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Context: We demonstrate that the minimum of the reaction force curve of a diatomic or polyatomic molecule undergoing bond dissociation is significant in several respects. As has been pointed out in the past, it is the point at which the force opposing dissociation is strongest. It marks the boundary between the primarily structural stage of a bond dissociation (stretching) and the transition region between the stretched bond and independent atoms. We now show that the reaction force minimum is also where the kinetic and potential energy curves tend to change direction abruptly. At this point, the total energy E(R) has increased by about 27% of the dissociation energy, for both diatomic and polyatomic molecules. Methods: Dissociation curves are analyzed at the UHF/daug-cc-pV5Z level of theory using Gaussian 16. [ABSTRACT FROM AUTHOR]

The boundaries of galaxy groups and clusters are defined by the interplay between the Newtonian attractive force and the decoupling from the local expansion of the Universe. This work extends the definition of a zero radial acceleration surface (ZRAS) and the turnaround surface (TS) for a general distribution of the masses in an expanding background, governed by the cosmological constant. We apply these definitions to different galaxy groups in the local Universe, mapping these groups up to ten megaparsec distances. We discuss the dipole and the quadrupole rate for the Local Group of Galaxies and the implementations on the Hubble diagram correction and galaxy groups virialization. With these definitions, we present the surfaces showing the interplay between the local expansion vs the local Newtonian attraction for galaxy groups in the local Universe. Further, we estimate the masses of different galaxy groups and show that the inclusion of the Cosmological Constant in the analysis predicts these masses to be higher by 5-10%. For instance the Local Group of galaxies is estimated to be (2.47±0.08)⋅1012M⊙. For the groups with enough tracers close to the TS, the contribution of the Cosmological Constant makes the masses to be even higher. The results show the importance of including the local cosmic expansion in analyzing the Cosmic Flow of the local Universe.

Disk galaxies have a typical dimension of a few tens of kiloparsecs. It follows from the theory of general relativity that any signal originating from the galactic center will be noticed at the outskirts of the galaxy only tens of thousands of years later. This retardation effect, however, is absent in modelling used to calculate rotation curves throughout entire galaxies and their external gas. The considerable differences between Newtonian predictions and observed velocities are currently removed either by assuming dark matter or by suggesting various modifications to the laws of gravity, MOND being a long standing alternative to Newtonian gravity. In previous papers we have shown that by applying general relativity in a rigorous fashion, without neglecting retardation, one can explain the rotational velocities of galactic matter without modifying gravity or adding dark matter. Moreover, it was shown that dark matter effects, as they appear in gravitational lensing, the Tully-Fisher relation, and mass estimations based on the virial theorem could also be explained as retarded-gravity effects. It must be noted, however, that the proposed theory relies on the existence of a mass flow (of a changing rate) into the galaxy; a requirement that was not directly observed. In the original paper on the subject only one galaxy (M33) was analysed in detail. This was later amended with a published study of eleven galaxies. Here we give a more comprehensive retardation analysis of 143 galaxies of different types from the SPARC Galaxy collection. We show that in most cases we obtain very accurate fits to the data. [ABSTRACT FROM AUTHOR]

Our understanding of the formation and interaction processes of astronomical structures in the Universe and in galaxies has significantly expanded due to recent progress in improving the quality of observations, brought about by the emergence of new telescopes. Such processes are associated with the release of a significant amount of energy and may influence our vital activities. One example of energy flow release is magnetic storms, which are associated with processes occurring on the Sun. Such storms lead to disruptions of communication facilities, impact people, and can cause satellite and aircraft failures. There are objects in both near and deep space that are much more powerful than our star. In such objects, enormous energy flows are generated and released in the form of space jets. These energy flows occur in quasars, black holes, protostars, and during supernova explosions of types Ia, Ib, Ic, and II. During a supernova explosion, an immense amount of energy is released in the form of energy and high-energy particles. Such processes can have a significant impact on the safety of space flights. In this paper, the processes of formation of stellar systems within a turbulent interstellar medium are considered. This process is the result of complex processes that occur in the interstellar medium and denser gas formations – molecular clouds. At the same time, there are nonlinear interactions between turbulence and gravity, collisions between molecular clouds, interactions with shock waves, and so on. Supersonic turbulence is one of the main reasons for the formation of dynamic prestellar structures. The evolution of superdense substances formation begins when they gather in turbulent flows or are formed by supersonic collisions between molecular clouds. This process continues until these areas reach prestellar density. Depending on various factors, these superdense formations can either collapse and form new star systems or disintegrate, returning the substance to the interstellar medium. Until recently, the prevailing opinion was that a significant portions of molecular clouds were not gravitationally bound. Conclusions about this were made based on the virial theorem, which expresses the relationship between gravitational and kinetic energy. However, recent studies have refuted this. In this paper, the results of a simulation carried out to study the collision of large ellipsoidal and spherical molecular clouds in a three-dimensional setting. The simulation was conducted on high-resolution grids and utilized the adaptive mesh refinement (AMR) method. • Morphological analysis of supersonic turbulence arising from the collision of spherical and prolate molecular clouds. • Modeling processes leading to the possible formation of new stellar systems. • Study of complex shock wave loading leading to the emergence of supersonic turbulence, Kelvin-Helmholtz instability and the formation of clumps. • Comparison of Chandrasekhar's analytical and numerical solution for an prolate molecular cloud. [ABSTRACT FROM AUTHOR]

We present dynamical scaling relations, combined with the stellar population properties, for a subsample of about 6000 nearby galaxies with the most reliable dynamical models extracted from the full Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) sample of 10 000 galaxies. We show that the inclination-corrected mass plane for both early-type galaxies (ETGs) and late-type galaxies (LTGs), which links dynamical mass, projected half-light radius R e, and the second stellar velocity moment σe within R e, satisfies the virial theorem and is even tighter than the uncorrected one. We find a clear parabolic relation between |$\lg (M/L)_{\rm e}$|⁠ , the total mass-to-light ratio (M / L) within a sphere of radius R e, and |$\lg \sigma _{\rm e}$|⁠ , with the M / L increasing with σe and for older stellar populations. However, the relation for ETGs is linear and the one for the youngest galaxies is constant. We confirm and improve the relation between mass-weighted total density slopes |$\overline{\gamma _{_{\rm T}}}$| and σe: |$\overline{\gamma _{_{\rm T}}}$| become steeper with increasing σe until |$\lg (\sigma _{\rm e}/{\rm km\, s^{-1}})\approx 2.2$| and then remain constant around |$\overline{\gamma _{_{\rm T}}}\approx 2.2$|⁠. The |$\overline{\gamma _{_{\rm T}}}\!-\!\sigma _{\rm e}$| variation is larger for LTGs than ETGs. At fixed σe the total density profiles steepen with galaxy age and for ETGs. We find generally low dark matter fractions, median f DM(< R e) = 8 per cent, within a sphere of radius R e. However, we find that f DM(< R e) depends on σe better than stellar mass: dark matter increases to a median f DM(< R e) = 33 per cent for galaxies with |$\sigma _{\rm e}\lesssim 100$|  km s−1. The increased f DM(< R e) at low σe explains the parabolic |$\lg (M/L)_{\rm e}\!-\!\lg \sigma _{\rm e}$| relation. [ABSTRACT FROM AUTHOR]

