An advanced multipole model of the (130) Elektra quadruple system

Context. The Ch-type asteroid (130) Elektra is orbited by three moons, making it the first quadruple system in the main asteroid belt.Aims. We aim to characterise the irregular shape of Elektra and construct a complete orbital model of its unique moon system.Methods. We applied the All-Data Asteroid Modelling (ADAM) algorithm to 60 light curves of Elektra, including our new measurements, 46 adaptive-optics (AO) images obtained by the VLT/SPHERE and Keck/Nirc2 instruments, and two stellar occultation profiles. For the orbital model, we used an advanced N-body integrator, which includes a multipole expansion of the central body (with terms up to the order l = 6), mutual perturbations, internal tides, and the external tide of the Sun acting on the orbits. We fitted the astrometry measured with respect to the central body and also relatively, with respect to the moons themselves.Results. We obtained a revised shape model of Elektra with the volume-equivalent diameter (201 +/- 2) km. Of two possible pole solutions, (lambda,beta) = (189; -88) deg is preferred, because the other one leads to an incorrect orbital evolution of the moons. We also identified the true orbital period of the third moon S/2014 (130) 2 as P2 = (1.642112 +/- 0.000400) days, which is in between the other periods, P-1 similar or equal to 1.212days, P-3 = 5.300 days, of S/2014 (130) 1 and S/2003 (130) 1, respectively. The resulting mass of Elektra, (6.606(-0.013)(+0.007)) x10(18) kg, is precisely constrained by all three orbits. Its bulk density is then (1.536 +/- 0.038) g cm(-3). The expansion with the assumption of homogeneous interior leads to the oblateness J2 = -C-20 similar or equal to 0.16. However, the best-fit precession rates indicate a slightly higher value, similar or equal to;0.18. The number of nodal precession cycles over the observation time span 2014-2019 is 14, 7, and 0.5 for the inner, middle, and outer orbits.Conclusions. Future astrometric or interferometric observations of Elektra's moons should constrain these precession rates even more precisely, allowing the identification of possible inhomogeneities in primitive asteroids.