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At least 19 records

The Electromagnetic Counterpart of the Binary Neutron Star Merger LIGO/Virgo GW170817. III. Optical and UV Spectra of a Blue Kilonova from Fast Polar Ejecta

We present optical and ultraviolet spectra of the first electromagnetic counterpart to a gravitational-wave (GW) source, the binary neutron star merger GW170817. Spectra were obtained nightly between 1.5 and 9.5 days post-merger, using the Southern Astrophysical Research and Magellan telescopes; the UV spectrum was obtained with the Hubble Space Telescope at 5.5 days. Our data reveal a rapidly fading blue component (T≈5500 K at 1.5 days) that quickly reddens; spectra later than ≳4.5 days peak beyond the optical regime. The spectra are mostly featureless, although we identify a possible weak emission line at ∼7900 Å at t≲4.5 days. The colors, rapid evolution, and featureless spectrum are consistent with a “blue” kilonova from polar ejecta comprised mainly of light r-process nuclei with atomic mass number A≲140. This indicates a sightline within θ{sub obs}≲45{sup ∘} of the orbital axis. Comparison to models suggests ∼0.03 M {sub ⊙} of blue ejecta, with a velocity of ∼0.3c. The required lanthanide fraction is ∼10{sup −4}, but this drops to <10{sup −5} in the outermost ejecta. The large velocities point to a dynamical origin, rather than a disk wind, for this blue component, suggesting that both binary constituents are neutron stars (as opposed to a binary consisting of a neutron star and a black hole). For dynamical ejecta, the high mass favors a small neutron star radius of ≲12 km. This mass also supports the idea that neutron star mergers are a major contributor to r-process nucleosynthesis.

79 ASTRONOMY AND ASTROPHYSICS↗

The Electromagnetic Counterpart of the Binary Neutron Star Merger LIGO/Virgo GW170817. VI. Radio Constraints on a Relativistic Jet and Predictions for Late-time Emission from the Kilonova Ejecta

We present Very Large Array (VLA) and Atacama Large Millimeter/submillimeter Array (ALMA) radio observations of GW170817, the first Laser Interferometer Gravitational-wave Observatory (LIGO)/Virgo gravitational wave (GW) event from a binary neutron star merger and the first GW event with an electromagnetic (EM) counterpart. Our data include the first observations following the discovery of the optical transient at both the centimeter (13.7 hr post-merger) and millimeter (2.41 days post-merger) bands. We detect faint emission at 6 GHz at 19.47 and 39.23 days after the merger, but not in an earlier observation at 2.46 days. We do not detect cm/mm emission at the position of the optical counterpart at frequencies of 10–97.5 GHz at times ranging from 0.6 to 30 days post-merger, ruling out an on-axis short gamma-ray burst (SGRB) for energies ≳10{sup 48} erg. For fiducial SGRB parameters, our limits require an observer viewer angle of ≳20°. The radio and X-ray data can be jointly explained as the afterglow emission from an SGRB with a jet energy of ∼10{sup 49}--10{sup 50} erg that exploded in a uniform density environment with n∼10{sup −4}--10{sup −2} cm{sup −3}, viewed at an angle of ∼20°–40° from the jet axis. Using the results of our light curve and spectral modeling, in conjunction with the inference of the circumbinary density, we predict the emergence of late-time radio emission from the deceleration of the kilonova (KN) ejecta on a timescale of ∼5–10 years that will remain detectable for decades with next-generation radio facilities, making GW170817 a compelling target for long-term radio monitoring.

79 ASTRONOMY AND ASTROPHYSICS↗

Evaluation of RMI Ejecta Sourcing from Reconstructed Interfaces in 2D FLAG Flyer-Plate Simulations

