Stellar Magnitude Formula (Pogson Scale)
The stellar magnitude formula explained — apparent magnitude, absolute magnitude, and the Pogson scale.
Includes worked examples with real stars.
The Formula
The stellar magnitude scale quantifies how bright a star appears. Devised by the ancient Greeks and formalised by Norman Pogson in 1856, the scale is logarithmic and inverted — lower numbers mean brighter objects. A difference of 5 magnitudes equals exactly a factor of 100 in brightness (flux).
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| m | Apparent magnitude (brightness as seen from Earth) | dimensionless |
| M | Absolute magnitude (brightness at 10 parsecs) | dimensionless |
| F | Flux (energy received per unit area per second) | W/m² |
| d | Distance to the star | parsecs |
Distance Modulus
This links apparent magnitude (m), absolute magnitude (M), and distance (d in parsecs). It allows astronomers to calculate a star's true luminosity once distance is known — or to find distance if luminosity is known from the star's type.
Example 1 — Comparing Two Stars
Sirius (m = −1.46) vs Polaris (m = +1.98): which is brighter and by how much?
Δm = 1.98 − (−1.46) = 3.44 magnitudes
Flux ratio = 10^(3.44 / 2.5) = 10^1.376
Sirius appears 23.8× brighter than Polaris in our sky
Example 2 — Distance Modulus
The Sun: m = −26.74, M = +4.83. What distance does this imply?
m − M = 5 log₁₀(d) − 5
−26.74 − 4.83 = −31.57 = 5 log₁₀(d) − 5
5 log₁₀(d) = −26.57 → log₁₀(d) = −5.314
d = 10^(−5.314) ≈ 4.85 × 10⁻⁶ parsecs ≈ 1 AU (confirms Earth-Sun distance)
Reference — Famous Objects
| Object | Apparent Magnitude | Note |
|---|---|---|
| Sun | −26.74 | Brightest object in the sky |
| Full Moon | −12.74 | Brightest night object |
| Venus (max) | −4.89 | Brightest planet |
| Sirius | −1.46 | Brightest star (after Sun) |
| Polaris | +1.98 | North Star |
| Faintest naked eye | +6.5 | Under perfect dark sky |
| Hubble telescope limit | +31.5 | Most distant detectable objects |
When to Use It
- Comparing star brightnesses in observational astronomy
- Calculating distances using standard candles (Cepheids, Type Ia supernovae)
- Planning telescope observations — determining if a target is visible
- Photometry — measuring brightness changes in variable stars