4 Spectral analysis

The aim of this study is to measure the various physical parameters, including effective temperature (T $_{\rm eff}$), surface gravity (log g), chemical composition and, for binary stars, the radius ratio directly from optical and near-infrared spectra. This is achieved by finding the best-fit model spectra within a grid of theoretical models using the method of $\chi^2$ minimization.

The methods for hot single stars have been described in detail (Jeffery et al. 2001), including the generation of model atmospheres (STERNE), the synthesis of model spectra (SPECTRUM) and the least-squares minimization ( SFIT). These techniques have been extended to model binary stars which consist of both a hot and a cool component. Although they have already been introduced (Jeffery & Aznar Cuadrado 2001), the method is described more formally here.

The fundamental assumption is that both cool and hot absorption sources are primarily stellar. We consider the observed normalized spectrum $S_{\lambda}$. Our aim is to reconstruct the best-fit model spectrum

$$s_{\lambda} = \frac{\theta_1^2 f_{\lambda 1} + \theta_2^2 f_{\lambda 2}} {\theta_1^2 f_{\rm c 1} + \theta_2^2 f_{\rm c 2} }$$

where $\theta$ are the stellar angular diameters, $f_{\lambda}$ is the theoretical emergent flux, $f_{\rm c}$ is the continuum flux and subscripts 1 and 2 refer to the individual stellar components. The fluxes are functions of each star's properties: $f_1 = f_1 (\mbox{\it T$_{\rm eff}$ }_1, \mbox{\,log $g$ }_1, \ldots)$ and $f_2 = f_2 (\mbox{\it T$_{\rm eff}$ }_2, \mbox{\,log $g$ }_2, \ldots)$. The angular diameters cannot be solved for explicitly from S, but their ratio gives $R_2/R_1 = \theta_2/\theta_1$.

The principal free parameters which govern the measured spectrum are thus:

• $\mbox{\it T$_{\rm eff}$ }_1, \mbox{\it T$_{\rm eff}$ }_2$: effective temperature;
• $\mbox{\,log $g$ }_1, \mbox{\,log $g$ }_2$: surface gravity;
• R₂/R₁: radius ratio;
• $\mbox{\,$v\,\sin i$ }_1, \mbox{\,$v\,\sin i$ }_2$: rotational velocity;
• v₁, v₂: radial velocity;
• $\mbox{\,$v_{\rm t}$ }_1, \mbox{\,$v_{\rm t}$ }_2$: microturbulent velocity;
• y₁, y₂: helium abundance;
• $\mbox{${\rm [Fe/H]}$ }_1, \mbox{${\rm [Fe/H]}$ }_2$: metallicity.

The radial velocities have already been discussed. 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 [Fe/H]}$ }_2$ cannot be uniquely determined from the given data. Similarly, given the magnitude of errors in $\mbox{\it T$_{\rm eff}$ }_1$, it is not practical to measure $\mbox{${\rm [Fe/H]}$ }_1$ in detail. Thus the following assumptions are made. The abundances of all elements other than hydrogen and helium are in proportion to their cosmic abundances, with $\mbox{${\rm [Fe/H]}$ }_1 = \mbox{${\rm [Fe/H]}$ }_2 = 0$. The helium abundance of the cool star is normal: $y_2 = y_\odot$. 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 [Fe/H]}$ }_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 (Paper I) where we used cool star models computed with $\mbox{\,$v_{\rm t}$ }_2=2\,\mbox{$\mbox{km}\,\mbox{s}^{-1}$ }$. Secondary effects on $\mbox{\it T$_{\rm eff}$ }_2$ and $\mbox{\,log $g$ }_2$ are not significant here.

4.1 Model grids

The model atmospheres and flux distributions used to analyse the hot star were computed with the line-blanketed plane-parallel LTE code STERNE. The high-resolution spectra were calculated with the LTE code SPECTRUM (see Jeffery et al. 2001).

