Mars (6800 km diameter) is viewed through a telescope on a...

Mars (6800 km diameter) is viewed through a telescope on a night when it is $1.1 \times 10^{8} \mathrm{km}$ from the earth. Its angular size as seen through the eyepiece is $0.50^{\circ},$ the same size as the full moon seen by the naked eye. If the eyepiece focal length is $25 \mathrm{mm}$, how long is the telescope?

Key Concepts

Angular Size

Angular size refers to the apparent size of an object as seen by an observer, measured as the angle that the object subtends at the observer’s eye. In astronomy and optics, this concept is often used with the small-angle approximation, where the angular size (in radians) can be estimated by dividing the actual physical size of the object by its distance from the observer. This allows for a simple relationship between an object's dimensions and how large it appears in the sky.

Small-Angle Approximation

The small-angle approximation is a useful assumption in optics and astronomy when dealing with objects that have a small angular size. By assuming that for small angles (in radians) sin(?) is approximately equal to ?, the calculation of the angular size becomes straightforward, typically expressed as the ratio of the object’s diameter to its distance. This approximation simplifies many computations involving distant celestial objects.

Telescope Magnification

Telescope magnification is the factor by which a telescope increases the apparent angular size of an object. It is determined by the ratio of the focal length of the telescope’s objective lens to that of the eyepiece. This magnification factor explains how a relatively small object can be made to appear larger in the eyepiece, enhancing details that are otherwise too minute to detect with the naked eye.

Objective and Eyepiece Focal Length Relationship

In a simple refracting telescope, the overall length of the telescope is largely determined by the focal length of the objective lens, since it is the primary element that collects light from distant objects. The eyepiece has a much shorter focal length, and its function is to magnify the image formed at the focus of the objective. The ratio between these two focal lengths not only dictates the magnification of the telescope but also influences the design and physical dimensions of the telescope itself.

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