Derivation and some applications of the scalar virial theorem to physical problems in classical mechanics, classical statistical mechanics, and non-relativistic quantum mechanics have been dealt with lucidly. Besides, brief comments about the history of its development and its versions in the special theory of relativity, in the systems enclosed in a container of finite volume, and in quantum statistical mechanics have been made. Also included are remarks bringing out its main limitations. [ABSTRACT FROM AUTHOR]

Available inventories of baryonic mass in the universe are based largely on galactic data and empirical calculations made >20 years ago. Values falling below cosmological estimates underlie proposals that certain rarified gassy regions could have extremely high T, which motivated absorption measurements and hydrodynamic models. Yet, the shortfall remains. We inventory the total baryonic mass, focusing on gravitational interactions and updated measurements. A recent analytical inverse method for analyzing galactic rotation curves quantified how baryon mass and associated volumetric density (ρ) depend on distance (r) from galactic centers. The model is based on the dynamical consequences of the observed oblate shape of galaxies and the Virial Theorem. The parameter-free solution provides ρ(r) ∝ 1/r2 which describes star-rich galactic interiors, gas-rich outer discoids, circumgalactic media, and gradation into intergalactic media. Independent observational determinations of baryonic ρ validate that our 1/r2 result describes baryons alone. This solution shows that total baryonic mass associated with any galaxy is 2.4 to 40 times detectable luminosity, depending on galaxy size and spacing. Luminosity data within 50 Mpc show that Andromeda equivalents separated by ~1 Mpc represent the local universe. Combining the above yields (6 ± 2) × 10−25 kg m−3 for the present-day universe. Three other approaches support this high density: (1) evaluating trends and luminosity data near 1000 Mpc; (2) using a recent estimate for the number of galaxies in the universe; (3) calculating an energy balance. We discuss uncertainties in the critical density. Implications of large baryonic ρ are briefly discussed. [ABSTRACT FROM AUTHOR]

A hydrodynamic model for small acoustic oscillations in a cloud of stars is built, taking into account the self-consistent gravitational field in equilibrium with a non-zero second correlation moment. It is assumed that the momentum flux density tensor should include the analog of the anisotropic pressure tensor and the second correlation moment of both longitudinal and transverse gravitational field strength. The non-relativistic temporal equation for the second correlation moment of the gravitational field strength is derived from the Einstein equations using the first-order post-Newtonian approximation. One longitudinal and two transverse branches of acoustic oscillations are found in a homogeneous and isotropic star cloud. The requirement for the velocity of transverse oscillations to be zero provides the boundary condition for the stability of the cloud. The critical radius of the spherical cloud of stars is obtained, which is precisely consistent with the virial theorem. [ABSTRACT FROM AUTHOR]

Feedback from active galactic nuclei (AGNs) and supernovae can affect measurements of integrated Sunyaev–Zeldovich (SZ) flux of haloes (Y SZ) from cosmic microwave background (CMB) surveys, and cause its relation with the halo mass (Y SZ– M) to deviate from the self-similar power-law prediction of the virial theorem. We perform a comprehensive study of such deviations using CAMELS, a suite of hydrodynamic simulations with extensive variations in feedback prescriptions. We use a combination of two machine learning tools (random forest and symbolic regression) to search for analogues of the Y – M relation which are more robust to feedback processes for low masses (⁠|$M\lesssim 10^{14}\, \mathrm{ h}^{-1} \, \mathrm{ M}_\odot$|⁠); we find that simply replacing Y → Y (1 + M */ M gas) in the relation makes it remarkably self-similar. This could serve as a robust multiwavelength mass proxy for low-mass clusters and galaxy groups. Our methodology can also be generally useful to improve the domain of validity of other astrophysical scaling relations. We also forecast that measurements of the Y – M relation could provide per cent level constraints on certain combinations of feedback parameters and/or rule out a major part of the parameter space of supernova and AGN feedback models used in current state-of-the-art hydrodynamic simulations. Our results can be useful for using upcoming SZ surveys (e.g. SO, CMB-S4) and galaxy surveys (e.g. DESI and Rubin) to constrain the nature of baryonic feedback. Finally, we find that the alternative relation, Y – M *, provides complementary information on feedback than Y – M. [ABSTRACT FROM AUTHOR]

We apply Newtonian fractional-dimension gravity (NFDG), an alternative gravitational model, to some notable cases of galaxies with little or no dark matter. In the case of the ultra-diffuse galaxy AGC 114905, we show that NFDG methods can effectively reproduce the observed rotation curve using a variable fractional dimension D R , as was performed for other galaxies in previous studies. For AGC 114905, we obtain a variable dimension in the range D ≈ 2.2–3.2, but our fixed D = 3 curve can still fit all the experimental data within their error bars. This confirms other studies indicating that the dynamics of this galaxy can be described almost entirely by the baryonic mass distribution alone. In the case of NGC 1052-DF2, we use an argument based on the NFDG extension of the virial theorem applied to the velocity dispersion of globular clusters showing that, in general, discrepancies between observed and predicted velocity dispersions can be attributed to an overall fractal dimension D < 3 of the astrophysical structure considered, and not to the presence of dark matter. For NGC 1052-DF2, we estimate D ≈ 2.9 , thus confirming that this galaxy almost follows standard Newtonian behavior. We also consider the case of the Bullet Cluster merger (1E0657-56), assumed to be one of the strongest proofs of dark matter existence. A simplified but effective NFDG model of the collision shows that the observed infall velocity of this merger can be explained by a fractional dimension of the system in the range D ≃ 2.4–2.5, again, without using any dark matter. [ABSTRACT FROM AUTHOR]

