6D Phase Space Research Articles

Close Binary Fractions in accreted and in-situ Halo Stars

Abstract The study of binary stars in the Galactic halo provides crucial insights into the dynamical history and formation processes of the Milky Way. In this work, we aim to investigate the binary fraction in a sample of accreted and in-situ halo stars, focusing on short-period binaries. Utilising data from Gaia DR3, we analysed the radial velocity (RV) uncertainty σRV distribution of a sample of main-sequence stars. We used a novel Bayesian framework to model the dependence in σRV of single and binary systems allowing us to estimate binary fractions F in a sample of bright (GRVS< 12) Gaia sources. We selected the samples of in-situ and accreted halo stars based on estimating the 6D phase space information and affiliating the stars to the different samples on an action-angle vs energy (Lz − E) diagram. Our results indicate a higher, though not significant, binary fraction in accreted stars compared to the in-situ halo sample. We further explore binary fractions using cuts in E and Lz, and find a higher binary fraction in both high-energy and prograde orbits that might be explained by differences in metallicity. By cross-matching our Gaia sample with APOGEE DR17 catalogue, we confirm the results of previous studies on higher binary fractions in metal-poor stars and find the fractions of accreted and in-situ halo stars consistent with this trend. Our finding provides new insights into binary stars’ formation processes and dynamical evolution in the primordial Milky Way Galaxy and its accreted dwarf Galaxies.

Scaling properties of particle density fluctuations at LHC energies

The availability of high-multiplicity events at LHC energies provides a unique opportunity to examine the nature of phase-transition (PT) in heavy-ion collisions. Within the framework of HYDJET++ model, a systematic study of particle density fluctuations in narrow phase-space bins in Pb-Pb collisions at and 5.02 TeV energies is performed using the method of scaled factorial moments (SFM). The findings reveal the presence of intermittency in the 2-dimensional (2D) phase space distribution of charged particles at the LHC energies. The anomalous fractal dimensions, d q , are observed to increase with the order of the moments q, suggesting the multifractal nature of the charged particle production. The parameter λ q is found to exhibit a monotonically decreasing trend with the order of the moments but with no clear minima in λ q values. Furthermore, the value of the critical exponent is estimated and compared with those predicted by other models reported at RHIC and LHC energies. The value of the critical exponent, ν, obtained in the present study, shows significant departure from the value ∼1.304, as expected for the Ginzberg-Landau (GL) type second-order PT but exhibits a good level of agreement with the results obtained by other models at LHC energies.

Considerations on time resolution of neutron irradited single pixel 3D structures at fuences up to1017 neq/cm2 using 120 GeV SPS pion beams

The proven radiation hardness of silicon 3D devices up to fluences of 1 × 1017 neq/cm2 makes them an excellent choice for next generation trackers, providing < 10 μm position resolution at a high multiplicity environment. The anticipated pile-up increase at HL-LHC conditions and beyond, requires the addition of < 50 ps per hit timing information to successfully resolve displaced and primary vertices. In this study, the timing performance, uniformity and efficiency of neutron and proton irradiated single pixel 3D devices is discussed. Fluences up to 1 × 1017 neq/cm2 in three different geometrical implementations are evaluated using 120 GeV SPS pion beams. A MIMOSA-26 type telescope is used to provide detailed tracking information with a ∼ 5 μm position resolution. Productions with single- and double-sided processes, yielding active thicknesses of 130 and 230 μm respectively, are examined with varied pixel sizes from 55 × 55 μm2 to 25 × 100 μm2 and a comparative study of field uniformity is presented with respect to electrode geometry. The question of electronics bandwidth is extensively addressed with respect to achievable time resolution, efficiency and collected charge, forming a 3D phase space to which an appropriate operating point can be selected depending on the application requirements.

