Fundamental parameters of B Supergiants from the BCD System
Key Words.:Stars: early-type; Stars: fundamental parameters; Stars: spectrophotometry
Context:Effective temperatures of early-type supergiants are important to test stellar atmosphere- and internal structure-models of massive and intermediate mass objects at different evolutionary phases. However, these values are more or less discrepant depending on the method used to determine them.
Aims:We aim to obtain a new calibration of the parameter for early-type supergiants as a function of observational quantities that are: a) highly sensitive to the ionization balance in the photosphere and its gas pressure; b) independent of the interstellar extinction; c) as much as possible model-independent.
Methods:The observational quantities that best address our aims are the () parameters of the BCD spectrophotometric system. They describe the energy distribution around the Balmer discontinuity, which is highly sensitive to and . We perform a calibration of the () parameters into using effective temperatures derived with the bolometric-flux method for 217 program stars, whose individual uncertainties are on average .
Results:We obtain a new and homogeneous calibration of the BCD () parameters for OB supergiants and revisit the current calibration of the () zone occupied by dwarfs and giants. The final comparison of calculated with obtained values in the calibration show that the latter have total uncertainties, which on average are for all spectral types and luminosity classes.
Conclusions:The effective temperatures of OB supergiants derived in this work agree on average within some 2 000 K with other determinations found in the literature, except those issued from wind-free non-LTE plane-parallel models of stellar atmospheres, which produce effective temperatures that can be overestimated by up to more than 5 000 K near K.
Since the stellar spectra needed to obtain the () parameters are of low resolution, a calibration based on the BCD system is useful to study stars and stellar systems like open clusters, associations or stars in galaxies observed with multi-object spectrographs and/or spectro-imaging devices.
The effective temperatures of early-type stars of luminosity classes V to III are similar whatever the method used to determine them. Among the most commonly used methods are those based on: fits of the observed absolute spectral energy distributions (hereafter ASEDs) with Kurucz (1979) line-blanketed stellar atmosphere models; line profile fittings; Strömgren and Geneva photometric color indices. Using Breger’s spectrophotometric catalogue (Breger, 1976a) for the ASED in the visual range, Morossi & Malagnini (1985) and Malagnini & Morossi (1990) obtained values with internal uncertainties below 5%. The uncertainties are of the order of 10%, either when the TD1 far-UV fluxes and IUE low resolution spectra are used in combination with fluxes from Breger’s catalogue (Malagnini et al., 1983, 1986; Gulati et al., 1989), or when H and He absorption line profiles are fitted with models (Morossi & Crivellari, 1980). In general, values derived using calibrated Strömgren and Geneva photometric indices present low uncertainties (Balona, 1984; Moon & Dworetsky, 1985; Castelli, 1991; Achmad et al., 1993). However Napiwotzki et al. (1993), based on a critical comparison of several calibrations of Strömgren intermediate-band uvby photometric indices, recommended the use of the calibration done by Moon & Dworetsky (1985), corrected for gravity deviations.
As regards B supergiants, Code et al. (1976) and Underhill et al. (1979) derived effective temperatures based on spectrophotometric observations in the far-UV, visible and near-IR spectral regions. Later, effective temperature determinations for these stars were made using the silicon lines in the optical spectral region, either by fitting the line profiles or by measuring their equivalent widths (Becker & Butler, 1990; McErlean et al., 1999; Trundle et al., 2004). Nowadays, methods based on adjustments of He i and He ii line profiles, and/or the line intensity ratio Si iv/Si iii are preferred (Herrero et al., 2002; Repolust et al., 2004; Martins et al., 2005; Crowther et al., 2006; Benaglia et al., 2007; Markova & Puls, 2008; Searle et al., 2008). Recent estimates of the effective temperature of early B-type supergiants have shown that the values obtained with non-LTE blanketed models including winds (hereafter non-LTE BW models) are systematically lower than those derived with models without winds (hereafter wind-free models) (Lefever et al., 2007; Crowther et al., 2006). Differences range roughly from 0 to 6 000 K and they tend to be lower the later the B-sub-spectral type (Markova & Puls, 2008). On the other hand, Morossi & Crivellari (1980) noted that the effective temperatures of B-supergiants derived with photometric methods are in general higher than those obtained spectroscopically. Physical characteristics and phenomena like activity and/or instabilities taking place in the extended atmospheric layers affect more significantly the spectral lines than the continuum spectrum. Thus, the genuine signature due to the carried by the lines could be somewhat blurred.
The calibrations of effective temperatures for B supergiants are of great importance since these stars are in a significant phase of the evolutionary sequence of massive stars. They are also the main contributors to the chemical and dynamical evolution of galaxies. Accurate effective temperatures are then needed to construct HR diagrams and to test the theories of stellar structure and evolution, as well as to estimate the chemical content of the stellar environment. Effective temperatures are also necessary to study the physical processes in the atmosphere, such as non-radial pulsations or stellar winds (radiative forces; changes in ionization). Regarding the stellar winds, the effective temperatures are particularly useful in discussing terminal velocities, mass loss rates, the bi-stability jump, and the wind momentum luminosity relationship (Kudritzki et al., 2003; Crowther et al., 2006; Markova & Puls, 2008).
