A non-local thermodynamic equilibrium spherical line-blanketed stellar atmosphere model of the early B giant β CMa

Abstract

The observed multiwavelength spectrum of the B1II–III star β CMa is successfully reproduced, including the extreme ultraviolet (EUV) continuum observed by Extreme Ultraviolet Explorer (EUVE), with a non-local thermodynamic equilibrium fully line-blanketed spherical hydrostatic model atmosphere. The available spectrophotometry of β CMa from 500 Å to 25 μm is best fitted with model parameters T_eff = 24000 K, log g = 3.5 and an angular diameter of θ_LD = 0.565 mas. We find that a neutral interstellar hydrogen column of N(H⁰) ≃ 2 × 10¹⁸ cm⁻² provides the best agreement between the model EUV flux and that observed by EUVE. We use model atmosphere fits together with Hipparcos distances to calculate radii, luminosities and ionizing fluxes for β CMa and α Vir. An investigation of spherical and plane-parallel models shows that the Lyman continuum predictions are quite sensitive to model geometry and surface gravity between effective temperatures 18 000 and 33 000 K. This result provides an explanation for the reported excesses between the observed EUV fluxes from β CMa and ∈ CMa and plane-parallel model atmosphere predictions.

methods: numerical, stars: atmospheres, stars: early, stars: fundamental parameters, stars: individual: β CMa, stars: individual: α Vir

1 Introduction

Extreme Ultraviolet Explorer (EUVE) spectroscopic observations of the B giants e CMa and β CMa below the Lyman edge (Cassinelli et al. 1995, 1996) revealed that previous local thermodynamic equilibrium (LTE) and non-local thermodynamic equilibrium (NLTE) plane-parallel stellar atmosphere models underpredicted the flux level of their extreme ultraviolet (EUV) continua by at least a factor of 2. These observations are of fundamental importance since they allow us the rare opportunity to test directly model stellar atmospheres of hot luminous stars in the EUV. Owing to the severe limitations on direct observations of the Lyman continuum from B stars imposed by neutral hydrogen in the local interstellar medium, e CMa and β CMa, which lie along the extremely rarefied β CMa tunnel (Welsh 1989), will likely remain the only hot luminous stars for which a direct measure of their EUV spectrum is available.

Less direct evidence for an EUV excess in the early B stars has been provided by Hα images of very diffuse early B star H II regions, such as the one surrounding αVir (Reynolds 1985), which suggest that the ionizing flux of the central star is more than a factor of 2 larger than previously predicted by model atmospheres. The detection of this very faint emission nebula puts a lower limit on the ionizing flux.

In addition, accurate predictions of EUV radiation are particularly important for accessing the contribution to the diffuse EUV background radiation responsible for the ionization of the warm interstellar medium (Cassinelli 1996). The observed He I λ5876/Hα line ratio in the warm ionized medium indicates that the ratio of helium to hydrogen ionizing photons in the EUV background corresponds to the ratio predicted for late O and early B stars (Reynolds & Tufte 1997) and suggests a more important role for early B stars than previously thought.

A solution to the EUV excess problem for e CMa was proposed by Aufdenberg et al. (1998b, hereafter AHSB). We found that the lack of a sufficiently strong EUV continuum in previous models is linked to the use of plane-parallel (PP) geometry in the line blanketed LTE and line blanketed NLTE hydrostatic models (Kurucz 1992; Hubeny & Lanz 1995) and the lack of full metal line blanketing in the spherically extended hydrodynamic models (Najarro et al. 1996; Schaerer & de Koter 1997). The spherically extended, line blanketed, hydrostatic models of AHSB are an intermediate step between these two approaches. Other authors (Fieldus, Lester & Rogers 1990; Rosenzweig & Anderson 1993; Drake, Plez & Smith 1993) have computed hydrostatic, spherically extended, line blanketed models for A, F and G type giants; however, to our knowledge no similar models have been published for early B giants.

A general concern for atmosphere modellers of stars earlier than mid-B is that hydrostatic equilibrium will only be an approximation as radiation pressure within the atmosphere becomes more important. Both e CMa and β CMa are X-ray sources and show observational evidence for possessing stellar winds (Drew, Denby & Hoare 1994; Cassinelli et al. 1996). In addition, β CMa resides in the β Cephei instability strip and pulsates, while the cooler e CMa does not. To what degree these factors affect the EUV continua of these stars remains unclear. In this paper we present atmosphere models for these stars which are hydrostatic approximations as a basis for comparison and as a step towards models that include a unified line blanketed wind.

Although the models of AHSB neglected the presence of the weak wind of e CMa, the combination of spherical geometry and line blanketing in these models leads to a significant improvement in the agreement with the observed spectral energy distribution. For stellar parameters T_eff=21 kK and logg=3.0, the combination of line blanketing and sphericity produces greater ‘back-warming’ in the model temperature structure relative to PP models. This in turn results in warmer temperatures at the formation depths of the H I and He I Lyman continua.

