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Relaxation methods in fluid mechanics

The present work considers the iterative solution of a coupled set of difference equations and examines methods that carry successive approximates to a state that is invariant with further iteration and independent of the initial guess. Methods are studied with regard to their efficiency and economy of computer resources. The basic principles of classical relaxation are set forth, with attention confined to linear elliptic equations. This discussion involves the evaluation of the spectral radius that is the magnitude of the eigenvalue with largest modulus. The subject of relaxation is then related to the study of ordinary differential equations and hyperbolic partial differential equations. Problems that occur when linearly dependent eigenvectors appear in the relaxation matrix are discussed, leading to multiply connected eigenvalues in the Jordan canonical form. Finally, a brief survey of relaxation methods used in aerodynamics is given.

Lomax, H.↗

Techniques for Estimating Emissions Factors from Forest Burning: ARCTAS and SEAC4RS Airborne Measurements Indicate Which Fires Produce Ozone

Previous studies of emission factors from biomass burning are prone to large errors since they ignore the interplay of mixing and varying pre-fire background CO2 levels. Such complications severely affected our studies of 446 forest fire plume samples measured in the Western US by the science teams of NASA's SEAC4RS and ARCTAS airborne missions. Consequently we propose a Mixed Effects Regression Emission Technique (MERET) to check techniques like the Normalized Emission Ratio Method (NERM), where use of sequential observations cannot disentangle emissions and mixing. We also evaluate a simpler "consensus" technique. All techniques relate emissions to fuel burned using C(sub burn) = delta C(sub tot) added to the fire plume, where C(sub tot) approximately equals (CO2 + CO). Mixed-effects regression can estimate pre-fire background values of Ctot (indexed by observation j) simultaneously with emissions factors indexed by individual species i, delta epsilon lambda tau alpha−x(sub i)/(C(sub burn))i,j., MERET and "consensus" require more than two emissions indicators. Our studies excluded samples where exogenous CO or CH4 might have been fed into a fire plume, mimicking emission. We sought to let the data on 13 gases and particulate properties suggest clusters of variables and plume types, using non-negative matrix factorization (NMF). While samples were mixtures, the NMF unmixing suggested purer burn types. Particulate properties (bscat, babs, SSA, AAE) and gas-phase emissions were interrelated. Finally, we sought a simple categorization useful for modeling ozone production in plumes. Two kinds of fires produced high ozone: those with large fuel nitrogen as evidenced by remnant CH3CN in the plumes, and also those from very intense large burns. Fire types with optimal ratios of delta-NOy/delta- HCHO associate with the highest additional ozone per unit Cburn, Perhaps these plumes exhibit limited NOx binding to reactive organics. Perhaps these plumes exhibit limited NOx binding to reactive organics.

fires↗

Inversion of Gravity Fields From the Spacecraft Orbital Data Using an Adjoint Operator Approach

In perturbation approximation, the forward problem of orbital dynamics (equations with initial conditions) is linear with respect to variations of coordinates and/or velocities of the spacecraft and to corresponding variations of the gravity field in the models used. The linear operator adjoint to the linear operator of such forward problem turns out to be instrumental in inversion of differences between observed and predicted coordinates/velocities in terms of the updates of harmonics in the initial gravity field model. Based on this approach, the solution of resulting adjoint problem of orbital dynamics can be used to directly evaluate the matrix of partial derivatives of observable differences with respect to the gravity field harmonics. General discussion of the adjoint problem of orbital dynamics is given and an example of a mathematical formalism for the practical retrieval algorithm is presented.

Ustinov, E. A.↗

A Simple Stochastic Model for Generating Broken Cloud Optical Depth and Top Height Fields

A simple and fast algorithm for generating two correlated stochastic twodimensional (2D) cloud fields is described. The algorithm is illustrated with two broken cumulus cloud fields: cloud optical depth and cloud top height retrieved from Moderate Resolution Imaging Spectrometer (MODIS). Only two 2D fields are required as an input. The algorithm output is statistical realizations of these two fields with approximately the same correlation and joint distribution functions as the original ones. The major assumption of the algorithm is statistical isotropy of the fields. In contrast to fractals and the Fourier filtering methods frequently used for stochastic cloud modeling, the proposed method is based on spectral models of homogeneous random fields. For keeping the same probability density function as the (first) original field, the method of inverse distribution function is used. When the spatial distribution of the first field has been generated, a realization of the correlated second field is simulated using a conditional distribution matrix. This paper is served as a theoretical justification to the publicly available software that has been recently released by the authors and can be freely downloaded from http://i3rc.gsfc.nasa.gov/Public codes clouds.htm. Though 2D rather than full 3D, stochastic realizations of two correlated cloud fields that mimic statistics of given fields have proved to be very useful to study 3D radiative transfer features of broken cumulus clouds for better understanding of shortwave radiation and interpretation of the remote sensing retrievals.