The model atmospheres were calculated on a three-dimensional rectangular grid defined by $\mbox{\it T$_{\rm eff}$ }= 18\,000 (2000) 40\,000 \,\mbox{K}$, $\mbox{\,log $g$ }= 4.5 (0.5) 7.0$, and composition $n_{\rm H}$ = 1  - $n_{\rm He}$, $\mbox{\,$n_{\rm He}$ }= 0.01, 0.10 (0.10) 0.60$ and ${\rm [Fe/H]}$ = 0. The larger value of $n_{\rm H}$ or $n_{\rm He}$ is reduced to compensate for trace elements.

Synthetic spectra were calculated on wavelength intervals 3800-5020 Å (blue) and 8450-8670 Å (CaT). Linelists comprising some 142 absorption lines of hydrogen, helium, carbon, magnesium and silicon 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 1993a), for $\mbox{\it T$_{\rm eff}$ }=3500 (500) 8000$, $\mbox{\,log $g$ }=2.0 (0.5) 4.5$, $\mbox{${\rm [Fe/H]}$ }=-0.5$, -0.3, 0.0 and $\mbox{\,$v_{\rm t}$ }=2.0\,\mbox{$\mbox{km}\,\mbox{s}^{-1}$ }$.

Grids of high resolution spectra were calculated in the same spectral regions as for the hot star using Kurucz' code SYNTHE (Kurucz 1993b; Jeffery et al. 1996) assuming a microturbulent velocity $\mbox{\,$v_{\rm t}$ }=2~\,\mbox{$\mbox{km}\,\mbox{s}^{-1}$ }$.

4.2 SFIT

For a given observation, an optimum fit was obtained by minimizing $\chi^2$,

$$\chi^2 = \sum_{\lambda} { (S_{\lambda} - s_{\lambda})^2 \over \sigma^2_{\lambda}},$$ (1)

the weighted square residual between the normalized observed spectrum, $S_{\lambda}$, and the theoretical spectrum,

$$s'_{\lambda} = s_{\lambda}(\mbox{\it T$_{\rm eff}$ }, \mbox{\... ... })\otimes I(\Delta \lambda) \otimes V(v~{\rm sin} i, \beta),$$ (2)

where $I(\Delta \lambda)$ and $V(v~{\rm sin} i, \beta)$ represent the instrumental and rotational broadening, respectively. Instrumental broadening is measured from the width of the emission lines in the copper-argon comparison lamp. The $\chi^2$ minimization was carried out using a variant of the algorithm AMOEBA (Press et al. 1989; Jeffery et al. 2001).

In the construction of $\chi^2$, each wavelength point was given a weight $w_{\lambda} = 1 / \sigma_{\lambda}$, the inverse of the standard deviation of the mean normalized flux in line-free regions. In our spectra $\sigma_{\lambda} \sim 0.01$. Some spectral regions needed to be excluded from the fit (e.g. bad columns or strong lines missing from the theoretical spectrum). In SFIT, such defects are masked by increasing $\sigma$ in appropriate wavelength intervals. We used $\sigma_{\lambda}$ = 0.1.

In any such fitting procedure, the normalization of the observed spectrum can be of crucial importance (Jeffery 1998). The problem is to normalize the observed spectrum correctly without, for example, compromising the wings of broad absorption lines. This is particularly difficult when there is an unknown contribution to the line opacity from metal lines in a cool star companion, so that there may be no "true'' continuum anywhere in the observed spectrum.

 

 

Table 4: Atmospheric properties of single-spectrum sdB stars measured spectroscopically using SFIT and previously.

