THE YELLOW AND RED SUPERGIANTS OF M33^*

We now turn to our yellow sample and begin by examining their radial velocities. In Section 3.2, we characterized the rotation curve of M33 based on the radial velocities of 55 H ii regions from Zaritsky et al. (1989). However, these radial velocities were obtained in a relative, as opposed to absolute (heliocentric), sense and often had high associated errors (10–15 km s⁻¹). Thus, to calculate the expected radial velocities for our yellow sample, we opt to use a characterization of M33's rotation curve based on our probable RSGs.

In Figure 11, we plot X/R versus observed radial velocity for our rank 1 RSGs. A linear least-squares fit to the sample yields a relation (in km s⁻¹) of

$$\begin{eqnarray*} &&V_{\rm expect} = -182 - 81.9 (X/R) \end{eqnarray*}$$

with a dispersion of 14.3 km s⁻¹. We see that the slope produced by this relation varies slightly from that produced by the H ii regions of Zaritsky et al. (1989). We also note that this relation accurately reproduces the systemic velocity of M33 (−180 ± 3 km s⁻¹; Kerr & Lynden-Bell 1986).

Using this relation, we calculate the expected radial velocity for each of our yellow stars, were it a genuine M33 member. In Figure 12, we plot the observed minus expected radial velocities versus X/R for our yellow sample. As in Figure 8 for our red sample, we expect the genuine M33 members to fall near the zero point of the y-axis. We note that although two bands of stars are evident in this figure, the separation is not nearly as distinct as seen in Figure 9 of Drout et al. (2009) for the case of M31 (where both the systematic and rotational velocities are larger than the modest values of M33). It is not until the value of X/R approaches zero that the two bands are noticeably distinct. Thus, if we were to restrict our sample of potential supergiants to those that Figure 12 indicates possess a modest separation from the bulk of Milky Way dwarfs, we would be left only with stars in the NW portion of M33.

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Additionally, the Besançon model for galactic dwarfs applied to the color and magnitude range of our yellow sample reveals that we can expect modest contamination from high-velocity foreground dwarfs across the entire face of M33. In Figure 13, we plot a histogram of the velocities predicted by the Besançon model (solid, black) and the observed velocities of our yellow sample (dashed, red). In this figure, the output of the foreground model has been scaled by the fraction of our initial yellow catalog we actually observed. We see that at a radial velocity of approximately −200 km s⁻¹ (corresponding to X/R = 0.22 for a typical M33 member in our formulation of M33's rotation curve) a majority of the stars in our sample should be genuine M33 members. However, at a velocity of −170 km s⁻¹ (X/R = −0.15 for a typical M33 member) a significant portion of our sample is likely composed of foreground dwarfs. Based on this model, we can predict 2 foreground stars with velocities beyond −300 km s⁻¹, 9 foreground stars beyond −200 km s⁻¹, and 17 foreground stars beyond −150 km s⁻¹ (the region in which we also expect to find 65% of our genuine supergiants). Thus, we conclude that even in M33, which possesses a modest radial velocity separation, it would be prudent to obtain an independent means of membership determination to complement the radial velocity data.

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To address this issue, we now examine the possibility of using the O i λ7774 triplet as a luminosity indicator at M33 metallicities. Osmer (1972) demonstrated that O i λ7774 is one of the strongest spectral features in high-luminosity A- to F-type supergiants, yet it is relatively weak in dwarfs at galactic metallicities. Przybilla et al. (2000) later showed that this feature is a result of non-LTE atmospheric effects, exacerbated by sphericity. Although promising, it is not yet understood whether this luminosity dependence extends to the cooler G-type supergiants and what effect, if any, metallicity creates.

As an initial test we measured the equivalent width of the deepest spectral feature in the wavelength range 7760–7785 Å. The continuum level was determined by averaging the intensity over a length of the normalized spectra at both longer and shorter wavelengths than the desired line. These ranges for continuum averaging were taken at least 10 Å away from the line, and the equivalent width was calculated in a wavelength range ±5 Å from the wavelength of minimum intensity. For any stars observed in multiple fields, we adopt the average of the measured equivalent widths. Figure 14 shows a histogram of the results. We see that the distribution is distinctly bimodal, with a separation at approximately 1 Å.

