The temperature of the Earth

Adapted from IOAA 2021 (Bogotá, held online), theory problem Q2. We have restated it in our own words and kept the original data, adding the Sun's temperature and radius and the astronomical unit. The solution was last checked on 20 October 2026.

For at least the last few million years, Earth has been roughly in balance with the sunlight it receives. Take the Sun's effective temperature as 5772 K, its radius as $6.957\times {10}^8$ m and 1 au as $1.496\times {10}^{11}$ m.

(a) Treating Earth as a perfect blackbody, find its equilibrium temperature in degrees Celsius. (4 points)

(b) Earth's albedo is about 30%. Find its temperature, in degrees Celsius, taking the albedo into account. (2 points)

(c) Earth radiates the energy it absorbs from its surface, but the atmosphere absorbs 58% of that radiation and sends it back to the ground, which warms the ground further. Treat this return as happening only once, and do not treat the atmosphere as a separate blackbody. Find the temperature of the ground, in degrees Celsius. (4 points)

AnswerEarth would be at about 5 °C as a blackbody and −19 °C with its albedo, and the returned heat raises the ground to about 12 °C.