Third Integral Of Motion Research Articles

The Nature of the Milky Way’s Stellar Halo Revealed by the Three Integrals of Motion

Abstract We developed a new selection method of halo stars in the phase-space distribution defined by the three integrals of motion in an axisymmetric Galactic potential (E, L z , I 3), where I 3 is the third integral of motion. The method is used to explore the general chemodynamical structure of the halo based on stellar samples from SDSS-SEGUE DR7 and APOGEE DR16, matched with Gaia DR2. We found the following. (a) Halo stars can be separated from disk stars by selecting over (1) 0 < L z < 1500 kpc km s−1, kpc km s−1 (orbital angle θ orb > 15–20 deg), and E < −1.5 × 105 km2 s−2, and (2) L z < 0 kpc km s−1. These selection criteria are free from kinematical biases introduced by the simple high-velocity cuts adopted in recent literature. (b) The averaged, or coarse-grained, halo phase-space distribution shows a monotonic, exponential decrease with increasing E and I 3 like the Michie–Bodenheimer models. (c) The inner stellar halo described in Carollo et al. is found to comprise a combination of Gaia Enceladus debris (GE), lowest-E stars (likely in situ stars), and metal-poor prograde stars missed by the high-velocity cut selection. (d) The very metal-poor outer halo, ([Fe/H] < −2.2), exhibits both retrograde and prograde rotation, with an asymmetric L z distribution toward high retrograde motions and larger θ orb than those possessed by the GE-dominated inner halo. (e) The Sgr dSph galaxy could induce a long-range dynamical effect on local halo stars. Implications for the formation of the stellar halo are also discussed.

Modification of photon trapping orbits as a diagnostic of non-Kerr spacetimes

Photon circular orbits, an extreme case of light deflection, are among the hallmarks of black holes and are known to play a central role in a variety of phenomena related to these extreme objects. The very existence of such orbits when motion is not confined in the equatorial plane, i.e. spherical orbits, is indeed a special property of the separable Kerr metric and may not occur, for instance, in the spacetime of other more speculative ultracompact objects. In this paper we consider a general stationary-axisymmetric spacetime and examine under what circumstances spherical or more general, variable-radius, `spheroidal' non-equatorial photon orbits may exist with the ultimate goal of using the modifications -- or even loss -- of photon trapping orbits as a telltale of non-Kerr physics. In addressing this issue, we first derive a general necessary condition for the existence of spherical/spheroidal orbits and then go on to study photon trapping orbits in a variety of known non-Kerr metrics (Johannsen, Johanssen-Psaltis, and Hartle-Thorne). The first of these is an example of a separable spacetime which supports Kerr-like spherical photon orbits. A more detailed analysis reveals a deeper connection between the presence of spherical orbits and the separability of a metric (that is, the existence of a third integral of motion). Specifically, a spacetime that does not admit spherical orbits in any coordinates is necessarily non-separable. The other two spacetimes considered here exhibit a clear non-Kerr behaviour by having spherical photon orbits replaced by spheroidal ones. More importantly, subject to the degree of deviation from Kerr, equatorial photon rings give place to non-equatorial ones with an accompanying loss of low-inclination spheroidal orbits. The implications of these results for the electromagnetic and gravitational wave signature of non-Kerr objects are briefly discussed.

Vertical stability of circular orbits in relativistic razor-thin disks

During the last few decades, there has been a growing interest in exact solutions of Einstein equations describing razor-thin disks. Despite the progress in the area, the analytical study of geodesic motion crossing the disk plane in these systems is not yet so developed. In the present work, we propose a definite vertical stability criterion for circular equatorial timelike geodesics in static, axially symmetric thin disks, possibly surrounded by other structures preserving axial symmetry. It turns out that the strong energy condition for the disk stress-energy content is sufficient for vertical stability of these orbits. Moreover, adiabatic invariance of the vertical action variable gives us an approximate third integral of motion for oblique orbits which deviate slightly from the equatorial plane. Such new approximate third integral certainly points to a better understanding of the analytical properties of these orbits. The results presented here, derived for static spacetimes, may be a starting point to study the motion around rotating, stationary razor-thin disks. Our results also allow us to conjecture that the strong energy condition should be sufficient to assure transversal stability of periodic orbits for any singular timelike hypersurface, provided it is invariant under the geodesic flow.

