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Readers may get an impression by reading the previous sections that we have solid and successful theories. Quite contrarily, there are several critical issues to be understood before we can ever claim so.

4.1. alpha-enhancement

Theorists (including myself) often interpret the UV vs Mg2 relation as a metallicity effect on the UV flux. However, it should be noted that Mg2 strength may not be representative of the overall metallicity. In fact, it has been known that elliptical galaxies are enhanced in alpha-elements with respect to iron. We then naturally wonder if it is not the overall metallicity but alpha-enhancement that generates the UV upturn. To perform this test, we need alpha-enhanced stellar models. The Y2 Isochrones group have released their alpha-enhanced stellar models for the main sequence (MS) through red giant branch (RGB) (Kim et al. 2002). But, no alpha-enhanced HB models are publicly available yet. alpha-enhancement can have severalimpacts on the galaxy spectral evolution. First, it changes the stellar evolutionary time scale, as CNO abundance affects the nuclear generation rates. Second, it changes opacities and thus the surface temperatures of stars. These two effects will make a change in the mass loss computed using a parameterised formula, such as the Reimers (1975) formula. For a fixed mass loss efficiency, we find the alpha-enhanced ([alpha/Fe] = 0.3-0.6) tracks yield approx 0.03 Modot smaller mass loss at ages 5-8Gyr but approx 0.03 Modot greater mass loss at ages gtapprox 8 Gyr, compared to the standard ([alpha/Fe] = 0) tracks. alpha-enhancement must have similar opacity effects on the HB evolution, while its effect on the mass loss on the HB should be negligible. Thus its effects are expected to be greater to the MS to RGB than to the HB phase. Considering this, I have decided to inspect the overall effects of alpha-enhancement by just adopting new alpha-enhanced MS through RGB tracks, ignoring the change in the HB models. My earlier review (Yi & Yoon 2004) shows the results for two metallicities and three values of alpha-enhancement. It can be summarised as follows. In old metal-poor models alpha-enhancement causes a positive effect to the relative UV strength because (1) it causes a slight increase in mass loss on the RGB and (2) it causes MS stars and red giants to be redder and fainter in V band. The [alpha/Fe] = 0.3 model roughly reproduces the SED of a typical UV-strong metal-poor globular cluster, which is satisfying. The metal-rich models on the other hand do not show any appreciable change in response to alpha-enhancement. Because giant elliptical galaxies are largely metal-rich (roughly solar) and the light contribution from metal-poor stars is not substantial, it is unlikely for alpha-enhancement to play a major role to the UV upturn.

4.2. EHB stars in star clusters

With the HST spatial resolution, a number of studies have found hot, extended horizontal branch (EHB) stars in globular clusters (e.g., Piotto et al. 1999). They are efficient UV sources and important candidates for the main UV sources in elliptical galaxies; but canonical population synthesis models have difficulty reproducing them as they are observed (number density, colours and brightness).

NGC 6791 is a particularly interesting case. This old (8-9 Gyr) metal-rich (twice solar) open cluster is unique resembling the stellar populations of the giant elliptical galaxies. Strikingly, 9 out of its 32 seemingly-HB stars have the properties of typical EHB stars (Kaluzny & Udalski 1992; Liebert et al. 1994), while canonical models do not predict any (Yong et al. 2000). It is critical to understand the origin of these old hot metal-rich stars. Landsman et al. (1998), based on UIT data, concluded that NGC 6791, if observed from afar without fore/background stellar contamination, would exhibit a UV upturn just like the ones seen in elliptical galaxies.

Through detailed synthetic HB modelling we found that it is impossible to generate an HB with such a severely-bimodal colour distribution as shown in this cluster, unless an extremely (and unrealistically) large mass dispersion is adopted. In the hope of finding a mechanism that produces such an HB Yong et al. (2000) explored the effect of mass loss on the HB. Yong et al. found that with some mass loss taking place on the HB (approx 10-9 - 10-10 Modot yr-1) HB stars born cool quickly become hot, suggesting that mass loss on the HB might be an effective mechanism of producing such stars. Vink & Cassisi (2002) however pointed out that the level of the mass loss assumed by Yong et al. is too high to justify in their radiation pressure calculations in the context of single-star evolution. Green et al. (2000) reported that most of these hot stars in NGC 6791 are in binary systems. If they are close binaries and experience mass transfer it would be an effective mechanism for mass loss. But at the moment it is difficult to conclude whether binarity had causality on their EHB nature or not.

