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Context. The 51 Eridani system harbors a complex architecture with itsprimary star forming a hierarchical system with the binary GJ 3305AB at aprojected separation of 2000 au, a giant planet orbiting the primary star at 13au, and a low-mass debris disk around the primary star with possibly a coldcomponent and a warm component inferred from the spectral energy distribution.Aims. We aim to better constrain the orbital parameters of the known giantplanet. Methods. We monitored the system over three years from 2015 to 2018with the VLT/SPHERE exoplanet imaging instrument. Results. We measure anorbital motion for the planet of ~130 mas with a slightly decreasing separation(~10 mas) and find a hint of curvature. This potential curvature is furthersupported at 3$\sigma$ significance when including literature GPI astrometrycorrected for calibration systematics. Fits of the SPHERE and GPI data usingthree complementary approaches provide broadly similar results. The datasuggest an orbital period of 32$^{+17}_{-9}$ yr (i.e. 12$^{+4}_{-2}$ au insemi-major axis), an inclination of 133$^{+14}_{-7}$ deg, an eccentricity of0.45$^{+0.10}_{-0.15}$, and an argument of periastron passage of87$^{+34}_{-30}$ deg [mod 180 deg]. The time at periastron passage and thelongitude of node exhibit bimodal distributions because we do not detect yet ifthe planet is accelerating or decelerating along its orbit. Given theinclinations of the planet's orbit and of the stellar rotation axis (134-144deg), we infer alignment or misalignment within 18 deg for the star-planetspin-orbit. Further astrometric monitoring in the next 3-4 years is required toconfirm at a higher significance the curvature in the planet's motion,determine if the planet is accelerating or decelerating on its orbit, andfurther constrain its orbital parameters and the star-planet spin-orbit.

Hint of curvature in the orbital motion of the exoplanet 51 Eridani b using 3 years of VLT/SPHERE monitoring