Angular Momentum Lz Research Articles

Positive Lynden-Bell derivative as a ticket to the bar trap?

ABSTRACT We have translated the results of N-body simulations of one barred model into the language of action variables and frequencies. Using this language, we analysed the behaviour of all orbits in the model on a large time-scale at the stage of a mature bar. We show that the orbits join the bar while preserving their adiabatic invariant, which takes into account the three-dimensional structure of the orbits. This allows us to apply the concept of the Lynden-Bell derivative for each of these orbits and trace how the sign of the derivative changes; i.e. how asynchronous changes in angular momentum Lz and orbital precession rate Ωpr (normal orbital mode) change to synchronous (abnormal mode). The transition to the abnormal mode occurs when Ωpr reaches the angular velocity of the pattern Ωp, after which the orbit becomes stuck in the bar trap. All this happens against the background of secular changes in actions (Lz decreases, JR and Jz increase). At the same time, corotating particles near two stable Lagrange points are also subject to secular changes in their actions. They increase Lz and drift to the periphery, shifting corotation outwards. We also show that a change in the orbital mode from normal to abnormal and the trapping of orbits in a bar are possible only when the bar speed decreases with time, regardless of what is causing the bar to slow down. Our findings clarify and expand the picture of bar formation and evolution in numerical models.

Nitrogen enrichment and clustered star formation at the dawn of the Galaxy

ABSTRACT Anomalously high nitrogen-to-oxygen abundance ratios [N/O] are observed in globular clusters (GCs), among the field stars of the Milky Way (MW), and even in the gas in a z ≈ 11 galaxy. Using data from the APOGEE Data Release 17 and the Gaia Data Release 3, we present several independent lines of evidence that most of the MW’s high-[N/O] stars were born in situ in massive bound clusters during the early, pre-disc evolution of the Galaxy. Specifically, we show that distributions of metallicity [Fe/H], energy, the angular momentum Lz, and distance of the low-metallicity high-[N/O] stars match the corresponding distributions of stars of the Aurora population and of the in situ GCs. We also show that the fraction of in situ field high-[N/O] stars, fN/O, increases rapidly with decreasing metallicity. During epochs when metallicity evolves from $\rm [Fe/H]=-1.5$ to $\rm [Fe/H]=-0.9$, the Galaxy spins up and transitions from a turbulent Aurora state to a coherently rotating disc. This transformation is accompanied by many qualitative changes. In particular, we show that high N/O abundances similar to those observed in GN-z11 were common before the spin-up ($\rm [Fe/H]\lesssim -1.5$) when up to $\approx 50~{{\ \rm per\ cent}}-70~{{\ \rm per\ cent}}$ of the in situ stars formed in massive bound clusters. The dramatic drop of fN/O at $\rm [Fe/H]\gtrsim -0.9$ indicates that after the disc emerges the fraction of stars forming in massive bound clusters decreases by two orders of magnitude.

Energy wrinkles and phase-space folds of the last major merger

ABSTRACT Relying on the dramatic increase in the number of stars with full 6D phase-space information provided by the Gaia Data Release 3, we resolve the distribution of the stellar halo around the Sun to uncover signatures of incomplete phase-mixing. We show that, for the stars likely belonging to the last massive merger, the (vr, r) distribution contains a series of long and thin chevron-like overdensities. These phase-space substructures have been predicted to emerge following the dissolution of a satellite, when its tidal debris is given time to wind up, thin out, and fold. Such chevrons have been spotted in external galaxies before; here, we report the first detection in our own Milky Way. We also show that the observed angular momentum Lz distribution appears more prograde at high energies, possibly revealing the original orbital angular momentum of the in-falling galaxy. The energy distribution of the debris is strongly asymmetric with a peak at low E – which, we surmise, may be evidence of the dwarf’s rapid sinking – and riddled with wrinkles and bumps. We demonstrate that similar phase-space and (E, Lz) substructures are present in numerical simulations of galaxy interactions, both in bespoke N-body runs and in cosmological hydrodynamical zoom-in suites. The remnant traces of the progenitor’s disruption and the signatures of the on-going phase-mixing discovered here will not only help to constrain the properties of our Galaxy’s most important interaction, but also can be used as a novel tool to map out the Milky Way’s current gravitational potential and its perturbations.

