Apparent Brightness (Inverse-Square Law)

A star radiates its luminosity L in every direction, so by the time the light has travelled a distance d it is spread over a sphere of area 4πd² — and the flux you receive falls as 1/d². The Sun makes the numbers vivid: L = 3.828 × 10²⁶ W spread over a sphere of radius 1 AU gives b = 3.828 × 10²⁶ / (4π × (1.496 × 10¹¹)²) ≈ 1 361 W/m², the measured "solar constant" above Earth's atmosphere. At Neptune, thirty times farther out, sunlight is 900 times weaker.

The d-rearrangement is one of astronomy's great distance tools. For a "standard candle" — a source whose true luminosity is known, like a Cepheid variable or a Type Ia supernova — measuring the apparent brightness immediately yields d = √(L/4πb). Henrietta Leavitt's Cepheid work made the method possible, Hubble used it to prove the spiral nebulae were separate galaxies, and Type Ia supernovae stretched it far enough to reveal the accelerating expansion of the Universe in 1998.