Integrability of motion around galactic razor-thin disks

We consider the three-dimensional bounded motion of a test particle around razor-thin disk configurations, by focusing on the adiabatic invariance of the vertical action associated with disk-crossing orbits. We find that it leads to an approximate third integral of motion predicting envelopes of the form $Z(R)\propto[\Sigma(R)]^{-1/3}$, where $R$ is the radial galactocentric coordinate, $Z$ is the z-amplitude (vertical amplitude) of the orbit and $\Sigma$ represents the surface mass density of the thin disk. This third integral, which was previously formulated for the case of flattened 3D configurations, is tested for a variety of trajectories in different thin-disk models.

Stellar kinematics using a third integral of motion: method and application on the Andromeda galaxy

We probe the feasibility of describing the structure of a multi-component axisymmetric galaxy with a dynamical model based on the Jeans equations while taking into account a third integral of motion. We demonstrate that using the third integral in the form derived by G. Kuzmin, it is possible to calculate the stellar kinematics of a galaxy from the Jeans equations by integrating the equations along certain characteristic curves. In cases where the third integral of motion does not describe the system exactly, the derived kinematics would describe the galaxy only approximately. We apply our method to the Andromeda galaxy, for which the mass distribution is relatively firmly known. We are able to reproduce the observed stellar kinematics of the galaxy rather well. The calculated model suggests that the velocity dispersion ratios ${\sigma}_z^2/{\sigma}_R^2$ of M31 decrease with increasing R. Moving away from the galactic plane, ${\sigma}_z^2/{\sigma}_R^2$ remains the same. The velocity dispersions ${\sigma}_{\theta}^2$ and ${\sigma}_R^2$ are roughly equal in the galactic plane.

Quasi integral of motion for axisymmetric potentials

We present an estimate of the third integral of motion for axisymmetric three-dimensional potentials. This estimate is based on a Staeckel approximation and is explicitly written as a function of the potential. We tested this scheme for the Besancon Galactic model and two other disc-halo models and find that orbits of disc stars have an accurately conserved third quasi integral. The accuracy ranges from of 0.1% to 1% for heights varying from z = 0~kpc to z= 6 kpc and Galactocentric radii R from 5 to 15kpc. We also tested the usefulness of this quasi integral in analytic distribution functions of disc stellar populations: we show that the distribution function remains approximately stationary and that it allows to recover the potential and forces by applying Jeans equations to its moments.

A SIMPLE FORMULA FOR THE THIRD INTEGRAL OF MOTION OF DISK-CROSSING STARS IN THE GALAXY

We present an analytical simple formula for an approximated third integral of motion associated with nearly equatorial orbits in the Galaxy: $I_{3}=Z\Sigma_{I}^{1/3}$, where $Z(R)$ is the vertical amplitude of the orbit at galactocentric distance $R$ and $\Sigma_{I}(R)$ is the integrated dynamical surface mass density of the disk, a quantity which has recently become measurable. We also suggest that this relation is valid for disk-crossing orbits in a wide variety of axially symmetric galactic models, which range from razor-thin disks to disks with non-negligible thickness, whether or not the system includes bulges and halos. We apply our formalism to a Miyamoto-Nagai model and to a realistic model for the Milky Way. In both cases, the results provide fits for the shape of nearly equatorial orbits which are better than the corresponding curves obtained by the usual adiabatic approximation when the orbits have vertical amplitudes comparable to the disk's scale height. We also discuss the role of this approximate third integral of motion in modified theories of gravity.

The cross-helicity effect on cascade processes in MHD turbulence

The cross helicity H = 〈 v · b 〉 characterizes the level of correlation between pulsations of the magnetic field b and the velocity field v . In the ideal three-dimensional magnetic hydrodynamics, it is the integral of motion alongside with the total energy E T = E v + E b , E v = , and E b = . The third integral of motion is the magnetic helicity; but within the framework of this study, we consider the fields in which the average magnetic helicity is close to zero and, thus, it is not considered here. The representation about the developed turbulence as a random process makes it possible to expect that, if there are no special reasons, the developed conductingfluid turbulence (the MHD turbulence) should be characterized by a low cross-helicity level. Exactly such a situation is usually considered. Interest in cross helicity arose after highly correlated pulsations of velocity and magnetic field were found in the solar wind [1]. Analysis of the energy and helicity evolution in a freely degenerating MHD turbulence showed that the helicity decays more slowly than the energy; hence, the degree of correlation of fields v and b determined by the correlation coefficient C = H/E T can increase in time for the free degeneration [2]. By itself, MHD turbulence gives the possibility of developing various scenarios. The specificity of the conducting-fluid hydrodynamics is the possibility of occurrence of Alfven waves; it is assumed they play a key role in the turbulent cascade, which leads to the Iroshnikov‐Kraichnan spectral law E b ( k ) ~ E v ( k ) ~ k –3/2 [3, 4]. In the calculations on the mesh ( 512 3 ) [5], it was shown that no Kraichnan‐Iroshnikov spectrum arises in the noncorrelated turbulence without an external field, and the turbulence with the spectrum close to E b ( k ) ~ v 2 2

