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Astron. Astrophys. 354, 193-215 (2000)

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4. Spectral analysis using plane-parallel, hydrostatic models

Fig. 5 shows the spectra of all observed stars between 4000 and 5000 Å, whereas Fig. 6 shows the spectra around [FORMULA].

[FIGURE] Fig. 5. The blue stellar spectra. The relative fluxes have been arbitrarily displaced in steps of 0.5 for the sake of clarity. Wavelengths are in Å.

[FIGURE] Fig. 6. As Fig. 5, however for the red spectra

Before performing the comparison with the theoretical model atmospheres, we have corrected the spectra from radial velocity displacements. Correction for the radial velocities is crucial in the very early type stars, because the H and He lines commonly used for the spectral analysis may be contaminated by wind effects that fill their red wings. However, these corrections are particularly difficult: the cores of strong lines may be affected by the wind and only a few weak metal lines are present. Rotation adds a new handicap, as it broadens lines, making them shallower and favouring blends. In addition the limited spectral range of each single frame limits the number of suitable lines on individual spectra. Thus the first difficulty is to find a set of lines appropriate for the measurement of the radial velocity.

We have discarded all H and He lines. This is already necessary, since sometimes we can see a trend in the measured radial velocity with the excitation potential of the line, indicating that the line cores are formed in higher layers moving faster as the excitation potential decreases.

Thus the radial velocity correction is based on lines of Si (Si IV and Si III ), N (N V , N IV and N III ), O (O II ) and C (C III ). Sometimes, a few lines are in emission and not in absorption. These lines are used only if they give results concordant with the rest of the metal lines (usually, this is the case). The typical uncertainties we obtain are [FORMULA]20 km s-1, which is about half of the spectral resolution. This large error is due to the existence of only a few lines, in addition with broad cores, and reflects the dispersion in the individual values. Within these limits, we have sometimes displaced the fitted lines (also in the analysis with spherical models), when it was clear that this was the only way to obtain a good fit. Table 2 lists the radial velocities in the kinematical Local Standard of Rest.


Table 2. Parameters determined for the programme stars using plane-parallel models. Note that the parameters of Cyg OB2 [FORMULA]7 and HD 15 570 are only indicative, and could not be determined with the plane parallel models. Temperatures are in thousands of Kelvin. [FORMULA], [FORMULA] and [FORMULA] are respectively, the present spectroscopic and evolutionary masses, and the initial evolutionary mass, in solar units. The last column indicates whether we have formally a mass discrepancy considering an error of 0.22 in log([FORMULA]). [FORMULA] sin i and [FORMULA] are given in km s-1. Semicolons in [FORMULA] indicate that the uncertainty in this value is larger than [FORMULA]20 kms-1

We then determined rotational velocities following the same procedure as indicated in Paper I. Our values compare well with those by Penny (1996) and Howarth et al. (1997) for objects in common (see discussion about individual objects). Rotational velocities are given in Table 2 together with the parameters determined for each star. In this table, temperatures are given in thousands of Kelvin; gravities are corrected for centrifugal force effects (see individual discussions for model parameters); [FORMULA] is the helium abundance by number with respect to the total number of H and He atoms; V is the integral of the stellar flux over [FORMULA], weighted by the V-filter function of Matthews & Sandage (1963), used to calculate stellar radii from the model atmospheres (see Kudritzki 1980, or Paper I); [FORMULA] is the absolute magnitude. This has been obtained using the photometry and colours from Hipparcos for HD 15 570, HD 15 629, HD 15 558, HD 14 947 and HD 210 839 combined with the extinction laws and distances from Garmany & Stencel (1992). For Cyg OB2 [FORMULA]7, [FORMULA] has been taken from Massey & Thompson (1991), and for HD 5 689 has been taken from Garmany & Stencel (1992). The evolutionary masses have been obtained from the evolutionary tracks by Schaller et al. (1992).

Except for the line-blocking, the models are the same ones used in Paper I. These are H/He,, plane-parallel models, in hydrostatic and radiative equilibrium. The line fit is also made in the same way. We first fit [FORMULA], obtaining the best gravity at a given [FORMULA]. The same procedure is followed for the helium lines, at a normal He abundance (i.e., 0.09 by number). The locus where all lines intersect in the log [FORMULA] - [FORMULA] diagram is taken as giving the stellar parameters. If the lines do not intersect, the helium abundance is changed. The helium abundance giving the smallest intersection region is adopted as the one appropriate for the star. In previous studies, the helium lines used were He II [FORMULA] 4541, 4200 and He I [FORMULA] 4387, 4922, whereas He I [FORMULA] 4471 was used only for dwarfs or high-temperature stars. As this is the case in the present study, this is the line we use here. In addition, for the reasons explained before, we give it a larger weight than for the singlet lines.

