Reprieve is over; I’m back.

Originally Posted by

**Hornblower**
Perhaps your thought process is being cluttered up by concepts of moment arms and leverage,…

Yep, too many FBDs from too long ago, perhaps. It’s the linear relation with distance to determine the barycenter that seems to be where I stumbled since it matches the moment arm equation (see saws).

ow consider a pair of bodies in free fall and orbiting one another. The line connecting them corresponds geometrically to the beam, but there actually is no beam present. The only forces acting on the bodies are gravitational ones on each one directed toward the other along the line, not crossways. The bodies have velocity vectors crossways and in opposite directions, and the gravitation causes the bodies to curve toward each other instead of flying off in straight lines. With just the right velocities they will be in a pair of stable orbits.

Using Newton and thinking only in terms of a two-body model with circular orbits about the barycenter, the orbital motion itself seems to be the answer to the radius linearity (inverse) in calculating the barycenter, as opposed to the inverse square associated with gravity. Namely, centrifugal force which is simply – mv^{2}/r. The two forces of gravity and centrifugal force are in balance, no doubt, and this too makes this a linear (inverse) relationship when equating each other. This gets us off the see-saw, where we belong. Twirling an unbalanced baton might be the better analogy.

Originally Posted by

** grapes**
Originally Posted by

**George**
) Saturn's Epimetheus and Janus exchange orbits close to every 4 years. As they near each other, excluding Saturn for a second, their mutual barycenter is easily determined, if we know their mass and locations, and we do. In a heavy-thinking (bull-headed) leverage model, the faster inner moon should swing past the slower moon and outward enough to become the outer orbiting moon. Vice-versa for the other moon. This is exactly what doesn't happen. And I don't think the kinematics improve by adding Saturn to this dynamic. The gravity/Kepler model, however, does work.

I've never looked into that, what does happen when they exchange orbits?

The inner moon gets boosted into the higher orbit *before* it is aligned with the outer moon. A “leverage” model would be opposite of this.

We know time flies, we just can't see its wings.