Astronomers still use a brightness scale invented over 2000 years ago. The Greek astronomer Hipparchus sorted the stars into six classes: the brightest were "first magnitude," the faintest he could see "sixth magnitude." So on this scale, smaller numbers mean brighter stars. It measures apparent brightness in a way that matches how our eyes judge it: by ratios.
Why it works
Our eyes respond to brightness by ratios, not differences: each step down the old scale looked about equally dimmer, and it turned out each step was the same factor. In 1856 Norman Pogson made this exact: $5$ magnitudes is a factor of exactly $100$. So one magnitude is a factor of $\sqrt[5]{100} = 2.512$, and a difference of $\Delta m$ magnitudes means a brightness ratio of $$\frac{b_1}{b_2} = 100^{\Delta m/5} = 2.512^{\Delta m}.$$ This is an exponential function: adding magnitudes multiplies brightness. Going the other way, $\Delta m = 2.5 \log_{10}\frac{b_1}{b_2}$. The scale extends past zero for very bright objects: Sirius is $-1.46$, Venus up to $-4.6$, the full Moon $-12.7$ and the Sun $-26.7$.
Key idea
Smaller (and negative) magnitudes are brighter. A difference of $5$ magnitudes is a factor of $100$ in brightness; $1$ magnitude is a factor of $2.512$.
Worked examples
Worked example: First against sixth
How much brighter is a first-magnitude star than a sixth-magnitude star?
- $\Delta m = 6 - 1 = 5$.
- Five magnitudes is a factor of exactly $100$.
Answer: $100$ times brighter.
Worked example: Sirius and Polaris
Sirius is magnitude $-1.46$ and Polaris magnitude $1.98$. How many times brighter does Sirius look?
- $\Delta m = 1.98 - (-1.46) = 3.44$.
- Ratio $= 2.512^{3.44} = 100^{3.44/5} = 10^{1.376} = 23.8$.
- Sirius looks about $24$ times brighter.
Answer: About $24$ times.
Worked example: From a ratio to magnitudes
A telescope can see stars $10{,}000$ times fainter than the eye can. How many magnitudes deeper does it reach?
- $\Delta m = 2.5\log_{10}(10{,}000) = 2.5 \times 4 = 10$.
- If your eye reaches magnitude $6$, the telescope reaches magnitude $16$.
Answer: $10$ magnitudes.
Common mistake
On most scales, bigger numbers mean more. On this one a magnitude-6 star is fainter than a magnitude-1 star, by a factor of $100$. Check with the brightest things in the sky: the Sun is $-26.7$, the most negative magnitude of all.
Apparent magnitude depends on distance. To compare stars fairly, astronomers also use absolute magnitude: how bright a star would look from a standard distance of $10$ parsecs.