JohnD: Before looking at your idealized situation (equally massive binary planets in locked rotation), I want to describe a simpler one.

Imagine the Earth is orbiting the Sun in a perfectly circular orbit. Imagine the Earth is in locked rotation, so that it keeps the same side facing the Sun at all times (rotating on its axis once a year). And imagine the Earth's axis is at 90 degrees to the plane of its orbit. Finally, imagine these are the only two bodies in the solar system.

The CoM of the Earth is in free-fall about the barycentre of the Earth-Sun system. Its orbital velocity is just right for it to remain permanently in free-fall (Kepler's Third Law) and maintain a constant distant from the barycentre.

Now, it seems clear to me that in such a situation the point on the Earth's surface closest to the Sun is also travelling around the barycentre in a perfectly circular path, but one whose radius is about 6,200 km shorter than that of the CoM's. Because it completes one revolution about the barycentre in one year, its speed is slightly less than that of the CoM. But being closer to the Sun, it is in a stronger gravitational field (or, as Einstein might have put it, the curvature of space is steeper). In order to be in free-fall orbit (and therefore maintain a constant distance from the barycentre), it ought to be going faster (Kepler, again). At its slower speed it begins to slip into the Sun's gravity well, and this produces the tidal bulge.

A similar argument explains the tidal bulge on the far side. There, the point farthest from the Sun is travelling on a circular path about the barycentre that is longer than the CoM's. But as it too completes one revolution about the barycentre in one year, it must be travelling at a greater speed. But to be in a free-fall orbit (and therefore maintain a constant distance from the barycentre), it ought to be going more slowly. At its greater speed it tries to climb up out of the Sun's gravity well (just like Pioneer 10), and this creates another tidal bulge.

What is wrong with this interpretation? Is it not a perfect description of your different orbits theory?

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