Sometimes trying to figure out the different aspects of your eyepiece/telescope  combination can be  confusing for the beginner or maybe more advanced amateurs  need a refresher in the subject. Either way it can be useful to calculate these aspects.

The specific variable names in this article will always be the same. The variables I will be using are:

D = diameter
M = magnification
F_o = focal length of objective
F_e = focal length of eyepiece
F = actual field of view
A = apparent field of view
P = exit pupil
f = focal ratio

It doesn't matter what type of telescope you are using the formulas will apply to all types.

Let's Get Started

To Start, you will need to know some basic information about your telescope. The first piece of data we need to define is the diameter of your objective lens or mirror. There may be a plate or sticker on your telescope tube that says something like:

D=60 F=600

The "D=" part is the diameter of the objective, and the "F=" part is the focal length of the objective, usually expressed in milimeters. If this information is not on the telescope, check your manual.

If this information is not in your manual, either, measuring the objective is easy enough. For a rough estimate of the focal length, measure the length from the objective to the end of the tube for a newtonian, or to the position of the eyepiece, including diagonal, for a refractor. Schmidt-Cassagrains are a little more difficult, but most of them are approximately 10 times the diameter of the objective.

Our first formula comes from these two variables. The focal ratio (f) is determined by the ratio of the objective focal length (F_o) to the diameter of the objective. (D)

f = F_O / D

In our example above, a 600mm focal length divided by a 60mm objective diameter gives us a focal ratio of f/10.

600 / 60 = 10

This may not seem important to you right now, but this value will be used in later calculations. It also tells you the photographic speed of the objective, if you use your telescope for astrophotography.

Magnification

The magnification formula is probably the one most used by astronomers. If we know the focal lengths of the objective and the eyepieces being used, we can calculate the magnification, or power (x) of an eyepiece:

M = F_o / F_e

The F_e variable is the focal length of the eyepiece, which is usually on the eyepiece

Staying true with our example, if we are using a 10mm eyepiece, (F_e) our formula is:

600 / 10 = 100x

This tells us how many times closer an object apears in the eyepiece. Using this formula with all of your eyepieces will tell you the range of magnifications available to you.

Barlow Lenses

The Barlow lens is a common accesory with many telescopes that doubles the number of magnifications available to you. Most barlows are marked with the amplification factor marked on the side, usually 2x or 3x. If we take the magnification from the previous formula and multiply by the amplification factor, (we will use a 2x barlow as an example) we get:

100x * 2 = 200x

When buying eyepieces, it is a good idea to make a list of all of the current magnifications that you have available to you, including with the barlow, so you are not buying eyepieces that will just duplicate what you already have. If you don't already have one, a barlow could be your best investment.

Field of View

While magnification lets us know how much closer something looks, if we know the field of view, we will know exactly how much sky area a particular eyepiece covers. But first we must define the two types of field of view:

Apparent Field of View - Apparent field of view is the circle you see when you look into an eyepiece. Hold your eyepiece up to your eye without the telescope. You will see a light circle. This is your apparent field of view. It will always be the same with this eyepiece, no matter what telescope or barlow lens you use with it.

Actual Field of View - the actual amout of sky coverage seen in the eyepiece. Since it looks closer, the actual field is much smaller than the apparent field, depending on the magnification being used.

For example, let's say something has an actual angular size of 1 degree. (twice the diameter of the full moon) If we use an eyepiece that gives us 100x, it will appear to span 100 degrees through the telescope, but since most eyepieces only have about 45 degrees apparent field, we will only be able to see less than half of it at a time!

This brings us to our next formula. The actual field of view (F) is equal to the apparent field of view (A) divided by the magnification. (M)

F = A / M

Most modern eyepiece manufacturers give you the apparent field in the specs for the eyepiece that came with the eyepiece or in the product catalog, but if you don't have this information, here is a list of popular eyepiece types and their approximate fields of view:

• Plossl - 50^o
• Orthoscopic - 45^o
• Ultrascopic - 50^o
• Modified Achromat (MA) - 45^o
• Erfle - 65^o
• Wide-angle - 65^o
• Super wide-angle - 80^o

We can also figure the apparent (A) field if we know the actual (F) field of view and the magnification, (M) with the formula:

A = F * M

One way to estimate the actual field of view is to use a star chart and compare what you see in the eyepiece with what is on the chart. This method is only as accurate as as you make your estimate, and can be off by quite a bit if the scale on the chart is not large enough, or if there aren't enough stars visible through the eyepiece or plotted on the chart.

Star Drift Method

A much more accurate method is to use the 'star drift' method. (This gets a little complicated, but it's easier to understand if you are careful to remember the difference between minutes and seconds of arc (angular measure), and minutes and seconds of time.

By placing a star just outside the field of view in the eyepiece so that as it drifts by in the eyepiece (with the motor drive off, if you have one), we can time the star's passage and get a very accurate measurement of the actual field of view.

The earth turns 360^o in 24 hours, so in one hour, the earth turns 15^o:

360 / 24 = 15

15^o divided by 60 minutes gives us .25^o, or 15' of arc. An angular degree is divided into 60 minutes of arc, (') and a minute of arc is divided into 60 seconds of arc, (") so there are 60' per degree, and 360" per degree.

If we take this relationship, and time the passage of a star through the field of view in any eyepiece in seconds, we will be able to measure exactly the actual field of view. Lets say that a star takes 2 minutes (120 seconds) from the time it enters the eyepiece field of view, passing through the center, until it disappears on the other side (t). We figure the field of view, (F) with the formula:

F = t * 15

This gives us 1800", or if we divide this by 60, (60 seconds in a minute) we get 30', or 1/2^o.

Lets try another example. If a star's passage takes 200 seconds, then we get:

200 * 15 = 3000" = 50' = .83^o

Exit Pupil

Our last calculation has to do with the size of the image that exits at the eyepiece. Point your telescope at a bright light source or the open sky with an eyepiece in the focuser. Now position a piece of white notebook paper against the eyepiece and move it out slowly. You will notice a white circle of light on the paper. This is the exit pupil. This is the circle of light that enters your eye when you look into the eyepiece.

The size of the exit pupil can be measured directly from the paper, or more accurately, from the formula:

P = F_e / f

The dark-adapted human eye has a maximum opening at the iris of about 8mm, and slightly less as we get older. If the exit pupil is larger than the iris, there is a loss of light entering the eye and subsequently a dimming of the image. For most telescope/eyepiece combinations, this light loss is negligable, but if we are trying to find our lowest usable power, rearrange the formula like this:

F_e = f * 8

Thus, if we have a telescope with a focal ratio of 5, (f/5) multiplied by 8mm (maximum eye pupil size) we get 40mm for the longest usable focal length.

It is not really necessary to calculate the exit pupil size for every eyepiece that you own, because the only inportant information you gain is the maximum usable eyepiece focal length that you can use with your telescope.