Determination of the age of Globular Clusters
By Geri Losekoot
Supervisor Ralph Wijers
(Additional) Bachelor thesis
9th June 2009
The age of Globular Clusters 3
Methods to compute the age of globular clusters 4
Recent publications 8
Nawoord (in Dutch) 10
There are several methods to determine the age of star clusters. But what if the estimated age of a star cluster is more than the age of the universe? Are the estimated ages consistent with our theory about the age of the universe?
In this report I show my study to the methods to determine the age of star clusters and the uncertainties in the outcomes. Because especially the oldest globular clusters have ages of about the age of the universe, I only take in consideration the older globular clusters.
The age of Globular Clusters
In this chapter I summarise some studies to ages of old globular clusters.
In 1984 Iben and Renzini  wrote an article about single star evolution of massive stars and early evolution of low and intermediate mass stars. In this article they estimate the age of M92 at 16 (±3,5) Gyr, where the uncertainty is only based on the uncertainty in ΔM(V). If they choose other values for the Helium abundance, the estimated age drops to about 14 (±3,5) Gyr.
Relying on only the absolute luminosities of the main sequence turnoff and of the horizontal branch, the age of a Galactic globular cluster cannot be determined to an accuracy better than ±4 Gyr.
VandenBerg, Bolte and Stetson  wrote in 1996 an article about the uncertainties in determining the ages of the oldest globular clusters. The most metal-poor globular clusters have ages near 15 Gyr. Bringing Helium diffusion into account, the estimated age decreased by 7%.
In 1996 Jimenez  publishes his findings on the study of globular clusters M22, M5, M68, M107, M72, M92, M3, 47Tuc using the Horizontal Branch Morphology method. The oldest clusters have estimated ages of 13,5 (±2) Gyr. A 1σ uncertainty in each of the parameters of mass and helium content, combined with the effects of helium diffusion, gives a lower limit for the age of the oldest clusters of 9,7 Gyr.
In 1998 Jimenez  obtained, using the luminosity function method, an age of 12 (±0,5) Gyr for M55.
Strader  publishes in 2005 an article about analysis of Keck spectra of globular clusters in eight galaxies: NGC 1407, NGC 4365, NGC 4594, NGC 3610, NGC 1052, NGC 7457, VCC 1087 and Fornax. He found that the globular clusters have ages of >~ 10 Gyr (uncertainty of ~2Gyr).
Methods to compute the age of globular clusters
There are several methods to compute the age of globular cluster.
The most common used methods are isochrone fitting and Horizontal Branch morphology. So the other methods I will take out of consideration from now on.
3.1 Isochrone fitting method
The path an individual star follows in the Hertzsprung-Russell Diagram differs depending on the mass of the star. The length of the lifetime of the star will also differ the mass of the star. Low mass stars will live longer than heavy stars.
When we look at a H-R Diagram of a very young star cluster, almost all stars would sit on the “zero-age main sequence”, the diagonal line running from the upper left to the lower right. In time the heavy stars evaluate quickly to the right in the diagram toward the lower temperatures. The small and cold stars evaluate from the main sequence toward the Red Giant Branch (RGB).
With the isochrone fitting method we do the assumption that all stars in a globular cluster have the same age. We plot the stars of the globular cluster in the plane Teff (temperature) versus L (luminosity). The result is a track in the plane for all masses, with the same chemical composition, at the same time. This track is the isochrone. The isochrone turns off at some point from the main-sequence to the giant branch. This “turn off” is a measure for the age of the cluster. The younger the cluster is, the hotter the temperature at the turn off.
Following figure represents the isochrones at certain times.
In the following figure is shown the color magnitude diagram for M92. Stars are shown as dots, and sequences of isochrones show the expected positions of stars in the cluster for a range of masses but the same age. The isochrones shown here have ages of (left-to-right) 12, 15, 18 and 21 Gyr.
3.2 The Horizontal Branch morphology method
The horizontal branch morphology method is based on the spread of stars along the horizontal branch in the color magnitude diagram compared to theoretical models.
The range of colors where zero-age horizontal branch stars are found is depending on the metallicity (“the first parameter”) and the range of zero-age horizontal branch masses.
The zero-age horizontal branch color at given metallicity depends on both the star’s total mass and the ratio of the core mass to the total mass. Since the core mass is fixed by the physics of the Helium flash, it is quite insensitive to the mass and metallicity. The “second parameter” affecting the horizontal branch could be found in the mass loss and in observations of the CNO abundances in the clusters.
