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NASA NTRS1994

Comparison of dynamical approximation schemes for non-linear gravitational clustering

We have recently conducted a controlled comparison of a number of approximations for gravitational clustering against the same n-body simulations. These include ordinary linear perturbation theory (Eulerian), the adhesion approximation, the frozen-flow approximation, the Zel'dovich approximation (describable as first-order Lagrangian perturbation theory), and its second-order generalization. In the last two cases we also created new versions of approximation by truncation, i.e., smoothing the initial conditions by various smoothing window shapes and varying their sizes. The primary tool for comparing simulations to approximation schemes was crosscorrelation of the evolved mass density fields, testing the extent to which mass was moved to the right place. The Zel'dovich approximation, with initial convolution with a Gaussian e(exp -k(exp 2)/k(exp 2, sub G)) where k(sub G) is adjusted to be just into the nonlinear regime of the evolved model (details in text) worked extremely well. Its second-order generalization worked slightly better. All other schemes, including those proposed as generalizations of the Zel'dovich approximation created by adding forces, were in fact generally worse by this measure. By explicitly checking, we verified that the success of our best-choice was a result of the best treatment of the phases of nonlinear Fourier components. Of all schemes tested, the adhesion approximation produced the most accurate nonlinear power spectrum and density distribution, but its phase errors suggest mass condensations were moved to slightly the wrong location. Due to its better reproduction of the mass density distribution function and power spectrum, it might be preferred for some uses. We recommend either n-body simulations or our modified versions of the Zel'dovich approximation, depending upon the purpose. The theoretical implication is that pancaking is implicit in all cosmological gravitational clustering, at least from Gaussian initial conditions, even when subcondensations are present.

Melott, Adrian L.↗
NASA NTRS1999

Cloud Thickness from Diffusion of Lidar Pulses in Clouds

Measurements of the distribution of reflected light from a laser beam incident on an aqueous suspension of particles or "cloud" with known thickness and particle size distribution are reported. The distribution is referred to as the "cloud radiative Green's function", G. In the diffusion domain, G is sensitive to cloud thickness, allowing that important quantity to be retrieved. The goal of the laboratory simulation is to provide preliminary estimates of sensitivity of G to cloud thickness,for use in the optimal design of an offbeam Lidar instrument for remote sensing of cloud thickness (THOR, Thickness from Offbeam Returns). These clouds of polystyrene microspheres suspended in water are analogous to real clouds of water droplets suspended in air. The microsphere size distribution is roughly lognormal, from 0.5 microns to 25 microns, similar to real clouds. Density of suspended spheres is adjusted so mean-free-path of visible photons is about 10 cm, approximately 1000 times smaller than in real clouds. The light source is a ND:YAG laser at 530 nm. Detectors are flux and photon-counting Photomultiplier Tube (PMTS), with a glass probe for precise positioning. A Labview 5 VI controls positioning, and data acquisition, via an NI Motion Control board connected to a stepper motor driving an Edmund linear slider, and a 16-channel 16-bit NI-DAQ board. The stepper motor is accurate to 10 microns, and step size is selectable from the VI software. Far from the incident beam, the rate of exponential increase as the direction of the incident beam is approached scales as expected from diffusion theory, linearly with the cloud thickness, and inversely as the square root of the reduced optical thickness, and is independent of particle size. Near the beam the signal begins to increase faster than exponential, due to single and low-order scattering near the backward direction, and here the distribution depends on particle size. Results are being used to verify 3D Monte Carlo radiative transfer simulations, used to estimate signal-to-noise ratios for remotely sensed off beam returns, for both homogeneous and inhomogeneous clouds. Signal-to-noise estimates show that unfiltered observations are straight forward at night, while narrow band pass filters are being studied for day.

