Inverse Square Law Explorer

Voyager 1 is now ~163 AU from the Sun (≈ 2.4 × 10¹² m). Plug that into the irradiance formula: E = 3.828 × 10²⁶ / (4π × (2.4 × 10¹²)²) ≈ 0.00053 W/m² — about 2.6 million times fainter than sunlight on Earth. Solar panels would be useless. Voyager runs on radioactive decay (a radioisotope thermoelectric generator) producing roughly 250 W of usable electricity. Its radio transmitter broadcasts at 22 W — about what a dim LED strip uses. By the time the signal reaches Earth's Deep Space Network dishes (5 km across), the 1/r² law has spread those 22 W over a sphere the size of the inner solar system. The received power is around 10⁻¹⁸ W. That is why the receiving antennas need to be enormous and why NASA must chill the receivers to near absolute zero to hear the signal at all. Same law, same geometry — whether it is sunlight warming your face or the whisper of humanity's most distant artefact.

The rule is pure geometry. Energy leaving a point source spreads evenly over the surface of an expanding sphere, and a sphere's area grows as 4πr². Double the distance and that same energy covers four times the area, so intensity drops to one quarter — not one half. That is why you cool off so fast when you step back from a campfire, and why this tool traces gravity, light, and electric force on the very same curve.

The exponent isn't arbitrary — it equals the number of spatial dimensions minus one. We get 1/r² because a sphere's area scales as r² in three dimensions. In a hypothetical 4D universe gravity would fall as 1/r³, and Paul Ehrenfest showed in 1917 that planetary orbits would then be unstable — planets would spiral into the star or fly away. Stable solar systems, and therefore life, may depend on space having exactly three dimensions.