Ernst Julius Öpik's (1916) note on the theory of explosion cratering on the Moon's surface—The complex case of a long‐overlooked benchmark paper

High‐velocity impact as a common phenomenon in planetary evolution was ignored until well into the twentieth century, mostly because of inadequate understanding of cratering processes. An eight‐page note, published in Russian by the young Ernst Julius Öpik, a great Estonian astronomer, was among the key selenological papers, but due to the language barrier, it was barely known and mostly incorrectly cited. This particular paper is here intended to serve as an explanatory supplement to an English translation of Öpik's article, but also to document an early stage in our understanding of cratering. First, we outline the historical–biographical background of this benchmark paper, and second, a comprehensive discussion of its merits is presented, from past and present perspectives alike. In his theoretical research, Öpik analyzed the explosive formation of craters numerically, albeit in a very simple way. For the first time, he approximated relationships among minimal meteorite size, impact energy, and crater diameter; this scaling focused solely on the gravitational energy of excavating the crater (a “useful” working approach). This initial physical model, with a rational mechanical basis, was developed in a series of papers up to 1961. Öpik should certainly be viewed as the founder of the numerical simulation approach in planetary sciences. In addition, the present note also briefly describes Nikolai A. Morozov as a remarkable man, a forgotten Russian scientist and, surprisingly, the true initiator of Öpik's explosive impact theory. In fact, already between 1909 and 1911, Morozov probably was the first to consider conclusively that explosion craters would be circular, bowl‐shaped depressions even when formed under different impact angles.     

Carbon and Oxygen Coronal Line Intensities for the Electron κ-DistributionI

Solar Physics - The ionization and excitation equilibrium of C and O for the electron κ-distribution in the solar corona is studied. The change of the distribution shape changes the...     

Palaeomagnetic directions (223-1.4 ka) recorded in the volcanites of Lipari, Aeolian Islands

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The island of Lipari is formed by Pleistocene volcanites emplaced during four main periods of volcanic activity. A study has been made of their magnetic properties, primarily with the aim of determining changes in the palaeomagnetic directions. Titanomagnetite is always the primary ferromagnetic mineral and its magnetic characteristics are common to the various lithotypes. However, the concentration of titanomagnetite and the degree of magnetic anisotropy vary systematically and correlate with the types of magma (basalt-andesite in the first and second volcanic periods; rhyolite in the third and fourth periods). All palaeomagnetic directions are of normal polarity (Brunhes epoch). Their mean overall palaeomagnetic pole (86N, 238E; dp = 5, dm = 6) is statistically indistinguishable from the geographic pole.
The variations in declination and inclination with age, however, are marked by some evidence of a discontinuity between 150 ± 10 and 127 ± 8 ka. This age can be correlated with the beginning of the Blake event of reverse polarity. The discontinuity might therefore correspond to magnetic excursions which occurred immediately before.     

Pollen‐inferred palaeoclimate reconstructions in mountain areas: problems and perspectives

Transfer functions are an efficient tool for the quantitative reconstruction of past climate from low to mid‐elevation pollen sites. However, the application of existing methods to high‐altitude pollen assemblages frequently leads to unrealistic results. In the aim of understanding the causes of these biases, the standard ‘best modern analogue’ method has been applied to two high‐altitude pollen sequences to provide quantitative climate estimates for the Lateglacial and Holocene periods. Both pollen sequences (Laghi dell'Orgials, 2130 m, SW aspect and Lago delle Fate, 2240 m, E aspect) are located in the subalpine belt, on opposing sides of the St. Anna di Vinadio Valley (Italian Maritime Alps). Different results were obtained from the two sequences. The largest differences occurred in palaeotemperature reconstruction, with notable differences in both the values and trends at each site. These biases may be attributed to: (1) a lack of high elevation ‘best modern analogues’ in the database of modern samples; (2) the problem of pollen taxa that have multiple climatic significance; (3) problems related to the complexity of mountainous ecosystems, such as the phenomenon of uphill transport of tree pollen by wind. Possible improvements to the reconstruction process are discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

On the Relationship Between Fast Lyapunov Indicator and Periodic Orbits for Continuous Flows

