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All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

New Atmospheric Turbulence Model for Shuttle Applications

An updated NASA atmospheric turbulence model, from 0 to 200 km altitude, which was developed to be more realistic and less conservative when applied to space shuttle reentry engineering simulation studies involving control system fuel expenditures is presented. The prior model used extreme turbulence (3 sigma) for all altitudes, whereas in reality severe turbulence is patchy within quiescent atmospheric zones. The updated turublence model presented is designed to be more realistic. The prior turbulence statistics (sigma and L) were updated and were modeled accordingly.

Justus, C. G.↗
NASA NTRS1973

User's instructions for the cardiovascular Walters model

The model is a combined, steady-state cardiovascular and thermal model. It was originally developed for interactive use, but was converted to batch mode simulation for the Sigma 3 computer. The model has the purpose to compute steady-state circulatory and thermal variables in response to exercise work loads and environmental factors. During a computer simulation run, several selected variables are printed at each time step. End conditions are also printed at the completion of the run.

Croston, R. C.↗
NASA NTRS1994

A Knowledge-Based Representation Scheme for Environmental Science Models

One of the primary methods available for studying environmental phenomena is the construction and analysis of computational models. We have been studying how artificial intelligence techniques can be applied to assist in the development and use of environmental science models within the context of NASA-sponsored activities. We have identified several high-utility areas as potential targets for research and development: model development; data visualization, analysis, and interpretation; model publishing and reuse, training and education; and framing, posing, and answering questions. Central to progress on any of the above areas is a representation for environmental models that contains a great deal more information than is present in a traditional software implementation. In particular, a traditional software implementation is devoid of any semantic information that connects the code with the environmental context that forms the background for the modeling activity. Before we can build AI systems to assist in model development and usage, we must develop a representation for environmental models that adequately describes a model's semantics and explicitly represents the relationship between the code and the modeling task at hand. We have developed one such representation in conjunction with our work on the SIGMA (Scientists' Intelligent Graphical Modeling Assistant) environment. The key feature of the representation is that it provides a semantic grounding for the symbols in a set of modeling equations by linking those symbols to an explicit representation of the underlying environmental scenario.

Keller, Richard M.↗
NASA NTRSDate not supplied

Shielding Considerations for CubeSat Structures During Solar Maximum

The purpose of this lessons learned is to communicate the utility of shielding in small spacecraft planning for the support of mission assurance and reliability. Numerous SmallSats have been flying in polar low earth orbit for scientific, communications, technology demonstrations, and imaging with academic, commercial, and government interests. Shielding has been part of mission assurance and reliability from the advent of long duration spacecraft missions. The Shields-1 CubeSat has been operating in polar low earth orbit since 16 December 2018 with atomic number (Z)-grade radiation shielding and demonstrates shielding effectiveness. Shields-1 has collected a representative example of solar minimum data in 2019 with 8 Teledyne dosimeters over varying shielding effectiveness. It serves as current experimental data and has been compared with NOVICE Shielding estimates using the AP8 –AE8 trapped radiation model with the Shields-1 CAD and generic CubeSat 3 unit (U) models. Using NOVICE model radiation analysis coding, the shielding effectiveness's, based on a generic CubeSat 3U structure with 4 electronic boards, were estimated for aluminum wall thicknesses ranging from 0.204 cmto4.44 cm (0.550 g/cm2–12.0 g/cm2) thick aluminum. For modeled polar orbiting spacecraft, solar maximum total ionizing dose (TID)increases by nearly a magnitude for thin-walled aluminum 0.550g/cm2-0.686 g/cm2(0.204 cm –0.254 cm) typical CubeSat structures. The shielding effectiveness by NOVICE Sigma estimates, which is a shielding sphere approximation around a detector, showed a linear relationship with wall thickness, which increased over the wall thickness by a ratio of 1.43 determined by linear regression analysis. Using NOVICE Adjoint Monte-Carlo Modeling of solar minimum and solar maximum with the inclusion of a worst-case solar particle event over a one year mission without geomagnetic shielding, the TID for minimum and maximum conditions for a generic 3U with a wall thickness of 0.254 cm is 158 RAD and 1540 RAD, respectively. The modeled total solar maximum TID is over estimated, because at low orbital latitudes a spacecraft will have shielding from Earth's magnetic field. However, TID will still be significant at high latitudes over the poles, where a spacecraft is exposed in a solar particle event. In contrast, to a thin walled generic 3U CubeSat, Shields-1 electronics enclosure has a shielding effectiveness of 21.3 g/cm2from NOVICE Sigma modeling and is expected to show reduced total ionizing dose increases during the present active Solar Cycle 25 period. Because solar particle events during solar maximum increase TID on electronic parts with thin-walled shielding in short periods of time, it is a mission assurance and reliability consideration on the spacecraft’s mission value versus adding shielding for risk reduction of premature spacecraft or instrument payload loss. Since the volumes of many instruments and system electronics have reduced with small spacecraft, shielding material costs and weight penalties have diminished. A small spacecraft project budget and schedule may limit traditional radiation-hardened part use and radiation testing requirements, where shielding can contribute to mission assurance and reliability with reduced costs.

