A new tool for the study of periodic orbits

We present the generalization of the method of rational Fourier-series approximations for nonlinear ordinary differential equations in galactic dynamics. In previous papers we presented the formulae which describe simple families of periodic orbits with very good accuracy. In this Letter we generalize our formulae to describe periodic orbits of any resonancem:n.     

Fuel optimization for low-thrust Earth–Moon transfer via indirect optimal control

The problem of designing low-energy transfers between the Earth and the Moon has attracted recently a major interest from the scientific community. In this paper, an indirect optimal control approach is used to determine minimum-fuel low-thrust transfers between a low Earth orbit and a Lunar orbit in the Sun–Earth–Moon Bicircular Restricted Four-Body Problem. First, the optimal control problem is formulated and its necessary optimality conditions are derived from Pontryagin’s Maximum Principle. Then, two different solution methods are proposed to overcome the numerical difficulties arising from the huge sensitivity of the problem’s state and costate equations. The first one consists in the use of continuation techniques. The second one is based on a massive exploration of the set of unknown variables appearing in the optimality conditions. The dimension of the search space is reduced by considering adapted variables leading to a reduction of the computational time. The trajectories found are classified in several families according to their shape, transfer duration and fuel expenditure. Finally, an analysis based on the dynamical structure provided by the invariant manifolds of the two underlying Circular Restricted Three-Body Problems, Earth–Moon and Sun–Earth is presented leading to a physical interpretation of the different families of trajectories.     

The axial momentum balance of Earth and its fluid envelope

The emergence of greatly improved data sets over the past decade has heightened awareness of the close relationship between changes in the axial component of the angular momentum of the atmosphere and that of the solid Earth, the latter being reflected in small, though detectable, changes in the planet's rate of rotation. Changes in the large-scale wind field, and hence in atmospheric angular momentum, on intraseasonal through interannual time scales can be associated with a number of identifiable meteorological phenomena, whose further study has been given new impetus by the discovery of their signals in Earth's rotation. Future advances in the subject are apt to occur in connection with new data sets that will help address questions remaining about rapid changes in Earth rotation and the torques responsible for the momentum changes. Also in the coming decade, both new data and modeling approaches should help clarify the role of the oceanic portion of Earth's fluid envelope in the planetary momentum balance.     

Silicic differentiates of abyssal oceanic magmas: Evidence for late-magmatic vapor transport of potassium

Massive, nearly holocrystalline dolerites from DSDP Hole 417D contain from 0.5 to 1.5% of granophyric patches composed mainly of Na-plagioclase and quartz. These patches are compositionally similar to other crystalline silicic rocks from oceanic spreading centers and differ from rarer abyssal silicic glasses. Crystalline varieties withSiO₂60wt.% generally haveNa/K>10, whereas silicic glasses have Na/K in the range 3–6. While crystal fractionation readily accounts for the Na₂O and K₂O contents of abyssal silicic glasses, both the 417D granophyres and other crystalline abyssal silicic rocks have much lower K₂O than that predicted by any reasonable crystal-liquid fractionation model. We propose that high-temperature vapor phase transport is responsible for removal of potassium during late-stage crystallization of these rocks. This allows for the formation of cogenetic silicic glassy and crystalline rocks with greatly different Na/K ratios. These observations and interpretations lead to a more confident assignment of high Na/K silicic rocks of oceanic and ophiolitic environments to a cogenetic origin with basaltic oceanic crust.     

Input-output models

Summary Elementary mass balance and export models are explored in relation to eutrophication as caused by phosphorus and nitrogen. New criteria for phosphorus loading are given in relation to the ratio ‘mean depth-water fill-in-time’ . The results suggest that lakes having long water renewal times are much more sensitive to phosphorus loading than would appear from mean depth only. Further, from comparison of the relative residence time of nitrogen and phosphorus, it is deduced that—with increasing eutrophication—the nitrogen metabolism is speeded up beyond the point of simple proportionality which would explain the transition from phosphorus to nitrogen limitation in highly eutrophied lakes. It is further suggested that the principles derived from eutrophication in regard to the metabolism of phosphorus and nitrogen in lakes are applicable also to other environmental compartments and stress factors.

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Zusammenfassung Elementare Massenbilanzen und Exportmodelle werden bezüglich ihrer Bedeutung für die Eutrophierung von Seen mit Phosphor und Stickstoff untersucht. Als neues Kriterium für die Phosphorbelastung wird das Verh?ltnis ?mittlere Wassertiefe-Zeit der Wassererneuerung? eingeführt. Die Beobachtungen lassen vermuten, dass Seen mit langen Wassererneuerungsperioden viel empfindlicher auf die Phosphorbelastung ansprechen, als aus ihrer mittleren Tiefe geschlossen werden k?nnte. Aus dem Vergleich der mittleren Aufenthaltszeiten von Stickstoff und Phosphor ergibt sich, dass mit zunehmender Eutrophie der Stickstoffumsatz weit st?rker beschleunigt wird, als der einfachen Proportionalit?t zu seiner Konzentrationszunahme entsprechen würde. Dies erkl?rt die Verschiebung von Phosphor zu Stickstoff als limitierendem Faktor in hocheutrophierten Seen. Es wird vermutet, dass die Prinzipien, welche bezüglich der Eutrophierung aus dem Umsatz von Phosphor und Stickstoff in Seen abgeleitet werden, auch für andere Lebensr?ume und andere Belastungsfaktoren angewendet werden k?nnen.

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Résumé Un équilibre massive élémentaire et des modèles de sortie ont été explorés en relation avec l’eutrophisation provoquée par le phosphore et l’azote. De nouveaux critères ont été décrits en relation avec le taux ?profondeur moyenne?—durée de remplissage d’eau? . Les résultats suggèrent que les lacs dont la durée de renouvellement de l’eau est longue, sont beaucoup plus sensibles aux charges de phosphore, que ne laisserait appara?tre le critère seul de la profondeur moyenne. De plus, en comparant les durées relatives de séjour de l’azote et du phosphore, on déduit que—avec la progression de l’eutrophisation—le métabolisme de l’azote est accéléré au delà du point de simple proportionalité qui expliquerait la transition du phosphore à l’azote en tant que nutriment limitant dans des lacs fortement eutrophes. En outre, il est suggéré que les principes, dérivant de l’eutrophisation des lacs en relation avec le métabolisme du phosphore et de l’azote, sont également applicables à d’autres compartiments et facteurs de containte de l’environnement.

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Paper given at the Conference on Chemical-Ecological Considerations for Defining the Goals of Water Pollution Control, Kastanienbaum, Switzerland, April 19–21, 1972.     

Correction: Guidelines and pitfalls of refraction microtremor surveys

Journal of Seismology -