In general the actual physical situation we are studying
is the motion of some number N of bodies under their mutual
gravitation and other forces (e.g. atmospheric drag).

A common special case is the approximate three-body problem in which:

- The other forces can be neglected
- The bodies can all be treated as perfect homogeneous spheres with spherically symmetric gravitational fields (i.e. polar flattening and gravitational harmonics can be neglected)
- Relative distances in the problem are such that
the gravitational effect of
all but 2 of the N bodies, body B1 of mass m
_{1}and body B2 of mass m_{2}, can be neglected; we are interested in the motion of a small body B0 of negligible mass.

In this case the orbit of body B0 is a conic section (ellipse, parabola or hyperbola)
with the barycenter of the body B1/B2 system as the conic focus,
and one can express the orbit using Keplerian elements
relative to this barycenter.

If the small terms are not neglected, they can be treated as
perturbations so that the barycentric Keplerian elements
are slowly varying functions of time.

In the limit m_{2}/m_{1} --> 0 the problem reduces to the
pure Keplerian one where the focus is the center of body 1
and we can say (if e<1) that body 2 is `in orbit around' body 1.

If m_{2}/m_{1} << 1 then we can still use body-1-centered
Keplerian elements, but they will vary with time more rapidly
than the barycentered ones.

For space probes in orbit around the Sun, there's a difficult
choice to make here. The effect of Jupiter is enough to
make the solar system barycenter (SSB) noticeably move relative
to the center of the Sun. The mathematically natural choice
is to express the orbital elements barycentrically. However,
one cannot **see** the SSB - the astronavigationally natural
choice is to ask `where am I relative to the Sun', even though
those elements are time-variable.
The same is true of the Earth-Moon system - choosing
geocentric or Earth-Moon-barycenter (EMB) coordinates.

In general I will choose heliocentric and geocentric coordinates
rather than barycentered ones.

See also the discussion of the Hill Sphere in ems.html