Bodily tides near the 1:1 spin-orbit resonance: correction to Goldreich’s dynamical model

Spin-orbit coupling is often described in an approach known as ??the MacDonald torque??, which has long become the textbook standard due to its apparent simplicity. Within this method, a concise expression for the additional tidal potential, derived by MacDonald (Rev Geophys 2:467?C541, 1994), is combined with a convenient assumption that the quality factor Q is frequency-independent (or, equivalently, that the geometric lag angle is constant in time). This makes the treatment unphysical because MacDonald??s derivation of the said formula was, very implicitly, based on keeping the time lag frequency-independent, which is equivalent to setting Q scale as the inverse tidal frequency. This contradiction requires the entire MacDonald treatment of both non-resonant and resonant rotation to be rewritten. The non-resonant case was reconsidered by Efroimsky and Williams (Cel Mech Dyn Astron 104:257?C289, 2009), in application to spin modes distant from the major commensurabilities. In the current paper, we continue this work by introducing the necessary alterations into the MacDonald-torque-based model of falling into a 1-to-1 resonance. (The original version of this model was offered by Goldreich (Astron J 71:1?C7, 1996). Although the MacDonald torque, both in its original formulation and in its corrected version, is incompatible with realistic rheologies of minerals and mantles, it remains a useful toy model, which enables one to obtain, in some situations, qualitatively meaningful results without resorting to the more rigorous (and complicated) theory of Darwin and Kaula. We first address this simplified model in application to an oblate primary body, with tides raised on it by an orbiting zero-inclination secondary. (Here the role of the tidally-perturbed primary can be played by a satellite, the perturbing secondary being its host planet. A planet may as well be the perturbed primary, its host star acting as the tide-raising secondary). We then extend the model to a triaxial primary body experiencing both a tidal and a permanent-figure torque exerted by an orbiting secondary. We consider the effect of the triaxiality on both circulating and librating rotation near the synchronous state. Circulating rotation may evolve toward the libration region or toward a spin faster than synchronous (the so-called pseudosynchronous spin). Which behaviour depends on the orbit eccentricity, the triaxial figure of the primary, and the mass ratio of the secondary and primary bodies. The spin evolution will always stall for the oblate case. For libration with a small amplitude, expressions are derived for the libration frequency, damping rate, and average orientation. Importantly, the stability of pseudosynchronous spin hinges upon the dissipation model. Makarove and Efroimsky (Astrophys J, 2012) have found that a more realistic tidal dissipation model than the corrected MacDonald torque makes pseudosynchronous spin unstable. Besides, for a sufficiently large triaxiality, pseudosynchronism is impossible, no matter what dissipation model is used.     

Bodily tides near spin–orbit resonances

Spin–orbit coupling can be described in two approaches. The first method, known as the “MacDonald torque”, is often combined with a convenient assumption that the quality factor Q is frequency-independent. This makes the method inconsistent, because derivation of the expression for the MacDonald torque tacitly fixes the rheology of the mantle by making Q scale as the inverse tidal frequency. Spin–orbit coupling can be treated also in an approach called “the Darwin torque”. While this theory is general enough to accommodate an arbitrary frequency-dependence of Q, this advantage has not yet been fully exploited in the literature, where Q is often assumed constant or is set to scale as inverse tidal frequency, the latter assertion making the Darwin torque equivalent to a corrected version of the MacDonald torque. However neither a constant nor an inverse-frequency Q reflect the properties of realistic mantles and crusts, because the actual frequency-dependence is more complex. Hence it is necessary to enrich the theory of spin–orbit interaction with the right frequency-dependence. We accomplish this programme for the Darwin-torque-based model near resonances. We derive the frequency-dependence of the tidal torque from the first principles of solid-state mechanics, i.e., from the expression for the mantle’s compliance in the time domain. We also explain that the tidal torque includes not only the customary, secular part, but also an oscillating part. We demonstrate that the lmpq term of the Darwin–Kaula expansion for the tidal torque smoothly passes zero, when the secondary traverses the lmpq resonance (e.g., the principal tidal torque smoothly goes through nil as the secondary crosses the synchronous orbit). Thus, we prepare a foundation for modeling entrapment of a despinning primary into a resonance with its secondary. The roles of the primary and secondary may be played, e.g., by Mercury and the Sun, correspondingly, or by an icy moon and a Jovian planet. We also offer a possible explanation for the “improper” frequency-dependence of the tidal dissipation rate in the Moon, discovered by LLR.     

Multi-wavelength observations of Asteroid 2100 Ra-Shalom

We observed near-Earth asteroid (NEA) 2100 Ra-Shalom over a six-year period, obtaining rotationally resolved spectra in the visible, near-infrared, thermal-infrared, and radar wavelengths. We find that Ra-Shalom has an effective diameter of Deff=2.3±0.2 km, rotation period P=19.793±0.001 h, visual albedo pv=0.13±0.03, radar albedo , and polarization ratio μc=0.25±0.04. We used our radar observations to generate a three-dimensional shape model which shows several structural features of interest. Based on our thermal observations, Ra-Shalom has a high thermal inertia of ∼10³ J m⁻² s^−0.5 K⁻¹, consistent with a coarse or rocky surface and the inferences of others [Harris, A.W., Davies, J.K., Green, S.F., 1998. Icarus 135, 441-450; Delbo, M., Harris, A.W., Binzel, R.P., Pravec, P., Davies, J.K., 2003. Icarus 166, 116-130]. Our spectral data indicate that Ra-Shalom is a K-class asteroid and we find excellent agreement between our spectra and laboratory spectra of the CV3 meteorite Grosnaja. Our spectra show rotation-dependent variations consistent with global variations in grain size. Our radar observations show rotation-dependent variations in radar albedo consistent with global variations in the thickness of a relatively thin regolith.     

Dynamical friction in modified Newtonian dynamics

We have tested a previous analytical estimate of the dynamical friction time-scale in modified Newtonian dynamics (MOND) with fully non-linear N -body simulations. The simulations confirm that the dynamical friction time-scale is significantly shorter in MOND than in equivalent Newtonian systems, i.e. systems with the same phase-space distribution of baryons and additional dark matter. An apparent conflict between this result and the long time-scales determined for bars to slow and mergers to be completed in previous N -body simulations of MOND systems is explained. The confirmation of the short dynamical-friction time-scale in MOND underlines the challenge that the Fornax dwarf spheroidal poses to the viability of MOND.     

Short-Period Fluctuations in Coronal Mass Ejection Activity during Solar Cycle 23

We have constructed a time series of the number of coronal mass ejections (CMEs) observed by SOHO/LASCO during solar cycle 23. Using spectral analysis techniques (the maximum entropy method and wavelet analysis) we found short-period (< one year) semiperiodic activity. Among others, we found interesting periodicities at 193, 36, 28, and 25 days. We discuss the implications of such short-period activity in terms of the emergence and escape of magnetic flux from the convection zone, through the low solar atmosphere (where these periodicities have been found for numerous activity parameters), toward interplanetary space. This analysis shows that CMEs remove the magnetic flux in a quasiperiodic process in a way similar to that of magnetic flux emergence and other solar eruptive activity.