A planet and its surface temperature

Adapted from IOAA 2007 (Chiang Mai), theory problem Q2. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

A fast-rotating planet of radius $R$ and albedo $a$ orbits a star of luminosity $L$ at distance $D$. Assume that all the energy the planet absorbs is radiated again as a blackbody.

(a) What is the flux of starlight at the planet? (1.5 points)

(b) At what rate does the planet absorb energy? (1.5 points)

(c) At what rate does the planet reflect light, its reflected luminosity? (2 points)

(d) What is the mean blackbody temperature of the planet's surface? (2 points)

(e) If the planet always kept one face towards the star, what would be the mean temperature of that face? (2 points)

(f) Find the temperature of part (d) in kelvin for $a=0.25$, $D=1.523$ au and $L=3.826\times {10}^{26}$ W. (1 point)

AnswerThe flux is $L/(4\pi D^2)$, the planet absorbs $(1-a)L{R}^2/(4D^2)$ and reflects $aL{R}^2/(4D^2)$, its temperature is ${((1-a)L/(16\pi \sigma D^2))}^{1/4}$, which is 210 K for the numbers given, and a face kept towards the star would be $2^{1/4}$ times warmer.