THE BLUE HOOK POPULATIONS OF MASSIVE GLOBULAR CLUSTERS^*

The stellar evolutionary models used in the present study were computed with a procedure similar to the one described in Paper I. For each cluster, we constructed grids of evolutionary sequences that followed the evolution of appropriate low-mass stars continuously from the zero-age MS through the end of the HB phase for two values of the initial helium abundance: a helium-normal abundance Y of 0.23, representing the helium abundance in stars born during the first stellar generation with standard cluster abundances, and a helium-rich abundance Y of 0.40, representing a reasonable upper limit on the helium enhancement in the second stellar generation born after a process of cluster self-enrichment. The hotter HB sequences in these grids were also evolved through the post-HB phases.

The heavy element abundance Z for these sequences was determined from the observed [Fe/H] value for each cluster, assuming an α-element enhancement of [α/Fe] = 0.3, as appropriate for metal-poor globular cluster stars (Carney 1996). Using the prescription given by Salaris et al. (1993) for converting an α-enhanced abundance into an equivalent scaled-solar abundance, we obtained scaled-solar heavy element abundances Z of 0.00023, 0.00064, 0.00081, 0.0015, 0.0077, and 0.0090, respectively, for the six clusters NGC 2419, NGC 6273, NGC 6715, NGC 2808, NGC 6388, and NGC 6441. We used the same Z value to compute both the helium-normal and helium-rich sequences for each cluster. The initial MS mass for each composition (Y, Z) was calculated for an age of 13 Gyr at the tip of the RGB, although ages a few Gyr younger would have little qualitative effect on the subsequent evolutionary behavior.

The sequences for each composition (Y, Z) differ from each other only in the amount of mass loss along the RGB, which we parameterize with the Reimers (1975, 1977) mass-loss formulation:

$$\begin{equation*} \dot{M} = -4 \times 10^{-13} \eta _R L / g R (M_\odot \,{\rm yr}^{-1}), \end{equation*}$$

where L, g, and R are the stellar luminosity, gravity, and radius, respectively, in solar units, and η_R is the well-known Reimers mass-loss parameter. Because our calculations follow the evolution through the helium-core flash, we are able to determine how the evolutionary path from the MS to the ZAHB depends on the assumed mass-loss rate (see Figure 1). The red end of the ZAHB is set by the sequence with η_R = 0.0 (no mass loss). Up to a given value of η_R (∼0.7 for intermediate [Fe/H] values), the models remain tightly bound to the RGB until the helium-core flash (Figure 1(a)). However, for higher values of η_R, the models leave the RGB and evolve to high effective temperatures before igniting helium (Castellani & Castellani 1993), either during the crossing of the CMD (Figure 1(b)) or while descending the WD cooling curve (Figure 1(c)). D'Cruz et al. (1996) suggested that such "hot flashers" might provide another channel for populating the EHB. Although the masses of the late hot flashers for each composition (Y, Z) fall within a narrow range, they are produced over a rather wide range in the mass-loss rate (Δη_R ∼0.1). Extremely fine tuning of the mass-loss rate is thus not required to produce a late hot flasher (see Paper I). For sufficiently high values of η_R, the models simply cross the HR diagram and descend the WD cooling curve without igniting helium, thus dying as helium WDs (see Figure 1(d)).

The RGB tip flashers and early hot flashers ignite helium at a time when the models are very luminous. The strong hydrogen-burning shell in these models forms a high entropy barrier that prevents the flash convection from penetrating into the envelope, and hence no mixing then occurs between the envelope and the core. Thus, any star that undergoes the helium-core flash before reaching the WD cooling curve will arrive on the ZAHB via canonical evolution (i.e., with a normal atmospheric composition). In contrast, stars that ignite helium on the WD cooling curve have much weaker hydrogen-burning shells and thus much lower entropy barriers (Sweigart 1997; Paper I). Under these conditions, penetration of the flash convection into the envelope becomes inevitable. This flash mixing will mix hydrogen-rich material from the envelope into the core and helium- and carbon-rich material from the core into the envelope, thereby greatly enhancing the surface helium and carbon abundances. A similar process may produce other extremely hydrogen-poor stars (also known as "born again" stars) during a very late helium-shell flash (Renzini 1990).

Due to the numerical difficulties associated with this mixing process (Paper I), we terminated our sequences as soon as they encountered flash mixing. Nevertheless, we can still predict the surface composition that these flash-mixed models should have when they arrive on the ZAHB. The available calculations clearly indicate that flash mixing will consume most, if not all, of the envelope hydrogen. Thus, we expect the surface composition of the flash-mixed models to be strongly depleted in hydrogen and enriched in helium. The surface carbon abundance should also be close to its value in the flash convection zone, namely, 4% by mass. For these reasons the flash-mixed models given in this paper will assume that the surface carbon abundance has been increased to 4% by mass and that all of the remaining surface hydrogen has been burned to helium. It is worth noting that Cassisi et al. (2003) and Miller Bertolami et al. (2009) were able to follow the flash-mixing process in detail, and their values for the flash-mixed surface composition are in good agreement with the composition adopted here. Note also that some of the carbon may be burned to nitrogen during the flash mixing, depending on the details of the nucleosynthesis; such an outcome would not have a significant effect on the stellar evolution models.