Prigarin, Sergei M.↗

Techniques for Estimating Emissions Factors from Forest Burning: ARCTAS and SEAC4RS Airborne Measurements Indicate which Fires Produce Ozone

Previous studies of emission factors from biomass burning are prone to large errors since they ignore the interplay of mixing and varying pre-fire background CO2 levels. Such complications severely affected our studies of 446 forest fire plume samples measured in the Western US by the science teams of NASA's SEAC4RS and ARCTAS airborne missions. Consequently we propose a Mixed Effects Regression Emission Technique (MERET) to check techniques like the Normalized Emission Ratio Method (NERM), where use of sequential observations cannot disentangle emissions and mixing. We also evaluate a simpler "consensus" technique. All techniques relate emissions to fuel burned using C(burn) = delta C(tot) added to the fire plume, where C(tot) approximately equals (CO2 = CO). Mixed-effects regression can estimate pre-fire background values of C(tot) (indexed by observation j) simultaneously with emissions factors indexed by individual species i, delta, epsilon lambda tau alpha-x(sub I)/C(sub burn))I,j. MERET and "consensus" require more than emissions indicators. Our studies excluded samples where exogenous CO or CH4 might have been fed into a fire plume, mimicking emission. We sought to let the data on 13 gases and particulate properties suggest clusters of variables and plume types, using non-negative matrix factorization (NMF). While samples were mixtures, the NMF unmixing suggested purer burn types. Particulate properties (b scant, b abs, SSA, AAE) and gas-phase emissions were interrelated. Finally, we sought a simple categorization useful for modeling ozone production in plumes. Two kinds of fires produced high ozone: those with large fuel nitrogen as evidenced by remnant CH3CN in the plumes, and also those from very intense large burns. Fire types with optimal ratios of delta-NOy/delta- HCHO associate with the highest additional ozone per unit Cburn, Perhaps these plumes exhibit limited NOx binding to reactive organics. Perhaps these plumes exhibit limited NOx binding to reactive organics

ozone↗

Cure kinetics of epoxy matrix resin by differential scanning calorimetry

A study was made on the cure kinetics of an epoxy neat-resin (Narmco 5208) using Differential Scanning Calorimetry (DSC). Two interrelated analytical methods were applied to dynamic DSC data for evaluating the kinetic parameters, such as activation energy, E, the order of reaction, n, and the total heat of polymerization (or crosslinking), delta H sub t. The first method was proposed by Ellerstein (1968), and uses a thorough differential-integral analysis of a single DSC curve to evaluate the kinetic parameters. The second method was proposed by Kissinger (1957), and uses multiple DSC curves obtained at various heating rates to evaluate E regardless of n. Kinetic analysis of Narmco 5208 epoxy resin showed that the reaction order, n, is substantially affected by the rate of heating; i.e., n is approximately 2 at slow scan rates but is reduced to 1.5 at higher scan rates. The activation energy, E, is not affected by the scan rate, and the average value of E is 25.6 + or - 1.8 kcal/mole.

Cizmecioglu, M.↗

Microstructural Evaluations of Baseline HSR/EPM Disk Alloys

Six alloys representing two classes of powder metallurgy nickel-based superalloys were examined by transmission electron microscopy (TEM) and phase extraction. Alloys KM4, CH98, IN-100 and 456 are based on a Ni-18Co-12Cr composition while alloys Rene' 88 DT and SR 3 have lower Al and Co and higher Cr contents. The lambda size distributions were determined from quantitative image analysis of the TEM images. The volume fraction of lambda and carbides and the composition of the phases were determined by a combination of phase extraction and TEM. The results showed many similarities in lambda size distributions, grain boundary serrations, and grain boundary carbide frequencies between alloys KM4, CH98, 456, Rene' 88 DT and SR 3 when heat treated to give an approximate grain size of ASTM 6. The density of grain boundary carbides in KM4 was shown to substantially increase as the grain size increased. IN-100 and 456 subjected to a serration cooling heat treatment had much more complex lambda size distributions with very large intergranular and intragranular secondary lambda as well as finer than average cooling and aging lambda. The grain boundary carbides in IN-100 were similar to the other alloys, but 456 given the serration cooling heat treatment had a more variable density of grain boundary carbides. Examination of the phases extracted from the matrix showed that there were significant differences in the phase chemistries and elemental partitioning ratios between the various alloys.