Анализ регулярности/хаотичности динамики шаровых скоплений в центральной области Млечного Пути

We analyzed the regularity/chaoticity of the orbits of 45 globular clusters in the central region of the Galaxy with a radius of 3.5 kpc, which are subject to the greatest influence from an elongated rotating bar. Various analysis methods were used, namely, methods for calculating the maximum characteristic Lyapunov exponents, the method of Poincaré sections, the frequency method based on the calculation of fundamental frequencies, as well as the visual assessment method. Bimodality was discovered in the histogram of the distribution of positive Lyapunov exponents, calculated in the classical version, without renormalization of the shadow orbit, which makes it possible to implement a probabilistic method of GC classification. To construct the orbits of globular clusters, we used a gravitational potential model with a bar in the form of a triaxial ellipsoid, described in detail in the work of Bajkova et al., Izvestiya GAO in Pulkovo, 2023, 228, 1. The following bar parameters were adopted: mass 1010M⊙, semimajor axis length 5 kpc, bar viewing angle 25o, rotation speed 40 km s−1 kpc−1. To form the 6D-phase space required for orbit integration, the most accurate astrometric data to date from the Gaia satellite (EDR3) (Vasiliev, Baumgardt, 2021), as well as new refined average distances to globular clusters (Baumgardt, Vasiliev, 2021) were used. A classification is made of globular clusters with regular and chaotic dynamics. As the analysis showed, globular clusters with small pericentric distances and large eccentricities are most susceptible to the influence of the bar and demonstrate the greatest chaos. It is shown that the results of classifying globular clusters by the nature of their orbital dynamics, obtained using the various methods of analysis considered in the work, correlate well with each other.

A Walk on the Retrograde Side (WRS) project

Relics of ancient accretion events experienced by the Milky Way are predominantly located within the stellar halo of our Galaxy. However, debris from different objects display overlapping distributions in dynamical spaces, making it extremely challenging to properly disentangle their contribution to the build-up of the Galaxy. To shed light on this chaotic context, we initiated a program aimed at the homogeneous chemical tagging of the local halo of the Milky Way, focusing on the component in retrograde motion, since this is expected to host a large fraction of stars accreted from past mergers. The A Walk on the Retrograde Side (WRS) project targets retrograde halo stars in the solar neighborhood with accurate 6D phase space information available, measuring the precise chemical abundance of several chemical elements from high-resolution spectroscopy. In this first paper, we present the project and the analysis of high-resolution spectra obtained with UVES at VLT and at LBT for 186 stars. We obtained accurate radial velocity and chemical abundances for several elements for all the target stars. In particular, we focus on the chemical composition of a specific subset of substructures that have been dynamically identified in the literature. Our study reveals that two among the more recently discovered structures in the retrograde halo, namely, Antaeus/L-RL64 and ED-3, have identical chemical patterns and similar integrals of motion, suggesting a common origin. In turn, the abundance patterns of this unified system differ from that of Gaia-Enceladus, confirming that it is an independent structure. Finally, Sequoia exhibits a different chemistry with respect to that of Gaia-Enceladus at [Fe/H] &lt; −1.5 dex, showcasing an excess of stars with lower Mg and Ca in the common metallicity range.

Radial and azimuthal gradients of the moving groups in Gaia DR3: The slow and fast bar degeneracy problem

The structure and dynamics of the central bar of the Milky Way (MW) are still under debate whilst being fundamental ingredients for the evolution of our Galaxy. The recent Gaia DR3 offers an unprecedented detailed view of the 6D phase space of the MW, allowing for a better understanding of the complex imprints of the bar on the phase space. We aim to identify and characterise the dynamical moving groups across the MW disc, and use their large-scale distribution to help constrain the properties of the Galactic bar. We used 1D wavelet transforms of the azimuthal velocity ($V_ distribution in bins of radial velocity to robustly detect the kinematic substructure in the Gaia DR3 catalogue. We then connected these structures across the disc to measure the azimuthal (phi ) and radial ($R$) gradients of $V_ of the moving groups. We simulated thousands of perturbed distribution functions using backward integration, sweeping a large portion of parameter space of feasible Galaxy models that include a bar, in order to compare them with the data and to explore and quantify the degeneracies. The radial gradient of the Hercules moving group ($ V_ R$ = 28.1pm 2.8 $) cannot be reproduced by our simple models of the Galaxy that show much larger slopes both for a fast and a slow bar. This suggests the need for more complex dynamics (e.g. a different bar potential, spiral arms, a slowing bar, a complex circular velocity curve, external perturbations, etc.). We measured an azimuthal gradient for Hercules of $ V_ = -0.63pm $deg$^ $ and find that it is compatible with both the slow and fast bar models. Our analysis points out that in using this type of analysis, at least two moving groups are needed to start breaking the degeneracies. We conclude that it is not sufficient for a model to replicate the local velocity distribution; it must also capture its larger-scale variations. The accurate quantification of the gradients, especially in the azimuthal direction, will be key for the understanding of the dynamics governing the disc.

