do not be misled . ALTHOUGH THE HEAT LEAKING SLOWLY OUT OF A PLANET'S INTERIOR DOES EVENTUALLY GET RADIATED AWAY BY ITS SURFACE , MOST OF THE RADIATION THAT A

Question

TERRESTRIAL PLANET EMITS IS SIMPLY RETURNING TO SPACE HEAT ENERGY THAT THE PLANET RECEIVED BY ABSORBING LIGHT FROM THE SUN- ULTRA VIOLET , VISIBLE AND INFRARED (BUT PRINCIPALLY VISIBLE) . THE ENERGY PER AREA THAT A PLANET RECEIVES FROM THE SUN IS PROPORTIONAL  TO 1/D SQUARED, WHERE D IS THE PLANETS DISTANCE FROM THE SUN. MAKE THE APPROXIMATION THAT ALL OF THE ENERGY A PLANET RADIATES INTO SPACE BALANCES THE RADIATION IT RECEIVES FROM THE SUN . DERIVE A FORMULA WHICH SHOWS TO WHAT POWER OF D THE TEMPERATURE OF A PLANET IS PROPORTIONAL . THIS FORMULA WILL BE GOOD ONLY IF THE PLANET DOESN'T HAVE A GREEN HOUSE EFFECT.

Answer 1

Power that a planet of radius R recieves from a star of luminosity L a distance D away:

(1) P = L*(πR²⁄4πD²)

since the solar flux is distributed evenly across a sphere of radius D away from the star, so the fraction of luminosity or power reaching the planet is just the ratio of the planets cross sectional area divided by the total area of a sphere a distance D away. Hence, the power is proportional to 1/D² as you state.

Now using the Stefan-Boltzmann Law we can argue that the power emitted by the planet at temperature T is given by:

(2) P = Aεσ T⁴ = 4πεσ R² T⁴

where A is the surface area of the planet, 4πR², σ is the Stefan-Boltzmann constant, and ε is the emissivity of the planet (some number typically from 0-1 corresponding to how well the planet radiates as a black body)

Equation equations (1) and (2) assuming perfect equilibrium without greenhouse effects gives the following result with little algebra:

(3) T = 1/2⋅(L / πεσ)^1/4⋅1/sqrt(D)

hence we find T ∝ D^-1/2.