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However, we caution that a strong spectral feature in the wavelength range given above does not necessarily imply a strong O i λ7774 line. This is demonstrated by Kirby et al. (2008), who generated synthetic spectra in the temperature range 4000–8000 K for a range of surface gravities well matched to our sample. Using the molecular transitions from Kurucz (1992), they find that CN is the most important molecule at red wavelengths for these stellar parameters. Most notably for our purposes, this includes a CN band head at ∼7774 Å along with companion features at ∼7865 Å and ∼7938 Å.⁶

Thus, in order to determine whether the population of stars with high equivalent widths represents a sample of M33 supergiants with a strong O i λ7774 feature, we examine all stars with an initial equivalent width greater than 0.5 Å, all stars with a velocity difference less than 100 km s⁻¹ in Figure 12, and all the stars observed in the 2009 fnt-4 field. We distinguish a CN band head near λ7774 from a strong O i λ7774 feature by (1) the presence or absence of the other two band heads and (2) the width of the feature. We find that our spectra fall roughly into four categories: spectra with no obvious features around λ7774; spectra with three broad CN features that possess a natural progression of line depths; spectra with features at all three band head wavelengths, but whose feature at λ7774 is deeper and narrower than the other two; and spectra with a single, narrow feature at λ7774. Examples of the latter three are shown in Figure 15.

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We believe that spectra falling in the latter two categories possess real O i λ7774 features. The red histogram in Figure 14 (right) represents the equivalent widths from these stars. We see that in almost every case the strongest equivalent widths represent real O i λ7774 features. We also note that with the exception of a handful of spectra (all of which possessed particularly high noise levels) the spectra with strong O i λ7774 are the same spectra that possess Paschen lines (see Section 3.1). This is promising, as the Paschen lines are expected to remain stronger to later spectral types for supergiants than for giants and dwarfs (Andrillat et al. 1995).

In Table 3 we list our measured equivalent widths for only the stars we believe have true O i λ7774 features. Propagating the errors in our calculation of the mean continuum level leads to errors in our measured equivalent widths on the order of 0.02 Å, while a comparison of the two equivalent widths measured for stars observed in both 2009 and 2010 reveals an average difference of 0.02 Å with a standard deviation of 0.36 Å.

In Figure 16 we again plot the observed minus expected velocities versus X/R for our yellow sample. The green points now represent stars with significant amounts of O i λ7774. We see that there is excellent agreement between stars with a distinguishable O i λ7774 feature and stars that have radial velocities consistent with the kinematics of M33. This indicates that the O i λ7774 triplet can, indeed, be used as a luminosity indicator for F- and G-type stars at the metallicity of M33.

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We do note that there are (1) two stars that seem to possess significant amounts of O i λ7774 but whose velocities place them in the middle of the strong band of foreground dwarfs and (2) a handful of stars that possess velocities consistent with, or more negative than, their expected M33 velocities, but which do not possess significant amounts of O i λ7774.

To explain the latter, we first checked our sample for possible non-stellar objects. Using the R-band images of the LGGS, we measured the FWHM of the point-spread function for objects in our sample with either an observed minus expected velocity less than 100 km s⁻¹ or significant amounts of O i λ7774. This process revealed seven previously known clusters in our sample (J013350.92+303936.9, J013419.85+303613.1, J013456.51+304100.4, J013400.19+303747.3, J013403.30+302756.1, J013256.08+303826.0, J013418.67+303137.7) and one additional extended object on the outskirts of the LGGS images (J013224.07+301242.9). Despite this fact, our sample still contains eight stars with −200 > V_obs > −300 km s⁻¹ and one star with V_obs < −300 km s⁻¹ that do not possess noticeable amounts of O i λ7774. As described above, however, this is consistent with the numbers predicted by the Besançon model for foreground stars. Thus, although it is not impossible for our sample to contain some bona fide M33 members without obvious O i λ7774, we find that our data are not inconsistent with all F- and G-type supergiants in M33 possessing significant amounts. A fuller examination of the temperature and metallicity dependence of this line will be addressed in a future paper.

For the purpose of comparing our sample to the current evolutionary models we assign the following ranks to our yellow sample: rank 1 stars possess both velocities consistent with M33 and significant amounts of O i λ7774, rank 2 stars have either a velocity difference less than 60 km s⁻¹ and a V_expect less than −150 km s⁻¹ or significant amounts of O i λ7774 and a velocity inconsistent with M33, rank 3 stars have a velocity difference less than 60 km s⁻¹ and V_expect greater than −150 km s⁻¹, and the rest of the sample is designated as rank 4. Even if it is possible for a genuine supergiant in our sample to lack a noticeable O i λ7774 feature, we expect our rank 3 sample to be heavily contaminated by foreground dwarfs (recall from Section 3.2 that M33 only stands out from the Galactic clutter at −150 km s⁻¹). With these definitions our sample contains 121 rank 1 stars, 14 rank 2 stars, 68 rank 3 stars, 705 rank 4 stars, and 8 clusters. This corresponds to a foreground contamination of ∼83%.