We derive simple bootstrap bounds on correlation functions of the BFSS matrix theory/D0-brane quantum mechanics. The result strengthens and extends Polchinski's virial theorem bound to finite energies and gives the first non-trivial bound on ⟨Tr X2⟩. Despite their simplicity, the bounds hint at some features of the dual black hole geometry. Our best lower bounds are already a factor of ∼ 2 from existing Monte Carlo results. [ABSTRACT FROM AUTHOR]

We analyse the bulge/spheroid size–(stellar mass), R e, Sph − M *, Sph, relation and spheroid structural parameters for 202 local (predominantly |$\lesssim 110~\rm Mpc$|⁠) galaxies spanning |$M_*\sim 3\times 10^{9}\!-\!10^{12}~\rm M_{\odot }$| and |$0.1 \lesssim R_\mathrm{e, Sph}\lesssim 32~\rm kpc$| from multicomponent decomposition. The correlations between the spheroid Sérsic index (n Sph), central surface brightness (μ0, Sph), effective half-light radius (R e, Sph), absolute magnitude (⁠|$\mathfrak {M}_\mathrm{Sph}$|⁠), and stellar mass (M *, Sph) are explored. We also investigate the consequences of using different scale radii, |$R_{z,\rm Sph}$|⁠ , encapsulating a different fraction (z , from 0 to 1) of the total spheroid luminosity. The correlation strengths for projected mass densities, Σ z and 〈Σ〉 z , vary significantly with the choice of z. Spheroid size (⁠|$R_\mathrm{z, \rm Sph}$|⁠) and mass (M *, Sph) are strongly correlated for all light fractions z. We find |$\log (R_\mathrm{e,Sph}/\rm kpc) = 0.88\log (M_\mathrm{*,Sph}/\rm M_{\odot })-9.15$| with a small scatter of |$\Delta _{rms} = 0.24~\rm dex$| in the log (R e, Sph) direction. This result is discussed relative to the curved size–mass relation for early-type galaxies due to their discs yielding larger galaxy radii at lower masses. Moreover, the slope of our spheroid size–mass relation is a factor of ∼3, steeper than reported bulge size–mass relations, and with bulge sizes at M *, sph ∼3 × 109 M⊙ which are 2–3 times smaller. Our spheroid size–mass relation present no significant flattening in slope in the low-mass end (⁠|$M_{\rm *,sph}\sim 10^9 - 10^{10}\rm ~M_{\odot }$|⁠). Instead of treating galaxies as single entities, future theoretical and evolutionary models should also attempt to recreate the strong scaling relations of specific galactic components. Additional scaling relations, such as log (n Sph) − log (M *, Sph), log (Σ0, Sph) − log (n Sph), and log (n Sph) − log (R e, Sph), are also presented. Finally, we show that the local spheroids align well with the size-mass distribution of quiescent galaxies at z ∼ 1.25–2.25. In essence, local spheroids and high- z quiescent galaxies appear structurally similar, likely dictated by the virial theorem. [ABSTRACT FROM AUTHOR]

In the absence of work, the exchange of heat of a sample of matter corresponds to the change of its internal energy, given by the kinetic energy of random translational motion of all its constituent atoms or molecules relative to the center of mass of the sample, plus the excitation of quantum states, such as vibration and rotation, and the energy of electrons in excess to their ground state. If the sample of matter is equilibrated it is described by Boltzmann's statistical thermodynamics and characterized by a temperature T. Monotonic motion such as that of the stars of an expanding universe is work against gravity and represents the exchange of kinetic and potential energy, as described by the virial theorem, but not an exchange of heat. Heat and work are two distinct properties of thermodynamic systems. Temperature is defined for the radiative cosmic background and for individual stars, but for the ensemble of moving stars neither temperature, nor pressure, nor heat capacities are properly defined, and the application of thermodynamics is, therefore, not advised. For equilibrated atomic nanoclusters, in contrast, one may talk about negative heat capacities when kinetic energy is transformed into potential energy of expanding bonds. [ABSTRACT FROM AUTHOR]

While dry mergers can produce considerable scatter in the (black hole mass, M bh)-(spheroid stellar mass, M *, sph) and M bh-(spheroid half-light radius, R e, sph) diagrams, the virial theorem is used here to explain why the scatter about the M bh–(velocity dispersion, σ) relation remains low in the face of such mergers. Its small scatter has been claimed as evidence of feedback from active galactic nuclei (AGNs). However, it is shown that galaxy mergers also play a significant role. The collision of two lenticular (S0) galaxies is expected to yield three types of merger product (a core-Sérsic S0, an ellicular ES,e or an elliptical E galaxy), depending on the remnant's orbital angular momentum. It is shown that the major merger of two S0 galaxies with M *, sph ∼ 1011 M⊙ advances the system along a slope of ∼5 in the M bh-σ diagram, while a major E+E galaxy merger moves a system slightly along a trajectory with a slope of ∼9. Mergers of lower-mass S0 galaxies with M *, sph ∼ 1010 M⊙ move slightly along a trajectory with a slope of ∼3, thereby further contributing to the steeper distribution for the E (and Es,e) galaxies in the M bh-σ diagram, reported here to have a slope of 7.27 ± 0.91, compared to the S0 galaxies that have a slope of 5.68 ± 0.60. This result forms an important complement to the AGN feedback models like that from Silk & Rees, providing a more complete picture of galaxy/(black hole) co-evolution. It also has important implications for nanohertz gravitational-wave research. [ABSTRACT FROM AUTHOR]

A specific modification of Newtonian dynamics known as MOND has been shown to reproduce the dynamics of most astrophysical systems at different scales without invoking non-baryonic dark matter (DM). There is, however, a long-standing unsolved problem when MOND is applied to rich clusters of galaxies in the form of a deficit (by a factor around two) of predicted dynamical mass derived from the virial theorem with respect to observations. In this article, we approach the virial theorem using the velocity dispersion of cluster members along the line of sight rather than using the cluster temperature from X-ray data and hydrostatic equilibrium. Analytical calculations of the virial theorem in clusters for Newtonian gravity + DM and MOND are developed, applying pressure (surface) corrections for non-closed systems. Recent calibrations of DM profiles, baryonic ratio, and baryonic (β model or others) profiles are used, while allowing free parameters to range within the observational constraints. It is shown that solutions exist for MOND in clusters that give similar results to Newton + DM – particularly in the case of an isothermal β model for β = 0.55–0.70 and core radii r c between 0.1 and 0.3 times r 500 (in agreement with the known data). The disagreements found in previous studies seem to be due to the lack of pressure corrections (based on inappropriate hydrostatic equilibrium assumptions) and/or inappropriate parameters for the baryonic matter profiles. [ABSTRACT FROM AUTHOR]

