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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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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

Ultraviolet observations of close-binary and pulsating nuclei of planetary nebulae; Winds and shells around low-mass supergiants; The close-binary nucleus of the planetary nebula HFG-1; A search for binary nuclei of planetary nebulae; UV monitoring of irregularly variable planetary nuclei; and The pulsating nucleus of the planetary nebula Lo 4

A brief summary of the research highlights is presented. The topics covered include the following: binary nuclei of planetary nebulae; other variable planetary nuclei; low-mass supergiants; and other IUE-related research.

Bond, Howard E.↗
DOE OSTI2025

Environmental Impacts of Closed-Loop Pumped Storage Hydropower

The goal of this report is to help license applicants, resource agencies, and other members of the hydropower community involved in closed-loop pumped storage hydropower permitting and licensing process, focus the scope of environmental reviews, and more quickly identify impacts with project nexus and potential mitigation measures for these impacts. Pumped storage hydropower (PSH) is an energy storage technology that uses energy to pump water up from a lower reservoir to an upper reservoir where water is stored until electricity is needed and the water is released to a lower reservoir passing through turbines. Closed-loop PSH—PSH that is not continuously connected to a naturally flowing water feature—is one of the lowest greenhouse gas emitting energy storage technologies and is therefore a critical part of the transition to renewable energy (Simon et al. 2023). Proposals for closed-loop PSH facilities in the United States currently account for more than 40% of original licenses and 99% of potential generation capacity in the Federal Energy Regulatory Commission (FERC) hydropower licensing pipeline (Johnson et al. 2023). While closed-loop PSH facilities can have lower environmental impacts than open-loop PSH facilities, no closed-loop facilities have been constructed in the United States to enable direct accounting of project impacts and efficacy of mitigations. Many proposals for closed-loop PSH submitted to FERC are abandoned early in the permitting and licensing process prior to license applications and environmental assessments, so there is little documentation describing potential project impacts and proposed mitigations. The newness of closed-loop PSH proposals in the United States may mean that tribal, federal, and state agencies with authorities for cultural and natural resources protection and management involved in the FERC licensing process may not have experience with closed-loop PSH regulation. Moreover, many proposed closed-loop PSH facilities are in areas that do not have high concentrations of conventional hydropower, so these agencies may also be unfamiliar with the FERC hydropower licensing process. The goal of this report is to help license applicants, resource agencies, and other members of the hydropower community focus the scope of environmental review for the closed-loop PSH development, licensing, and federal authorization process enabling quicker identification of potential impacts, mitigations, and situations where mitigation may not be possible. We found that environmental impacts of closed-loop PSH are highly site-specific, and generalizations about the types of environmental impacts across closed-loop PSH projects are difficult to make. Environmental impacts of closed-loop PSH are like those for open-loop PSH with a few exceptions including water sourcing, which can lead to delays and contention due to potential complexities with water rights, impacts to aquatic resources, and greenhouse gas emission potential. Cultural resource impacts were commonly reported in National Environmental Policy Act (NEPA) documents reviewed and discussed in interviews, but in many cases such impacts cannot be mitigated.

13 HYDRO ENERGY↗
DOE OSTI2020

A Comparison of the Environmental Effects of Open-Loop and Closed-Loop Pumped Storage Hydropower

