A generalization of Szebehely's equation for three dimensions

Szebehely's equation for the potential generating a prescribed family of orbits in two dimensions is generalized for three-dimensional orbits. A simultaneous system of first-order linear partial differential equations is derived for the determination of the potential in the three-dimensional case. Solutions of this system are found in several cases including Kepler's problem too.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.This paper is dedicated to Professor Victor Szebehely on the occasion of his 60th birthday.     

A further study of the solution of Poisson's equation for three-dimensional spiral galaxies

We consider a three-dimensional model of spiral arms, which are in the form of logarithmic spirals in the galactic plane and falling off exponentially in the perpendicular direction. We derive an analytic solution of the gravitational potential corresponding to this density perturbation. We discuss the effect of the finite thickness on the spiral pattern and the question of stability of the spiral arms.     

Existence of the solution of Szebehely's equation in three dimensions using a two-parametric family of orbits

The three-dimensional inverse problem is investigated. A quasi-linear system of partial differential equations is derived for the determination of the potential. The solution of this system is studied by a method of differential geometry. A necessary condition for the solution is derived and the determination of the potential is reduced to algebraic equations written in vectorial form. A few examples are also given.     

A general algorithm for the solution of Kepler's equation for elliptic orbits

An efficient algorithm is presented for the solution of Kepler's equationf(E)=E–M–e sinE=0, wheree is the eccentricity,M the mean anomaly andE the eccentric anomaly. This algorithm is based on simple initial approximations that are cubics inM, and an iterative scheme that is a slight generalization of the Newton-Raphson method. Extensive testing of this algorithm has been performed on the UNIVAC 1108 computer. Solutions for 20 000 pairs of values ofe andM show that for single precision (10^–8) 42.0% of the cases require one iteration, 57.8% two and 0.2% three. For double precision (10^–18) one additional iteration is required. Single- and double-precision FORTRAN subroutines are available from the author.     

A classification of galactic collisions for galaxies of unequal dimensions

The earlier work on classification of galactic collisions for identical galaxies is extended to the situations where the two colliding galaxies differ in mass and dimension. We consider collisions between galaxies of masses 10⁷, 10⁹, 10¹¹ and 10¹³ M and let their radii be 0.1, 1, 10 and 100 kpc, respectively. The density distribution in both galaxies is represented by that of a polytrope of indexn=4. The type of collision is determined for head-on collisions for initial relative velocities 500, 1000, 2000 and 3000 km s^–1. The corresponding final velocities are also indicated.     

The evolution of disk galaxies in clusters

We are carrying out a programme to measure the evolution of the stellar and dynamical masses and M/L ratios for a sizeable sample of morphologically-classified disk galaxies in rich galaxy clusters at 0.2 < z < 0.9. Using FORS2 at the VLT we are obtaining rotation curves for the cluster spirals so that their Tully-Fisher relation can be studied as a function of redshift and compared with that of field spirals. We already have rotation curves for ∼ 10 cluster spirals at z = 0.83, and 25 field spirals at lower redshifts and we plan to increase this sample by one order of magnitude. We present here the first results of our study, and discuss the implications of our data in the context of current ideas and models of galaxy formation and evolution. This revised version was published online in September 2006 with corrections to the Cover Date.     

Mass model of spiral galaxies disk

The study of rotation curves is the most powerful tool for determining mass distributions inside spiral galaxies. Since the first trial by Burbidgeet al. (1959), followed by Brandt (1960), few methods have been proposed for the determination of the masses of galaxies (Faber and Gallagher, 1979). Monnet and Simien (1977) recently proposed a method based on a two component model (bulge and disk) which permits to compute the total mass providing that the photometry is well-known. We suggest here a method based on a new simple parametrization of the optical rotation curve. This method indicates disk masses substantially higher than the provious estimates and suggests an exponential increase of theM/L ratio in the disk toward the outer parts of the Galaxies.     

Formation and evolution of disk galaxies

I review several of the current issues in the theory of disk galaxy formation. There is still much to be done, observationally and theoretically, before we can expect to approach an understanding of disk galaxies that is reliable enough to make robust predictions about the high redshift universe. This revised version was published online in August 2006 with corrections to the Cover Date.     

