Dashed light blue line: true oblateness (ɛ) of Varda as afunction of its density (ρ), as derived from our occultation results and knowledge of the mass of Varda (see Sect. 5.2.2 for details). The lowest point corresponds to the equator-on geometry (B = 0) for Varda, while the uppermost point (outside the figure) corresponds to B = arcsin(1 − ɛ′) ~ 69° for ɛ = 1. The continuous blue lines to the left and right of the ɛ(ρ) curve delimit the 1σ uncertainty domain for the density. This domain stems from the uncertainties on M and a′ (Eq. (3)). The MacLaurin solutions (Eq. (4)) corresponding to the rotational periods 4.76, 5.91, and 7.87 h (Thirouin et al. 2014) are shown as solid purple (solution 1), green (solution 2), and red (solution 3) curves, respectively, using the Varda mass from Sect. 5.2.2. The red circles along the curve ɛ(ρ) indicate some planeto-centric sub-observer latitudes B. The shaded region corresponds to one of the possible Ilmarë’s orbit opening angle at epoch, B_open = −59 . ° . A solution compatible with an equatorial orbit for Ilmarë would be close to solution 2 and encompass solution 1 (shaded area). For each MacLaurin solution, we plot the corresponding solutions with the double period as dashed lines of the same colours, i.e. assuming double-peaked rotational light-curves (see Sect. 5.2.3 for details).