## Abstract

The problems of equilibrium structures and periods of oscillations of rotationally and/or tidally distorted polytropic models of stars in binary systems have been studied previously. In the studies, only the effects of gravitational and centrifugal forces while obtaining the Roche equipotential of such stars were considered. The Coriolis force were not considered. In the present study, we have included the effects of the Coriolis force also. The objective of this study is to investigate as to how the inclusion of the Coriolis force in the Roche equipotential affects the equilibrium structures and periods of oscillations of rotationally and/or tidally distorted stars. The results of our study show that the Coriolis force has no appreciable effect on the equilibrium structures and periods of small oscillations of polytropic models of stars in binary systems. The present study also confirms the earlier conclusions that uniform rotation decreases the eigenfrequencies of various modes of oscillation.

## 1. Introduction

Equilibrium structures and the periods of oscillations of stars in the binary systems are influenced by the presence of gravitational, centrifugal and Coriolis forces. The problem of determining the effects of rotation and/or tidal distortions on the equilibrium structures and the eigenfrequencies of small adiabatic oscillations of stars are quite complex. Approximate methods have, therefore, been usually used in literature to study such problems. Whereas authors such as Chandrasekhar (1933, 1939), Chan & Chau (1979), Deupree & Karakas (2005), James (1964), Landin *et al*. (2009), Linnell (1981), Naylor & Anand (1972), Singh & Singh (1983), Song *et al*. (2009) and Tassoul & Tassoul (1982) have addressed themselves to the problems of equilibrium structures of rotationally and/or tidally distorted stars (from hereafter, we will refer to rotationally and/or tidally distorted stars as RTD stars), Chandrasekhar & Ferrari (1991), Clement (1965), Cox (1980), Hurley *et al*. (1966), Lovekin *et al*. (2009), Marten & Smeyers (1986), Reese *et al*. (2006), Reyniers & Smeyers (2003), Robe (1968), Rocca (1989) and Saio (1981) have discussed the problems of the oscillations of rotating stars and stars in binary systems.

Mohan & Saxena (1983, 1985) and Mohan *et al*. (1990, 1991) used the averaging technique of Kippenhahn & Thomas (1970)—in conjunction with certain results on Roche equipotential, as given by Kopal (1972)—to investigate the problems of stars in binary systems. However, most of their studies only account for the effect of gravitational and centrifugal forces. The effect of the Coriolis force has not been taken into account as it was expected to complicate the analysis. Moreover, this effect was expected to be small when compared with the effects of other two forces.

Lal *et al*. (2008, 2009) have included the effects of the Coriolis force while determining the equilibrium structures of rotating stars and stars in binary systems. However, certain errors were noted in some of their expressions (but these were not of a type to alter the conclusions reached). In Lal *et al*. (2008), a wrong assumption (angular velocity of rotation is equal to angular velocity of revolution) was made while determining the Roche equipotential in the presence of the Coriolis force. It was corrected in the work of Lal *et al*. (2009), where the authors studied the effect of the Coriolis force on the Roche limits of certain non-synchronous binary systems. The formalism for obtaining the Roche equipotential in the presence of the Coriolis force is the same in the present paper and in the work by Lal *et al*. (2009) except for the expression for rotation parameter (*n*_{1}). In Lal *et al*. (2009), the parameter of rotation (*n*_{1}) also depends on angular velocity of revolution. In the present study, this error has been rectified. The parameter of rotation now strictly depends on the angular velocity of rotation. To make our study more extensive, we have also studied in detail the effect of Coriolis force on the periods of barotropic modes of oscillations of RTD stars.

While studying the oscillations of RTD polytropic models of stars, Mohan & Saxena (1985) used the collocation method to find the modes of oscillations. This method involves the determination of the unknown coefficients by using a trial function to satisfy the equations of motion at arbitrary discrete points in the stellar model. They concluded that the effect of inclusion of uniform rotation is to decrease the eigenfrequencies of oscillations of all modes. These results are in accordance with the earlier results of Saio (1981), who used the perturbation method to study the effect of rotation and tidal forces on the non-radial modes of oscillation of polytropic models of stars. However, using the variational method, Clement (1984) has concluded that rotation increases the values of eigenfrequencies of oscillations of *g*-modes, even though values of eigenfrequencies of *f* and *p*-modes decrease because of rotation. In a more recent study, Lovekin *et al*. (2009) have used the more realistic two-dimensional stellar evolution code and have concluded that the eigenfrequencies for all modes decrease because of uniform rotation. In our present study, we also wanted to check whether inclusion of the Coriolis force will in any way affect our earlier conclusion in this regard.

