Tag Archives: Rotation

Tilting and retilting our Moon

Hi there! The Moon is so close and so familiar to us, but I realize this is my first post on it. Today I present you a paper entitled South Pole Aitken Basin magnetic anomalies: Evidence for the true polar wander of Moon and a lunar dynamo reversal, by Jafar Arkani-Hamed and Daniel Boutin, which will be published soon in Journal of Geophysical Research: Planets. The idea is to track the variations of the magnetic field of the Moon along its history, as a signature of the motion of its rotation pole, i.e. of a polar wander.

Our Moon’s facts

The Moon is a fascinating object, as it is the only known natural satellite of the Earth, and we see it as large as the Sun in our sky. It orbits around the Earth at a distance of almost 400,000 km in 27.3 days. It shows us always the same face, as a result of a tidal locking of its rotation, making it synchronous, i.e. its spin period is equal to its orbital period.

Moonset over Paris, France. Copyright: Josselin Desmars.

Something interesting is its pretty large size, i.e. its radius is one fourth of the one of the Earth. It is widely admitted that the Moon and the Earth have a common origin, i.e. either a proto-Earth has been impacted by a Mars-sized impactor, which split it between the Earth and the Moon, or the Earth-Moon system results from the collision of two objects of almost the same size. In both cases, the Earth and the Moon would have been pretty hot just after the impact, which also means active… and this has implications for the magnetic field.

A very weak magnetic field has been detected for the Moon, but which is very different from the Earth’s. The magnetic field of the Earth, or geomagnetic field, has the signature of a dipolar one, in the sense that it has a clear orientation. This happens when the rotating core acts as a dynamo. The north magnetic pole is some 10° shifted from the spin pole of the Earth, and has an amplitude between 25 and 65 μT (micro-teslas). However, the magnetic field of the Moon, measured at its surface, does not present a clear orientation, and never reaches 1 μT. Its origin is thus not obvious, even if we could imagine that the early Moon was active enough to harbor a dynamo, from which the measured magnetic field would be a signature… But the absence of preferred orientation is confusing.

The core dynamo

The core of the Earth spins, it is surrounded by liquid iron, which is conductive, and there is convection in this fluid layer, which is driven by the heat flux diffusing from the core to the surface of the Earth. This process creates and maintains a magnetic field.

For the Earth, the core dynamo is assumed to account for 80 to 90% of the total magnetic field. This results in a preferred orientation. Other processes that could create a magnetic field are a global asymmetry of the electric charges of the planet, or the presence of an external magnetic field, for instance due to a star.

A dynamo could be expected for many planetary objects, which would be large enough to harbor a global fluid layer. It is usually thought that the detection of a magnetic field is a clue for the presence of a global ocean. Such a magnetic field has been detected for Jupiter’s moon Ganymede, which is probably due to an outer liquid layer coating its iron core.

The Moon has probably no dynamo, but could have had one in the past. The measured magnetic field could be its signature. A question is: what could have driven this dynamo? The early Moon was hotter than the current one, so a magnetic field existed at that time. And after that, the Moon experienced intense episodes of bombardment, like the Late Heavy Bombardment. The resulting impacts affected the orientation of the Moon, its shape, and also its temperature. This could have itself triggered a revival of the magnetic field, particularly for the biggest impact.

The study I present today deals with measurements of the magnetic field in the South Pole-Aitken Basin, not to be confused with the Aitken crater, which is present in its region. The South Pole-Aitken Basin is one of the largest known impact crater in the Solar System, with a diameter of 2,500 km and a depth of 13 km. This basin contains other craters, which means that it is older than all of them, its age is estimated to be 4.1 Gyr (gigayears, i.e. billions of years). Measurements of the magnetic field in each of these craters could give its evolution over the ages. But why is it possible?

