Why is the absolute temperature of planetary surfaces on average inversely proportional to the square root of their distances from the Sun?

The illumination of a planet by the Sun is inversely proportional to the square of the distance from the Sun. If we consider the planet as a black or gray body, then its emissivity according to the Stefan-Boltzmann law will be proportional to the fourth power of the absolute temperature. From these ratios it follows that T = 1 / r ^ 0.5. Using the Earth’s surface temperature as a normalization, the formula for the planet’s surface temperature takes the form T (K) = 277 / t ^ 0.5, where the heliocentric distance r is expressed in astronomical units.

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