Polytropic Structure Research Articles

Light curves from symmetric, polytropic contact binaries

ABSTRACT We study the structure of contact binaries assuming a polytropic relation between pressure and density, restricting ourselves to the case of equal-mass components, i.e. symmetric contact binaries. In this case, matter is at rest in the corotating reference frame making this problem far simpler than the general case of non-symmetric contact binaries. We compute these structures assuming values of the polytropic index of n = 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, and 3.5 employing a self-consistent technique due to Hachisu. As a part of the results, we find the shape of their surfaces. While for the case of n = 3.5, such surfaces are very close to those corresponding to equipotentials of the restricted Lagrangian three-body problem, for lower n values the departure is remarkable. We propose a generalized function to fit these surfaces, which allows us to perform an accurate integration of the light curve due to the object. Then, for values of n > 0.0 we computed a family of light curves considering different inclinations and values for the width of the neck connecting the components (or equivalently, the fillout factor) of the pair. We compare our calculations with the solution found for the symmetric contact binary V803 Aquilae by employing the popular phoebe code, that assumes the above-mentioned Lagrangian equipotential surfaces. We conclude that considering polytropic structures leads to parameters that may be appreciably different from those deduced by assuming that their surfaces correspond to equipotentials of the restricted three-body problem.

Structure of polytropic stars in General Relativity

The inner structure of a star or a primordial interstellar cloud is a major topic in classical and relativistic physics. The impact that General Relativistic principles have on this structure has been the subject of many research papers. In this paper we consider within the context of General Relativity a prototype model for this problem by assuming that a star consists of polytropic gas. To justify this assumption we observe that stars undergo thermodynamically irreversible processes and emit heat and radiation to their surroundings. Due to the emission of this energy it is worthwhile to consider an idealized model in which the gas is polytropic. To find interior solutions to the Einstein equations of General Relativity in this setting we derive a single equation for the cumulative mass distribution of the star and use Tolman-Oppenheimer-Volkoff equation to derive formulas for the isentropic index and coefficient. Using these formulas we present analytic and numerical solutions for the polytropic structure of self-gravitating stars and examine their stability. We prove also that when the thermodynamics of a star as represented by the isentropic index and coefficient is known, the corresponding matter density within the star is uniquely determined.

Stability of oscillating gaseous masses in massive Brans–Dicke gravity

This paper explores the instability of gaseous masses for the radial oscillations in post-Newtonian correction of massive Brans–Dicke (BD) gravity. For this purpose, we derive linearized perturbed equation of motion through Lagrangian radial perturbation which leads to the condition of marginal stability. We discuss radius of instability of different polytropic structures in terms of the Schwarzschild radius. It is concluded that our results provide a wide range of difference with those in general relativity and BD gravity.

The equilibrium of overpressurized polytropes

We investigate the impact of an external pressure on the structure of self-gravitating polytropes for axially symmetric ellipsoids and rings. The confinement of the fluid by photons is accounted for through a boundary condition on the enthalpy $H$. Equilibrium configurations are determined numerically from a generalised "Self-Consistent-Field"-method. The new algorithm incorporates an intra-loop re-scaling operator ${\cal R}(H)$, which is essential for both convergence and getting self-normalised solutions. The main control parameter is the external-to-core enthalpy ratio. In the case of uniform rotation rate and uniform surrounding pressure, we compute the mass, the volume, the rotation rate and the maximum enthalpy. This is repeated for a few polytropic indices $n$. For a given axis ratio, over-pressurization globally increases all output quantities, and this is more pronounced for large $n$. Density profiles are flatter than in the absence of an external pressure. When the control parameter asymptotically tends to unity, the fluid converges toward the incompressible solution, whatever the index, but becomes geometrically singular. Equilibrium sequences, obtained by varying the axis ratio, are built. States of critical rotation are greatly exceeded or even disappear. The same trends are observed with differential rotation. Finally, the typical response to a photon point source is presented. Strong irradiation favours sharp edges. Applications concern star forming regions and matter orbiting young stars and black holes.

Perturbative solution to the Lane–Emden equation: an eigenvalue approach

Under suitable scaling, the structure of self-gravitating polytropes is described by the standard Lane-Emden equation (LEE), which is characterised by the polytropic index $n$. Here we use the known exact solutions of the LEE at $n=0$ and $1$ to solve the equation perturbatively. We first introduce a scaled LEE (SLEE) where polytropes with different polytropic indices all share a common scaled radius. The SLEE is then solved perturbatively as an eigenvalue problem. Analytical approximants of the polytrope function, the radius and the mass of polytropes as a function of $n$ are derived. The approximant of the polytrope function is well-defined and uniformly accurate from the origin down to the surface of a polytrope. The percentage errors of the radius and the mass are bounded by $8.1 \times 10^{-7}$ per cent and $8.5 \times 10^{-5}$ per cent, respectively, for $n\in[0,1]$. Even for $n\in[1,5)$, both percentage errors are still less than $2$ per cent.

