Nebular Plasma

The NebularPlasma class is a more complex description of the Plasma state than the LTEPlasma. It takes a dilution factor (W) into account, which deals with the dilution of the radiation field due to geometric, line-blocking and other effects.

The calculations follow the same steps as LTEPlasma; however, the calculations are different and often take into account if a particular level is meta-stable or not. NebularPlasma will start calculating the partition functions.

\[Z_{i,j} = \underbrace{\sum_{k=0}^{max(k)_{i,j}} g_k \times e^{-E_k / (k_\textrm{b} T)}}_\textrm{metastable levels} + \underbrace{W\times\sum_{k=0}^{max(k)_{i,j}} g_k \times e^{-E_k / (k_\textrm{b} T)}}_\textrm{non-metastable levels}\]

where Z is the partition function, g is the degeneracy factor, E the energy of the level, T the temperature of the radiation field and W the dilution factor.

The next step is to calculate the ionization balance using the Saha ionization equation and then calculate the number density of the ions (and an electron number density) in a second step. In the first step, we calculate the ionization balance using the LTE approximation (\(\Phi_{i, j}(\textrm{LTE})\)). Then we adjust the ionization balance using two factors: \(\zeta\) and \(\delta\).

Calculating Zeta

\(\zeta\) is read in for specific temperatures and then interpolated for the target temperature.

Calculating Delta

\(\delta\) is a radiation field correction factors which is calculated according to Mazzali & Lucy 1993 ([MazzaliLucy93]; henceforth ML93)

In ML93, the radiation field correction factor is denoted as \(\delta\) and is calculated in Formula 15 & 20.

The radiation correction factor changes according to a ionization energy threshold \(\chi_\textrm{T}\) and the species ionization threshold (from the ground state) \(\chi_0\).

For \(\chi_\textrm{T} \ge \chi_0\)

\[\delta = \frac{T_\textrm{e}}{b_1 W T_\textrm{R}} \exp(\frac{\chi_\textrm{T}}{k T_\textrm{R}} - \frac{\chi_0}{k T_\textrm{e}})\]

For \(\chi_\textrm{T} < \chi_0\)

\[\delta = 1 - \exp(\frac{\chi_\textrm{T}}{k T_\textrm{R}} - \frac{\chi_0}{k T_\textrm{R}}) + \frac{T_\textrm{e}}{b_1 W T_\textrm{R}} \exp(\frac{\chi_\textrm{T}}{k T_\textrm{R}} - \frac{\chi_0}{k T_\textrm{e}}),\]

where \(T_\textrm{R}\) is the radiation field temperature, \(T_\textrm{e}\) is the electron temperature and W is the dilution factor.

Now, we can calculate the ionization balance using equation 14 in [MazzaliLucy93]:

\[ \begin{align}\begin{aligned}\begin{split}\Phi_{i,j} &= \frac{N_{i, j+1} n_e}{N_{i, j}} \\\end{split}\\\begin{split}\Phi_{i, j} &= W \times[\delta \zeta + W ( 1 - \zeta)] \left(\frac{T_\textrm{e}}{T_\textrm{R}}\right)^{1/2} \Phi_{i, j}(\textrm{LTE}) \\\end{split}\end{aligned}\end{align} \]

In the last step, we calculate the ion number densities according using the methods in LTEPlasma

Finally, we calculate the level populations (NebularPlasma.calculate_level_populations()), by using the calculated ion species number densities:

\[\begin{split}N_{i, j, k}(\textrm{not metastable}) &= W\frac{g_k}{Z_{i, j}}\times N_{i, j} \times e^{-\beta_\textrm{rad} E_k} \\ N_{i, j, k}(\textrm{metastable}) &= \frac{g_k}{Z_{i, j}}\times N_{i, j} \times e^{-\beta_\textrm{rad} E_k} \\\end{split}\]

This concludes the calculation of the nebular plasma. In the code, the next step is calculating the \(\tau_\textrm{Sobolev}\) using the quantities calculated here.

Example Calculations