Keeping these factors in mind, in the present paper, we have computed the equilibrium structures and periods of small adiabatic barotropic modes of oscillations of RTD stars in binary systems by taking into account the effect of the Coriolis force in addition to the effects of gravitational and centrifugal forces. The paper is organized as follows: the expression for the Roche equipotential of an RTD star in a binary system is obtained in §2. This expression accounts for the effect of the Coriolis force in addition to the effects of gravitational and centrifugal forces. This modified expression of the Roche equipotential is used in §3 to obtain the equations governing the equilibrium structures of RTD polytropic models of the stars. In §4, expressions for the volumes and surface areas of such polytropic models of the stars are obtained. Eigenvalue boundary problems that determine the eigenfrequencies of small adiabatic pseudo-radial and non-radial modes of oscillations of these polytropic models of stars have been formulated in §5. These eigenvalue boundary problems incorporate the effect of the Coriolis force in addition to the effects of gravitational and centrifugal forces. Numerical computations have been performed in §6 and §7 to obtain the inner structures, values of some physical parameters and eigenfrequencies of small adiabatic pseudo-radial and non-radial modes of oscillations of certain RTD polytropic models of stars. The numerical results thus obtained have been compared with the earlier results of Mohan & Saxena (1983, 1985) in which the effect of the Coriolis force was not considered and with other similar results available in the literature. Finally, certain conclusions of astrophysical significance have also been drawn in §8.

## 2. Expression for the Roche equipotential of a rotationally and tidally distorted primary component in a binary system

A rotating star is a star that is rotating about an axis passing through its centre. In the case of a binary system of stars, we have two stars that are rotating about their axes passing through their centres as well as revolving about the common centre of mass of the system. In the case of most of the observed binary stars, one star (usually called the primary component) is more massive than the other star (called the secondary component).

In a binary system, let *M*_{0} and *M*_{1} be the masses of the two components separated by distance *D*. The primary component of mass *M*_{0} of this binary system is supposed to be much more massive than its companion star of mass *M*_{1}(*M*_{0}≫*M*_{1}), which—for all practical purposes—is regarded as a point mass. The primary component is supposed to have its normal configuration (that is, a star in a binary system distorted by rotation and tidal effects). However, for convenience in the analysis, its inner structure is approximated by the Roche model while computing the potential of the system. (In the case of the Roche model, it is assumed that the total mass of a star is concentrated at its centre and this point mass is surrounded by an evanescent envelope in which density varies inversely as the square of the distance from its centre.) This approximation is reasonably valid for the majority of the realistic stars of main sequence and post-main-sequence stages (cf. Chandrasekhar 1939).

Now suppose that the position of the two components of such a binary system is referred to a rectangular system of cartesian coordinates with origin at the centre of gravity of the primary star of mass *M*_{0}, *x*-axis along the line joining the mass centres of the two stars, and *z*-axis perpendicular to the plane of the orbit of the two components.

Let , and represent the distances of a point *P*(*x*,*y*,*z*) from the centres of gravity of the primary star with centre at *O*, secondary star with centre at *O*_{1} and the centre of gravity *C* ((*d*_{1}, 0, 0), where *d*_{1}=*M*_{1}*D*/(*M*_{0}+*M*_{1})) of the system, respectively. Let *Ω* denote the angular velocity of revolution of the system about a line parallel to *z*-axis that passes through the centre of gravity C of the system and is perpendicular to the *XY*-plane. Also, let *Ω*_{1} be the angular velocity of rotation of the primary component about the *z*-axis. Here we have assumed that the primary component of the binary system is rotating uniformly (solid body rotation). This assumption is not valid in the case of rotating stars and stars in binary systems. However, to avoid complexity in obtaining the final expression of Roche equipotential surfaces, the star in question is assumed to be rotating uniformly. In addition, the angular velocities of rotation and revolution are assumed to be small so that spherical symmetry of the star in question is not distorted to a large extent.

