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

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3. Analysis

3.1. Radial velocity determinations

The velocity shifts between spectra were measured using the cross correlation package FXCOR in IRAF . For each measurement the velocities found from 28 of the 35 orders of the echelle spectra were used to find a weighted average velocity. Weights for the averaging were the inverse of the velocity errors reported by FXCOR . The unused spectral orders either had no strong absorption lines or had strong interstellar lines which did not allow a reliable stellar velocity determination.

The procedure took two iterations. In the first iteration the best results were obtained by using the sum of all the second night's LSS 3184 spectra as the cross correlation template for the first night's spectra and vice versa. The spectra from both nights were then shifted by the velocities thus found and co-added to provide the template for the second iteration. The absorption lines in the second template were much sharper, as the velocity smearing due to the stellar pulsations was effectively removed. A third iteration was performed with the spectra shifted by the velocities found in the second iteration before co-adding to make the template, but the velocities from the third iteration were effectively identical to those from the second. The velocities from the second iteration, shifted so that the mean velocity is zero, are shown in Fig. 1. Error bars show the weighted standard deviations of the velocity from the echelle orders. Gaps are present in the data where clouds prevented observations. From the the data in Fig. 1, we find that the peak-to-peak radial velocity variation is [FORMULA] km [FORMULA], which is larger than 30 km [FORMULA], the value found by Kilkenny et al. (1999).

[FIGURE] Fig. 1. Plot showing velocity and phase coverage for LSS 3184 spectroscopic observations, with error bars shown for velocity. Note that the velocity here is not corrected for projection effects

Velocity data from the two nights were phased to the pulsation cycle using the ephemeris of Kilkenny et al. (1999): [FORMULA], [FORMULA]. The measured radial velocities were multiplied by the factor -1.42 to correct for projection effects and give the surface velocity through the pulsation cycle in the stellar rest frame. The projection factor was chosen based on preliminary work by Montañés Rodriguez et al. (2000) (See also Albrow & Cottrell 1994; Gautschy 1987; and references therein). We will discuss later the effects of choosing a different projection factor. A smooth curve (high order polynomial) was fit to the velocity data (Fig. 2). The velocity values on this curve were used to calculate change in stellar radius ([FORMULA]) and surface acceleration. The phase bin centers for acceleration and [FORMULA] are shifted by half a bin with respect to the velocity bins, i.e. the end of an acceleration bin is the center of the next velocity bin and the end of a velocity bin is the center of the next [FORMULA] bin.

[FIGURE] Fig. 2. Fit (solid line) to velocity data corrected for projection effects (points). The data have been folded over by 0.2 cycles in Figs. 2, 4, and 5

3.2. Temperature determination

To find temperature variations through the pulsation cycle, synthetic spectra were fit to the ground-based BV photometry of Kilkenny et al. (1999) and our HST ultraviolet spectrophotometry. Before the fitting, the spectra were binned in wavelength. No information is lost through the binning since the fits are to the shape of the spectrum, not to individual lines. The spectra are noisy at short wavelengths, reflecting a drop in detector sensitivity. To avoid possible problems caused by this increased noise, only the part of the spectra with [FORMULA] Å was used.

The pulsation period was divided into phase bins and the spectra and photometry within each bin were averaged. No significant difference in the results was found with the period divided into 15, 25, 50, or 99 bins. We will report the results found for 25 bins.

A grid of line-blanketed model atmospheres was calculated under the assumption of plane-parallel geometry, hydrostatic equilibrium and local thermodynamic equilibrium using the code STERNE described by Jeffery & Heber (1992) and by Drilling et al. (1998). Following the latter, we assumed a composition for LSS 3184 given by [FORMULA] and [FORMULA], where n represents fractional abundance by number, and all other elements were assumed to have solar-like relative abundances. The grid extended between 15 000 and 30 000 K, [FORMULA] and 4.00 (cgs units). As will be shown later, changing the [FORMULA] used in the model atmospheres makes only a minor difference in the temperatures and angular radii derived.

In the fitting procedure [FORMULA], angular radius ([FORMULA]), and [FORMULA] were allowed to vary and the downhill simplex program AMOEBA (Press et al. 1992) was used to find the minimum [FORMULA] difference between the synthetic spectrum and the observed spectral and photometric data at each phase (Jeffery et al. 2000). An example of the fit is shown in Fig. 3.

[FIGURE] Fig. 3. Best fit synthetic spectrum (dashed curve) plotted with ultraviolet spectrum (histogram) and BV photometry values. This is for the phase bin centered at 0.580

When the extinction [FORMULA] was allowed to vary we found the mean to be [FORMULA]. Because the extinction is not expected to vary with pulsation phase, we chose to use [FORMULA] throughout the cycle and did a second iteration allowing only temperature and [FORMULA] to vary.

3.3. Radius determination

In determining the radius of LSS 3184, we make two assumptions. First, we assume that the temperature (and thus [FORMULA]) and radial velocity (and thus [FORMULA]) were measured at approximately the same layer in the stellar atmosphere. Second, because we measure [FORMULA] perpendicular to the line of sight and [FORMULA] parallel to the line of sight, we are assuming that the pulsation is spherically symmetric. Thus we get [FORMULA]. With these assumptions we can use a modified Baade's method (Baade 1926) to find [FORMULA], using our previously determined values of [FORMULA] and [FORMULA].

There are two ways we can do this. In the first, we choose two phase bins, determine [FORMULA] and [FORMULA] between them and find [FORMULA], with [FORMULA] and [FORMULA] defined at one of the chosen phase bins. This method has the disadvantage of using data from only two phase bins and thus ignoring additional information available from the rest of the pulsation cycle. In the second method we plot [FORMULA] versus [FORMULA]. The slope of a linear fit to the data points is then [FORMULA]. We use the second method.

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© European Southern Observatory (ESO) 2000

Online publication: June 20, 2000