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NASA NTRS1994

Identification and control of structures in space

During the last phase of the project, emphasis has changed to flexible space robotics, by mutual agreement between Dr. R. C. Montgomery, NASA Technical Officer, and the Principal Investigator. Significant advances have been achieved over the period covered by this report. Research has been concerned with two main subjects: (1) the maneuvering and control of freely floating flexible space robots, and (2) the development of a theory for the motion of flexible multibody systems. Work on the first subject has resulted in two papers, both of them concerned with planar maneuvers. The first is concerned with the maneuvering and delivery of a payload to a certain point and in a certain orientation in space. The second is concerned with the docking maneuver with a target whose motion is not known a priori. Both papers will appear in the 'Journal of Guidance, Control, and Dynamics.' The second subject is concerned with the development of hybrid (ordinary and partial) differential equations for the three dimensional motion of flexible multibody systems, a subject of vital interest in flexible space robotics. The paper will appear in the 'Journal of Guidance, Control and Dynamics' in an issue dedicated to the memory of Lawrence W. Taylor, Jr. Abstracts and copies of the papers are hereby included.

Meirovitch, Leonard↗

NASA NTRS2016

The Stability of Hydrogen-Rich Atmospheres of Earth-Like Planets

Understanding hydrogen escape is essential to understanding the limits to habitability, both for liquid water where the Sun is bright, but also to assess the true potential of H2 as a greenhouse gas where the Sun is faint. Hydrogen-rich primary atmospheres of Earth-like planets can result either from gravitational capture of solar nebular gases (with helium), or from impact shock processing of a wide variety of volatile-rich planetesimals (typically accompanied by H2O, CO2, and under the right circumstances, CH4). Most studies of hydrogen escape from planets focus on determining how fast the hydrogen escapes. In general this requires solving hydro- dynamic equations that take into account the acceleration of hydrogen through a critical transonic point and an energy budget that should include radiative heating and cooling, thermal conduction, the work done in lifting the hydrogen against gravity, and the residual heat carried by the hydrogen as it leaves. But for planets from which hydrogen escape is modest or insignificant, the atmosphere can be approximated as hydrostatic, which is much simpler, and for which a relatively full-featured treatment of radiative cooling by embedded molecules, atoms, and ions such as CO2 and H3+ is straightforward. Previous work has overlooked the fact that the H2 molecule is extremely efficient at exciting non-LTE CO2 15 micron emission, and thus that radiative cooling can be markedly more efficient when H2 is abundant. We map out the region of phase space in which terrestrial planets keep hydrogen-rich atmospheres, which is what we actually want to know for habitability. We will use this framework to reassess Tian et al's hypothesis that H2-rich atmospheres may have been rather long-lived on Earth itself. Finally, we will address the empirical observation that rocky planets with thin or negligible atmospheres are rarely or never bigger than 1.6 Earth radii.

Earth-like planets↗

DOE OSTI2025

Effects of artificial collisions, filtering, and nonlocal closure approaches on Hermite-based Vlasov–Poisson simulations

Kinetic simulations of collisionless plasmas are computationally challenging due to phase-space mixing and filamentation, resulting in fine-scale velocity structures. This study compares three methods developed to reduce artifacts related to limited velocity resolution in Hermite-based Vlasov–Poisson simulations: artificial collisions, filtering, and nonlocal closure approaches. We evaluate each method's performance in approximating the linear kinetic response function and suppressing recurrence in linear and nonlinear regimes. Numerical simulations of Landau damping demonstrate that artificial collisions, particularly higher orders of the Lenard-Bernstein collisional operator, most effectively recover the correct damping rate across a range of wavenumbers. Moreover, Hou-Li filtering and nonlocal closures underdamp high wavenumber modes in linear simulations, and the Lenard-Bernstein collisional operator overdamps low wavenumber modes in both linear and nonlinear simulations. This study demonstrates that hypercollisions offer a robust approach to kinetic simulations, accurately capturing collisionless dynamics with limited velocity resolution.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