In order to produce theoretical spectra for the evolutionary tracks, we calculated a series of 60 non-LTE (NLTE) line-blanketed model atmospheres with our model atmosphere program, Tlusty, version 201.⁶ For each globular cluster, we used the appropriate metallicities and helium abundances for the various evolutionary scenarios explored. Tlusty computes stellar model photospheres in a plane-parallel geometry, assuming radiative and hydrostatic equilibria. Departures from LTE are explicitly allowed for a large set of chemical species and arbitrarily complex model atoms, using our hybrid Complete Linearization/Accelerated Lambda Iteration method (Hubeny & Lanz 1995). More specifically, the model atmospheres allow for departures from LTE for 1132 levels and superlevels of 52 ions, H i, He i, He ii, C i–C iv, N i–N v, O i–O vi, Ne i–Ne iv, Mg ii, Al ii, Al iii, Si ii–Si iv, P iv, P v, S ii–S vi, and Fe ii–Fe vi. Details of the model atom setup are provided in Lanz & Hubeny (2003, Lanz & Hubeny 2007), and in Cunha et al. (2006) for updated neon models.

The model atmospheres span the range of parameters suitable for most EHB stars: 20,000 K⩽ T_eff ⩽50,000 K (with a 5,000 K step), 4.5 ⩽ log g ⩽ 6.0 (0.5 dex step), and V_turb = 5 km s⁻¹. We assumed three different helium compositions: standard helium (Y = 0.23), helium-rich (Y = 0.4), and flash-mixed (Y = 0.96, with carbon and nitrogen mass fractions of 3% and 1%, respectively; Lanz et al. 2004, hereafter Paper II). Note that the assumption of 4% carbon or 4% nitrogen (or some intermediate mix) makes little difference in broadband UV colors (see Paper I).

With our spectrum synthesis code, Synspec, we then calculated the detailed emergent ultraviolet spectrum of each model atmosphere. Quadratic interpolations in the spectra database were performed to derive theoretical far-UV and near-UV magnitudes for the specific stellar parameters along each evolutionary track. The parameters of the quadratic interpolation are wavelength-dependent. Detailed tests show that this interpolation process yields colors that do not differ by more than 0.01 mag, compared to using a model atmosphere calculated with the exact stellar parameters. This approach saves significant computing time without significantly affecting the final accuracy.

With these models in hand, we transformed the evolutionary models for each cluster to the observable plane. For each cluster, we interpolated in the grid of synthetic spectra having the appropriate abundance pattern (cluster metallicity with Y = 0.23, cluster metallicity with Y = 0.4, or the flash-mixed composition). For stars cooler than the lowest effective temperature in our NLTE grids, we interpolated in the Castelli & Kurucz (2003) spectral grids at enhanced [α/Fe]. The interpolated spectrum for each stellar evolutionary model was reddened using the average Galactic extinction curve of Fitzpatrick (1999), and then passed through the relevant near-UV and far-UV bandpasses using the IRAF synphot package, resulting in magnitudes in the STMAG system, where m = −2.5 × log₁₀f_λ − 21.1 mag. Note that this method appropriately applies the reddening in a bandpass-dependent and T_eff-dependent manner.

We first compare our CMDs to the expectations from canonical stellar evolutionary models with a standard helium abundance. We use the BHB as a standard candle to guide the normalization of these models. Specifically, we adopt metallicity and distance values that are representative of values in the literature, and then make small adjustments to the extinction so as to place the BHB of a canonical model at the expected location. Note that the reddening vector in our m_FUV versus m_FUV–m_NUV CMDs is nearly vertical, so one could adjust the distance instead of the reddening, or some combination thereof; a change of 0.01 mag in E(B − V) is equivalent to a change of 0.079 mag in distance modulus. For the four metal-poor clusters, the theoretical HB distribution should be coincident with the observed HB locus at the BHB. For the two metal-rich clusters, Rich et al. (1997) have shown that the HB slopes strongly upward as one moves from the reddest HB stars to the BHB (see also Busso et al. 2007), such that the BHB is ∼0.5 mag brighter in the optical than one would expect for a canonical locus normalized to the red HB. It would be simpler to consistently align our models to the red HB in all of our clusters, but it falls outside of our UV CMDs, driving our use of the BHB; thus, for the two metal-rich clusters we choose an extinction that places the canonical locus 0.5 mag fainter in the far-UV than the observed BHB locus. We are assuming that a star at a given reference point on the BHB (defined at a particular effective temperature) has changed in bolometric luminosity, such that the offset in the far-UV is the same 0.5 mag observed in the optical. The physical motivation for this assumption comes from the fact that the bolometric luminosities of all HB stars in the range 6000 ≲ T_eff ≲ 18, 000 K are increased by nearly the same amount when Y is enhanced, and that the effect on the UV-to-optical color in the emergent spectrum is minimal. As we shall see in the next section, our choice of extinction is equivalent to aligning canonical BHB models of intermediate Y (∼0.3) to the observed BHB locus.

