This is what Sawicki says about solar tides:
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Consider the point C on Earth closest to the Sun and the point F on a far side of Earth. The Sun pulls harder on a unit mass at the point C, not as hard on a unit mass at Earth center O, and weaker yet on a unit mass at point F. The acceleration "a" of Earth as a whole in free fall towards the Sun is determined by the gravitational pull of the Sun on Earth’s center. Hence the unit mass at C has a tendency to accelerate towards the Sun with acceleration a + Delta a, i.e. more than the center of Earth, while a mass at the far side F has a tendency to accelerate towards the Sun with acceleration a - Delta a, i.e. to lag behind the center of Earth.
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I thought this amounted to what JohnD was saying in his "different orbit" theory. If not, I obviously misinterpreted it. Sorry again.
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The Earth's centre of mass (CoM) is in free-fall about the barycentre of the Earth-Sun system (which is located near the centre of the Sun), orbiting at 108,000 kph, just the right speed to stay in orbit.
The point on the Earth's surface closest to the Sun is also travelling about the Earth-Sun barycentre at this speed
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No, it follows a similar path, but it is not centered on the barycenter. It is centered on a point that is offset from the barycenter by the radius of the Earth. Of course, when the Earth completes a half revolution about the Sun, the point will no longer be the closest to the Sun, but will be the farthest.
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I don't follow. At the moment that the point in question is closest to the Sun, it is travelling on an orbit about the Earth-Sun barycentre whose radius is less than that of the Earth's CoM by the radius of the Earth - not offset by the radius of the Earth.
Incidentally, why wait six months for it to be the farthest point? Isn't twelve hours just as good?
It might be easier if we considered a situation in which the Earth is imagined to be travelling on a circular orbit and to be in a locked rotation, so that the point in question will always be closest to the Sun, and will surely be "orbiting" the Earth-Sun barycentre in a circular path, but at a speed that is too slow for free-fall. The tidal bulges will not move then. Like the Moon, the Earth will be permanently stretched.