Characterizing skyrmion flow phases with principal component analysis

Principal component analysis (PCA) is a powerful method that can identify patterns in large, complex data sets by constructing low-dimensional order parameters from higher-dimensional feature vectors. There are increasing efforts to use space-and-time-dependent PCA to detect transitions in nonequilibrium systems that are difficult to characterize with equilibrium methods. Here, we demonstrate that feature vectors incorporating the position and velocity information of driven skyrmions moving through random disorder permit PCA to resolve different types of disordered skyrmion motion as a function of driving force and the ratio of the Magnus force to the dissipation. Since the Magnus force creates gyroscopic motion and a finite Hall angle, skyrmions can exhibit a greater range of flow phases than what is observed in overdamped driven systems with quenched disorder. We show that in addition to identifying previously known skyrmion flow phases, PCA detects several additional phases, including different types of channel flow, moving fluids, and partially ordered states. Guided by the PCA analysis, we further characterize the disordered flow phases to elucidate the different microscopic dynamics and show that the changes in the PCA-derived order parameters can be connected to features in bulk transport measures, including the transverse and longitudinal velocity-force curves, differential conductivity, topological defect density, and changes in the skyrmion Hall angle as a function of drive. We discuss how asymmetric feature vectors can be used to improve the resolution of the PCA analysis, and how this technique can be extended to find disordered phases in other nonequilibrium systems with time-dependent dynamics.

36 MATERIALS SCIENCE↗

NASA NTRS1986

C.A.D. representation of ternary and quaternary phase diagrams

This work is concerned with the utilization of C.A.D. solid-modeling software for the computer representation of three-dimensional phase diagrams. The work was undertaken in two parts. First, the C.A.D. software (I-DEAS, by Structural Dynamics Research Corp.) was integrated with a variety of auxiliary Fortran 77 and I-DEAS language programs which were written specifically for the purpose of phase diagram representation. The capabilities of the resulting suite of software for three-dimensional phase diagram representation were developed and illustrated by the construction, display and manipulation of solid-model phase diagrams for a hypothetical quaternary eutectic system. The results of this work are discussed in some detail in the attached publication ('Solid-modeling: a C.A.D. Alternative for Three-dimensional Phase Diagram Representation'). Such a technique is of general applicability, having utility in both research and education. Secondly, using the C.A.D. technique, data from the literature (gleaned from some 70 separate publications), which represent experimentally determined phase boundaries, were combined to form solid-model representations of the CMS2-M2S-S ternary space diagram and the CMS2-CAS2-M2S-S quaternary liquidus projection (where C=CaO, M=MgO, A=Al2O3, and S=SiO2). These diagrams were utilized in a concurrent study of solidification in the CMAS system.

Delao, James D.↗

NASA NTRS2008

Fiber Optic Sensing Monitors Strain and Reduces Costs

In applications where stress on a structure may vary widely and have an unknown impact on integrity, a common engineering strategy has been overbuilding to ensure a sufficiently robust design. While this may be appropriate in applications where weight concerns are not paramount, space applications demand a bare minimum of mass, given astronomical per-pound launch costs. For decades, the preferred solution was the tactic of disassembly and investigation between flights. Knowing there must be a better way, Dr. Mark Froggatt, of Langley Research Center, explored alternate means of monitoring stresses and damage to the space shuttle. While a tear-it-apart-and-have-a-look strategy was effective, it was also a costly and time consuming process that risked further stresses through the very act of disassembly and reassembly. An alternate way of monitoring the condition of parts under the enormous stresses of space flight was needed. Froggatt and his colleagues at Langley built an early-warning device to provide detailed information about even minuscule cracks and deformations by etching a group of tiny lines, or grating, on a fiber optic cable five-thousandths of an inch thick with ultraviolet light. By then gluing the fiber to the side of a part, such as a fuel tank, and shining a laser beam down its length, reflected light indicated which gratings were under stress. Inferring this data from measurements in light rather than in bonded gauges saved additional weight. Various shuttle components now employ the ultrasonic dynamic vector stress sensor (UDVSS), allowing stress detection by measuring light beamed from a built-in mini-laser. By measuring changes in dynamic directional stress occurring in a material or structure, and including phase-locked loop, synchronous amplifier, and contact probe, the UDVSS proved especially useful among manufacturers of aerospace and automotive structures for stress testing and design evaluation. Engineers could ensure safety in airplanes and spaceships with a narrower, not overbuilt, margin of safety. For this development, in 1997, Discover Magazine named Froggatt a winner in the "Eighth Annual Awards for Technological Innovation" from more than 4,000 entries.

