Up: BI Lyncis: A hydrogen-deficient G

3 Analysis

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{H2958_F2.PS} \end{figure}$

Figure 2: Ultraviolet and visual spectrophotometry of BI Lyn (histogram) together with the best fitting theoretical flux distribution (polyline and horizontal bars). The latter represents the sum of two model atmospheres (dashed lines) with effective temperature and angular radii $\mbox{~\em T$_{\rm eff}$ }_1=28~600~\mbox{K}$ , $\theta _1=0.55\times 10^{-11}$ rad, $\mbox{~\em T$_{\rm eff}$ }_2=5~840~\mbox{K}$ and $\theta _2=4.09\times 10^{-11}$ rad. Interstellar reddening is negligible.

The available spectra of BI Lyn suggest that it comprises a hot helium-rich source and a cooler source since we see absorption lines due to plasma in two quite different ionization states. In constructing a model to fit these spectra, a number of assumptions - or approximations - influence our conclusions. Many of these are necessary because both the number of free parameters and the limited number of constraints preclude an exhaustive search of solution space.

The fundamental assumption is that both cool and hot absorption sources are primarily stellar. This seems to hold well for the hot source. This remains a good working hypothesis even though high rotational broadening in the cool source spectrum and the presence of H $\alpha$ emission makes a purely stellar identification less secure.

For two stellar sources, the principal free parameters which govern the measured spectrum are as follows (subscripts refer to the hot and cool source respectively):

$\mbox{~\em T$_{\rm eff}$ }_1, \mbox{~\em T$_{\rm eff}$ }_2$ : effective temperature,
$\mbox{~$E_{B-V}$ }$ : interstellar extinction,
$\mbox{~log $g$ }_1, \mbox{~log $g$ }_2$ : surface gravity,
$\theta_1, \theta_2$ : angular diameter providing the relative radii R₂/R₁,
$\mbox{~$v~\sin i$ }_1, \mbox{~$v~\sin i$ }_2$ : rotational velocity,
v₁, v₂: radial velocity,
$\mbox{~$n_{\rm He}$ }_1, \mbox{~$n_{\rm He}$ }_2$ : helium abundance,
$\mbox{~${\rm [\beta_{\rm Fe}]}$ }_1, \mbox{~${\rm [\beta_{\rm Fe}]}$ }_2$ : metallicity,
$\mbox{~$v_{\rm t}$ }_1, \mbox{~$v_{\rm t}$ }_2$ : microturbulent velocity.

Ideally, stellar composition entails many more free parameters than metallicity and helium abundance alone, but requires high-resolution spectroscopy to measure, as does the microturbulent velocity. Helium abundance cannot be measured directly for cool sources. Tests showed that $\mbox{~${\rm [\beta_{\rm Fe}]}$ }_2$ cannot be uniquely determined from the given data. Similarly, given the magnitude of errors in $\mbox{~\em T$_{\rm eff}$ }_1$ , it is not practical to measure $\mbox{~${\rm [\beta_{\rm Fe}]}$ }_1$ in detail.

Thus the second group of assumptions are as follows. The abundances of all elements other than hydrogen and helium are in proportion to their cosmic abundances, with $\mbox{~${\rm [\beta_{\rm Fe}]}$ }_1 = \mbox{~${\rm [\beta_{\rm Fe}]}$ }_2 = 0$ . The helium abundance of the cool star is normal: $\mbox{~$n_{\rm He}$ }_2 = \mbox{~$n_{\rm He}$ }_\odot$ . The adopted microturbulent velocities are typical for early-type stars $\mbox{~$v_{\rm t}$ }_1=5~\mbox{$\mbox{km}~\mbox{s}^{-1}$ }$ and main-sequence late-type stars $\mbox{~$v_{\rm t}$ }_2=2~\mbox{$\mbox{km}~\mbox{s}^{-1}$ }$ . The latter assumption is very important as it affects both the metallicity $\mbox{~${\rm [\beta_{\rm Fe}]}$ }_2$ (see above) and the derived radius ratio R₂/R₁. We have adopted $\mbox{~$v_{\rm t}$ }_2=2~\mbox{$\mbox{km}~\mbox{s}^{-1}$ }$ in order that the latter quantity as derived from spectral fitting be as consistent as possible with $\theta_2/\theta_1$ derived from spectrophotometry where we have also used cool star models computed with $\mbox{~$v_{\rm t}$ }_2=2~\mbox{$\mbox{km}~\mbox{s}^{-1}$ }$ . Secondary effects on $\mbox{~\em T$_{\rm eff}$ }_2$ and $\mbox{~log $g$ }_2$ are not significant here.

