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Approximation 1: Neglecting the transverse Doppler effect when $\ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}\ll \ensuremath {c}$

Astronomers have very little access to the velocity component perpendicular to the line of sight. They will thus assign the observed Doppler shift to the line-of-sight motion, i.e., they will neglect the transverse Doppler effect.

The formula 2 indicates that the dependency on \ensuremath{v_{\ensuremath{\mathrm{\perp}}}^{\ensuremath{\mathrm{obs}}}} is of second order in $v/\ensuremath{c}$, while the dependency on \ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}} is only of first order in $v/\ensuremath{c}$. Hence, neglecting the transverse Doppler effect is correct as long as

\begin{displaymath}
(\ensuremath{v_{\ensuremath{\mathrm{\perp}}}^{\ensuremath...
...suremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}})^2.
\end{displaymath} (3)

In this case, the Doppler equation can be approximated to first order to
\begin{displaymath}
\ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{o...
...uremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}}}.
\end{displaymath} (4)

In other words, the frequency change under a change of frame is attributed only to the radial velocity between the rest and observation frame. We note that condition 3 is enforced if the velocities are non-relativistic, i.e.
\begin{displaymath}
\ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}\ll \ensuremath{c}.
\end{displaymath} (5)


next up previous contents
Next: Approximation 2: First order Up: Various approximations of the Previous: The Special Relativity formula   Contents
Gildas manager 2015-03-19