Asymptotic analysis of displaced lunar orbits

The design of spacecraft trajectories is a crucial task in space mission design. Solar sail technology appears as a promising form of advanced spacecraft propulsion which can enable exciting new space science mission concepts such as solar system exploration and deep space observation. Although solar sailing has been considered as a practical means of spacecraft propulsion only relatively recently, the fundamental ideas are by no means new (see McInnes1 for a detailed description). A solar sail is propelled by reflecting solar photons and therefore can transform the momentum of the photons into a propulsive force. Solar sails can also be utilised for highly non-Keplerian orbits, such as orbits displaced high above the ecliptic plane (see Waters and McInnes2). Solar sails are especially suited for such non-Keplerian orbits, since they can apply a propulsive force continuously. In such trajectories, a sail can be used as a communication satellite for high latitudes. For example, the orbital plane of the sail can be displaced above the orbital plane of the Earth, so that the sail can stay fixed above the Earth at some distance, if the orbital periods are equal (see Forward3). Orbits around the collinear points of the Earth-Moon system are also of great interest because their unique positions are advantageous for several important applications in space mission design (see e.g. Szebehely4, Roy,5 Vonbun,6 Thurman et al.,7 Gomez et al.8, 9). Several authors have tried to determine more accurate approximations (quasi-Halo orbits) of such equilibrium orbits10. These orbits were first studied by Farquhar11, Farquhar and Kamel10, Breakwell and Brown12, Richardson13, Howell14, 15.If an orbit maintains visibility from Earth, a spacecraft on it (near the L2 point) can be used to provide communications between the equatorial regions of the Earth and the lunar poles. The establishment of a bridge for radio communications is crucial for forthcoming space missions, which plan to use the lunar poles.McInnes16 investigated a new family of displaced solar sail orbits near the Earth-Moon libration points.Displaced orbits have more recently been developed by Ozimek et al.17 using collocation methods. In Baoyin and McInnes18, 19, 20 and McInnes16, 21, the authors describe new orbits which are associated with artificial Lagrange points in the Earth-Sun system. These artificial equilibria have potential applications for future space physics and Earth observation missions. In McInnes and Simmons22, the authors investigate large new families of solar sail orbits, such as Sun-centered halo-type trajectories, with the sail executing a circular orbit of a chosen period above the ecliptic plane. We have recently investigated displaced periodic orbits at linear order in the Earth-Moon restricted three-body system, where the third massless body is a solar sail (see Simo and McInnes23). These highly non-Keplerian orbits are achieved using an extremely small sail acceleration. It was found that for a given displacement distance above/below the Earth-Moon plane it is easier by a factor of order 3.19 to do so at L4=L5 compared to L1=L2 - ie. for a fixed sail acceleration the displacement distance at L4=L5 is greater than that at L1=L2. In addition, displaced L4=L5 orbits are passively stable, making them more forgiving to sail pointing errors than highly unstable orbits at L1=L2.The drawback of the new family of orbits is the increased telecommunications path-length, particularly the Moon-L4 distance compared to the Moon-L2 distance

Invariant manifolds and orbit control in the solar sail three-body problem

In this paper we consider issues regarding the control and orbit transfer of solar sails in the circular restricted Earth-Sun system. Fixed points for solar sails in this system have the linear dynamical properties of saddles crossed with centers; thus the fixed points are dynamically unstable and control is required. A natural mechanism of control presents itself: variations in the sail's orientation. We describe an optimal controller to control the sail onto fixed points and periodic orbits about fixed points. We find this controller to be very robust, and define sets of initial data using spherical coordinates to get a sense of the domain of controllability; we also perform a series of tests for control onto periodic orbits. We then present some mission strategies involving transfer form the Earth to fixed points and onto periodic orbits, and controlled heteroclinic transfers between fixed points on opposite sides of the Earth. Finally we present some novel methods to finding periodic orbits in circumstances where traditional methods break down, based on considerations of the Center Manifold theorem

Designing displaced lunar orbits using low-thrust propulsion

The design of spacecraft trajectories is a crucial task in space mission design. Solar sail technology appears as a promising form of advanced spacecraft propulsion which can enable exciting new space science mission concepts such as solar system exploration and deep space observation. Although solar sailing has been considered as a practical means of spacecraft propulsion only relatively recently, the fundamental ideas are by no means new (see McInnes1 for a detailed description). A solar sail is propelled by re ecting solar photons and therefore can transform the momentum of the photons into a propulsive force. This article focuses on designing displaced lunar orbits using low-thrust propulsion

