Magnitude is a ratio wearing the costume of a quantity. Nearly every error I have made in twelve years of measuring faint things came from forgetting that sentence.
A magnitude is defined against a reference flux, never against zero:
is the flux received in some passband , and is the zero point flux for that same band. Change the band, change the number. The minus sign is inherited from the Greeks, who called the brightest stars first class and the faintest sixth, and we have been apologising for it since.
The form you actually use is the difference, because the zero point cancels and you no longer need to know it:
Pogson, in 1856, wanted the six inherited classes kept and the scale made arithmetic. He fixed a first magnitude star as one hundred times as bright as a sixth magnitude star. Five steps, factor of one hundred.
Then require that the scale be geometric, which is what "equal steps look equal" means for a logarithmic sense organ. One step is a fixed ratio , and five steps multiply:
That is the entire origin of the 2.512 you see quoted everywhere. It is also where the 2.5 in the definition comes from, since exactly, so of a flux ratio counts steps of and nothing else.
Two consequences worth writing on the inside of your eyelids.
Magnitudes add when fluxes multiply, so you can never average magnitudes. To combine two sources you convert both to flux, add the fluxes, convert back. People do this wrong with stacked frames and with sky readings constantly.
And 0.1 magnitudes is a flux ratio of . Nine and a half percent. So 0.1 mag/arcsec^2 of extra sky brightness is not a rounding error, it is a tenth of your exposure time.
mag/arcsec^2 is a magnitude per unit solid angle. It is not a flux and it is not a total. An object's integrated magnitude and its mean surface brightness are related through its area:
with in square arcseconds. This is why a magnitude 10 galaxy can be far harder than a magnitude 12 star. The star puts all of its photons in one place and the galaxy spreads them across ninety thousand square arcseconds.
The best modelling I know of for visual thresholds is Crumey (2014)1, who replaced the Hecht formula that light pollution studies had used since 1947. For a sky background of surface brightness in mag/arcsec^2 above 21:
and for the two lower ranges,
is a field factor covering the observer and the viewing situation. Crumey speculated it lies between about 1.4 and 2.4 for most people, with 2 typical. Larger gives a shallower limit.
First thing I did was check the three expressions against each other in their overlaps, because a set of piecewise fits that disagree at the seams is a set of fits I am not going to trust. At the first two give 5.956 and 5.965. At 20.0 the last two give 5.447 and 5.467. They agree to about two hundredths. Good.
Second thing I did was put them next to the naked eye limits in the Bortle table, and this is where it gets awkward. Bortle published his scale in the February 2001 issue of Sky and Telescope. The sky brightness column that circulates alongside it now comes from a nomogram, not from meters pointed at the same skies at the same time as the star counts. Here is the comparison at , using each formula in its own valid range and taking the midpoint of each Bortle class:
| Bortle class | SQM (mag/arcsec^2) | Bortle's naked eye limit | Crumey, F = 2 | Gap |
|---|---|---|---|---|
| 1 excellent | 21.76 to 22.0 | 7.6 to 8.0 | 6.20 | 1.4 to 1.8 |
| 2 truly dark | 21.6 to 21.75 | 7.1 to 7.5 | 6.12 | 1.0 to 1.4 |
| 3 rural | 21.3 to 21.6 | 6.6 to 7.0 | 6.02 | 0.6 to 1.0 |
| 4 brighter rural | 20.8 to 21.3 | 6.3 to 6.5 | 5.87 | 0.4 to 0.6 |
| 4.5 transition | 20.3 to 20.8 | 6.1 to 6.3 | 5.68 | 0.4 to 0.6 |
| 5 suburban | 19.25 to 20.3 | 5.6 to 6.0 | 5.39 | 0.2 to 0.6 |
Even taking the most generous , which adds 0.39 to every model figure, class 1 still comes out more than a magnitude short of the claimed 7.6 to 8.0. I am not going to tell you Bortle was wrong. I will tell you that the two columns are measuring different acts. One is a physical threshold for a target you are looking for. The other is a summary of what experienced observers, dark adapted for an hour, on fields they know by heart, using averted vision, report as their faintest star. Those are not the same number and the difference between them is somewhere between half a magnitude and nearly two.
