Lectures in Applied Mathematics. Volume VI - Space Mathematics, Part 2
Applied mathematics in celestial mechanics - theory of librational motions, earth shape, and satellite orbit calculations
CELESTIAL MECHANICS↗
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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.
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The precession and nutation of deformable bodies. II - The form of distorted configurations
Derivation of explicit equations governing deformations of self-gravitating viscous bodies in external force field for application to physical librations of moon
NUTATIONAL OSCILLATION↗
Some periodic solutions of elliptical and restricted four-body problems.
Periodic solutions of elliptical and restricted four-body problems about libration points of restricted three-body problem
FOUR-BODY METHOD↗
Existence of four families of commensurability- related periodic Trojan orbits.
Oscillating libration orbits with period rigorously commensurable in rational fraction to basic long period
OSCILLATION FREQUENCY↗
Resonant oscillations of a flexible satellite in an elliptic orbit.
Librational and flexural resonances induced in satellite whose center of mass is moving in planar elliptic orbit
RESONANT FREQUENCY↗
Nonlinear resonance affecting gravity-gradient stability.
Nonlinear resonance effect on attitude librations of undamped rigid gravity gradient stabilized satellite in circular Earth orbit
GRAVITY GRADIENT SATELLITE↗
The influence of material elasticity on satellite motions.
Material elasticity effects on planar librational motion of rigid satellite under action of gravity gradients
GRAVITY GRADIENT SATELLITE↗
Photography of the Earth's Cloud Satellites from an Aircraft
Photography from aircraft of cloud-like objects in vicinity of libration points L4 and L5 of earth- moon system
AERIAL PHOTOGRAPHY↗
Three-dimensional nonlinear stability analysis of the sun-perturbed earth-moon equilateral points.
Nonlinear analysis of earth-moon system motion stability in three dimensions near L4 libration point when perturbed by sun
PERTURBATION THEORY↗
Far-infrared spectra and space group of crystalline hydrazine and hydrazine-d sub 4.
Far IR spectra and space group of crystalline hydrazine and hydrazine-d4 noting coupling of translational and librational motions
LIBRATIONAL MOTION↗
Stationkeeping Monte Carlo Simulation for the James Webb Space Telescope
The James Webb Space Telescope (JWST) is scheduled to launch in 2018 into a Libration Point Orbit (LPO) around the Sun-Earth/Moon (SEM) L2 point, with a planned mission lifetime of 10.5 years after a six-month transfer to the mission orbit. This paper discusses our approach to Stationkeeping (SK) maneuver planning to determine an adequate SK delta-V budget. The SK maneuver planning for JWST is made challenging by two factors: JWST has a large Sunshield, and JWST will be repointed regularly producing significant changes in Solar Radiation Pressure (SRP). To accurately model SRP we employ the Solar Pressure and Drag (SPAD) tool, which uses ray tracing to accurately compute SRP force as a function of attitude. As an additional challenge, the future JWST observation schedule will not be known at the time of SK maneuver planning. Thus there will be significant variation in SRP between SK maneuvers, and the future variation in SRP is unknown. We have enhanced an earlier SK simulation to create a Monte Carlo simulation that incorporates random draws for uncertainties that affect the budget, including random draws of the observation schedule. Each SK maneuver is planned to optimize delta-V magnitude, subject to constraints on spacecraft pointing. We report the results of the Monte Carlo simulations and discuss possible improvements during flight operations to reduce the SK delta-V budget.
Stationkeepting↗
James Webb Space Telescope Orbit Determination Analysis
The James Webb Space Telescope (JWST) is designed to study and answer fundamental astrophysical questions from an orbit about the Sun-Earth/Moon L2 libration point, 1.5 million km away from Earth. This paper describes the results of an orbit determination (OD) analysis of the JWST mission emphasizing the challenges specific to this mission in various mission phases. Three mid-course correction (MCC) maneuvers during launch and early orbit phase and transfer orbit phase are required for the spacecraft to reach L2. These three MCC maneuvers are MCC-1a at Launch+12 hours, MCC-1b at L+2.5 days and MCC-2 at L+30 days. Accurate OD solutions are needed to support MCC maneuver planning. A preliminary analysis shows that OD performance with the given assumptions is adequate to support MCC maneuver planning. During the nominal science operations phase, the mission requires better than 2 cm/sec velocity estimation performance to support stationkeeping maneuver planning. The major challenge to accurate JWST OD during the nominal science phase results from the unusually large solar radiation pressure force acting on the huge sunshield. Other challenges are stationkeeping maneuvers at 21-day intervals to keep JWST in orbit around L2, frequent attitude reorientations to align the JWST telescope with its targets and frequent maneuvers to unload momentum accumulated in the reaction wheels. Monte Carlo analysis shows that the proposed OD approach can produce solutions that meet the mission requirements.
