Tidal dissipation within an elongated asteroid with satellite, and application to asteroid (216) Kleopatra
Introduction
Asteroid (216) Kleopatra presents several striking peculiarities. Firstly it is extremely elongated, in such a way that its shape resembles the «dumbbell » equilibrium shapes studied by Descamps (2015). Computing the effective potential including gravity and rotation, Marchis et al. (2021) have shown that Kleopatra's shape extends to a distance that is very close to the critical L1 equipotential. Another peculiarity is that Kleopatra possesses two identified satellites on close orbits. Broz et al. (2022) have been able to measure an increase of the period P2 of the outer satellite Alexhelios, with a rate =(1.8 ± 0.1)x10−8dd−1, and attributed it to the effect of tidal dissipation within Kleopatra produced by that satellite. Assuming a tidal quality factor Q ≈ 40, which is typical for rubble pile asteroid, they found that the rate of increase of the satellite's rotation period was consistent with a Love number k2 ≈ 0.3. As noticed by Broz et al. (2022), for uniform and spherical bodies, the Love number is related to the material rigidity μ of the asteroid through (Goldreich and Sari, 2009):where μ is the material rigidity, M, R and SO are mass, radius and surface area of the asteroid, and G (=6.672 × 10−11 m3kg−1 s−2) is gravitational constant.
From this equation, taking R as the maximal radius R ≈ 135 km for Kleopatra, they obtained μQ ≈ 2.7 × 107 Pa.
On another side, Marchis et al. (2008) reported that μ ≈ 108 Pa is a typical value for a moderately fractured asteroid, and μ > 1010 Pa for consolidated rocky material. Assuming Q ≈ 100 from Yoder (1982), they concluded that μQ can be in a large range from 1010 to 1012 Pa. Then by computing the evolution time scales of the most documented binary asteroid systems, and comparing them to the age of the solar system as an upper limit (4.5 Gyears), they were able to reduce that range, and concluded that μQ ≈ 1010 Pa should be a realistic value for these binary systems.
The estimate μQ ≈ 2.7 × 107 Pa by Broz et al. (2022) is therefore about three orders of magnitude smaller than the estimate μQ ≈ 1010 Pa proposed by Marchis et al. (2008) for 100 km asteroids.
This paper is an attempt to go further in the comparison between the inferred elastic properties of Kleopatra with what is usually expected for 100 km asteroids, by revisiting the relation between Love number and rigidity for the specific case of a highly elongated asteroid such as Kleopatra.
In our formalism, we will follow the usual approach by considering the asteroid as an elastic solid. It should be pointed out, however, that such an object is not meant to be truly solid, but should be regarded as a weak medium consisting presumably of a pile of rubble. Dissipation in these weak media is not determined by their rigidity, but rather by their viscosity. The formalism developed here is however relevant provided that the asteroid can be considered as a Maxwell material and for frequencies well above the inverse of Maxwell time. This point will be discussed in the conclusion.
Section snippets
Interaction between a rigid elongated asteroid and its satellite
As shown by Marchis et al. (2021), a visual inspection of the global shape of Kleopatra shows a very elongated object, with the presence of two lobes separated by a neck, resulting in a dumb-bell appearance. In this paper we will simplify the problem by considering the asteroid as an homogeneous beam, or rod, whose model geometry will be presented below. The asteroid is spinning at angular frequency Ω. The satellite will be supposed to be on a circular orbit, in the plane perpendicular to the
Normal stress and bending moment for an elastic solid beam (static approach)
The asteroid is now regarded as a solid elastic beam, which is subject to variable normal stress as well as shear stress due to the presence of the gravitational forces produced by the satellite. In this section we assume that the frequencies of gravitational excitations of the asteroid by the satellite are much slower than the natural frequencies of the asteroid. Therefore a static approach is taken here. The oscillatory behaviour of the system will be addressed in the next sections. Three
Natural frequencies of oscillations of the asteroid
In Section 3, compression and bending were treated in a static approach. Since the gravitational excitation by the satellite is oscillatory, such approach may be justified only if the forcing frequency is small compared to the natural frequencies of the asteroid. It is therefore necessary to determine the natural resonance frequencies of the asteroid, with respect to both axial compression and bending. Exact solutions require solving intricate partial differential equations. However for most
Coupling between bending and compression under Coriolis acceleration
The expressions for accelerations Γx and Γy were given in Section 2, including a first Coriolis term related to asteroid libration, given by Eq. (9). In addition, since the distortions of the asteroid vary with time, those also imply velocities with respect to the spinning frame, and thus additional Coriolis acceleration.
Explicitly writing the dependence with time, we may express the complex amplitudes and of the elastic displacement:
The phases ζ
Amplitude of elastic energy stored by the forced oscillator
The amplitudes of the longitudinal and bending elastic energies ESx and EB stored within the asteroid, given in Eqs. (17), (32), were obtained in a static approach, characterized by the assumption that the rate of variations are much slower than the natural frequencies of the asteroid. The tidal perturbation, on the other hand, produces a forced oscillation at the semi-diurnal angular rate ωS = 2(Ω-n2), which needs to be compared to the natural resonance rates of the asteroid. When the forcing
Kleopatra's rigidity μ inferred from period migration rate
Three parameters are relevant to describe the elastic properties of a material. Those are (i) Young's modulus E which has been used up to now in the paper, (ii) shear modulus or rigidity μ, and (iii) Poisson's ratio ν. These are however not independent since they are linked by the relation E = 2(1 + ν)μ. Moreover the value of Poisson ratio ν is fairly well constrained for fractured rocky material. The estimates of Poisson ratio performed by Davy et al. (2018) for fractured rock masses span a
Conclusion
The classical theory of planetary tidal dissipation is usually given for spherical or near spherical planets (Goldreich and Sari, 2009). Although such an approach is most often fully appropriate, it may become irrelevant for exotic planetary bodies whose shapes deviate significantly from a sphere. This is the case for some highly elongated asteroids, among which asteroid (216) Kleopatra is exemplary. The aim of this paper was to compute the tidal dissipation within such highly elongated
Declaration of Competing Interest
The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (17)
Dumb-bell-shaped equilibrium figures for fiducial contact-binary asteroids and EKBOs
Icarus
(2015)Near-equilibrium dumb-bell-shaped figures for cohesionless small bodies
Icarus
(2016)- et al.
Main belt binary asteroidal systems with circular mutual orbits
Icarus
(2008) - et al.
Scaling forces to asteroid surfaces: the role of cohesion
Icarus
(2010) Tidal rigidity of Phobos
Icarus
(1982)- et al.
An advanced multipole model for (216) Kleopatra triple system
Astron. Astrophys.
(2021) - et al.
Observed tidal evolution of Kleopatra’s outer satellite
Astron. Astrophys.
(2022) - et al.
Elastic properties of fractured rock masses with frictional properties and power law fracture size distributions
J. Geophys. Res. Solid Earth
(2018)
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2024, Monthly Notices of the Royal Astronomical Society