The magnetic field as a signature of the history

When a material is surrounded by a magnetic field, it can become magnetic itself. This phenomenon is known as induced magnetization, and depends on the magnetic susceptibility of the material, i.e. the efficiency of this process depends on the material. Once the surrounding magnetic field has disappeared, the material might remain magnetic anyway, i.e. have its remanent magnetic field. This is what has been measured by the Lunar Prospector mission, whose data originated this study.
An issue is the temperature. The impact should be hot enough to trigger the magnetic field, which implies that the material would be hot, but it cannot be magnetized if it is too hot. Below a Curie temperature, the process of induced magnetization just does not work. You can even demagnetize a material in heating it. For the magnetite, which is a mineral containing iron and present on the Moon, the Curie temperature is 860 K, i.e. 587°C, or 1089°F.

Lunar Prospector

This study uses data of the Lunar Prospector mission. This NASA mission has been launched in January 1998 from Cape Canaveral and has orbited the Moon on a polar orbit during 18 months, until July 1999. It made a full orbit in a little less than 2 hours, at a mean altitude of 100 km (60 miles). This allowed to cover the whole surface of the Moon, and to make measurements with 6 instruments, related to gamma rays, electrons, neutrons, gravity… and the magnetic field.

Results of this study

This study essentially consists of two parts: a theoretical study of the temperature evolution of the Moon over its early ages, including after impacts, and the interpretation of the magnetic field data. These data are 14 magnetic anomalies in the South Pole-Aitken Basin, which the theoretical study helps to date. And the data show two orientations of the magnetic field in the magnetic in the past, giving an excursion of more than 100° over the ages.

Now, if we consider that in the presence of a core dynamo, the magnetic field should be nearly aligned with the spin pole, this means that the Moon has experienced a polar wander of more than 100° in its early life. More precisely, the two orientations are temporally separated by the creation of the Imbrium basin, 3.9 Gyr ago. In other words, the Moon has been tilted. This is not the only case in the Solar System, see e.g. Enceladus.

To know more

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

An asteroid pair

Hi there! Today I present you the study of an asteroid pair. Not a binary, a pair. A binary asteroid is a couple of asteroids which are gravitationally bound, while in pair, the asteroids are just neighbors, they do not live together… but have. The study is entitled Detailed analysis of the asteroid pair (6070) Rheinland and (54287) 2001 NQ8, by Vokrouhlický et al., and it has recently been published in The Astronomical Journal.

Asteroid pairs

I have presented asteroid families in a previous post. These are groups of asteroids which present common dynamical and physical properties. They can be in particular identified from the clustering of their proper elements, i.e. you express their orbital elements (semimajor axis, eccentricity, inclination, pericentre, …), you treat them properly so as to get rid off the gravitational disturbance of the planets, and you see that some of these bodies tend to group. This suggests that they constitute a collisional family, i.e. they were a unique body in the past, which has been destroyed by collisions.
An asteroid pair is something slightly different, since these are two bodies which present dynamical similarities in their osculating elements, i.e. before denoising them from the gravitational attraction of the planets. Of course, they would present similarities in their proper elements as well, but the fact that similarities can be detected in the osculating elements means that they are even closer than a family, i.e. the separation occurred later. Families younger than 1 Myr (1 million of years) are considered to be very young; the pair I present you today is much younger than that. How much? You have to read me before.
A pair suggests that only two bodies are involved. This suggests a non-collisional origin, more particularly an asteroid fission.

Asteroid fission

Imagine an asteroid with a very fast rotation. A rotation so fast that it would split the asteroid. We would then have two components, which would be gravitationally bound, and evolving… Depending on the energy involved, it could remain a stable binary asteroid, a secondary fission might occur, the two or three components may migrate away from each other… and in that case we would pair asteroid with very close elements of their heliocentric orbits.
It is thought that the YORP (Yarkovsky – O’Keefe – Radzievskii – Paddack) could trigger this rotational fission. This is a thermic effect which alter the rotation, and in some cases, in particular when the satellite has an irregular shape, it could accelerate it. Until fission.
Thermic effects are particularly efficient when the Sun is close, which means that NEA (Near Earth Asteroids) are more likely to be destroyed by this process than Main Belt asteroids. Here, we deal with Main Belt asteroids.

