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NASA NTRS2018

Variability Timescale and Spectral Index of Sgr A* in the Near Infrared: Approximate Bayesian Computation Analysis of the Variability of the Closest Supermassive Black Hole

Sagittarius A* (Sgr A* ) is the variable radio, near-infrared (NIR), and X-ray source associated with accretion onto the Galactic center black hole. We present an analysis of the most comprehensive NIR variability data set of Sgr A* to date: eight 24 hr epochs of continuous monitoring of Sgr A* at 4.5 μm with the IRAC instrument on the Spitzer Space Telescope, 93 epochs of 2.18 μm data from Naos Conica at the Very Large Telescope, and 30 epochs of 2.12 μm data from the NIRC2 camera at the Keck Observatory, in total 94,929 measurements. A new approximate Bayesian computation method for fitting the first-order structure function extracts information beyond current fast Fourier transformation (FFT) methods of power spectral density (PSD) estimation. With a combined fit of the data of all three observatories, the characteristic coherence timescale of Sgr A* is b 243 57 82 t = - + minutes (90% credible interval). The PSD has no detectable features on timescales down to 8.5 minutes (95% credible level), which is the ISCO orbital frequency for a dimensionless spin parameter a = 0.92. One light curve measured simultaneously at 2.12 and 4.5 μm during a low flux-density phase gave a spectral index αs = 1.6 ± 0.1 (F µ n s n -a ). This value implies that the Sgr A* NIR color becomes bluer during higher flux-density phases. The probability densities of flux densities of the combined data sets are best fit by log-normal distributions. Based on these distributions, the Sgr A* spectral energy distribution is consistent with synchrotron radiation from a nonthermal electron population from below 20 GHz through the NIR.

accretion↗

NASA NTRS1993

Using harmonic oscillators to determine the spot size of Hermite-Gaussian laser beams

The similarity of the functional forms of quantum mechanical harmonic oscillators and the modes of Hermite-Gaussian laser beams is illustrated. This functional similarity provides a direct correlation to investigate the spot size of large-order mode Hermite-Gaussian laser beams. The classical limits of a corresponding two-dimensional harmonic oscillator provide a definition of the spot size of Hermite-Gaussian laser beams. The classical limits of the harmonic oscillator provide integration limits for the photon probability densities of the laser beam modes to determine the fraction of photons detected therein. Mathematica is used to integrate the probability densities for large-order beam modes and to illustrate the functional similarities. The probabilities of detecting photons within the classical limits of Hermite-Gaussian laser beams asymptotically approach unity in the limit of large-order modes, in agreement with the Correspondence Principle. The classical limits for large-order modes include all of the nodes for Hermite Gaussian laser beams; Sturm's theorem provides a direct proof.

Steely, Sidney L.↗

NASA NTRS1992

Probabilistic evaluation of uncertainties and risks in aerospace components

This paper summarizes a methodology developed at NASA Lewis Research Center which computationally simulates the structural, material, and load uncertainties associated with Space Shuttle Main Engine (SSME) components. The methodology was applied to evaluate the scatter in static, buckling, dynamic, fatigue, and damage behavior of the SSME turbo pump blade. Also calculated are the probability densities of typical critical blade responses, such as effective stress, natural frequency, damage initiation, most probable damage path, etc. Risk assessments were performed for different failure modes, and the effect of material degradation on the fatigue and damage behaviors of a blade were calculated using a multi-factor interaction equation. Failure probabilities for different fatigue cycles were computed and the uncertainties associated with damage initiation and damage propagation due to different load cycle were quantified. Evaluations on the effects of mistuned blades on a rotor were made; uncertainties in the excitation frequency were found to significantly amplify the blade responses of a mistuned rotor. The effects of the number of blades on a rotor were studied. The autocorrelation function of displacements and the probability density function of the first passage time for deterministic and random barriers for structures subjected to random processes also were computed. A brief discussion was included on the future direction of probabilistic structural analysis.

