Easy Math For Astronomers

This is easy, I promise… no calculus required! Just a few simple formulae to help you get the most out of your telescope and binoculars and find your way around the sky.

The Basics

• The first useful number is the focal ratio of your telescope

Focal Ratio (F#) = Focal Length of Objective/Diameter of Objective

If, for example, you have a 6-inch (150 mm) scope with a focal length of 36 inches (900 mm), then the focal ratio is f/6. The focal ratio is also called the F# (“F-Number), much like the number you see on a camera lens.

• To find out how much your telescopes magnifies an image:

Magnification=Focal Length of Telescope/Focal Length of Eyepiece

So take the above example with a 900 mm focal length, using an eyepiece with 15 mm focal length, you get 900/15=60x (60 power).

• Here’s an important one: the exit pupil. This is the diameter of the beam of light coming out of your eyepiece. If the exit pupil of your eyepiece exceeds the size of your eye’s pupil, which is 6-7 mm at best, then you waste light from your telescope. The exit pupil is:

Exit Pupil = Focal Length of Eyepiece /Focal Ratio of Telescope

With our example of an f/6 scope, if we use the 15 mm eyepiece, the exit pupil is 15/6=2.5 mm. That’s good… all the light will enter your eye. But a 45 mm eyepiece gives you a 7.5 mm exit pupil. Your eye can’t take that in, so using such an eyepiece with an f/6 telescope gives you a low magnification but it’s a waste of light.

A Deeper Look

• The resolution of your telescope tells you how good your telescope is at seeing fine detail. The resolving power of a telescope:

Resolving Power (in arc-sec) = 116/Diameter of Objective (in mm)

A 150 mm telescope can resolve features, such as a tightly-spaced double star or fine details on a planet), as close as 116/150=0.77 arc-seconds apart. That’s pretty good… but it takes near-perfect atmospheric seeing conditions to get close to such resolution.

• Can you stand one more? The true field of view tells you how much of the sky you can see with a particular eyepiece through a particular telescope. Each type of eyepiece has an apparent field of view, which is what you see when you look through the lone eyepiece. This number is often printed on the eyepiece itself. To get the “true field of view”:

True Field of View = Apparent Field of View/Magnification

So if a 15 mm eyepiece has an apparent field of view 50 degrees, and you use it with our scope with 900 mm focal length to get 60x, then the true field of view is 50/60=0.83 degrees, which is about 1.6x the size of the full moon. You should calculate the true field of view of each of your eyepieces… it helps you know how big a circle of sky you can see with each eyepiece.

A Bit of History

Once upon a time, a wide true field of view meant you needed low magnification. But modern eyepieces like the Nagler and Ethos from Televue have a huge apparent field, which means you get high magnification AND wide field at the same time. Quite impressive.

Personal View

I’m pretty good at math, but my memory is really, really, bad. So I have a small card taped to my eyepiece box to remind me of the true field-of-view and the magnification of all my eyepieces. It’s a great help in star-hopping around the sky.