Analyzing the spatial motion of a rigid body subjected to constant body-fixed torques and gyrostatic moment

This paper aims to explore the rotatory spatial motion of an asymmetric rigid body (RB) under constant body-fixed torques and a nonzero first component gyrostatic moment vector (GM). Euler's equations of motion are used to derive a set of dimensionless equations of motion, which are then proposed for the stability analysis of equilibrium points. Specifically, this study develops 3D phase space trajectories for three distinct scenarios; two of them are applied constant torques that are directed on the minor and major axes, while the third one is the action of applied constant torque on the body’s middle axis. Novel analytical and simulation results for both scenarios of constant torque applied along the minor and middle axes are provided in the context of separatrix surfaces, equilibrium manifolds, periodic or non-periodic solutions, and periodic solutions’ extreme. Concerning the scenario of a directed torque on the major axis, a numerical solution for the problem is presented in addition to a simulation of the graphed results for the angular velocities' trajectories in various regions. Moreover, the influence of GM is examined for each case and a full modeling for the body's stability has been present. The exceptional impact of these results is evident in the development and assessment of systems involving asymmetric RBs, such as satellites and spacecraft. It may serve as a motivating factor to explore different angles within the GM in similar cases, thereby influencing various industries, including engineering and astrophysics applications.

A Theory for the Balance between Warm Rain and Ice Crystal Processes of Precipitation in Mixed-Phase Clouds

Abstract Mixed-phase clouds contain both supercooled cloud liquid and ice crystals. In principle, precipitation may be initiated either by the liquid phase or by the ice phase. Ice crystals may grow by vapor diffusion to become snow (“ice crystal process”), forming “cold” precipitation. Equally, cloud droplets, when large enough, coalesce to form “warm” precipitation by the “warm rain process.” Warm rain could be supercooled and freeze as “warm” graupel. In the present paper, a new simplified theoretical analysis is provided to examine the microphysical system consisting of three species of hydrometeor, namely, cloud liquid, “cold ice” (crystals, snow), and “warm rain” (frozen or supercooled). This is obtained by nondimensionalizing and simplifying the evolution equations for the mass of each species. Analytical formulas are given for equilibria. Feedback analysis shows that the sign of the feedback is linked to the abundance of precipitation, with a neutral surface in the 3D phase space. The system’s precipitation amount explodes while in the initial unstable regime, crossing the neutral surface and approaching the equilibrium point that is a stable attractor. Positive and negative feedbacks are elucidated. In a standard case, the cold ice mass is about 1000 times larger than the warm rain mass. To illustrate the physical behavior of the theory, sensitivity tests are performed with respect to environmental conditions (e.g., aerosol, updraft speed) and microphysical parameters (e.g., riming and sedimentation rates for cold ice). Cold ice prevails, especially in fast ascent, due to its low bulk density, favoring slow sedimentation and a wide cross-sectional area for riming. Significance Statement The theory elucidates how the ice phase can prevail in the precipitation from any mixed-phase clouds with supercooled cloud liquid and crystals. The ice phase radically suppresses cloud liquid by riming when active and “wins” the competition against coalescence. This prevalence of ice is shown to arise from the low bulk density of snow. The cloud is viewed as a system of negative and positive feedbacks that prevail in realms of stability and instability in a 3D phase space.

Dynamic aperture studies for vertical fixed field accelerators

Vertical orbit excursion Fixed Field Accelerators (vFFAs) feature highly nonlinear magnetic fields and strong transverse motion coupling. The detailed study of their Dynamic Aperture (DA) requires computation codes allowing long-term tracking and advanced analysis tools to take the transverse motion linear and nonlinear coupling into account. This coupling completely transforms the beam dynamics compared to a linear uncoupled motion, and an explicit definition of the DA is needed to characterize the performance and limitations of these lattices. A complete study of the DA in the 4D phase space in highly nonlinear and strongly coupled machines must give a measure of the stability domain but also means to assess the operating performance in the physical coupled space. This work presents a complete set of methods to perform such detailed analysis. These methods were explored and compared to compute and characterize the DA of an example vFFA lattice. The whole procedure can be further applied to evaluate DA using realistic models of the magnetic fields, including fringe fields and errors.