The 5 log D everybody quotes falls straight out of Pogson. Flux collected is proportional to entrance pupil area, area is proportional to , so substitute into the difference form:
which is why the standard shape is
with the entrance pupil in centimetres. is empirical and it is where all the honesty lives. Values from 6.8 to 8.7 appear in the literature. Crumey, choosing parameters he considered typical of normal dark site observing (eye pupil 0.7 cm, ), derived .
| Aperture | 5 log D | m at N = 6.8 | m at N = 7.69 | m at N = 8.7 |
|---|---|---|---|---|
| 50 mm | 3.49 | 10.29 | 11.19 | 12.20 |
| 80 mm | 4.52 | 11.32 | 12.21 | 13.22 |
| 100 mm | 5.00 | 11.80 | 12.69 | 13.70 |
| 150 mm | 5.88 | 12.68 | 13.57 | 14.58 |
| 200 mm | 6.51 | 13.31 | 14.20 | 15.21 |
| 250 mm | 6.99 | 13.79 | 14.68 | 15.69 |
| 300 mm | 7.39 | 14.19 | 15.08 | 16.09 |
| 400 mm | 8.01 | 14.81 | 15.70 | 16.71 |
Read the row for your telescope and then read it across. The literature band is 1.9 magnitudes wide, a flux factor of 5.8. Anyone who tells you their 200 mm reaches magnitude 14.2 has picked a value of and not mentioned it.
The middle column is optimistic and here is the list of what it assumes, all of which I have failed at least once. A dark site, because silently contains the sky brightness and 7.69 was derived for dark site conditions, so under my class 4 sky it does not apply at all. A fully dark adapted observer, meaning an hour with no phone. A known field, because you find a marginal star far more easily when you know where it is. Transmittance, which the black sky form of the equation makes explicit as with around 0.75 typical, and a refractor with a star diagonal and six months of pollen on the objective is not at 0.75. Seeing good enough that the star stays a point when you push the magnification, which on my median 2.7 arcsecond nights it does not.
This is a point source table. It predicts nothing whatsoever about extended objects, and treating it as though it does is the single most common error I see. Raising magnification spreads the sky background over more retina, so its surface brightness in the eyepiece drops, while a star stays essentially a point delivering the same total flux. Contrast improves, which is why high power finds faint stars. An extended object dims at exactly the same rate as the sky does, so the contrast ratio between galaxy and background is invariant under magnification. What changes is the target's angular size, and the eye's contrast threshold falls as targets get larger, up to a limit. So there is an optimum power for a galaxy, not a monotonic gain, and the aperture table has nothing to say about it. Crumey's own paper makes the point about Hecht's older formula: it "is applicable only to point sources and is shown to be of limited accuracy".
On 3 October 2025 my meter read 20.90 at the zenith and I counted stars in a field I have my own photometry for, from my own frames. Faintest I could hold with averted vision: 6.1.
Put that into the applicable expression and solve for the field factor rather than for :
Which sits inside Crumey's stated range. So the model needed no defending and neither did I. My is 1.53 because my garden gives an unobstructed zenith and because I have memorised where the faint stars are, and memorising them is cheating in precisely the direction the model already allows for.
None of the above tells you what you will see tonight. It gives you a ceiling. The only number actually worth collecting is the gap between that ceiling and your real result, because the gap is made of specific, nameable, fixable things. Mine is about 0.4 magnitudes and I can list all four of them. I spent three winters finding that out on a single galaxy: @dust/two-hundred-and-eighteen-hours-on-one-galaxy.
Crumey, A. "Human contrast threshold and astronomical visibility", Monthly Notices of the Royal Astronomical Society 442(3), 2600 to 2619, 2014. doi:10.1093/mnras/stu992. The abstract is worth reading even if the fits are not what you came for. ↩