> The bulge on the Earth induced by the Moon accelerates the Moon
More precisely, it exerts a torque on the Moon, which is equal and opposite to the torque exerted by the Moon on the bulge. So does the bulge on the Earth induced by the Sun.
> the bulge on the Earth induced by the Sun accelerates the Sun.
This statement is correct in principle (or more precisely, the statement that the bulge on the Earth induced by the Sun exerts a torque on the Sun), and you are right that I left it out of my previous post.
Also, as I noted above, the bulge on the Earth induced by the Sun also exerts a torque on the Moon. And, for that matter, the bulge on the Earth induced by the Moon also exerts a torque on the Sun. I believe the net torque on the Sun is about 20% of the net torque on the Moon.
However, that does not mean the effect on the Earth's orbit is 20% of the effect on the Moon's orbit. The relative magnitudes of the effects depend on how large the torque is compared to the relevant orbital angular momentum. Since we are working in an Earth-centered frame, that means we need to compare the Moon's orbital angular momentum around the Earth with the Sun's orbital angular momentum around the Earth. If you do the math, you find that the latter is about 300 billion times larger than the former. Multiply that by another factor of 5 (for the 20% ratio of torques) and the effect on the Earth's year is about 1.5 trillion times smaller than the effect on the Moon's month.
> the bulge on the Earth induced by the Sun exerts a torque on the Sun
There's a further point here, though. The torque exerted on the Moon by the bulge is constant over time, because the position of the Moon relative to the bulge is constant (the bulge "leads" the Moon by a constant angle which is the result of a balance between the torque on the bulge exerted by the Moon and the pull on the bulge exerted by the Earth's spin). But the position of the Sun relative to the bulge is not constant: it goes through a complete cycle over the course of a month. So it seems to me that, at least to first order, the net torque of the bulge on the Sun over the course of a month should cancel out, so there will be no net exchange of angular momentum between the Earth's spin and the Sun-Earth orbit.
More precisely, it exerts a torque on the Moon, which is equal and opposite to the torque exerted by the Moon on the bulge. So does the bulge on the Earth induced by the Sun.
> the bulge on the Earth induced by the Sun accelerates the Sun.
This statement is correct in principle (or more precisely, the statement that the bulge on the Earth induced by the Sun exerts a torque on the Sun), and you are right that I left it out of my previous post.
Also, as I noted above, the bulge on the Earth induced by the Sun also exerts a torque on the Moon. And, for that matter, the bulge on the Earth induced by the Moon also exerts a torque on the Sun. I believe the net torque on the Sun is about 20% of the net torque on the Moon.
However, that does not mean the effect on the Earth's orbit is 20% of the effect on the Moon's orbit. The relative magnitudes of the effects depend on how large the torque is compared to the relevant orbital angular momentum. Since we are working in an Earth-centered frame, that means we need to compare the Moon's orbital angular momentum around the Earth with the Sun's orbital angular momentum around the Earth. If you do the math, you find that the latter is about 300 billion times larger than the former. Multiply that by another factor of 5 (for the 20% ratio of torques) and the effect on the Earth's year is about 1.5 trillion times smaller than the effect on the Moon's month.