Astron. Astrophys. 336, 539-544 (1998)

6. Concluding remarks

Asymptotic Expansions (23) and (24) for and differ from the corresponding asymptotic expansions (164) and (165) in Paper I. Moreover, boundary Condition (25) and matching Conditions (40) stand for Conditions (154), (155), and (166) of Paper I.

We have verified the validity of asymptotic Expansion (23) for the polytropic models with indices in the range of values of from 0.01 to 0.001. The relative errors of the values of found in comparison to the exact values are presented in Fig. 2. For the three models, the relative errors are smaller than from on and decrease below as becomes smaller than 0.001.

Fig. 2. Relative errors of the values in comparison to the exact values determined by integration of the full fourth-order system of equations governing forced oscillations.

A main motivation for the construction of the asymptotic representation of low-frequency, non-resonant dynamic tides has been the derivation of an asymptotic expansion for the Eulerian perturbation of the gravitational potential that is generated at the star's surface by the tidal distortion. This asymptotic expansion is useful for studies of dynamic effects of tides in close binaries as the apsidal motion.

The asymptotic expansion for the Eulerian perturbation of the gravitational potential at the star's surface can be derived by means of Expression (32) of Paper I and Expression (23) of this paper. It follows that, to order ,

The Eulerian perturbation of the gravitational potential of order that is generated at the star's surface by a low-frequency, non-resonant dynamic tide turns out to be determined simply by the values of the function and its gradient at the star's surface.

From the asymptotic representation of low-frequency, non-resonant dynamic tides to order , one passes on to the first asymptotic representation of low-frequency free oscillation modes in the star by setting the mass of the companion equal to zero. According to Definition (2) of Paper I, the small parameter then becomes equal to zero. Consequently, boundary Condition (40) of Paper I, which relates the function and its first derivative at the star's surface, becomes homogeneous and can generally no more be satisfied by a solution different from zero. The fact that the solution for is identically zero greatly simplifies the asymptotic representation in the various regions of the star.

First, from Condition (25) it now results that

Secondly, matching Condition (40) remains valid and is equivalent with Condition (129) of Smeyers et al. (1995). From the matching condition, it follows that

so that the function too is identically zero, and

Hence, the asymptotic solutions for the divergence and the radial component of the Lagrangian displacement are purely oscillatory. From to a sufficiently large distance from , they take the form

and, in the boundary layer near , the form

From asymptotic Expansion (41), it follows that the Eulerian perturbation of the gravitational potential at the star's surface is equal to zero at the order of asymptotic approximation considered, as was observed by Smeyers et al. (1995).

The same authors also noted that asymptotic Solutions (45) and (46) even apply to the -modes of the equilibrium sphere with uniform mass density when one replaces the frequency by and by from the starting equations on. The asymptotic solutions can then be compared with the known analytical solutions (see Ledoux and Walraven 1958, Sect. 76). From their analysis, Smeyers et al. concluded that the asymptotic approximation is excellent.

One point may be stressed here. From the analytical solution for the radial component of the Lagrangian displacement, the following approximation of the lowest order in can be derived:

Hence, in the lowest-order approximation, the radial component of the Lagrangian displacement is equal to zero at , while the divergence of the Lagrangian displacement is different from zero at that point. This is reproduced by our asymptotic Solutions (20) and (21) at the same point. The lowest-order solution for a free -mode of the equilibrium sphere with uniform mass density is given by the terms of order . At that order, it is seen that indeed and .

© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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