In Figure 3, we show the CMD of each cluster compared to the expected locus for stars in the core helium-burning stage, assuming standard cluster metallicity and helium abundance. The comparison is made over a wide color range that includes stars much redder than the EHB. In Figure 5, we restrict the comparison to those stars hotter than ∼9, 000 K, and shade the HB locus to indicate the timescales expected for HB evolution. We expect most of the HB stars to follow close to the low-luminosity edge of the canonical helium-burning locus, because the evolution accelerates toward the latter stages of the HB phase (i.e., from the bottom to the top of the canonical HB locus). Because we use the BHB as a standard candle to normalize the models, we can make an appropriate comparison of the observed EHB locus and the predicted EHB locus.

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NGC 2419. We have assumed a metallicity of [Fe/H] = −2.12 (Harris 1996)⁷ and a distance of 83.2 kpc (Ripepi et al. 2007). For the reddening, our assumption of E(B − V) = 0.03 mag is somewhat lower than that assumed elsewhere (0.08–0.11 mag; Harris 1996; Piotto et al. 2002; Ripepi et al. 2007), but the upward shift in the models is required to align the observed and theoretical loci at the BHB. As one progresses blueward in m_FUV–m_NUV color from 1.0 mag to −0.8 mag, the observed HB appears to slope slightly upward relative to canonical HB, but it is worth noting that optical CMDs spanning the B, V, and I bandpasses (Piotto et al. 2002; Ripepi et al. 2007) do not show a sloping HB that would justify normalizing the canonical locus significantly below the BHB. As one moves to stars bluer than the BHB, the observed HB locus drops abruptly to fainter magnitudes, such that nearly all of the observed EHB stars are subluminous (i.e., BH stars).

NGC 6273. We have assumed a metallicity of [Fe/H] = −1.68 and a distance of 8.7 kpc (Piotto et al. 2002). For the reddening, our assumption of E(B − V) = 0.43 mag is larger than that assumed by Piotto et al. (2002; 0.37 mag), shifting the theoretical locus fainter by ∼0.5 mag in the far-UV to give alignment with the observed BHB. As with NGC 2419, the HB slope appears somewhat steeper than expected as one moves blueward from 0.5 mag to −0.5 mag in m_FUV–m_NUV color, even though the optical CMD (e.g., Piotto et al. 2002) shows no indication of an upward HB slope from the red HB to the BHB. At the hot end of the HB, the observed population is roughly split between normal EHB stars and BH stars.

NGC 6715. We have assumed a metallicity of [Fe/H] = −1.58 and a distance of 26.8 kpc (Harris 1996). For the reddening, our assumption of E(B − V) = 0.16 mag is slightly higher than that assumed elsewhere (0.15 mag; Harris 1996; Siegel et al. 2007), shifting the models fainter by 0.08 mag to align with the observed BHB. The observed HB locus and the theoretical locus agree well over the entire BHB range, but at the EHB, the observed population is roughly split between normal EHB stars and BH stars. Note that NGC 6715 may actually represent the core of the disrupted Sgr dwarf galaxy (e.g., Layden & Sarajedini 2000; Siegel et al. 2007). Even if it is not the core of the Sgr dwarf, the earlier optical CMDs in this vicinity exhibit populations more complex than one would usually associate with a globular cluster, which may explain the large scatter in our UV CMD.

NGC 2808. This cluster was examined extensively in Paper I. The cluster is included here for comparison, and we assume the same cluster parameters used previously. Although the HB sample in NGC 2808 is smaller than that in NGC 6715, the morphology of the observed HB locus is similar. The similarities between the NGC 6715 and NGC 2808 CMDs may be related to the similarities in metallicity and central concentration; NGC 6273 is also close in metallicity, but is less centrally concentrated and less luminous than NGC 6715 and NGC 2808 (Harris 1996).

