CFHT Information Bulletin, number 38, First Semester 1998

Abstract

Introduction

Not long after the great antiquity of the globular clusters had been proven (Sandage and Schwarzschild 1952; Hoyle and Schwarzschild 1955), it was also learned that they differ from the Sun - and from each other - in their metal abundance. It was shown by the use of slit spectroscopy (e.g., Kinman 1959, Morgan 1959) and by photometric measurements of line blanketing (e.g., Arp 1959) that (a) globular clusters are in general more metal-poor than the Sun and nearby stars, (b) within any one cluster no range of metal abundance can be perceived (one or two exceptions are now known), and (c) different globulars have different chemical compositions: it was soon concluded that the intrinsic abundances of elements heavier than helium span nearly two orders of magnitude among the globular clusters, from more than 1/10 to less than 1/100 of the Solar value.

The cottage industry of obtaining photographic photometry for stars in globular clusters gained steam during the epoch of the commissioning of the 5- and 4-meter class telescopes - from the early 1950s into the 1970s - and certain correlations between the metallicity of a cluster and the appearance of its Hertzsprung-Rusell (or, more specifically, "color-magnitude") diagram soon became apparent. In particular, the more metal-poor the cluster, the taller (Sandage and Wallerstein 1960), bluer (Sandage and Smith 1966), and steeper (Hartwick 1968) the giant branch. The position of the main sequence of unevolved stars at the bottom of the color-magnitude diagram also shifted toward bluer colors as clusters of decreasing metal content were considered. At first glance, then, the set of globular-cluster color-magnitude diagrams seemed to form a one-parameter family, with overall metal abundance serving as the independent variable.

However, the Hertzsprung-Russell diagram of a globular cluster or some other very old population also displays the so-called "horizontal branch" that bridges the "Hertzsprung gap" normally found in the center of the diagram for nearby stellar populations. In the mid-1960's, at almost the same time as this feature was explained as consisting of evolved, core-helium-burning stars (Faulkner 1966), it was also found that the appearance of the horizontal branch was largely, but not uniquely correlated with the metal abundance of the cluster. In general, the horizontal branch ranges from being red and jammed up against the side of the giant branch in the most metal-rich clusters (like 47 Tucanae), to being stretched all the way across the diagram and containing a multitude of RR Lyrae variables in clusters of moderate metal deficiency (like M5), to being almost wholly to the blue of the RR Lyrae instability strip in the most metal-poor clusters (like M92). Van den Bergh (1965, 1967a,b) and Sandage and Wildey (1967) pointed out striking exceptions to this trend. Van den Bergh first noted that the Draco dwarf galaxy, the intergalactic "tramp" globular cluster Abell 4, and the Small Magellanic Cloud globular NGC 121 all displayed the steep, tall giant branch of a metal-poor system, and at the same time had the stubby, red horizontal branch of a metal-rich one. Sandage and Wildey reported that the "anomalous" color-magnitude diagram of the Galactic globular cluster NGC 7006 showed the same seemingly irreconcilable combination of traits. Thus, it was established that some "second parameter" in addition to overall metal abundance differed from one cluster to another and was capable of affecting the way in which the clusters' constituent stars aged.

The first suspect was the helium abundance, Y. Theoretical models of Faulkner and of Faulkner and Iben (1966) showed that altering the assumed abundance of helium affects the horizontal branch, in the sense that for any assumed chemical composition, there is a critical point redward of which no self-consistent core-helium-burning configuration can exist, and that critical point itself moves farther to the red as lower values of Y are considered. At the same time, theoretical models of giant-branch stars by Hoyle and Schwarzschild (1955) and Demarque and Geisler (1963) showed that the morphology of the giant branch is strongly dependent on overall metal abundance, Z, but is very nearly independent of helium abundance, Y. The two parameters, Y and Z, then, do a very good job of explaining the observed range of combinations of giant-branch type with horizontal-branch type - a model which was brought to full fruition by Hartwick (1968). But where the did variations in Y come from? Were pockets of variously helium-rich material formed during the Big Bang, or were separate regions of space randomly enriched in helium by the first generations of stars? An answer was never found.

