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# eyepiece magnifications and significant figures...

46 replies to this topic

### #1 BDS316

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Posted 22 October 2021 - 08:28 AM

In high school chemistry and physics we learn to use significant figures in our calculations and some teachers even deducted points if we failed to do so.

In calculating magnifications on this forum we see things like "46.4x" and the like.  If you have an 13mm Nagler, that's 2 significant figures,  and a scope of 1200mm is 3 significant figures, so when doing the math the answer should be reported in 2 significant figures, ie 92x, not 92.3x.

I'm interested to see what responses, if any, I get to this post.

### #2 John Gauvreau

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Posted 22 October 2021 - 08:42 AM

The purpose of using significant digits is to prevent the implication of a level of precision we don’t have, but it is never meant to eliminate a level that we do have.  It must be used judiciously.

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### #3 Ernest_SPB

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Posted 22 October 2021 - 08:43 AM

13 mm Nagler, actually has focal length 13.0 (with appr. 0.1 tolerance), so it is 3 digits.

I do not know what scope do you mean, but focal length of generic scope has 1-2% tolerance and this 1200 mm must be in range 1180-1220 mm, again 3 digits.

Magnification  1200/13.0 = 92.3x looks for me adequate enough.

But more precise tolerance would be 92.3*sqrt((0.1/13)2+(20/1200)2) = 1.7 and result 92.3+/-1.7

Edited by Ernest_SPB, 22 October 2021 - 08:48 AM.

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### #4 sixela

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Posted 22 October 2021 - 08:43 AM

I never post magnifications beyond the decimal point (or comma for Europeans). The magnification is not even constant in the entire field for most wide field eyepieces anyway. On the other hand, I'm not going to round it to even less significant digits either; there are implied error bars but the expected value need not be made more imprecise than necessary.

1200mm is 3 significant figures

Written like that 1200 has either two or four significant figures depending on context (given that focal lengths are often expressed to the decimal point, my hunch here would be four; I have a mirror with a focal lenght of 1787mm and one with a focal length of 1200mm -- yes, the ROC is not half a millimetre removed from 2400mm, measured). But three?

Edited by sixela, 22 October 2021 - 08:47 AM.

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### #5 Jon Isaacs

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Posted 22 October 2021 - 08:46 AM

In high school chemistry and physics we learn to use significant figures in our calculations and some teachers even deducted points if we failed to do so.

In calculating magnifications on this forum we see things like "46.4x" and the like.  If you have an 13mm Nagler, that's 2 significant figures,  and a scope of 1200mm is 3 significant figures, so when doing the math the answer should be reported in 2 significant figures, ie 92x, not 92.3x.

I'm interested to see what responses, if any, I get to this post.

Since some TeleVue eyepieces are spec'd to 0.1mm, it's not really possible to know whether 13mm = 13 mm or 13.0mm.  If one is using telescopes with fixed focal lengths like Newtonians and refractors, I think one is generally justified in using 3 significant figures though I would round off 92.3x to 92x.  But 12.7x is meaningfully different than 13x.

I truss the focal length of a telescope with a custom mirror to 0.1".  My 13.1 inch has 71.9" written on it by Robert Royce.  That works out to 1826mm, I think it was measured during the testing phase.

Jon

### #6 cst4

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Posted 22 October 2021 - 09:03 AM

I'm not sure why anybody would describe a magnification with decimals at all... from a practical standpoint, I'm sure nobody can tell a difference between 92x and 92.3x.  If adding more significant figures does not change the real world results then they are not significant.  Really, I doubt most people could see a difference between 90x and 95x so why not just round off... it's good enough to have a meaningful conversation or comparison.  As the quote goes, "any difference that makes no difference is not a difference at all."

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### #7 Starman1

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Posted 22 October 2021 - 11:16 AM

Indeed.  Rounding to the nearest 5x is fine.

After all, no one can see a difference between 90x and 92.3X.

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### #8 Jon Isaacs

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Posted 22 October 2021 - 11:43 AM

Indeed.  Rounding to the nearest 5x is fine.