In Figure 17, we present the location of our newly determined RSGs and YSGs on the CMD of M33. Black dots represent the LGGS sample, red ×'s our rank 1 RSGs, green ×'s our rank 1 YSGs, and blue ×'s the M33 W-R population, determined by Neugent & Massey (2011). In Figure 18, we also show the spatial distribution of all three populations in the disk of M33. We note that the lack of RSGs in the central region of the galaxy is an observational bias, created by our restriction to isolated red stars as described in Section 2.2.

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For the YSGs, we follow the same procedure as Drout et al. (2009) and Neugent et al. (2010), using the Kurucz (1992) Atlas 9 model atmospheres to transform B − V colors into effective temperatures and luminosities. The B- and V-band photometries for all of our stars were taken from the LGGS, and we apply a constant reddening correction of E(B − V) = 0.12, in keeping with the median value found by Massey et al. (2007).

Before choosing which Atlas 9 model to use for the transformations, we first note that the metallicity of M33 has long been a topic of debate. As summarized in detail in Neugent & Massey (2011), there is strong evidence for a metallicity gradient within the plane of M33. Through the years strong gradients (e.g., Kwitter & Aller 1981; Zaritsky et al. 1989; Garnett et al. 1997), weak gradients (e.g., Crockett et al. 2006; Rosolowsky & Simon 2008; Bresolin 2011), and two-component models (Magrini et al. 2007) have all been suggested. By examining the ratio of WC to WN W-R stars in M33, Neugent & Massey (2011) suggest that the central regions may lie closer to solar metallicity while the outer regions are closer to that of the LMC (∼0.5 solar).

Based on this fact, we initially compute a transformation from B − V to effective temperature using the Z = 0.6 Z_☉ Atlas 9 models. A least-squares fit to the low surface gravity models over the temperature range 4000–10,000 K yields the relation

$$\begin{eqnarray*} \log {T_{\rm eff}} &=& 3.936 - 0.5272(B-V)_0\nonumber\\ && + 0.4752(B-V)_0^2 - 0.1749(B-V)_0^3. \end{eqnarray*}$$

In order to assess any errors that may result from the metallicity gradient present in M33, we also compute an analogous relation using the Z = Z_☉ Atlas 9 models. We find that the difference between the two transformations is negligible. For a star with (B − V)₀ = 0.6 the Z = 0.6 Z_☉ relation gives log T_eff = 3.753 while the Z = Z_☉ relation gives log T_eff = 3.759. As there is only a slight (0.06 dex) difference, we conclude that our original transformation is acceptable over the entire area of M33. Since we restricted our initial fit to the temperature range 4000–10,000 K, the relation for log T_eff given above is only valid for the range −0.11 ⩽ (B − V)₀ ⩽ 1.57 or 0.01 ⩽ (B − V) ⩽ 1.69. We find, however, that this range encompasses our entire yellow sample.

The bolometric corrections for our yellow sample are modest (a few tenths of a magnitude) and were also computed using the Atlas 9 model atmospheres. The models yield a relation

$$\begin{eqnarray*} &&{\rm BC} = -260.05 + 135.205 \times \log {T_{\rm eff}} - 17.5746 \times (\log {T_{\rm eff}})^2, \end{eqnarray*}$$

which is also valid over the temperature range 4000–10,000 K. Adopting a distance modulus of 24.60 (van den Bergh 2000), we calculate effective temperatures and bolometric luminosities for our rank 1 and rank 2 YSGs. These derived properties are listed in Table 4.

Deriving a place on the H-R diagram for RSGs based on photometry is a more uncertain process than that described above for YSGs.

