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Radial Velocity Prospects Current and Future: A White Paper Report prepared by the Study Analysis Group 8 for the Exoplanet Program Analysis Group (ExoPAG)

[Abridged] The Study Analysis Group 8 of the NASA Exoplanet Analysis Group was convened to assess the current capabilities and the future potential of the precise radial velocity (PRV) method to advance the NASA goal to "search for planetary bodies and Earth-like planets in orbit around other stars.: (U.S. National Space Policy, June 28, 2010). PRVs complement other exoplanet detection methods, for example offering a direct path to obtaining the bulk density and thus the structure and composition of transiting exoplanets. Our analysis builds upon previous community input, including the ExoPlanet Community Report chapter on radial velocities in 2008, the 2010 Decadal Survey of Astronomy, the Penn State Precise Radial Velocities Workshop response to the Decadal Survey in 2010, and the NSF Portfolio Review in 2012. The radial-velocity detection of exoplanets is strongly endorsed by both the Astro 2010 Decadal Survey "New Worlds, New Horizons" and the NSF Portfolio Review, and the community has recommended robust investment in PRVs. The demands on telescope time for the above mission support, especially for systems of small planets, will exceed the number of nights available using instruments now in operation by a factor of at least several for TESS alone. Pushing down towards true Earth twins will require more photons (i.e. larger telescopes), more stable spectrographs than are currently available, better calibration, and better correction for stellar jitter. We outline four hypothetical situations for PRV work necessary to meet NASA mission exoplanet science objectives.