Polar wander on tidally deformed rotating bodies – sensitivity study for various load cases

Polar wander is the reorientation of a body following a perturbation in the moment of inertia due to a mass anomaly. It has occurred on Mars as a result of the formation of the Tharsis region (e.g. Matsuyama and Manga 2010), on Pluto due to volatile ice deposition (Keane et al. 2016) and has also been proposed for icy moons such as Europa (Schenk et al. 2020). Pluto and icy moons have a tidal bulge in addition to being flattened due to rotation. In that case the polar wander path is more complicated and involves both a reorientation around the tidal axis (pointing to the central body) and one around the rotational axis. The path and speed of the anomaly depend on the size and location of the load and the mantle viscosity, with the end location determined by the strength of the outer shell with very high viscosity. Thus, if there are observational clues about polar wander, different scenarios in terms of size and timing of load, and internal structure can be tested. Most studies assume approximate solutions or focus on the end points of polar wander. Sometimes small angle polar wander is assumed, such as for polar wander due to Pleistonene deglaciation for Earth. In that case the linearized form of the Liouville equation is used where it is assumed that the rotational axis is the z-axis and the equatorial bulge is perpendicular to that. Here we present sensitivity studies of the complete  path for large angle polar wander to changes in loading.

We use a semi-analytical solution for reorientation of a rotating tidally deformed body with a high-viscosity visco-elastic shell (Hu et al. 2019) valid for large polar wander (>10 degrees). The method does not assume that the body is fully relaxed at any moment of the reorientation (the so-called fluid limit solution). Therefore, it works for faster loads (e.g. due to impacts) and can provide the complete reorientation path. Results of this method have recently been reproduced by Patočka and Kihoulou (2023) using a simpler approach.

We use Triton as case study. Polar wander on Triton can occur because of volatile migration (Rubincam 2003). Here we assume a point load. For Triton we use a 5-layer model as given in Hu et al. (2019) with a mantle viscosity of 10¹⁹ Pa s, overlain by ice shells with a viscosity of 10²¹ Pa s.

Figure 1: The movement of a mass anomaly emplaced on the Triton model. The rotational axis is at the origin pointing out of plane, the tidal axis pointing to the central body is at 0° longitude, pointing to the right. The starting position of the mass anomaly is at 15 ° colatitude and 15 ° longitude. Circles: Heaviside load for a point mass of 1.5×10¹⁷ kg, open squares: Heaviside load for a point mass of 3×10¹⁷ kg which roughly corresponds to a 100m thick disc of ice of 100 km radius. Crosses: load increasing linearly in time over 15 Ma to a magnitude of 3×10¹⁷ kg.

We apply a point mass with a magnitude of 3×10¹⁷ kg which roughly corresponds to a 100 m thick disc of ice of 100 km radius. The load is applied as instantaneous (Heaviside) forcing or linearly increasing (ramp). The path of a mass anomaly in the so-called bulge-fixed reference frame is shown in figure 1. For a larger mass the anomaly ends up closer to the tidal axis, as is expected. The ramp load leads to a path that is not very different from that of the Heaviside load. However, if the time over which the load increases is increased, the path is closer to that of the smaller Heaviside load (not shown). Thus, the path of the mass anomaly is an important constraint on the magnitude and timing of the load.

In future work we aim to study Pluto. The Sputnik Planitia region is the remnant of an impact. This by itself would cause a negative mass anomaly which would induce polar wander such that it moves towards the north pole. However, if the impact basin that was compensated by uplift in the subsurface ocean (Keane et al. 2016) possible compounded with the buried remnant of the impactor (Ballantyne et al 2024) it would shift towards the sub-Charon point. We will investigate various initial positions and thickness of a nitrogen ice layer and study possible end locations.

References

Ballantyne, H.A., Asphaug, E., Denton, C.A., Emsenhuber, A. and Jutzi, M., 2024. Sputnik Planitia as an impactor remnant indicative of an ancient rocky mascon in an oceanless Pluto. Nature Astronomy, pp.1-8.

Hu, H.X.S., van der Wal, W. and Vermeersen, L.L.A., 2019. Rotational dynamics of tidally deformed planetary bodies and validity of fluid limit and quasi-fluid approximation. Icarus, 321, pp.583-592.

Keane, J.T., Matsuyama, I., Kamata, S. and Steckloff, J.K., 2016. Reorientation and faulting of Pluto due to volatile loading within Sputnik Planitia. Nature, 540(7631), pp.90-93.

Matsuyama, I. and Manga, M., 2010. Mars without the equilibrium rotational figure, Tharsis, and the remnant rotational figure. Journal of Geophysical Research: Planets, 115(E12).

Patočka, V. and Kihoulou, M., 2023. Dynamic reorientation of tidally locked bodies: Application to Pluto. Earth and Planetary Science Letters, 617, p.118270.

Rubincam, D.P., 2003. Polar wander on Triton and Pluto due to volatile migration. Icarus, 163(2), pp.469-478.

Schenk, P., Matsuyama, I. and Nimmo, F., 2020. A very young age for true polar wander on Europa from related fracturing. Geophysical Research Letters, 47(17), p.e2020GL088