1 Introduction

In spite of advances in theoretical formulations and observational capabilities, the details of the inner workings of the cool H-rich pulsating white dwarfs (DAVs or ZZ Cetis) continue to elude us.

Early attempts to explain the nature of the driving of the pulsations of ZZ Cetis postulated that driving occurred via the $\kappa$-mechanism - i.e. in a manner akin to that in Cepheids and $\delta$-Scuti stars - (e.g. Dolez & Vauclair 1981; Dziembowski & Koester 1981; Winget et al. 1982) thereby ignoring the effects of pulsation on the convection zone (also known as the "frozen-in'' approximation) even though radiative flux transport is negligible in these regions. Brickhill (1991) realised that the response of the convection zone to the perturbation is almost instantaneous (${\sim}1$ s) compared to the periods of the g-modes (hundreds of seconds). He found that as a result, the convection zone itself can drive the pulsations: part of the flux perturbations entering the convection zone from the largely radiative interior are absorbed by the convection zone and are released half a cycle later. This interaction between the perturbation and the response of the convection zone drives the g-modes.

Within the framework of this "convective-driving'' mechanism, recently confirmed analytically by Goldreich & Wu (1999a), the blue edge of the instability strip i.e. the hottest temperature for which pulsations are excited and at which pulsations are expected to be observable, is set by the condition that $\omega \tau_{\rm c} \approx 1$ for radial order, n = 1 and spherical degree $\ell=1$; $\omega$ is the radian frequency and $\tau_{\rm c}$ the thermal time constant of the convection zone. (Note that $\tau_{\rm c}$ is different from the "global'' thermal time scale, $t_{\rm t}$ in Brickhill 1991. Where necessary, we follow the notation of Goldreich & Wu 1999a.)

The location of the red edge is less clearly defined. As long as a mode is driven, its intrinsic amplitude is likely to remain roughly constant as it is most likely set by parametric resonance with stable daughter modes (Wu & Goldreich 2001). As the white dwarf cools, the depth of the convection zone increases, and damping in the shear layer at the base of the convection zone becomes stronger. At some stage, damping will exceed the driving due to the convection zone. This sets the physical red edge, beyond which pulsations are no longer excited.

Observationally, though, the red edge may appear at higher temperatures. This is because, as the convection zone deepens, the amplitude of flux variations at the photosphere becomes an ever smaller fraction of the amplitude in deeper layers; the convection zone acts as a low-pass, frequency-dependent filter leading to a reduction in flux given by (Goldreich & Wu 1999a)

 

$$\left(\frac{\Delta F}{F}\right)_{\rm ph} = \left(\frac{\Delta F}{F}\right)_{\rm b}\frac{1} {\sqrt{1 + (\omega\tau_{\rm c})^{2}}}$$ (1)

where the subscripts "ph'' and "b'' refer to the photospheric flux variation and that at the base of the convection zone respectively, for a particular eigenmode. From Eq. (1) one expects that the observed amplitude of a given mode will decrease smoothly with increasing $\tau_{\rm c}$, and thus with decreasing temperature. This might seem inconsistent with the observations, which show a rather sharply defined red edge. However, $\tau_{\rm c}$ depends very steeply on temperature, and hence it may also be that the observations do not yet resolve the transition from easily to barely visible (see Fig. 8 in Wu & Goldreich 1999).

Given the uncertainties in the physical processes leading to damping, it is not clear whether the observed red edge corresponds to the physical red edge, or whether it is just an apparent red edge where pulsations are still driven, but do not give rise to observable flux variations at the photosphere. The implication, then, is that a perfectly constant white dwarf might actually still be pulsating!

Theoretical uncertainties aside, there are also uncertainties in interpreting the observations. Indeed, over the years, the location and extent of the observed ZZ Ceti instability strip have undergone several transformations (e.g., Greenstein 1982; Robinson et al. 1995; Giovannini et al. 1998). This is due in part to the difficulty in accurately determining the atmospheric parameters of objects having convectively unstable atmospheres, as some version of the mixing length prescription usually has to be assumed. Additionally, at optical wavelengths, the Balmer lines attain their maximum strengths close to the instability strip. The unfortunate consequence is that varying the atmospheric parameters gives rise to only slight changes in the appearance of the spectra, making it difficult to uniquely fit observed spectra (e.g., Koester & Vauclair 1996). Supplementary constraints in the form of uv spectra, parallaxes, gravitational redshifts, have almost become a prerequisite.