The errors are similar to those quoted in Paper I, with [FORMULA]1500 K in [FORMULA], [FORMULA]0.1 in [FORMULA] and [FORMULA]0.03 in [FORMULA]. This produces errors of [FORMULA]0.06, [FORMULA]0.19 and [FORMULA]0.22 in [FORMULA], [FORMULA] and [FORMULA], respectively, when adopting an uncertainty of [FORMULA]0.3 mag for [FORMULA].

We now describe the line fits of each star independently, and the spectral features, if appropriate. In Fig. 7 we show the theoretical HR diagram for these stars, with the parameters listed in Table 2.

[FIGURE] Fig. 7. The programme stars on the Hertzsprung-Russell diagram after the plane-parallel analysis. The meanings of the symbols are those indicated in the lower left corner, together with typical error bars. The theoretical tracks are from Schaller et al. (1992). The numbers to the left of the zero-age main sequence indicate the initial stellar masses in units of solar masses. L is in solar units.

4.1. HD 5 689, O6 V

This is the only star for which we can clearly see lines of He I other than He I [FORMULA]4471. Its line fit, shown in Fig. 8, is obtained at [FORMULA] = 40 000 K, [FORMULA] = 3.40, [FORMULA] = 0.25 and [FORMULA] sin i = 250 km s-1. The rotational velocity, however, is uncertain, as in this cases we only have the He lines (especially He I ). The given value can be seen as an upper limit to the projected rotational velocity. We must correct for centrifugal force by adding the term ([FORMULA] sin i)2/R, since the measured value is the effective gravity reduced by the centrifugal acceleration, which has to be added here in order to derive the masses. Corrected for centrifugal forces the gravity in Table 2 is [FORMULA] = 3.57. As usual for fast rotators, the helium abundance obtained is very large. The radius, however, is very small, and the derived spectroscopic mass is much lower than the predicted evolutionary one. These facts reflect certain problems in the set of parameters for this star. The star has been classified as O6 V by Garmany & Vacca (1991). This classification is not in complete agreement with the star belonging to Cas OB7 (see Humphreys 1978, or Garmany & Stencel 1992), since the absolute magnitude derived from the association distance corresponds to a less luminous object than an O6 V star by one magnitude, which causes the small radius derived (see Table 5 of Vacca et al. 1996). The situation is of course much worse if, based on the low gravity, we associate this star with a giant and adopt parameters characteristics of a luminosity class III object. Thus we have two possibilities: 1) the star is a main sequence star that belongs to Cas OB7, with a smaller radius than usual for its spectral classification, in spite of the large rotational velocity; in this case, the low gravity is difficult to explain, or the models are giving us completely wrong parameters for this star; or 2) the star has parameters typical for a luminosity class III object. In this case, the star would probably not belong to Cas OB7, although we have found no indications in the literature about this possibility. Other combinations of the above arguments are also possible, but the question that something is non-standard with the star remains. This case is very similar to the one we had in Paper I for HD 24 912, which is a known runaway star. Using the Oort constants given by Lang (1980), we obtain a peculiar velocity of 39 km s-1 for HD 5689, which, given our uncertainties, does not allow us to decide clearly whether it is a runaway or not, if we adopt the conventional limit of 30 km s-1 for runaway stars.

[FIGURE] Fig. 8. The fit to the spectral lines in HD 5 689 using the plane-parallel, hydrostatic models, with the parameters given in Table 2. From top to bottom and left to right: [FORMULA], [FORMULA], He I [FORMULA]4387, 4922, He II [FORMULA]4200, 4541.

