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%.     

Gravitational stability and dynamical overheating of galactic stellar disks

Stellar velocity dispersion data at galactocentric distance of two disk radial scale lengths (R = 2h), available in the literature allowed us to determine the upper limits of disk local surface densities at a given R and (by extrapolation) total masses of disks proceeding from the marginal gravitational stability condition. A comparison of the obtained disk masses with the photometric estimates based on the stellar population models indicates the absence of strong dynamical overheating inmost spiral galaxies and hence the absence of significant major merging events, which were able to heat dynamically the inner parts of disks. The same conclusion is valid for some of S0 galaxies. However, a significant part of the latter possesses stellar velocity dispersion, which exceeds the threshold value needed for gravitational stability. Dynamically overheated disks occur both among paired and isolated galaxies. Disk to total mass ratios within R = 4h found for marginally stable disks in most cases lie in the range 0.5–0.8 with the absence of the clearly defined correlation of this ratio with color index or morphological type.     

Numerical experiments on the stability of preplanetary disks

Gravitational stability of gaseous protostellar disks is relevant to theories of planetary formation. Stable gas disks favor formation of planetesimals by the accumulation of solid material; unstable disks allow the possibility of direct condensation of gaseous protoplanets. We present the results of numerical experiments designed to test the stability of thin disks against large-scale, self-gravitational disruption. The disks are represented by a distribution of about 6 × 10⁴ point masses on a two-dimensional (r, φ) grid. The motions of the particles in the self-consistent gravity field are calculated, and the evolving density distributions are examined for instabilities. Two parameters that have major influences on stability are varied: the initial temperature of the disk (represented by an imposed velocity dispersion), and the mass of the protostar relative to that of the disk. It is found that a disk as massive as 1M_⊙, surrounding a 1M_⊙ protostar, can be stable against long-wavelength gravitational disruption if its temperature is about 300°K or greater. Stability of a cooler disk requires that it be less massive, but even at 100°K a stable disk can have an appreciable fraction ($\text{～}\text{1}\text{3}$) of a solar mass.     

The gas content in galactic disks: Correlation with kinematics

We consider the relationship between the total HI mass in late-type galaxies and the kinematic properties of their disks. The mass M_HI for galaxies with a wide variety of properties, from dwarf dIrr galaxies with active star formation to giant low-brightness galaxies, is shown to correlate with the product V_cR₀ (V_c is the rotational velocity, and R₀ is the radial photometric disks cale length), which characterizes the specific angular momentum of the disk. This correlation, along with the decrease in the relative mass of the gas in a galaxy with increasing V_c, can be explained in terms of the previous assumption that the gas density in the disks of most galaxies is maintained at a level close to the threshold (marginal) stability of a gaseous layer to local gravitational perturbations. In this case, the regulation mechanism of the star formation rate associated with the growth of local gravitational instability in the gaseous layer must play a crucial role in the evolution of the gas content in the galactic disk.     

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.     

Numerical study of the relaxation of one-dimensional gravitational systems

The effective relaxation of one-dimensional gravitational systems towards micro-canonical equilibrium is demonstrated in a series of computer experiments, the Kolmogorov-Smirnov goodness of fit test being used to give a measure of the evolution of the distribution function. The relaxation timescalet _R is shorter thanNt _C, and thus much shorter than the generally accepted estimatet _RN ²t_C,N being the number of particles in the system andt _C a characteristic dynamical time.     

Disk Planet Interactions And Early Evolution in Young Planetary Systems

We study and review disk protoplanet interactions using local shearing box simulations. These suffer the disadvantage of having potential artefacts arising from periodic boundary conditions but the advantage, when compared to global simulations, of being able to capture much of the dynamics close to the protoplanet at high resolution for low computational cost. Cases with and without self sustained MHD turbulence are considered. The conditions for gap formation and the transition from type I migration are investigated and found to depend on whether the single parameter M _p R ³/(M_* H ³), with M _p, M_*, R, and H being the protoplanet mass, the central mass, the orbital radius and the disk semi-thickness, respectively, exceeds a number of order unity. We also investigate the coorbital torques experienced by a moving protoplanet in an inviscid disk. This is done by demonstrating the equivalence of the problem for a moving protoplanet to one where the protoplanet is in a fixed orbit which the disk material flows through radially as a result of the action of an appropriate external torque. For sustainable coorbital torques to be realized a quasi steady state must be realized in which the planet migrates through the disk without accreting significant mass. In that case, although there is sensitivity to computational parameters, in agreement with earlier work by Masset and Papaloizou [2003, ApJ, 588, 494] based on global simulations, the coorbital torques are proportional to the migration speed and result in a positive feedback on the migration, enhancing it and potentially leading to a runaway. This could lead to fast migration for protoplanets in the Saturn mass range in massive disks and may be relevant to the mass period correlation for extrasolar planets which gives a preponderance of sub Jovian masses at short orbital periods.     