Star	T $_{\rm eff}$	log g	y	Reference
PG0004+133	$^\dagger$25025 $\pm$ 400	4.97 $\pm$ 0.10	0.01 $\pm$ 0.01	SFIT
(Fig. 1)	25025 $\pm$ 400			Paper I
	24700 $\pm$ 1300	4.5 $\pm$ 0.2	0.028	Moehler et al. (1990a)
PG0229+064	18000 $\pm$ 1025	4.35 $\pm$ 0.10	0.33 $\pm$ 0.01	SFIT
	20100 $\pm$ 400			Paper I
	19000 $\pm$ 950	4.55 $\pm$ 0.10	0.16	Ramspeck et al. (2001)
	22000 $\pm$ 1000	4.65 $\pm$ 0.15	0.137	Heber et al. (1999)
PG0240+046	36200 $\pm$ 400	6.25 $\pm$ 0.10	1.94 $\pm$ 0.02	SFIT
	34800 $\pm$ 1850			Paper I
	37000 $\pm$ 2000	5.3 $\pm$ 0.3	1.222	Thejll et al. (1994)
PG0342+026	24000 $\pm$ 375	5.17 $\pm$ 0.10	0.01 $\pm$ 0.01	SFIT
	27900 $\pm$ 975			Paper I
	24000 $\pm$ 1200	4.90 $\pm$ 0.20	0.003	Heber et al. (1999)
	25000 $\pm$ 2500	5.25 $\pm$ 0.20	0.000	Theissen et al. (1995)
	26220 $\pm$ 1000	5.67 $\pm$ 0.15	0.004	Saffer et al. (1994)
	22300 $\pm$ 1000	5.00 $\pm$ 0.30	0.000	Lamontagne et al. (1987)
PG0839+399	37300 $\pm$ 500	6.02 $\pm$ 0.10	<0.01 $\pm$ 0.01	SFIT
	35600 $\pm$ 1800			Paper I
	36100 $\pm$ 1000	5.91 $\pm$ 0.15	0.002	Saffer et al. (1994)
PG1233+426	25560 $\pm$ 550	5.52 $\pm$ 0.10	<0.01 $\pm$ 0.01	SFIT
	28750 $\pm$ 900			Paper I
	26500 $\pm$ 1000	5.60 $\pm$ 0.15	0.005	Saffer et al. (1994)
	26200 $\pm$ 1500	5.30 $\pm$ 0.30	0.000	Lamontagne et al. (1985)
PG2259+134	28500 $\pm$ 600	5.93 $\pm$ 0.10	0.02 $\pm$ 0.01	SFIT
	28350 $\pm$ 750			Paper I
	22500 $\pm$ 2500	5.00 $\pm$ 0.20	0.000	Theissen et al. (1995)
	28500 $\pm$ 1600	5.30 $\pm$ 0.20	0.022	Theissen et al. (1993)
$^\dagger$ T $_{\rm eff}$ assumed from Paper I.

The initial normalization was performed by fitting a low-order spline function to a series of pseudo-continuum points, usually the highest points in the spectrum. SFIT includes two re-normalization algorithms which may be used to optimize the fit (cf. Jeffery et al. 1998). One computes a low-order polynomial fit to the residual, the other applies a low-pass Gaussian filter. A second order polynomial was used to renormalize the spectrum in wavelength ranges 4200-4650 Å and 8000-8850 Å while a third order polynomial was used in the region of high order Balmer lines (H₉, H₈, H $_{\epsilon}$, etc.). Because low-order polynomials are used, individual line profiles are unaffected.

In principle and for suitable data with negligible noise, SFIT can solve simultaneously for as many parameters as required. In practice, it is necessary to restrict the free solution to between two and three parameters at a time, keeping others fixed, and to iterate until the optimum solution is obtained. SFIT requires a set of initial estimates for the free parameters. Results from the flux distribution analysis (Paper I) were used for $\mbox{\it T$_{\rm eff}$ }_1, \mbox{\it T$_{\rm eff}$ }_2$ and the radius ratio. Standard values were assumed for $\mbox{\,log $g$ }_1, \mbox{\,log $g$ }_2$ and y₁.

4.3 Analysis: single stars

For single sdB stars, SFIT was applied to the blue spectra. The first parameter derived was the composition y of the sdB star. Within the T $_{\rm eff}$ range of sdB stars, the strength of helium lines depends far more on helium abundance than T $_{\rm eff}$ or log g. T $_{\rm eff}$ and log g were found next by finding the best fit to the Balmer lines. Rotational broadening is small compared with the instrumental profiles in these spectra.

Table 4 presents the results of the spectral analysis of single sdB stars (labeled SFIT), together with the results of the flux distribution analysis (Paper I) and results from literature.

