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Astron. Astrophys. 356, 665-675 (2000)

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3. NLTE analysis

Since the spectral analysis presented by Kudritzki et al. (1982) and Heber et al. (1988) our (plane-parallel, hydrostatic) NLTE model atmospheres have been significantly improved, e.g. by the consideration of level dissolution following the Hummer-Mihalas (1988) occupation probability formalism for H I and He II (Hubeny et al. 1994, Werner 1996). The employed model atoms are much more detailed (Rauch 1997) and we consider additionally Stark line broadening for the H I and He II [FORMULA] series (Werner 1996). The line opacities are considered up to a distance to the line center of 3 000 Doppler widths. In the following we describe our new spectral analysis.

3.1. The rotational velocity of LB 3459

Since the orbital period of LB 3459 is short ([FORMULA]), both, the rotation of the primary and its orbital motion during the relatively long exposure times (Table 1, 2) will contribute to the line broadening. HHH have determined a velocity amplitude [FORMULA] for the primary of LB 3459. In Fig. 2 we show the observations in relation to the orbital phase [FORMULA] as defined by HHH .

For a precise determination we use our CES spectra (Table 1) of He II  [FORMULA]Å which have the best resolution and (at least the 18/03 spectrum) a relatively small contribution from the orbital motion. In Fig. 1 we show fits of theoretical profiles to the observation with [FORMULA], [FORMULA], different He/H abundance ratios, and different [FORMULA]. Obviously, the observed line profiles in the 18/03 and 19/03 spectrum are slightly different due to the different orbital phase during the observation (Fig. 2). [FORMULA] and [FORMULA] are necessary in order to achieve a sufficient fit, respectively. From Fig. 2 we derive velocity coverages of 18.7 and [FORMULA] during the 18/03 and 19/03 observation, respectively. With these values and the rough approximation [FORMULA], [FORMULA] and [FORMULA] ([FORMULA]) is calculated for the primary. At the given quality of the spectra, these values appear to be in good agreement. From their mean of [FORMULA] and the minimum value of the velocity difference of [FORMULA] for a 3 600 sec exposure (Fig. 2), we can estimate a minimum of [FORMULA] which is the best case. However, the exact starting time of the CASPEC observations (Table 1) is not known and we do not expect best conditions. Thus, we adopt [FORMULA] for our further analysis. This value appears suitable for the IUE spectra SWP 17821 and 17822 as well (Fig. 2).

[FIGURE] Fig. 1. Comparison of theoretical He II  [FORMULA]Å line profiles to the CES spectra of the 18/03 and 19/03 (Table 1). The models are calculated with [FORMULA], [FORMULA], and He/H = 0.0007, 0.0010, 0.0013. The theoretical profiles are convolved with rotational profiles with [FORMULA] = 8 (thin), 40 (thick), and [FORMULA] (dashed) and with a Gaussian (FWHM = 0.1 Å). In the case of the 19/03 spectrum, [FORMULA] is used

[FIGURE] Fig. 2. The radial velocity curve of LB 3459. The phase is given with respect to the maximum positive velocity. The CES observations (Table 1) and the IUE SWP exposures (Table 2) are indicated. The thin curve indicates the maximum velocity coverage during an exposure of 3 600 sec (small dashes: 6 000 sec, long dashes: 10 680 sec, respectively) for a given start phase

3.2. Effective temperature and surface gravity

Firstly we carried out test calculations with [FORMULA] and [FORMULA] (Kudritzki et al. 1982) and found that [FORMULA] (by number) fits the observation (around He II  [FORMULA]Å) sufficiently well. With this preliminary helium abundance (Sect. 3.4.1), we calculated a grid of H+He models ([FORMULA] = 30 000 - 47 000 K in 250 K steps, [FORMULA] = 4.50 - 5.90 (cgs) in steps of 0.05 dex). From these models, synthetic spectra were calculated using the line-broadening tables calculated by Lemke (1997) and by Schöning & Butler (1989), based upon the VCS theory (Vidal et al. 1973)

In a coarse approach, the effective temperature [FORMULA] and the surface gravity g were determined by fitting the synthetic spectra which are convolved firstly with a rotational profile ([FORMULA]) and then with a Gaussian with a FWHM according to the spectral resolution (Table 1) to the normalized EMMI and CASPEC spectra (Table 1). A [FORMULA] test (cf. Bevington & Robinson 1992) was performed and the results are [FORMULA] and [FORMULA], and [FORMULA] and [FORMULA], respectively. Although these agree within their error limits, it turned out that it is impossible to achieve a perfect simultaneous fit of H [FORMULA] - H [FORMULA] at these values: We are running into a mild form of the well-known Balmer-line problem (Napiwotzki & Rauch 1994). This indicates that the photospheric temperature structure of LB 3459 is not well modeled due to the neglected metal-line blanketing in our H+He models (Werner 1996, Fig. 3).

