Figure: Multi astro-body systems and centers of mass

Image 1 Caption: An image of triple gravitationally-bound system of astro-bodies. Since it is a perspective view from an oblique direction, it is difficult to know if the orbits are circular orbits or elliptical orbits, but we will assume they are circular orbits.

Although the system of astro-bodies is general, it is most obviously realized by a triple star system.

Further explication:

1. The cross marks the overall center of mass and defines the origin of center-of-mass inertial frame of the system which is an reference frame unrotating with respect to the observable universe in with the center of mass in free fall with respect to the external gravitational field. There are negligible tidal forces, and so the internal motions (i.e., those relative to the center of mass) are only determined by internal gravitational fields.

2. The two blue astro-bodies in a close circular orbit by the themselves constitute their own center-of-mass inertial frame with the center-of-mass marked the star symbol.

   The red astro-body is far enough away from the blue bodies that its tidal force on the blue bodies is small. But not negligible. So the two blue bodies do not constitute an ideal isolated center-of-mass inertial frame.

   The upshot is that to 1st order approximation the internal motions of the blue astro-bodies can be treated as independent of red astro-body, but not to 2nd order approximation.

3. Since the center of mass (i.e., the mass-weighted average position) is the origin (i.e., the zero location) of overall center-of-mass inertial frame, we have

   0 = m blue r blue + m red r red

   which implies

   r red = ( m blue m red ) r blue ≅ 3 r blue

   where the 3 is just estimated from Image 1 itself. The blue bodies dominate the mass, and this why the overall center of mass is relatively near them and relatively far from the red body just as the above formula shows.

   

4. Image 2 Caption: An image of quadruple gravitationally-bound system of astro-bodies. Since it is a perspective view from an oblique direction, it is difficult to know if the orbits are circular orbits or elliptical orbits, but we will assume they are circular orbits.

   Although the system of astro-bodies is general, it is most obviously realized by a quadruple star system.

5. In the case of the Image 2 system, the overall system is an isolated center-of-mass inertial frame: i.e., no tidal forces from outside act.

6. The red astro-body subsystem and the orange-astro-body subsystem are well enough separated that to 1st order approximation both can be treated as ideal isolated systems (i.e., ideal isolated center-of-mass inertial frames), but not to 2nd order approximation. There will be tidal forces between the two subsystems.

7. Recall, there is only an exact solution (i.e., one that can be written as a formula) for the two-body system (i.e., a gravitationally bound system of two spherically symmetric astro-bodies which is ideally isolated so that no tidal forces affect it and the external graviational only determines the center of mass acceleration).

   In all other cases, one must solve numerically on the computer for the motions of system of astro-bodies or use perturbation theory.