We make use of optical and UV spectra that have been presented elsewhere (Herrero et al. 1999; Herrero et al. 2001). The optical spectra have been newly rectified, resulting in a less pronounced bump in the 4630-4700 Å region, which, however, does not affect our present analysis. Basic data, together with previously determined parameters adopted here are listed in Table 1, while the new parameters determined in this work are provided in Table 2. Note that the gravities in that table have been corrected for the effect of centrifugal acceleration (in the same way as in Herrero et al. 1992). This effect is small (modifying only the last digit of the entries in our table) and the gravities actually used in our calculations are always given by the nearest lower value ending with "0'' or "5'' in the second decimal (i.e., a corrected value of 3.52 corresponds to a model value of 3.50). The corrected gravities have been used for the calculation of stellar masses.

Errors adopted for the parameters given in Table 2 have been taken from the errors obtained for Cyg OB2 #7 (see below) or from those for 10 Lac, depending on the stellar parameters, the spectrum quality and the fit conditions. A summary of all errors is given in Table 3.

For Cyg OB2 #7, errors have been estimated "by eye'' from a microgrid of models around the final one. At fixed $\beta$ (the exponent of the velocity law), this resulted in $\pm$ 1000 K in $T_{\rm eff}$ , ^+0.15_-0.10 dex in $\log g$ , $\pm$ 0.05 dex in $\epsilon$ and ^+0.05_-0.10 dex in $\dot{M}$ . $\beta$ is determined from the form of the H $_{\rm\alpha}$ wings, and its uncertainty is estimated to be $\pm$ 0.10. This has an influence on the derived effective temperature and mass-loss rates. Therefore, the adopted error for $T_{\rm eff}$ has been increased to $\pm$ 1500 K and that for $\dot{M}$ to ^+0.10_-0.15.

The errors for radii and masses depend on the error in the absolute magnitude. This is assumed to be $\pm$ 0.1 from the work by Massey & Thompson (1991). (The influence of the error in $T_{\rm eff}$ on the stellar radius is marginal and has been neglected here.) For the analysis we have adopted a microturbulence of 10 km s^-1. Tests indicate that this parameter is of no relevance for the results presented here.

In the following we will comment on the individual analyses. Further discussions about the individual stars were presented in Herrero et al. (2001).

Table 1: Cyg OB2 supergiants studied in this work. All numerical identifications are taken from Schulte (1958). Magnitudes have been adopted from Massey & Thompson (1991), as well as spectral types, except for Cyg OB2 $\char93$ 11 and $\char93$ 4, that are taken from Walborn (1973). S/N values have been estimated from the rms of the continuum at several wavelength intervals. Velocities are given in km s^-1.
Ident V Spectral M_v S/N $V_{\rm r}$ sini $v_{\rm\infty}$

mag. Type

7 10.55 O3 If $^\star$ -5.91 140 105 3080

11 10.03 O5 If⁺ -6.51 190 120 2300

8C 10.19 O5 If -5.61 195 145 2650

8A 9.06 O5.5 I(f) -7.09 135 95 2650

4 10.23 O7 III((f)) -5.44 230 125 2550

10 9.88 O9.5 I -6.86 145 85 1650

2 10.64 B1 I -4.64 195 50 1250

**Table 1:** Cyg OB2 supergiants studied in this work. All numerical identifications are taken from Schulte (1958). Magnitudes have been adopted from Massey & Thompson (1991), as well as spectral types, except for Cyg OB2 $\char93$ 11 and $\char93$ 4, that are taken from Walborn (1973). S/N values have been estimated from the rms of the continuum at several wavelength intervals. Velocities are given in km s^-1.
Ident	V	Spectral	M_v	S/N	$V_{\rm r}$ sini	$v_{\rm\infty}$
	mag.	Type
7	10.55	O3 If $^\star$	-5.91	140	105	3080
11	10.03	O5 If⁺	-6.51	190	120	2300
8C	10.19	O5 If	-5.61	195	145	2650
8A	9.06	O5.5 I(f)	-7.09	135	95	2650
4	10.23	O7 III((f))	-5.44	230	125	2550
10	9.88	O9.5 I	-6.86	145	85	1650
2	10.64	B1 I	-4.64	195	50	1250

