A&A 409, 677-688 (2003)
DOI: 10.1051/0004-6361:20031098
K. Reyniers - P. Smeyers
Instituut voor Sterrenkunde, Katholieke Universiteit Leuven, Belgium
Received 7 May 2003 / Accepted 7 July 2003
Abstract
The paper is devoted to a verification of the validity of a first-order perturbation method that was developed in an earlier paper and allows one to determine the effects of an equilibrium tide on linear, isentropic oscillations of a component in a close binary.
The verification is done by a comparison between results obtained by the perturbation method and results obtained in other ways, in the cases of two simple models: the compressible equilibrium sphere with uniform mass density and the polytropic model with index n = 3.
In the first case, a comparison is made with second-harmonic oscillations in compressible Jeans spheroids with a small eccentricity, and in the second case, with results determined by Saio (1981) for a rotationally and tidally distorted polytrope.
For the comparison, the second-harmonic oscillations of the incompressible and the compressible Jeans spheroids are redetermined by means of a method of direct integration of the governing equations which has the advantage of yielding exact analytical solutions of the eigenfrequency equations.
Key words: stars: binaries: close - stars: oscillations - methods: analytical
Here our aim is to test the validity of the perturbation method by comparing results derived by means of the perturbation method with results derived by other means, in the cases of two simple models: the compressible equilibrium sphere with uniform mass density, shortly called the compressible homogeneous model, and the polytropic model with index n = 3. In the first case, we compare with second-harmonic oscillations established for compressible Jeans spheroids with a small eccentricity, and in the second case, with results determined by Saio (1981) for a rotationally and tidally distorted polytrope by means of a perturbation method that has been developed earlier by Smeyers & Denis (1971) and Denis (1972).
The Jeans spheroids are static equilibrium configurations with uniform mass density that are subject to the equilibrium tide generated by another body, which is considered to be a point mass. For our purpose, we redetermine the second-harmonic oscillations successively of the incompressible and the compressible Jeans spheroids by a method of direct integration of the governing equations similar to the method that has been developed by Smeyers (1986) for the determination of oscillation modes of the incompressible MacLaurin spheroids. Compared to the virial method of Chandrasekhar & Lebovitz (1963), our method has the advantage of yielding exact analytical solutions of the eigenfrequency equations, which are of particular interest here in the limiting case of weak equilibrium tides. The redetermination of the second-harmonic oscillations of the Jeans spheroids is done in Sect. 2.
Section 3 is devoted to the comparison of results derived by means of the perturbation method in the cases of the compressible homogeneous model and the polytropic model with index n = 3. Within the framework of this comparison, we regard the Jeans spheroids as tidally distorted equilibrium configurations that rotate uniformly in synchronization with an orbiting body without being affected by the rotation.
In Sect. 4, concluding remarks are presented.
We adopt the same notations as in Paper I. In particular, we use two different systems of spherical coordinates with origin at the star's mass centre: a system of spherical coordinates r, , whose polar axis coincides with the star's rotation axis, and a system of spherical coordinates r, , whose polar axis coincides with the axis joining the star's mass centre and the companion. With the first system of spherical coordinates, spherical harmonics of degree are associated, and with the second system, spherical harmonics of the same degree.
Figure 1: An orthogonal system of confocal ellipses and hyperbolas, which generates coordinate surfaces of constant and coordinate surfaces of constant by rotation around the axis joining the common foci. | |
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For the construction of the Jeans spheroids, we apply a procedure similar to the one adopted by Smeyers (1986) for the construction of uniformly rotating MacLaurin spheroids. We suppose that the surface of the Jeans spheroid coincides with the spheroidal coordinate surface .
The internal gravitational potential is derived from Poisson's integral formula
After integration over the azimuthal angle
,
the expansion for the internal gravitational potential reduces to
Figure 2: The variation of as a function of the eccentricity e of the surface of the Jeans spheroid. | |
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Adapting the procedure of Smeyers (1986), we start constructing solutions from the centre of the Jeans spheroid in terms of spherical coordinates and then express the solutions in terms of the prolate spheroidal coordinates and in order to impose the boundary conditions at the surface of the Jeans spheroid. We concentrate on the second-harmonic oscillations in view of the comparison of our results with those of Chandrasekhar & Lebovitz (1963).
Because of the axial symmetry of the Jeans spheroids with regard to the x-axis, it is appropriate to use a system of spherical coordinates r,
,
whose polar axis coincides with this axis. The transformation formulae from the Cartesian coordinates x, y, z defined in the inertial frame of reference to the spherical coordinates r,
,
are
By expanding the scalar functions S, ,
in terms of spherical harmonics as
For
,
solutions of Eqs. (33) which satisfy the regularity conditions at r = 0 are
The requirements that the pressure vanishes and the gravitational potential and its gradient are continuous at the perturbed surface of the Jeans spheroid lead to the conditions at the equilibrium surface S
We distinguish between the cases , , .
