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At least 343 records · Page 19

A Monte Carlo Based Ray Tracing Tool for Evaluating Wolter-Type Prescriptions

Evaluating Wolter-like X-ray mirror prescriptions via ray tracing is useful for selecting and optimizing the right mirror prescriptions for a specified application. Moreover, incorporating real metrology data into a ray trace and simulating point spread functions (PSF) allows for performance predictions representative of real manufacturing errors and tolerances. In fulfillment of an internship project, an X-ray ray trace routine using a Monte-Carlo method has been developed to examine different Wolter-like prescriptions and characterize their theoretical performances over a specified field of view. This routine includes the ability to use real metrology data to evaluate the impact of figure error on imaging performance. As a test case, the Marshall Grazing Incidence X-ray Spectrometer (MaGIXS) Wolter-I mirror prescription and an equivalent Wolter-Schwarzschild prescription were traced and imaging performance of a specified field of view were mapped. Here we present the approach used in this routine, showcase example results, and discuss future goals for expanding the routine to address azimuthally varying figure errors and surface roughness.

X-ray optics↗

X-57 Wing Structural Load Testing

The X-57 flight project will provide an opportunity to assess the benefits of distributed electric propulsion. The plan is to use a TECNAM P2006T twin-engine light aircraft (Aeronautiche TECNAM S.p.A., Capua, Italy) as the baseline aircraft, but design and fabricate a new wing to test the technology. The wing when fully integrated onto the X-57 TECNAM P2006T fuselage will incorporate two wingtip cruise electric motors and 12 high-lift electric motors along the wing span. The testing described in this paper confirmed the strength of the X-57 wing for flight and provided an opportunity to calibrate the wing flight strain gages for monitoring loads in flight. The X-57 wing was qualification tested in the National Aeronautics and Space Administration Armstrong Flight Research Center Flight Loads Laboratory. This paper documents the airworthiness approach, test setup, instrumentation, and preliminary results. The X-57 ground load testing lessons learned are also discussed.

Wing loading↗

MULTI-WAVELENGTH STUDY OF HESS J1741–302

We present the results of two Chandra X-ray Observatory (CXO) observations of TeV γ-ray source HESS J1741–302. We investigate whether there is any connection between HESS J1741−302 and the sources seen at lower energies. One of the brightest X-ray sources in the HESS J1741–302 field, CXOU J174112.1−302908, appears to be associated with a low-mass star (possibly representing a quiescent low-mass X-ray binary or cataclysmic variable (CV)), hence, it is unlikely to be a source of TeV γ-rays. In the same field we have potentially detected X-rays from WR 98a, which is likely to be a colliding wind binary with massive stars. No TeV emission has been reported so far from such systems although predictions have been made. Finally, we found that the previously reported Suzaku source, Suzaku J1740.5–3014 (which is not covered by the CXO observations), appears to be a hard X-ray source detected by INTERGAL ISGRI, which supports the magnetized CV classification but makes its association with the TeV emission unlikely. The young pulsar PSR B1737–30, so far undetected in X-rays and projected on the sky near the CV, may be the contributor of relativistic particles responsible for the TeV emission.

79 ASTRONOMY AND ASTROPHYSICS↗

SU-F-18C-13: Low-Dose X-Ray CT Reconstruction Using a Hybrid First-Order Method

Purpose: To develop a novel reconstruction method for X-ray CT that can lead to accurate reconstruction at significantly reduced dose levels combining low X-ray incident intensity and few views of projection data. Methods: The noise nature of the projection data at low X-ray incident intensity was modeled and accounted by the weighted least-squares (WLS) criterion. The total variation (TV) penalty was used to mitigate artifacts caused by few views of data. The first order primal-dual (FOPD) algorithm was used to minimize TV in image domain, which avoided the difficulty of the non-smooth objective function. The TV penalized WLS reconstruction was achieved by alternated FOPD TV minimization and projection onto convex sets (POCS) for data fidelity constraints. The proposed FOPD-POCS method was evaluated using the FORBILD jaw phantom and the real cadaver head CT data. Results: The quantitative measures, root mean square error (RMSE) and contrast-to-noise ratio (CNR), demonstrate the superior denoising capability of WLS over LS-based TV iterative reconstruction. The improvement of RMSE (WLS vs. LS) is 15%∼21% and that of CNR is 17%∼72% when the incident counts per ray are ranged from 1×10{sup 5} to 1×10{sup 3}. In addition, the TV regularization can accurately reconstruct images from about 50 views of the jaw phantom. The FOPD-POCS reconstruction reveals more structural details and suffers fewer artifacts in both the phantom and real head images. The FOPD-POCS method also shows fast convergence at low X-ray incident intensity. Conclusion: The new hybrid FOPD-POCS method, based on TV penalized WLS, yields excellent image quality when the incident X-ray intensity is low and the projection views are limited. The reconstruction is computationally efficient since the FOPD minimization of TV is applied only in the image domain. The characteristics of FOPD-POCS can be exploited to significantly reduce radiation dose of X-ray CT without compromising accuracy for diagnosis or treatment planning.