In a binary system, the primary star (which is of interest to us in our present study) is rotating about axis OZ with angular velocity *Ω*_{1} and revolving about an axis parallel to the *z*-axis passing through the common centre of mass C with angular velocity *Ω*. A point *P* inside the primary component will experience the effects of the Coriolis force besides the gravitational and centrifugal forces. For the system as described already, the total potential at a point *P* inside the primary component, which experiences the effect of the Coriolis force besides the gravitational and centrifugal forces, is given by
2.1Here ** V** denotes the velocity of particle of unit mass at point

*P*(

*x*,

*y*,

*z*) with respect to a frame of reference that is rotating with the angular velocity

*Ω*. The first two terms in equation (2.1) correspond to the gravitational potential, which arises owing to the primary and secondary components of the binary system, and the third term is due to centrifugal force. These three terms are the same as obtained earlier by Kopal (1972) in his studies on the problems of the Roche model and its applications to close binary systems. The fourth term

**⋅(**

*V***×**

*Ω***) represents the contribution of the Coriolis force to the potential at point**

*r**P*, where

**is the tangential component of velocity of this particle in the rotating frame of reference. Points inside the rotating star will be subjected to the Coriolis force even when these points do not have any external velocity. This is due to differences in the angular velocity of rotation of the primary and angular velocity of revolution of the non-synchronous binary system. (This difference of course vanishes in the case of synchronous binary stars where velocity of rotation is the same as that of revolution.) In the earlier studies carried out on Roche equipotentials by Kopal (1972), Mohan & Saxena (1983, 1985) and Mohan**

*V**et al*. (1990, 1991), the contribution of this last term

**⋅(**

*V***×**

*Ω***), which arises on account of the Coriolis force, has not been considered, assuming its effect will be negligible. In the present study, we have incorporated it.**

*r*If we write to represent the position vector of the point *P*(*x*,*y*,*z*) inside the star rotating about its axis with angular velocity *Ω*_{1}, the term ** V**⋅(

**×**

*Ω***) can be simplified to give 2.2which is the contribution to potential (at point**

*r**P*(

*x*,

*y*,

*z*) inside the star) owing to the Coriolis force (for a point

*P*outside the primary star, this will be zero unless

*P*has some external velocity). Using equation (2.2) in equation (2.1), we get the modified expression for potential at a point

*P*inside the star in cartesian form as 2.3Following Kopal (1972), equation (2.3) may be expressed in non-dimensional form as follows: 2.4where 2.5In these expressions,

*r**=

*r*/

*D*is the non-dimensional form of

*r*and , , (

*r*,

*θ*,

*ϕ*being the polar spherical coordinates of the point

*P*). Moreover,

*q*=

*M*

_{1}/

*M*

_{0}is a non-dimensional parameter representing the ratio of the mass of the secondary component over the primary component (we assume

*q*≪1).

Equation (2.4) of the potential *ψ** at point *P* incorporates the effect of the Coriolis force in addition to the effect of gravitational and centrifugal forces. If we assume that the angular velocity of revolution *Ω* is identical to Keplerian angular velocity *Ω*_{k} (where ), then in terms of the non-dimensional variables used by us, we get a relation *Ω*^{*2}=*q*+1=2*n*. Using this relation, equation (2.4) can be written in a more simplified form as follows:
2.6where
2.7

In this equation, *n* and *n*_{1} are the parameters that represent the distortions owing to revolution and rotation, respectively, whereas *q* is parameter of tidal distortion. For a binary system rotating synchronously, the angular velocity due to rotation and revolution are the same (that is, *Ω*_{1}=** Ω**). Hence, there will be no explicit term for the Coriolis force. In such cases, equation (2.4) reduces to
2.8Expression (2.8) is same as that obtained earlier by Kopal (1972) for synchronous binaries. Expression (2.8) can also be obtained by substituting

*α*=1 and

*β*=1 in equation (2.6).

In the case of pure rotation (single rotating star) the Coriolis force is not generated as there is no revolution of the centre of the star and hence no rotating frame of reference. In such a case, the expression for Roche equipotential for a purely rotating star that is not subject to tidal effects of the companion star becomes
2.9and it can be obtained from equation (2.6) by setting *q*=0, *β*=1. This expression is the same as that given earlier in Kopal (1972) for pure rotating stars.

Thus, the effect of the Coriolis force is explicitly present in the expression of potential *ψ*^{**} only in the case of non-synchronous binaries. In synchronous binaries and single rotating stars, no explicit term of its effect is present in the expression for potential *ψ*^{**}.