It is important to note that while the spherical, NLTE, line blanketed models reproduce the EUV spectrum of e CMa, it is the line blanketing-sphericity combination, not exclusively the NLTE aspect, that is the critical ingredient for the enhanced EUV continuum flux. In fact, at T_eff=21 kK, LTE models predict more EUV flux than otherwise similar NLTE models because of the non-negligible departure coefficients in the ground states of H I and He I in the NLTE models, which produce more prominent ionization edges and cooler Lyman continua relative to LTE models. As shown by AHSB, non-line blanketed hydrostatic models, irrespective of model geometry, have nearly identical temperature structures and significantly underpredict the strength of the observed EUV continuum. In other words, when line blanketing is absent, geometrical effects on the temperature structure are significantly reduced or absent for temperatures and gravities typical of early B giants.

These results are in contrast to the differences in the predicted EUV continua between hydrostatic PP models and spherically extended hydrodynamic models of early O stars and central stars of planetary nebulae (CSPN) (Gabler et al. 1989). This work shows that geometric and dynamical differences predict that the EUV continuum below 228 Å is significantly enhanced by the underpopulation of the He Ii ground state in the extended hydrodynamic models relative to the PP hydrostatic models. In addition, these model predictions show little sensitivity to details of the temperature structure. In contrast, it appears that the predicted flux enhancement in the H I and He I Lyman continua of early B giants results from the sensitivity of the temperature structure to the model geometry in the presence of metal line blanketing, while the predicted enhancement in the He ii continuum flux of early O stars and CSPN results from non-LTE effects in an extended hydrodynamic structure.

The bright B1II–III star β CMa (HD 44743, HR 2294) also shows an EUV excess relative to PP models (Cassinelli et al. 1996). Cassinelli et al. (1996) also find flux discrepancies between their NLTE model atmospheres and the measured fluxes in the ultraviolet (UV) and infrared (IR), at 2200 Å, 12 μm and 25 μm. The effective temperature of β CMa has been ambiguous because of the inability of previous LTE and NLTE stellar atmosphere models to match simultaneously the observed flux from the EUV through the IR. Cassinelli et al. (1996) found that the EUV continuum is fitted by an LTE atlas9 model atmosphere with an effective temperature of T_eff=24 800 K; however, this model predicts V-band fluxes 10 per cent larger than those observed. A cooler LTE atlas9 model (T_eff=23 250 K) is in much better agreement with the observed visual fluxes, but the EUV continuum of β CMa observed by EUVE exceeds the flux predicted by this model by a factor of 5. In addition, both models predict that β CMa has an IR excess at 12 and 25 μm and fail to reproduce the flux observed by OAO-2 near 2200 Å. These problems pose a challenge for new models of the spectral energy distribution of this star.

In this paper we present a NLTE, spherical, line-blanketed model atmosphere and synthetic spectrum for β CMa, which closely reproduces the spectrophotometric observations from the EUV to the IR. Furthermore, we derive fundamental parameters for β CMa using the Hipparcos parallax together with the effective temperature and angular diameter derived from the comparison of our model atmosphere to the spectrophotometric observations. We also present a grid of early B star ionizing fluxes and compare them to previously published values. We compare the spherical and PP model temperature structures from this grid of B star models to investigate the different ionizing flux predictions for the two geometries and the magnitude of these differences over a range of effective temperature and surface gravity. In Section 2 the computation of the model atmospheres is described. In Section 3 we compare the synthetic spectra to the observational data for β CMa. Fundamental parameters for β CMa, e CMa and αVir are discussed in Section 4. In Section 5 we analyse the differences between plane-parallel and spherical model atmosphere temperature structures and ionizing flux predictions. Ionising fluxes from a grid of spherical early B star model atmospheres are presented in Section 6. We summarize our results and conclusions in Section 7. An analysis of the 2200-Å flux discrepancy in β CMa and the relative flux calibration of the OAO-2 spectrophotometric data is left for Appendix A.

2 Description of Stellar Atmosphere Models

We have used Version 8.1 of PHOENIX (Hauschildt, Baron & Allard 1997, and references therein) for the computation of the model atmospheres presented here. The hydrostatic model atmosphere temperature structure is computed from the condition of radiative equilibrium. The temperature corrections are based on the absolute error in both the flux and the flux derivative for the plane-parallel models, and in both the luminosity and the luminosity derivative for the spherical models. Full line blanketing is treated in both the computation of the model atmosphere and the synthetic spectrum. The standard outer boundary conditions for our model atmospheres are a continuum optical depth of τ₂₀₀₀ = 10⁻¹⁰ and an outer gas pressure of P_gas = 10⁻⁴ dyn cm⁻². The inner optical depth boundary is τ₂₀₀₀ = 10², well below the thermalization depth. Models are computed on an optical depth grid of 50 points. The most important adjustable model parameters are the effective temperature T_eff, surface gravity logg, and the model geometry. For the spherical models, where T_eff is not well defined, the model T_eff refers to the temperature at a reference radius, R_⋆, such that L=4R_⋆²σT_eff⁴, where τ₂₀₀₀(R_⋆) = 1. We include the effects of statistical (random) velocity fields using a Gaussian broadening velocity of ξ=2 km s⁻¹ in addition to pressure broadening. All models assume solar abundances (Anders & Grevesse 1989). Further details on the model physics, model parameters, boundary conditions and computational information can be found in AHSB.

We do not consider here the effects of stellar winds, rotation, X-ray heating or magnetic fields. Our results from AHSB suggest that these processes are not required to produce the level of the EUV continuum and the magnitude of ionizing flux for the stars we discuss here. Model atmosphere calculations that are intended to reproduce accurately the detailed line spectrum of β CMa, particularly in the EUV (Cassinelli et al. 1996), will have to model in considerable detail a line blanketed dynamic atmosphere with possible heating from shocks caused by wind instabilities. Our experience has been that, as a first step, the observed continuum spectral energy distribution of β CMa longward of 500 Å can be represented by an extended hydrostatic model.

