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Next: Solution 2: Special relativity Up: Going from the frequency Previous: Going from the frequency   Contents

Solution 1: The radio convention is used to define both \ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}} and \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}}

This is the simplest case, which is currently implemented in CLASS. It is defined as

\begin{displaymath}
\ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm...
...{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}}.
\end{displaymath} (42)

The radio convention is the result of the first order term of the Taylor expansion at $x=0$ of

\begin{displaymath}
\frac{1}{\ensuremath{\mathrm{f_{SR}}}(x)} = \sqrt{\frac{1+x}{1-x}}.
\end{displaymath} (43)

To check the accuracy of the approximation on \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{RC}}}^{\ensuremath{\mathrm{obs}}}}}, let's use the next term in this Taylor expansion, i.e.,
\begin{displaymath}
\sqrt{\frac{1+x}{1-x}} \simeq 1+x+\frac{x}{2}.
\end{displaymath} (44)

Differentiating this equation, we obtain the first order correction to the velocity channel width
\begin{displaymath}
\Delta \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathr...
...th{d_{\ensuremath{\mathrm{RC}}}^{\ensuremath{\mathrm{obs}}}}}.
\end{displaymath} (45)

Let's now assume that we want an accumulated error over \ensuremath{n_{\ensuremath{\mathrm{}}}} channel to be less than a given tolerance \ensuremath{T_{\ensuremath{\mathrm{}}}}. This yields
\begin{displaymath}
\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mat...
...math{\mathrm{}}}}}{\ensuremath{T_{\ensuremath{\mathrm{}}}}}+1}
\end{displaymath} (46)

This is the criterion to change from the local universe representation (using \ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}) to the high redshift universe (using \ensuremath{z_{\ensuremath{\mathrm{}}}}). If we want a tolerance of one tenth of channel (i.e., $\ensuremath{T_{\ensuremath{\mathrm{}}}}=1/10$), we obtain This does not seem enough.


next up previous contents
Next: Solution 2: Special relativity Up: Going from the frequency Previous: Going from the frequency   Contents
Gildas manager 2015-03-01