Radial Frequency Research Articles

Force Fields for Deep Eutectic Mixtures: Application to Structure, Thermodynamics and 2D-Infrared Spectroscopy.

Parametrizing energy functions for ionic systems can be challenging. Here, the total energy function for an eutectic system consisting of water, SCN-, K+ and acetamide is improved vis-a-vis experimentally measured properties. Given the importance of electrostatic interactions, two different types of models are considered: the first (model M0) uses atom-centered multipole whereas the other two (models M1 and M2) are based on fluctuating minimal distributed charges (fMDCM) that respond to geometrical changes of SCN-. The Lennard-Jones parameters of the anion are adjusted to best reproduce experimentally known hydration free energies and densities, which are matched to within a few percent for the final models irrespective of the electrostatic model. Molecular dynamics simulations of the eutectic mixtures with varying water content (between 0 and 100%) yield radial distribution functions and frequency correlation functions for the CN-stretch vibration. Comparison with experiments indicates that models based on fMDCM are considerably more consistent than those using multipoles. Computed viscosities from models M1 and M2 are within 30% of measured values and their change with increasing water content is consistent with experiments. This is not the case for model M0.

Mass and spin measurements of black hole and neutron stars X-ray binaries through axisymmetric oscillation modes of thick torus

Abstract This paper examines the oscillatory behaviour of relativistic, non-self-gravitating, charged-fluid toroidal structures within the context of the Kerr metric. The primary objective is to explore how thick accretion discs influence the mass and spin measurements of black holes and neutron stars in Low-Mass X-ray Binaries (LMXBs) through Quasi-Periodic Oscillations (QPOs) models. To achieve this, we conduct a local analysis within a general relativistic framework, determining the radial epicyclic and orbital frequencies in a perfect fluid disk. The tori are modelled using a non-Keplerian distribution of specific angular momentum, and we analyse how the oscillation properties depend on the model’s angular momentum distribution parameters. Subsequently, we connect these oscillatory frequencies to high-frequency QPOs observed in low-mass X-ray binaries, enabling us to calculate the optimal mass and spin values for each studied source.

Epicyclic oscillations and particle collision with trajectories around quantum corrected black holes

The present study deals with the orbital, oscillatory motion, and trajectories of test particles around quantum-corrected black holes. The angular momentum within specific energy is calculated to discuss stable circular orbits. The effective potential through Hamiltonian is calculated around quantum-corrected black holes. The trajectories scenario around particles of considered black hole solutions are discussed. Further, we provide the mathematical formulas for both latitudinal and radial frequencies. The primary properties of test particle quasi-periodic oscillations in stable circular orbits in the black hole’s equatorial plane are investigated. We provide a Markov Chain Monte Carlo analysis of the restrictions on quantum-corrected black hole attributes. We analyze QPOs detected in X-ray binaries, namely the cores of the galaxies Sgr A∗, M82 X-1, microquasars XTE J1550-564, GRS 1915+105 and GRO J1655-40. Additionally, the process of Periastron precession is examined. We show that the velocity of particles around black holes is strongly dependent on the parameters of the model. Notably, the pictorial behavior that describes our results feasibility.

Dual effect of string cloud and dark matter halos on particle motions, shadows and epicyclic oscillations around Schwarzschild black holes

In this work, we study the particle dynamics, shadows and quasi-periodic oscillations around the Schwarzschild black hole with three different dark matter halos and string cloud parameters. The motion of particles is considered with the string cloud parameter α and investigated for massive and massless particles. The photon orbits and inner stable circular orbits of these black holes are obtained from the effective potential. The black hole shadows are calculated using three different dark matter halos. The shadow radius of these black holes increases compared to a pure Schwarzschild black hole for different values of the string cloud parameter. Further, quasi-periodic oscillations are also discussed in the current analysis. Our study examines how the radial profiles of orbital and radial harmonic oscillation frequencies change with the string cloud parameter. This study examines the major properties of test particle's quasi-periodic oscillations near stable circular orbits in the black hole equatorial plane. Our findings indicate that observational tests for black holes influenced by dark matter halos and cloud strings are also feasible and viable.

