Estimating age of a star cluster

How can we determine the age of a star cluster using its H-R diagram.

Background

Star Clusters are groups of stars that are gravitationally bound ranging from just ten to over ten million in number e.g. Pleiades cluster (Krittika) or M45 in Taurus constellation. An H-R diagram is a scatter plot of Luminosity vs. surface temperature or absolute magnitude vs. spectral type of stars. Main sequence is the track of stars that runs diagonally across the H-R diagram from high luminosity, hot stars to low luminosity, cool stars.

Need

The star clusters are useful in checking theories of stellar evolution, for calibrating absolute luminosities, and
for the study of the initial mass function (IMF). It is also well known that open clusters are vital for determining galactic structure. Owing to these reasons, the study of star clusters becomes very important and its age being one of the important properties, age determination is also crucial.

Idea

A star spends most of its life time on the main sequence in the steady state (hydrostatic equilibrium). The main sequence turnoff point (point on the H-R diagram of a cluster at which stars just begin to leave the main sequence) is a measure of cluster age since the older the cluster becomes, fainter is the turnoff point. This forms the basic idea behind this method of determining the cluster age.

Concept

The lifetime of a star varies as ratio of total fuel available to the rate of fuel consumption i.e. ratio of stellar mass to its luminosity (M/L). Since there exists a known relationship between Luminosity and Mass of stars, the age of a star is expressed as its luminosity raised to an exponent (L^b). So by measuring the luminosity of the most luminous star on the main sequence (L_{MS_max}) in a cluster, one can get an upper limit for the age of the cluster. It is an upper limit of the age since absence of stars brighter than the observed could be due to no stars being formed in the appropriate mass range. But for clusters with thousands of stars, such a gap in the mass function is very unlikely.

In Practice

Stellar Isochrones are theoretical calculations of ageing stellar populations. These stellar isochrones are matched against the observed H-R diagram of the star cluster, thus enabling us to measure the age of a star cluster.

References:
[1] Carroll, Bradley W., Dale A. Ostlie, and M. Friedlander. An introduction to modern astrophysics. Vol. 523. New York: Addison-Wesley, 1996.
[2] Kaufmann, William J. "Discovering the universe." New York: WH Freeman, c1993. 3rd ed. 1 (1993).
[3] Böhm-Vitense, Erika. Introduction to stellar astrophysics. Vol. 3. Cambridge University Press, 1992.