A Ruler Running Backward
Apparent magnitude is a backward logarithmic scale for brightness: smaller numbers mean brighter lights, and five magnitudes mean exactly a factor of one hundred. · 11 min
Around 150 BC the Greek astronomer Hipparchus cataloged the stars by eye and sorted them into six brightness classes. The brightest he called first magnitude; the faintest his eye could catch, sixth. The word meant rank, not amount — first class, second class — and the ranking stuck. Modern astronomy kept his scale, stretched it in both directions, and made it exact. That history explains the scale's one strange feature before you meet it: brighter objects get smaller numbers. The magnitude scale runs backward because it began as a ranking.
Guess before you learn
A magnitude 1 star stands next to a magnitude 6 star, right at the naked-eye limit. How many times brighter is the magnitude 1 star?
One hundred times — exactly. The scale is defined so that five magnitudes always mean a factor of 100 in brightness, which makes each single magnitude a factor of about 2.512. If you guessed something like 5 or 6, keep the pencil mark: reading magnitudes as a plain count is the most natural mistake there is, and this folio exists to replace it.
9–12
3–5
Star brightness is scored like places in a race: the brightest stars took first place — first magnitude — ordinary bright stars second and third, and the faintest visible stars sixth. Smaller number, brighter star. That is the whole trick to reading the scale.
Each step of one magnitude means about two and a half times the brightness. And a few lights — Venus, the full Moon — outshine every first-place star, so their scores dive past zero into negative numbers. The brightest things in the sky carry minus signs.
6–8
Apparent magnitude measures how bright an object looks from Earth. The scale runs backward — smaller means brighter — because it began as Hipparchus's ranking, first class down to sixth. Modern astronomers kept the classes but fixed the steps: a difference of 5 magnitudes is defined as exactly 100 times in brightness, so each single magnitude is a factor of about 2.512.
The scale extends both ways. Under a truly dark sky your eye reaches about +6. Brighter than magnitude 1, the numbers pass through zero and go negative: Sirius shines at −1.5, Venus can reach −4.6, the full Moon −12.7, and the Sun −26.7.
9–12
The rule in symbols: a brightness ratio b₁/b₂ corresponds to a magnitude difference m₂ − m₁ = 2.5 log₁₀(b₁/b₂). Pogson chose the constant in 1856 so that 5 magnitudes make exactly 100; one step is therefore 100^(1/5) ≈ 2.512. A logarithm fits because starlight spans an enormous range — the Sun outshines the faintest naked-eye star by a factor near 10¹³.
Ratios turn into differences: 2 magnitudes is 2.512², about 6.3 times; 2.5 magnitudes is exactly 10 times. Note also what apparent magnitude does not say — how bright the star truly is. A dim-looking point may be a brilliant star far away; folio 13 splits apparent brightness from absolute.
K–2
Long ago a stargazer sorted the stars. The brightest ones he called number 1. Fainter ones got 2, then 3, all the way to 6 — the faintest of all.
So star numbers work like race places. Number 1 is the winner — the brightest. Number 6 is the faintest star your eyes can find.
Undergrad
Define m = −2.5 log₁₀(F/F₀): flux on a logarithmic scale, anchored by a zero-point flux F₀ fixed by convention — historically Vega's flux sets m ≈ 0. The scale's oddities all follow: the minus sign preserves Hipparchus's backward ranking, and the factor of 100 per 5 magnitudes preserves his six classes.
In practice a magnitude is meaningless without a bandpass: a star has a V magnitude, a B magnitude, and so on, each integrating flux through a filter. Differences between bands — the color index B−V — carry physics, and folio 12 reads temperature from them. The hard part of photometry is not the logarithm; it is calibrating F₀.
Postgrad
Two zero-point conventions coexist: Vega-based systems, tied to one inconveniently variable A0 star, and the AB system, m_AB = −2.5 log₁₀(f_ν) − 48.60 with f_ν in erg s⁻¹ cm⁻² Hz⁻¹ — a pure flux-density scale. Cross-survey work lives or dies on the small offsets between the two.
The logarithm also misbehaves at low signal-to-noise, where measured flux can scatter to zero or below and log flux diverges. Modern surveys answer with asinh magnitudes (Lupton, Gunn, and Szalay, 1999), which track classical magnitudes at high flux and stay finite through zero — a reminder that the magnitude is an interface, and the flux is the measurement.
apparent magnitude
How bright an object looks from Earth, on the backward scale: smaller numbers mean brighter, and a step of 5 magnitudes means exactly 100 times in brightness.
Now stretch the scale across everything you can see. The faintest star a dark-sky site shows the naked eye sits near +6. A typical suburban yard, with its haze of artificial light, cuts you off near +4. From there the scale runs down through the bright stars, past zero, and into the negatives, where the planets and the Moon live. Before the ink places them, commit your own guesses: four lights, four magnitudes.
Why is this true?
Why is five magnitudes exactly a factor of 100 — not roughly?
Because Pogson defined it that way in 1856. Hipparchus's first-to-sixth span was close to a hundredfold to begin with, so the modern scale pinned the old ranking to a clean number and made every step exactly the fifth root of 100.
How many times brighter is Sirius (−1.5) than Polaris (+2.0)? — the steps fade as you master them
2.0 − (−1.5) = 3.5
3.5 ÷ 2.5 = 1.4
10^1.4 ≈ 25
One number now travels with every light you observe, and it will keep working for you. Folio 13 splits looks-bright from is-bright and turns the difference into a distance. Folio 16 uses the faintest magnitude your sky shows — your limiting magnitude — to score the sky itself. Next folio, though, the question is not how bright but where: what a constellation really is, and why the answer makes the whole sky navigable.
Practice — new ink and old, interleaved
1.Low in the west just after sunset hangs a steady, lamp-bright light at about magnitude −4. What are you looking at?
2.Name, with rough magnitudes, one object brighter than −10, one near −1, and the faintest thing your eye can catch under real darkness.
The full Moon near −12.7; Sirius near −1.5; and a +6 star, the naked-eye limit at a dark site.
How close were you? Grade yourself honestly — it sets your review date.
3.How many times brighter is a +1 star than a +6 star at the naked-eye limit?
4.A steady, bright light shines halfway up the northern sky, nowhere near the ecliptic. Could it be a planet?
5.Roughly how many times brighter is the full Moon than the first-quarter Moon?
6.From your yard the faintest stars you can find are about +4. Which of these can you see tonight?
7.In one sentence: why can Mars stand opposite the Sun in our sky while Venus never can?