4.2. HD 210 839, O6 I(n)fp

This is [FORMULA] Cep, a well known fast rotator. The line fit (see Fig. 9) is obtained at [FORMULA] = 41 500, [FORMULA] = 3.40, [FORMULA] = 0.25 and [FORMULA] sin i = 250 km s-1. The rotational velocity is again uncertain and could be lower, as we suggest later. This is also indicated by the values found by Penny (1996; 214 km s-1) and Howarth et al. (1997; 219 km s-1). With the centrifugal force correction, [FORMULA] increases to 3.47 (see Table 2). The line fit is the best agreement we could find between all the He lines. The parameters are very similar to those of HD 5 689, (except that now the mass discrepancy is only the usual factor 2) but the spectrum shows important differences. [FORMULA] Cep displays Of features, and has strong emission in [FORMULA]. This has to be attributed to the large difference in luminosity (see Fig. 7 and Table 2). In Fig. 7, HD 210 839 occupies the position of an evolved luminous star, already within the instability strip predicted by Kiriakidis et al. (1993). In agreement with this location, [FORMULA] Cep is known to show profile variations due to non-radial pulsations, which however are below the accuracy of our optical spectra (cf. Fullerton et al. 1996, de Jong et al. 1999). (Note, that we have seen temporal variations in the [FORMULA] profile which was observed several times. In contrast, however, two spectra taken in the region from 4200 to 4600 Å showed no significant variations in the lines used for the spectral analysis.) Finally, let us remark that the IR spectrum of [FORMULA] Cep has been analysed recently by Najarro et al. (1998), who obtained parameters very similar to ours (except in the temperature, for which they find a value lower by 4 000 K, see discussion).

[FIGURE] Fig. 9. As Fig. 8, for HD 210 839. Here we show He I [FORMULA] 4471 instead of He I [FORMULA] 4922.

4.3. HD 14 947, O5 If+

The spectral lines of this star are fitted at [FORMULA] = 45 000 K, [FORMULA] = 3.50, [FORMULA] = 0.15 and [FORMULA] sin i = 140 km s-1, which agrees with the value of 133 km s-1 given by Penny (1996) and Howarth et al. (1997). The spectral line fits are shown in Fig. 10. The star is an extreme Of (without being a transition object, see Conti et al. 1995), with a large emission in N III [FORMULA]4630-40 and He II [FORMULA]4686, and also in [FORMULA]. We can also begin to see Si IV [FORMULA]4116 in emission and the N V [FORMULA]4604, 4620 lines in absorption. The line fit, shown in Fig. 10, can be considered as acceptable. The fit of the He I [FORMULA]4922 line indicates that the predicted line is a little too strong as compared to the observations, but the difference is very small compared to variations of the line within the error box. The He abundance is not as large as for the two previous objects, but is still larger than solar. The spectroscopic mass is more than a factor of two smaller than the evolutionary one.

[FIGURE] Fig. 10. As Fig. 8, for HD 14 947. Here we show He I [FORMULA] 4471 instead of He I [FORMULA] 4387.

4.4. HD 15 558, O5 III(f)

This star is a binary, but we expect the spectrum not to be contaminated, as it is a single component in a well separated system (Mason et al. 1998). The best line fit, shown in Fig. 11, is obtained at [FORMULA] = 46 500 K, [FORMULA] = 3.70, [FORMULA] = 0.07 and [FORMULA] sin i = 120 km s-1. The rotational velocity value agrees with the 123 km s-1 of Howarth et al. (1997) although it departs slightly from the one given by Penny (1996) of 147 km s-1. The Of features are weaker than in the previous star, as is the emission in Si IV [FORMULA]4116, but note that the gravity is now larger. The line fit with hydrostatic models is very difficult. Only the wings of [FORMULA] and [FORMULA] are fitted, together with He II [FORMULA]4541. The singlet He I lines are again too weak and noisy, but in this case also He I [FORMULA]4471 cannot be fitted. We cannot simply attribute the distortion in the blue wing of He I [FORMULA]4471 to binarity. If this were the case, we should see the singlet lines too (if the companion is relatively cool) or distortions in the He II lines (if the companion is relatively hot). The lack of a good fit in any He I line makes the parameter determination much less accurate. From the plane-parallel hydrostatic models this is the most massive star, and in fact the mass discrepancy is comparatively low, the difference between the spectroscopic and evolutionary masses being only 30[FORMULA].

[FIGURE] Fig. 11. As Fig. 9, for HD 15 558.

