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# Limiting Magnitude Calculation

10 replies to this topic

### #1 HotRod217

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Posted 11 April 2021 - 06:31 AM

Is there a formula that allows you to calculate the limiting magnitude of your telescope with different eyepieces and also under different bortle scale skies?

### #2 beggarly

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Posted 11 April 2021 - 07:17 AM

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### #3 TOMDEY

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Posted 11 April 2021 - 07:35 AM

The one thing these formulae seem to ignore is that we are using only one eye at the monoscopic telescope. Just going true binoscopic will recover another 0.7 magnitude penetration.    Tom

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### #4 Jon Isaacs

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Posted 11 April 2021 - 07:45 AM

The one thing these formulae seem to ignore is that we are using only one eye at the monoscopic telescope. Just going true binoscopic will recover another 0.7 magnitude penetration.    Tom

That's mighty optimistic,  that assumes using two eyes is nearly as effective as doubling the light gathering and using it all in one eye.. The brain is not that good..

Close one eye while using binoculars.. how much less do you see???

These equations are just rough guesses, variation from one person to the next are quite large.

Jon

### #5 John Fitzgerald

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Posted 11 April 2021 - 08:24 AM

I have always used 8.8+5log D (d in inches), which gives 12.7 for a 6 inch objective.  Somewhat conservative, but works ok for me without the use of averted vision.

### #6 Astrojensen

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Posted 11 April 2021 - 08:44 AM

An easy way to calculate how deep you should at least be able to go, is to simply calculate how much more light your telescope collects, convert that to magnitudes, and add that to the faintest you can see with the naked eye.

It doesn't take the background-darkening effect of increased magnification into account, so you can usually go a bit deeper. How much deeper depends on the magnification.

Clear skies!

Thomas, Denmark

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### #7 TOMDEY

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Posted 11 April 2021 - 12:53 PM

That's mighty optimistic,  that assumes using two eyes is nearly as effective as doubling the light gathering and using it all in one eye.. The brain is not that good..

Close one eye while using binoculars.. how much less do you see???

These equations are just rough guesses, variation from one person to the next are quite large.

Jon

You need to perform that experiment the other way around. Get a great binoscope and view a a random field with one eye, sketching the stars from bright to dim to subliminal. Only then view with both. The gain will be doubled!    Tom

### #8 Starman1

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Posted 12 April 2021 - 01:17 PM

An easy way to calculate how deep you should at least be able to go, is to simply calculate how much more light your telescope collects, convert that to magnitudes, and add that to the faintest you can see with the naked eye.

It doesn't take the background-darkening effect of increased magnification into account, so you can usually go a bit deeper. How much deeper depends on the magnification.

Clear skies!

Thomas, Denmark

I'm glad you said "at least".

Let's say the pupil of the eye is 6mm wide when dark adapted (I used that for easy calculation for me).

My 12.5" mirror gathers 2800x as much light as my naked eye (ignoring the secondary shadow light loss).

Since 2.512x =2800, where x= magnitude gain, my scope should go about 8.6 magnitudes deeper than my naked eye (about NELM 6.9 at my observing site) = magnitude 15.5

That is quite conservative because I have seen stars almost 2 magnitudes fainter than that, no doubt helped by magnification, spectral type, experience, etc.

It's a good way to figure the "at least" limit.

[one flaw: as we age, the maximum pupil diameter shrinks, so that would predict the telescope would gain MORE over the naked eye.  I don't think most people find that to be true,

that limiting magnitude gets fainter with age.]

But, I like the formula because it shows how much influence various conditions have in determining the limit of the scope.

And it gives you a theoretical limit to strive toward.

Edited by Starman1, 12 April 2021 - 01:20 PM.

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### #9 mikemarotta

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Posted 12 April 2021 - 07:19 PM

Is there a formula that allows you to calculate the limiting magnitude of your telescope with different eyepieces and also under different bortle scale skies?