Models A–D are models for the individual stars β CMa, αVir, e CMa and η UMa. These models are listed in Table 1. In addition, a set of LTE models with effective temperatures from 15 000 to 33 000 K and gravities between logg=3.0 and 6.0 were computed for both PP and spherical geometry. We have computed three classes of models, which are described below:

• (i)

  Models that treat in LTE approximately 2×10⁶ of the most important blanketing lines from 39 elements with up to 26 ionization stages. Model D and the LTE grid models are of this type.

• (ii)

  More sophisticated models that treat in NLTE the ∼12 000 strongest lines from the ions H I, He I–ii, C i–iv, N i–vi and O i–vi in addition to the millions of weaker LTE background lines. Models A and B are of this type.

• (iii)

  More complete NLTE models that, in addition to the ions above, include in detailed NLTE the ∼28 000 strongest lines from the ions Ca II, Mg II and Fe I–III. By number, the vast majority of lines are still treated in LTE; however, the bulk of the total opacity is computed in detailed NLTE. Model C is of this type.

Table 1.

Models. Name(s) Type T_eff (K) log gξ (km s⁻¹) Geometry Comment

We note that the structure in some of these models is only approximate since we compute the pressure structure using only the Newtonian gravity. We compute the pressure gradient using  

where g_rad is the radiative acceleration. For consistency, this is done for models B, C and D, plus all the LTE grid models. As such, the logg parameter listed for these models specifies the (constant) Newtonian surface gravity, not the total effective surface gravity, which would be slightly lower. Observationally, one can at best determine the effective gravity and cannot easily separate g from g_rad. Only for Model A is the pressure structure computed using the full radiation pressure. We find that computing the structure using the effective gravity leads to slightly warmer temperatures in the EUV and IR continuum forming region.

We find in our higher-temperature, lower-gravity models that the local radiative acceleration can exceed the local surface gravity [g_rad(r)>g(r)] at certain depths in the atmosphere. Models with T_eff> 25 kK and logg < 4.0 show some degree of gravity inversion in the outermost layers (τ₂₀₀₀ < 10⁻⁶); however, our tests show that these layers have a negligible effect on the temperature structure (see AHSB). We find that Models A–D all have low enough effective temperatures and high enough surface gravities to be hydrostatically stable at depth, yet the radiative accelerations are non-negligible in some of these models. For example, Model A, with a surface gravity of logg=3.5, has an effective gravity of log(g_eff)=3.3 at τ₂₀₀₀ = 1. The LTE B giant grid models with T_eff> 23 kK and logg=3.0 have g_rad>g in the outermost layers and in layers below τ₂₀₀₀ = 1. We are currently extending our study of the limits of hydrostatic stability in early B giants and are working on the integration of dynamical stellar wind models into our current models.

3 Model Comparisons to Spectrophotometric Data

Here we compare the theoretical spectral energy distribution of β CMa with EUV, UV, optical and IR spectrophotometric data. We emphasize here that the data have not been normalized—all comparisons are made based on absolute fluxes.

The observational data used for comparison with the synthetic spectra are from the following sources: The EUVE data are from Cassinelli et al. (1996). The archival UV data are from the Orbiting Astronomical Observatory 2 (OAO-2) (Code & Meade 1979) and the International Ultraviolet Explorer (IUE). The OAO-2 data were obtained from the on-line archival catalogues of the Astronomical Data Center at Goddard Space Flight Center (GSFC). The IUE data were obtained from the National Space Science Data Center at GSFC. The Hipparcos parallax measurements were obtained from the on-line catalogue established by Centre de Données astronomiques de Strasbourg (CDS). With the exception of the long-wavelength primary (LWP) camera high-dispersion spectra, which were processed using IUESIPS, the IUE data were all processed with the NEWSIPS software. Optical spectrophotometric data for β CMa are from Bastiaansen (1992). The optical spectrophotometric data are placed on an absolute scale by adopting the Hayes (1985) absolute calibration of α Lyr: 4.65×10⁻⁹ erg cm⁻² s⁻¹Å⁻¹ at 5000 Å. Infrared JHKL photometric data are from Bouchet, Manfroid & Schmider (1991) together with the absolute calibration of Bersanelli, Bouchet & Falomo (1991). Far-infrared fluxes are from IRAS (1988). The IRAS fluxes were colour-corrected following Cassinelli et al. (1995). We have applied the colour corrections of α CMa (Cohen et al. 1992) to the β CMa IRAS non-colour-corrected fluxes assuming any differences in the corrections for the two stars are negligible.

A synthetic spectrum from Model A is compared with the observed spectrum of β CMa from 500 Å to 25 μm in Fig. 1. The effective temperature of Model A, T_eff=24 000 K, is in good agreement with the measured value from Smalley & Dworetsky (1995) (T_eff=24 020±1150&emsp13; K) and is just below the value from Code et al. (1976) (T_eff=25 180±1130 K). We find the best agreement between the synthetic spectrum from Model A and the observed spectrophotometry for an angular diameter of θ_LD=0.565 mas, just above the value from Hanbury Brown, Davis & Allen (1974) (θ_LD=0.52±0.03 mas). We compare our model spectrum for β CMa with the EUV, UV/optical and IR spectrophotometry below.

