Question

The temperature of a spherical planet is related to the distance from sun as :

• A. $T\propto 1/d^2$
• B. $T\propto \frac{1}{\sqrt{d}}$
• C. $T\propto d$
• D. $T\propto d^2$

Answer 1

The correct option is B $T\propto \frac{1}{\sqrt{d}}$

A planet reaches its equilibrium temperature when the amount of heat it absorbs from the sun is equal to the amount that it radiates back to space.

The energy absorbed is equal to the insolation at planet's distance (which is Sun's bolometric output divided by $4\pi$ times the distance squared) times the planet's profile area times the planet's Bond aldedo. The amount radiated (assuming that the thermal radiation is approximately black body radiation) is proportional to the planet's surface area times its emmisivity times the forth power of its temperature.

Now, throwing away 2, $\pi$ and other constants that do not vary with temperature or distance.

$T^4\alpha \frac{1}{d^2}\Rightarrow T\alpha \frac{1}{\sqrt{d}}$