The spectral line model

Spectuner implements the one-dimensional LTE spectral line model. For a single velocity component, the spectrum of one vibration state is given by

\[\begin{split}\begin{aligned} J_\nu &= \eta_\nu(\theta, \nu) \left(S_\nu - J^{bg}_\nu \right) \left(1 - e^{-\tau_\nu}\right), \label{eqn:slm_first} \\ \eta(\theta, \nu) &= \frac{\theta^2}{\theta^2 + \theta^2_\text{T}(\nu)} \\ S_\nu &= \frac{h\nu}{k} \frac{1}{e^\frac{h\nu}{kT} - 1} \\ \tau_\nu &= \sum_t \tau^t_\nu, \\ \tau^t_\nu &= \frac{c^2}{8\pi \nu^2} N_\text{tot} \frac{ A^t_{ul} g^t_u}{Q(T_\text{ex})} e^{-\frac{E^t_l}{k T_\text{ex}}} (1 - e^{-\frac{h\nu^t}{k T_\text{ex}}}) \phi^t_\nu, \\ \phi^t_\nu &= \frac{1}{\sqrt{2\pi}\sigma^t} \exp \left[ -\frac{1}{2} \left( \frac{\nu - \delta \nu^t}{\sigma^t}\right)^2 \right], \\ \delta \nu^t &= \left( 1 - \frac{v_\text{offset}}{c} \right) \nu^t, \\ \sigma^t &= \frac{1}{2 \sqrt{2 \ln 2}} \frac{\Delta v}{c} \delta \nu_t, \end{aligned}\end{split}\]

where \(J^{bg}_\nu\) is the background intensity, \(c\) is the speed of light and \(k\) is the Boltzmann constant. For single dish telescopes, the beam size is calculated by

\[\theta_\text{T} = 1.22 \frac{c}{\nu D} \frac{180}{\pi} \; \text{deg},\]

where \(D\) is the diameter of the telescope. For interferometers, the beam size is given by

\[\theta_\text{T} = \sqrt{\theta_\text{maj}\theta_\text{min}},\]

where \(\theta_\text{maj,min}\) is the major (minor) axis of the synthesis beam.

The model includes five fitting parameters:

  • \(\theta\): Source size.

  • \(T_\text{ex}\): Excitation temperature.

  • \(N_\text{tot}\): Column density.

  • \(\Delta v\): Velocity width.

  • \(v_\text{offset}\): Velocity offset.

The following properties should be loaded from a spectroscopic database:

  • \(\nu^t\): Transition frequency.

  • \(A^t_\text{ul}\): Einstein A cofficient.

  • \(g^t_u\): Upper state degeneracy.

  • \(E^t_\text{l}\): Energy of the lower state.

  • \(Q(T_\text{ex})\): Partition function.

Furthermore, the code takes into the instrumental resolution effect according to Möller et al. (2017). The following integral is applied to computing the output model spectrum:

\[J'(\nu) = \frac{1}{\Delta \nu_\text{c}} \int^{\nu + \Delta \nu_\text{c}/2}_{\nu - \Delta \nu_\text{c}/2} J(\nu') \, d\nu',\]

where \(\Delta \nu_\text{c}\) is the channel width.

References

  • Möller, T., Endres, C., & Schilke, P. (2017), eXtended CASA Line Analysis Software Suite (XCLASS), Astronomy and Astrophysics, 598, A7.