Cahalan, Robert F.↗
NASA NTRS2012

Satellite Material Type and Phase Function Determination in Support of Orbital Debris Size Estimation

In performing debris surveys of deep-space orbital regions, the considerable volume of the area to be surveyed and the increased orbital altitude suggest optical telescopes as the most efficient survey instruments; but to proceed this way, methodologies for debris object size estimation using only optical tracking and photometric information are needed. Basic photometry theory indicates that size estimation should be possible if satellite albedo and shape are known. One method for estimating albedo is to try to determine the object's material type photometrically, as one can determine the albedos of common satellite materials in the laboratory. Examination of laboratory filter photometry (using Johnson BVRI filters) on a set of satellite material samples indicates that most material types can be separated at the 1-sigma level via B-R versus R-I color differences with a relatively small amount of required resampling, and objects that remain ambiguous can be resolved by B-R versus B-V color differences and solar radiation pressure differences. To estimate shape, a technique advanced by Hall et al. [1], based on phase-brightness density curves and not requiring any a priori knowledge of attitude, has been modified slightly to try to make it more resistant to the specular characteristics of different materials and to reduce the number of samples necessary to make robust shape determinations. Working from a gallery of idealized debris shapes, the modified technique identifies most shapes within this gallery correctly, also with a relatively small amount of resampling. These results are, of course, based on relatively small laboratory investigations and simulated data, and expanded laboratory experimentation and further investigation with in situ survey measurements will be required in order to assess their actual efficacy under survey conditions; but these techniques show sufficient promise to justify this next level of analysis.

Hejduk, M. D.↗
NASA NTRS2003

Constraints on the Energy Content of the Universe from a Combination of Galaxy Cluster Observables

We demonstrate that constraints on cosmological parameters from the distribution of clusters as a function of redshift (dN/dz) are complementary to accurate angular diameter distance (D(sub A)) measurements to clusters, and their combination significantly tightens constraints on the energy density content of the Universe. The number counts can be obtained from X-ray and/or SZ (Sunyaev-Ze'dovich effect) surveys, and the angular diameter distances can be determined from deep observations of the intra-cluster gas using their thermal bremsstrahlung X-ray emission and the SZ effect. We combine constraints from simulated cluster number counts expected from a 12 deg(sup 2) SZ cluster survey and constraints from simulated angular diameter distance measurements based on the X-ray/SZ method assuming a statistical accuracy of 10% in the angular diameter distance determination of 100 clusters with redshifts less than 1.5. We find that Omega(sub m), can be determined within about 25%, Omega(sub lambda) within 20% and w within 16%. We show that combined dN/dz+(sub lambda) constraints can be used to constrain the different energy densities in the Universe even in the presence of a few percent redshift dependent systematic error in D(sub lambda). We also address the question of how best to select clusters of galaxies for accurate diameter distance determinations. We show that the joint dN/dz+ D(lambda) constraints on cosmological parameters for a fixed target accuracy in the energy density parameters are optimized by selecting clusters with redshift upper cut-offs in the range 0.55 approx. less than 1. Subject headings: cosmological parameters - cosmology: theory - galaxies:clusters: general

Molnar, Sandor M.↗
NASA NTRS2003

Constraints on the Energy Density Content of the Universe Using Only Clusters of Galaxies

We demonstrate that it is possible to constrain the energy content of the Universe with high accuracy using observations of clusters of galaxies only. The degeneracies in the cosmological parameters are lifted by combining constraints from different observables of galaxy clusters. We show that constraints on cosmological parameters from galaxy cluster number counts as a function of redshift and accurate angular diameter distance measurements to clusters are complementary to each other and their combination can constrain the energy density content of the Universe well. The number counts can be obtained from X-ray and/or SZ (Sunyaev-Zeldovich effect) surveys, the angular diameter distances can be determined from deep observations of the intra-cluster gas using their thermal bremsstrahlung X-ray emission and the SZ effect (X-SZ method). In this letter we combine constraints from simulated cluster number counts expected from a 12 deg2 SZ cluster survey and constraints from simulated angular diameter distance measurements based on using the X-SZ method assuming an expected accuracy of 7% in the angular diameter distance determination of 70 clusters with redshifts less than 1.5. We find that R, can be determined within about 25%, A within 20%, and w within 16%. Any cluster survey can be used to select clusters for high accuracy distance measurements, but we assumed accurate angular diameter distance measurements for only 70 clusters since long observations are necessary to achieve high accuracy in distance measurements. Thus the question naturally arises: How to select clusters of galaxies for accurate diameter distance determinations? In this letter, as an example, we demonstrate that it is possible to optimize this selection changing the number of clusters observed, and the upper cut off of their redshift range. We show that constraints on cosmological parameters from combining cluster number counts and angular diameter distance measurements, as opposed to general expectations, will not improve substantially selecting clusters with redshifts higher than one. This important conclusion allow us to restrict our cluster sample to clusters closer than one, in a range where the observational time for accurate distance measurements are more manageable. Subject headings: cosmological parameters - cosmology: theory - galaxies: clusters: general - X-rays: galaxies: clusters