In the case of the one-electron Dirac equation with a point nucleus, the virial theorem (VT) states that the ratio of the kinetic energy to potential energy is exactly − 1 , a ratio that can be an independent test of the accuracy of a computed solution. This paper studies the virial theorem for subshells of equivalent electrons and their interactions in many-electron atoms. This shows that the linear scaling of the dilation is achieved through the balancing of the contributions to the potential of an electron from inner and outer regions that some Slater integrals impose conditions on a single subshell, but others impose conditions between subshells. The latter slows the rate of convergence of the self-consistent field process in which radial functions are updated one at a time. Several cases are considered. Results are also extended to the nonrelativistic case. [ABSTRACT FROM AUTHOR]

The boundaries of galaxy groups and clusters are defined by the interplay between the Newtonian attractive force and the decoupling from the local expansion of the Universe. This work extends the definition of a zero radial acceleration surface (ZRAS) and the turnaround surface (TS) for a general distribution of the masses in an expanding background, governed by the cosmological constant. We apply these definitions to different galaxy groups in the local Universe, mapping these groups up to ten megaparsec distances. We discuss the dipole and the quadrupole rate for the Local Group of Galaxies and the implementations on the Hubble diagram correction and galaxy groups virialization. With these definitions, we present the surfaces showing the interplay between the local expansion vs the local Newtonian attraction for galaxy groups in the local Universe. Further, we estimate the masses of different galaxy groups and show that the inclusion of the Cosmological Constant in the analysis predicts these masses to be higher by 5-10%. For instance the Local Group of galaxies is estimated to be (2.47 ± 0.08) ⋅ 10 12 M ⊙. For the groups with enough tracers close to the TS, the contribution of the Cosmological Constant makes the masses to be even higher. The results show the importance of including the local cosmic expansion in analyzing the Cosmic Flow of the local Universe. [ABSTRACT FROM AUTHOR]

In this work, we apply virial analysis to the model of self-gravitating turbulent cloud ensembles introduced by Donkov & Stefanov in two previous papers, clarifying some aspects of turbulence and extending the model to account not only for supersonic flows but for trans- and subsonic ones as well. Making use of the Eulerian virial theorem at an arbitrary scale, far from the cloud core, we derive an equation for the density profile and solve it in approximate way. The result confirms the solution ϱ(ℓ) = ℓ−2 found in the previous papers. This solution corresponds to three possible configurations for the energy balance. For trans- or subsonic flows, we obtain a balance between the gravitational and thermal energy (Case 1) or between the gravitational, turbulent, and thermal energies (Case 2) while for supersonic flows, the possible balance is between the gravitational and turbulent energy (Case 3). In Cases 1 and 2, the energy of the fluid element can be negative or zero; thus the solution is dynamically stable and shall be long lived. In Case 3, the energy of the fluid element is positive or zero, i.e. the solution is unstable or at best marginally bound. At scales near the core, one cannot neglect the second derivative of the moment of inertia of the gas, which prevents derivation of an analytic equation for the density profile. However, we obtain that gas near the core is not virialized and its state is marginally bound since the energy of the fluid element vanishes. [ABSTRACT FROM AUTHOR]

We investigate changes in stellar population age and metallicity ([Z/H]) scaling relations for quiescent galaxies from intermediate redshift (0.60 ≤   |$z$| ≤ 0.76) using the LEGA-C Survey to low redshift (0.014 ≤   |$z$| ≤ 0.10) using the SAMI Galaxy Survey. Specifically, we study how the spatially integrated global age and metallicity of individual quiescent galaxies vary in the mass–size plane, using the stellar mass M * and a dynamical mass proxy derived from the virial theorem M D  ∝ σ2  R e. We find that, similarly to at low redshift, the metallicity of quiescent galaxies at 0.60 ≤   |$z$| ≤ 0.76 closely correlates with M / R e (a proxy for the gravitational potential or escape velocity), in that galaxies with deeper potential wells are more metal-rich. This supports the hypothesis that the relation arises due to the gravitational potential regulating the retention of metals by determining the escape velocity for metal-rich stellar and supernova ejecta to escape the system and avoid being recycled into later stellar generations. Conversely, we find no correlation between age and surface density (⁠|$M/R_\mathrm{e}^2$|⁠) at 0.60 ≤   |$z$| ≤ 0.76, despite this relation being strong at low redshift. We consider this change in the age– |$M/R_\mathrm{e}^2$| relation in the context of the redshift evolution of the star-forming and quiescent mass–size relations, and find our results are consistent with galaxies forming more compactly at higher redshifts and remaining compact throughout their evolution. Furthermore, galaxies appear to quench at a characteristic surface density that decreases with decreasing redshift. The |$z$| ∼  0 age– |$M/R_\mathrm{e}^2$| relation is therefore a result of building up the quiescent and star-forming populations with galaxies that formed at a range of redshifts and therefore a range of surface densities. [ABSTRACT FROM AUTHOR]

In this paper, we theoretically investigate the stability of spin-wave solitons in Bose-Einstein condensates of repulsive magnons, confined by an inhomogeneous external magnetic field described by a Gaussian well. For this purpose, we use the quasi-one-dimensional Gross-Pitaevskii equation to describe the behavior of the condensate. In order to solve the Gross-Pitaevskii equation, we used two different approaches: one analytical (variational method) and another numerical (split-step Crank-Nicolson method). The stability of the solutions and the validation of the numerical results were confirmed, respectively, through the anti-VK criterion and the virial theorem. Furthermore, the simulations described the behavior of physical quantities of interest such as chemical potential, energy per magnon and central density as a function of the nonlinearity of the model (magnon-magnon interactions). The theoretical results provide subsidies for a better understanding of the nonlinear phenomena related to the Bose-Einstein condensates of magnons in ferromagnetic films. [ABSTRACT FROM AUTHOR]

Based on the virial theorem, the influence of the following factors to characteristics of mechanical support systems of toroidal magnetic energy storage is considered: the configuration of the coils, the support system elements location under compressive forces, the mechanical properties of the material, the distribution of the cross-sectional areas of the support rings and the corresponding distribution of radial forces. It is established that the volume of structure requirement depends only on the radii from which the coils transmit forces to the support rings. The found general relation, in contrast to the existing ones, establishes a relationship between all the parameters that determine the volume of the support structures. The characteristics of support systems are analyzed both for identical mechanical stresses and for identical mechanical properties of all support elements. The characteristics for specific examples of toroidal systems with circular coils are found. It is shown that systems with support elements in compression located on the side of the vertical axis of the torus have a significantly smaller volume of structure compared to other locations. The influence of the relative radial size of the torus cross-section and the distribution of the cross-sectional areas of the support rings on the structure requirement is analyzed. The relationship between dimensionless characteristics that determine the volume of elements subjected to mechanical stresses of compression and tension is noted. References 37, figures 8, table 1. [ABSTRACT FROM AUTHOR]

