Gravitational potential energy of a disk-sphere pair of galaxies

Algebraic expressions are obtained for the interaction potential energy of a pair of galaxies in which one is disk shaped and the other spherical. The density distribution in the disk galaxy is represented by a polynomial in ascending powers of the distance from the centre of the disk while the density distribution in the spherical galaxy is represented by the superposition of spherical polytropes of integral indices. The basic functions required for obtaining the interaction potential energy of a coplanar disk-sphere pair of galaxies are tabulated. The forces of attraction between a coplanar disk-sphere pair of galaxies are shown graphically for two density models of disk and spherical galaxies. An overlapping coplanar disk-sphere pair of galaxies attract just like two mass-points at a certain separation,r _c, of their centres. The force of attraction is less than that of two mass-points having masses equal to the masses of the two galaxies, if the separation of the centres is less thanr _c, and greater if the separation is greater thanr _c.For a typical coplanar disk-sphere pair of galaxies (the density of the disk is represented by Model II and of the sphere by a polytropic indexn=4) of equal radii, we note the following. At a separation of 0.79R, R being the common radius of the two galaxies, the force of attraction between the pair is the same as if the entire mass of each galaxy is concentrated at its centre. The mass-point model for the two galaxies will overestimate the force of attraction by more than a factor of 10 if the separation is less than 0.36R. For separation greater than the radii of the galaxies the mass-point model will underestimate the force but the departure in this case is less than 33%.     

The formation of faint ring structures

In a previous paper (Chatterjee, 1984), we studied the formation of ring galaxies due to head-on collisions between disk and spherical galaxies. The rings so formed are quite prominent and well defined. However, in the domain of rings, there are many faint rings which are ill defined and are so faint that they lie on the borderline of detectability. In this paper the formation of such rings is investigated.Using the impulsive approximation technique, in this paper we study an off-center collision between an exponential model disk galaxy and a polytropic (n=4) model spherical galaxy, in which the spherical galaxy moves normal to the plane of the disk galaxy and grazes its periphery. The radii of the two galaxies are taken to be equal, but due to the high central concentration of the spherical galaxy its radius is effectively one-third that of the disk galaxy. It is found that, as a result of the collision, the disk galaxy expands and attains an oblate shape. If the fractional change in binding energy of the disk galaxy, U/|U|, lies between 0.001 to 0.01, then a faint ring structure is formed which does not close on itself. If U/|U|0.01, then in addition to the faint ring a bridge and tail are formed. The ability of such low-energy collisions to affect even minor structural changes in the disk is due to the preferentially higher energy changes occurring in the side nearer to the perturbing galaxy than in the opposite side. Consequently, the structural changes produced are asymmetric.Such a faint ring may elude detection on photographs, but it will be revealed by sensitive photometric observations. There are several examples of such faint ring structures embedded in disks, which have been detected by observation.     

The formation of a ring galaxy during a head-on collision between a disk and a spherical galaxy

The tidal force effects of a spherical galaxy passing head-on through a disk galaxy have been studied at various regions of the disk galaxy and for various orientations of the disk galaxy with respect to the direction of relative motion of the two galaxies. The density distribution of the disk galaxy is taken to be, (r)=_ce^–4r/R , where _c is the central density andR is the radius of the disk. The density distribution of the spherical galaxy is taken to be that of a oolytrope of indexn=4. It is found that as a result of the collision, through the central parts and the outer parts of the disk galaxy remain intact, the region in between these two regions disrupts. Thus a ring galaxy with a nucleus embedded in the ring-i.e., a ring galaxy of the RN-type, is formed.     

