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NASA NTRS2013

Smart-Divert Powered Descent Guidance to Avoid the Backshell Landing Dispersion Ellipse

A smart-divert capability has been added into the Powered Descent Guidance (PDG) software originally developed for Mars pinpoint and precision landing. The smart-divert algorithm accounts for the landing dispersions of the entry backshell, which separates from the lander vehicle at the end of the parachute descent phase and prior to powered descent. The smart-divert PDG algorithm utilizes the onboard fuel and vehicle thrust vectoring to mitigate landing error in an intelligent way: ensuring that the lander touches down with minimum- fuel usage at the minimum distance from the desired landing location that also avoids impact by the descending backshell. The smart-divert PDG software implements a computationally efficient, convex formulation of the powered-descent guidance problem to provide pinpoint or precision-landing guidance solutions that are fuel-optimal and satisfy physical thrust bound and pointing constraints, as well as position and speed constraints. The initial smart-divert implementation enforced a lateral-divert corridor parallel to the ground velocity vector; this was based on guidance requirements for MSL (Mars Science Laboratory) landings. This initial method was overly conservative since the divert corridor was infinite in the down-range direction despite the backshell landing inside a calculable dispersion ellipse. Basing the divert constraint instead on a local tangent to the backshell dispersion ellipse in the direction of the desired landing site provides a far less conservative constraint. The resulting enhanced smart-divert PDG algorithm avoids impact with the descending backshell and has reduced conservatism.

Carson, John M.↗

NASA NTRS2017

Board Level Proton Testing Book of Knowledge for NASA Electronic Parts and Packaging Program

This book of knowledge (BoK) provides a critical review of the benefits and difficulties associated with using proton irradiation as a means of exploring the radiation hardness of commercial-off-the-shelf (COTS) systems. This work was developed for the NASA Electronic Parts and Packaging (NEPP) Board Level Testing for the COTS task. The fundamental findings of this BoK are the following. The board-level test method can reduce the worst case estimate for a board's single-event effect (SEE) sensitivity compared to the case of no test data, but only by a factor of ten. The estimated worst case rate of failure for untested boards is about 0.1 SEE/board-day. By employing the use of protons with energies near or above 200 MeV, this rate can be safely reduced to 0.01 SEE/board-day, with only those SEEs with deep charge collection mechanisms rising this high. For general SEEs, such as static random-access memory (SRAM) upsets, single-event transients (SETs), single-event gate ruptures (SEGRs), and similar cases where the relevant charge collection depth is less than 10 μm, the worst case rate for SEE is below 0.001 SEE/board-day. Note that these bounds assume that no SEEs are observed during testing. When SEEs are observed during testing, the board-level test method can establish a reliable event rate in some orbits, though all established rates will be at or above 0.001 SEE/board-day. The board-level test approach we explore has picked up support as a radiation hardness assurance technique over the last twenty years. The approach originally was used to provide a very limited verification of the suitability of low cost assemblies to be used in the very benign environment of the International Space Station (ISS), in limited reliability applications. Recently the method has been gaining popularity as a way to establish a minimum level of SEE performance of systems that require somewhat higher reliability performance than previous applications. This sort of application of the method suggests a critical analysis of the method is in order. This is also of current consideration because the primary facility used for this type of work, the Indiana University Cyclotron Facility (IUCF) (also known as the Integrated Science and Technology (ISAT) hall), has closed permanently, and the future selection of alternate test facilities is critically important. This document reviews the main theoretical work on proton testing of assemblies over the last twenty years. It augments this with review of reported data generated from the method and other data that applies to the limitations of the proton board-level test approach. When protons are incident on a system for test they can produce spallation reactions. From these reactions, secondary particles with linear energy transfers (LETs) significantly higher than the incident protons can be produced. These secondary particles, together with the protons, can simulate a subset of the space environment for particles capable of inducing single event effects (SEEs). The proton board-level test approach has been used to bound SEE rates, establishing a maximum possible SEE rate that a test article may exhibit in space. This bound is not particularly useful in many cases because the bound is quite loose. We discuss the established limit that the proton board-level test approach leaves us with. The remaining possible SEE rates may be as high as one per ten years for most devices. The situation is actually more problematic for many SEE types with deep charge collection. In cases with these SEEs, the limits set by the proton board-level test can be on the order of one per 100 days. Because of the limited nature of the bounds established by proton testing alone, it is possible that tested devices will have actual SEE sensitivity that is very low (e.g., fewer than one event in 1 × 10(exp 4) years), but the test method will only be able to establish the limits indicated above. This BoK further examines other benefits of proton board-level testing besides hardness assurance. The primary alternate use is the injection of errors. Error injection, or fault injection, is something that is often done in a simulation environment. But the proton beam has the benefit of injecting the majority of actual SEEs without risk of something being missed, and without the risk of simulation artifacts misleading the SEE investigation.

