Plasma

The role of the plasma module is to determine the ionisation and excitation states of the elements of the supernova ejecta, given the basic structure, including the elemental abundances, densities and radiation temperature. After the calculation of the plasma state, the \(\tau_{\textrm{sobolev}}\) values can be calculated.

The TARDIS plasma structure inherits from the BasePlasma class. The code currently uses the LegacyPlasmaArray for generating a plasma from the information provided by model. A variety of different plasmas can be generated depending on the options selected in the plasma section of the TARDIS config. file. The options currently considered by the Legacy Plasma when creating the plasma calculation structure include:

plasma:

• ionization: lte/nebular

• excitation: lte/dilute-lte

• line_interaction_type: scatter/downbranch/macroatom

• helium_treatment: dilute-lte/recomb-nlte

• nlte: [can provide list of ion species to be treated in NLTE, as well as specifying the use of the coronal_approximation/classical_nebular settings.

LegacyPlasmaArray uses these options to construct a map of the necessary plasma parameters that demonstrates how these parameters are dependent on one another (using NetworkX). Each time a particular parameter of the plasma is updated, all of the parameters dependent (directly or indirectly) on that particular one can be easily updated automatically, without requiring that all the plasma calculations are repeated.

Properties, Inputs and Outputs

Each TARDIS plasma possesses an array of plasma properties, which are used to calculate plasma parameter values. Most plasma properties have a single output, e.g.

• GElectron: (g_electron,)

• HeliumNLTE: (helium_population,)

but some have two or more, e.g.

• IonNumberDensity: (ion_number_density, electron_densities)

• Levels: (levels, excitation_energy, metastability, g)

Every property has a calculate function that returns the values of its outputs. The arguments required for that function become the property inputs. TARDIS will raise an error if it does not have all of the required inputs for a particular property. It will also raise an error if there is an output loop, i.e. if two properties are dependent on each other. Some different properties share output names; for example, PhiSahaLTE and PhiSahaNebular both have an output called phi. That is because the phi value is calculated differently depending on the ionization method selected, but once calculated, both values interact in the same way with the rest of the plasma. TARDIS will import only one of the phi properties when initialising the plasma.

The Plasma Graph

If the necessary Python modules (PyGraphviz and dot2tex) are available, TARDIS can output a .dot and a .tex file at the beginning of each run that can be compiled to produce a PDF image of the plasma module graph via the write_to_dot() and write_to_tex() functions, respectively. The nodes on this graph are the names of plasma properties, e.g. Levels, TauSobolev, IonNumberDensity, along with a list of outputs from those properties. These nodes are connected by edges linking them with the sources of their inputs. The .tex file contains the name of the input/output linking the properties (e.g. levels, \(\tau_{\textrm{sobolev}}\), \(n_{e}\)), as well as the equations used to calculate them, written in LaTeX. See the tutorial below!

Updating the Plasma

During each iteration of the main code, TARDIS updates the plasma using the update_radiationfield function. This requires, at minimum, new values for t_rad (the radiation temperature), w (the dilution factor) and j_blues (the intensity in the blue part of each line).

Plasma Calculations

Note

In this documentation we use the indices \(i, j, k\) to mean atomic number, ion number and level number respectively.

BasePlasma serves as the base class for all plasmas and can just calculate the atom number densities for a given input of abundance fraction.

\[N_{atom} = \rho_\textrm{total} \times \textrm{Abundance fraction} / m_\textrm{atom}\]

In the next step the line and level tables are purged of entries that are not represented in the abundance fractions are saved in BasePlasma.levels and BasePlasma.lines. Finally, the function BasePlasma.update_t_rad is called at the end of initialization to update the plasma conditions to a new \(T_\textrm{radiation field}\) (with the give t_rad). This function is the same in the other plasma classes and does the main part of the calculation. In the case of BasePlasma this is only setting BasePlasma.beta_rad to \(\frac{1}{k_\textrm{B}T_\textrm{rad}}\).

The next more complex class is LTEPlasma which will calculate the ionization balance and level populations in Local Thermal Equilibrium conditions (LTE). The NebularPlasma-class inherits from LTEPlasma and uses a more complex description of the BasePlasma.

• LTE Plasma
  • Example Calculations
• Nebular Plasma
  • Calculating Zeta
  • Calculating Delta
  • Example Calculations

TARDIS also allows for NLTE treatments of specified species, as well as special NLTE treatments for Helium.

Note

The NLTE treatment of specified species is currently incompatible with the NLTE treatment for helium and cannot be used simultaneously.

• NLTE treatment
• Helium NLTE
  • Recombination He NLTE
  • Numerical He NLTE

Sobolev optical depth

After the above calculations, TARDIS calculates the Sobolev optical depth \(\tau_\textrm{Sobolev}\) with the following formula:

\[ \begin{align}\begin{aligned}C_\textrm{Sobolev} = \frac{\pi e^2}{m_e c}\\\tau_\textrm{Sobolev} = C_\textrm{Sobolev}\, \lambda\, f_{\textrm{lower}\rightarrow\textrm{upper}}\, t_\textrm{explosion}\, N_\textrm{lower} (1 - \frac{g_\textrm{lower}}{g_\textrm{upper}}\frac{N_\textrm{upper}}{N_\textrm{lower}})\end{aligned}\end{align} \]

Macro Atom Line Interaction Treatment

The following page describes the macro atom treatment of line interactions:

• Macro Atom