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Case 2: Local universe

In contrast to the previous case, the observer is used to express the Doppler as a function of the source systemic velocity in a local frame (observatory or Local Standard of Rest). Changing the frame to interpret the frequency axis must thus use a given approximation for the Doppler effect. As long as $\ensuremath{d_{\ensuremath{\mathrm{vsys}}}^{\ensuremath{\mathrm{obs}}}} \ll 1$, we can use the Special Relativity formula, i.e.,

\begin{displaymath}
\ensuremath{\mathrm{f_{SR}}}(x) = \sqrt{\frac{1-x}{1+x}}.
\end{displaymath} (37)

Using the radio convention is an additional approximation in which
\begin{displaymath}
\ensuremath{\mathrm{f_{RC}}}(x) = \frac{1}{1+x}.
\end{displaymath} (38)

The difference in the rest frequency between the two approximations is
\begin{displaymath}
\ensuremath{f_\ensuremath{\mathrm{SR}}^{\ensuremath{\mathrm...
...ath{d_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}})].
\end{displaymath} (39)

This just means that the same observed spectrum will be assigned two different systemic velocities, \ensuremath{v_{\ensuremath{\mathrm{SR}}}^{\ensuremath{\mathrm{obs}}}} and \ensuremath{v_{\ensuremath{\mathrm{RC}}}^{\ensuremath{\mathrm{obs}}}}, depending on the level of approximation used. In other words, if an observer uses \ensuremath{v_{\ensuremath{\mathrm{SR}}}^{\ensuremath{\mathrm{obs}}}} as systemic velocity to interpret a spectrum observed with the radio velocity convention, he will find that the lines appear at slightly different rest frequencies in a way that can be interpreted as a wrong systemic velocity. He will thus fit another systemic velocity and find \ensuremath{v_{\ensuremath{\mathrm{RC}}}^{\ensuremath{\mathrm{obs}}}}. The relation between the two associated doppler factors ( \ensuremath{d_{\ensuremath{\mathrm{SR}}}^{\ensuremath{\mathrm{obs}}}} and \ensuremath{d_{\ensuremath{\mathrm{RC}}}^{\ensuremath{\mathrm{obs}}}}) is of course
\begin{displaymath}
\ensuremath{d_{\ensuremath{\mathrm{RC}}}^{\ensuremath{\math...
...d_{\ensuremath{\mathrm{SR}}}^{\ensuremath{\mathrm{obs}}}}}}-1,
\end{displaymath} (40)

and, using the Taylor expansion in $x=0$,
\begin{displaymath}
\frac{\ensuremath{d_{\ensuremath{\mathrm{RC}}}^{\ensuremath...
...th{d_{\ensuremath{\mathrm{SR}}}^{\ensuremath{\mathrm{obs}}}}).
\end{displaymath} (41)

As $\ensuremath{d_{\ensuremath{\mathrm{vsys}}}^{\ensuremath{\mathrm{obs}}}} \ll 1$, this is a negligible difference in the systemic velocities, i.e., the radio convention is good enough.

Nevertheless, the main use case is the following one. A ``naive'' observer got time at a given observatory that uses the radio convention to make a follow-up from an observation acquired in another observatory that uses the special relativity formula. The difference in rest frequency can easily be measurable as it is proportional to the tuned frequency. He probably does not understand the subtleties between the different approximations of the Doppler effect. He will then start to ask around what is wrong. This means that the problem is mainly an interface issue between the different observatories. The easiest solution to this problem would be to use the current standard in (radio-)observatories (What about ALMA, VLA, GBT, APEX). If most of them uses the special relativity formula, IRAM should probably also adopt the special relativity formula.


next up previous contents
Next: Going from the frequency Up: Going from to Previous: Case 1: High redshift   Contents
Gildas manager 2015-03-01