Final answer:
The measurement of the period and apparent brightness of RR Lyrae stars in each cluster allows astronomers to use the period-luminosity relation to determine the distance from us, as it relies on comparing known intrinsic brightness with observed brightness.
option a is the correct
Step-by-step explanation:
To determine each cluster's distance from us, the correct option would be (d) the period and apparent brightness of RR Lyraes in each cluster. The reason is that RR Lyrae stars, just like Cepheid variables, exhibit a relation between their luminosity (intrinsic brightness) and their periodicity—meaning that the period of their light variations is directly related to their intrinsic brightness.
RR Lyrae stars have nearly the same intrinsic luminosity which is known to be about 50 Lsun. By measuring the period of pulsation, we can determine their absolute luminosity. Then, by comparing this with the apparent brightness of the RR Lyraes, which is how bright they appear to be from Earth, we can calculate the distance to these stars using the inverse-square law of light. This method is part of the distance ladder used in astronomy to measure distances within our Galaxy and to nearby galaxies, extending up to around 60 million light-years.
Options (a), (b), and (c) do not provide the information required to use the period-luminosity relation, thereby making them less useful for determining distances to star clusters based on RR Lyrae stars.