Compact dwarfs made of light-quark nuggets

Utilizing an equiparticle model with both linear confinement and leading-order perturbative interactions, we obtain systematically the properties of strangelets and nonstrange quark matter (udQM) nuggets at various baryon (A) and charge (Z) numbers, where the detailed single-quark-energy levels are fixed by solving Dirac equations in mean-field approximation. We then examine the structures of compact dwarfs made of light strangelets or udQM nuggets forming body-centered cubic lattices in a uniform electron background. Despite that the strangelets and udQM nuggets generally become more stable at larger A, the compact dwarfs are still stable since the fusion reactions between those objects do not take place in the presence of a Coulomb barrier, which is similar to the cases of light nuclei in normal white dwarfs. If udQM dwarfs or strangelet dwarfs are covered with normal matter, their masses and radii become larger but do not exceed those of ordinary white dwarfs. Finally, we investigate the radial oscillation frequencies of udQM dwarfs and strangelet dwarfs and find that their frequencies are typically higher than traditional white dwarfs. The stability of compact dwarfs are then analyzed by examining radial oscillation frequencies of the fundamental mode, where compact dwarfs covered by normal matter are still stable. Published by the American Physical Society 2024

Fast inspirals and the treatment of orbital resonances

Abstract Extreme mass ratio inspirals (EMRIs), where a compact object orbits a massive black hole, are a key source of gravitational waves for the future Laser Interferometer Space Antenna (LISA). Due to their small mass ratio, ε ~ 10-4 -- 10-7 , the binary evolves slowly and EMRI signals will be in-band for years. Additionally, astrophysical EMRIs are expected to have complex dynamics featuring both spin-precession and eccentricity. A standard approach to modelling these inspirals is via the method of osculating geodesics (OG) which we employ along with a toy model for the gravitational self-force. Using this method requires resolving tens of thousands radial and polar orbital librations over the long duration of the signal which makes the inspiral trajectory expensive to compute.In this work we accelerate these calculations by employing Near-Identity (averaging) Transformations. 
However, this averaging technique breaks down at orbital resonances where the radial and polar frequencies are an integer ratio of each other. Thus, we switch to a partial averaging transformation in the vicinity of the resonance where the dynamics are characterised by the slow evolution of the so-called ``resonant phase". Additionally, we develop an optimal switching criterion to minimise the computation time while maximising accuracy. We find the error in the waveform phase is improved from O(ε-1/2) in the fully averaged scheme to O(ε4/7) in the switching scheme. At the same time, this scheme improves the scaling of the computation time from being inversely proportional to ε using OG, to a very weak scaling with ε. This results in a speed-up of at least two orders of magnitude for LISA EMRIs with room for further optimisation.

Which shape features determine detectability of camouflaged radial frequency patterns?

Which shape features determine detectability of camouflaged radial frequency patterns?

Stability Analysis of Stable Circular Orbit in Multi-Static Black Hole Spacetime

We herein study the circular orbit stability of a static black hole system composed of multiple Reissner–Nordstrom (RN) black holes. By comparing the circular orbits of two static black holes, three static black holes (TBHs), four static black holes and five static black holes at different spacetime, we find that the continuity of their stable circular orbits changes, i.e., the peaks of the effective potentials are transformed from single-peaked to bi-peaked, and that the distance a between the black holes is the main reason for this change. This characteristic is completely different from the continuity of the stable circular orbit interval of any kind of single black hole in the past. After calculation, we obtain several critical values that lead to the change in circular orbit stability. The three fundamental frequencies (orbital frequency, radial local frequency, and vertical local frequency) are derived and compared for two different spacetimes of double and three black holes. We also analyse the effect of the black hole distance a on the three fundamental frequencies of circular orbits.

