In terms of telescopes, resolving power is crucial for which of the following tasks?

Resolving power refers to the ability of a telescope to distinguish between two closely spaced objects. This characteristic is particularly significant when trying to identify and analyze fine details in celestial images, such as the surface features of planets, the structure of distant galaxies, or the individual stars in a cluster. High resolving power allows astronomers to see these details more clearly, facilitating a deeper understanding of the physical processes taking place in these celestial bodies.

In contrast, measuring distances to astronomical objects often relies on other methods, such as parallax or redshift measurements, rather than the resolving power of a telescope. Similarly, determining the brightness of stars and calculating the orbital paths of planets are tasks that hinge more on photometric and astrometric techniques respectively, rather than the ability to resolve fine details in images. Therefore, the task that most directly benefits from high resolving power is identifying and analyzing fine details in celestial images.