Determining the character of motion in quiet and active galaxies with a satellite companion

A galaxy model with a satellite companion is used to study the character of motion for stars moving in the x ‐y plane. It is observed that a large part of the phase plane is covered by chaotic orbits. The percentage of chaotic orbits increases when the galaxy has a dense nucleus of massM_n. The presence of the dense nucleus also increases the stellar velocities near the center of the galaxy. For small values of the distance R between the two bodies, low energy stars display a chaotic region near the centre of the galaxy, when the dense nucleus is present, while for larger values of R the motion in active galaxies is regular for low energy stars. Our results suggest that in galaxies with a satellite companion, the chaotic character of motion is not only a result of galactic interaction but also a result caused by the dense nucleus. Theoretical arguments are used to support the numerical outcomes. We follow the evolution of the galaxy, as mass is transported adiabatically from the disk to the nucleus. Our numerical results are in satisfactory agreement with observational data from M51‐type binary galaxies (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The formation of ring galaxies

The conditions under which a head-on collision between a disk galaxy and a spherical galaxy can lead to ring formation are investigated, using the impulsive approximation. The spherical galaxy is modeled as a polytrope of indexn=4 and radiusR _S and the disk galaxy as an exponential disk whose surface density is given by \(\sigma (r) = \sigma _c e^{ - 4r/R_D } \) , where σ _c is the central density andR _D is the radius of the disk. The formation and properties of the rings are closely related to the fractional change in binding energy of the disk galaxy, given by ΔU/?U?=γ _D β _D , where (GM _S ² R _D )/(V ² M _D R _S ² ),M _S andM _D being the masses of the spherical and disk galaxies, respectively, and β _D ≡β _D (n, σ, ?,i) is a function of the models of the two galaxies, the ratio of the radii of the two galaxies ?=R _S /R _D , and the angle of inclinationi, of the disk to the direction of relative motion of the two galaxies. Calculations are made for the caseR _S =R _D . Since practically the entire mass of the spherical galaxy, for the chosen model, lies within 1/3 of its radius, the radius of the spherical galaxy is effectively \(\tfrac{1}{3}\) that of the disk galaxy. It is found that as a result of the collision, the innermost and the outer parts of the disk galaxy are not much affected, but the intermediate region expands and gets evacuated, leading to the crowding of stars in a preferential region forming a ring structure. The rings are best formed for a normal, on-axis collision. For this case, rings form when ΔU/|U| lies between \(\tfrac{1}{2}\) and 2, while they are very sharp and bright when ΔU/|U| lies between \(\tfrac{1}{2}\) and 1. Within this range, as ΔU/|U| increases, the rings become sharper and their positions shift outwards with respect to the centre of the disk galaxy. The relationship $$\gamma _D = 0.0016 + 0.045s_{{\text{max}}}^2 ,$$ wheres _max is the radial distance of the density maximum of the ring from the centre of the disk galaxy (measured in terms of the radius of the disk galaxy as unit) enables us to finds _max from γ _D and vice versa, and interpret some prominent ring galaxies. The effect of introducing a bulge to the disk is to distribute the tidal disruptive effects more evenly and, hence, reduce the sharpness of the ring.     

Tidal disruption of a disk galaxy

The tidal force effects of a spherical galaxy passing head-on through a disk galaxy have been studied for various orientations of the disk galaxy with respect to the direction of relative motion of the two galaxies. The density distribution of the spherical galaxy is taken to be that of a polytrope of indexn=4 and that of the disk galaxy is taken to be, (r)=_ce^–4r/R, where _c is the central density andR the radius of the disk. It is found that the disruptive effects due to the tidal force are minimum when the plane of the disk lies along the direction of relative motion, but are maximum when the plane of the disk is slightly inclined to this direction (about 15°). The tidal force effects at the median radius have also been computed. The tadal force effects are much higher in the interior region of the disk.     