Figure 1.

The continuous energy distribution of β CMa from 500 Å to 25 μm compared with the 24 000 K NLTE model (Model A) and an atlas9 23 250 K model. The EUVE data are corrected adopting a neutral hydrogen column density of 2×10¹⁸ cm⁻². The observed spectrum below 504 Å is not a realistic stellar continuum because of attenuation by the interstellar He⁰ column, which is not corrected for here.

3.1 EUV data

A synthetic spectrum from Model A is compared to the observed EUV spectrum of β CMa in Fig. 2. The EUV spectrum of Model A is tabulated in Table 2. The synthetic spectrum is scaled to match the observed optical and near-IR spectrophotometry by adopting an angular diameter of θ_LD= 0.565 mas. A plane-parallel LTE atlas9 model (T_eff=23250 K, logg=3.4, ξ = 2 km s⁻¹, θ_LD= 0.59 mas) (Kurucz 1992), the previously best-fitting model to the optical and UV data (Cassinelli et al. 1996), is shown for comparison.

Figure 2.

The EUVE LW spectrum of β CMa from 500 to 700 Å compared with the 24 000 K NLTE model (Model A) and an atlas9 23 250 K model. The EUVE data are corrected for the neutral interstellar column adopting a neutral hydrogen column density of 2×10¹⁸ cm⁻². The observed spectrum below 504 Å is not corrected for the attenuation by the interstellar He⁰ column and is probably scattered light within the EUVE spectrograph. The error bar represents the EUVE spectrum absolute flux calibration uncertainty.

Table 2.

Model A EUV surface flux. λ (Å) Log F_λ^aλ (Å) Log F_λ^a

The EUVE absolute calibration uncertainty, which awaits a complete analysis of the in-orbit calibration, is estimated to be 25 per cent, while the relative uncertainty is better than 10 per cent (Vallerga & Welsh 1996). We correct the EUVE data for transmission through the neutral hydrogen interstellar column based on cross-sections given by Rumph, Bowyer & Vennes (1994) and Morrison & MacCammon (1983), using the the on-line ISM Transmission Tool available at the Center for Extreme Ultraviolet Astronomy. We do not correct the data for the interstellar He⁰ column (see below).

We find that a neutral hydrogen column density of N(H⁰) = 2×10¹⁸ cm⁻² provides the best match between the observed and synthetic continua, consistent with the column density of N(H⁰) =2×10¹⁸ cm⁻² from Gry, York & Vidal-Madjar (1985). We find that, when the EUVE data are corrected using neutral hydrogen column densities less than 2×10¹⁸ cm⁻², the observed spectrum has an unrealistic shape, in agreement with Cassinelli et al. (1996).

Our model spectra closely reproduce the observed H I Lyman continuum. The flux level of the Lyman continuum longward of 504 Å, predicted in the model spectra, is sensitive to the model surface gravity at the 20 per cent level, equivalent to the level of uncertainty in the absolute calibration of the EUVE spectrum. Therefore we cannot definitively constrain the surface gravity from this comparison. The best match to the observed Lyman continuum is provided by logg=3.5. We discuss the fundamental stellar parameters for β CMa in Section 4.

It is likely that the stellar He I continuum was not detected by EUVE, but instead the observed signal below the He I 504 Å edge is the result of scattered light from longer wavelengths within the EUVE spectrograph (Cassinelli et al. 1996). The uncertainty in logg and its effect on the predicted He I continuum make a derivation of a lower limit on the interstellar neutral helium column uncertain. Comparing the observed flux below the 504 Å edge to the flux predicted by the best-fitting Model A, we find a lower limit on the interstellar neutral helium column towards β CMa of N(He⁰) =6×10¹⁷ cm⁻²; however, since the predicted level of the He I continuum decreases with increasing gravity, the lower limit is further reduced for higher model gravities.

3.2 Ultraviolet and optical data

The β CMa OAO-2 data, IUE data and optical spectrophotometry from Bastiaansen (1992) are compared with Model A in Fig. 3. Models with Δlogg = ±0.1 differ by less than 3 per cent in absolute flux in this region. We plot the data and synthetic spectrum as λ versus λ³F_λ to accentuate small differences in the slope and absolute levels of these spectral energy distributions in the ultraviolet region. To the OAO-2 data we have applied the correction of Bohlin & Holm (1984), which helps to resolve a discrepancy regarding the relative shapes of the OAO-2 data and model spectra. A more detailed discussion can be found in Appendix A.

Figure 3.

A comparison of the IUE SWP data, corrected OAO-2 data and optical spectrophotometry of β CMa to the 24 000 K NLTE model (Model A). The IUE high-dispersion spectrum has been rebinned to 5- Å resolution.

The upturn in the OAO-2 data longward of 2700 Å may be explained by the fact that the longer-wavelength OAO-2 data were calibrated separately from the shorter-wavelength data and normalized to overlapping ground-based data (Strongylis & Bohlin 1979). The optical spectrophotometry from both Breger (1976) and Bastiaansen (1992) placed on the absolute scale from Hayes (1985) are in very good agreement with the model flux.