Table 2: Results obtained using FASTWIND plus line blocking/blanketing. Effective temperatures have been derived from the He ionization equilibrium. Gravities include the correction for centrifugal acceleration. $\epsilon$ is the He abundance by number relative to H plus He; R, L and M are given in solar units. $M_{\rm s}$ , $M_{\rm ev}$ and $M_{\rm0}$ are the spectroscopic, present evolutionary and initial evolutionary masses, the two latter from the models by Schaller et al. (1992). $\dot{M}$ is given in solar masses per year, and $\log MWM$ means the logarithm of the Modified Wind Momentum rate, $\dot{M}v_{\rm \infty}~R^{0.5}$
Ident Spectral $T_{\rm eff}$ $\log g$ $\epsilon$ R $\beta$ $\dot{M}$ $\log L$ $M_{\rm s}$ $M_{\rm ev}$ $M_{\rm0}$ $\log MWM$

Type

7 O3 If^* 45.5 3.71 0.23 14.6 0.90 9.86e-6 5.91 39.7 67.4 69. 29.864

11 O5 If⁺ 37.0 3.61 0.09 22.2 0.90 9.88e-6 5.92 73.0 58.1 63. 29.829

8C O5 If 41.0 3.81 0.08 13.3 0.90 2.25e-6 5.66 42.2 46.1 48. 29.137

8A O5.5 I(f) 38.5 3.51 0.09 27.9 0.70 1.35e-5 6.19 90.5 78.4 95. 30.076

4 O7 III((f)) 35.5 3.52 0.09 13.6 1.00 8.60e-7 5.41 21.8 32.6 34. 28.708

10 O9.5 I 29.0 3.11 0.09 30.3 1.00 3.09e-6 5.77 43.1 43.7 48. 29.248

2 B1 I 28.0 3.21 0.09 11.3 1.00 6.92e-8 4.85 7.5 18.1 19. 27.263

**Table 2:** Results obtained using FASTWIND plus line blocking/blanketing. Effective temperatures have been derived from the He ionization equilibrium. Gravities include the correction for centrifugal acceleration. $\epsilon$ is the He abundance by number relative to H plus He; R, L and M are given in solar units. $M_{\rm s}$ , $M_{\rm ev}$ and $M_{\rm0}$ are the spectroscopic, present evolutionary and initial evolutionary masses, the two latter from the models by Schaller et al. (1992). $\dot{M}$ is given in solar masses per year, and $\log MWM$ means the logarithm of the Modified Wind Momentum rate, $\dot{M}v_{\rm \infty}~R^{0.5}$
Ident	Spectral	$T_{\rm eff}$	$\log g$	$\epsilon$	R	$\beta$	$\dot{M}$	$\log L$	$M_{\rm s}$	$M_{\rm ev}$	$M_{\rm0}$	$\log MWM$
	Type
7	O3 If^*	45.5	3.71	0.23	14.6	0.90	9.86e-6	5.91	39.7	67.4	69.	29.864
11	O5 If⁺	37.0	3.61	0.09	22.2	0.90	9.88e-6	5.92	73.0	58.1	63.	29.829
8C	O5 If	41.0	3.81	0.08	13.3	0.90	2.25e-6	5.66	42.2	46.1	48.	29.137
8A	O5.5 I(f)	38.5	3.51	0.09	27.9	0.70	1.35e-5	6.19	90.5	78.4	95.	30.076
4	O7 III((f))	35.5	3.52	0.09	13.6	1.00	8.60e-7	5.41	21.8	32.6	34.	28.708
10	O9.5 I	29.0	3.11	0.09	30.3	1.00	3.09e-6	5.77	43.1	43.7	48.	29.248
2	B1 I	28.0	3.21	0.09	11.3	1.00	6.92e-8	4.85	7.5	18.1	19.	27.263

Table 3: Errors adopted for the parameters given in Table 2, in the same units as in that table. If only one value is provided, it should be preceeded by the $\pm$ sign. See text for comments.
Ident Spectral $\Delta$ $\Delta$ $\Delta$ $\Delta$ $\Delta$ $\Delta$ $\Delta$ $\Delta$ $\Delta$ $\Delta$ $\Delta$

