TheSkySearchers

Resolving power (Rayleigh limit) = 138/D arc seconds
Resolving power (Dawes limit) = 116/D arc seconds
Theoretical visual limiting magnitude (optical system 100% efficient) = 2 + 5 log10D
In practice, it is likely that the constant 2 in the above equation could be replaced by a value between 3 and 4, particularly when higher magnifications are used.
(where D = diameter of aperture in millimetres)
https://britastro.org/computing/pdf/Con ... rmulae.pdf

Resolution = 5.45 / D where D=aperture in inches.
Dawes Limit = 4.56 /D where D=aperture in inches.
Starware: The Amateur Astronomer’s Guide to Choosing, Buying, and Using Telescopes, 4th edition, by Philip S. Harrington John Wiley & Sons, Inc. 2007. Pg 6-8.

Light grasp “the ratio of light flux intercepted by a telescope’s objective lens or mirror to that intercepted by the human eye having a 7-mm-diameter entrance pupil.
Limiting visual magnitude m = 2.7 + 5 log D
Smallest Resolvable Angle = wavelength / diameter; 555 nanometers / D (nm) = 114/D in seconds of arc.
Useful magnification range: approximately between 0.2 D and 2.0 D. (“… atmospheric turbulence usually limits the maximum magnification to 400x or less.”)
“Optics and Observing: Telescope Parameters” by Roy Bishop. RASC Observer’s Handbook 2021, page 49. (Royal Astronomical Society of Canada)

Angular Resolution = 0.25 * (wavelength micrometers / mirror diameter meters) with a micron = 10^-6 meters.
Astronomy Today, Chaisson and McMillan, 6th edition, Pearson Addison Wesley, 2008 page119.

So, just for example:
The formulae all work out the same. The RASC parameters are a bit more generous. To figure magnitude limits from the RASC numbers, I took the standard deviation [0.682 * (20+.02) * D] as an approximation.

NATIONAL GEOGRAPHIC 70 mm refractor
Rayleigh Limit: 1.97 arc seconds
Dawes Limit: 1.65 arc seconds
Magnitude Limits: 11.2 to 11.9
Magnification 105X

EXPLORE SCIENTIFIC 102 mm refractor
Rayleigh Limit: 1.35 arc seconds
Dawest Limit: 1.13 arc seconds
Magnitude Limits: 12 to 12.74
Magnification 153 X

In practice, the results are somewhat different. I use epsilon Lyrae as my test case. On three different nights over a couple of weeks, the 70 mm would not split the doubles, but on the second night with better viewing, the 102 mm would.
Also, with the ES-102, I was working at 165X using a 8-mm eyepiece and a 2X Barlow. The National Geographic has a longer focal length, but the additional magnification at 175X did not help. The aperature was just not wide enough to gather the light needed.

Note that in all of these references, they always caution that better seeing conditions is often the more important stricture.

From my experience with the Explore Scientific AR102, the limiting magnitude is well below what you have listed. I have seen the quasar 3C 273 in Virgo, listed at mag 12.9, and it wasn't overly difficult. 13.5 would be a more reasonable limit under excellent conditions.

Hi Mike and Gordon. A very informative thread here. And, I remembered an interesting article from the June 2019 Sky & Telescope magazine. I have scanned the 5 page article and hope that these add to this thread discussion.

Thanks Marshall,

It is interesting that the 84yo author was able to consistently reduce Dawes' Limit by around 23%, to come up with 3.5/d, vs the conventional figure of 4.56/d

Mind you, he was using 3.5" and 7" Questars which are arguably as good as you can get! It does agree pretty much with my personal observations using a top quality 4" scope. Noting of course that the original formula was empirically derived, we should really see it as an approximation for a good telescope on a night of good seeing.

happy viewing folks!

- Dean

DeanD wrote: ↑Sat Mar 27, 2021 9:57 am ... Mind you, he was using 3.5" and 7" Questars which are arguably as good as you can get! It does agree pretty much with my personal observations using a top quality 4" scope. Noting of course that the original formula was empirically derived....

I went to the Questar website and I read about them on Wikipedia. They seem to be the Maybach-Brabus of telescopes.

mikemarotta wrote: ↑Sat Mar 27, 2021 6:40 pm

DeanD wrote: ↑Sat Mar 27, 2021 9:57 am ... Mind you, he was using 3.5" and 7" Questars which are arguably as good as you can get! It does agree pretty much with my personal observations using a top quality 4" scope. Noting of course that the original formula was empirically derived....

I went to the Questar website and I read about them on Wikipedia. They seem to be the Maybach-Brabus of telescopes.

Yep! I looked up Maybach-Brabus and I agree. They must be the Questar of cars!

Bear in mind that these limits are theoretical. At any given site, the seeing may never be good enough to reach the actual limit. Bad seeing spreads out images, making the faintest stars bloat and seem to disappear.

Limiting magnitude of a telescope is a far more complex thing than a simple formula.
There are many factors influencing the limit:
Diameter of objective
Magnification
Darkness of sky
Seeing
Transparency (extinction)
altitude above the horizon for the star
The spectrum of the star
the visual acuity of the observer
the experience of the observer
the cleanliness of the optics
the transmission of the scope (i.e. the type of optics and coatings)
Any formula that does not take those factors into account would at best be within 2 magnitudes of the actual limit.
This calculator DOES take all those things into account and is far more accurate than a simple formula:
https://www.cruxis.com/scope/limitingmagnitude.htm

To show you how great a difference there can be from the data entered, my 12.5" aperture varies from 13.1-17.9 depending on the factors I enter.
That would simply make any fixed formula an almost laughable idea.