Pumped storage hydropower (PSH) capabilities are generally characterized as either open-loop or closed-loop. The U.S. Department of Energy (DOE) defines open-loop PSH as “continuously connected to a naturally flowing water feature,” and closed-loop PSH as “not continuously connected to a naturally flowing water feature.” All the PSH projects constructed in the United States to date are open-loop, so the potential environmental effects of closed-loop systems are not as well documented as the effects of open-loop systems. To address this knowledge gap, the DOE Water Power Technologies Office, under its HydroWIRES Initiative, has prepared this report to: (1) compare the potential environmental effects of open-loop PSH projects with those of closed-loop PSH projects; (2) describe how these effects are being avoided, minimized, or mitigated at existing projects in other countries and proposed projects in the United States; and (3) discuss the relative significance of the environmental issues. The report begins with a brief introduction on the current status of PSH development in the United States. Next, the report describes the current Federal Energy Regulatory Commission (FERC) hydropower licensing process, emphasizing some recent regulatory changes that affect closed-loop PSH development. The report then provides a summary comparison of the environmental effects of constructing and operating open-loop and closed-loop PSH systems and discusses the relative significance of those issues. This comparison and discussion are based on the results of a literature review and a review of FERC licensing records. The literature review includes journal articles, technical reports, and presentations on the environmental effects of PSH systems. It includes literature from the United States as well as countries where closed-loop PSH projects have been constructed. The FERC records review examines the FERC licensing record (e.g., National Environmental Policy Act documents and license orders) to identify the environmental effects anticipated and mitigation measures proposed for six of the closed-loop PSH projects currently licensed or permitted in the United States. For comparison, the FERC records review also discusses the environmental effects and mitigation measures for four open-loop PSH projects proposed or currently operating in the United States. The comparison of effects between open-loop and closed-loop projects is relative; that is, it characterizes the impacts of each project type as generally lower or higher than another project type. The comparison reflects general trends among project types because there are sometimes exceptions to the examples cited. The comparison is based on both spatial (location) and temporal (duration) factors and reflects both the severity and likelihood of effects. The report concludes that the environmental effects of closed-loop PSH projects are generally lower (i.e., more localized and of shorter duration) than those of open-loop PSH projects because they are located “off-stream,” minimizing aquatic and terrestrial impacts, and often have greater siting flexibility than open-loop PSH projects. For certain closed-loop project designs, however, impacts to groundwater may be higher relative to open-loop projects and should be considered in any environmental review.

13 HYDRO ENERGY↗
DOE OSTI2022

Long-term Evolution of Tightly Packed Stellar Black Holes in AGN Disks: Formation of Merging Black Hole Binaries via Close Encounters

We study the long-term evolution of two or more stellar black holes (BHs) on initially separated but unstable circular orbits around a supermassive BH (SMBH). Such a close-packed orbital configuration can naturally arise from BH migrations in the AGN disk. Dynamical instability of the orbits leads to recurring close encounters between two BHs, during which the BH separation r p becomes less than the Hill radius R H . In rare very close encounters, a tight merging BH binary can form with the help of gravitational wave emission. We use N-body simulations to study the time evolution of close encounters of various degrees of closeness. For a typical "SMBH+2BH" system, the averaged cumulative number of close encounters (with r p ≲ R H ) scales approximately as ∝ t 0.5 . The minimum encounter separation r p follows a cumulative distribution P( p ) ∝ r p for r p « R H . We obtain a semi-analytical expression for the averaged rate of binary captures that lead to BH mergers. Our results suggest that close-packed BHs in AGN disks may take a long time (≳10 7 orbits around the SMBH) to experience a sufficiently close encounter and form a bound binary. This time can be shorter if the initial BH orbits are highly aligned. The BH binary mergers produced in this scenario have high eccentricities when entering the LIGO band and broad distribution of orbital inclinations relative to the original AGN disk. We explore the effects of the gas disk and find that simple gas drags on the BHs do not necessarily lead to an enhanced BH binary capture rate.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Half-closed discontinuous Galerkin discretisations

Here we introduce the concept of half-closed nodes for nodal discontinuous Galerkin (DG) discretisations. Unlike more commonly used closed nodes in DG, where on every element nodes are placed on all of its boundaries, half-closed nodes only require nodes to be placed on a subset of the element's boundaries. The effect of using different nodes on DG operator sparsity is studied and we find in particular for there to be no difference in the sparsity pattern of the Laplace operator whether closed or half-closed nodes are used. On quadrilateral/hexahedral elements we use the Gauss-Radau points as the half-closed nodes of choice, which we demonstrate is able to speed up DG operator assembly in addition to leverage previously known superconvergence results. We also discuss in this work some linear solver techniques commonly used for Finite Element or discontinuous Galerkin methods such as static condensation and block-based methods, and how they can be applied to half-closed DG discretisations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Neutral leakage, power dissipation and pedestal fueling in open vs closed divertors

