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All sourcesNASA NTRSDOE OSTI
NASA NTRS1999

Static Properties and Stark Effect of the Ground State of the HD Molecular Ion

We have calculated static properties of the ground state of the HD(+) ion and its lowest-lying P-state without making use of the Born-Oppenheimer approximation, as was done in the case of H2(+) and D2(+) [Phys. Rev. A 58, 2787 (1998)]. The ion is treated as a three-body system whose ground state is spherically symmetric. The wavefunction is of generalized Hylleraas type, but it is necessary to include high powers of the internuclear distance to localize the nuclear motion. We obtain good values of the energies of the ground S-state and lowest P-state and compare them with earlier calculations. Expectation values are obtained for various operators, the Fermi contact parameters, and the permanent quadrupole moment. The cusp conditions are also calculated. The polarizability was then calculated using second-order perturbation theory with intermediate P pseudostates. Since the nuclei in HD(+) are not of equal mass there is dipole coupling between the lowest two rotational states, which are almost degenerate. This situation is carefully analyzed, and the Stark shift is calculated variationally as a function of the applied electric field.

Bhatia, A. K.↗
NASA NTRS1999

Effective Forces Between Colloidal Particles

Colloidal suspensions have proven to be excellent model systems for the study of condensed matter and its phase behavior. Many of the properties of colloidal suspensions can be investigated with a systematic variation of the characteristics of the systems and, in addition, the energy, length and time scales associated with them allow for experimental probing of otherwise inaccessible regimes. The latter property also makes colloidal systems vulnerable to external influences such as gravity. Experiments performed in micro-ravity by Chaikin and Russell have been invaluable in extracting the true behavior of the systems without an external field. Weitz and Pusey intend to use mixtures of colloidal particles with additives such as polymers to induce aggregation and form weak, tenuous, highly disordered fractal structures that would be stable in the absence of gravitational forces. When dispersed in a polarizable medium, colloidal particles can ionize, emitting counterions into the solution. The standard interaction potential in these charged colloidal suspensions was first obtained by Derjaguin, Landau, Verwey and Overbeek. The DLVO potential is obtained in the mean-field linearized Poisson-Boltzmann approximation and thus has limited applicability. For more precise calculations, we have used ab initio density functional theory. In our model, colloidal particles are charged hard spheres, the counterions are described by a continuum density field and the solvent is treated as a homogeneous medium with a specified dielectric constant. We calculate the effective forces between charged colloidal particles by integrating over the solvent and counterion degrees of freedom, taking into account the direct interactions between the particles as well as particle-counterion, counterion-counterion Coulomb, counterion entropic and correlation contributions. We obtain the effective interaction potential between charged colloidal particles in different configurations. We evaluate two- and three-body forces in the bulk as well as study the influence of soft walls. We qualitatively explain the effects of the walls on the forces and demonstrate that many-body effects are negligible in our system. With adjustments in the parameters, the DLVO pair-potential can describe the results quantitatively. Besides electrostatic interactions, entropic depletion effects that arise from (hard-core) exclusion play an important role in determining the behavior of multi-component colloidal suspensions. A standard theory for depletion forces is due to Asakura and Oosawa and is based on the ideal gas approximation. To go beyond this approximation, we have studied entropic forces in molecular dynamics simulations of systems of hard spheres (the effects of the solvent have been ignored). The effective depletion forces for these systems can be found either from equilibrium distribution functions or from direct momentum transfer calculations. Our results obtained by either method show qualitative differences from the Asakura-Oosawa forces, indicating a longer range, higher value at contact and most importantly a more complicated structure, comprising of several maxima and minima. Our calculations include the determination of effective forces between two spheres, a hard sphere and a wall, and the behavior of a hard sphere near a step-edge and a corner. We also demonstrate that such entropic forces do not necessarily satisfy pairwise additivity.