4.4 Analysis: binary stars

For composite sdB stars, SFIT was applied separately to both blue and red spectra. Again, the first parameter to be fixed from the analysis of the blue spectrum is the composition of the sdB star, i.e. y₁. Afterwards, $\mbox{\it T$_{\rm eff}$ }_1$, $\mbox{\,log $g$ }_1$ of the sdB star and R₂/R₁ are found by fitting the observed Balmer lines. The lower limit of the model grid was occasionally too large to fit the observed helium lines. In these cases only an upper limit to y₁ can be given.

The radius ratio R₂/R₁ is directly related to the effective temperatures of both components, so must be a "free'' parameter whenever either T $_{\rm eff}$ is free. The contribution of the cool companion in the blue spectrum is reflected in the presence and strength of some metallic lines, being good indicators of the temperature of the cool star and the radius ratio of the system. Therefore, the blue spectrum is also used to fix $\mbox{\it T$_{\rm eff}$ }_2$ and R₂/R₁. It was frequently difficult to find a solution for $\mbox{\it T$_{\rm eff}$ }_1$, $\mbox{\it T$_{\rm eff}$ }_2$ and R₂/R₁ consistent with the flux distribution analysis (Paper I). In cases of conflict, we attempted to keep R₂/R₁ consistent between the two studies, although even this was not always possible (Table 5).

 

 

Table 5: Atmospheric properties of composite sdB stars measured spectroscopically using SFIT and previously. Subscript 1 refers to the hot subdwarf, subscript 2 refers to the cool companion.

Star	$\mbox{\it T$_{\rm eff}$ }_1$	$\mbox{\,log $g$ }_1$	y1	$\mbox{\it T$_{\rm eff}$ }_2$	$\mbox{\,log $g$ }_2$	R2/R1	Ref.$^\dagger$
PG0110+262	21000 $\pm$ 750	5.17 $\pm$ 0.17	<0.01 $\pm$ 0.01	5250 $\pm$ 800	4.53 $\pm$ 0.21	3.2 $\pm$ 1.9	SFIT
	21050 $\pm$ 575			5485 $\pm$ 200		4.2 $\pm$ 0.2	1
	21000 $\pm$ 1000	< $5.90\pm 0.10$		5000 $\pm$ 500		6.0	2
	22000 $\pm$ 1000	< $5.50\pm 0.10$		5500 $\pm$ 500		4.4	3
	22000 $\pm$ 1500			4500 $\pm$ 500		7.8	4
PG0749+658	25400 $\pm$ 500	5.70 $\pm$ 0.11	<0.01 $\pm$ 0.02	5000 $\pm$ 500	4.58 $\pm$ 0.24	3.5 $\pm$ 1.2	SFIT
	25050 $\pm$ 675			5600 $\pm$ 300		3.9 $\pm$ 0.3	1
	24600 $\pm$ 1000	5.54 $\pm$ 0.15	0.004				5$^\ast$
	23500 $\pm$ 1500			4125 $\pm$ 500		6.3	4
PG1104+243	32850 $\pm$ 1550	5.40 $\pm$ 0.12	0.01 $\pm$ 0.02	6400 $\pm$ 1000	4.30 $\pm$ 0.31	5.9 $\pm$ 1.1	SFIT
	28000 $\pm$ 875			5735 $\pm$ 150		6.1 $\pm$ 0.2	1
	27500 $\pm$ 1500			4300 $\pm$ 500		10.6	4
	27200 $\pm$ 1500	5.50 $\pm$ 0.30					6
	28000 $\pm$ 5000			4600 $\pm$ 1000		9.8	7
PG1701+359	32500 $\pm$ 1325	5.75 $\pm$ 0.12	<0.01 $\pm$ 0.01	6000 $\pm$ 1000	4.60 $\pm$ 0.23	2.7 $\pm$ 1.8	SFIT
	36075 $\pm$ 700			6450 $\pm$ 230		4.8 $\pm$ 0.2	1
	30000 $\pm$ 2500	5.00 $\pm$ 0.20	0.000				8
	28500 $\pm$ 1500			4000 $\pm$ 500		6.2	4
	26250 $\pm$ 1250	5.80 $\pm$ 0.20					9
PG1718+519	29000 $\pm$ 1550	6.00 $\pm$ 0.14	<0.01 $\pm$ 0.01	5200 $\pm$ 400	4.55 $\pm$ 0.23	4.8 $\pm$ 1.6	SFIT
	29950 $\pm$ 1100			5925 $\pm$ 70		8.2 $\pm$ 0.3	1
	30000 $\pm$ 2500	5.00 $\pm$ 0.20	0.000	5125 $\pm$ 500			8
	25000 $\pm$ 1500			4300 $\pm$ 500		10.7	4
	23300 $\pm$ 1000	4.25 $\pm$ 0.20					9
PG2110+127	26500 $\pm$ 1700	5.20 $\pm$ 0.18	<0.01 $\pm$ 0.06	5400 $\pm$ 400	4.40 $\pm$ 0.24	4.7 $\pm$ 1.1	SFIT
(Fig. 2)	24900 $\pm$ 6500			5500 $\pm$ 575		5.5 $\pm$ 0.3	1
	30000 $\pm$ 2500	5.00 $\pm$ 0.20	0.000	5375 $\pm$ 500			8
	33700 $\pm$ 1000	5.33 $\pm$ 0.15	0.004				5
	26000 $\pm$ 1500			4500 $\pm$ 500		10.4	4
PG2135+045	28400 $\pm$ 800	4.80 $\pm$ 0.22	<0.01 $\pm$ 0.01	5000 $\pm$ 500	4.40 $\pm$ 0.30	3.1 $\pm$ 0.5	SFIT
	26325 $\pm$ 9950			4375 $\pm$ 1790		4.7 $\pm$ 0.6	1
	32100 $\pm$ 1000	4.79 $\pm$ 0.15	0.016				5$^\ast$
	27000 $\pm$ 1500			4400 $\pm$ 500		6.5	4
PG2148+095	30000 $\pm$ 860	4.90 $\pm$ 0.16	<0.01 $\pm$ 0.01	5700 $\pm$ 400	4.40 $\pm$ 0.31	3.0 $\pm$ 0.8	SFIT
(Fig. 4)	22950 $\pm$ 825			4375 $\pm$ 200		5.0 $\pm$ 0.2	1
	25000 $\pm$ 1000	< $5.80\pm 0.10$		5000 $\pm$ 500		6.0	2
	26000 $\pm$ 1500			4300 $\pm$ 500		7.9	4
$^\dagger$References: 1 = Paper I; 2 = Ulla & Thejll (1998); 3 = Thejll et al. (1995); 4 = Allard et al. (1994);
5 = Saffer et al. (1994); 6 = Lamontagne et al. (1987); 7 = Ferguson et al. (1984); 8 = Theissen et al. (1995);
9 = Theissen et al. (1993).
$^\ast$ Saffer et al. (1994) did not recognise the composite nature of these stars.

