Effects of rotation and tidal distortions on the periods of adiabatic oscillations of composite models of stars

Mohan and Saxena's approach of using the averaging technique of Kippenhahn and Thomas in conjunction with Kopal's method of evaluating various parameters on the Roche equipotentials has been used to compute the effects of rotation and tidal distortions on the periods of small adiabatic radial and nonradial modes of oscillations of a series of composite models of stars. In these stars the density decreases slowly in the core from the centre to the interface and then falls of rapidly in the envelope from the interface to the outer surface.     

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.     

Effect of rotation and tidal distortions on the structure of polytropic stars

The averaging technique of Kippenhahn and Thomas has been used in conjunction with Kopal's method of evaluating various parameters on the Roche equipotentials, to determine the effects of rotation and tidal distortions on the shapes and structures of the polytropic models of the stars.     

Effects of rotation and tidal distortions on the shapes of radial velocity curves of polytropic models of pulsating variable stars

Anharmonic oscillations of rotating stars have been studied by various authors in literature to explain the observed features of certain variable stars. However, there is no study available in literature that has discussed the combined effect of rotation and tidal distortions on the anharmonic oscillations of stars. In this paper, we have created a model to determine the effect of rotation and tidal distortions on the anharmonic radial oscillations associated with various polytropic models of pulsating variable stars. For this study we have used the theory of Rosseland to obtain the anharmonic pulsation equation for rotationally and tidally distorted polytropic models of pulsating variable stars. The main objective of this study is to investigate the effect of rotation and tidal distortions on the shapes of the radial velocity curves for rotationally and tidally distorted polytropic models of pulsating variable stars. The results of the present study show that the rotational effects cause more deviations in the shapes of radial velocity curves of pulsating variable stars as compared to tidal effects.     

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.     

Structure-distortion of polytropic stars by differential rotation: Third-order perturbation theory

In order to specify the structure of a rapidly and differentially rotating gaseous polytrope, we extend Chandrasekhar's perturbation theory to include third-order terms in the perturbation parameter. In the present paper, the theory developed is required for a subsequent numerical treatment of the structure-determination.This research was supported by the Research Development Project of the University of Patras, Greece.     

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 ²=b ₀+b ₁ s ²+b ₂ s ⁴, 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 parametersb ₀,b ₁, andb ₂ 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.     

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.     

A comment on some oscillatory models of adiabatic stars

Three different oscillatory models of adiabatic stars are reinvestigated. These are the homogenous model, the inverse square model and the Roche model. The ratio between the amplitude of the oscillations and the distance from the center is developed in a power series. For physical conclusions to be drawn, it turns out to be crucial if the power series is divergent or convergent. Mathematical arguments are given which show that the power series are really divergent for all three models.     

The effect of toroidal magnetic field and differential rotation on self-gravitating polytropic models

In this paper we construct a polytropic model distorted by toroidal magnetic field and differential rotation. We then compute states of critical rotation of this model. In the computations we implement the so-called complex-plane strategy and multiple partition technique which are numerical methods deviced recently by the first author.     

Effects of rotation on internal structure of the stars

The aim of the present paper will be to establish the explicit form of the equations which govern the internal structure of stars rotating with constant angular velocity formulated in terms of Clairaut coordinates (cf. Kopal, 1980) in which the radial coordinate is replaced by the total potential, which for equilibrium configurations remains constant over distorted level surfaces. The introductory Section 1 contains an account of previous work on rotating stars, commencing with Milne (1923), von Zeipel (1924) and Chandrasekhar (1933), who all employed orthogonal coordinates for their analysis. In Section 2 we shall apply to this end the curvilinear Clairaut coordinates introduced already in our previous work (cf. Kopal, 1980, 1981); and although these are not orthogonal, this disadvantage is more than offset by the fact that, in their terms, the fundamental equation of our problem will assume the form of ordinary differential equations, subject to very simple boundary conditions. The explicit form of these equations — exact to terms of fourth order in surficial distortion caused by centrifugal force—will be obtained in Section 3; while in the concluding Section 4 these will be particularized (for the sake of comparison with work of previous investigators) to stars of initially polytropic structure. These will prove to be much simpler in Clairaut coordinates than they were in any previously used frame of reference. Lastly, in Appendix A we shall present the explicit forms, in Clairaut coordinates, of the differential operators which were needed to establish the results given in Sections 3–4; while Appendix B will summarize other auxiliary algebraic relations of which use was made to formulate our fourth-order theory developed in Section 3.     

Effect of the rotation and tidal dissipation history of stars on the evolution of close-in planets

Since 20 years, a large population of close-in planets orbiting various classes of low-mass stars (from M-type to A-type stars) has been discovered. In such systems, the dissipation of the kinetic energy of tidal flows in the host star may modify its rotational evolution and shape the orbital architecture of the surrounding planetary system. In this context, recent observational and theoretical works demonstrated that the amplitude of this dissipation can vary over several orders of magnitude as a function of stellar mass, age and rotation. In addition, stellar spin-up occurring during the Pre-Main-Sequence (PMS) phase because of the contraction of stars and their spin-down because of the torque applied by magnetized stellar winds strongly impact angular momentum exchanges within star–planet systems. Therefore, it is now necessary to take into account the structural and rotational evolution of stars when studying the orbital evolution of close-in planets. At the same time, the presence of planets may modify the rotational dynamics of the host stars and as a consequence their evolution, magnetic activity and mixing. In this work, we present the first study of the dynamics of close-in planets of various masses orbiting low-mass stars (from \(0.6~M_\odot \) to \(1.2~M_\odot \)) where we compute the simultaneous evolution of the star’s structure, rotation and tidal dissipation in its external convective envelope. We demonstrate that tidal friction due to the stellar dynamical tide, i.e. tidal inertial waves excited in the convection zone, can be larger by several orders of magnitude than the one of the equilibrium tide currently used in Celestial Mechanics, especially during the PMS phase. Moreover, because of this stronger tidal friction in the star, the orbital migration of the planet is now more pronounced and depends more on the stellar mass, rotation and age. This would very weakly affect the planets in the habitable zone because they are located at orbital distances such that stellar tide-induced migration happens on very long timescales. We also demonstrate that the rotational evolution of host stars is only weakly affected by the presence of planets except for massive companions.     

