In practice one aims to derive pulsation parameters from observed spectra. To enable a comparison of our results with that of observational studies, we analyze the artificial time series of spectra with two diagnostic methods, each closely related to techniques discussed in the literature (Gies & Kullavanijaya 1988; Aerts et al.\ 1992).
We calculated amplitude and phase diagrams as diagnostic tools for studying line-profile variations. The diagrams are equivalent to the power and phase diagrams obtained by the popular method of Fourier analysis proposed by Gies & Kullavanijaya (1988).
Figure 1: Line-profile variability for a pulsation mode with input parameters = 5, m = -5, = 0.1 , = 0.0, W = 0.1 , i = , and = 0.0 (zero-rotation model).
Left: Line profiles during one pulsation cycle, with time increasing upwards. The first and the last profile are identical. The dotted line indicates the mean of the time series. The separation between the tick marks at the velocity axis is . The two arrows at the bottom refer to V = 0 and V = 0.7 .
Right: Top: Intensity variations in line center (V = 0). The variations are shown during two pulsation cycles. The time tick marks correspond to those at the vertical axis of the left plot. The intensity variation is indicated by a thick line and is measured with respect to the mean profile. The thin lines show the decomposition of the signal into sinusoids with once and twice the apparent frequency. Bottom: Same as top panel but at velocity V = 0.7 . The intensity scales are equal for top and bottom panel
The apparent frequency of observed line-profile variability
() has to be determined by Fourier techniques. For
synthetic line profiles it is equal to the input frequency of the
variations. Since the intensity variations in the line profile are
not strictly sinusoidal (see Fig. 8b in Gies 1991), there will be
also some fraction of the total variational power distributed over
harmonics. Once the main frequency of the line-profile variation is
known, the variability of the normalized intensity can be decomposed
into its harmonic contributions, which is in our case conveniently
achieved by fitting a combination of sinusoids of the form
to the intensity variations of a time series of generated spectra. In general, the amplitudes of the intensity variations decrease for higher harmonics. Harmonics higher than the second (3) are not taken into account in the fits since their amplitudes are too small to be of importance in practice.
In Fig. 1 (click here) we illustrate the importance of harmonics in the analysis line profile variations. It appears that in general at least one harmonic is needed to properly describe the variability. The phases of the sinusoids, as a function of position in the line profile, describe the motion of the bumps that cross the profile. Through the line profile the phases and amplitudes of the sinusoids show independent behavior, which is a clear indication that the inclusion of a harmonic provides additional information to the analysis of the line-profile variability. Taking into account one or more harmonics can therefore improve the identification of pulsation parameters from observed spectra.
For the apparent frequency and its first two harmonics, we calculate the amplitudes (, , ) and phases (, , ) for each position in the line profile. We remove the wrap from , and to create continuous blue-to-red phase diagrams. In the figures which result from our grid calculations (Figs. 3-15 (click here)) we only plot the first-harmonic amplitude and phase diagrams, because of the relatively low amplitude of the second harmonic.
Figure 2: Example diagram for a pulsation mode with input parameters = 4, m = -4, = 0.15 , = 0.15, W = 0.1 , i = , and = 0.0.
Left: Line profiles, and first (thick line, M1) and second (thin line, M2) velocity moments during one whole pulsation cycle, with time increasing upwards. The last profile and the first are identical. The tick marks at the velocity axis indicate . The first moment varies sinusoidally around zero with amplitude M1. The second moment varies between M2 and M2.
Right: Top: Three pulsation cycles of residual (mean subtracted) spectra are shown in a grey-scale image. Intensities less than average are indicated black; bright regions in the profile are indicated by lighter shades. Middle and bottom: Amplitude distribution and phase diagram. From thick to thin: the amplitude and phase of the line-profile variations at the input frequency ( and ), the first harmonic frequency ( and ), and the second harmonic frequency ( and ). In the middle panel, the maximum values of the amplitude distributions are given in units of the average central line depth . The total blue-to-red phase differences , , and are given inside the bottom panel, in radians. In all other figures we show only , , ,
For a Gaussian intrinsic profile, the amplitudes of the intensity variations (in continuum units) are proportional to the depth of the intrinsic profile, , and consequently also to the time-averaged depth of the time-series. We give the maxima of the amplitudes across the line profile (, , and ) in units of the central depth of the mean of the time series, i.e. we present them in units of = 0) (see Fig. 2 (click here)), where P(V)=1-I(V) is the profile function. Therefore, our results are independent of the line depth , and thus applicable to a wide range of absorption lines.
In Fig. 2 (click here) we see that the line-profile variations, and therefore the amplitude and phase distributions, extend significantly beyond . In fact, the variability extends to the maximum projected pulsation velocity, and reading off the phase diagrams at can lead to a significant underestimate of the blue-to-red phase difference. We measure the blue-to-red phase difference over the full range at which variability is found. The advantage of this is that one does not need to have a priori knowledge of ; the disadvantage is that the investigated lines must not be affected by line blending. For blended lines, the phase diagram will continue from one line into the other, and can therefore not be used to determine , , .
The blue-to-red phase differences , , and (printed inside the bottom panel at the right in Fig. 2 (click here)) are obtained by taking the maximum phase difference between the outermost velocity values at which the corresponding amplitude exceeds . For real observations, this cut-off velocity will always be smaller because of noise which makes the phase diagram indeterminate at the wings of the profile.
In most cases, the phase diagrams are monotonic over the entire line profile, so that the maximum phase difference is found between the outer ends of the region in which variability is found. In some cases, however, there are slope reversals in , so that the maximum difference is found in a smaller region in the line profile (see e.g. the middle row of Fig. 10 (click here)b).
Another important diagnostic for the determination of the properties of line-profile variations is the evaluation of the first few velocity moments of the line profile (Balona 1986; Aerts et al. 1992). It is not our aim to present a detailed study of the moment variations as a function of the different parameters. Such an analysis is presented by De Pauw et al. (1993) in the case of the zero-rotation model. In this work we compute the first and second velocity moment to allow a comparison with De Pauw et al. and Aerts et al. (1992), and to show what global line-profile variations (line shifts, line-width changes) go along with the intensity variations as expressed by the amplitude and phase distributions.
We calculate the first and second velocity moment and ,
by a weighted summation of the normalized intensity across the line
where P(V)=1-I(V) is the profile function and is a reference velocity. The zeroth moment (equivalent width) is constant since local variations of the intrinsic profile are not taken into account here. The first moment is calculated with the rest wavelength of the line as reference, and corresponds to the centroid velocity of the line profile. The second moment is calculated using the first moment as reference velocity , and is a measure of the squared width of the line. We normalize the velocity moments with respect to the equivalent width , and scale the normalized first and second moments to and respectively. The first velocity moment varies sinusoidally with the pulsation frequency; the second moment shows variations with the pulsation frequency as well as its harmonics.
Our definition of the moments and is related to the
definition of velocity moments used by Aerts et al. (1992)