In the case of the one-electron Dirac equation with a point nucleus, the virial theorem (VT) states that the ratio of the kinetic energy to potential energy is exactly −1, a ratio that can be an independent test of the accuracy of a computed solution. This paper studies the virial theorem for subshells of equivalent electrons and their interactions in many-electron atoms. This shows that the linear scaling of the dilation is achieved through the balancing of the contributions to the potential of an electron from inner and outer regions that some Slater integrals impose conditions on a single subshell, but others impose conditions between subshells. The latter slows the rate of convergence of the self-consistent field process in which radial functions are updated one at a time. Several cases are considered. Results are also extended to the nonrelativistic case.

We present high spatial resolution (≈12 pc) Atacama Large Millimeter/submillimeter Array 12CO(J  = 3–2) observations of the nearby lenticular galaxy NGC 4429. We identify 217 giant molecular clouds within the 450 pc radius molecular gas disc. The clouds generally have smaller sizes and masses but higher surface densities and observed linewidths than those of Milky Way disc clouds. An unusually steep size–linewidth relation (⁠|$\sigma \propto R_{\rm c}^{0.8}$|⁠) and large cloud internal velocity gradients (0.05–0.91 km s−1 pc−1) and observed virial parameters (〈αobs,vir〉 ≈ 4.0) are found, which appear due to internal rotation driven by the background galactic gravitational potential. Removing this rotation, an internal virial equilibrium appears to be established between the self-gravitational (U sg) and turbulent kinetic (E turb) energies of each cloud, i.e. |$\langle \alpha _{\rm sg,vir}\equiv \frac{2E_{\rm turb}}{\vert U_{\rm sg}\vert }\rangle \approx 1.3$|⁠. However, to properly account for both self and external gravity (shear and tidal forces), we formulate a modified virial theorem and define an effective virial parameter |$\alpha _{\rm eff,vir}\equiv \alpha _{\rm sg,vir}+\frac{E_{\rm ext}}{\vert U_{\rm sg}\vert }$| (and associated effective velocity dispersion). The NGC 4429 clouds then appear to be in a critical state in which the self-gravitational energy and the contribution of external gravity to the cloud's energy budget (E ext) are approximately equal, i.e. |$\frac{E_{\rm ext}}{\vert U_{\rm sg}\vert }\approx 1$|⁠. As such, 〈αeff,vir〉 ≈ 2.2 and most clouds are not virialized but remain marginally gravitationally bound. We show this is consistent with the clouds having sizes similar to their tidal radii and being generally radially elongated. External gravity is thus as important as self-gravity to regulate the clouds of NGC 4429. [ABSTRACT FROM AUTHOR]

In this paper, we study, in the Newtonian limit, the virial theorem in the context of a scalar tensor fourth order gravity. In particular, we show, that for a isolated galaxy in viral equilibrium, a specific class of scalar tensor fourth order gravity, i.e. f (R , ϕ) + ω (ϕ) ϕ ; α ϕ ; α in not suitable to explain the large fraction of dark matter necessary to have the flatness of the galaxies rotation curves experimentally observed. [ABSTRACT FROM AUTHOR]

Utilizing the Atacama Large Millimeter/submillimeter Array, we present CS line maps in five rotational lines (Ju = 7, 5, 4, 3, 2) toward the circumnuclear disk (CND) and streamers of the Galactic center. Our primary goal is to resolve the compact structures within the CND and the streamers, in order to understand the stability conditions of molecular cores in the vicinity of the supermassive black hole (SMBH) Sgr A*. Our data provide the first homogeneous high-resolution (1.″3 = 0.05 pc) observations aiming at resolving density and temperature structures. The CS clouds have sizes of 0.05–0.2 pc with a broad range of velocity dispersion (σFWHM = 5–40 km s−1). The CS clouds are a mixture of warm (Tk ≥ 50–500 K, = 103–105 cm−3) and cold gas (Tk ≤ 50 K, = 106–108 cm−3). A stability analysis based on the unmagnetized virial theorem including tidal force shows that of the total gas mass is tidally stable, which accounts for the majority of gas mass. Turbulence dominates the internal energy and thereby sets the threshold densities 10–100 times higher than the tidal limit at distance ≥1.5 pc to Sgr A*, and therefore it inhibits the clouds from collapsing to form stars near the SMBH. However, within the central 1 pc, the tidal force overrides turbulence and the threshold densities for a gravitational collapse quickly grow to ≥ 108 cm−3. [ABSTRACT FROM AUTHOR]

The electron spin angular momentum features are revealed to be identical to those of a uniformly charged ring. The electric field for this configuration can capture the electron being accelerated toward a proton, thus explaining the stability of the hydrogen atom. In addition, the proposed atom structure describes the bonding of two hydrogen atoms, which illuminates the covalent bond formation, the electronic state, bond length, and binding energy. Additionally, important wave properties ascribed to photons are assigned to particles with mass such as wave energy, angular frequency, phase velocity, and quantum spin. [ABSTRACT FROM AUTHOR]

The discovery of 19 dwarf galaxies without dark matter (DM) provides, counterintuitively, strong support for the ΛCDM standard model of cosmology. Their presence is well accommodated in a scenario where the DM is in the form of cold dark particles. However, it is interesting to explore quantitatively what is needed from modified gravity models to accommodate the presence of these galaxies and what extra degree of freedom is needed in these models. To this end, we derive the dynamics at galaxy scales (Virial theorem) for a general class of modified gravity models. We distinguish between theories that satisfy the Jebsen–Birkhoff theorem, and those that do not. Our aim is to develop tests that can distinguish whether DM is part of the theory of gravity or a particle. The 19 dwarf galaxies discovered provide us with a stringent test for models of modified gravity. Our main finding is that there will always be an extra contribution to the Virial theorem coming from the modification of gravity, even if a certain galaxy shows very small, if not negligible, trace of DM, as has been reported recently. Thus, if these and more galaxies are confirmed as devoid (or negligible) of DM, while other similar galaxies have abundant DM, it seems interesting to find modifications of gravity to describe DM. Our result can be used by future astronomical surveys to put constraints on the parameters of modified gravity models at astrophysical scales where DM is described as such. [ABSTRACT FROM AUTHOR]