Warped polar ring in the Arp 212 galaxy

The Fabry-Perot scanning interferometer mounted on the 6-m telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences is used to study the distribution and kinematics of ionized gas in the peculiar galaxy Arp 212 (NGC 7625, IIIZw 102). Two kinematically distinct subsystems—the inner disk and outer emission filaments—are found within the optical radius of the galaxy. The first subsystem, at galactocentric distances r < 3.5 kpc, rotates in the plane of the stellar disk. The inner part of the ionized-gas disk (r<1.5–2 kpc) exactly coincides with the previously known disk consisting of molecular gas. The second subsystem of ionized gas is located at galactocentric distances 2–6 kpc. This subsystem rotates in a plane tilted by a significant angle to the stellar disk. The angle of orbital inclination in the outer disk increases with galactocentric distance and reaches 50° at r ≈ 6 kpc. The ionized fraction of the gaseous disk does not show up beyond this galactocentric distance, but we believe that the HI disk continues to warp and approaches the plane that is polar with respect to the inner disk of the galaxy. Hence Arp 212 can be classified as a galaxy with a polar ring (or a polar disk). The observed kinematics of the ionized and neutral gas can be explained assuming that the distribution of gravitational potential in the galaxy is not spherically symmetric. Most probably, the polar ring have formed via accretion of gas from the dwarf satellite galaxy UGC 12549.     

Head-on Collision of Disk–Sphere Galaxies

Numerical simulations are performed to study the tidal effects of non-merging rapid head-on collision between a disk galaxy and a spherical galaxy. The disk consists of three components – a disk, a bulge and a halo – and the spherical galaxy is a Plummer model. The galaxies have the same dimensions with different mass ratios viz., 2, 1 and 0.5. They move in a rectilinear orbit with a relative velocity of 1000 km s⁻¹. None of the simulations leads to the merger of the galaxies by tidal capture. The results of our simulations indicate that although tidal effects are sensitive to both the mass ratio and the inclination of the disk to the orbital plane, it is the mass ratio which is more important in producing tidal damage to the less massive galaxy. The spherical galaxy undergoes considerable tidal effects if the mass of the disk is same or larger. On the other hand the collisions in which the mass of the spherical galaxy is more, result in the formation of a ring structure after the closest approach and the structure disappears by the end of the simulations.     

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.     

Newtonian and post Newtonian expansionfree fluid evolution in f(R) gravity

We consider a collapsing sphere and discuss its evolution under the vanishing expansion scalar in the framework of f(R) gravity. The fluid is assumed to be locally anisotropic which evolves adiabatically. To study the dynamics of the collapsing fluid, Newtonian and post Newtonian regimes are taken into account. The field equations are investigated for a well-known f(R) model of the form R+δR ² admitting Schwarzschild solution. The perturbation scheme is used on the dynamical equations to explore the instability conditions of expansionfree fluid evolution. We conclude that instability conditions depend upon pressure anisotropy, energy density and some constraints arising from this theory.     

The relative orientation of the accretion disk and host galaxy disk in Seyferts

We present the results of the study of the orientation of the accretion disk relative to the host galaxy disk in Seyfert galaxies. We used a sample selected by a mostly isotropic property, the flux at 60 μm, with radio and optical data homogeneously observed and analyzed, to avoid selection effects. We found that the observed i and δ values, galaxy inclination and difference between the position angle of the jet and the galaxy major axis, respectively, are consistent with a random β-distribution, the angle between the jet and the galaxy plane axis. We also found that the previously suggested Zone of Avoidance disappears and was probably due to a selection effect. We suggest several explanations for the misalignment of the accretion disk relative to the galaxy disk.     

Exchange of gravitational energy during a collision of galaxies

The problem of the change in internal energy of a colliding galaxy due to tidal effects is considered, assuming that the galaxies may be regarded as spherical stellar systems whose over-all structure remains unchanged during the collision and that the stars move in circular orbits. The numerical estimates thus made for the energy gained by the stars during the collision are compared with those derived on the basis of the assumption that the motions of the stars may be neglected during the encounter (the impulsive approximation) to test the adequacy of the latter approximation. If the two galaxies are of 10¹¹ M , of radii 10 kpc and of mass distribution that of a polytrope of indexn=4; and if the relative distance and velocity at their closest approach are taken as 2 kpc and 1000 km/sec respectively, the mass of escaping stars from a galaxy is estimated to be roughly 4% of the total mass of the galaxy and the total increase in the internal energy of a galaxy during the collision due to the tidal acceleration of all its stars is equal in magnitude to approximately 25% of its initial internal energy, about one-fifth of which is associated with the escaping stars.     