Bifurcation analysis, chaotic analysis and diverse optical soliton solutions of time-fractional (2+1) dimensional generalized Camassa-Holm Kadomtsev-Petviashvili equation arising in shallow water waves

This paper presents a study on (2+1) generalized Camassa-Holm Kadomtsev-Petviashvili equation, which is used to describe the behavior of shallow water waves in nonlinear media. The considered equation provides a more accurate description of wave behavior compared to linear wave equations and can account for wave breaking and other nonlinear effects. This model can be used to describe and study the behavior of nonlinear waves such as rogue waves in complex fluid dynamics scenarios. This includes the behavior of waves in stratified fluids, nonlinear dispersive media and wave interactions in fluid flows with varying velocities and densities. The bifurcation analysis of the governing equation has been performed using the planar dynamical system method. The chaotic behavior of the dynamical system has been examined by utilizing various techniques such as time series analysis and the construction of 2D and 3D phase space trajectories. Furthermore, the introduction of a perturbed term has resulted in the observation of chaotic and quasi-periodic behaviors across a range of parameter values. The considered equation has been reduced to ordinary differential equation by performing symmetry reduction. The Kudryashov method has been used to obtain the exact solution of reduced equation. The single soliton solution of governed equation has been obtained by using Hirota method and impact of fractional parameter on the obtained solution has been studied using graphical representation. The extended sinh-Gordon equation expansion method and modified generalized exponential rational function method have been exploited to obtain dark, bright and singular soliton solutions of considered equation. The motivation for this study arises from the need to understand and analyze the complex dynamics of shallow water waves in nonlinear media with a particular focus on the (2+1) generalized Camassa-Holm Kadomtsev-Petviashvili equation. By performing symmetry reduction and applying various analytical methods, we aim to unravel the intricate behavior and soliton solutions of considered equation, contributing to the broader understanding of nonlinear wave phenomena.

The Sound Field Intensity Distribution in the Deep Sea in the “Depth—Angle–Time” Phase Space

The transition from the traditional representation of the wave field in the vertical section of an underwater sound channel as a function of depth and time to the distribution of this field in the 3D phase space “depth–angle–time” is considered. For this purpose, the method of coherent states developed in quantum theory is used. The meaning of the proposed transition is that the field intensity distribution in the specified phase space is less sensitive to sound velocity fluctuations than in the original 2D depth–time space. This fact can be used in solving inverse problems. As an example, we consider the reconstruction of the coordinates of a source in a waveguide from measurements of the field intensity distribution of this source in phase space.

Шаровые скопления в центральной области Млечного Пути. II. Частотный анализ орбит, построенных по данным Gaia EDR3

This work represents the second stage of research aimed at studying the influence of the bar on the orbital dynamics of globular clusters (Bajkova, Smirnov, and Bobylev, 2023). The orbits of 45 globular clusters in the central galactic region with a radius of 3.5 kpc have been analyzed by spectral dynamics methods in order to identify objects captured by the bar. Since the parameters of the bar of the Milky Way are known with a very large uncertainty, orbits were constructed and their frequency analysis was carried out with varying the mass, length and angular velocity of rotation of the bar in a wide range of values with a fairly dense grid. The orbits were integrated 2.5 billion years backward. As a result, globular clusters supporting the bar were identified for each set of bar parameters. To form the 6D phase space required for orbit integration, we used the most accurate astrometric data from the Gaia satellite (Vasiliev and Baumgardt, 2021), as well as new refined average distances to globular clusters (Baumgardt and Vasiliev, 2021).

Ongoing hierarchical massive cluster assembly: The LISCA II structure in the Perseus complex

We report on the identification of a massive (∼105 M⊙) substructured stellar system in the Galactic Perseus complex likely undergoing hierarchical cluster assembly. This system comprises nine star clusters (including the well-known clusters NGC 654 and NGC 663) and an extended and low-density stellar halo. Gaia-DR3 and available spectroscopic data show that all its components are physically consistent in 6D phase-space (position, parallax, and 3D motion), and homogeneous in age (14–44 Myr) and chemical content (half-solar metallicity). In addition, the system’s global stellar density distribution is that of typical star clusters and shows clear evidence of mass segregation. We find that the hierarchical structure is mostly contracting toward the center with a speed of up to ≃4 − 5 km s−1, while the innermost regions expand at a lower rate (about ≃1 km s−1) and are dominated by random motions. Interestingly, this pattern is dominated by the kinematics of massive stars, while low-mass stars (M &lt; 2 M⊙) are characterized by contraction across the entire cluster. Finally, the nine star clusters in the system are all characterized by a relatively flat velocity dispersion profile possibly resulting from ongoing interactions and tidal heating. We show that the observational results are generally consistent with those found in N-body simulations following the cluster violent relaxation phase, strongly suggesting that the system is a massive cluster in the early assembly stages. This is the second structure with these properties identified in our Galaxy and, following the nomenclature of our previous work, we named it LISCA II.