The third level of assumptions concerns the physics. The stars are assumed to be spherically symmetric and not to vary significantly over time. The approximations of plane-parallel geometry, local thermodynamic, radiative and hydrostatic equilibrium have been assumed valid for modelling their atmospheres (rapid rotation and H $\alpha$ emission may compromise these).

However, with these assumptions and approximations, it becomes possible, in principle, to solve for the remaining 12 (!) free parameters by modelling the overall flux distribution and the intermediate-resolution spectra of BI Lyn.

3.1 Spectral fitting: TFIT and SFIT

The methods for fitting flux distributions and high-resolution spectra by least-squares minimization within grids of models have been described in detail elsewhere (Jeffery et al. 2001). The primary codes used are BINFIT and SFIT.

BINFIT is an extension of the single star code TFIT used to model the flux distribution of a binary system containing a hot and a cool star. Fitting of each component is done within a one-dimensional grid, normally $\mbox{~\em T$_{\rm eff}$ }$ , convolved with an extinction curve $a_{\lambda}(\mbox{~$E_{B-V}$ })$ and angular radius $\theta$ . The flux distributions used in the model grids are of low resolution and taken directly from the output of LTE model atmosphere codes STERNE (Jeffery et al. 2001) or ATLAS (Kurucz 1979).

SFIT is used to model intermediate and high-resolution stellar spectra by interpolation in three-dimensional model grids, normally $\mbox{~\em T$_{\rm eff}$ }$ , $\mbox{~log $g$ }$ and an abundance parameter, e.g. $n_{\rm He}$ or ${\rm [\beta_{\rm Fe}]}$ . Versions exist for both single and binary stars, and appropriate allowance is made for velocity shifts, rotational broadening, and instrumental broadening. Given the relative radii of stars in a binary, the spectra are added correctly at the absolute flux level and then normalized to the true total continuum. Provision is made to renormalize the observed spectrum to the true continuum in order to optimize the fit. SFIT may be used to measure any or all of $\mbox{~\em T$_{\rm eff}$ }$ , $\mbox{~log $g$ }$ , abundance, $\mbox{~$v~\sin i$ }$ , and R₂/R₁. In practise, only two or three variables should be solved for simultaneously, although R₂/R₁ should always be a free parameter. Consequently, the use of SFIT is iterative, particularly for binaries.

The radial velocity shifts v may be found by cross-correlating the observed spectrum with individual components of a synthetic spectrum which is a sufficiently good approximation to the final solution. Providing the hot and cool star spectra are sufficiently different, cross-correlation will automatically identify the individual components in the observed spectrum, although it is important to exclude strong features present in both spectra (e.g. Balmer lines).

3.2 The model grids

The model atmospheres and flux distributions used to analyse the hot star are computed with the plane-parallel LTE code STERNE, which is adapted for dealing with hydrogen-deficient stellar atmospheres. The high-resolution spectra are calculated with the LTE code SPECTRUM. Both codes are described more fully by Jeffery et al. (2001).

Model atmospheres used in this investigation were calculated on a three-dimensional rectangular grid defined by $\mbox{~\em T$_{\rm eff}$ }= 10~000 (5000) 40~000 ~\mbox{K}$ , $\mbox{~log $g$ }= 1.0 (0.5) 6.0$ , and composition $[\mbox{~$n_{\rm H}$ },\mbox{~$n_{\rm He}$ },\mbox{~${\rm [\beta_{\rm Fe}]}$ }] = [0.0,1.0,0.0]$ , [0.01,0.99,0.0], [0.05,0.95,0.0]and [0.1,0.9,-1.0]. The larger value of $n_{\rm H}$ or $n_{\rm He}$ is reduced to compensate for the trace elements. Coarse model grids for $\mbox{~$n_{\rm H}$ }=0.3$ , 0.5, 0.7, 0.9, 0.95, 0.99 and 1.0, with corresponding $\mbox{~$n_{\rm He}$ }=1-\mbox{~$n_{\rm H}$ }$ and ${\rm [\beta_{\rm Fe}]}$ , are also available, as are fine grids with $\delta\mbox{~\em T$_{\rm eff}$ }=1000~\mbox{K}$ and $\delta\mbox{~log $g$ }=0.1$ for selected areas of ( ${\mbox{~\em T$_{\rm eff}$ },\mbox{~log $g$ }}$ ) space.