Shah, A. R.↗

DOE OSTI2025

Control-Affine Schrödinger Bridge and Generalized Bohm Potential

From a stochastic control perspective, the Schrödinger bridge is a density-valued continuous curve parameterized by time that connects a given pair of initial and terminal probability densities via minimum effort controlled Brownian motion. The control-affine Schrödinger bridge extends this idea to a generic control-affine Itô diffusion, possibly with an additive state cost. Here, in this letter, we recast the necessary conditions of optimality for the control-affine Schrödinger bridge problem as a two point boundary value problem for a quantum mechanical Schrödinger PDE with complex potential. This complex-valued potential is a generalization of the real-valued Bohm potential in quantum mechanics. Our derived potential is akin to the optical potential in nuclear physics where the real part of the potential encodes elastic scattering (transmission of wave function), and the imaginary part encodes inelastic scattering (absorption of wave function). The key takeaway is that the process noise that drives the evolution of probability densities induces an absorbing medium in the evolution of wave function. These results make new connections between control theory and non-equilibrium statistical mechanics through the lens of quantum mechanics.

Markov processes↗

NASA NTRS1993

PDF approach for compressible turbulent reacting flows

The objective of the present work is to develop a probability density function (pdf) turbulence model for compressible reacting flows for use with a CFD flow solver. The probability density function of the species mass fraction and enthalpy are obtained by solving a pdf evolution equation using a Monte Carlo scheme. The pdf solution procedure is coupled with a compressible CFD flow solver which provides the velocity and pressure fields. A modeled pdf equation for compressible flows, capable of capturing shock waves and suitable to the present coupling scheme, is proposed and tested. Convergence of the combined finite-volume Monte Carlo solution procedure is discussed, and an averaging procedure is developed to provide smooth Monte-Carlo solutions to ensure convergence. Two supersonic diffusion flames are studied using the proposed pdf model and the results are compared with experimental data; marked improvements over CFD solutions without pdf are observed. Preliminary applications of pdf to 3D flows are also reported.

Hsu, A. T.↗

NASA NTRS1975

Mean and fluctuating flow measurements of a fully-developed, non-adiabatic, hypersonic boundary layer

Extensive boundary-layer measurements have been made on a cone-ogive-cylinder model at a free-stream Mach number of 7.0 and momentum-thickness Reynolds number of 8500. Mean flow transformations and calculated turbulence correlations are presented which are in good agreement with previous incompressible results. New quantitative turbulence measurements including measurements of the first higher moment and probability density of fluctuations in mass flow and total temperature in hypersonic flow are also presented. The higher moment and probability density data show that the characters of the fluctuation modes of the mass flow and total temperature are significantly different in the wall region and in the outer part of the boundary layer. These differences together with data on the turbulence scale and lifetime obtained from autocorrelation and space-time correlation measurements are discussed.

Owen, F. K.↗

NASA NTRS1999

The Use of the Satellite Breakup Risk Assessment Model (SBRAM) to Characterize Collision Risk to Manned Spacecraft

NASA uses environment models such as ORDEM96 to characterize the long-term orbital debris collision hazard for spacecraft in LEO. Occasionally, however, there are breakups of satellites or rocket bodies that create enhanced collision hazard for a period of time. This enhanced collision hazard can pose increased risks to space operations - especially those involving manned missions where the tolerance for risk is very low. NASA has developed SBRAM to simulate the enhanced debris environment in the days and weeks that follow such a breakup. This simulation provides the kind of risk probabilities that can be used by mission planners to consider if changes are warranted for the mission. Announcements of breakups come to NASA from US Space Command as soon as they are identified. The pre-breakup orbit and time of breakup are used to determine the initial conditions of the explosion. SBRAM uses the latest explosion models developed at NASA to simulate a debris cloud for the breakup. The model uses a Monte Carlo technique to create a random debris cloud from the probability distributions in the breakup model. Each piece of debris randomly created in the cloud is propagated in a deterministic manner to include the effects of drag and other orbital perturbations. The detailed geometry of each simulated close approach to the target spacecraft is noted and logged and the collision probability is computed using an estimated probability density in down-range and cross-range positions of both the target spacecraft and debris object. The collision probability is computed from the overlap of these probability densities for each close-approach geometry and summed over all computed conjunctions. Cloud propagation runs over the desired time interval are then repeated until the scale of the collision risk can be estimated to a desired precision. This paper presents an overview of the SBRAM model and a number of examples, both real and hypothetical, to demonstrate its use. In addition, a number of different examples are shown how the data can be used by decision makers on issues such as spacecraft orientation and timing of EVAs.