First, the derivation of the effective temperature is complicated by the fact that RSGs have so many atomic and molecular lines that one is not observing the true continuum, but rather regions of relative transparency between overlapping absorption. Thus, B  −  V is no longer a color indicator but is instead primarily sensitive to surface gravity, due to line blanketing in the B band (see discussion in Massey 1998, and references therein). Furthermore, their surfaces appear to be covered in large spots (Freytag et al. 2002), resulting in slight differences in the effective temperatures and radii one derives from different bandpasses (Young et al. 2000). Levesque et al. (2006) found, using the state-of-the-art MARCS models (Gustafsson et al. 2003, 2008), that the effective temperatures derived from V  −  K were hotter than those derived by fitting spectral features (i.e., the TiO bands) by 105 K for the LMC and by 170 K for the SMC. They found, however, good agreement between the temperatures derived from V  −  R and the spectral fitting. Massey et al. (2009) attribute the problem with V  −  K to the inherent limitations of the one-dimensional models, given the presence of cooler and warmer regions on the surface.

Second, the bolometric corrections are both a large and a strong function of the assumed effective temperature, while for the YSGs the bolometric corrections were small and fairly insensitive to the effective temperature. This problem can be partially alleviated by using the K-band photometry and applying a bolometric correction to K, as discussed by Levesque et al. (2006). There is an additional advantage to using K to derive the luminosity, namely, that it is less affected by uncertainties in the interstellar (and circumstellar) reddening. Massey et al. (2009) rederived the bolometric luminosities of RSGs in the SMC, LMC, Milky Way, and M31 and found good agreement on average with the values derived from V and K when the same effective temperatures were used, although the individual scatter was large.

All of the RSGs in our sample have BVR data from the LGGS, and hence we can use the V  −  R photometry to derive effective temperatures. We adopt the conversions given in Section 3.3 of Levesque et al. (2006) for the LMC, since the metallicity of M33 is similar. Some of the RSGs in our sample also have 2MASS photometry, and for those stars we compute the bolometric luminosity from K, after converting the 2MASS K_s to standard K following Levesque et al. (2006). For stars without 2MASS, we compute the bolometric correction to the LGGS V following the transformations in Levesque et al. (2006) and adopting a reddening of E(B − V) = 0.12, typical for OB stars in M33 (Massey et al. 2007).

For the stars with 2MASS data, the bolometric luminosity we derive from the K-band is, on average (median), 0.2 mag fainter than that from V. Increasing the assumed reddening to E(B − V) = 0.5 would remove most of the discrepancy. This is consistent with our earlier findings that RSGs invariably have circumstellar reddening, attributable to dust formation (Massey et al. 2005).⁸ Of course, changing the assumed reddening would also change the derived effective temperatures. For now we have taken the conservative approach of adopting the lower reddening (E(B − V) = 0.12) but note that this probably underestimates the effective temperature for some stars. It also means that the V-band bolometric correction is more positive for V but more negative for K.⁹

Thus, again adopting a distance modulus of 24.60, we calculate effective temperatures and bolometric luminosities for our rank 1 and rank 2 RSGs. These derived properties are listed in Table 5.

In the recent literature, single-star evolutionary models have been found to be fairly effective at reproducing the locations of RSGs and YSGs on the H-R diagram.¹¹ Despite this historic agreement, we find that much can be learned from the locations of our RSGs and YSGs in Figures 20 and 21.

We first note that our RSG population contains a large number of stars below log L/L_☉ ∼ 5.2 and relatively sparse numbers above this point, similar to the upper luminosity limit for RSGs in the Milky Way and Magellanic Clouds found by Levesque et al. (2005, 2006). The presence of an RSG branch greatly increases the amount of time spent and, hence, the number of stars observed in a given region of the H-R diagram. Thus, our distribution of RSGs is consistent with the results of the Geneva Z = 0.014 models (Figure 20, solid lines), which predict an RSG branch up to log L/L_☉ ∼ 5.2, while tracks above this point only briefly loop though the RSG portion of the H-R diagram. By contrast, the previous generation of Geneva models predicts RSG branches above log L/L_☉ ∼ 5.2.

We next draw attention to the five most luminous stars in our sample: two YSGs and three RSGs. These stars appear to be inconsistent with the new generation S4 models and both the S0 and S3 old generation models, all of which loop back to the blue portion of the H-R diagram without extending to such cool temperatures. Further examination reveals that two of the three RSGs (J013418.56+303808.6 and J013339.28+303118.8) are moderately crowded in the B band (as described in Section 3.2), but that the most luminous star (J013312.26+310053.3) is particularly isolated. Additionally, the two most luminous YSGs (J013358.05+304539.9 and J013349.86+303246.1) both possess particularly strong and isolated O i λ7774 features. Thus, for at least three of these five stars, we see no reason to suspect that they were falsely identified as rank 1 supergiants.