Edge‐on disk galaxies in the SDSS DR6: Fractions of bulgeless and other disk galaxies

The aim of this study is to determine the fractions of different spiral galaxy types, especially bulgeless disks, from a complete and homogeneous sample of 15 127 edge‐on disk galaxies extracted from the sixth data release from the Sloan Digital Sky Survey. The sample is divided in broad morphological classes and sub types consisting of galaxies with bulges, intermediate types and galaxies which appear bulgeless. A small fraction of disky irregulars is also detected. The morphological separation is based on automated classification criteria which resemble the bulge sizes and the flatness of the disks. Each of these broad classes contains about 1/3 of the total sample. Using strict criteria for selecting pure bulgeless galaxies leads to a fraction of 15% of simple disk galaxies. We compare this fraction to other galaxy catalogs and find an excellent agreement of the observed frequency of bulgeless galaxies. Although the fraction of simple disk galaxies in this study does not represent a “cosmic” fraction of bulgeless galaxies, it shows that the relative abundance of pure disks is comparable to other studies and offers a profound value of the frequency of simple disks in the local Universe. This fraction of simple disks emphasizes the challenge for formation and evolution models of disk galaxies since these models are hard pressed to explain the observed frequency of these objects (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Potential energy of gravitationally interacting disk galaxies

Methods are developed for analysing the gravitational properties of disks having circularly symmetric distribution of matter. It is shown how this can be conveniently done by assuming that the surface density distribution may be approximated by a polynomial in ascending powers of the distance from the centre of the configuration. A theory has been developed to determine the gravitational potential of a single disk at any point in space in terms of the coefficients of the polynomial defining the surface distribution of matter, and the potential energy of two disks of arbitrary separation and orientation due to their mutual gravitational attraction. The basic functions, required for obtaining the potential in the plane of the disk and the mutual potential energy of two coplanar disks, have been tabulated. Two overlapping coplanar disks attract just like mass-points at a certain separation,r _c , of their centres. The force of attraction of disks is less than the force of attraction of mass-points having masses equal to the masses of the disks, if the separation of the centres is less thanr _c , and greater if the separation is greater thanr _c . For typical galaxies of equal radiiR,r _c ≈R.     

Spatial orientation of the spin vectors of the disk galaxies in the Virgo cluster I. A catalogue of the disk galaxies in the Virgo cluster

Based on four UK Schmidt Telescope (UKST) IIIa-j direct plates which had been digitized by the Automated Plate Measuring System (APM), a detailed identification of the disk galaxies in the Virgo cluster has been carried out. The certain and possible member galaxies are mainly selected from the updated Catalogue of 2096 Galaxies in the Virgo Cluster Area (VCC). The area of this identification covers a 85 square degree region centered on this cluster. As a result, we compiled a new catalogue of 340 certain and possible member disk galaxies, including 85 lenticulars (SO), 136 spirals (S) and 119 irregulars (Irr), where the APM-measured values of the position angle (P.A.) and diameters of 299 disk galaxies are provided. For at least 149 galaxies, the values of the P.A. and diameters are published for the first time.Comparison between the APM-measured P.A. and diameters of galaxies and those given by the Uppsala General Catalog of Galaxies (UGC) demonstrates the reliability of our data. The statistical properties and completeness of this database are also discussed. Compared with the UGC-given P.A. and diameters which were determined by eyes, our data has no selection effect owing to the instrument measurements. Within our searching region, this might be the largest and relatively complete photometric database published so far for investigating the spatial orientation of the spin vectors of galaxies in the Virgo cluster.     

Recent investigations on disk galaxies in massive halos

Numerical experiments undertaken to investigate the longevity and behavior of dark-lane elliptical galaxies are described. This is dynamically the same problem as a disk galaxy in a massive halo. Spiral galaxies are disks from a dynamical point of view. A disk of particles embedded in a self-consistent galaxy provides the basic model used for the experiments. This model is applicable to ordinary disk galaxies if the disk is interpreted as the visible galaxy and the galaxy is interpreted as the massive halo thought to be present around disk galaxies. Fully three-dimensional fully self-consistentn-body computer programs that can handle 100,000 particles are used for the experiments. The background galaxy is oblate, and the disk is inclined to the axis of the oblate galaxy, so the disk precesses differentially to produce a warp. A surprising result is that the galaxy center shifted, leaving the disk center orbiting around the galaxy center. This produces interesting phenomena reminiscent of observations in the region of the Galactic center.     