Libration Point Orbit↗
James Webb Space Telescope Initial Mid-Course Correction Monte Carlo Implementation using Task Parallelism
The James Webb Space Telescope will be launched into a highly elliptical orbit that does not possess sufficient energy to achieve a proper Sun-Earth/Moon L2 libration point orbit. Three mid-course correction (MCC) maneuvers are planned to rectify the energy deficit: MCC-1a, MCC-1b, and MCC-2. To validate the propellant budget and trajectory design methods, a set of Monte Carlo analyses that incorporate MCC maneuver modeling and execution are employed. The first analysis focuses on the effects of launch vehicle injection errors on the magnitude of MCC-1a. The second on the spread of potential V based on the performance of the propulsion system as applied to all three MCC maneuvers. The final highlights the slight, but notable, contribution of the attitude thrusters during each MCC maneuver. Given the possible variations in these three scenarios, the trajectory design methods are determined to be robust to errors in the modeling of the flight system.
James Webb Space Telescope↗
James Webb Space Telescope Initial Mid-Course Correction Monte Carlo Implementation using Task Parallelism
The James Webb Space Telescope will be launched into a highly elliptical orbit that does not possess sufficient energy to achieve a proper Sun-Earth L2 libration point orbit. Three mid-course correction (MCC) maneuvers are planned to rectify the energy deficit: MCC-1a, MCC-1b, and MCC-2. To validate the propellant budget and trajectory design methods, a set of Monte Carlo analyses that incorporate MCC maneuver modeling and execution are employed. The first analysis focuses on the effects of launch vehicle injection errors on the magnitude of MCC-1a. The second on the spread of potential V based on the performance of the propulsion system as applied to all three MCC maneuvers. The final highlights the slight, but notable, contribution of the attitude thrusters during each MCC maneuver. Given the possible variations in these three scenarios, the trajectory design methods are determined to be robust to errors in the modeling of the flight system.
Libration Point Orbit↗
James Webb Space Telescope (JWST) Stationkeeping Monte Carlo Simulations
The James Webb Space Telescope (JWST) will launch in 2018 into a Libration Point Orbit (LPO) around the Sun-EarthMoon (SEM) L2 point, with a planned mission lifetime of 11 years. This paper discusses our approach to Stationkeeping (SK) maneuver planning to determine an adequate SK delta-V budget. The SK maneuver planning for JWST is made challenging by two factors: JWST has a large Sunshield, and JWST will be repointed regularly producing significant changes in Solar Radiation Pressure (SRP). To accurately model SRP we employ the Solar Pressure and Drag (SPAD) tool, which uses ray tracing to accurately compute SRP force as a function of attitude. As an additional challenge, the future JWST observation schedule will not be known at the time of SK maneuver planning. Thus there will be significant variation in SRP between SK maneuvers, and the future variation in SRP is unknown. We have enhanced an earlier SK simulation to create a Monte Carlo simulation that incorporates random draws for uncertainties that affect the budget, including random draws of the observation schedule. Each SK maneuver is planned to optimize delta-V magnitude, subject to constraints on spacecraft pointing. We report the results of the Monte Carlo simulations and discuss possible improvements during flight operations to reduce the SK delta-V budget.