Shields-1↗
NASA NTRSDate not supplied

Shielding Considerations for CubeSat Structures During Solar Maximum

The purpose of this lessons learned paper is to communicate the utility of shielding in small spacecraft planning for the support of mission assurance and reliability. Numerous SmallSats have been flying in polar low Earth orbit for science, communications, technology demonstrations, and imaging with academic, commercial, and government interests. Shielding has been part of mission assurance and reliability from the advent of long duration spacecraft missions. The Shields-1 CubeSat has been operating in polar low Earth orbit since 16 December 2018 with atomic number (Z)-grade radiation shielding and demonstrates shielding effectiveness. Shields-1 has collected a representative example of solar minimum data in 2019 with eight Teledyne µDosimeters over varying shielding effectivenesses. It serves as current experimental data and has been compared with NOVICE Shielding estimates using the AP8 – AE8 trapped radiation model with the Shields-1 CAD and generic CubeSat three unit (3U) models. Using NOVICE model radiation analysis coding, the shielding effectivenesses, based on a generic CubeSat 3U structure with four electronic boards, were estimated for aluminum wall thicknesses ranging from 0.204 cm to 4.44 cm (0.550 g/cm 2 to 12.0 g/cm 2 ). For modeled polar orbiting spacecraft, solar maximum total ionizing dose (TID) increases by nearly a magnitude for thin-walled aluminum 0.550-g/cm 2 - 0.686-g/cm 2 (0.204-cm – 0.254-cm) typical CubeSat structures. The shielding effectiveness estimated by NOVICE Sigma, which is a shielding sphere approximation around a detector, showed a linear relationship with wall thickness, which increased over the wall thickness by a ratio of 1.43 determined by linear regression analysis. Using NOVICE Adjoint Monte-Carlo Modeling of solar minimum and solar maximum with the inclusion of a worst-case solar particle event over a one-year mission without geomagnetic shielding, the TID for minimum and maximum conditions for a generic 3U with a wall thickness of 0.254 cm is 158 RAD and 1540 RAD, respectively. The modeled total solar maximum TID is over estimated, because at low orbital latitudes a spacecraft will have shielding from the Earth’s magnetic field. However, TID will still be significant at high latitudes over the poles, where a spacecraft is exposed in a solar particle event. In contrast to a thin-walled generic 3U CubeSat, the Shields-1 electronics enclosure has a shielding effectiveness of 21.3 g/cm 2 from NOVICE Sigma modeling and is expected to show reduced total ionizing dose increases during the present active Solar Cycle 25 period. Because solar particle events during solar maximum increase TID on electronic parts with thin-walled shielding in short periods of time, it is a mission assurance and reliability consideration on the mission value of the spacecraft versus adding shielding for risk reduction of premature spacecraft or instrument payload loss. Since the volumes of many instruments and system electronics have reduced with small spacecraft, shielding material costs and weight penalties have diminished. A small spacecraft project budget and schedule may limit traditional radiation-hardened part use and radiation testing requirements, where shielding can contribute to mission assurance and reliability with reduced costs.