Guertin, Steven M.↗

DOE OSTI2020

Examination of the inversion of isobaric analogue states in mirror nuclei

A recent study by Hoff and collaborators [Nature (London) 580, 52 (2020)] presented evidence that the ground-state spin of [Math Processing Error] is different from that of its mirror, [Math Processing Error], likely due to an inversion of the ground- and first-excited states. Here, we assess the likelihood of such an inversion arising from normal isospin-symmetry breaking and, more broadly, whether such phenomena challenge our understanding of charge-symmetry-breaking forces in atomic nuclei. By placing the result within the context of previous experimental and theoretical work we demonstrate that this inversion lies entirely within the bounds of normal isospin-symmetry-breaking behavior. We further note that, in the context of isospin, neither level inversions nor the nuclear ground state hold any special significance.

59 ≤ A ≤ 89↗

NASA NTRS2019

Reexamination of the Edge Crack Torsion Test for Determining the Mode III Delamination Toughness of Laminated Composites

The edge crack torsion (ECT) test is reexamined to address two primary concerns with this method. The first of these relates to inconsistencies in the data reduction procedure that was originally proposed and is most commonly used to compute the mode III interlaminar fracture toughness, G(sub IIIc), from the experimental data. The second relates to the observation that plies neighboring the preimplanted insert in the standard ECT layup are unable to contain initiation and growth of intralaminar cracks, and that growth of these cracks contributes to errors in the apparent values of G(sub IIIc). To address these issues, an alternative data reduction technique, which uses a polynomial expression to represent the ECT specimen’s compliance versus crack length relation, is proposed. The accuracy and potential drawbacks of this expression are investigated using FE simulations. Next, seven groups of ECT specimens are fabricated and tested. Each of these groups contains specimens with preimplanted inserts that are bounded by plies at different +q and -q orientations. The seven groups consist of values of q that vary from 0° to 90° in 15° increments. Tests on these specimens show that intralaminar cracking occurs in specimens from all batches except those for which q equals 15° or 30°. However, values of G(sub IIIc) measured using specimens with these two ply orientations were not equal and, for q = 30°, the values of G(sub IIIc) that were obtained were not completely independent of initial starter crack length. Potential issues accounting for these observations and other challenges associated with measuring G(sub IIIc) using the ECT test are discussed.

Polymer matrix composites↗

NASA NTRSDate not supplied

Resilience Modeling in Complex Engineered Systems with Human-Machine Interactions

In recent times, there has been a growing interest in resilience-based design. Resilience-based design operates on the concept that failures and unexpected events will happen, and when they occur, complex engineered systems should be able to operate within acceptable bounds and recover reasonably. Humans can contribute to the resilience of a system by quickly detecting unforeseen events and taking corrective measures. To this effect, researchers have proposed guidelines and design approaches that can help promote human-system resilience. However, there is no early design stage tool to validate if a system is indeed resilient after applying these guidelines and design methods. In this research, we integrate the Human Error and Functional Failure Reasoning (HEFFR) framework into the fmdtools toolkit to enable designers to model the combined (machine, human, and joint) failures, including their propagation and dynamic effects, during early design stages. This integrated tool also allows designers to model the effects of performance shaping factors, team dynamics, and human-machine interactions in systems of systems. A demonstrative example of a remotely operated rover is explored to demonstrate how this approach can be applied to understand resilience in complex engineered systems with human interactions.

