Spin axis evolution of two interacting bodies
Introduction
We consider here two rigid bodies orbiting each other. The main purpose of this work is to determine the long term evolution of their spin orientation and to a lower extent, the orientation of the orbital plane. Examples of such systems are binary asteroids or a planet with a massive satellite.
If the two bodies are spherical, then the translational and the rotational motions are independent (e.g. Duboshin, 1958). In that case, the orbit is purely Keplerian and the proper rotation of the bodies are uniform. General problems with triaxial bodies are more complicated, and usually non-integrable. Even formal expansions of the gravitational potential or the proof of their convergence can be an issue (Borderies, 1978, Paul, 1988, Tricarico, 2008). In some cases, especially for slow rotations close to low order spin–orbit resonances, the spin evolution of rigid bodies of irregular shape can be strongly chaotic (Wisdom et al., 1984, Wisdom, 1987), but we will not consider this situation in the present paper where we focus on regular and quasiperiodic motions.
Stationary solutions of spin evolution are known in the case of a triaxial satellite orbiting a central spherical planet (Abul'naga and Barkin, 1979). In their paper, Abul'naga and Barkin used canonical coordinates, based on the Euler angles, to set the orientation of the satellite. On the contrary, in 1991, Wang et al. also studied relative equilibria but with a vectorial approach that enabled them to analyze easily the stability of those solutions. For a review of different formalisms that can be used in rigid body problems, see Borisov and Mamaev (2005).
The vectorial approach turned out to be also powerful for the study of relative equilibria of two triaxial bodies orbiting each other (Maciejewski, 1995). General motions of this problem were studied by Fahnestock and Scheeres (2008) in the case of the typical binary asteroid system called 1999 KW4. For that, the authors expanded the gravitational potential up to the second order only. In this approximation, there is no direct interaction between the orientation of the two bodies. Ashenberg gave in 2007 the expression of the gravitational potential expanded up to the fourth order but did not study the solutions.
In Boué and Laskar (2006) we gave a new method to study the long term evolution of solid body orientations in the case of a star–planet–satellite problem where only the planet is assumed to be rigid. This method used a similar vectorial approach as Wang et al. (1991), plus some averaging over the fast angles. We showed that the secular evolution of this system is integrable and provided the general solution.
In the present paper, we show that the problem of two triaxial bodies orbiting each other is very similar to the star–planet–satellite problem and thus can be treated in the same way.
In Section 2, we compute the Hamiltonian governing the evolution of two interacting rigid bodies. The gravitational potential is expanded up to the fourth order and averaged over fast angles. The resulting secular Hamiltonian is a function of three vectors only: the orbital angular momentum and the angular momenta of the two bodies.
In a next step (Section 3), we show that the secular problem is integrable but not trivially (i.e. it cannot be reduced to a scalar first order differential equation that can be integrated by quadrature). The general solution is the product of a uniform rotation of the three vectors (global precession around the total angular momentum) by a periodic motion (nutation). We prove also that in a frame rotating with the precession frequency, the nutation loops described by the three vectors are all symmetric with respect to a same plane containing the total angular momentum. We then derive analytical approximations of the two frequencies of the secular problem with their amplitudes. These formulas need averaged quantities that can be computed recursively. However we found that the first iteration already gives satisfactory results.
In Section 5, we consider the general case of a n-body system of rigid bodies in gravitational interaction, and we demonstrate that the regular quasiperiodic solutions of these systems can, in a similar way, be decomposed into a uniform precession, and a quasiperiodic motion in the precessing frame.
Finally, we compare our results with those of Fahnestock and Scheeres (2008) on the typical binary asteroid system 1999 KW4. We show that their analytical expression of the precession frequency corresponds to the simple case of a point mass orbiting an oblate body treated in Boué and Laskar (2006). We then integrate numerically from the full Hamiltonian, an example of a doubly asynchronous system where the Fahnestock and Scheeres (2008) expression of the precession frequency does not apply. We compare the results with the output of the averaged Hamiltonian and with our numerical approximation and show that they are in good agreement.
Section snippets
Fundamental equations
We are considering a two rigid body problem in which the interaction is purely gravitational with no dissipative effects. Let and be the masses of the two solids. Hereafter the mass is called the satellite or the secondary and the mass the primary. It should be stressed that this notation does not imply any constraint on the ratio of the masses which can even be equal to one.
The configuration of the system is described by the position vector r of the satellite barycenter relative to
Secular equations
The secular Hamiltonian (28) is similar to the one obtained in Boué and Laskar (2006) although its expression is slightly more complicated. The difference with Boué and Laskar (2006) is that the secular Hamiltonian is not anymore the equation of an ellipsoid in . A few results in Boué and Laskar (2006) were proved for this special surface. We recall here the main steps of the derivation of the solutions adapted to the new surface defined by the current secular Hamiltonian.
The
Analytical approximation
In this section we give an analytical approximation of the secular evolution. So far, only general features of the solutions have been obtained. Here analytical approximations of the two frequencies that appear in the problem as well as their amplitudes are computed. The two frequencies being the global precession and the nutation.
In an invariant frame where the third axis is aligned with the direction of the total angular momentum, we can write where
Global precession of a n-body system
We have seen that the secular motion of a two solid body system can, as in Boué and Laskar (2006), be decomposed in a uniform precession of angular motion Ω, and a periodic motion of frequency ν. In fact, this can be extended to a very general system of n solid bodies in gravitational interaction. The following result, which is of very broad application, is a consequence of the general angular momentum reduction in case of regular, quasiperiodic, motion.
Proposition 3 Let be a system of bodies of mass
Application
In this section we compare our rigorous results on the averaged system and our analytical approximations of the solutions of the same system with the integration of the full Hamiltonian (2), (7), (12), (13), (14) on two different binary systems I and II (see Table 1, Table 2). The physical and orbital parameters of the system II are those of the binary Asteroid 1999 KW4 studied in Fahnestock and Scheeres (2008). We choose this system in order to compare our results with Fahnestock and Scheeres
Conclusions
We have shown here that the general framework developed in Boué and Laskar (2006) applies as well to the problem of two rigid bodies orbiting each other. This formalism enables us to obtain the long term evolution of the spin axis of the two bodies as well as the evolution of the orientation of the orbital plane. The two bodies can be very general, with strong triaxiality, and their rotation vector is not necessary aligned with their axis of maximum inertia. The gravitational potential is
Acknowledgments
We thank Franck Marchis for discussions on binary asteroids observations, and Alain Albouy for his comments. The authors largely benefited from the interactions and discussions inside the Astronomy and Dynamical System group at IMCCE.
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