As a consequence of the large discrepancies found in the estimates of B supergiants, the current temperature scale is being revisited (Markova & Puls, 2008; Searle et al., 2008). In this context, we present an independent and homogeneous temperature calibration for B-type dwarfs to supergiants, based on the use of the BCD spectrophotometric system (Barbier & Chalonge, 1941; Chalonge & Divan, 1952). This method has numerous advantages (see §2), mainly because it is based on : a) measurable quantities that are strongly sensitive to the ionization balance in the stellar atmosphere and to its gas pressure, thus being excellent indicators of and ; b) parameters that describe the visible continuum spectrum, whose atmospheric formation layers are on average deeper than those for spectral lines.
Our first step will be to determine the effective temperatures of our sample of Galactic B-type supergiants. However, to perform a consistent calibration of the BCD parameters for supergiants, we re-determine the calibration into effective temperature of the BCD domain corresponding to B-type dwarfs and giants ( size of the Balmer jump; mean spectral position of the Balmer discontinuity (BD); see further explanations on these parameters in §2 and Appendix §A). We use a large and homogeneous sample of B stars observed in the BCD system, and newly-derived effective temperatures for all of them, based on the bolometric-flux method (hereafter BFM). We leave for another contribution the discussion of BCD calibrations related to , visual and bolometric absolute magnitudes.
The present paper is organized as follows: In §2 we briefly describe the BCD spectrophotometric system and the advantages of its use. In §3 we present the BFM on which the determinations of the stellar effective temperature and angular diameter () are based. Observations and the values determined with the BFM are presented in §4. The uncertainties of the effective temperatures and angular diameters obtained with the BFM are discussed in §5. Comparisons of our and determinations with those obtained by other authors are given in §6. In §7, we present the empirical temperature calibration curves and discuss the accuracy of the values obtained. A discussion and global conclusions are presented in §8 and §9, respectively.
2 The BCD system
The Paris spectrophotometric classification system of stellar spectra, best known as BCD (Barbier-Chalonge-Divan), was defined by Barbier & Chalonge (1941) and Chalonge & Divan (1952). The original presentation of this system is given in French; explanations in English can be found in Dufay (1964); Underhill (1966); Underhill & Doazan (1982); Divan (1992). A short overview of the system is given in Appendix §A. The BCD system is based on four measurable quantities in the continuum spectrum around the BD: D, the Balmer jump given in dex and determined at 3700 Å; , the mean spectral position of the BD, usually given as the difference -3700 Å; , the gradient of the Balmer energy distribution in the near-UV from 3100 to 3700 Å, given in m; , the gradient of the Paschen energy distribution in the wavelength interval 4000-6200 Å given in m. The sole BCD parameters that are relevant to the present work are: , which is a strong function of , and that is very sensitive to .
The use of the (,) pair to determine the spectral classification and the stellar fundamental parameters presents numerous advantages, not only because the BD is a well visible spectral characteristic for stars ranging from early O to late F spectral types but also because:
a) The parameters (,) are obtained from direct measurement on the stellar continuum energy distribution. This implies that, on average, they are relevant to the physical properties of photospheric layers which are deeper than those described by spectral lines;
b) Each MK (Morgan & Keenan) spectral type-luminosity class (SpT/LC) is represented by wide intervals of and values, which implies high SpT/LC classification resolution: ranges from about 75 Å for dwarfs to -5 Å for supergiants, while D ranges from near 0.0 dex, for the hottest O stars and F9 stars, to about 0.5 dex, for the A3-4 stars;
c) Typical 1 measurement uncertainties affecting and are: 0.02 dex and 2 Å, respectively. Thus, from b) and c) we find that hardly any other classification system has reached such a high resolution, especially concerning the luminosity class;
d) The parameter is independent of the interstellar medium (ISM) extinction, while has a low E(B-V) color excess dependence, roughly = 0.03 E(B-V) dex, which is almost insensitive to the selective absorption ratio . The difference is produced by extrapolation of the Paschen energy distribution from 4000 Å to 3700 Å, which carries the ISM reddening of the Paschen continuum. The low ISM extinction dependence is however of great interest for the study of early-type stars, since they are frequently distant and strongly reddened;
e) Spectra needed to obtain the (,) measurements are of low resolution (8 Å at the BD). This means that numerous faint stars can be observed with short exposure times. Furthermore, the flux calibrations required to derive the BCD parameters rely on spectral reduction techniques which are easy-to-use and of common practice;
f) The BCD system is generally used for ‘normal’ stars, i.e. objects whose atmospheres can be modeled in the framework of hydrostatic and radiative equilibrium approximations. However, since both the photospheric and the circumstellar components of the BD are spectroscopically well separated, it can also be used to study some ‘peculiar’ objects, like: i) Be stars (Divan & Zorec, 1982; Zorec, 1986; Zorec & Briot, 1991; Chauville et al., 2001; Zorec et al., 2005; Vinicius et al., 2006); ii) objects with the B[e] phenomenon (Cidale et al., 2001); iii) chemically peculiar stars (He-W group) (Cidale et al., 2007).
