Z-component Of Angular Momentum Research Articles

Orbits of globular clusters computed with dynamical friction in the Galactic anisotropic velocity dispersion field

ABSTRACT We present a preliminary analysis of the effect of dynamical friction on the orbits of part of the globular clusters in our Galaxy. Our study considers an anisotropic velocity dispersion field approximated using the results of studies in the literature. An axisymmetric Galactic model with mass components consisting of a disc, a bulge, and a dark halo is employed in the computations. We provide a method to compute the dynamical friction acceleration in ellipsoidal, oblate, and prolate velocity distribution functions with similar density in velocity space. Orbital properties, such as mean time-variations of perigalactic and apogalactic distances, energy, and z-component of angular momentum, are obtained for globular clusters lying in the Galactic region R ≲ 10 kpc, |z| ≲ 5 kpc, with R, z cylindrical coordinates. These include clusters in prograde and retrograde orbital motion. Several clusters are strongly affected by dynamical friction, in particular Liller 1, Terzan 4, Terzan 5, NGC 6440, and NGC 6553, which lie in the Galactic inner region. We comment on the more relevant implications of our results on the dynamics of Galactic globular clusters, such as their possible misclassification between the categories ‘halo’, ‘bulge’, and ‘thick disc’, the resulting biasing of globular-cluster samples, the possible incorrect association of the globulars with their parent dwarf galaxies for accretion events, and the possible formation of ‘nuclear star clusters’.

THE LOSS-CONE PROBLEM IN AXISYMMETRIC NUCLEI

We consider the problem of consumption of stars by a supermassive black hole (SBH) at the center of an axisymmetric galaxy. Inside the SBH sphere of influence, motion of stars in the mean field is regular and can be described analytically in terms of three integrals of motion: the energy E, the z-component of angular momentum L_z, and the secular Hamiltonian H. There exist two classes of orbits, tubes and saucers; saucers occupy the low-angular-momentum parts of phase space and their fraction is proportional to the degree of flattening of the nucleus. Perturbations due to gravitational encounters lead to diffusion of stars in integral space, which can be described using the Fokker-Planck equation. We calculate the diffusion coefficients and solve this equation in the two-dimensional phase space (L_z, H), for various values of the capture radius and the degree of flattening. Capture rates are found to be modestly higher than in the spherical case, up to a factor of a few, and most captures take place from saucer orbits. We also carry out a set of collisional N-body simulations to confirm the predictions of the Fokker-Planck models. We discuss the implications of our results for rates of tidal disruption and capture in the Milky Way and external galaxies.

Long-time limit behavior of the solution to an atom's master equation

We study the long-time limit behavior of the solution to an atom's master equation. For the first time we derive that the probability of the atom being in the α-th (α = j + 1 − jz, j is the angular momentum quantum number, jz is the z-component of angular momentum) state is {(1 − K/G)/[1 − (K/G)2j+1]}(K/G)α−1 as t → +∞, which coincides with the fact that when K/G > 1, the larger the α is, the larger the probability of the atom being in the α-th state (the lower excited state) is. We also consider the case for some possible generalizations of the atomic master equation.

A new mass model describing motion in axially symmetric galaxies with dark matter

We present a new analytical model in order to describe motion in axially symmetric galaxies containing dark matter. The model has a finite mass and can describe motion in disc as well as in elliptical galaxies. The advantage of this new model is an additional parameter that characterizes the percentage of dark matter in the system. This fractional portion of dark matter affects some basic physical quantities of the galaxy, such as the rotational velocity, the mass density and the vertical force. Adding to this model a dense spherical nucleus, we obtain a composite mass model, which can be used to describe motion in galaxies with central mass concentrations, where, for low values of the star’s z-component of angular momentum, regular motion together with large chaotic regions in the r–pr phase plane is observed. We find that the presence of the dark matter causes an increase in the rotational velocity, the mass density and the vertical force, both in the discussed disc model and in the oblate elliptical galaxy model. The significant difference is that the increase in the elliptical galaxy model is larger than that observed in the disc model. On the contrary, the fraction A per cent in the r–pr phase plane covered by chaotic orbits, which is measured empirically, for the same fractional portion of dark matter, is much larger in the disc model than that found in the elliptical model. This result may depend on the particular way the dark matter is built-in into the model. The numerical experiments indicate that the distribution of dark matter in our model is such that the amount of dark matter increases as the distance from the centre increases, which seems to be in agreement with the observational data.

