A BAKE-OFF BETWEEN cmfgen AND fastwind: MODELING THE PHYSICAL PROPERTIES OF SMC AND LMC O-TYPE STARS

The primary driver behind the current undertaking was to see how well the physical properties agreed when the same spectra were fit by the two codes. Table 3 provides the answer, at least for this particular sample of 10 O-type stars.

In general, there is no difference in the average effective temperatures determined from the two codes: the mean difference (in the sense of fastwind minus cmfgen) is −80 K, and the median difference is 0 K. However, what we think is quite significant is the scatter in the effective temperatures, with a sample standard deviation of 1300 K. This scatter is 2.6× greater than our fitting precision with each spectrum, which we (perhaps overly optimistically) estimate as 500 K. The external accuracy of fastwind is usually taken as 1000 K (see, for example, Repolust et al. 2004; and Papers I–III), but our confidence fitting the same spectra is considerably better than that. In Section 3.2 we discuss part of the reason for the occasionally large differences in our temperature determinations.

In contrast, we find a systematic difference of −0.12 dex in the determination of log g between the two codes, with a scatter (0.07 dex) that is quite similar to our fitting precision (0.05 dex). We discuss this below in Section 3.3, along with the implications for the mass discrepancy.

Other fitting parameters, i.e., the mass-loss rates and the He/H ratios, agree well between the two codes, as shown in Table 3. The cmfgen mass-loss values are slightly higher than those of fastwind as we might expect for the difference in how the stratification of clumping is treated, as discussed above. Still, the differences are small, and the scatters are similar to the uncertainties, both for the mass-loss rates and the He/H ratios.

To better understand how well our results from cmfgen and fastwind agreed, it may be instructive to examine the differences in the results we find by using two different versions of the same code; i.e., comparing our results with the latest version of fastwind to those we had previously determined with the same spectra but older versions of the code. We show the results in Table 4. We see that there is little difference between our old and new values. It is interesting to note that the scatter in the effective temperature determinations, σ = 560 K, is essentially the same as what we consider our typical uncertainty in determining these values, about 500 K. In our comparison between fastwind and cmfgen in Table 3 we find a σ that is 2.5× larger. This underscores that the differences we find between fastwind and cmfgen, while not systematic, are much greater than our precision in determining these quantities. As discussed below (Section 3.2) we feel these differences are due to the different behaviors of the code with regard to the He i singlets and triplets. Note too, in Table 4, the utter consistency in the determination of the surface gravities, which significantly bears on the discussion in Section 3.3 concerning the mass discrepancy. We also note that the new values of the mass-loss rates are a bit higher than we would get by decreasing the old values by a factor of 3.3, $1/\sqrt{f}$, as previously described.

Table 4. Model Fits: fastwind New versus fastwind Old^a

Star	Type	Teff (K)			log g (cgs)			$\dot{M}$ (10−6 M☉ yr−1)b			He/H (by number)
		New	Old	Δ	New	Old	Δ	New	Old	Δ	New	Old	Δ
AzV 177	O4 V((f))	44,000	44,000c	0	3.80	3.85c	−0.05	0.2	0.1c	0.1	0.15	0.15c	0.00
AzV 388	O5.5 V((f))	41,500	42,500	−1,000	3.85	3.90	−0.05	0.1	0.1	0.0	0.10	0.10	0.00
AzV 75	O5.5 I(f)	40,000	39,500	500	3.65	3.50	0.15	1.3	0.9	0.4	0.10	0.10	0.00
AzV 26	O6 I(f)	38,000	38,000d	0	3.50	3.50d	0.00	1.3	0.8d	0.4	0.15	0.15d	0.00
NGC 346-682	O8 V	35,500	36,000	−500	4.10	4.10	0.00	0.0	0.0	0.0	0.10	0.10	0.00
AzV 223	O9.5 II	31,600	32,000	−400	3.45	3.50	−0.05	0.2	0.2	0.0	0.10	0.10	0.00
LH 81:W28-23	O3.5 V((f+))	48,000	47,500c	500	3.75	3.80c	−0.05	0.9	0.8c	0.1	0.25	0.25c	0.00
Sk −70°69	O5.5 V((f))	39,500	40,500	−1,000	3.70	3.70	0.00	0.3	0.2	0.1	0.10	0.10	0.00
BI I70	O9.5 I	31,500	31,500	0	3.18	3.15	0.03	0.3	0.2	0.1	0.15	0.15	0.00
Sk −69°124e	O9.7 I	29,000	29,250	−250	3.12	3.10	0.02	0.3	0.2	0.1	0.10	0.10	0.00
Mean difference	...	...	...	−210	...	...	0.00	...	...	0.1	...	...	0.00
Median difference	...	...	...	0	...	...	0.00	...	...	0.1	...	...	0.00
σ	...	...	...	560	...	...	0.06	...	...	0.2	...	...	0.00
Typical uncertainty	...	...	...	500	...	...	0.05	...	...	0.2	...	...	0.02

Notes. ^aNew = fastwind version 10.1, Old = fastwind version 9.2, unless otherwise noted. Δ = New result minus Old result. Old values from Papers I, II, and III. ^bAll mass loss-rates $\dot{M}$ were determined using β = 0.8, except for BI 170 and Sk −69°124, for which we adopted β = 1.2. The (unclumped) $\dot{M}$ from old fastwind fits have been divided by 3.3 to allow direct comparisons with the clumped $\dot{M}$ rates from the new fastwind fits. ^cOld = fastwind version 8.3.1. ^dOld = fastwind version 6.6. ^eSk −69°124 is excluded from the averages as we conclude it is likely a binary. See text.