## 3. Effect of the Coriolis force on the equilibrium structures of rotationally and tidally distorted polytropic models of stars

In order to compute the equilibrium structures of RTD stars in binary systems, we assume that the primary component is a uniformly rotating polytropic model. Polytropic models have frequently been used in literature to depict the inner structures of stars. (A polytropic model is a model in which the quantity of heat supplied (say d*q*) is directly proportional to the instantaneous change of temperature (say d*t*), so that d*q*/d*t* is constant.) As the primary component is subject to rotational and tidal distortions, its structure becomes an RTD polytropic model.

Following Mohan & Saxena (1983), if *P*_{ψ} denotes the pressure and *ρ*_{ψ} the density on the equipotential surface (*ψ*^{**}=const.) of an RTD polytropic model of a star, then *ρ*_{ψ} and *P*_{ψ} may be assumed to be connected through the polytropic relation of the type
3.1where *N* is the polytropic index and *P*_{cψ}, *ρ*_{cψ} the values of *P*_{ψ} and *ρ*_{ψ}, respectively, at the centre. Here, *θ*_{ψ} represents some average of the value of the polytropic parameter *θ* at various points on the equipotential surface (*ψ*^{**}=const.). Following Mohan & Saxena (1983), the differential equation governing the equilibrium structure of such an RTD primary component with polytropic structure can be written in non-dimensional form as follows:
3.2where *A* and *B* become
In these series expansions, terms up to the second order of smallness in distortion parameters *n*, *n*_{1}, *q* and up to in *r*_{0} have been retained. The dimensionless constant *K* in equation (3.2) is the ratio of the undistorted radius *R*_{ψ} of the primary to the separation *D* between the centres of the primary and secondary star. In fact,
3.3where
Here, *ξ*_{u} is the value of *ξ* (where *ξ* is the Lane–Emden variable; specifically we have the value of *ξ*_{u}=3.65375, 6.89685 corresponding to polytropic index *N*=1.5 and 3.0, respectively) at the outermost surface of the undistorted polytropic model, *N* is the polytropic index and *G* is the universal gravitational constant.

In the absence of the Coriolis force, that is, on setting *α*=*β*=1, the expressions *A*(*r*_{0},*n*,*n*_{1},*q*) and *B*(*r*_{0},*n*,*n*_{1},*q*) reduce to their corresponding expressions reported earlier in the literature (Mohan & Saxena 1983).

The boundary conditions that equation (3.2) has to satisfy are 3.4aand 3.4b

where *r*_{0s} is the value of *r*_{0} at the surface. Equation (3.2) subject to the boundary conditions (3.4*a*,*b*) determines the equilibrium structures of the RTD polytropic models of stars in the presence of the Coriolis force.

## 4. Computation of volumes and surface areas of rotationally and tidally distorted polytropic models of stars

Following Kopal (1972) and Mohan & Saxena (1983), explicit expressions for the volume *V* _{ψ} and the surface area *S*_{ψ} of an RTD polytropic model that takes into account the effect of the Coriolis force beside the gravitational and centrifugal force become
4.1and
4.2In these expressions, terms up to the second order of smallness in *n*, *n*_{1}, *q* and up to *r*^{10}_{0s} in *r*_{0s} are retained.

In the absence of the Coriolis force (that is, *α*=*β*=1), the expressions for *V* _{ψ}, *S*_{ψ} reported in equations (4.1) and (4.2) reduce to their corresponding expressions as obtained earlier by Mohan & Saxena (1983).

## 5. Effect of the Coriolis force on the eigenfrequencies of pseudo-radial and non-radial modes of oscillations of rotationally and tidally distorted polytropic models of stars

Using the modified expression for potential as obtained in §2 and the approach adopted by Mohan & Saxena (1985), we have also computed the eigenfrequencies of small adiabatic pseudo-radial and non-radial modes of oscillations of RTD stars in binary systems.