So it was necessary to round up some other suspects. Hartwick and McClure (1972) observed giants in the anomalous cluster NGC 7006 photoelectrically with the DDO photometric system, and found that absorption due to the CN molecule was unusually strong. Therefore, the relative abundance of carbon plus nitrogen (plus oxygen) to iron could be the second parameter, altering the structure of a horizontal-branch star either by affecting the opacity in the stellar envelope, or by catalyzing the conversion of hydrogen to helium in the hydrogen-burning shell (Hartwick and VandenBerg 1973). On the other hand, McClure and Norris (1974) obtained CN measurements in the southern globular cluster NGC 362, which also displays a horizontal branch somewhat redder than normal for its metal abundance, and found no such enhancement.

That left age as the outstanding obvious suspect. The horizontal-branch models of Faulkner and of Faulkner and Iben showed that the temperature of a core-helium-burning star is dependent upon its total mass: since the mass of the degenerate helium core produced by previous stages of evolution should be virtually independent of the original mass of the star, a larger total mass means a larger envelope mass on the horizontal branch, and therefore a bigger and redder star. Having shown that the classical second-parameter effect (i.e., too red a horizontal branch) occurs predominantly in the outer halo of the Galaxy, Searle and Zinn (1978) assumed that the second parameter was (younger) age, and proposed a new paradigm for the formation of our Galaxy. Until 1978, the commonly accepted vision of the early days of our Galaxy was that of Eggen, Lynden-Bell, and Sandage (1962), who employed the observed eccentricities, inclinations, and angular momenta of the orbits of the most metal-poor field stars to argue that the proto-Galaxy had collapsed in free-fall from a much larger configuration to its present form in a very brief period of time, ~10⁸ years. Searle and Zinn argued, conversely, that for second-parameter (i.e., relatively young) objects to exist at large Galactocentric radii, the proto-Galaxy must have remained large for at least several × 10⁹ years; their proposed model was a large number of comparatively small gas clouds, evolving in isolation and moving independently through space, until over the course of time they eventually collided with each other, and gradually coalesced into the present Milky Way Galaxy, leaving some of their earlier generations of stars and star clusters to wander forever through the remote spaces of the Galactic halo. The present-day dwarf spheroidal galaxies like Draco might even be the last remaining survivors of this population of primordial galaxylets.

The modern-day computer modelling of Lee (1989) and colleagues (e.g., Lee, Demarque, and Zinn 1992) has placed this hypothesis on a firm theoretical basis. Proponents of age as the second parameter argue that it is possible, from knowledge of metal abundance and horizontal-branch morphology alone, to pin down the age of a stellar population to of order 1 Gyr (cf. Figure 1), and to trace in quantitative detail the formation history of the Milky Way Galaxy (e.g., op. cit. and Lee 1992). It is undeniably true that some globular clusters are younger than most (e.g., Gratton and Ortolani 1988, Stetson et al. 1989, Buonanno et al. 1994), but still there has been sporadic resistance to the acceptance of age as the sole, or even the principal second parameter (e.g., age differences sufficient to explain horizontal-branch morphologies are not consistent with age differences derived from turnoffs: Catelan and de Freitas~Pacheco 1993, Ferraro et al. 1997, Johnson and Bolte 1997; the second-parameter effect exists even among "young" globulars: Buonanno et al. 1994; environmental factors are important: Buonanno et al. 1997). Nevertheless the age explanation of the second-parameter phenomenon and its implications for the formation of the Galaxy have become widely accepted.