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3.2.1. The Problem with Triplets and the Effects of Microturbulence

First, consider the He i triplet fastwind problem. Repolust et al. (2004) invoke the "generalized dilution effect" (Voels et al. 1989) to explain the weakness of the fastwind He i λ4471 model profile for mid-to-late type giants and supergiants. The explanation offered by Voels et al (1989) would only apply to plane-parallel models without winds, and so would not strictly be applicable to fastwind's problem. Repolust et al. (2004) of course realized this, but could offer no physical explanation and kept the terminology to describe the effect. Here we confirm that the problem is not present in cmfgen models. Compare the He i λ4471 profile fits in our three late O-type supergiants, AzV 233 (O9.5 II; Figure 6), BI 170 (O9.5 I; Figure 9) and Sk −69°124¹⁰ (O9.7 I; Figure 10). The fastwind profile (solid red) for He i λ4471 is significantly weak in all three cases, but the cmfgen profiles (dashed blue) are spot on. Thus, the problem cannot be due to something common to both codes. In most other cases, the fastwind and the cmfgen λ4471 profiles agree, and match the spectra well, except for the O5.5 I(f) star AzV 75 (Figure 3), where neither give a particularly good fit. Whatever the problem is with the fastwind λ4471 profile, the problem seems to begin to occur after O6 I, as fastwind still does a good job for the O6 I(f) star AzV 26 (Figure 4).

Is the problem that fastwind has with He i λ4471 restricted to just this one line, or does it affect other triplets? The He i λ4471 transition is 1s2p³P^o − 1s4d³D. (For convenience we have reproduced the Moore & Merrill (1968) version of the Grotrian diagram for He i in Figure 11.) The He i λ4026 line is a similar transition (1s2p³P^o − 1s5d³D), but is a blend with the He ii λ4026 line. The He i λ5876 line is also similar (1s2p³P^o − 1s3d³D) and is usually quite strong, but fastwind does not include this line in its formal solution.¹¹ In fact, the only other He i triplet line profile that is normally computed by fastwind is He i λ4713 (1s2p³P^o − 1s4s³S), and that is relatively weak. (It is for that reason that we did not include it in our assessment of the fits.) We show the fits of fastwind and cmfgen for He i λ4713 in Figures 12 and 13, along with the cmfgen fit for He i λ5876.

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We see in Figure 12 that both fastwind and cmfgen do a relatively good job of fitting the He i triplets for the early and intermediate O dwarfs and supergiants.¹² None of that is surprising. When we turn to the late-type supergiants in Figure 13 we are also not surprised to find that when fastwind produces too weak a model profile for He i λ4471 it also produces too weak a profile for He i λ4713. What we do find surprising is that although cmfgen does a good job of matching the He i λ4471 profile, it also produces too weak a profile for the He i λ4713 and He i λ5876 profiles for the three late-type supergiants, AzV 223, BI 170, and Sk −69°124.

In experimenting with possible cures, we investigated the effect of increasing the microturbulence velocity in computing the emergent profiles. In this temperature regime, we had used 10 km s⁻¹ both for the fastwind and cmfgen formal calculations. A few experiments showed that increasing the microturbulence velocity strengthened He i λ5876 significantly more than it strengthened He i λ4471. We thus tried using the same fastwind and cmfgen atmosphere models but recomputed the synthetic spectra using values of 15 and 20 km s⁻¹. In Figures 14 through 16 we compare these with the original fits for the He i triplet lines. Indeed, increasing the microturbulence to 15 km s⁻¹ fixes the problem with the cmfgen fits (shown in dashed blue) of the He i λ5876 lines in AzV 223 (Figure 14) and BI 170 (Figure 15), while still maintaining excellent fits for He i λ4471 and λ4713. For Sk −69°124 we would have to use a slightly higher value (but not as high as 20 km s⁻¹), and the solution would still be a bit of a compromise, with He i λ4471 slightly too strong and He i λ5876 slightly too weak, but still quite acceptable. (We argue below that Sk −69°124 is a binary.)

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Even more surprising, however, is the effect that the microturbulence has on the He i λ4471 profile problem fastwind has with the late O-type supergiants. With a value of 20 km s⁻¹ for the microturbulence, the fastwind fits (shown in red) are excellent. In the next section we will show that this does not come at the expense of any other lines, and in fact it improves the fits in some cases. It remains an unexplored mystery why the He i λ4471 profile is more affected by microturbulence with the fastwind formal solution than with that of cmfgen. Note, however, that the improvement of the cmfgen fit for He i λ5876 with a ∼15 km s⁻¹ value for the microturbulence does suggest that a large value (i.e., 15–20 km s⁻¹) may be a more appropriate choice when modeling late-type O supergiants.

Hubeny et al. (1991) have argued that the presence of microturbulence should result in a turbulence pressure term in the equation of hydrostatic equilibrium, which would lead to increased surface gravities. (See also the three-dimensional convective simulations of the Sun by Trampedach et al. 2013, which shows that turbulence affects the Sun's hydrostatic structure.) Neither cmfgen nor fastwind take this approach. It is not clear at this time if the need for microturbulence is due to real turbulence, or if it is due to a combination of pulsations, wind-velocity fields, or even simply errors in our still rather simplified treatment of the complex physics involved.

3.2.2. Singlets

Next, we turn to the He i singlet fits. Consider the He i λ4388 model profiles in the same late O-type supergiants stars (i.e., AzV 233 in Figure 6, BI 170 in Figure 9, and Sk −69°124 in Figure 10). cmfgen produces too strong a profile, and while the match by fastwind is better, it is certainly not spectacular, at least for the second two stars. Najarro et al. (2006) explained the poor fits to He i λ4388 by cmfgen as being due to uncertainties in the atomic data of the Fe iv lines overlapping with the He i 1s^{2 1}S − 1s2p¹P^o resonance line at 584 Å, whose upper level is the lower level of He i λ4388 (1s2p¹P^o − 1s5d¹D), as shown in Figure 11. According to Najarro et al. (2006), a similar effect is also seen for the singlet He i λ4922 (1s2p¹P^o − 1s4d¹D) by Najarro et al. (2006), which is not surprising given that both lines have the same lower level. We can test this explanation further, as we might expect the He i λ5016 (1s2s¹S − 1s3p¹P^o) fit to be better, since it has a different lower level, as argued by Najarro et al. (2006). In Figures 17 and 18 we compare the profile fits by cmfgen and fastwind to these three He i singlet lines. We find that even when the cmfgen fits to λ4388 and λ4922 are very poor (e.g., for the late O supergiants in Figure 18), the fit to He i λ5016 is quite good. Unfortunately, the He i λ5016 line is not computed in the formal solution by fastwind.

