Spectral Function

Note

The symbols in these equations are defined on the Definition of Symbols page.

The spectral radiance of \(\text{H}_3^+\) at a particurlar wavelength can be calculated as a sum of guaussians, each representing an indivudial emission line.

\[I(\lambda, T) = b(\lambda) + N \sum_{i=0}^{n_{lines}}\frac{I_{i}(T)}{\sigma_{i}\sqrt{2\pi}}\exp{\left(-\frac{(\lambda-(\lambda_i+s(\lambda)))^{2}}{2\sigma_{i}^{2}}\right)}\]

where \(I_i(T)\) is the radiance of emission line \(i\). It is given by:

\[I_i(T) = \frac{ g_{ns}(2J+1)100 \times hcw_{if}A_{if}}{4\pi Q(T)}\exp{\left[-\frac{100 \times hcw_{upper}}{kT}\right]}\]

The \(\text{H}_3^+\) partiation function \(Q(T)\) is taken from Miller et al., (2013) and is expressed as a polynomial function:

\[\log{Q} = \sum_{i=0} a_n (\log{T})^n\]

where the constants \(a_n\) are provided in Table 1 of Miller et al., (2013).

The \(\text{H}_3^+\) line list is from Neale et al., (1996) provided in the package in a reduced form that removes emision lines with negligble radiance at temperatures < 2000 K.