In The Magnitude Scale you met the astronomer's way of saying how bright a star looks: smaller numbers are brighter, and 5 magnitudes means a factor of 100 in brightness. That rule came from the naked-eye catalogs of ancient Greece, which sorted stars into six classes. In 1856 Norman Pogson made it exact. Now we turn it into an equation you can solve, using properties of logarithms.
What a magnitude measures is the flux $F$: the power of starlight arriving per square meter of telescope, in W/m², usually through a filter that passes one band of colors.
Why it works
Key idea
| $\Delta m$ | Flux ratio |
|---|---|
| $1$ | $10^{0.4} \approx 2.512$ |
| $2$ | $\approx 6.31$ |
| $2.5$ | $10$ |
| $5$ | $100$ |
| $10$ | $10^{4}$ |
| $15$ | $10^{6}$ |
Worked examples
Worked example: From magnitudes to a flux ratio
- Vega has the smaller magnitude, so it is the brighter one. The difference is $\Delta m = 1.98 - 0.03 = 1.95$.
- $F_{\text{Vega}}/F_{\text{Polaris}} = 10^{0.4 \times 1.95} = 10^{0.78}$.
- $10^{0.78} \approx 6.03$.
Answer: Vega delivers about $6.0$ times as much flux.
Worked example: From a flux ratio to magnitudes
- $m_B - m_A = -2.5\log_{10}(F_B/F_A) = -2.5\log_{10}(1/40)$.
- $\log_{10}(1/40) = -\log_{10}40 = -1.602$.
- $m_B - m_A = -2.5 \times (-1.602) = +4.0$.
Answer: $m_B - m_A \approx 4.0$: B, the fainter star, has the larger magnitude.
Worked example: Light adds, magnitudes don't
- The fluxes add: the pair sends $2F$, where $F$ is one star's flux.
- Compare the pair with one star: $m_{\text{pair}} - 3.0 = -2.5\log_{10}(2F/F) = -2.5\log_{10}2 = -0.75$.
- $m_{\text{pair}} = 3.0 - 0.75 = 2.25$.
Answer: $m \approx 2.25$. Adding the magnitudes (to get 6.0) would make the pair look fainter than one star, which can't be right.
Common mistake