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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