Source record↗

DOE OSTI2022

Photoinduced evolution of lattice orthorhombicity and conceivably enhanced ferromagnetism in LaMnO 3 membranes

Ultrashort laser pulses have been utilized to dynamically drive phase transitions in correlated quantum materials. Of particular interest is whether phases not achievable in thermal equilibrium can be induced in complex oxides with intricately coupled lattice, electron and spin degrees of freedom. Here, we tracked atomic motions in LaMnO 3 following photoexcitation with MeV ultrafast electron diffraction (MeV-UED) technique. We found that the light excited state exhibits numerous signatures different from thermal equilibrium ones, including nearly conserved Bragg intensities, strongly suppressed La cation and oxygen anion displacements, and the long-range lattice orthorhombicity evolution. Furthermore, using first-principles calculations, we predict that the ferromagnetic ordering and conductivity are both enhanced upon laser excitation due to the reduction of the lattice orthorhombicity. This work benefits from recent advance in fabrication of membrane films with high epitaxial quality and in MeV-UED with large momentum space access and high temporal resolution.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Coherent correlation imaging for resolving fluctuating states of matter

Fluctuations and stochastic transitions are ubiquitous in nanometre-scale systems, especially in the presence of disorder. However, their direct observation has so far been impeded by a seemingly fundamental, signal-limited compromise between spatial and temporal resolution. Here we develop coherent correlation imaging (CCI) to overcome this dilemma. Our method begins by classifying recorded camera frames in Fourier space. Contrast and spatial resolution emerge by averaging selectively over same-state frames. Temporal resolution down to the acquisition time of a single frame arises independently from an exceptionally low misclassification rate, which we achieve by combining a correlation-based similarity metric with a modified, iterative hierarchical clustering algorithm. We apply CCI to study previously inaccessible magnetic fluctuations in a highly degenerate magnetic stripe domain state with nanometre-scale resolution. We uncover an intricate network of transitions between more than 30 discrete states. Our spatiotemporal data enable us to reconstruct the pinning energy landscape and to thereby explain the dynamics observed on a microscopic level. CCI massively expands the potential of emerging high-coherence X-ray sources and paves the way for addressing large fundamental questions such as the contribution of pinning and topology in phase transitions and the role of spin and charge order fluctuations in high-temperature superconductivity.

36 MATERIALS SCIENCE↗

NASA NTRS1987

The Jeffcott equations in nonlinear rotordynamics

The Jeffcott equations are a system of coupled differential equations representing the behavior of a rotating shaft. This is a simple model which allows investigation of the basic dynamic behavior of rotating machinery. Nolinearities can be introduced by taking into consideration deadband, side force, and rubbing, among others. The properties of the solutions of the Jeffcott equations with deadband are studied. In particular, it is shown how bounds for the solution of these equations can be obtained from bounds for the solutions of the linearized equations. By studying the behavior of the Fourier transforms of the solutions, we are also able to predict the onset of destructive vibrations. These conclusions are verified by means of numerical solutions of the equations, and of power spectrum density (PSD) plots. This study offers insight into a possible detection method to determine pump stability margins during flight and hot fire tests, and was motivated by the need to explain a phenomenon observed in the development phase of the cryogenic pumps of the Space Shuttle, during hot fire ground testing; namely, the appearance of vibrations at frequencies that could not be accounted for by means of linear models.

Zalik, R. A.↗

NASA NTRS2011

The Roles of Transport and Wave-Particle Interactions on Radiation Belt Dynamics

Particle fluxes in the radiation belts can vary dramatically during geomagnetic active periods. Transport and wave-particle interactions are believed to be the two main types of mechanisms that control the radiation belt dynamics. Major transport processes include substorm dipolarization and injection, radial diffusion, convection, adiabatic acceleration and deceleration, and magnetopause shadowing. Energetic electrons and ions are also subjected to pitch-angle and energy diffusion when interact with plasma waves in the radiation belts. Important wave modes include whistler mode chorus waves, plasmaspheric hiss, electromagnetic ion cyclotron waves, and magnetosonic waves. We investigate the relative roles of transport and wave associated processes in radiation belt variations. Energetic electron fluxes during several storms are simulated using our Radiation Belt Environment (RBE) model. The model includes important transport and wave processes such as substorm dipolarization in global MHD fields, chorus waves, and plasmaspheric hiss. We discuss the effects of these competing processes at different phases of the storms and validate the results by comparison with satellite and ground-based observations. Keywords: Radiation Belts, Space Weather, Wave-Particle Interaction, Storm and Substorm

Fok, Mei-Ching↗

NASA NTRS2014

How Well Does the Latest Anthropomorphic Test Device Mimic Human Impact Responses?