NGC 6388. We have assumed a metallicity of [Fe/H] = −0.6 and a distance of 11.5 kpc (Piotto et al. 2002). For the reddening, our assumption of E(B − V) = 0.41 mag is slightly larger than that of Piotto et al. (2002; 0.4 mag), placing the canonical locus 0.5 mag fainter than the observed BHB, as found by Busso et al. (2007) and discussed above. However, Busso et al. (2007) assumed a distance of 11.2 kpc and E(B − V) = 0.45 mag; if we adopted these values, the canonical HB locus would be an additional 0.3 mag fainter than we have assumed. In contrast, assuming the shorter distance (10 kpc) and lower reddening (0.37 mag) of Harris (1996) would align the canonical locus with the observed HB locus at the BHB, as we did for the metal-poor clusters. Although we have chosen an extinction that places the canonical BHB 0.5 mag below the observed BHB, nearly all of the observed EHB stars are subluminous with respect to the theoretical locus, as also seen in NGC 2419. However, unlike the situation in NGC 2419, the subluminous EHB stars of NGC 6388 cover a much wider range of color than that seen in any of the metal-poor clusters; the subluminous EHB stars in NGC 2419 are well separated in color from the BHB stars, but there is no such color gap in the NGC 6388 population. Despite their extension to unusually red colors, the subluminous EHB stars in NGC 6388 are much bluer than the BSS (see Figure 3), and thus there is no confusion between these two classes of stars. Moreover, the subluminous EHB stars in NGC 6388 comprise a contiguous and distinct group, which suggests that they may all be BH stars, including the reddest stars in this group.

NGC 6441. We have assumed a metallicity of [Fe/H] = −0.53, and a distance of 11.7 kpc (Harris 1996). For the reddening, our assumption of E(B − V) = 0.53 mag is larger than that of Harris (1996; 0.47 mag), placing the canonical locus 0.5 mag fainter than the observed BHB, as found by Busso et al. (2007) and discussed above. However, assuming the distance (15.2 kpc) and reddening (0.48 mag) of Busso et al. (2007) would place the canonical locus an additional 0.2 mag fainter than we have assumed. As with NGC 6388, assuming the Harris (1996) distance and reddening would align the canonical locus with the observed HB locus on the BHB, which is the normalization we used with the metal-poor clusters. Piotto et al. (2002) assumed a shorter distance (11.3 kpc) and less reddening (0.44 mag); assuming these values would place our models above the observed HB locus for all temperatures (i.e., not simply the potential BH stars). Regardless of the normalization, the colors imply that there are almost no stars hotter than ∼15, 000 K in the cluster on or below the canonical EHB. There are stars below the canonical HB in the temperature range of 10,000–15,000 K, which is hotter than the BSS. When considered in the context of the NGC 6388 BH population (which extends far to the red), it is possible that these subluminous stars in NGC 6441 are BH stars, despite their relatively red color. We inspected archival WFPC2 images in the F336W (U) and F439W (B) bands to determine where these unusually red BH candidates would fall in optical CMDs. Although the core of the cluster is crowded at these longer wavelengths, five such stars were sufficiently isolated to obtain clean photometry, and in the B versus U − B CMD they clearly fall within the low-luminosity end of the long blue HB tail, reinforcing the need for UV photometry to distinguish between temperature and luminosity variations in hot stars.

In all six clusters, the hottest HB stars observed are subluminous with respect to canonical models (see Figures 3 and 5). Thus, there are BH stars in each cluster, with the possible exception of NGC 6441, where these subluminous stars are all relatively red. It is clear that canonical models at Y = 0.23 cannot explain the HB populations of these clusters. Regardless of how such models are normalized via varying the assumed reddening or distance, they cannot reproduce the observed luminosity difference between the BHB and the subluminous EHB stars.

We next consider models with the cluster metallicity but with the helium abundance increased to Y = 0.4. This value of helium enhancement is at the upper end of the plausible range for stars that have been enhanced at birth due to cluster self-enrichment. In Figure 6, we compare these models with Y = 0.4 to those with Y = 0.23, assuming the same cluster parameters ([Fe/H], distance, and reddening) used previously.

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Figure 6 shows that an enhanced helium abundance produces a more pronounced dropoff in luminosity between the BHB and the EHB. Relative to the Y = 0.23 locus, the Y = 0.4 locus is fainter on the EHB but much brighter on the BHB. The fainter EHB luminosity of the Y = 0.4 locus is due to a slightly smaller core mass while the brighter BHB luminosity is due to a greater energy output of the H-burning shell. This dependence of the luminosity difference between the BHB and EHB on the helium abundance is well established and was evident even in the early HB calculations of Sweigart & Gross (1976). We note in Figure 6 that the Y = 0.4 models for the very metal-poor cluster NGC 2419 are restricted to hot temperatures, even for those HB stars that lost no mass on the RGB. This is because the MSTO mass for a population at Y = 0.4, [Fe/H] = −2.12, and an age of 13 Gyr is only 0.614 M_☉; this low turnoff mass, combined with this metal-poor composition, will only produce blue HB stars, even with no mass loss.

