Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque

Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of the eccentricity of its orbit. However, the mathematical model originally introduced to study its spin-orbit evolution proved not to be entirely convincing, because of the expression commonly used for the tidal torque. Only recently, in a series of papers mainly by Efroimsky and Makarov, a different model for the tidal torque has been proposed, which has the advantages of being more realistic, and of providing a higher probability of capture in the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and consists of a superposition of kinks, so that both analytical and numerical computations turn out to be rather delicate: indeed, standard perturbation theory based on power series expansion cannot be applied and the implementation of a fast algorithm to integrate the equations of motion numerically requires a high degree of care. In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model: In particular, we present numerical and analytical results about the nature of the librations of Mercury's spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time.Comment: 32 pages, 8 figures, 5 table

MOSES AND THE SEVENTY ELDERS: MOSAIC AUTHORITY IN NUMBERS 11 AND THE LEGEND OF THE SEPTUAGINT

This thesis seeks an exegesis of Numbers 11:16-17, 24-25, the so-called “elders story,” within a larger wilderness episode involving Moses’ bitter complaint, the people’s great craving for meat, and the enigmatic Eldad and Medad. While most recent interpreters have considered the elders a curious side-show, occurring nearly inexplicably in both their narrative setting and pentateuchal position, pre-modern interpreters have often drawn more from their configuration. As the first full-length study into the elders of Numbers 11, this thesis seeks to explore what the elders contribute to their own biblical setting by tracing their impact on later generations of Jews and Christians. In particular, it explores the possible links between these seventy elders and the seventy translators of the Legend of the Septuagint in its Hellenistic versions in Letter of Aristeas, Philo, and Josephus. The first two chapters examine the recent history of interpretation of the passage and re-appraise typical interpretative stances toward both the elders’ climactic activity as “speaking in ecstasy (ויתנבאו)” and their designation as not only elders of Israel but “their officers (שטריו).” “Prophesying” and “scribes” are presented, respectively, as preferred terms, both philologically and contextually. The next two chapters critically examine the relationship between Moses and the elders of Numbers 11, vis-à-vis their symbolic presentation as “seventy (שבעים)” (or, with Eldad and Medad included, as “seventy-two”) and their potential ability to inherit, represent, and interpret Moses’ law-giving authority. In both cases, Moses’ burden and cry for his own death in Numbers 11:11-14, brings the necessity of inheritors of his authority closer to the concerns of Numbers 11 and Exodus-Joshua. The final main chapter examines the many ways the seventy elders of Numbers 11 may be understood as foundational to the framing of the Legend of the Septuagint. As those drawn closer to Moses than any other biblical persona, the seventy elders are uniquely imbued with Moses’ authority, biblically and beyond.

Basins of attraction in forced systems with time-varying dissipation

We consider dissipative periodically forced systems and investigate cases in which having information as to how the system behaves for constant dissipation may be used when dissipation varies in time before settling at a constant final value. First, we consider situations where one is interested in the basins of attraction for damping coefficients varying linearly between two given values over many different time intervals: we outline a method to reduce the computation time required to estimate numerically the relative areas of the basins and discuss its range of applicability. Second, we observe that sometimes very slight changes in the time interval may produce abrupt large variations in the relative areas of the basins of attraction of the surviving attractors: we show how comparing the contracted phase space at a time after the final value of dissipation has been reached with the basins of attraction corresponding to that value of constant dissipation can explain the presence of such variations. Both procedures are illustrated by application to a pendulum with periodically oscillating support.Comment: 16 pages, 13 figures, 7 table

Quasi-periodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems

We consider a class of ordinary differential equations describing one-dimensional analytic systems with a quasi-periodic forcing term and in the presence of damping. In the limit of large damping, under some generic non-degeneracy condition on the force, there are quasi-periodic solutions which have the same frequency vector as the forcing term. We prove that such solutions are Borel summable at the origin when the frequency vector is either any one-dimensional number or a two-dimensional vector such that the ratio of its components is an irrational number of constant type. In the first case the proof given simplifies that provided in a previous work of ours. We also show that in any dimension $d$, for the existence of a quasi-periodic solution with the same frequency vector as the forcing term, the standard Diophantine condition can be weakened into the Bryuno condition. In all cases, under a suitable positivity condition, the quasi-periodic solution is proved to describe a local attractor.Comment: 10 page

Illustrating field emission theory by using Lauritsen plots of transmission probability and barrier strength

This technical note relates to the theory of cold field electron emission (CFE). It starts by suggesting that, to emphasize common properties in relation to CFE theory, the term 'Lauritsen plot' could be used to describe all graphical plots made with the reciprocal of barrier field (or the reciprocal of a quantity proportional to barrier field) on the horizontal axis. It then argues that Lauritsen plots related to barrier strength (G) and transmission probability (D) could play a useful role in discussion of CFE theory. Such plots would supplement conventional Fowler-Nordheim (FN) plots. All these plots would be regarded as particular types of Lauritsen plot. The Lauritsen plots of -G and lnD can be used to illustrate how basic aspects of FN tunnelling theory are influenced by the mathematical form of the tunnelling barrier. These, in turn, influence local emission current density and emission current. Illustrative applications used in this note relate to the well-known exact triangular and Schottky-Nordheim barriers, and to the Coulomb barrier (i.e., the electrostatic component of the electron potential energy barrier outside a model spherical emitter). For the Coulomb barrier, a good analytical series approximation has been found for the barrier-form correction factor; this can be used to predict the existence (and to some extent the properties) of related curvature in FN plots.Comment: Based on a poster presented at the 25th International Vacuum Nanoelectronics Conference, Jeju, S. Korea, July 2012. Version 3 incorporates small changes made at proof stag