Motion of Particles around Time Conformal Dilaton Black Holes

In this paper, the geodesic motion of neutral and test particles around the time conformal (TC) Dilaton black hole (BH) is investigated using the eϵg(t) as the time conformal factor in which g(t) is an arbitrary function of time and ϵ is a perturbation parameter. The function g(t) leads to (ta) by utilizing the well-known approximate Noether symmetry (ANS). Furthermore, we discuss the effect of magnetic fields and find the location of stable and unstable orbits w. r. t time, graphically. After that, in the presence and absence of a magnetic field, we interrogate the crucial physical parameters such as effective potential (Ueff), effective force (Feff) and escape velocity (ν⊥). We find the unstable and stable regions of particles for different values of angular momentum (Lz) and magnetic field (B) near the TC Dilaton BH. Moreover, the effects of the Dilaton parameter (μ) on neutral and charged particles are also discussed, which provide some new features. The important results in this study could estimate the powerful relativistic jets originating from the BH.

Fundamental relations for the velocity dispersion of stars in the Milky Way

ABSTRACT We explore the fundamental relations governing the radial and vertical velocity dispersions of stars in the Milky Way, from combined studies of complementary surveys including GALAH, LAMOST, APOGEE, the NASA Kepler and K2 missions, and Gaia DR2. We find that different stellar samples, even though they target different tracer populations and employ a variety of age estimation techniques, follow the same set of fundamental relations. We provide the clearest evidence to date that, in addition to the well-known dependence on stellar age, the velocity dispersions of stars depend on orbital angular momentum Lz, metallicity, and height above the plane |z|, and are well described by a multiplicatively separable functional form. The dispersions have a power-law dependence on age with exponents of 0.441 ± 0.007 and 0.251 ± 0.006 for σz and σR, respectively, and the power law is valid even for the oldest stars. For the solar neighbourhood stars, the apparent break in the power law for older stars, as seen in previous studies, is due to the anticorrelation of Lz with age. The dispersions decrease with increasing Lz until we reach the Sun’s orbital angular momentum, after which σz increases (implying flaring in the outer disc) while σR flattens. For a given age, the dispersions increase with decreasing metallicity, suggesting that the dispersions increase with birth radius. The dispersions also increase linearly with |z|. The same set of relations that work in the solar neighbourhood also work for stars between 3 < R/kpc < 20. Finally, the high-[α/Fe] stars follow the same relations as the low-[α/Fe] stars.

The influence of the spiral arm parameters on radial migration in late-type-like galaxies

ABSTRACT We performed test-particle simulations of a spiral late-type-like galaxy to study the radial migration in a steady potential. Our model includes a bulge, a disc, a dark matter halo, and spiral arms represented by a cosine potential. We varied the structural and dynamical parameters of the spiral arms to study the influence of each configuration on the stellar orbits, such as the pitch angle, the radial scale length, and the pattern speed. We analysed the changes in the z component of the angular momentum (Lz), radius (R), and eccentricity of individual stars at different times. We found that the stellar orbits are affected by radial migration in all cases. Also, we found that the most relevant parameter to the radial migration phenomenon, in co-rotation, is the radial scale length of the spiral arms.

N-body Self-consistent Stellar-halo Modeling of the Fornax Dwarf Galaxy

Abstract We present nearly self-consistent stellar-halo models of the Fornax dwarf spheroidal galaxy associated with the Milky Way galaxy. Such galaxies are dominated by dark matter and have almost no gas in the system. Therefore, they are excellent objects for N-body modeling that takes into account visible and dark matter halo components. In order to model the dark matter halo inferred from the analysis of the measured velocities of Fornax's stars, we constructed several self-consistent quasi-equilibrium models based on two source code sets. One of them (GalactICS Software, NEMO) deals with the self-consistent distribution function modeling that depends on the energy E and vertical component of the angular momentum L z . The other is included in the AGAMA framework and is based on Schwarzschild's calculation of orbits. It can reproduce the nonspherical self-consistent structure of Fornax as the weighted sum of orbit contributions to the galactic density even though the inferred dark halo parameters come from Jeans analysis, which does not require that any distribution functions be positive. To guess the parameters that make the N-body models close to the visible object, we use the stellar dark matter model of the Fornax galaxy based on hydrodynamic axisymmetric Jeans equations taking into account the velocity anisotropy parameter. Then, we studied the evolution of the models by performing N-body simulations with the falcON code in order to test their stability. The variability of the model parameters over time was obtained during simulations. The AGAMA models show the best agreement of the resulting velocity dispersion profiles with the observed data.

