## 2. ResultsKjeldsen et al. (1998) compared the frequencies of a rotating model
that matched the observed stellar luminosity ( Before we go on, we note that there is a natural scaling of the
frequency of a star with mass
(this is essentially a scaling by the root of the mean density ). The second-order effects of rotation have a natural scaling with where 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 for the dimensional frequency and for the scaled dimensionless frequency; whereas Saio uses for the dimensional frequency and 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 between our rotating and nonrotating models (for radial modes). Three models of masses , and 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 times To make a proper comparison with the results of Saio (1981) we need to evaluate this scaled difference from his tables.
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 this is plotted in Fig. 1 with stars. (Actually Saio's Table 1
only provides results for the first three
modes: we have been able to
extrapolate to higher orders by treating Saio's quantity where is the difference in mean density between Saio's rotating and nonrotating models. As already noted, Saio keeps central density and central pressure (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) where is the polytropic radius variable, its value at the surface, is the gradient of the Lane-Emden function at the surface and the function is given in Table VI of Chandrasekhar (1933). Noting that the central density is , this expression computes for a polytrope of index 3 to After multiplying by , and taking
account of the 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
() end of the range of comparison.
The results agree less well at low © European Southern Observatory (ESO) 1999 Online publication: October 14, 1999 |