4.5. HD 15 629, O5 V((f))

The best line fit for this star is obtained at [FORMULA] = 48 000 K, [FORMULA] = 3.80, [FORMULA] = 0.09, with [FORMULA] sin i = 90 km s-1. This last value agrees within [FORMULA]1 km s-1 with the values quoted by Penny (1996) and Howarth et al. (1997). In spite of the high temperature, the larger gravity allows us the calculation of hydrostatic models. The line fit is shown in Fig. 12. We can see that the fit is good for the wings of [FORMULA] and [FORMULA], and for He I [FORMULA]4471 and He II [FORMULA]4541, but it is bad for He I [FORMULA]4387, and also for He I [FORMULA]4922, that is not shown in Fig. 12. We have given much less weight to these two lines, however, because they are very weak and noisy. In the case of He I [FORMULA]4922, the local continuum has also been placed too low. Correction of this would bring the calculated line into agreement with the observations. More worrying is the lack of fit in the He II [FORMULA]4200 line. The fit of this line is bad, and comparable to that of the He II [FORMULA]4686 line, for which we expect a bad fit with hydrostatic models. For this reason we preferred to give more weight to the fit of He II [FORMULA]4541 than to He II [FORMULA]4200. In spite of the large gravity and the luminosity class V, this star also shows the mass discrepancy, which was not the case for less luminous stars in Paper I, where we found no significant mass discrepancy for stars of high gravities. This indicates the increasing role of radiation pressure.

[FIGURE] Fig. 12. As Fig. 9, for HD 15 629.

It is interesting to note the case of HD 15 629, HD 14 947 and HD 210 839. Within our error bars, these stars could represent different evolutionary stages of a star of initially around 70 [FORMULA] following standard evolutionary tracks with mass loss but without rotation. Within this scenario, it is impossible to explain the higher He abundances of the two cooler stars. This can be an indication that rotation plays a strong role in stellar evolution, since there is no other known mechanism that might substantially modify the atmospheric He content of a single star like HD 15 629 in only two Myr. (Another possible scenario for an increased He abundance, close binary interaction, can most probably be discarded in the case of these three particular objects, see Mason et al. 1998). Turning the argument around, if rotation does not play a significant role in the evolution of single massive stars, HD 15 629 could become rather similar to first HD 14 947 and then HD 210 839 in one to two Myr. Unfortunately, we cannot wait to confirm this hypothesis.

4.6. HD 15 570, O4 If+

This star is hotter than the former one, and its Of signatures are stronger (again, without being a transition object, see Conti et al. 1995), indicating a lower gravity. This is confirmed by inspection of the Balmer series, especially the [FORMULA] profile, which is strong in emission. From 4600 to 4750 Å we see a broad emission feature with emission lines typical of Of stars, and both Si IV  lines neighbouring [FORMULA] are in emission, indicating a large luminosity, which is confirmed. There is also absorption in N V [FORMULA]4604, 4620. It is by far the most luminous star in the sample, and we were not able to fit the spectrum properly for this object. The extreme character of the features allows only a crude guess of the stellar parameters, and we do not show any line fit. Apart from the projected rotational velocity of 105 km s-1, we can only say that the temperature is close to 50 000 K and the gravity should be of the order of [FORMULA] [FORMULA] 3.50 or even less. With the given luminosity, the evolutionary mass is very large, indicating an initial mass in excess of 140 [FORMULA], which would make HD 15 570 one of the most massive and luminous stars known in the Milky Way. The spectroscopic mass is much lower, although we have to consider that the gravity has only been "guessed" so far (see next section). There is a chance that the star is helium enhanced, and we adopt [FORMULA] = 0.15.

4.7. Cyg OB2 #7, O3 If*

This is a very hot star. He I [FORMULA]4471 is only marginally seen, and the emission in N IV [FORMULA]4058 and the absorptions in N V [FORMULA]4604, 4620 are strong. We again find the broad emission feature with emission lines typical for Of stars, and both Si IV  lines neighbouring [FORMULA] in emission. In spite of our efforts, we were unable to fit the star with plane-parallel models, the required gravities being too low. As for HD 15 570, we had to extrapolate the plane-parallel parameters, and we do not show a line fit. The final parameters we adopt are [FORMULA] = 51 000 K, [FORMULA] = 3.65, [FORMULA] = 0.12 and [FORMULA] sin i = 105 km s-1. We did not succeed in calculating models in this range ([FORMULA] had to be larger at least by 0.1), and spherical models including mass-loss are obviously required. This star is also very luminous and massive, with an initial mass in excess of 110 [FORMULA], but with a large mass discrepancy.

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Online publication: January 31, 2000