Molnar, Sandor M.↗
NASA NTRS1999

Simulating the Formation of Lunar Floor-Fracture Craters Using Elastoviscoplastic Relaxation

Lunar floor-fracture craters formed during the height of mare basalt emplacement. Due to a general temporal and spatial relation with the maria, these craters, numbering some 200, may be diagnostic of the thermal structure of the crust during this time. As the name suggests, these craters exhibit brittle failure, generally limited to the central floor region. That, and a shallower depth than fresh lunar craters, has led to two main theories as to their formation: laccolith emplacement under the crater and viscous relaxation. The implications of each model for the state of the Moon's crust during this time are quite different, so the viability of each model must be checked. Laccolith emplacement has been treated elsewhere. However, previous attempts to study the relaxation of the craters have assumed only a uniform, Newtonian viscous response of the near surface to the topographic driving forces, and simply postulated that the fractures resulted from tensile stresses associated with floor uplift. Here, we use a more sophisticated rheological model that includes not only non-Newtonian viscous behavior (i.e., the viscosity is stress-dependent), but also incorporates elastic behavior and a plastic component to the rheology to directly simulate the formation of the floor fractures. The results of our simulations show that while elastoviscoplastic relaxation is potentially viable for larger floor-fracture craters, it is not viable for craters with diameters < or = 60 km, the size of the majority of floor-fracture craters. We employ the finite element method, a numerical technique well suited for boundary-value problems, via the commercially available MARC software package. To test the viability of topographic relaxation, our goal is to prepare the simulations as to maximize the amount of relaxation. We take advantage of the natural axisymmetry of craters, simulating one radial plane. Initial shapes are based on data for fresh craters from Pike. To simplify implementation, a fourth order polynomial is used for the basin, while a third order inverse function is used for the rim. This form closely approximates the long-wavelength behavior of complex craters, while ignoring higher-frequency topography, save the rim. This approximation is appropriate because crater relaxation is strongly controlled by long-wavelength topography. Loading is accomplished assuming a uniform gravity field (1.62 m/s-square) and a uniform density of 2900 kg/cubic m. The initial stress state is set to be hydrostatic, with an additional pressure term to account for any overlying topography. This additional pressure term is tapered exponentially with depth. While the simulations quickly settle on a preferred stress state, and while the final solution is fairly insensitive to the choice of the e-folding depth of the taper, selecting an e-folding depth close to the diameter of the crater sets the initial stress state near the preferred state. We assume a diuranally averaged surface temperature of -20 C, and allow temperature to increase with depth at a rate of 50 K//km. Assuming a thermal conductivity of 2 W/in/K, this gradient translates to a heat flow of 100mW/square m, an extremely high value for the Moon. Temperature, of course, will not increase without bound. To maximize relaxation, we allow our temperature profile to increase linearly until it reaches the solidus (assumed to be 1200C) at a depth of 24.4 km, at which point it is kept constant. The presence of melt will drop the bulk viscosity; however, we have no rheological control for partial melts. Therefore, we make no attempt to simulate this situation. Elastoviscoplastic rheological model. In general, geologic materials can behave in three main ways: elastically, viscously (via solid-state creep), and brittly (plasticity is a continuum approach to simulate this phenomenon). We combine these three deformation mechanisms in an extended Maxwell solid, where the total strain can be broken down into a simple summation of the elastic, creep, and plastic strains. In relaxation phenomena in general, the system takes advantage of any means possible to eliminate deviatoric stresses by relaxing away the topography. Previous analyses have only modeled the viscous response. Comparatively, the elastic response in our model can augment the relaxation, to a point. This effect decreases as the elastic response becomes stiffer; indeed, in the limit of infinite elastic Young's modulus (and with no plasticity), the solution converges on the purely viscous solution. Igneous rocks common to the lunar near-surface have Young's modulii in the range of 10-100 GPa. To maximize relaxation, we use a Young's modulus of 10 GPa. (There is negligible sensitivity to the other elastic modulus, the Poisson's ratio; we use 0.25.) For the viscous response, we use a flow law for steady-state creep in thoroughly dried Columbia diabase, because the high plagioclase (about 70 vol%) and orthopyroxene (about 17 vol%) content is similar to the composition of the lunar highland crust as described by remote sensing and sample studies: noritic anorthosite. This flow law is highly non-Newtonian, i.e., the viscosity is highly stress dependent. That, and the variability with temperature, stands in strong contrast to previous examinations of lunar floor-fracture crater relaxation. To model discrete, brittle faulting, we assume "Byerlee's rule," a standard geodynamical technique. We implement this "rule" with an-angle of internal friction of about 40 deg, and a higher-than-normal cohesion of about 3.2 MPa (to approximate the breaking of unfractured rock). The actual behavior of geologic materials is more complex than in our rheological model, so the uncertainties in the plasticity do not represent the state-of-the-art error. Additional information is contained in the original.