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In Paper III we argued that because of its approximate treatment of line blanketing and blocking, fastwind should be less sensitive to the problem with the uncertain atomic data for the Fe iv lines that overlap with the 584 Å He i resonance line, and hence should give better fits to He i λ4388 than with the more exact treatment by cmfgen. Indeed this seems to be the case. Nevertheless, fastwind is not immune to the problem. For BI 170 (Figure 9) and Sk −69°124 (Figure 10) we get adequate fits to He i λ4922 but the He i λ4388 fastwind profile lines are too strong, albeit not as badly as with cmfgen. We note that in both cases we adopted a significantly hotter temperature based on the fastwind fits, as we knew to expect the triplets (such as He i λ4471) to give inconsistent answers and we tried to get the singlet lines weak enough to fit the profile. However, we can see that for some stars cmfgen and fastwind both give satisfactory fits to the singlet lines, i.e., the O5.5 I(f) star AzV 75 (Figure 3), the O6 I(f) star AzV 26 (Figure 4), the O8 V star NGC 346-682 (Figure 5), and the O5.5 V((f)) star Sk −70°69 (Figure 8). The fact that the He i singlet fits only fail in cmfgen for the late-type supergiants in our sample is not surprising, as the influence of the iron lines depends upon the effective temperature and surface gravities.

Finally, we investigate the effect of microturbulence on the He i singlet profiles. We argued above that a higher value than 10 km s⁻¹ was needed in order to obtain satisfactory fits for all of the He i triplet lines (particularly He i λ5016) for the three late-O supergiants. In Figures 19 through 21 we show the effect of increasing the microturbulence velocity on the He i singlets for these three stars. For AzV 223 (Figure 19) there is actually a slight improvement in the reasonably good fits obtained with cmfgen and fastwind. For BI 170 and Sk −69°124, where the singlet fits for He i λ4388 and λ4922 were already poor, increasing the microturbulence certainly does not improve the situation, as the model lines were already too deep. Note, however, that the cmfgen fit for He i λ5016 remains quite good even at the higher microturbulence values.

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For convenience, we show the fits for the H, He i, and He ii lines used when originally modeling these three stars but now computed using a 15 km s⁻¹ microturbulence (see Figures 22–24). The improvement in the fastwind He i λ4471 fits are dramatic. Note that we did not have to change the effective temperatures or any other physical parameters to achieve this improvement. This improvement is not completely surprising, as it is well known that macroturbulence is larger in supergiants (see, e.g., Howarth et al. 1997), and hence it might be expected that microturbulence be larger as well.

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3.2.3. Summary

The prevailing wisdom has been that fastwind does a poor job of fitting the He i triplets (particularly He i λ4471) for late O-type supergiants for poorly understood reasons (Repolust et al. 2004, and Papers I–III). At the same time, Najarro et al. (2006) recommend eschewing the He i singlets because of the overlap of the Fe iv lines with the He i 1s^{2 1}S − 1s2p¹P^o resonance line at 584 Å, whose upper level is the lower level of the He i λ4388 and λ4922 lines. What we found here modifies this significantly, at least for our sample of SMC and LMC stars¹³:

• 1.  
  The cmfgen problem with the He i singlets occurs only for the late O-types; we obtain quite satisfactory fits for earlier types. This is not really surprising as the strengths of the Fe iv transitions will also be temperature dependent. However, even when the fits for He i λ4388 and λ4922 fail, the He i singlet λ5016 is very well fit by cmfgen, in accordance with Najarro et al. (2006), as the lower level of this line will be less affected. Although fastwind does a better job of matching the He i λ4388 and λ4922 lines (since its blanketing calculation is not exact), it is nevertheless affected. Including He i λ5016 in the formal solution for fastwind would be a useful addition.
• 2.  
  The fastwind problem with the He i triplets (such as He i λ4471) for the late O-type supergiants largely goes away if we use a microturbulence of 15–20 km s⁻¹ in computing the line profiles. At the same time, a value of 15 km s⁻¹ for the late O-type supergiants causes cmfgen to give more consistent results for the He i λ5876 triplet, which is otherwise too weak compared to He i λ4471, suggesting that a large value (15–20 km s⁻¹) may be more appropriate for late O-type supergiants. Inclusion of He i λ5876 in the formal solution for fastwind would also be useful.

We emphasize that we cannot achieve the same good effect by simply lowering the effective temperature for the fastwind models to strengthen the He i λ4471 line. To do so renders the He ii lines ridiculously weak. We demonstrate this in Figure 25. For AzV 223, if we keep the microturbulence velocity at 10 km s⁻¹ and lower the effective temperature of the model from 31,600 K (shown in dashed red) to 29,000 K (shown in dashed green) the He i λ4471 line gets a bit stronger, but still much weaker than the actual observed spectrum. The lower two panels though show the drastic change in the He ii lines. Thus, our temperatures for the late O-type supergiants (where the He i singlet/triplet behaviors are undependable) are nevertheless well constrained by the He ii strengths.

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Given these difficulties, we would conservatively estimate the actual uncertainty in the determination of the effective temperature of a star with either code to be 1000 K, rather than our 500 K fitting precision. This is consistent with the errors usually quoted, e.g., Papers I–III. Still, this 2%–3% uncertainty is remarkable, given the physical complexities involved.

We turn next to our results for the surface gravity. Our experience in the fitting is that the wings of the Hγ profiles allow fitting log g to a precision of about 0.05 dex. Yet, it is clear from Table 3 that there is a systematic difference between the cmfgen and fastwind results, with fastwind consistently yielding a surface gravity that is about 0.10 dex lower. This problem was first reported by Neugent et al. (2010) based upon preliminary results of the current project, and a similar systematic effect can be inferred from Najarro et al. (2011) using a smaller sample of Galactic stars.

In general, cmfgen does a better job of fitting the Hγ wings, because it includes metal lines that might affect the wings, such as O ii λ4350. But, that can not be the entire explanation, as one would expect to find O ii lines primarily among the later O-type supergiants, and the systematic difference in log g seems to be independent of effective temperature or luminosity class.¹⁴ Consider the excellent fits shown from Hγ for the O5.5 V((f)) star Sk −70°69 in Figure 8. The cmfgen and fastwind Hγ profiles are nearly indistinguishable, and yet the former corresponds to log g = 3.80 and the latter to log g = 3.70, albeit with rather different effective temperatures. Perhaps a more convincing case is that of the O6 I(f) star AzV 26, where the adopted effective temperatures are the same and the Hγ profile fits are hard to distinguish in the wings (Figure 4), but the cmfgen fit is based on log g = 3.60 and the fastwind fit on log g = 3.50.

