3 Results

Several kinds of features are visible in the observed spectra: i) Broad emission features formed in the stellar wind. The lines of He , C and O are broadened due to the global motion of the expanding stellar wind. Some of the emissions are accompanied by blue-shifted absorption troughs (P-Cygni profiles); ii) Narrow [N II] and He I emission lines which are formed in the planetary nebula. These emission features appear here very weak given the small size of the used apertures and the relatively short exposure times (note that the dimmest parts of NGC 40 are mainly located close to the central star); iii) Interstellar Na I absorption features. See Fig. 1. The rest of this paper will concentrate on i) and their variability.

Figure 2: Residuals from the mean of C III$\lambda$5696 in HD 826 for 1996 November, 16 and 17. The mean was calculated based on all 22 nights of data. The segment in the lower right corner of the upper panel indicates the amplitude for 1.0 continuum unit in the residuals. Velocities are given relative to $\lambda _0=5695.3$ Å. The mean profile in the bottom panels is based on all thespectra in 22 nights

  
3.1 Line profile variations

Among the broad emission features in our spectra, the C III$\lambda$5696 line and the C IV $\lambda \lambda$5801/12+C III$\lambda$5826 blend dominate in strength and hence merit more intense study. In particular, the C III$\lambda$5696 line is known to be particularly sensitive to changes in density (e.g. D. J. Hillier, priv. comm.). In Fig. 2, differences from the mean profile (calculated from the whole set of spectra obtained in 22 nights; see lower panels) of the C III$\lambda$5696 emission line are shown as a function of time for two typical observing nights at the OMM (upper panels). The mean profile in the bottom panel refers to the global mean profile from all 22 nights for this star. In order to reveal the precise appearance of the moving blobs, it is necessary to subtract the smoothest mean profile as possible. For that purpose, we favor the use of the 22-night global mean profile (rather than the nightly mean). The segment in the lower right corner of the upper panel indicates the amplitude for 1.0 continuum unit in the residuals. This amplitude does not take into account the increased variability due to Poisson statistics as the intensity changes across the emission line relative to the adjacent continuum. True amplitudes relative to the continuum will be estimated in Sect. 3.2.

The characteristic time scale for significant variations is confirmed to be a few hours. Ejection times and starting wavelengths of individual blobs appear at random (this will be clearer by inspecting Fig. 6 in a subsequent section). The strongest, most obvious features appear to last longer and move throughout the C III line with apparently constant acceleration (see Sect. 3.3).

In order to emphasize the trajectories of subpeaks on the top of the C III$\lambda$5696 line, Fig. 3 shows grayscale plots of nightly differences from the global mean profile for the 22 nights, which is presented in the lower panels. These plots were obtained in a manner similar to that presented in Paper I. Gaps within the time series appear as a black horizontal bar. In these plots, we also show the trajectories of subpeaks on the top of the nearby C IV $\lambda \lambda$5801/12 emission line. In this complex carbon line, moving subpeaks appear with ghost images on their side. This is likely due to the line blending within this emission line. Unfortunately, the blending of the C IV $\lambda \lambda$5801/12 (+C III$\lambda$5826) emission feature prevents us from clearly identifying moving features. On the other (weak) lines, the situation is even worse; most of the subpeaks arising from noise can be erroneously associated with true manifestations of local overdensities because of the low S/N ratios in the lines.

  
3.2 Level of variability

The precise characterization of the variations showed in Figs. 2 and 3 could be greatly influenced by photon statistics and other sources of error. In order to rigorously estimate the significance level of the line profile variations, we have applied the "temporal variance spectrum'' analysis (TVS) of Fullerton et al. (1996). For details, we refer the reader to Fullerton et al.'s original paper and to Grosdidier et al. (2000; Paper I). Roughly speaking, the values of the TVS give a statistical assessment of the variability level at a given wavelength. Another outcome of this technique is the possibility of comparing time series of spectroscopic data obtained with different instrumentation and/or inhomogeneous quality.