After all, no one can see a difference between 90x and 92.3X.

I wouldn't round up from 15.6x to 20x but I will round up from 15.6x to 16x.

Along these lines..  Just how closely can one estimate the actual magnification?  If I gave someone 5 eyepieces ranging from 16mm to 7 mm with various AFoVs, how long would it take them to sort them out??

Jon

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### #9 Starman1

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Posted 22 October 2021 - 01:33 PM

That wouldn't be too hard.

But, give them a 30mm eyepiece and a 31mm eyepiece and if they were chosen to yield the same true field,

I doubt anyone could tell without designing a test to check for differences in magnification.

As for the 15.6x, I'd round down to 15, the nearest 5x multiple.

### #10 Jon Isaacs

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Posted 22 October 2021 - 09:24 PM

That wouldn't be too hard.

But, give them a 30mm eyepiece and a 31mm eyepiece and if they were chosen to yield the same true field,

I doubt anyone could tell without designing a test to check for differences in magnification.

As for the 15.6x, I'd round down to 15, the nearest 5x multiple.

Rounding off to the nearest 5x might make sense at 150x but at 15x, it doesn't.. In the NP-101, both the 35 mm Panoptic (15.4x) and the 31mm Nagler (17.4x) would round off to 15x while the 30mm UFF (18x) would round up to 20x..

The 31mm Nagler and 35 mm Pan are clearly different in terms of image brightness and exit pupil while the 30 mm UFF and 31mm Nagler are very similar.. 20x for one and 15x for the other is clearly misleading.

As far as the telling eyes apart, a Plossl and an Ethos, one the Plossl being 90% of the Ethos focal length.

I'm not saying you would have difficulty but I think many would and it wouldn't be obvious at a first glance.

And too, if we don't really care about magnification precision, I'd say the same thing about TFoV. Just round off to the nearest half degree.. just use AFoV/mag and round it off..

That doesn't work for the same reason rounding 17.4x down to 15x doesn't work,

Jon

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### #11 russell23

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Posted 23 October 2021 - 07:30 AM

In high school chemistry and physics we learn to use significant figures in our calculations and some teachers even deducted points if we failed to do so.

In calculating magnifications on this forum we see things like "46.4x" and the like.  If you have an 13mm Nagler, that's 2 significant figures,  and a scope of 1200mm is 3 significant figures, so when doing the math the answer should be reported in 2 significant figures, ie 92x, not 92.3x.

I'm interested to see what responses, if any, I get to this post.

Yeah - my chem students always dislike significant figures.  They’d rather carry every single digit the calculator provides or just round to the tenth.

It is an interesting question.  For example, is a 102mm f/7 rounded to f/7 instead of 7.00 or is the focal length of 714mm calculated by multiplying 102mm x 7?  In other words is the scope actually 714mm and they simplify the f/7 because nobody cares about the .00 or is the scope about f/7 and they calculate it to 714mm even though it is not necessarily that exact?

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### #12 Jon Isaacs

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Posted 23 October 2021 - 08:02 AM

Yeah - my chem students always dislike significant figures.  They’d rather carry every single digit the calculator provides or just round to the tenth.

It is an interesting question.  For example, is a 102mm f/7 rounded to f/7 instead of 7.00 or is the focal length of 714mm calculated by multiplying 102mm x 7?  In other words is the scope actually 714mm and they simplify the f/7 because nobody cares about the .00 or is the scope about f/7 and they calculate it to 714mm even though it is not necessarily that exact?

If I am not mistaken, my AT-102 ED had 714mm written on the objective cell.  My AT-80LE has FL 480 mm written on the cell, my WO Megrez II FD has 555 mm written on the cell.

But if one is doing science, the question of the accuracy of that measurement comes into play, how trustworthy is that number.  If one were doing research, some sort of measurement to verify that number would be needed.