Shields-1↗
DOE OSTI2021

Measurement of beam asymmetry for π – Δ ++ photoproduction on the proton at E γ = 8.5 GeV

We report a measurement of the $\pi^-$ photoproduction beam asymmetry for the reaction $\vec{\gamma} p \rightarrow \pi^- \Delta^{++}$ using data from the GlueX experiment in the photon beam energy range 8.2--8.8 GeV. The asymmetry $\Sigma$ is measured as a function of four-momentum transfer $t$ to the $\Delta^{++}$ and compared to phenomenological models. We find that $\Sigma$ varies as a function of $t$: negative at smaller values and positive at higher values of $|t|$. The reaction can be described theoretically by $t$-channel particle exchange requiring pseudoscalar, vector, and tensor intermediaries. In particular, this reaction requires charge exchange, allowing us to probe pion exchange and the significance of higher-order corrections to one-pion exchange at low momentum transfer. Constraining production mechanisms of conventional mesons may aid in the search for and study of unconventional mesons. Furthermore, this is the first measurement of the process at this energy.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS1984

X-ray spectra of clusters of galaxies

The X-ray emission from luminous clusters of galaxies is dominated by thermal bremsstrahlung from an intergalactic medium. The central density of the gas is strongly correlated with the X-ray surface brightness. The X-ray surface brightness S(sigma) of many clusters is well modeled by a law of the form S(sigma) alpha S(o) 1 + R(2)/A(2) to the minus 3 beta + 1/2 power with beta approximately equal to 0.66. However, this model does not fit the X-ray spectral or optical galaxy counts well. In clusters with cooling flows in their center there is a strong correlation between the cooling rates of X-ray emitting material and optical H alpha emission. It is not clear, at present, what percentage of the virial mass of the cluster is in hot gas but if beta = 0.66, it is possible for the values to be of the order of 1/2. Spatially resoled X-ray spectroscopy is necessary to determine this value with any accuracy.

Mushotzky, R. F.↗
NASA NTRS1994

The motions of clusters and group of galaxies

The distributions of peculiar velocities of rich clusters and of groups of galaxies are investigated for different cosmological models and are compared with observations. Four cosmological models are studied: standard cold dark matter (CDM) (omega = 1); low-density CDM (omega = 0.3); hot dark matter (HDM) (omega = 1); and primeval baryonic isocurvature (PBI) (omega = 0.3). All models are normalized to the microwave background fluctuations observed by Cosmic Background Explorer (COBE). We find that rich clusters of galaxies exhibit a Maxwellian distribution of peculiar velocities in all models, as expected from a Gaussian initial density field. The clusters appear to be fundamental and efficient tracers of the large-scale velocity field. The cluster three-dimensional velocity distribution typically peaks at v approximately 600 km/s and extends to high cluster velocities of v approximately 2000 km/s. The low-density CDM model exhibits somewhat lower velocities: it peaks at approximately 400 km/s and extends to approximately 1200 km/s. Approximately 10% (approximately 1% for low-density CDM) of all model rich clusters move with high peculiar velocities of V greater than or = 10(exp 3) km/s. The highest velocity clusters frequently originate in dense superclusters. The model velocity distributions of rich clusters are compared with the model velocity distributions of small groups of galaxies, and of the total matter. The group velocity distribution is, in general, similar to the velocity distribution of the rich clusters. The matter velocity distribution is similar to that of the rich clusters for the omega = 0.3 models; these models exhibit Maxwellian velocity distributions for clusters, for groups, and for matter that are all similar to one another. The mass distribution in the omega = 1 models, however, exhibits a longer tail of high velocities than do the clusters. This high-velocity tail originates mostly from the high velocities that exist within rich clusters. The model velocity distributions of groups and clusters of galaxies are compared with observations. The data exhibit a larger high-velocity tail, to v(sub r) greater than or = 2000 km/s, than is seen in the model simulations (except for HDM). Because of the large observational uncertainties, however, the data are consistent at a approximately 1-3 sigma level with the model predictions and with a Gaussian initial density field. Accurate observations of cluster peculiar velocities, especially at the high-velocity tail, shoud provide powerful constraints on the cosmological models.