Dynamics of molecular clouds on the galactic scale

We have carried out an extensive investigation into the dynamics of the molecular clouds in the disk of the Galaxy. We have used both computational methods and physical arguments to try to understand how the ensemble of molecular clouds interacts, how the clouds are affected by the gravitational field of the Galaxy and also the circumstances under which they can aggregate into giant molecular clouds (GMC's).The dynamical model is three dimensional and consists of 120,000 spherical clouds, each having a mass of 10⁴ M . It allows for the mutual gravitation between clouds, up to a cut-off distance; when two clouds collide they rebound, with a specified coefficient of restitutione. We have also developed a physically more realistic model for a cloud, supported by a magnetic field, and used it to select a suitable range of values fore. Our first paper deals with the case of an axially symmetrical galaxy. The clouds are distributed initially in a disk extending 100 pc on either side of the Galactic plane. As it evolves the system of clouds loses energy, and the disk grows thinner at a rate which depends on the value ofe. GMC's start to form once the disk is thin enough. We believe this result to be valid more generally, and that it holds also in models with spiral structure.     

Suspected OB stars near the andromeda nebula

An investigation of suspected OB stars on plates of the Schmidt telescope of Karl Schwarzschild Observatory Tautenburg gave the following results: I. There exists a sharp boundary of the M 31 disk. For distances larger than R = 30 kpc from the centre no OB stars of the Andromeda galaxy could be found. 2. The isolated OB stars in the outer regions of M 31 fit into the spiral structure. 3. The suspected OB stars outside M 31 (R>30 kpc) in all probability belong to our own galaxy. It is to be expected that they partly are blue horizontal branch stars of extreme population II, partly white dwarfs rather far left of the black body line in the two colour diagram.     

The irregular galaxy Cassiopeia 1 – a new member of the Local group

A photometry in the V, R, I (Cousins) system has been performed for stars in the galaxy Cassiopeia 1. The resulting colour-agnitude (CM) diagram of Cas 1 reveals the existence of blue stars and the absence of red ones. From an analysis of the CM diagram of the galaxy it is concluded that A_v = 4.0 mag, and the distance modulus (m – M)_o = 24.5 mag, corresponding to a distance of 790 kpc. The visual magnitude of the galaxy is V = 14^m.62 (M_v = −13^m.8) and the colour index (V – R) = 0.89 mag. From the distance determination the galaxy Cassiopeia 1 is a member of the Local group.     

On the mass and density of the stellar disk of M33

The disk surface density of the nearby spiral galaxy M33 is estimated assuming that it is marginally stable against gravitational perturbations. For this purpose we used the radial profile of line-of-sight velocity dispersion of the disk planetary nebulae obtained by Ciardullo et al. (2004). The surface density profile we obtained is characterized by the radial scalelength which is close to the photometrical one and is in a good agreement with the rotation curve of M33 and with the mass-to-light ratio which follows from the stellar population model. However at the galactocentric distance r > 7 kpc the dynamical overheating of the disk remains quite possible. The thickness of the stellar disk of M33 should increase outwards from the center. The dark halo to total mass ratio is estimated as a function of r. The effective oxygen yield obtained in the frame of instantaneous recycling approximation using the disk surface density and the observed gradient of O/H increases with radius. It may indicate that the role of accretion of metalpoor gas in the chemical evolution of interstellar medium decreases outwards.     