This study investigates the performance of RMI ejecta sourcing from a reconstructed interface in two-dimensional FLAG ejecta flyer-plate simulations. Enabling the RMI ejecta model to operate on reconstructed interfaces has been a longstanding challenge in FLAG, limiting its applicability to experiments in which interface reconstruction is not required. Addressing this limitation is therefore an important step toward expanding the model’s predictive capability. The first phase of this effort focused on identifying a flyer-plate/target configuration that produces large ejecta clouds from two distinct shocks using the standard RMI ejecta model. A series of simulations examined the effects of flyer composition, layer thickness, and mesh resolution on shock structure and ejecta formation. VISAR traces were used to evaluate shock timing and identify configurations that produced a clear double-shock response. A copper middle-layer flyer with a doubled tin layer yielded the most favorable shock characteristics and was selected as the baseline configuration. After confirming this configuration performed as expected when running a reference FLAG version, it became the basis for three simulations: (1) a standard two-dimensional ejecta flyer-plate calculation using a reference FLAG version, (2) the same calculation with the reconstructed interface option enabled, and (3) a calculation employing ALE strategies to force the ejecta model to operate on the reconstructed interface. While the baseline simulation produced the expected ejecta response, the reconstructed interface approach did not result in measurable ejecta, indicating that this capability remains under development. These results highlight both the progress made and the remaining challenges in extending the RMI ejecta model to reconstructed interfaces.

36 MATERIALS SCIENCE↗

A mathematical framework for ejecta cloud dynamics with application to source models and piezoelectric mass measurements

We present a mathematical framework for describing the dynamical evolution of an ejecta cloud generated by a generic ejecta source model. We consider a piezoelectric sensor fielded in the path of an ejecta cloud, for experimental configurations in which the ejecta are created at a singly shocked planar surface and fly ballistically through vacuum to the stationary sensor. To do so, we introduce the concept of a time- and velocity-dependent ejecta “areal mass function.” We derive expressions for the analytic (“true”) accumulated ejecta areal mass at the sensor and the measured (“inferred”) value obtained via the standard method for analyzing piezoelectric voltages. In this way, we derive an exact expression and upper bound for the error imposed upon a piezoelectric ejecta mass measurement (in a perfect system) by the assumption of instantaneous creation, which is commonly required for momentum diagnostic analyses. This error term is zero for truly instantaneous source models; otherwise, the standard piezoelectric analysis is guaranteed to overestimate the true mass. When combined with a piezoelectric dataset, this framework provides a unique solution for the ejecta particle velocity distribution, subject to the assumptions inherent in the data analysis. The framework also leads to strong boundary conditions that any ejecta source model must satisfy in order to be consistent with apparently global properties of piezoelectric measurements from a wide range of experiments. We demonstrate this methodology by applying it to the Richtmyer–Meshkov instability+self-similar velocity distribution ejecta source model currently under development at Los Alamos National Laboratory.

97 MATHEMATICS AND COMPUTING↗

Analytically Derived Piezoelectric Ejecta Data Transformations, with Implications for Claims of Experimentally Observed Self-Similarity

Building upon a previously documented analytic formalism, we derive a set of data transformations that relate piezoelectrically inferred time-dependent ejecta masses, dynamical ejecta cloud densities, and "radiographic" ejecta cloud density distributions. These transformation rules are derived rigorously, with careful attention to the assumption of instant ejecta production which is inherent in the piezoelectric mass inference procedure. Some of these relationships have not appeared previously in the literature, including solutions for the unique ejecta velocity distribution and radiographic ejecta cloud density distribution consistent with the instant-production assumption for a given piezoelectrically inferred mass dataset. In deriving the latter, we prove that—again subject to the ubiquitous instant-creation assumption—measurements at a single sensor are sufficient to determine properties of the ejecta cloud everywhere at all times. Other data transformation relationships have appeared in the literature without a rigorous derivation, or have been stated in forms that obscure the key mathematical relationships. We apply these results to arguments for self-similar ejecta cloud evolution, thereby deriving mathematical expressions by which claims of self-similarity may be examined. We prove that one analysis used in the literature to claim experimental evidence of a short ejecta production timescale—and consequently of self-similar cloud evolution—in fact places no constraint the ejecta production interval and therefore does not support the associated conclusion. We also provide a potentially useful uniqueness proof.

