Context : The 51 Eridani system harbors a complex architecture with its primary star forming a hierarchical system with the binary GJ 3305AB at a projected separation of 2000 au , a giant planet orbiting the primary star at 13 au , and a low-mass debris disk around the primary star with possible cold and warm components inferred from the spectral energy distribution .

Aims : We aim to better constrain the orbital parameters of the known giant planet .

Methods : We monitored the system over three years from 2015 to 2018 with the Spectro-Polarimetric High-contrast Exoplanet REsearch ( SPHERE ) instrument at the Very Large Telescope ( VLT ) .

Results : We measure an orbital motion for the planet of \sim 130 mas with a slightly decreasing separation ( \sim 10 mas ) and find a hint of curvature .
This potential curvature is further supported at 3 \sigma significance when including literature Gemini Planet Imager ( GPI ) astrometry corrected for calibration systematics .
Fits of the SPHERE and GPI data using three complementary approaches provide broadly similar results .
The data suggest an orbital period of 32 ^ { +17 } _ { -9 } yr ( i.e. , 12 ^ { +4 } _ { -2 } au in semi-major axis ) , an inclination of 133 ^ { +14 } _ { -7 } deg , an eccentricity of 0.45 ^ { +0.10 } _ { -0.15 } , and an argument of periastron passage of 87 ^ { +34 } _ { -30 } deg [ mod 180 ^ { \circ } ] .
The time at periastron passage and the longitude of node exhibit bimodal distributions because we do not yet detect whether the planet is accelerating or decelerating along its orbit .
Given the inclinations of the orbit and of the stellar rotation axis ( 134–144 ^ { \circ } ) , we infer alignment or misalignment within 18 ^ { \circ } for the star–planet spin-orbit .
Further astrometric monitoring in the next 3–4 years is required to confirm at a higher significance the curvature in the motion of the planet , determine if the planet is accelerating or decelerating on its orbit , and further constrain its orbital parameters and the star–planet spin-orbit .

Conclusions :