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The focal length of the objective lens in an astronomical telescope is an important factor that determines how much light the telescope can collect and how clearly you can see distant objects. In simple terms, the objective lens is the big lens at the front of the telescope. It collects light from the object you are observing and focuses it to a point inside the tube of the telescope. This focal length, denoted as \(f_o\), is crucial because it helps in forming the initial clear image of the distant object. The longer the focal length of the objective lens, the better it can capture light, which is especially important for observing faint astronomical objects.
In our specific problem, an objective lens has a focal length of 75.5 cm. This means that if parallel rays from a distant star are entering the lens, they will converge at a point 75.5 cm from the lens. Understanding this means grasping how critical the objective lens is for the functionality of the telescope, providing the primary focus needed to observe celestial objects.

The eyepiece lens is another essential part of an astronomical telescope. It serves the role of magnifying the image that is formed by the objective lens. The focal length of the eyepiece, denoted as \(f_e\), plays a pivotal role in determining the size of the image you see.The shorter the focal length of the eyepiece, the more it magnifies the image created by the objective lens. This is key to the overall magnification power of the telescope. The shorter focal length forces your eye to see the image like it is much bigger than it really is.
In our problem, once we've used the objective lens's focal length to capture light and form a clear image, we then use an eyepiece with a focal length of 2.5 cm. Given that the total distance between these lenses is 78.0 cm, this eyepiece's focal length is perfectly calculated to provide a magnification of 30.2 times. Remember, magnification is affected by the relationship between the focal lengths of the objective and eyepiece lenses. Hence, the eyepiece plays a key role in refining the focus for you to see a larger, more detailed image of distant celestial bodies.

A relaxed eye observation setting is vital for comfortable and long-lasting viewing through an astronomical telescope. But what does it mean exactly?
When we talk about relaxed eye observation, we refer to the setup of the telescope where the final image it produces is at infinity. This makes it easier for your eye because it minimizes eye strain, allowing you to focus naturally as if looking into the distance without constant adjustments. For a telescope to be in this mode, the eyepiece is adjusted such that the image produced by the objective lens appears to be at an infinite distance. With relaxed eye observation, viewers can comfortably stargaze for extended periods since their eyes do not have to keep readjusting focus. In the given exercise, the relaxation state ensures that the measured magnification takes into consideration the most comfortable viewing experience, which, in this case, results in a clear, steady magnification of 30.2 times.