As described in detail by Meynet et al. (2011), two main factors that affect if, and where, a single star will loop back toward the blue are mass loss during the RSG phase and rotationally induced mixing (both of which have been modified in the new generation models; see above). Higher values of either parameter tend to favor blueward evolution. Indeed, we see that the new generation S0 tracks (Figure 20, bottom panel) proceed to cooler temperatures than the S4 tracks (Figure 20, top panel) before looping back to the blue. Although the distribution of stellar rotational velocities has yet to be fully decoupled from the effect of inclination (since one can only measure vsin i), there is increasing evidence that some massive stars are born as genuine slow rotations (e.g., Huang et al. 2010). Thus, although the location of our five most luminous supergiants could likely be explained by a change in the mass-loss prescriptions used by the Geneva models, it is possible that they are simply slow rotators. If this were the case, their locations would be fully consistent with the new generation models presented here.

Finally, we call attention to the effective temperatures of a majority of our RSG population. As can be seen in Figure 20, the predicted location of the RSG branch moves to warmer temperatures as metallicity decreases. Recalling that M33 may possess a strong metallicity gradient, in Figure 22 we again plot the locations of our RSGs on the H-R diagram, but now separated into bins based on the value of ρ (the distance from the center within the plane of M33, normalized by the D₂₅ isophotal radius of 308).¹² Moving from left to right, the panels are arranged innermost to outermost. The evolutionary models present in this figure are the S4 models from the top panel of Figure 20. From this we see that our RSG sample is consistent with a metallicity gradient within the disk of M33. The supergiants in the outermost region of M33 are systematically warmer than those in the innermost region, and the two populations are consistent with the Z = 0.006 and Z = 0.014 Geneva models, respectively. The inner two panels appear to show a progression (albeit with some scatter) between the two.

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We do note that a handful of our RSGs appear to be inconsistently cool compared to both the Z = 0.014 and Z = 0.006 Geneva evolutionary models. Most notable is the group at log T_eff ∼ 3.53 with 4.9 < log L/L_☉ < 5.2. Further examination of these stars reveals that they are all known variables, as discussed in Section 3.4. Thus, it is possible that our sample contains several of the unusually cool, and potentially unstable, RSGs described by Levesque et al. (2007). Or, it could be that these stars are like the "too cool" LMC RSGs discussed by Neugent et al. (2012), with high circumstellar reddening.

Having examined the locations of our RSGs and YSGs on the H-R diagram, we now assess the relative numbers of our YSGs at various luminosities. If we assume that the star formation rate averaged over the entire disk of M33 has been relatively constant over the past 20 Myr (the lifetime of a 12 M_☉ star), then the number of stars expected between masses m₁ and m₂ will be proportional to

$$\begin{eqnarray*} &&N_{m_1}^{m_2} \propto [m^\Gamma ]_{m_1}^{m_2} \times \bar{\tau }, \end{eqnarray*}$$

where Γ is the slope of the IMF, taken here to be −1.35 (Salpeter 1955), and $\bar{\tau }$ is the average duration of the evolutionary phase for masses m₁ and m₂.

In Table 6, we list the theoretical YSG lifetimes obtained from the new and old generation Geneva models. Note that (1) we include the lifetime of the 32 M_☉ track for the new non-rotating models, which was omitted from the figure for clarity, and (2) for the old generation of Z = 0.008 models, there was an insufficient number of rotating tracks available to make a comparison to our data. Recalling that our observed sample should be complete down to log L/L_☉ ∼ 4.8 (which corresponds to ∼15 M_☉), we use the luminosity of the evolutionary tracks with M ⩾ 15 M_☉ to define mass bins on the H-R diagram. In Table 7, we list the total number of stars we observe in each mass bin (for each set of tracks), as well as the number relative to the 15–20 M_☉ bin. These relative numbers are then compared to those predicted by the theoretical YSG lifetimes.

For the new Z = 0.006 models we present results based on a 25–40 M_☉ bin in addition to 25–32 M_☉ and 32–40 M_☉ bins. Both the rotating and non-rotating Z = 0.006 32 M_☉ tracks actually terminate in the YSGs regime, and we wish to assess what effect this produces on the predicted number of YSGs with luminosity.

