- Newtonian physics
shows that
two
gravitationally bound
spherically-symmetric
bodies absolutely isolated will
orbit their
mutual center of mass
(called the barycenter
for orbits)
in exact elliptical orbits
with the barycenter
at focus
of each of the elliptical orbits---the
other focuses are just empty
points in space---as is the
barycenter itself.
If the mass of body
is overwhelmingly dominant, that body's
center becomes effectively the barycenter
that the other body orbits
in an elliptical orbit.
- There is, in fact, an exact analytic solution (i.e.,
a formula) for
the gravitational two-body system
just described.
There is, in fact, no other
gravitationally bound
system
with an exact solution.
All others must treated by
perturbation theory or
by numerical solution
ground out by
computers.
- But you say,
gravitational two-body systems
(as described) do
**NOT**actually exist, and so how can the exact solution be useful.The exact gravitational two-body system is an ideal limit that is actually approached very often in the astrophysical realm, and so the exact solution is often an excellent approximation to 1st order or a good starting point for perturbation theory.

- Let's expand a bit on the foregoing by considering the internal interactions
of a localized multi-body
gravitationally bound system
with a overwhelmingly dominant body by mass:
e.g.,
a planetary system
with a dominant star
(like the Solar System)
or a moon system
with a dominant planet
(like Jupiter moon system).
In such a gravitationally bound system, the gravitational interaction between the dominant body and each other body individually is approximately a gravitational two-body system to 1st order. The other bodies cause gravitational perturbations and those have to be accounted for to 2nd order in detailed analyses often by perturbation theory.

- But what of the all the enormous massive bodies throughout the
observable universe?
They must have an effect on the localized multi-body
gravitationally bound system.
Yes. Gravity in the classical limit is an inverse-square law force. Now in one sense, inverse-square law forces fall off rapidly with distance: e.g., double distance and the force decreases by a factor of 4. However, in another and important sense they are long-range forces since formally they do

**NOT**go to zero until you reach infinity. So Newtonian physics predicts all the enormous massive bodies throughout the observable universe do have an effect on the localized multi-body gravitationally bound system. General relativity (the more adequate theory) gives the same result.However, general relativity also tells us that if the bodies outside of the localized multi-body gravitationally bound system are sufficiently far off relative to size scale of the gravitationally bound system, then they create a sufficiently uniform gravitational field that internal motions of the localized multi-body gravitationally bound system relative to its center of mass (AKA their barycenter) are determined only by internal forces. The external forces only determine the trajectory of the center of mass of the localized multi-body gravitationally bound system. This situation often holds: e.g., for example for most planetary systems with a dominant star (like the Solar System) and most moon systems with a dominant planet (like the Jupiter moon system).

The upshot is that the exact solution from Newtonian physics for the gravitational two-body system is verified again as a useful approximation.

Caption: The Sun and a general Solar-System as a gravitational two-body system.

Features:

Image link: Itself.

File: Orbit file: two_body_system_unexact.html.