The effect of divertor closure on neutral leakage, power dissipation and pedestal fueling is explored using the SOLPS code for different densities including attached and detached divertor conditions for DIII-D open and closed divertors. Here, a metric for divertor closure in terms of the percentage of neutrals escaping the divertor is presented. At the detachment onset the population of neutrals escaping the divertor is 12% for the closed divertor and 35% for the open divertor. This results in a lower contribution of the convective term to the total heat flux and increased power losses in the closed divertor. The comparison between the two geometries reveals that for the same upstream density, the upstream temperature is 23% higher in the closed divertor. The physical mechanisms for detachment at lower upstream density in the closed divertor are also identified with the reduction of the convective term directly resulting in 40% decrease and the increased power losses contributing with a factor of 2. The baffling also lowers the neutral flux refueling the pedestal resulting in further reduction of the core density for achieving detachment. The results introduced here show that the divertor recycling is the primary fueling mechanism in DIII-D. It is shown that fueling occurring in the region between the X-point and the outer midplane is at least four times lower in the closed divertor due the baffling structure which enhances target neutral trapping and consequently reduced upstream leakage. Due to the openness of the inner leg to the common flux and HFS regions both open and closed divertor experience almost identical fueling at the HFS highlighting the need to also consider the inboard divertor for shape optimization.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

The Baby Universe is Fine and the CFT Knows It: On Holography for Closed Universes

Big bang/big crunch closed universes can be realized in AdS/CFT, even though they lack asymptotically AdS boundaries. With enough bulk entanglement, the bulk Hilbert space of a closed universe can be holographically encoded in the CFT. We clarify the relation of this encoding to observer-clone proposals and refute recent arguments about the breakdown of semiclassical physics in such spaces. In the limit of no bulk entanglement, the holographic encoding breaks down. The oft-cited one-dimensional nature of the closed universe Hilbert space represents the limitation of the external (CFT) Hilbert space to access the quantum information in the closed universe, similar to the limitations imposed on observers outside a perfectly isolated quantum lab. We advocate that the CFT nevertheless continues to determine the physical properties of the closed universe in this regime, showing how to interpret this relationship in terms of a final state projection in the closed universe. We provide a dictionary between the final state wavefunction and CFT data. We propose a model of the emergence of an arrow of time in the universe with a given initial or final state projection. Finally, we show that the conventional EFT in the closed universe, without any projection, can be recovered as a maximally ignorant description of the final state. This conventional EFT is encoded in CFT data, and it can be probed by computing coarse-grained observables. We provide an example of one such observable. Taken together, these results amount to a clean bill of health for baby universes born of AdS/CFT.

FOS: Physical sciences↗
NASA NTRS1991

Closed-loop transfer recovery with observer-based controllers. I - Analysis

The description of a novel problem in loop transfer recovery is presented. The problem of closed-loop recovery (CLTR) is examined using a measurement feedback-control law where the closed-loop transfer function from the external signal to the controlled output can be made either exactly equal or approximately close to the so-called target closed-loop transfer function achieved under full-state feedback. The analysis of CLTR applies specifically to the general class of observer-based controllers and focuses on three fundamental issues. The first issue is concerned with what can and cannot be achieved for a given system and for an arbitrary target closed-loop transfer function. The second issue is to develop necessary and/or sufficient conditions for a target closed-loop transfer function to be recoverable either exactly or approximately. The third issue deals with the necessary and/or sufficient conditions on a given system such that it has at least one recoverable target closed-loop transfer function. The result is applied to the lateral autopilot design for a commercial transport airplane.

Chen, Ben M.↗
NASA NTRS1994

Man-Made Closed Ecological Systems as Model of Natural Ecosystems and as Means to Provide High Quality of Human Life in Adverse Environment

