Emden-Chandrasekhar axisymmetric,rigid-body rotating polytropes

In connection with the basic theory reported in a previous paper (Paper I) for EC1 (rigidly rotating) polytropes, we define exact configurations as configurations for which the equilibrium equation has solutions which are infinitely close to some analytical function and the related gravitational potential coincides, in fact, with the gravitational potential due to mass distribution, at any point not outside the system. Then we restrict to the special casen=5 and divide the related polytropes into two components, a massive body where each mass element has a finite (polytropic) distance from the centre, and a massless atmosphere where each mass element has an infinite (polytropic) distance from te centre. It is found a single exact configuration exists, which under some assumptions may be related to Roche systems. In the special casen=0 it is shown a particular configuration, the spheroidal one, is an exact configuration and evidence is given that spheroidal configurations are the stablest among all the allowed (axisymmetric) configurations. It is also pointed out that EC1 polytropes withn=0 and incompressible MacLaurin spheroids belong to different sequences, even if they exhibit some common features. In the special casen=1 it is shown each allowed configuration is expressible by a convenient series development, which reduces to the relatedn=0 configuration by maintaining only the first two or the first one terms of the sum. It is also deduced, by analogy with the casen=0, that pseudospheroidal configurations are exact and the stablest among all the allowed (axisymmetric) configurations.     

Emden-chandrasekhar axisymmetric,solid-body rotating polytropes

According to the general results of a previous work (Caimmi, 1980; hereafter referred to as Paper I), solutions to EC equation, which expresses a necessary and sufficient condition for equilibrium of Emden-Chandresekhar axisymmetric, solid-body rotating polytropes (EC polytropes), are taken into consideration, of the type $$\vartheta (\xi ,\mu ) = A_0 \vartheta _0 (\upsilon ,\xi ) + \sum\limits_l^\infty {_l {\rm A}_{2l} (\upsilon )\vartheta _{2l} (\xi )P_{2l} (\mu ),} $$ with ? _2l later defined as the EC associated function of degree 2l. Thus the EC equation, involving (?, μ), is found to be equivalent to the infinite set of EC associated equations, involving ? _2l (μ). We approximate g (?, μ) by neglecting all terms of degree higher than 2 which appear in the above expression, and then search power series solutions to EC associated equations of degree 0 and 2, corresponding to any choice ofn (polytropic index, related to density distribution) andv (related to rotational distorsion). To this aim, we extend the methods used by Seidov and Kuzakhmedov (1977), and Mohan and Al-Bayaty (1980), to construct power series of the type outlined above, related to solid-body rotating configurations and originating both inside and outside the radial boundary (defined as the first zero of ?₀(μ)=0). The corresponding expressions of ?₀ and ?₂ may serve to derive an approximate expression of, and future work becomes possible concerning the determination of some physical parameters (such as volume, mass, potential energy, angular momentum) related to any choice ofn andv. Computations have been performed forn=k/4 (0≤k≤20, i.e. 0≤n≤5) andv=0,v≈v _R/2,v≈v _R, withv _R lowest value ofv leading to balance between gravitation and centrifugal force at the equator of the system. An upper limit to the error, ε^*(μ), done in computing ? _2l , ?? _2l , and ?? _2l at any point ? for a given choice ofn andv, is estimated, ranging from large values (ε^*=1E-2) forn close enough to 0 and ? close enough or outside the radial boundary, to low values (ε^*=1E-10) forn far enough from 0 and no constraint on ?. Comparison between results of this paper and the accurate results by Linnell (1977, 1981) obtained using a different approach and available forn=2,v=0, andn=3,v=0, lead to a fair agreement (up to (1E?5?1E?6). It is apparent that the method followed here continues to hold when the first EC associated functions up to degree 2l are taken into account, leading — at least in way of principle — to a more refined approximation to the EC function; this would only make the related calculations much more complicated.     

Physical characteristics ofN-dimensional,radially-symmetric polytropes

The physical characteristics radius, mass, mean density, gravitational potential and acceleration, gravitational and internal energy are presented with the aid of the gamma function forN-dimensional, radially-symmetric polytropes. The virial theorem with external pressure is derived in the relativistic limit, with Newtonian gravitation still valid. The gravitational energy of polytropes obeying the generalized Schuster—Emden integral is shown to be finite. Finiteness of mass and radius is discussed for the cases of practical interestN=1 (slab),N=2 (cylinder), andN=3 (sphere). Uniform contraction or expansion ofN-dimensional polytropes is considered in the last section.     