Dynamical evolution of rotating globular clusters with embedded black holes

AbstractEvolution of rotating globular clusters with embedded black holes is presented. The interplay between velocity diffusion due to relaxation and black hole star accretion is followed together with cluster rotation, using 2-dimensional, in energy and z-component of angular momentum, Fokker Planck numerical methods. Gravogyro and gravothermal instabilities drive the system to a faster evolution leading to shorter collapse times and a faster cluster dissolution in the tidal field of a parent galaxy.Angular momentum transport and star accretion support the development of central rotation in relaxation time scales. Two-dimensional distribution (in the meridional plane) of kinematical and structural parameters (density, dispersions, rotation) are reproduced, with the aim to enable the use of set of models for comparison with observational data.

Instabilities of a family of oblate stellar spheroids

We have examined the stability of a sequence of oblate elliptical galaxy models having the Stackel form suggested by Kuz'min & Kutuzov. We have employed the 2-integral DFs given by Dejonghe & de Zeeuw for which flattened non-rotating models are characterized by counter-streaming motion and are radially cool; we introduce net rotation in some models by changing the sign of the z-component of angular momentum for a fraction of the particles. We have found that all non-rotating and slowly rotating members of this sequence rounder than E7 are stable, and that even maximally rotating models rounder than E4 are stable. In the absence of strong rotation, the most disruptive instability, and the last to be stabilized by increasing thickness, is a lopsided (m=1) mode. This instability appears to be driven by counter-rotation in radially cool models. Its vigour is lessened as rotation is increased, but it remains strong even in models with net angular momentum 90% of that of a maximally rotating model before finally disappearing in maximally rotating models. Strongly rotating models are more unstable to bar-forming modes which afflict maximally rotating models with c/a >~ 0.5, but this mode is quickly stabilized by moderate fractions of counter-rotating particles. Bending instabilities appear not to be very important; they are detectable in the inner parts of the flatter models, but are less vigorous and more easily stabilized than the lop-sided or bar modes in every case. We briefly discuss the possible relevance of the lop-sided instability to the existence of many lop-sided disk galaxies.

Some comments on the electrostatic potential of a molecule

In this article we discuss several principles and tools which should expedite description of the electrostatic potentials and electrostatic interactions of molecules, and show that these also lead to some rather remarkable results in the theory of the irreducible representations of the full rotation group SO(3). First, by representing a molecule's charge-density matrix over a basis of atomic-like orbitals (on the various atoms), we observe that outside its charge distribution the molecule's electrostatic potential is exactly the same as if that charge distribution were merely a sum (and in the case of a finite orbital basis, this is a finite sum) of point multipoles on each of the atomic centers and line multipoles on the line segments joining each of those atomic centers. Possible methods of approximating the field of these line charges and line multipoles, as if they were due to point charges and point multipoles, are discussed. The calculation of the interaction of point multipoles of high order, as is necessary for this procedure to successfully calculate the interaction of arbitrarily oriented molecules, motivates our second topic. Here we present a differential operator which, when acting on the 3-dimensional delta function, produces the source density for a scalar field that is exactly an (l,m) multipole field. Using the Hermitian adjoint of this operator, we express the interaction of this (l,m) multipole with an external scalar field as the result of this differential operator acting on that external field at the location of this multipole source. Irreducible representation matrices of the full rotation group are then used, together with these relations, to simplify the interaction of two arbitrarily oriented multipoles of any orders. Finally, we use the representation of the Condon and Shortley “raising and lowering” relations on eigenstates of the z-component of angular momentum, in an orientation that is not aligned with its fundamental basis states, to generate recursion relations that allow simple calculations of the irreducible representation matrices of the full rotation group, SO(3), and the special unitary group, SU(2). From these recursion relations we display some useful symmetry properties of our parameterization of these matrices, that allow the entire matrix to be very simply generated from an explicit calculation of only about 1/8 of its elements. © 1992 John Wiley & Sons, Inc.