Category Archives: Theoretical celestial mechanics

Modeling the shape of a planetary body

Hi there! Do you know the shape of the Moon? You say yes of course! But up to which accuracy? The surface of the Moon has many irregularities, which prompted Christian Hirt and Michael Kuhn to study the limits of the mathematics, in modeling the shape of the Moon. Their study, entitled Convergence and divergence in spherical harmonic series of the gravitational field generated by high-resolution planetary topography — A case study for the Moon, has recently been accepted for publication in Journal of Geophysical Research: Planets.

The shape of planetary bodies

If you look at a planetary body from far away (look at a star, look at Jupiter,…), you just see a point mass. If you get closer, you would see a sphere, if the body is not too small. Small bodies, let us say smaller than 100 km, can have any shape (may I call them potatoids?) If they are larger, the material almost arranges as a sphere, which gives the same gravity field as the point mass, provided you are out of the body. But if you look closer, you would see some polar flattening, due to the rotation of the body. And for planetary satellites, you also have an equatorial ellipticity, the longest axis pointing to the parent planet. Well, in that case, you have a triaxial ellipsoid. You can say that the sphere is a degree 0 approximation of the shape, and that the triaxial ellipsoid is a degree 2 approximation… but still an approximation.

A planetary body has some relief, mountains, basins… there are explanations for that, you can have, or have had, tectonic activity, basins may have been created by impacts, you can have mass anomalies in the interior, etc. This means that the planetary body you consider (in our example, the Moon), is not exactly a triaxial ellipsoid. Being more accurate than that becomes complicated. A way to do it is with successive approximations, in the same way I presented you: first a sphere, then a triaxial ellipsoid, then something else… but when do you stop? And can you stop, i.e. does your approximation converge? This study addresses this problem.

The Brillouin sphere

This problem is pretty easy when you are far enough from the body. You just see it as a sphere, or an ellipsoid, since you do not have enough resolution to consider the irregularities in the topography… by the way, I am tempted to make a confusion between topography and gravity. The gravity field is the way the mass of your body will affect the trajectory of the body with which it interacts, i.e. the Earth, Lunar spacecrafts… If you are close enough, you will be sensitive to the mass distribution in the body, which is of course linked to the topography. So, the two notions are correlated, but not fully, since the gravity is more sensitive to the interior.

But let us go back to this problem of distance. If you are far enough, no problem. The Moon is either a sphere, or a triaxial ellipsoid. If you get closer, you should be more accurate. And if you are too close, then you cannot be accurate enough.

This limit is given by the radius of the Brillouin sphere. Named after the French-born American physicist Léon Brillouin, this is the circumscribing sphere of the body. If your planetary body is spherical, then it exactly fills its Brillouin sphere, and this problem is trivial… If you are a potatoidal asteroid, then your volume will be only a fraction of this sphere, and you can imagine having a spacecraft inside this sphere.

The asteroid Itokawa in its Brillouin-sphere. Credit: JAXA.
The asteroid Itokawa in its Brillouin-sphere. Credit: JAXA.

The Moon is actually pretty close to a sphere, of radius 1737.4±1 km. But many mass anomalies have been detected, which makes its gravity field not that close to the one of the sphere, and you can be inside the equivalent Brillouin sphere (if we translate gravity into topography), in flying over the surface at low altitude.

Why modeling it?

Why trying to be that accurate on the gravity field / topography of a planetary object? I see at least two good reasons, please pick the ones you prefer:

  • to be able to detect the time variations of the topography and / or the gravity field. This would give you the tidal response (see here) of the body, or the evolution of its polar caps,
  • because it’s fun,
  • to be able to control the motion of low-altitude spacecrafts. This is particularly relevant for asteroids, which are somehow potatoidal (am I coining this word?)

You can object that the Moon may be not the best body to test the gravity inside the Brillouin sphere. Actually we have an invaluable amount of data on the Moon, thanks to the various missions, the Lunar Laser Ranging, which accurately measures the Earth-Moon distance… Difficult to be more accurate than on the Moon…

The goal of the paper is actually not to find something new on the Moon, but to test different models of topography and gravity fields, before using them on other bodies.

Spherical harmonics expansion

Usually the gravity field (and the topography) is described as a spherical harmonics expansion, i.e. you model your body as a sum of waves with increasing frequencies, over two angles, which are the latitude and the longitude. This is why the order 0 is the exact sphere, the order 2 is the triaxial ellipsoid… and in raising the order, you introduce more and more peaks and depressions in your shape… In summing them, you should have the gravity field of your body… if your series converge. It is usually assume that you converge outside the Brillouin sphere… It is not that clear inside.

To test the convergence, you need to measure a distance between your series and something else, that you judge relevant. It could be an alternative gravitational model, or just the next approximation of the series. And to measure the distance, a common unit is the gal, which is an acceleration of 1 cm/s2 (you agree that gravity gives acceleration?). In this paper, the authors checked differences at the level of the μgal, i.e. 1 gal divided by 1 million.

Methodology

In this study, the authors used data from two sources:

  • high-resolution shape maps from the Lunar Orbiter Laser Altimeter (LOLA),
  • gravity data from the mission GRAIL (Gravity Recovery And Interior Laboratory),

and they modeled 4 gravity fields:

  1. Topography of the surface,
  2. Positive topographic heights, i.e. for basins the mean radius was considered, while the exact topography was considered for mountains,
  3. “Brillouin-sphere”, at a mean altitude of 11 km,
  4. “GRAIL-sphere”, at a mean altitude of 23 km.

In each of these cases, the authors used series of spherical harmonics of orders between 90 (required spatial resolution: 60.6 km) and 2,160 (resolution: 2.5 km). In each case, the solution with spherical harmonics was compared with a direct integration of the potential of the body, for which the topography is discretized through an ensemble of regularly-shaped elements.

Results

And here are the results:

Not surprisingly, everything converges in the last two cases, i.e. altitudes of 11 and 23 km. However, closer to the surface the expansion in spherical harmonics fails from orders 720 (case 1) and 1,080 (case 2), respectively. This means that adding higher-order harmonics does not stabilize the global solution, which can be called divergence. The authors see from their calculations that this can be predicted from the evolution of the amplitude of the terms of the expansion, with respect to their orders. To be specific, their conclusion is summarized as follows:

A minimum in the degree variances of an external potential model foreshadows divergence of the spherical harmonic series expansions at points inside the Brillouin-sphere.

 

My feeling is that this study should be seen as a laboratory test of a mathematical method, i.e. testing the convergence of the spherical harmonics expansion, not on a piece of paper, but in modeling a real body, with real data. I wonder how the consideration of time variations of the potential would affect these calculations.

To know more…

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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

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