Gabb, Timothy P.↗

Methodology for sensitivity analysis, approximate analysis, and design optimization in CFD for multidisciplinary applications

In this study involving advanced fluid flow codes, an incremental iterative formulation (also known as the delta or correction form) together with the well-known spatially-split approximate factorization algorithm, is presented for solving the very large sparse systems of linear equations which are associated with aerodynamic sensitivity analysis. For smaller 2D problems, a direct method can be applied to solve these linear equations in either the standard or the incremental form, in which case the two are equivalent. Iterative methods are needed for larger 2D and future 3D applications, however, because direct methods require much more computer memory than is currently available. Iterative methods for solving these equations in the standard form are generally unsatisfactory due to an ill-conditioning of the coefficient matrix; this problem can be overcome when these equations are cast in the incremental form. These and other benefits are discussed. The methodology is successfully implemented and tested in 2D using an upwind, cell-centered, finite volume formulation applied to the thin-layer Navier-Stokes equations. Results are presented for two sample airfoil problems: (1) subsonic low Reynolds number laminar flow; and (2) transonic high Reynolds number turbulent flow.

Taylor, Arthur C., III↗

Some optimal considerations in attitude control systems

The conventional six-engine reaction control jet relay attitude control law with deadband is shown to be a good linear approximation to a weighted time-fuel optimal control law. Techniques for evaluating the value of the relative weighting between time and fuel for a particular relay control law is studied along with techniques to interrelate other parameters for the two control laws. Vehicle attitude control laws employing control moment gyros are then investigated. Steering laws obtained from the expression for the reaction torque of the gyro configuration are compared to a total optimal attitude control law that is derived from optimal linear regulator theory. This total optimal attitude control law has computational disadvantages in the solving of the matrix Riccati equation. Several computational algorithms for solving the matrix Riccati equation are investigated with respect to accuracy, computational storage requirements, and computational speed.

Boland, J. S., III↗

Comparison of some optimal control methods for the design of turbine blades

This paper attempts a comparative study of some numerical methods for the optimal control design of turbine blades whose vibration characteristics are approximated by Timoshenko beam idealizations with shear and incorporating simple boundary conditions. The blade was synthesized using the following methods: (1) conjugate gradient minimization of the system Hamiltonian in function space incorporating penalty function transformations, (2) projection operator methods in a function space which includes the frequencies of vibration and the control function, (3) epsilon-technique penalty function transformation resulting in a highly nonlinear programming problem, (4) finite difference discretization of the state equations again resulting in a nonlinear program, (5) second variation methods with complex state differential equations to include damping effects resulting in systems of inhomogeneous matrix Riccatti equations some of which are stiff, (6) quasi-linear methods based on iterative linearization of the state and adjoint equation. The paper includes a discussion of some substantial computational difficulties encountered in the implementation of these techniques together with a resume of work presently in progress using a differential dynamic programming approach.

Desilva, B. M. E.↗

Mechanical, Chemical and Microstructural Characterization of Monazite-Coated Silicon Carbide Fibers

Tensile strengths of as-received Hi-Nicalon and Sylramic fibers and those having monazite surface coatings, deposited by atmospheric pressure chemical vapor deposition, were measured at room temperature and the Weibull statistical parameters determined. The average tensile strengths of uncoated Hi-Nicalon and Sylramic fibers were 3.19 +/- 0.73 and 2.78 +/- 0.53 GPa with a Weibull modulus of 5.41 and 5.52, respectively. The monazite-coated Hi-Nicalon and Sylramic fibers showed strength loss of approx. 10 and 15 percent, respectively, compared with the as-received fibers. The elemental compositions of the fibers and the coatings were analyzed using scanning Auger microprobe and energy dispersive X-ray spectroscopy. The LaPO4 coating on Hi-Nicalon fibers was approximately stoichiometric and about 50 nm thick. The coating on the Sylramic fibers extended to a depth of about 100 to 150 nm. The coating may have been stoichiometric LaPO4 in the first 30 to 40 nm of the layer. However, the surface roughness of Sylramic fiber made this profile somewhat difficult to interpret. Microstructural analyses of the fibers and the coatings were done by scanning electron microscopy, transmission electron microscopy, and selected area electron diffraction. Hi-Nicalon fiber consists of fine beta-SiC nanocrystals ranging in size from 1 to 30 mn embedded in an amorphous matrix. Sylramic is a polycrystalline stoichiometric silicon carbide fiber consisting of submicron beta-SiC crystallites ranging from 100 to 300 nm. Small amount of TiB2 nanocrystallites (approx. 50 nm) are also present. The LaPO4 coating on Hi-Nicalon fibers consisted of a chain of peanut shape particles having monazite-(La) structure. The coating on Sylramic fibers consisted of two layers. The inner layer was a chain of peanut shape particles having monazite-(La) structure. The outer layer was comprised of much smaller particles with a microcrystalline structure.