Synthetic spectra were calculated on wavelength intervals 3900-5000 Å (blue), 6360-6770 Å (red/H $\alpha$ ) and 8400-8800 Å (CaT). Linelists were taken from the list of assessed data for hot stars LTE/_LINES (Jeffery 1991). Microturbulent velocity $\mbox{~$v_{\rm t}$ }=5.0~\mbox{$\mbox{km}~\mbox{s}^{-1}$ }$ and solar abundances for all elements other than hydrogen and helium were assumed (see above).

Model atmospheres and flux distributions used to analyse the cool star were taken from the Kurucz' standard grid of ATLAS models (Kurucz 1993), for $\mbox{~\em T$_{\rm eff}$ }=3500 (500) 8000$ , $\mbox{~log $g$ }=2.0 (0.5) 4.5$ , $\mbox{~${\rm [\beta_{\rm Fe}]}$ }=-0.5, -0.3, 0.0$ and $\mbox{~$v_{\rm t}$ }=2.0~\mbox{$\mbox{km}~\mbox{s}^{-1}$ }$ .

High resolution spectra were calculated in the same spectral regions as for the hot star using Kurucz' code SYNTHE (Kurucz 1991; Jeffery et al. 1996). Grids with microturbulent velocities $\mbox{~$v_{\rm t}$ }=2, 5$ and 10.0 $\mbox{km}~\mbox{s}^{-1}$ were computed.

Table 3: Best solution for BI Lyn. See text for explanation of model fit parameters.
Star 1 $\pm$ 2 $\pm$

Spectrophotometry

$\mbox{~$E_{B-V}$ }$ 0.00 0.02

$\mbox{~\em T$_{\rm eff}$ }$ 28.6 1.0 5.84 0.96 kK

$\theta$ 0.55 0.01 4.09 0.10 10^-11 rad

$\mbox{~$v_{\rm t}$ }$ 5 ^a 2 ^a $\mbox{km}~\mbox{s}^{-1}$

R/R₁ 1 7.44 0.03

Spectroscopy

$({\it T}_{\rm eff}$ 30.1 0.01 kK)^b

$\mbox{~\em T$_{\rm eff}$ }$ 28.6 ^a 5.84 ^a kK

$\mbox{~log $g$ }$ 3.6 0.1 3.2 0.3 (cgs)

$\mbox{~$n_{\rm He}$ }$ 0.95 0.01 0.1 ^a

$\mbox{~${\rm [\beta_{\rm Fe}]}$ }$ 0.0 ^a 0.0 ^a

$\mbox{~$v~\sin i$ }$ 0 ^a 120 20 $\mbox{km}~\mbox{s}^{-1}$

$\mbox{~$v_{\rm t}$ }$ 5 ^a 2 ^a $\mbox{km}~\mbox{s}^{-1}$

R/R₁ 1 4.9 0.5

^a Assumed value.

^b Free solution not used.

**Table 3:** Best solution for BI Lyn. See text for explanation of model fit parameters.
Star	1	$\pm$	2	$\pm$
Spectrophotometry
$\mbox{~$E_{B-V}$ }$	0.00	0.02
$\mbox{~\em T$_{\rm eff}$ }$	28.6	1.0	5.84	0.96	kK
$\theta$	0.55	0.01	4.09	0.10	10^-11 rad
$\mbox{~$v_{\rm t}$ }$	5	^a	2	^a	$\mbox{km}~\mbox{s}^{-1}$
R/R₁	1		7.44	0.03
Spectroscopy
$({\it T}_{\rm eff}$	30.1	0.01			kK)^b
$\mbox{~\em T$_{\rm eff}$ }$	28.6	^a	5.84	^a	kK
$\mbox{~log $g$ }$	3.6	0.1	3.2	0.3	(cgs)
$\mbox{~$n_{\rm He}$ }$	0.95	0.01	0.1	^a
$\mbox{~${\rm [\beta_{\rm Fe}]}$ }$	0.0	^a	0.0	^a
$\mbox{~$v~\sin i$ }$	0	^a	120	20	$\mbox{km}~\mbox{s}^{-1}$
$\mbox{~$v_{\rm t}$ }$	5	^a	2	^a	$\mbox{km}~\mbox{s}^{-1}$
R/R₁	1		4.9	0.5
^a Assumed value.
^b Free solution not used.