Galactic scale gas flows in colliding galaxies: 3-Dimensional,N-body/hydrodynamics experiments

We present some results from three dimensional computer simulations of collisions between models of equal mass galaxies, one of which is a rotating, disk galaxy containing both gas and stars and the other is an elliptical containing stars only. We use fully self consistent models in which the halo mass is 2.5 times that of the disk. In the experiments we have varied the impact parameter between zero (head on) and 0.9R (whereR is the radius of the disk), for impacts perpendicular to the disk plane. The calculations were performed on a Cray 2 computer using a combined N-body/SPH program. The results show the development of complicated flows and shock structures in the direction perpendicular to the plane of the disk and the propagation outwards of a density wave in both the stars and the gas. The collisional nature of the gas results in a sharper ring than obtained for the star particles, and the development of high volume densities and shocks.     

On-Going Galaxy Formation

We investigate the process of galaxy formation as can be observed in the only currently forming galaxies - the so-called Tidal Dwarf Galaxies, hereafter TDGs - through observations of the molecular gas detected via its CO (Carbon Monoxide) emission. These objects are formed of material torn off of the outer parts of a spiral disk due to tidal forces in a collision between two massive galaxies. Molecular gas is a key element in the galaxy formation process, providing the link between a cloud of gas and a bona fide galaxy. We have detected CO in 8 TDGs (Braine, Lisenfeld, Duc and Leon, 2000: Nature 403, 867; Braine, Duc, Lisenfeld, Charmandaris, Vallejo, Leon and Brinks: 2001, A&A 378, 51), with an overall detection rate of 80%, showing that molecular gas is abundant in TDGs, up to a few 10⁸ M _⊙. The CO emission coincides both spatially and kinematically with the HI emission, indicating that the molecular gas forms from the atomic hydrogen where the HI column density is high. A possible trend of more evolved TDGs having greater molecular gas masses is observed, in accord with the transformation of HI into H₂. Although TDGs share many of the properties of small irregulars, their CO luminosity is much greater (factor ∼ 100) than that of standard dwarf galaxies of comparable luminosity. This is most likely a consequence of the higher metallicity (≳sim 1/3 solar) of TDGs which makes CO a good tracer of molecular gas. This allows us to study star formation in environments ordinarily inaccessible due to the extreme difficulty of measuring the molecular gas mass. The star formation efficiency, measured by the CO luminosity per Hα flux, is the same in TDGs and full-sized spirals. CO is likely the best tracer of the dynamics of these objects because some fraction of the HI near the TDGs may be part of the tidal tail and not bound to the TDG. Although uncertainties are large for individual objects, as the geometry is unknown, our sample is now of eight detected objects and we find that the ‘dynamical’ masses of TDGs, estimated from the CO line widths, seem not to be greater than the ‘visible’ masses (HI + H₂ + a stellar component). Although higher spatial resolution CO (and HI) observations would help reduce the uncertainties, we find that TDGs require no dark matter, which would make them the only galaxy-sized systems where this is the case. Dark matter in spirals should then be in a halo and not a rotating disk. Most dwarf galaxies are dark matter-rich, implying that they are not of tidal origin. We provide strong evidence that TDGs are self-gravitating entities, implying that we are witnessing the ensemble of processes in galaxy formation: concentration of large amounts of gas in a bound object, condensation of the gas, which is atomic at this point, to form molecular gas and the subsequent star formation from the dense molecular component. This revised version was published online in September 2006 with corrections to the Cover Date.     

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.     

Dynamics of the Interstellar Gas in the Vicinity of a Bar

A numerical simulation of the disk dynamics of a flat stellar-gaseous galaxy is presented. This simulation is based on N-body modelling for a stellar disk, together with integration of the two-dimensional hydrodynamic equations for a gaseous disk. The existence of a quasi-periodic regime of disk evolution found earlier in a purely hydrodynamic simulation is confirmed. Intense gas flows in the central area of the disk due to the saddle point of the bar potential can support the active galactic nuclei in Seyfert galaxies and form double nuclei.