In 1850 Roche gave a mathematical investigation of this process of tidal break-up. His discussion dealt only with solid or liquid bodies, but the underlying mechanism is the same whether the bodies are solid, liquid or gaseous. We have seen that the smaller of the two bodies involved in a tidal encounter suffers the most. Roche dealt only with the case in which one body was very small in comparison with the other; in such a case the small body was completely broken up, while the larger one remained unscathed. Roche imagined the small body to describe an orbit of gradually decreasing size around the big body. If the two bodies were of equal density, he calculated that the small body would be broken up as soon as the radius of its orbit fell to 2·45 times the radius of the large body. If the bodies are of different density the matter is slightly more complicated. We must imagine the larger body to expand or contract until it has the same average density as the smaller body; the critical distance is then 2·45 radii of the larger body in its imaginary expanded or contracted state. This distance is generally known as Roche’s limit. A satellite can with safety describe a circular orbit about its primary so long as this orbit lies beyond Roche’s limit, but it is broken into fragments as soon as it trespasses within the limit. The following figures confirm Roche’s mathematical analysis: Radius of Saturn’s outermost ring 2·30 radii of Saturn _Roche’s limit_ 2·45 _radii of primary_ Radius of orbit of Saturn’s innermost satellite 3·11 radii of Saturn Radius of orbit of Jupiter’s innermost satellite 2·54 radii of Jupiter Radius of orbit of Mars’ innermost satellite 2·79 radii of Mars At the same time they suggest very forcibly that Saturn’s rings are the broken-up fragments of a former satellite which ventured into the danger-zone marked out by Roche’s limit. We have seen how our own moon is destined in time to contract its orbit, until it is finally drawn within the Roche’s limit surrounding the earth and broken into fragments. After this the earth will have no moon, but will be surrounded by rings like Saturn.