Bansal, N. P.↗

Zircon, Baddeleyite, and Reidite Found in Ries Crater Suevite

Introduction: Impact events can generate superheated impact melts and even vapor [1]. In the past decade, evidence of the high temperatures and high pressures of the impact process has been found in the impact melt from terrestrial craters (e.g., Mistastin Lake [2], Meteor Crater [3]) and even on the Moon [4–5]. Often, these studies involve zircon or zirconium-bearing phases. Zircon is a particularly useful mineral, due to its robustness and durability against weathering. It is used principally for chronology, but it has a multitude of geologic applications, including geothermometry and fingerprinting magma sources. Here we focus on its capability of recording impact conditions. Grains of zircon (ZrSiO4) are converted at high temperature and/or pressure during an impact event to reidite (a high-pressure polymorph of ZrSiO4) or to tetragonal-ZrO2 + SiO2 [6]. Reidite and tertragonal-ZrO2 leave identifying relicts in the rocks, markers of the high pressures and/or temperatures these rocks underwent. Ries Crater is a 26-km-diameter peak ring crater in southern Germany that formed approximately 15 Ma [7–9]. Ries is the type locality of the polymict impact breccia known as suevite. Here, we report a microanalytical study of suevite from the Ries impact structure. These analyses are used to inform our understanding of the pressure and temperature conditions involved in creating a polymict impact breccia such as suevite in a peak ring crater. Sample Description: The sample, denoted 16RS08, originates from Otting Quarry, at 48.8777° N, 10.7921° E, approximately 17 km from the center of the Ries Crater in Germany (approximately 4 km outside the crater rim). The material recorded in sample 16RS08 is therefore considered an outer suevite. The outer suevite is a discontinuous layer of polymict impact breccia that occurs outside the central ring of Ries, up to 22 km from the center of the crater [9]. Lithic clasts in the outer suevite consist primarily of crystalline basement rocks (gneiss, granite, amphibolite), with less than 5% of lithic clasts being overlying sedimentary rocks (limestone, sandstone, shale) [9]. The thin section studied contains variably shocked lithic and mineral clasts, impact glass, and interstitial minerals that make up the matrix of the breccia (Fig. 1) [9]. Methods: We used a Cameca SX100 electron probe microanalyzer (EPMA) located in the Kuiper Materials Imaging and Characterization Facility (KMICF) at the University of Arizona to obtain 15 elemental X-ray maps of 16RS08. We next used the JEOL 7900F SEM at the Astromaterials Research & Exploration Science (ARES) at NASA Johnson Space Center (JSC) to obtain electron backscatter diffraction (EBSD) maps and energy dispersive X-ray spectroscopy (EDS) maps of select portions of the section. The EBSD data were collected under beam conditions of 20 kV, and ~9 μA, with step sizes varying from 0.05 to 2 μm. Following EBSD data collection, we processed the data using AZtecCrystal and MTEX, a free MATLAB toolbox. Results: We used the elemental X-ray maps to identify the areas of interest in the section, particularly phosphates and Zr-bearing grains. These areas of interest were then targeted for follow up EBSD and EDS analyses. We have identified, through combined EBSD and EDS analysis, the presence of zircon, reidite, and monoclinic-ZrO2 (baddeleyite) in 16RS08. In 16RS08, we have found singular grains of zircon, zircon with a vermicular baddeleyite halo (Fig. 2a), and granular zircon with reidite (Fig. 2b). The different Zr-rich phases and their corresponding textures signifies that this sample underwent a broad spectrum of pressure and temperature conditions during the impact event. For example, zircons surrounded by a vermicular baddeleyite and SiO2 intergrowth (i.e., Fig. 2a) have been shown to preserve evidence of the extremely high temperatures of impact melt, upwards of 2370 °C [2, 6]. Similarly, reidite and granular zircon (i.e., Fig. 2b) have been shown to preserve evidence of high pressure, as the transition to reidite occurs >30 GPa [3, 6]. Future Work: Next, we will process the EBSD and EDS data for these Zr-rich grains, specifically looking for indicators of cubic- or tetragonal-ZrO2 in the baddeleyite remnants and of shock-precursors to the reidite. To further inform our work, we will also obtain BSE images of these grains using a Hitachi S-4800 SEM in KMICF at the University of Arizona, as well as geochemical spot analyses via EPMA. The data collected will be used to constrain the formation conditions of the Ries Crater outer suevite. Acknowledgments: We thank Ken Domanik and Jerry Chang for their support with data collection. This work was supported by a University of Arizona RII Core Facilities Pilot Program grant and start-up funds to JJB. TME thanks A. Cavosie and N. Timms for assistance during field sampling. We acknowledge support from NASA’s Planetary Science Research program for analysis performed at JSC. References: [1] Melosh H. J. (1989) Oxf. U. Press. [2] Timms et al. (2017) EPSL 477, 52–58. [3] Cavosie et al. (2016) Geology 44:9, 703–706. [4] White et al. (2020) Nature Astr. 4, 974–978. [5] Crow C. A. et al. (2017) GCA 202, 264–284. [6] Timms et al. (2017) Earth-Sci. Rev. 165, 185–202. [7] Schmieder M. et al. (2018) GCA 220, 146–157. [8] Schwarz W. H. et al. (2020) M&PS 55:2, 312–325. [9] Stöffler et al. (2013) M&PS 43:4, 515–589.