Can elliptical galaxies be equilibrium systems?

This paper deals with the question of whether elliptical galaxies can be considered as equilibrium systems (i.e., the gravitational+centrifugal potential is constant on the external surface). We find that equilibrium models such as Emden-Chandrasekhar polytropes and Roche polytropes withn=0 can account for the main part of observations relative to the ratio of maximum rotational velocity to central velocity dispersion in, elliptical systems. More complex models involving, for example, massive halos could lead to a more complete agreement. Models that are a good fit to the observed data are characterized by an inner component (where most of the mass is concentrated) and a low-density outer component. A comparison is performed between some theoretical density distributions and the density distribution observed by Younget al. (1978) in NGC 4473, but a number of limitations must be adopted. Alternative models, such as triaxial oblate non-equilibrium configurations with coaxial shells, involve a number of problems which are briefly discussed. We conclude that spheroidal oblate models describing elliptical glaxies cannot be ruled out until new analyses relative to more refined theoretical equilibrium models (involving, for example, massive halos) and more detailed observations are performed.     

Rotational distortion of polytropes by analytic approximation

This paper modifies the first-order perturbation theory of Chandrasekhar, for rotational distortion of polytropes. Comparison with numerical integrations by other authors demonstrates that the present analytic theory is as accurate as other published first-order theories. The present theory is in a form permitting rapid calculation of boundary shapes as a function of the rotation parameter,v, and the polytrope index,n. Results are presented for a critically rotating polytrope, for the casen=3.     

Structure of polytropic models of stars containing poloidal magnetic fields

Polytropic models of axially-symmetric equilibrium stars of infinite conductivity with poloidal magnetic fields are constructed by numerical integration of the exact equations governing internal structure. The mathematical method used, a further generalization and improvement of Stoeckly's method, allows the construction of a sequence of equilibrium models starting with a spherically symmetric star (when no magnetic field is present) and terminating with a doughnut-shaped object (for a very strong magnetic field) — a fact already shown by Monaghan. Detailed results are given only for two polytropes with the indexn=1.5 and 3.0, although any other value ofn greater than or equal to one could have been selected. Contrary to Monaghan's results, it is found that along the sequence of configurations forn=3.0 the ratio of the magnetic and gravitational energy peaks out before a doughnut-shaped configuration is reached; but this effect does not characterize then=1.5 sequence. The calculations confirm, however, another result of Monaghan asserting that the magnetic field is a fairly insensitive function of the polytropic index.     

The equilibrium and stability of magnetopolytropes

The theory of the oscillations of axisymmetric gaseous configurations with a prevalent magnetic field is presented. The virial tensor method is used to obtain the nine second harmonic modes of oscillations of the system. It is found that out of the nine modes, three are neutral, four are non-radial, and two are coupled. For the Prendergast spherical model it is found that one of the coupled modes is radial and the other non-radial. Both the radial and the non-radial modes obtained in this case agree with the corresponding formulae obtained byChandrasekhar andLimber (1954) andWoltjer (1962).The equilibrium structure of gaseous polytropes with toroidal magnetic fields is also investigated in detail for values of the polytropic indexn=1, 1.5, 2, 3 and 3.5. For this model the components of the moment of intertia and potential energy tensors together with the non-zero components of the supermatrix potential are obtained. The final results in terms of the effect of weak toroidal magnetic fields on the characteristic frequencies of distorted polytropes are presented in the form of tables.     

Periodic solutions in the commensurable three-body problem

Various families of periodic solutions are shown to exist in the three body problem, in which two of the bodies are close to a commensurability in mean motions about the third body, the primary, which is considerably more massive than the other two. The cases considered are

1. The non-planar circular restricted problem (in which one of the secondary bodies has zero mass, and the other moves in a fixed circular orbit about the primary).
2. The planar non-restricted problem (in which the three bodies move in a plane, and both secondaries have finite mass).
3. The planar elliptical restricted problem (in which the three bodies move in a plane, one of the secondary bodies has zero mass, and the other moves in a fixed elliptical orbit about the primary).