Theoretical Study of Dynamic Stark-Induced π-Electron Rotations in Low-Symmetry Aromatic Ring Molecules beyond the Frozen Nuclear Approximation.

The effects of vibrational motions on dynamic Stark-induced π-electron rotations in a low-symmetry aromatic ring molecule are theoretically studied in the adiabatic approximation. We adopt a simplified three-electronic state model with a few vibronic states. A pair of the lowest vibronic states in two electronic excited states is set degenerate by irradiation of two linearly polarized UV lasers. The resultant degenerate state is named the dynamic Stark-induced degenerate vibronic state (DSIDVS). The laser parameters (intensities and central frequencies) are determined under the conditions of DSIDVS formation. The aromatic ring molecules of interest are supposed to belong to the weak coupling case. The analytical expressions for the DSIDVS and coherent angular momentum LZ(t) are derived in the displaced harmonic oscillator (DHO) model. Two horizontal potential displacements (δα, δβ) between the two electronic excited states (α and β) and the ground state are the parameters in the DHO model. The LZ(t) calculated with δα = δβ is characterized by a regular sequence of the angular momentum pulses with a positive (or negative) constant. For a more general case with δα ≠ δβ, the regular sequence is broken down because of the contribution of the first excited vibronic state in each electronic state to LZ(t).

Exploring the Galactic Warp through Asymmetries in the Kinematics of the Galactic Disk

Abstract Previous analyses of large databases of Milky Way stars have revealed the stellar disk of our Galaxy to be warped and that this imparts a strong signature on the kinematics of stars beyond the solar neighborhood. However, due to the limitation of accurate distance estimates, many attempts to explore the extent of these Galactic features have generally been restricted to a volume near the Sun. By combining the Gaia DR2 astrometric solution, StarHorse distances, and stellar abundances from the APOGEE survey, we present the most detailed and radially expansive study yet of the vertical and radial motions of stars in the Galactic disk. We map velocities of stars with respect to their Galactocentric radius, angular momentum, and azimuthal angle and assess their relation to the warp. A decrease in vertical velocity is discovered at Galactocentric radius R = 13 kpc and angular momentum L z = 2800 kpc km s−1. Smaller ripples in vertical and radial velocity are also discovered superposed on the main trend. We also discovered that trends in the vertical velocity with azimuthal angle are not symmetric about the peak, suggesting the warp is lopsided. To explain the global trend in vertical velocity, we built a simple analytical model of the Galactic warp. Our best fit yields a starting radius of and precession rate of . These parameters remain consistent across stellar age groups, a result that supports the notion that the warp is the result of an external, gravitationally induced phenomenon.

Ground state properties of trapped boson system with finite-range Gaussian repulsion: exact diagonalization study

We use exact diagonalization to study an interacting system of N spinless bosons with finite-range Gaussian repulsion, confined in a quasi-two-dimensional harmonic trap with and without an introduced rotation. The diagonalization of the Hamiltonian matrix using Davidson algorithm in subspaces of quantized total angular momentum Lz is carried out to obtain the N-body lowest eigenenergy and eigenstate. To bring out the effect of quantum (Bose) statistics and consequent phase stiffness (rigidity) of the variationally obtained many-body wavefunction on various physical quantities, our study spans from few-body (N = 2) to many-body (N = 16) systems. Further, to examine the finite-range effect of the repulsive Gaussian potential on many-body ground state properties of the Bose-condensate, we obtain the lowest eigenstate, the critical angular velocity of single vortex state and the quantum correlation (measured) in terms of von Neumann entanglement entropy and degree of condensation. It is found that for small values of the range (measured by the parameter σ) of Gaussian potential, the ground state energy increases for few-boson (2 ⩽ N ⩽ 8) systems but decreases for many-boson (N > 8) systems. On the other hand for relatively large values of the range of Gaussian potential, the ground state energy exhibits a monotonic decrease, regardless of the number of bosons N. For a given N, there is found an optimal value of the range of Gaussian potential for which the first vortex (with Lz = N) nucleates at a lower value of the rotational angular velocity Ωc1 compared to the zero-range (δ-function) potential. Further, we observe that the inter-particle interaction and the introduced rotation are competing effects with latter being dominant over the former. With increase in the range of Gaussian potential, the value of von Neumann entropy decreases and the degree of condensation increases implying an enhanced quantum correlation and phase rigidity.

