Use of Roche coordinates in the problems of small oscillations of tidally distorted stellar models

The system of Roche coordinates developed by Kopal to study the problems of stars in close binary systems has been used to study the problems of small oscillations of tidally distorted stars.     

Use of Roche coordinates in the problems of small oscillations of tidally-distorted stellar models

Kopal's method of Roche coordinates used by us in an earlier paper (Mohan and Singh, 1978) to study the problems of small oscillations of tidally-distorted stars has been extended further to take into account the effect of second-order terms in tidal distortion. Our results show that the effect of including terms of second order of smallness in tidal distortion in the metric coefficients of the Roche coordinates of tidally distorted stars is quite significant, especially in case of stars with extended envelopes and (or) larger values of the mass ratio of the companion star producing tidal distortion. Some of the models which were earlier found stable against small perturbations now become dynamically unstable with the inclusion of the terms of second order of smallness in tidal effects.At present on leave of absence with the department of Mathematics, College of Science, Baghdad, Iraq.     

Equilibrium structures of rotationally and tidally distorted stellar models

In this paper we present a method for computint the equilibrium structures of rotationally distorted stars as well as rotationally and tidally distorted primary components of the stars in binary systems. The method is based on the averaging technique of Kippenhahn and Thomas (1970) and utilizes the concepts of Roche equipotentials (Kopal, 1972). The method takes into account terms up to second-order of smallness in the rotational and tidal distortion parameters. The use of the method in obtaining the equilibrium structures of certain rotationally and (or) tidally distorted models of Main-Sequence stars is also illustrated.     

Roche harmonics for tidally-distorted stellar models

Kopal's analysis for Roche harmonics associated with the Roche model of a star, distorted by the tidal forces of a companion star, has been extended further to obtain the explicit expressions for Roche harmonics when terms up to second order of smallness in tidal effects are considered.On leave from University of Roorkee, Institute of Paper Technology, Saharampur (U.P.), India.     

Nonradial oscillations of models of the generalized Roche series

Nonradial oscillations of certain models of the generalized Roche series have been investigated numerically to further understand the nature of the eigen-values and the eigen-functions of the modes of nonradial oscillations of stellar models.     

Eigenfrequencies of rotationally and tidally distorted white dwarf models of stars

In the present paper we have studied the eigenfrequencies of small adiabatic barotropic pseudo-radial and nonradial modes of oscillations of the white dwarf models of rotating stars in binary systems. In this work the methodology of Mohan and Saxena (in Astrophys. Space Sci. 113:155, 1985) has been used that utilizes the averaging technique of Kippenhahn and Thomas (in Proc. IAU Colloq., vol. 4, p. 20, 1970) and certain results on Roche equipotential as that given by Kopal (in Advances in Astronomy and Astrophysics, Academic Press, 1972). The objective of this study is to investigate the effects of rotation and/or tidal distortion on the periods of oscillations of rotationally and/or tidally distorted white dwarf models of stars assuming it to be the primary component of the binary system and rotating uniformly. The results of present study show that the eigenfrequencies (both radial and nonradial modes) of the rotationally distorted and rotationally and tidally distorted white dwarf model of stars in binary systems tend to decrease under the influence of rotational distortions and rotational and tidal distortions, respectively. However, results are contrary for tidally distorted white dwarf model of stars.     

Effect of rotational and tidal distortions on the periods of small adiabatic oscillations of stellar models

Kopal's method of representing the inner structure of a rotationally and tidally-distorted star by a rotationally-and tidally-distorted Roche model has been used in conjunction with an averaging concept introduced by Kippenhahn and Thomas of representing the equipotential surfaces of a rotationally and tidally distorted stellar model by the equipotential surfaces of an equivalent spherical model to determine the combined effects of rotation and tidal distortion on the periods of small adiabatic oscillations of a stellar model.     

Linear and nonlinear oscillations of simple stellar models

A general property for the stability of the one zone model is derived in the case of linear nonadiabatic radial oscillations. The problem of the nonlinear adiabatic radial oscillations of the homogeneous model with a slow rotational velocity is discussed.     

Radiative transfer effects in rotationally distorted stars

A general expression for the gravity darkening of the tidally and non-uniformly rotating Roche components of close binary systems is used to calculate the uniform rotational effects on line profiles in an expanding atmosphere. We consider a non-local thermodynamic equilibrium (non-LTE) two-level atom approximation in an extended atmosphere, and use Von Zeipel's theorem for the incident radiation at the maximum optical depth  (τ=τ_max)  in the atmosphere. These calculations are performed with uniform rotational velocities of 1, 4 and 8 mtu (mean thermal units). It is found that rotation dilutes the radiation field which is similar to the expansion velocity.
We also study rotational aspects, which make the outer layers of the star distorted. The equation of line transfer is solved in the comoving frame of the expanding atmosphere of the primary using complete redistribution in the line. We use a linear law for the velocity of expansion such that the density varies as r ⁻³, where r is the radius of the star, satisfying the law of conservation of mass. It is found that rotation broadens the line profile, and P-Cygni-type line profiles are obtained.     

