Date: March 22, 2012
Title: An Overview of the n-Body Problem
Podcaster: Dr. Kyle Kneisl
Description: Anyone interested in astronomy understands that two bodies under the force of gravitation will orbit in ellipses or other familiar shapes. Many people, until they begin formally studying the subject, believe that many bodies, interacting under gravity, will trace out those same familiar shapes; this assumption is supported by the picture most people have in their mind of the solar system, where planets orbit, predictably, in familiar shapes. However, one quickly discovers that as soon as three (or more) bodies are considered, not only does this simple description generally break down, but it breaks down in spectacular fashion, and our familiar notions are not at all the truth. In this podcast, we describe the so-called n-Body problem, giving an informal description of the chaotic trajectories of several bodies under gravity, and one way scientists deal with modeling such systems.
Bio: Dr. Kneisl is a mathematician and a lifelong astronomy buff, and this is his third podcast for 365DoA. He lives near Washington, DC, with his wife and two young children, who he hopes to introduce to the beauty of astronomy in the coming years.
Sponsor: This episode of the “365 days of Astronomy” is sponsored by — NO ONE. please consider sponsoring a day or two in 2012 so we can continue to bring you daily “infotainment”.
Transcript:
This is Dr. Kyle Kneisl, and today we’ll be talking about a fact of physics that is as surprising as it is fascinating. Necessarily, as the topic would otherwise be too complicated for this podcast series, some of the things I say will be incomplete or not perfectly correct in some minor details. Should you wish a complete understanding of this topic and all of its details, you will find many advanced texts and web pages that will cater to that intention. But, this topic is already rich enough, assuming ideal point masses and classical physics. Keep in the back of your mind though that the situation is actually somewhat more complicated by effects from modern physics and relativity which, again, we will neglect for the podcast.
There is no shortage in the sciences of cases where some system governed by extremely simple mathematics is completely understood–every aspect of it–up until a certain number of dimensions, or objects, or exponent, or what have you. But, after that special tipping point, adding just one more to the system results in something which is suddenly so vastly complex that not only is it not fully understood, but it is, often times, not understandable. There are also in mathematics and physics many systems whose evolution is described by relatively few and/or easy equations that nonetheless result in unbelievably rich and difficult-to-understand behaviors. An example of the latter is weather modeling; that is to say, we can effectively write down all of the mathematics governing every aspect of the weather, but it still too complicated to understand precisely what will happen just a day, or even an hour, into the future.
One place that remarkably shows both properties in one of the strongest ways is the trajectories objects take under nothing more than the completely understood influence of gravity. Just one simple concept: that the force of gravity is inversely proportional to the square of the distance between massive bodies is all that is needed to drive a system of extraordinary, almost incomprehensible complexity.
Certainly anyone interested in astronomy has in their mind the concept of an object orbiting around another. Indeed when two objects orbit each other, this is called the “two-body problem”, and it is, as you would expect, completely understood. Any college student with a few semesters of calculus behind him can easily follow a proof that for a closed system of two bodies interacting with each other gravitationally, each body’s path describes a “conic section”, that is to say, a familiar shape like a circle or ellipse (if they are “in orbit”), or a parabola or hyperbola if they are just passing by each other. By “completely understood”, what we mean to say is that if we know the exact mass, positions, and velocities of the two objects at any time, we can immediately calculate their precise positions at any time in the future or past with a reasonably simple equation. Any system two bodies, of any masses, at any distance, with any velocity can be “solved” in this sense.
(And now, we get to one of those cases where maybe I’m lying just a little bit.) For three (or more) bodies interacting, aside from very special, artificial, cases, we have almost no ability to describe the orbits in any familiar mathematical way. Specifically, there is no useful function we can plug some point of time into, and get the precise location of one of three (or more) bodies interacting gravitationally, except in very special cases.