Lukman Irshad↗

DOE OSTI2022

Conservative Numerical Schemes with Optimal Dispersive Wave Relations: Part II. Numerical Evaluations

A new energy and enstrophy conserving scheme (EEC) for the shallow water equations is proposed and evaluated using a suite of test cases over the global spherical or bounded domain. The evaluation is organized around a set of pre-defined properties: accuracy of individual operators, accuracy of the whole scheme, conservation of key quantities, control of the divergence variable, representation of the energy and enstrophy spectra, and simulation of nonlinear dynamics. The results confirm that the scheme is between the first and second order accurate, and conserves the total energy and potential enstrophy up to the time truncation errors. Here, the scheme is capable of producing more physically realistic energy and enstrophy spectra, indicating that it can help prevent the unphysical energy cascade towards the finest resolvable scales. With an optimal representation of the dispersive wave relations, the scheme is able to keep the flow close to being non-divergent, and maintain the geostrophically balanced structures with large-scale geophysical flows over long-term simulations.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

A Particle-in-Cell Method for Plasmas with a Generalized Momentum Formulation, Part II: Enforcing the Lorenz Gauge Condition

In a previous paper Christlieb et al. (A particle-in-cell method for plasmas with a generalized momentum formulation, part I: Model formulation, 2024), we developed a new particle-in-cell (PIC) method for the relativistic Vlasov–Maxwell system in which the electromagnetic fields and the equations of motion for the particles were cast in terms of scalar and vector potentials through a Hamiltonian formulation. This new method evolved the potentials under the Lorenz gauge using integral equation methods. New methods to construct spatial derivatives of the potentials that converge at the same rates as the fields were also presented. The new particle method was compared against standard explicit discretizations, including the well-known FDTD-PIC method, for a range of applications involving sheaths and particle beams. Here, this paper extends this new class of methods by focusing on the enforcement the Lorenz gauge condition in both exact and approximate forms using co-located meshes. A time-consistency property of the proposed field solver for the vector potential form of Maxwell’s equations is established, which is shown to preserve the equivalence between the semi-discrete Lorenz gauge condition and the analogous semi-discrete continuity equation. Using this property, we present three methods to enforce a semi-discrete gauge condition. The first method introduces an update for the continuity equation that is consistent with the discretization of the Lorenz gauge condition. Both the finite difference and spectral implementations satisfy this discrete gauge condition to machine precision. The second approach we propose enforces a semi-discrete continuity equation using the boundary integral solution to the field equations. The potential benefit of this approach is that it eliminates spatial derivatives that appear on the particle data, namely the current density, which is often calculated by linear combinations of low-order spline basis functions. This method is ideally suited to boundary integral equation methods that invert multi-dimensional operators without dimensional splitting techniques and will be the subject of future work. The third approach introduces a gauge correcting method that makes direct use of the gauge condition to modify the scalar potential and uses local maps for both the charge and current densities. This results in a gauge error, as the maps do not enforce the continuity equation. The vector potential coming from the current density is taken to be exact, and using the Lorenz gauge, we compute a correction to the scalar potential that makes the two potentials satisfy the gauge condition. This method also enforces the gauge condition to machine precision. We demonstrate two of the proposed methods in the context of periodic domains. Problems defined on bounded domains, including those with complex geometric features remain an ongoing effort. However, this work shows that it is possible to design computationally efficient methods that can effectively enforce the Lorenz gauge condition in a non-staggered PIC formulation.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1976

The sodium chloride primary pressure gauge

The failure of a central force model for sodium chloride is discussed. It is noted that it does not closely satisfy the Cauchy conditions at low temperatures, and that it fails the central force requirement of the Love condition. The available shock data for sodium chloride and its analysis is examined, and two reasons why the Hugoniot transformation pressure is likely to be less than 231 kbar are discussed. The important (but unjustified) theoretical assumptions made in converting Hugoniot to isothermal data is discussed; it is noted that serious error can enter for very large pressures for a given material and that at such high pressures the isothermal data should thus be considered only semiquantitative even if the Hugoniot data itself is accurate. An alternate method of estimating the isothermal transformation pressure from the Hugoniot transformation pressure is used. This method is based on the temperature derivative of the transformation pressure. On this basis it is concluded that an upper bound for the isothermal transformation of NaCl (to a CsCl-type structure) at room temperature is 257 kbar; it is noted that the actual value may be considerably less than this.