Human Visual System applied to Facial Recognition

Research in the field of biometrics is constantly expanding. Numerous studies have been carried out to develop different techniques with the aim of ensuring reliable and efficient recognition systems. Of our five senses, vision takes up the most of the neurons in our brain. This makes the visual approach challenging. This paper proposes to develop a new facial recognition technique based on the concept of Human Visual System in order to simulate or imitate human perception. Our approach exploits the behavior of the Human Visual System in biometric systems to improve individual recognition. It focuses on Perceptual Channel Decomposition in order to generate images, confined around a certain frequency range, perfectly uncorrelated. For the extraction of characteristic vectors, our technique uses the Principal Component Analysis. The principle of the visual approach consists of exploiting one or more characteristics of the peripheral parts of the Human Visual System. These characteristics can integrate the sensitivity of the Human Visual System to spatial frequencies, its sensitivity to local contrast. The Principal Component Analysis uses a set of seventeen channels, each adjusted to a band of given radial frequencies and orientation. The seventeen output images contain the same spatial information but are perfectly uncorrelated from a spectral point of view. In the implementation phase, we limited ourselves to explore this technique by using only four frequency rings. The results obtained are conclusive and satisfactory.

Development of radial frequency pattern perception in macaque monkeys

Development of radial frequency pattern perception in macaque monkeys

Corrections to: Development of radial frequency pattern perception in macaque monkeys.

Corrections to: Development of radial frequency pattern perception in macaque monkeys.

Diagnostic efficacy of complexity metrics from cardiac MRI myocardial segmental motion curves in detecting late gadolinium enhancement in myocardial infarction patients

BackgroundMyocardial segmental motion is associated with cardiovascular pathology, often assessed through myocardial strain features. The stability of the motion can be influenced by myocardial fibrosis. This research aimed to explore the complexity metrics (CM) of myocardial segmental motion curves, observe their correlation with late gadolinium enhancement (LGE) transmural extension (TE), and assess diagnostic efficacy combined with segmental strains in different TE segments. MethodsWe included 42 myocardial infarction patients, dividing images into 672 myocardial segments (208 remote, 384 viable, and 80 unviable segments based on TE). Radial and circumferential segmental strain, along with CM for motion curves, were extracted. Correlation between CM and LGE, as well as the potential distinguishing role of CM, was evaluated using Pearson correlation, univariate linear regression (F-test), multivariate regression analysis (T-test), area under curve (AUC), machine learning models, and DeLong test. ResultsAll CMs showed significant linear correlation with TE (P < 0.001). Six CMs were correlated with TE (r > 0.3), with radial frequency drift (FD) displayed the strongest correlation (r = 0.496, P < 0.001). Radial and circumferential FD significantly differed in higher TE myocardium than in remote segments (P < 0.05). Radial FD had practical diagnostic efficacy (remote vs. unviable AUC = 0.89, viable vs. unviable AUC = 0.77, remote vs. viable AUC = 0.65). Combining CM with segmental strain features boosted diagnostic efficacy than models using only segmental strain features (DeLong test, P < 0.05). ConclusionsThe CM of myocardial motion curves has been associated with LGE infarction, and combining CM with strain features improves the diagnosis of different myocardial LGE infarction degrees.

Analyzing the Active Site and Predicting the Overall Activity of Alloy Catalysts.

As one of the potential catalysts, disordered solid solution alloys can offer a wealth of catalytic sites. However, accurately evaluating their activity localization structure and overall activity from each individual site remains a formidable challenge. Herein, an approach based on density functional theory and machine learning was used to obtain a large number of sites of the Pt-Ru alloy as the model multisite catalyst for the hydrogen evolution reaction. Subsequently, a series of statistical approaches were employed to unveil the relationship between the geometric structure and overall activity. Based on the radial frequency distribution of metal elements and the distribution of ΔGH, we have identified the surface and subsurface sites occupied by Pt and Ru, respectively, as the most active sites. Particularly, the concept of equivalent site ratio predicts that the overall activity is highest when the Ru content is 20-30%. Furthermore, a series of Pt-Ru alloys were synthesized to validate the proposed theory. This provides crucial insights into understanding the origin of catalytic activity in alloys and thus will better guide the rational development of targeted multisite catalysts.