Bahcall, Neta A.↗
NASA NTRS2005

The electron excited ultraviolet spectrum of HD : cross sections and transition probabilities

We have analyzed the high-resolution ultraviolet (UV) emission spectrum of molecular deuterium hydride (HD) excited by electron impact at 100 eV under optically thin, single-scattering experimental conditions. The high-resolution spectrum (FWHM=160 mA) spans the wavelength range from 900 to 1650 A and contains the two Rydberg series of HD: (sup 1)Sigma(sub u)(sup +)1s(sigma), np(si n=2, 3, 4) --> X(sup 1)Sigma(sub g)(sup +) and (sup 1)Pi(sub u)(sup +)1s(sigma), np(pi)(C,D,D',D'', n=2, 3, 4, 5) -->X(sup 1)Sigma(sub g)(sup +). A model spectrum of HD, based on newly calculated tra rovibrational coupling for the strongest band systems, B (sup 1)Sigma(sub u)(sup +)-X(sup 1)Sigma(sub g)(sup +),B'(sup 1)Sigma(sub g)(sup +)-X(sup 1)Sigma(sub g)(sup +),C(sup 1)Pi(sub u)-X(sup 1)Sigm sections for direct excitation at 100 eV of the B (sup 1)Sigma(sub u)(sup +), B' (sup 1)Sigma(sub u)(sup +), C(sup 1)Pi(sub u), and D(sup 1)Pi(sub u) states were derived from a model analysis of the state. The absolute cross section values for excitation to the B (sup 1)Sigma(sub u)(sup +), B' (sup 1)Sigma(sub u)(sup +), C(sup 1)Pi(sub u), and D(sup 1)Pi(sub u) states were found to be (2.57+/-0. and (0.17+/-0.04)x10(exp -17) sq cm, respectively. We have also determined the dissociative excitation cross sections at 100 eV for the emission of Ly(alpha) at 1216 A and Ly(Beta) at 1025 A lines, which are (7.98+/-1.12)x10(exp -18) and (0.40+/-0.10)x10(exp -18) sq cm, respectively. The summed excitation function of the closely spaced pair of lines, H Ly(alpha) and D Ly(Beta), resulting from excitation of HD, has been measured from the threshold to 800 eV and is analytically modeled with a semiempirical relation. The model cross sections are in good agreement with the corrected Ly(alpha) cross sections of Mohlmann et al. up to 2 keV. Based on measurements of H, D (2s) production cross section values by Mohlmann et al., the H, D (n=2) cross section is estimated to be 1.6 x 10(exp -17) sq cm at 100 eV.

molecular data↗
DOE OSTI2021

A multishock model for the density variance of anisotropic, highly magnetized, supersonic turbulence

ABSTRACT Shocks form the basis of our understanding for the density and velocity statistics of supersonic turbulent flows, such as those found in the cool interstellar medium (ISM). The variance of the density field, $\sigma ^2_{\rho /\rho _0}$, is of particular interest for molecular clouds (MCs), the birthplaces of stars in the Universe. The density variance may be used to infer underlying physical processes in an MC, and parametrizes the star formation (SF) rate of a cloud. However, models for $\sigma ^2_{\rho /\rho _0}$ all share a common feature – the variance is assumed to be isotropic. This assumption does not hold when a trans-/sub-Alfvénic mean magnetic field, ${B}_0$, is present in the cloud, which observations suggest is relevant for some MCs. We develop an anisotropic model for $\sigma _{\rho /\rho _0}^2$, using contributions from hydrodynamical and fast magnetosonic shocks that propagate orthogonal to each other. Our model predicts an upper bound for $\sigma _{\rho /\rho _0}^2$ in the high Mach number $(\mathcal {M})$ limit as small-scale density fluctuations become suppressed by the strong ${B}_0$. The model reduces to the isotropic $\sigma _{\rho /\rho _0}^2\!-\!\mathcal {M}$ relation in the hydrodynamical limit. To validate our model, we calculate $\sigma _{\rho /\rho _0}^2$ from 12 high-resolution, three-dimensional, supersonic, sub-Alfvénic magnetohydrodynamical (MHD) turbulence simulations and find good agreement with our theory. We discuss how the two MHD shocks may be the bimodally oriented overdensities observed in some MCs and the implications for SF theory in the presence of a sub-Alfvénic ${B}_0$. By creating an anisotropic, supersonic density fluctuation model, this study paves the way for SF theory in the highly anisotropic regime of interstellar turbulence.

Beattie, James R.↗
NASA NTRS1994

Amplitude of primeval fluctuations from cosmological mass density reconstructions

We use the POTENT reconstruction of the mass density field in the nearby universe to estimate the amplitude of the density fluctuation power spectrum for various cosmological models. We find that sigma(sub 8) Omega(sub m sup 0.6) = 1.3(sub -0.3 sup +0.4), almost independently of the power spectrum. This value agrees well with the Cosmic Background Explorer (COBE) normalization for the standard cold dark matter model, while alternative models predict an excessive amplitude compared with COBE. Flat, low Omega(sub m) models and tilted models with spectral index n less than 0.8 are particularly discordant.