For its more than thirty year long history, the experimental creation of closed ecological systems has from its very sources been distinctly and strongly motivated by the development of human life-support systems for space. As the trend developed its fundamental significance and broad opportunities of terrestrial applications of the technologies under development were coming to the foreground. Nowadays, it can be argued that development of closed ecosystems is experimental foundation of a new branch of ecology biospherics, the goal of which is to comprehend the regularities of existence of the biosphere as a unique in the Universe (in that part of it that we know, at least) closed ecosystem. Closed technologies can be implemented in life-support systems under adverse conditions of life on the Earth - in Arctic and Antarctic latitudes, deserts, high mountains or deep in the ocean, as well as under the conditions of polluted water and air. In space where the environment is hostile for life all around the cell of life should be sealed and the life-support system as close to the ideally closed cyclic turnover of the matter as possible. Under terrestrial conditions designers should strive for maximum closure of the limiting factor: water - in deserts, oxygen - in high mountains, energy - in polar latitudes, etc. Essential closure of a life-support systems withstands also pollution of the environment by the wastes of human vital activity. This is of particular importance for the quarantine of visited planets, and on the Earth under the conditions of deficient heat in high latitudes and water in and areas. The report describes experimental ecosystem 'BIOS' and exohabitats being designed on its basis, which are adapted to various conditions, described capacities of the Center for Closed Ecosystems in Drasnoyarsk for international collaboration in research and education in this field.

Gitelson, I. I.↗
NASA NTRS2005

Probabilistic Analysis of Impact of Wake Vortices on Closely-Spaced Parallel Approaches

One of the primary constraints on the capacity of the nation's air transportation system is the landing capacity of its largest airports. Many airports with closely spaced parallel runways suffer a severe runway acceptance rate when the weather conditions do not allow full utilization of these parallel runways. The present requirement for simultaneous independent landings in Instrument Meteorological Conditions, IMC, is at least 4300 feet of lateral runway spacing (as close as 3000 feet for runways with a Precision Runway Monitor). Operations in Visual Meteorological Conditions, VMC, to Closely Spaced Parallel Approaches only require a lateral runway spacing greater than 750 feet. A study by Hardy and Lewis integrated and extended earlier studies and concepts in lateral traffic separation, longitudinal station keeping, wake prediction, wake display, and the concepts of R N P into a preliminary system concept for Closely Spaced Parallel Approaches in IMC. This system allows IMC airport acceptance rates to approach those for VMC. The system concept that was developed, presented traffic and wake information on the NAVigation Display, NAV, and developed operational procedures for a mix of conventional and Runway Independent Aircraft with different approach speeds to Closely Spaced Parallel Runways. This paper first describes some improvements made on the technology needed to better predict and formulate a probabilistic representation for the time-dependent motion and spreading of the hazardous region associated with the lift-generated vortex wakes of preceding aircraft. In this way, the time at which the vortex wakes of leading aircraft intrude into the airspace of adjacent flight-corridor/runway combinations can be more reliably predicted. Such a prediction is needed because it determines restraints to be placed on in-trail separation distances; or, the allowable time intervals between aircraft executing nearly simultaneous landings or takeoffs on very closely-spaced runways. Improved estimates of wake spreading are achieved by inclusion of representations in the equations for wake spreading due to ambient turbulence and due to the long-wave instability of a vortex pair. Wake motion and spreading due to the time-averaged wind and its variations with time, are retained. The more detailed representation of wake spreading presented here permits the development of probabilistically-based uncertainty estimates for wake spreading. Measurements needed within actual aircraft wake vortices to validate and support this analysis are also described. The second part of the paper uses the improvements in the accuracy of the location of wake vortices to extend the preliminary system concept for Closely Spaced Parallel Approaches described earlier with more robust operational procedures. Additionally, improvements in longitudinal station keeping, wake display, and risk assessment methodologies are incorporated and described.

Hardy, Gordon H.↗
NASA NTRS2010

A General Closed-Form Solution for the Lunar Reconnaissance Orbiter (LRO) Antenna Pointing System

