A big subject

We describe how to do something that’s hard.

Our astronomer is engaged in working out how to explain perturbations without resorting to mathematics.  Why that is difficult may require some background.

Mathematics, including mathematical problems found or inspired by science, can be fairly easily divided into problems that can be solved exactly and those that cannot.  The former are the “easy” problems, and are almost exclusively what our tutor’s students see.  (They may disagree with the “easy” label.)  The latter are “hard” problems, and constitute the overwhelming majority of the universe.  In that sense, school mathematics is very deceptive.  It’s only with the occasional typo that students come up against, for instance, polynomials that cannot be factored, even though those vastly outnumber those that can be.

When mathematical scientists come upon a hard problem, one line of approach is to find an easy problem that is almost the same (in a very specific sense of “almost”) and solve that.  Then perturb it; that is, work out what a small dose of reality, a perturbation, would do to the solution.  The adjusted solution is not exact, but is closer to being right, and often is close enough for the purpose at hand.  Sometimes there’s a second iteration, or more.

The motion of the planets in the Solar System is a hard problem.  There are nine major masses (ignoring dwarf planets and smaller) all interacting under gravity, and it has been proven that there is no closed-form solution for as few as three masses.  However, almost all the mass in the System lies in the Sun.  So we start with the easy problem of two bodies (each planet, taken separately, and the Sun).  In that case, each planet follows an elliptical orbit whose size and shape can be worked out with good accuracy.  So do the other bodies in the Solar System, some of which we may be very interested in for various reasons.

Then we add in the effects of the other planets.  These are worked out for the original orbit (“first-order perturbations”) and are often expressed as familiar elliptical orbits whose parameters vary with time.  That is, the orientation and size of this fictitious orbit change at every moment.  However, they don’t change very much, and retaining the elliptical picture allows a reasonably tractable set of calculations.

The question our astronomer now faces is: what is the overall effect of the perturbations?  The answer is not simple.  The orbit of an asteroid may be changed drastically, sending it out of the Sun’s influence entirely or into the inner System, where it might cross Earth’s orbit.  On the other hand, there are asteroids kept in place by the periodic tugs of Jupiter and Saturn.  Astronomers in the nineteenth century thought that the stability of the main planets had been proved, but more complicated work with digital computers has cast doubt on that conclusion.

Our astronomer’s immediate task is to explain to science-fiction authors how to make their star-systems stable (or otherwise).  He has Sir John Herschel’s 1833 treatise on the subject, supposedly written for the general public.  However, the chapter in question runs to 60 small-type pages.  We’re not optimistic on the chances of a simple answer.