LearnLight & spectraNaked eye
The inverse-square law of brightness
The apparent brightness of a source falls off as the square of its distance, because the same light is spread over an expanding sphere.
Why it matters
It is the bridge between how bright something looks and how bright it really is — and therefore the foundation of every distance measurement that uses standard candles.
The maths, in layers
1 · Intuition
The surface area of a sphere is 4πr². Twice as far away means four times the area, so a quarter the light per unit area.
2 · The equation
b = L / (4π·d²)
- b
- apparent brightness, watts per square metre
- L
- intrinsic luminosity, watts
- d
- distance
Rests on
Nothing. This is a starting point — you can begin here.
Leads to
- Extinction and airmassThe dimming and reddening of starlight by the atmosphere, which increases with how much air it has to cross.
- The magnitude scaleA logarithmic, backwards brightness scale in which smaller numbers are brighter and five magnitudes is a factor of exactly 100.
- The cosmic distance ladderThe chain of overlapping techniques by which distances are measured, each calibrated against the shorter rung below it.
- Equilibrium temperatureThe temperature a world settles at when the sunlight it absorbs exactly balances the heat it radiates away.
- Photon noiseThe irreducible grain in any measurement of light, arising because photons arrive at random and counting N of them carries an uncertainty of √N.