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[I have re-asked this, because it was a good question by @Banyan, which was deleted whilst I was composing an answer.]

Most exoplanets that are found by the radial velocity ("Doppler wobble") technique are quite massive - Jupiter/Saturn-like objects.

What are the principle challenges in extending this technique to finding Earth-like exoplanets?

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Assuming a circular orbit of a planet of mass $m_p$ around a star of mass $m_*$, with an orbital period of $P$ and we can assume that $m_* \gg m_p$ for the case of an Earth-like planet.

Newton's second law tells us that $$ m_p a \frac{4\pi^2}{P^2} = G\frac{m_* m_p}{a^2}\ , \tag*{(1)}$$ where $a$ is the orbital radius.

The observed radial velocity amplitude of the star will be $$ K \simeq \frac{2\pi a}{P} \left(\frac{m_p}{m_*}\right)\sin i\ , $$ where $i$ is the orbital inclination. Replacing $a$ using eqn. (1) $$ K \simeq \left( \frac{2\pi G}{Pm_*^2}\right)^{1/3} m_p \sin i\ .$$ Putting in some typical numbers $$ K = 0.09 \left(\frac{P}{1\ {\rm year}}\right)^{-1/3} \left( \frac{m_*}{M_\odot}\right)^{-2/3}\left(\frac{m_p}{M_{\rm Earth}}\right)\sin i\ {\rm m/s}\ . $$

This equation shows that to detect an Earth-like planet in an Earth-like orbit around a sun-like star we would need to detect a sinusoidal wiggle of amplitude 9 cm/s, with a period of 1 year. Things become more favourable if (i) $m_*$ is lower or (ii) $P$ is shorter.

Indeed Earth-sized (Earth-mass) planets have already been found around very low mass stars (e.g. a planet with $m_p \sin i = 1.35$ Earth masses in a 10-day orbit around the M-dwarf Ross 128; Bonfils et al. 2018).

The state of the art fibre-fed, temperature and pressure-controlled high resolution spectrograph at the moment is known as ESPRESSO at the VLT. It has a demonstrated precision capability of 25 cm/s and an instrument stability that is likely to be at the 10 cm/s level over several years (Pepe et al. 2020). ESPRESSO may be able to start picking up many Earth-mass planets, with orbital periods of up to 1 year around solar-type and less massive stars.

However, just achieving the necessary instrument stability is only part of the problem. The spectra of sun-like and lower-mass stars are not well behaved because of (i) convective motions in their photospheres and (ii) the presence of dark spots and bright plages that are inhomogeneously distributed over their surfaces. Both of these effects can caused velocity "jitter" that are an order of magnitude (or more) larger than 10 cm/s. The two approaches to deal with this are avoidance - where you try to choose targets that are old, quiet and magnetically inactive; or mitigation - where you try to work out what perturbations are being inflicted on your velocity measurements by looking for other signatures of the jitter like asymmetric spectral line profiles or by decorrelating with magnetic activity indicators (e.g. Wright 2017).

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