# Naked-eye limiting magnitude: mismatch between Wikipedia, online converter and Shaefer's formula

I'm trying to make sense of the difference between the reference Sky Quality Meter (SQM) readings given in the Wikipedia article on Bortle scale, online converters like [1] that are based on Shaefer's paper [2], and an equation from [2] itself.

Wikipedia article gives naked-eye limiting magnitude (NELM), as well as corresponding SQM readings for each Bortle class. The original paper [3] by Bortle only specifies NELM, so the SQM column in Wikipedia must have been derived from the NELM values.

But in the very first value, SQM reading for Bortle class 1, the Wikipedia article fails to match what the online converter yields (other such converters use the same formula). Namely, the "excellent dark-sky site" NELM is 7.6–8.0, for which Wikipedia gives lower bound of SQM reading $$21.99\,\mathrm{mag/arcsec}^2$$, while the converter [1] yields $$25.50\,\mathrm{mag/arcsec}^2$$. The upper bound, NELM=8.0, is converted to $$22\,\mathrm{mag/arcsec}^2$$ in Wikipedia, while the online converter simply fails, yielding NaN. This failure is because the formula used by the converter has 7.93 as the maximum possible NELM.

If we use equation $$(18)$$ from [2] instead (with $$k_v=0.3$$ and $$F_s=1$$, as suggested in the text under the equation), we'll get the range of SQM readings $$24.35\,\mathrm{mag/arcsec}^2$$ to $$26.32\,\mathrm{mag/arcsec}^2$$, which doesn't match Wikipedia either.

Also, these values $$\ge22\,\mathrm{mag/arcsec}^2$$ seem to also conflict with [4]:

The natural level is around $$21.6\,\mathrm{mag/sec}^2$$ at a solar activity minimum.

Besides, an answer on Astronomy.SE implies that values of NELM larger than 6 don't make any sense at all.

So, how to make sense of the conflicting results? Which of these are right, which are wrong? Or do they correctly describe different things?

### References

1. Schaefer, B. E.. "Telescopic limiting magnitudes". Astronomical Society of the Pacific, Publications (ISSN 0004-6280), vol. 102, Feb. 1990, p. 212-229.

2. Bortle, John E. (February 2001). "Gauging Light Pollution: The Bortle Dark-Sky Scale". Sky & Telescope. Sky Publishing Corporation.

3. Dark Skies Awareness: "Sky Brightness Nomogram"

• Although Wikipedia is usually pretty good with hard science & mathematics, when in doubt, be very cautious of trusting data and formulae on Wikipedia. FWIW, a couple of years ago, a Stack Exchange member discovered some errors in astronomical info tables on Wikipedia, affecting several articles, IIRC. It took a little while to get that fixed. Sorry, I can't remember any details. Apr 8, 2021 at 13:53
• I don’t own an SQM myself, and I actually never even saw one in real life, but from what I remember from discussions with/between actual users, an SQM above around 22.5 is impossible as there’s always a natural glow to the sky. Maybe someone else can confirm? Apr 8, 2021 at 17:25
• The user manual for Stellarium 0.21.0 gives for a Bortle 1 sky an SQM of 21.7–22.0. Apr 8, 2021 at 17:57
• @PierrePaquette Stellarium is another story. It's the point where I started this research :) Apr 8, 2021 at 19:18
• “Wikipedia is usually pretty good with hard science & mathematics, when in doubt, be very cautious of trusting” No, just be very cautious in general. Wikipedia runs by consensus, yet places ZERO qualification on net (overall) editor body (low barrier to entry… close to zip). Then, a preponderance of high school-/BS-level “facts” can drown out one or even a few PhDs, let alone a PhD in that field who even bothers to read WP, let alone spend the time editing. The result: wikipedia is vulnerable to False Consensus Bias- ‘we’re right, because we feel right (and we are we , you’re just you)’. Aug 3 at 12:12

Being by no means an expert of light pollution, I still find my intermediate results worth sharing.

On the one hand, there is the Bortle Scale, and I find the flow chart from darksky.org the most accesible way (and probably it is also a reliable source):

This flow chart refers to the original publication by Bortle from 2001 which was mentioned in the question. For all objects mentioned in a flow chart, one could determine their magnitude and obtain a magnitude-Bortle-scale conversion table (I might do that in a later edit).

On the other hand, there is a conversion site directly by a manufacturer of SQMs which converts $${\rm visual\,mags}/{\rm arcsecond}^2$$ to $${\rm cd/m^2}$$ according to the following formula:

$$[{\rm value\,in\,cd/m^2}]= 10.8\cdot 10^4 \cdot 10^{-0.4 \cdot [{\rm value\,in\, mag/arcsec^2}])}$$

The impression I got from my research is that there is no single, golden source publication linking NELM with physical properties such as $${\rm cd/m^2}$$.

Last but not least, I stumbled up Roy Bishop's visibility factor (see German wikipedia) as more advanced (aka quantitiative) approach.

• The chart doesn't give any numeric values: it's purely qualitative. The original paper at least cites some NELM values, converting which to photometric units is my main problem. The formula you cite comes from the same site I cite in my ref. 1 (unihedron.com), which does nothing to explain the discrepancy. Apr 28, 2021 at 8:20
• @Ruslan: I tried to explain that it is not purely qualitative since it basically it is a table of visible objects (M31, M33, M4, M5, M15, M22) which would actually correspond to a magnitude. Apr 28, 2021 at 8:29
• OK, but we already have limiting magnitude values from the Bortle paper, this isn't something I'm lacking. The problem is at the next stage: conversion of NELM to cd/m². It's this stage where I get conflicting results. Apr 28, 2021 at 8:32
• About the references you are right, the ones you collected are pretty much the essential ones, I think. The impression I got from my research on the topic and which I wanted to share is that SQM readings might depend on who build it (and uses what conversion formula). Apr 28, 2021 at 8:32
• The table here is even more complete. Mar 4 at 22:07