Exploring disc galaxy dynamics using integral field unit data

In order to test the basic equations believed to dictate the dynamics of disc galaxies, we present and analyse deep two-dimensional spectral data obtained using the PPAK integral field unit for the early-type spiral systems NGC 2273, NGC 2985, NGC 3898 and NGC 5533. We describe the care needed to obtain and process such data to a point where reliable kinematic measurements can be obtained from these observations, and a new more optimal method for deriving the rotational motion and velocity dispersions in such disc systems. The data from NGC 2273 and NGC 2985 show systematic variations in velocity dispersion with azimuth, as one would expect if the shapes of their velocity ellipsoids are significantly anisotropic, while the hotter discs in NGC 3898 and NGC 5533 appear to have fairly isotropic velocity dispersions. Correcting the rotational motion for asymmetric drift using the derived velocity dispersions reproduces the rotation curves inferred from emission lines reasonably well, implying that this correction is quite robust, and that the use of the asymmetric drift equation is valid. NGC 2985 is sufficiently close to face on for the data, combined with the asymmetric drift equation, to determine all three components of the velocity ellipsoid. The principal axes of this velocity ellipsoid are found to be in the ratio σ z : σ o : σ R ≈0.7:0.7:1, which shows unequivocally that this disc distribution function respects a third integral of motion. The ratio is also consistent with the predictions of epicyclic theory, giving some confidence in the application of this approximation to even fairly early-type disc galaxies.

Discrete Schwarzschild Models: Constraining Dark Halos and Black Holes

AbstractWe have developed a code for modeling 3-dimensional kinematical data (i.e., both radial velocity and proper motions) of individual tracers in stellar systems. The code relies on the Schwarzschild method of orbit superposition to find the distribution function (a function of energy, angular momentum, and third integral of motion) that best reproduces all the observations (the positions and velocities of the tracers as well as the overall light distribution) in a given potential. Previous implementations of this method have generally focused on fitting integrated light measurements, instead of data for individual tracers. However, the latter is the only data that are generally available in a large class of problems (e.g., kinematics of tracers in nearby globular clusters, kinematics of planetary nebulae or globular clusters in galaxy halos, etc.). We have performed extensive tests of the fitting process by making use of data generated by a well-tested N-body simulation of an axisymmetric galaxy, quantifying how well the code can recover the known input distribution function and potential.

Rydberg states of the hydrogen atom interacting with a metal surface and a perpendicular magnetic field

The interaction between Rydberg atoms and a metal surface can be described by the instantaneous van der Waals potential for larger atom–surface distances. This interaction breaks the spherical symmetry of free atom and, due to the lack of a third integral of motion, the problem is non-integrable. A similar situation occurs for the hydrogen atom in a constant magnetic field. We show, however, that in a combined case with the magnetic field perpendicular to the surface, this system becomes integrable at a given atom–surface distance for a specific value of the field.

Non-integrability of the anisotropic Stormer problem with angular momentum

We consider the Hamiltonian of the Stormer problem reduced with respect to the rotational symmetry about the magnetic dipole. Considering an anisotropy in the masses, it turns out to be a classical model for the hole-mediated ferromagnetism in Ga 1− x Mn x As. We study the anisotropic Stormer problem from the point of view of the existence of a third integral of motion in involution with the angular momentum p φ and the Hamiltonian H, and that is independent of both. For p φ =0 the anisotropic Stormer potential is homogeneous. Almeida et al. [Rev. Bras. Fı́sica 28 (4) (1998)] found that it is non-integrable for all values of the anisotropy parameter outside the interval [0, 2/3]. In this paper, we are able to extend non-integrability to this interval except for two points, viz., 5/12 and 2/3. For these two values, the integrability remains undecided. If p φ ≠0, the anisotropic Stormer problem is non-integrable for any value of the anisotropy.

