Measuring the Magnification of a Telescope

"What magnification does this eyepiece give?" A question asked all too frequently and usually answered in as casual a fashion. "Well, let me see, the eyepiece focal length is 24mm and the telescope has a focal length of 90 inches, so the magnification is roughly 90 times."

Voi la! A triumph of mental arithmetic. I converted 24mm to the nearest inch and divided it into the objective focal length.

How often have we all done this sort of thing, especially at observing sessions where there is a queue of beginners at the telescope? Some of us, blessed with good memories, go a stage further. We sit down beforehand and calculate each and every eyepiece power, taking the manufacturer's stated focal length for the telescope and dividing it by that stated for each eyepiece. Some of us even write it neatly on the eye-cap.

There seems nothing wrong with such a seemingly meticulous attention to detail. After all, is it not helpful to know the precise magnification of each eyepiece we use with our telescope?

The obvious answer is, "Yes, it does." But, what is "precise", and how
do we know the magnification calculated is correct? It is all predicated on the assumption
that the manufacturer's stated focal length is *exactly* what they
say they are. But how do you know that is so? And, although saying the power is, say
x90, carries with it the implication that this is a ball park figure; the real
value could be anything between x85 and x95, by putting the calculated value
as x86, we convey the aura of precision. We imply it is not x85 or x87, or any other
value. But how do we *know*, and does it matter?

Well, given that we are all guilty of this particular sin, and also that
manufacturer's quoted focal lengths are never *exactly* what they
claim, "Yes, it does matter." So what can be done to determine the eyepiece magnifications
more accurately, and how can we estimate the errors?

Errors in objective focal lengths are typically less than ±1%. For example a 6-inch f/8 mirror will most probably have a true focal length of 48 inches ± 1/2-inch. However Schmidt-Cassegrain telescopes with moving primary focusing can have an effective focal length that differs significantly from the nominal value given by the manufacturer. For example, consider the classic 8-inch f/10 SCT. The efl varies with the distance the eyepiece is mounted behind the backplate. The nominal 80-inch focal length applies at the backplate only. In fact the nominal 80-inch can become in excess of 88 inches. The error begins to increase!

Errors in eyepiece focal lengths are permitted to vary by standardized agreement by up to ±10%. So the 24mm eyepiece could in reality have a true focal length between 21.6mm and 26.4mm.

Returning to the original question, the ball park value I gave was x90. According to the focal lengths quoted by the manufacturer it should have been x95.25, but it could be anything between x86 and x107.

Clearly, just relying on the quoted focal lengths is hopelessly inaccurate, even at low to medium powers. At high powers the errors are even greater. The probable error of that confidently quoted x233, could be ±x25 or more, depending on the type of telescope.

So how is the true eyepiece power to be determined? Is there a better way?
The answer is that there is a much better way, *but* it calls for accurate
measuring devices. Either an engineer's eyepiece comparator, or, for the utmost
precision, an eyepiece dynameter.

I shall describe these instruments and how they are used shortly, but before I do so I wish to go into a little optics and arithmetic, which underlies the reasons why these devices lead to a more realistic estimate of eyepiece magnification.

The magnification is given by:- M = F/fe & also M = D/ep

where F = objective focal length

fe = eyepiece focal length

& D = objective effective aperture

ep = exit pupil diameter

The exit pupil is the image of the objective formed by the eyepiece, projected from the eyepoint (where you position your eye to see the whole field of view) onto the sky. In daylight, when you point your telescope at clear sky you will see a small disc of light within the eyepiece when you stand back and look directly into it. If we measure both the clear aperture of the objective and the diameter of the exit pupil, then it is possible to obtain a more accurate estimate of the eyepiece power.

The cheapest way of accurately measuring the exit pupil is be purchasing an engineer's eyepiece comparator. This is an eyepiece with a reticule graduated in both angular and linear scales. The one I use was made by Edmund Scientific. With it I can measure exit pupil diameters to an accuracy of about ±5thou [±1/10mm]. The comparator has a clear plastic skirt that enables it to be placed over the eye lens and focused onto the exit pupil. The sharp circular image is then measured off the reticule.

For example. my Quantum 6 Maksutov has a clear aperture of 6 inches, and the 24mm eyepiece has an exit pupil of 1.5mm as measured with my comparator. Hence the magnification is 152.4/1.5 = x101.6. However the exit pupil could be 1.45mm to 1.55mm, so the magnification could be between x98.3 and x105.1. To be on the safe side I shall quote x100 when asked!

Knowing the objective focal length, and a reasonably accurate estimate of the magnification we can determine the eyepiece's true focal length. The actual focal length of my Q6 is almost exactly 92 inches, i.e. 2336.8mm. If I divide this by the measured magnification I obtain the real eyepiece focal length of 23mm.

An engineer's eyepiece comparator is sufficiently accurate to enable low and medium pwers to be measured within ±5%, but the measurement errors become significant at high powers because the diameter of the exit pupil shrinks to almost microscopic size.

For the ultimate accuracy you must measure the exit pupil with an eyepiece dynameter. This is a measuring microscope with a divided lens and a micrometer screw, capable of measurement accuracies in the order of ±40 micro-inches.

Unfortunately these instruments have not been made for well over a century and are now to be found only in museum or private scientific instrument collections. I found a fine example at a market in Preston in 1993 and payed the princely sum of £50 for it! It was manufactured c1858 by Troughton & Simms. The instrument is in fact a double-image micrometer designed for double star measurement. In doing some background research needed to restore this particular micrometer I came across an account of its use by none other than the celebrated double star observer, William Rutter Dawes, in the 1868 R.A.S. Monthly Notices. He mentions en passant, his adaptation of the micrometer to the purpose of calibrating eyepiece powers and states he intends to describe this in a follow-up article. Unfortunately Dawes died before he could do so. Despite a thorough search no such article was unearthed.

The Troughton & Simms double-image micrometer is fitted with five interchangeable first lenses, in a microscope type unit, each yielding a different effective power. I discovered that with the lowest power it was possible to focus the exit pupil, so I had H.N. Irving & Son make up a brass sleeve piece to locate the micrometer over any eyepiece eye lens, thereby enabling the aquisition of a steady and measureable image.

Being mid C19th, and English, the dynameter reads in inches, and the 24mm eyepiece had an exit pupil diameter of 0".0603±0".000 05, as determined from the mean of five independent measures. The true power was therefore x99.5, and the true eyepiece focal length 23.49mm (-2.1% of the quoted value).

I have since calibrated all my eyepiece powers, including those obtained with my selection of Barlow lenses and extension tubes, for each of my telescopes.

[f/10.6 Calver Eyepiece Magnications]

Next time you are asked what the eyepiece magnification is, will you be
able to answer in all candour, x86, or will you be as ill-informed and misguided
as all those who rely on the focal lengths quoted by the manufacturer? And next
time someone informs you the power is x86, ask them how they know! The only
way to* know* the precise eyepiece power is by rigorous calibration,
as I have done. If you have not done so, and cannot be bothered doing so, don't fall into
the same trap. Stick to ball park figures, and say, "Roughly x90."

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