It is clear that the Y = 0.4 models do not agree well with the observed photometry when we assume the same parameters ([Fe/H], distance, and reddening) used in the Y = 0.23 models. The hot end of the Y = 0.4 locus is not as faint nor as red as the observed BH stars, and the cooler part of the Y = 0.4 locus is much brighter than the observed BHB stars. While a combination of the Y = 0.23 and Y = 0.4 models would lead to a larger luminosity width for the EHB, the predicted range in luminosity would still not extend to the faint magnitudes reached by the BH stars. It is also clear that any range in Y along the BHB must be fairly small, given the narrow luminosity width of the BHB loci in Figure 6. Note that the larger scatter in NGC 6715 may be due to the complex populations associated with the Sgr dwarf galaxy (Layden & Sarajedini 2000; Siegel et al. 2007).

At first glance, one might think that the agreement between Y = 0.4 locus and the observed BHB and EHB stars could be improved by assuming a larger reddening or distance, because such adjustments to the cluster parameters would shift the models to fainter magnitudes. Indeed, if the Y = 0.4 locus were shifted until it fit the observed BHB, the lower edge of the EHB for these models would be in much better agreement with the observed lower edge of the BH stars. However, such adjustments to the reddening or distance would violate important observational constraints.

If we attempt to fit the BHB with the Y = 0.4 locus, we will have to confront the following observational difficulties. Optical CMDs of NGC 2419 (Sandquist & Hess 2008), NGC 6273 (Piotto et al. 1999), NGC 6715 (Layden & Sarajedini 2000), or NGC 2808 (Bedin et al. 2000) do not show a significant difference in the V-band luminosity of the red HB and BHB stars, implying that any difference in the helium abundance between these stars must be small. Thus, the cluster parameters we derived in fitting the BHBs of these clusters to the canonical Y = 0.23 locus should also apply to the red HB stars. In the case of the two metal-rich clusters NGC 6388 and NGC 6441, it is well known that the HB slopes upward (Rich et al. 1997; Busso et al. 2007), with the V-band luminosity of the BHB being ∼0.5 mag brighter than that of the red HB clump. The BHB stars in these clusters may thus have an intermediate helium abundance (Y ∼ 0.3). However, as discussed above, we took this offset in the BHB luminosity into account when fitting the Y = 0.23 model to our UV CMDs. Thus, our parameters for these two clusters should also apply to their red HB clumps. If the BHB should be dominated by stars with Y = 0.4 and the red HB by stars with Y = 0.23, one would therefore predict a luminosity difference between the BHB and red HB clump that is much larger than observed in any of our six clusters. To avoid this contradiction, one might argue alternatively that the red HB stars are themselves helium-rich, but this would imply a much brighter red HB luminosity and hence a much younger (and untenable) cluster age from the observed luminosity difference between the red HB and the MSTO (e.g., see Martín-Franch et al. 2010). As one example, Catelan et al. (2006) have compared the CMDs of NGC 6388 and 47 Tuc and concluded that both of these clusters have similarly old ages.

In short, while the Y = 0.4 locus exhibits a sharp drop in luminosity between the BHB and EHB, these models cannot by themselves explain the BH phenomenon. Given the lack of helium enhancement on the red HB and the relative luminosities of the BHB and red HB, we cannot align models of arbitrarily high Y to the BHB without violating other observational constraints. If we assume the BHB has a standard helium abundance (Y = 0.23, in the case of the four metal-poor clusters) or an intermediate helium abundance (Y ∼ 0.3, in the case of the two metal-rich clusters), then even a wide range of helium abundance (0.23 ≲ Y ≲ 0.4) on the EHB would not explain either the faint luminosities or the wide color range observed in the BH stars.

We next consider models which undergo flash mixing due to high mass loss on the RGB. In Figure 7, we plot these flash-mixed models in our UV CMDs, assuming the same reddening and distance values obtained in Section 4.1. Those values align a canonical Y = 0.23 locus to the BHB in the four metal-poor clusters, but place this Y = 0.23 locus 0.5 mag fainter (in m_FUV) than the BHB in the two metal-rich clusters. For each cluster in Figure 7 we show the HB evolutionary tracks for three representative flash-mixed models; two of these tracks fall at the extremes of the range in the mass-loss rate leading to flash mixing, as measured by the value of η_R, while one falls halfway between these extremes. We plot these triplets of flash-mixed tracks for both Y = 0.23 and Y = 0.4 in order to bracket the range of behavior expected from any sub-population in the cluster that undergoes enough mass loss for flash mixing. All of the flash-mixed tracks for each cluster fall within a narrow temperature range. The offset in luminosity among these tracks is due to small differences in the helium-core mass.

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As can be seen in Figure 7, flash-mixed models can reproduce the faint luminosities observed in the BH population, particularly the flash-mixed models originating in a subpopulation with enhanced helium. Flash mixing thus offers a way of reproducing the faint luminosities observed in the BH stars without causing problems at the BHB or red HB clump, as discussed in the last section. Furthermore, the flash-mixed models, when combined with the canonical HB locus, are better able to reproduce the large luminosity width observed at the hot end of the EHB.