M3 and M13: New Data, New Analysis

Don VandenBerg and I obtained photometric imagery of the globular clusters M2, M3, M5, M13 and M92 and a number of standard fields with the CFHT on the nights 1994 June 4/5 and 5/6. Images were obtained with the Loral 3 2k × 2k CCD and the standard B and I filters, and conditions were photometric for the entirety of both nights. For M5, M13, and M92 we obtained series of exposures in each filter with the center of the cluster in the southwest, northwest, northeast, and southeast corner of the chip, to cover a field roughly 13´ square in each case, with the crowded central region of the cluster appearing in all images¹. However, on the second of our two nights - the same night that M13 was observed -we also obtained data for much smaller fields in the clusters M2 (= NGC 7089) and M3. M2 was an end-of-the-night fill-in object for us. Although it has been less thoroughly studied than either M3 or M13, according to all available evidence it is very similar to these two clusters in its chemical composition (according to Harris (1996), its metal abundance is within 0.08 dex of the abundances of M3 and M13), and resembles the latter, but not the former, in its horizontal-branch morphology. Here, I will use only the data for the better-studied clusters, M3 and M13, to probe the nature of the second-parameter phenomenon, at least among nearby clusters in the inner Galactic halo.

Figure 2 is a composite of our full mosaic for M13, spanning roughly 13´ × 13´. Figure 3 is a blowup of a small portion of this image, including the center of the cluster just within the top edge of the strip, shown at two different gray-level stretches. The dynamic range of these data is enormous: we were able to obtain precise photometry for more than 90,000 stars spanning an I-magnitude range of 11 magnitudes; these will be used in our future luminosity-function analyses. For my present purposes, however, I want merely to define the cluster loci in the color-magnitude diagram with the utmost accuracy. Therefore I have selected out the most isolated stars possible, using an algorithm which is insensitive to the colors of the stars: although the faintest stars are preferentially selected against (as being most likely to be affected by the presence of other stars) at any given magnitude the color distribution will be faithfully reproduced.

Figure 4 is a color-magnitude diagram for the 6,624 M13 stars that were deemed to be the least crowded in the field; these are the ones that will be employed in the analysis below. Stetson and VandenBerg (1998, in preparation) will provide a fuller analysis, including a luminosity function based on the full set of ~10⁵ stars. Note that this diagram does not reach all the way to the tip of the giant branch, because these stars were saturated in our images. For M3 we observed only one field, about 5´ from the center of the cluster. Because these data are less extensive than those for M13, I have included data from other CFHT and Kitt Peak observing runs for the same M3 field; these additional data increase the number of stars by only about 10%, but they have been rigorously referred to the same photometric system and do increase the overall precision of the photometry. The present discussion will be based on the 2,648 M3 stars that are minimally crowded.

Let us suppose that M3 and M13 differ in age only. Judging from the difference in their horizontal-branch morphologies (Figure 1), we would expect M13 to be some 2-2.5 Gyr older than M3. How would we expect their color-magnitude arrays to differ? The one feature of the color-magnitude diagram which is most simply and uniquely related to age (for fixed chemical-composition parameters) is the magnitude and color of the main-sequence turnoff. As Figure 5 shows, with advancing age the main-sequence turnoff of a star cluster becomes fainter and redder. I have determined the magnitude and color of the turnoff of each cluster using a completely automatic and impersonal computer algorithm that compares theoretically generated isochrones directly to the observed cluster sequences. I make no a priori restrictions on the distance, reddening, chemical abundance, or age of the cluster: the software simply takes the full set of available isochrones (Figure 6) and compares them all to the data, with completely arbitrary vertical and horizontal shifts, and finds the one that provides the best match. The uncertainties of the model physics (convection, equation of state, detailed chemical abundance ratios) and of the observations (distance, reddening, general metallicity) are so significant that I do not in any way pretend that we are determining absolute ages, distances, abundances or anything else. We are simply performing objective fits of drafting-spline-like curves that happen to resemble the data more closely than, say, low-order polynomials. Having done that, I adopt the turnoff color and magnitude of the best-fitting shifted isochrone as valid for the cluster sequence².