The oscillations involve appropriate linear combinations of
and
terms. For
these are
Figure 3: The squares of the eigenfrequencies of the second-harmonic oscillations in the incompressible Jeans spheroids as functions of the eccentricity of the Jeans spheroid. | |
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The eigenvalue corresponds to the eigenvalue denoted as by Chandrasekhar & Lebovitz (1963) and Chandrasekhar (1969, Table 19). It is positive for e < 0.8830265, vanishes at e = 0.8830265, and is negative for e > 0.8830265. Hence, the second-harmonic, axisymmetric mode is dynamically unstable for e > 0.8830265. The critical value of the eccentricity corresponds to the value at which the parameter determined by Eq. (14) attains its maximum value.
Near e = 0, the following Taylor series applies:
For
,
the solutions reduce to solutions associated with
and take the form
The eigenvalue corresponds to the eigenvalue denoted as by Chandrasekhar & Lebovitz (1963) and Chandrasekhar (1969, Table 19) and is positive for all values of the eccentricity of the Jeans spheroid.
Near e=0, the following Taylor series applies:
For
,
the solutions also involve only
terms, so that they take the form
The eigenvalue corresponds to the eigenvalue denoted as by Chandrasekhar & Lebovitz (1963) and Chandrasekhar (1969, Table 19) and is positive for all values of the eccentricity of the Jeans spheroid.
Near e = 0, the following Taylor series applies:
(48) |
The linear, isentropic oscillations of the compressible Jeans spheroids that depend on time by a factor
are also governed by Eqs. (22)-(27).
The Lagrangian displacement field is however generally not divergence-free and is described by the more general representation
By expanding the scalar functions ,
S, T, and the Eulerian perturbations ,
,
in terms of spherical harmonics in the form given in Equality (32), setting
One readily verifies that the non-axisymmetric second-harmonic solutions for the incompressible Jeans spheroids are also solutions for the compressible Jeans spheroids. Therefore, the search for second-harmonic solutions for the compressible Jeans spheroids can be restricted to a search for axisymmetric second-harmonic solutions. We proceed as follows.
The infinite number of systems of Eqs. (52)-(56) and (58) must be satisfied. To this end, we set
First, the system of equations associated with
then reduces to
Secondly, from the system of equations associated with ,
it follows that
For
and Jeans spheroids with a small eccentricity, one derives the following Taylor series for the roots
and
:
The variations of
and
as functions of the eccentricity of the Jeans spheroid are represented in Figs. 4 and 5 for various values of
different from 1.6.
Figure 4: The eigenvalues of the second-harmonic R-mode in the compressible Jeans spheroid as functions of the eccentricity of the Jeans spheroid, for various values of . | |
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Figure 5: The eigenvalues of the second-harmonic S-mode in the compressible Jeans spheroid as functions of the eccentricity of the Jeans spheroid, for various values of . | |
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In the case of the compressible homogeneous model that is perturbed by an equilibrium tide, we compare the results determined by means of the first-order perturbation method for the fundamental radial mode and the second-degree Kelvin modes with the analytical solutions established in the previous section for the second-harmonic oscillations of compressible Jeans spheroids with a small eccentricity. For this comparison, we regard the Jeans spheroids as tidally distorted equilibrium configurations that rotate uniformly around an axis perpendicular to the orbital plane and synchronously with the accompanying body, without being perturbed by the axial rotation.
In the homogeneous model, the pressure distribution is given by
(74) |
From the perturbation method developed in Paper I, it follows that, for any radial mode n of an arbitrary stellar model, the first-order perturbation of the eigenfrequency due to an equilibrium tide is identically zero. The expansion for the square of the eigenfrequency of a radial mode is thus of the form
Here the subscript n introduced in Paper I denotes the second-degree Kelvin modes of the compressible homogeneous model.
For these modes, the functions
and
,
which appear in Eqs. (71) of Paper I for the components of the Lagrangian displacement at order zero, are simply given by
(82) |
After comparison, one reaches the conclusion that the foregoing expansions for agree, up to order e^{2}, with the expansions given by Eqs. (70), (44), and (47) for the squares of the eigenfrequencies of the second-harmonic oscillations of the compressible Jeans spheroids.
We also verify that the zero-order components of the Lagrangian displacement determined for the second-degree Kelvin modes of the compressible homogeneous model are equivalent to the components of the Lagrangian displacement derived, at the same degree of approximation, for the second-harmonic oscillations of the compressible Jeans spheroids.
In the perturbation method developed in Paper I, we initially determined the zero-order components of the Lagrangian displacement with respect to a system of spherical coordinates whose polar axis coincides with the star's rotation axis perpendicular to the orbital plane. Subsequently, we moved on to the zero-order components with respect to a system of spherical coordinates whose polar axis coincides with the axis pointing from the star's mass centre towards the companion. In the latter case, the zero-order components of the Lagrangian displacement in a second-degree oscillation mode are given by Eqs. (123) of Paper I. We here apply these equations to the second-degree Kelvin modes of the compressible homogeneous model that is perturbed by an equilibrium tide.
For the second-degree Kelvin mode associated with
,
the zero-order components of the Lagrangian displacement are
(83) |
On the other hand, the components of the Lagrangian displacement in the second-harmonic oscillations of the compressible Jeans spheroids associated with
are given by the first three Eqs. (65).
