The Isochrone Fitting Method

We will now summarize the technique for estimating the age of globular clusters. This provides an opportunity to add slightly more detail to the discussion of stellar evolution that was started above. With our introduction to GCs we are now in a better position to appreciate one of the main tools for the study of stellar astrophysics, the color- magnitude or Hertzsprung-Russell diagram. A CMD or H-R diagram is a scatterplot where each star is represented by a point according to its effective surface temperature,

Tef f in Kelvin, (x-axis, increasing toward the origin) and its luminosity, L in watts,

(y-axis, decreasing toward the origin). Figure 1.5 gives an example H-R diagram for the GC M55. The x-axis is labeled with both effective surface temperature and the equivalent B −V color index. Likewise, the y-axis is labeled with both the absolute magnitudeMV and luminosity relative to the Sun, L/L. The effective temperature of

a star is the temperature for which a blackbody emitter of the same radius would have the same luminosity. Hotter stars emit more blue light than red, which corresponds to a smaller B−V index. Cooler stars emit more red light than blue, corresponding to a higherB−V index. Luminosity, absolute magnitude, apparent magnitude (m), and the distance to a star are interrelated. Characterization of stars by apparent magnitude, still in use today, is a practice pioneered by the ancient Greeks. The details, while fascinating, are peripheral to the scope of this work. When discussing H-R diagrams

Figure 1.5: Example H-R diagram of globular cluster M55 (image from The Center for Cosmology and Particle Physics at NYU).

we will refer to quantities, luminosity and effective temperature, that can be expressed in SI units [Murdin, 2001].

The major stages of stellar evolution for low and intermediate-mass stars are illus- trated on the H-R diagram shown in Figure 1.5. The following discussion will apply only to low-mass stars since these are the stars observed in the oldest globular clus- ters. A star on the main sequence generates energy through hydrogen burning,i.e., the pp-chain or CN-cycle, in its core. As hydrogen fuel is consumed, the star increases its effective temperature and gradually moves toward the turnoff point (TO). The turnoff is the bluest and hottest position on the main sequence track and marks the point where hydrogen is exhausted in the core. The star begins to switch over to the CN- cycle as it approaches the turnoff point, so the turnoff luminosity is mostly regulated

by the 14N(p, γ)15O reaction rate. As the star climbs the red giant branch, it is grad- ually building up an ever-larger core of inert helium ash from hydrogen burning and the site of the CN-cycle is now a thick shell around the core. Eventually, when all of the hydrogen in the shell is converted to helium, the star experiences the helium flash and commences converting helium into carbon in the core. We leave this summary of nuclear burning at this point as further stages are less germane to the topic at hand. A few general remarks are in order before returning to age estimation. The evolutionary track depicted in an H-R diagram is known as anisochrone as it represents a snapshot in the life of a cluster. The sharpness of turnoff points in observed H-R diagrams is evidence for the nearly simultaneous creation of all the stars in the cluster from the same proto-stellar cloud. Likewise, the narrowness of the observed collection of points is evidence for the nearly homogeneous initial chemical composition of the stars.

Figure 1.6 shows theoretical isochrones for a globular cluster at three time stages: the zero-age main sequence (ZAMS), 10.4 Myr, and 112.5 Myr, from top to bottom panels respectively. The ZAMS corresponds to the point in time where all stars in the cluster have just started core hydrogen burning. Larger mass stars have higher effective temperature and thus lead shorter lives than lower mass stars. For this reason, the stars with higher Tef f will begin to leave the main sequence before those with lower

Tef f. As a result, the main sequence turnoff point will be seen to decrease in effective

temperature and luminosity as the cluster increases in age. Using stellar models, it is possible to establish a relationship between the turnoff luminosity of the observed H-R diagram for a GC and the age of the cluster. The procedure is, in effect, to “fit” a theoretical isochrone to an observed isochrone and extract the cluster age from the main sequence turnoff point. An example is shown in Figure 1.7. This is a very complicated and detailed process with a great deal of input physics, each with their own inherent uncertainties. The main sequence turnoff (MSTO) luminosity is one of

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Figure 1.6: Time sequence of simulated globular cluster H- R diagrams from ZAMS - 10.4 Myr - 112.5 Myr, respectively (http://www.astro.ubc.ca/ scharein/a311/Sim/hr/HRdiagram.html).

628 G. Imbriani et al.: The bottleneck of the CNO and Globular Cluster age

Fig. 4.Test of the CMD of the metal-poor cluster NGC 6397. The new isochrones with 13, 14 and 15 Gyr are reported. Their metallicity is

Z=0.0003 ([M/H]=₋1.8). We adopt (m−M)V=12.58 andE(B−V)=0.18. The data are from Rosenberg et al. (2000).

Fig. 5.Test of the CMD of the intermediate metallicity cluster NGC 5904. The new isochrones with 13, 14 and 15 Gyr are presented. Their metallicity isZ=0.001 ([M/H]=₋1). We adopt (m−M)V=14.41 andE(B−V)=0.02. The data are from Rosenberg et al. (2000).

Figure 1.7: Globular cluster NGC 6397 CMD with 12, 14, and 15 Gyr fitted isochrones to main sequence turnoff region (reproduced from [Imbriani et al., 2004]).

several age indicators for globular clusters. However, [Chaboyer et al.,1996] argues that methods other than isochrone fitting of the MSTO region are less desirable because of larger uncertainties arising from certain model considerations, such as the treatment of convection.