Astron. Nachr. / AN 333, No. 8, 744 – 753 (2012) / DOI 10.1002/asna.201111715 Numerical solutions of the equation concerning small adiabatic oscillations of pulsating components in double star systems B. Ulas ¸ 1,2,⋆ 1 Department of Astronomy and Space Sciences, University of Ege, ˙ Izmir, 35100, Turkey 2 Department of Physics, Onsekiz Mart University of C ¸ anakkale, C ¸ anakkale, 17100, Turkey Received 2011 Dec 2, accepted 2012 Jul 11 Published online 2012 Oct 2 Key words binaries: close – methods: numerical – stars: oscillations 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. c  2012 WILEY-VCH Verlag GmbH& Co. KGaA, Weinheim 1 Introduction Oscillating components in binary stars play important role to understand the nature of the oscillations inside rotation- ally and tidally distorted stars. The geometric shape of the distorted stars may cause an additional change in the ampli- tude and the frequency of the wave travelling inside when compared to similar undistorted stars. Eddington (1926) de- rived the second order differential equation concerning os- cillation for a sphere of adiabatic gas. The author solved the equation numerically for various values of the adiabatic in- dex. Eddington then obtained the relative wave amplitude from the centre to the surface of a gaseous sphere. Edding- ton also concluded that the solution of the equation is very sensitive to small changes in the dimensionless frequency. Chandrasekhar & Lebovitz (1962a) investigated the os- cillation of rotating gaseous masses using the tensor form of the virial theorem. They derived the mathematical repre- sentations for neutral, radial and non-radial modes of oscil- lations under rotation. Chandrasekhar & Lebovitz (1962b) determined the pulsation frequencies of rotating gases that can be represented by Maclaurin spheroids. The authors, in addition, derived the pulsation frequencies for various val- ues of the adiabatic index. Kopal (1972) solved the equation governing small adia- batic oscillations near the centre and the surface of undis- torted stars. Kopal also determined a hypergeometric se- ries that represents the amplitude of the oscillation near the centre. Sidorov (1982) derived the equation of small adia- batic oscillations for homogeneous stars. The author con- cluded that the radial oscillation frequency of a rotating star ⋆ Corresponding author: bulash@gmail.com is smaller than that of the non-rotating one. His results show that, the rotation is more effective on the radial oscillations than the tidal effect between the components. Mohan & Singh (1982) used the concept of topo- logically equivalent spheres developed by Kippenhahn & Thomas (1970) and solved the equation by using the Roche model for the stars distorted by tidal and and rotational ef- fects. They found that the eigenvalues of small adiabatic oscillations were decreased by the effects of rotational and tidal distortions. Theoretical relation between the oscilla- tion frequencies of distorted and undistorted stars was also shown by Ulas ¸ & Demircan (2008) for polytropic model of stars. Willems & Aerts (2002) studied on the tidally induced amplitude of radial-velocity variations in close binary stars. They pointed out that the orbital eccentricity and inclina- tion are the key factors to variations seen by the observer. Similar idea is supported by present study by explaining the dependency of the observed eigenvalue and relative ampli- tude in geometrical parameters such as surface potential and mass ratio. Willems & Aerts (2002) also show that the higher mode second-degree g-mode is excited for the eccentric binary HD 177863 which supports the idea that some modes are effected by tidal interaction generally seen in close binaries. In addition, during the last decade observational studies es- pecially on δ Sct-type pulsators in binary systems have been increased (Mkrtichian et al. 2002; Mkrtichian et al. 2003; Mkrtichian et al. 2004; Soydugan et al. 2006). Recently, Zhou (2010) gave the number of candidate systems as 434. In this study, we solved the equation concerning small adiabatic oscillations given by Mohan & Singh (1982). The assumptions of Roche model (Kallrath & Milone 1999) are c  2012 WILEY-VCH Verlag GmbH& Co. KGaA, Weinheim