There is one obvious problem, however, which was also seen in the previous sections; like canonical models, flash-mixed models (whether originating in a population with enhanced or standard Y) cannot reproduce the red colors in the BH population. Paper I explored various ways of producing such red BH stars, including enhanced line blanketing in the atmosphere, due to an increase in the atmospheric Fe abundance via radiative diffusion. In the case of NGC 2808, an increase in the [Fe/H] from −1.36 to +1.0 could shift the BH stars ∼0.3 mag to red. However, this color shift is not enough to explain the full color range of the BH population. Furthermore, the effect would be even smaller for the metal-rich clusters, which are the very clusters exhibiting the reddest BH stars. Because the metal-rich clusters already incur significant line blanketing, increasing the [Fe/H] via diffusion does not help much; for example, increasing the [Fe/H] in NGC 6388 from −0.6 to +1.4 only shifts the BH color redward by 0.26 mag. Thus, at this time, the mechanism for producing subluminous HB stars with such red colors remains a puzzle, and it suggests that the formation mechanism for these stars is not yet fully understood.

In this paper, we have compared the UV CMDs of six massive globular clusters with the expectations from stellar evolution theory. We find that in each of these clusters, the hottest HB stars are subluminous with respect to canonical models, whether or not those models assume a standard helium abundance (Y = 0.23) or an enhanced helium abundance (Y = 0.4). The most straightforward interpretation is that all of these clusters host BH stars. However, in the two metal-rich clusters, the subluminous stars either extend to relatively red colors (NGC 6388) or are restricted to these red colors (NGC 6441), which might imply that an additional or distinct phenomenon is at work at high metallicity. Indeed, the existence of BH stars in NGC 6441 has been questioned (e.g., Dieball et al. 2009). The BH population of NGC 6388 may be the link between the BH populations in the metal-poor clusters and that in NGC 6441, because it spans the full color range observed in these clusters. The bluest stars in the NGC 6388 BH population appear to be analogs of the BH stars in the metal-poor clusters, while the reddest stars in the NGC 6388 BH population appear to be analogs of the BH stars in the NGC 6441 population. Moreover, the entire BH population in NGC 6388 appears to be contiguous.

In some massive clusters, all of the EHB stars may have originated in a sub-population with enhanced helium. For example, in NGC 2808, nearly all of the EHB stars falling within the Y = 0.23 canonical locus (Figure 6; pink shaded region) also fall within the Y = 0.4 canonical locus (Figure 6; green shaded region); such stars are normal in the sense that they are not flash-mixed. However, there is a dearth of stars falling in the brighter part of the Y = 0.23 EHB locus, which may indicate that the dominant Y = 0.23 population in this cluster only produced BHB and red HB stars. In NGC 2419, the situation is even more extreme. Few EHB stars fall in either the Y = 0.23 or Y = 0.4 canonical loci, and thus all of the EHB stars in this cluster may have been produced via flash mixing (see Figure 7). As with NGC 2808, the dominant Y = 0.23 population in NGC 2419 may only have produced BHB and red HB stars.

The faint luminosities observed in the BH stars can be reproduced by models that undergo flash mixing during a late helium-core flash on the WD cooling curve. Helium enhancements may also play an important role in the formation of BH stars, primarily because a higher helium abundance lowers the MSTO mass at a given age so that less mass loss is required on the RGB to populate the hot HB. Helium enhancements also depress the luminosity somewhat in canonical EHB models, but not by the ∼1 mag of the faintest BH stars.

Several recent studies have argued against flash mixing as the origin of the BH stars. For example, Lee et al. (2005) attribute the BH stars to the evolution of a population with a high initial helium abundance (Y = 0.4). However, as we demonstrate in Section 4.2, an enhancement in Y cannot by itself explain the BH phenomenon. While an enhanced Y decreases the EHB luminosity and increases the BHB luminosity (thus providing a hook at the hot end of the HB), one cannot arbitrarily adjust the cluster reddening and distance to match the theoretical HB locus to an observed CMD without regard to the relation between the BHB, the red HB, and the MSTO. Furthermore, the models of Lee et al. (2005) do not predict any carbon enhancement in the BH stars, while optical spectra of BH stars in ω Cen and NGC 2808 do suggest such an enhancement may be present (Moehler et al. 2004, Moehler et al. 2007)—an enhancement naturally produced by flash mixing.

D'Antona et al. (2010) go further, postulating that MS stars born at Y ≈ 0.4 could undergo deep mixing on the RGB that elevates their helium abundance to Y ≈ 0.8 before arrival on the EHB. The mechanism that drives this deep mixing is unclear, unlike the mechanism for flash mixing that follows from the basic stellar structure equations and which is the inevitable outcome of a helium-core flash on the WD cooling curve. However, the effect of the D'Antona et al. (2010) "RGB mixing" is to generate EHB stars with very high helium abundances that are hotter than the canonical HB, and in this way their models are similar to those produced by flash mixing. An important distinction, however, is that their RGB mixing precedes the helium-core flash, and thus does not produce an enhancement in the carbon abundance.