The method used is to eliminate all short period terms from the Hamiltonian of the motion by means of a von Zeipel transformation, leaving only the long period terms which are due to the commensurability. Hence only the long period part of the motion is considered, and the variables used differ from the variables describing the full motion by a series of short-period trigonometric terms of the order of the ratio of the mass of the secondaries to that of the primary body. It is shown that solutions of the long-period problem in which the variables remain constant are equivalent to solutions in the full motion in which the bodies periodically return to the same configuration, and these are the types of periodic solution that are shown to exist. The form of the disturbing function, and hence of the equations of motion, is found up to the fourth powers of the eccentricities and inclination by considering the d'Alembert property. The coefficients of the terms appearing in this expansion are functions of the semi-major axes of the orbits of the secondary bodies. Expressions for these coefficients are not worked out as they are not required. Lete, n, m be the orbital eccentricity, mean motion and mass of one of the secondary bodies, and lete′, n′, m′ be the corresponding quantities for the other. (The mass of the primary is taken as unity). In cases (a) and (c) we will havem=0. In case (a)e′ will be zero, and in case (c) it will be a constant. Leti be the mutual inclination of the orbits of the secondary bodies. Suppose the commensurability is of the form(p+q) n =pn′, wherep andq are relatively prime integers, and put γ=(p+q) n/n′?p. The families of periodic solutions shown to exist are as follows. For q=1 No periodic solutions are found withi≠0 in case (a), and none withe′≠0, in case (c). In case (b) periodic solutions are found in whiche=0 (m′/γ),e′=0 (m/γ) for values of γ away from the exact commensurability. As γ approaches zero thene ande′ become 0 (1). For q≠1 Case (a). Families of periodic solutions bifurcating from the family withe=0, i=0 are shown to exist. Families in whichi=0 ande becomes non-zero exist for all values ofq. Families in whiche=0 andi becomes non-zero exist for even values ofq. Families in whiche andi become non-zero simultaneously exist for odd values ofq. Case (b). No families are found other than those withe=e′=0. Case (c). Families are found bifurcating from the familye=e′=0 in whiche ande′ become non-zero simultaneously. For all these solutions existence is only demonstrated close to the point of bifurcation, where all the variables are small, as the method uses series expansions ine, e′ andi. From the form of the solutions it is clear that the non-zero variables will become large for values of γ away from the bifurcation point.     

Emden-chandrasekhar axisymmetric,rigidly rotating polytropes

We determine equilibrium configuration of Emden-Chandrasekhar axisymmetric, solid-body rotating polytropes, defined as EC polytropes, for polytropic indices ranging from 0 (homogeneous bodies) to 5 (Roche-type bodies). To this aim, we improve Chandrasekhar's method to determine equilibrium configurations on two respects: namely, (a) no distinction exists between undistorted and distorted terms in the expression of the potential, and (b) the comparison between the expressions of gravitational potential and its first derivatives inside and outside the body has to be made on the boundary of a sphere of radius Ξ_E, which does not necessarily coincide with the undistorted Emden's sphere of radius \(\bar \xi _0 \geqslant \Xi _{\text{E}} \) . We also allow different values of \(\bar \xi _0 \) for different physical parameters, and choose a special set which best fits more refined results (involving more complicated and more expensive computer codes) by James (1964). We find an increasing agreement with increasing values of polytropic indexn and vice-versa, while a large discrepancy arises for 0≤n<1, which makes the approximations used here too much rough tobe accepted in this range. A real slight non-monotonic trend is exhibited by axial rations and masses related to rotational equilibrium configurations — i.e., when gravity at the equator is balanced by centrifugal force-with extremum points for 4.8<n<4.85 in both cases. The same holds for masses related to spherical configurations, as already pointed out by Seidov and Kuzakhmedov (1978). Finally, it is shown that isotrophic, one-component models of this paper might provide the required correlation between the ratio of a typical rotation velocity to a typical peculiar velocity and the ellipticity, for about \(\tfrac{3}{4}\) of elliptical systems for which observations are available.     

Potential energy of gravitationally interacting spherical galaxies

A theory has been developed for obtaining the potential energy of two interpenetrating spherically symmetric galaxies of unequal dimensions due to their mutual gravitational interaction. The mass distribution in both the galaxies is assumed to be that of a polytrope of integral index. A basic function that occurs in the theory has been tabulated for the cases of polytropes of indicesn=0 and 4 for four ratios of the radii.     

