**34th Meeting of the AAS Division on Dynamical Astronomy, May 2003**

* 9 Standards and Gauges*

Oral, Tuesday, May 6, 2003, 1:00-3:05pm,
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## [9.06] Adventures in Coordinate Space

*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.