Recently, two-component spherical galaxy models have been presented, where the stellar profile is described by a Jaffe law, and the total density by another Jaffe law, or by an r −3 law at large radii. We extend these two families to their ellipsoidal axisymmetric counterparts: the JJe and J3e models. The total and stellar density distributions can have different flattenings and scale lengths, and the dark matter halo is defined by difference. First, the analytical conditions required to have a nowhere negative dark matter halo density are derived. The Jeans equations for the stellar component are then solved analytically, in the limit of small flattenings, also in the presence of a central BH. The azimuthal velocity dispersion anisotropy is described by the Satoh k -decomposition. Finally, we present the analytical formulae for velocity fields near the centre and at large radii, together with the various terms entering the virial theorem. The JJe and J3e models can be useful in a number of theoretical applications, e.g. to explore the role of the various parameters (flattening, relative scale lengths, mass ratios, rotational support) in determining the behaviour of the stellar kinematical fields before performing more time-expensive integrations with specific galaxy models, to test codes of stellar dynamics and in numerical simulations of gas flows in galaxies. [ABSTRACT FROM AUTHOR]

Generally the virial theorem provides a relation between various components of energy integrated over a system. This helps us to understand the underlying equilibrium. Based on the virial theorem we can estimate, for example, the maximum allowed magnetic field in a star. Recent studies have proposed the existence of highly magnetized white dwarfs (B-WDs), with masses significantly higher than the Chandrasekhar limit. Surface magnetic fields of such white dwarfs could be more than |$10^{9}$|  G with the central magnitude several orders higher. These white dwarfs could be significantly smaller in size than their ordinary counterparts (with surface fields restricted to about |$10^9$|  G). In this paper, we reformulate the virial theorem for non-rotating B-WDs in which, unlike in previous formulations, the contribution of the magnetic pressure to the magnetohydrostatic balance cannot be neglected. Along with the new equation of magnetohydrostatic equilibrium, we approach the problem by invoking magnetic flux conservation and by varying the internal magnetic field with the matter density as a power law. Either of these choices is supported by previous independent work and neither violates any important physics. They are useful while there is no prior knowledge of field profile within a white dwarf. We then compute the modified gravitational, thermal, and magnetic energies and examine how the magnetic pressure influences the properties of such white dwarfs. Based on our results we predict important properties of these B-WDs, which turn out to be independent of our chosen field profiles. [ABSTRACT FROM AUTHOR]

We study the modelling of the halo occupation distribution (HOD) for the eBOSS DR16 emission line galaxies (ELGs). Motivated by previous theoretical and observational studies, we consider different physical effects that can change how ELGs populate haloes. We explore the shape of the average HOD, the fraction of satellite galaxies, their probability distribution function (PDF), and their density and velocity profiles. Our baseline HOD shape was fitted to a semi-analytical model of galaxy formation and evolution, with a decaying occupation of central ELGs at high halo masses. We consider Poisson and sub/super-Poissonian PDFs for satellite assignment. We model both Navarro–Frenk–White and particle profiles for satellite positions, also allowing for decreased concentrations. We model velocities with the virial theorem and particle velocity distributions. Additionally, we introduce a velocity bias and a net infall velocity. We study how these choices impact the clustering statistics while keeping the number density and bias fixed to that from eBOSS ELGs. The projected correlation function, w p, captures most of the effects from the PDF and satellites profile. The quadrupole, ξ2, captures most of the effects coming from the velocity profile. We find that the impact of the mean HOD shape is subdominant relative to the rest of choices. We fit the clustering of the eBOSS DR16 ELG data under different combinations of the above assumptions. The catalogues presented here have been analysed in companion papers, showing that eBOSS RSD+BAO measurements are insensitive to the details of galaxy physics considered here. These catalogues are made publicly available. [ABSTRACT FROM AUTHOR]

Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrödinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this work, we address this issue by solving the particle in a finite box and show that the virial theorem is fulfilled when the appropriate Cauchy boundary conditions are taken into account. We also illustrate how, in the limit of the infinite potential box, the virial theorem holds as well. As a consequence, it is possible to determine the averaged force exerted by the walls on the particle. Finally, a discussion of these results in the classical limit is provided. [ABSTRACT FROM AUTHOR]

Based on the virial theorem, the influence of the following factors to characteristics of mechanical support systems of toroidal magnetic energy storage is considered: the configuration of the coils, the support system elements location under compressive forces, the mechanical properties of the material, the distribution of the cross-sectional areas of the support rings and the corresponding distribution of radial forces. It is established that the volume of structure requirement depends only on the radii from which the coils transmit forces to the support rings. The found general relation, in contrast to the existing ones, establishes a relationship between all the parameters that determine the volume of the support structures. The characteristics of support systems are analyzed both for identical mechanical stresses and for identical mechanical properties of all support elements. The characteristics for specific examples of toroidal systems with circular coils are found. It is shown that systems with support elements in compression located on the side of the vertical axis of the torus have a significantly smaller volume of structure compared to other locations. The influence of the relative radial size of the torus cross-section and the distribution of the cross-sectional areas of the support rings on the structure requirement is analyzed. The relationship between dimensionless characteristics that determine the volume of elements subjected to mechanical stresses of compression and tension is noted. References 37, figures 8, table 1.

The ground state solution of hydrogen molecule and ions are numerically obtained as an application of our scheme to solve many-electron multi-center potential Schrödinger equation by using one-dimensional hydrogen wavefunctions as basis functions. The all-electron sparse Hamiltonian matrix for the given system is generated with the standard order finite-difference method, then the electronic trial wavefunction to describe the ground state is constructed based on the molecular orbital treatment, and finally an effective and accurate iteration process is implemented to systematically improve the result. Many problems associated with the evaluation of the matrix elements of the Hamiltonian in more general basis and potential are circumvented. Compared with the standard results, the variationally obtained energy of H2+ is within 0.1 mhartree accuracy, while that of H2 and H3+ include the electron correlation effect. The equilibrium bond length is highly consistent with the accurate results and the virial theorem is satisfied to an accuracy of −V/T = 2.0. [ABSTRACT FROM AUTHOR]