Following Mohan & Saxena (1985), the equation governing the pseudo-radial modes of oscillations of an RTD polytropic model that incorporates the effects of the Coriolis force besides the gravitational and centrifugal forces can be expressed in the non-dimensional form as
5.1Here *ω*^{2}=*D*^{3}*r*^{3}_{0s}*σ*^{2}/*GM* and *H*_{1}, *H*_{2}, *H*_{3}, *H*_{4} are nonlinear functions of structure and distortion parameters *α*, *β*, *n*, *n*_{1} and *q* (explicit expressions of these are given in Pathania (2010)). Whereas *ω*^{2} is the non-dimensional form of the actual eigenfrequency of oscillation *σ*, *ς* denotes a suitable average of the relative amplitudes of pulsation of the fluid elements on the equipotential surface (*ψ*^{**}=const.). Also *r*_{0s} is the value of *r*_{0} at the surface of the model, *G* is the universal gravitational constant, *M* the total mass of the star and *D* is the separation between the two components of a binary system.

Equation (5.1), which determines the eigenfrequencies of pseudo-radial modes of oscillations of RTD polytropic models, is of Sturm–Liouville type. It has to be solved subject to the boundary conditions that require *ς* to be finite at points corresponding to the centre (*r*_{0}=0) and the free surface (*r*_{0}=*r*_{0s}). In the absence of the Coriolis force (that is, *α*=1, *β*=1), equation (5.1) reduces to the corresponding equation reported earlier by Mohan & Saxena (1985).

Again following Mohan & Saxena (1985), the system of differential equations governing the non-radial modes of oscillations of RTD polytropic models of stars which incorporate the effects of the Coriolis force besides the effects of gravitational and centrifugal forces can be expressed as
5.2where
5.2aHere *δr*_{ψ} being the amplitude of Lagrangian variation in *r*_{ψ}, *P*′_{ψ} the amplitude of variation of *P*_{ψ}, *ψ*′_{g} the amplitude of variation of the gravitational potential *ψ*_{g} at a point on the topologically equivalent spherical equipotential surface (*ψ*^{**}=const.) and *l* is the pulsation mode which represents the total number of node lines on the stellar surface. *B*_{1}, *B*_{2}, *E*_{1}, *E*_{2}, etc., are again nonlinear functions of structure and distortion parameters (explicit expressions of these are given in Pathania (2010)).

The eigenvalue problem (5.2) which determines the eigenfrequencies of non-radial modes of oscillations of RTD polytropic models has to be solved subject to the boundary conditions:
5.3at the centre *x*=0 and
5.4aand
5.4bat the surface *x*=1.

In the absence of the Coriolis force (that is, *α*=1, *β*=1), equation (5.2) reduces to corresponding equation reported earlier by Mohan & Saxena (1985).

## 6. Numerical computations for the equilibrium structures of rotationally and tidally distorted polytropic models of stars

To obtain the inner structure, the volume and the surface area of an RTD polytropic model, equation (3.2) has to be integrated numerically subject to the boundary conditions (3.4*a*,*b*) for the specified values of the parameters *N*, *ξ*_{u}, *n*_{1}, *q* and *K*. In the case of a polytropic model, the value of *K* has been chosen such that the outermost surface of the primary component lies well within the Roche lobe; otherwise the two components of the binary will coalesce (cf. Kopal 1972, p. 11).

For obtaining the numerical solutions, equation (3.2) has been integrated numerically using the fourth-order Runge–Kutta method for the specified values of the input parameters. To start integration from the centre, a series solution similar to that available for undistorted polytropic models (see Chandrasekhar 1939, p. 95) was developed at a point near the centre. Taking starting values from this series solution at *r*_{0}=0.005, numerical integration of equation (3.2) was then carried forward using the Runge–Kutta method of order four. Using a step length of 0.005, numerical integration was continued till *θ*_{ψ} first became zero. Relations (4.1) and (4.2) were then used to determine the volume and surface area of the distorted polytropic model. The value of the parameter *K* has been taken as one for the rotationally distorted models and 0.5 for rotationally and tidally distorted models. (The chosen value of *K* provides the outermost surface of the model well within the Roche lobe for each considered case.)

The values of the volume and surface area obtained for certain distorted models of polytropic index 1.5 and 3.0 are presented in table 1. For comparison, we have also presented the corresponding results of Mohan & Saxena (1983) in this table.

