A Simple Model

For this comparison between the new ALMA data and our experimental results, we adopt a simple model: Each species is characterized by (1) a column density, with terrestrial isotopic abundances, (2) a single temperature, (3) a lineshape that is characteristic of its velocity distribution within the beam, (4) an optical depth parameter (for the strongest species), and (5) a constant continuum that is subtracted from the astronomical data.

For our simulation in the optically thin case, to convert laboratory absorbance $A(n)$ (Equation 1) to Jansky/beam,\begin{equation}I = \int{A(\nu)K'S(\nu-\nu')d\nu'}\end{equation} Where $S(n)$ is a normalized lineshape function based on the numerical lineshape functions derived from the astronomical data. $K’$ was adjusted empirically so that the ALMA spectra in units of Jansky/beam matched the intensity of the simulation and was .0025, .002, and .16 for methyl cyanide, ethyl cyanide, and vinyl cyanide respectfully. This factor, which absorbs $nL/Q$, is related to the astronomical column density and molecular partition function and has units of Jansky/beam.

If $I_{max}$ is the intensity for an optically thick line, then \begin{equation}I = I_{max}\left(1-e^{-\left(\frac{I}{I_{max}}\right)}\right)\end{equation} After we present the results of such a comparison, we will discuss prospects for straightforward extensions of this model.