Matney, Mark J.↗

NASA NTRS1995

On the joint spectral density of bivariate random sequences

For univariate random sequences, the power spectral density acts like a probability density function of the frequencies present in the sequence. This dissertation extends that concept to bivariate random sequences. For this purpose, a function called the joint spectral density is defined that represents a joint probability weighing of the frequency content of pairs of random sequences. Given a pair of random sequences, the joint spectral density is not uniquely determined in the absence of any constraints. Two approaches to constraining the sequences are suggested: (1) assume the sequences are the margins of some stationary random field, (2) assume the sequences conform to a particular model that is linked to the joint spectral density. For both approaches, the properties of the resulting sequences are investigated in some detail, and simulation is used to corroborate theoretical results. It is concluded that under either of these two constraints, the joint spectral density can be computed from the non-stationary cross-correlation.

Aalfs, David D.↗

NASA NTRS1975

Polarization tracking of a partially coherent signal using a double loop

We consider the problem of simultaneously estimating the phase and polarization angles of a linearly polarized electromagnetic wave. The solution of this problem has applications to space communications and ionospheric probing. The nonlinear stochastic characteristics of a cross-coupled double-loop implementation for the tracker are evaluated using the multidimensional Fokker-Planck equation. The exact stationary probability density is found for a special case of equal signal-to-noise ratio (SNR) in the two loops. The stationary probability density is obtained for a general class of double loops when the two loop bandwidths differ greatly.

Ohlson, J. E.↗

NASA NTRS2005

Surveillance system and method having an adaptive sequential probability fault detection test

System and method providing surveillance of an asset such as a process and/or apparatus by providing training and surveillance procedures that numerically fit a probability density function to an observed residual error signal distribution that is correlative to normal asset operation and then utilizes the fitted probability density function in a dynamic statistical hypothesis test for providing improved asset surveillance.

Herzog, James P.↗

NASA NTRS2006

Surveillance system and method having an adaptive sequential probability fault detection test

Bickford, Randall L.↗

NASA NTRS2008

Surveillance System and Method having an Adaptive Sequential Probability Fault Detection Test

Bickford, Randall L.↗

NASA NTRS1992

Asteroid orbital error analysis: Theory and application

We present a rigorous Bayesian theory for asteroid orbital error estimation in which the probability density of the orbital elements is derived from the noise statistics of the observations. For Gaussian noise in a linearized approximation the probability density is also Gaussian, and the errors of the orbital elements at a given epoch are fully described by the covariance matrix. The law of error propagation can then be applied to calculate past and future positional uncertainty ellipsoids (Cappellari et al. 1976, Yeomans et al. 1987, Whipple et al. 1991). To our knowledge, this is the first time a Bayesian approach has been formulated for orbital element estimation. In contrast to the classical Fisherian school of statistics, the Bayesian school allows a priori information to be formally present in the final estimation. However, Bayesian estimation does give the same results as Fisherian estimation when no priori information is assumed (Lehtinen 1988, and reference therein).

Muinonen, K.↗

NASA NTRS1985

Birds and insects as radar targets - A review

A review of radar cross-section measurements of birds and insects is presented. A brief discussion of some possible theoretical models is also given and comparisons made with the measurements. The comparisons suggest that most targets are, at present, better modeled by a prolate spheroid having a length-to-width ratio between 3 and 10 than by the often used equivalent weight water sphere. In addition, many targets observed with linear horizontal polarization have maximum cross sections much better estimated by a resonant half-wave dipole than by a water sphere. Also considered are birds and insects in the aggregate as a local radar 'clutter' source. Order-of-magnitude estimates are given for many reasonable target number densities. These estimates are then used to predict X-band volume reflectivities. Other topics that are of interest to the radar engineer are discussed, including the doppler bandwidth due to the internal motions of a single bird, the radar cross-section probability densities of single birds and insects, the variability of the functional form of the probability density functions, and the Fourier spectra of single birds and insects.