60 APPLIED LIFE SCIENCES↗

WE-AB-207A-09: Optimization of the Design of a Moving Blocker for Cone-Beam CT Scatter Correction: Experimental Evaluation

Purpose: A moving blocker based strategy has shown promising results for scatter correction in cone-beam computed tomography (CBCT). Different geometry designs and moving speeds of the blocker affect its performance in image reconstruction accuracy. The goal of this work is to optimize the geometric design and moving speed of the moving blocker system through experimental evaluations. Methods: An Elekta Synergy XVI system and an anthropomorphic pelvis phantom CIRS 801-P were used for our experiment. A blocker consisting of lead strips was inserted between the x-ray source and the phantom moving back and forth along rotation axis to measure the scatter signal. Accoriding to our Monte Carlo simulation results, three blockers were used, which have the same lead strip width 3.2mm and different gap between neighboring lead strips, 3.2, 6.4 and 9.6mm. For each blocker, three moving speeds were evaluated, 10, 20 and 30 pixels per projection (on the detector plane). Scatter signal in the unblocked region was estimated by cubic B-spline based interpolation from the blocked region. CBCT image was reconstructed by a total variation (TV) based algebraic iterative reconstruction (ART) algorithm from the partially blocked projection data. Reconstruction accuracy in each condition is quantified as CT number error of region of interest (ROI) by comparing to a CBCT reconstructed image from analytically simulated unblocked and scatter free projection data. Results: Highest reconstruction accuracy is achieved when the blocker width is 3.2 mm, the gap between neighboring lead strips is 9.6 mm and the moving speed is 20 pixels per projection. RMSE of the CT number of ROIs can be reduced from 436 to 27. Conclusions: Image reconstruction accuracy is greatly affected by the geometry design of the blocker. The moving speed does not have a very strong effect on reconstruction result if it is over 20 pixels per projection.

60 APPLIED LIFE SCIENCES↗

SU-C-207-04: Reconstruction Artifact Reduction in X-Ray Cone Beam CT Using a Treatment Couch Model

Purpose: to mitigate artifacts induced by the presence of the RT treatment couch in on-board CBCT and improve image quality Methods: a model of a Varian IGRT couch is constructed using a CBCT scan of the couch in air. The model is used to generate a set of forward projections (FP) of the treatment couch at specified gantry angles. The model couch forward projections are then used to process CBCT scan projections which contain the couch in addition to the scan object (Catphan phantom), in order to remove the attenuation component of the couch at any given gantry angle. Prior to pre-processing with the model FP, the Catphan projection data is normalized to an air scan with bowtie filter. The filtered Catphan projections are used to reconstruct the CBCT with an in-house FDK algorithm. The artifact reduction in the processed CBCT scan is assessed visually, and the image quality improvement is measured with the CNR over a few selected ROIs of the Catphan modules. Results: Sufficient match between the forward projected data and the x-ray projections is achieved to allow filtering in attenuation space. Visual improvement of the couch induced artifacts is achieved, with a moderate expense of CNR. Conclusion: Couch model-based correction of CBCT projection data has a potential for qualitative improvement of clinical CBCT scans, without requiring position specific correction data. The technique could be used to produce models of other artifact inducing devices, such as immobilization boards, and reduce their impact on patient CBCT images.

