Tidal dissipation within an elongated asteroid with satellite, and application to asteroid (216) Kleopatra

(216) Kleopatra is a highly elongated dumbbell shaped asteroid, which is spinning rapidly (spin period 5.38 h), and has two satellites. The tidal migration rate of its outer satellite (related to asteroid tidal despinning) has been measured by Broz et al. (2021), and is used here to deduce the elastic properties of the asteroid, in particular its rigidity mu. For this purpose the satellite is modeled as an homogeneous elongated axisymmetric body, whose shape is described either as a cylinder or more realistically as a dumbbell. Such model asteroid is regarded as a beam (or rod) that undergoes tidal disturbances from the satellite. Due to the deformability of the asteroid, the tidal stresses produced within the body rise a compression tide in the direction of the asteroid's long axis, and a bending tide in the perpendicular direction. We compute the tidal amplitude of the total elastic energy stored within the asteroid, as a function of Young's modulus E. Provided that the tidal quality factor Q is known, this permits to deduce the power tidally dissipated within the asteroid. This is compared to the tidal dissipation deduced from Broz et al. (2022) observations of the tidal migration of Kleopatra's outer satellite. This permits to deduce the asteroid's Young's modulus E (or equivalently the rigidity mu through mu approximate to E/(2.6 +/- 0.1)). Using our dumbbell model for Kleopatra, we obtain a rigidity mu approximate to 1.94 x 107 Pa if one assumes Q approximate to 40, or mu approximate to 1.40 x 107 Pa if one assumes Q = 100. Such tidal dissipation is found to be >2 orders of magnitude higher than the tidal dissipation which would occur in a hypothetical spherical asteroid of same density and rigidity as Kleopatra, with radius equal to the volume equivalent radius (RV approximate to 59.1 km). Among model asteroids with same long axis length as Kleopatra (2 L approximate to 267 km), dissipation is also found to be strongly dependent upon the shape of the asteroid (dumbbell, cylinder, or ellipsoid). Here the asteroid is regarded as an elastic solid, whereas it is presumably a weak rubble pile medium. The formalism developed here is however relevant provided that the asteroid may be regarded as a Maxwell material, because tidal frequencies are expected to be much higher than the inverse Maxwell time.