Defined in this way, the turnoff colors for M3 and M13, respectively, are (B-I)_TO = 0.996 and 0.982, with random errors expected to be only ~0.002 mag; since both clusters were observed on the same night with the same equipment, systematic calibration errors are irrelevant (though they are believed to be comparably small). The reddenings to these clusters are small and fairly well known: in his catalog of Milky Way globular clusters, Harris (1996) lists E(B-V) = 0.01 and 0.02 for the two clusters, which translates to E(B-I) = 0.027 and 0.054. Thus, the intrinsic turnoff colors are E(B-I)_,TO 0.97 and 0.93 for M3 and M13, respectively. To the limits of the available data, the allegedly older cluster has a bluer turnoff, contrary to the most straightforward and unambiguous of theoretical predictions. This result is fairly robust, because M13's turnoff is directly observed to be bluer than M3's; with every indication that the reddening toward M13 is higher than that toward M3 (including the appearance of slight differential reddening among the M13 stars in our data), the discrepancy can only increase.

Apart from the problem with the turnoff color, are the shapes of the turnoffs and subgiant branches of M3 on the one hand and M13 on the other consistent with age being the only parameter distinguishing them? The only way to answer this question is by a critical comparison to theoretical isochrones. While the uncertainties in the model physics preclude us from estimating absolute distances, abundances, or ages from the comparison of cluster sequences to isochrones, we are permitted to make differential comparisons between clusters, such as whether a given morphological difference does or does not appear to resemble the anticipated effects of a change of age as predicted by theory. Figure 7 is an example of this comparison for M3, where in the left-hand panel the observed stars are represented by points, and the best-fitting isochrone selected from among the ones in Figure 6 is plotted twice, at +0.10 mag and -0.10 mag in color from its best-fitting position (to have plotted the actual fitted locus would have lost the curve in the points). The right-hand panel shows the horizontal residual of each individual star from the fitted curve. To fit M13, I required the solution to utilize an isochrone representing the same chemical abundance as M3, but gave the program the freedom to choose the best-fitting age. The result is shown in Figure 8. The formal answer from this analysis is that, under the given assumptions, M13 appears to be about 12% older than M3 (specifically 17 as compared to 15 isochrone time units, which some call gigayears). This is slightly less than the 14%-20% age difference predicted by Lee, Demarque, and Zinn (1994), and seriously less than the ~35% age difference predicted by Catelan and de Freitas Pacheco (1995). However, the residuals show that even this age difference does not in fact provide a good fit to the M13 data: that isochrone which correctly matches the main-sequence turnoff of M13 completely mispredicts the slopes of the subgiant and giant branches (there may also be a bit of a systematic trend in the residuals on the main sequence fainter than about magnitude 20). The base of M13's giant branch is too blue for the postulated age difference, but it also has too shallow a slope, so the observed giant branch eventually reaches and even crosses over the predicted giant branch at the brightest magnitudes modeled. It is true that the comparison of M3 to the theoretical isochrones (Figure 7) is not perfect, with maximum systematic deviations of order 0.02 mag in B-I (this corresponds to less than 0.01 mag in B-V) indicating that some tweaking of the model stellar parameters is called for, but the residuals in the case of M13 are far more striking. A direct comparison of M13 to M3's best-fitting locus is shown in Figure 9, where we can see that M13's subgiant branch also rises in luminosity too rapidly as it changes color; if age were the only variable factor, the two subgiant branches should be more nearly parallel (e.g., VandenBerg, Bolte, and Stetson 1990).

I conclude that the assertion that age is the only relevant difference between M3 and M13 is not consistent with modern theoretical models, as used in the present, strictly differential, fashion.

A somewhat better match to the M13 principal sequence is achieved with a 16 Gyr isochrone for a chemical composition ~0.4 dex more metal-poor than M3, but such a metallicity difference is completely ruled out by the available spectroscopy. Furthermore, even at the lower metal abundance the sequence shows a similarly poor match to the slope of the giant branch. So what could be the difference betwee M3 and M13? One possibility is stellar rotation. VandenBerg, Larson, and De Propris (1998, submitted) have produced simple stellar evolution models incorporating the first-order effects of stellar rotation. Their results show a striking similarity to the observed anomaly in M13: in their models (see their Fig. 4) the nearly horizontal part of the subgiant branch rises faster in rotating models than in non-rotating ones, then the main part of the giant branch turns sharply upward at a bluer than normal color, but the giant-branch slope is a bit shallower, so the giant branch in the rotating models approaches (but does not actually reach or cross) the giant branch of non-rotating models. In Figure 8, the observed data cross over the giant branch because, in its attempt to predict the colors of as many M13 stars as possible, my computerized algorithm selected an isochrone with a giant branch that was too blue (i.e., an age that was too great). If the method had been instructed to allow the observed giant branch to approach the predicted one from the blue without actually reaching it, a younger age -- an age more like that of M3 - would have been preferred.