A fluid dynamical flow model for the central peak in the rotation curve of disk galaxies

The rotation curve of the central region in some disk galaxies shows a linear rise, terminating at a peak (primary peak) which is then followed by a deep minimum. The curve then again rises to another peak at more or less half-way across the galactic radius. This latter peak is considered as the peak of the rotation curve in all large-scale analysis of galactic structure. The primary peak is usually ignored for the purpose. In this work an attempt has been made to look at the primary peak as the manifestation of the post-explosion flow pattern of gas in the deep central region of galaxies. Solving hydrodynamical equations of motion, a flow model has been derived which imitates very closely the actually observed linear rotational velocity, followed by the falling branch of the curve to minimum. The theoretical flow model has been compared with observed results for nine galaxies. The agreement obtained is extremely encouraging. The distance of the primary peak from the galactic centre has been shown to be correlated with the angular velocity in the linear part of the rotation curve. Here also, agreement is very good between theoretical and observed results. It is concluded that the distance of the primary peak from the centre not only speaks of the time that has elapsed since the explosion occurred in the nucleus, it also speaks of the potential capability of the nucleus of the galaxy for repeating explosions through some efficient process of mass replenishment at the core.     

Expansion of the stellar subsystems of disk galaxies

Based on archival Hubble Space Telescope images, we have performed stellar photometry for eight edge-on spiral and irregular galaxies. We have identified stars with ages of 20, 50, 80, 160, and 500 Myr in the derived Hertzsprung-Russell diagrams and constructed their number density distributions perpendicularly to the plane of the galactic disk. We have determined the sizes of the stellar subsystems and constructed the size-age diagrams for the stars constituting these subsystems. The stellar subsystems have been found to expand in all of the investigated galaxies within the range of ages studied (from 20 to 500 Myr). The expansion velocity of the subsystems decreases as one recedes from the galactic plane. The subsystems with ages of 1.5 and 6 Gyr also exhibit an increase in their sizes with age. The sizes of these subsystems approach those of the thick disk consisting of red giants. Our results confirm the model of thick-disk formation in irregular and low-mass spiral galaxies through thin-disk expansion.     

Kinematics and stellar disk modeling of lenticular galaxies

We present the results of spectroscopic observations of three S0-Sa galaxies: NGC 338, NGC 3245, and NGC 5440 at the SAO RAS 6-m BTA telescope. The radial distributions of the line-ofsight velocities and radial velocity dispersions of stars and ionized gas were obtained, and rotation curves of galaxies were computed. We construct the numerical dynamic N-body galaxy models with N ?? 10⁶ points. The models include three components: a ??live?? bulge, a collisionless disk, dynamically evolving to the marginally stable state, and a pseudo-isothermal dark halo. The estimates of radial velocities and velocity dispersions of stars obtained from observations are compared with model estimates, projected onto the line of sight. We show that the disks of NGC 5440 and the outer regions of NGC 338 are dynamically overheated. Taking into account the previously obtained observations, we conclude that the dynamic heating of the disk is present in a large number of early-type disk galaxies, and it seems to ensue from the external effects. The estimates of the disk mass and relative mass of the dark halo are given, as well as the disk mass-to-luminosity ratio for seven galaxies, observed at the BTA.     

Mass accretion by the nuclei of disk galaxies

In this work a model has been proposed to explain how the nucleus of a Galaxy can accumulate mass and becomes supermassive — ultimately giving way to gravitational instability leading to an explosion in the nucleus. The process may be repeated many times during the life-span of a Galaxy. The mass shed by the evolved stars populating the central region of the Galaxy can be attracted toward the nuclear core by gravitational pull. Since the incident gas possesses rotational velocity, the centrifugal repulsion of the gas may balance the gravitational pull of the nucleus; thus infall of mass into the nucleus will ordinarily be inhibited. But dissipative agents — such as the prevailing magnetic field and the viscosity of gas — may be sufficient to destroy the rotational velocity of the incident gas and keep the accretion process efficient. The correlation between rotational velocity of gas and its distance from the centre of the Galaxy has been deduced. The radial equation of motion has been solved and the time-scale during which the nucleus accumulates mass sufficient for explosion, has been estimated.     

Evolution of disk galaxies regulated by supernova remnants

Assuming that a disk galaxy is composed of an ambient pervasive gas, small clouds, molecular clouds and stars, its evolution is studied through examining the interchange processes among them. Main results obtained are: (1) The star formation rate is directed by the formation process of molecular clouds. (2) Depending upon the parameters there may be three or four types of evolution of disk galaxies: the no star formation case, the active in the past and inactive at present star formation case, the burst-like star formation case and the very active in star formation case.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan between 30 September–6 October, 1984.     

Dynamics of disk galaxies (invited review)

Astrophysics and Space Science - In order to detect the gravitational effect of the dark corona component of disk galaxies, it is necessary to have surface photometry and rotation data that extend...