From Table 7, we see that both the Z = 0.014 and Z = 0.006 new generation S4 models relatively accurately predict our observed ratios of YSGs with luminosity. In fact, the only main variation we find is the prediction of a few high-mass (M > 32 M_☉) YSGs that are not observed. The deviations are only on the order of a handful (2–5) of stars, however. To this end, we emphasize that our observations did not include every potential YSG in M33. As argued in Section 3.5, our sample should be relatively complete and representative of the relative number of stars at varying luminosities down to log L/L_☉ ∼ 4.8. This does not, however, preclude variations on the order of a handful of stars between our observed luminosity distribution and the true luminosity distribution for M33 YSGs. Indeed, when considering variations of only a few stars, small fluxuations in star formation rate and possible stochastic IMF sampling (see Leitherer & Ekström 2011) may become significant.

We note that this effect is very different than that observed by Drout et al. (2009) and Neugent et al. (2010) when comparing their populations of M31 and SMC YSGs to the old generation models. In those cases, tens to hundreds of additional high-luminosity YSGs would be required to bring their observations into agreement with the evolutionary models, an effect not consistent with the completeness of their samples.

The non-rotating Geneva models fared slightly worse than the rotating models, overpredicting the number of high-luminosity (M > 25 M_☉) stars in comparison to low-luminosity stars. They would require on order of 10–15 additional stars between 25 and 40 M_☉ to bring our observations in line with the predicted ratios. We do admit, however, that the mass bins become slightly confused at higher luminosities due to the number of times the tracks seem to loop though the YSG region. This result may also be indicative of the fact that, although some massive stars are expected to be genuine slow rotators, a much larger percentage of the population likely possesses an initial rotational velocity closer to 0.4v_crit. In fact, the value 0.4v_crit was chosen to correspond to the peak in the distribution of initial rotation speeds for young B-type stars in Huang et al. (2010). In a more rigorous comparison one would compute an average YSG lifetime, weighted by a velocity distribution for each initial mass.

We note that for both the S0 and S4 Z = 0.006 models better results are obtained by ignoring the 32 M_☉ tracks (which, recall, terminated in the YSG regime). Including the 32 M_☉ track leads to the prediction of ∼15 and ∼31 stars between 25 M_☉ and 40 M_☉ (for S0 and S4 models, respectively). On the other hand, ignoring the 32 M_☉ tracks leads to predictions of ∼9 and ∼11 stars between 25 M_☉ and 40 M_☉, and we observe 5 and 6 (S0 and S4, respectively). Thus, our observed sample does not seem to contain an increase in stars at intermediate luminosities that one would expect as a result of these tracks that terminate in the YSG regime (although we emphasize that even these deviations are far less than those observed in our previous works). This will be discussed briefly in the context of YSG supernova progenitors below.

Given these considerations, we find that the relative number of YSGs at various luminosities predicted by the new generation of Geneva evolutionary tracks is relatively consistent, within our uncertainties, with our sample of observed M33 YSGs. From Table 7 we see that this agreement does reflect a real improvement over previous models. At best, the previous generation of Geneva models requires an additional 10–15 stars at high luminosity (Z = 0.008, S0 models) and at worst 400+ additional stars at high luminosity (Z = 0.02, S3 models). Similar results (i.e., a significant improvement in the performance of the new models in comparison to the old) are found by Neugent et al. (2012, a companion paper to this manuscript), who compare the YSG population of the LMC to the new Geneva Z = 0.006 models described here (V. Chomienne et al. 2012, in preparation).

One additional concern raised by our initial studies of YSG populations was the actual duration of the YSG phase. By comparing the (albeit uncertain) number of unevolved OB-type stars to the number of YSGs in M31 and the SMC, Drout et al. (2009) and Neugent et al. (2010) estimated that the duration of the YSG phase should be on the order of 3000 yr for M > 20 M_☉. As can be seen from Table 6, this value is an order of magnitude below the lifetimes predicted by both the new and old generation Geneva models. At this time, we do not possess accurate enough information on the number of unevolved OB-type stars in M33 to directly assess whether this discrepancy is due to remaining deficiencies in the models or with the lifetimes predicted from observations (which should be taken as lower limits due to the effects of crowding; see Massey 2003). However, in a companion paper examining the YSG and RSG populations of the LMC Neugent et al. (2012) are able to revise this estimate of the YSG lifetime. By comparing the number of observed LMC YSGs to the number of unevolved LMC OB-type stars from Massey (2010), they estimate that the YSG lifetime is ∼17,000 yr for M > 20 M_☉. This number, although smaller than those listed in Table 6, is, in our opinion, within the uncertainties of the number of OB-type stars.