When we were drift timing the field of view of the 30mm APM UFF, I eventually decided to use my 13.1 inch F5.5 without the Paracorr because the focal length was written by Robert Royce on the back of the mirror.  As I said previously, I believe they measure that during the bench testing phase so I am reasonably confident that the focal length is 71.9 inches = 1826mm.  In that situation, the drift times were approximately 270 seconds and measured to 0.1 seconds, several measurements were made. 3 digits of significant figures in the field stop was justified.

In the case of one of my refractors, I am not so confident the focal lengths are accurate.  I would guess that all refractors of a type have the same focal length within less than a millimeter because refractors require very accurate radii on the lenses but I am not so sure that the 480mm means 480mm +/- 1mm.

In general, it really doesn't matter much.  It's not really the significant digits in the magnifications, it's what differences in magnification are significant to the observer.  If I say I split a double at 822x or 820x, it really doesn't matter, the fact that I was using a Barlow means 8.2 x 102 x is the most I could possibly justify.

Jon

### #13 Jon Isaacs

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Posted 23 October 2021 - 09:44 AM

Here's something to consider:

With a Paracorr, the focal length of a 10 inch F/5 is 1437.5 mm.  I found this up to 1440 mm.

1440 is a very nice number because it's 10 x 12 x12 = 5 x 32 x 25

There are many magnifications that come out even numbers, many that end in 0. Doing the math in my head can be easy.  A 5 mm eyepiece provides 288x, 2 times 144x.

I use 288x instead of 290x because it's very easy to remember. For me, when it comes to magnifications, easy to remember is way more important than the correct usage of the least significant digits. "Around 300x" is really good enough but a little more precision is good too.

jon

### #14 Starman1

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Posted 23 October 2021 - 10:13 AM

When selecting an eyepiece for use at the scope, though, the "around 300" is the operant figure.

Whether it's 288x or 287.5x or even 295x isn't really that important.  I don't have enough eyepieces to choose one at every 10X magnification (though some CNers do ).

I have eyepieces that yield 388x and 493x, but I think of them as "around 400" and "around 500" because those are the significant figures to remember.

Plus, I regularly observe with others, and they don't want to know the magnification is 388.4x.  If they ask, I say 400x, which also gives them an idea what to use

in their own scopes if they want a similarly-sized view.

I don't have a lot of scopes I have to remember magnifications for, so a rounding off is sufficient for me to remember the magnifications off the top of my head.

What is interesting, though, is that I tend to select eyepieces below 250x by focal length ("I think I should use the 12.5mm on this object") but above 250x,

I tend to choose based on magnification ("I think this object needs about 300x").  I know all the magnifications, but I don't select eyepieces that way.

Perhaps it's simply familiarity with the eyepieces and what views they offer.

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### #15 Jon Isaacs

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Posted 23 October 2021 - 10:36 AM

When selecting an eyepiece for use at the scope, though, the "around 300" is the operant figure.

Whether it's 288x or 287.5x or even 295x isn't really that important.  I don't have enough eyepieces to choose one at every 10X magnification (though some CNers dogrin.gif ).

The fact that I can easily remember 288x is what is important.  The fact that a 9mm provides 230x exactly in my 16 inch is what is important simply because it is easy to remember.  When one is dealing with telescopes with a wide range of focal lengths, keeping the magnifications straight is something of a challenge.  There's a lot of mental gears to shift. This makes it easier.

Jon

### #16 SteveG

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Posted 23 October 2021 - 03:15 PM

I used to spend way too much time crunching numbers and trying to have every possible magnification. Now that I'm older, I just grab an eyepiece and scope combo that I know I will like. Most of the time I don't even know the magnification factor, and guess what? Jupiter and Saturn still look great!

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### #17 areyoukiddingme

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Posted 23 October 2021 - 03:56 PM

It would be interesting to actually find out how sensitive we are to magnification changes.

I've often found myself surprised when using a small refractor with zoom on the local mountains. Both the apparent field and magnification can be good bit off where I think I am before I notice.