60 APPLIED LIFE SCIENCES↗

Ares I-X Roll Control System Development

Project Managers often face challenging technical, schedule and budget issues. This presentation will explore how the Ares I-X Roll Control System Integrated Product Team (IPT) mitigated challenges such as concurrent engineering requirements and environments and evolving program processes, while successfully managing an aggressive project schedule and tight budget. IPT challenges also included communications and negotiations among inter- and intra-government agencies, including the US Air Force, NASA/MSFC Propulsion Engineering, LaRC, GRC, KSC, WSTF, and the Constellation Program. In order to successfully meet these challenges it was essential that the IPT define those items that most affected the schedule critical path, define early mitigation strategies to reduce technical, schedule, and budget risks, and maintain the end-product focus of an "unmanned test flight" context for the flight hardware. The makeup of the IPT and how it would function were also important considerations. The IPT consisted of NASA/MSFC (project management, engineering, and safety/quality) and contractors (Teledyne Brown Engineering and Pratt and Whitney Rocketdyne, who supplied heritage hardware experience). The early decision to have a small focused IPT working "badgelessly" across functional lines to eliminate functional stove-piping allowed for many more tasks to be done by fewer people. It also enhanced a sense of ownership of the products, while still being able to revert back to traditional roles in order to provide the required technical independence in design reviews and verification closures. This presentation will highlight several prominent issues and discuss how they were mitigated and the resulting Lessons Learned that might benefit other projects.

Unger, Ronald J.↗

Final Report for Isee/3ICE Data Restoration NAG5-7188 Restoration of ISEE-3/ICE X-Ray Data

The objective of this project is to take solar flare X-ray data obtained by ISEE-3/ICE (Third International Sun Earth Explorer/ International Cometary Explorer) spacecraft and convert it from its current non-standard format into FITS binary tables. These files will then be made available to the public at the Solar Data Analysis Center (SDAC).

Fisher, George H.↗

Generation of Bright Soft X-ray Laser Beams (Final Report)

The project goal was to demonstrate new compact soft x-ray lasers emitting high energy ultrashort pulses at shorter wavelengths and increased repetition rates by amplification in atomic transitions in high density laser-created plasmas. To demonstrate these lasers we developed new high power solid state lasers. The project succeeded in demonstrating the shortest wavelength compact soft x-ray laser to date, 6.85 nm wavelength, and the highest repetition rate table-top soft x-ray laser, 400 Hz. We also applied these compact soft x-ray laser sources to determine the composition of materials with nanoscale resolution using a new technique that combines soft x-ray laser ablation with mass spectrometry, and we demonstrated that these lasers are capable of acquiring nanoscale resolution holograms with picosecond time resolution. The combination of an increased pumping efficiency with increased repetition rates can yield soft x-ray laser beams with an unprecedented average power on a table-top for applications demanding a high photon flux. We have demonstrated that the efficiency of plasma-based laser-pumped soft x-ray lasers can be significantly increased by detailed pulse shaping of the pump laser using a new pulse synthesizer. Results include the demonstration of 18.9 nm wavelength laser operation at a record 400 Hz repetition rate, and the extension of repetitive gain-saturated plasma-based table-top x-ray lasers to the shortest wavelength to date, 6.85 nm. Laser amplification was also observed at wavelengths as short at 5.8 nm by isolectronic scaling in Ni-like lanthanide ions. We have also conducted simulations to explore the possibility of demonstrating gain saturated atomic x-ray lasers near the carbon K-edge, in the “carbon window” for the first time. The work conducted under this program also includes the use of compact soft x-ray lasers in applications to problems of chemical interest, such as isotopic composition mapping of micron-size uranium particles with nanoscale resolution, and nanoscale resolution depth profiling analysis of CoNCN-coated electrodes for water oxidation catalysis. We also demonstrated single-shot high resolution soft x-ray Fourier transform holography over a broad 7 µm diameter field of view with ~ 5 ps temporal resolution.