How sensitive are the fits to log g? In Figure 26 we show two fastwind models, one computed with the value of log g = 3.50 we obtained using fastwind and one computed with the log g = 3.60 value we derived using cmfgen. There is a clear difference, and based on the (uncontaminated) blue side of the line, there is no question of mistaking which fit is better with fastwind. Thus, the difference between the two codes in the resulting value for the surface gravity is quite real. At the same time, the exact value we obtain with either code is clearly dependent upon the normalization, and it would take a very brave person to say that we can do better (in an absolute sense) than 0.1 dex in log g. Thus, for the purposes of computing errors (below) we will adopt 0.1 dex as σ_log g, about twice the precision we achieved with a particular spectrum. (Note that the 0.1 dex value is consistent with what is usually quoted anyway; see, for example, Repolust et al. 2004 and Papers I–III.)

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We were naturally curious as to the source of the discrepancy in log g between the fits made with the two codes. D.J.H. computed a set of cmfgen models (pure H and He) spanning the relevant range in surface gravities and effective temperatures with various Stark broadening profiles, and the theoretical H lines were then compared to those of fastwind, computed by J.P. with the same physical parameters. The emergent profiles were computed both with no rotation and with a projected rotational broadening of vsin i = 80 km s⁻¹. It was immediately evident that there was a systematic difference in the wings of Hγ (and other H lines) in the same sense shown by our fitting methods: the wings of Hγ were always, and mostly significantly, lower with the fastwind models, requiring lower surface gravities (by typically 0.1 dex, consistent with our findings from the profile fitting) to achieve the same effect.¹⁵ Our Magellanic Cloud star fitting with cmfgen used the Lemke (1997) Stark profiles, while fastwind adopts the Schöning & Butler (1989a, 1989b) formulation, but the differences in our pure H/He tests were still prominent—though slightly smaller—even when the Schöning & Butler Stark profiles were used with cmfgen.¹⁶ Investigation showed that the difference between the two codes probably traces to the treatment of electron scattering, at least for objects with log g below 4.0. As described in the Appendix, cmfgen uses coherent scattering to compute the level populations, but incoherent scattering when computing the spectrum. In contrast, fastwind does not explicitly use any scattering model, but instead uses constant Thomson scattering emission over the line (with the mean intensity from the neighboring continuum) to approximate the incoherent scattering approach (see Mihalas et al. 1976). However, these tests showed that the constant emission method used by fastwind actually is closer to that of coherent scattering, and does not have the intended effect on the emergent line profiles; i.e., fastwind's surface gravities are probably too low. Further work is planned on investigating this problem, particularly with respect to fully blanketed model atmospheres as used in our current analyses.

Does this have any implication for the "mass discrepancy"? Groenewegen et al. (1989) and Herrero et al. (1992) called attention to the fact that the masses for O-type stars derived from spectroscopic analysis via stellar atmosphere modeling were often smaller than those derived by evolutionary tracks using the same effective temperatures and bolometric luminosities, sometimes by as much as a factor of two. The "spectroscopic mass" depends upon the derived surface gravity and the stellar radius (m_spect ∼ gR²), with the value of R derived from the bolometric luminosity and effective temperatures, the same quantities used with the evolutionary models to find the masses. Thus, the determination of the surface gravity has long been considered a possible culprit if the problem lay with the atmosphere models. Improvements both in the stellar evolutionary models and stellar atmosphere models have resulted in reducing the mass discrepancy. For instance, Lanz et al. (1996) found that using fully blanketed atmosphere models greatly alleviated the problem. However, most modern studies find that the problem has not been completely eliminated; see discussion in Paper II and Repolust et al. (2004) for instance. The analysis of the current sample of stars in Papers I–III found stars with significant mass discrepancies and stars without, with the stars with the largest discrepancies tending to be those of earliest type.

Consider the O3.5 V((f+)) star LH 81:W28-23. The spectroscopic mass derived in Paper II was 24 M_☉, while its evolutionary mass was 55 M_☉. Increasing the surface gravity by 0.15 dex (as indicated by the cmfgen fit) by itself would raise the spectroscopic mass to 34 M_☉, significantly reducing the mass discrepancy. The fact that for this star we also find a significantly cooler effective temperature from the cmfgen fit also helps reduce the mass discrepancy, as it both lowers the derived evolutionary mass (as the bolometric correction becomes less negative, lowering the luminosity), and it also increases the stellar radius, further increasing the spectroscopic mass.

In Table 6 we reconsider the mass discrepancy for all of these stars. We begin by computing the bolometric luminosities (log L, in solar units) using the approximation that the bolometric correction BC = −6.90log T_eff + 27.99 from Paper II: M_bol = M_V + BC, where the values for M_V come from Table 2. Then log L = −(M_bol − 4.75)/2.5, where we have taken the bolometric magnitude of the Sun to be 4.75. The effective radius R (in solar units) then follows from the Stefan–Boltzmann equation: log L = 4log T_eff − 15.05 + 2log R, where we have adopted the effective temperature of the Sun as 5778 K. In computing the spectroscopic mass m_spec (also in solar units) we have taken the surface gravity we have measured (Table 3) and corrected this for the effects of centrifugal acceleration, as the measured (effective) surface gravities have been lessened by the rotation of the star. For this, we used

$$\begin{eqnarray*} &&g_{\rm true}=g_{\rm eff} + \frac{(v \sin i)^2}{6.96R}, \end{eqnarray*}$$

following Repolust et al. (2004), where g is in cgs units and vsin i is in km s⁻¹. Two caveats are worth noting: first, this correction is valid only in a statistical sense, as it assumes randomly distributed rotational axis orientations. Secondly, the rotational velocities vsin i we use from Table 2 are really too large on average, as the broadening of our profiles include (1) the actual rotation, (2) the instrumental resolutions (of order 100–150 km s⁻¹), and (3) the macroturbulence velocity¹⁷ (of order 30–100 km s⁻¹). Still, the corrections are small (typically 0.01–0.02 dex, and 0.05 dex at the most extreme), and over-estimating vsin i would increase the value of m_spect, reducing the mass discrepancy. Finally,

$$\begin{eqnarray*} &&m_{\rm spec}= (g_{\rm true}/g_\odot) R^2, \end{eqnarray*}$$

where we take log g_☉ = 4.438.