Zircon↗

Launch Vehicle Sizing Benefits Utilizing Main Propulsion System Crossfeed and Project Status

To meet the goals for a next generation Reusable Launch Vehicle (RLV), a unique propulsion feed system concept was identified using crossfeed between the booster and orbiter stages that could reduce the Two-Stage-to-Orbit (TSTO) vehicle weight and Design, Development, Test and Evaluation (DDT&E) costs by approximately 25%, while increasing safety and reliability. The Main Propulsion System (MPS) crossfeed water demonstration test program addresses all activities required to reduce the risks for the MPS crossfeed system from a Technology Readiness Level (TRL) of 2 to 4 by the completion of testing and analysis by June 2003. During the initial period, that ended in March 2002, a subscale water flow test article was defined. Procurement of a subscale crossfeed check valve was initiated and the specifications for the various components were developed. The fluid transient and pressurization analytical models were developed separately and successfully integrated. The test matrix for the water flow test was developed to correlate the integrated model. A computational fluid dynamics (CFD) model of the crossfeed check valve was developed to assess flow disturbances and internal flow dynamics. Based on the results, the passive crossfeed system concept was very feasible and offered a safe system to be used in an RLV architecture. A water flow test article was designed to accommodate a wide range of flows simulating a number of different types of propellant systems. During the follow-on period, the crossfeed system model will be further refined, the test article will be completed, the water flow test will be performed, and finally the crossfeed system model will be correlated with the test data. This validated computer model will be used to predict the full-scale vehicle crossfeed system performance.

