= ELODIE spectrograph
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Built by <André Baranne>, <Michel Mayor>, <Didier Queloz> and colleagues, the ELODIE <fiber-fed spectrograph> carried light from Haute-Provence's 1.93-metre telescope through <optical fibers>. Its <echelle grating> dispersed light in high <diffraction orders>; an <optical prism> and <grism> <cross-disperser> separated them perpendicularly on a <charge-coupled device>. This <cross-dispersed echelle spectrograph> achieved <spectral resolving power> $R=42000$. A simultaneous <thorium> reference tracked <instrumental drift>. <Wavelength calibration> mapped detector positions to <wavelengths>; <spectral cross-correlation> combined many <spectral lines> into a fitted profile. A <barycentric velocity correction> removed the observer's motion.<ELODIE: A spectrograph for accurate radial velocity measurements by André Baranne and others>[Baranne et al. (1996), abstract; section 3.1 and figure 6; section 3.7]{ref}
\Image[/elodie-spectrum-to-radial-velocity.png]
{title=ELODIE measurement chain and synthetic radial-velocity readout}
{description=Original schematic; the correlation profile and periodic signal are synthetic.}
Why can a <spectrograph> measure a <Doppler shift> much smaller than one <spectral resolution element>? Resolution concerns separating neighbouring features, whereas locating the centre of a known profile uses information across that profile and across many <spectral lines>. For a chosen example, at $\lambda=500\,\mathrm{nm}$ a recession <velocity> of $60\,\mathrm{m\,s^{-1}}$ gives
$$
\Delta\lambda\simeq\lambda\frac{v_r}{c}=1.00069\times10^{-4}\,\mathrm{nm}.
$$
At $R=42000$, one <spectral resolution element> corresponds to $c/R\simeq7.14\,\mathrm{km\,s^{-1}}$, about $119$ times larger. A well measured centre can move by a small fraction of that width. More <photons> and more useful <spectral lines> improve the estimate, but do not remove <wavelength calibration> errors or <stellar activity>.
A simple mathematical picture of <spectral cross-correlation> makes the readout concrete. Define $x=c\log(\lambda/\lambda_0)$, with fixed reference <wavelength> $\lambda_0$, and let $d(x)$ be continuum-normalized line absorption. For small <velocities>, a common <Doppler shift> approximately translates $d(x)$. A positive absorption mask $m(x)$ then gives
$$
C(u)=\int d(x)m(x-u)\,dx.
$$
If $d(x)=m(x-v_r)$, the maximum occurs at $u=v_r$; a convention correlating the received <observed radiative flux> instead produces a dip. A fitted correlation centre is an effective collective line position, not a claim that each individual <spectral line> has been resolved into a narrower feature. The lamp tracks <instrumental drift>, while the <barycentric velocity correction> removes the observer's changing line-of-sight <velocity>; neither removes changes in the <star> itself.
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