Dombard, A. J.↗
NASA NTRS2015

Rao-Blackwellization for Adaptive Gaussian Sum Nonlinear Model Propagation

When dealing with imperfect data and general models of dynamic systems, the best estimate is always sought in the presence of uncertainty or unknown parameters. In many cases, as the first attempt, the Extended Kalman filter (EKF) provides sufficient solutions to handling issues arising from nonlinear and non-Gaussian estimation problems. But these issues may lead unacceptable performance and even divergence. In order to accurately capture the nonlinearities of most real-world dynamic systems, advanced filtering methods have been created to reduce filter divergence while enhancing performance. Approaches, such as Gaussian sum filtering, grid based Bayesian methods and particle filters are well-known examples of advanced methods used to represent and recursively reproduce an approximation to the state probability density function (pdf). Some of these filtering methods were conceptually developed years before their widespread uses were realized. Advanced nonlinear filtering methods currently benefit from the computing advancements in computational speeds, memory, and parallel processing. Grid based methods, multiple-model approaches and Gaussian sum filtering are numerical solutions that take advantage of different state coordinates or multiple-model methods that reduced the amount of approximations used. Choosing an efficient grid is very difficult for multi-dimensional state spaces, and oftentimes expensive computations must be done at each point. For the original Gaussian sum filter, a weighted sum of Gaussian density functions approximates the pdf but suffers at the update step for the individual component weight selections. In order to improve upon the original Gaussian sum filter, Ref. [2] introduces a weight update approach at the filter propagation stage instead of the measurement update stage. This weight update is performed by minimizing the integral square difference between the true forecast pdf and its Gaussian sum approximation. By adaptively updating each component weight during the nonlinear propagation stage an approximation of the true pdf can be successfully reconstructed. Particle filtering (PF) methods have gained popularity recently for solving nonlinear estimation problems due to their straightforward approach and the processing capabilities mentioned above. The basic concept behind PF is to represent any pdf as a set of random samples. As the number of samples increases, they will theoretically converge to the exact, equivalent representation of the desired pdf. When the estimated qth moment is needed, the samples are used for its construction allowing further analysis of the pdf characteristics. However, filter performance deteriorates as the dimension of the state vector increases. To overcome this problem Ref. [5] applies a marginalization technique for PF methods, decreasing complexity of the system to one linear and another nonlinear state estimation problem. The marginalization theory was originally developed by Rao and Blackwell independently. According to Ref. [6] it improves any given estimator under every convex loss function. The improvement comes from calculating a conditional expected value, often involving integrating out a supportive statistic. In other words, Rao-Blackwellization allows for smaller but separate computations to be carried out while reaching the main objective of the estimator. In the case of improving an estimator's variance, any supporting statistic can be removed and its variance determined. Next, any other information that dependents on the supporting statistic is found along with its respective variance. A new approach is developed here by utilizing the strengths of the adaptive Gaussian sum propagation in Ref. [2] and a marginalization approach used for PF methods found in Ref. [7]. In the following sections a modified filtering approach is presented based on a special state-space model within nonlinear systems to reduce the dimensionality of the optimization problem in Ref. [2]. First, the adaptive Gaussian sum propagation is explained and then the new marginalized adaptive Gaussian sum propagation is derived. Finally, an example simulation is presented.