3 The bolometric-flux method (BFM)
To obtain the calibration of the BCD (,) parameters into effective temperature, we could simply adopt for each star the average of all the values found in the literature. Nevertheless, these quantities were obtained with several heterogeneous methods which carry more or less systematic differences on the estimates of . Since most determinations are based on adjustment of the observed ASEDs with theoretical ASEDs, or on fittings of line profiles with models, the properties of the studied stars are implicitly assumed to be in accordance with the physical characteristics of the best fitted stellar model atmosphere. Instead, in the present contribution, we preferred to determine the effective temperature by using the total amount of radiated energy, so that the details of its distribution are of marginal importance and the dependence on models of stellar atmospheres are kept to a minimum. While models of stellar atmospheres produce Balmer jumps which are close to the observed ones, the theoretical parameter may differ somewhat. In fact, depends on the distribution of the emergent radiation fluxes near the limit of the Balmer line series, where the theoretical uncertainties concern the treatment of the non-ideal effects in the hydrogen upper level populations (Rohrmann et al., 2003).
We decided to adopt a single method for all program stars and to estimate their effective temperatures using its definition, where the incidence of the model-dependence is in principle strongly minimized.
By definition, the effective temperature of a star is:
where is the Štefan-Boltzmann constant; is the stellar bolometric radiation flux received at the Earth, corrected for the ISM extinction; is the angular diameter of the star. If the radiation field coming from the stellar interior were the sole energy source in the atmosphere, the effective temperature deduced from the bolometric flux would be the same as that derived from the analysis of stellar spectra with model atmospheres in radiative and hydrostatic equilibrium. If the bolometric flux and the angular diameter were issued entirely from observations, the deduced from Eq. (1) could then also be considered a genuine observational parameter. The BFM was applied in this way by Code et al. (1976) using stellar fluxes observed with the OAO-2 satellite and the angular diameters determined interferometrically (Hanbury Brown et al., 1974). Unfortunately, this could not happen in our case, as we do not have observed fluxes over the entire spectrum, nor do we have angular diameters measured for all program stars. In what follows the effective temperature and the angular diameter derived with the ‘bolometric flux method’ are called and , respectively.
Blackwell & Shallis (1977) showed that a stellar angular diameter can be well reproduced with observed fluxes in near-IR and model atmospheres. The monochromatic stellar angular diameter is thus given by:
where is the absolute monochromatic flux received at the Earth corrected for the ISM extinction, is the emitted monochromatic flux at the stellar surface, is the so called ‘astrophysical’ flux predicted by a model atmosphere. To represent we have used the grids of ATLAS9 model atmospheres calculated by Castelli & Kurucz (2003). In wavelengths lying in the Rayleigh-Jeans tail of the energy distribution, not only are the theoretical fluxes nearly independent of model characteristics, but the observed ASEDs also are mildly affected by the ISM extinction. However, they have the inconvenience of being frequently marred by infrared flux excesses of non stellar origin. We decided to calculate Eq. (2) in the red extreme of the Paschen continuum: m. In this spectral region, the theoretical radiative fluxes are still only slightly dependent on the particular model characteristics and the layers where this radiation field is formed have local electron temperatures close to , so that the color temperature of the energy distribution of OB stars in this wavelength interval also approaches closely. While derived with Eq. (1) is insensitive to within the characteristic uncertainties of its determination, the parameter does depend slightly on models. Therefore, in Eq.(2) model atmospheres giving are chosen for gravity parameters = ), where is the H-line index of the uvby Strömgren photometry. The parameters used are from the Hauck & Mermilliod (1975) compilation and the relations used are from Castelli & Kurucz (2006). An extensive use of the BFM was made by Underhill et al. (1979), where unfortunately Eqs. (1) and (2) were iterated only twice, which left the derived effective temperatures strongly correlated with their initial approximate values.
3.1 Effective temperatures of B-type dwarfs to giants
The ASEDs of our program stars, taken from the literature to calculate , are in most cases observed in the wavelength interval ranging from 1200-1300 Å to some in the far-IR. Many times they present intermediate gaps, according to the observational method or the instruments used. On the other hand, the angular diameter has been measured only for a few bright stars. Nevertheless, as in Eq. (1) depends on , the effective temperature can still be reliably determined even though the unobserved spectral regions are represented using ‘modestly realistic’ model atmospheres, as discussed in §5. The calculation of the effective temperature for dwarfs to giants is based on the following iteration:
1) Adopt initial values of and ;
2) Interpolate the model fluxes in the far-UV and IR spectral regions for the adopted values ;
4) Calculate the bolometric flux as follows:
where and represent, respectively, the contribution of the observed and unobserved spectral regions (extreme-UV, far-UV and IR) to the bolometric flux; and delimit the spectral range of ASEDs actually observed; and have the same meaning as given in Eqs. (1) and (2);
6) Use the new estimates of ( to continue the iteration in step 2).