Explosive disruption of polytropes: a one-dimensional hydrodynamic calculation

We study explosions of stellar models using a one-dimensional Lagrangian hydrodynamics code. We calculate how much mass is liberated as a function of the energy of explosion for a variety of pre-explosion polytropic structures and for equations of state with a range of radiation-to-gas pressure ratios. The results show that simple assumptions about the amount of mass lost in an explosion can be quite inaccurate, and that even one-dimensional stellar models exhibit a rich phenomenology. The mass loss fraction rises from about 50 to 100 per cent as a function of the explosion energy in an approximately discontinuous manner. Combining our results with those of other, more realistic models, we suggest that Nova Scorpii (J1655 - 40) may have experienced significant mass fallback because the explosion energy was less than the critical value. We infer that the original progenitor was less than twice the mass of today's remnant.

Equilibrium Structures of Differentially Rotating Primary Components of Binary Stars

In this paper a method is proposed for computing the equilibrium structures and various other observable physical parameters of the primary components of stars in binary systems assuming that the primary is more massive than the secondary and is rotating differentially about its axis. Kippenhahn and Thomas averaging approach (1970) is used in a manner earlier used by Mohan, Saxena and Agarwal (1990) to incorporate the rotational and tidal effects in the equations of stellar structure. Explicit expressions for the distortional terms appearing in the stellar structure equations have been obtained by assuming a general law of differential rotation of the typeω2 = b0+b1s2+b2s4, where ω is the angular velocity of rotation of a fluid element in the star at a distance s from the axis of rotation, and b0, b1, b2 are suitably chosen numerical constants. The expressions incorporate the effects of differential rotation and tidal distortions up to second order terms. The use of the proposed method has been illustrated by applying it to obtain the structures and observable parameters of certain differentially rotating primary components of the binary stars assuming the primary components to have polytropic structures.

Equilibrium structure of differentially rotating polytropic models of the stars

In this paper we propose a method for computing the equilibrium structure of differentially rotating polytropic models of the stars. A general law of differential rotation of the type ω2=b0+b1s2+b2s4, which can account for a reasonably large variety of possible differential rotations in the stars has been used. The distortional effects have been incorporated in the structure equations up to second order of smallness in distortion parametersb0,b1, andb2 using Kippenhahn and Thomas' averaging approach in conjunction with Kopal's results on Roche equipotentials in manner similar to the one earlier used by Mohan and Saxena for computing the equilibrium structure of polytropes having solid body rotation. Numerical results have been obtained for various types of differentially rotating polytropic models of stars of polytropic indices 1.5, 3, and 4. Certain differentially rotating models of the Sun which are possible with such a type of law of differential rotation, have also been computed.

Gravitation in 2+1 dimensions.

We investigate gravitational field theories in 2+1-spacetime dimensions. The consequences of the lack of a Newtonian limit to general relativity are reviewed. Further insight into the implications of this fact is gained by considering a new, general class of exact hydrostatic solutions. We show that all self-gravitating polytropic structures have the same gravitational mass and produce matter-filled spaces of finite spatial volume. Other theories of gravitation are also considered and the behavior of one such theory with a Newtonian limit is studied. Cosmological solutions of these gravitational theories are also studied in detail.

