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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
, we can use the Special Relativity
formula, i.e.,
 |
(37) |
Using the radio convention is an additional approximation in which
 |
(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}](img71.png) |
(39) |
This just means that the same observed spectrum will be assigned two
different systemic velocities,
and
, depending on the
level of approximation used. In other words, if an observer uses
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
. The relation between the two associated doppler factors
(
and
) is of course
 |
(40) |
and, using the Taylor expansion in
,
 |
(41) |
As
, 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: Going from the frequency
Up: Going from to
Previous: Case 1: High redshift
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2015-03-19