The iterations were performed until the difference between two consecutive values was smaller than 1 K. Depending on the star, this implies roughly 10 to 30 iterations. From (3) it is obvious that the estimates of the effective temperatures may in principle be more uncertain the higher the value of ; i.e. for the hottest stars. In Table 1 are listed the fractions and for different effective temperatures and gravities. As seen in this table, has a low dependence on .
|For and the temperature is simply the effective temperature written in the first column of the table.|
|Parameters and are values of and calculated with and fluxes , which do not undergo the transforma-|
|tion given by Eq. 6. To calculate them we have used Å and Å. Note that .|
3.2 Effective temperatures of B-type supergiants
The theoretical ASEDs predicted for supergiants by the wind-free plane-parallel model atmospheres with the effective temperatures issued directly from Eq. (1), do not always fit well to the observed ASEDs in the near- and far-UV. Generally, the predicted fluxes are higher than the observed ones in wavelengths Å, while they are lower in the near-UV. For the obtained they yield somewhat larger Balmer discontinuities than those observed. This might be partially due to the plane-parallel approximation of model atmospheres that probably is not suited to the extended atmospheric layers of these stars, to an incomplete treatment of the spectral line formation in such diluted atmospheres, to the omission of the effects produced on the photosphere by the stellar winds (Abbott & Hummer, 1985; Gabler et al., 1989; Smith et al., 2002; Morisset et al., 2004), and also to an insufficient line blocking in the model atmospheres, as discussed by Remie & Lamers (1981). Thus, to obtain the parameter of supergiants, we have slightly modified the use of relations (1) and (2) through the following iteration procedure:
1) Adopt approximate values of and ;
2) Interpolate the model fluxes in the far-UV and IR for the adopted () parameters;
3) Use a least square procedure to search for an enhancement parameter of the line blocking in the Å wavelength interval, and correct the failure of the wind-free model fluxes used to fit the far-UV spectral region. The validity of is then extended to the entire Å spectral region. The empirical method used to calculate and to modify the line-blocking is explained in §3.2.1;
4) Define a new effective temperature, , to account for the lack of energy in the far-UV, produced by the increased absorption induced by and for the redistribution of this energy in the longer wavelengths. The calculation of is explained in §3.2.2;
5) Interpolate the model fluxes in the far-UV and IR for the and parameters;
6) Calculate the angular diameter using Eq. (2) and the IR fluxes dependent on the  pair of fundamental parameters. With the fluxes interpolated in step 5), calculate also the bolometric corrections and as indicated in Eq. (3). The bolometric correction is calculated using the -modified far-UV fluxes defined in §3.2.1 by Eq. (6), where is replaced by :
8) Continue the iteration in step 3) by searching for a new enhancement parameter using the far-UV fluxes interpolated in step 5).
3.2.1 Modification of theoretical fluxes
Since we are not interested here in reproducing detailed energy distributions to fit the observed ones, but rather in estimating the integrated amount of the emitted energy over the whole spectrum, we use an empirical method to minimize the disagreement between the predicted and the observed far-UV energy distributions. However, the method is not able to simulate the consequences due to the presence of winds which become conspicuous in the extreme-UV (Smith et al., 2002). Nevertheless, this fact cannot significantly change the estimate of , as we shall see in §5.1.
Thus, we have proceeded in a similar way as previously attempted by Remie & Lamers (1981) and Zorec & Mercado-Ibanez (1987), i.e. we have modified the blocking degree of spectral lines in the Å region by writing the radiative flux emitted by a star at a given as:
where is the continuum flux for a given set of parameters , and is the line blocking factor. The line blocking factor easily can be calculated using the and fluxes listed by Castelli & Kurucz (2003) (http://kurucz.harvard.edu/grids.html). The enhanced line blocking factor was then calculated by multiplying by a parameter , which we assumed constant over all the wavelength interval Å. The modified theoretical flux at a given is then:
For each star and at each iteration step of its , we looked for the value of that produced the best possible fit between the observed and theoretical fluxes, in the Å wavelength interval. The model fluxes are calculated for a new effective temperature , which is different from , and for a , which change as the iteration of the effective temperature continues.
