3 Atmospheric parameters and projected rotational velocities

To derive atmospheric parameters (effective temperature, surface gravity, and photospheric helium abundance) and projected rotational velocities all Balmer lines and the He  I lines $\lambda\lambda$ 4026 Å, 4388 Å, 4438 Å, 4472 Å, 4713 Å, 4922 Å, 5016Å, 5048 Å, 5678 Å, in the observed spectra were fitted with synthetic line profiles calculated from model atmospheres.

We computed LTE model atmospheres using the program of Heber et al. (2000), which calculates plane parallel, chemically homogeneous and fully line blanketed models, using the opacity distribution functions for metal line blanketing by Kurucz (1979, ATLAS6). From these model atmospheres synthetic spectra were calculated with Lemke's version of the LINFOR program (developed originally by Holweger, Steffen, and Steenbock at Kiel University). The spectra include the Balmer lines H$_\alpha$ to H₂₂ and the He I lines listed above and the grid covers the range 11000 K$\leq$  ${T_{\rm eff}}$ $\leq$ 40000 K, 3.5 $\leq$ $\log {g}$ $\leq$ 6.5 and $-4.0 \leq$  ${{\rm\log}{\frac{n_{\rm He}}{n_{\rm H}}}}$ $\leq -0.5$ at solar metallicity.

The fit procedure is based on a $\chi^{2}$ test using the routines developed by Bergeron et al. (1992) and Saffer et al. (1994) and modified by Heber et al. (1997) to derive also the rotational velocity. The theoretical spectra are convolved with the instrument profiles (Gaussian with the appropriate instrumental FWHM) and a rotational profile. The fit program then normalizes theoretical and observed spectra using the same continuum points. Example fits for a rapidly rotating and a slowly rotating programme star are shown in Fig. 1 for hydrogen and helium lines, whilst Fig. 2 compares the metal line spectra of slowly rotating (PHL 159, BD-15$^\circ$115) stars and a rapidly rotating star (PG 1533+467).

Figure 1: Fit examples for a slowly rotating star (BD-15$^\circ$115, left hand side) and a rapidly rotating star (PG 1533+467, right hand side).

Figure 2: Wavelength range with strong N  II and O  II lines to show examples for spectra of slowly rotating (bottom, middle) and rapidly rotating (top) stars.

  

 

Table 2: Atmospheric parameters and rotational velocities for the progamme stars as derived from high and low resolution spectroscopic data and comparison of these data with the effective temperatures calculated from Strømgren photometry. The rotational velocities derived from the high resolution data were used to fit the low resolution spectra.

	High Resolution				Low Resolution		Photometry
Name	${T_{\rm eff}}$	${\log(\frac{g}{\rm cm~s^{-2}})}$	${\log\frac{N({\rm He})}{N(\rm H)}}$	$v \sin i$	${T_{\rm eff}}$	${\log(\frac{\it g}{\rm {cm~s}^{-2}})}$	${T_{\rm eff}}$	E(b-y)
	(K)			(km s-1)	(K)		(K)
PG 0122+214	18300	3.86	-0.98	117	18700	3.90	18500 (1)	0.0
PG 1511+367	16100	4.15	-1.16	77	15600	4.20	15900 (1)	0.0
PG 1533+467	18500	4.09	-0.94$^\star$	215	17700	3.93	17700 (1)	0.020
PG 1610+239	15500	3.72	-0.84$^\star$	75	15400	3.69	18600 (1)	0.082
PG 2219+094	19500	3.58	-1.00$^\star$	225	18200	3.52	16700 (2)	0.037
							19500 (3)	0.081
PHL 159	18500	3.59	-0.84	21	-	-	20900 (4)	0.025
PHL 346	20700	3.58	-1.00	45	-	-	22300 (7)	0.037
SB 357	19700	3.90	-1.00$^\star$$^\star$	180	-	-	19700 (5)	0.052
							19700 (8)	0.061
							19800 (9)	0.037
BD-15$^\circ$115	20100	3.81	-0.97	35	-	-	19800 (5)	0.0
							20200 (6)	0.0
HS 1914+7139	17600	3.90	-0.99	250	18100	3.60	-	-

$\parbox{15.8cm}{ $^\star$\space $ {T_{\rm eff}}$ , $\log {g}$ { }fix... ...e{kihi77}); (8) Kilkenny D. (\cite{kil95}); (9) Graham et al. (\cite{grsl73}).}$

The fit procedure was executed for all high and low resolution spectra and the results are listed in Table 2. Formal fitting errors are very small for the high resolution spectra, on average $\Delta {T_{\rm eff}=100~\rm K}$, $\Delta {\log g=0.02}$. Systematic errors (e.g. continuum placement, uncertainties in line broadening theory) are certainly larger and therefore dominate the error budget. We estimate errors in effective temperatures conservatively as 5% and adopted an error of $\pm$0.1dex for the gravities. The fitting errors for the Helium abundance are on average $\Delta {\log (n({\rm He})/n(\rm H))=0.05}$. Since sharp Helium lines as well as broad Helium lines are well reproduced (see Fig. 1) systematic errors due to He  I line broadening theory appear to be small and we adopted an error of $\pm$0.1dex in all cases.

For rapidly rotating stars the $\chi^{2}$ minimum is too poorly defined to allow a reliable determination of the He abundance simultaneously. Therefore, in a first step the helium abundance was kept fixed at -1.00 (i.e. solar) for the fit procedure. In a second iteration step the helium abundance was determind by fitting the helium lines while keeping the ${T_{\rm eff}}$ and $\log {g}$ fixed at those values determined in the first iteration step. For all stars (except HS 1914+7139) Strømgren photometry is available, which allowed an independent determination of the effective temperature. We used the program of Moon (1985) as modified by Napiwotzki et al. (1993) to derive the effective temperature and the reddening and compare the photometric temperatures to the spectroscopic ones in Table 2. There is a good agreement between results from low and high resolution spectra and photometry, except for PG 1610+239, PHL 159 and PHL 346. The spectrum of SB 357 shows the presence of emission in $\rm {H_{\beta}}$ and $\rm {H_{\gamma}}$ but not in $\rm {H_{\delta}}$. Therefore the effective temperature were obtained from Strømgren photometry and the surface gravity from fitting the far wings of the hydrogen lines. The helium lines of this object were difficult to fit, but the observation is compatible with normal abundance and there is no indication of emission in any of the helium lines observed. The parameters used for further analyses were taken from the high resolution spectra, because of the larger wavelength coverage and the excellent quality of the fits. In the case of PG 1533+467, however, the wavelength coverage of the low resolution spectrum is larger than that of the high resolution one and therefore we used the average. The finally adopted parameters are listed in Table 6. Results are shown in a ( ${T_{\rm eff}}$, $\log {g}$) diagram (Fig. 3).