The pair 6070-54827 (Rheinland – 2001 NQ8)

The following table present properties of Rheinland and 2001 NQ8. The orbital elements are at Epoch 2458000.5, i.e. September 4th 2017. They come from the JPL Small-Body Database Browser.

(6070) Rheinland (54827) 2001 NQ8
Semimajor axis (AU) 2.3874015732216 2.387149297807496
Eccentricity 0.2114524962733347 0.211262507795103
Inclination 3.129675305535938° 3.128927421642917°
Node 83.94746016534368° 83.97704257098502°
Pericentre 292.7043398319871° 292.4915004062336°
Orbital period 1347.369277588708 d (3.69 y) 1347.155719572348 d (3.69 y)
Magnitude 13.8 15.5
Discovery 1991 2001

Beside their magnitudes, i.e. Rheinland is much brighter than 2001 NQ8, this is why it was discovered 10 years earlier, we can see that all the slow orbital elements (i.e. all of them, except the longitude) are very close, which strongly suggests they shared the same orbit. Not only their orbits have the same shape, but they also have the same orientation.

Shapes and rotations from lightcurves

A useful tool for determining the rotation and shape of an asteroid is the lightcurve. The object reflects the incident Solar light, and the way it reflects it will tell us something on its location, its shape, and its orientation. You can imagine that the surfaces of these bodies are not exclusively composed of smooth terrain, and irregularities (impact basins, mountains,…) will result in a different Solar flux, which also depends on the phase, i.e. the angle between the normale of the surface and the asteroid – Sun direction… i.e. depends whether you see the Sun at the zenith or close to the horizon. This is why recording the light from the asteroid at different dates tell us something. You can see below an example of lightcurve for 2001 NQ8.

Example of lightcurve for 2001 NQ8, observed by Vokrouhlický et al.

Recording such a lightcurve is not an easy task, since the photometric measurements should be denoised, otherwise you cannot compare them and interpret the lightcurve. You have to compensate for the variations of the luminosity of the sky during the observation (how far is the Moon?), of the thickness of the atmosphere (are we close to the horizon?), of the heterogeneity of the CCD sensors (you can compensate that in measuring the response of a uniform surface). And the weather should be good enough.

Once you have done that, you get a lightcurve alike the one above. We can see 3 maxima and 2 minima. Then the whole set of lightcurves is put into a computational machinery which will give you the parameters that best match the observations, i.e. periods of rotation, orientation of the spin pole at a given date, and shape… or at least a diameter. In this study, the authors already had the informations for Rheinland but confirmed them with new observations, and produced the diameter and rotation parameters for 2001 NQ8. And here are the results:

Spin pole(124°,-87°)(72°,-49°) or (242°,-46°)

(6070) Rheinland (54827) 2001 NQ8
Diameter (km) 4.4 ± 0.6 2.2 ± 0.3
Spin period (h) 4.2737137 ± 0.0000005 5.877186 ± 0.000002

We can see rapid rotation periods, as it is often the case for asteroids. The locations of the poles mean that their rotations
are retrograde, with respect to their orbital motions. Moreover, two solutions best match the pole of 2001 NQ8.

Dating the fission

The other aspect of this study is a numerical simulation of the orbital motion of these two objects, backward in time, to date their separation. Actually, the authors considered 5,000 clones of each of the two objects, to make their results statistically relevant.
They not only considered the gravitational interactions with other objects of the Solar System, but also the Yarkovsky effect, i.e. a thermal pull due to the Sun, which depends on the reflectivity of the asteroids, and favors their separation. For that, they propose new equations implementing this effect. They also simulated the variations of the spin pole orientation, since it affects the thermal acceleration.

And here is the result: the fission probably occurred 16,340 ± 40 years ago.


Why doing that? Because what we see is the outcome of an asteroid fission, which occurred recently. The authors honestly admit that this result could be refined in the future, depending on

  • Possible future measurements of the Yarkovsky acceleration of one or two of these bodies,
  • The consideration of the mutual interactions between Rheinland and 2001 NQ8,
  • Refinements of the presented measurements,
  • Discovery of a third member?