3.2.2 Effective temperature of models
It can readily be understood that the estimate of with the modified fluxes given by Eq. (6), where in most cases , leads to an effective temperature lower than the nominal value, , of the model used. Thus, when using Eq. (6) in the wavelength range Å, we can recover the bolometric flux initially represented by the model characterized by , by means of a new effective temperature, named hereafter , which is larger than if . Mathematically, this is due to the lack of energy produced by an increased absorption in the far-UV. Physically, this accounts for a redistribution of the excess of absorbed energy in the far- and extreme-UV, towards the near-UV, visible and IR spectral regions produced by the back-warming induced by the enhanced line blocking. To find the relation between and , we write the same bolometric flux, , in terms of both temperatures. On the one hand, it can be calculated using model fluxes dependent on as:
and, on the other hand, the same can be obtained with models that are a function of , as follows:
where is given by the relation (6). Since
where we have not made explicit the dependence on , but have written that in the Å wavelength interval, the unmodified fluxes and modified fluxes , are both calculated for the effective temperature . It is obvious that to have the corresponding at each iteration step of the stellar effective temperature we put and iterate the relation (11). Then, replacing Eq. (6) into , we derive:
where the function is given by:
which can be calculated as a function of and used to derive the required value of by interpolation. The function is given in Table 1, where we also give the values of derived from (12). In this table appears as the entering temperature. Actually, in the calculation of the stellar effective temperatures, we enter relation (12) with and deduce the corresponding value of at each iteration step. At each iteration step of , obtained by means of Eqs. (1) and (2), and calculated with (6), we changed the gravity parameter accordingly using the tables of given by Castelli & Kurucz (2006). For instance, in Fig. 1 we show the results obtained at the final step of the analysis carried out for HD 41117. For this particular star we have obtained K, K, and . All fluxes in Fig.1, observed and modelled, are normalized to a given flux in the visible wavelengths where the angular diameter is calculated. However, in the plot the logarithm of the fluxes is shifted by a constant value. We can also see that for and the model fluxes in the far-UV are higher than the observed ones. In spite of the roughness of our approach, the fluxes from models computed with and improve the fit of the observed energy distribution. It is also seen that the -modified flux is slightly higher in Å than the fluxes calculated with , which leads to a smaller Balmer discontinuity as desired, i.e. dex, dex, while it is dex.
It is important to note that produces a lowering of fluxes in the far- and extreme-UV, while the effect carried by the stellar wind on the photosphere, neglected here, increases the emitted fluxes (see Smith et al. (2002)). As we shall see in §5.1, this increase happens at global flux levels that may have an effect on the estimate of at effective temperatures higher than 20 000 K, as shown in §5.1. In general, the values of obtained with wind-free models can differ significantly from those derived with non-LTE BW models if . The variation of with can be obtained by noting that we can write:
where is given by Eq. (6). We introduce the bolometric corrections and defined as:
which are written in terms of fluxes calculated for that do not undergo the transformation used for in the far- and extreme-UV. Subtracting from (14) and taking into account the definition of written as:
by making use of (13), we readily obtain:
where , and the function are given in Table 1.
In the visible wavelengths used to calculate the stellar angular diameter, the blocking factor is and the enhancement parameter is reduced to . However, the model fluxes employed to obtain in Eq. (2) are for the effective temperature , i.e. . In general, when the effective temperature issued from (1) is higher than for . Figure 2 shows for our sample of supergiants. Thus, even though can sometimes be as high as , we see that the differences in the final value of are less than 300 K, which is much lower than the average expected error affecting the determination of (see §5). Finally, we note that for the stars in common with Remie & Lamers (1981), our values are systematically smaller by than theirs, probably because in the models used here the line blocking effect is more realistic than in the Kurucz (1979) models.
4 The program stars and the observed quantities
4.1 BCD parameters
The program B-type stars, dwarfs to supergiants, are simply those for which both the BCD () parameters and calibrated fluxes from the far-UV to the IR, at least up to m were available. In this work we excluded stars with the Be phenomenon, but included some A0 to A2 type stars in the cold extreme of the hot fold of the BCD (,) diagram. The list of the program stars and their (,) are given in Table 2.
Our sample contains 217 stars with MK luminosity classes from V to Ia. Observations in the BCD system were carried out over different periods. Most of them were observed with the Chalonge spectrograph (Baillet et al., 1973) attached to several telescopes at the Haute Provence Observatory from 1977 to 1987 and at the ESO (La Silla) from 1978 to 1988. More recently, on January 2006, low resolution spectra in the optical wavelength range 3400-5400 Å were obtained at CASLEO (Argentina), with the Boller & Chivens spectrograph. These spectra were wavelength and flux calibrated. Since the parameter depends on the spectral resolution, we selected a resolution of 7 Å at 3760 Å, which is similar to the resolution required in the original BCD system. From the spectra obtained at CASLEO, the BCD ( and ) parameters were directly measured on the spectrograms. We have determined the parameter following the original BCD prescriptions (Chalonge & Divan, 1952). We also controlled that corresponds to the flux level where the higher members of Balmer lines merge. Most stars have been observed many times, so that for each star the adopted () set comes from 2 to about 50 determinations. The uncertainties that characterize these quantities are then dex and Å, respectively.