We performed a [FORMULA] test, this time in order to fit H [FORMULA] - H [FORMULA] individually. In the CASPEC spectrum, the observed profiles of H [FORMULA] and H [FORMULA] (Fig. 18) appear asymmetric (most likely due to problems during the data reduction) and thus, we achieve different results if we split the region in two parts, bluewards and redwards of the line centers. Hence these lines are not useful. The results of H [FORMULA] - H [FORMULA] are shown in Fig. 4. These three lines (both, CASPEC and EMMI spectrum) indicate an average surface gravity of [FORMULA] within a very small error range.Thus, we finally adopt this value.

[FIGURE] Fig. 3. Formation depths of the line cores of H [FORMULA] - H 9, He I  [FORMULA]Å , and He II  [FORMULA]Å . In the model ([FORMULA], [FORMULA]) which accounts for metal-line blanketing of C+N+O+Mg+Si, the temperature is much lower at the formation depths of the lower member of the H I Balmer series, e.g. [FORMULA] at H [FORMULA]. The dotted line shows the temperature of our final model with Fe+Ni in addition. m is the column mass in the stellar atmosphere. Model parameters see Table 4

[FIGURE] Fig. 4. Results of [FORMULA] tests performed in order to determine [FORMULA] and [FORMULA]. [FORMULA] - [FORMULA] denote the results if we use [FORMULA] of the CASPEC and EMMI spectra for the Balmer lines, respectively. The contours given indicate the formal 1 [FORMULA], 2 [FORMULA], and 3 [FORMULA] error ranges

The derived [FORMULA] increases towards the higher series members which, again, is most likely due to the neglect of metal opacities and the resulting Balmer line problem (Werner 1986). Since the higher members of the Balmer series form deeper inside the atmosphere (Fig. 3) where the influence of metal-line blanketing on the photospheric structure is smaller, [FORMULA] derived from those is more reliable. The formation depths of the Balmer lines (Fig. 3) indicate that H [FORMULA] forms deep enough in the atmosphere to be suited for a reliable determination of [FORMULA]. From this line (CASPEC spectrum) we derive [FORMULA]. However, for a more precise determination we evaluate the ionization equilibrium of He I / He II which is a sensitive indicator of [FORMULA] (Fig. 7). Moreover, the He lines form deeper in the atmosphere than H [FORMULA] and H [FORMULA] (Fig. 3) and thus, are more reliable than those. For [FORMULA] kK He I  [FORMULA]Å appear too strong. At [FORMULA] the theoretical He I  [FORMULA]Å line profile fits well the observation. Within this [FORMULA] range, He II  [FORMULA]Å show only very small changes in the inner line cores. We preliminarily adopt this value which will be verified in the following by the examination of the ionization equilibria of other elements which are identified in the spectrum of LB 3459.

3.3. Interstellar absorption

We have used the co-added IUE spectrum of LB 3459 in order to determine the neutral hydrogen column density from the Ly [FORMULA] interstellar absorption. An inspection of the observation shows that the background correction around Ly [FORMULA] is not perfect. Thus, we subtracted the estimated remaining background and normalized the spectrum. A theoretical spectrum with [FORMULA] fits well the observation (Fig. 5). A color excess [FORMULA] is calculated using the formula given by Groenewegen & Lamers (1989).

[FIGURE] Fig. 5. Theoretical line profile of H I Ly [FORMULA] from our final H+He+C+N+O+Mg+Si model (Table 4) compared with the IUE spectrum of LB 3459. Interstellar absorption has been considered with H I column densities of [FORMULA], [FORMULA], and [FORMULA]. Note that the photospheric Si III  [FORMULA]Å is blended by interstellar absorption (Sect. 2)

3.4. Photospheric abundance ratios

In the spectrum of LB 3459 we have identified H, He, C, N, O, Mg, and Si lines. We will now determine the photospheric abundance ratios of these elements. Therefore we calculated new models and considered element by element in addition, i.e. the He/H ratio is determined from H+He models, the C/H ratio from H+He+C models, and so on. In each case, we checked the influence of the newly considered element on the previous results.