Figure 3: Grayscale plots for HD 826 of C III$\lambda$5696 & C IV $\lambda \lambda$5801/12 residuals for 22 nights. Bottom panels show the 22-night mean. The range of the grayscale plots is -1.0 (black, lack of emission) to 1.0 (white, excess of emission) continuum units

The temporal variance spectra have been calculated for each of the 21 nights made up of at least 8 individual spectra, in order to secure statistical significance. Figures 4 and 5 have been obtained in the way described in Paper I: they show the related TVS^1/2(i.e. reflecting the amplitude of variability rather than the variance) along with contour levels for significant variability at the 1% and 5% levels. To facilitate the identification of the variable zones, the nightly mean spectra are superposed. The main results are the following:

1.

All obvious stellar emission lines within our spectral range are variable at the 1% level;

2.

In the case of the faint C II $\lambda\lambda\lambda$5641, 48, 63 stellar emission complex, we report only one significant intensification of the activity on 1996 January 15 (up to about 4% of the adjacent continuum flux);

3.

The stellar oxygen complex at $\approx$5590 Å (O V$\lambda$5579, O III$\lambda$5592 and O V $\lambda\lambda\lambda$5598, 5604, 5608) is always variable (about 3-7% of the adjacent continuum flux), with a burst (9-10% variability) of emission occuring on the 1998 January run;

4.

The He I$\lambda$5876 stellar emission feature generally exhibits variability from 4-12%, up to about 17% of the continuum flux on 1998 January 21. Note that the variability in this line is dominated by its P-Cygni absorption component. However, the emission component variability is clearly detected on 1996 January 15, 1996 May 26, 1996 July 28, 1996 September 30, 1996 November 16, and 1996 November 17;

5.

The stellar complex C IV $\lambda \lambda$5801/12+C III$\lambda$5826 is always variable with amplitudes reaching 6-13% of the continuum flux, and sometimes up to 17%. Note that the entire line shows prominent variability (although marginally detected during the 1997 March run because of poor S/N ratio), suggesting blobs/inhomogeneities propagating into the whole line emission region;

6.

The C III$\lambda$5696 stellar line always shows significant variability of 10-17% of the continuum flux, the maximum ($\approx$29%) being detected on 1998 January 21 and correlated with bursts of C IV $\lambda \lambda$5801/12+C III$\lambda$5826, He I$\lambda$5876 and the oxygen complex at $\approx$5590 Å. Note that like the C IV $\lambda \lambda$5801/12+C III$\lambda$5826 emission lines, the entire C III line shows prominent variability. On the whole, the 1998 January run appears as a particular epoch of high activity for this central star;

7.

As was already noticed for massive WR stars (Robert 1992) and the low-mass [WC 9] star BD +30$^\circ$3639 (Grosdidier et al.; Paper I), the blue-shifted absorption component of the lines exhibiting P-Cygni profiles in HD 826 is significantly more variable than the emission component. This is likely mainly due to the small volume of matter in front of the stellar "disk'', making it more sensitive to relative fluctuations.

Note that the variability of the He I$\lambda$5876 nebular line centred on the broad emission is illusory. It is caused by imperfect guiding and variable seeing (typically 2-3 $^{\prime\prime}$), both of which remove stellar light but not nebular light from the slit, in combination with rectification of the stellar continuum to unity. This effect appears only marginally for the [N II]$\lambda$5755 nebular line. The nebular line raw data show no significant variability.

  
3.3 Kinematics of the C III$\lambda$5696 subpeaks

The clearly visible C III$\lambda$5696 subpeaks always show measurable velocity shifts during their lifetime. Given the typical error in measuring the radial velocity (about 10-40 km s^-1) the trajectories of the subpeaks (or gaps) appear virtually related to features accelerating at an apparently constant acceleration. Thus, for the intense features (representing an apparent excess of emission as well as an apparent deficit of emission in the difference spectra) seen on at least three consecutive spectra, we measured mean radial velocities $v_{\rm R}$ and computed the related mean radial accelerations $a_{\rm R}={\rm d}v_{\rm R}/{\rm d}t$ through linear fits. Figure 6 summarizes the results for 120 extracted features. Horizontal error bars reflect the observed range of radial velocities for a single blob, whereas vertical error bars show the range ( $\pm1\sigma$) of possible accelerations derived from the linear fits. The spread in $v_{\rm R}$ values suggests that the starting and ending wavelengths appear at random. In contrast with Balick et al. (1996), we report a significant spread in the apparent acceleration values.