Tehver, Riina↗
NASA NTRS2001

Polarizabilities and Other Properties of the td Muons Molecular Ion

Wavefunctions of Hylleraas type were used earlier to calculate energy levels of muonic systems. Recently, we found in the case of the molecular ions H2+, D2+ and HD+ that it was necessary to include high powers of the internuclear distance in the Hylleraas functions to localize the nuclear motion when treating the ions as three-body systems without invoking the Born-Oppenheimer approximation. We try the same approach in a muonic system, td(mu-). Improved convergence is obtained for J = 0 and 1 states for shorter expansions when we use this type of generalized Hylleraas function, but as the expansion length increases the high powers are no longer useful. We obtain good energy values for the two lowest J = 0 and J = 1 states and compare them with the best earlier calculations. Expectation values are obtained for various operators, the Fermi contact parameters, and the permanent quadrupole moment. The cusp conditions are also calculated. The polarizability of the ground state is then calculated using second-order perturbation theory with intermediate J = 1 pseudostates. It should be possible to measure the polarizability by observing Rydberg states of atoms with td(mu-) acting as the nucleus.

Bhatia, A. K.↗
NASA NTRS2001

High Earth Orbit Design for Lunar-Assisted Medium Class Explorer Missions

This study investigates the application of high-Earth orbit (HEO) trajectories to missions requiring long on-target integration times, avoidance of the Earth's radiation belt, and minimal effects of Earth and Lunar shadow periods which could cause thermal/mechanical stresses on the science instruments. As used here, a HEO trajectory is a particular solution to the restricted three-body problem in the Earth-Moon system with the orbit period being either 1/2 of, or 1/4 of, the lunar sidereal period. A primary mission design goal is to find HEO trajectories where, for a five-year mission duration, the minimum perigee radius is greater than seven Earth radii (R(sub E)). This minimum perigee radius is chosen so that, for the duration of the mission, the perigee is always above the relatively heavily populated geosynchronous radius of 6.6 R(sub E). A secondary goal is to maintain as high an ecliptic inclination as possible for the duration of the mission to keep the apsis points well out of the Ecliptic plane. Mission design analysis was completed for launch dates in the month of June 2003, using both direct transfer and phasing loop transfer techniques, to a lunar swingby for final insertion into a HEO. Also provided are analysis results of eclipse patterns for the trajectories studied, as well as the effects of launch vehicle errors and launch delays.

McGiffin, Daniel A.↗
NASA NTRS2001

A Non-Linear Approach to Spacecraft Formation Control in the Vicinity of a Collinear Libration Point

An expanding interest in mission design strategies that exploit libration point regions demands the continued development of enhanced, efficient, control algorithms for station-keeping and formation maintenance. This paper discusses the development of a non-linear, formation maintenance, control algorithm for trajectories in the vicinity of a libration point. However, the formulation holds for any trajectory governed by the equations of motion for the restricted three body problem. The control law guarantees exponential convergence, based on a Lyaponov analysis. FreeFlyer and MATLAB provide the simulation environment for controller performance evaluation. The simulation, modeled after the MAXIM Pathfinder mission, maintains the relative position of a "follower" spacecraft with respect to a "leader" spacecraft, stationed near the L2 libration point in the Sun-Earth system. Evaluation metrics are fuel usage and tracking accuracy.

Luquette, Richard J.↗
NASA NTRS1994

Autonomic Responses to Microgravity

The purpose of this report is to describe how changes in autonomic nervous system responses may be used as an index of individual differences in adaptational capacity to space flight. During two separate Spacelab missions, six crewmembers wore an ambulatory monitoring system which enabled continuous recording of their physiological responses for up to twelve hours a day for 3 to 5 mission days. The responses recorded were electrocardiography, respiration wave form, skin conductance level, hand temperature, blood flow to the hands and triaxial accelerations of the head and upper body. Three of these subjects had been given training, before the mission, in voluntary control of these autonomic responses as a means of facilitating adaptation to space. Three of these subjects served as Controls, i.e., did not receive this training but took anti-motion sickness medication. Nearly 300 hours of flight data are summarized. These data were examined using time-series analyses, spectral analyses of heart rate variability, and analyses of variance. Information was obtained on responses to space motion sickness, inflight medications, circadian rhythm, workload and fatigue. Preliminary assessment was made on the effectiveness of self-regulation training as a means of facilitating adaptation, with recommendations for future flights.