Paper presents the results of the in-silico experiments concerning simulation of the tension and failure dynamics of two-dimensional (2D) titanium carbide Ti2C under different types of tensile loading. The behavior of 2D nanosheet was studied within classical molecular dynamics (MD) methods. Two different loading methods, namely axial displacement and uniform tensile strain were considered in experiments. The first loading method consists in a consecutive shift of atoms in the right edge of the sample along the x-axis while atoms in the left edge of the sample are held fixed. The uniform tensile strain was performed by shifting left and right parts of the sample in opposite directions. During the simulations, atomistic configurations of the 2D Ti2C nanosheet at different strain values were built for both loading methods. As it follows from the obtained data, different loading procedures lead to different fracture dynamics and crack formation in the studied sample. As calculated atomistic configurations show, in the case of axial displacement the fracture begins from the formation of cracks at the lateral edges of the sample. Cracks appear along the layers of constrained atoms at both fixed and shifted edges of the nanosheet, while at uniform tensile strain Ti2C sample undergoes uniform stretching up to the critical strain where the crack starts to form. The strain-stress curves for both axial displacement and homogenous strain were calculated through the virial theorem. Strain-stress dependencies obtained for different loading procedures for Ti2C sample overlap in the area of elastic deformation. Calculated data also show that plastic deformation and following destruction of the Ti2C sample occur at strain ε > 0.04 for both methods of loading. [ABSTRACT FROM AUTHOR]

In this paper we explore the effects that the presence of a fraction of binary stars has in the determination of a star cluster mass via the virial theorem. To reach this aim in an accurate and consistent way, we run a set of simulations using the direct summation, high precision, code NBODY7. By means of this suite of simulations we are able to quantify the overestimate of open-star-cluster-like models' dynamical masses when making a straight application of the virial theorem using available position and radial velocity measurements. The mass inflation caused by the binary "heating" contribution to the measured velocity dispersion depends, of course, on the initial binary fraction, fb0 and its following dynamical evolution. For an fb (evolved up to 1.5 Gyr) in the range 8%–42% the overestimate of the mass done using experimentally sounding estimates for the velocity dispersion can be up to a factor of 45. We provide a useful fitting formula to correct the dynamical mass determination for the presence of binaries, and underline how neglecting the role of binaries in stellar systems might lead to erroneous conclusions about their total mass budget. If this trend remains valid for larger systems like dwarf spheroidal galaxies, which are still far out of reach for high-precision dynamical simulations taking their binaries into account, it would imply an incorrect overestimation of their dark matter content, as inferred by means of available velocity dispersion measurements. [ABSTRACT FROM AUTHOR]

Early-type galaxies – slow and fast rotating ellipticals (E-SRs and E-FRs) and S0s/lenticulars – define a Fundamental Plane (FP) in the space of half-light radius Re , enclosed surface brightness Ie , and velocity dispersion σ e . Since Ie and σ e are distance-independent measurements, the thickness of the FP is often expressed in terms of the accuracy with which Ie and σ e can be used to estimate sizes Re. We show that: (1) The thickness of the FP depends strongly on morphology. If the sample only includes E-SRs, then the observed scatter in Re is |$\sim 16{{\ \rm per\ cent}}$|⁠ , of which only |$\sim 9{{\ \rm per\ cent}}$| is intrinsic. Removing galaxies with M * < 1011 M⊙ further reduces the observed scatter to |$\sim 13{{\ \rm per\ cent}}$| (⁠|$\sim 4{{\ \rm per\ cent}}$| intrinsic). The observed scatter increases to |$\sim 25{{\ \rm per\ cent}}$| usually quoted in the literature if E-FRs and S0s are added. If the FP is defined using the eigenvectors of the covariance matrix of the observables, then the E-SRs again define an exceptionally thin FP, with intrinsic scatter of only 5 per cent orthogonal to the plane. (2) The structure within the FP is most easily understood as arising from the fact that Ie and σ e are nearly independent, whereas the Re − Ie and Re −σ e correlations are nearly equal and opposite. (3) If the coefficients of the FP differ from those associated with the virial theorem the plane is said to be 'tilted'. If we multiply Ie by the global stellar mass-to-light ratio M */ L and we account for non-homology across the population by using Sérsic photometry, then the resulting stellar mass FP is less tilted. Accounting self-consistently for M */ L gradients will change the tilt. The tilt we currently see suggests that the efficiency of turning baryons into stars increases and/or the dark matter fraction decreases as stellar surface brightness increases. [ABSTRACT FROM AUTHOR]

We study a sample of 148 early-type galaxies in the Coma cluster using SDSS photometry and spectra, and calibrate our results using detailed dynamical models for a subset of these galaxies, to create a precise benchmark for dynamical scaling relations in high-density environments. For these galaxies, we successfully measured global galaxy properties, modelled stellar populations, and created dynamical models, and support the results using detailed dynamical models of 16 galaxies, including the two most massive cluster galaxies, using data taken with the SAURON IFU. By design, the study provides minimal scatter in derived scaling relations due to the small uncertainty in the relative distances of galaxies compared to the cluster distance. Our results demonstrate low (≤55 per cent for 90th percentile) dark matter fractions in the inner 1 R e of galaxies. Owing to the study design, we produce the tightest, to our knowledge, IMF–σe relation of galaxies, with a slope consistent with that seen in local galaxies. Leveraging our dynamical models, we transform the classical Fundamental Plane of the galaxies to the Mass Plane. We find that the coefficients of the Mass Plane are close to predictions from the virial theorem, and have significantly lower scatter compared to the Fundamental Plane. We show that Coma galaxies occupy similar locations in the (M *– R e) and (M *−σe) relations as local field galaxies but are older. This, and the fact we find only three slow rotators in the cluster, is consistent with the scenario of hierarchical galaxy formation and expectations of the kinematic morphology–density relation. [ABSTRACT FROM AUTHOR]

The dissemination of models across disciplinary lines has become a phenomenon of interest to philosophers of science. To account for this phenomenon, philosophers have invented two units of analysis. The first identifies to the thing that transfers, model templates. The second identifies the thing to which transferable templates apply, landing zones. There exists a dynamic between the thing that is transferred and the thing to which transferrable templates apply. The use of a transferable template in a new domain requires reconception of domain-specific phenomena. This paper examines two cases of model transfer, the use of the ideal gas law in biology by R.A. Fisher and the use of the virial theorem in chemistry by Richard Bader. These two stories of model transfer in biology and chemistry indicate a dimension to conceptual progress related to this dynamic. Using discourse on model transfer affords philosophers a novel approach for depicting the invention of, for instance, chemical concepts and resulting disputes. [ABSTRACT FROM AUTHOR]

