The age of stellar clusters luminositytemperature relationships

Beside the radioactive-isotope systematics there are several other methods of estimating the age T when nucleosynthesis started. Most of these methods employ so-called stellar isochrons (lines of equal age) within the luminosity-temperature H-R diagram. The luminosity L depends on the rate of energy production in the stellar interior. In a solar-like star this is proportional to the rate of hydrogen burning, f (dH/dt), where f is the yield of nuclear energy per hydrogen atom (Shklovsky, 1977):

Approximating the solution of Eqn (7.3) allows us to estimate the age of a main-sequence star, TMS:

Zero-age stars, for which T = 0, having an initial hydrogen abundance H0 — 1057 atoms for a solar-like star and the highest effective temperatures, are situated along the left-hand boundary of the main sequence (approximately along the left-hand edge of the shaded band in Fig. 5.1). This edge determines a zero isochron. As a star evolves it moves toward the right-hand edge of the main sequence; the rate and the trajectory of the movement depend on the stellar parameters. Figure 5.1 illustrates the trajectory for a solar-mass star. By the time such a star turns off the main sequence, its amount of hydrogen has dropped to 0.9 of the initial amount.

The energy produced by the fusion of each hydrogen atom is 1.12 x 10-5 erg (4He being the final product, Section 1.2), whereas the solar luminosity is 3.86 x 1033 erg s-1. Substituting the above values in Eqn (7.4) gives the age of a solar-like star situated on the right-hand edge of the main sequence, TMS — 10 Gyr.

Modelling gives the position of 10 Gyr stars for other masses, thus determining the 10 Gyr isochron within the H-R diagram. Clusters of "syngenetic" stars of different mass, generally formed almost simultaneously from the same cold molecular cloud complex, indeed define such isochrons (Peacock, 1999; Hartmann et al., 2001). These cluster isochrons thus allow an age determination with more confidence than for individual stars, and values of 13 ± 1 Gyr were obtained for the two most ancient clusters (Salaris et al., 1997).

Further, modelling of the evolution of very-low-mass main-sequence stars and white dwarf cooling rates gives — 12 Gyr and — 10 Gyr respectively (Peacock, 1999). Obviously these objects can only record the time that has passed since their formation and thus provide minimum estimates for the age of the earliest stars.

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