Power spectra for the obliquity and inclination evolution (left) for Kepler-62 f, in the secular resonances in Figure 5. The left panel corresponds to point b, the right to point d. The vertical dotted lines are the natural axial precession frequencies from Equation (14) at the minimum and maximum obliquities, and the vertical dashed lines are the inclination eigenvalues from the Laplace–Lagrange solution. In the left panel, the natural precession rate falls nearly on top of the largest inclination frequency, s₄. In the right panel, the precession rate lies close to a secondary cluster of frequencies associated with s₅. Despite the fact that the argument ﹩\psi +{\rm{\Omega }}﹩ is not seen to librate in this case (Figure 8, right panels), the power spectrum reveals the secular resonance.