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Astron. Astrophys. 358, 1001-1006 (2000)

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4. Results and discussion

The velocities, effective temperatures, and angular radii measured through the pulsation cycle of LSS 3184 are listed in Table 1. The velocities are not corrected for projection effects and are reported for the center of the phase bins. Temperatures and angular radii are reported for the beginning of the phase bins. Data are missing where HST observations did not fall in the affected phase bins. Typical uncertainties are indicated.


Table 1. Velocity, temperature, and angular radius through the pulsation cycle of LSS 3184. Velocity is not corrected for projection effects.

The surface acceleration, surface velocity, and change in radius determined for LSS 3184 using the AAT spectra are shown in Fig. 4. Note that the surface velocity reported here is the velocity measured from the spectra multiplied by -1.42 to correct for projection effects and make positive velocity be away from the star's center.

[FIGURE] Fig. 4. Surface acceleration, surface velocity, and change in radius of LSS 3184 through its pulsation cycle

The change in radius has been set so that it is zero at photometric maximum, [FORMULA].

Fig. 5 shows the integrated flux between 1270 and 2508 Å and the temperature and angular radius determined by fitting synthetic spectra to HST UV spectra and ground-based BV photometry through the pulsation cycle.

[FIGURE] Fig. 5. Flux [FORMULA], effective temperature, and angular radius of LSS 3184 through its pulsation cycle. The curves simply connect the data points and are not fits to the data

[FORMULA] is plotted against [FORMULA] in Fig. 6. The `0' subscript indicates that the reference bin is at [FORMULA].

[FIGURE] Fig. 6. [FORMULA] versus [FORMULA] through the pulsation cycle of LSS 3184. Points are connected in order of phase with [FORMULA] at the origin. The dashed line is a linear least squares fit to all data points. Error bars for [FORMULA] are smaller than the symbols. Numbers next to symbols indicate the corresponding pulsation phase

As is seen in Figs. 4 and 5, the shapes of the [FORMULA] and [FORMULA] curves are not identical, which is the cause of the non-linear, looped shape of the [FORMULA] versus [FORMULA] curve. If the [FORMULA] and [FORMULA] curves are normalized and placed on the same plot (Fig. 7), the differences in the curves are more noticeable. Fig. 12 from Kilkenny et al. (1999) shows a similar loop on the right hand side of the angular radius versus stellar linear radius plot. Our work cannot be used as an independent confirmation of the non-linear angular versus stellar radius curve, however, as we have used the same BV photometry as Kilkenny et al.

[FIGURE] Fig. 7. [FORMULA] (solid curve) and [FORMULA] (dashed curve) versus pulsation phase

The non-linear shape makes determining the radius more difficult. It raises questions about the assumption made in using Baade's method to find [FORMULA], that [FORMULA] and [FORMULA] are measurements of the same quantity, with [FORMULA] decreased by a factor proportional to the distance to LSS 3184. One possible explanation of the discrepancy is that we are measuring [FORMULA] and [FORMULA] at different layers in the atmosphere. [FORMULA] is measured using optical spectra, while [FORMULA] is measured using ultraviolet spectra and optical photometry. It is possible that the layer where the optical lines are formed expands and contracts a bit differently than the layer where the ultraviolet continuum is formed. It is also possible that the star and/or its pulsations are not spherically symmetric, as might occur if the star were flattened by rotation, so that the measured [FORMULA], which is perpendicular to the line of sight, and [FORMULA], which is parallel to the line of sight, act differently. Further, the atmosphere is not static. The atmosphere's temperature is constantly changing, and the pulsating layers undergo a substantial compression at minimum radius, as shown by the spike in the acceleration in Fig. 4, which may cause nonadiabatic effects. So the temperature may be acting differently in the expanding part of the phase than in the contracting part.

However, it is encouraging that the slopes of the expanding (the upper half of the loop in Fig. 6) and contracting (lower half of the loop) parts of the [FORMULA] versus [FORMULA] curve are not too different. [FORMULA] and [FORMULA] are still correlated.

