The fitting procedure

According to assumptions A3 and A4, the opacity of the $i$th component is written:

$$\tau_i(v) = \tau_i\cdot\exp\left[ -4\ln{2}\left( \frac{v-v_{0,i}}{p_3} \right)^2 \right]$$ (2.1)

where $p_3$ is the common FWHM of all components. The central velocity of component $i$ is $v_{0,i}=v_i+p_2$, where $p_2$ is the velocity of the reference component (i.e. the one with $v_i=0$).

The opacity of the multiplet is the sum of the $N$ opacities:

$$\tau(v) = p_4 \displaystyle\sum_{i=1}^{N} r_i\cdot\exp\left[ -4\ln{2}\left( \frac{v-v_i-p_2}{p_3} \right)^2 \right]$$ (2.2)

Given the opacity $\tau(v)$, the antena temperature is given by

$$T_{\rm ant}(v) = \frac{p_1}{p_4} \left( 1 - e^{-\tau(v)} \right)$$ (2.3)

From these equations, we deduce:

$$\begin{eqnarray*} \tau(v_i+p_2) &=& p_4 \cdot r_i\qquad (\rm assumption\, A4)\\... ...p_4 r_i}\right)\\ T_{\rm ant}(v_i+p_2) &\approx& p_1 \cdot r_i \end{eqnarray*}$$

where the last equality holds in the optically thin regime.

This implies that the physical meaning of $p_4$ depends on the value of $S$: If the relative intensities are normalized to unity, then $S=1$ and $p_4$ equals the sum of all centerline opacities.

The results of the HFS fitting procedure are:

   Line     T ant * Tau           V lsr          Delta V            Tau main

   1    1.313 ( 0.018)    3.783 ( 0.001)    0.589 ( 0.003)    0.230 ( 0.006)

where T ant * Tau=$p_1$, V lsr=$p_2$, Delta V=$p_3$ and Tau main=$p_4$.

According to assumptions A1 and A5, the hyperfine structure fitting procedure allows you to deduce the excitation temperature, since (assuming the Rayleigh-Jeans regime is valid, which is not true at $\lambda<3$mm...):

$$T_{\rm ant}(v) = T_A^*(v) = \frac{B_{\rm eff}}{F_{\rm eff}} [T_{\rm ex} - T_{\rm bg}] (1-e^{-\tau(v)})$$ (2.4)

Finally, the excitation temperature is given by:

$$T_{\rm ex} = T_{\rm bg} + \frac{F_{\rm eff}}{B_{\rm eff}} \frac{p_1}{p_4}$$ (2.5)

Note: the main group opacity is limited to the range $0.1-30$ since outside these limits, the problem becomes degenerate because the opacity no longer appears in the equations (in the optically thin limit, line ratio no longer depend on the opacity, and in the thick limit, $\exp(-\tau)\ll 1$).