [ZingaleNonakaAlmgren+11]. It models carbon burning in a regime appropriate for a simmering white dwarf, and captures the effects of a much larger network by setting the ash state and energetics to the values suggested in [ChamulakBrownTimmesDupczak08].

Energy generation.

The binding energy, \(q\), in this network is interpolated based on the density. It is stored as the binding energy (ergs/g) per nucleon, with a sign convention that binding energies are negative. The energy generation rate is then:

\[\edot = q \frac{dX(\isotm{C}{12})}{dt} = q A_{\isotm{C}{12}} \frac{dY(\isotm{C}{12})}{dt}\]

(this is positive since both \(q\) and \(dY/dt\) are negative)

ignition_reaclib

This contains several networks designed to model C burning in WDs. They include:

• C-burn-simple : a version of ignition_simple built from ReacLib rates. This just includes the C+C rates and doesn’t group the endpoints together.

• URCA-simple : a basic network for modeling convective Urca, containing the \({}^{23}\mathrm{Na}\)-\({}^{23}\mathrm{Ne}\) Urca pair.

• URCA-medium : a more extensive Urca network than URCA-simple, containing more extensive C burning rates.

ignition_simple

This is the original network used in our white dwarf convection studies [ZingaleAlmgrenBell+09]. It includes a single-step \(^{12}\mathrm{C}(^{12}\mathrm{C},\gamma)^{24}\mathrm{Mg}\) reaction. The carbon mass fraction equation appears as

\[\frac{D X(^{12}\mathrm{C})}{Dt} = - \frac{1}{12} \rho X(^{12}\mathrm{C})^2 f_\mathrm{Coul} \left [N_A \left <\sigma v \right > \right]\]

where \(N_A \left <\sigma v\right>\) is evaluated using the reaction rate from (Caughlan and Fowler 1988). The Coulomb screening factor, \(f_\mathrm{Coul}\), is evaluated using the general routine from the Kepler stellar evolution code (Weaver 1978), which implements the work of (Graboske 1973) for weak screening and the work of (Alastuey 1978 and Itoh 1979) for strong screening.

nova networks

The nova and nova2 networks both are intended for modeling classical novae.

• nova focuses just on CNO/hot-CNO:

  

• nova2 expands nova by adding the pp-chain nuclei:

  

powerlaw

This is a simple single-step reaction rate. We will consider only two species, fuel, \(f\), and ash, \(a\), through the reaction: \(f + f \rightarrow a + \gamma\). Baryon conservation requires that \(A_f = A_a/2\), and charge conservation requires that \(Z_f = Z_a/2\). We take our reaction rate to be a powerlaw in temperature. The standard way to write this is in terms of the number densities, in which case we have

\[\frac{d n_f}{d t} = -2\frac{d n_a}{d t} = -r\]

with

\[r = r_0 n_X^2 \left( \frac{T}{T_0} \right )^\nu\]

Here, \(r_0\) sets the overall rate, with units of \([\mathrm{cm^3~s^{-1}}]\), \(T_0\) is a reference temperature scale, and \(\nu\) is the temperature exponent, which will play a role in setting the reaction zone thickness. In terms of mass fractions, \(n_f = \rho X_a / (A_a m_u)\), our rate equation is

\[\begin{split}\begin{align} \frac{dX_f}{dt} &= - \frac{r_0}{m_u} \rho X_f^2 \frac{1}{A_f} \left (\frac{T}{T_0}\right)^\nu \equiv \omegadot_f \\ \frac{dX_a}{dt} &= \frac{1}{2}\frac{r_0}{m_u} \rho X_f^2 \frac{A_a}{A_f^2} \left (\frac{T}{T_0}\right)^\nu = \frac{r_0}{m_u} \rho X_f^2 \frac{1}{A_f} \left (\frac{T}{T_0}\right)^\nu \end{align}\end{split}\]

We define a new rate constant, \(\rt\) with units of \([\mathrm{s^{-1}}]\) as

\[\begin{split}\rt = \begin{cases} \dfrac{r_0}{m_u A_f} \rho_0 & \text{if $T \ge T_a$} \\[1em] 0 & \text{if $T < T_a$} \end{cases}\end{split}\]

where \(\rho_0\) is a reference density and \(T_a\) is an activation temperature, and then our mass fraction equation is:

\[\frac{dX_f}{dt} = -\rt X_f^2 \left (\frac{\rho}{\rho_0} \right ) \left ( \frac{T}{T_0}\right )^\nu\]

Finally, for the energy generation, we take our reaction to release a specific energy, \([\mathrm{erg~g^{-1}}]\), of \(\qburn\), and our energy source is

\[\edot = -\qburn \frac{dX_f}{dt}\]

There are a number of parameters we use to control the constants in this network. This is one of the few networks that was designed to work with gamma_law as the EOS.

rprox

This network contains 10 species, approximating hot CNO, triple-\(\alpha\), and rp-breakout burning up through \(^{56}\mathrm{Ni}\), using the ideas from [WallaceWoosley81], but with modern reaction rates from ReacLib [CAF+10] where available. This network was used for the X-ray burst studies in [MaloneZingaleNonaka+14], [ZingaleMaloneNonaka+15], and more details are contained in those papers.

triple_alpha_plus_cago

This is a 2 reaction network for helium burning, capturing the \(3\)-\(\alpha\) reaction and \(\isotm{C}{12}(\alpha,\gamma)\isotm{O}{16}\). Additionally, \(^{56}\mathrm{Fe}\) is included as an inert species.

subch networks

The subch networks recreate an aprox13 alpha-chain + including a bypass rate for \(\isotm{C}{12}(\alpha, \gamma)\isotm{O}{16}\) discussed in [ShenBildsten09]. This is appropriate for explosive He burning.

