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Astron. Astrophys. 350, 852-854 (1999)

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2. Results

Kjeldsen et al. (1998) compared the frequencies of a rotating model that matched the observed stellar luminosity (L) and effective temperature ([FORMULA]) with a nonrotating model with the same luminosity and temperature. For fixed stellar mass and age, rotation increases the radius of the star and decreases its central temperature and hence luminosity. Thus, as pointed out in that paper, a rotating star must be younger and slightly more massive than a nonrotating star at the same L and [FORMULA]. However, the adjustment in mass is very small, at least for the stars considered by Kjeldsen et al., since the luminosity has a sensitive dependence on mass. Moreover, the radius is fixed since the luminosity and effective temperature both are. This means in fact that the mean density [FORMULA] is almost the same in the rotating and nonrotating models that have the same luminosity and effective temperature.

Before we go on, we note that there is a natural scaling of the frequency [FORMULA] of a star with mass M and radius R of the star through the dynamical timescale [FORMULA]. Thus it is often convenient to think in terms of the dimensionless frequency

[EQUATION]

(this is essentially a scaling by the root of the mean density [FORMULA]). The second-order effects of rotation have a natural scaling with [FORMULA] where [FORMULA] is the (uniform) rotation rate, so this is also often conveniently factored out. It may also be noted that, following the common modern notation, we use [FORMULA] for the dimensional frequency and [FORMULA] for the scaled dimensionless frequency; whereas Saio uses [FORMULA] for the dimensional frequency and [FORMULA] for the dimensionless frequency scaled by properties of the nonrotating model.

Fig. 1 (adapted from Kjeldsen et al. 1998) compares scaled relative differences in dimensionless frequency [FORMULA] between our rotating and nonrotating models (for radial modes). Three models of masses [FORMULA], [FORMULA] and [FORMULA] are included, corresponding approximately to three delta Scuti stars in the Praesepe cluster, observed by Arentoft et al. (1998); further details on the models are provided in the figure caption. Thus what is plotted (lines without symbols) is [FORMULA] times

[EQUATION]

To make a proper comparison with the results of Saio (1981) we need to evaluate this scaled difference from his tables.

[FIGURE] Fig. 1. Scaled relative changes in the dimensionless frequencies of radial modes. The right axis shows these changes in terms of observational quantities: [FORMULA] and [FORMULA] are radius and mass in solar units and [FORMULA] (the equatorial rotational velocity) is in [FORMULA] ([FORMULA]); also [FORMULA] is dimensionless cyclic frequency. Line with stars are the original results of Saio (1981) for a polytrope of index 3 (see text) whereas the line with diamonds is the result of shifting these results by 0.33 (in the units of the left-hand ordinate). The remaining lines are for realistic models, of masses [FORMULA] (solid), [FORMULA] (dashed), [FORMULA] (dot-dashed). The lower panel shows the same results, on an expanded scale. [Adapted from Fig. 6 of Kjeldsen et al. (1998).]

Saio's formulae estimate the relative change in frequency scaled by the mean density of the nonrotating model. Thus the quantities provided by Saio essentially determine

[EQUATION]

this is plotted in Fig. 1 with stars. (Actually Saio's Table 1 only provides results for the first three [FORMULA] modes: we have been able to extrapolate to higher orders by treating Saio's quantity Z as a function of frequency independent of degree - indeed it is almost a linear function of the squared frequency over the range of interest.) Now [FORMULA] in Eq. (3) is evaluated in Saio's rotating polytropic model, but the mean density has changed relative to the nonrotating model. Thus to obtain a quantity directly comparable to our results, where the mean density is essentially constant, we must add to what we get from Saio's table a correction

[EQUATION]

where [FORMULA] is the difference in mean density between Saio's rotating and nonrotating models. As already noted, Saio keeps central density [FORMULA] and central pressure [FORMULA] (and hence polytropic constant) fixed between the nonrotating and rotating models; the corresponding relative change in mean density is obtained as (Chandrasekhar 1933; Eq. 44)

[EQUATION]

where [FORMULA] is the polytropic radius variable, [FORMULA] its value at the surface, [FORMULA] is the gradient of the Lane-Emden function at the surface and the function [FORMULA] is given in Table VI of Chandrasekhar (1933). Noting that the central density is [FORMULA], this expression computes for a polytrope of index 3 to

[EQUATION]

After multiplying by [FORMULA], and taking account of the [FORMULA] scaling, we get that the immediate result from Saio's table must be increased by 0.33. This is in close accord (see the figure) with what is required to bring Saio's and Kjeldsen et al.'s results into agreement at the high ([FORMULA]) end of the range of comparison. The results agree less well at low n, but here the frequency changes no doubt depend on the detailed structure of the model. Indeed, one might expect that the response for high-order modes is essentially homologous, and hence less sensitive to the details of the structure.

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© European Southern Observatory (ESO) 1999

Online publication: October 14, 1999
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