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Astron. Astrophys. 356, 895-902 (2000)

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4. Astrophysical considerations

4.1. The nature of the association

From the mean colour indices in the Geneva Photometric System and the corresponding calibrations for A-F type stars (Hauck 1973; Künzli et al. 1997), we derive the physical parameters presented in the upper part of Table 5. An estimation of the masses is obtained utilising the calibrations from Kobi & North (1990) and North (private comm.). Spectral types were determined by Gray & Garrison (1989). Rotational velocities are from Levato (1975). Bolometric corrections have been taken from Flower (1996). In addition, we have the Hipparcos trigonometric parallaxes and proper motions, useful to establish the nature of the association between both stars: the values of the parallaxes differ by only 1 to 1.5 [FORMULA] and the resemblance of the proper motions is striking (bottom part of Table 5) (ESA 1997). A very small relative proper motion of magnitude 0.0029"/yr in the direction of 315o accompanies the large angular separation of 26.5", which is the reason of its classification as a common proper motion pair. For this reason and because both parallaxes are in reasonable agreement, the physical association of the pair is probable (van de Kamp 1982). The fact that both stars share the same location in space augments the probability that they were formed at the same time from the same parent cloud. Adopting the mean of both values as the system's parallax ([FORMULA] = [FORMULA], in good agreement with [FORMULA]), we obtain a real separation of the order of 3300 AU between the two components (neglecting the [FORMULA] effect). For a mass sum of 4.1 [FORMULA], the orbital period is very long, [FORMULA] [FORMULA] years. Both stars also share the same projected rotational velocity. We checked for radial velocity data as a further evidence of the wide association (i.e. we expect a small radial velocity difference). Grenier et al. (1999) published radial velocities for both stars only very recently: they determined 17.2 [FORMULA] 0.69 km/s for HD 220392 and 10.75 [FORMULA] 4.06 km/s for HD 220391 (while Barbier-Brossat et al. 1994listed +6.7 km/s for HD 220392). Thus, not only is there a good agreement between both values, in addition it seems that component B has a variable radial velocity (No further conclusion can be drawn for the latter component as this is based on three measurements only). Again making use of the Hipparcos parallax and of the definition of distance modulus, one can derive an absolute magnitude, [FORMULA], but - due to the relative error of 20-25% on the parallaxes - the absolute magnitudes thus derived are too imprecise.


Table 5. Physical parameters for HD 220392/1

We give preference to the absolute magnitudes derived from the photometric calibration, [FORMULA], to fit a model of stellar evolution of solar chemical composition (Schaller et al. 1992) in a theoretical H-R diagram. The same isochrone with an estimated age of [FORMULA] [FORMULA] years for the system appears to fit both stars well (Fig. 5), as was also verified by Tsvetkov (1993). We conclude that both stars form a common origin pair and probably even a true binary system.

[FIGURE] Fig. 5. Isochrone fit in the H-R diagram (Z=0.020). Filled symbols illustrate the locations after the correction for rotation

4.2. The effects of rotation

In this section we want to investigate whether rotation could have an influence on the derived physical quantities from Table 5 and on the previously determined age and evolutionary phases. Both stars indeed seem to present rapid rotation and their photometric indices might be affected by the rotation effects such as described by Pérez Hernández et al. (1999) (hereafter PH99). In some cases these effects appear to be larger than the errors from the calibration: corrections for rotation have been considered by Michel et al. (1999) when analysing several fast rotating [FORMULA] Scuti stars of the Praesepe cluster.

We recall here that the calibration of the multicolour Geneva colour indices in terms of various physical stellar parameters rests on a large sample of stars with well known spectroscopic characteristics (i.e. with known abundances, vsini , spectral classification, etc) that have been measured in this photometric system. Such calibrations are therefore in the first place empirical (Golay 1980). They are based on real stars and do not exactly correspond to non-physical objects (such as zero-rotating stars). Spectroscopically calibrated parameters (such as [FORMULA] and log g) will not suffer too much from the effects of rotation however, mainly because slow rotators will preferentially be chosen as reference objects because of a higher precision of the stellar parameters. On the other hand, one must also recall that the mean rotational velocity of normal A9V and F0IV stars is [FORMULA] 130 km/s (Schmidt-Kaler 1982). When we thus wish to correct the photometric indices for the effects of rotation, we will not need to apply the full range of proposed colour differences: the true correction in the sense observed minus reference object will be smaller than the corrections computed by comparing a uniformly rotating model (represented by the observed star) to a non-rotating model (represented by the zero-rotation "copartner").

