Astron. Astrophys. 341, 151-162 (1999)

2. Regularities of frequency spacing

The values of the observed frequencies and regularities in their patterns can be an excellent intital tool for mode identifications, if enough frequencies are excited and detected. For high-order, low-degree p-mode pulsation, the different radial orders show uniform frequency spacing, with a mode of order n and of degree being shifted from the corresponding mode (n, ) by half of the frequency difference between the (n,) and (,) modes (Vandakurov 1967). In  Scuti stars, the excited pulsation modes are of low order (n up to 7), so that the asymptotic relations do not apply exactly. Nevertheless, they also show some regularities. Additionally g-modes invade the p-mode region and decrease the spacing in a small frequency region of about two radial orders. This effect, known as avoided crossing (Osaki 1975, Aizenman, Smeyers & Weigert 1977), complicates the theoretical frequency spectra, but can provide information about the stellar interior (Dziembowski & Pamyatnykh 1991). Moreover, stellar rotation splits multiplets and this splitting is non-symmetric, if second-order effects of rotation and effects of rotational mode coupling are taken into account (Dziembowski & Goode 1992, Soufi et al. 1998, Pamyatnykh et al. 1998). Nevertheless, the spacing of adjacent radial orders as well as the rotational splitting is still regular enough to be detectable, if complete multiplets are excited and identified. We will demonstrate this by using a pulsation model of a 1.85 star with K, , and km/s. This model will be referred to as model 1. The parameters for the model were not chosen at random, but can be regarded as an estimate for FG Vir.

To investigate the period regularities, Winget et al. (1991) have successfully applied the method of the Fourier transform of the period spacing to the star PG 1159+035. This method requires coherence over a large frequency range. Handler et al. (1997) also found frequency regularities from Fourier transformations of the frequency spectrum of the unevolved  Scuti star XX Pyx. Since strict equidistant frequency or period spacing is not expected for FG Vir, the method is not optimal for this  Scuti star. Instead, we use a method which does not require such a coherence: an examination of a histogram of the observed frequency differences between all detected frequencies. In such a diagram, regularities in the frequency spacing of adjacent radial orders of modes with the same degree, , should show up as a peak. Furthermore, modes of different degree are shifted in frequency relatively to each other, but would still have similar patterns and, therefore, contribute to the peaks in the histogram.

The frequency spacing is examined in Fig. 1 with both the theoretically predicted and observed spacings. Pulsation models show a typical frequency spacing of c/d for adjacent radial orders of p-modes, independent of the degree of the modes. The leftmost peaks in the top panels of Fig. 1 are caused by rotationally split multiplets. A similar diagram for (not plotted separately) does not show such strong peaks in the expected region. The reason is that both the presence of g-modes in addition to the p-modes and non-equidistant rotational splitting significantly disturb the regularity in the distribution of quadrupole mode frequencies (see Fig. 7 below.) As a result, the combined pattern of frequency spacings for all modes becomes much less clear. Moreover, due to the fact that only low-order oscillations are present in this frequency range, there is no additional peak at c/d as might be expected from the asymptotic spacing between p-modes of adjacent degrees (see Fig. 7 for more details).

Fig. 1. Histograms of frequency spacing between all specified pulsation modes. Top left: The diagram demonstrates that for high orders the patterns of frequency spacing clearly show adjacent radial orders ( 4 c/d) and the effects of rotational splitting, which is extremely asymmetric even at  km/s. Top right: The frequency spacing predicted from model 1 for in the observed frequency range of 11 - 35 c/d. Note that the patterns from adjacent orders and rotational splitting are still present. Bottom panels: Observed frequency spacings in the observed range from 11 to 35 c/d. Although these are a mixture of = 0, 1 and 2 modes, the effects of adjacent radial orders and a small peak in the range of rotational splitting can be seen. To demonstrate that the results are not sensitive to which observed frequencies are included, two different choices (see text) are shown

Next, we turn to the observed frequency spacing for the 24 certain and 8 probable frequency detections of FG Vir (Table 1). The most cautious approach would be to use the 24 certain frequencies with a few exceptions: the 2 term at 25.4 c/d (reflecting the departure from a pure sinusoidal light curve shape of ), the two combination frequencies (the pulsation models cannot yet predict which combinations and resonances are excited), and the two low-frequency modes for which the p-mode character can definitely be excluded from the assumption that is the radial fundamental mode. To show that the agreement between the theoretically predicted and observed frequency spacing is not based on the choice of frequencies, the analysis was repeated by including the 8 additional `probable' modes listed in Table 1.

To conclude, the theoretical and observed frequency spacings agree quite well. In particular, for frequency differences in the 0 - 5 c/d range, two features near 3.9 and 0.8 c/d stand out, suggesting an identification with the spacing of successive radial orders and rotational splitting, respectively.

© European Southern Observatory (ESO) 1999

Online publication: November 26, 1998
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