= Doppler spectroscopy
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= Radial-velocity method
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<Doppler spectroscopy> detects an <exoplanet> through the host <star>'s changing <Doppler shift>. Both bodies orbit their shared <astronomical barycenter>: the <star> undergoes <stellar reflex motion> even when the <exoplanet> itself is unseen. Its <velocity> along the line of sight shifts its <spectral lines>; at speeds small compared with the <speed of light>, $\Delta\lambda/\lambda\simeq v_r/c$, with positive $v_r$ for recession. Repeated measurements can fit a <Kepler orbit>, rather than merely detect one shifted spectrum. <Starspots> and other changes in stellar <spectral lines> must be distinguished from actual orbital motion.
For one companion in an isolated <two-body problem>, the <stellar radial-velocity semi-amplitude> is
$$
K=\left(\frac{2\pi G}{P}\right)^{1/3}\frac{M_p\sin i}{(M_\star+M_p)^{2/3}}\frac{1}{\sqrt{1-e^2}}.
$$
Here $G$ is the <gravitational constant>, $P$ the <orbital period>, $e$ the <orbital eccentricity>, and $M_\star,M_p$ the <star> and <exoplanet> <masses>. The <orbital inclination> $i$ is measured relative to the sky plane: $i=0$ is face-on and $i=90^\circ$ is an <edge-on orbit>. The exact <binary mass function> measured from the fitted <orbit> is
$$
f=\frac{PK^3(1-e^2)^{3/2}}{2\pi G}=\frac{M_p^3\sin^3i}{(M_\star+M_p)^2}.
$$
With known $M_\star$ and $M_p\ll M_\star$, this is usually reported as $M_p\sin i$. Taking $\sin i=1$ gives the minimum companion <mass>; a smaller <orbital inclination> allows a larger <mass>. A detected <exoplanet transit> constrains $i$ close to $90^\circ$ for a compact <planet> orbiting well outside the <star>'s surface, substantially resolving this ambiguity. Combining <Doppler spectroscopy> and <exoplanet transit photometry> then supplies a <mass> and <radius> for <exoplanet interior> studies.
For a computed circular example, let $q=M_p/M_\star=0.001$ and choose a relative orbital <speed> $V_{\rm rel}=100\,\mathrm{km\,s^{-1}}$: this is the speed of the <planet> relative to the <star>, not its speed about the <astronomical barycenter>. <Conservation of momentum> gives
$$
V_\star=\frac{q}{1+q}V_{\rm rel}=99.9001\,\mathrm{m\,s^{-1}},\qquad V_p=\frac{1}{1+q}V_{\rm rel}=99.9001\,\mathrm{km\,s^{-1}}.
$$
The barycentric <velocities> point in opposite directions, so their <speeds> add to $V_{\rm rel}$. Here $e=0$ and $K=V_\star\sin i$: $K=99.9001\,\mathrm{m\,s^{-1}}$ at $i=90^\circ$, $49.9500\,\mathrm{m\,s^{-1}}$ at $i=30^\circ$, and zero at $i=0$. The <star> still orbits in the face-on case, but its orbital <velocity> has no line-of-sight component. The plotted circular signal is $v_r(t)-\gamma=K\sin(2\pi t/P)$ for a chosen phase origin and constant system <velocity> $\gamma$.
\Image[/stellar-reflex-radial-velocity.png]
{title=A planet makes its star orbit too: inclination changes the measured velocity amplitude}
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