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J.E. Chambers (NASA Ames/SETI Institute)
A variety of coordinate systems have been used to study the N-body problem for cases involving a dominant central mass. These include the traditional Keplerian orbital elements and the canonical Delaunay variables, which both incorporate conserved quantities of the two-body problem. Recently, Cartesian coordinate systems have returned to favour with the rise of mixed-variable symplectic integrators, since these coordinates prove to be more efficient than using orbital elements.
Three sets of canonical Cartesian coordinates are well known, each with its own advantages and disadvantages. Inertial coordinates (which include barycentric coordinates as a special case) are the simplest and easiest to implement. However, they suffer from the disadvantage that the motion of the central body must be calculated explicitly, leading to relatively large errors in general. Jacobi coordinates overcome this problem by replacing the coordinates and momenta of the central body with those of the system as a whole, so that momentum is conserved exactly. This leads to substantial improvements in accuracy, but has the disadvantage that every object is treated differently, and interactions between each pair of bodies are now expressed in a complicated manner involving the coordinates of many bodies. Canonical heliocentric coordinates (also known as democratic heliocentric coordinates) treat all bodies equally, and conserve the centre of mass motion, but at the cost of introducing momentum cross terms into the kinetic energy. This complicates the development of higher order symplectic integrators and symplectic correctors, as well as the development of methods used to resolve close encounters with the central body.
Here I will re-examine the set of possible canonical Cartesian coordinate systems to determine if it is possible to (a) conserve the centre of mass motion, (b) treat all bodies equally, and (c) eliminate the momentum cross terms. I will demonstrate that this is indeed possible using a new coordinate system, and I will briefly describe the properties and advantages of these coordinates.
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Bulletin of the American Astronomical Society, 35 #4
© 2003. The American Astronomical Soceity.