Reorienting a non-rigid body

Hi there! Today’s post deals with the following problem: Imagine you have a planetary body, in the Solar System, which orbits either the Sun or a massive planet. This body has its own rotation. And, for some reason, for instance a mass anomaly, its orientation changes dramatically. This is a pretty complex problem when the body is not rigid, i.e. its shape is not constant. This problem is addressed in A numerical method for reorientation of rotating tidally deformed visco-elastic bodies, by a Dutch team of the University of Delft, composed of Haiynag Hu, Wouter van der Wal, and Bert Vermeersen. This paper has recently been published in Journal of Geophysical Research: Planets.

Shaping a planetary body

The main difficulty of the problem comes from the fact that the involved body is not rigid, i.e. its shape might change.
Beside a catastrophic event like an impact, 2 physical effects are likely to shape a planetary body: its rotation, and the tides.
The deformation due to the rotation is easy to understand. Imagine a body which rotates about one axis. The centrifugal force will tend to repel the masses, especially at the equator, creating a symmetric polar flattening.
The tides are the differential gravitational attraction created by a massive object, on every mass element of the involved body. Not only that would result in a loss of energy because of the internal frictions created by the tides, but that would also alter its shape. If the body has a rotation rate which has no obvious connection with its orbital rate around its parent body, which would be the Sun for a planet, or a planet for a satellite, then the tidal deformation essentially results in an oscillatory, quasi-periodic variation of the shape. However, if the body has a rotation which is synchronous with its orbit, as it is the case for many planetary satellites (the Moon shows us always the same face), then the tides would raise a permanent equatorial bulge, pointing to the massive perturber. Consequently, the satellite would be triaxial.
When there is no remnant deformation, for instance due to a mass anomaly, then the shape of the satellite is rendered by the so-called hydrostatic equilibrium.
The intensity of the deformation is given by Love numbers, the h number being related to the shape, and the k number to the gravity field. The most commonly used is the second-order Love number k2, which is the lowest-order relevant Love number. It permits to render the triaxiality of a synchronous body.

All this means that, when a satellite or a planet undergoes a brutal reorientation, then its shape is altered. Modeling this transition is challenging.

The True Polar Wander in the Solar System

Several Solar System bodies are thought to have undergone Polar Wander in the past. The reason for that is, when a mass anomaly is created, for instance due to a collision, or because of the liquefaction of water ice in the body, then the shape of the body, i.e. its mass balance, does not match with its rotation and the undergone tides anymore. The natural response is then a reorientation, which is accompanied by reshaping, since the body is not rigid.

Clues of Polar Wander are present in the Solar System, such as

  • Enceladus presents a subsurface water diapir at its South Pole. Since this is an equilibrium configuration, the diapir has probably been created at another orientation, and then Enceladus was out of balance, and reoriented,
  • the orientation of Sputnik Planitia on Pluto, which is aligned with the direction Pluto-Charon, can result from reorientation, since Sputnik Planitia corresponds to a mass anomaly,
  • a past Polar Wander is suspected for Mars, from the presence of similar volatiles elements at the equator and at the poles, from the distribution of the impact basins, and from the magnetic field,
  • Polar Wander has been proposed to explain the retrograde rotation of Venus.

Modeling the dynamics of True Polar Wander for a visco-elastic body is a true challenge, one of the issues being: how do you model the evolution of the orientation and of the shape simultaneously?

Some approximations have been proposed in the past to answer this question:

  • the quasi-fluid approximation: the shape if the body is supposed to relax almost instantaneously, i.e. over a timescale, which is very fast with respect to the timescale of the reorientation,
  • the small angles approximation (linear true polar wander): the reorientation angle is assumed to be small enough, so that the equations ruling the rotation of the body can be linearized, which makes them much easier to solve. Of course, this does not work for large reorientation angles,
  • the equilibrium approximation: the idea is here to not try to simulate the process of True Polar Wander, but only its outcome. This would assume that the reorientation is now finished, and the shape is relaxed. But we cannot be sure that the bodies we observe are in this new equilibrium state.

The study I present here is the first paper of a series, which aims at going beyond these approximations, to criticize their validity, and to be more realistic on the evolution of the involved Solar System bodies. Before presenting its results, I will briefly present the Finite Elements Method (FEM).

Numerical computation with finite elements

In such a problem, you have to model both the orientation of the rotation axis of the body, which depends on the time, and the distribution of masses in the body, which are interconnected to each others and are ruled by the centrifugal and tidal forces. This would result in a time-dependent tensor of inertia. This is basically a 3×3 matrix, which contains all the information on the mass repartition.
For that, a common way is to split the body into finite elements, i.e. split its volume into small volume elements, and propagate the deformations from one to another. Proceeding this way is far from easy, since it is very time-consuming, and the accuracy is a true issue. It is tempting to reduce the size of the volume elements to improve the accuracy, which should work… until they are too small and generate too many numerical errors. Moreover, smaller elements means more elements, and a longer computation time… In this study, the authors borrow the finite elements solver from a commercial software.

This study

To test these approximations, the authors propose 3 algorithms:

  • Algorithm 1, suitable for small-angle polar wander without addressing its cause,
  • Algorithm 2, suitable for large-angle polar wander without addressing its cause,
  • Algorithm 3, which models the response to a mass anomaly.

Comparing the Algorithms 1 with 2 and 1 with 3 tests the limit of the small-angle approximation, while comparing 2 and 3 tests the validity of the quasi-fluid approximation. And here are the results:

  • the small angles approximations (linear theory) gives the worst results when the cause of the mass anomaly causing the reorientation is close to the equator or to one of the poles,
  • the quasi-fluid approximation is reliable only when the body is close to its final state, i.e. equilibrium rotation and relaxed shape.

More results are to be expected, since the authors announce to be working on the effects of lateral heterogeneity on True Polar Wander.

Some links

That’s all for today. Please feel free to comment, to follow the Planetary Mechanics Blog on Twitter (@planetmechanix), and to subscribe to the RSS feed.

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