Figure 4: HD 826 nightly mean spectra (solid lines) and the computed square root of the TVSs (dashed curves), for 6 nights (see text). Contours of statistical significance for 1% and 5% levels are indicated by horizontal dotted lines (see arrows). Our calculations account for pixel-to-pixel and spectrum-to-spectrum differences in the noise distribution

The distribution does not appear symmetric in each of the two occupied quadrants, a few blobs with significant higher acceleration being observed in the wind receding region. We suspect that, if more blobs had been secured for a larger sample of spectra, the distribution would have been more symmetric. Therefore, we interpret this fact as a statistical effect. As already noticed for BD +30$^\circ$3639 (Paper I), the large majority of the blobs in Fig. 6 satisfies $a_{\rm R}\times v_{\rm R} \ge 0$. Therefore, the assumption of outwardly radially accelerating features is quite reasonable. However, 4 structures (only two at more than 2$\sigma$) move towards line centre. These features are likely spurious, being the tail end of a statistical distribution.

For comparison, the theoretical $(a_{\rm R},v_{\rm R})$-relation derived from the $\beta$velocity field, $v(r)=v_{\infty}(1-R_{\ast}/r)^{\beta}$, is also plotted in Fig. 6 for different angles $\theta$ between the line of sight and blob directions of movement ( $v_{\rm R}=v(r)\cos\theta$, $a_{\rm R}={\rm d}v_{\rm R}/{\rm d}t$). Adopting $v_{\infty}=$ 1000 km s^-1and the value $R_{\ast}=$ 0.33 $R_{\odot}$ for HD 826 (Leuenhagen et al. 1996), the kinematics are consistent with a $\beta$ velocity law with $\beta\approx 10$, in contrast to the value $\beta=1$ adopted in the atmosphere model (Leuenhagen et al. 1996). A $\beta$ value as small as 1 is ruled out because it would imply accelerations ranging up to about 0.65 km s^-2, which are not observed (see also Sect. 3.2.4 of Paper I). The line formation region is evaluated to span radial distances 10-100 $R_{\ast }$ from the nucleus, judging from the distribution of the data in Fig. 6. Therefore, the line formation region appears much more extended than previously reported by Balick et al. (1996). Since the lifetime of the subpeaks is a few hours, they would cross, at speed $\stackrel{<}{\approx}$1000 km s^-1, a zone limited to about a few tenths of the line formation region in radial extension. Thus the wind of HD 826 is highly variable on a very short time-scale, which supports a turbulent origin. Note that Lépine et al. (2000) find lifetimes for C III$\lambda$5696 blobs in the wind of the pop. I WC 8 star WR 135 to be of the order of the crossing time in the C III$\lambda$5696 formation zone, thus implying relatively long lasting blobs, that still could be turbulent.

Figure 5: HD 826 nightly mean spectra (solid lines) and the computed square root of the TVSs (dashed curves), for 6 other nights (see text and Fig. 4)

Figure 6: Kinematics in the form of projected mean acceleration vs projected mean velocity for each subpeak/gap on top of the C III$\lambda$5696 emission line (120 points). Filled (open) symbols correspond to an excess (deficit) of emission. The projected $\beta$-velocity law is shown for $\theta$ = 0 $\hbox {$^\circ $ }$ (towards the observer, lower left corner) to 180 $\hbox {$^\circ $ }$ (away from the observer, upper right corner), in steps of 10 $\hbox {$^\circ $ }$ ( $R_{\rm min} \le r \le R_{\rm max}$: solid curves; $r< R_{\rm min}$ and $r>R_{\rm max}$: dotted lines). We use the stellar parameters given by Leuenhagen et al. (1996); see text

Judging from the distribution of the data in Fig. 6, we estimate $\beta R_{\ast}$for HD 826 to be around 3.3 $R_{\odot}$. For nine massive WR stars, Lépine & Moffat (1999) found $\beta R_{\ast} \stackrel{\approx}{>} 20$. Our lower $\beta R_{\ast}$ is likely mainly related to the very small radius of HD 826, as expected for PN nuclei.