The classical Lenz vector and the two-dimensional quantum harmonic oscillator

Both the two-dimensional harmonic oscillator and the Newton potential allow particular solutions for the orbits which are ellipses with center of attraction in the center, in the first case, and in one focus, in the second. The same complex map which allows to go from Kepler’s to Hooke’s orbits, and back, is used to transform the Lenz vector, defined for the Kepler orbit, into two conserved quantities for the harmonic motion. Upon quantization, the resulting operators, together with the angular momentum Lz, are found to correspond to the generators of the SU(2) internal symmetry of the two-dimensional quantum oscillator and the connection to the Schwinger model of angular momentum is made apparent. We give a self-contained new look on this topic.

Angular momentum in the fractional quantum Hall effect

Suppose a classical electron is confined to move in the xy plane under the influence of a constant magnetic field in the positive z direction. It then traverses a circular orbit with a fixed positive angular momentum Lz with respect to the center of its orbit. It is an underappreciated fact that the quantum wave functions of electrons in the ground state (the so-called lowest Landau level) have an azimuthal dependence ∝ exp (−imϕ) with m≥0, seemingly in contradiction with the classical electron having positive angular momentum. We show here that the gauge-independent meaning of that quantum number m is not angular momentum, but that it quantizes the distance of the center of the electron's orbit from the origin, and that the physical angular momentum of the electron is positive and independent of m in the lowest Landau levels. We note that some textbooks and some of the original literature on the fractional quantum Hall effect do find wave functions that have the seemingly correct azimuthal form ∝ exp (+imϕ) but only on account of changing a sign (e.g., by confusing different conventions) somewhere on the way to that result.

On the Chemical and Kinematic Consistency between N-rich Metal-poor Field Stars and Enriched Populations in Globular Clusters

Abstract Interesting chemically peculiar field stars may reflect their stellar evolution history and their possible origin in a different environment from where they are found now; this is one of the most important research fields in Galactic archeology. To explore this further, we have used the CN–CH bands around 4000 Å to identify N-rich metal-poor field stars in LAMOST DR3. Here we expand our N-rich, metal-poor field star sample to ∼100 stars in LAMOST DR5, where 53 of them are newly found in this work. We investigate light elements of common stars between our sample and APOGEE DR14. While Mg, Al, and Si abundances generally agree with the hypothesis that N-rich metal-poor field stars come from enriched populations in globular clusters, it is still inconclusive for C, N, and O. After integrating the orbits of our N-rich field stars and a control sample of normal metal-poor field stars, we find that N-rich field stars have different orbital parameter distributions compared to the control sample—specifically, apocentric distances, maximum vertical amplitude (Z max), orbital energy, and z-direction angular momentum (Lz ). The orbital parameters of N-rich field stars indicate that most of them are inner-halo stars. The kinematics of N-rich field stars support their possible GC origin. The spatial and velocity distributions of our bona fide N-rich field star sample are important observational evidence to constrain simulations of the origin of these interesting objects.

