On a bright clear day, you are at the top of a mountain and looking at a city \(12 \mathrm{km}\) away. There are two tall towers \(20.0 \mathrm{m}\) apart in the city. Can your eye resolve the two towers if the diameter of the pupil is 4.0 mm? If not, what should be the minimum magnification power of the telescope needed to resolve the two towers? In your calculations use \(550 \mathrm{nm}\) for the wavelength of the light.

Angular separation is a crucial concept in optics that helps us understand how far apart two objects appear when viewed through an optical device like a telescope or even with the naked eye. It's measured in angles, typically in degrees or radians, and it's defined as the angle formed by lines originating from the observer's eye to two separate points in the field of view.

For instance, when you gaze at twin towers from a distance, the angular separation would be the angle your line of sight makes when moving from the top of one tower to the top of the other. Mathematically, if the towers are close and the distance to them is much greater than the distance between them, we can use what we call the 'small-angle approximation'. In essence, the angular separation, denoted as \( \theta \), can be approximately calculated by dividing the distance between the towers by the distance from the observer to the towers. It allows us to estimate how objects that are spatially apart will be perceived at a given distance.

The Rayleigh criterion is a valuable rule of thumb to assess the limit at which an optical system can distinguish two points as separate. Essentially, it defines the 'angular resolution' possible with a limited diameter of the system's aperture, such as the pupil of an eye or a telescope's lens.

According to the Rayleigh criterion, two points are resolved when the principal diffraction maximum of one image coincides with the first minimum of the other. If two light sources are closer together than this limit, they will appear as a single blurred point instead of two distinct points. This phenomenon is attributed to diffraction, a consequence of the wave nature of light. The equation derived from the Rayleigh criterion expresses the minimum angular resolution \( \delta\theta \) that can be achieved, given by the formula \( \delta\theta = \dfrac{1.22\lambda}{D} \). Here, \( \lambda \) represents the wavelength of light, and \( D \) denotes the diameter of the aperture.

Angular resolution describes the ability of an imaging system to resolve detail in the object that is being imaged. An optical system with a higher angular resolution can distinguish objects that are closer together than a system with lower resolution. When comparing the angular separation and the minimal angular resolution of an optical system, if the angular separation is larger than the angular resolution, the system can distinguish between the two objects as separate entities.

For example, when looking at two towers from a distance with the human eye, if the angular separation between the towers is greater than the eye's angular resolution, we can see the two as distinct structures. This is precisely what happens in the exercise provided. The angular separation is larger than the calculated angular resolution based on the Rayleigh criterion, implying that the human eye should be able to resolve the two towers without the aid of a telescope or any additional magnification.