Search NASA⌕ Search

SEARCH · Search NASA

Results for “ORBIT EQUATION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 181 records · Page 10
All sourcesNASA NTRSDOE OSTI
NASA NTRS2001

Trajectory Design Strategies for the NGST L2 Libration Point Mission

The Origins' Next Generation Space Telescope (NGST) trajectory design is addressed in light of improved methods for attaining constrained orbit parameters and their control at the exterior collinear libration point, L2. The use of a dynamical systems approach, state-space equations for initial libration orbit control, and optimization to achieve constrained orbit parameters are emphasized. The NGST trajectory design encompasses a direct transfer and orbit maintenance under a constant acceleration. A dynamical systems approach can be used to provide a biased orbit and stationkeeping maintenance method that incorporates the constraint of a single axis correction scheme.

Folta, David↗
NASA NTRS2003

Evaluation of a Drag-Free Control Concept for Missions in Low Earth Orbit

Atmospheric drag causes the greatest uncertainty in the equations of motion for spacecraft in Low Earth Orbit (LEO). If atmospheric drag eflects can be continuously and autonomously counteracted through the use of a drag-fee control system, drag may essentially be eliminated from the equations of motion for the spacecraft. The main perturbations on the spacecraft will then be those due to the gravitational field, which are much more easily predicted Through dynamical analysis and numerical simulation, this paper presents some potential costs and benefits associated with the fuel used during continuous drag compensation. In light of this cost-benefit analysis, simulation results are used to validate the concept of drag-free control for LEO spacecraft missions having certain characteristics.

Fleck, Melissa E.↗
NASA NTRS1991

Mechanism test bed. Flexible body model report

The Space Station Mechanism Test Bed is a six degree-of-freedom motion simulation facility used to evaluate docking and berthing hardware mechanisms. A generalized rigid body math model was developed which allowed the computation of vehicle relative motion in six DOF due to forces and moments from mechanism contact, attitude control systems, and gravity. No vehicle size limitations were imposed in the model. The equations of motion were based on Hill's equations for translational motion with respect to a nominal circular earth orbit and Newton-Euler equations for rotational motion. This rigid body model and supporting software were being refined.

Compton, Jimmy↗
NASA NTRS2006

On the design of a science orbit about Europa

A science mission about Europa requires high-inclination low-altitude orbits. However, perturbations of Jupiter on the orbiter result in instability. Previous approaches to maximize the lifetime of the orbiter use the doubly averaged problem. We work with the unaveraged equations and find unstable periodic orbits with long lifetimes. These low-altitude repeat ground track solutions exist at all inclinations, making them suitable for mapping missions. The governing dynamics include Hill's model and a Europa gravity field based on synchronous moon theory. Inclusion of additional gravity terms is trivial to the solution method, and for the case of J3, we find a marginal impact on orbit lifetime. The science orbits are found to last on the order of 1 year when the initial conditions are achieved to 11 significant digits and 4 months when only 3 significant digits are achieved. Finally, we demonstrate that the solutions are robust in a realistic ephemeris model, finding average lifetimes of 3 to 4 months for wide range of initial conditions with peak lifetimes of up to 6 months.

unstable↗
NASA NTRS1976

Optimal perturbation models for averaged orbit generation

Averaging techniques applied to the variation of parameters (VOP) formulation of the equations of motion are being investigated as methods for long-term prediction of artificial satellite orbits. Analytically averaged equations were compared with numerically averaged equations with respect to accuracy and efficiency for computation of zonal and nonresonant third-body perturbations. Numerically averaged equations were also evaluated for computation of long-period effects from resonant third-body, tesseral harmonic, and atmospheric drag perturbations. Guidelines will be presented for application of averaged VOP equations to a broad class of orbits.