Table 6. Derived Quantities and the Mass Discrepancy

Star	fastwind							cmfgen
	Teff	R	log gtrue	log L	mspect	mevol	log mspec/mevol	Teff	R	log gtrue	log L	mspect	mevol	log mspec/mevol
	(K)	(R☉)	(cgs)	(L☉)	(M☉)	(M☉)	(dex)	(K)	(R☉)	(cgs)	(L☉)	(M☉)	(M☉)	(dex)
AzV 177	44,000	8.9	3.85	5.43	$20.7^{+4.8}_{-3.9}$	$41.6^{+5.1}_{-4.5}$	−0.30 ± 0.12	44,500	9.0	3.99	5.45	$28.6^{+6.6}_{-5.3}$	$42.7^{+4.1}_{-3.8}$	−0.18 ± 0.12
AzV 388	41,500	11.0	3.88	5.51	$33.6^{+7.7}_{-6.3}$	$40.8^{+4.0}_{-3.6}$	−0.08 ± 0.12	44,000	10.6	4.12	5.58	$54.4^{+12.5}_{-10.2}$	$44.3^{+4.3}_{-3.9}$	+0.09 ± 0.12
AzV 75	40,000	23.1	3.67	6.09	$91.4^{+21.0}_{-17.1}$	$71.3^{+5.3}_{-4.9}$	+0.11 ± 0.12	39,500	23.2	3.71	6.07	$100.6^{+23.2}_{-18.8}$	$70.0^{+5.2}_{-4.8}$	+0.16 ± 0.12
AzV 26	38,000	27.2	3.52	6.14	$89.1^{+20.5}_{-16.7}$	$75.8^{+6.2}_{-5.7}$	+0.07 ± 0.04	38,000	27.2	3.61	6.14	$109.6^{+25.2}_{-20.5}$	$75.8^{+6.2}_{-5.7}$	+0.16 ± 0.12
NGC 346-682	35,500	8.0	4.11	4.96	$30.1^{+6.9}_{-5.6}$	$23.2^{+1.6}_{-1.4}$	+0.11 ± 0.12	35,800	8.0	4.11	4.97	$29.8^{+6.9}_{-5.6}$	$23.6^{+1.3}_{-1.4}$	+0.10 ± 0.12
AzV 223	31,600	16.4	3.48	5.38	$29.5^{+6.8}_{-5.5}$	$28.9^{+2.2}_{-2.1}$	+0.01 ± 0.12	31,000	16.6	3.62	5.36	$42.0^{+9.7}_{-7.9}$	$28.4^{+2.0}_{-1.9}$	+0.17 ± 0.12
LH 81:W28-23	48,000	10.0	3.77	5.68	$21.6^{+5.0}_{-4.0}$	$57.5^{+5.3}_{-4.8}$	−0.43 ± 0.12	46,500	10.2	3.91	5.64	$30.9^{+7.1}_{-5.8}$	$52.8^{+4.9}_{-4.4}$	−0.23 ± 0.12
Sk −70°69	39,500	9.8	3.73	5.32	18.7$^{+4.3}_{-3.5}$	$32.2^{+2.2}_{-2.0}$	−0.24 ± 0.12	41,000	9.6	3.83	5.37	$22.8^{+5.3}_{-4.3}$	$35.2^{+2.5}_{-2.4}$	−0.19 ± 0.12
BI 170	31,500	16.5	3.23	5.38	$16.8^{+3.9}_{-3.1}$	...	...	30,000	17.0	3.33	5.32	$^{+5.2}_{-4.2}$	...	...
Sk −69°124a	29,000	21.1	3.17	5.45	$24.0^{+5.5}_{-4.5}$	$29.4^{+1.9}_{-1.7}$	−0.09 ± 0.12	27,800	21.7	3.24	5.40	$29.7^{+6.8}_{-5.6}$	$25.3^{+1.8}_{-1.7}$	+0.07 ± 0.12
Mean difference	...	...	...	...	...	...	−0.09 ± 0.04	...	...	...	...	...	...	+0.01 ± 0.04
Median difference	...	...	...	...	...	...	−0.04	...	...	...	...	...	...	+0.09

Note. ^aSk −69°124 is excluded from the averages as we conclude it is likely a binary. See text.

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To determine the evolutionary masses for the LMC stars, we used the newest Geneva models of V. Chomienne et al. (2013, in preparation) based upon the latest improvements described by Ekström et al. (2012) but computed for a metallicity of z = 0.006. For the SMC stars, we used the older Geneva models described by Maeder & Meynet (2001). Both models incorporate the effects of stellar rotation, albeit with somewhat different starting assumptions. To obtain the evolutionary masses, we performed a linear interpolation for log m_evol using the observed log L and whichever two tracks bounded the star's luminosity. This process was mainly straightforward,¹⁸ but for BI 170 there was an ambiguity in the mass predicted by the evolutionary tracks, as the tracks zig-zag at the end of core H-burning, which left us unsure as to its evolutionary mass.