The breakup of self-gravitating rings, tori, and thick accretion disks

view Abstract Citations (66) References (46) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Breakup of Self-gravitating Rings, Tori, and Thick Accretion Disks Tohline, Joel E. ; Hachisu, Izumi Abstract Using Hachisu's self-consistent-field technique, we have constructed equilibrium sequences of self-gravitating, axisymmetric rings (or, tori) having n = 3/2 polytropic structures and radial angular velocity profiles of the form {OMEGA} is proportional to R^-1/2^, with l = 3.0, 3.5, and 4.0. Then, using Tohline's three-dimensional hydrodynamic computer code, we have identified the position along the equilibrium sequences where the rings first become dynamically unstable toward the development of an m = 2 (ellipsoidal), nonaxisymmetric distortion. (Principally for computational convenience, we have limited this study to nonaxisymmetric distortions with even azimuthal mode numbers.) In terms of the ratio of rotational kinetic energy T to gravitational potential energy W, the instability arises in systems having T/|W| >= 0.16. Although the dynamics of odd azimuthal modes has not been examined here, we suspect that tori having smaller values of T/|W| are dynamically stable to all nonaxisymmetric modes because the only odd mode having a longer azimuthal wavelength--the m = 1 mode---cannot develop without requiring a shift in the center of mass of the torus. We conclude that the onset of a dynamical instability in self- gravitating rings occurs at a value of T/|W| that is substantially lower than the value (T/|W|~0.27) at which an analogous instability arises in centrally condensed, self-gravitating configurations. We have quantitatively measured the growth rate and the pattern speed of the m = 2 mode in eight models having values of T/|W| in the range 0.167 <= T/|W| <= 0.271. The character of the unstable eigenmode in these models is clearly different from the unstable, nonaxisymmetric eigenmode that was first shown by Papaloizou and Pringle to arise in zero mass, accretion tori. We are therefore convinced that the instability we have identified here is driven by the self-gravity of the torus and is distinctly different from the Papaloizou-Pringle instability. Although the early nonlinear development of this instability causes the torus to deform into an ellipsoidal configuration with noticeable density enhancements arising on opposite sides of the ring, further evolution shows that the instability does not lead to fragmentation of the torus. We therefore call into question the perceived notion that self-gravitating rings are generally unstable toward fragmentation along their length. Publication: The Astrophysical Journal Pub Date: October 1990 DOI: 10.1086/169205 Bibcode: 1990ApJ...361..394T Keywords: Accretion Disks; Galactic Evolution; Galactic Structure; Star Formation; Angular Velocity; Self Consistent Fields; Astrophysics; ACCRETION; GALAXIES: FORMATION; GALAXIES: STRUCTURE; HYDRODYNAMICS; STARS: FORMATION full text sources ADS |

The formation and evolution of tidal binary systems

view Abstract Citations (96) References (38) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Formation and Evolution of Tidal Binary Stytems McMillan, Stephen L. W. ; McDermott, P. N. ; Taam, Ronald E. Abstract Cross sections for the tidal capture binary formation process are calculated for a variety of stellar models. The formalism used in the determination of the energy dissipated by a close encounter between two unbound stars and the associated capture cross sections are reviewed. The case of an n = 3/2 polytropic structure is calculated with the formalism, and the behavior of realistic stellar models is considered, including Population II main-sequence stars with masses of 0.4, 0.8, and 1.5 solar. The calculation is repeated for a 'slightly evolved' 0.8 solar mass star just as it begins to leave the main sequence, and the behavior of more evolved stars is discussed. A quasi-adiabatic analysis is used to estimate the time scale on which the pulsation energy is actually dissipated internally or radiated away. This analysis also indicates where in the star most of the dissipation takes place, allowing the stellar response to be estimated by including the heating in the equations of stellar structure. Publication: The Astrophysical Journal Pub Date: July 1987 DOI: 10.1086/165365 Bibcode: 1987ApJ...318..261M Keywords: Binary Stars; Main Sequence Stars; Star Formation; Stellar Evolution; Stellar Oscillations; Tides; Energy Dissipation; Globular Clusters; Polytropic Processes; Stellar Envelopes; Stellar Models; Stellar Structure; Astrophysics; STARS: BINARIES; STARS: EVOLUTION; STARS: PULSATION; STARS: STELLAR DYNAMICS full text sources ADS |

The structure of tidally distorted polytropes

A numerical study of the structure of tidally-distorted polytropes has been carried out by using the Monaghan and Roxburgh (1965) method.

A Convective Model of Solar Granulation

Some years ago the full non-linear equations, within the one mode approximation, describing finite amplitude convection in a compressive medium with polytropic structure were derived (Van der Borght 1977). It is the purpose of the present paper to report on the numerical integrations of these equations and to show how these can be used to construct a convective model of granulation. In this way one may gain some insight into the structure of the outer layers of the Sun.

A numerical study of non-linear convection in a compressible medium

In this paper we consider the problem of non-linear convection in a compressible layer with polytropic structure. After deriving the appropriate forms of the basic conservation equations a single mode expansion is used to obtain a simpler model, within the framework of the anelastic approximation. These equations are integrated, using a band-matrix algorithm, for two characteristic density stratifications. The results are given in the form of graphs and are discussed in detail. An outline of the numerical algorithm is given in an appendix together with a discussion of its effectiveness.