In the initial analysis of the blue spectrum, the cool companion is assumed to have $\mbox{\,log $g$ }_2 = 4.5$ (cf. Paper I). Applying SFIT gives new values, first for y₁, then for $\mbox{\it T$_{\rm eff}$ }_1$, $\mbox{\,log $g$ }_1$, $\mbox{\it T$_{\rm eff}$ }_2$ and R₂/R₁.

With these improved estimates for the sdB star properties, the red spectrum is analyzed. In particular, $\mbox{\,log $g$ }_2$ is determined by fitting the calcium triplet.

A second analysis is now performed in the blue in order to refine the fit to the hot star spectrum, using the new parameters for the cool star. The above procedure is repeated until the solutions converge.

Table 5 presents the results for composite spectrum sdB stars, together with previous results from the literature. Solar metallicity was adopted for all stars except PG1104+243, for which we adopted $\mbox{${\rm [Fe/H]}$ }_2=-0.5$. The instrumental profile is large compared with any rotational broadening except in the cases of PG1701+359 and PG1718+519, where $\mbox{\,$v\,\sin i$ }=5$ and 10  $\,\mbox{$\mbox{km}\,\mbox{s}^{-1}$ }$, respectively, were adopted.

Table 5 includes values for R₂/R₁ for some previous papers. These have been computed from cited flux ratios at 5500 Å, effective temperatures and/or spectral types and an appropriate bolometric correction.