From the relations given by Eqs. (64) and the Taylor series for
near e=0,
one can derive the relation between A_{2,0} and A_{0,0}. At the lowest order of approximation, this is
For the second-degree Kelvin modes associated with
,
the zero-order components of the Lagrangian displacement are
(86) |
These solutions are equivalent to those given by Eqs. (42) for the components of the Lagrangian displacement in the second-harmonic oscillations associated with of a Jeans spheroid with a small eccentricity.
For the second-degree Kelvin modes associated with
,
the zero-order components of the Lagrangian displacement are
(87) |
The solutions are now equivalent to those given by Eqs. (45) for the components of the Lagrangian displacement in the second-harmonic oscillations associated with of a Jeans spheroid with a small eccentricity.
In the case of the polytropic model with index n = 3, we compare the tidal corrections to the eigenfrequencies that are determined by means of the perturbation method developed in Paper I, with results that were obtained by Saio (1981) for several lower-order modes n belonging to the degrees . In his investigation, Saio used a perturbation method developed earlier by Smeyers & Denis (1971) and Denis (1972), whose applicability is restricted to polytropic models.
In Table 1, we present the dimensionless squares of the eigenfrequencies and the ratios of the single first-order correction to the zero-order eigenfrequency . One observes that the ratios are all negative.
Table 1: The relative first-order correction for various modes of the polytropic model with index n = 3 and the ratio of this relative correction to its value in Saio's paper.
According to Saio's paper, the eigenfrequency of an oscillation mode that is perturbed by an equilibrium tide can be expressed as follows up to the second order in the star's angular velocity of rotation :
By using these relations, we have derived the following equations for the relative first-order corrections to the eigenfrequency: for modes belonging to ,
(91) |
(92) |
(93) |
The foregoing equations lead to relations between the first-order corrections to the eigenfrequency of a mode n that fully agree with the relations we derived in Paper I by means of the perturbation method.
Next, we have used the values of X_{2} tabulated in Saio's paper in order to determine the values of the relative first-order corrections
for the low-degree and low-order modes considered by that author. The ratios of the relative first-order correction
determined by means of our perturbation method to the corresponding first-order correction
are listed in the last column of Table 1.
From these ratios, it appears that our first-order corrections are nearly equal to minus two times the first-order corrections based on Saio's results:
(94) |
The discrepancy seems to be related to the determination of our quantity H_{n,0;n,0} or/and that of Saio's quantity X_{2}. Despite repeated verifications of our derivations of the quantities H_{n,0;n,0} and of our numerical code, we have not been able to remove the discrepancy.
We may observe that, in the case of the compressible homogeneous model, the quantities H_{n,0;n,0} and our numerical code did lead to first-order corrections to the eigenfrequency that all agreed with the corrections derived for the second-harmonic oscillations of the compressible Jeans spheroids at the same degree of approximation, as shown above. We also observe that the quantities H_{n,0;n,0} determined for the polytropic model with index n = 3 are positive as are those determined for the compressible homogeneous model. However, according to the results given by Saio, the quantities H_{n,0;n,0} determined for the polytropic model with index n = 3 should be negative.
First, in the case of the compressible homogeneous model, we have compared the results that are obtained by the application of the perturbation method to the fundamental radial mode and the second-degree Kelvin modes, with the second-harmonic oscillations of compressible Jeans spheroids with a small eccentricity. To this end, we have previously derived exact analytical solutions for the squares of the eigenfrequencies and for the eigenfunctions of the second-harmonic oscillations of the compressible Jeans spheroids, for any value of the eccentricity. From our severe test, we have concluded that the perturbed eigenfrequencies of the fundamental radial mode and the second-degree Kelvin modes of the compressible homogeneous model fully agree with the eigenfrequencies of the second-harmonic oscillations of the compressible Jeans spheroids, up to the second order in the small eccentricity of the spheroids. For the same modes, we have also concluded that the lowest-order Lagrangian displacements in the tidally perturbed homogeneous model are equivalent to those in the Jeans spheroids.
Secondly, in the case of the polytropic model with index n = 3, we have compared the results that are obtained by the application of our perturbation method to various low-order and low-degree modes with results given by Saio (1981) for the same modes in a rotationally and tidally polytropic model. Saio's results were obtained by the use of an earlier perturbation method due to Smeyers & Denis (1971) and Denis (1972). By an analysis of Saio's results, we have recovered the relations between the first-order tidal corrections to the eigenfrequencies established in Paper I for modes of any stellar model that belong to one of the low degrees . However, from the comparison of the values of the first-order corrections to the eigenfrequency, it appears that the first-order corrections determined by means of our perturbation method differ by nearly a factor - 2 from those of Saio. We have been unable to identify the source of the discrepancy. Nevertheless, we believe our results to be correct if only because they are consistent with the eigenfrequencies and eigenfunctions for the compressible Jeans spheroids with a small eccentricity.
Acknowledgements
The authors acknowledge the stimulating remarks of the referee Dr. M. Clement, which allowed them to improve the presentation of the paper.