74 ATOMIC AND MOLECULAR PHYSICS↗

Modeling hot gas flow in the low-luminosity active galactic nucleus of NGC 3115

Based on the dynamical black hole (BH) mass estimates, NGC 3115 hosts the closest billion solar mass BH. Deep studies of the center revealed a very underluminous active galactic nucleus (AGN) immersed in an old massive nuclear star cluster. Recent 1 Ms Chandra X-ray visionary project observations of the NGC 3115 nucleus resolved hot tenuous gas, which fuels the AGN. In this paper we connect the processes in the nuclear star cluster with the feeding of the supermassive BH. We model the hot gas flow sustained by the injection of matter and energy from the stars and supernova explosions. We incorporate electron heat conduction as the small-scale feedback mechanism, the gravitational pull of the stellar mass, cooling, and Coulomb collisions. Fitting simulated X-ray emission to the spatially and spectrally resolved observed data, we find the best-fitting solutions with χ{sup 2}/dof = 1.00 for dof = 236 both with and without conduction. The radial modeling favors a low BH mass <1.3 × 10{sup 9} M {sub ☉}. The best-fitting supernova rate and the best-fitting mass injection rate are consistent with their expected values. The stagnation point is at r {sub st} ≲ 1'', so that most of the gas, including the gas at a Bondi radius r{sub B} = 2''-4'', outflows from the region. We put an upper limit on the accretion rate at 2 × 10{sup –3} M {sub ☉} yr{sup –1}. We find a shallow density profile n∝r {sup –β} with β ≈ 1 over a large dynamic range. This density profile is determined in the feeding region 0.''5-10'' as an interplay of four processes and effects: (1) the radius-dependent mass injection, (2) the effect of the galactic gravitational potential, (3) the accretion flow onset at r ≲ 1'', and (4) the outflow at r ≳ 1''. The gas temperature is close to the virial temperature T{sub v} at any radius.

79 ASTRONOMY AND ASTROPHYSICS↗

Coal as Value-Added for Lithium Battery Anodes (Final Report)

Graphite is a basic material used for energy storage in lithium-ion (Li-ion) batteries. More than 70% of worldwide graphite production and nearly 50% of the annual graphite supply for the United States comes from China. Potential economic and national security concerns exist because Li-ion batteries are used not only for the fast-growing electrical vehicle market (15.6% compound annual growth rate [CAGR] from 2018 to 2028 in North America), but also for defense applications including unmanned aerial and ground vehicles. Development of an alternative formulation for Li-ion batteries is key. Building upon previous research confirming the ability of silicon alloys to improve the energy density of anodes, Semplastics developed a novel material based on our X-MAT® polymer-derived ceramic (PDC) technology. The X-MAT anode material is a composite of chemically tailored silicon oxycarbide (SiOC) and domestically sourced coal powder, designed to be a drop-in replacement for graphite within Li-ion battery anodes. Preliminary tests of this material showed more than twice the reversible capacity of graphite anodes (1,000 mAh/g vs. 372 mAh/g), with excellent stability and capacity retention, almost 100% coulombic efficiency, low voltage potential, and low thermal expansion. At full adoption in 2024, we expect the coal utilization to reach 27,300 tons. Through this project, Semplastics proposed to complete development and begin commercialization of this material. The objective was to determine the best formulation for technical performance and economic viability. At the end of the project, the X-MAT anode material is ready for implementation into existing battery manufacturing processes and can have a significant impact on the utilization of coal, with positive effects for the mining sector and the mitigation of carbon dioxide (CO 2 ) emissions.

01 COAL, LIGNITE, AND PEAT↗

High-Energy X-ray Diffraction Microscopy for Nuclear Forensics FY2022 Project Report

Morphological information on nuclear material has been identified using visible light and scanning electron microscopy. These identify qualitative differences in particle morphology. Three-dimensional imaging of materials through alternating scanning electron microscopy imaging and focused ion beam milling has also been used. Unfortunately, these techniques are time- and labor-intensive, with significant sample preparation required and lengthy analysis times. Further, the resulting 3D images are qualitative, require manual identification, and do not capture statistically-representative populations. High energy X-ray 3D imaging using a direct-beam or diffracted-beam (High-Energy Diffraction Microscopy) have been developed at the Advanced Photon Source and can produce quantitative information on grains (phase, location, etc.) and pores (size distribution, sphericity) in a material. These techniques require only minutes to characterize a sample volume and are non-destructive, thus suitable for a wide range of existing samples and for confirmatory analyses to be carried out using conventional microscopy techniques. In this first year of the project, all uranium oxide samples were synthesized and characterized using conventional analyses by the analytical chemistry laboratory. Conventional analysis methods included powder x-ray diffraction, scanning electron microscopy, impurity analysis via inductively coupled plasma mass spectrometry, and infrared spectroscopy. Impurity analysis shows a drop in boron content from UO 3 to the lowest U 3 O 8 calcination temperature, but otherwise no appreciable difference in any sample. Analysis of diffraction data shows a flip of peaks from UO 3 dominated for the 600 °C calcined sample to U 3 O 8 dominated at 700 °C and 800 °C. Analysis of scanning electron microscopy images shows that with increased calcination temperature the size distribution of particles seems to increase and broaden. Both of these last findings are in line with previously published data, though this work used significantly fewer particles to simply show similar trends instead of getting truly quantitative particle analysis. Infrared analysis similarly shows ingrowth of U 3 O 8 as calcination temperature is increased, along with depression of peaks associated with UO 3 and water. Samples were prepared for analysis at the Advanced Photon Source at beamline 1-ID. It is anticipated that analysis will occur in November of 2022. AI/ML techniques to de-noise data coming out of 1-ID during the analyses was also developed during this time using previously gathered data. Preliminary results using a self-supervision technique called Noise2Selfshow good de-noising of data. Once the uranium oxide samples are analyzed, real data will be used to test the de-noising and other AI/ML techniques that may be developed in the second year of the project.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Test Capability Enhancements to the NASA Langley 8-Foot High Temperature Tunnel