The extremely low column density, N(H⁰) ≃2×10¹⁸ cm⁻², towards β CMa produces negligible dust extinction in the ultraviolet and visible portions of the spectrum. Consequently, no such corrections were applied to these data. No low-dispersion IUE data exist for β CMa as a result of its brightness. Of the high-dispersion data, only three long-wavelength (LW) exposures (LWR 4531, LWR 4532 and LWP 15384) were taken, all with the small aperture. As a result the absolute flux of β CMa in the LW range is not constrained by IUE. There are 25 SWP large-aperture exposures, three of which are not overexposed (SWP 40453, SWP 42724 and SWP 42725). There are 21 SWP small-aperture exposures, three of which are overexposed. We compare the well-exposed IUE spectrum SWP 42725 with Model A in Fig. 4. The IUE spectrum LWP 15384 is scaled to the model flux level since an absolute calibration is not possible in the small aperture. The relative shapes of Model A and the IUE data agree very well. The IUE LWP data and uncorrected OAO-2 data longward of 2000 Å have different relative shapes. We suggest that this peculiarity can be accounted for by applying the correction of Bohlin & Holm (1984) (see Appendix A).

Figure 4.

IUE high-dispersion data compared to 24 000 K NLTE Model A. The small-aperture LWP camera data (not absolutely flux-calibrated) have been multiplied by a factor of 2.6 to match the model flux. The IUE data have been binned to 1 Å to match the synthetic spectrum resolution.

3.3 Infrared data

A synthetic spectrum from Model A is compared with the JHKL band fluxes and IRAS 12 and 25 μm colour-corrected fluxes in Fig. 5. We find that there is no significant infrared excess between Model A and the infrared data. The model spectral energy distribution is just consistent within the uncertainties of the absolute IR fluxes for a stellar angular diameter of θ_LD= 0.565 mas. Uncertainties in the JHKL photometry are dominated by the uncertainty in the absolute calibration, about 5 per cent (Bersanelli et al. 1991). IRAS relative flux uncertainties are 4 and 9 per cent for the 12 and 25 μm bands respectively. The uncertainty in the absolute calibration is harder to judge, but by using the colour corrections of α CMa, we hope to avoid the uncertainties in the IRAS zero-point flux calibration and absolute colour corrections. Nevertheless, the colour corrections from IRAS (1988) for a pure Rayleigh-Jeans spectral energy distribution are smaller than those for α CMa, by 4 and 10 per cent in bands 1 and 2 respectively, which, if correct, yield correspondingly larger fluxes at 12 and 25 μm. Ultimately, the IRAS monochromatic fluxes are only as precise as our a priori understanding of the intrinsic IR spectral energy distribution of β CMa, but these absolute fluxes are unlikely to be uncertain by more than 20 per cent. A lower angular diameter and systematically higher IRAS fluxes would lead to a significant IR excess; however, we find with our selected colour corrections and our best-fitting model, which is constrained to match the observed continuum from the EUV to the IR, that β CMa shows no significant IR excess.

Figure 5.

Absolute monochromatic infrared fluxes in the JHKL bands and IRAS bands 1 and 2. The JHKL flux error bars represent the absolute calibration uncertainty of ±5 per cent (Bersanelli et al. 1991), the dominant source of error. The 12 and 25 μm flux errors are ±4 and ±9 per cent respectively (IRAS 1988).

4 Fundamental Stellar Parameters

The accurate distances to nearby hot, luminous stars from Hipparcos coupled with the fundamental effective temperatures and angular diameters from Code et al. (1976) and Smalley & Dworetsky (1995) provide direct luminosities and error boxes on a theoretical Hertzsprung-Russell (HR) diagram. The error boxes for β CMa, e CMa and αVir are plotted with the evolutionary tracks given by Schaller et al. (1992) in Fig. 6.

Figure 6.

The symbols, a square (for β CMa), a diamond (for e CMa) and a triangle (for αVir), are the best-fitting positions, based on the synthetic spectra comparisons to the spectrophotometry, for these stars on the theoretical HR diagram. The best-fitting positions adopt the mean Hipparcos parallax. A pair of observational error boxes, each connected by a dotted line, is plotted near each symbol. The size of each error box is calculated from the 1σ measurement errors in the total flux and the angular diameter, and the standard error in the Hipparcos parallax. Smalley & Dworetsky (1995) (lower box of each pair) find a systematically lower fundamental effective temperature for each of the three stars compared with Code et al. (1976) (upper box of each pair). Also plotted are the evolutionary tracks given by Schaller et al. (1992).

We have used these bolometric fluxes measured above the Earth's atmosphere, angular diameters and trigonometric parallaxes with their associated 1σ errors to calculate the values and propagated errors (added in quadrature) in the effective temperature T_eff, the stellar radius R and luminosity L. The mean effective temperatures and luminosities from Smalley & Dworetsky (1995) are consistently smaller than those from Code et al. (1976). With the exception of e CMa, the effective temperatures derived from the model fits are in very good agreement with the mean values for T_eff from Smalley & Dworetsky.

Table 3 lists our best estimates for the radii, luminosities and ionizing fluxes for β CMa, αVir and e CMa (from AHSB) based on the comparison of our best model fits and the Hipparcos parallaxes. The uncertainties presented with these fundamental parameters represent only the uncertainty in the Hipparcos parallax since the angular diameter and effective temperature are fixed by the model fits. These results confirm the previously understood similarities and distinctions between β CMa and e CMa. Both stars have essentially the same bolometric luminosity; however, β CMa is the hotter and more compact star.