If we find the slope of the curve using a least squares fit to all of the data points (dashed line in Fig. 6) then the radius at photometric maximum ([FORMULA]), the inverse of the slope, is [FORMULA], where the uncertainty is derived from the standard error in the least squares fit to the slope. The average [FORMULA] over the pulsation cycle, with [FORMULA] at [FORMULA], is 28 190 km, or [FORMULA]. So if we use all the [FORMULA] versus [FORMULA] data points, we find the mean [FORMULA] to be [FORMULA]. This is larger than the [FORMULA] mean radius found by Kilkenny et al. (1999) for LSS 3184 and is closer to the [FORMULA] mean radius found for V652 Her by Lynas-Gray et al. (1984).

We tested the effects of using only a portion of the [FORMULA] versus [FORMULA] curve to determine [FORMULA], though we have no reason to reject any particular data points. If we use the points on the upper part of the curve, with phase [FORMULA], then we find [FORMULA]. If we use the points on the lower part of the curve, ignoring the points to the left of the change in slope at [FORMULA] km, so that [FORMULA], we find [FORMULA]. If we use only the points to the right of [FORMULA] km then we find [FORMULA]. In all cases, even the extreme one where we reject all points but those on the lower part of the curve, our [FORMULA] is larger than [FORMULA], the value found by Kilkenny et al. (1999).

Drilling et al. (1998) estimated [FORMULA] for LSS 3184. If we use this with our [FORMULA] in the formula [FORMULA], we find [FORMULA]. This is larger than [FORMULA], the mass found by Kilkenny et al. (1999) for LSS 3184 and closer to [FORMULA], the estimated mass of V652 Her (Lynas-Gray et al. 1984). Our larger estimate results mainly from the larger peak-to-peak range of radial velocities we measured. It is likely that the smaller velocity amplitude measured by Kilkenny et al. resulted from a combination of the lower spectral resolution and the lower signal to noise in the spectra they used. This meant that even the best cross correlation template still contained some velocity broadening, thus diluting the velocity amplitude.

There are several sources of uncertainty in our mass determination. Uncertainty in the [FORMULA] used is a major contributor. We note that it is impossible to determine the pulsation phase of LSS 3184 when the spectrum used in the analysis of Drilling et al. (1998) was taken in 1985, as [FORMULA] is unknown. They estimated the temperature of LSS 3184 at [FORMULA] K. Our data yield a similar value, [FORMULA] K, if we use [FORMULA], the extinction they used. However, because temperature, line strengths, and other parameters presumably varied throughout the 1-hour exposure for their spectrum, it is possible that the errors in their temperature and gravity determinations are larger than the formal errors quoted. If we assume that LSS 3184 has the same [FORMULA] as V652 Her, [FORMULA] (Jeffery et al. 1999), instead of [FORMULA], then we find that it has [FORMULA]

As mentioned earlier, the model atmospheres used to calculate temperature and angular radius from HST UV spectra and ground-based BV photometry assumed [FORMULA]. The assumed gravity enters into the stellar parameter calculations in the stellar atmospheres used and in calculating the mass from the radius. Changing [FORMULA] in the model atmospheres by 0.50 dex (to 3.00 or 4.00), but using [FORMULA] to calculate the mass as before, changes the [FORMULA], [FORMULA], and [FORMULA] derived by about 1 per cent, and thus has only a small effect ([FORMULA] per cent) on the mass derived. The mass finally derived is proportional to g.

Uncertainty in the extinction also adds uncertainty to the stellar parameters calculated. Using [FORMULA] instead of 0.24 increases the derived [FORMULA] by 510 K, or about 2 per cent, but changes [FORMULA] by only 0.2 per cent, and thus has a very minor effect on the derived stellar radius and mass.

The derived radius varies proportionally with the projection factor used to transform the measured stellar radial velocities into surface velocities. For example, using a projection factor of 1.31, as Lynas-Gray et al. (1984) used in their analysis of V652 Her, instead of 1.42 would give [FORMULA] instead of [FORMULA]. The smaller radius would give [FORMULA] instead of [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: June 20, 2000