3.4.1. He/H

We calculated H+He models ([FORMULA], [FORMULA]) with different abundance ratios and performed a [FORMULA] test using our optical spectra (Table 1) and He II  [FORMULA]Å . The result is shown in Fig. 6.

[FIGURE] Fig. 6. Reduced [FORMULA] values of our fitting procedure to the observed He II  [FORMULA]Å line profile. From the CES, CASPEC, and EMMI spectra (Table 1), we derive [FORMULA] values of 0.0009, 0.0008, and 0.0004, respectively. However, since the resolution of the EMMI spectrum (Table 1) is comparable to the line width of He II  [FORMULA]Å , this result is less reliable

Due to the much better resolution of the CES and CASPEC spectra (Table 1) and thus a larger error range for the EMMI result, we have finally adopted [FORMULA].

The fit of He I  [FORMULA]Å and He II  [FORMULA]Å at [FORMULA] shows both theoretical He I lines too strong while He II  [FORMULA]Å appears too shallow. Since the ionization equilibrium is a very sensitive indicator of [FORMULA], [FORMULA] has to be higher: At [FORMULA] we achieve good agreement with the observation (Fig. 7).

[FIGURE] Fig. 7. Fit to the line profiles of He I  [FORMULA]Å and of He II  [FORMULA]Å at [FORMULA] (dashed), [FORMULA] (fully drawn), [FORMULA] (dotted), [FORMULA], and [FORMULA] compared with the CASPEC (upper panel), co-added IUE (lower panel left, cf. Table 2) and CES spectrum (lower panel, right) of LB 3459. In the case of He I  [FORMULA]Å and He II  [FORMULA]Å , we show also the abundance dependency of their line cores ([FORMULA]). For clarity, these profiles are shifted by +2Å. In the case of He II  [FORMULA]Å , the thin line shows the normalized flux of our final H+He+C+N+O+Mg+Si+Fe+Ni model (Table 4). Note the different flux scales

3.4.2. C/H

In order to determine the C/H abundance ratio, we calculated models with [FORMULA], [FORMULA], [FORMULA], and carbon in addition. We used the most prominent carbon lines in the co-added IUE spectrum, the C III  [FORMULA]Å complex, and the C IV  [FORMULA]Å resonance doublet. The best fit is achieved with [FORMULA] (Fig. 8).

[FIGURE] Fig. 8. Fit to the line profiles of C III  [FORMULA]Å and C IV  [FORMULA]Å (model parameters see Table 4) compared with the IUE (SWP 03729) spectrum of LB 3459. In the right panel, the thin line shows the normalized flux of our final H+He+C+N+O+Mg+Si+Fe+Ni model (Table 4). The observation is smoothed using a Savitzky-Golay filter (Press et al. 1992)

3.4.3. N/H

The N/H abundance ratio is determined analogously to the C/H ratio (Sect. 3.4.2). We used the model parameters from that analysis and added nitrogen. The best fit to lines from the ionization stages N III - N V is achieved at [FORMULA] (Fig. 9).

[FIGURE] Fig. 9. Fit to the line profiles of N III  [FORMULA]Å , N IV  [FORMULA]Å , and N V  [FORMULA]Å (model parameters see Table 4) compared with the spectrum of LB 3459

3.4.4. O/H

In the UV spectrum, O IV  [FORMULA]Å and O V  [FORMULA]Å are identified. We used these lines to determine the O/H ratio (Fig. 10). A sufficient fit is achieved at [FORMULA] The available IUE spectra in the O V  [FORMULA]Å wavelength region show a broad absorption trough. However, the O V line is clearly present, and is reproduced well if it is normalized to the local continuum (Fig. 10). The synthetic spectrum of our final H+He+C+N+O+Mg+Si+Fe+Ni (Fig. 10) shows that the continuum around O V  [FORMULA]Å is dominated by Fe and Ni lines.

[FIGURE] Fig. 10. Fit to the line profiles of O IV  [FORMULA]Å (left) and O V  [FORMULA]Å (right) (H+He+C+N+O model parameters see Table 4) compared with the spectrum (SWP 17821 only) of LB 3459. At the bottom of the right panel, the theoretical spectrum is normalized to the local continuum: O V  [FORMULA]Å matches the local feature. The thin line (right) shows the spectrum of our final model with Mg+Si+Fe+Ni in addition which improves the overall fit to the observation. However, no perfect fit of the region around O V  [FORMULA]Å is achieved

3.4.5. Mg/H

Unfortunately, neither Mg II nor Mg III (dominant ionization stage in the line forming region) lines are unambiguously identified in the IUE spectrum due to its S/N ratio and the bulk of metal lines which are present therein. Thus, the Mg/H ratio has to be determined from Mg II  [FORMULA]Å alone (Fig. 11). At a surprisingly high ratio of [FORMULA], which is six times the solar ratio, we achieve a sufficient fit to the observation.