Toscano, W. B.↗
NASA NTRS2000

Currents and Flows in Distant Magnetospheres

Space scientists have explored, described, and explained the terrestrial magnetosphere for four decades. Rarely do they point out that the planetary and solar wind parameters controlling the size, shape, and activity of Earth's magnetosphere map out only a small portion of the space of dimensionless parameters that govern magnetospheric properties. With the discovery of Ganymede's magnetosphere, the range of parameters relevant to magnetospheric studies has grown by orders of magnitude. Consider the extremes of Ganymede's and Jupiter's magnetospheres. Jupiter's magnetosphere forms within a plasma flowing at super-Alfvenic speed, whereas Ganymede's forms in a sub-Alfvenic flow. The scale sizes of these magnetospheres, characterized by distances to the magnetopause of order 7x10(exp 6) km and 5x10(exp 3) km, respectively, differ by three orders of magnitude, ranging from 100 to 0.1 times the scale of Earth's magnetosphere. The current systems that control the structure and dynamics of a magnetosphere depend on specific plasma and field properties. Magnetopause currents at Ganymede differ greatly from the forms familiar for Earth and Jupiter, principally because the Mach number of the ambient plasma flow greatly influences the shape of the magnetosphere. A magnetodisk current, present at Jupiter because of its rapid rotation, is absent at Earth and Ganymede. The ring current, extensively investigated at Earth, is probably unimportant at Ganymede because the dynamical variations of the external flow are slow. The ring current is subsumed within the magnetodisk current at Jupiter. This paper describes and contrasts aspects of these and other current systems for the three bodies.

Kivelson, Margaret Galland↗
NASA NTRS2003

Preliminary Planar Formation: Flight Dynamics Near Sun-Earth L2 Point

NASA's Goddard Space Flight Center is planning a series of missions in the vicinity of the Sun-Earth L2 libration point. Some of these projects will involve a distributed space system of telescope spacecraft acting together as a single telescope for high-resolution. The individual telescopes will be configured in a plane, surrounding a hub, where the telescope plane can be aimed toward various astronomical targets of interest. In preparation for these missions, it is necessary to develop an improved understanding of the dynamical behavior of objects in a planar configuration near L2. The classical circular restricted three body problem is taken as the basis for the analysis. At first order, the motion of such a telescope relative to the hub is described by a system of linear second order differential equations. These equations are identical to the circular restricted problem's linear equations describing the hub motion about L2. Therefore, the fundamental frequencies, both parallel to and normal to the ecliptic plane, are the same for the relative telescope motion as for the hub motion. To maintain the telescope plane for the duration necessary for the planned observations, a halo-type orbit of the telescopes about the hub is investigated. By using a halo orbit, the individual telescopes remain in approximately the same plane over the observation duration. For such an orbit, the fundamental periods parallel to and normal to the ecliptic plane are forced to be the same by careful selection of the initial conditions in order to adjust the higher order forces. The relative amplitudes of the resulting oscillations are associated with the orientation of the telescope plane relative to the ecliptic. As in the circular restricted problem, initial conditions for the linearized equations must be selected so as not to excite the convergent or divergent linear modes. In a higher order analysis, the telescope relative motion equations include the effects of the position of the hub relative to L2. In this paper, the differential equations are developed through second order in the distance of the hub from the libration point. A modified Lindstedt-Poincad perturbation method is employed to construct the solution of these differential equations through that same order of magnitude. In the course of the solution process, relationships are determined between the initial conditions of the telescopes, selected in order to avoid resonance excitation. As the differential equations include the hub position, it is necessary to simultaneously develop the solution for the hub. As has been done in past analyses of the circular restricted problem, the hub position is written in a power series formulation in terms of its distance from L2. Then, in order to be included in the telescope equations, the hub solution is cast in terms of the nonlinear frequency of the relative telescope motion. In the course of the analysis, it is determined that the hub should also maintain a halo orbit - about L2. Additionally, relationships are formed between the initial conditions of the telescopes and the hub. These relationships may be used to associate sets of initial conditions with particular orientations of the telescope plane. The accuracy of the analytical solution is verified through various simulations and comparison to numerical integration of the differential equations. The results of the simulations are presented, along with a graphical representation of the relationships between the initial conditions of the telescopes and hub.