A second-order perturbation theory for differentially rotating gaseous polytropes

The aim of the present investigation has been to specify the structure of a differentially rotating gaseous polytrope, by extending Chandrasekhar's method to include second-order terms in the perturbation parameter. The formal results obtained permit the determination of the polytropic structure for all types of differential rotation of cylindrical symmetry. The formalism developed works efficiently in the limiting case of uniform rotation and the results obtained are compared with those of previous investigations of uniformly rotating polytropes.

OSO 8 X-ray spectra of clusters of galaxies. II - Discussion

The Goddard Space Flight Center 2--20 keV OSO 8 data on X-ray clusters are examined for information which will restrict models for hot intracluster gas structures. From the correlation between X-ray temperature and velocity dispersion, we conclude that the X-ray core radius should be less than the galaxy core radius. Thus fits of X-ray profiles of clusters to isothermal spheres yield a core radius approximately equal to the density cutoff radius rather than the true core radius. If we instead assume a generalized polytropic structure, from the same correlation we may conclude that the gas is more nearly isothermal than adiabatic within a few core radii of the center. Therefore we require thermal conduction, radiative cooling, or both in cluster atmospheres. In both of these interpretations, the Coma cluster and possibly Abell 2319 A are quite discrepant.From the correlation between bremsstrahlung emission integral and temperature, we conclude that hotter clusters contain a larger fraction of their virial mass in the form of gas emitting in the 2--20 keV band. We find that the '/sup 7/central concentration factor'' for the emission cannot vary with temperature steeply enough to account for the correlation.We closely examine relations among the various measures of galaxymore » density and optical cluster morphology. There is evidence for real correlations between optical central cluster properties and X-ray properties which are not ''artifacts'' of underlying richness correlations as suggested by Jones and Forman.We also predict a population of cool, dim X-ray cluster which has not been observed.From consideration of the errors in published atomic physics data and spectrum calculations, we conclude that the iron abundance determinations recently quoted for intracluster gas are uncertain by from 50% to greater than 100% from this nonstatistical cause alone.« less

A polytropic model for a two-component galaxy

A model is presented consisting of two different axially deformed polytropic spheroids, homocentric and coaxial — with arbitrary values for the two masses, the two equatorial radii and the two polytropic indices — interacting with each other only gravitationally. The model represents the two main components, halo and bulge plus disk, of a galaxy. The flattening of the two spheroids is assumed to be due to rigid-body rotation and tidal interaction, and the treatment follows closely the method of Chandrasekhar and Lebovitz for single polytropic structures. All useful quantities are evaluated up to first order in the two rotation frequencies. The main properties of sequences of models intended to mimic evolution at constant masses and constant angular momenta are presented.

A Model for a Two Component Galaxy.

As most of today approaches to the problem of galactic structure are based on the consideration of the two main components of a galaxy, halo and disk, it was thought that even simple models of such kinds of double structures, based only on the application of general theorems, could be of some use, at least for simplified outlooks concerning galac tic evolution. With this aim in mind, we have undertaken the treatment of bodies constituted by two different axially deformed polytropic structures, homocentrical and coaxial, with arbitrary values for the two masses MIand MII, the two radii aIand aIIof the undeformed objects (aIaII) and polytropic indices nIand nII, interacting with each other only gravitationally. The flattening of the spheroids is assumed to be due to solid body rotation and tidal interaction, and the treatment follows closely the method used for such single structures by Chandrasekhar and Lebovitz.

On formation of close binaries by two-body tidal capture

We calculate in detail the two-body tidal capture mechanism of Fabian, Pringle, and Rees: when two unbound stars have a close encounter, they may become bound by the energy that each deposits into nonradial oscillations of the other. After dimensional scalings are removed, the process depends only on a single dimensionless parameter, and on the dimensionless envelope structure of the stars. General formulae are derived; for definiteness, we apply them to the specific case of stars with an n=3 polytropic structure. Capture cross sections as a function of velocity and capture rates for an isothermal distribution are given for the case of equal-mass stars; other cases can easily be computed from the formulae given.

Finite Amplitude Convection in a Compressible Layer with Polytropic Structure

In this paper are derived the basic equations for finite amplitude convection in a compressible medium with polytropic structure within the framework of the one-mode anelastic approximation. Only the case of non self-interacting planforms is considered, and the effects of viscous dissipation are not taken into account. These equations are solved numerically to illustrate the effects of compressibility and density stratification on the flow pattern and on the thermodynamic variables and their fluctuations.