4.3. Binaries

SdB/O stars, the central objects of this conference, may be the field counterparts of the EHB stars in clusters. They have the properties similar to those of the UV sources in the UV-upturn galaxies. Surprisingly, more than 70% of sdB stars are found to be in binary systems (Saffer et al. 2000; Maxted et al. 2001).

Han et al. (2003) used a binary population synthesis technique to study the effects of binary evolution and found that 75-90% of sdB stars should be in binaries. SdBs are detected to be in a small mass range centred at 0.5 Modot, but Han et al. found that the range should be in truth as wide as 0.3 through 0.8 Modot. They predict a birthrate of 0.05 yr-1 for Population I stars and 6 million sdB stars in the disc. Assuming the Galactic Disc mass of 5 × 1010 Modot, this means roughly 100 sdB stars per 106 Modot. In a back-of-the-envelope calculation, there are roughly a few thousand HB stars per million solar mass in globular cluster populations. A comparison between the sdB rate (100 per 106 Modot) and that of the HB (say, 5000 per 106 Modot) suggests that an old disc population may develop 1 sdB star for 50 HB stars (2%). This sounds by and large reasonable from the EHB-to-HB number ratio found in globular clusters. But it is hardly impressive from the perspective of searching for copious UV sources in galaxies. For comparison, NGC 6791 has roughly 30% (8 EHB-like stars out of 32 HB-like stars) and the UV-brightest Galactic globular cluster omegaCen has 20%. These two examples show an order of magnitude higher values of EHB-to-HB ratio than deduced from a simple estimation based on the binary population synthesis models. Yet, even omegaCen does not exhibit a UV upturn as observed in giant elliptical galaxies: FUV-V is comparable but FUV-NUV is 1-2 magnitudes redder than found in ellipticals. If this calculation is realistic at least within an order, binary mass transfer may not be sufficient to provide the origin of the majority of the UV sources in UV-upturn galaxies. On the other hand, a larger sdB production rate might be plausible in elliptical galaxy environment due to large age and/or large metallicity.

A considerably more detailed investigation was presented by Han et al. (2007). They constructed the population synthesis models including binaries of varied properties (in mass ratio and separation). The conclusions from their prediction can be summarised as (1) most of the UV light of ellipticals comes from binary sdB stars (2) a UV upturn starts to appear as early as when the galaxy is 1.5 Gyr old (3) and the FUV-V colour stays virtually constant since then. This is an important prediction because this is the first study that realistically consider binary products in population models. One immediately notices that the item (3) contradicts the single-star population models of Yi et al. (1999) discussed in Section 3.3 and Figure 2.

4.4. Other issues

There are other important issues as well. For example, the late-stage flash mixing scenarios and the like (D'Cruz et al. 1996; Brown et al. 2001) may also be effective ways of producing hot stars (such as sdB stars) in old populations. Their typical temperature range (Teff >> 20,000 K) and the predicted birthrate may not be entirely consistent with the UV upturn shown in elliptical galaxies, however.

Another important observational constraint comes from the HST UV images of M32. First, Brown et al. (2000) found that PAGB stars are two orders of magnitudes fewer than predicted by simple stellar evolution theory. This is significant as PAGB stars are thought to account for 10-30% of the UV flux in the UV-upturn galaxies (Ferguson & Davidsen 1993). More importantly, they find too many faint hot HB stars to reproduce with standard population models that are based on the mass loss rate calibrated to the globular cluster HB morphology. It is possible to reproduce the observed number of hot stars in M32-type populations if a greater mass loss rate is used, which would be consistent with the variable mass loss hypothesis (Willson et al. 1996; Yi et al. 1997b, 1998). But theoretical justification is a problem again.

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