Segerman, Alan M.↗
NASA NTRS2003

Formation and Evolution of Binary Systems Containing Collapsed Stars

This research includes theoretical studies of the formation and evolution of five types of interacting binary systems. Our main focus has been on developing a number of comprehensive population synthesis codes to study the following types of binary systems: (i) cataclysmic variables (#3, #8, #12, #15), (ii) low- and intermediate-mass X-ray binaries (#13, #20, #21), (iii) high-mass X-ray binaries (#14, #17, #22), (iv) recycled binary millisecond pulsars in globular clusters (#5, #10, #ll), and (v) planetary nebulae which form in interacting binaries (#6, #9). The numbers in parentheses refer to papers published or in preparation that are listed in this paper. These codes take a new unified approach to population synthesis studies. The first step involves a Monte Carlo selection of the primordial binaries, including the constituent masses, and orbital separations and eccentricities. Next, a variety of analytic methods are used to evolve the primary star to the point where either a dynamical episode of mass transfer to the secondary occurs (the common envelope phase), or the system evolves down an alternate path. If the residual core of the primary is greater than 2.5 solar mass, it will evolve to Fe core collapse and the production of a neutron star and a supernova explosion. In the case of systems involving neutron stars, a kick velocity is chosen randomly from an appropriate distribution and added to the orbital dynamics which determine the state of the binary system after the supernova explosion. In the third step, all binaries which commence stable mass transfer from the donor star (the original secondary in the binary system) to the compact object, are followed with a detailed binary evolution code. Finally, we include all the relevant dynamics of the binary system. For example, in the case of LMXBs, the binary system, with its recoil velocity from the supernova explosion, is followed in time through its path in the Galactic potential. For our globular cluster study, detailed three-body calculations are used to follow the interactions of the binary with passing field stars.

Rappaport, Saul↗
NASA NTRS2003

Spacecraft Formation Design Near the Sun-Earth L(sub 2) Point

Over the next two decades international space agencies including the National Aeronautics and Space Administration and the European Space Agency are proposing space missions which employ distributed spacecraft technologies to enable vast improvements in remote sensing performance as compared to fundamental performance limitations associated with fairing sizes of even the largest launch vehicles. A key initial step towards enabling such challenging missions is the development of processes and algorithms for designing the desired motion of the spacecraft formation subject to simultaneous gravitational and fuel constraints. In this paper we develop analogous methodologies for designing trajectories of relative motion near the L(sub 2) point as have been thoroughly developed for the Earth-orbiting regime. In this preliminary study, we confine ourselves to the basic assumptions of the Circular Restricted Three-Body Problem where disturbances, non-gravitational effects, and fourth and greater body affects are ignored. The focus is on determining formations that are defined primarily by the natural gravitational effects on the vehicles, such that maintenance over long-term will not require significant fuel consumption.

Collange, Guillaume↗
NASA NTRS2004

Thorium Anomalies in the NW Quadrant of the South Pole-Aitken Basin

The relatively high concentrations of Th near the Imbrium antipode in the South Pole-Aitken (SPA) basin might represent Imbrium ejecta, a consequence of convergence of Th-rich material ejected by the Imbrium impact that occurred in the Th-rich Procellarum KREEP Terrane. Here, we present landing positions for 7500 fragments ejected from Imbrium obtained by three-body (Earth-Moon-fragment) numerical integration for uniformly selected azimuthal launch positions, ejection angles of 45 deg, and velocities from 0.95 to 0.99 lunar escape. This provides an estimate of the density of infalling ejecta fragments to be expected in the vicinity of the Imbrium antipode. Similar calculations for 35 and 50 deg leave large empty regions surrounding the antipode.