The geometry of the Roche coordinates

The exact geometry of the Roche curvilinear coordinates (, , ) in which corresponds to the zero-velocity surfaces is investigated numerically in the plane, as well as in the spatial, case for various values of the mass-ratio between the two point-masses (m ₁,m ₂) constituting a binary system.The geometry of zero-velocity surfaces specified by -values at the Lagrangian points are first discussed by taking their intersections with various planes parallel to thexy-, xz- andyz-planes. The intersection of the zero-velocity surface specified by the -value at the Lagrangian equilateral-triangle pointsL _4,5 with the planex=1/2 discloses two invariable curves passing through the pointsL _4,5 and situated symmetrically with respect to thexy-plane whose form is independent of the mass-ratio.The geometry of the remaining two coordinates (, ) orthogonal to the zero-velocity surfaces is investigated in thexy- andxz-planes from extensive numerical integrations of differential equations generated from the orthogonality relations among the coordinates. The curves (x, y)=constant in thexy-plane are found to be separated into three families by definite envelopes acting as boundaries whose forms depend upon the mass-ratio only: the inner -constant curves associated with the masspointm ₁, the inner -constant curves associated with the mass-pointm ₂ and the outer -constant curves. All the -constant curves in thexy-plane coalesce at either of the Lagrangian equilateraltriangle pointsL _4,5, except for a limiting case coincident with thex-axis. The curves (x, z)=constant in thexz-plane are also separated by definite envelopes depending upon the mass-ratio into different families: the inner -constant curves associated with the mass-pointm ₁, the inner -constant curves associated with the mass-pointm ₂ and the outer -constant curves on both sides out of the envelopes. For larger values ofz, the curves =constant tend asymptotically to the line perpendicular to thex-axis and passing through the centre of mass of the system, except for a limiting case coincident with thex-axis. The geometrical aspects of the envelopes for the curves (x, y)=constant in thexy-plane and the curves (x, z)=constant in thexz-plane are also discussed independently.In the three-dimensional space, the Roche coordinates can be conveniently defined in such a way as to correspond to the polar coordinates in the immediate neighbourhood of the origin, and to the cylindrical coordinates at great distances. From numerical integrations of simultaneous differential equations generating spatial curves orthogonal to the zero-velocity surfaces, the surfaces (x, y, z)=constant and the surfaces (x, y, z)=constant are constructed as groups of such spatial curves with common values of some parameters specifying the respective surfaces.On leave of absence from the University of Tokyo as an Honorary Fellow of the Victoria University of Manchester.     

On the integrability of the Roche coordinates

The aim of the present paper is to prove that the system of partial differential equations, which define a set of curvilinear coordinates , , that are orthogonal to the Roche equipotentials (r, , ) incorporating the effects of both rotationaland tidal distortion, does not admit of any formal integrals; and can be solved only numerically in an asymptotic manner. This fact is related with analytic properties of the problem of three bodies, in which represents the potential.     

The Roche coordinates in non-synchronous binaries

The Cayley-Darboux problem for the Roche model of binaries is reinvestigated. Generalised Roche coordinates are then defined and calculated in the form of power series of potential for the general case of non-synchronous binaries with eccentric orbits.     

The structure of the tidally and rotationally distorted polytropes

A study of the structure of tidally and rotationally distorted polytropes has been carried out using Monaghan and Roxburgh (1965) method for rapidly rotating polytropes, at the new interfacial points (Singh and Singh, 1982), ξ_f = 3.025, 3.470, and 4.950 for the polytropic models with the indicesn=1.5, 2.0, and 3.0, respectively. The structure has been studied for various mass ratios too.     

The Roche coordinates for the rotational problem

Curvilinear coordinates in three dimensions associated with the Roche model distorted by centrifugal force alone constitute a Lamé family, of which one (-) coordinate can be defined by equipotential surfaces which are known in closed algebraic form; the other () becomes identical with the meridional planes of the rotationally distorted Roche model; while the third () then follows from the requirements of orthogonality to the others. The explicit form of such coordinates in terms of the polar or cartesian systems has already been established by the author (Kopal, 1970) correctly to quantities of the first order in superficial distortion of the respective Roche model. In the present paper this latter restriction on accuracy will be removed, and expressions constructed for the -coordinate in the form of infinite series which are exact and converge rapidly for any distortion below that which entails equatorial break-up.     

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).     

Eigenfrequencies of small adiabatic barotropic modes of oscillations of rotationally and tidally-distorted stars

In this paper a method is proposed for computing the eigenfrequencies of small adiabatic barotropic modes of oscillations of rotationally and tidally-distorted stars. The method utilizes Kippenhahn and Thomas approach and concepts of Roche equipotentials to incorporate up to second-order the effects of rotation and tidal distortion terms on the eigenfrequencies. The proposed method has also been used to compute the eigenfrequencies of certain barotropic modes of oscillation of some rotationally and tidally distorted models of 10M , and 2.5M Main-Sequence stars.     

Excitation of solar-like oscillations: From PMS to MS stellar models

The amplitude of solar-like oscillations results from a balance between excitation and damping. As in the sun, the excitation is attributed to turbulent motions that stochastically excite thep modes in the uppermost part of the convective zone. We present here a model for the excitation mechanism. Comparisons between modeled amplitudes and helio and stellar seismic constraints are presented and the discrepancies discussed. Finally the possibility and the interest of detecting such stochastically excited modes in pre-main sequence stars are also discussed.     

Effect of central condensation on the modes of nonradial oscillations of stellar models

Modes of nonradial oscillations of six composite polytropic models have been investigated numerically to study the effect of central condensation parameter being the density at the centre and the mean density of a stellar model) on the modes of nonradial oscillations of stellar models.