The NASA Langley 8-Foot High Temperature Tunnel produces true enthalpy environments simulating flight from Mach 4 to Mach 7, primarily for airbreathing propulsion and aerothermal/thermo-structural testing. Flow conditions are achieved through a methane-air heater and nozzles producing aerodynamic Mach numbers of 4, 5 or 7 and have exit diameters of 8 feet or 4.5 feet. The 12-ft long free-jet test section, housed inside a 26-ft vacuum sphere, accommodates large test articles. Recently, the facility underwent significant upgrades to support hydrocarbon fueled scramjet engine testing and to expand flight simulation capability. The upgrades were required to meet engine system development and flight clearance verification requirements originally defined by the joint NASA-Air Force X-43C Hypersonic Flight Demonstrator Project and now the Air Force X-51A Program. Enhancements to the 8-Ft. HTT were made in four areas: 1) hydrocarbon fuel delivery; 2) flight simulation capability; 3) controls and communication; and 4) data acquisition/processing. The upgrades include the addition of systems to supply ethylene and liquid JP-7 to test articles; a Mach 5 nozzle with dynamic pressure simulation capability up to 3200 psf, the addition of a real-time model angle-of-attack system; a new programmable logic controller sub-system to improve process controls and communication with model controls; the addition of MIL-STD-1553B and high speed data acquisition systems and a classified data processing environment. These additions represent a significant increase to the already unique test capability and flexibility of the facility, and complement the existing array of test support hardware such as a model injection system, radiant heaters, six-component force measurement system, and optical flow field visualization hardware. The new systems support complex test programs that require sophisticated test sequences and precise management of process fluids. Furthermore, the new systems, such as the real-time angle of attack system and the new programmable logic controller enhance the test efficiency of the facility. The motivation for the upgrades and the expanded capabilities is described here.

Harvin, S. F.↗

Understanding the Evolution of Biomass Burning Particles in the Atmosphere: Analysis of BBOP Data, Laboratory Experiments, and Model Integration