Experimental study on flow-induced vibration under laminar flow and turbulent flow transition state in helical coiled tube

This study investigates the vibration characteristics induced by single-phase flow at low velocities within helical coiled tube, encompassing laminar and turbulent flow transition regimes. Modal experiments were conducted to ascertain the natural frequencies in the radial and vertical directions within the helical coiled tube. The influence of fluid mass in tube on vibration was considered, revealing the fluid mass has a greater influence on the vibration frequency in the vertical direction. When the coil was filled with water, the radial natural frequency decreased by 0.1 Hz, while the vertical natural frequency decreased by 0.8 Hz.By simplifying the spiral coil into a spring system, the intrinsic causes of the influence of mass on the natural frequency are studied and verified with the experimental data. The results show that the radial natural frequency is only related to the local mass change of the pipe, while the vertical natural frequency is related to the static deformation of the system under the influence of gravity.By conducting single-phase flow induced vibration experiments inside the pipeline, the radial and vertical vibration response characteristics were obtained, and the modal experimental results were analyzed. Findings indicated larger vibration displacements when laminar flow prevailed within the tube, with radial RMS (Root Mean Square) at 10.1 μm and vertical RMS at 14.5 μm. After the transition to turbulent flow, significant reductions in vibration displacements were observed, reducing the radial RMS to 2.3 μm and vertical RMS to 5.5 μm.Based on statistical and frequency-domain analysis methods, the influence mechanism of secondary flow on the vibration of helical coils at low flow rates was revealed, and the importance of friction coupling was clarified. Moreover, this study considered the impact of fluid viscosity on vibration response and experimentally validated the critical role of friction coupling in exciting secondary flows.

Strong gravitational lensing, quasi-periodic oscillations and constraints from EHT observations for quantum-improved charged black hole

We investigate strong gravitational lensing by quantum-improved charged black holes characterized by an additional parameter denoted as ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}, in addition to the mass M and charge Q. Our findings reveal that when both the parameters Q/2M and ω/4M2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega /4M^2$$\\end{document} increase simultaneously, various astrophysical consequences, such as the deflection angle αD(u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _D(u)$$\\end{document} and angular image separation increase. Concurrently, the angular position θ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta _{\\infty }$$\\end{document}, relative magnification rmag\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_{mag}$$\\end{document}, and the time delay ΔT2,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta T_{2,1}$$\\end{document} between the first and second relativistic images also decrease with the growing values of the parameters Q/2M and ω/4M2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega /4M^2$$\\end{document}. It is also observed that the Einstein ring θ1E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta _1^E$$\\end{document} for the quantum-improved charged black hole is more significant than those for Schwarzschild, quantum-improved Schwarzschild, and Reisner–Nordström black holes. As with supermassive black holes such as M87∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M87^*$$\\end{document} and SgrA∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SgrA^*$$\\end{document}, it is observed that to be a viable astrophysical black hole, the EHT results constrain the parameter space (ω/4M2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega /4M^2$$\\end{document}, Q/2M). Remarkably, the EHT results for SgrA∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SgrA^*$$\\end{document} impose more stringent limits on the parameter space of quantum-improved charged black holes compared to those established by the EHT results forM87∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M87^*$$\\end{document}. We investigate the radial profiles of orbital and radial harmonic oscillation frequencies as a function of the dimensionless coupling constants and black hole mass. The main characteristics of test particle quasi-periodic oscillations close to stable circular orbits in the black hole equatorial plane are also examined. We study the positioning of resonant radii in the background of quantum-improved charged black holes for high-frequency quasi-periodic oscillations models: warped disc (WD) models, relativistic precession (RP) and its types, and epicyclic resonance (ER) and its variants.

Auxetic honeycomb core layer effect on vibrations of annular plates based on shear deformation theory

Light composite materials such as auxetic honeycomb structures offer a unique combination of strength, lightweight, and durability, making them ideal for a wide range of industrial and structural applications. Consequently, it is crucial to study the effect of these materials on the dynamic behavior of structures. In this study, an analytical solution based on the multiple-scale method has been proposed to solve equations of motion of a multilayer axisymmetric annular plate. The plate includes two top and bottom isotropic layers and one auxetic honeycomb core layer. By considering both radial and transverse deflections, the governing equations are derived based on the first-order shear deformation theory. The radial and transverse frequencies of the plate are determined from the order zero of equations and a correction coefficient for each frequency has been extracted from the order one and solvability conditions. The response of the plate under harmonic transverse excitation has been determined analytically and by conducting a parametric study, the effect of different geometrical and mechanical parameters on the free and forced vibration responses has been studied. It is observed that employing a honeycomb core in the multilayer plate allows us to significantly decrease the plate weight while the dynamic response and the frequencies remain nearly the same or vary slightly. The results are validated by comparing them with the finite element method and some other references.

