Astron. Astrophys. 355, 1031-1040 (2000)

3. Discussion

In Fig. 5 we show the distribution of roAp and noAp stars in the special version of the H-R diagram proposed by Arenou & Luri (1999) and called by them "astrometric H-R diagram": instead of considering the absolute visual magnitude (or the logarithm of the luminosity), we use what these authors call the Astrometry-Based Luminosity (ABL) which is written

where is the absolute magnitude, the apparent one and is the visual interstellar absorption. This quantity has the advantage that the error bars are essentially symmetrical (the error on the apparent visual magnitude may be neglected) and that no Lutz-Kelker bias (Lutz & Kelker 1973) occurs. Therefore, all stars may be represented, since there is no reason to impose any limit on the relative error of the parallaxes.

Fig. 5. "Astrometric" HR diagram (with , see text) of the roAp (full dots) and of the noAp stars (open dots); the recently discovered roAp stars HD 99563 and HD 122970 are included in this plot. The horizontal segment at the lower left indicates the typical error on ( K in ), its total length being . The lower continuous curve (ZAMS) is an isochrone at based on the models of Schaller et al. 1992for while the upper curve (TAMS) links the 11th points of Schaller et al.'s evolutionary tracks. The dotted curves are the isochrones at the indicated , while the dashed curves are the main sequence parts of the evolutionary tracks for masses between 1.5 and 2.5 .

The evolutionary tracks for theoretical stars of 1.5, 1.7, 2.0 and 2.5 solar masses and the isochrones , 9.0 and 9.2 computed by Schaller et al. (1992) are also shown.

Because the errors on the Hipparcos parallaxes are gaussian (Arenou et al. 1995), the average ABL may be estimated using the weighted mean

while the average absolute magnitude is (Arenou & Luri 1999)

The result that we obtain in this way is , and , , including the two new roAp stars HD 99563 and HD 122970 (Dorokhova & Dorokhov 1998; Handler & Paunzen 1999). HD 99563 is a close visual double with and , so its visual magnitude and colours are affected. Its colours have been used to estimate  K, neglecting the influence of the companion: this estimate should rather be a lower limit. The apparent magnitude has been corrected for the presence of the companion, which results in instead of 8.32, and a very uncertain was assumed from estimated from the colours. This value of should be regarded as an upper limit, considering the rather high galactic latitude () of the star, which in any case has a very uncertain parallax (). HD 122970 has a much better parallax, and its colours (the only ones available) yield  K - in good agreement with its F0 type - and , hence . If these two stars are not taken into account, the average absolute magnitude of the sample is not significantly affected: one gets and .

Although noAp stars are similar to roAp stars in their colour indices, abundances and magnetic fields, the roAp stars as a group are less luminous and less evolved, consistently with the results obtained by North et al. (1997). The noAp stars are also more massive on average than the roAp stars, since their masses range from about 1.6 to instead of 1.5 to (see Fig. 5). The absolute magnitude difference between roAp and noAp stars found in this study is 0.86 mag.

From comparison of the kinematical characteristics calculated from Hipparcos data, we conclude that both groups are very similar. We see in Table 4 that kinematical study for the set 4 gives somewhat higher values for the dispersion of the space velocity components for roAp stars compared to that for noAp stars. It can be understood as some hint of older kinematics for roAp stars. On the other hand, within the uncertainties, this result can also be consistent with the view that both roAp and noAp stars are of the same or only slightly different age, approximately that described by the isochrone .

A plot of the cumulative distributions of for roAp and noAp stars shows very clearly two parallel curves (Fig. 6), and the Kolmogorov-Smirnov test indicates that the two distributions differ at the significance level of 99.98%. Although this seems highly significant, it is necessary to keep in mind two possible observational biases:

Fig. 6. Cumulative distributions of the "Astrometry-Based Luminosities" of the roAp stars (continuous line) and of the noAp stars (broken line). The "prob." number is the probability given by the KS test, that both distributions are drawn from the same parent distribution.

(1) Most of the roAp stars were discovered by Kurtz and Martinez. They found that looking for roAp stars among candidates with a negative value of was especially efficient. is sensitive to line blanketing, which is heavy for cool Ap stars, but also to luminosity through the Balmer jump. Therefore, such a selection criterion for detecting roAp stars would certainly lead to select also lower luminosity stars. One may fear, then, that we simply have detected this bias.

The question is whether most noAp stars have suffered the same selection bias than the roAp stars. The simplest way to answer this is to plot the cumulative distribution of for both groups. The result is that both distributions do not differ, according to the Kolmogorov-Smirnov test (probability level: 64% that both distribution are drawn from the same parent distribution). A slightly higher proportion of noAp stars have positive values of , but removing them would make no significant difference on Fig. 6, the probability level becoming 0.0005 instead of 0.00012. In conclusion, this possible bias does not really exist.