Monte Carlo↗
James Webb Space Telescope Orbit Determination Analysis
The James Webb Space Telescope (JWST) is designed to study and answer fundamental astrophysical questions from an orbit about the Sun-EarthMoon L2 libration point, 1.5 million km away from Earth. Three mid-course correction (MCC) maneuvers during launch and early orbit phase and transfer orbit phase are required for the spacecraft to reach L2. These three MCC maneuvers are MCC-1a at Launch+12 hours, MCC-1b at L+2.5 days and MCC-2 at L+30 days. Accurate orbit determination (OD) solutions are needed to support MCC maneuver planning. A preliminary analysis shows that OD performance with the given assumptions is adequate to support MCC maneuver planning. During the nominal science operations phase, the mission requires better than 2 cmsec velocity estimation performance to support stationkeeping maneuver planning. The major challenge to accurate JWST OD during the nominal science phase results from the unusually large solar radiation pressure force acting on the huge sunshield. Other challenges are stationkeeping maneuvers at 21-day intervals to keep JWST in orbit around L2, frequent attitude reorientations to align the JWST telescope with its targets and frequent maneuvers to unload momentum accumulated in the reaction wheels. Monte Carlo analysis shows that the proposed OD approach can produce solutions that meet the mission requirements.
James Webb Space Telescope↗
Lunar Cube Transfer Trajectory Options
Contingent upon the modification of an initial condition of the injected or deployed orbit. Additionally, these designs can be restricted by the selection of the Cubesat subsystem design such as propulsion or communication. Nonetheless, many trajectory options can be designed with have a wide range of transfer durations, fuel requirements, and final destinations. Our investigation of potential trajectories highlights several design options including deployment into low Earth orbit (LEO), geostationary transfer orbits (GTO), and higher energy direct lunar transfer orbits. In addition to direct transfer options from these initial orbits, we also investigate the use of longer duration Earth-Moon dynamical systems. For missions with an intended lunar orbit, much of the design process is spent optimizing a ballistic capture while other science locations such as Sun-Earth libration or heliocentric orbits may simply require a reduced Delta-V imparted at a convenient location along the trajectory. In this article we examine several design options that meet the above limited deployment and subsystem drivers. We study ways that both impulsive and low-thrust Solar Electric Propulsion (SEP) engines can be used to place the Cubesat first into a highly eccentric Earth orbit, enter the Moon's Sphere of Influence, and finally achieve a highly eccentric lunar orbit. We show that such low-thrust transfers are feasible with a realistic micro-thruster model, assuming that the Cubesat can generate sufficient power for the SEP. Two examples are shown here: (1) A Cubestat injected by Exploration Mission 1 (EM-1) then employing low thrust; and (2) a CubSat deployed in a GTO, then employing impulsive maneuvers. For the EM-1 injected initial design, we increase the EM-1 targeted lunar flyby distance to reduce the energy of the lunar flyby to match that of a typical lMoon system heteroclinic manifold. Figure 1 presents an option that encompasses the similar dynamics as that of the ARTEMIS mission design. Low-thrust maneuvers are used along the manifold trajectory to raise perigee to that of a lunar orbit, adjust the timing with respect to the Moon, rotate the line of apsides, and target a ballistic lunar encounter. In this design a second flyby decreases the orbital energy with respect to the Moon, so that C3 -0.1 km2s2. Another design, shown in Figure 2 emanates from a GTO then uses impulsive maneuvers to phase onto a local Earth-Moon manifold, which then transfers the CubeSat to a lunar encounter.
Dynamical System↗
Lunar Cube Transfer Trajectory Options
Numerous Earth-Moon trajectory and lunar orbit options are available for Cubesat missions. Given the limited Cubesat injection infrastructure, transfer trajectories are contingent upon the modification of an initial condition of the injected or deployed orbit. Additionally, these transfers can be restricted by the selection or designs of Cubesat subsystems such as propulsion or communication. Nonetheless, many trajectory options can be considered which have a wide range of transfer durations, fuel requirements, and final destinations. Our investigation of potential trajectories highlights several options including deployment from low Earth orbit (LEO), geostationary transfer orbits (GTO), and higher energy direct lunar transfers and the use of longer duration Earth-Moon dynamical systems. For missions with an intended lunar orbit, much of the design process is spent optimizing a ballistic capture while other science locations such as Sun-Earth libration or heliocentric orbits may simply require a reduced Delta-V imparted at a convenient location along the trajectory.
Libration Orbits↗