30 DIRECT ENERGY CONVERSION↗

On the Jet–Ejecta Interaction in 3D GRMHD Simulations of a Binary Neutron Star Merger Aftermath

Short γ-ray burst (sGRB) jets form in the aftermath of a neutron star merger, drill through disk winds and dynamical ejecta, and extend over four to five orders of magnitude in distance before breaking out of the ejecta. We present the first 3D general-relativistic magnetohydrodynamic sGRB simulations to span this enormous scale separation. They feature three possible outcomes: jet+cocoon, cocoon, and neither. Typical sGRB jets break out of the dynamical ejecta if (i) the bound ejecta's isotropic equivalent mass along the pole at the time of the BH formation is ≲10 -4 M ⊙ , setting a limit on the delay time between the merger and BH formation, otherwise, the jets perish inside the ejecta and leave the jet-inflated cocoon to power a low-luminosity sGRB; (ii) the postmerger remnant disk contains a strong large-scale vertical magnetic field, ≳1015 G; and (iii) if the jets are weak (≲10 50 erg), the ejecta's isotropic equivalent mass along the pole must be small (≲10 -2 M ⊙ ). Generally, the jet structure is shaped by the early interaction with disk winds rather than the dynamical ejecta. As long as our jets break out of the ejecta, they retain a significant magnetization (≲1), suggesting that magnetic reconnection is a fundamental property of sGRB emission. The angular structure of the outflow isotropic equivalent energy after breakout consistently features a flat core followed by a steep power-law distribution (slope ≳3), similar to hydrodynamic jets. In the cocoon-only outcome, the dynamical ejecta broadens the outflow angular distribution and flattens it (slope ~1.5).

Jets↗

SPATIAL DISTRIBUTION OF MG-RICH EJECTA IN LMC SUPERNOVA REMNANT N49B

The supernova remnant (SNR) N49B in the Large Magellanic Cloud is a peculiar example of a core-collapse SNR that shows the shocked metal-rich ejecta enriched only in Mg without evidence for a similar overabundance in O and Ne. Based on archival Chandra data, we present results from our extensive spatially resolved spectral analysis of N49B. We find that the Mg-rich ejecta gas extends from the central regions of the SNR out to the southeastern outermost boundary of the SNR. This elongated feature shows an overabundance for Mg similar to that of the main ejecta region at the SNR center, and its electron temperature appears to be higher than the central main ejecta gas. We estimate that the Mg mass in this southeastern elongated ejecta feature is ∼10% of the total Mg ejecta mass. Our estimated lower limit of >0.1 M {sub ⊙} on the total mass of the Mg-rich ejecta confirms the previously suggested large mass for the progenitor star ( M ≳ 25 M {sub ⊙}). We entertain scenarios of an SNR expanding into a nonuniform medium and an energetic jet-driven supernova in an attempt to interpret these results. However, with the current results, the origins of the extended Mg-rich ejecta and the Mg-only-rich nature of the overall metal-rich ejecta in this SNR remain elusive.

79 ASTRONOMY AND ASTROPHYSICS↗

Microscopic ejecta measurements from hypervelocity impacts on aluminum and powdered regolith targets

Meteoroids and space debris pose a two-fold threat to spacecraft due to the risk of both mechanical damage from the impact and electrical damage from the impact-generated plasma, which produces electromagnetic emissions. Analysis of impact products, however, has the potential to provide useful information about planetary rings and small solar system bodies, such as asteroids and comets. Understanding the plasma dynamics and radiation mechanism is essential to addressing the impact threat and extracting science data from impacts. Charge attachment to condensed-phase debris may lead to a dusty plasma under certain conditions, so interpreting hypervelocity impacts requires knowledge of both the nature of the plasma and of the ejecta. We address the latter with measurements of the microscopic ejecta distribution produced by light gas gun impacts on solid aluminum and powdered regolith simulant targets. We examine both the size and shape of the ejecta using thin-film witness plates and find that the size distributions of microscopic ejecta follow power laws. Observations from the powdered regolith simulant match well with disruption experiments of solid basalt, while measurements from the aluminum impacts diverge from extrapolations based on studies of macroscopic debris. Analysis of the ejecta shape shows a transition from smooth to irregularly shaped ejecta at the material grain size, and the shape of debris may provide insight on the phase of the ejected material. Finally, based on our results, we predict a significant number of secondary impacts on plasma sensors due to ejecta, even at considerable distance from the target, supporting the hypothesis that charged dust affects plasma measurements in the ejecta curtain.