Type $T_{\rm eff}$ $\log g$ $\epsilon$ $\log R$ $\beta$ $\log \dot M$ $\log L$ $M_{\rm s}$ $M_{\rm ev}$ $M_{\rm0}$ $\log MWM$

7 O3 If^* 1.5 ^+0.15_-0.10 ^+0.10_-0.05 0.02 0.10 ^+0.10_-0.15 0.10 ⁺¹⁷_-13 ⁺¹¹_-8 ⁺¹¹_-8 ^+0.12_-0.17

11 O5 If⁺ 1.5 ^+0.15_-0.10 0.03 0.02 0.10 ^+0.10_-0.15 0.11 ⁺³²_-24 ⁺¹⁰_-9 ⁺¹¹_-10 ^+0.12_-0.17

8C O5 If 1.5 0.10 0.03 0.02 0.10 ^+0.10_-0.15 0.10 14 5 5 ^+0.12_-0.17

8A O5.5 I(f) 1.5 0.10 0.03 0.02 0.10 ^+0.10_-0.15 0.11 29 ⁺⁴_-7 15 ^+0.12_-0.17

4 O7 III((f)) 1.0 0.10 0.03 0.02 0.10 0.10 0.09 7 4 5 0.12

10 O9.5 I 1.0 0.10 0.03 0.02 0.10 0.10 0.10 14 ⁺¹²_-6 ⁺¹⁶_-6 0.13

2 B1 I 1.0 0.10 0.03 0.04 0.10 0.10 0.14 3 3 3 0.13

**Table 3:** Errors adopted for the parameters given in Table 2, in the same units as in that table. If only one value is provided, it should be preceeded by the $\pm$ sign. See text for comments.
Ident	Spectral	$\Delta$	$\Delta$	$\Delta$	$\Delta$	$\Delta$	$\Delta$	$\Delta$	$\Delta$	$\Delta$	$\Delta$	$\Delta$
	Type	$T_{\rm eff}$	$\log g$	$\epsilon$	$\log R$	$\beta$	$\log \dot M$	$\log L$	$M_{\rm s}$	$M_{\rm ev}$	$M_{\rm0}$	$\log MWM$
7	O3 If^*	1.5	^+0.15_-0.10	^+0.10_-0.05	0.02	0.10	^+0.10_-0.15	0.10	⁺¹⁷_-13	⁺¹¹_-8	⁺¹¹_-8	^+0.12_-0.17
11	O5 If⁺	1.5	^+0.15_-0.10	0.03	0.02	0.10	^+0.10_-0.15	0.11	⁺³²_-24	⁺¹⁰_-9	⁺¹¹_-10	^+0.12_-0.17
8C	O5 If	1.5	0.10	0.03	0.02	0.10	^+0.10_-0.15	0.10	14	5	5	^+0.12_-0.17
8A	O5.5 I(f)	1.5	0.10	0.03	0.02	0.10	^+0.10_-0.15	0.11	29	⁺⁴_-7	15	^+0.12_-0.17
4	O7 III((f))	1.0	0.10	0.03	0.02	0.10	0.10	0.09	7	4	5	0.12
10	O9.5 I	1.0	0.10	0.03	0.02	0.10	0.10	0.10	14	⁺¹²_-6	⁺¹⁶_-6	0.13
2	B1 I	1.0	0.10	0.03	0.04	0.10	0.10	0.14	3	3	3	0.13