Vaughn, C. R.↗

NASA NTRS2002

Optimization of Systems with Uncertainty: Initial Developments for Performance, Robustness and Reliability Based Designs

This paper presents a study on the optimization of systems with structured uncertainties, whose inputs and outputs can be exhaustively described in the probabilistic sense. By propagating the uncertainty from the input to the output in the space of the probability density functions and the moments, optimization problems that pursue performance, robustness and reliability based designs are studied. Be specifying the desired outputs in terms of desired probability density functions and then in terms of meaningful probabilistic indices, we settle a computationally viable framework for solving practical optimization problems. Applications to static optimization and stability control are used to illustrate the relevance of incorporating uncertainty in the early stages of the design. Several examples that admit a full probabilistic description of the output in terms of the design variables and the uncertain inputs are used to elucidate the main features of the generic problem and its solution. Extensions to problems that do not admit closed form solutions are also evaluated. Concrete evidence of the importance of using a consistent probabilistic formulation of the optimization problem and a meaningful probabilistic description of its solution is provided in the examples. In the stability control problem the analysis shows that standard deterministic approaches lead to designs with high probability of running into instability. The implementation of such designs can indeed have catastrophic consequences.

Crespo, Luis G.↗

NASA NTRS1989

Probabilities and statistics for backscatter estimates obtained by a scatterometer with applications to new scatterometer design data

The values of the Normalized Radar Backscattering Cross Section (NRCS), sigma (o), obtained by a scatterometer are random variables whose variance is a known function of the expected value. The probability density function can be obtained from the normal distribution. Models for the expected value obtain it as a function of the properties of the waves on the ocean and the winds that generated the waves. Point estimates of the expected value were found from various statistics given the parameters that define the probability density function for each value. Random intervals were derived with a preassigned probability of containing that value. A statistical test to determine whether or not successive values of sigma (o) are truly independent was derived. The maximum likelihood estimates for wind speed and direction were found, given a model for backscatter as a function of the properties of the waves on the ocean. These estimates are biased as a result of the terms in the equation that involve natural logarithms, and calculations of the point estimates of the maximum likelihood values are used to show that the contributions of the logarithmic terms are negligible and that the terms can be omitted.

Pierson, Willard J., Jr.↗

DOE OSTI2026

The 2025 Evaluation of Experimental Thermonuclear Reaction Rates (ETR25)

This work describes the formalism for estimating thermonuclear reaction rates for astrophysical applications, emphasizing modern statistical approaches such as Monte Carlo sampling and Bayesian models. We discuss related topics including the calculation of resonance energies from nuclear Q values, indirect estimates of particle partial widths, and matching of reaction rates at elevated temperatures to statistical model results. We have evaluated available experimental data on cross sections, resonance energies and strengths, partial widths, lifetimes, spin-parities, and spectroscopic factors. Based on these results, we have estimated numerical values of 78 experimental charged-particle thermonuclear reaction rates for target nuclei in the A = 2–40 mass region, for temperatures ranging from 1 MK to 10 GK. For each reaction, three rate values are provided: low, median, and high, corresponding to the 16th, 50th, and 84th percentiles, respectively, of the cumulative reaction rate probability density distribution. Additionally, we present the factor uncertainty of each rate at each temperature grid point. These results enable users to sample the reaction rate probability density in nucleosynthesis calculations, facilitating uncertainty estimates of nuclidic abundances. The rates presented here refer to their laboratory values. For use in stellar model simulations, these values need to be corrected for the effects of thermal excitations of the interacting nuclei. For each reaction, we include graphs that illustrate the fractional contributions to the overall reaction rate along with the associated uncertainty. These visuals are designed to assist both stellar modelers and nuclear experimentalists by identifying the primary sources of rate uncertain=^texttx);ty at specific stellar temperatures. A graphical comparison with earlier Monte Carlo rates is also provided.

Nuclear astrophysics↗

NASA NTRS1977

Radar backscattering from a sea having an anisotropic large-scale surface, part 2

A two scale scattering model was derived that combines specular reflections from sea waves and Bragg scattering in a manner consistent with energy conservation. The effect of the tilting of the small scale roughness by the large scale roughness was included, which accounted for the reduction of reflected power. The special case of backscattering for which the transmitted polarization equaled the received polarization was considered. An anisotropic large scale surface was used to specify the probability density function of the large scale surface normal. In order to isolate the azimuthal variation of the normalized radar cross section produced by the anisotropic probability density function, an isotropical small scale spectrum was assumed.

Wentz, F. J.↗