42 ENGINEERING↗

(U) An Analytic Study of Piezoelectric Ejecta Mass Measurements

We consider the piezoelectric measurement of the areal mass of an ejecta cloud, for the specific case where ejecta are created by a single shock at the free surface and fly ballistically through vacuum to the sensor. To do so, we define time- and velocity-dependent ejecta “areal mass functions” at the source and sensor in terms of typically unknown distribution functions for the ejecta particles. Next, we derive an equation governing the relationship between the areal mass function at the source (which resides in the rest frame of the free surface) and at the sensor (which resides in the laboratory frame). We also derive expressions for the analytic (“true”) accumulated ejecta mass at the sensor and the measured (“inferred”) value obtained via the standard method for analyzing piezoelectric voltage traces. This approach enables us to derive an exact expression for the error imposed upon a piezoelectric ejecta mass measurement (in a perfect system) by the assumption of instantaneous creation. We verify that when the ejecta are created instantaneously (i.e., when the time dependence is a delta function), the piezoelectric inference method exactly reproduces the correct result. When creation is not instantaneous, the standard piezo analysis will always overestimate the true mass. However, the error is generally quite small (less than several percent) for most reasonable velocity and time dependences. In some cases, errors exceeding 10-15% may require velocity distributions or ejecta production timescales inconsistent with experimental observations. These results are demonstrated rigorously with numerous analytic test problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Analytic solutions for Asay foil trajectories with implications for ejecta source models and mass measurements

We consider the trajectory of an Asay foil ejecta diagnostic for scenarios where ejecta are produced at a singly shocked planar surface and fly ballistically through a perfect vacuum to the sensor. We do so by building upon a previously established mathematical framework derived for the analytic study of stationary sensors. First, we derive the momentum conservation equation for the problem, in a form amenable to accelerating sensors, in terms of a generic ejecta source model. The result is an integrodifferential equation of motion for the foil trajectory. This equation yields an easily calculable closed-form implicit solution for the foil trajectory in instant-production scenarios. From there, we derive a boundary condition that particle velocity distributions must satisfy if their associated foil trajectories are to exhibit a smooth initial acceleration, as occurs in some experiments. This condition is identical to one derived previously from a consideration of piezoelectric voltage data obtained in similar experiments. We also compare techniques for inferring accumulated ejecta masses from foil trajectories, first by deriving the exact solution, and then by quantifying the error imposed by a frequently used approximate solution (both subject to the assumption of instantaneous ejecta production). Finally, we examine the common practice of presenting inferred cumulative ejecta masses as a function of implied ejecta velocity, establishing the conditions under which this methodology is most meaningful.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Nonthermal Ionization of Kilonova Ejecta: Observable Impacts

The characteristic rapid rise and decline at optical wavelengths of a kilonova is the product of the low ejecta mass (≲0.05 M ⊙ ) and high ejecta velocity (≳0.1c). We show that, even at very early times (≲2 days), regions of ejecta fall below critical density and temperature thresholds at which nonlocal thermodynamic equilibrium (NLTE) effects become important. Here, we present an approximate method for calculating the ionization state of the ejecta that accounts for the NLTE impact of high-energy electrons produced in the beta decay of freshly synthesized r-process elements. We find that incorporating ionization from high-energy electrons produces an “inverted” and “blended” ionization structure, where the most highly ionized species are located in the fastest moving homologous ejecta, and multiple ionization states coexist. In radiation transport calculations, the higher degree of ionization reduces line blanketing in optical bands, leading to improved agreement with the light-curve properties of AT 2017gfo, such as the duration, decay rates, brightness, and colors. Our quasi-NLTE implementation helps to alleviate tensions in kilonova modeling: for high-velocity (∼0.3 c) ejecta components, our models require less mass for a given peak brightness in optical bands, by as much as a factor of 3; our models can explain the presence of observed features associated to Sr II, W III, Se III, and Te III under conditions where LTE models would predict only neutral species; and we naturally predict the coexistence of species like Sr II and Ce III without the need for fine-tuning of the ejecta properties.