[ShenBildsten09] discuss the sequences:

• \(\isotm{C}{14}(\alpha, \gamma)\isotm{O}{18}(\alpha, \gamma)\isotm{Ne}{22}\) at high temperatures (T > 1 GK). We don’t consider this.

• \(\isotm{N}{14}(\alpha, \gamma)\isotm{F}{18}(\alpha, p)\isotm{Ne}{21}\) is the one they consider important, since it produces protons that are then available for \(\isotm{C}{12}(p, \gamma)\isotm{N}{13}(\alpha, p)\isotm{O}{16}\).

This leaves \(\isotm{Ne}{21}\) as an endpoint, which we connect to the other nuclei by including \(\isotm{Na}{22}\).

For the \(\isotm{C}{12} + \isotm{C}{12}\), \(\isotm{C}{12} + \isotm{O}{16}\), and \(\isotm{O}{16} + \isotm{O}{16}\) rates, we also need to include:

• \(\isotm{C}{12}(\isotm{C}{12},n)\isotm{Mg}{23}(n,\gamma)\isotm{Mg}{24}\)

• \(\isotm{O}{16}(\isotm{O}{16}, n)\isotm{S}{31}(n, \gamma)\isotm{S}{32}\)

• \(\isotm{O}{16}(\isotm{C}{12}, n)\isotm{Si}{27}(n, \gamma)\isotm{Si}{28}\)

Since the neutron captures on those intermediate nuclei are so fast, we leave those out and take the forward rate to just be the first rate. We do not include reverse rates for these processes.

subch_simple

subch_simple uses the ideas above but approximates some of the rates by combining some of the \(A(\alpha,p)X(p,\gamma)B\) links with \(A(\alpha,\gamma)B\), allowing us to drop the intermediate nucleus \(X\). We do this for \(\isotm{Cl}{35}\), \(\isotm{K}{39}\), \(\isotm{Sc}{43}\), \(\isotm{V}{47}\), \(\isotm{Mn}{51}\), and \(\isotm{Co}{55}\).

Further simplifications include:

• The reverse rates of \(\isotm{C}{12}+\isotm{C}{12}\), \(\isotm{C}{12}+\isotm{O}{16}\), \(\isotm{O}{16}+\isotm{O}{16}\) are neglected since they’re not present in the original aprox13 network

• The \(\isotm{C}{12}+\isotm{Ne}{20}\) rate is removed

• The \((\alpha, \gamma)\) links between \(\isotm{Na}{23}\), \(\isotm{Al}{27}\) and between \(\isotm{Al}{27}\) and \(\isotm{P}{31}\) are removed, since they’re not in the original aprox13 network.

The network appears as:

The nuclei in gray are those that have been approximated about, but the links are effectively accounted for in the approximate rates.

Warning

Due to inclusion of the rate sequence, \({}^{14}\mathrm{N}(\alpha, \gamma){}^{18}\mathrm{F}(\alpha, \mathrm{p}){}^{21}\mathrm{Ne}\), there is an artificial end-point at \({}^{22}\mathrm{Na}\).

subch_base

subch_base is the simplest subch network. It is created to reconcile the artificial end-point at \(\isotm{Na}{22}\). This is done by excluding \(\isotm{N}{14}\), \(\isotm{F}{18}\), \(\isotm{Ne}{21}\), and \(\isotm{Na}{22}\). These nuclei were added to include \(\isotm{N}{14}(\alpha, \gamma)\isotm{F}{18}(\alpha, p)\isotm{Ne}{21}\) rate sequence, which allows an enhancement to the \(\isotm{C}{12}(p, \gamma)\isotm{N}{13}(\alpha, p)\isotm{O}{16}\) rate due to the additional proton release. However, we find the effect is not extremely significant.

disabling rates

For all subch networks, there are 2 runtime parameters that can be used to disable rates:

• network.disable_p_c12__n13 : if set to 1, then the rate \(\isotm{C}{12}(p,\gamma)\isotm{N}{13}\) and its inverse are disabled.

• network.disable_he4_n13__p_o16 : if set to 1, then the rate \(\isotm{N}{13}(\alpha,p)\isotm{O}{16}\) and its inverse are disabled.

Together, these parameters allow us to turn off the sequence \(\isotm{C}{12}(p,\gamma)\isotm{N}{13}(\alpha, p)\isotm{O}{16}\) that acts as a bypass for \(\isotm{C}{12}(\alpha, \gamma)\isotm{O}{16}\).