Modelling continuum

For continuum states spectra calculations, i.e. for transitions between bound and unbound (usually upper) states, the corresponding line lists are represented by discretized transitions with their intensities, which need to be re-distributed across the simulation wavenumber regions. This is typically done with a Gaussian line profile with a larger width (50-200 cm-1) to cover gaps between these discrete lines. For example,

Temperature  5000
Range    45000 85000

npoints  40000

absorption
gaussian

hwhm 200

cutoff 5000

output NH_box_5000K_L20

States NH.states

Transitions NH.trans

It is also important to set cutoff (wing cut-off) to a large number.

‘Particle in the box’

In the case of a very large spectroscopic range, the distance between discrete continuum lines rapidly increases. For example, for the case of the particle-in-a-box solution, the energy separation increases with the excitation number linearly:

\[\Delta \tilde{E}_n^{\rm box} = \tilde{E}_{n+1}^{\rm box} - \tilde{E}_n^{\rm box} = \frac{h (2n+1)}{8 \mu L^2 c}.\]

where \(\mu\) is the reduced mass, \(L\) is the box size) and \(n\ge 1\) is the state counting number, \(c\) is the speed of light in cgs. Due to the linear dependence on \(n\), there is no a single optimal value of \(\alpha_{\rm G}\) (Gaussian HWHM) for the entire region. In order to account for such an increase, the ‘recipe’ Box'' for continuum states spectra calculations can be used as part of the ``Species construct. The corresponding line lists are represented by discretized transitions with their intensities to be re-distributed across the simulation wavenumber regions. In the Box recipe, we define \(\alpha_{\rm G}\) to be \(\gamma_n^{\rm box} = \gamma_0 \Delta \tilde{E}_n^{\rm box}\), i.e.

\[\gamma_n^{\rm box} = \gamma_0 \frac{h (2n+1)}{8 \mu L^2 c},\]

where \(n\) is the counting number of the continuum states (from the lowest) and \(\gamma_0\) is the initial value of HWHM for \(n=1\).

Here is an example of the ExoCross input:

Temperature  5000
Range    45000 85000

npoints  40000

QN
  K 8
end

absorption
gaussian

cutoff 5000

species
     particle  gamma 1.0 mass 0.937  Lbox 20  model box
end

output NH_box_5000K_L20

States NH.states

Transitions NH.trans

Here model box defines the recipe type box, Lbox is the size of the box and mass specifying the reduced mass, particle is an arbitrary dummy name for the broadening and gamma defines the initial HWHM value \(\gamma_0\).