High energy astrophysics↗

Estimating the Contribution of Dynamical Ejecta in the Kilonova Associated with GW170817

The source of the gravitational-wave (GW) signal GW170817, very likely a binary neutron star merger, was also observed electromagnetically, providing the first multi-messenger observations of this type. The two-week-long electromagnetic (EM) counterpart had a signature indicative of an r-process-induced optical transient known as a kilonova. This Letter examines how the mass of the dynamical ejecta can be estimated without a direct electromagnetic observation of the kilonova, using GW measurements and a phenomenological model calibrated to numerical simulations of mergers with dynamical ejecta. Specifically, we apply the model to the binary masses inferred from the GW measurements, and use the resulting mass of the dynamical ejecta to estimate its contribution (without the effects of wind ejecta) to the corresponding kilonova light curves from various models. The distributions of dynamical ejecta mass range between ${M}_{\mathrm{ej}}={10}^{-3}-{10}^{-2}\,{M}_{\odot }$ for various equations of state, assuming that the neutron stars are rotating slowly. In addition, we use our estimates of the dynamical ejecta mass and the neutron star merger rates inferred from GW170817 to constrain the contribution of events like this to the r-process element abundance in the Galaxy when ejecta mass from post-merger winds is neglected. We find that if ≳10% of the matter dynamically ejected from binary neutron star (BNS) mergers is converted to r-process elements, GW170817-like BNS mergers could fully account for the amount of r-process material observed in the Milky Way.

79 ASTRONOMY AND ASTROPHYSICS↗

Foreword to the Special Issue on Ejecta

We report that ejecta physics is a young field, having developed over the last 60 years or so. Essentially, ejecta forms as a spray of dense particles generated from the free surface of metals subjected to strong shocks, but the detailed mechanisms controlling the properties of this particulate ejecta are only now being fully elucidated. The field is dynamic and rapidly growing, with military and industrial applications, and applications to areas such as fusion research. This Special Issue on Ejecta reports the current state of the art in ejecta physics, describing experimental, theoretical and computational work by research groups around the world. While much remains to be done, the dramatic recent progress in the field, some of it first reported here, means that this volume provides a particularly timely review. In this foreword, we provide a brief historical overview of the development of ejecta physics, to define the context for the work in the rest of this Special Issue.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

(U) Physics Validation of the RMI-Based Ejecta Source Model Implementation in FLAG: L2 Milestone #6035 Report

The Los Alamos Physics and Engineering Models (PEM) program has developed a model for Richtmyer-Meshkov instability (RMI) based ejecta production from shock-melted surfaces, along with a prescription for a self-similar velocity distribution (SSVD) of the resulting ejecta particles. We have undertaken an effort to validate this source model using data from explosively driven tin coupon experiments. The model’s current formulation lacks a crucial piece of physics: a method for determining the duration of the ejecta production interval. Without a mechanism for terminating ejecta production, the model is not predictive. Furthermore, when the production interval is hand-tuned to match time-integrated mass data, the predicted time-dependent mass accumulation on a downstream sensor rises too sharply at early times and too slowly at late times because the SSVD overestimates the amount of mass stored in the fastest particles and underestimates the mass stored in the slowest particles. The functional form of the resulting m(t) is inconsistent with the available time-dependent data; numerical simulations and analytic studies agree on this point. Simulated mass tallies are highly sensitive to radial expansion of the ejecta cloud. It is not clear if the same effect is present in the experimental data but if so, depending on the degree, this may challenge the model’s compatibility with tin coupon data. The current implementation of the model in FLAG is sensitive to the detailed interaction between kinematics (hydrodynamic methods) and thermodynamics (material models); this sensitivity prohibits certain physics modeling choices. The appendices contain an extensive analytic study of piezoelectric ejecta mass measurements, along with test problems, excerpted from a longer work (LA-UR-17-21218).

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

The Role of Helium on Ejecta Production in Copper

The effect of helium (He) concentration on ejecta production in OFHC-Copper was investigated using Richtmyer–Meshkov Instability (RMI) experiments. The experiments involved complex samples with periodic surface perturbations machined onto the surface. Each of the four target was implanted with a unique helium concentration that varied from 0 to 4000 appm. The perturbation’s wavelengths were λ ≈ 65 μ m, and their amplitudes h 0 were varied to determine the wavenumber ( 2 π / λ ) amplitude product k h 0 at which ejecta production beganfor Cu with and without He. The velocity and mass of the ejecta produced was quantified using Photon Doppler Velocimetry (PDV) and Lithium-Niobate (LN) pins, respectively. Our results show that there was an increase of 30% in the velocity at which the ejecta cloud was traveling in Copper with 4000 appm as compared to its unimplanted counterpart. Our work also shows that there was a finer cloud of ejecta particles that was not detected by the PDV probes but was detected by the early arrival of a “signal” at the LN pins. While the LN pins were not able to successfully quantify the mass produced due to it being in the solid state, they did provide information on timing. Our results show that ejecta was produced for a longer time in the 4000 appm copper.