Table 4: Fundamental properties for stellar components of BI Lyn derived from Table 3 assuming two different values for the radius ratio R₂/R₁, the hot star mass M₁ and the cool star gravity $\log g_2$ .
Assumed values

$\pm$ $\pm$ $\pm$

R₂/R₁ 4.9^a 0.5 7.4^b 0.03 4.9^a 0.5

$M_1/\mbox{$M_{\odot}$ }^c$ 0.5 0.05 0.5 0.05 1.0 0.1

Derived values

$R_1/\mbox{$R_{\odot}$ }$ 1.85 0.34 1.85 0.34 2.62 0.48

$\log L_1/\mbox{$L_{\odot}$ }$ 3.32 0.10 3.32 0.10 3.62 0.10

$R_2/\mbox{$R_{\odot}$ }$ 9.09 1.89 13.80 2.51 12.85 2.68

$\log L_2/\mbox{$L_{\odot}$ }$ 1.93 0.11 2.29 0.10 2.23 0.11

d/kpc 5.01 1.05 7.61 1.40 7.09 1.49

$M_2/\mbox{$M_{\odot}$ }(\mbox{~log $g$ }_2=3.2)$ 4.8 1.8 11.0 3.9 9.6 3.5

$M_2/\mbox{$M_{\odot}$ }(\mbox{~log $g$ }_2=2.5^c)$ 1.0 0.4 2.2 0.8 1.9 0.7

^a R₂/R₁ from spectroscopy.

^b R₂/R₁ from spectrophotometry.

^c Trial values.

**Table 4:** Fundamental properties for stellar components of BI Lyn derived from Table 3 *assuming* two different values for the radius ratio R₂/R₁, the hot star mass M₁ and the cool star gravity $\log g_2$ .
Assumed values
		$\pm$		$\pm$		$\pm$
R₂/R₁	4.9^a	0.5	7.4^b	0.03	4.9^a	0.5
$M_1/\mbox{$M_{\odot}$ }^c$	0.5	0.05	0.5	0.05	1.0	0.1
Derived values
$R_1/\mbox{$R_{\odot}$ }$	1.85	0.34	1.85	0.34	2.62	0.48
$\log L_1/\mbox{$L_{\odot}$ }$	3.32	0.10	3.32	0.10	3.62	0.10
$R_2/\mbox{$R_{\odot}$ }$	9.09	1.89	13.80	2.51	12.85	2.68
$\log L_2/\mbox{$L_{\odot}$ }$	1.93	0.11	2.29	0.10	2.23	0.11
d/kpc	5.01	1.05	7.61	1.40	7.09	1.49
$M_2/\mbox{$M_{\odot}$ }(\mbox{~log $g$ }_2=3.2)$	4.8	1.8	11.0	3.9	9.6	3.5
$M_2/\mbox{$M_{\odot}$ }(\mbox{~log $g$ }_2=2.5^c)$	1.0	0.4	2.2	0.8	1.9	0.7
^a R₂/R₁ from spectroscopy.
^b R₂/R₁ from spectrophotometry.
^c Trial values.

3.3 Spectrophotometry

The method for measuring T $_{\rm eff}$ and angular radii $\theta$ for both components in composite systems containing a hot subdwarf and a cool companion using IUE spectrophotometry and optical-IR broad-band photometry has been described elsewhere (Aznar Cuadrado & Jeffery 2001). In fact PG0900+400 appeared in the sample analyzed, yielding $\mbox{~\em T$_{\rm eff}$ }_1=25~000~\mbox{K}$ , $\mbox{~\em T$_{\rm eff}$ }_2=5150~\mbox{K}$ , with R₂/R₁=8.0. However it had been assumed that $\mbox{~$n_{\rm He}$ }_1=0.0$ and $\mbox{~log $g$ }_1=5.0$ . For this paper, the data were re-analyzed iteratively with the optical data (see below). Consequently, quite different model atmosphere grids were adopted in the final analysis, with $\mbox{~$n_{\rm He}$ }_1=0.95$ and $\mbox{~log $g$ }_1=3.6$ . Interstellar extinction was still found to be negligible, but the different distribution of opacity in the hydrogen-deficient atmosphere of the hot star resulted in a higher T $_{\rm eff}$ being obtained for both components.