Ruoff, A. L.↗

DOE OSTI2024

Ion-chain sympathetic cooling and gate dynamics

Sympathetic cooling is a technique often employed to mitigate motional heating in trapped-ion quantum computers. However, choosing system parameters such as number of coolants and cooling duty cycle for optimal gate performance requires evaluating trade-offs between motional errors and other slower errors such as qubit dephasing. The optimal parameters depend on cooling power, heating rate, and ion spacing in a particular system. In this study, we aim to analyze best practices for sympathetic cooling of long chains of trapped ions using analytical and computational methods. We use a case study to show that optimal cooling performance is achieved when coolants are placed at the center of the chain and provide a perturbative upper bound on the cooling limit of a mode given a particular set of cooling parameters. In addition, using computational tools, we analyze the trade-off between the number of coolant ions in a chain and the center-of-mass mode heating rate. We also show that cooling as often as possible when running a circuit is optimal when the qubit coherence time is otherwise long. These results provide a roadmap for how to choose sympathetic cooling parameters to maximize circuit performance in trapped-ion quantum computers using long chains of ions.

Cooling & trapping↗

DOE OSTI2024

Learning Quantum States and Unitaries of Bounded Gate Complexity

While quantum state tomography is notoriously hard, most states hold little interest to practically minded tomographers. Given that states and unitaries appearing in nature are of bounded gate complexity, it is natural to ask if efficient learning becomes possible. In this work, we prove that to learn a state generated by a quantum circuit with G two-qubit gates to a small trace distance, a sample complexity scaling linearly in G is necessary and sufficient. We also prove that the optimal query complexity to learn a unitary generated by G gates to a small average-case error scales linearly in G . While sample-efficient learning can be achieved, we show that under reasonable cryptographic conjectures, the computational complexity for learning states and unitaries of gate complexity G must scale exponentially in G . We illustrate how these results establish fundamental limitations on the expressivity of quantum machine-learning models and provide new perspectives on no-free-lunch theorems in unitary learning. Together, our results answer how the complexity of learning quantum states and unitaries relate to the complexity of creating these states and unitaries. Published by the American Physical Society 2024

Zhao, Haimeng (ORCID:0000000166751489)↗

DOE OSTI2024

The Effective Fragment Molecular Orbital Method: Achieving High Scalability and Accuracy for Large Systems

The effective fragment molecular orbital (EFMO) method has been developed to predict the total energy of a very large molecular system accurately (with respect to the underlying quantum mechanical method) and efficiently by taking advantage of the locality of strong chemical interactions and employing a two-level hierarchical parallelism. The accuracy of the EFMO method is partly attributed to the accurate and robust intermolecular interaction prediction between distant fragments, in particular, the many-body polarization and dispersion effects, which require the generation of static and dynamic polarizability tensors by solving the coupled perturbed Hartree–Fock (CPHF) and time-dependent HF (TDHF) equations, respectively. Solving the CPHF and TDHF equations is the main EFMO computational bottleneck due to the inefficient (serial) and I/O-intensive implementation of the CPHF and TDHF solvers. In this work, the efficiency and scalability of the EFMO method are significantly improved with a new CPU memory-based implementation for solving the CPHF and TDHF equations that are parallelized by either message passing interface (MPI) or hybrid MPI/OpenMP. Here, the accuracy of the EFMO method is demonstrated for both covalently bonded systems and noncovalently bound molecular clusters by systematically examining the effects of basis sets and a key distance-related cutoff parameter, R cut . R cut determines whether a fragment pair (dimer) is treated by the chosen ab initio method or calculated using the effective fragment potential (EFP) method (separated dimers). Decreasing the value of Rcut increases the number of separated (EFP) dimers, thereby decreasing the computational effort. It is demonstrated that excellent accuracy (<1 kcal/mol error per fragment) can be achieved when using a sufficiently large basis set with diffuse functions coupled with a small R cut value. With the new parallel implementation, the total EFMO wall time is substantially reduced, especially with a high number of MPI ranks. Given a sufficient workload, nearly ideal strong scaling is achieved for the CPHF and TDHF parts of the calculation. For the first time, EFMO calculations with the inclusion of long-range polarization and dispersion interactions on a hydrated mesoporous silica nanoparticle with explicit water solvent molecules (more than 15k atoms) are achieved on a massively parallel supercomputer using nearly 1000 physical nodes. In addition, EFMO calculations on the carbinolamine formation step of an amine-catalyzed aldol reaction at the nanoscale with explicit solvent effects are presented.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

