# Change in Luminosity of a planet

When viewed from the sun, the brightness of the planet with a given size and albedo changes according to the fourth power of inverse distance.

I found the statement in one encyclopedia but I can't find any mathematical proof of this.

• Look for "inverse square law" Jul 27, 2022 at 15:42
• Physics doesn't prove. It measures ;) Aug 1, 2022 at 16:45

Light obeys an inverse square law (think of light as spreading out in a sphere) So the brightness of the planet is inversely proportional to the distance from the planet to the observer (B is proportional to $$\frac{1}{d^2}\$$) (assuming things like the planet is fully illuminated etc)
But the brightness of the incident light also obeys an inverse square law with respect to the distance from the planet to the sun (r). The brightness of the incident light is proportional to $$\frac{1}{r^2}\$$
So the brightness is proportional to $$\frac{1}{d^2}\$$×$$\frac{1}{r^2}\$$ But if your observer is near the sun, then d=r and so the brightness is proportional to $$\frac{1}{r^4}\$$
• To paraphrase: as a planet moves, $\frac{1}{r^2}\$ light from the Sun reaches the planet. Then $\frac{1}{r^2}\$ is of the reflected light reaches back to the Sun. Jul 27, 2022 at 18:35