Chandler, Frank↗

The Mineralogy of Circumstellar Silicates Preserved in Cometary Dust

Interplanetary dust particles (IDPs) contain a record of the building blocks of the solar system including presolar grains, molecular cloud material, and materials formed in the early solar nebula. Cometary IDPs have remained relatively unaltered since their accretion because of the lack of parent body thermal and aqueous alteration. We are using coordinated transmission electron microscope (TEM) and ion microprobe studies to establish the origins of the various components within cometary IDPs. Of particular interest is the nature and abundance of presolar silicates in these particles because astronomical observations suggest that crystalline and amorphous silicates are the dominant grain types produced in young main sequence stars and evolved O-rich stars. Five circumstellar grains have been identified including three amorphous silicate grains and two polycrystalline aggregates. All of these grains are between 0.2 and 0.5 micrometers in size. The isotopic compositions of all five presolar silicate grains fall within the range of presolar oxides and silicates, having large (17)O-enrichments and normal (18)O/(16)O ratios (Group 1 grains from AGB and RG stars). The amorphous silicates are chemically heterogeneous and contain nanophase FeNi metal and FeS grains in a Mg-silicate matrix. Two of the amorphous silicate grains are aggregates with subgrains showing variable Mg/Si ratios in chemical maps. The polycrystalline grains show annealed textures (equilibrium grains boundaries, uniform Mg/Fe ratios), and consist of 50-100 nm enstatite and pyrrhotite grains with lesser forsterite. One of the polycrystalline aggregates contains a subgrain of diopside. The polycrystalline aggregates form by subsolidus annealing of amorphous precursors. The bulk compositions of the five grains span a wide range in Mg/Si ratios from 0.4 to 1.2 (avg. 0.86). The average Fe/Si (0.40) and S/Si (0.21) ratios show a much narrower range of values and are approximately 50% of their solar abundances. The latter observation may indicate a decoupling of the silicate and sulfide components in grains that condense in stellar outflows. The amorphous silicate grains described here were not extensively affected by irradiation, sputtering, or thermal processing and may represent relatively pristine circumstellar grains. They are strong candidates for the "dirty silicates" in astronomical observations of circumstellar dust shells. The polycrystalline grains were originally amorphous silicate grains that were likely annealed in the early solar nebula but the processing was not sufficient to erase their anomalous oxygen isotopic compositions.

Keller, L. P.↗

Meteoritic constraints on processes in the solar nebula

The origin of chondritic meteorites is discussed. The characteristics of the chondrite subtypes are shown, and the fine structure of chondrites is described in terms of the three principal categories of aggregated objects present in chondrites: chondrules, Ca,Al-rich inclusions (CAIs), and matrix. Four possibilities for the origin of these components are considered, leading to four important constraints on the chondrite-forming processes, based on mineralogy and astrophysics. It is concluded that the chondrules and CAIs were formed in the solar nebula at a radial distance of about 3 AU. Some type of transient high-energy event or process, operating in zones of dust enrichment, furnished the high temperatures needed to process the chondrules and CAIs. These objects cooled on a time scale of minutes, then on a similarly short time scale began to aggregate, together with dust, into chondritic masses having approximately the solar proportions of major condensable elements.

Wood, J. A.↗

Application of the pseudostate theory to the three-body Lambert problem

The pseudostate theory, which approximates three-body trajectories by overlapping the conic effects of both massive bodies on the third body, has been used to solve boundary-value problems. Frequently, the approach to the secondary is quite close, as in interplanetary gravity-assist or satellite-tour trajectories. In this case, the orbit with respect to the primary is radically changed so that perturbation techniques are time consuming, yet higher accuracy than point-to-point conics is necessary. This method reduces the solution of the three-body Lambert problem to solving two conic Lambert problems and inverting a 7 x 7 matrix, the components of which are all found analytically. Typically 90-95 percent of the point-to-point conic error, with respect to an integrated trajectory, is eliminated.

Byrnes, Dennis V.↗

In-flight demonstration of a Real-Time Flush Airdata Sensing (RT-FADS) system

A prototype real-time flush airdata sensing (RT-FADS) system has been developed and flight tested at the NASA Dryden Flight Research Center. This system uses a matrix of pressure orifices on the vehicle nose to estimate airdata parameters in real time using nonlinear regression. The algorithm is robust to sensor failures and noise in the measured pressures. The RT-FADS system has been calibrated using inertial trajectory measurements that were bootstrapped for atmospheric conditions using meteorological data. Mach numbers as high as 1.6 and angles of attack greater than 45 deg have been tested. The system performance has been evaluated by comparing the RT-FADS to the ship system airdata computer measurements to give a quantitative evaluation relative to an accepted measurement standard. Nominal agreements of approximately 0.003 in Mach number and 0.20 deg in angle of attack and angle of sideslip have been achieved.

Whitmore, Stephen A.↗

Application of the pseudostate theory to the three-body Lambert problem

The pseudostate theory, which approximates three-body trajectories by overlapping the conic effects of both massive bodies on the third body, has been used to solve boundary value problems. Frequently, the approach to the secondary is quite close, as in interplanetary gravity assist trajectories or satellite tour trajectories. In this case the orbit with respect to the primary is radically changed so that perturbation techniques are time consuming, yet higher accuracy than point-to-point conics (V-infinity matching) is necessary. This method reduces the solution of the three-body Lambert problem to solving two conic Lambert problems and inverting a 7 x 7 matrix, the components of which are all found analytically. Typically 90-95% of the point-to-point conic error with respect to an integrated trajectory is eliminated.

Byrnes, D. V.↗