Telescopes: Mathematical Eyes to See the Universe

The telescope was first invented in the Netherlands around 1608, with Hans Lippershey being the first to apply for a patent. Soon after, Galileo Galilei improved the design and famously used it for astronomical observations, including discovering Jupiter's moons. How does this marvel, which revolutionized astronomy, work?

At its heart, a basic telescope uses lenses (or mirrors) to gather and focus light from distant objects, making them appear magnified. There are several key concepts to understand here:

Telescope's Objective

This is the large lens (or mirror) at the front of the telescope that gathers light from the object being viewed. It has a specific focal length, f_o , described below.

Focal Length, f_o

This is the distance from the telescope's objective to the point where the light rays converge after passing through the objective. Think of it like a magnifying glass focusing sunlight to a small, bright spot – that spot is the focal point, and its distance from the magnifying glass is the focal length.

Eyepiece 

The eyepiece is a smaller lens you look through. It also has a focal length, f_{e .}

Magnification 

The telescope's magnification (how much bigger the object appears) is determined by the ratio of the objective's focal length to the eyepiece's focal length. The formula is straightforward: M =  f_o/f_e.

For example, if your telescope's objective has a 900mm (900 millimeter) focal length and you use a 25mm eyepiece, the magnification is 900/25 = 36x (36 times). Different eyepieces allow you to change the magnification with the same objective.

Credit: skyandtelescope

Aperture and light-gathering power

The aperture is the diameter of the objective lens or mirror. A larger aperture collects more light, resulting in brighter and more detailed images, which is especially useful for faint objects like distant galaxies. The light-gathering ability is proportional to the square of the aperture. For instance, a 200mm aperture telescope collects four times as much light as a 100mm aperture telescope.

Focal ratio (f/stop)

The focal ratio is another important number and is calculated by dividing the focal length of the objective by its aperture: Focal Ratio =  f_o/Aperture.

A low focal ratio (e.g., f/stop value f/5) gives a wider field of view and brighter images of faint and distant objects like nebulae, while a high focal ratio (e.g., f/stop value f/10) gives higher magnification and is better for observing brighter objects like the planets or the moon.

Galileo's original telescope. Credit: Alessandro Nassiri via wikimedia commons

We have come a long way since Galileo's original telescope from the 1600s. Now, we have optical telescopes, radio telescopes, and specialized infrared, UV, X-ray, and other types of telescopes, some of which are even stationed in space to eliminate atmospheric interference (like the James Webb Space Telescope)! What's more, many of these telescopes are backed by powerful computers with impressive image- and data-processing capabilities.

James Webb Space Telescope. Credit: NASA

While the number and variety of telescopes has changed tremendously since Galileo's time, what has not changed is the fact that mathematics continues to be fundamental to their design and operations!