Submicron particles in the atmosphere can have a profound influence on Earth’s climate [Stocker et al., 2013]. Particulate matter (PM) affects climate via both direct effects (scattering or absorption of radiation) and indirect effects (changes to the formation and properties of clouds). Biomass burning (BB) is the largest global source of submicron, carbonaceous particles [Wiedinmyer et al., 2011]. Field campaigns have shown that BB aerosol (BBA) can evolve through rapid transformations and removal processes in the atmosphere, with changing radiative and hygroscopic properties, but these processes are not well characterized. Better understanding of the evolution of biomass burning PM in the atmosphere is critical for improving the accuracy of regional to global climate models. In order to address uncertainties in the initial physical, chemical and radiative forcing of BBA and the evolution of these parameters, DOE ASR conducted an airborne field campaign, the Biomass Burning Observation Project (BBOP), during the summer and fall of 2013 to make measurements in the plumes of northwest U.S. wildfires and southeast U.S. agricultural burns. BBOP provided unique, extremely detailed trace gas and particle measurements in fresh biomass burning plumes and captured large, rapid downwind transformations within a few hours of the sources. These fast changes in properties have also been observed in laboratory-generated BB smoke in a smog chamber, but in general the controlling factors and most likely outcomes in the real atmosphere are still not clear. The goals of this project were to 1) analyze and interpret the BBOP measurements in detail, 2) conduct laboratory experiments to explore the formation of tar balls, and 3) to incorporate our findings into improved models of biomass burning emissions and aging. This research focuses on understanding the chemical, physical and optical properties and transformations of BB emissions in the near-field within hours downwind of the burn and on a more regional scale and, in turn, how BB emissions relate to climate forcing. Analysis of the BBOP data paid particular attention to relating SOA production, changes in size distribution, and mixing state to the environmental conditions (gas-phase precursors, relative humidity, radiation, oxidants, NO x , etc.). This project included subcontracts and unfunded collaborations with many of the key participants in the BBOP campaign. Professor Robert Yokelson at the University of Montana is an expert on biomass burning fuels and emissions and participated in BBOP as an unfunded collaborator. Professor Peter Buseck at Arizona State University collected particle samples during BBOP and analyzed some of them as part of this project. Unfunded collaborators include Dr. Arthur Sedlacek, III at Brookhaven National Laboratory (BNL) and Dr. Rahul Zaveri at the Pacific Northwest National Laboratory (PNNL) who were both principal investigators in the BBOP campaign, focusing on measurements and modeling, respectively. We are also working closely with Prof. Qi Zhang at UC Davis to coordinate analysis of BBOP flight data and data from the Mt. Bachelor Observatory (MBO), and with Dr. Kouji Adachi at the Meteorological Institute (Japan) to analyze laboratory-generated tar ball samples.

09 BIOMASS FUELS↗

Coordinated Broadband Monitoring of Variable X-Ray Sources in NGC 1313

The nearby (3.7 Mpc) late-type barred spiral galaxy NGC 1313 contains three ultraluminous X-ray sources (ULXs) that are offset from the center of the galaxy. Two of these have been associated with optical ionized nebulae, and their spectral and variability properties indicate that they contain black holes. The third ULX is SN 1978K, which was the second supernova to be detected and recognized as a supernova from its radio emission, and the first from its X-rays. In this project, we have started a long-term program of X-ray (XMM-Newton), optical (VLT), and radio (ATCA) observations of these and other sources in NGC 1313. This program has been continued through Cycle 3. XMM-Newton observations in cycle 2 consisted of eight 10 ksec priority A monitoring observations between 2003 November 25 and 2004 January 16. The data obtained from these observations have been fully analyzed, and the results have been published in Smith et al. (2004). The results included the identification of the optical counterpart of ULX X-2, and firm upper limits for the radio counterparts of ULXs X-1 and X-2. Previously unknown X-ray spectral variability has been detected in ULXs X-1 and X-2. For SNR 1978K, we were able to extend and refine the X-ray decay light curve. In tandem, we developed a semi-analytical model for the spectral variability of microquasar jets, which will be refined and applied to the combined results from our cycle 2 + 3 observations after the cycle-3 data have been fully analyzed.

Boettcher, Markus↗

Improving NO x Entitlement with Axial Staging

Manufacturers are constantly under pressure to increase the efficiency of air breathing gas turbines for the next generation of ground based combined cycle power plants without a significant increase in NO x emissions. Axially staging some of the fuel and air is one method manufacturers are exploring to increase the firing temperature with a minimal NO x increase. This project used a high-pressure combustion facility to systematically explore the axial stage design space at conditions relevant to engine manufacturers. Chemiluminescence and PIV images of the reacting jets were obtained as well as exit emissions. This data was used to create correlations for reacting jets-in-crossflow, analogous to what currently exists for non-reacting jets-in-crossflow. Two CFD programs were used to model these results to quantifying the accuracy of the codes and help elucidate the underlying physics that governs the high-pressure reacting jet in crossflow. This report indicates a change in flame/flow-field behavior, and emissions characteristics with increasing pressure from 1-5.5 atm. Additionally, the effects of preheating and premixing the axial jet are explored at various conditions and show increased jet liftoff leads to a decrease in NO x emissions.

03 NATURAL GAS↗

Inverting x,y grid coordinates to obtain latitude and longitude in the vanderGrinten projection

The latitude and longitude of a point on the Earth's surface are found from its x,y grid coordinates in the vanderGrinten projection. The latitude is a solution of a cubic equation and the longitude a solution of a quadratic equation. Also, the x,y grid coordinates of a point on the Earth's surface can be found if its latitude and longitude are known by solving two simultaneous quadratic equations.

Rubincam, D. P.↗