Spread of matter over a neutron-star surface during disk accretion: Deceleration of rapid rotation

The accretion of hot slowly rotating gas onto a supermassive black hole is considered. The important case where the velocities of turbulent pulsations at the Bondi radius r _B are low, compared to the speed of sound c _s, is studied. Turbulence is probably responsible for the appearance of random average rotation. Although the angular momentum at r _B is low, it gives rise to the centrifugal barrier at a depth r _c = l ² /GM _BH ≪ r _B, that hinders supersonic accretion. The numerical solution of the problem of hot gas accretion with finite angular momentum is found taking into account electron thermal conductivity and bremsstrahlung energy losses of two temperature plasma for density and temperature near Bondi radius similar to observed in M87 galaxy. The saturation of the Spitzer thermal conductivity was also taken into account. The parameters of the saturated electron thermal conductivity were chosen similar to the parameters used in the numerical simulations of interaction of the strong laser beam radiation with plasma targets. These parameters are confirmed in the experiments. It is shown that joint action of electron thermal conductivity and free-free radiation leads to the effective cooling of accreting plasma and formation of the subsonic settling of accreting gas above the zone of a centrifugal barrier. A toroidal condensation and a hollow funnel that separates the torus from the black hole emerge near the barrier. The barrier divides the flow into two regions: (1) the settling zone with slow subKeplerian rotation and (2) the zone with rapid supersonic nearly Keplerian rotation. Existence of the centrifugal barrier leads to significant decrease of the accretion rate Ṁ in comparison with the critical Bondi solution for γ = 5/3 for the same values of density and temperature of the hot gas near Bondi radius. Shear instabilities in the torus and related friction cause the gas to spread slowly along spirals in the equatorial plane in two directions.As a result, outer (r > r _c) and inner (r < r _c) disks are formed. The gas enters the immediate neighborhood of the black hole or the zone of the internal ADAF flow along the accretion disk (r < r _c). Since the angular momentum is conserved, the outer disk removes outward an excess of angular momentum along with part of the matter falling into the torus. It is possible, that such outer Keplerian disk was observed by Hubble Space Telescope around the nucleus of the M87 galaxy in the optical emission lines. We discuss shortly the characteristic times during which the accretion of the gas with developed turbulence should lead to the changes in the orientation of the torus, accretion disk and, possibly, of the jet.     

Radial migration of planetesimals

Planetesimals orbiting a protostar in a circumstellar disk are affected by gravitational interaction among themselves and by gas drag force due to disk gas. Within the Kyoto model of planetesimal accretion, the migration rate is interpreted as the inverse of the planetary formation time scale. Here, we study time scales of gravitational interaction and gas drag force and their influence on planetesimal migration in detail. Evaluating observations of 86 T Tauri stars (Beckwithet al., 1990), we find the mean radial temperature profile of circumstellar disks. The disk mass is taken to be 0.01M in accordance with minimum mass models and observed T Tauri disks. The time scale of gravitational interaction between planetesimals is studied analogously to Chandrasekhar's stellar dynamics. Hence, Chandrasekhar's coefficient , defined as the fraction between the mean separation of planetesimals and the impact parameter, plays an important role in determining the migration rate. We find ln to lie between 5 and 10 within the protosolar disk. Our result is that, at the stage of disk evolution considered here, gas drag force affects the radial migration of planetesimals by a few orders of magnitude more than gravitational interaction.Paper presented at the Conference on Planetary Systems: Formation, Evolution, and Detection held 7–10 December, 1992 at CalTech, Pasadena, California, U.S.A.     

Compact objects with large masses

It is shown that gravitational theory allows stable equilibrium configurations of the degenerated Fermi gas, whose masses M ≫ M_⊙ and sizes are less than α = 2GM/c. They have no events horizon at the distance r = α from the center. Orbits of test particles near the above objects are considered.     

Structure and stability of rotating fluid disks around massive objects. I. Newtonian formulation

In this paper we have presented a very general class of solutions for rotating fluid disks around massive objects (neglecting the self gravitation of the disk) with density as a function of the radial coordinate only and pressure being nonzero. Having considered a number of cases with different density and velocity distributions, we have analysed the stability of such disks under both radial and axisymmetric perturbations. For a perfect gas disk with γ= 5/3 the disk is stable with frequency (MG/r³)^1/2 for purely radial pulsation with expanding and contracting boundary. In the case of axisymmetric perturbation the critical γ_c for neutral stability is found to be much less than 4/3 indicating that such disks are mostly stable under such perturbations. On leave of absence from Government College, Jagdalpur 494005.     

