Final answer:
The age of a star cluster is best determined by finding the main-sequence turnoff point on its H-R diagram. This point reflects the evolution of the cluster's most massive stars into later stages, such as red giants, indicating the cluster's age.
Step-by-step explanation:
The age of a star cluster can be determined by determining the turnoff point on the main sequence of its H-R (Hertzsprung-Russell) diagram. The H-R diagram plots stars according to their luminosity and temperature, showing the different stages of stellar evolution. As a cluster ages, the most massive and luminous stars evolve off the main sequence first, becoming red giants or other evolved types of stars. This 'turnoff' point from the main sequence to the red giant branch indicates the point at which stars of a certain mass are transitioning out of the main sequence phase of their lifecycle.
By comparing the luminosities of the brightest stars that are still on the main sequence in different clusters and their calculated H-R diagrams, astronomers can estimate the cluster's age. The younger the cluster, the more massive and luminous the stars at the turnoff point will be. Over time, stars of progressively lower masses will evolve off the main sequence. In clusters like globular clusters, which are over 11 billion years old, there are no high-mass luminous blue stars left on the main sequence.
This method of estimating a cluster's age is more accurate than relying on other less direct measures such as measuring its speed of motion relative to the Sun, carrying out a star count of the stars in the cluster, or observing its position in the sky with respect to the Sun. Therefore, the correct answer is (d) determining the turnoff point on the main sequence of its HR diagram.