4.2 ASED data
The ASED data were collected in the CDS astronomical database. The far-UV fluxes used in this work were obtained with the IUE spectra in low resolution mode and the 59 narrow-band fluxes, between 1380 and 2500 Å measured during the S2/68 experiment from the TD1 satellite (Jamar et al., 1976; Macau-Hercot et al., 1978). We have compared the TD1 fluxes with the low resolution IUE spectra calibrated in absolute fluxes, but we have not detected any systematic deviation neither in the far-UV ASED nor in the values of obtained. The TD1 fluxes give a homogeneous far-UV flux data set, for they are laboratory based calibrations.
In the visible and near-IR spectral region, fluxes are from Breger’s catalogue (Breger, 1976a, b) and the 13-color photometry calibrated in absolute fluxes (Johnson & Mitchell, 1975) (hereinafter JM). The normalized 13-color fluxes of the stars in common were compared to Breger (1976a, b) spectrophotometry and to the monochromatic fluxes observed by Tüg (1980). The comparison of JM’s absolute fluxes with those of known flux standards Lyr (HD 172167), 109 Vir (HD 130109) and UMa (HD 120315) given in the Hayes & Latham (1975) system revealed no noticeable differences in the 0.58-0.80 m spectral region, which was chosen in this work to obtain the stellar angular diameter . Nevertheless, there are small differences near the BD that lie within the uncertainties of other calibrations, but they do not affect the estimate of . We also noticed that JM’s fluxes give consistent continuations to the IUE and ANS fluxes (Wesselius et al., 1982).
4.3 Correction for interstellar extinction
The adopted ISM extinction law is from Cardelli et al. (1989) and O’Donnell (1993). The ratio was adopted in this work according to the galactic region, as specified in Gulati et al. (1987) and Gulati et al. (1989). The adopted color excess is the average of several more or less independent determinations based on the following methods: a) UBV photometry with the standard intrinsic colors given by Lang (1992); b) BCD system with the intrinsic gradients taken from the calibration done by Chalonge & Divan (1973) from where we obtain that = 0.55 ; c) depths of the 2200 Å ISM absorption band (Beeckmans & Hubert-Delplace, 1980; Zorec & Briot, 1985); d) profile parameters of the 2200 Å band (Guertler et al., 1982; Friedemann et al., 1983; Friedemann & Roeder, 1987) calibrated in (Moujtahid, 1993); e) diagrams of vs. distance obtained with ‘normal’ stars surrounding the program stars within less than 2. The UBV photometry used is from the CDS compilation and the distances are either from the hipparcos satellite or spectroscopy.
The list of the program stars is given in Table 2: 1) HD number of the star; 2) the MK SpT/LC determined with the BCD system; 3) the parameter in Å; 4) the Balmer discontinuity in dex; 5) the adopted average color excess , in mag; 6) the effective temperature with its estimated uncertainty, given in K; 7) the angular diameter with its estimated uncertainty, given in mas (milliarcseconds); 8) the parameter used to fit the observed far-UV ASED (see §3.2); 9) the read on the new calibration curves (see Fig. 10a).
5 Comments on the uncertainties affecting the and determinations
5.1 Systematic deviations
Hot supergiants have massive winds whose optical depths can be high enough to heat somewhat the photosphere back to the continuum formation region. Thus, both spectral lines and the emitted continuum energy distribution can in principle be modified (Abbott & Hummer, 1985; Gabler et al., 1989; Smith et al., 2002; Morisset et al., 2004). Since the models used in this work are wind-free, doubts can be raised about whether the use of these models introduces systematic effects on the estimate of . As is seen in §6, this question may be of particular interest at 25 000 K. We then compared the absolute energy distribution produced by a non-LTE BW model atmosphere for supergiants (Smith et al., 2002) with that predicted by an LTE wind-free model atmosphere for 25 000 K and 2.95, similar to those used in the present work (Castelli & Kurucz, 2003). This comparison is shown in Fig. 3 where we see [Fig. 3(a)] that the most significant differences appear only in the extreme-UV energy distribution, at Å. In this spectral region the fluxes are between two and three orders of magnitude lower than in slightly longer wavelengths, where the most significant contribution to the value of the the filling factor in relation (3) arises. Although the differences seen in Å can certainly be important for the excitation/ionization of the stellar environment, as we shall see below, they seem to have a limited incidence on the estimate of the stellar bolometric flux on which the estimate of relies.
In Fig. 3(b) we also see that the fluxes produced by both types of models are similar in the visible wavelengths where the angular diameter is calculated. This ensures that does not suffer from the use of plane-parallel model atmospheres either. The same conclusion is reached from Fig. 5 by comparing the obtained values with the measured angular diameters.
