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Astron. Astrophys. 362, 1020-1040 (2000)

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2. Stellar parameters revisited

Before moving on to the observational results of EW Lac, we will reconsider stellar parameters such as radius and mass for this rapidly rotating star. It is well known (Collins & Sonneborn 1977) that rapid rotators viewed edge-on are shifted in the HR diagram with respect to non-rotators so that they seem less massive and more evolved than they really are.

The BCD stellar spectrophotometric parameters ([FORMULA] and [FORMULA], respectively, the mean position and value of the stellar Balmer discontinuity) are calibrated in terms of fundamental stellar parameters (Divan & Zorec 1982, Zorec 1986) which obviously correspond to the observed hemisphere-averaged photospheric radiation field. For EW Lac [FORMULA] Å and [FORMULA] dex, which lead to [FORMULA] mag; [FORMULA] mag; [FORMULA]. From them we obtain a first estimate of stellar radius and mass: [FORMULA] and [FORMULA], which imply a critical rotational velocity [FORMULA] km s-1. As the star is assumed to rotate with an angular velocity as high as [FORMULA] (Slettebak et al. 1992, Moujtahid et al. 1999), its equatorial velocity then would not exceed [FORMULA] km s-1, which nevertheless disagrees with the observed [FORMULA] = 340 km s-1.

Due to its high [FORMULA], finding out the true nature of the central object of EW Lac means we have to interpret the observed apparent photospheric parameters by taking into account the effects of stellar rotation on spectra. First of all, we note that the Hipparcos distance d = [FORMULA] pc, together with the apparent visual magnitude [FORMULA] mag measured at the epoch when total Balmer discontinuity was [FORMULA] (Moujtahid et al. 1999) and E(B-V) = 0.12 mag, produce a visual absolute magnitude [FORMULA](Hipp) = [FORMULA]] mag, where [FORMULA] is the magnitude excess due to the circumstellar envelope. Analysis of EW Lac energy distribution from [FORMULA] to [FORMULA] (Moujtahid & Zorec 2000) shows that [FORMULA] mag. Within the parallax uncertainty, this implies that [FORMULA] [FORMULA](Hipp) and thus, that [FORMULA] is a good approximation of the radiation field in the V-band emitted by the observed rotationally distorted stellar hemisphere. Knowing that as a function of rotational effects [FORMULA] varies in the same way as [FORMULA], we can assume that the emitted bolometric luminosity of EW Lac is reliably given by [FORMULA] = [FORMULA]. Hence, to relate the observed quantities to those the star would have if it were not rotating, we adopt the following representations:


where [FORMULA] and [FORMULA] are functions of the angular velocity ratio [FORMULA] and the inclination angle i, [FORMULA] and [FORMULA] are the bolometric luminosity and the Balmer discontinuity of the star without rotation. The functions [FORMULA] and [FORMULA] were derived using the Collins & Sonneborn (1977) and Collins et al. (1991) models of rigidly rotating B stars. Inclination averaged [FORMULA] functions are shown in Zorec & Briot (1997). By parametrizing [FORMULA] and i we therefore obtain [FORMULA] and [FORMULA], from which it follows that therefore [FORMULA] = [FORMULA]) and [FORMULA] = ([FORMULA] of the non-rotating star. The observed [FORMULA] parameter of EW Lac implies an apparent luminosity class III. Since we assumed that this luminosity class may be partially due to rotational effects (Slettebak et al. 1980), we adopted only main sequence to subgiant [FORMULA]) relations. Then, for each [FORMULA] we interpolated the corresponding "rest" mass [FORMULA] =[FORMULA]) in Maeder & Meynet's (1988) stellar evolutionary tracks. These values are represented in Fig. 1 by the "vertical" curves.

[FIGURE] Fig. 1. The mass of EW Lac derived from relations (1) ("vertical urves") and from [FORMULA] ("horizontal curves"). The dots are the solutions for the given [FORMULA] and the corresponding inclinations i. The vertical dashed line indicates that solutions are possible only for [FORMULA]. The horizontal dashed line stands for the mean mass [FORMULA] solution

Another estimate of [FORMULA] as a function of [FORMULA] and i was obtained by using the observed value of [FORMULA] and the relation [FORMULA] km s-1. The [FORMULA] function used is given in Moujtahid et al. (1999). The masses [FORMULA] thus obtained are represented in Fig. 1 by the "horizontal" curves. The intersections of both series of curves give, for each possible inclination i and the respective [FORMULA], the estimates of the mass we are looking for. The mean value of these determinations is [FORMULA]. Moreover, we note that [FORMULA] solutions are possible for 0.89 [FORMULA] 1.0 ([FORMULA]). In Table 1 we reproduce several stellar parameters obtained as a function of [FORMULA]: [FORMULA]; [FORMULA] = "rest" stellar radius; [FORMULA] = equatorial radius distorted by rotation; [FORMULA] = equatorial rotational velocity and [FORMULA] = critical equatorial velocity. In Table 1 we see that the new velocities [FORMULA] obtained are consistent with the observed [FORMULA].


Table 1. Fundamental parameters of EW Lac as a function of [FORMULA]

Another straightforward, though less consistent, way of deriving [FORMULA], uses the relation [FORMULA] = [FORMULA] that leads to an estimate of [FORMULA] which gives [FORMULA] combined with [FORMULA]). The masses [FORMULA] thus obtained produce curves which are similar to the "vertical" ones shown in Fig. 1. The i-averaged parameters thus obtained are [FORMULA] and [FORMULA].

From these estimations we conclude that the central object of EW Lac most probably corresponds to a B1.5-B2 rotating main sequence star, which contrasts with the spectral type B3 III derived from the apparent photospheric parameters.

For this star seen under [FORMULA], we adopt [FORMULA], [FORMULA] and [FORMULA] km s-1 (an error of 10% is usual for rapidly rotating stars) corresponding to [FORMULA]. The rotational period is estimated as [FORMULA] day ([FORMULA] c/d) and the acceleration of a corotating feature on the stellar equator as [FORMULA] km s-1/d.

It should be kept in mind that the above error bars are obtained adopting Collins & Sonneborn (1977) and Collins et al. (1991) models and Maeder & Meynet (1988) evolutionary tracks.

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Online publication: October 30, 2000