Based on high-resolution high S/N observations we confirmed earlier results obtained by Savanov et al. (1999), who discovered strong RV variations of doubly ionized REE lines in the spectrum of $\gamma$ Equ. In addition to refining the measurements of variations of the third REE ions, we were also able to obtain precise radial velocity amplitudes and phases for singly ionized REE. Our data give an indication for the existence of a phase lag of $\approx$ $80\hbox{$^\circ$ }$ between RV variations of singly (average phase $\langle\varphi\rangle=0.935\pm0.053$ ) and doubly ionized ( $\langle\varphi\rangle=0.160\pm0.009$ ) REE. The lines of all other elements but Na I do not show RV variations with amplitudes greater than 100 ms^-1. Below this threshold some RV variations are possible for Fe II lines and are definitely present for Ba II line (which varies in phase with singly ionized REE), while strong Ca I lines are stable to within 30-50 ms^-1. These amplitudes are consistent with earlier RV analyses (Libbrecht 1988; Kanaan & Hatzes 1998).

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h2579f7.eps}\end{figure}$

Figure 7: Amplitudes of RV variations of spectral lines shown as a function of equivalent width. Filled symbols correspond to unblended lines of Fe (circles), Cr (triangles) and Ti (squares) from the study of Kanaan & Hatzes (1998), while open circles represent results of our study for all lines except Ba II, Na I and REE. The solid line shows a weighted least-squares fit by a straight line ${\mathrm {Amp}} = a \cdot {\it EW} +b$ ( $a=-0.14\pm 0.07$ , $b=39.66\pm 5.31$ , $\chi ^2=69.6$ ). The low $\chi ^2$ probability of this linear fit ( $P(\chi ^2)=18.5\%$ ) together with the value of the weighted linear correlation coefficient r=-0.16 and the relatively high false-alarm probability $P(r)=20.1\%$ indicate the absence of significant correlation.

We did not confirm a correlation of the RV amplitudes with the line intensity found in the latter paper. Looking through the line list published by Kanaan & Hatzes we found that many spectral lines with high RV amplitudes were not identified properly and in reality belong to REE lines in different ionization stages. We performed a careful spectral synthesis of all lines published by Kanaan & Hatzes (1998) with the best atmospheric abundances of $\gamma$ Equ to check for blending effects. First, blends with REE lines were removed from Kanaan & Hatzes' list of Ti, Cr and Fe lines. Next, a few lines were found to be too strong or shifted in wavelength for the proposed identification ( $\lambda\lambda$ 5151.84, 5566.00, 6044.50, 5379.00 Å). The first two unidentified lines are very strong in the famous, extremely REE-rich, Fe-weak Przybylski's star (HD 101065 - see line identification list from Cowley et al. 2000), which provides additional evidence for their wrong identification by Kanaan & Hatzes. Finally 54 out of 70 lines of Ti, Cr and Fe were chosen as unblended or partially blended by the same species. We add 8 lines of Ca, Cr and Fe from our study and plot RV amplitudes versus equivalent widths in Fig. 7. There is no statistically significant correlation between line intensity and pulsational amplitude.

Taking into account our results we looked through RV pulsational analyses of $\alpha$ Cir (Baldry et al. 1998a, 1999) and HD 83368 (Baldry et al. 1998b; Baldry & Bedding 2000). Low resolution spectra were used for both stars. There are 24 spectral regions in $\alpha$ Cir with RV amplitudes higher than 150 ms^-1 and exceeding three individual rms estimates. The lines of doubly ionized REE are identified in 9 of them. Two telluric bands, 6864-6881 Å and 6903-6920 Å, used as a fiducial reference for H $\alpha$ RV analysis, contain rather strong Pr III lines, 6866.73 Å and 6910.15 Å. The influence of the last line on RV analysis is easily seen, because a phase of the band ( $263\hbox{$^\circ$ }$ ) is close to the phases of the bands with other Pr III and Nd III lines (e.g. band 18 with Pr III 6195.63 Å and band 33 with Nd III 6327.27 Å lines, see Table 3 in Baldry et al. 1998a). Strong RV variations of REE lines may also be responsible for producing small phase difference between light and H $\alpha$ RV variations found by Baldry et al. (1998b) for HD 83368. A presence of strong Nd III 6550.33 Å line in the blue wing of H $\alpha$ is the most probable reason for the jump in amplitude and phase near the intensity 0.8 of H $\alpha$ in $\alpha$ Cir (Baldry et al. 1999). Our knowledge of the second spectra of the REE is very scarce, therefore it will not be surprising if their future identification will explain a significant part of the observational features of the roAp stellar pulsations. A weak correlation between line intensity and RV pulsational amplitude found by Baldry et al. (1998a) in $\alpha$ Cir can be easily explained by the low spectral resolution of their observations.