Let us now explore what systematic deviation can be expected in our estimates due to the use of fluxes predicted by wind-free models of stellar atmospheres. In what follows, we use the notation and to designate the parameters that should be derived with non-LTE BW model atmospheres. In §6.1 and Fig. 5 it is demonstrated that the angular diameters obtained agree well with the observed ones. This means that they cannot introduce a systematic deviation in the derivation of with Eq. (1). Then, the ratio between and is given by:
where and are the bolometric corrections calculated with wind-free models for different values of the enhancement parameter . The value of is obtained from Eq. (17), while it is reasonable to assume that . Concerning the non-LTE BW model bolometric correction, we ask what extreme-UV flux excess carried by the wind-related effects must exist to explain the underestimations suggested by the comparison shown in Fig. 7 that possibly affect our determinations for supergiants. To this end we simply write:
where the factor mimics the extreme-UV flux excess due to wind-related effects. Using for and the values given in Table 1, we readily obtain the estimates of for a series of imposed underestimations which are displayed in Table 3.
|The estimates are done for|
From the non-LTE BW models published by Smith et al. (2002) we obtain,
for 25 000 K, 2.95 and 32 200 K, 3.14,respectively, where 25 000 K corresponds to the effective temperature where there are strong deviations in the diagram of Fig. 7. Unfortunately Smith et al. (2002) have not made available fluxes for 25 000 K. Then, by extrapolation and approximate calculation with our codes for extended spherical atmospheres (Cruzado et al., 2007), we get the value . The values of derived here are close to those in Table 3 for . Since for the superginats that are in common with those in the category (see §6.2), we have , it means that we may expect our to be systematically smaller by less than 470 K, 480 K and 640 K, for objects whose temperatures are 20 000 K, 25 000 K and 30 000 K, respectively. In addition, for later B sub-spectral types, the underestimates can be even smaller, as it is difficult to believe that the parameters are larger than those quoted above when K. For (see Table 3) differences easily can approach 1 000 K. Nevertheless, there is only one common late type supergiant with (HD 202850, ), for which curiously the difference between our estimate and that in the group is K. We note that to obtain the imposed underestimation when for a given factor , the non-LTE BW models should predict lower extreme-UV fluxes than wind-free models do.
The model atmospheres used in this work to obtain are interpolated here for parameters that were estimated using the uvby photometry. As the index is calibrated mainly for dwarf and giant stars, to see whether its extrapolation induces significant systematic effect for supergiants, we compare in Fig. 4 the adopted in the present work, with the (lines) parameters derived in the literature by detailed fitting of spectral lines with model atmospheres, mainly hydrogen H and H lines (McErlean et al., 1999; Repolust et al., 2004; Martins et al., 2005; Crowther et al., 2006; Benaglia et al., 2007; Markova & Puls, 2008; Searle et al., 2008). From this comparison we can conclude that our estimates of do not deviate either strongly or in a systematic way from those based on spectral lines. Although in the cited works the authors claim that their (lines) are determined with errors ranging from 0.10 to 0.15 dex, they differ among them by 0.10 to 0.29 dex. The comparison in Fig. 4 of our estimates with (lines) shows that most of the points deviate randomly from the first diagonal within dex. This implies that we do not expect that the values used in the present work will induce systematic errors in the and parameters.
In conclusion, we do not see how we can attribute to our BFM method systematic effective temperature deviations attaining 2 000 K, or more than 5 000 K, as observed in Fig. 7.
5.2 Random errors
The values issued from relation (1) and the parameters derived with Eq. (2) have the following sources of error: a) the ISM color excess ; b) the line blocking enhancement parameter ; c) the parameter on which depend the model fluxes used to estimate the angular diameter and the filling factor due to the unobserved spectral region; d) the filling factor used to calculate the bolometric flux in relation (3).
In Appendix §B we explain in detail how we have calculated the uncertainties of and by taking into account the combined effect of all error sources mentioned above. Since the most important uncertainty on and is produced by the error on the ISM color excess estimate , in this section we consider only the effect caused by this parameter as if it were the sole error source. Let us recall that the affects the bolometric flux estimate through in (3) of §3.1, and the monochromatic fluxes on which the calculation of the stellar angular diameter depends. However, from (1) and (2) we find that:
which implies that the effect on the estimate of by an error on is reduced by the error of the monochromatic fluxes entering the calculation of . In Table 4 are given the estimates of the errors produced on and by the uncertainty in the ISM color excess, assuming that it is the only source of error. We note the asymmetric propagation of uncertainties in and due to those in . Since for us , from Table 4 we find that on average the corresponding errors are of 5% in and 2% in .
6 Comparison with other and determinations
6.1 Apparent angular diameters
In spite of the fact that for the wavelengths where is calculated, all models predict roughly the same flux level, the angular diameter is in principle the quantity among those estimated in this work that can be the most strongly model-dependent. To test our angular diameter determinations, we have compared them with those found in the literature. We used the data collected in Pasinetti Fracassini et al. (2001), where we discarded the very old determinations and privileged those determined by interferometry. The comparison is shown in Fig. 5, where we put on the ordinate axis the average values from the data in Pasinetti Fracassini et al. (2001) and the respective error-bars. In this figure we can see that there is no systematic deviation of points from the first bisecting line and that they distribute around it with a fairly uniform dispersion: mas. The uncertainty that can affect the values due to errors in the angular diameter estimates ranges then from 1.5% to 9% as the angular diameter goes from mas to mas.