On Integrable Quantum Group Invariant Antiferromagnets

A new open spin chain hamiltonian is introduced. It is both integrable (Sklyanin`s type $K$ matrices are used to achieve this) and invariant under ${\cal U}_{\epsilon}(sl(2))$ transformations in nilpotent irreps for $\epsilon^3=1$. Some considerations on the centralizer of nilpotent representations and its representation theory are also presented.Comment: IFF-5/92, 13 pages, LaTex file, 8 figures available from author

The Inverse Amplitude Method and Adler Zeros

The Inverse Amplitude Method is a powerful unitarization technique to enlarge the energy applicability region of Effective Lagrangians. It has been widely used to describe resonances from Chiral Perturbation Theory as well as for the Strongly Interacting Symmetry Breaking Sector. In this work we show how it can be slightly modified to account also for the sub-threshold region, incorporating correctly the Adler zeros required by chiral symmetry and eliminating spurious poles. These improvements produce negligible effects on the physical region.Comment: 17 pages, 4 figure

First clear evidence of quantum chaos in the bound states of an atomic nucleus

We study the spectral fluctuations of the $^{208}$Pb nucleus using the complete experimental spectrum of 151 states up to excitation energies of $6.20$ MeV recently identified at the Maier-Leibnitz-Laboratorium at Garching, Germany. For natural parity states the results are very close to the predictions of Random Matrix Theory (RMT) for the nearest-neighbor spacing distribution. A quantitative estimate of the agreement is given by the Brody parameter $\omega$, which takes the value $\omega=0$ for regular systems and $\omega \simeq 1$ for chaotic systems. We obtain $\omega=0.85 \pm 0.02$ which is, to our knowledge, the closest value to chaos ever observed in experimental bound states of nuclei. By contrast, the results for unnatural parity states are far from RMT behavior. We interpret these results as a consequence of the strength of the residual interaction in $^{208}$Pb, which, according to experimental data, is much stronger for natural than for unnatural parity states. In addition our results show that chaotic and non-chaotic nuclear states coexist in the same energy region of the spectrum.Comment: 9 pages, 1 figur

Exceptional orthogonal polynomials and the Darboux transformation

We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape-invariance of these new polynomial families is a direct consequence of the permutability property of the Darboux-Crum transformation.Comment: corrected abstract, added references, minor correction

Large Scale Morphological Segregation in Optically Selected Galaxy Redshift Catalogs

We present the results of an exhaustive analysis of the morphological segregation of galaxies in the CfA and SSRS catalogs through the scaling formalism. Morphological segregation between ellipticals and spirals has been detected at scales up to 15-20 h$^{-1}$ Mpc in the CfA catalog, and up to 20-30 h$^{-1}$ Mpc in the SSRS catalog. Moreover, it is present not only in the densest areas of the galaxy distribution, but also in zones of moderate density.Comment: 9 pages, (1 figure included), uuencode compressed Postscript, (accepted for publication in ApJ Letters), FTUAM-93-2

Quasi-exact solvability beyond the SL(2) algebraization

We present evidence to suggest that the study of one dimensional quasi-exactly solvable (QES) models in quantum mechanics should be extended beyond the usual \sla(2) approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the \sla(2) Lie algebraic method allow for a new larger portion of the spectrum to be obtained algebraically. This is done via another algebraization in which the algebraic hamiltonian cannot be expressed as a polynomial in the generators of \sla(2). We then show an example of a new quasi-exactly solvable potential which cannot be obtained within the Lie-algebraic approach.Comment: Submitted to the proceedings of the 2005 Dubna workshop on superintegrabilit

Cauchy-characteristic Evolution of Einstein-Klein-Gordon Systems

A Cauchy-characteristic initial value problem for the Einstein-Klein-Gordon system with spherical symmetry is presented. Initial data are specified on the union of a space-like and null hypersurface. The development of the data is obtained with the combination of a constrained Cauchy evolution in the interior domain and a characteristic evolution in the exterior, asymptotically flat region. The matching interface between the space-like and characteristic foliations is constructed by imposing continuity conditions on metric, extrinsic curvature and scalar field variables, ensuring smoothness across the matching surface. The accuracy of the method is established for all ranges of $M/R$, most notably, with a detailed comparison of invariant observables against reference solutions obtained with a calibrated, global, null algorithm.Comment: Submitted to Phys. Rev. D, 16 pages, revtex, 7 figures available at http://nr.astro.psu.edu:8080/preprints.htm