Consider, for example, a planet like ours orbiting a binary star system. It is possible, under certain configurations, that such a planet might more-or-less stably orbit around one of the two stars (generally the smaller of the two), while the smaller star itself orbited around the very distant center of the gravity of the two star system. Such a system was envisioned in the science fiction Helliconia Trilogy by Brian Aldiss in the early 1980s. In this fascinating trilogy, there are two types of year on the planet Hellicon, the lesser year, similar to ours, with seasons similar to ours, and the greater year (corresponding to a revolution around the greater star) lasting thousands of times longer causing terrific, generation-spanning seasonal changes on a longer time scale. It would be quite possible, indeed, to make mathematics inroads in the 3-body problem in a system like this, as these arrangements are mostly “stable” and periodic.
However, when multiple bodies are given random positions, velocities, and masses, the more likely result is an impossible-to-guess wild dance, where it is truly impossible to predict the future positions of the bodies. A small planet orbiting two comparably sized stars that were relatively close together might orbit first one star for several orbits, and then might transition into the orbiting the other, or even orbiting them both like a figure-eight, or any other possibility. It would be a truly crazy dance that would have to be seen to be believed.
How can scientists study such things? It is really quite simple. Even though the n-body problem results in extremely complicated paths in the general case, these paths still arise from the application of remarkably simple gravitational laws. In a nutshell, the simple statement that the acceleration due to gravity exerted on object A by object B is inversely proportional to the square of the distance between them is all that is needed. Having this information, you can then “simulate” the system (in a computer, for example), and see how the objects behave.
But, how, exactly? The least complicated way to do this is for each object in the system, add all of the accelerations acting upon it by all of the other objects, one by one. This sum of accelerations will be the true acceleration acting upon the object at that instant. Knowing this, we can adjust the velocity accordingly, and let the object move along the new trajectory for some (generally small) period of time. We repeat this for all other objects in the system, calculating their new positions after this small period of time. We do this millions and millions of times, and thereby trace out the paths of our objects. This is called, as you might expect, “n-body simulation”. Methods like these (this simple one formally called the Euler Method) have been known since the 18th century!
But of course there are problems. There is always loss of precision when systems are numerically simulated just from the errors inherent to representing numbers in computers at all. However, that small period of time where we let the objects go in a straight line is a larger issue. As the objects move, the precise direction that they should be moving will change ever-so-slightly (for, as example, the precise direction to the other bodies has changed slightly since the objects have moved). Therefore, the assumption we make that the new velocity holds for some small period of time is only approximately correct. Therefore, a bit of precision is lost at each stage in the simulation. When the velocities and accelerations are large, quite a lot of precision can be lost, quite quickly, simulating in this fashion. As you can imagine, mathematicians have come up with other, better, methods to simulate the system, but these all have these same challenges.
Nonetheless, no matter what method we use, numerical simulation will eventually become inaccurate, not to mention, taking much longer to simulate, say, a century into the future versus simply plugging that time into an equation and getting an answer immediately (as we would be able to do with the 2-body problem).
A number of scientists and mathematicians have determined “special” solutions to the n-body problem, where, for example, three exactly equal mass objects can each trace out the same figure-8 pattern, not to mention a variety of more complicated, but stable and predictable patterns. Such solutions are called “choreographies”. But these are always “non-physical”, meaning, that the conditions one requires to achieve these solutions do not actually arise in any realistic physical system.
In any case, the n-body gravitational problem is hopelessly complicated. When, however, one of the masses is overwhelmingly massive compared to the others, and when they are sufficiently spaced apart from each, as we have for the solar system, we can make extremely good approximations into the past and future; but this is really all they are, approximations. There are other factors at play as well: the real universe consists of non-ideal masses, dark matter, relativistic effects, and so on.
If you are interested in further information on the n-body problem, a good starting point is the Wikipedia article, and the many links to other scholarly pages which give you more precise details.
Thank you for listening. This is Kyle Kneisl from Baltimore, and we’ll see you next time.
End of podcast:
365 Days of Astronomy
=====================
The 365 Days of Astronomy Podcast is produced by the Astrosphere New Media Association. Audio post-production by Preston Gibson. Bandwidth donated by libsyn.com and wizzard media. Web design by Clockwork Active Media Systems. You may reproduce and distribute this audio for non-commercial purposes. Please consider supporting the podcast with a few dollars (or Euros!). Visit us on the web at 365DaysOfAstronomy.org or email us at info@365DaysOfAstronomy.org. Until tomorrow…goodbye.