More than just a wrinkle: a wave-like pattern in Ug versus Lz from Gaia data

ABSTRACT We present a newly found wave-like pattern in mean Galactocentric radial velocity ${\overline{U}_\mathrm{ g}}$ versus guiding centre radius Rg or angular momentum Lz of stars in the RV subsample of Gaia DR2. The short-wave pattern has a wavelength of order $1.2 \, {\rm kpc}$ in Rg or $285 \, {\rm kpc}\, {\rm km}\, {\rm s}^{-1}$ in Lz. The pattern shows only weak changes with Galactocentric radius R and little change in strength in particular with the vertical energy Ez of the stars or the distance to the Galactic plane |z|. The pattern is to first order symmetric around the plane, i.e. has no significant odd terms in z. There is a weak phase shift with the pattern moving towards slightly lower Lz (i.e. trailing) with |z| and Ez. However, we observe a highly significant phase shift in Galactic azimuth ϕ, which is different for different peaks. The peak around $L_{z}\sim 2100 \, {\rm kpc}\, {\rm km}\, {\rm s}^{-1}$ only shows a weak change with ϕ, while the rest of the pattern shows a clearly detectable shift of $\text{d} L_{z}/\text{d} \phi = (200 \pm 22) \, {\rm kpc}\, {\rm km}\, {\rm s}^{-1}\, {\rm rad}^{-1}$. If we consider all peaks to belong to the same pattern, this would suggest a wavenumber m = 4. We further find that the wave-like pattern in Ug appears to be related to the ${\overline{W}}$ versus Lz pattern detected in Gaia DR1. A comparison of the ${\overline{U}_\mathrm{ g}}\!-\!L_{z}$ wave pattern with changes of ${\overline{U}_\mathrm{ g}}$ versus R, which have been previously discussed, suggests that the latter can be understood as just the ${\overline{U}_\mathrm{ g}}\!-\!L_{z}$ pattern washed out by blurring (i.e. orbital excursions around their guiding centre) of disc stars.

Ground state of spin-2 dipolar rotating Bose–Einstein condensates

We study the ground state of spin-2 dipolar rotating Bose–Einstein condensates in a harmonic potential. As the strength of dipolar interaction enhances, our results show that the vortex number becomes smaller and smaller at a fixed component. In addition, the vortex stripe occurs and its number increases. The components m[Formula: see text]1 or ±2 display the same density and the total density shows layered distribution. Increasing the rotational frequency to a higher value, the vortex number and vortex row number increase. We also plot the dependence of angular momentum Lz as a function of dipole strength c[Formula: see text], showing that Lz decreases with increasing c[Formula: see text].

Constraining Solar Position and Velocity with a nearby Hypervelocity Star

Abstract Gravitational three-body interaction among binary stars and the supermassive black hole at the center of the Milky Way occasionally ejects a hypervelocity star (HVS) with a velocity of ∼ 1000 km s − 1 . Due to the ejection location, such an HVS initially has negligible azimuthal angular momentum L z ≃ 0 kpc km s − 1 . Even if the halo is mildly triaxial, L z of a recently ejected nearby HVS remains negligible, since its flight time from the Galactic center is too short to accumulate noticeable torque. However, if we make a wrong assumption about the solar position and velocity, such an HVS would apparently have noticeable nonzero azimuthal angular momentum, due to the wrong reflex motion of the Sun. Conversely, with precise astrometric data for a nearby HVS, we can measure the solar position and velocity by assuming that the HVS has zero azimuthal angular momentum. Based on this idea, here we propose a method to estimate the Galactocentric distance of the Sun R 0 and the Galactocentric solar azimuthal velocity V ⊙ by using an HVS. We demonstrate with mock data for a nearby HVS candidate that the Gaia astrometric data, along with the currently available constraint on V ⊙/R 0 from the proper motion measurement of Sgr A*, can constrain R 0 and V ⊙ with uncertainties of ∼0.27 kpc and ∼7.8 km s−1 (or fractional uncertainties of 3%), respectively. Our method will be a promising tool to constrain (R 0, V ⊙), given that Gaia is expected to discover many nearby HVSs in the near future.

Locality and nonlocality in the interaction-free measurement

We present a paradox involving a particle and a mirror. They exchange a nonlocal quantity, modular angular momentum Lz mod 2ћ, but there seems to be no local interaction between them that allows such an exchange. We demonstrate that the particle and mirror do interact locally via a weak local current 〈Lz mod 2ћ〉w. In this sense, we transform the “interaction-free measurement” of Elitzur and Vaidman, in which two local quantities (the positions of a photon and a bomb in the two arms of a Mach-Zehnder interferometer) interact nonlocally, into a thought experiment in which two nonlocal quantities (the weak modular angular momentum of the particle and of the mirror) interact locally.