Gaia Data Release 3

Context. With the most recent Gaia data release, the number of sources with complete 6D phase space information (position and velocity) has increased to well over 33 million stars, while stellar astrophysical parameters are provided for more than 470 million sources, and more than 11 million variable stars are identified. Aims. Using the astrophysical parameters and variability classifications provided in Gaia DR3, we selected various stellar populations to explore and identify non-axisymmetric features in the disc of the Milky Way in configuration and velocity space. Methods. Using more about 580 000 sources identified as hot OB stars, together with 988 known open clusters younger than 100 Myr, we mapped the spiral structure associated with star formation 4−5 kpc from the Sun. We selected over 2800 Classical Cepheids younger than 200 Myr that show spiral features extending as far as 10 kpc from the Sun in the outer disc. We also identified more than 8.7 million sources on the red giant branch (RGB), of which 5.7 million have line-of-sight velocities. This later sample allows the velocity field of the Milky Way to be mapped as far as 8 kpc from the Sun, including the inner disc. Results. The spiral structure revealed by the young populations is consistent with recent results using Gaia EDR3 astrometry and source lists based on near-infrared photometry, showing the Local (Orion) Arm to be at least 8 kpc long, and an outer arm consistent with what is seen in HI surveys, which seems to be a continuation of the Perseus arm into the third quadrant. The subset of RGB stars with velocities clearly reveals the large-scale kinematic signature of the bar in the inner disc, as well as evidence of streaming motions in the outer disc that might be associated with spiral arms or bar resonances. A local comparison of the velocity field of the OB stars reveals similarities and differences with the RGB sample. Conclusions. This cursory study of Gaia DR3 data shows there is a rich bounty of kinematic information to be explored more deeply, which will undoubtedly lead us to a clearer understanding of the dynamical nature of the non-axisymmetric structures of the Milky Way.

Шаровые скопления в центральной области Млечного Пути I. Влияние бара на параметры орбит по данным Gaia EDR3

The work is devoted to the analysis of the influence of the galactic bar on the orbital motion of globular clusters in the central region of the Galaxy. For this task, 45 globular clusters were selected, 34 of which belong to the bulge and 11 to the disk. The most accurate astrometric data from the Gaia satellite (Vasiliev and Baumgardt, 2021), as well as new refined average distances (Baumgardt and Vasiliev, 2021), were used to form the 6D phase space required for orbit integration. The orbits of globular clusters are obtained both in an axisymmetric potential and in a potential including a bar. In this case, the mass, rotation speed, shape and scale length of the bar were varied. A comparison is made of such orbital parameters as apocentric and pericentric distances, eccentricity and maximum distance from the galactic plane. It is shown that the mass of the bar exerts the greatest influence on the orbital motion, which is expressed mainly in an increase in both the apocentric and pericentric distances in the vast majority of globular clusters. The eccentricities of the orbits in the overwhelming majority also change significantly, and there is a change both upward and downward, especially in the range of values from 0.2 to 0.8. The greatest changes in parameters are observed in globular clusters with high radial velocities and small pericentric distances. The change in orbital parameters depending on the bar rotation speed is less pronounced. The influence of the geometric parameters of the bar is insignificant in the accepted range of their changes. Several examples show that globular clusters in the bulge are more affected by the bar than those belonging to the disk.

Adaptive autoencoder latent space tuning for more robust machine learning beyond the training set for six-dimensional phase space diagnostics of a time-varying ultrafast electron-diffraction compact accelerator.

