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The Special Relativity formula

For sources lying at cosmological distances, General Relativity should be used to express the Doppler effect in order to take into account the effects of curvature of space. For a Euclidian cosmology where $q_0=0$, the Doppler effect can be expressed with Special Relativity.

In this case, the relation between \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}} and \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}} is given by

\begin{displaymath}
\ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{o...
...suremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}},
\end{displaymath} (2)

with ${\ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}}^2 = {\ens...
...^2+{\ensuremath{v_{\ensuremath{\mathrm{\perp}}}^{\ensuremath{\mathrm{obs}}}}}^2$, where \ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}} and \ensuremath{v_{\ensuremath{\mathrm{\perp}}}^{\ensuremath{\mathrm{obs}}}} are respectively the velocity components along and perpendicular to the line of sight. \ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}} and \ensuremath{v_{\ensuremath{\mathrm{\perp}}}^{\ensuremath{\mathrm{obs}}}} are often called the radial and transverse velocity components.



Gildas manager 2015-03-01