(2) The noAp stars are systematically fainter - in apparent magnitudes - than the roAp stars, by roughly one magnitude on average (the difference, judged from cumulative distributions of , ranges from mag around to mag around ). This difference is seen also in the Hipparcos parallaxes, which are systematically smaller for the noAp stars than for the roAp ones. Therefore, one may fear that minute variations of 1-2 mmag may have simply escaped detection in these fainter stars, which we would, then, have unduly put into the "noAp" category. Non-oscillating Ap stars are also farther away on average, and more numerous: then, they have a good chance to span the whole width of the main sequence. The roAp stars, on the contrary, are rarer, and since the lifetime is longer near the ZAMS than near the terminal-age main sequence (TAMS), these few objects would tend to cluster near the ZAMS. In the end, one may have the impression that noAp stars are intrinsically brighter on average than roAp stars.

The only way to decide whether or not this bias holds is to show that the noise does not increase with apparent magnitude in a significant way. This test is delicate, because the detection of oscillations depends on many factors, such as telescope aperture, sky transmission stability, total duration of the monitoring, and even rotational phase of the star. We have estimated the noise level on the periodograms published by Martinez & Kurtz (1994) for frequencies larger than 1 mHz and plotted it against the apparent visual magnitude to see whether any correlation appears. In cases where there are several periodograms (i.e. several observing runs) per star, the one giving the smallest noise was retained. The result is shown on Fig. 7 as full dots. The open dots in Fig. 7 are taken directly from Nelson & Kreidl (1993), who give the noise in tabulated form and have about the same criteria of noise definition.

Fig. 7. Noise level versus apparent magnitudes for noAp stars. Full dots: data from Martinez & Kurtz (1994). Open dots: data from Nelson & Kreidl (1993)

A slight correlation emerges, showing that the above mentioned bias might be real. A more thorough investigation is required in order to check its significance for the detection of rapid oscillations in noAp stars.

The difference between the masses of roAp stars and noAp stars may be important for the understanding of the origin of their oscillations. Plausibly, convection starts becoming efficient for the roAp stars. More generally, the difference of internal structure associated with the mass difference can probably explain why oscillations are observed only in the roAp stars. On the other hand, the domains of the roAp and noAp stars in the H-R diagram largely overlap. This shows that mass and internal structure differences between the roAp and noAp stars cannot be the only decisive factor in their respective evolution.

As mentioned above, none of the roAp stars is known to be a spectroscopic binary. With respect to this, it is noteworthy that also no pulsating white dwarf is known to be a spectroscopic binary (Koester 1999). In one case, GW Lib, the dwarf primary of a cataclysmic variable star shows non-radial pulsations (Warner & van Zyl 1998). However, this is a special case where the white dwarf has been pumped in into the instability strip by accretion heating.

On general grounds, the issue of whether duplicity affects pulsation through tidal interaction is unsettled. From the theoretical point of view, while some authors (e.g., Cowling 1941; Zahn 1977) have conjectured that tides in close binary systems may act as an external perturbing force driving oscillations, the question whether tidal interaction may also be efficient in damping already existing pulsations does not seem to have ever been addressed. Observationally, in the same region of the parameter space in which pulsations were detected, there is only one binary system with a noAp primary presently known, in which the two components are close enough so that significant tidal interaction occurs between them (Giuricin et al. 1984): HD 200405 (SB1, days, North 1994). This star does not appear in Table 2 because its proper motion and parallax were not measured by Hipparcos.

Tidal forces might conceivably also play a non-negligible rôle in systems with a larger average separation, provided that their eccentricity is large enough. Interaction would then occur mostly on the part of the orbit when the components are closest, since tidal forces are strongly dependent on the distance between the components. At present, though, almost nothing is known about the orbital eccentricities of the noAp binaries.

In other words, neither theoretically nor observationally is our present knowledge sufficient to decide confidently whether tidal interaction in binaries may reduce the amplitude of or inhibit pulsation in cool Ap stars. To establish this, a necessary condition would be to show that essentially all noAp stars are binaries. Although this is not inconsistent with the information available so far, the latter is too incomplete to draw any more definite conclusion. To gain further insight, it will be important to establish if no roAp star is a binary (except for very wide visual binaries). Another potentially fruitful investigation would be to search for binarity among the noAp stars in the region of overlap, since among the stars of this region in which pulsations have been sought and not found, only three are not definite binaries.

Answering those questions will require a major additional observational effort. Future observations aimed at determining the orbital elements of the noAp binaries should also contribute to a better knowledge of the interaction between binarity and pulsation. Such observations will help to establish which conditions must prevail for the appearance of rapid oscillations in cool Ap stars.

© European Southern Observatory (ESO) 2000

Online publication: March 21, 2000
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