One of the goals of the NASA Occupant Protection Group is to understand the human tolerance to dynamic loading. This knowledge has to come through indirect approaches such as existing human response databases, anthropometric test devices (ATD), animal testing, post‐mortem human subjects, and models. This study investigated the biofidelity of the National Highway Traffic Safety Administration's ATD named the THOR (test device for human occupant restraint). If THOR responds comparably to humans, then it could potentially be used as a human surrogate to help validate space vehicle requirements for occupant protection. The THOR responses to frontal and spinal impacts (ranging from 8 to 12 G with rise times of 40, 70, and 100 ms) were measured and compared to human volunteer responses (95 trials in frontal and 58 in spinal) previously collected by the U. S. Air Force on the same horizontal impact accelerator. The impact acceleration profiles tested are within the expected range of multi‐purpose crew vehicle (MPCV) landing dynamics. A correlation score was calculated for each THOR to human comparison using CORA (CORrelation and Analysis) software. A two‐parameter beta distribution model fit was obtained for each dependent variable using maximum likelihood estimation. For frontal impacts, the THOR head x‐acceleration peak response correlated with the human response at 8 and 10‐G 100 ms but not 10‐G 70 ms. The phase lagged the human response. Head z‐acceleration was not correlated. Chest x‐acceleration was in phase, had a higher peak response, and was well correlated with lighter subjects (Cora = 0.8 for 46 kg vs. Cora = 0.4 for 126 kg). Head x‐displacement had a leading phase. Several subjects responded with the same peak displacement but the mean of the group was lower. The shoulder x‐displacement was in phase but had higher peaks than the human response. For spinal impacts, the THOR head x‐acceleration was not well correlated. Head and chest z‐acceleration was in phase but had a higher peak response. Chest z‐acceleration was highly correlated with heavier subjects at lower G pulses (Cora = 0.86 for 125 kg at 8 G). The human response was variable in shoulder z‐displacement but the THOR was in phase and was comparable to the mean peak response. Head xand z‐displacement was in phase but had higher peaks. Seat pan forces were well correlated, were in phase, but had a larger peak response than most subjects. The THOR does not respond to frontal and spinal impacts exactly the same way that a human does. Some responses are well matched and others are not. Understanding the strengths and weaknesses of this ATD is an important first step in determining its usefulness in occupant protection at NASA

Newby, N.↗

NASA NTRS2014

How Well Does the Latest Anthropomorphic Test Device Mimic Human Impact Responses?

One of the goals of the NASA Occupant Protection Group is to understand the human tolerance to dynamic loading. This knowledge has to come through indirect approaches such as existing human response databases, anthropometric test devices (ATD), animal testing, post-‐mortem human subjects, and models. This study investigated the biofidelity of the National Highway Traffic Safety Administration's ATD named the THOR (test device for human occupant restraint). If THOR responds comparably to humans, then it could potentially be used as a human surrogate to help validate space vehicle requirements for occupant protection. The THOR responses to frontal and spinal impacts (ranging from 8 to 12 G with rise times of 40, 70, and 100 ms) were measured and compared to human volunteer responses (95 trials in frontal and 58 in spinal) previously collected by the U. S. Air Force on the same horizontal impact accelerator. The impact acceleration profiles tested are within the expected range of multi-‐purpose crew vehicle (MPCV) landing dynamics. A correlation score was calculated for each THOR to human comparison using CORA (CORrelation and Analysis) software. A two-‐parameter beta distribution model fit was obtained for each dependent variable using maximum likelihood estimation. For frontal impacts, the THOR head x-‐acceleration peak response correlated with the human response at 8 and 10-‐G 100 ms but not 10-‐G 70 ms. The phase lagged the human response. Head z-‐acceleration was not correlated. Chest x-‐acceleration was in phase, had a higher peak response, and was well correlated with lighter subjects (Cora = 0.8 for 46 kg vs. Cora = 0.4 for 126 kg). Head x-‐displacement had a leading phase. Several subjects responded with the same peak displacement but the mean of the group was lower. The shoulder x-‐displacement was in phase but had higher peaks than the human response. For spinal impacts, the THOR head x-‐acceleration was not well correlated. Head and chest z-‐acceleration was in phase but had a higher peak response. Chest z-‐acceleration was highly correlated with heavier subjects at lower G pulses (Cora = 0.86 for 125 kg at 8 G). The human response was variable in shoulder z-‐displacement but the THOR was in phase and was comparable to the mean peak response. Head x-‐ and z-‐displacement was in phase but had higher peaks. Seat pan forces were well correlated, were in phase, but had a larger peak response than most subjects. The THOR does not respond to frontal and spinal impacts exactly the same way that a human does. Some responses are well matched and others are not. Understanding the strengths and weaknesses of this ATD is an important first step in determining its usefulness in occupant protection at NASA