Because our targets have such similar properties, both in temperature and in projected rotational velocity, we determined the corrections for the secondary (a MS star) and applied identical corrections to the more evolved component. To do this, we have estimated the break-up velocity and the rate of rotation for each of them. Using [FORMULA] and Eq. (27) (PH99) with a polar radius Rp= 1.5 [FORMULA] we find that [FORMULA] = 41-42 µHz. Since [FORMULA] = 165 km/s and [FORMULA] = 140 km/s, we determine a rotation rate [FORMULA] = [FORMULA]/[FORMULA] smaller than 40% for both. In addition, we may deduce that the inclination is probably [FORMULA]. We applied the (excessive) colour differences corresponding to [FORMULA] = 50%, i = [FORMULA], log ge = 4.34 and log Te = 3.89, where


(Eqs. (21) and (22) in PH99). The (over)corrected photometric parameters then are: B2-V1=0.042,d=1.335, m2=-0.489 for star A and B2-V1=0.031,d=1.281, m2=-0.492 for star B. The corresponding new locations of both stars in the H-R diagram are represented by the filled symbols in Fig. 5. The differences are of the order of the respective errors but somewhat larger in [FORMULA]: 0.04-0.05 dex in log g (or -0.03 to -0.07 in [FORMULA]) and 100 K in temperature. One may therefore safely state that the application of realistic corrections for the rotation of both stars does not really affect the previous conclusions re their physical properties, their age and evolutionary phases.

4.3. The nature of the variability

The mean (d, B2-V1)-values place both stars well within the [FORMULA] Scuti instability strip as observed in the Geneva Photometric System. We note the interesting situation that two stars having such similar characteristics behave quite differently from the variability point-of-view. In the previous sections we have shown that the brighter component has a [FORMULA] Scuti type of variability with a total amplitude of 0.05 mag while the fainter component presents no short-period variability of amplitude larger than 0.01 mag. What could the cause(s) be for this observed difference in variability? From the Geneva colour indices, it appears that the brightest component has [FORMULA], thus it is more evolved than its companion. From the isochrone fit, one may also notice the probable core hydrogen burning evolutionary phase of HD 220391 and the overall contraction or shell hydrogen burning phase of the brighter component, HD 220392. Evolution appears here to be the most probable cause for the diversity in variability (in period and/or amplitude) between the two stars.

Many [FORMULA] Scuti stars are evolved objects (e.g. North et al. 1997). It is further known that many [FORMULA] Scuti stars in the advanced shell H burning stage showing single or double-mode pulsation with high amplitudes (semi-amplitude [FORMULA] mag) are confined to the cooler part of the instability strip (Andreasen 1983). In addition, these are slow rotators. We here have a case of an evolved [FORMULA] Scuti star of low amplitude (with a semi-amplitude of 0.014 mag if one considers only the main frequency - which is disputable), presenting the signature of multiple frequencies and of rapid axial rotation. This is not surprising since low-amplitude pulsators cover the entire instability strip (Liu et al. 1997). We might conjecture that, in this case, the amplitude of the pulsation could be limited due to fast rotation. In fact, from the point-of-view of pulsation versus rotation, Solano & Fernley (1997) tend to believe that fast rotation favours the [FORMULA] Scuti type of pulsation. One could wonder why there is no evidence for short-period variability of this type in the less evolved companion star. (A possible explanation might be that the companion is an even faster rotator with a different (smaller) inclination than the more evolved star and that the amplitude(s) of the pulsation are further damped, possibly beyond photometric detectability.)

Can we identify any pulsation mode for HD 220392? Expected values for a [FORMULA] 2 [FORMULA] standard Population I model are 0.033 days (F), 0.025 days (1H), 0.020 days (2H) or 0.017 days (3H) in the case of radial modes (l=0). For non-radial pressure modes (l=1), these values may be slightly larger: 0.036 days (f), 0.029 days (p1), 0.022 days (p2) ... (Fitch 1981; Andreasen et al. 1983). The physical parameters of Table 5 may be used for the computation of the pulsation constant Q:

log Q = log([FORMULA]) + 0.5 log([FORMULA]) + 0.3 [FORMULA] + 3 log([FORMULA]) -12.697, where f is the frequency in cpd.

The propagation of errors shows that the error on the pulsation constant is of order 0.003 days (0.07 on [FORMULA]). The results are given in Table 6. The values thus computed are on the high side for a definitive mode identification: one could draw the conclusion that the frequency [FORMULA] possibly corresponds to the fundamental radial mode (F). We wish to remark that non-radial g modes as well as undetected binarity are possible reasons for higher values of Q (There is however no indication for the latter from the Hipparcos results). The frequency ratio f2/f1, 0.84, is not very helpful in this case. We stress the fact that additional photometric observations for this interesting couple of stars are highly recommended. The obtained data are not sufficiently numerous to allow unambiguous solutions nor to solve for the multiple frequencies. Radial velocities would be needed too.


Table 6. Q-values for HD 220392

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Online publication: April 17, 2000