[FIGURE] Fig. 11. Fit to the line profile of Mg II  [FORMULA]Å at different Mg/H ratios: solar (thin line), 5 (dotted), 6 (thick line), and 7 (dashed) times solar (other model parameters see Table 4) compared with the CASPEC spectrum of LB 3459

3.4.6. Si/H

Unfortunately, Si III  [FORMULA]Å (Fig. 5) seems to be contaminated by an interstellar component (Sect. 2) and we can use the Si IV  [FORMULA]Å resonance doublet for this analysis, only. The best fit is achieved at [FORMULA] (Fig. 12). At this ratio, the Si lines which are identified in the optical spectrum are well reproduced, e.g. Si IV  [FORMULA]Å which are shown in Fig. 18.

[FIGURE] Fig. 12. Fit to the line profiles of Si IV  [FORMULA]Å (model parameters see Table 4) compared with the IUE spectrum of LB 3459. The strong absorption features which are not reproduced by this model are Fe and Ni lines (Fig. 14)

3.4.7. Fe and Ni

Helium and the light metals C, N, O, and Si appear underabundant in the photosphere of LB 3459 compared to solar values while Mg is enriched. In a last step we determine the Fe and Ni abundances. We calculated a final model (parameters see Table 4) with Fe and Ni in addition. Their opacities are considered following Deetjen et al. (1999). The statistics of our model atoms for these calculations are summarized in Table 3.


[TABLE]

Table 3. Statistics of the model atoms used for the calculation of final model of LB 3459. The notation is: NLTE = levels treated in NLTE, LTE = LTE levels, RBB = radiative bound-bound transitions. In the case of Fe and Ni, [FORMULA] and [FORMULA] denote individual levels and transitions used in the statistical NLTE line-blanketing approach



[TABLE]

Table 4. Parameters of LB 3459 compared to the results of Kudritzki et al. 1982 (K 82). The solar abundance ratios are given by Holweger (1979) and Stürenburg & Holweger (1990)


We used a fixed (solar) Fe/Ni ratio and compared theoretical spectra with different Fe/H ratios with a low-resolution IUE spectrum (Fig. 13) which is well suited for this purpose. At the solar Fe/H ratio, the overall agreement with the spectral shape is much improved compared to a "pure" H+He+C+N+O+Mg+Si model. Fe/H ratios which are higher or lower by 0.5 dex provide worse results.

[FIGURE] Fig. 13. Synthetic spectra of H+He+C+N+O+Mg+Si+Fe+Ni models with Fe/H = 0 (thin line), [FORMULA] (thick line, [FORMULA] solar value), and [FORMULA] (dotted) compared with the UV spectrum (IUE SWP 04887, selected because it has the longest exposure time of 437 sec of the available low resolution spectra) of LB 3459. The three synthetic spectra are scaled by [FORMULA] in order to match the observed flux level at 1700 Å with the Fe/H = 0 spectrum. Note that the flux from the model with eight times solar abundance is obviously too low. The theoretical flux is reddened according to our result. Parameters see Table 4

The comparison to the available high-resolution IUE spectra of LB 3459 allows to identify the strongest Fe and Ni lines (Fig. 14, see also Figs. 7, 8, and 10). From the fit to the observation, a Ni/Fe ratio much different from solar can be excluded. The comparison suggests that Ni might be mildly overabundant. However, the low S/N ratio and the high rotational velocity prevent a more precise analysis.