Having demonstrated that the new generation of Geneva evolutionary models offers a dramatic improvement in predicting both the observed locations and relative numbers of YSGs as a function of luminosity, we now spend a moment discussing the natural question that arises: What is the physical cause of this improvement? As described above, the new models include several substantial modifications over the previous generation. However, despite the complex and somewhat degenerate nature of the problem, we are able to offer some modest logical arguments.

One modification that almost certainly impacts the behavior of the evolutionary tracks in this region of the H-R diagram is the adoption of a new prescription for RSG mass loss. In the new prescription, mass loss is artificially increased by a factor of three whenever any layer in the envelope of the star reaches a luminosity greater than five times the Eddington luminosity (an effect that is relevant for RSGs with masses above ∼15 M_☉). We emphasize that this is an ad hoc prescription, but one that is not without both theoretical and observational precedent (e.g., Heger et al. 1997; Vanbeveren et al. 2007; Yoon & Cantiello 2010; van Loon et al. 2005, 2008). As discussed in Ekström et al. (2012), this higher mass-loss rate favors blueward evolution, thus lowering the minimum mass required for blueward evolution and increasing the ratio of the blue to red lifetimes for high-mass tracks (see Section 5.5 of Ekström et al. 2012). In order to investigate whether this new mass-loss prescription can explain the improved behavior of the models in the YSG regime specifically, we again examine the YSG lifetimes listed in Table 6.

From this we see that the improved YSG ratios predicted by the new models are not simply a result of a decrease in the YSG lifetimes predicted for the highest mass tracks, nor simply an increase in the YSG lifetimes predicted for the lower mass tracks, but rather a complex modification of the predicted lifetimes at all masses. We do observe significant differences in the lifetimes predicted for some tracks that experience RSG mass loss before evolving back to the blue portion of the H-R diagram (most notably the 20 M_☉ rotating solar mass track, which only evolves back to the blue in the new generation models). However, we also observe significant differences in the predicted lifetimes for tracks that evolve straight across the H-R diagram and end their lives as RSGs (e.g., all the 15 M_☉ tracks and the 20 M_☉ non-rotating, low-metallicity tracks). These tracks do not possess blue loops, and, hence, their YSG lifetimes should be unaffected by the new prescription for RSG mass loss. The fact that we observe significant modifications to the predicted YSG lifetimes for even these tracks indicates to us that, although the modified RSG mass loss undoubtedly affects the behavior of the evolutionary tracks in this region of the H-R diagram, other effects must also be behind the modified behavior of the new models.

Thus, we emphasize a few other modifications made to the new set of models. With respect to rotation, the new models adopt the shear diffusion coefficient proposed by Maeder (1997) as opposed to that of Talon & Zahn (1997), and the initial rotation velocities included in the models have been modified. As mentioned above, the effects of rotation can also play a significant role in dictating the behavior of evolutionary tracks in this region. This is evidenced by the 20 M_☉ Z = 0.014 tracks, which only evolve back to the blue in the rotating models. Additionally, from Table 7 we see that, although the predictions made by both the rotating and non-rotating tracks are improved in the new models, the rotating tracks now perform vastly better than the old, indicating that some of the improved behavior is likely due to the modified descriptions of rotation.

However, even these modifications cannot explain the contrast between the 15 M_☉ non-rotating Z = 0.008 and Z = 0.006 tracks, the 20 M_☉ non-rotating Z = 0.008 and Z = 0.006 tracks, and the 15 M_☉ non-rotating Z = 0.014 and Z = 0.02 tracks. All three pairs show a significant change in their predicted YSG lifetimes, and none should be affected by either RSG mass loss or rotation. To this end we note that the new models contain modified initial compositions and opacities. We are considering a transitionary phase during which the star is changing shape (inflating when evolving from the blue to the red), and thus changes in the opacities may actually have important implications.

It is therefore our conclusion that no single modification in the new Geneva models can be pointed to as the main cause for their improved overall behavior in the YSG regime. Rather, when put together, the various modifications produce a set of models that are, at least heuristically, able to reproduce the observed trends. This emphasizes the role of the YSG regime as a magnifying glass for models of stellar evolution, and that much work can still be done to understand these objects from a theoretical point of view.