It is already known (Froeschlé et al., 1997a) that the fast Lyapunov indicator (hereafter FLI), i.e. the computation on a relatively short time of a quantity related to the largest Lyapunov indicator, allows us to discriminate between ordered and weak chaotic motion. Using the FLI many results have been obtained on the standard map taken as a model problem. On this model we are not only able to discriminate between a short time weak chaotic motion and an ordered one, but also among regular motion between non resonant and resonant orbits. Moreover, periodic orbits are characterised by constant FLI values which appear to be related to the order of periodic orbits (Lega and Froeschlé, 2001). In the present paper we extend all these results to the case of continuous dynamical systems (the Hénon and Heiles system and the restricted three-body problem). Especially for the periodic orbits we need to introduce a new value: the orthogonal FLI in order to fully recover the results obtained for mappings.     

Neutron-loaded outflows in gamma-ray bursts

The force-free limit of magnetohydrodynamics (MHD) is often a reasonable approximation to model black hole and neutron star magnetospheres. We describe a general relativistic force-free (GRFFE) formulation that allows general relativistic magnetohydrodynamic (GRMHD) codes to directly evolve the GRFFE equations of motion. Established, accurate and well-tested conservative GRMHD codes can simply add a new inversion piece of code to their existing code, while continuing to use all the already-developed facilities present in their GRMHD code. We show how to enforce the   E · B = 0  constraint and energy conservation, and we introduce a simplified general model of the dissipation of the electric field to enforce the   B ²− E ² > 0  constraint. We also introduce a simplified yet general method to resolve current sheets, without much reconnection, over many dynamical times. This formulation is incorporated into an existing GRMHD code ( harm ), which is demonstrated to give accurate and robust GRFFE results for Minkowski and black hole space–times.     

Period Change of CU Pegasi

On the basis of the published times of minima and our own observations, we analysed the period change of the Algol-type eclipsing binary CU Pegasi. Over almost seventy years of observations, the parabolic period change has been clearly seen as dP/dt = 1.38 × 10⁻⁶ d/year. The estimated mass transfer in the system is about 1 × 10⁻⁷ M_M⊙/year.     

Transition spectra of dynamical systems

The spectra of ‘stretching numbers’ (or ‘local Lyapunov characteristic numbers’) are different in the ordered and in the chaotic domain. We follow the variation of the spectrum as we move from the centre of an island outwards until we reach the chaotic domain. As we move outwards the number of abrupt maxima in the spectrum increases. These maxima correspond to maxima or minima in the curve a(θ), where a is the stretching number, and θ the azimuthal angle. We explain the appearance of new maxima in the spectra of ordered orbits. The orbits just outside the last KAM curve are confined close to this curve for a long time (stickiness time) because of the existence of cantori surrounding the island, but eventually escape to the large chaotic domain further outside. The spectra of sticky orbits resemble those of the ordered orbits just inside the last KAM curve, but later these spectra tend to the invariant spectrum of the chaotic domain. The sticky spectra are invariant during the stickiness time. The stickiness time increases exponentially as we approach an island of stability, but very close to an island the increase is super exponential. The stickiness time varies substantially for nearby orbits; thus we define a probability of escape P_n(x) at time n for every point x. Only the average escape time in a not very small interval Δx around each x is reliable. Then we study the convergence of the spectra to the final, invariant spectrum. We define the number of iterations, N, needed to approach the final spectrum within a given accuracy. In the regular domain N is small, while in the chaotic domain it is large. In some ordered cases the convergence is anomalously slow. In these cases the maximum value of a_k in the continued fraction expansion of the rotation number a = [a₀,a₁,... a_k,...] is large. The ordered domain contains small higher order chaotic domains and higher order islands. These can be located by calculating orbits starting at various points along a line parallel to the q-axis. A monotonic variation of the sup {q}as a function of the initial condition q₀ indicates ordered motions, a jump indicates the crossing of a localized chaotic domain, and a V-shaped structure indicates the crossing of an island. But sometimes the V-shaped structure disappears if the orbit is calculated over longer times. This is due to a near resonance of the rotation number, that is not followed by stable islands. This revised version was published online in July 2006 with corrections to the Cover Date.