[FIGURE] Fig. 14. Synthetic spectra of H+He+C+N+O+Mg+Si (thin line, in the middle) and H+He+C+N+O+Mg+Si+Fe+Ni models (parameters see Table 4) with Fe/Ni = 2 (dashed), 20 (thick line, solar ratio), and 200 (thin line) compared with a section of the IUE spectrum of LB 3459 (smoothed with a Savitzky-Golay filter, cf. Press et al. 1992). At solar Fe/H and Ni/H ratios, the observation is well matched. The Ni lines around [FORMULA] might suggest a Fe/Ni ratio lower than solar. For clarity, three synthetic spectra with different Fe/Ni ratio are artificially shifted to the top. At the bottom, [FORMULA] values of Fe IV , Fe V , Ni IV , and Ni V are indicated (1 cm = [FORMULA]) as a hint for line identification

3.4.8. Mass, radius, and luminosity

The mass and the luminosity of the primary component of LB 3459 is determined by comparison with recently presented models departing from the first giant branch prior to core helium ignition and evolving into low-mass white dwarfs (Driebe et al. 1998). We derive [FORMULA] (Fig. 15) and [FORMULA]. The mass is 10% higher than determined by Kudritzki et al. (1982), who assumed an upper limit of [FORMULA].

[FIGURE] Fig. 15. Position of the primary component of LB 3459 in the [FORMULA] - [FORMULA] plane compared with theoretical evolutionary tracks of post-RGB stars (Driebe et al. 1998). The ellipse represents the derived errors in [FORMULA] and [FORMULA] (Table 4). The tracks are labeled with the respective stellar masses (in [FORMULA])

The radius of the primary can be calculated using

[EQUATION]

We calculate [FORMULA].

The mass of the secondary is calculated from the mass function given by HHH :

[EQUATION]

([FORMULA], Sect. 1). The solution is shown in Fig. 16. The mass of the secondary is [FORMULA]. This value is almost twice the value determined by Kudritzki et al. (1982).

[FIGURE] Fig. 16. Determination of the secondary's mass. From the mass function (Eq. 2) and [FORMULA], we derive [FORMULA]

In order to compare our results with those from the analyses of the light curve and orbital parameters, we adopt [FORMULA], [FORMULA], and [FORMULA] the primary orbit is [FORMULA]. The separation [FORMULA] of the two components can be calculated using [FORMULA]. Kilkenny et al. 1979 have given the relative radii: [FORMULA] and [FORMULA]. In Fig. 17 the comparison is shown.

[FIGURE] Fig. 17. Mass-radius relation for the primary of LB 3459. Obviously, the solution from f(m) & light curve does not intersect with our result ([FORMULA]) and the mass value determined from comparison to evolutionary models (Fig. 15). The dashed lines indicate the error ranges

The results of our analysis are not in agreement with the solution of [FORMULA] & light curve analysis. The reason for this is unclear. A surface gravity [FORMULA] which is necessary for a fit at around [FORMULA] can be excluded from the line profile fits to the available spectra. Only less strict (statistical) error ranges in both cases could provide a marginal agreement. We conclude that new spectra are necessary which are optimized in the view of orbital motion. This could definitely improve the spectral analysis.

3.5. Distance

Kudritzki et al. (1982) determined the distance [FORMULA] of LB 3459 from its radius and the flux normalization.

From our final model, we can determine the spectroscopic distance of LB 3459 using the flux calibration of Heber et al. (1984):

[EQUATION]

with [FORMULA], [FORMULA] (Fig. 3.3), [FORMULA] and [FORMULA], the distance is derived from

[EQUATION]

With the Eddington flux at [FORMULA] our final model atmosphere [FORMULA] we derive a distance of [FORMULA].

The trigonometric parallax measured by TYCHO (TYC 9166-00716-1) is 46.5 mas. Thus, the distance would be 21.5 pc, however, this value is meaningless because LB 3459 has not been treated as a binary in the TYCHO data reduction (Wicenec priv. comm.).

3.5.1. Summary

The parameters of LB 3459 are summarized in Table 4. LB 3459 is He deficient by a factor of 125. C, N, O, Si are depleted by factors of about 265, 33, 12, 6, and 5, respectively. Fe and Ni are present in the photosphere at roughly solar abundances. Mg is enriched by a factor of 6. A fit of our final H+He+C+N+O+Mg+Si+Fe+Ni model to the optical observation is shown in Fig. 18. It is worthwhile to note that the slight changes in the temperature structure due to the consideration of iron-group elements (Fig. 3) results in small changes in the inner line cores only.

[FIGURE] Fig. 18. Theoretical line profiles calculated from our final H+He+C+N+O+Mg+Si+Fe+Ni model, compared with the observation (Table 1): H [FORMULA] (CES), H [FORMULA] - H [FORMULA] (CASPEC), He II  [FORMULA]Å (CES). Model model parameters see Table 4. The emission in the line cores of H [FORMULA] and H [FORMULA] is due to the reflection from the secondary

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Online publication: April 10, 2000
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