Long, A. C.↗
NASA NTRS1965

Finite beta effects in a weakly unstable plasma

Finite beta effects on interchange mode in weakly unstable plasma calculated in finite Larmor radius limit - differential equation to obtain stability properties of arbitrary systems

PLASMA INSTABILITY↗
NASA NTRS1973

The early history of the lunar inclination

The effect of tidal friction on the inclination of the lunar orbit to the earth's equator for earth-moon distances of less than 10 earth radii is examined. The results obtained bear on a conclusion drawn by Gerstenkorn and others which has been raised as a fatal objection to the fission hypothesis of lunar origin, namely, that the present nonzero inclination of the moon's orbit to the ecliptic implies a steep inclination of the moon's orbit to the earth's equatorial plane in the early history of the earth-moon system. This conclusion is shown to be valid only for particular rheological models of the earth. The earth is assumed to behave like a highly viscous fluid in response to tides raised in it by the moon. The moon is assumed to be tideless and in a circular orbit about the earth. The equations of tidal friction are integrated numerically to give inclination of the lunar orbit as a function of earth-moon distance.

Rubincam, D. P.↗
NASA NTRS1981

Determination of the potential in a synodic system

Determination of the potential field in a fixed (inertial) system may be accomplished by the solution of a homogeneous linear partial differential equation when a family of orbits of a body moving in the field is given. This partial differential equation was presented and thoroughly analyzed earlier. The present paper discusses the same problem in a rotating system where the centrifugal and Coriolis effects render the pertinent partial differential equation in general non-homogeneous and non-linear. A linear, though non-homogeneous, partial differential equation for the determination of the synodic potential is obtained only in the special case of iso-energetic families of orbits.

Szebehely, V.↗
NASA NTRS1977

Scale-covariant theory of gravitation and astrophysical applications

A scale-covariant theory of gravitation is presented which is characterized by a set of equations that are complete only after a choice of the scale function is made. Special attention is given to gauge conditions and units which allow gravitational phenomena to be described in atomic units. The generalized gravitational-field equations are derived by performing a direct scale transformation, by extending Riemannian geometry to Weyl geometry through the introduction of the notion of cotensors, and from a variation principle. Modified conservation laws are provided, a set of dynamical equations is obtained, and astrophysical consequences are considered. The theory is applied to examine certain homogeneous cosmological solutions, perihelion shifts, light deflections, secular variations of planetary orbital elements, stellar structure equations for a star in quasi-static equilibrium, and the past thermal history of earth. The possible relation of the scale-covariant theory to gauge field theories and their predictions of cosmological constants is discussed.

Canuto, V.↗
NASA NTRS1978

An introduction to shuttle/LDEF retrieval operations: The R-bar approach option

Simulated orbiter direct approaches during long duration exposure facility (LDEF) retrieval operations reveal that the resultant orbiter jet plume fields can significantly disturb LDEF. An alternate approach technique which utilizes orbital mechanics forces in lieu of jets to brake the final orbiter/LDEF relative motion during the final approach, is described. Topics discussed include: rendezvous operations from the terminal phase initiation burn through braking at some standoff distance from LDEF, pilot and copilot activities, the cockpit instrumentation employed, and a convenient coordinate frame for studying the relative motion between two orbiting bodies. The basic equations of motion for operating on the LDEF radius vector are introduced. Practical considerations of implementing an R-bar approach, namely, orbiter/LDEF relative state uncertainties and orbiter control system limitations are explored. A possible R-bar approach strategy is developed and demonstrated.

Hall, W. M.↗
NASA NTRS1981

Derivation of the collision probability between orbiting objects The lifetimes of Jupiter's outer moons

A general form is derived for Opik's equations relating to the probability of collision between two orbiting objects to their orbital elements, and used to determine the collisional lifetime of the eight outer moons of Jupiter. The derivation is based on a concept of spatial density, or average number of objects found in a unit volume, and results in a set of equations that are easily applied to a variety of orbital collision problems. When applied to the outer satellites, which are all in irregular orbits, the equations predict a relatively long collisional lifetime for the four retrograde moons (about 270 billon years on the average) and a shorter time for the four posigrade moons (0.9 billion years). This short time is suggestive of a past collision history, and may account for the orbiting dust detected by Pioneers 10 and 11.

