6 Pulsational broadening of spectral lines

As we mentioned in Sect. 3 there exists an unexplained extra broadening of spectral lines in $\gamma$ Equ. The pulsations themselves may introduce such a broadening. Hao (1998) noticed that the average line profile in the spectrum of pulsating stars is wider than in the spectrum of non-pulsating stars. M. Montgomery (private communication) calculated the line profile for $\ell=1$, m=1, v_p=10.5 kms^-1 in the spectrum of the $\delta$ Scuti-type variable FG Vir. This line profile corresponds to a synthesized one in a non-pulsating star with a macroturbulent broadening of 10 kms^-1 (Mittermayer 2000). Figure 6 shows a comparison between the observed and computed $\gamma$ Equ spectrum for two values of macroturbulence: 2 and 10 kms^-1. A macroturbulence of 10 kms^-1 is required to fit the Nd III and Pr III lines, while we need much lower macroturbulence to fit most of the other lines. Our pulsational velocity v_p$\approx10$ kms^-1, estimated from the Pr III and Nd III lines, is in a good agreement with the macroturbulent velocity inferred from these lines. The lower macroturbulence for other lines is consistent with the lower observable RV amplitudes. Of course, this effect should be investigated for a larger number of roAp and related stars, but we may conclude that a study of differential macroturbulent-like line broadening may give us important information on RV pulsation amplitudes.

Non-radial RV oscillations can introduce a line broadening in the average spectrum through two main effects. First, in the process of coaddition of the line profiles with different RV shifts one produces an average spectrum which is somewhat wider than the individual line profiles. However, careful inspection of our $\gamma$ Equ observations showed that the time-resolved profiles of Nd III and Pr III lines are not significantly sharper in comparison with the average spectrum. We suggest that the second broadening effect related to non-radial oscillations is responsible for the large observed width of doubly ionized REE lines. At any given pulsation phase the individual line profiles contain information about the velocity distribution on the stellar surface. If this velocity distribution is sufficiently inhomogeneous and the influence of the rotational Doppler broadening is negligible, individual time-resolved line profiles will appear broader in comparison with a non-pulsating star. Note that for a given pulsational velocity non-radial modes with lower $\ell$ numbers will produce higher amplitudes of RV variations, since in the case of slow rotation RV shifts due to higher $\ell$ modes will tend to cancel out more efficiently in the disk-integrated profiles. Thus, the very fact of significant broadening of Nd III and Pr III lines together with the value of the most probable pulsational velocity ( v_p$\approx10$ kms^-1) and high RV amplitudes indicates that $\gamma$ Equ pulsates in a dominant low-$\ell$ non-radial mode.

We performed a rough study of the influence of non-radial oscillations on the shape of average line profiles using the code lnprof2 developed by Balona (2000). This programme makes it possible to calculate line profile variations of non-radially pulsating star due to the changes in local RV, temperature and gravity under the assumption of a constant intrinsic local line profile. This simplification is definitely not correct for roAp stars for which we expect the local line profile to undergo strong changes from one point on the stellar surface to another due to the variations of the local elemental abundance as well as magnetic field strength and direction. These complications associated with the nature of magnetic Ap stars preclude us from directly fitting the line profile variations using lnprof2. Nevertheless, Balona's code is useful to illustrate the pulsation broadening of average line profiles described above. In the lower panels of Fig. 6 we compare high-resolution observations of Nd III and Pr III lines with average synthetic line profiles computed by lnprof2 for sectoral modes with $\ell =2$ and $\ell =4$, $\alpha=130\hbox{$^\circ$ }$ and v_p=10 kms^-1. For the intrinsic local profile we adopted synthetic disk-average Synthmag spectra (dashed line in lower panels of Fig. 6) broadened by Gaussian with ${\it FWHM}=1.8$ kms^-1 to take into account instrumental effects. Figure 6 shows that including non-radial pulsations allows us to improve the agreement between spectrum synthesis and observations. (The discrepancy between observations and synthetic spectra seen in the blue wing of Nd III 6145.07 Å line is due to the blending by Si I 6145.02 Å, which cannot be included in the local intrinsic profile and broadened by v_p=10 kms^-1 because we do not observe strong RV variations for other Si I lines.) In general we found that the width of the average profiles broadened by pulsations is rather insensitive to the pulsation mode, but shows strong dependence on v_p and $\alpha$, which again supports the idea that v_macro, required to fit doubly ionized REE lines in $\gamma$ Equ spectrum, provides a useful estimate of the pulsation velocity amplitude.