36 MATERIALS SCIENCE↗

Evaluating the Thickness and Stratigraphy of Ejecta Materials at the Chang’e-4 Landing Site

The landing site of China’s Chang’e-4 (CE-4) probe is located on the mare basalts on the floor of the Von Kármán crater on the lunar far side. The Von Kármán crater is inside the ancient and highly cratered South Pole–Aitken basin, which has experienced complex emplacement sequences of both near and distant ejecta materials. These issues complicated the interpretation of the CE-4 surface in situ measurements of the visible and near-infrared spectrometer and the lunar penetrating radar (LPR) onboard Yutu-2 rover. To evaluate the sources and amounts of all principle foreign materials at the CE-4 landing site, we thoroughly examine the ejecta delivered by crater-forming events that occurred later than the formation of the mare basalts at the CE-4 landing site. We found a total of 16 craters that may have delivered ejecta thicker than 10 cm level superposed on the mare basalts at the CE-4 landing site. Crater Finsen, Von Kármán L, Von Kármán L′, and Maksutov are the top four major foreign material sources, and each of them contributed ejecta thicker than 1 m. Our surveys confirm that the ejecta from Finsen crater are the most dominant foreign materials in the uppermost few meters at the CE-4 landing site and the total impact ejecta deposited upon the mare basalts at the landing site is estimated to be thinner than 30 m. We found that the estimations from Pike’s model are the most consistent with the Yutu-2 LPR observations.

58 GEOSCIENCES↗

The Three-dimensional Expansion of the Ejecta from Tycho's Supernova Remnant

We present the first 3D measurements of the velocity of various ejecta knots in Tycho’s supernova remnant, known to result from a Type Ia explosion. Chandra X-ray observations over a 12 yr baseline from 2003 to 2015 allow us to measure the proper motion of nearly 60 “tufts” of Si-rich ejecta, giving us the velocity in the plane of the sky. For the line-of-sight velocity, we use two different methods: a nonequilibrium ionization model fit to the strong Si and S lines in the 1.2–2.8 keV regime, and a fit consisting of a series of Gaussian lines. These methods give consistent results, allowing us to determine the redshift or blueshift of each of the knots. Assuming a distance of 3.5 kpc, we find total velocities that range from 2400 to 6600 km s{sup −1}, with a mean of 4430 km s{sup −1}. We find several regions where the ejecta knots have overtaken the forward shock. These regions have proper motions in excess of 6000 km s{sup −1}. Some SN Ia explosion models predict a velocity asymmetry in the ejecta. We find no such velocity asymmetries in Tycho, and we discuss our findings in light of various explosion models, favoring those delayed-detonation models with relatively vigorous and symmetrical deflagrations. Finally, we compare measurements with models of the remnant’s evolution that include both smooth and clumpy ejecta profiles, finding that both ejecta profiles can be accommodated by the observations.

79 ASTRONOMY AND ASTROPHYSICS↗

A stress-based fracture model for reacting metal ejecta

The evolution of reacting metal ejecta continues to be a topic of interest at the forefront of metals in reactive and extreme environments. Ejecta are small particles formed when the surface of a metal undergoes Richtmyer–Meshkov instability from a strong shock. Experiments have shown that in the case where ejecta are in ambient conditions that induce a reaction, the ejecta behave irregularly. The ejecta temperature rises and then plateaus, and the acceleration profile shows unexpected jumps. These variations are assumed to be related to the exothermic heat release and particle mass loss caused by the reaction. To explain this phenomenon, efforts to model this in simulations have increased. While current models can capture many of these physical processes, they currently assign a constant reaction shell thickness with little physical reasoning. This work remedies this problem by assigning a dynamic physically informed shell thickness to the reacting particles, using solid analysis. The shell thickness of the particles impacts the rate of change of reacted mass in the system, as well as the rate at which the particles react. The model is based on a simple stress–strain relationship and gives a dynamic assignment for when the reacting particle should begin to fracture. We compare our model to the previous computational and simulation data to analyze the effects of different model parameters.

42 ENGINEERING↗