David Peat

From Clockwork to Chaos 119of the gravitational pull of the asteroid belt on the orbit of Jupiter?These tiny effects are analogous to the perturbations that coastline irregularitieshave on tide predictions. Scientists call this astronomicalproblem the three-body problem. It asks: How do three or more bodiesmove under their mutual attractions of gravity?While the two-body problem can be solved exactly, there is nosimple solution to the three-body problem. No single equation can bewritten down directly and used to calculate numerical answers to anydegree of accuracy. This does not mean that Newton’s laws are incorrector approximate. Rather, the corresponding mathematical equationspresent insurmountable difficulties that make it impossible for ageneral solution to be written down in a direct way. In the case of thesimpler two-body problem, it is just a matter of inserting the numericalvalues for the position, speed, and mass of the earth and sun intothe relevant equation, and the answer pops out. But when the mutualpulls of earth, sun, and moon act together on each other this simpleapproach no longer works.Astronomers found a way around this problem using an approachcalled “perturbation theory.” In perturbation theory you begin withthe reasonable assumption that the moon’s effect on the earth’s orbitaround the sun is very small. Start with the simple two-body problem,the earth’s orbit around the sun (neglecting the moon), and then applya small correction (called the “perturbation”) to take into account themuch smaller pull of the moon. To this first correction apply another,even smaller, correction. And then a third correction, and so on adinfinitum. In practice scientists don’t need to add too many of thesecorrections because, after the first, the size of successive correctionsbecomes so tiny as to make no practical difference to the value of earth’sorbit of the sun.It is a case of “wheels within wheels.” 2 Using perturbation theoryastronomers made tiny corrections to the orbits of the planets to accountfor the gravitational pulls of smaller third and fourth bodies.2The complex machinery of cogs I had seen at Bidston Observatory was themechanical analogy of perturbation theory, in which the motion of ever smaller cogsadds in tiny corrections to the tide predictions.