Explicit exact expressions for the thermodynamic functions of the electron liquid via the single-particle Green's function are obtained and self-consistency conditions for these functions are formulated using the virial theorem and the equations of motion for the single-particle Green's function. [ABSTRACT FROM AUTHOR]

The dynamical properties of spherically symmetric galaxy models, where a Jaffe stellar density profile is embedded in a total mass density decreasing as r −3 at large radii, are presented. The orbital structure of the stellar component is described by the Osipkov–Merritt anisotropy; the dark matter halo is isotropic, and a black hole is added at the centre of the galaxy. First, the conditions for a nowhere negative and monotonically decreasing dark matter halo density profile are derived; this profile can be made asymptotically coincident with an NFW profile at the centre and large radii. Then, the minimum value of the anisotropy radius for phase-space consistency is derived as a function of the galaxy parameters. The Jeans equations for the stellar component are solved analytically; the projected velocity dispersion at the centre and large radii is also obtained, for generic values of the anisotropy radius. Finally, analytical expressions for the terms entering the Virial Theorem are derived, and the fiducial anisotropy limit required to prevent the onset of Radial Orbit Instability is determined as a function of the galaxy parameters. The presented models, built following an approach already adopted in our previous works, can be a useful starting point for a more advanced modelling of the dynamics of elliptical galaxies, and can be easily implemented in numerical simulations requiring a realistic dynamical model of a galaxy. [ABSTRACT FROM AUTHOR]

The force virial is a construct combining information about the current configuration of a mechanical system in motion with information about the acting forces; its long-time average turns out to be proportional to the long-time average of the system's kinetic energy, that is, the virial theorem holds true; a version of this theorem is obtained when a connection between kinetic energy and temperature is established, as it happens in the kinetic theory of gases or in classical equilibrium statistical mechanics. In fact, a virial theorem holds in whatever mechanics. This paper is an exposition of its various formulations, from the simplest deterministic formulation for a single massy particle in Newtonian motion to the fairly more complicated statistical formulation, which makes the theorem a corollary of the equipartition theorem. [ABSTRACT FROM AUTHOR]

We evaluate the performance of multipole, linear Paul traps for the purpose of studying cold ion–atom collisions. A combination of numerical simulations and analysis based on the virial theorem is used to draw conclusions on the differences that result, by considering the trapping details of several multipole trap types. Starting with an analysis of how a low energy collision takes place between a fully compensated, ultracold trapped ion and an stationary atom, we show that a higher order multipole trap is, in principle, advantageous in terms of collisional heating. The virial analysis of multipole traps then follows, along with the computation of trapped ion trajectories in the quadrupole, hexapole, octopole and do-decapole radio frequency traps. A detailed analysis of the motion of trapped ions as a function of the amplitude, phase and stability of the ion’s motion is used to evaluate the experimental prospects for such traps. The present analysis has the virtue of providing definitive answers for the merits of the various configurations, using first principles.

In this study, we present low mass to intermediate‐mass star‐forming hydrostatic cores in a self‐gravitating molecular cloud. Considering that the gravitational energy released during the collapse is the energy difference between the initial and final conditions. According to the virial theorem half the change in gravitational energy goes into internal energy the other half gets radiated away. However, half the change in gravitational energy is not completely (100%) emitted in the form of radiation. Therefore, we impose a dimensionless thermodynamic efficiency factor ϵ (where 0 < ϵ < 1) and quantify it as ϵ=4πRc2σTc4Lgf. Using these preconditions, we formulate equations of mass, density, and radius for the first hydrostatic core and the second hydrostatic core. The model shows there is a close relationship between the first hydrostatic core properties and the prestellar core properties. Moreover, the first hydrostatic core properties can influence the second hydrostatic core formation. [ABSTRACT FROM AUTHOR]

We consider the Gross-Pitaevskii equation describing a dipolar Bose-Einstein condensate without external confinement. We first consider the unstable regime, where the nonlocal nonlinearity is neither positive nor radially symmetric and standing states are known to exist. We prove that under the energy threshold given by the ground state, all global in time solutions behave as free waves asymptotically in time. The ingredients of the proof are variational characterization of the ground states energy, a suitable profile decomposition theorem and localized virial estimates, enabling to carry out a Concentration/Compactness and Rigidity scheme. As a byproduct we show that in the stable regime, where standing states do not exist, any initial data in the energy space scatters. [ABSTRACT FROM AUTHOR]

We report the computational study of mechanical properties of gold, silver and core-shell Au@Ag nanorods. The dynamical behavior of the samples under tensile deformation was studied within classical molecular dynamics simulation. Interactions between atoms in the samples were described within embedded atom method (EAM) and EAM alloy model for gold – silver interaction in core-shell Au@Ag nanorod. To study the mechanical properties of the nanorods, the tensile deformation procedure was applied to the samples. Stretching of the samples was simulated by displacement of the fixed atoms located in the two external atomic layers at the opposite edges of the samples from each other at constant strain rate of 0.004 ps–1. Temperature of the samples was maintained at 300 K using the Berendsen thermostat through the total simulation time. During the deformation process, mechanical stresses were calculated according to virial theorem for each sample and strain-stress curves were built from obtained data. Obtained strainstress curves have typical shape with a linear section at the beginning and a further nonlinear part corresponding to plastic deformation. Young moduli of the studied samples were estimated through linear regression of the elastic part of the strain-stress curves. The obtained values are equal to 57.3 GPa, 80.1 GPa and 70.9 GPa for Au@Ag, Au and Ag samples, respectively. Analysis of the snapshots of atomistic configurations shows that considered samples are characterized by different failure dynamics, namely necking was observed only for Au@Ag core-shell nanorod, while pristine gold and silver samples are characterized by failure without clearly visible necking. Comparative analysis of the obtained results shows a good agreement with the similar data available in the literature. [ABSTRACT FROM AUTHOR]

Kepler's rescaling becomes, when "Eisenhart-Duval lifted" to 5-dimensional "Bargmann" gravitational wave spacetime, an ordinary spacetime symmetry for motion along null geodesics, which are the lifts of Keplerian trajectories. The lifted rescaling generates a well-behaved conserved Noether charge upstairs, which takes an unconventional form when expressed in conventional terms. This conserved quantity seems to have escaped attention so far. Applications include the Virial Theorem and also Kepler's Third Law. The lifted Kepler rescaling is a Chrono-Projective transformation. The results extend to celestial mechanics and Newtonian Cosmology. [ABSTRACT FROM AUTHOR]