The Genesis Trajectory and Heteroclinic Cycles

Genesis will be NASA's first robotic sample return mission. The purpose of this mission is to collect solar wind samples for two years in an L_1 halo orbit and return them to the Utah Test and Training Range (UTTR) for mid-air retrieval by helicopters. To do this, the Genesis spacecraft makes an excursion into the region around L_2 . This transfer between L_1 and L_2 requires no deterministic maneuvers and is provided by the existence of heteroclinic cycles defined below. The Genesis trajectory was designed with the knowledge of the conjectured existence of these heteroclinic cycles. We now have provided the first systematic, semi-analytic construction of such cycles. The heteroclinic cycle provides several interesting applications for future missions. First, it provides a rapid low-energy dynamical channel between L_1 and L_2 such as used by the Genesis Discovery Mission. Second, it provides a dynamical mechanism for the temporary capture of objects around a planet without propulsion. Third, interactions with the Moon. Here we speak of the interactions of the Sun-Earth Lagrange point dynamics with the Earth-Moon Lagrange point dynamics. We motivate the discussion using Jupiter comet orbits as examples. By studying the natural dynamics of the Solar System, we enhance current and future space mission design

A Multipartite Hajnal-Szemer\'edi Theorem

The celebrated Hajnal-Szemer\'edi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K_k-packing. Fischer's conjecture states that the analogous result holds for all multipartite graphs except for those formed by a single construction. Recently, we deduced an approximate version of this conjecture from new results on perfect matchings in hypergraphs. In this paper, we apply a stability analysis to the extremal cases of this argument, thus showing that the exact conjecture holds for any sufficiently large graph.Comment: Final version, accepted to appear in JCTB. 43 pages, 2 figure

Free-living marine nematode communities: In San Jorge gulf, Argentina

The aim of this study was to investigate the patterns of nematode diversity and community structure in San Jorge Gulf, Argentina, in order to improve knowledge of this key group of organisms. Free-living marine nematodes were sampled at 13 stations in February 2014 during an expedition aboard R/V Coriolis II. We found a total of 188 species (101 of which were new to science) belonging to 98 genera. The statistical results indicated the presence of three different assemblages of free-living marine nematodes distributed spatially in three distinct zones in the gulf: the central part, the outer thermal front at both sides of the entrance, and the south thermal front area. Diversity increased from the coast to the entrance of the gulf, and the highest diversity was found in areas with coarser sediment. Sediment and salinity were the environmental parameters that best matched nematode community distribution.Fil: Pastor de Ward, Catalina T.. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Centro Nacional Patagónico. Instituto de Diversidad y Evolución Austral; ArgentinaFil: Lo Russo, Virginia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Centro Nacional Patagónico. Instituto de Diversidad y Evolución Austral; ArgentinaFil: Varisco, Martin Alejandro. Universidad Nacional de la Patagonia Austral. Centro de Investigaciones y Transferencia Golfo San Jorge. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigaciones y Transferencia Golfo San Jorge. Universidad Nacional de la Patagonia "San Juan Bosco". Centro de Investigaciones y Transferencia Golfo San Jorge; Argentin

Constructing a Low Energy Transfer Between Jovian Moons

There has recently been considerable interest in sending a spacecraft to orbit Europa, the smallest of the four Galilean moons of Jupiter. The trajectory design involved in effecting a capture by Europa presents formidable challenges to traditional conic analysis since the regimes of motion involved depend heavily on three-body dynamics. New three-body perspectives are required to design successful and efficient missions which take full advantage of the natural dynamics. Not only does a three-body approach provide low-fuel trajectories, but it also increases the flexibility and versatility of missions. We apply this approach to design a new mission concept wherein a spacecraft "leap-frogs" between moons, orbiting each for a desired duration in a temporary capture orbit. We call this concept the "Petit Grand Tour." For this application, we apply dynamical systems techniques developed in a previous paper to design a Europa capture orbit. We show how it is possible, using a gravitional boost from Ganymede, to go from a jovicentric orbit beyond the orbit of Ganymede to a ballistic capture orbit around Europa. The main new technical result is the employment of dynamical channels in the phase space - tubes in the energy surface which naturally link the vicinity of Ganymede to the vicinity of Europa. The transfer V necessary to jump from one moon to another is less than half that required by a standard Hohmann transfer

Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L1 and the other around L2, with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L1 and L2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the "interior" and "exterior" Hill's regions and other resonant phenomena

Exploring the relationship between mathematics content knowledge and pedagogical content knowledge among pre-service teachers

This study replicates and extends a previous study (Nathan & Petrosino, 2003) that explored the relationship between the content knowledge and pedagogical content knowledge of pre-service mathematics teachers. In that study, Nathan and Petrosino (2003) examined and reported evidence supporting the counterintuitive hypothesis that, in some situations, having a high degree of content knowledge may be associated with the expert blind spot (symbol-precedence) in pedagogical content knowledge. Pre-service teachers with various levels of expertise in mathematics subject matter were given a series of mathematics problems and asked to rank order their difficulty. Nathan and Petrosino (2003) reported that, on the average, pre-service teachers with more advanced mathematics education courses and fewer pedagogical content knowledge courses rank ordered the problems in ways that were inconsistent with actual patterns of student performance. This suggested that they had less insight into how students think about and solve these problems. In contrast, students who had taken fewer advanced mathematics courses but more pedagogical content knowledge courses rank ordered the problems in a way that was more consistent with actual student performance. The present study builds upon this work. Forty pre-service teachers majoring or minoring in mathematics education were surveyed to assess their knowledge of algebra and aspects of their knowledge regarding how to teach algebra. They were asked to rank order Nathan and Petrosino\u27s (2003) problems in the Difficulty Factor Analysis task. Based on their assessment scores, teacher candidates in three categories were selected for follow-up interviews. The categories were: high content knowledge (CK) and symbol-precedence pedagogical content knowledge (PCK), low CK and verbal-precedence PCK, and high CK and verbal-precedence PCK. The importance of both elements (CK and PCK) for pre-service education majors and professional development programs was also investigated, although no causal relationship was implied. Quantitative results replicated previous findings---students with higher CK showed they used symbol-precedence to teach algebra significantly more than students with lower CK. Follow-up interview data suggest a more complicated relationship between content and pedagogical content knowledge. Those findings revealed that (a) the high CK pre-service teacher with symbol-precedence was knowledge-centered in her teaching perspectives; (b) the low CK pre-service teacher with verbal-precedence was problem-centered in her teaching perspectives; and (c) the high CK pre-service teacher with verbal-precedence was response-based in his teaching perspectives

Statistical Theory of Asteroid Escape Rates

Transition states in phase space are identified and shown to regulate the rate of escape of asteroids temporarily captured in circumplanetary orbits. The transition states, similar to those occurring in chemical reaction dynamics, are then used to develop a statistical semianalytical theory for the rate of escape of asteroids temporarily captured by Mars. Theory and numerical simulations are found to agree to better than 1%. These calculations suggest that further development of transition state theory in celestial mechanics, as an alternative to large-scale numerical simulations, will be a fruitful approach to mass transport calculations

The Mars mapper science and mission planning tool

The Mars Mapper Program (MOm) is an interactive tool for science and mission design developed for the Mars Observer Mission (MO). MOm is a function of the Planning and Sequencing Element of the MO Ground Data System. The primary users of MOm are members of the science and mission planning teams. Using MOm, the user can display digital maps of Mars in various projections and resolutions ranging from 1 to 256 pixels per degree squared. The user can overlay the maps with ground tracks of the MO spacecraft (S/C) and footprints and swaths of the various instruments on-board the S/C. Orbital and instrument geometric parameters can be computed on demand and displayed on the digital map or plotted in XY-plots. The parameter data can also be saved into files for other uses. MOm is divided into 3 major processes: Generator, Mapper, Plotter. The Generator Process is the main control which spawns all other processes. The processes communicate via sockets. At any one time, only 1 copy of MOm may operate on the system. However, up to 5 copies of each of the major processes may be invoked from the Generator. MOm is developed on the Sun SPARCStation 2GX with menu driven graphical user interface (GUI). The map window and its overlays are mouse-sensitized to permit on-demand calculations of various parameters along an orbit. The program is currently under testing and will be delivered to the MO Mission System Configuration Management for distribution to the MO community in 3/93