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Interpretation 3: At fixed \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}} and fixed \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{sou}}}}

Let's assume that we are interested to detect a given bright line, e.g., CO(1-0), at high redshift but the redshift of the source is unknown and/or their would be a forest of this line at different redshift for the same line of sight. We can also assume that the parcel of gas is at the same local velocity (an interesting particular case being when the parcel of gas is at rest) in the different source frames corresponding to the different redshift. We thus wish to associate a redshift axis to the set of brightnesses, $T(\ensuremath{i_{\ensuremath{\mathrm{}}}})$. To do this, we use

\begin{displaymath}
\ensuremath{z_{\ensuremath{\mathrm{}}}}{} = \frac{\ensurema...
...th{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}} - 1,
\end{displaymath} (30)

and, after differentiation,
\begin{displaymath}
\ensuremath{\delta \ensuremath{z_{\ensuremath{\mathrm{}}}}{...
...th{\mathrm{}}}^{\ensuremath{\mathrm{sou}}}}}}{\ensuremath{c}}.
\end{displaymath} (31)

This yields
\begin{displaymath}
\ensuremath{z_{\ensuremath{\mathrm{}}}}{}(\ensuremath{i_{\e...
...{}}}}{}}}{1+\ensuremath{z_{\ensuremath{\mathrm{tuned}}}}}} - 1
\end{displaymath} (32)


\begin{displaymath}
\mbox{with} \quad
\ensuremath{z_{\ensuremath{\mathrm{tuned...
...f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}}.
\end{displaymath} (33)

The sign in the definition of \ensuremath{\delta \ensuremath{z_{\ensuremath{\mathrm{}}}}{}} ensures that an increase of \ensuremath{i_{\ensuremath{\mathrm{}}}} implies an increase of \ensuremath{z_{\ensuremath{\mathrm{}}}}( \ensuremath{i_{\ensuremath{\mathrm{}}}}). Moreover, the non-linear character of this axis comes from the fact that the spectral axis is regularly sampled in frequency unit while the adopted redshift definition is the optical one. We retrieve a linear axis
\begin{displaymath}
\ensuremath{z_{\ensuremath{\mathrm{}}}}{}(\ensuremath{i_{\e...
...th{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{obs}}}}.
\end{displaymath} (34)

With current heterodyne detector in millimeter radioastronomy, the last condition is difficult to satisfy as the radiofrequence bandwidth can cover a significant fraction of the tune frequency, at least in the 3 \ensuremath{\, \ensuremath{\mathrm{mm}}} band. The non-linear formula is thus useful in CLASS.


next up previous contents
Next: Validity of the radio Up: Interpreting the spectral axis Previous: Interpretation 2: At fixed   Contents
Gildas manager 2015-03-01