Effects of rotation on the colours and line indices of stars

The intrinsicuvby and HΒ indices of member stars of α-Persei, Pleiades and Scorpio-Centaurus association have been analysed in detail for rotation effects. These stars range in spectral type from B0 to F0 and the observed effects of rotation are found to be in agreement with photometric effects calculated by Collins & Sonneborn (1977) for rigidly rotating B0 to F0 stars On leave of absence from Assumption College, Changanacherry, Kerala.     

The effect of toroidal magnetic field and differential rotation on self-gravitating polytropic models,II

In this paper we compute differentially rotating polytropic models distorted by toroidal magnetic field. In particular, we study rotating sequences, which do not terminate with a critical rotation. In the computations we use the so-called complex-plane strategy and multiple partition technique, which are numerical methods developed recently by the first author.     

The Monitor project: rotation periods of low-mass stars in M50

We report on the results of a time-series photometric survey of M50 (NGC 2323), a  ∼130 Myr  open cluster, carried out using the Cerro Tololo Inter-American Observatory (CTIO) 4-m Blanco telescope and Mosaic-II detector as part of the Monitor project. Rotation periods were derived for 812 candidate cluster members over the mass range  0.2 ≲ M /M_⊙≲ 1.1  . The rotation period distributions show a clear mass-dependent morphology, statistically indistinguishable from those in NGC 2516 and M35 taken from the literature. Due to the availability of data from three observing runs separated by ∼10 and 1 month time-scales, we are able to demonstrate clear evidence for evolution of the photometric amplitudes, and hence spot patterns, over the 10 month gap. We are not able to constrain the time-scales for these effects in detail due to limitations imposed by the large gaps in our sampling, which also prevent the use of the phase information.     

Effects of rotation on colours and line indices of stars

We have analysed the broad-bandUBV colours and the intermediate banduvby colours of Persei, Pleiades, and the Scorpio-Centaurus association for rotation effects. An attempt was made to see if we can discriminate normal single stars from that of binary and peculiar stars after taking the observed rotation effects into account. It is found that the spread in the observed colours does not allow in general such a discrimination except that the objects with large reddening are double-lined binaries, peculiar stars or emission-lined objects. The few normal stars in these three clusters with such large reddening are listed as they are likely to belong to one of the above classes.On leave of absence from Assumption College, Changanacherry, Kerala.     

Effect of interaction of the various modes on the radial velocity curves of the polytropic models of rotationally and tidally distorted pulsating variable stars

In our previous work, we developed a model to study the effects of rotation and/or tidal distortions on anharmonic radial oscillations and hence on the radial velocity curves of the polytropic models of pulsating variable stars.We considered the first three modes(fundamental and the next two higher modes) for the polytropic models of index 1.5 and 3.0 in that work.In the present paper, we are further extending our previous work to study the effect of the interaction of various modes on anharmonic radial oscillations and hence on radial velocity curves of the rotationally and/or tidally distorted polytropic models of pulsating variable stars.For this purpose, we have considered the following cases:(i) fundamental mode(ii) fundamental and the first mode,(iii) fundamental and the next two modes and finally(iv) fundamental and the next three higher modes of pulsation in our study.The objective of this paper is also to investigate whether the interaction of various modes affects the results of our previous study or not.The results of this study show that the interaction of the fundamental mode with higher modes appreciably changes the shape of the radial velocity curve of rotationally distorted and rotationally and tidally distorted polytropic models of pulsating variable stars.     

Thermal and tidal effect on the rotation of Mercury

It is shown that the influences of the thermal and tidal effects on Mercury's libration are in equilibrium with the periods of rotation and revolution of Mercury locked in the 32 resonant state. The suggestion by Liu that the solar gravitational couple on the thermal bulges accelerates Mercury's rotation is investigated and the production of mechanical energy to balance the dissipation of the bodily tides is discussed. It is possible for Mercury to rotate with two bulges as a solar thermal engine; the tidal effect causes this engine to function and its maximum power is close to 10¹⁶ ergs per sec.     

Numerical solutions of the equation concerning small adiabatic oscillations of pulsating components in double star systems

The equation governing small adiabatic radial oscillations for pulsating components in close binary stars modelled by tidally and rotationally distorted Roche geometry is solved numerically. With assumed initial conditions, solutions for systems with different mass ratios are presented. The changes in relative wave amplitude with various parameters are shown. The variation of the ratio of the pulsation frequencies of distorted to undistorted stars for given mass ratio of the binary systems is also investigated. Observational evidence is examined by using two data sets that show the modelled effects, by taking into account likely practical factors. The results show that the measured frequencies and amplitudes of surface waves can vary slightly for distorted stars in comparison to undistorted ones (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)