Castellani et al. (2006) have argued that canonical HB models can reach the high T_eff observed for BH stars, but their EHB stars reach a maximum T_eff of ∼32, 000 K. This is much less than that found by Moehler et al. (2004, 2007) for the BH stars (>35, 000 K), and much less than the temperatures reached by either flash mixing or RGB mixing models. Furthermore, canonical HB models would not produce any enhancements in helium or carbon, as observed for such stars (Moehler et al. 2004, 2007).

In Paper II, we found that a delayed helium-core flash could sometimes lead to a more shallow form of flash mixing, where the envelope hydrogen is only mixed into the outer layers of the core. When this happens, flash-mixed stars can retain some residual envelope hydrogen, and consequently would be distributed closer to the canonical EHB. Our models imply that shallow mixing may be a relatively rare occurrence, especially at low metallicities. Our CMDs do not show any systematic variation in the number of BH stars falling close to the canonical locus, suggesting that shallow mixing is not important in the metallicity range covered by our clusters. It may be, however, that we simply do not have enough BH stars to detect such a trend. With the current sample, it appears that the BH stars in each cluster fall up to ∼1 mag below the canonical HB, regardless of metallicity.

The sdB and sdO stars of the Galactic field are the analogs of the EHB and post-EHB stars found in clusters. One would expect some of these stars to also show evidence of flash mixing. This expectation motivated far-UV spectroscopy of helium-rich sdB stars in the Galactic field using the Far Ultraviolet Spectroscopic Explorer, to see if these stars exhibit the helium and carbon abundance enhancements expected from flash mixing. Those observations do in fact lend support to the flash-mixing scenario. The spectrum of PG1544+488 (archetype of the He-sdB class) shows incredibly strong carbon lines, indicating a surface that is ≈2% carbon by mass (Paper II).

Further insight into the nature of the BH stars may come from upcoming far-UV spectroscopy of NGC 2808. In the current observing cycle, we will use the Space Telescope Imaging Spectrograph on HST to obtain spectra of BH stars, normal EHB stars, and BHB stars selected from the NGC 2808 CMD (Figure 5). With these spectra, we will be able to search for the atmospheric signatures of flash mixing (e.g., enhanced carbon, diminished H) and perhaps shed light on the origin of the wide color range exhibited in BH stars.

Our understanding of the BH phenomenon is far from complete. None of the models discussed in this paper (canonical at standard Y, canonical at enhanced Y, or flash-mixed) are able to explain the red colors found in some or all of the subluminous stars in each cluster. This is particularly true in the two metal-rich clusters in our sample. Furthermore, the near-total lack of normal EHB stars in half of our clusters is puzzling. If we consider the categories of HB stars (red HB, BHB, normal EHB, BH) as a sequence governed by cluster parameters (mass loss, helium abundance, cluster mass, etc.), why do some clusters (NGC 2419, NGC 6388, and NGC 6441) exhibit BHB stars and BH stars but few normal EHB stars? The dearth of such stars may imply a gap in one or more cluster parameters, but the mechanism for such a gap is unclear.

A remarkable feature of all clusters either known or suspected to host BH stars is that they are among the most massive clusters in the Galaxy. Dieball et al. (2009) also noted that BH stars only appear to exist in massive clusters, and suggested that this could simply be a statistical effect, because "intrinsically rare objects can only be found by searching a sufficiently large number of stars." To examine this possibility, Dieball et al. (2009) apply a statistical test by comparing the locations of clusters with and without significant BH populations within the cumulative mass distribution of the clusters in their sample. Although Dieball et al. (2009) state that their analysis is "effectively combining many low-mass clusters to form aggregate high-mass clusters," their test, in fact, ignores the actual numbers of BH stars in each cluster, and instead only considers whether clusters do or do not host significant numbers of BH stars. This is a difficult test, given the small number of clusters in their comparison. Although Dieball et al. (2009) suggested that it would be worthwhile to investigate how the numbers of BH stars scale with cluster properties, they did not actually perform this test due to the non-uniformities in the extant data. A uniform data sample would be important if one were trying to address the total size of the BH population as a function of cluster mass. However, the Dieball et al. (2009) sample already contains sufficient data to explore the relationship between BH population size and cluster mass. If the UV imaging is sufficient to produce a statistically significant sample of EHB stars in each cluster, one can ask if a significant fraction of the EHB stars are BH stars, and can then compare this fraction in clusters of different masses. This comparison is meaningful even if different clusters are observed in distinct ways (e.g., differences in region sampled, bandpasses used, depth); the important point is to observe a large enough sample of EHB stars to determine the relative numbers of normal and subluminous (BH) EHB stars.