Of course, in evaluating the significance of these comparisons we must understand the size of the errors. Our fitting uncertainties in the effective temperatures are 500 K; we assume that the actual error is 1000 K, in keeping with the above discussion. This leads to an error in log T_eff of ±0.015 dex for our coolest star and ±0.009 dex for our hottest star. The uncertainty in log L will include a component that is due to the propagation of the error in log T_eff due to the bolometric correction (ΔBC = −6.9Δlog T_eff; i.e., ∼0.10 mag) but will be dominated by the uncertainty in the absolute visual magnitude, which we estimate to be 0.18 mag, based upon a 0.15 mag uncertainty in A_V and the uncertainty in the distances to the LMC and SMC, which we take to be 0.1 mag based upon discussion in van den Bergh (2000). Thus the typical uncertainty in log L is 0.08 dex. This error in log L dominated the uncertainty in our determination of m_evol, and results in a typical error of 0.04 dex in log m_evol. The uncertainty in log m_spec is then

$$\begin{eqnarray*} &&\sigma _{\log m_{\rm spec}}=\sqrt{4\sigma ^2_{\log R}+\sigma ^2_{\log g}}. \end{eqnarray*}$$

The uncertainty in log R then follows from propagation of errors through the Stefan–Boltzmann equation, i.e.,

$$\begin{eqnarray*} &&\sigma _{\log R}= \sqrt{0.25 \sigma ^2_{\log L} + 4\sigma ^2_{\sigma _{\log T_{\rm eff} }} }, \end{eqnarray*}$$

or σ_log R ∼0.04 dex for all of our stars. Our fitting uncertainty in log g is 0.05 dex, but as argued above, we will assume an "accuracy" of 0.10 dex for the surface gravities determined by each code. This leads to a nearly constant uncertainty in log m_spec of 0.13 dex.

However, in measuring the size of the mass discrepancy we do better to consider the quantity Δlog m = log m_spec/m_evol, and its associated error. This is because both m_spec and m_evol depend upon log L_☉, and any error in the luminosity will be the same for a given star, be it due to the uncertainty in A_V or in the BC. In evaluating the error we assumed that the error Δm_evol was roughly 0.5 ΔL. We can calculate this error straightforwardly, since

$$\begin{eqnarray*} &&\log m_{\rm spec} \sim \log g + \log R^2 \sim \log g + \log L - 4 \log T_{\rm eff} \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&\log m_{\rm evol} \sim 0.5 \log L \end{eqnarray*}$$

then

$$\begin{eqnarray*} &&\log \frac{m_{\rm spec}}{m_{\rm evol}} \sim \log g - 0.5 \log L - 4 \log T_{\rm eff} \end{eqnarray*}$$

and hence

$$\begin{eqnarray*} &&\sigma _{\log \frac{m_{\rm spec}}{m_{\rm evol}}} = \sqrt{\sigma ^2_{\log g} + 0.25\sigma ^2_{\log L} + 16 \sigma ^2_{\log T_{\rm eff}}}, \end{eqnarray*}$$

i.e., greatly reducing the dependence on the error in log L. The errors on log m_spec/m_evol are thus about 0.12 dex. In other words, mass discrepancies smaller than ∼30% for a single object are not significant. If we had instead adopted our fitting precision (rather than twice the precision) on the uncertainties, then the error on this quantity would have been 0.07 dex rather than 0.12 dex.

We should also comment further on the actual "error" in the evolutionary models. What if we had chosen different models? The LMC models assume an initial rotation velocity that is 40% of the breakup speed (Ekström et al. 2012) as this agrees well with the peak rotational velocities observed in young B stars (Huang et al. 2010). (Of course, this is a statistical average, and any individual star may have an initial rotation that is higher or lower than this value.) However, some evidence suggests that this rotation speed may be a bit high; see discussion in Neugent et al. (2012). For the mass range and ages considered here, there is generally little difference in the evolutionary masses we would compute using the tracks with and without rotation: for instance, rather than the m_evol = 35.2 M_☉ we obtain with the cmfgen results and the rotating models, we would instead obtain m_evol = 35.1 M_☉, a completely negligible difference. If instead we had used the older rotation models at LMC metallicity described by Meynet & Maeder (2005), we would have obtained $m_{\rm evol}=37.9^{+3.0}_{-2.9}\ M_\odot$, slightly higher that what we obtain from the newer models, but still within the errors. We note that Massey et al. (2012) recently compared the evolutionary masses to the masses determined in two binary systems, finding very good agreement, with the evolutionary masses higher by about 11% (i.e., 0.05 dex) compared to the dynamical masses.

The comparisons are shown in Figure 27, where we have plotted log m_spec/m_evol against various properties. Consider the left panels in the figure. We see that with our fastwind results there is a systematic tendency for the masses to be too low. The average difference in log m between the spectroscopic mass and the evolutionary mass is −0.09 ± 0.04 dex, where the uncertainty quoted is the error-weighted sigma of the mean. The median difference (which helps avoid outliers) is also −0.09 dex. This is in the classic sense of the mass discrepancy, with the model atmosphere results deriving too small a mass compared to that predicted by the evolutionary tracks. By contrast, the values in the right panel, which show the cmfgen results, are more evenly distributed about the 0 line, and the average difference in log m is +0.01 ± 0.04, with a median of +0.09 dex. (Recall that we expect a small positive bias due to overestimating vsin i.) Another way to consider these data is from the point of view of outliers: in the case of fastwind, there are two stars whose masses are more than 2.5σ away from the masses of the evolutionary tracks (both in a negative direction, i.e., in the classic sense of the mass discrepancy), while there are none more than 1.9σ away with the cmfgen results. Thus, the higher surface gravities found with cmfgen are more consistent with the evolutionary masses. Still, it should be noted that even in the case of the two fastwind "outliers" the spectroscopic masses found by cmfgen also show relatively large differences with the evolutionary masses. Note from the bottom panel that these two stars are the ones with the highest temperatures, consistent with the findings of Paper III that the fastwind mass discrepancy was most significant for the hotter stars in the sample. Further investigation of these stars may be instructive.

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cmfgen automatically computes the entire spectrum, including lines of C, N, and O. These lines are not as useful for determining the fundamental physical properties of the stars (such as effective temperatures), as the behavior even of the relative strength of two ions of the same element (i.e., N iii versus N iv) depends strongly not only on the effective temperature but also surface gravity, mass-loss rates, and especially the assumed N abundance (Rivero González et al. 2012a). However, it is reassuring to see how well these metal lines are fit. After we had obtained a good model fit with cmfgen paying attention to only the He and H lines (as we did with fastwind) we then varied the CNO abundances (if needed) to improve the fit to selected spectral lines of these elements. We list the final adopted CNO mass fractions in Table 5, and illustrate the fits in Figures 28 through 37. The final cmfgen fits are shown by the dashed blue lines; if these reflected a change in abundances, we show the original fits in red.