Table 3.

Stellar parameters. Star T_eff (K) θ_LD(mas) d (pc) Radius (R) log(L/L_⊙) log(Q₀)^alog(Q₁)^b

The location of the error box of β CMa on the HR diagram relative to the evolutionary tracks puts the mass of β CMa at ∼13 M. This mass requires a larger gravity and/or a larger angular diameter than previously established in the literature: logg=3.4±0.15 (Drew et al. 1994) and θ_LD=0.52±0.03 mas (Hanbury Brown et al. 1974), since these values yield a mass of only 6.7±2.8 M. A gravity of logg=3.4 gives a reasonable mass when combined with the previously assumed distance of 206 pc (Bohlin, Savage & Drake 1978); however, the Hipparcos distance of 157±17 pc is 25 per cent closer. Using the Hipparcos distance lowers the measured radius and therefore the mass. The location of β CMa on the HR diagram favours a gravity of logg=3.5±0.1 based on the evolutionary track mass, angular diameter and distance. This is consistent with the gravity of logg=3.6±0.1 from the Hβ profile fit of Smalley & Dworetsky (1995). Additionally, our best-fitting value for the angular diameter, θ_LD=0.565 mas, suggests that the measured angular diameter may be too small.

The model fit to the observed spectral energy distribution of β CMa is not very sensitive to logg, and therefore we cannot establish a mass independent from the evolutionary tracks. The sensitivity of the absolute flux level in the Balmer continuum to changes in logg of 0.2 dex is ∼3 per cent. As such, the Balmer continuum does not constrain the gravity. The absolute flux level of the Lyman continuum is more sensitive to the same change in surface gravity, ∼20 per cent, but, not to a degree in excess of the current absolute calibration uncertainty in the EUV spectrum.

For αVir, the dynamical mass of 10.8±1.3 M (Popper 1980), the spectroscopic mass of 9.2±1.5 M from the distance, angular diameter and logg, and the evolutionary mass of ∼11 M from the HR diagram are basically consistent. The slightly larger angular diameter predicted by our model fit, relative to the measured value of θ_LD=0.87±0.04 mas (Hanbury Brown et al. 1974), would bring the spectroscopic mass into better agreement with the dynamical and evolutionary masses. Our best model for αVir (see Fig. A2) is consistent with Smalley & Dworetsky (1995) who compute a lower effective temperature than Code et al. (1976). A model for the H II region of αVir using the ionizing spectrum from Model B is being developed (Sankrit & Aufdenberg, in preparation).

Figure A2.

A comparison of the IUE low-dispersion data, corrected OAO-2 data and optical spectrophotometry of αVir with the 23 070 K NLTE model (Model B). The IUE data in the ranges 1215–1400 Å, 1800–1850 Å and 2485–2940 Å are from overexposed regions on the SWP and LWP cameras.

5 Effects Of Sphericity on Early B Star Lyman Continua

Here we systematically explore the sensitivity of the Lyman continuum to the model atmosphere geometry. We compare spherical and PP models over a range of effective temperatures and surface gravities from a grid of PHOENIX LTE fully metal line blanketed model atmospheres. NLTE computations, which require significantly larger amounts of computer time, are required for accurate predictions of ionizing fluxes, but in order to study the first-order effects of sphericity over a large parameter space in effective temperature and surface gravity we computed a grid of LTE models. Between effective temperatures 15 000 and 33 000 K and surface gravities below logg=3.5, we find to first order that the model geometry has a greater impact on Lyman continuum flux than NLTE effects.

It is reasonable to expect that, as the stellar surface gravity increases and the atmosphere becomes less extended, the approximation of a PP geometry will become more valid and the spherical and PP solutions should come into agreement. We have computed a grid of models, for both PP and spherical geometries, over a range of surface gravities from logg=3.0 to 6.0 at a single effective temperature of T_eff=21 kK. Although surface gravities in excess of logg=4.0 are not applicable to B giants, these models confirm that our model atmosphere solutions for the two geometries indeed converge at high surface gravities. Fig. 7 shows that at low surface gravities (logg=3.0) the spherical models produce over twice as much hydrogen ionizing flux compared to the PP models. The spherical-to-PP ionizing flux ratio approaches unity with increasing surface gravity. For gravities above logg=4.5 the spherical and PP model ionizing flux predictions differ by less than 6 per cent.

Figure 7.

The ratio of the number of the hydrogen ionizing photons predicted by spherical models to the photons predicted by plane-parallel models as a function of surface gravity at T_eff=21 000 K. At lower gravities, the spherical effects become important and the spherical models predict stronger Lyman continua compared with the plane-parallel models.

Fig. 8 compares four model temperature structures from the PP and spherical models at logg=3.0 and 6.0. At logg=3.0 there is a clear separation between the PP and spherical temperature structures; the spherical model temperature is warmer beginning near a column mass of 1 g cm⁻². This warmer temperature structure has important consequences for the strength of the Lyman continuum. The H I continuum becomes optically thin (τ₉₁₂≃1) at a column mass of 10⁻² g cm⁻², where the spherical and PP temperature structures differ by 1000 K. As a result, the Lyman fluxes produced by the two models are quite different. In contrast, the much less geometrically extended logg=6.0 models show nearly identical temperature structures. Clearly, a PP geometry is a good approximation at logg=6.0; however, the spherical model temperature structure shows significant departures from the PP model at surface gravities below logg=3.5.

