Days in a year long ago

Adapted from IOAA 2011 (Katowice), theory problem 6. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

Tides exert a torque on Earth. Assume that for the last several hundred million years this torque and the length of the sidereal year have stayed the same, at $6.0\times {10}^{16}$ N m and $3.15\times {10}^7$ s. How many days were there in a year 600 million years ago? The moment of inertia of a uniform solid sphere of mass $m$ and radius $R$ is $I=\frac{2}{5}m{R}^2$. Take Earth's mass as $5.972\times {10}^{24}$ kg and its radius as 6371 km.

AnswerA year held about 424 sidereal days, or 423 solar days, 600 million years ago. The official solution writes the earlier spin as the present spin minus the change, which would make the day longer in the past, though its final figure of about 420 days uses the right sign. Growth bands in fossil corals of about 400 million years ago show about 400 days in a year.