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General case

Let's assume that the user wishes to observe a line that appears at the rest frequency, \ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}, for a source that has a systemic velocity, \ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{1}}}}}^{\ensuremath{\mathrm{}}}}, in a given velocity convention #1. The rest frame and observatory frame frequencies are linked by the Doppler effect through

\begin{displaymath}
\ensuremath{f_\ensuremath{\mathrm{tuned,conv1}}^{\ensuremat...
...mathrm{\ensuremath{\mathrm{1}}}}}^{\ensuremath{\mathrm{}}}})},
\end{displaymath} (74)

where \ensuremath{D_{\ensuremath{\mathrm{1}}}()} is the Doppler function that can be deduced from Eq. [*], [*], or [*], depending on the assumed convention. For instance, the Doppler function of the optical convention is
\begin{displaymath}
D_\ensuremath{\mathrm{opt}}(\ensuremath{v_{\ensuremath{\mat...
...th{\mathrm{opt}}}^{\ensuremath{\mathrm{}}}}}{\ensuremath{c}}}.
\end{displaymath} (75)

Now, let's assume that the analysis exactly uses the same observed frequency, \ensuremath{f_\ensuremath{\mathrm{tuned,1}}^{\ensuremath{\mathrm{obs}}}}, and systemic velocity, \ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{1}}}}}^{\ensuremath{\mathrm{}}}}, but it interprets it using another Doppler convention and thus Doppler function, \ensuremath{D_{\ensuremath{\mathrm{2}}}()}. The spectral axis in the rest frame is then computed as

\begin{displaymath}
\ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{r...
...\ensuremath{\mathrm{apparent}}^{\ensuremath{\mathrm{rest}}}}},
\end{displaymath} (76)

with
\begin{displaymath}
\ensuremath{f_\ensuremath{\mathrm{apparent}}^{\ensuremath{\...
...athrm{\ensuremath{\mathrm{1}}}}}^{\ensuremath{\mathrm{}}}})}}.
\end{displaymath} (77)

This means that the lines will seem to appear at a wrong rest frequency with a wrong full width at half maximum.


next up previous contents
Next: The IRAM-30m case as Up: Mixing conventions between observation Previous: Mixing conventions between observation   Contents
Gildas manager 2023-06-01