Dynamics of a Rydberg hydrogen atom near a topologically insulating surface

We investigate the classical dynamics of a Rydberg hydrogen atom near the surface of a planar topological insulator. The system is described by a Hamiltonian consisting of the free-hydrogen part and the hydrogen-surface potential. The latter includes the interactions between the electron and both image electric charges and image magnetic monopoles. Owing to the axial symmetry, the z component of the angular momentum lz is conserved. Here we consider the case. The structure of the phase space is explored extensively by means of numerical techniques and Poincaré surfaces of section for the recently discovered topological insulator TlBiSe2. The phase space of the system is separated into regions of vibrational and rotational motion. We show that vibrational-rotational-vibrational–type transitions can be tuned with the topological magnetoelectric polarizability.

Vortex patterns in moderately rotating Bose-condensed gas

Using exact diagonalization, we investigate the many-body ground state for regular vortex patterns in a rotating Bose-condensed gas of N spinless particles, confined in a quasi-two-dimensional harmonic trap and interacting repulsively via finite-range Gaussian potential. The N-body Hamiltonian matrix is diagonalized in given subspaces of quantized total angular momentum Lz, to obtain the lowest-energy eigenstate. Further, the internal structure of these eigenstates is analyzed by calculating the corresponding conditional probability distribution. Specifically, the quantum mechanically stable as well as unstable states in a co-rotating frame are examined in the moderately rotating regime corresponding to angular momenta for N = 16 bosons. In response to externally impressed rotation, the patterns of singly quantized vortices are formed, shaping into canonical polygons with a central vortex at the trap center. The internal structure of unstable states reveals the mechanism of entry, nucleation and pattern formation of vortices with structural phase transition, as the condensate goes from one stable vortical state to the other. The stable polygonal vortex patterns having discrete p-fold rotational symmetry with p = 5 and p = 6 are observed. The hexagonal vortex pattern with p = 6 symmetry is a precursor to the triangular vortex lattice of singly quantized vortices in the thermodynamic limit. For unstable states, quantum melting of vortex patterns due to uncertainty in positions of individual vortices, is also briefly discussed.

Exploring the nature of orbits in a galactic model with a massive nucleus

In the present article, we use an axially symmetric galactic gravitational model with a disk–halo and a spherical nucleus, in order to investigate the transition from regular to chaotic motion for stars moving in the meridian (r,z) plane. We study in detail the transition from regular to chaotic motion, in two different cases: the time independent model and the time evolving model. In both cases, we explored all the available range regarding the values of the main involved parameters of the dynamical system. In the time dependent model, we follow the evolution of orbits as the galaxy develops a dense and massive nucleus in its core, as mass is transported exponentially from the disk to the galactic center. We apply the classical method of the Poincaré (r,pr) phase plane, in order to distinguish between ordered and chaotic motion. The Lyapunov Characteristic Exponent is used, to make an estimation of the degree of chaos in our galactic model and also to help us to study the time dependent model. In addition, we construct some numerical diagrams in which we present the correlations between the main parameters of our galactic model. Our numerical calculations indicate, that stars with values of angular momentum Lz less than or equal to a critical value Lzc, moving near to the galactic plane, are scattered to the halo upon encountering the nuclear region and subsequently display chaotic motion. A linear relationship exists between the critical value of the angular momentum Lzc and the mass of the nucleus Mn. Furthermore, the extent of the chaotic region increases as the value of the mass of the nucleus increases. Moreover, our simulations indicate that the degree of chaos increases linearly, as the mass of the nucleus increases. A comparison is made between the critical value Lzc and the circular angular momentum Lz0 at different distances from the galactic center. In the time dependent model, there are orbits that change their orbital character from regular to chaotic and vise versa and also orbits that maintain their character during the galactic evolution. These results strongly indicate that the ordered or chaotic nature of orbits, depends on the presence of massive objects in the galactic cores of the galaxies. Our results suggest, that for disk galaxies with massive and prominent nuclei, the low angular momentum stars in the associated central regions of the galaxy, must be in predominantly chaotic orbits. Some theoretical arguments to support the numerically derived outcomes are presented. Comparison with similar previous works is also made.