With $\mbox{~\em T$_{\rm eff}$ }_1=28~600~\pm~1000~\mbox{K}$ , the earlier measurement of $\sim$ $31~000~\mbox{K}$ (Ferguson et al. 1984) was approximately recovered. However, with $\mbox{~\em T$_{\rm eff}$ }_2=5840~\pm~960~\mbox{K}$ , the cool star appears to be somewhat hotter than the K3 spectral type indicated before. The broader spectral range covered by our data should lead to a more robust result than the flux ratio method (Wade 1980). It is noted that the errors are formal errors and do not allow for systematic errors as may be introduced, for example, by an inappropriate choice for unconstrained model atmosphere parameters. The relative radii of the two stars, given by their angular diameters, is $R_2/R_1=7.44\pm0.03$ . Other parameters of the fit are given in Table 3. The best-fit model flux distribution is shown together with the data in Fig. 2.

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{H2958_F3.PS} \end{figure}$

Figure 3: Normalized blue spectrum of BI Lyn (bottom: d)) together with a best fit composite model spectrum c) formed by adding models with a) $\mbox{~\em T$_{\rm eff}$ }_2=5~840~\mbox{K}$ , $\mbox{~log $g$ }_2=3.2$ , $\mbox{~${\rm [\beta_{\rm Fe}]}$ }_2=0.00$ (top) and b) $\mbox{~\em T$_{\rm eff}$ }_1=28~600~\mbox{K}$ , $\mbox{~log $g$ }_1=3.6$ , $\mbox{~$n_{\rm He}$ }_1=0.95$ assuming that the relative radii R₂/R₁=4.9. The model spectra have been velocity shifted and degraded to match the observed spectral resolution (1 Å) (WHT images: 337312 + 337313 + 337318 + 337319 + 337416 + 337418 + 337419). A detailed comparison between the observed spectrum and best-fit model is shown in Fig. 4.

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{H2958_F4.PS} \end{figure}$

Figure 4: Normalized blue spectrum of BI Lyn (histogram) together with the best fit composite model spectrum (polyline: see Fig. 3c) showing the fit in detail.

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{H2958_F5.PS} \end{figure}$

Figure 5: As Fig. 3 in the region of the infared calcium triplet (INT image: 155728).

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{H2958_F6.PS} \end{figure}$

Figure 6: As Fig. 3 in the region of H $\alpha$ (WHT images: 337414 + 337415 + 337417).

3.4 Spectrum synthesis

Much more information is available in the line spectrum than can be obtained from photometry alone. The object of spectrum synthesis is to find, given assumptions that have been introduced already, a model for the overall spectrum that best matches the observation by, for example, minimizing the square of residuals between model and data. Techniques for doing this with single stars have been described already (Jeffery et al. 2001). The extension to binary stars has been developed by Aznar Cuadrado & Jeffery (in preparation) and described by Aznar Cuadrado (Aznar Cuadrado 2001).

Under perfect conditions (e.g. noise-free data), a residual-minimization procedure could solve for many free parameters simultaneously. In practice, it is necessary to hold most parameters fixed while solving for two or three at a time, and iterating around several parameters until an optimum solution is obtained. With the assumptions introduced already, the final solution for BI Lyn is given in Table 3. The errors cited in fitted quantities are formal errors from the $\chi^2$ minimization. Other errors have been propagated from these. Systematic errors (e.g. choice of ${\rm [\beta_{\rm Fe}]}$ , $v_{\rm t}$ ) can significantly affect these.

A free spectroscopic solution for effective temperature gave $\mbox{~\em T$_{\rm eff}$ }_1=30.1$ kK, but could not provide $\mbox{~\em T$_{\rm eff}$ }_2$ because of a lack of temperature-sensitive diagnostics. To maintain consistency in the radius ratio, the spectrophotometric solutions for $\mbox{~\em T$_{\rm eff}$ }_1$ and $\mbox{~\em T$_{\rm eff}$ }_2$ were retained.

The surface gravity and helium abundance of the hot star were well determined spectroscopically; there is negligible rotational broadening (at the instrumental resolution) in the line profiles. It is clear that the hot component is a hydrogen-deficient giant with $\mbox{~log $g$ }_1=3.6\pm0.1$ with an atmosphere containing some 5 per cent hydrogen and 95 per cent helium (by number).

The cool star is more difficult to analyse, mainly because of substantial rotation broadening $\mbox{~$v~\sin i$ }_2=120\pm20~\mbox{$\mbox{km}~\mbox{s}^{-1}$ }$ . Metallicity, microturbulent velocity, helium abundance and gravity all affect the strength of the metal line spectrum, which is then heavily smeared by the rotation. The infrared calcium triplet could provide a good gravity indicator if ${\rm [\beta_{\rm Fe}]}$ were known. Assuming $\mbox{~${\rm [\beta_{\rm Fe}]}$ }_2=0.0$ we find $\mbox{~log $g$ }_2=3.2\pm0.3$ .