Embedded Error Bayesian Calibration of Thermal Decomposition of Organic Materials

Organic materials are an attractive choice for structural components due to their light weight and versatility. However, because they decompose at low temperatures relative to tradiational materials they pose a safety risk due to fire and loss of structural integrity. To quantify this risk, analysts use chemical kinetics models to describe the material pyrolysis and oxidation using thermogravimetric analysis. This process requires the calibration of many model parameters to closely match experimental data. Previous efforts in this field have largely been limited to finding a single best-fit set of parameters even though the experimental data may be very noisy. Furthermore the chemical kinetics models are often simplified representations of the true de- composition process. The simplification induces model-form errors that the fitting process cannot capture. In this work we propose a methodology for calibrating decomposition models to thermogravimetric analysis data that accounts for uncertainty in the model-form and experimental data simultaneously. The methodology is applied to the decomposition of a carbon fiber epoxy composite with a three-stage reaction network and Arrhenius kinetics. The results show a good overlap between the model predictions and thermogravimetric analysis data. Uncertainty bounds capture devia- tions of the model from the data. The calibrated parameter distributions are also presented. In conclusion, the distributions may be used in forward propagation of uncertainty in models that leverage this material.

36 MATERIALS SCIENCE↗

NASA NTRS1988

Estimating Rates Of Single-Event Upsets

Mathematical model yields estimates of upper and lower bounds on rates of single-event upsets (SEU's) in logic circuits. SEU's are reversible changes in logic states, caused by single high-energy ions. Energetic ions passed through device along tracks perpendicular to surface, to determine parameters of model. Used to estimate rates caused by ions of various energies incident at arbitrary angles. Useful in design of integrated circuits as high-density memories for minimal susceptibility to bit errors.

Zoutendyk, John A.↗

NASA NTRS1992

Minimax confidence intervals in geomagnetism

The present paper uses theory of Donoho (1989) to find lower bounds on the lengths of optimally short fixed-length confidence intervals (minimax confidence intervals) for Gauss coefficients of the field of degree 1-12 using the heat flow constraint. The bounds on optimal minimax intervals are about 40 percent shorter than Backus' intervals: no procedure for producing fixed-length confidence intervals, linear or nonlinear, can give intervals shorter than about 60 percent the length of Backus' in this problem. While both methods rigorously account for the fact that core field models are infinite-dimensional, the application of the techniques to the geomagnetic problem involves approximations and counterfactual assumptions about the data errors, and so these results are likely to be extremely optimistic estimates of the actual uncertainty in Gauss coefficients.

Stark, Philip B.↗

DOE OSTI2023

Neural networks for large eddy simulations of wall-bounded turbulence: numerical experiments and challenges

Here, we examine the application of neural network-based methods to improve the accuracy of large eddy simulations of incompressible turbulent flows. The networks are trained to learn a mapping between flow features and the subgrid scales, and applied locally and instantaneously—in the same way as traditional physics-based subgrid closures. Models that use only the local resolved strain rate are poorly correlated with the actual subgrid forces obtained from filtering direct numerical simulation data. We see that highly accurate models in a priori testing are inaccurate in forward calculations, owing to the preponderance of numerical errors in implicitly filtered large eddy simulations. A network that accounts for the discretization errors is trained and found to be unstable in a posteriori testing. We identify a number of challenges that the approach faces, including a distribution shift that affects networks that fail to account for numerical errors.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

S -factor and scattering-parameter extractions from ${}^{3}\mathrm{He}+{}^{4}\mathrm{He}{ \rightarrow }^{7}\mathrm{Be}+\gamma $

