Introduction to the N-body problem
The Book: Gravitational N-body Simulations
20 Jul 2005 : Tigers!
21 Feb 2005 : Sverre.com official opening
Dr. Sverre Aarseth
Institute of Astronomy
Cambridge CB3 0HA
About the site
© Sverre Aarseth 2005
Welcome to sverre.com - N-body Simulations online
The N-body problem:
The classical N-body problem is defined as a generalization of Newton's Law of gravitation for a system of N interacting bodies. Although analytical solutions for N = 2 can be obtained in an elegant manner, extension to the case N = 3 already presents intractable difficulties. A comprehensive description of the many historical developments of this rich problem can be found here.
The modern astronomer makes use of computers to obtain the complete solutions as functions of the time. Increasingly fast processor speed facilitates the laborious calculation of all the individual interactions needed in order to advance the solutions by a suitably small time interval (see "Introduction to the N-body problem)). More efficient numerical integration schemes have also been developed in response to new challenges. Consequently, progress over the last 40-odd years has enabled astronomers to extend their investigations from a few dozen stars to systems containing up to about 100,000 members.
The aim of numerical experiments is to describe the entire evolutionary history of self-contained stellar systems orbiting the Galaxy. This has only become possible by combining speed and reliability of the hardware with sophisticated software development, while still obtaining the solutions by essentially direct brute-force means.
Such simulations therefore constitute a stellar laboratory where time spans of millions of years are compressed into hours. Repeated experiments then allow for a range of effects to be explored systematically in order to improve the models, which should ideally account for the observational properties. Working in this way, the numerical simulator provides an important interface between theory and the real world. Hence the computer code becomes a powerful tool which can be employed for both practical and theoretical purposes.
Planets and Star Clusters:
Direct N-body simulations can be used to study planetary systems and a variety of star clusters. The latter vary in size from small groups to rich systems of a few million stars. Using the most powerful supercomputers, current technology is still not adequate to attack the largest observed clusters. However, the basic process of gravitational interaction is already exhibited by the three-body problem which is particularly suitable for graphical illustrations. Such deceptively simple systems are in fact chaotic and their description has attracted the attention of many eminent mathematicians since Newton's days.
Since analytical solutions have proved elusive, numerical studies are of both scientific and educational value. Observing the intricate orbits in space and time of just three interacting bodies yields an insight into the general behaviour of gravitational systems where the processes of exchange and escape are demonstrated.
Adding a fourth body (i.e. binary--binary collisions) introduces further complexity which also allows for the possibility of stable triples to be formed. Beyond that, simple orbit visualization of chaotic systems becomes less useful and other means of graphical representation would be more appropriate.
Planetary motions provide another type of few-body system which can also be illustrated graphically. However, because the central Sun dominates, such systems are intrinsically stable and movie displays of conventional solar systems with nearly circular orbits do not have much entertainment value. For educational purposes the initial conditions for an idealized Earth and Jupiter orbit are provided on the simulations page. Note that the combination of the Earth's faster speed (factor 2.28) and its smaller distance (factor 5.2) results in nearly a factor of 12 difference in the periods.
The book and the code:
Many practical aspects of the N-body problem are discussed in The Book. The code descriptions are based on previously published papers with collaborators who provided key ideas (see the references in the bilbiography).
My main occupation over more than four decades has been concerned with the implementation of algorithms for a variety of dynamical problems, ranging from planetary formation to cosmological simulations. However, the efficient treatment of strong few-body interactions has taken pride of place. It is therefore appropriate that such movies are featured on my personal website. The whole dedicated effort has resulted in many computer codes that are freely available. Writing the book proved a natural conclusion -- a project which took ten years.
I hope you enjoy the site!