Kessler, D. J.↗
NASA NTRS1977

Gravitational harmonics from shallow resonant orbits

Gravitational constraint (lumped coefficient) equations are derived from GEOS-2 data and a detailed analysis of the shallow resonance problem. The equations follow from elementary perturbation theory and show that all such lumped coefficients are harmonic in the argument of perigee. The best along-track constraints derived from them account for all but about 2% of the 13th-order resonant information in the tracking data. The equations are also in good agreement with recent comprehensive gravity models which use substantial amounts of GEOS-2 data.

Wagner, C. A.↗
NASA NTRS1984

An analytic solution for the orbital perturbations of the Venus Radar Mapper due to gravitational harmonics

Hill's variational equations are solved analytically for the orbital perturbations of a spacecraft nominally in an elliptic orbit around a non-spherical body. The rotation of the central planet about its spin-axis is not considered in the analysis. The perturbations are restricted to the planetary gravitational harmonics only. An extremely simple algorithm is derived to transform the spherical harmonic potentials to the orbital coordinate system, and the resulting accelerations are shown to be simply trigonometric functions of the true anomaly. With the principal matrix solution for the differential equations of the adjoint system given in closed form, the orthogonality of the trigonometric functions makes it possible to obtain an analytic solution for the non-homogeneous problem, at intervals of 2 pi in true anomaly. The solution for orbital perturbations can be extended over several revolutions by applying well-known results from Floquet's theory. The technique is demonstrated with results presented on the spacecraft periapsis altitude for the forthcoming Venus Radar Mapper Mission.

Vijayaraghavan, A.↗
NASA NTRS1975

Passive three-axis stabilization of the Long Duration Exposure Facility

This paper presents an analysis of the attitude dynamics of the Long Duration Exposure Facility (LDEF). LDEF is a large cylindrical gravity gradient stabilized earth satellite which is planned to be delivered to a 270-n mi circular orbit by the space shuttle. The fundamental linear stability, capture requirements, and pitch bias constraints generated by the Garber instability are discussed. Numerical simulations, based on the full nonlinear equations for the coupled orbital and attitude motion of the vehicle and the viscous magnetic damper, show stable behavior of the spacecraft and a damping time constant of 30 to 70 orbits.

Huckins, E. K., III↗
NASA NTRS1990

Model Of Orbital Density Of Air For Computing Drag

Simple, Orbital Density Model for Drag Equations program useful for computing effect of drag over one or more orbits. Mathematical model embodied in program incorporates major changes in density due to solar activity and magnetic activity of Earth. Diurnal (day/night) effects on orbit averaged out. Based on Jacchia daily-average density, evaluated at average time of year. Advantages, right ascension and declination of Sun not needed and computation time much reduced. Written in FORTRAN 77.

Lear, W. M.↗
NASA NTRS1991

Single wall penetration equations

Five single plate penetration equations are compared for accuracy and effectiveness. These five equations are two well-known equations (Fish-Summers and Schmidt-Holsapple), two equations developed by the Apollo project (Rockwell and Johnson Space Center (JSC), and one recently revised from JSC (Cour-Palais). They were derived from test results, with velocities ranging up to 8 km/s. Microsoft Excel software was used to construct a spreadsheet to calculate the diameters and masses of projectiles for various velocities, varying the material properties of both projectile and target for the five single plate penetration equations. The results were plotted on diameter versus velocity graphs for ballistic and spallation limits using Cricket Graph software, for velocities ranging from 2 to 15 km/s defined for the orbital debris. First, these equations were compared to each other, then each equation was compared with various aluminum projectile densities. Finally, these equations were compared with test results performed at JSC for the Marshall Space Flight Center. These equations predict a wide variety of projectile diameters at a given velocity. Thus, it is very difficult to choose the 'right' prediction equation. The thickness of a single plate could have a large variation by choosing a different penetration equation. Even though all five equations are empirically developed with various materials, especially for aluminum alloys, one cannot be confident in the shield design with the predictions obtained by the penetration equations without verifying by tests.

Hayashida, K. B.↗