Greater rotation speeds among M13 stars than among M3 stars is consistent with other observational data, as well. Peterson (1985) directly measured greater values of v sin i among blue horizontal branch stars in M13 than among similar stars in M3. M13 giants are known to show greater depletions of oxygen and enhancements of sodium than giants of similar luminosity in M3 (e.g., Kraft 1994) - effects now attributed to deep envelope mixing, which is commonly believed to be especially effective in rotating stars. The same deep mixing that destroys the oxygen and enhances the sodium must also mix freshly generated helium into the envelope as well. This alone would tend to increase the temperatures and luminosity of the resulting horizontal-branch stars, but higher envelope helium abundance would also increase the efficiency of mass loss on the giant branch, producing still bluer horizontal-branch progeny, just as is seen in M13.

But we have not yet explained why M13's turnoff is bluer than M3's. Simple centrifugal support of the stellar interior against gravity, plus envelope mixing on the giant branch would not be able to explain this anomaly. Perhaps we must revert to one of the original suggestions, that M13 has a higher helium abundance Y or a lower abundance of CNO elements than M3. Don VandenBerg has begun computing new sets of isochrones with different values of the helium and alpha-element abundance. It turns out, in fact, that an enhanced-helium isochrone (Y=0.272 rather than the 0.236 previously assumed) does a much better job of fitting the observed M13 locus (Figure 10), yielding maximum systematic deviations of the same order of magnitude as found in the fit of the M3 data to a standard helium abundance. The best fits to M3 and M13 from among all the available isochrones suggest that M3 is perhaps 1 Gyr younger than M13, but with the lower helium abundance. However, these are just the best fits from among the isochrones I have presently at hand - note that the residual errors in the fit to the M13 sequence have not been entirely removed even in Figure 10. When Don computes isochrones for, say, Y = 0.30 we may find that these provide an even better fit to M13, with the same age as M3. That is work for the future.

But if M13 has a helium abundance higher than that produced by the Big Bang, where did it come from? Is it possible that deep mixing can actually occur on the main sequence as well, mixing some nucleosynthesized helium into the envelope and some fresh fuel into the core, and producing a main sequence which is bluer and a turnoff a bit brighter than predicted for a given physical age? If such mixing does occur on the main sequence, it would help explain the CN anomalies seen among main-sequence stars in 47 Tuc (Briley et al. 1994). However, mixing on the main sequence would add fuel to the stellar core and would significantly change the internal variation of helium abundance with radius in comparison to non-mixed stars. This would seriously affect the cluster's luminosity function, because the luminosity evolution of a red giant depends upon the run of chemical abundance that the hydrogen-burning shell encounters as it eats its way outward within the star. If our future luminosity-function studies rule out this possibility, then it will become necessary to reconsider van den Bergh's original (1965, 1967a,b) suggestion, namely that the proto-Galaxy was inhomogeneous in its abundance of helium as well as of heavy elements.

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² For present purposes I have adopted as the working definition of the turnoff magnitude of an isochrone that magnitude V_TO such that the color at V_TO - 0.2 mag equals the color at VTO + 0.2 mag - this makes derived quantities slightly less sensitive to numerical imprecision in either the stellar evolution calculations or in the interpolations required to go from evolutionary tracks to isochrones. The color of the turnoff is taken to be the color of the isochrone at V_TO, which typically differs from the bluest color actually achieved along the isochrone by less than 0.001 mag.