We present a general adaptive latent space tuning approach for improving the robustness of machine learning tools with respect to time variation and distribution shift. We demonstrate our approach by developing an encoder-decoder convolutional neural network-based virtual 6D phase space diagnostic of charged particle beams in the HiRES ultrafast electron diffraction (UED) compact particle accelerator with uncertainty quantification. Our method utilizes model-independent adaptive feedback to tune a low-dimensional 2D latent space representation of ∼1 million dimensional objects which are the 15 unique 2D projections (x,y),...,(z,p_{z}) of the 6D phase space (x,y,z,p_{x},p_{y},p_{z}) of the charged particle beams. We demonstrate our method with numerical studies of short electron bunches utilizing experimentally measured UED input beam distributions.

Phase Space Reconstruction from Accelerator Beam Measurements Using Neural Networks and Differentiable Simulations.

Characterizing the phase space distribution of particle beams in accelerators is a central part of understanding beam dynamics and improving accelerator performance. However, conventional analysis methods either use simplifying assumptions or require specialized diagnostics to infer high-dimensional (>2D) beam properties. In this Letter, we introduce a general-purpose algorithm that combines neural networks with differentiable particle tracking to efficiently reconstruct high-dimensional phase space distributions without using specialized beam diagnostics or beam manipulations. We demonstrate that our algorithm accurately reconstructs detailed 4D phase space distributions with corresponding confidence intervals in both simulation and experiment using a limited number of measurements from a single focusing quadrupole and diagnostic screen. This technique allows for the measurement of multiple correlated phase spaces simultaneously, which will enable simplified 6D phase space distribution reconstructions in the future.

Clouds and Turbulence Theory: Peculiar Self-Similarity, 4/3 Fractal Exponent and Invariants

In 1982 Lovejoy has published an illustration to Mandelbrot proposal how to characterize the area-perimeter ratio of complicated planar forms and it was found that exponent \(\beta \) for the satellite- and radar-determined cloud and rain areas of such a fractal is 1.35 close to 4/3. Later on it was notified that the same exponent was found also for noctilucent clouds. Such a value might be related to classic turbulence theory of 1941. This text demonstrates this relation using two basic papers by Kolmogorov and Obukhov. The role of prefractal multipliers is revealed, they form a couple of the peculiar invariants for cloud fields and a non-dimensional self-similarity numbers for these fields of sizes \(1 - {{10}^{6}}\,\,{\text{k}}{{{\text{m}}}^{2}}.\) The peculiarity is in their dimensional dependence and in the presence of few invariants, not usual invariants in cloud forms. Further research on random walk of a fluid particle in the 6D phase-space may lead to new discoveries.

6D Phase space diagnostics based on adaptively tuned physics-informed generative convolutional neural networks

A physics-informed generative convolutional neural network (CNN)-based 6D phase space diagnostic is presented which generates all 15 unique 2D projections (x, y), (x, y′),...,(z, E) of a charged particle beam’s 6D phase space (x, y, z, x′, y′, E). The CNN is trained by supervised learning over a wide range of input beam distributions, accelerator parameters, and the associated 6D beam phase spaces at multiple accelerator locations. The CNN is applied in an un-supervised adaptive manner without knowledge of the input beam distribution or accelerator parameters and is robust to their unknown time variation. Adaptive feedback automatically tunes the low-dimensional latent space of the encoder-decoder CNN to predict the 6D phase space based only on 2D (z, E) longitudinal phase space measurements from a device such as a transverse deflecting RF cavity (TCAV). This method has the potential to provide diagnostics beyond the existing state of the art at many accelerator facilities. Studies are presented for two very different accelerators: the 5-meter-long ultra-fast electron diffraction (UED) HiRES compact accelerator at LBNL and the kilometer long plasma wakefield accelerator FACET-II at SLAC.

Transverse phase space tomography in an accelerator test facility using image compression and machine learning

We describe a novel technique, based on image compression and machine learning, for transverse phase space tomography in two degrees of freedom in an accelerator beamline. The technique has been used in the CLARA accelerator test facility at Daresbury Laboratory: results from the machine learning method are compared with those from a conventional tomography algorithm (algebraic reconstruction) and applied to the same data. The use of machine learning allows reconstruction of the 4D phase space distribution of the beam to be carried out much more rapidly than using conventional tomography algorithms and also enables the use of image compression to reduce significantly the size of the data sets involved in the analysis. Results from the machine learning technique are at least as good as those from the algebraic reconstruction tomography in characterizing the beam behavior, in terms of the variation of the beam size in response to variation of the quadrupole strengths.