The fact that the lines of the second ions of Pr and Nd show the highest pulsational amplitudes in roAp stars was supported by the recent time-series analysis of another roAp star HD 83368 (Baldry & Bedding 2000). Six out of eight spectral bands with metallic lines presented in their Table 4, which show the highest amplitudes of RV pulsations (Nos. 13, 14, 18, 33, 54, 90), contain known spectral lines of Pr III and Nd III. It is important to note that RV amplitudes are higher for those bands where Pr III and Nd III lines contribute more than 60% to the total band intensity (Nos. 18 and 33), while RV amplitudes of the bands where contribution of Pr III and Nd III lines is less than 30% (Nos. 13, 14, 54) are smaller. Rather strong lines of iron-peak elements with low RV amplitudes in these bands decrease the net value of the RV amplitude derived by cross-correlation analysis of low resolution spectra.

Weiss et al. (2000) and Ryabchikova et al. (2001) analysed Pr III and Nd III lines in the spectra of a dozen roAp and non-pulsating Ap stars and found that for roAp stars elemental abundances obtained from Pr III and Nd III lines are up to two orders of magnitude higher than abundances derived from the lines of singly ionized Pr and Nd. In non-pulsating Ap stars the same anomaly is marginal if present at all. Ryabchikova et al. (2001) proposed that REE are concentrated in the uppermost atmospheric layers where a transition from singly to doubly ionized REE occurs due to a pressure drop. Unfortunately no quantitative diffusion calculations are available for the REE. However, qualitatively, proposed REE distribution is supported by diffusion calculations for another heavy element, Hg (Michaud et al. 1974). An analysis of the abundance stratification in the atmosphere of $\gamma$ Equ and the distribution of the pulsational amplitudes will be given in a separate paper. The phase shifts of RV variations (cf. Na I, singly and doubly ionized REE) as well as the amplitude difference may be connected with inhomogeneous surface and vertical abundance and pulsation amplitude distributions or indicate non-adiabatic pulsations. The extremely slow rotation of $\gamma$ Equ does not allow us to investigate the distribution of the different elements over the surface of the star and correlate it with pulsation amplitudes, but at least two other roAp stars, HD 24712 (Ryabchikova et al. 2000) and HD 83368 (Polosukhina et al. 2000), are known to be spotty pulsators. The latter star is of particular interest for simultaneous time-resolved photometry and spectroscopy over the rotational period due to its substantial rotation.

An application of the relation between velocity and luminosity amplitudes for the solar-like p-mode oscillations to $\delta$ Scuti and roAp stars was discussed by Kjeldsen & Bedding (1995). If we adopt the pulsational amplitude of approximately 100 ms^-1 in the atmosphere of $\gamma$ Equ, as it follows from the lines of iron-peak elements, Si I and Ba II, and effective temperature of 7700 K (Ryabchikova et al. 1997), we get a prediction for the light variation amplitude of about 1.3 mmag at 440 nm (Johnson B) using Eq. (5) of Kjeldsen & Bedding (1995). This value agrees reasonably well with the semi-amplitude of 0.5-1.3 mmag observed for $\gamma$ Equ by Kurtz (1983). But if we apply the pulsational amplitude of 500 to 800 ms^-1 as derived from Nd III and Pr III lines then the expected light variation amplitude would be greater than 6.0 mmag. Either the relation between velocity and luminosity amplitudes is not valid for roAp stars or high amplitude RV pulsations occur in the atmospheric layers whose contribution to the total luminosity is negligible.

7 Discussion