SQM to Bortle and NELM.

What the three numbers measure

A Sky Quality Meter reads how bright the sky is per patch of sky, in magnitudes per square arcsecond. Bigger means darker, and the scale is logarithmic, so a one-point gain is about 2.5 times less light. Naked-eye limiting magnitude (NELM) is something a person does: the faintest star you can see. The US National Park Service puts it simply, “NELM is the magnitude of the faintest star we can see in the sky with naked eyes,” and gives anchors: 6.6 is near pristine under average conditions, 7.0 needs good seeing and dark adaptation, and a number lower than 6.3 usually means degraded sky. The Bortle class is a 1 to 9 label devised by amateur astronomer John Bortle and published in Sky & Telescope in February 2001.

The two equations

The Unihedron page, written by K. Fisher (revised August 2006) and crediting Schaefer’s 1990 paper on telescopic limiting magnitude, gives NELM = 7.93 − 5 log₁₀(10^{(4.316 − SQM/5)} + 1) and the inverse, SQM = 21.58 − 5 log₁₀(10^{(1.586 − NELM/5)} − 1). The page is titled as a conversion between a V-band NELM system and a B-band MPSAS system, which is a reminder that these are fitted relationships, not physical identities. The tool also shows a second estimate from Crumey’s fit on Wikipedia’s limiting magnitude page, with a field factor of 2, so you can see how far two published models differ for the same sky. At 21.00 they differ by about a quarter of a magnitude.

Where the Bortle table fits

The classes use the approximate SQM and NELM columns on Wikipedia’s Bortle scale page: class 1 is 21.76 to 22.0 and NELM 7.6 to 8.0, class 4 is 20.8 to 21.3 and 6.3 to 6.5, class 4.5 is 20.3 to 20.8 and 6.1 to 6.3, class 8 is below 18.00, and class 9 has no SQM value at all on that page. The page says the classes “correspond closely” with SQM and NELM, not exactly, and notes a 2014 study (Crumey) suggesting Bortle may have overstated what typical observers see, with a NELM of 6.0 to 6.5 more often cited than the 7.1 and up listed for class 2. When your SQM and NELM land in different classes, trust neither more than the other.

A warning about very dark readings

The National Park Service notes that the SQM “does not reliably measure the sky brightness when the sky is darker than 21.5 mag/arcsec²,” and treats 22.0 as the darkest part of pristine skies. If your reading is above 21.5 the result below is flagged, because the meter itself is the weak link.

Worked example

Take an SQM reading of 21.00. The exponent is 4.316 − 21.00/5 = 0.116, and 10^0.116 is 1.306. Add 1 to get 2.306; log₁₀ of that is 0.363; times 5 is 1.815; and 7.93 − 1.815 = 6.12 for the NELM. Feeding the unrounded 6.1155 back through the inverse returns 21.00. On the Bortle table, 21.00 sits between 20.8 and 21.3, so class 4 by SQM, while 6.12 falls in the 6.1 to 6.3 band, so class 4.5 by NELM. Crumey’s fit for 21.00 gives 5.85. These are worked from the published equations, not measurements of a real site.