## 7. Numerical computations for the eigenfrequencies of pseudo-radial and non-radial modes of oscillations of rotationally and tidally distorted polytropic models of stars

Eigenvalue problems developed in §5 have been solved numerically to compute the eigenvalues of pseudo-radial and non-radial modes of oscillations of certain RTD polytropic models of stars. The eigenvalue problem (5.1) is of Sturm–Liouville type. In order to compute the eigenfrequencies of small adiabatic pseudo-radial modes of oscillations of RTD polytropic models, equation (5.1) has been integrated numerically subject to the boundary conditions that require *ς* to be finite at points corresponding to the centre and the free surface of the model. The values of *θ*_{ψ} and d*θ*_{ψ}/d*x* needed for this purpose at various points were taken from the numerical solution of the structure equation (3.2). For numerical work, series expansions of nonlinear functions *H*_{1}, *H*_{2}, *H*_{3}, *H*_{4} were obtained. In these expansions, terms up to second order smallness in distortion parameters *n*, *n*_{1}, *q* and up to in *r*_{0} have been retained. Computations were started with some trial value of *ω*^{2}. For this chosen value of *ω*^{2}, a series solution was first developed at a point close to the centre (*x*=0.005). This solution was then used to carry the integration of the pulsation equation (5.1) outwards using the fourth-order Runge–Kutta method. Using the same numerical value of *ω*^{2}, a series solution was also developed at a point near the surface (*x*=0.01), which was then used to carry the integration of the equation (5.1) inwards. The value of *ς*/(d*ς*/d*x*) obtained from the outward and inward integrations of equation (5.1) was then matched at some preselected point in the interior of the model. The process was continued iteratively with different choices of the value of *ω*^{2}, till a value of *ω*^{2} was found for which the two solutions agree to a specified accuracy.

Computations have been performed to compute the eigenfrequencies of the fundamental and the first mode of pseudo-radial modes of oscillations of RTD polytropic models of polytropic index 1.5 and 3.0 for different choices of distortion parameters *n*, *n*_{1} and *q*. The obtained eigenfrequencies are presented in table 2. For comparison, we also present in this table the corresponding results of Mohan & Saxena (1985).

Eigenfrequencies of the non-radial modes of oscillations have also been computed numerically using eigenvalue problem (5.2). For this purpose, the Chebyshev polynomial expansion technique used earlier by Mohan & Saxena (1985) was used. For polytropic models of indices 1.5 and 3.0, we have used 10 and 15 collocation points, respectively. However, for determining the eigenfrequencies of certain higher modes of non-radial oscillations, the number of collocation points was further increased to achieve the desired accuracy of 0.0001 in getting the discriminant condition satisfied. The number of collocation points used in determining a specific mode of non-radial oscillation of a distorted polytropic model was, however, kept the same as used in determining the corresponding mode for the corresponding undistorted model. The numerical results for the non-radial modes of oscillations of certain RTD polytropic models of polytropic indices 1.5 and 3.0 are presented in table 3. The corresponding results of Mohan & Saxena (1985) are also presented in this table for comparison.

In table 6, we have compared the present eigenfrequencies for non-radial modes of oscillations of uniformly rotating stars with the eigenfrequencies as obtained earlier by Saio (1981), Clement (1984) and Christensen-Dalsgaard & Mullan (1994). We have used polytropic models with index 3.0, *l*=2 and the rotation parameter *α*=Ω^{2}/8*πG* as defined in Clement (1984) for studying the effect of rotation on eigenfrequencies. The eigenfrequencies are presented in the units of (*GM*/*R*^{3})^{1/2}. We have taken *G*=6.67232×10^{−8} dyne cm g^{−2}, *M*=1.989×10^{33} g and *R*=6.9599×10^{10} cm as given in Christensen-Dalsgaard & Mullan (1994) so that (*GM*/*R*^{3})^{1/2} has a value of 627.41 μHz. We have converted the eigenfrequencies of Clement (1984) in the above units using the formula where *ω*_{g}=(*GM*/*R*^{3})^{1/2}=627.41 μHz and *ω*_{c} is the eigenfrequency computed by Clement (1984) in its own units.

In tables 4 and 5, we have studied the effect of various parameters *n*, *n*_{1}, *q* on the eigenfrequencies of primary component of non-synchronous binaries by varying their values for certain cases. It may be mentioned that for the cases of practical interest, in a binary system the angular velocity of rotation (*n*_{1}) is greater than the angular velocity of revolution (*n*). We have kept this fact in view while concluding the main results in the next section. However, from the analytical point of view, we have also considered the cases when *n*_{1}<*n*. Results are presented in tables 4 and 5.