Previous studies of the reaction ${}^{3}\mathrm{He}+{}^{4}\mathrm{He}\to {}^{7}\mathrm{Be}+\gamma $ have mainly focused on providing the best central value and error bar for the S factor at solar energies. Experimental measurements of this capture reaction at higher energies, the ${}^{3}\mathrm{He}$–${}^{4}\mathrm{He}$ scattering phase shifts, as well as properties of ${}^{7}\mathrm{Be}$ and its excited state, have been used to constrain the theoretical models employed for this purpose. Here we show that much more information than was previously appreciated can be extracted from angle-integrated capture data alone. We use the next-to-leading-order (NLO) amplitude in an effective field theory (EFT) for ${}^{3}\mathrm{He}+{}^{4}\mathrm{He}\to {}^{7}\mathrm{Be}+\gamma $ to perform the extrapolation. At this order the EFT describes the capture process using an s-wave scattering length and effective range, the asymptotic properties of ${}^{7}\mathrm{Be}$ and its excited state, and short-distance contributions to the E1 capture amplitude. We extract the multi-dimensional posterior of all these parameters via a Bayesian analysis that uses capture data below 2 MeV. We find that properties of the ${}^{7}\mathrm{Be}$ ground and excited states are well constrained. The total S factor $S(0)\,={0.577}_{-0.016}^{+0.015}$ keV b, while the branching ratio for excited- to ground-state capture at zero energy, ${Br}(0)={0.406}_{-0.011}^{+0.013}$, both at 68% degree of belief. This S(0) is broadly consistent with other recent evaluations, and agrees with the previously recommended value $S(0)=0.56\pm 0.03\,\,\mathrm{keV}$ b, but has a smaller error bar. We also find significant constraints on ${}^{3}\mathrm{He}$–${}^{4}\mathrm{He}$ scattering parameters, and we obtain constraints on the angular distribution of capture gamma rays, which is important for interpreting experiments. The path forward for this reaction seems to lie with better measurements of the scattering phase shifts and S(E)'s angular dependence away from zero energy, together with better understanding of the asymptotic normalization coefficients of the ${}^{7}$Be bound states' wave functions. Lastly, data on these could further reduce the uncertainty on S(0).

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2021

Statistical and systematic uncertainties in extracting the source properties of neutron star-black hole binaries with gravitational waves

Gravitational waves emitted by neutron star black hole mergers encode key properties of neutron stars - such as their size, maximum mass and spins - and black holes. However, the presence of matter and the high mass ratio makes generating long and accurate waveforms from these systems hard to do with numerical relativity, and not much is known about systematic uncertainties due to waveform modeling. In this work, we simulate gravitational waves from neutron star black hole mergers by hybridizing numerical relativity waveforms produced with the SpEC code with a recent numerical relativity surrogate NRHybSur3dq8Tidal. These signals are analyzed using a range of available waveform families, and statistical and systematic errors are reported. We find that at a network signal-to-noise ratio (SNR) of 30, statistical uncertainties are usually larger than systematic offsets, while at an SNR of 70 the two become comparable. The individual black hole and neutron star masses, as well as the mass ratios, are typically measured very precisely, though not always accurately at high SNR. At a SNR of 30 the neutron star tidal deformability can only be bound from above, while for louder sources it can be measured and constrained away from zero. All neutron stars in our simulations are non-spinning, but in no case we can constrain the neutron star spin to be smaller than ~0.4 (90% credible interval). Waveform families whose late inspiral has been tuned specifically for neutron star black hole signals typically yield the most accurate characterization of the source parameters. Their measurements are in tension with those obtained using waveform families tuned against binary neutron stars, even for mass ratios that could be relevant for both binary neutron stars and neutron star black holes mergers.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1978

An energy-dependent map of the X-ray emission of the Perseus cluster

Results are discussed for X-ray scans of the Perseus cluster with the SAS 3 slat detectors. Counting rates in two broadband channels (1.5 to 5.5 keV and 5.5 to 15.5 keV) are compared as a function of the instrument scan angle measured from the center of emission, and it is found that the emission in the higher energy range is broader in extent. Tests for a systematic energy-dependent error in the background demonstrate that the discovered energy dependence of the Perseus cluster emission is valid. Iron line emission is determined by two methods, various models are fitted to the data, and the total cluster emission in the two energy ranges is computed. The iron line emission is determined to be 0.003 photon/sq cm per sec. It is concluded that the diffuse emission increases in the range from 1.5 to 15.5 keV within the bounds of a model involving a point source with a radius not exceeding about 3 arcmin plus a diffuse source.

Ulmer, M. P.↗