4.5 Errors

The formal errors associated with the best fit parameters x_i are given by the diagonal elements $\delta_i = (\alpha^{-1})_{ii}$ of the inverse of the covariance matrix $\alpha$, whose elements are given by

$$\alpha_{ij} = \sum_{\lambda} \left( {{\partial s_{\lambda}}... ...} \over {\partial x_{j}}} / \sigma^{2}_{\lambda}\right)\cdot$$ (3)

Since SFIT is never run with all parameters free, the full covariance matrix is never computed. Thus the total errors $\sigma_i$ need to be obtained from a careful analysis of the partial errors $\delta_i$.

For a single-star spectrum, only the derivatives between T $_{\rm eff}$, log g and y need to be calculated. In the case of a binary system, the derivatives between the physical parameters of both components of the system are required. These have to be evaluated numerically, e.g.

$${\partial x_{1} \over \partial x_{2}} = {{x'_{1} - x_{1}} \over \Delta x_{2}},$$ (4)

where x₁ represents parameters of the sdB star and x₂ represents parameters of the cool star. $\Delta x_{2}$ is an increment used to compute a change in x₁, usually 1$\%$ of x₂. x'₁ are parameters derived when using $x_{2}+\Delta x_{2}$ as input to the fit.

The errors given in Tables 4 and 5 are total errors. For example the total error $\sigma$ in $\mbox{\it T$_{\rm eff}$ }_{\rm 1}$ is given by

$$\sigma^2_{\mbox{\it T$_{\rm eff}$ }_{1}} \delta^2_{\mbox{\it T$_{\rm eff}$ }_{1}} + \left({\partial ~\mbox... ...over \partial ~\mbox{\,log $g$ }_{1}}\right)^2 \delta^2_{\mbox{\,log $g$ }_{1}}$$ =

$$+ \left({\partial ~\mbox{\it T$_{\rm eff}$ }_{1} \over \partial ~... ...over \partial ~\mbox{\,log $g$ }_{2}}\right)^2 \delta^2_{\mbox{\,log $g$ }_{2}}$$

$$+ \left({\partial ~\mbox{\it T$_{\rm eff}$ }_{1} \over \partial ~(R_2/R_1)}\right)^2 \delta^2_{R_2/R_{1}}.$$ (5)

In addition to the formal errors cited, there are additional systematic errors. Principal amongst these are in the metallicity of the cool star, $\mbox{${\rm [Fe/H]}$ }_2$. This has a strong influence on the measurement of $\mbox{\,log $g$ }_2$ from the calcium triplet, and hence on the radius ratio R₂/R₁ because the latter is primarily fixed by the strength of the metal-lines relative to Balmer lines in the blue spectrum. High-resolution spectra will be required to address this problem.

There are also systematic differences between the methods used to obtain R₂/R₁ in this paper and in Paper I. To compare these methods we have constructed a simple test. The energy distribution and normalized spectrum of a binary system containing a typical sdB star and a main-sequence star were computed. These were resampled to mimic the observational data available to us in each investigation. The data were then analyzed using BINFIT (to fit the flux distribution, Paper I) and SFIT independently. The model parameters and the results of the $\chi^2$ analysis are shown in Table 6.

 

 

Table 6: Comparison of BINFIT and SFIT for a test binary.

Parameter	Model	BINFIT	SFIT
$\mbox{\it T$_{\rm eff}$ }_1$ (K)	24000	24060 $\pm$ 260	24190 $\pm$ 300
$\mbox{\,log $g$ }_1$	6.0		6.02 $\pm$ 0.05
y1	0.10		0.11 $\pm$ 0.01
$\mbox{\it T$_{\rm eff}$ }_2$ (K)	4500	4550 $\pm$ 150	4500 $\pm$ 200
$\mbox{\,log $g$ }_2$	4.5		4.54 $\pm$ 0.05
R2/R1	6.29	5.90 $\pm$ 0.16	6.27 $\pm$ 0.20

The results are all consistent with the test model except the value of R₂/R₁ given by BINFIT. The errors associated with this parameter are possibly underestimated, since they are only derived from the formal error in the angular diameters.