Partial separability of the Laplace equation in Roche coordinates

The Laplace equation in the coordinatesu, v, w is calledu-separable if there are solutions of the formF(u)G(v, w). If the surfacesv = constant andw = constant are orthogonal tou = constant the coordinate system is called semi-orthogonal. The Laplace equation is notu-separable for the rotation problem semi-orthogonal Roche coordinate system (n0, q=0) or the general problem (n0, q0) ifv andw are analytic functions ofn andq and the coordinate system is proper in some region of then, q plane including the origin,n=q=0 (u is the Roche potential).     

Roche limit for homogeneous incompressible masses

The aim of the present investigation has been to establish the minimum distance (commonly referred to as the ‘Roche limit’), to which a small satellite can approach its central star without the loss of its stability. In order to do so, we shall depart from hydrodynamical equations governing small oscillations of stellar structures, and set out to establish the limit at which their distorted form of equilibrium can no longer vibrate periodically in response to arbitrary perturbations. To this end, such equations will be rewritten in terms of curvilinear Clairaut coordinates (Kopal, 1980) in which the gravitational potential defining equilibrium surfaces plays the role of the radial coordinate; and their solution constructed for the classical Roche problem in which the oscillating satellite of infinitesimal mass consists of material which is homogeneous and incompressible, while its primary component acts gravitationally as a mass-point. The outcome of such a solution agrees satisfactorily with that previously established by Chandrasekhar (1963) on the basis of the virial theorem; but the method employed by us lends itself more readily to a generalization of the Roche limit to systems of finite mass ratios and consisting of the components of finite size.     

Epstein weight functions for non-radial oscillations of polytropes

Weight functions for the determination of the periods of linear adiabatic non-radial oscillations have been calculated in the same manner as Epstein's classic treatment of purely radial oscillations. Quadrupole (l=2) oscillations for thef and lower orderp andg-modes were considered. One group of static models were polytropes in the range 1.0n4.0 with ; thus included were configurations that were convectively stable, unstable and neutrally stable throughout. Another group consisted ofn=3.0 polytropes with convective shells or convective cores; ₁ was set at different values in each region in order to produce stability ( ) or instability ( ). The weight function provides a pictorial means for assessing the relative importance of each region of a given static model with respect to generating a given non-radial mode.     

Equilibrium of self-gravitating polytropic cylinders with a magnetic field

The effect of a prevalent magnetic field on static and uniformly rotating self-gravitating cylinders of infinite length is examined. The magnetic field is assumed to consist ofH andH _z components, which are taken to be functions of the radial coordinate alone. A variety of magnetic-field configurations are shown to be admissible solutions of equations of motion, from which some feasible cases are presented. A particular magnetic-field configuration having bothH andH _z components is studied in detail. The configuration is such that the assumption of a polytropic equation of state reduces the equation governing the density function to a non-homogeneous cylindrical analogue of the Lane-Emden equation for spherical polytropes. The homogeneous case is also studied and shows interesting magnetic-field patterns.     

General integral transform of the exponential integrals

General integral transform of the exponential integralsE _n is considered and will be denoted asB _(k) ⁿ (). Different expressions and the equations satisfied byB _(k) ⁿ are developed. Two-term recurrence formula forB _(k) ⁿ (0) and three-term recurrence formula forB _(k) ⁿ (); 0 will be established for a givenk1 andn=2,3, ...,N. The computational algorithms based on these formulae are also constructed for the casesk=1,2,3, andn2. Finally the numerical results fork=2,3 andn=2(1)25 are presented to 15-digit accuracy     

Intrinsic variational equations for conservative dynamical systems withN degrees of freedom

For a conservative dynamical system withn deg. of freedom we show that the equations of variation along an orbit may be written with respect to an orthonormal moving frame (a generalized Frenet frame) in which the tangential variation is given by a quadrature and the normal andn-2 binormal variations are solutions ofn-1 coupled second order equations of the form of Hill's equation.     

Collisions of galaxies of different masses, forms and sizes

We study the tidal effects of close collisions between two spherical galaxies of various mass and mass distributions by numerical simulations. The galaxies are represented by polytropes of indices n=4 and n=0 which denote cases of highcentral concentration and uniform mass distribution respectively.The initial relative velocity of the galaxies is chosen to be 700 km s^-1.The results indicate that the tidal effects are quite sensitive to both mass and mass distribution of the galaxies. The dependence oftidal capture and tidal disruption on the choice of the model and mass ratio is investigated. The classification of collisionis given for each simulation. The results also indicate that in a collision between two identical galaxies relatively more spin is imparted to the galaxies ifthey are centrally concentrated than if they are homogeneous.