Haskin, Larry A.↗
NASA NTRS2001

Chaos in the Solar System

The physical basis of chaos in the solar system is now better understood: In all cases investigated so far, chaotic orbits result from overlapping resonances. Perhaps the clearest examples are found in the asteroid belt. Overlapping resonances account for its kirkwood gaps and were used to predict and find evidence for very narrow gaps in the outer belt. Further afield, about one new "short-peroid" comet is discovered each year. They are believed to come from the "Kuiper Belt" (at 40 AU or more) via chaotic orbits produced by mean-motion and secular resonances with Neptune. Finally, the planetary system itself is not immune from chaos. In the inner solar system, overlapping secular resonances have been identified as the possible source of chaos. For example, Mercury in 1012 years, may suffer a close encounter with Venus or plunge into the Sun. In the outer solar system, three-body resonances have been identified as a source of chaos, but on an even longer time scale of 109 times the age of the solar system. On the human time scale, the planets do follow their orbits in a stately procession, and we can predict their trajectories for hundreds of thousands of years. That is because the mavericks, with shorter instability times, have long since been ejected. The solar system is not stable; it is just old!

Lecar, Myron↗
NASA NTRS1986

The maximum momentum transfer in proton-hydrogen collisions

The upper limit of momentum transfer by a proton to K-shell electrons is calculated in a restricted three-body classical model. The model shows that the infinite upper limit used in practice, is generally good except for low energy protons passing through an extremely rarefied gas.

NASA Discipline Number 22-70↗
NASA NTRS2005

Perturbed Equations of Motion for Formation Flight Near the Sun-Earth L2 Point

NASA is planning missions to the vicinity of the Sun-Earth L(sub 2) point, some involving a distributed system of telescope spacecraft, configured in a plane about a hub. Several sets of differential equations are written for the formation flight of such telescopes relative to the hub, with varying levels of fidelity. Effects are cast as additive perturbations to the circular restricted three-body problem, expanded in terms of the system distanced, to an accuracy of 10-20 m. These include Earth's orbital eccentricity, lunar motion, solar radiation pressure, and small thrusting forces. Simulations validating the expanded differential equations are presented.

Luquette, Richard↗
NASA NTRS2005

Perturbed Equations of Motion for Formation Flight Near the Sun-Earth L2 Point

This Memorandum Report consists of a compilation of three individual reports, of increasing complexity, describing investigations of formation flight of spacecraft in the vicinity of the L2 Sun-Earth 1ibration point. The individual reports form the following parts of this compilation: - Introduction to the relative motion of spacecraft about the Sun-Earth L2 Point - Linear and quadratic modelling and solution of the relative motion - Modelling the Perturbations - Elliptical Earth Orbit, Lunar Gravity, Solar Radiation Pressure, Thrusters. The three parts are self-contained, with somewhat, varying notation and terminology. After fair1y significant literature searches: this new work (of Parts 2 and 3) is deemed to be unique because it describes the primary perturbations to the description of relative motion between nearby spacecraft. The effect of the elliptical motion of the Earth about the Sun was verified to be the dominant perturbation to the circular restricted three body problem. Contributions due to lunar gravity and solar radiation pressure are seen to have much smaller effect.

Segerman, Alan M.↗
NASA NTRS2005

Investigations of Spacecraft Orbits Around the L(2) Sun-Earth Libration Point

Space agencies are planning missions to the vicinity of the Sun-Earth L sub 2 point, some involving a distributed system of telescope spacecraft, configured in a plane about a hub. An improved understanding is developed of their relative motion. First, the telescope equations of motion are written relative to L sub 2 in the context of the classical circular restricted three-body problem, and expanded in terms of the distance from L sub 2. A basic examination is presented of the spacecraft configuration requirements and effects of small orbit insertion errors. Next, the telescope equations of motion relative to the hub are written and further expanded in terms of the hub-L sub 2 and hub-telescope distances. An analytical solution is developed and a halo telescope orbit investigated, with appropriate initial conditions. Then, the force model is extended to include perturbations to an accuracy of 10 to 20 m, east as additive contributions to the circular restricted problem. Perturbations include Earth's orbital eccentricity, lunar motion, solar radiation pressure, and small thrusting forces. Simulations are presented, along with solution sensitivity to errors in hub position knowledge.

RELATIVE MOTION↗
NASA NTRS1994

A Search for Low V Earth-to-Moon Trajectories

A search for low V Earth-to-Moon trajectories has been initiated. Numerical integration of the equations of motion from the circular restricted three-body problem has resulted in the computation of a trajectory that saves more that 100 m/s over a Hohmann transfer, although the flight time is almost ten months.

trajectory↗