Seljak, Uros↗
NASA NTRS1995

Use of an adjoint model for finding triggers for Alpine lee cyclogenesis

The authors propose a new procedure, designated the adjoint-based genesis diagnostic (AGD) procedure, for studying triggering mechanisms and the subsequent genesis of the synoptic phenomena of interest. This procedure makes use of a numerical model sensitivity to initial conditions and the nonlinear evolution of the initial perturbations that are designed using this sensitivity. The model sensitivity is evaluated using the associated adjoint model. This study uses the dry version of the National Center for Atmospheric Research Mesoscale Adjoint Modeling System (MAMS) for the numerical experiments. The authors apply the AGD procedure to two cases of Alpine lee cyclogenesis that were observed during the Alpine Experiment special observations period. The results show that the sensitivity fields that are produced by the adjoint model and the associated initial perturbations are readily related to the probable triggering mechanisms for these cyclones. Additionally, the nonlinear evolution of these initial perturbations points toward the physical processes involved in the lee cyclone formation. The AGD experiments for a weak cyclone case indicate that the MAMS forecast model has an underrepresented topographic forcing due to the sigma vertical coordinate and that this model error can be compensated by adjustments in the initial conditions that are related to the triggering mechanisms, which is not associated with the topographic blocking mechanism.

Vukicevic, Tomislava↗
NASA NTRS1981

A new search for insterstellar H2O absorption in the spectrum of Zeta Ophiuchi

Intensive Copernicus scans of the 1(11)-0(00) line of the F-X band of H2O were carried out, resulting in a 2 sigma upper limit of 1.04 mA for this line at 1114.225 A. The recently published f-value implies a column density limit of 5.3 x 10 to the 12th/sq cm, marginally above the level expected from recent chemical model calculations. A 1.8 sigma feature appears at the expected position of the H2O line, implying a column density for the ground rotational state of 2.8 x 10 to the 12th/sq cm.

Snow, T. P., Jr.↗
DOE OSTI2020

Measurement of the Photon Beam Asymmetry in yP -> K+S0 at Ey=8.5 GeV with GlueX

In this work the photon beam asymmetry Sigma for the reaction yp ? K+S0 (1193) is measured using the GlueX experiment in Hall D at Jefferson Lab. The analysis used data that were collected using a linearly polarized photon beam in the energy range (8.2 - 8.8) GeV incident on a liquid hydrogen target. The beam asymmetries are measured as a function of the Mandelstam variable t and as a single value for the low u region. These are the first exclusive measurements of the photon beam asymmetry Sigma in this reaction at energies above the baryon resonance region. For the t-channel, results are close to unity and show that the reaction is dominated by the natural-parity exchange of the K? (892) Regge trajectory as predicted by theoretical models. A value of Sigma = 0.410±0.090 is obtained for the u-channel and suggests the exchange of both Sigma(J = 1/2) and Y*(J = 3/2) trajectory contributions to the production of K+S0.

Wickramaarachchi, Nilanga↗
DOE OSTI2022

Measurement of the anomalous precession frequency in the Muon $g-2$ experiment at Fermilab

The muon anomalous magnetic moment, $a_\mu=\frac{g-2}{2}$, is a low energy observable that can be both measured and computed with high precision, therefore it can provide an important test of the Standard Model and it is a sensitive probe for new physics. In particular, any discrepancy between the experimental value and the theoretical prediction can be due to Beyond Standard Model physics contribution. The E989 Muon $g-2$ Experiment at Fermilab aims to measure $a_\mu$ with a precision of 140 parts per billion, four time more precisely than the previous experiment at Brookhaven National Laboratory (BNL). E989 seeks to either resolve or confirm the observed discrepancy of $3.7\sigma$ between the Standard Model value and the experimental one. The $a_\mu$ measurement requires a precise determination of both the muon spin anomalous precession frequency and the average magnetic field seen by the muons as they circulate in a storage ring. The anomalous precession frequency measurement is b ased on the time distribution of high-energy decay positrons observed by 24 electromagnetic calorimeters placed around the inner circumference of the ring, while the magnetic field is constantly monitored by NMR probes. Last spring, E989 has published a new measurement of $a_\mu$ from the 2018 data-taking campaign, confirming the previous BNL value with comparable precision, bringing the discrepancy with the theoretical value to $4.2\sigma$. The candidate worked on the measurement of the anomalous precession frequency and on the estimate of the related systematic uncertainties of the 2018 data, giving an important contribution to the data analysis. The result presented in chapters 4 and 5 of this dissertation was included, among three other analyses, to determine the estimate of the anomalous magnetic moment by the Muon g-2 experiment. This work will present the anomalous precession frequency analysis technique and its result, the related systematics and finally the latest $a_\mu$ result.

