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Christer Watson edited sectionResults_Eight.tex
over 8 years ago
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\end{equation*}
where
\begin{align} \begin{align*}
g_K= g_I &=
1\nonumber\\
\mu^2_{el} 1\notag\\
%\mu^2_{el} S &= 3.8 \;\mathrm{Debye}^2\nonumber\\
Z_{rot} %Z_{rot} &= .8556\; T_{ex}-0.10\nonumber\\
F(T) %F(T) &=
\frac{1}{e^{h\nu/k_BT}-1}.\nonumber\\
\end{align} \frac{1}{e^{h\nu/k_BT}-1}.\nonumber
\end{align*}
{\bf where $\epsilon_0$ is the vacuum permittivity, $\mu_{el}$ is the permanent electric dipole moment, S is the line strength, Z$_{rot}$ is the rotational partition function, g$_K$ is the K-level degeneracy, g$_I$ is the reduced nuclear spin degeneracy, E$_u$ is the energy of the upper-transition state, T$_{ex}$ is the excitation temperature and T$_{bg}$ is the background temperature.} The dipole moment line strength ($\mu^2_{el}$ S) is taken from the JPL spectral line catalog \citep{Pickett1998}. The partition function (Z$_{rot}$) is a linear fit to JPL data between T=37 to 75 K. T$_{bg}$ was taken to be the cosmic microwave background temperature, 2.725 K. The uncertainty in the fit amplitudes and derived column densities is dominated by our flux-calibration uncertainty. Since the relationships are linear, we estimate the uncertainty in both as 20\%.