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In all cases the final fits are better than the original, but the final adopted CNO abundances in Table 5 should be taken as uncertain by factors of several. In principle, changing the CNO abundances is not an unreasonable thing to do, particularly for the supergiants, as we expect evolution of the surface composition of these elements as a star ages. Certainly in cases where we needed a higher He/H to fit the He lines, we also see strong evidence of changes in the surface composition. Still, there are complexities to using many of these lines for abundance determinations; Martins & Hillier (2012) demonstrate complexities of using the C iii 4647-51-50 complex, and argue that abundances determined using this line are suspect. We suspect these cautions may extend to other lines.

Rivero González et al. (2012a) derive nitrogen abundances¹⁹ using a modified version of fastwind for two of our stars (using the identical spectra), namely, AzV 177 and LH 81:W28-23. For AzV 177 they find an enhancement of about a factor of three, while we find no enhancement necessary with cmfgen for a good fit. For LH81:W28 they find an enhancement of a factor of four, while we find a similar enhancement with cmfgen of a factor of five. Clearly more tests are needed to know if fastwind N abundances are consistent with those of cmfgen or not.

Sk −69°124 (Figure 37) represents a special case. In our initial modeling (red) we did not change the abundances but simply noted that all the CNO lines were poorly fit, and are inconsistent. D.J.H. attempted to refit the star and found improved fits by lowering the CNO abundances to ∼1/3 that of the LMC, almost SMC-like.²⁰ This provides some improvement to the optical CNO lines, but it cannot be understood from the point of stellar evolutionary theory, as may be inferred from below. An intriguing possibility would be that this star formed out of gas that was accreted from the SMC (see, e.g., Olsen et al. 2011). The radial velocity of Sk −69°124 is also abnormally small (190 km s⁻¹) for LMC membership, about 90 km s⁻¹ more negative than what is expected given the star's position in the LMC. This might support the possibility that it originated from gas drawn in from the SMC, but even so, the velocity is quite abnormally small.²¹ We think a more likely explanation is that the star is a binary and that the difficulties in fitting the CNO abundances and the strain to fit even the He lines (Section 3.2) are due to that. We therefore have not included Sk −69°124 in computing the differences in the tables.

The Geneva evolutionary models also predict the evolution of the surface abundances; in Figure 38 we show the expected change with age over the course of main-sequence evolution. The exact details depend upon how much initial rotation is assumed, but generally we expect to see a strong N enhancement (along with a modest He enhancement) at the expense of C and N. For the SMC results, things are pretty much as expected for the dwarfs, which all have normal SMC abundances. For the two supergiants, the enhancement for AzV 75 makes sense; we are surprised though that AzV 233 does not require a higher N abundances, but inspection of Figure 33 shows that at this cooler temperature N iv λ4058 is nonexistence, and the only constraint comes from N iii λλ4634, 42. For the LMC stars, there are three surprising cases. LH81:W28-23 shows a very depleted C abundance relative to O, and that is hard to understand from an evolutionary point of view. With a high He enhancement, one would expect both C and O to be similarly depleted, and in any case, such enhancements are not expected early in a star's life, which is what the effective temperature and luminosity of LH81:W28-23 suggests. For BI 170 it is unexpected to see C and O depletion without increased N. Better constraints on the abundances may be obtained by including the UV lines, and such studies are planned.

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cmfgen is a general purpose non-LTE radiative transfer code whose properties have been outlined by Hillier (1990), Hillier & Miller (1998), and Hillier & Dessart (2012). In the context of OB stars, it solves the time-independent spherical radiative transfer code in the co-moving frame subject to the constraints on the level populations imposed by the equations of statistical equilibrium. To simplify the atomic models, super levels are utilized. Levels within a super level are assumed to have the same departure coefficient (for a few cases a slightly more sophisticated assumption is utilized). The number of super levels is model and ion dependent and is easily changed. To compute atomic level populations we adopt a Doppler profile of fixed width (usually defined by the microturbulent velocity). The temperature is computed using the assumption of radiative equilibrium. The accuracy of the temperature structure is checked using flux conservation, and the electron-energy balance. The later two are not exactly satisfied because of numerical discretization errors, and the use of super levels (Hillier 2003b). To treat electron scattering we assume coherent scattering in the co-moving frame. Test models in which this assumption is relaxed (in cmfgen only) show very similar spectra to models computed assuming coherent scattering.

cmfgen computes the hydrostatic structure using an iterative procedure which is undertaken during the general cmfgen procedure. For O stars, only a few hydrostatic iterations are needed to obtain a well converged hydrostatic structure. The initial hydrostatic structure is generally taken from an earlier model (but can be generated), although in earlier versions of cmfgen we often utilized the density structure from a tlusty (Lanz & Hubeny 2003) model. We attach a velocity law to the hydrostatic structure at 0.5 times the isothermal sound speed, although this was recently adjusted to 0.75 times the isothermal sound speed. Below the sonic point, the vdv/dr term (see, for example, Lamers & Cassinelli 1999) is taken into account. While possible, a contribution to the hydrostatic structure from turbulent pressure was not taken into account in the models presented in this paper.

With only a few exceptions, atomic data and super-level assignments are made available to cmfgen via ASCII files. For many species, more than one data set is available. Photoionization cross-sections are smoothed by a Gaussian (FWHM of 3000 km s⁻¹) although for most species this can be varied. For dipole-forbidden transitions, without published data, we adopt a fixed collision strength of 0.1. In general, D.J.H. has not found major changes in the He i triplet or H lines when different atomic data for Fe has been adopted, although this may be regime dependent. Level dissolution is taken into account for levels higher than n_eff > 2z where z is the effective charge on the core (e.g., 3 for C iii). Collision rates for He i are from Berrington & Kingston (1987). For H we have three different data sets, but for the present calculations we utilized collision strengths adapted from Mihalas et al. (1975), which are similar to those utilized in tlusty.