Evolutionary processes in protoplanetary accretion disks: the propagation of axisymmetric shock waves

In this paper we investigate both the global and the local hydrodynamics of axisymmetric accretion disks around young stellar objects under the simultaneous action of viscosity, self-gravity and pressure forces. For simplicity, we take for the global model a polytropic equation of state, make the infinitely thin disk approximation and characterize the surface density and temperature profiles in the disk as power laws in the radial distance r from the protostar. We solve the problem of the general density profile of a Keplerian disk showing that self-gravity could not be an important factor for the fast formation of the rocky cores of giant gaseous planets in our solar system. Under the hypothesis that the unperturbed rotation curve of the disk is nearly Keplerian throughout the radial extent, we can estimate with our polytropic model a lower limit for the resulting masses M_d(r) of stable disks up to 100 AU. These masses are in the range of the so-called minimum mass solar nebular (_d/M_s ≈ 0.01–0.02).By adopting a simplified viscosity model, where the height-integrated turbulent dynamical viscosity ν is a function of the surface density σ like η ∝ σ^Γ, we derive in the local shearing sheet model linearized evolution equations for small density perturbations describing both a diffusion process and the propagation of acoustic density waves. We solve a special initial value problem and calculate the appropriate Green's function. The analytical solutions so obtained describe in the case Γ < 0 the successive formation of quasi-stationary ring-shaped density structures in a disk with a definite mode of maximum instability, whereas in the case Γ > Γ_c the density wave equation describes the propagation of an “overstable” ring-shaped acoustic density wavelet to the outer ranges of the accretion disk. Whereas the group velocity of the wave packet is subsonic, the phase velocities of individual wave crests in the wave packet are supersonic. The mode of maximum instability, the growth rate and the number of growing waves in the wavelet are controlled by Γ and α. Our present knowledge concerning turbulent viscosity in protoplanetary disks is not sufficient to decide whether or not the case Γ > Γ_c is realized.The suggested structuring processes in the linear theory should initiate in the non-linear regime the formation of narrow ring-shaped density shock waves moving through the protoplanetary disk. These non-linear waves could produce extremely spatially and temporally heterogeneous temperature regions in the disk. We speculate that ring-shaped density waves, excited by inner boundary conditions and which have dominated the disk's evolution at early times, are responsible both for the fast growth of dust to planetesimals and at least for the rapid accretion of the rocky cores of giant gaseous planets in the protoplanetary accretion disk (shock wave trigger hypothesis). We derive provisional scaling rules for planetary systems regarding the spacing of orbits as a function of the mass ratio of the protoplanetary disk to the protostar. However, further analytical work and linear as well as nonlinear numerical simulations of density waves excited by inner boundary conditions are needed to consolidate the results and speculations of our linear wave mechanics in the future.     

A theory of galactic mass and rotation curves

The goal of this paper is to account for the complete observed rotation curves of disk galaxies without dark matter. To attain that goal, use is made of a conservation law from stability theory of linear waves, leading to a vector-based theory of gravitation. In the theory, galactic centers are sites of strong gravitational fields. The new theory predicts extra matter at the center of disk galaxies, which is well-known to be consistent with intergalactic dynamics. For given disk radiusr ₀ and edge tangential speedv, the greater the deviation of a rotation curve from linear (solid disk rotation), the greater the mass of the galaxy as a multiple of Newtonian massr ₀v²/G, up to a factor of about 1000. In an approximate calculation it turns out that disk density (r) (in kg m^–2) is proportional to 1/r for typical rotation curves. Rotation is characterized by two constants which in turn are determined by the edge speed and mass distribution. Not just any curve shape can be so obtained; in fact, the theoretically possible curves correspond to observed curves.     

Cosmogony of the solar system

In Sections 1–6, we determine an approximate analytical model for the density and temperature distribution in the protoplanetary could. The rotation of the planets is discussed in Section 7 and we conclude that it cannot be determined from simple energy conservation laws.The velocity of the gas of the protoplanetary cloud is found to be smaller by about 5×10³ cm s^–1 in comparison to the Keplerian circular velocity. If the radius of the planetesimals is smaller than a certain limitr ₁, they move together with the gas. Their vertical and horizontal motion for this case is studied in Sections 8 and 9.As the planetesimals grow by accretion their radius becomes larger thanr ₁ and they move in Keplerian orbits. As long as their radius is betweenr ₁ and a certain limitr ₂ their gravitational interaction is negligible. In Section 10, we study the accretion for this case.In Section 11, we determine the change of the relative velocities due to close gravitational encounters. The principal equations governing the late stages of accretion are deduced in Section 12, In Section 13 there are obtained approximate analytical solutions.The effect of gas drag and of collisions is studied in Sections 14 and 15, respectively. Numerical results and conclusions concerning the last and principal stage of accretion are drawn in Section 16.     

