2 Observations and data reduction

Time-resolved spectra of three alleged non-pulsators, G 1-7, G 67-23, and G 126-18 (Kepler et al. 1995) and one known ZZ Ceti-type white dwarf, HL Tau 76, were acquired using the Low Resolution Imaging Spectrometer (Oke et al. 1995) on the Keck II telescope during a service run on the nights of 18th and 20th October 1997. In addition to the above, we took time-resolved spectra of G 126-18 during an observing run with the same instrument and set-up on 11th December 1997. Details of the observations together with other quantities are summarized in Table 1. An 8 $\hbox{$.\!\!^{\prime\prime}$ }$7 slit was used with a 600 line mm^-1 grating, the resulting dispersion being $\sim$1.25 Å pix^-1. For the service run, the seeing was around 1 $\hbox{$.\!\!^{\prime\prime}$ }$0, yielding a resolution of 6 Å, while the December 1997 data were taken in slightly worse seeing conditions (1 $\hbox{$.\!\!^{\prime\prime}$ }$2), corresponding to a resolution of about 7 Å. A problem with the Active Control System meant that 22 frames of G 67-23 taken during the beginning of the run had to be discarded and 4 frames taken at the very end of the run also had to be disregarded as the position of the target in the slit seemed to have altered. The last frame of HL Tau 76 was not used as it was clear that light was being lost. For the observations taken during the service run, we were unfortunately unable to obtain a sufficient number of halogen frames so we decided to forgo flatfielding the spectra altogether.

Shifts in the positions of the Balmer lines are introduced by instrumental flexure and differential atmospheric refraction. As in van Kerkwijk et al. (2000), we corrected for the former using the position of the O I $\lambda$ 5577 Å sky emission line, and for the latter by computing the wavelength-dependent shifts using the recipe of Stone (1996). As the spectra were taken through a wide slit, the positions of the Balmer lines will also depend on the exact position of the target in the slit. In principle, this position should be tied to the position of the guide star, but in practice we found that there was substantial random jitter, probably due to guiding errors and differential flexure between the guider and the slit.

One way of accounting for the random movements in the slit is to in effect "tag'' the movements of the target to those of a reference star that can also be accommodated in the slit. For the Dec. 1997 observations, the slit was set at a position angle such that a G-type star was in the slit together with G 126-18. The wandering of G 126-18 could then be corrected for using the positions of H$\beta$ of the former as fiducial points.

An estimate of the scatter due to random wander in the slit can be obtained by measuring the shifts along the slit of the spatial profiles and translating these into scatter in the dispersion direction. To do this, we simply fit Gaussian functions to the spatial profiles, and fit a low order polynomial to the resulting centroid positions (if no reference star was available). The standard deviation of the centroid positions from the smooth curve is then a proxy of the scatter due to wander in the slit. We compute this scatter at the representative wavelength of H$\gamma$. The resulting estimates are shown in Table 1. As expected, the use of a reference object for G 126-18 dramatically lowers the scatter in the measured line-of-sight velocities (see Table 1). However, the value for G 67-23 shows that commensurate results can also be obtained without a local calibrator if the observing conditions and guiding are sufficiently stable.

Figure 1: Fourier transforms of the light (top panels) and velocity curves (bottom panels) for all objects. The marked peaks have been included in the Monte Carlo simulations in Sect. 4.1. We find no significant peaks in either the flux or the velocity FTs for G 1-7, G 126-18, and G 67-23. All FTs are shown up to the Nyquist Frequency except that of HL Tau 76 for clarity (see also caption to Fig. A.1).