The ESA Euclid mission has been designed to map the geometry of the dark Universe. Scheduled for launch in 2020, it will conduct a six-years visible and NIR imaging and spectroscopic survey over 15,000 deg 2 down to mag~24.5. Although the survey will avoid low ecliptic latitudes, the survey pattern in repeated sequences of four broad-band filters seems well-adapted to Solar System objects (SSOs) detection and characterization. We aim at evaluating Euclid capability to discover SSOs, and measure their position, apparent magnitude, and SED. Also, we investigate how these measurements can lead to the determination of their orbits, morphology, physical properties, and surface composition. We use current census of SSOs to estimate the number of SSOs detectable by Euclid. Then we estimate how Euclid will constrain the SSOs dynamical, physical, and compositional properties. With current survey design, about 150,000 SSOs, mainly from the asteroid main-belt, should be observed by Euclid. These objects will all have high inclination. There is a potential for discovery of several 10,000 SSOs, in particular KBOs at high declination. Euclid observations will refine the spectral classification of SSOs by extending the spectral coverage provided by, e.g. Gaia and the LSST to 2 microns. The time-resolved photometry, combined with sparse photometry will contribute to the determination of SSO rotation period, spin orientation, and shape model. The sharp and stable point-spread function of Euclid will also allow to resolve KBO binary systems and detect activity around Centaurs. The depth of Euclid survey, its spectral coverage, and observation cadence has great potential for Solar System research. A dedicated processing for SSOs is being set in place to produce catalogs of astrometry, multi-color and time-resolved photometry, and spectral classification of some 10$^5$ SSOs, delivered as Legacy Science.
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