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Register a new userTidal Evolution of Asteroidal Binaries. Ruled by Viscosity. Ignorant of Rigidity
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Michael Efroimsky
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The rate of tidal evolution of asteroidal binaries is defined by the dynamical Love numbers divided by quality factors. Common is the (often illegitimate) approximation of the dynamical Love numbers with their static counterparts. As the static Love numbers are, approximately, proportional to the inverse rigidity, this renders a popular fallacy that the tidal evolution rate is determined by the product of the rigidity by the quality factor: $,k_l/Qpropto 1/(mu Q),$. In reality, the dynamical Love numbers depend on the tidal frequency and all rheological parameters of the tidally perturbed body (not just rigidity). We demonstrate that in asteroidal binaries the rigidity of their components plays virtually no role in tidal friction and tidal lagging, and thereby has almost no influence on the intensity of tidal interactions (tidal torques, tidal dissipation, tidally induced changes of the orbit). A key quantity that determines the tidal evolution is a product of the effective viscosity $,eta,$ by the tidal frequency $,chi,$. The functional form of the torques dependence on this product depends on who wins in the competition between viscosity and self-gravitation. Hence a quantitative criterion, to distinguish between two regimes. For higher values of $,etachi,$ we get $,k_l/Qpropto 1/(etachi);$; $,$while for lower values we obtain $,k_l/Qpropto etachi,$. Our study rests on an assumption that asteroids can be treated as Maxwell bodies. Applicable to rigid rocks at low frequencies, this approximation is used here also for rubble piles, due to the lack of a better model. In the future, as we learn more about mechanics of granular mixtures in a weak gravity field, we may have to amend the tidal theory with other rheological parameters, ones that do not show up in the description of viscoelastic bodies.
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We address the expressions for the rates of the Keplerian orbital elements within a two-body problem perturbed by the tides in both partners. The formulae for these rates have appeared in the literature in various forms, at times with errors. We reconsider, from scratch, the derivation of these rates and arrive at the Lagrange-type equations which, in some details, differ from the corresponding equations obtained previously by Kaula (1964). We also write down detailed expressions for $da/dt$, $de/dt$ and $di/dt$, to order $e^4$. They differ from Kaulas expressions which contain a redundant factor of $M/(M+M^{prime}),$ with $M$ and $M^{prime}$ being the masses of the primary and the secondary. As Kaula was interested in the Earth-Moon system, this redundant factor was close to unity and was unimportant in his developments. This factor, however, must be reinstated when Kaulas theory is applied to a binary composed of partners of comparable masses. We have found that, while it is legitimate to simply sum the primarys and secondarys inputs in $da/dt$ or $de/dt$, this is not the case for $di/dt$. So our expression for $di/dt$ differs from that of Kaula in two regards. First, the contribution due to the dissipation in the secondary averages out when the apsidal precession is uniform. Second, we have obtained an additional term which emerges owing to the conservation of the angular momentum: a change in the inclination of the orbit causes a change of the primarys plane of equator.
We have calculated the coherence and detectable lifetimes of synthetic near-Earth object (NEO) families created by catastrophic disruption of a progenitor as it suffers a very close Earth approach. The closest or slowest approaches yield the most violent `s-class disruption events. We found that the average slope of the absolute magnitude (H) distribution, $N(H)propto10^{(0.55pm0.04),H}$, for the fragments in the s-class families is steeper than the slope of the NEO population citep{mainzer2011} in the same size range. The families remain coherent as statistically significant clusters of orbits within the NEO population for an average of $bartau_c = (14.7pm0.6)times10^3$ years after disruption. The s-class families are detectable with the techniques developed by citet{fu2005} and citet{Schunova2012} for an average duration ($bartau_{det}$) ranging from about 2,000 to about 12,000 years for progenitors in the absolute magnitude ($H_p$) range from 20 to 13 corresponding to diameters in the range from about 0.5 to 10$km$ respectively. The short detectability lifetime explains why zero NEO families have been discovered to-date. Nonetheless, every tidal disruption event of a progenitor with D$>0.5km$ is capable of producing several million fragments in the $1meter$ to $10meter$ diameter range that can contribute to temporary local density enhancements of small NEOs in Earths vicinity. We expect that there are about 1,200 objects in the steady state NEO population in this size range due to tidal disruption assuming that one $1km$ diameter NEO tidally disrupts at Earth every 2,500 years. These objects may be suitable targets for asteroid retrieval missions due to their Earth-like orbits with corresponding low $v_{infty}$. The fragments from the tidal disruptions at Earth have $sim5times$ the collision probability with Earth compared to the background NEO population.
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Stellar radiation has conservatively been used as the key constraint to planetary habitability. We review here the effects of tides, exerted by the host star on the planet, on the evolution of the planetary spin. Tides initially drive the rotation period and the orientation of the rotation axis into an equilibrium state but do not necessarily lead to synchronous rotation. As tides also circularize the orbit, eventually the rotation period does equal the orbital period and one hemisphere will be permanently irradiated by the star. Furthermore, the rotational axis will become perpendicular to the orbit, i.e. the planetary surface will not experience seasonal variations of the insolation. We illustrate here how tides alter the spins of planets in the traditional habitable zone. As an example, we show that, neglecting perturbations due to other companions, the Super-Earth Gl581d performs two rotations per orbit and that any primordial obliquity has been eroded.
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