Cyg OB2 #7

The final fit to the observed H/He spectrum of Cyg OB2 #7 is shown in Fig. 4. The good fit to the H $_{\rm\alpha}$ profile is accompanied by a much poorer fit to the other two Balmer lines (and also to H $_{\rm\delta}$ , not displayed here). This behaviour reproduces the one found in Herrero et al. (2000): for stars with strong winds we could not obtain a consistent fit for all Balmer lines at a given mass-loss rate. The situation has improved with the new version of our code, but the discrepancy still reaches a 30 $\%$ effect, by which the mass-loss rate has to be reduced (from 10^-5 to 7.7 $\times$ 10^-6) in order to fit H $_{\rm\beta}$ and H $_{\rm\gamma}$ . The other stellar parameters are not affected by this modification, as the fit to the other lines does not change. We adopt the mass-loss rate indicated by H $_{\rm\alpha}$ as this line is much more sensitive to changes in this parameter and there is good general agreement between the mass-loss rates from H $_{\rm\alpha}$ and radio fluxes (Scuderi et al. 1998). Our result also supports this conclusion, as the mass-loss rate derived here agrees with the upper limit of 1.5 $\times$ 10 $^{-5}~M_{\odot}$ yr^-1 quoted by Bieging et al. (1989) (1.6 $\times$ 10^-5 if we use our values for distance and $v_{\rm\infty}$ ) in case the star is a thermal emitter (the authors classify the object among the probable thermal emitters). However, in addition to the error quoted, $\dot{M}$ could be lowered by an additional 20 $\%$ if we would adopt the value indicated by H $_{\rm\gamma}$ .

The singlet He I lines ( $\lambda\lambda$ 4387, 4922) give a poor fit to the observed spectrum, partly due to the difficulties in the continuum rectification. Therefore we have given a low weight to these lines when determining the stellar parameters. However, since these lines react strongly to changes in stellar parameter, it is always possible to find a reasonable fit within the error box. The error in $\epsilon$ is larger than for other stars and $\epsilon$ itself is not well constrained towards higher He abundances, because the already large He abundance produces a saturation effect.

The derived $T_{\rm eff}$ is much lower than the one obtained by Herrero et al. (2000) using the same code as here but without line-blanketing. The derived luminosity is also lower by more than 0.2 dex, as the radius does not change very much. The reason can be seen in Fig. 5, where we compare the emergent energy of two models for Cyg OB2 #7. The first model (the dash-dotted line in the figure) is the one adopted here, with a $T_{\rm eff}$ of 45 500 K, a radius of 14.6 $R_{\odot}$ and metal line opacity. The second model (solid line) corresponds to the model adopted by Herrero et al. (2001), with a $T_{\rm eff}$ of 50 000 K, a radius of 14.8 $R_{\odot}$ and pure H/He opacities. Both models give the same optical luminosity and thus fit equally well the observed visual magnitude of Cyg OB2 #7. We additionally show in the figure the CMFGEN luminosities calculated with the same parameters and conditions (dashed lines). The good agreement supports our approximated treatment of the metal line opacity.

The reason for the similarity in derived radii is that the emergent flux is strongly blocked in the UV by the metal line opacity and thus emerges at higher wavelengths, including the optical. Therefore, at lower temperatures we obtain larger optical fluxes for models that include metal line opacity than for pure H/He models of the same temperature.

The radius needed to fit the observed visual magnitude is then significantly smaller for models with metal line opacities (compared to unblanketed models at the same $T_{\rm eff}$ ), however roughly similar to the "old'' value derived from unblanketed models at higher $T_{\rm eff}$ . In consequence, the reduction in luminosity is mostly due to the change in the effective temperature. Note, however, the dramatic difference of the ionizing fluxes in the (E)UV.

$\begin{figure} \par\includegraphics[width=11.5cm,clip]{h3745f5.eps}\end{figure}$

Figure 5: Emergent energy of two models for Cyg OB2 #7, each one calculated both with FASTWIND and CMFGEN. The first pair of models corresponds to the one adopted here including metal line opacity and is represented by the lower dashed (CMFGEN) and dash-dotted (FASTWIND) lines. The second pair corresponds to the model adopted by Herrero et al. (2001) that did not include metal line opacity and is represented by the upper dashed (CMFGEN) and solid (FASTWIND) lines. We see that all models give the same optical flux, but very different UV fluxes (see text). We also see the good agreement between FASTWIND and CMFGEN, although individual strong UV resonance lines are not visible in the former due to the approximate method applied.

The helium abundance derived here is even larger than the one obtained by Herrero et al. (2000), although the error bars overlap significantly. Cyg OB2 #7 is thus confirmed as the only star in our sample for which we derive an enhanced He abundance. The $\beta$ value we obtain (0.9) is slightly larger than the one obtained from the UV analysis presented in Herrero et al. (2001), a behaviour which has been found already in previous investigations (e.g., Puls et al. 1996). However, our mass-loss rate is similar to the one derived in that work, resulting again from the fact that the optical fluxes are similar.