With the assumptions given and parameters deduced, a free solution gives $R_2/R_1=4.9\pm0.5$ in both blue and infared spectral regions. This is determined solely by the strength of the cool star absorption spectrum, so is clearly assumption dependent. Given the nature of these, it is satisfactory that R₂/R₁ is within 40% of the value obtained from photometry.

The best-fit model spectra for both hot and cool stars, their sum according to the given relative radii and a comparison with the observed spectrum are given for all spectra regions in Figs. 3-6.

3.5 Stellar dimensions

Given the measured dimensions of BI Lyn, a number of other properties including masses, luminosities and distance may be estimated. However these depend on which value for R₂/R₁ is adopted, an estimate for the hot star mass M₁ and the quality of the $\mbox{~log $g$ }_2$ measurement. Possible values, given choices for each of these parameters, are shown in Table 4. The given distance is derived from the measured angular diameter and the derived radius of the cool star ( $d=R_2/\theta_2$ ). A similar result is obtained from the apparent visual magnitude and derived luminosities.

These solutions must be reconciled with the position of BI Lyn which, with $l^{\rm II}=182\mbox{$^{\circ}$ }, b^{\rm II}=+42\mbox{$^{\circ}$ }$ , is substantially ( $\mathrel{\raise1.16pt\hbox{$>$ }\kern-7.0pt \lower3.06pt\hbox{{$\scriptstyle \sim$ }}}$ 3 kpc) out of the Galactic plane in the anticentre direction. It therefore seems preferable to seek the lowest possible mass for the cool star in which case a solution with $M_1\sim0.5~\mbox{$M_{\odot}$ }$ and $M_2\sim5~\mbox{$M_{\odot}$ }$ is indicated. The latter is comparable with the companion star mass estimated for another hydrogen-deficient binary - $\upsilon$ Sgr (Schönberner & Drilling 1983; Dudley & Jeffery 1990).

A better solution might be achieved if we have overestimated g₂ by, for example, placing the continuum around the infrared calcium triplet too low or underestimating the hot star flux at these wavelengths. With $M_1\sim0.5~\mbox{$M_{\odot}$ }$ and $\mbox{~log $g$ }_2\sim2.9$ or 2.5, a cool star mass $M_2\sim2.4$ or 1 $M_{\odot}$ would be easier to reconcile with the Galactic position.

In any event, the components must be highly evolved. A solution comprising a 0.5 $M_{\odot}$ B-type helium star and a 1-5 $M_{\odot}$ G-type giant (luminosity class II - III) is consistent with the observational data presented in this paper. It is suggested that BI Lyn is a post common-envelope binary in which the primary (helium star) has almost completely shed its outer hydrogen-rich envelope together with substantial angular momentum. A large fraction of this has been transferred to the cooler star. The envelope of the cool star may not yet be in thermal equilibrium - there may still be a main-sequence star underneath giving it the semblance of a giant. This would account for the apparently advanced evolutionary state of the cool star at the same time as the hot star is in a relatively short-lived phase of evolution.

The surface gravity for the hot star places it on an evolutionary track for a post-AGB star of $\sim$ $0.5~\mbox{$M_{\odot}$ }$ (Schönberner 1983).

The referee has rightly pointed out that the detailed results of our analysis are subject to the LTE assumption used in the analysis of the early-type star. Being a giant, departures from LTE can be significant and will affect, in particular, the predicted equivalent widths of the He I lines. Other spectral features may also be affected. Non-LTE calculations for hydrogen-poor hydrogen-helium atmospheres have been available for some years, but fully line-blanketed NLTE models for hydrogen-poor atmospheres with $\mbox{~\em T$_{\rm eff}$ }\sim30~000~\mbox{K}$ have not yet been successfully computed (Rauch 1996). Their eventual arrival will affect the detailed results presented here but not the overall conclusions concerning the dimensions of BI Lyn.

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{H2958_F7.PS} \end{figure}$

Figure 8: Residuals of four spectra with respect to the mean WHT spectrum in the region of H $\alpha$ , labelled by image number. Only the INT spectrum shows a significant change in the absorption strength.

Up: BI Lyncis: A hydrogen-deficient G