Impact of the equation of state on the vanishing of vertical precession frequency. Left: Profiles of the vertical precession frequency of the slender torus for a particular choice of space-time parameter values, along with the related NS radii calculated using the L NS EoS (Pandharipande & Smith 1975) and denoted by dashed vertical lines. The top (bottom) panel corresponds to a rotational frequency of 400 Hz (500 Hz). Right: Range of the Hartle-Thorne space-time parameters in which the vertical precession can change orientation. The thick black line marked β = 0 indicates the limiting NS parameters relevant for the case of a slender torus, which are the same as for the geodesic (thin-disc) case explored by Tsang & Pappas (2016). The shaded region below the β = 0 line is the area in which this effect can occur, but the corresponding torus overlaps with the NS surface. The coloured region above the β = 0 line corresponds to tori that do not overlap with the star, and the colour scale indicates the radii of their centres for the slender-torus case. In full analogy, the lines for β = 0.1 and β = 0.2 show the limiting parameters where the torus of the corresponding thickness would touch the NS surface. In these cases, the corresponding r₀ colour maps would be slightly different but still almost identical to the β = 0 map shown in this figure.