Figure 8.

Comparison of model temperature structures for spherical and plane-parallel (PP) geometries at T_eff=21 000 K for surface gravities logg=3.0 and 6.0. At high surface gravity the spherical and PP temperature structures are basically identical, while at low surface gravity the spherical and PP temperature structures differ by as much as 1000 K at depths corresponding to the formation of the EUV continuum (depths between 10⁻³ and 10⁻⁵ g cm⁻² in column mass).

The ratio of hydrogen ionizing flux predicted by the spherical models to that predicted by the PP models is plotted as a function of effective temperature in Fig. 9. At logg=4.0, the hydrogen ionizing flux ratio is near unity and varies little with effective temperature. In contrast, the logg=3.0 grid shows a pronounced variation in the q₀ ratio, which reaches a maximum of 2.5 at T_eff=25 kK, then falls steeply to 1.1 at T_eff=33 kK. The logg=3.5 grid shows a similar variation, but less pronounced. The differences between the PP and spherical model Lyman fluxes are concentrated between T_eff=18 and 28 kK and are significant below logg=3.5. The limited range of the geometrical effects on the Lyman continuum can be understood by examining the temperature structures of the models.

Figure 9.

The ratio of the number of hydrogen ionizing photons (q₀) predicted by the spherical models to the number predicted by plane-parallel models as a function of effective temperature for surface gravities logg=3.0 and 4.0. The q₀ ratio is more pronounced for the lower surface gravity, more extended atmospheres, and largest between T_eff=19 000 and 28 000 K.

Fig. 10 compares the spherical and PP model temperature structures at logg=3.5, a surface gravity comparable to that of β CMa. At logg=3.0 the temperature structure differences are larger, up to 2 kK in the plateau region, while at logg=4.0 the temperature structure differences are less pronounced. The q₀ ratio falls to nearly unity above T_eff=30 kK at all three gravities even though the temperature difference is still large in the plateau because, as the effective temperature increases, the depth of the Lyman continuum formation is pushed to greater depths (to larger column masses), where the temperature difference between the spherical and PP temperature structures is small.

Figure 10.

Comparison of model temperature structures for spherical and plane-parallel (PP) geometries between effective temperatures of 15 and 31 kK with a surface gravity of logg=3.5. Spherical and PP temperature structures bifurcate above T_eff=21 kK in the plateau region for column masses less than 1 g cm⁻². ‘X’ marks the depth and temperature at which the optical depth just shortward of the Lyman edge is unity (τ₉₁₁ = 1.0) for each model.

Therefore, we find that the greatest differences between the spherical and plane-parallel EUV flux predictions occur over a range in effective temperature at surface gravities below logg=3.5 where the formation depth of the Lyman continuum coincides with depths where the difference between the spherical and PP temperature structures is largest. With this understanding, in retrospect it is not surprising that e CMa and β CMa should show EUV excesses relative to the predictions of PP line blanketed model atmospheres since these stars fall into a range in effective temperature where the predicted Lyman continuum is quite sensitive to model geometry at low surface gravities. In addition, the smaller EUV excess reported for β CMa relative to e CMa is explained by the larger surface gravity of β CMa, where spherical effects, while still significant, are not as important.

6 Comparison of Model Ionizing Fluxes

The computation of the LTE model grid for the study of the effects of sphericity on the EUV continuum of early B stars also allows us to compare the ionizing fluxes from this grid with previously published values. More accurate predictions of ionizing fluxes require grids of NLTE models.

From the LTE spherical models we have calculated hydrogen ionizing fluxes, q₀ and Q₀, where  

These absolute ionizing fluxes from PHOENIX are compared with the model predictions from Panagia (1973) and atlas9 in Fig. 11. Stellar radii from the evolutionary tracks in Schaller et al. (1992) are used for the calculation of absolute ionizing fluxes for the logg=3.5 and 4.0 model grids, which correspond to luminosity classes II, III and IV, V respectively.

Figure 11.

Comparison of LTE model hydrogen ionizing fluxes from ATLAS9, Panagia (1973) and the PHOENIX spherical grids as a function of effective temperature and luminosity class. The temperature calibrations are provided at the top.

The atlas9 Q₀ values are systematically lower, by more than a factor of 2 below 25 kK. We attribute these differences to the warmer temperature structures present in the spherical models; however, the result that the PHOENIX and atlas9 values continue to diverge below T_eff=20 kK is not predicted by our plane-parallel versus spherical comparisons. The disagreement with the Panagia (1973) values at logg=4.0 is the result of the different stellar radii adopted, as the q₀ values are nearly identical. The radii from Schaller et al. (1992) are in better agreement with the observed radii of main-sequence B stars in eclipsing binaries (Andersen 1991). The Panagia values are from the LTE plane-parallel models of Morton and collaborators, which include ∼100 blanketing lines from H I Lyman series, C iii, N ii–iii, Cl iii–iv, Si iii and Ar ii.

