= Solution
\b[<Stellar reflex motion>.] A <planet> and its host orbit their common <barycentre>. The <star>'s line-of-sight velocity shifts its <absorption lines> by $\Delta\lambda/\lambda\simeq v_r/c$ at low speeds. A time series can be fitted with
$$
v_r(t)=\gamma+K[\cos(\nu(t)+\varpi)+e\cos\varpi],\qquad
K=\left(\frac{2\pi G}{P}\right)^{1/3}
\frac{M_p\sin i}{(M_\star+M_p)^{2/3}\sqrt{1-e^2}}.
$$
Here $\nu(t)$ is <true anomaly>, $\varpi$ the argument of periastron and $\gamma$ an arbitrary systemic velocity offset. At small companion mass and known stellar mass, the orbit supplies $M_p\sin i$, not the true <planet> mass without an inclination constraint. Long monitoring helps distinguish a reproducible Keplerian signal from stellar oscillations, magnetic activity, spots and instrumental trends.
\b[The Lick reference-cell implementation.] Starlight passes through a temperature-controlled cell of molecular <iodine> before entering the Hamilton high-resolution echelle <spectrograph>. The cell imprints a dense pattern of reference <absorption lines> on the stellar spectrum, predominantly in the approximately $500$–$630\ {\rm nm}$ interval. These lines share the same exposure and downstream optical path as the stellar light. A calibrated high-resolution Fourier-transform <iodine> spectrum and an iodine-free stellar template allow the <wavelength> solution and instrumental line-spread function to be fitted along with the stellar <Doppler shift>. The method corrects shifts and line-shape changes that a separate <calibration> exposure would not necessarily follow.<Precision Radial Velocities with an Iodine Absorption Cell by Geoffrey W. Marcy and R. Paul Butler>[Marcy and Butler (1992), sections 3–4, printed pages 271–274: Hamilton setup, iodine wavelength interval, templates and in-situ instrumental profile]{ref}
A useful forward model for one short spectral segment is
$$
\boxed{f_{\rm obs}(\lambda)=
\left\{S\!\left(\frac{\lambda}{1+v/c}\right)T_{\rm I_2}(\lambda)\right\}
*\mathcal P(\lambda),}
$$
followed by pixel integration, continuum scaling and a fitted <wavelength> <calibration>. Here $S$ is the intrinsic stellar spectrum, $T_{\rm I_2}$ the cell transmission and $\mathcal P$ the instrumental line-spread function. The spectra multiply before <convolution>; simply dividing a recorded blended spectrum by the laboratory <iodine> spectrum is not equivalent to this model. The <iodine> pattern is stationary in the observatory frame while stellar lines move relative to it. Separate spectral segments accommodate <wavelength> dependence of the instrumental profile.
\Image[/past-exam-of-the-frontiers-of-observational-astrophysics-course-of-the-university-of-cambridge/2015/paper-1-iodine-forward-model.png]
{title=Original synthetic iodine-cell forward model showing multiplication before instrumental convolution; all line positions and strengths are illustrative}
Earth's rotational and orbital velocities must then be removed with a barycentric correction using accurate timing, ideally weighted by <photon> arrival rate. Many stellar lines constrain a shift smaller than one instrumental resolution element; spectral resolution alone is not the velocity error. <Photon> statistics, template quality, instrumental-profile modelling and intrinsic stellar variability limit the precision. The method measures changes relative to a template and does not by itself provide a perfectly absolute centre-of-mass velocity. No particular discovery, modern precision floor or historical observing performance is assumed here.
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