Finally, the $\log g$ value obtained here is slightly larger than the one obtained by Herrero et al. (2000). This results in a spectroscopic mass of 39.7 $M_{\odot}$ , to be compared with an evolutionary mass of 67.4 $M_{\odot}$ . Accounting for maximum errors, masses of 56.7 and 59.4 $M_{\odot}$ , respectively, are possible, still not overlapping. However, we have to remember the large He abundance derived for Cyg OB2 #7. This is an indication that this star might be evolved or be affected by rotational mixing. Taken together, there is no (clear) evidence that evolutionary and spectroscopic masses really disagree.

Walborn et al. (2000) and Walborn et al. (2002) have recently studied Cyg OB2 #7 and HD 93 129A. They conclude, from a comparison of their spectra, that Cyg OB2 #7 has to be cooler than HD 93 129A, and in fact HD 93 129A has been reclassified as the prototype of the new O2 If^* class. Taresch et al. (1997) have analyzed this latter star and obtained an effective temperature of 52 000 $\pm$ 1000 K, based on the N V/N IV/N III ionization equilibrium. This is similar to the Herrero et al. (2000) value and again much higher than the temperature we obtain here for Cyg OB2 #7. Both stars display simultaneously N V, N IV and N III lines in their spectra, and thus we would not expect a very large temperature difference. Clearly, a cross-calibration of He and N blanketed temperature scales for the earliest stars is an urgent task.

Cyg OB2 #11

The fit to Cyg OB2 #11 is given in Fig. 6. It shows the same problems as the fit to Cyg OB2 #7, except for the fit to the He I singlet lines, which are again affected by continuum rectification problems. However their depths relative to the depressed local continuum are now well predicted.

Therefore we adopt the same errors, except for $\epsilon$ , for which we adopt $\pm$ 0.03. It is interesting that in spite of the extreme Of character of both stars and the similarity of the problems found, we do not derive an enhanced He abundance for Cyg OB2 #11. This star also shows the same trend as Cyg OB2 #7 towards cooler temperatures and lower luminosities, but now the spectroscopic and evolutionary masses (73.0 and 58.1 $M_{\odot}$ , respectively) invert their usual ratio. When considering the formal errors presented in Table 3, the mass ranges of both stars overlap significantly.

The $\beta$ value we have used is again 0.9. The mass loss rate we derive is consistent with the upper limit given by Bieging et al. (1989) (1.4 $\times$ 10 $^{-5}~M_{\odot}$ yr^-1, or 1.2 $\times$ 10^-5 using again our values for distance and wind terminal velocity).

Cyg OB2 #8C

The fit to Cyg OB2 #8C is presented in Fig. 7. The only problem is a serious failure in the predicted He II $\lambda$ 4686 line (which is not used in the fit procedure). To fit this line one had to increase the mass-loss rate by at least a factor of 1.7, although we note that the observed line lies at top of a broad emission feature that we cannot reproduce. The same comments as for Cyg OB2 #11 apply for the He I singlet lines.

$\beta$ is not well constrained from the wings of H $_{\rm\alpha}$ , as these are in absorption. We have adopted the same value as for Cyg OB2 #7 and #11 (0.9), as well as the same errors.

The resulting mass-loss rate is consistent with the upper limit quoted by Bieging et al. (1989) of 8.8 $\times$ 10 $^{-6}~M_{\odot}$ yr^-1. The gravity is large for a supergiant (a lower gravity is prohibited by the Balmer line wings), but we find good agreement between the spectroscopic and evolutionary masses.

Cyg OB2 #8A

The final fit can be seen in Fig. 8. The fit to He II $\lambda$ 4686 is problematic, but much less than for Cyg OB2 #8C, while the same comments apply for the He I singlet lines. Errors are the same as for Cyg OB2 #8C. The temperature is cooler and the luminosity lower than quoted in Herrero et al. (2001). However, the spectroscopic mass is again very large (90.5 $M_{\odot}$ ), larger than the evolutionary one (78.4 $M_{\odot}$ ), but with significant overlap when considering the errors.