The temperature calibration of Morton & Adams (1968) that is used by Panagia (1973) differs substantially from the calibration of Underhill (Underhill & Doazan 1982, their table 3.5). Care should be taken when choosing the ionizing flux based solely on the spectral type. For example, the spectral type B1V is 4230 K warmer on the Underhill scale, which is a factor of 16 higher in Q₀, compared to B1V on the Morton & Adams scale.

7 Summary

We find a good match between the model spectrum for β CMa and the EUV, UV, optical and IR spectrophotometry. This model com- putes NLTE line opacities for ∼12 000 of the strongest atomic lines in addition to the over 2×10⁶ weak LTE background lines. The important improvement over previous models is a combination of spherical geometry and line blanketing, which yields a higher EUV flux. Hipparcos parallaxes coupled with our model fits provide radii, luminosities and ionizing fluxes for β CMa, e CMa and αVir, which are important for modelling the ionization state of the local interstellar medium and the diffuse H II region of αVir.

We find that for surface gravities below logg=3.5, model ionizing fluxes are quite sensitive to model geometry over a range of effective temperatures T_eff=18–28 kK. This provides an explanation for relative reported excesses between the observed EUV fluxes from β CMa and ∈ CMa and plane-parallel model atmosphere predictions.

From our LTE model atmosphere grid of early B stars, we find hydrogen ionizing fluxes in reasonable agreement with the values presented by Panagia (1973) when the different stellar radii adopted are taken into account. We find significantly larger ionizing fluxes compared with the predictions of atlas9 models below T_eff=25 kK. These results are consistent with a larger role for early B stars in the ionization of warm interstellar medium as suggested by observations.

Acknowledgments

We thank D. H. Cohen for providing us with the reduced EUVE data, and S. Shore, R. Sankrit, S. Starrfield, S. Pistinner, D. Eichler and I. Hubeny for dicussions and useful comments on the draft. In addition we thank our anonymous referee for helpful comments. This work made use of the SIMBAD database, Strasbourg, France. JPA acknowledges support from an ASU NASA Space Grant Fellowship. This work was supported in part by NASA ATP grant NAG 5-3018, LTSA grant NAG 5-3619 and BSF grant 1802504 to the University of Georgia; by NASA LTSA grants NAGW 4510 and NAGW 2628, NASA ATP grant NAG 5-3067 and NSF grant AST-9417057 to Arizona State University; and NSF grant AST-9417242, NASA grant NAG5-3505 and an IBM SUR grant to the University of Oklahoma. Some of the calculations presented in this paper were performed on the IBM SP2 of the UGA UCNS, at the San Diego Supercomputer Center (SDSC) and the Cornell Theory Center (CTC), with support from the National Science Foundation, and at the NERSC with support from the DoE. We thank all these institutions for a generous allocation of computer time.

Appendix

Appendix A

In Aufdenberg et al. (1998b), it was shown that the OAO-2 spectrophotometric data of e CMa longward of 2000 Å were not in agreement with the Goddard High Resolution Spectrograph (GHRS) data and model predictions. The OAO-2 flux in this region is ∼30 per cent lower than the GHRS flux. A very similar discrepancy between OAO-2 data and model atmosphere predictions for β CMa was found by Cassinelli et al. (1996) who suggested that this may be the result of a lack of opacity in the model atmospheres in this wavelength region. We suggest an alternative reason for this discrepancy.

In Fig. A1 we show a comparison between our LTE model D for η UMa (T_eff=17 000 K, logg=4.25, θ_LD=0.89 mas), the uncorrected OAO-2 spectrum of η UMa, and the UV energy distribution of η UMa from Bohlin et al. (1980) as compiled in Underhill & Doazan (1982). These comparisons use OAO-2 data that do not include the wavelength-dependent correction of Bohlin & Holm (1984), which is based on the choice of η UMa (HR 5191, HD 120315) as a fundamental UV standard star. In the UV, model D is in agreement with the observed energy distribution of Bohlin & Holm; however, the OAO-2 flux is higher between 1280 and 2040 Å, peaking at ∼30 per cent higher near 1550 Å. Bohlin & Holm correct for this OAO-2 flux excess. Applying this same correction to the OAO-2 data of β CMa improves the relative agreement between Model A and the OAO-2 data.

Figure A1.

Model D for UV standard η UMa compared with the observed energy distribution from Bohlin et al. (1980) and OAO-2.

With no correction, the match between the β CMa OAO-2 data and Model A is quite good below 2000 Å, but between 2000 and 2700 Å the model flux is systematically ∼20 per cent higher. With the correction, the agreement between the relative shapes of the model and OAO-2 energy distribution is much improved. Differences between Model A and the corrected OAO-2 data appear to be indicative of a problem with the zero-point in absolute calibration rather than an opacity effect in the model atmosphere. Here we note that the optical spectrophotometric flux for β CMa from Burnashev (1985) are systematically ∼15 per cent higher than the fluxes from Breger (1976) and Bastiaansen (1992).

Furthermore, Fig. A2 shows the comparison of the spectrophotometic data for αVir with model B, which supports the correction of Bohlin & Holm (1984). Optical spectrophotometric data for αVir are from Davis & Webb (1974) compiled by Breger (1976). A synthetic spectrum of the B1.5IV–V primary in the αVir system (HD 116658, HR 5056) is in very good agreement with the IUE, corrected OAO-2 and optical spectrophotometric data from 1200 to 8000 Å (Aufdenberg, Hauschildt & Baron 1998a).