You got some good replies. I made a chart for my observing log. Just to note on that last point about the Bortle scale of your sky. No, it is not a formula, more of a rule of thumb. It will vary from night-to-night, also, as the sky changes. I live in a city and some nights are Bortle 6 and others are Borte 8. But even on a night (early morning) when I could not see the Milky Way (Bortle 7-8), I still viewed Ptolemy's Nebula (M7) and enjoyed splitting Zubenelgenubi (Alpha Libra), among other targets.

### #10 PKDfan

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Posted 13 April 2021 - 03:16 AM

Totally off topic, just wanted to say I love that name Zubenelgenubi!
Just rolls off the tongue!

Clear skies & Good seeing

Edited by PKDfan, 13 April 2021 - 03:16 AM.

### #11 Redbetter

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Posted 13 April 2021 - 04:49 AM

While the OP asks a simple question, the answers are far more complex because they cover a wide range of sky brightness, magnification, aperture, seeing, scope types, and individuals.

I am not keen on trying to estimate telescopic limiting magnitude (TLM) using naked eye limiting magnitude (NELM), pupil diameter and the like.  There are too many assumptions and often they aren't good ones for the individual's eye(s).

• Best TLM is determined at small exit pupil (best is around 0.5 to 1.0mm depending on the seeing and scope), while NELM is at the opposite end, the eye's widest pupil.  This results in a host of differences that vary across individuals.
• Small exit pupils increase the contrast for stars, even in pristine sky.  Naked eye the contrast is poor and the eye is operating in a brighter/less adapted regime even in the darkest sky.
• NELM estimates tend to be very approximate unless you spend some time doing this regularly and have familiar sequences of well placed stars to work with.  Typically people report in half magnitude steps.  Often people underestimate bright sky NELM.
• The focuser of a telescope allows an observer to find the best distance correction for the eye.  Naked...well...not so much, so naked eye acuity can suffer.
• Speaking of acuity, astigmatism has the greatest impact at large exit pupil, even if one has only very mild levels of astigmatism.  If one does not have a lot of astigmatism, it becomes a non-factor at small exit pupil.    This is another negative for NELM.
• NELM is binocular vision, the scope is mono.  Translating one to the other is a matter of some debate (as seen in the discussion above) and differs among individuals.
• Some folks have one good eye and one not so good eye, or some other issues that make their binocular vision poor.
• Assumptions about pupil diameter with age, etc. are of questionable validity.  It really doesn't matter for TLM, only for NELM, so it is an unnecessary source of error.

The most useful thing I did for my own observing, was to use a small ED refractor in dark sky on a sequence of known magnitude stars in a cluster at high magnifications (with the cluster well placed in the sky.)  This allowed me to find the dimmest possible star for my eye and aperture.  After a few tries I found some limits that I couldn't seem to get past.  This is powerful information, as it is applicable to the individual's eye under dark sky conditions.  With it I can estimate to high precision the magnitude limit of other refractors for my eye, and with some corrections, other types of scopes.

I didn't know if my original result would scale, so from there I tested other refractor apertures the same way at the same site in similar conditions, and empirically determined that I was seeing nearly perfectly scaled results.  I had a sequence of stars with enough steps that I had some precision/redundancy and it almost looked like I had "dry-labbed" the other tests.

So I can easily scale results to find what are limits for my eye under very dark sky, but this is for detecting stars in known positions.  Spotting stars that aren't already known, generally results in some discounting of a few tenths of a magnitude even if you spend the same amount of time studying a position.

Going deeper for known stars isn't necessarily "confirmation bias" if an observer does some cross checks, instead it is more a measure of recognizing and looking for things that are already there.  It is easy to overlook something near threshold in the field if you aren't even aware to look for it, or where to look.  But if you know roughly where to look, or that there might be something there at all, then you are far more likely to see it.  If you compare views with a larger scope, you will be surprised how often something you missed at first in the smaller scope is there or real when you either see it first in the larger scope or confirm it in the larger scope.

Difficulty comes in discounting for bright skies, or for low magnification (large or moderate exit pupil.)  There are some complex relations for this, but they tend to be rather approximate.

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