Gravitational potential energy of a pair of overlapping galaxies

The interaction of the gravitational potential energy of a pair of overlapping Plummer-model galaxies is determined exactly for various separationsr of their centres. It is shown that the results can be well represented by the simple relationW(r)=–GM ₁ M ₂/(r ²+ ²)^1/2, where 1/ is the average reciprocal distance between the stars of two galaxies of massesM ₁ andM ₂ when they have zero separation.     

Relationship between the thickness of stellar disks and the relative mass of a dark galactic halo

We analyze the R-and K _s-band photometric profiles for two independent samples of edge-on galaxies. The thickness of old stellar disks is shown to be related to the relative masses of the spherical and disk components of galaxies. The radial-to-vertical scale length ratio for galactic disks increases (the disks become thinner) with increasing total mass-to-light ratio of the galaxies, which reflects the relative contribution of the dark halo to the total mass, and with decreasing central deprojected disk brightness (density). Our results are in good agreement with numerical models of collisionless disks that evolved to a marginally stable equilibrium state. This suggests that, in most galaxies, the vertical stellar-velocity dispersion, on which the equilibrium-disk thickness depends, is close to a minimum value that ensures disk stability. The thinnest edge-on disks appear to be low-brightness galaxies in which the dark-halo mass far exceeds the stellar-disk mass.     

The Caledonian Symmetrical Double Binary Four-Body Problem I: Surfaces of Zero-Velocity Using the Energy Integral

The Caledonian four-body problem introduced in a recent paper by the authors is reduced to its simplest form, namely the symmetrical, four body double binary problem, by employing all possible symmetries. The problem is three-dimensional and involves initially two binaries, each binary having unequal masses but the same two masses as the other binary. It is shown that the simplicity of the model enables zero-velocity surfaces to be found from the energy integral and expressed in a three dimensional space in terms of three distances r ₁, r ₂, and r ₁₂, where r ₁ and r ₂ are the distances of two bodies which form an initial binary from the four body systems centre of mass andr ₁₂ is the separation between the two bodies.     

Galactic nuclei

Application of standard accretion-disk theory to the central parts of galactic disks implies that galactic nuclei should pass quasi-periodically through stages of (i) gravitational instability (Starburst), (ii) non-thermal disk activity (LINER), and (iii) central-plane nuclear burning (AGN). Nuclear disks can have large mass-to-light ratios, and store large masses at high (Keplerian) rotation velocities. Aburning disk can explain all the phenomena commonly attributed to a supermassive Black Hole.     

The in-plane motion of a Geosynchronous satellite under the gravitational attraction of the sun,the moon and the oblate earth

The in-plane motion of a Geosynchronous satellite under the gravitational effects of the sun, the moon and the oblate earth has been studied. The radial deviation (Δr) and the tangential deviation (r _cΔθ) have been determined. Herer _c represents the synchronous altitude. It has been seen that the sum of the oscillatory terms in Δr for different inclinations is a small finite quantity whereas the sum of the oscillatory terms inr _cΔθ for different inclinations is quite large due to the presence of the low-frequency terms in the denominator     

On the Dynamics of Dust Grains in a Hierarchical Environment

Most main sequence stars are binaries or higher multiplicity Systems and it appears that at birth most stars have circumstellar disks. It is commonly accepted that planetary systems arise from the material of these disks; consequently, binary and multiple systems may have a main role in planet formation. In this paper, we study the stage of planetary formation during which the particulate material is still dispersed as centimetre-to-metre sized primordial aggregates. We investigate the response of the particles, in a protoplanetary disk with radius R_D = 100 AU around a solar-like star, to the gravitational field of bound perturbing companions in a moderately wide (300–1600 AU) orbit. For this purpose, we have carried out a series of simulations of coplanar hierarchical configurations using a direct integration code that models gravitational and viscous forces. The massive protoplanetary disk is around one of the components of the binary. The evolution in time of the dust sub-disk depends mainly on the nature (prograde or retrograde) of the relative revolution of the stellar companion, and on the temperature and mass of the circumstellar disk. Our results show that for binary companions near the limit of tidal truncation of the disk, the perturbation leads to an enhanced accretion rate onto the primary, decreasing the lifetime of the particles in the protoplanetary disk with respect to the case of a single star. As a consequence of an enhanced accretion rate the mass of the disk decreases faster, which leads to a longer resultant lifetime for particles in the disk. On the other hand, binary companions may induce tidal arms in the dust phase of protoplanetary disks. Spiral perturbations with m = 1 may increase in a factor 10 or more the dust surface density in the neighbourhood of the arm, facilitating the growth of the particles. Moreover, in a massive disk (0.01M⊙) the survival time of particles is significantly shorter than in a less massive nebula (0.001M⊙) and the temperature of the disk severely influences the spiral-in time of particles. The rapid evolution of the dust component found in post T Tauri stars can be explained as a result of their binary nature. Binarity may also influence the evolution of circumpulsar disks. This revised version was published online in July 2006 with corrections to the Cover Date.