4 Line of sight velocities

Velocity curves were constructed by measuring the Doppler shifts of three Balmer lines: H$\beta$, H$\gamma$, and H$\delta$. This was done by fitting a sum of a Gaussian and Lorentzian profile and approximating the continuum by a line. The centroids of the Gaussian and Lorentzian functions were forced to be the same. The wavelength intervals chosen for our fits were $\sim$388 Å, 300 Å, and 164 Å centred approximately on H$\beta$, H$\gamma$, and H$\delta$ respectively. We computed the velocity curve by simply averaging the velocities derived from each line.

The measurement errors are of the order of the scatter due to random wander in all but one of the series of spectra (Table 1). The one exception is the series taken of G 126-18 in Dec. 1997, where the scatter is much smaller because of the tie to a reference object. Our measurements are most precise for G 67-23. We include our estimates for the scatter due to random wander in the least squares fit of the velocity curve (see below); the resulting fits all have $\chi_{\rm red}^2 \simeq 1$.

The Fourier Transforms of the velocity curves are shown in Fig. 1. None of the red edge objects show a clear peak well above the noise level at any frequency.

For HL Tau 76, two of the stronger peaks have corresponding peaks in the light curve. However, there are also two other peaks evident in Fig. 1 at 6651 $\mu$Hz and 6928 $\mu$Hz that are not present in the light curve at amplitudes greater than $0.9\pm0.8$% and $0.6\pm0.8$% respectively i.e. well below our detection limit. We will return to this point in Appendix A.

Although it is clear from Fig. 1 that none of our objects show a large ($\simgt$5  $\rm km~s^{-1}$) peak at any frequency, given the disparities in the quality of our various data sets we must carry out tests in order to be able to place meaningful limits. We describe below two different Monte Carlo tests we devised to obtain quantitative estimates.

   
4.1 Significance of possible detections

We first looked for periodicities in each of the velocity curves by concentrating on the few peaks of relatively high amplitude in the Fourier transform of the velocity curves and fitting these successively with sinusoids, the frequencies, amplitudes, and phases all being free parameters. We included a low order polynomial in the fit to remove slow variations for all objects except G 126-18 (Dec. 1997) for which the calibration with respect to the reference star will have already removed any slow variations.

The first test we carry out aims to determine the likelihood of the peaks seen in the Fourier Transforms (Fig. 1) as being simply due to random noise. We shuffle the measured velocities randomly with respect to their measurement times after removing slow variations by fitting a 3rd-order polynomial for all objects (except G 126-18, Dec. 1997). We then determine the amplitude of the highest peak in the Fourier transform at any frequency between 700 $\mu$Hz up to the Nyquist frequency. (This choice reflects an extended ZZ Ceti period range.) We repeat this procedure 1000 times, counting the number of artificial data sets in which the highest peak exceeded our maximum measured $A_{\rm V}$. The higher this number, the higher the chance that even the strongest peak is not genuine. Table 2 shows our results. The strongest peaks in all the red edge objects can be attributed to noise. Only in HL Tau 76, the one known pulsator, is there an indication that the velocity signal is real. We show in Appendix A that external information (from the light curve) is necessary to confirm this.

 

 

Table 2: Results from Monte Carlo simulations of velocity amplitudes.

Largest Peak in F.T.				Detection
Object	$\nu$	$A_{\rm V}$	$P({>}A_{\rm V})$	Limit ($\sim$95%)
	($\mu$Hz)	( $\rm km~s^{-1}$)	(%)	( $\rm km~s^{-1}$)
G 1-7 (s)	753	6.0 $\pm$ 1.5	42	8.8
G 126-18	6518	3.4 $\pm$ 1.0	67	5.2
G 126-18 (s)	2430	4.1 $\pm$ 1.2	35	5.9
G 67-23 (s)	3988	2.0 $\pm$ 0.7	60	3.0
HL Tau 76 (s)	1933	4.1 $\pm$ 1.0	17	5.1

Note. $A_{\rm V}$ is the maximum measured velocity amplitude for each object at the frequency ($\nu$) specified in the first column. $P({>}A_{\rm V}$) is the percentage probability that the largest peak (i.e. of amplitude $A_{\rm V}$) is present solely by chance. The third column lists the velocity amplitude necessary to constitute a detection at the 95% confidence level - at any frequency - as estimated from our second Monte Carlo test (see also Fig. 2). Note that the frequency corresponding to the maximum velocity peak is different for the two G 126-18 data sets.

Our second objective is to determine the minimum velocity amplitude in each data set for which we could have had a high ($\ga$95%) degree of confidence that, had such a signal been present, we would have detected it. To do this, we inject an artificial signal in our data of a given amplitude and random frequency and phase. The random frequencies are chosen from 700 $\mu$Hz up to the Nyquist frequency, while the phases are chosen from 0 to $2\pi$. We repeat the procedure 1000 times and count the number of artificial data sets in which the highest peak in the Fourier transform is located within a range corresponding to the input random frequency plus or minus the frequency resolution of the data. We repeat the procedure, varying the input amplitude, until we recover the artificial peak as the maximum peak in at least 95% of the trials within the expected frequency range. We expect that the velocity amplitude we obtain in this manner will be somewhat higher than the detection limit derived above. This is because for peaks having an amplitude exactly at the detection limit, there is an equal chance of a noise peak either enhancing or diminishing it. Thus for a peak exactly at the detection limit, we cannot expect >50% confidence. This test implicitly assumes that none of the signal in our Fourier transform is real. For HL Tau 76, we show in the appendix that the higher peaks in the velocity Fourier Transform that are coincident with peaks in the Fourier Transform of the light curve are real. Hence, the limits for any signal not associated with flux variations will be slightly lower than those inferred from Fig. 2.

The results of the above simulation are displayed in Fig. 2. We can rule out velocity amplitudes larger than 5.4  $\rm km~s^{-1}$ - as detected by van Kerkwijk et al. (2000) for ZZ Psc - for all objects except G 1-7. Our best limit for a minimum significant amplitude (3  $\rm km~s^{-1}$) comes from G 67-23.

Figure 2: Probabilities that the velocity signals with the given amplitudes that would have been detected in the different data sets had they been present. The dashed line shows the 95% confidence level, at or above which measured velocities would be taken to be significant. For our best data set (G 67-23), we can exclude periodic signals with amplitudes greater than 3  $\rm km~s^{-1}$ with 95% confidence.