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Astron. Astrophys. 334, 181-187 (1998)

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3. The [FORMULA] vs. [FORMULA] diagram

Fig. 4 shows the distribution of stars according to their rotational period and surface gravity. There is of course an intrinsic scatter, but on average the width of the period distribution is relatively narrow and there are clearly longer periods among the more evolved stars. Stars with both a small [FORMULA] and a very short period are lacking. There are two stars falling below the lower envelope in a significant way: HD 115599 and HD 150035. HD 115599 was measured photometrically by Moffat (1977) only once a night near culmination, so that the published period might very well be an alias of the real one. The photometric measurements of HD 150035 made by Borra et al. (1985) do not seem very precise, judging from the low S/N lightcurve they published. The period of this star appears to remain highly uncertain.

[FIGURE] Fig. 4. Rotational period versus surface gravity. Full symbols represent stars with a reliable period, open symbols are for possibly ambiguous periods. Round dots (and triangles) represent stars with a spectroscopic value of [FORMULA], while diamonds are for stars with [FORMULA] determined from Hipparcos data. The three triangles are for stars with a rotational period newly determined from Hipparcos photometry (the upside-down triangle has [FORMULA] determined from Hipparcos, the others from spectroscopy). The continuous and broken lines represent the evolution of the period predicted from that of the moment of inertia, under the assumption of rigid-body rotation and for initial periods of 0.5 and 4 days. The dotted lines show the ideal case of conservation of angular momentum in independent spherical shells.

It is interesting to consider the case of CU Vir or HD 124224, because in the literature a very small [FORMULA] is sometimes quoted: for instance, Hiesberger et al. (1995) quote values as small as 3.45 to 3.60 (obtained from spectrophotometric scans), but also 4.2 and 3.71. The latter two values come from the same [FORMULA] photometric indices but through two different calibrations. The Hipparcos data, together with [FORMULA]  K obtained from Geneva photometry, point to [FORMULA], i.e. the star is very close to the ZAMS. If a higher effective temperature is adopted, like [FORMULA]  K, the result becomes worse, with [FORMULA] (the error on [FORMULA] was computed assuming an error of only 400 K on [FORMULA]). The conclusion that CU Vir is unevolved seems unescapable and is coherent with the fact that no Bp or Ap star has a rotational period significantly shorter than 0.5 days (the record is held by HD 60431, with [FORMULA]  days, see North et al. 1988). This may bear some importance in view of the fact that CU Vir is the only Ap star for which a period change has been unambiguously identified (Pyper et al. 1998). Any explanation for this intriguing discovery will have to take into account the unevolved state of the star.

The full and broken lines drawn in Fig. 4 are kinds of evolutionary tracks: assuming an initial period of 0.5 days (respectively 4.0 days), they show how a star rotating as a rigid body will evolve, if no loss of angular momentum occurs. These lines essentially reflect how the moment of inertia changes with evolution for stars having 2.5 and [FORMULA]. They depend in a negligible way on the mass and are entirely compatible with the observations. They were established starting from the conservation of angular momentum:


where [FORMULA] is the star's angular velocity, I the moment of inertia and the subscript 0 indicates initial value (i.e. on the ZAMS). For the period, one has


How the moment of inertia changes with evolution is provided by the models of Schaller et al. (1992), through a code kindly provided by Dr. Georges Meynet.

The two steep, straight dotted lines illustrate the extreme case of conservation of angular momentum in concentric shells which would rotate rigidly but glide one over the other without any viscosity, i.e. without the least radial exchange of angular momentum. In such a case, the moment of inertia of each shell of mass [FORMULA] and radius r reads


and in particular, the outermost shell having [FORMULA] and being the only one observed, one gets


Surely this case is an ideal and not very realistic one, but it is shown for illustrative purpose.

Do Si stars undergo any rotational braking during their life on the Main Sequence? Because of the decreasing number of stars with decreasing [FORMULA], the statistics remains a bit small, and doubling the number of stars in the range [FORMULA] would be very useful. Nevertheless, the data are entirely compatible with nothing more than conservation of angular momentum for a rigidly rotating star. They may be marginally consistent with the dotted lines whose slope is 1 (conservation of angular momentum for independent spherical shells): if these lines are interpreted as betraying some loss of angular momentum through some braking mechanism yet to be understood, then this loss cannot increase the period by more than about


meaning a relative increase of no more than 82 percent during the whole Main Sequence lifetime. This is only a fraction of the increase due to angular momentum conservation alone (for a rigid sphere).

The whole reasoning has been applied to a mix of stars with various masses (between 2.2 and [FORMULA]), but if any magnetic breaking exists, its efficiency might well be a sensitive function of mass. Then, one would need a larger sample, allowing [FORMULA] vs [FORMULA] diagrams to be built separately for stars in narrow mass ranges. The sample as a whole would not need to be enlarged in an unrealistic way: it is especially the evolved stars which are crucial for the test, so increasing their number from 13 (for [FORMULA]) to about 50 or 70 would probably be enough to answer the question on firmer grounds. Spectroscopic observations would be needed to estimate [FORMULA] (and hopefully [FORMULA]!) and photometric ones to determine the periods.

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© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998