To date the fission, they dated a close approach between these two bodies. They also investigated the possibility that that
close approach, some 16,000 years from now, could have not been the right one, and that the fission could have been much older. For that, they ran long-term simulations, which suggest that older close approaches should have been less close: if the pair were older, Yarkovsky would have separated it more.

To know more

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

Enceladus lost its balance

Hi there! Today I will present you True polar wander of Enceladus from topographic data, by Tajeddine et al., which has recently been published in Icarus. The idea is this: Enceladus is a satellite of Saturn which has a pretty stable rotation axis. In the past, its rotation axis was already stable, but with a dramatically different orientation, i.e. 55° shifted from the present one! The authors proposed this scenario after having observed the distribution of impact basins at its surface.

Enceladus’s facts

Enceladus is one of the mid-sized satellites of Saturn, it is actually the second innermost of them. It has a mean radius of some 250 km, and orbits around Saturn in 1.37 day, at a distance of ~238,000 km. It is particularly interesting since it presents evidence of past and present geophysical activity. In particular, geysers have been observed by the Cassini spacecraft at its South Pole, and its southern hemisphere presents four pretty linear features known as tiger stripes, which are fractures.

Enceladus seen by Cassini (Credit: NASA / JPL / Space Science Institute).
Enceladus seen by Cassini (Credit: NASA / JPL / Space Science Institute).

Moreover, analyses of the gravity field of Enceladus, which is a signature of its interior, strongly suggest a global, subsurfacic ocean, and a North-South asymmetry. This asymmetry is consistent with a diapir of water at its South Pole, which would be the origin of the geysers. The presence of the global ocean has been confirmed by measurements of the amplitude of the longitudinal librations of its surface, which are consistent with a a crust, that a global ocean would have partially decoupled from the interior.

The rotation of a planetary satellite

Planetary satellites have a particularly interesting rotational dynamics. Alike our Moon, they show on average always the same face to a fictitious observer, which would observe the satellite from the surface of the parent planet (our Earth for the Moon, Saturn for Enceladus). This means that they have a synchronous rotation, i.e. a rotation which is synchronous with their orbit, but also that the orientation of their spin axis is pretty stable.
And this is the key point here: the spin axis is pretty orthogonal to the orbit (this orientation is called Cassini State 1), and it is very close to the polar axis, which is the axis of largest moment of inertia. This means that we have a condition on the orientation of the spin axis with respect to the orbit, AND with respect to the surface. The mass distribution in the satellite is not exactly spherical, actually masses tend to accumulate in the equatorial plane, more particularly in the satellite-planet direction, because of the combined actions of the rotation of the satellites and the tides raised by the parent planet. This implies a shorter polar axis. And the study I present today proposes that the polar axis has been tilted of 55° in the past. This tilt is called polar wander. This result is suggested by the distribution of the craters at the surface of Enceladus.

Relaxing a crater

The Solar System bodies are always impacted, this was especially true during the early ages of the Solar System. And the inner satellites of Saturn were more impacted than the outer ones, because the mass of Saturn tends to attract the impactors, focusing their trajectories.
As a consequence, Enceladus got heavily impacted, probably pretty homogeneously, i.e. craters were everywhere. And then, over the ages, the crust slowly went back to its original shape, relaxing the craters. The craters became then basins, and eventually some of them disappeared. Some of them, but not all of them.
The process of relaxation is all the more efficient when the material is hot. For material which properties strongly depend on the temperature, a stagnant lid can form below the surface, which would partly preserve it from the heating by convection, and could preserve the craters. This phenomenon appears preferably at equatorial latitudes.
This motivates the quest for basins. A way for that is to measure the topography of the surface.