## 8. Conclusions

Our results in table 1 show that with the inclusion of Coriolis force, the values of volume and surface area for non-synchronous binaries do not change to any appreciable extent when compared with the corresponding values of Mohan & Saxena (1983) (the maximum change is 0.8%). For pure rotation and synchronous binaries, the present results are more or less identical to the corresponding values of Mohan & Saxena (1983). (Theoretically these should have been identical. The marginal difference is due to truncation errors.)

The results in tables 2 and 3 show that inclusion of Coriolis force does not appreciably affect the eigenfrequencies of RTD primary components of non-synchronous binary systems (the maximum change is 2%). For the synchronously rotating primary component of a binary system and for purely rotating stars, the present values are same as obtained earlier by Mohan & Saxena (1985).

On comparing our results with the corresponding results of undistorted models in tables 2 and 3, we find that with the inclusion of the Coriolis force there is a decrease in the eigenfrequencies of radial modes and *f*, *g*, *p*-modes of non-radial oscillations. These results are in accordance with the results obtained earlier by Mohan & Saxena (1985). In other words, the inclusion of the Coriolis force does not affect the original pattern.

The results in table 6 show that our present results are in good agreement with the results of Saio (1981) and Christensen-Dalsgaard & Mullan (1994). However, we notice that although in Clement (1984) the values of *g*_{1}-modes decrease in the presence of rotation, our results are in agreement with the pattern obtained by Mohan & Saxena (1985). It can also be concluded from the table that on account of rotation, for *f* and *p* modes, there is an increase in the percentage difference between the presently obtained values and the values of Clement. This difference increases for higher modes. However, the percentage difference in case of *g*_{1}-modes is not large (<4%).

Our results in tables 4 and 5 show that when the parameters *n* and *q* are kept fixed, then with an increase in the value of parameter *n*_{1}, there is a decrease in the values of the eigenfrequencies of polytropic models of stars. These values are even greater than the corresponding values of the undistorted model when *n*_{1}≤*n*/4. However, for *n*_{1}≥*n*, the values are always smaller than the values of the undistorted model. Now when the parameters *n*_{1} and *q* are kept fixed, then for values of *n*≤*n*_{1}, with an increase in the value of parameter *n*, there is a decrease in the values of eigenfrequencies but for values of *n*>*n*_{1} there is an increase in the eigenfrequencies. For *n*≥4*n*_{1}, the present values even become greater than the values of the undistorted model. However, for *n*≤*n*_{1} the values are always smaller than the values of the undistorted model. Again when both the rotational parameters *n* and *n*_{1} are kept fixed, then with an increase in the value of *q*, there is a decrease in the values of eigenfrequencies but not to an appreciable extent (in this case, the value of eigenfrequencies is always less than value of undistorted model).

Thus, our present study has shown that as expected there is no effect of the Coriolis force on the equilibrium structures and eigenfrequencies of small adiabatic pseudo-radial and non-radial modes of oscillations of synchronous binaries and purely rotating stars. However, some effect is observed for non-synchronous binaries. But, again this effect is not very appreciable. Also, our study has confirmed that uniform rotation decreases the value of eigenfrequencies of various pseudo-radial and non-radial modes of oscillations.

It may be mentioned here that the analysis used in the present study for analysing the effect of the Coriolis force on the equilibrium structures and periods of oscillations of rotating stars and stars in binary systems is applicable only to those stars for which the angular velocity of rotation and revolution and/or mass of the companion star causing tidal distortions is not unduly large so that deviation of the distorted model from its original equilibrium configuration is not too large so as to entirely change the pattern of its modes of oscillations.

We have assumed uniform rotation in our present study; however, one can expect differential rotation in some oscillating stars. Using the finite difference technique, Lovekin *et al.* (2009) have concluded that differential rotation does not have significant effect on the frequencies except for the most extreme case of differentially rotating models. So it will be of interest to study this problem using Mohan & Saxena (1983, 1985) methodology as well as the effect of Coriolis force on the equilibrium structures and periods of small oscillations of RTD differentially rotating stars. We intend to undertake this study in our next paper.

## Acknowledgements

A.P. expresses his gratitude to the North-West University, Mafikeng for the award of Post-doctoral Fellowship. We are also very thankful to the valuable comments of learned referees that have enabled us to bring the paper in its present form.

- Received March 22, 2011.
- Accepted September 12, 2011.

- This journal is © 2011 The Royal Society