Newby, Nate↗

NASA NTRS2015

Cryogenic Orbital Testbed (CRYOTE) Ground Test Article, Final Report

Liquid propulsion has been used since Robert Goddard started developing a liquid oxygen (LO2) and gasoline powered rocket and fired it in 1923 (Ref. 1). In the following decades engineers settled on the combination of liquid hydrogen (LH2) and LO2 as the most efficient propellant combination for in-space travel. Due to their low temperatures (LH2 at 20 K and LO2 at 90 K), they require special handling and procedures. General Dynamics began developing LO2 and LH2 upper stages in 1956 in the form of Centaur, these efforts were soon funded by the Department of Defense in conjunction with NASA (beginning in 1958) (Ref. 2). Meanwhile NASA also worked with McDonnell Douglas to develop the SIV-B stage for the Saturn V rocket. In the subsequent years, the engineers were able to push the Centaur to up to 9 hr of orbital lifetime and the SIV-B to up to 6 hr. Due to venting the resultant boil-off from the high heat loads through the foam insulation on the upper stages, both vehicles remained in a settled configuration throughout the flights, thus the two phases of propellant (liquid and vapor) were separated at a known location. The one exception to this were the Titan/Centaur missions, which thanks to the lower boil-off using three layers of multilayer insulation (MLI), were able to coast unsettled for up to 5.25 hr during direct geosynchronous orbit insertion missions. In the years since there has been a continuous effort to extend the life of these upper stages from hours to days or even months.

Joule-Thompson Effect↗

NASA NTRS1998

Numerical Modeling of Physical Vapor Transport in Contactless Crystal Growth Geometry

Growth from the vapor under conditions of limited contact with the walls of the growth ampoule is beneficial for the quality of the growing crystal due to reduced stress and contamination which may be caused by interactions with the growth container. The technique may be of a particular interest for studies on crystal growth under microgravity conditions: elimination of some factors affecting the crystal quality may make interpretation of space-conducted processes more conclusive and meaningful. For that reason, and as a part of our continuing studies on 'contactless' growth technique, we have developed a computational model of crystal growth process in such system. The theoretical model was built, and simulations were performed using the commercial computational fluid dynamics code, (CFD) ACE. The code uses an implicit finite volume formulation with a gray discrete ordinate method radiation model which accounts for the diffuse absorption and reflection of radiation throughout the furnace. The three-dimensional model computes the heat transfer through the crystal, quartz, and gas both inside and outside the ampoule, and mass transport from the source to the crystal and the sink. The heat transport mechanisms by conduction, natural convection, and radiation, and mass transport by diffusion and convection are modeled simultaneously and include the heat of the phase transition at the solid-vapor interfaces. As the thermal boundary condition, temperature profile along the walls of the furnace is used. For different thermal profiles and furnace and ampoule dimensions, the crystal growth rate and development of the crystal-vapor and source-vapor interfaces (change of the interface shape and location with time) are obtained. Super/under-saturation in the ampoule is determined and critical factors determining the 'contactless' growth conditions are identified and discussed. The relative importance of the ampoule dimensions and geometry, the furnace dimensions and its temperature, and the properties of the grown material are analyzed. The results of the simulations are compared with related experimental results on growth of CdTe, CdZnTe, ZnTe, PbTe, and PbSnTe crystals by this technique.

