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The generic Doppler formula

The Doppler effect expresses that a source emitting in its rest frame at a frequency \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}} will appear to emit at a frequency \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}} in the observation frame, which depends on the velocity of the source in the observation frame, \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}. 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} (66)

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 perperdicular 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.

For objects far from the observer, $\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}\gg \ensuremath{v_{\ensuremath{\mathrm{\perp}}}^{\ensuremath{\mathrm{obs}}}}$ because of the universe expansion. For close-by objects, Eq. [*] shows 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, for all cases, the Doppler equation is simplified to

\begin{displaymath}
\ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{o...
...uremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}}}.
\end{displaymath} (67)

In other words, the frequency change under a change of frame is attributed only to the radial velocity between the rest and observation frame.


next up previous contents
Next: The radio velocity convention Up: Velocity conventions Previous: Velocity conventions   Contents
Gildas manager 2023-06-01