6.2 Effective temperatures
In order to test the accuracy of the effective temperatures obtained with the BFM method, we compared our values to those found in the literature. The values were gathered according to the method used to determine them:
The temperature is obtained by means of the BFM by Remie & Lamers (1981).
The temperature is computed by means of: a) NLTE line-blanketed model analysis of ionization balances due to He i/ ii, Si iii/ iv and Si ii/ iii on moderate resolution spectra, as done by McErlean et al. (1999) using the code TLUSTY (Hubeny, 1988), or b) LTE line-blanketed model fitting of the optical region and H profile using ATLAS9 model atmospheres as in Adelman et al. (2002).
The temperature is determined from synthetic fits to the optical spectral range of B supergiants employing unified NLTE line- and non-LTE BW extended model atmosphere codes, such as FASTWIND (Puls et al., 2005) or CMFGEN (Hillier et al., 2003), as in Searle et al. (2008); Markova & Puls (2008); Crowther et al. (2006).
For each program star, Table 5 lists our determinations of together with the corresponding uncertainties and, in the following columns, the ( 1,…,6) obtained by other authors with the methods described in items: 1) to 6). When more than one determination of exists for the same star, obtained using similar techniques, we adopted the mean value.
The difficulties in determining can be better shown by separating dwarfs and giants from supergiants. Figure 6 displays the comparison of (stars), (open circles), (open triangles), (diamonds), (open squares), and (plus signs) with our values (abscissa) for dwarfs and giants. From this figure, we can see that the differences in the estimates seem to be related to the absolute value of temperatures rather than to the adopted techniques. For K, all methods produce temperatures within K, while for K, K. We notice, however, that the derived with the BFM agree best with those obtained using line-blanketed LTE model atmospheres (Malagnini et al., 1983; Morossi & Malagnini, 1985; Malagnini & Morossi, 1990).
The same type of comparison, but for supergiants, is shown in Fig. 7 which appears less ordered than the plot for dwarfs and giants. However, the values do deviate strongly and in a systematic way. The values for supergiants are taken only from McErlean et al. (1999), and they were obtained with non-LTE wind-free model atmospheres. The observed deviations form a kind of ‘temperature-step’ rising at K and K, i.e. at temperatures identified as specific to the bi-stability phenomenon (Lamers et al., 1995; Vink et al., 1999; Crowther et al., 2006; Markova & Puls, 2008). This fact may reveal that there can be effects related to the presence of stellar winds which have not been taken into account in their models. Moreover, the noted deviations also reveal a dependence on the method used to determine the effective temperature. The values from McErlean et al. (1999) determined by He ii line profile fits deviate on average from our temperatures by K at abscissa K; the temperatures determined by the Si iii/Si iv ionization balance deviate by K at K; those determined from the Si ii/Si iii ioniation balance deviate by K at K. The values from McErlean et al. (1999) have random errors K. McErlean et al. (1999) pointed out that their effective temperatures could in fact be 10% lower, which means that for the stars lying in the ellipse of Fig. 7 the temperatures are on average 3 000 K too high. Thus, excluding the values, the average dispersions with the remaining sources cited in Fig. 7 are K for K, and K for K.
Stars in the group are from several sources, but their temperatures were derived using the same models, although corresponding to different implementation generations. Crowther et al. (2006) note that the current non-LTE BW model atmospheres lead to lower effective temperatures by 1 000 to 2 000 K, as compared to some earlier determinations in Kudritzki et al. (1999); Crowther et al. (2002); Repolust et al. (2004), and that their latest effective temperature estimates have uncertainties of about 1 000 K. However, three of the stars lying in the ellipse of Fig. 7: HD 30614, HD 37128 and HD 38771, were assigned temperatures that are from 2 300 K to 3 700 K higher than those obtained in the present work. From the discussion in Sect. 5.1 it appears that our method of determining the effective temperature does not introduce systematic deviations larger than 500 K to 700 K as the effective temperature goes from 20 000 K to 30 000 K. We suspect then that the spectral lines used to derive some may still undergo a perturbation that the existing non-LTE BW model atmospheres do not account for entirely. Among the phenomena that may explain the perturbed lines are: non-isothermal multicomponent plasma effects (Springmann & Pauldrach, 1992), shocks and/or the presence of exo-photospheric density clumps. Regarding the latter, it is worth noting that the use of an average continuous stellar wind to mask a possible clumpy environment, where a mass-loaded wind must appear (Hartquist et al., 1986), can lead to misleading conclusions. The radiation transfer effects expected from the resulting average opacity are quite different from those the clumpy environment is able to produce (Boisse, 1990). Furthermore, the actual opacity of the medium is higher than that expected from line diagnostic and the back warming on the photospheric layers could have different characteristics than those expected from winds with regular density and temperature structures.