Atomic data is taken from a wide variety of sources with the major sources being NIST (Ralchenko et al. 2010; Kramida et al. 2012), the Opacity Project (Seaton 1987; Cunto et al. 1993), Bob Kurucz,²² and Sultana Nahar's NORAD Web site.²³

Once the atmospheric structure and level populations are known, we undertake a spectrum calculation using cmf_flux (Busche & Hillier 2005)—this is typically done only for the observable wavelength range of 1000 Å to 4 μm. Stark broadening is considered for H and He, Voigt profiles are used for UV resonance lines, while pure Doppler profiles are adopted for all other lines. For this calculation, we compute the opacities and emissivities in the co-moving frame. We also compute the radiation field in the co-moving frame which allows us to compute the frequency-dependent emissivity due to electron scattering. To take into account thermal broadening due to the electron motions, we need to iterate—only two iterations (if that) are needed for O stars. For the computation of the observer's frame spectrum, we adopt the usual (p, z) coordinate system. Along each ray, we have enough points (typically every 0.25 Doppler widths) to fully sample the rapidly varying line opacity and emissivity, and to adequately sample optical depth space. We use the same spatial grid for all frequencies. The opacity and emissivity are taken from the co-moving frame calculation, and are interpolated into the observer's frame.

The H and He atomic models used in cmfgen can be briefly described as follows. H i has 30 levels with 20 super levels; its super levels begin at level 16. Similarly, He ii has 30 levels with 22 super levels; its super levels begin at level 21. The He i model atom has 69 levels (up to n = 20). Levels with n < 8 have their l states split, except that l = 3 and higher l states are treated as a single state. Singlet and triplet states are distinct in the model atom, and 45 super levels were used with the present calculations.

In general, cmfgen provides excellent matches to observed optical and UV spectra as can be seen, for example, in fits to O supergiants (Bouret et al. 2012), O dwarfs in the Galaxy and SMC (Martins et al. 2012; Bouret et al. 2013), and to the luminous blue variable AG Carinae (Groh et al. 2009).

fastwind is specifically designed for modeling the optical and NIR spectra of O-type stars; details are given in Santolaya-Rey et al. (1997), Puls et al. 2005, and Rivero González et al. (2012b). The latter reference is especially pertinent for the most recent version of fastwind, which has been used in the present study. fastwind consists of two program packages. The first one calculates the atmospheric structure and the NLTE occupation numbers. To set up the atmospheric stratification, fastwind adopts a smooth transition between an analytical wind structure, described by a β velocity law (Castor & Lamers 1979; for the actual formulation; see Santolaya-Rey et al. 1997), and a quasi-hydrostatic photosphere with a velocity law following from the equation of continuity. The transition between wind and photosphere can be specified by the user, with a default value (used here) of 0.1 of the local sound speed, which results in a hydro-structure that compares well with hydrodynamic models (e.g., Puls 2009). Also the photosphere is calculated in spherical symmetry, i.e., allows for a potential extension, and accounts for photospheric line and continuum acceleration and a consistent temperature structure. In the first iterations, the radiative acceleration is calculated from the Rosseland mean, later updated by using the flux mean from the current NLTE opacities. The photospheric stratification is calculated without accounting for any turbulent pressure.

To solve the NLTE rate equations, a distinction is made between two groups of elements, called explicit and background elements.

The explicit ones (mainly H and He but also, depending on the purpose of the analysis, other elements such as N) are those to be used as diagnostic tools. They are treated with high precision, i.e., by a complete NLTE approach and a co-moving frame radiative transfer (only Doppler-broadening), by means of an Accelerated Lambda Iteration with an Accelerated Lambda Operator designed for co-moving frame calculations (Puls 1991). The present version of fastwind does not account for level dissolution.

For these explicit elements, a detailed description of all required atomic data (levels, transition types, cross-sections in various parameterizations) needs to be provided in an external file, following the so-called DETAIL input format (Butler & Giddings 1985).

The background elements are used to obtain the background radiation field, accounting for line-blocking and blanketing effects. The occupation numbers for these elements are also calculated in NLTE, but using a Sobolev line transfer for the bulk of weaker lines, and a co-moving frame transfer for only those lines which are still strong in the wind. The opacities and emissivities from these background elements, together with those from the explicit ones, are used to calculate the radiation field and the temperature structure. The atomic data for the background elements are provided in a fixed way, from the compilation by Pauldrach et al. (1994, 2001).²⁴

The decisive, time-saving approximation within fastwind regards the calculation of the line opacities and emissivities required for the calculation of the background radiation field. In brief, suitable means for the line opacities/emissivities are used, averaged over a frequency interval on the order of the terminal wind speed (∼1000–500 km s⁻¹); see Puls et al. (2005) for further details.

The temperature structure is calculated from the thermal electron balance (e.g., Kubát et al. 1999 and references therein), accounting for all processes from explicit and background elements. This is done in parallel withe iteration cycle rate-equations and radiative transfer.

The second program package allows to calculate the emergent line profiles. The radiative line transfer is solved using the occupation numbers from the explicit elements and the background opacities/emissivities from the converged NLTE solution. The integrations are performed in the observer's frame, and on a radial micro-grid. Basic features are a separation between line and continuum transport (see Santolaya-Rey et al. 1997 and references therein), and various line broadening options (most important: Stark and Voigt broadening). An external input file with all information needs to be provided, notably the broadening functions or parameters.

Finally, let us briefly describe the H and He atomic models used in the present work. Our H i and He ii models consist of 20 levels each until principal quantum number n = 20, and He i includes 49 levels until n = 10, where levels with n = 8...10 have been packed.

H and He atomic data are similar to those used in DETAIL (see above), and have been described, though for "smaller" model atoms, in Santolaya-Rey et al. (1997). The only exception are the hydrogen collisional strengths, where the dataset provided by Giovanardi et al. (1987) is now used as a default, though other data are possible as well. The effects of using different collision strengths have been tested by Przybilla & Butler (2004) and Najarro et al. (2011), and both publications conclude that the differences are negligible for UV and optical lines, but may become important at infrared wavelengths (from the Paschen series on).

The line-blanketed version of fastwind has been used with good success to model the physical parameters of Galactic O stars in both the optical (Repolust et al. 2004) and NIR (Repolust et al. 2005), Magellanic Cloud O-type stars (e.g., Papers I–III), B-type supergiants in M33 (Urbaneja et al. 2005), and a very low-metallicity Of type star in IC 1613 (Herrero et al. 2012), to name just a few applications. Usage with an automatic fitting algorithm resulted in the analysis of many hundreds of stars as part of the VLT-FLAMES project (e.g., Mokiem et al. 2006, 2007).