Solitons and breathers from the third integral of motion in galaxies

We show how it is possible to derive partial differential equations from the third integral of motion of galactic orbits. An example is given in the case of orbits near the inner Lindblad resonance. In this case, a Sine–Gordon equation is derived. Applying the third integral on a string of stars having as initial conditions the successive consequents of one orbit on a Poincare surface of section, a discrete set of Sine–Gordon equations is derived corresponding to a Frenkel–Kontorova Hamiltonian system. Analytical soliton solutions are derived. In particular, we show that breathers can be formed in galaxies by collections of orbits. A good agreement between analytical and numerical solutions is found. The new formulation opens the possibility that, instead of single orbits, collections of stars forming solitons and breathers, which are more dispersion resistant, can be superimposed to construct non-linear density waves in galaxies. Such collections are more natural building blocks of density waves, as they take into account a more realistic distribution and correlation of phases.

Motion in the field of a rotationally symmetric potential: Exact use of an approximate equation for the derivative of the field of directions along the normal to a trajectory

The previously derived equation (Agekyan 1974) for the derivative ∂f/∂n of the field of directions along the normal to a trajectory is approximate, because differentiating along the normal takes the point out of the orbit and changes the third integral of motion. However, on the envelope of the trajectory, i.e., on the contour of an orbit or a fold, ∂f/∂n undergoes a discontinuity of the second kind. Many authors have used this property to find points of the contours of orbits and folds. Although the integrable equation is approximate, the envelope points are determined accurately.

Stäckel potentials and the relevance of the spheroidal coordinates

We study some consequences of the use of spheroidal coordinate frames to describe axisymmetric gravitational fields. In particular, we shall explain the meaning of the third integral of motion, and we shall analyse torque effects measured in a Stackel potential field.

The third integral of motion

Abstract An analytical expression for a third integral of motion in an rotationally symmetric potential is derived. This expression is constant along trajectories.

Hydrogen atom in a strong magnetic field: calculation of the energy levels by quantising the normal form of the regularised Kepler Hamiltonian

The energy levels of the hydrogen atom in a magnetic field are obtained by quantising the Birkhoff-Gustavson normal form of the diamagnetically perturbed, regularised Kepler Hamiltonian. The method enables the authors to go beyond the approximations quadratic in the field and to calculate quartic and higher terms in the power expansion for the energy levels. (Only the case m=0 is considered). The result is equivalent to finding the field-dependent third integral of motion.

Hydrogen atom in a strong magnetic field: on the existence of the third integral of motion

The problem of the existence of the third integral of motion for the classical Hamiltonian describing a hydrogen atom in a magnetic field is studied by numerical methods. It is found that the third integral is isolating for all initial conditions for which the energy is lower than a critical energy, beyond which the phase orbits are unstable and the Hamilton system can behave stochastically. This critical energy depends upon the strength of the magnetic field and the value of the z component of the angular momentum. The critical energy approaches the (classical) ionisation energy in the weak-field and strong-field limits, while it is lowest in the transition region. The consequences for the quantum mechanical energy spectrum of the hydrogen atom are discussed: the existence of this approximate dynamical symmetry would allow for close anti-crossings of levels, and might facilitate the analytic calculations of the energy levels below the critical energy. In discussion of the correspondence diagram a criticism of an earlier paper is given.

Normal form for mirror machine Hamiltonians

A systematic algorithm is developed for performing canonical transformations on Hamiltonians which govern particle motion in magnetic mirror machines. These transformations are performed in such a way that the new Hamiltonian has a particularly simple normal form. From this form it is possible to compute analytic expressions for gyro and bounce frequencies. In addition, it is possible to obtain arbitrarily high order terms in the adiabatic magnetic moment expansion. The algorithm makes use of Lie series, is an extension of Birkhoff’s normal form method, and has been explicitly implemented by a digital computer programmed to perform the required algebraic manipulations. Application is made to particle motion in a magnetic dipole field and to a simple mirror system. Bounce frequencies and locations of periodic orbits are obtained and compared with numerical computations. Both mirror systems are shown to be insoluble, i.e., trajectories are not confined to analytic hypersurfaces, there is no analytic third integral of motion, and the adiabatic magnetic moment expansion is divergent. It is expected also that the normal form procedure will prove useful in the study of island structure and separatrices associated with periodic orbits, and should facilitate studies of breakdown of adiabaticity and the onset of ’’stochastic’’ behavior.

Particle dynamics in spheroidal mass distributions, the third integral of motion, and angular momentum

Particle dynamics in spheroidal mass distributions, the third integral of motion, and angular momentum