The National Aeronautics and Space Administration s (NASA) Lunar Reconnaissance Orbiter (LRO) launched on June 18, 2009 from the Cape Canaveral Air Force Station aboard an Atlas V launch vehicle into a direct insertion trajectory to the Moon LRO, designed, built, and operated by the NASA Goddard Space Flight Center in Greenbelt, MD, is gathering crucial data on the lunar environment that will help astronauts prepare for long-duration lunar expeditions. During the mission s nominal life of one year its six instruments and one technology demonstrator will find safe landing site, locate potential resources, characterize the radiation environment and test new technology. To date, LRO has been operating well within the bounds of its requirements and has been collecting excellent science data images taken from the LRO Camera Narrow Angle Camera (LROC NAC) of the Apollo landing sites have appeared on cable news networks. A significant amount of information on LRO s science instruments is provided at the LRO mission webpage. LRO s Attitude Control System (ACS), in addition to controlling the orientation of the spacecraft is also responsible for pointing the High Gain Antenna (HGA). A dual-axis (or double-gimbaled) antenna, deployed on a meter-long boom, is required to point at a selected Earth ground station. Due to signal loss over the distance from the Moon to Earth, pointing precision for the antenna system is very tight. Since the HGA has to be deployed in spaceflight, its exact geometry relative to the spacecraft body is uncertain. In addition, thermal distortions and mechanical errors/tolerances must be characterized and removed to realize the greatest gain from the antenna system. These reasons necessitate the need for an in-flight calibration. Once in orbit around the moon, a series of attitude maneuvers was conducted to provide data needed to determine optimal parameters to load onboard, which would account for the environmental and mechanical errors at any antenna orientation. The nominal geometry for the HGA involves an outer gimbal axis that is exactly perpendicular to the inner gimbal axis, and a target direction that is exactly perpendicular to the outer gimbal axis. For this nominal geometry, closed-form solutions of the desired gimbal angles are simple to get for a desired target direction specified in the spacecraft body fame. If the gimbal axes and the antenna boresight are slightly misaligned, the nominal closed-form solution is not sufficiently accurate for computing the gimbal angles needed to point at a target. In this situation, either a general closed-form solution has to be developed for a mechanism with general geometries, or a correction scheme has to be applied to the nominal closed-form solutions. The latter has been adopted for Solar Dynamics Observatory (SDO) as can be seen in Reference 1, and the former has been used for LRO. The advantage of the general closed-form solution is the use of a small number of parameters for the correction of nominal solutions, especially in the regions near singularities. Singularities here refer to cases when the nominal closed-form solutions have two or more solutions. Algorithm complexity, however, is the disadvantage of the general closed-form solution.

Shah, Neerav↗
DOE OSTI2021

Close substellar-mass companions in stellar wide binaries: discovery and characterization with APOGEE and Gaia DR2

We present a search for close, unresolved companions in a subset of spatially resolved Gaia wide binaries containing main-sequence stars within 200 pc of the Sun, utilizing the APOGEE–Gaia Wide Binary Catalog. A catalogue of 37 wide binaries was created by selecting pairs of stars with nearly identical Gaia positions, parallaxes, and proper motions, and then confirming candidates to be gravitationally bound pairs using APOGEE radial velocities. We identify close, unresolved stellar and substellar candidate companions in these multiple systems using (1) the Gaia binary main sequence and (2) observed periodic radial velocity variations in APOGEE measurements due to the influence of a close substellar-mass companion. The studied wide binary pairs reveal a total of four stellar-mass close companions in four different wide binaries, and four substellar-mass close companion candidates in two wide binaries. The latter are therefore quadruple systems, with one substellar-mass companion orbiting each wide binary component in an S-type orbit. Taken at face value, these candidate systems represent an enhancement of an order of magnitude over the expected occurrence rate of ~2 percent of stars having substellar companions >2 M Jup within ~100-d orbits; we discuss implications and possible explanations for this result. Finally, we compare chemical differences between the components of the wide binaries and the components of the candidate higher-order systems and find that any chemical influence or correlation due to the presence of close companions to wide binary stars is not discernible.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1992

System identification from closed-loop data with known output feedback dynamics

This paper presents a procedure to identify the open loop systems when it is operating under closed loop conditions. First, closed loop excitation data are used to compute the system open loop and closed loop Markov parameters. The Markov parameters, which are the pulse response samples, are then used to compute a state space representation of the open loop system. Two closed loop configurations are considered in this paper. The closed loop system can have either a linear output feedback controller or a dynamic output feedback controller. Numerical examples are provided to illustrate the proposed closed loop identification method.