Modeling the topography

The surface of planetary body can be written as a sum of trigonometric series, known as spherical harmonics, in which the radius would depend on 2 parameters, i.e. the latitude and the longitude. This way, you have the radius at any point of the surface. Classically, two terms are kept, which allow to represent the surface as a triaxial ellipsoid. This is the expected shape from the rotational and tidal deformations. If you want to look at mass anomalies, then you have to go further in the expansion of the formula. But to do that, you need data, i.e. measurements of the radius at given coordinates. And for that, the planetologists dispose of the Cassini spacecraft, which made several flybys of Enceladus, since 2005.
Two kinds of data have been used in this study: limb profiles, and control points.
Limb profiles are observations of the bright edge of an illuminated object, they result in very accurate measurements of limited areas. Control points are features on the surface, detected from images. They can be anywhere of the surface, and permit a global coverage. In this study, the authors used 41,780 points derived from 54 limb profiles, and 6,245 control points.
Measuring the shape is only one example of use of such data. They can also be used to measure the rotation of the body, in comparing several orientations of given features at different dates.
These data permitted the authors to model the topography up to the order 16.

The result

The authors identified a set of pretty aligned basins, which would happen for equatorial basins protected from relaxation by stagnant lid convection. But the problem is this: the orientation of this alignment would need a tilt of 55° of Enceladus to be equatorial! This is why the authors suggest that Enceladus has been tilted in the past.

The observations do not tell us anything on the cause of this tilt. Some blogs emphasize that it could be due to an impact. Why not? But less us be cautious.
Anyway, the orientation of the rotation axis is consistent with the current mass distribution, i.e. the polar axis has the largest moment of inertia. Actually, mid-sized planetary satellites like Enceladus are close to sphericity, in the sense that there is no huge difference between the moments of inertia of its principal axes. So, a redistribution of mass after a violent tilt seems to be possible.

To know more

And now the authors:

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

Mathematics of the spin-orbit resonance

Hi there! Today things are a little bit different. The paper I present you is not published in a journal of astronomy, nor of planetary sciences, but of mathematics. It is entitled Hamiltonian formulation of the spin-orbit model with time-varying non-conservative forces, by Ioannis Gkolias, Christos Efthymiopoulos, Giuseppe Pucacco and Alessandra Celletti, and it has been recently published in Communications in Nonlinear Science and Numerical Simulation. It deals with a mathematical way to express and solve the spin-orbit problem. This mathematical way is the Hamiltonian formulation.

The spin-orbit problem

It is something I already discussed on this blog, but never mind. Imagine you have a triaxial body orbiting a largest one… e.g. the Moon orbiting the Earth… or a satellite orbiting a giant planet. Usually the satellite always show the same face to the planet, which is a consequence of a synchronous rotation, which you can call 1:1 spin-orbit resonance. It can be shown that this synchronous resonance is a dynamical equilibrium, i.e. the fact that the angular momentum of the satellite is almost orthogonal to its orbit, and the long axis always points to the parent planet, is a stable position. This is makes the synchronous rotation ubiquitous in the Solar System. Initially the satellite had some rotation, which could have had any spin and orientation. And then, the dissipations of energy, mostly tides raised by the planet, have damped the rotation until reaching the synchronous rotation. At this point, the energy given by the gravitational torque of the planet is large enough to compensate the tides. Since it is a stable equilibrium, then the system stays there, i.e. the rotation remains synchronous.

Hamiltonian formulation

Let us start from conservative mechanics, i.e. in the absence of dissipation. Neglecting the dissipation might be a priori surprising, but this approximation is used since centuries. In planetary systems, dissipation can be easily seen from geysers, volcanoes…, but its effects on the orbital and rotational dynamics are very small, and hence difficult to measure. Lunar Laser Ranging have shown us that the Earth-Moon distance is increasing by some 3.9 cm / yr, as a consequence of the dissipation. We have measurements of such an effect in the system of Jupiter since 2009, and in the system of Saturn since 2011. Moreover, if we assume that the equilibrium has been reached, then we can consider that the loss of energy is compensated by the energy exchanges between the parent planet and the satellite. This is why neglecting the dissipation is sometimes allowed… even if the paper I present you does not neglect it.