We refer to Table 2 of Dieball et al. (2009). Their table lists five clusters hosting significant numbers of BH stars (NGC 2419, NGC 2808, ω Cen, NGC 6388, and NGC 6715). All of these clusters have masses ranging from 1.42 × 10⁶ M_☉ (NGC 2808) to 3.35 × 10⁶ M_☉ (ω Cen). We would add NGC 6273 to this list; because it is 0.2 mag fainter in M_V than NGC 2808 (Harris 1996), its mass is ≈1.2 × 10⁶ M_☉. Table 2 of Dieball et al. (2009) also lists 11 clusters that do not have significant numbers of BH stars. Of course, besides these 11 clusters, there are many more low-mass and intermediate-mass clusters without known BH stars, but these 11 were considered by Dieball et al. (2009) to have sufficient imagery to rule one way or the other. Among that list of 11 clusters is NGC 6441, with a mass of 1.57 × 10⁶ M_☉; we would move it to the list of clusters hosting BH stars. Also in that list of 11 clusters is NGC 104 (47 Tuc); it is massive (1.5 × 10⁶ M_☉), but it is the only cluster in the entire table that has no BHB stars (its HB population is entirely red). The total mass of the nine remaining clusters in this list is 5.1 × 10⁶ M_☉. The stellar mass in this ensemble of nine clusters exceeds the mass in any individual cluster, and is twice the mass of any cluster we have explored in our own study. Although this ensemble is comprised of nine clusters hosting BHB stars and in some cases (e.g., M13, M80, NGC 6752) substantial populations of EHB stars, this ensemble has few if any BH stars (possibly 3–5; Dieball et al. 2009; Sandquist et al. 2010). If finding BH stars simply required the searching of enough stellar mass, we would expect to find tens of BH stars in the ensemble mass of nine clusters totaling 5.1 × 10⁶ M_☉. The implication is that any cluster hosting BHB stars will also host BH stars if its mass exceeds ∼1.2 × 10⁶ M_☉. However, the fraction of EHB stars that are subluminous varies from cluster to cluster in our sample, and that fraction does not appear to correlate with either cluster mass or metallicity, implying additional factors are at work (including perhaps the dynamical history of interactions between the cluster and the Galaxy). In half of our sample (NGC 2419, NGC 6388, and possibly NGC 6441), virtually all of the EHB stars appear to be BH stars, with very few stars on the canonical EHB, whereas in the remaining three clusters (NGC 6273, NGC 6715, and NGC 2808), the split between normal and subluminous EHB stars appears to be roughly equal.

Note that low-mass clusters can have large samples of EHB stars without having any BH stars. Consider NGC 6752, which is one of the nine clusters discussed above that hosts no BH stars; it is listed in Table 2 of Dieball et al. (2009) with a mass of 0.32 × 10⁶ M_☉. For NGC 6752 (Landsman et al. 1996) and NGC 2808 (Paper I), there exist CMDs that include a far-UV bandpass. In the case of NGC 6752, the CMD is sampling a less massive cluster outside of the core, while in the case of NGC 2808, the CMD is sampling a more massive cluster within the core. Despite these distinctions, the samples of EHB stars and their descendents are comparable in size. In each CMD, there are ∼80 HB stars hotter than 16,000 K, along with four post-EHB stars, i.e., the shorter-lived AGBM stars. However, for the case of NGC 2808, 46 of the EHB stars are BH stars (lying well below the canonical EHB) while in NGC 6752, none of them are. Given that the sizes of the EHB samples are approximately the same, the numbers of BH stars in the two clusters are discrepant at a high statistical significance. This argument is not restricted to NGC 6752; other low-mass clusters, such as M13 and M80 (Ferraro et al. 1998) host dozens of EHB stars but apparently few if any BH stars (M13 may host two BH stars; Sandquist et al. 2010). The large numbers of EHB stars in both the low-mass cluster sample and the high-mass cluster sample provide strong support for the notion that the fraction of EHB stars that are subluminous (i.e., BH stars) is nearly zero in low-mass clusters and ∼50%–100% in high-mass clusters.

There are several possible reasons why BH stars may only appear in massive globular clusters. One possibility is that only massive clusters can retain the helium-rich ejecta from their first generation of stars. Besides this helium enrichment, such self-pollution might account for many of the abundance variations in O, Na, Mg, and Al observed in MS and RGB stars (see Gratton et al. 2004, and references therein).

An alternative hypothesis is that massive clusters have many close binaries, which produce the high mass loss needed for flash mixing. Of the massive clusters known to host BH stars, none are core-collapsed, and all are of intermediate central concentration (Harris 1996), so that one might expect a significant fraction of surviving binaries (Ivanova et al. 2005). However, the lack of a radial gradient in the NGC 2808 BH population argues against a binary origin (Bedin et al. 2000).