Palosz, W.↗

NASA NTRS2010

Detection of the Secondary Eclipse of Exoplanet HAT P-11b

We have successfully conducted secondary eclipse observations of exoplanet HAT-P-11b using the Spitzer Space Telescope. HAT-P-11b was, until very recently, the smallest transiting extrasolar planet yet found and one of only two known exo-Neptunes. We observed the system at 3.6 microns for a period of 22 hours centered on the anticipated secondary eclipse time, to detect the eclipse and determine its phase. Having detected the secondary eclipse, we are at present making a more focused series of observations in both the 3.6 and 4.5 micron bands to fully characterize it. HAT-P-11b has a period of 4.8878 days, radius of 0.422 RJ, mass of 0.081 MJ and semi-major axis 0.053 AU. Measurements of the secondary eclipse will serve to clarify two key issues; 1) the planetary brightness temperature and the nature of its atmosphere, and 2) the eccentricity of its orbit, with implications for its dynamical evolution. A precise determination of the orbit phase for the secondary eclipse will also be of great utility for Kepler observations of this system at visible wavelengths.

Barry, R. K.↗

NASA NTRS2015

Observation Platform for Dynamic Biomedical and Biotechnology Experiments Using the International Space Station (ISS) Light Microscopy Module (LMM)

Techshot, Inc., has developed an observation platform for the LMM on the ISS that will enable biomedical and biotechnology experiments. The LMM Dynamic Stage consists of an electronics module and the first two of a planned suite of experiment modules. Specimens and reagent solutions can be injected into a small, hollow microscope slide-the heart of the innovation-via a combination of small reservoirs, pumps, and valves. A life science experiment module allows investigators to load up to two different fluids for on-orbit, real-time image cytometry. Fluids can be changed to initiate a process, fix biological samples, or retrieve suspended cells. A colloid science experiment module conducts microparticle and nanoparticle tests for investigation of colloid self-assembly phenomena. This module includes a hollow glass slide and heating elements for the creation of a thermal gradient from one end of the slide to the other. The electronics module supports both experiment modules and contains a unique illuminator/condenser for bright and dark field and phase contrast illumination, power supplies for two piezoelectric pumps, and controller boards for pumps and valves. This observation platform safely contains internal fluids and will greatly accelerate the research and development (R&D) cycle of numerous experiments, products, and services aboard the ISS.

Kurk, Michael A. (Andy)↗

NASA NTRS2015

Observation Platform for Dynamic Biomedical and Biotechnology Experiments Using the International Space Station (ISS) Light Microscopy Module (LMM)

Kurk, Michael A. (Andy)↗

DOE OSTI2024

Directional Superradiance in a Driven Ultracold Atomic Gas in Free Space

Ultracold atomic systems are among the most promising platforms that have the potential to shed light on the complex behavior of many-body quantum systems. One prominent example is the case of a dense ensemble illuminated by a strong coherent drive while interacting via dipole-dipole interactions. Despite being subjected to intense investigations, this system retains many open questions. A recent experiment carried out in a pencil-shaped geometry [Ferioli Nat. Phys. 19, 1345 (2023)] has reported measurements that have seemed consistent with the emergence of strong collective effects in the form of a “superradiant” phase transition in free space, when looking at the light-emission properties in the forward direction. Motivated by the experimental observations, we carry out a systematic theoretical analysis of the steady-state properties of the system as a function of the driving strength and atom number N . We observe signatures of collective effects in the weak-driving regime, which disappear with increasing drive strength as the system evolves into a single-particle-like mixed state comprised of randomly aligned dipoles. Although the steady state features some similarities to the reported superradiant-to-normal nonequilibrium transition, also known as cooperative resonance fluorescence, we observe significant qualitative and quantitative differences, including a different scaling of the critical drive parameter (from N to N ). We validate the applicability of a mean-field treatment to capture the steady-state dynamics under currently accessible conditions. Furthermore, we develop a simple theoretical model that explains the scaling properties by accounting for interaction-induced inhomogeneous effects and spontaneous emission, which are intrinsic features of interacting disordered arrays in free space. Published by the American Physical Society 2024

Agarwal, Sanaa (ORCID:0000000336863804)↗