The backwards magnitude scale

Why do astronomers still call the brightest stars first magnitude and faint ones sixth?

Every other measurement in science runs the sensible way. More mass, bigger number. More energy, bigger number. In astronomy, more light gives you a smaller number — and it does not stop at zero. Sirius is magnitude -1.46. Venus at its best is near -4.9. The full Moon is about -12.7 and the Sun about -26.7. Meanwhile the faintest thing your eye can pick out on a dark night is around magnitude 6, and the deepest telescope exposures reach past 30.

It looks like a mistake nobody got round to fixing. Before assuming that, though, it is worth asking a different question: what was the first version of this scale actually measuring? Because a number can only be backwards if it was trying to count the thing you think it was counting.

Go back to the earliest surviving form. Ptolemy's Almagest, around the middle of the second century, lists roughly a thousand stars sorted into six classes — the brightest called stars of the first magnitude, the faintest visible ones of the sixth. Tradition credits the scheme to Hipparchus three centuries earlier, though our evidence for that is indirect and historians are cautious about it.

Now notice what those numbers are. They are not measurements of anything. There was no instrument. They are ranks — first class, second class, sixth class — in a language where "of the first magnitude" meant foremost, chief, most important. Ask yourself: in a ranking, does first mean most or least? First is best. The scale is not backwards; it never counted light in the first place.

So the direction was set before anyone could measure brightness at all. Then what happened when they could?

By the nineteenth century photometry was good enough to put real ratios on the old classes, and the discovery was suggestive: each step of the ancient scale corresponded roughly to a constant ratio of brightness, not a constant difference. That is what you would expect if the eye's response is roughly logarithmic, and it meant the ancient classes were accidentally sitting on a sensible mathematical footing.

Here comes the decision that locked everything in. In 1856 Norman Pogson proposed defining the scale so that a difference of exactly five magnitudes corresponds to a factor of exactly 100 in brightness. One magnitude step is therefore the fifth root of 100, about 2.512.

Ask why 100 and five. Not because either is natural. Because it was close to what the existing measurements already showed, so the enormous body of star catalogues built on the old ranks stayed approximately valid. Pogson was not designing a scale; he was ratifying one, choosing the parameter that disturbed the fewest existing numbers.

And once you write a ranking as a continuous logarithmic quantity, the sign is set. If brightness rises as the number falls, then a star brighter than the ancient first class must be given a number below one, and then below zero. The negatives are not a quirk added later — they are the unavoidable consequence of extending a rank ordering into a real measurement without flipping its direction.

Then the lock tightens further. Everything downstream inherits the convention. Colour index is a difference of magnitudes, so a larger number means redder. Distance modulus, extinction, surface brightness, the entire calibration chain of photometric systems, and well over a century of published measurements all live in magnitudes. What would it cost to reverse the sign now? Not the arithmetic — that is trivial — but every catalogue, every archived measurement, every equation and every reader's trained intuition, all at once, for a gain that is purely cosmetic.

One caveat about the psychophysics, because it is often overstated. The eye's roughly logarithmic response, the Weber-Fechner relation, made Pogson's constant-ratio step a decent fit to the ancient classes, but human sensory scaling is better described in general by a power law than by a logarithm, and the ancient steps are not in fact a perfectly constant ratio. The fit was good enough to be convenient, not exact.

The correction to make is to stop reading magnitude as a broken brightness scale. It is a rank order that was later given metric clothing, and the strangeness lives entirely at the seam between those two things.

It also helps to separate three decisions that get blamed on each other. The direction was fixed in antiquity by ranking, and had nothing to do with light. The ratio was fixed by Pogson in 1856, chosen for continuity with existing catalogues rather than for elegance. The zero point is a third, separate choice, and it has been revised repeatedly — once anchored to Polaris until Polaris turned out to be variable, later to a standard sequence and effectively to Vega, and now, in the AB system, defined directly from flux density with no reference star at all. That is the honest limit of the path-dependence story: the parts that could be changed without stranding the archive were changed. Only the sign, which is the one part nothing else depends on numerically but everything is written in, survived.

• How the AB magnitude system defines a zero point from physical flux rather than from a reference star.
• Why absolute magnitude and the distance modulus inherit the same reversed sign, and what that does to the equation.
• Which other scientific scales are frozen accidents of their first users — and which ones did get replaced.

Every term the collection defines is gathered in the glossary.