Phan, Minh↗
DOE OSTI2022

Closed-loop control of meltpool temperature in directed energy deposition

The objective of this work is to mitigate flaw formation in powder and laser-based directed energy deposition (DED) additive manufacturing process through close-loop control of the meltpool temperature. In this work, the meltpool temperature was controlled by modulating the laser power based on feedback signals from a coaxial two-wavelength imaging pyrometer. The utility of closed-loop control in DED is demonstrated in the context of practically inspired trapezoid-shaped stainlesssteel parts (SS 316L). We demonstrate that parts built under closed-loop control have reduced variation in porosity and uniform microstructure compared to parts built under open-loop conditions. For example, post-process characterization showed that closed-loop processed parts had a volume percent porosity ranging from 0.036% to 0.043%. In comparison, open-loop processed parts had a larger variation in volume percent porosity ranging from 0.032% to 0.068%. Further, parts built with closed-loop processing depicted consistent dendritic microstructure. By contrast, parts built with open-loop processing showed microstructure heterogeneity with the presence of both dendritic and planar grains, which in turn translated to large variation in microhardness

36 MATERIALS SCIENCE↗
DOE OSTI2023

Revealing the closed pore formation of waste wood-derived hard carbon for advanced sodium-ion battery

Abstract Although the closed pore structure plays a key role in contributing low-voltage plateau capacity of hard carbon anode for sodium-ion batteries, the formation mechanism of closed pores is still under debate. Here, we employ waste wood-derived hard carbon as a template to systematically establish the formation mechanisms of closed pores and their effect on sodium storage performance. We find that the high crystallinity cellulose in nature wood decomposes to long-range carbon layers as the wall of closed pore, and the amorphous component can hinder the graphitization of carbon layer and induce the crispation of long-range carbon layers. The optimized sample demonstrates a high reversible capacity of 430 mAh g −1 at 20 mA g −1 (plateau capacity of 293 mAh g −1 for the second cycle), as well as good rate and stable cycling performances (85.4% after 400 cycles at 500 mA g −1 ). Deep insights into the closed pore formation will greatly forward the rational design of hard carbon anode with high capacity.

25 ENERGY STORAGE↗
DOE OSTI2020

Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed

The Equations of Motion (EOM) for the Heavy Symmetrical Top with One Point Fixed are highly non-linear. The literature describes the numerical methods that are used to resolve this Classical system including modern tools i.e. the Runge-Kutta Fourth Order method. It is more difficult to derivate closed-form solutions for the EOM and as mentioned in the literature it is not always possible to find the close-form solution for all the EOM. Fortunately, there are a few examples available that will serve as a guide to move further on in this topic. It is the purpose of this paper to find a methodology that will produce the solutions for a given subset of EOM that fulfill certain requisites. The report is organized as follows: it starts with a very short summary of the literature available on this topic and quickly follows into the derivation of the EOM using the Euler-Lagrange method. The Routhian will be used to reduce the size of the expression. It continues with the formulation of the Classical cubic function (f(u)) through a novel process. The roots of f(u) are of the outmost importance to be able to find the EOM closed-form solution, and once the final roots are selected the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1972

Studies of effects of closed microbial ecology. Report of 180-day test period

Experiments were performed to determine the influence closed microbial ecologies have on modification or simplification of natural intestinal flora of ferrets in a closed environmental system. On the basis of previous tests in which certain species (Salmonella and Bacteroides) were decreased at 90 days of enclosure, a second trial was constructed for 180-day tests. In this trial there was little difference in the 8 major classes of intestinal flora between animals in the Open and Closed environmental groups except for the level of Lactobacillus. It is of extreme importance to note that when both Open and Closed groups contracted hemorrhagic gastritis, the interrelationship of this agent with other intestinal flora produced a more profound effect on animals from the Closed Group, particularly with reference to Lactobacillus levels.

Kenyon, A. J.↗
NASA NTRS1972

Closed-form solutions for atomspheric flight with applications to shuttle guidance

Closed-form solutions for the motion of a rocket-powered vehicle during atmospheric ascent and closed-form solutions for unpowered atmospheric reentry are developed. These closed-form solutions are then used to develop a simplified guidance scheme and to develop a variation-of-parameters integration of more accurate equations of motion with the closed-form solutions as base solutions. The variation-of-parameters integration of the more accurate equations of motion also allows the transition partial derivative matrices associated with these equations to be easily developed. Then the partial derivative transition matrices are used to develop a guidance scheme based on the more accurate equations of motion instead of the less accurate closed-form solutions.

Ingram, H. L.↗