The mass-loss rate we derive here is nearly a factor of two lower than the one given in Herrero et al. (2001). Note, that the latter was not derived from spectrum analysis, however was calculated from the luminosity (believed to be larger at that time) and the Galactic WLR.

Our new mass-loss rate in Table 2 (1.35 ^+0.35_-0.39 $\times$ 10 $^{-5}~M_{\odot}$ yr^-1) agrees well with the radio mass-loss rate given by Waldron et al. (1998) (1.97 $\times$ 10^-5), and lies between the extreme values one would derive from the fluxes given by Bieging et al. (1989) (1.1-6.1 $\times$ 10^-5) assuming free-free emission. Although Cyg OB2 #8A is a known non-thermal emitter, with variable radio flux and spectral index (Waldron et al. 1998; Bieging et al. 1989), our H $_{\rm\alpha}$ mass-loss rate is of the same order of magnitude as the radio mass-loss rates and consistent with their lower limit. This consistency contradicts the suggestion by Waldron et al. (1998) that the X-ray emission might originate from an X-ray source deeply embedded in the stellar wind, i.e., a base corona model scenario, which would imply a much lower mass-loss rate ( $\approx$ 1.5 $\times$ 10 $^{-6}~M_{\odot}$ yr^-1).

Cyg OB2 #4

The fit to Cyg OB2 #4 is presented in Fig. 9. The predicted He II $\lambda$ 4686 and the singlet He I line at $\lambda$ 4922 are too strong in the core, although we note the large scale in the corresponding plots. The fit of He I $\lambda$ 4387 is acceptable taking the normalization into account. $\beta$ is again not well constrained from the H $_{\rm\alpha}$ wings and we adopt a similar value as for the cooler stars in our sample ( $\beta=$ 1). However, the influence of $\beta$ on the other stellar parameters begins to decrease and therefore we adopt the same errors as for 10 Lac.

The mass loss rate is not well constrained towards lower values, because the profiles react only slightly. In this case, as also for the next two Cyg OB2 stars, there are no radio mass-loss rates available to compare with (which is an indication of a rather low value). The derived effective temperature is still cooler than in Herrero et al. (2001), although the differences begin to decrease. The evolutionary and spectroscopic mass ranges agree within the large error bars.

Cyg OB2 #10

The fit to Cyg OB2 #10 is given in Fig. 10. As for Cyg OB2 #4, the main difficulties appear in the fit of the He II $\lambda$ 4686 and the He I lines, where from the three He I lines one is slightly too strong, the second slightly too weak and the third one fits well. The errors adopted are the same as for Cyg OB2 #4. The derived effective temperature is still lower than in Herrero et al. (2001), but the difference is only 2000 K. The spectroscopic and evolutionary masses agree well.

Cyg OB2 #2

The final fit to the star is shown in Fig. 11. The adopted errors are the same as for Cyg OB2 #8C (because of the lower S/N compared to 10 Lac), although again the mass loss rate is not well constrained towards lower values. Here, the derived effective temperature is hotter than in Herrero et al. (2001), but also the derived He abundance has decreased significantly. The masses, however, do not agree, with the spectroscopic mass being lower than the evolutionary one by a factor of two.

Herrero et al. (2001) indicated some problems with the stellar classification as B1 I and its Cyg OB2 membership, because this would result in a rather faint absolute magnitude for its spectral class. However, the large reddening quoted by Massey & Thompson (1991) indicates that the star is probably related to or lies beyond Cyg OB2. On the other hand, this large reddening is comparatively low when compared to other Cyg OB2 members, which additionally weakens the above argument. (Cyg OB2 #2 has the fourth lowest reddening in Table 7 of Massey & Thompson 1991, who list a total 64 Cyg OB2 stars. The stars with the three lowest values lie in the same region of the association as Cyg OB2 #2). Thus, we adopt the absolute magnitude derived from the canonical distance to Cyg OB2 and assume a larger error, $\pm$ 0.2 instead of $\pm$ 0.1, which also doubles the error in the (logarithmic) radius.