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.     

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.     

A new strategy for global climate protection

This essay proposes an innovative institutional strategy for global climate protection, quite distinct from but ultimately complementary to the UNFCCC climate treaty negotiations. Our “building block” strategy relies on a variety of smaller-scale transnational cooperative arrangements, involving not only states, but also subnational jurisdictions, firms, and civil society organizations, to undertake activities whose primary goal is not climate mitigation but which will achieve greenhouse gas reductions as a byproduct. This strategy avoids the problems inherent in developing an enforceable, comprehensive treaty regime by mobilizing other incentives—including economic self-interest, energy security, cleaner air, and furtherance of international development— to motivate a range of actors to cooperate on actions that will also produce climate benefits. The strategy uses three specific models of regime formation (club, linkage, and dominant actor models) which emerge from economics, international relations, and organizational behavior, to develop a variety of transnational regimes that are generally self-enforcing and sustainable, avoiding the free rider and compliance problems endemic in collective action to provide public goods. These regimes will contribute to global climate action not only by achieving emissions reductions in the short term, but also by creating global webs of cooperation and trust, and by linking the building block regimes to the UNFCCC system through greenhouse gas monitoring and reporting systems. We argue that the building blocks regimes would thereby help secure eventual agreement on a comprehensive climate treaty.     

Stability functions based upon shear functions

The stability functions for momentum and heat under a Richardson number formulation are derived from the nondimensional shear functions under a Monin-Obukhov formulation. The Prandtl number is also derived as a function of the Richardson number. Previously, this has been done only in a limited sense. Because the Richardson number formulation is expressed in closed form, iterative techniques are no longer needed in numerical models that use Monin-Obukhov similarity theory. This time-saving approach is made possible by deriving expressions for the friction velocity and temperature in terms of the Richardson-number-dependent stability functions. In addition, the Richardson number approximation in the lowest layer is made to depend explicitly upon the surface roughness.     

AN HISTORICAL APPROACH TO MOBILITY RESEARCH *

Most existing work on residential mobility has assumed that the household relocation decision is an intrinsically significant object of inquiry. In contrast, we argue that mobility derives its significance primarily from the particular historical and locational contexts within which it occurs. We suggest, therefore, that future mobility research should be directed away from the development of a theory of mobility per se and toward a more explicit articulation of mobility studies to existing theories of urbanization and social change.     

Geomorphic constraints on the active tectonics of southern Taiwan

Taiwan is a region of rapid active tectonics, yet the study of the tectonic processes that shape the interior of the island is difficult due to the high rates of erosion and dense vegetation. We use digital topography to look for indications of active deformation preserved in the local geomorphology. In particular, anomalies in the regional pattern of drainage are used to infer zones of enhanced tectonic activity. The apparent anticlockwise rotation of major river systems in plan view indicates the presence of a diffuse zone of left-lateral shear running down the southeastern side of Taiwan. Asymmetries in the catchments of individual drainage basins show the influence of varying rates of uplift across southern Taiwan, with the most rapid uplift close to Taitung at the indentation point of the Luzon arc with the Chinese continental margin. Our interpretations, though based predominantly on remote-sensing observations, are consistent with the available field evidence. This study demonstrates the usefulness of drainage basins as tectonic markers in the quantification of regional strain and uplift, which may have wider applicability in other deforming parts of the world.