Recall that the expected maximum acceleration within a $\beta$-velocity field is proportional to $k(\beta) v^2_\infty/R_\ast$, with the function k (see Paper I) being only slightly dependent on $\beta$ for $\beta$above 2-3. Thus, fitting the observed maximum acceleration with the theoretical $(a_{\rm R},v_{\rm R})$-relation gives a constraint on the ratio $v^2_\infty/R_\ast$. Therefore, we expect that reliable values on $v_{\infty}$and $R_{\ast }$ should follow the relation: $v^2_\infty/R_\ast\sim 4$-5 km s^-2.

Absolute values of the acceleration in Fig. 6 range from nearly 0 to 70 m s^-2. The mean radial acceleration in the line formation zone (calculated from the 120 observed points) is $14.7\pm11.7$ m s^-2 (compared to only $3.6 \pm 0.8$ m s^-2 for BD +30$^\circ$3639; Paper I). Within the line formation region, the spread in acceleration values appears quite large. Overall, these values in HD 826 are very similar to those observed in the massive WC 8 star WR 135 (Robert 1992; Lépine & Moffat 1999). However, especially impressive are the 3-4 times larger observed maximum $a_{\rm R}$ values (HD 826: $\approx$66 m s^-2) compared to those already reported for massive WC 5-9 stars or low-mass [WC 9] stars (4-20 m s^-2: see Paper I; Lépine & Moffat 1999; Robert 1992). Note that in massive WR stars the amplitudes of the accelerations do not show correlations with either the stellar effective temperature, or the stellar luminosity (see Table 4 in Lépine & Moffat 1999). Thus, conjecturing that this property still holds for low-mass WR stars, and although HD 826's massive counterparts have larger terminal velocities, large maximum $a_{\rm R}$ values are mainly a consequence of its very small core radius.

3.4 The velocity gradient in the line formation region

We have considered the whole set of blobs satisfying $a_{\rm R}\times v_{\rm R} \ge 0$, and calculated all their related corresponding $a_{\rm R}/v_{\rm R}={\rm d}v/{\rm d}r$. Note that the relative error on ${\rm d}v/{\rm d}r$ is often quite large: see Fig. 7.

Figure 7: Relative error on ${\rm d}v/{\rm d}r$, $\sigma ({\rm d}v/{\rm d}r)/({\rm d}v/{\rm d}r)$, as a function of ${\rm d}v/{\rm d}r$ for the blobs in Fig. 6, satisfying $a_{\rm R}\times v_{\rm R} \ge 0$. The horizontal axis is in s-1

However, the relatively high number of subpeaks encouraged us to perform a statistical analysis of the ${\rm d}v/{\rm d}r$ derived from our kinematic measurements. Figure 8 shows the frequency distribution of the ${\rm d}v/{\rm d}r$ values (solid histogram). Large bins have been chosen in order to compensate for the errors encountered in evaluating the ${\rm d}v/{\rm d}r$.

Figure 8: Frequency distribution of the ${\rm d}v/{\rm d}r$ values (solid histogram). For comparison, the distribution expected from a true $\beta$-velocity field is also plotted (dotted histogram). The line formation region in the second histogram is assumed to span radial distances 10-100 $R_{\ast }$from the central star, with $\beta =10$ (see text). The horizontal axis is in s-1

For comparison, the distribution expected from a purely $\beta$ velocity field is also shown (dotted histogram), for the radial distances between $R_{\rm min}=10~R_{\ast}$ and $R_{\rm max}=100~R_{\ast}$ found before, assuming $\beta =10$, $R_{\ast}=0.33$ $R_{\odot}$, and $v_{\infty} = 1000$ km s^-1. The latter histogram has been normalized to the observed histogram at their maxima, occurring in the same bin. Note the huge excess of features occurring at ${\rm d}v/{\rm d}r> 2~ 10^{-5}$ s^-1, compared to the number expected from a pure $\beta$-velocity field. Since the majority of the features showing ${\rm d}v/{\rm d}r$ above 2  10^-5 s^-1 have relative errors $\stackrel{<}{\approx}$30%, this result appears to be reliable. This suggests that the $\beta$velocity field underestimates the true gradient within the flow. Such a result has been also established in Paper I for the [WC 9] nucleus BD +30$^\circ$3639. This is in line with other studies concerning the high efficiency of the acceleration in the optically thin regions of pop. I WR winds (Marchenko & Moffat 1999).