Sorbara, Matteo↗
NASA NTRS1993

Studies of Greenland using the SEASAT scatterometer

We have used Ku-band (14.6 GHz) Seasat scatterometer (SASS) measurements to study changes in the backscatter of the Greenland ice sheet in response to seasonal changes in the surface and subsurface characteristics of the snow and glacial ice. To overcome the limitations imposed by the 50 km resolution of the SASS measurement, a resolution enhancement algorithm was used to generate 10 km resolution images of the backscatter response. To generate the images, the observed sigma(sup 0) (in dB) was modeled as a linear function of incidence angle theta, i.e., sigma(sup 0)(theta) = A + B(theta - 40 degrees) where theta epsilon (23 degrees, 55 degrees). The variable A can be thought of as the 'incidence angle-normalized sigma(sup 0)', while the gradient B describes the incidence angle dependence of sigma(sup 0). A time-series (one image every two weeks) of vertical and horizontally-polarized A and B was generated spanning a 100 day period between Julian Dates 180 and 280 (i.e. 29 June and 7 October) in 1978. This time series dramatically shows the effects of the summer melt and a clear transition with fall cooling and snowfall towards a typical winter scattering situation. The images also clearly delineate a number of principal snow and ice 'facies' which are identifiable on the surface.

Long, David G.↗
DOE OSTI2024

On the statistical theory of self-gravitating collisionless dark matter flow: Scale and redshift variation of velocity and density distributions

The statistics of velocity and density fields are crucial for cosmic structure formation and evolution. Here, this paper extends our previous work on the two-point second-order statistics for the velocity field [Phys. Fluids 35, 077105 (2023)] to one-point probability distributions for both density and velocity fields. The scale and redshift variation of density and velocity distributions are studied by a halo-based non-projection approach. First, all particles are divided into halo and out-of-halo particles so that the redshift variation can be studied via generalized kurtosis of distributions for halo and out-of-halo particles, respectively. Second, without projecting particle fields onto a structured grid, the scale variation is analyzed by identifying all particle pairs on different scales $r$. We demonstrate that: (i) Delaunay tessellation can be used to reconstruct the density field. The density correlation, spectrum, and dispersion functions were obtained, modeled, and compared with the N-body simulation; (ii) the velocity distributions are symmetric on both small and large scales and are non-symmetric with a negative skewness on intermediate scales due to the inverse energy cascade on small scales with a constant rate $\varepsilon_u$; (iii) On small scales, the even order moments of pairwise velocity $\Delta u_L$ follow a two-thirds law $\propto{(-\varepsilon_ur)}^{2/3}$, while the odd order moments follow a linear scaling $\langle(\Delta u_L)^{2n+1}\rangle=(2n+1)\langle(\Delta u_L)^{2n}\rangle\langle\Delta u_L\rangle\propto{r}$; (iv) The scale variation of the velocity distributions was studied for longitudinal velocities $u_L$ or $u_L^{'}$, pairwise velocity (velocity difference) $\Delta u_L$=$u_L^{'}$-$u_L$ and velocity sum $\Sigma u_L$=$u^{'}_L$+$u_L$. Fully developed velocity fields are never Gaussian on any scale, despite that they can initially be Gaussian; (v) On small scales, $u_L$ and $\Sigma u_L$ can be modeled by a $X$ distribution to maximize the entropy of the system. The distribution of $\Delta u_L$ can be different; (vi) On large scales, $\Delta u_L$ and $\Sigma u_L$ can be modeled by a logistic or a $X$ distribution, while $u_L$ has a different distribution; (vii) the redshift variation of the velocity distributions follows the evolution of the $X$ distribution involving a shape parameter $\alpha(z)$ decreasing with time.

79 ASTRONOMY AND ASTROPHYSICS↗