Most studies to date have only considered flux variations. However, velocity variations are necessarily associated with these flux variations and though small - of the order of a few  $\rm km~s^{-1}$ - have nevertheless been measured in ZZ Ceti white dwarfs: securely in ZZ Psc (van Kerkwijk et al. 2000), and probably also in HS 0507+0434B (Kotak et al. 2002). That there are negligible vertical velocity gradients in the convection zone due to damping by turbulent viscosity is a central tenet of the convective-driving theory (Brickhill 1990; Goldreich & Wu 1999b). Indeed for shallow convection zones, this simplification has been shown to hold in the 2D hydrodynamical simulations of Gautschy et al. (1996). This means that the horizontal velocity is nearly independent of depth in the convection zone so that although photometric variations become difficult to detect around the red edge of the instability strip as mentioned above, velocity variations pass virtually undiminished through the convection zone.

 

 

Table 1: Observations.

	WD	$T_{\rm eff}$	$\log g$	V	Start	End	Exposure	No. of	Scatter	Measurement
	number	(K)			UT	UT	Time (s)	frames	( $\rm km~s^{-1}$)	error ( $\rm km~s^{-1}$)
HL Tau 76 (s)	0416+272	11 440	7.89 $\pm$ 0.05	15.2	12:58:55	14:59:06	20	213	9.0	6.7
G 1-7 (s)	0033+013	11 214	8.70 $\pm$ 0.06	15.5	08:37:49	10:39:50	74	83	9.8	7.7
G 67-23 (s)	2246+223	10 770	8.78 $\pm$ 0.03	14.4	07:33:57	08:48:15	74	50	3.5	2.4
G 126-18	2136+229	10 550	8.17 $\pm$ 0.05	15.3	04:46:06	06:27:25	30	136	1.4	7.6
G 126-18 (s)					06:58:18	08:58:03	74	80	5.2	4.0

Note. The effective temperature and $\log g$ for each of the 3 red edge objects are taken from Kepler et al. (1995) and are inferred from optical data only using the ML2, $\alpha=1$ prescription for describing the convective flux. For HL Tau 76, these quantities are taken from Bergeron et al. (1995) and are constrained using both uv and optical spectra for ML2/ $\alpha=0.6$. The scatter due to wander is calculated at a representative wavelength of 4341 Å i.e. at H$\gamma$. The measurement error is the typical internal error on the line-of-sight velocity associated with the fits to the line profiles as described in Sect. 4. "(s)'' refers to the Oct. 1997 service run data. The number of frames refers to the number of useful frames.

Observationally, the consequence would be that the putative, perfectly constant white dwarf might reveal its pulsating nature by velocity variations. If objects below the red edge follow the same trend as the known pulsators i.e. longer pulsation periods and higher amplitudes with decreasing effective temperature (Clemens 1993), then we expect the highest velocity amplitudes at the longer ($\ga$600 s) periods. For ZZ Psc (a.k.a. G 29-38), a well-known pulsator close to the red edge, van Kerkwijk et al. (2000) measured a velocity amplitude of 5.4  $\rm km~s^{-1}$ for the strongest mode at 614 s. For all our objects, which are cooler, we expect velocity amplitudes at least as large as those measured for ZZ Psc - lower values would constitute a non-detection.

Armed with this testable theoretical expectation, we look for variations in the line-of-sight velocities in objects that lie just below the photometric red edge of the instability strip. Our three targets are chosen - subject to visibility constraints during the scheduled run - from the list of Kepler et al. (1995) who find a handful of non-pulsating ZZ Cetis close to the red edge of the instability strip. Kepler et al. (1995) inferred relatively high masses for two of our three targets (see Table 1). This inference, though dependent on the assumed convective prescription, would imply that these objects have instability strips at higher temperatures.

In addition to the above, we search for line-of-sight velocity variations in the first ZZ Ceti to be discovered, HL Tau 76 (Landolt 1968), using exactly the same instrumental setup and reduction techniques so that it serves as a comparison case. Treating it subsequently as a ZZ Ceti variable (Appendix A) allows us to indirectly constrain the spherical degree of the eigenmodes.