Estimation of Radial Plasma Frequency Values for Polymer Thread with Aluminum Nanoislands

The article evaluates the values of the radial plasma frequency of a polymer thread made of fibers of carbon benzene rings coated with chains of irregular aluminum nanoislands as a metamaterial with a negative dielectric constant (ENG-material).

Multiple Bloch surface wave excitation with gratings

We study the coupling of a finite number of Bloch Surface Waves (BSWs) propagating in different directions at the surface of a dielectric multilayer. These surface waves arise from a set of diffraction orders associated to a grating on the bottom surface of the substrate that is illuminated by a normally incident beam. Simultaneous excitation of multiple BSWs is possible with a set of diffraction orders having the same radial spatial frequency. Using rigorous electromagnetic theory, we design gratings for simultaneous excitation of two, four and six BSWs propagating in directions separated by π, π/2 and π/3 azimuthal intervals, respectively.

3D stellar motion in the axisymmetric Galactic potential and the e–z resonances

Context. The full phase-space information on the kinematics of a huge number of stars provided by Gaia Data Release 3 increases the demand for a better understanding of the 3D stellar dynamics. Aims. In this paper, we investigate the possible regimes of motion of stars in the axisymmetric approximation of the Galactic potential, applying a 3D observation-based model developed elsewhere. The model consists of three components: the axisymmetric disc, the central spheroidal bulge, and the spherical halo of dark matter. The axisymmetric disc model is divided into thin and thick stellar discs and H I and H2 gaseous disc subcomponents, by combining three Miyamoto-Nagai disc profiles of any model order (1, 2, or 3) for each disc subcomponent, to reproduce a radially exponential mass distribution. The physical and structural parameters of the Galaxy components are adjusted by observational kinematic constraints. Methods. The phase space of the two-degrees-of-freedom model was studied by means of the Poincaré and dynamical mapping, the dynamical spectrum method, and the direct numerical integrations of the Hamiltonian equations of motion. Results. For the chosen physical parameters, the nearly circular (close to the rotation curve) and low-altitude stellar behaviour is composed of two weakly coupled simple oscillations, radial and vertical motions. The amplitudes of the vertical oscillations of these orbits gradually increase with the growing Galactocentric distances, in concordance with the exponential mass decay assumed. However, for increasing planar eccentricities, e, and the altitudes over the equatorial disc, z, new regimes of stellar motion emerge as a result of the beating between the radial and vertical oscillation frequencies, which we refer to as e–z resonances. The corresponding resonant motion produces the characteristic sudden increase or decrease in the amplitude of the vertical oscillation, bifurcations in the dynamical spectra, and the chains of islands of stable motion in the phase space. Conclusions. The results obtained can be useful in understanding and interpreting the features observed in the stellar 3D distribution around the Sun.

Quark models and radial oscillations: decoding the HESS J1731-347 compact object’s equation of state

We investigate the peculiar nature of strange stars through an analysis of different quark models, i.e. vBag model and CFL model equation of states at different parameter sets, and focus on understanding the equation of state governing the intriguing central compact object (CCO) within the supernova remnant HESS J1731-347, with a mass and radius of M=0.77-0.17+0.20M⊙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M = 0.77^{+0.20}_{-0.17} M_{\\odot }$$\\end{document} and R=10.4-0.78+0.86\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R = 10.4^{+0.86}_{-0.78}$$\\end{document} km, respectively. Additionally, we compare the radial oscillations of two models to determine the frequency of the HESS J1731-347 compact object at its maximum mass. The frequencies of radial oscillations are computed for each of the four EoSs considered. In total, the 10 lowest radial frequencies for each of those EoSs have been computed. By delving into these aspects, we aim at deepening our understanding of strange stars and their connection to the observed HESS J1731-347 mass-radius relationship.