Formula for the Chandler Period (Free Wobble of Planetary Bodies)

If the rotational equilibrium of a planetary body is disturbed, the rotation pole responds with a cyclical motion. The duration of one cycle is referred to as the Chandler period, and, when viewed from space, the body wobbles. Because planets are not rigid, the wobble period differs from the Euler period by the factor 1-kX/kf $\left(1-{k}_{\mathrm{X}}/{k}_{\mathrm{f}}\right)$, where kX/kf ${k}_{\mathrm{X}}/{k}_{\mathrm{f}}$ is a ratio of two Love numbers. Here, we perform numerical simulations in which viscoelastic deformation of the planet and the Liouville equation hence polar motion are self-consistently coupled. We show that kX ${k}_{\mathrm{X}}$ is not the Love number at the frequency of the Chandler wobble itself, as is commonly assumed, but rather that it is close to ke ${k}_{\mathrm{e}}$, the elastic Love number. This result is important when the Chandler periods of Earth and Mars are interpreted, because the measured frequency is related to the internal rheological structure in a different way than previously thought.