How do you weigh a planet?

It's so easy to look up the mass of planets and other astronomical objects that it's rarely obvious where these values come from. Even quite young pupils may well very reasonably ask, for example, 'How do we know the mass of the Earth if you can't put it on a set of scales?'. The only decent method is, perhaps surprisingly, that you need to measure the orbit of a satellite (natural or artificial) around the object and calculate its mass from that data, basically by using Kepler III.


This means, amongst other things, that we had nothing better than vague estimates of the masses of Mercury and Venus (which have no natural satellites) until we started putting artificial satellites in orbit around them in the 1970s. For example, an undergraduate astronomical textbook from 1944 states that the mass of Mercury is 1/27th that of Earth, which is a whopping 50% too small (probably due to assuming Mercury has a similar density to the Earth, whereas in fact it's so Iron-rich that it's much denser). By comparison, the same book gives the mass of Jupiter to within 1% of the modern value. Fundamentally, this is because Mercury has no natural satellites but Jupiter has plenty.

Part of the calculation also requires an accurate value for G. This is why the first experiments to measure G (carried out in the 18th century by Cavendish in Cambridge and Maskelyne with the Scottish mountain Schiehallion) were often described as 'weighing the Earth'.

This worksheet gives you data for Charon's orbit around Pluto which is purely obtainable from Earth-bound telescope observations and invites the reader to use nothing more than trigonometry, circular motion equations and Newton's Law of Gravitation to calculate the mass of Pluto. A solution follows on the second page, so make sure you don't include this when giving it as a task!