4.1 Isochrone fitting method
With this method the assumption is used that all stars are born at the same time, and only their masses differ from one another. In reality, not all stars in the clusters are born at the same time. Also isochrones depend not only on the masses of the stars, but also on their chemical abundances.
VandenBerg, Bolte and Stetson  claimed that the main dependence on age is the turnoff luminosity, LTO. The major uncertainties in LTO are:
Convection theory (calibration of the mixing-length parameter αMLT).
Using OPAL opacities the turnoff luminosity can be determined more realistic. Opacities are uncertain by about 10% to 20%.
The mixing-length theory of convection has an influence on the LTO. Changes in the mixing-length parameter αMLT cause changes in the age-color (luminosity-color) relation such that the luminosity of the hottest point on the track significantly shifts. For values of αMLT in the range of 1,0 to 3,0, Chaboyer  found that convection leads to an uncertainty in globular cluster ages of about a 10%.
4.2 Horizontal Branch morphology
Jimenez  found that using the horizontal branch morphology method the uncertainty in age determinations is 2 Gyr. He claims the major uncertainty is the reddening, causing 10% uncertainty in age.
At the end of 2007 Fregeau  published an paper about his findings using results of NASA’s Chandra X-ray observatory. Galactic globular clusters were overabundant by orders of magnitude in bright X-ray sources.
Conventional theories claimed that globular clusters pass through three phases of structural development: adolescence, middle age and old age. During adolescence, the stars near the centre of the cluster collapse inward. The middle age has reached when interactions of double stars prevent the cluster from further collapse. Old age describes when these central binaries are disrupted or ejected, and the centre of the cluster collapses inwards. It has long been thought that most globular clusters are middle-aged with a few heading towards the end of their evolution.
Fegeau studied 13 globular clusters in the Milky Way and shows that three of them have unusually large X-ray sources, which result from the interaction of stars in the crowded centres of the globular cluster, and are thought to be characteristic of middle age clusters. Previously, these clusters had been classified as old age because they had very tight concentrations of stars in their centres, but the implication of Fregeau’s study is that most globular clusters, including the other ten that Fregeau studied, are in fact still in adolescence.
Over the past years more and more parameters have been studied and have lead to more and more realistic values on fitting data. Where Iben and Renzini in 1984 derived an age of 14 (±3,5) Gyr for M92, Jimenez found in 1996 an age of 13,5 (±2) Gyr.
The latest findings of Fregeau concerning cluster dynamics have lead to the conclusion that the oldest galactic globular clusters may be not as old as we thought before.
After all it seems to be that the oldest globular clusters are not estimated to be older than the age of the universe (13 Gyr). More studies on the dynamics of globular clusters are required.
In 2005 heb ik samen met Gijs Mulders het kandidaatsproject gedaan. De titel van ons verslag is “N-body simulation of black holes in globular clusters”. Voor dit kandidaatsproject kreeg ik 8.6 EC aan studiepunten: destijds de eisen voor een kandidaatsdiploma. Graag zou ik mijn Bachelor-diploma (nieuwe structuur) halen. Echter één van de eisen is het Bachelorproject van 12 EC aan studiepunten.
Dit project is een aanvulling op mijn kandidaatsproject om met deze twee projecten samen aan de eisen voor het Bachelor project te voldoen.
Hierbij wil ik van de gelegenheid gebruik maken om Ralph Wijers te bedanken voor de begeleiding bij dit project.
 VandenBerg, Bolte & Stetson, The age of the galactic globular cluster system, 1996 ARAA 34, 461-510;
 Jimenez, Globular cluster ages, 1998 PNAS vol. 95 pp. 13-17;
 Jimenez, Ages of globular clusters: a new approach, Mon. Not. R. Astron. Soc. 282, 926-942 (1996);
 Iben & Renzini, Single star evolution I. Massive stars and early evolution of low and intermediate mass stars, Physics Reports 105, No. 6 (1984) 329-406;
 Strader et al., Extragalactic globular clusters: old spectroscopic ages and new views on their formations, The Astronomical Journal 130:1315-1323 2005 October;
 Rood, Metal-poor stars V. Horizontal-Branch Morphology, The Astrophysical Journal, 184: 815-837, 1973 September 15;
 Jørgensen & Lindegren, Determination of stellar ages from isochrones: Bayesian estimation versus isochrone fitting, A&A 436, 127-143 (2005);
 Chaboyer, Absolute ages of globular clusters and the age of the universe, The Astrophysical Journal, 444:L9-L12, 1995 May 1;
 Fregeau, X-ray binaries and the current dynamical states of galactic globular clusters, The Astrophysical Journal, Volume 673, Issue 1, pp. L25-L28, 2007 December 18.