NSE

Important

NSE is only supported with the simplified-SDC method for coupling hydrodynamics and reactions. We do not support operator-splitting (Strang) coupling with NSE.

Nuclear statistical equilibrium (NSE) describes a state in which all strong nuclear reactions are in chemical equilibrium, with each forward reaction balanced by its reverse. The NSE condition is a state where the chemical potential of any isotope can be expressed as the sum of the chemical potentials of the its constituent neutrons and protons. which happens when \(T \gtrsim 6 \times 10^9 \ \mathrm{K}\). Mathematically it means:

\[\mu_i = Z_i \mu_p + N_i \mu_n\]

Applying this assumption to the Maxwell-Boltzmann gas, the mass abundance of the system in NSE is uniquely defined as:

\[X_i^{\mathrm{NSE}} = \dfrac{(m_i)^{5/2}}{\rho} g_i \left( \dfrac{k_B T}{2\pi \hbar^2} \right)^{3/2} \exp{\left( \dfrac{Z_i (\mu_p^\mathrm{kin} + \mu_p^c) + N_i \mu_n^{\mathrm{kin}} + B_i - \mu^c_i}{k_B T} \right)}\]

A more detailed derivation can be found in many literature, e.g., [20].

There are three possible constraint equations that may be required to solve the NSE equation. Depending on the specified input mode, either two or all three constraints are needed.

  1. Conservation of mass, or equivalently a constraint on the input density, \(\rho\).

    \[f_\rho = \sum_i X_i^{\mathrm{NSE}} - 1 = 0\]

  2. Conservation of charge for strong reactions, or equivalently a constraint on the input electron fraction, \(Y_e\)

    \[f_{Y_e}\sum_i \frac{Z_i}{A_i} X_i^{\mathrm{NSE}} - Y_e = 0\]

  3. Conservation of energy, or equivalently a constraint on the input internal energy, \(e\)

    \[f_e = \frac{e\left(\rho, T_\mathrm{NSE}, X_i^\mathrm{NSE}\right)}{e_0} - 1 = 0\]

Two input modes are currently supported:

  1. Given the input \((\rho, T, Y_e)\), the system has two unknowns, the chemical potential of proton and neutron, i.e. \((\mu_p^\mathrm{kin}, \mu_n^\mathrm{kin})\). In this mode, the first two constraint equations are sufficient to solve the system.

  2. Given the input \((\rho, e_0, Y_e)\), the above equation has three unknowns, the chemical potential of proton and neutron and temperature, i.e. \((\mu_p^\mathrm{kin}, \mu_n^\mathrm{kin}, T_9)\). In this mode, all three constraint equations are needed to solve the system.

NSE Evolution Modes

The reaction networks in Microphysics have the ability to use NSE instead of integrating the entire network when the conditions are appropriate. There are 2 different implementations of NSE in Microphysics, that have slightly different use cases.

  • Tabulated NSE and aprox19 : this uses a table of NSE abundances given \((\rho, T, Y_e)\) generate from a large network (96 isotopes). The table also returns \(dY_e/dt\) resulting from electron-captures, to allow for the NSE state to evolve. This is meant to be used in the cores of massive stars and works only with the aprox19 reaction network.

    Furthermore, since the table can achieve \(Y_e\) and \(\bar{A}\) that are not representable by the 19 isotopes in aprox19, this table requires that we use the auxiliary composition and advect \(Y_e\), \(\bar{A}\), and \(\langle B/A\rangle\). All of the EOS calls will work with these quantities.

    This algorithm was described in [46].

    This is enabled via USE_NSE_TABLE

  • Self-consistent NSE : this adds an NSE solver to the network that can be called to find the equilibrium abundances of each of the species defined in the network. It only works pynucastro-generated networks. Note that there are requirements in order for networks to be intrinsically compatible with NSE, see more detail in Self-consistent NSE. Unlike the tabulated NSE, there is no need to advect an auxiliary composition, since this only deals with the isotopes defined in the main reaction network.

    This is enabled via USE_NSE_NET

Both solvers define a number of preprocessor variables, and both will provide a function in_nse() that can be used to determine if a state is currently in NSE.

make option

preprocessor variables set

USE_NSE_TABLE

NSE, NSE_TABLE, AUX_THERMO

USE_NSE_NET

NSE, NSE_NET

The directive NSE should be used whether the specific implementation of NSE does not matter.

These two NSE solvers are in the next sections.