state estimation↗
NASA NTRS2002

Microscopic Foundation and Simulation of Coupled Carrier-Temperature Diffusions in Semiconductor Lasers

A typical semiconductor-based optoelectronic device, such as a diode laser, consists of three subsystems: an optical field, an electron-hole plasma (EHP), and a host crystal lattice. The physics of such a device involves the interplay of optical, electrical and thermal processes. A proper description of such a device requires that all three processes are treated on equal footing and in a self-consistent fashion. Furthermore, since a semiconductor laser has intrinsic spatial inhomogeneity, such a self-consistency naturally leads to a set of partial differential equations in space and time. There is a significant lacking of research interest and results on the transport aspects of optical devices in the literature with only a few exceptions. Even the most important carrier diffusion coefficient has not been properly derived and studied so far for optically excited plasma, while most of the work adopted results from electronics community where heavily doped semiconductors with mainly one type of carriers are dealt with. The corresponding transport equation for plasma energy or temperature has received even less attention. In this talk we describe our recent results on such a self-consistent derivation of temperature and carrier-density diffusion equations coupled with the lasing process. Starting from the microscopic semiconductor Bloch equations (SBEs) including the Boltzmann transport terms in the distribution function equations for electrons and holes, we derived a closed set of diffusion equations for carrier densities and temperatures with self-consistent coupling to Maxwell's equation and to an effective optical polarization equation. The coherent many-body effects are included within the screened Hartree-Fock approximation, while scatterings are treated within the second Born approximation including both the in- and out-scatterings. Microscopic expressions for electron-hole (e-h) and carrier-LO (c-LO) phonon scatterings are directly used to derive the momentum and energy relaxation rates. These rates expressed as functions of temperatures and densities lead to microscopic expressions for self- and mutual-diffusion coefficients in the coupled density-temperature diffusion equations. Approximations for reducing the general two-component description of the electron-hole plasma (EHP) to a single-component one are discussed. In particular, we show that a special single-component reduction is possible when e-h scattering dominates over c-LO phonon scattering. The ambipolar diffusion approximation is also discussed and we show that the ambipolar diffusion coefficients are independent of e-h scattering, even though the diffusion coefficients of individual components depend sensitively on the e-h scattering rates. Our discussions lead to new perspectives into the roles played in the single-component reduction by the electron-hole correlation in momentum space induced by scatterings and the electron-hole correlation in real space via internal static electrical field. Finally, the theory is completed by coupling the diffusion equations to the lattice temperature equation and to the effective optical polarization which in turn couples to the laser field. The equations derived above are implemented in various limiting cases to a typical diode laser to study the consequences of nonlinear diffusion and the cross diffusion terms on laser behavior, especially the dynamic behavior of a diode laser under modulation. Detailed results will be presented by comparing with the standard rate equation results.

Li, J.↗
NASA NTRS1999

Aeolian Sand Transport in the Planetary Context: Respective Roles of Aerodynamic and Bed-Dilatancy Thresholds