So, in conservative mechanics, the total energy of the system is conserved. The total energy of the system is the sum of the kinetic and potential energies of all of the bodies involved. This total energy depends on the variables of the system, i.e. the orbital and rotational variables. It can be shown that convenient sets of variables exist, i.e. canonical variables, which time derivatives are the partial derivatives of the total energy, written with this set of variables, which respect to their conjugate variables. In that case, the formulation of the total energy is called Hamiltonian of the system, and the ensuing equations are the Hamilton equations.

The Hamiltonian formulation is very convenient from a mathematical point of view. Its properties make the dynamics of the system easier to interpret. For instance, in manipulating the Hamiltonian, you can determine its equilibrium, their stability, and the small oscillations (librations) around it. This mathematical structure can also be used to construct dedicated numerical integrators, called symplectic integrators, which solve the equations numerically. Symplectic integrators are reputed for their numerical stability.

Viscoelasticity and tides

Let us talk now on the dissipation. The main source of dissipation is the tides raised by the parent planet. Since its gravitational torque felt by the satellite is not homogeneous over its volume, as distance-dependent, then the satellite experiences stress and strains which alter its shape and induces energy loss. So, the tides have two consequences: loss of energy and variation of the shape. The paper proposes a way to consider these effects in a Hamiltonian formalism.

This paper

As the authors honestly admit, it is somehow inaccurate to speak of Hamiltonian formulation when you have dissipation. Their paper deals with the dissipative spin-orbit problem, so their “Hamiltonian” function is not an Hamiltonian strictly speaking, but the ensuing equations have a symplectic structure.

They assume that the dissipation is contained in a function F, which depends on the time t, and discuss the resolution of the problem with respect to the form of F: either a constant dissipation, or a quasi-periodic one, or the sum of a constant and a quasi-periodic one.

Of course, this paper is very technical, and I do not want to go too deep into the details. I would like to mention their treatment of the quasi-periodic case. Quasi-periodic means that the function F, i.e. the dissipation, can be written under a sum of sines and cosines, i.e. oscillations, of different frequencies. This is physically realistic, in the sense that the material constituting the satellite has a different response with respect to the excitation frequency, and the time evolution of the distance planet-satellite and a pretty wide spectrum itself.
In that case, the dissipation function F depends on the time, which is a problem. But it is classically by-passed in assuming the time to be a new variable of the problem, and in adding to the Hamiltonian a dummy conjugate variable. This is a way to transform a non-autonomous (time-dependent) Hamiltonian into an autonomous one, with an additional degree of freedom.
Once this is done, the resolution of the problem is made with a perturbative approach. It is assumed, which is physically realistic, that the amplitudes of the oscillations which constitute the F function are of different orders of magnitudes. This allows to classify them from the most important to the less important ones, with the help of a virtual book-keeping parameter λ. This is a small parameter, and the amplitude of the oscillations will be normalized by λq, q being an integer power. The largest is q, the smallest is the amplitude of the oscillations. The resolution process is iterative, and each iteration multiplies the accuracy by λ.

It is to be noted that such algorithms are usually written as formal processes, but their convergence is not guaranteed, because of potential resonances between the different involved frequencies. When two frequencies become too close to each other, the process might be destabilized. But usually, this does not happen before a reasonable order, i.e. before a reasonable number of iterations, and this is why such methods can be used. The authors provide numerical tests, which prove the robustness of their algorithm.

Potential applications

Such a study is timely, since dissipation can now be observed. For instance, the variations of the shapes of planetary bodies have been observed by measurements of variations of their gravity fields, which give the tidal Love number k2. k2 has been measured for Mercury, Venus, the Earth, the Moon, Mars, Saturn, and Titan, thanks to space missions. Moreover, its dissipative counterpart, i.e. k2/Q, has been measured for the Earth, Mars, Jupiter and Saturn. This means that conservations models for the spin-orbit problem are not sufficient anymore.

To know more

That’s all for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter.

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

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