### #18 jokrausdu

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Posted 23 October 2021 - 09:19 PM

When I worked at a telescope store, we would measure the FL of telescopes using the drift method in a reticle eyepiece. It was similar to this 12.5mm one. https://optcorp.com/...ticle-eyepiece

We would report the telescope FL to an exact mm. For example, I measured my scope to be 1449mm. But, if the reticle eyepiece is actually 12.467mm (or whatever) instead of exactly 12.500mm, then my scope could be anywhere from 1440 to 1460mm (give or take). So, my estimate of magnification should probably be rounded appropriately, since I don't know the exact FL of my scope nor the exact FL of the eyepiece I am using.

But, in a printout of eyepiece mags that I bring with me to public events, I will still keep my scope at 1449mm, since people always ask what magnification they are using.

I also like that 1449 is not a prime number, it is a factor of 3x3, 7, and 23. That explains why my 7 and 9 Naglers give me exact magnification numbers. But, now that I know that 1449 probably isn't the exact FL, now those two eyepieces no longer give me an exact mag number. Bummer.

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### #19 MitchAlsup

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Posted 25 October 2021 - 05:14 PM

It is threads like this that make me happy I quit using magnification decades ago, and simply use Exit Pupil.

### #20 Jon Isaacs

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Posted 25 October 2021 - 09:02 PM

It is threads like this that make me happy I quit using magnification decades ago, and simply use Exit Pupil.

I think in terms of magnification, exit pupil and true field of view.. it all depends on the situation.

Jon

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### #21 Jeff Morgan

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Posted 25 October 2021 - 10:46 PM

And how many of you have experienced an observation that could not be made without that last decimal place of magnification?

Integers are just fine.

### #22 luxo II

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Posted 25 October 2021 - 11:50 PM

13 mm Nagler, actually has focal length 13.0 (with appr. 0.1 tolerance),

On what basis ?

IMHO its more likely the 13mm figure is purely nominal and actual focal lengths of random samples could be anywhere between +/- 10% either way.

There are older threads indicating variances around 1mm were not uncommon and even 2mm.

Edited by luxo II, 25 October 2021 - 11:54 PM.

### #23 Jon Isaacs

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Posted 26 October 2021 - 08:06 AM

On what basis ?

IMHO its more likely the 13mm figure is purely nominal and actual focal lengths of random samples could be anywhere between +/- 10% either way.

There are older threads indicating variances around 1mm were not uncommon and even 2mm.

Ernest is an optical designer who happens to like to measure optical parameter of eyepieces using an optical bench.

What evidence do you have that the focal lengths of the eyepieces vary by +/- 10%?

To me, that seems very unlikely.  Small lenses are easy to make in a repeatable fashion and to achieve any sort of consistent field correction, the components would have to be essentially identical.

If you look up the April 1996 issue of Sky and Telescope, you will find an article on page 38 where they measured and tested 30 Plossls.  The focal lengths of the TeleVue Plossls were exact, that is with in the error of measurement.  Nearly all the eyepieces were within the +/- 0.1 mm error bars of the measurements.

Of the 30 eyepieces, only 5 were greater than 0.2mm off and three of those 5 were by the same manufacturer.  And that only means they deviated from spec, its likely, that all the eyepieces of a given model were very close.

Jon

### #24 tog

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Posted 26 October 2021 - 08:44 AM

In high school chemistry and physics we learn to use significant figures in our calculations and some teachers even deducted points if we failed to do so.

In calculating magnifications on this forum we see things like "46.4x" and the like.  If you have an 13mm Nagler, that's 2 significant figures,  and a scope of 1200mm is 3 significant figures, so when doing the math the answer should be reported in 2 significant figures, ie 92x, not 92.3x.

I'm interested to see what responses, if any, I get to this post.

1200mm. Zeros at the end of a number having no decimal point are not significant so 1200mm has only two sig figs. To show three sig figs you have to use sci notation: 1.20x10^3mm.

Edited by tog, 26 October 2021 - 08:48 AM.

### #25 pweiler

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Posted 26 October 2021 - 01:37 PM

It would be usual for the human eye to detect a 1x power different between eyepieces even at the low power end of a telescope’s focus.
If trying to decide a purchase between two brands of eyepieces’, a difference of 1x power is probably near the bottom of deciding factors.

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