The traditional view of aeolian sand transport generally estimates flux from the perspective of aerodynamic forces creating the airborne grain population, although it has been recognized that "reptation" causes a significant part of the total airborne flux; reptation involves both ballistic injection of grains into the air stream by the impact of saltating grains as well as the "nudging" of surface grains into a creeping motion. Whilst aerodynamic forces may initiate sand motion, it is proposed here that within a fully-matured grain cloud, flux is actually governed by two thresholds: an aerodynamic threshold, and a bed-dilatancy threshold. It is the latter which controls the reptation population, and its significance increases proportionally with transport energy. Because we only have experience with terrestrial sand transport, extrapolations of aeolian theory to Mars and Venus have adjusted only the aerodynamic factor, taking gravitational forces and atmospheric density as the prime variables in the aerodynamic equations, but neglecting reptation. The basis for our perspective on the importance of reptation and bed dilatancy is a set of experiments that were designed to simulate sand transport across the surface of a martian dune. Using a modified sporting crossbow in which a sand-impelling sabot replaced the bolt-firing mechanism, individual grains of sand were fired at loose sand targets with glancing angles typical of saltation impact; grains were projected at about 80 m/s to simulate velocities commensurate with those predicted for extreme martian aeolian conditions. The sabot impelling method permitted study of individual impacts without the masking effect of bed mobilization encountered in wind-tunnel studies. At these martian impact velocities, grains produced small craters formed by the ejection of several hundred grains from the bed. Unexpectedly, the craters were not elongated, despite glancing impact; the craters were very close to circular in planform. High-speed photography showed them to grow in both diameter and depth after the impactor had ricochetted from the crater site. The delayed response of the bed was "explosive" in nature, and created a miniature ejecta curtain spreading upward and outward for many centimeters for impact of 100-300 micron-diameter grains into similar material. Elastic energy deposited in the bed by the impacting grain creates a subsurface stress regime or "quasi-Boussinesq" compression field. Elastic recovery of the bed occurs by dilatancy; shear stresses suddenly convert the grains from closed to open packing, and grains are consequently able to eject themselves forcefully from the impact site. Random jostling of the grains causes radial homogenization of stress vectors and a resulting circular crater. There is a great temptation to draw parallels with cratering produced by meteorite impacts, but a rigorous search for common modelling ground between the two phenomena has not been conducted at this time. For every impact of an aerodynamically energized grain, there are several hundred grains ejected into the wind for the high-energy transport that might occur on Mars. Many of these grains will themselves become subject to the boundary layer's aerodynamic lift forces (their motion will not immediately die and add to the creep population), and these grains will become indistinguishable from those lifted entirely by aerodynamic forces. As each grain impacts the bed, it will eject even more grains into the flow. A cascading effect will take place, but because it must be finite in its growth, damping will occur as the number of grains set in motion causes mid-air collisions that prevent much of the impact energy from reaching the surface of the bed -thus creating a dynamic equilibrium in a high-density saltation cloud. It is apparent that for a given impact energy, the stress field permits a smaller volume of grains to convert to open packing as the size of the bed grains increases, or as the energy of the "percussive" grain decreases (by decrease in velocity or mass). Thus, the mass of the "repercussive" grain population that is ejected from the impact site becomes a function of the scale of the stress field in relation to the scale of the bed material (self-similarity being applicable if both bed size and energy are simultaneously adjusted). In other words, in a very high energy aeolian system where an aerodynamically raised grain can ballistically raise many more grains, the amount of material lifted into the wind becomes largely a function of a dilatancy threshold. If this threshold is exceeded, grains are repercussively injected into the saltation cloud. The "dilatancy threshold" may be defined in terms of the saltation percussive force required to convert the bed, through elastic response, from a closed to an open packing system. If open packing cannot be created, the grains cannot escape from the impact site, even though the elastic deformation and percussive force may be able to reorganize the grains with respect to one another. As the crossbow experiments showed, for an ever-increasing bed grain size, a point is reached when no material can be moved because the energy of the percussive grain is insufficient to dilate the relatively coarse bed. Although this seems to be stating the obvious -- that too little energy will not cause the bed to splash -- the consequences of exceeding the "splash threshold" by dilatancy are not so obvious for high-energy aeolian transport. It is noted that the force required to elastically dilate the bed has to overcome Coulombic grain attractions such as dipole-dipole coupling, dielectric, monopole, contact-induced dipole attractions, van der Waals forces, molecular monolayer capillary forces, as well as the mechanical interlocking frictional resistance of the grains. On Mars, it is predicted that the dilatancy threshold may be the prime control of grain flux. Additional information is contained in the original.

Marshall, J. R.↗