Physics education can be pretty well described as a series of lies of ever-decreasing size. Force is not exactly equal to mass times acceleration, it turns out you can push a rope (a little!)… So is it “factually true” that the Earth orbits the sun?

Yes. Almost exactly. Compared with the previous belief (that the Sun revolves around the Earth), the statement is very close to entirely true.

Most importantly, for the vast majority of people, it is true for all practical purposes that the earth revolves around the Sun. In fact, the word “true” doesn’t have a stronger meaning than this in the context of physics. In this world, there is a very small number of people who must consider, for practical reasons, the inaccuracy of the statement.

These people would include the folks who track asteroids, the rocket scientists who compute orbits for space probes (and maybe not even them!) and perhaps a few others. If you were to ask any of those people if their model is “true”, they would say “No, but it is more accurate”.

Who told you the Earth doesn’t orbit the Sun? “The Earth orbits the sun” is a factually true statement, according to any reasonable interpretation of that sentence in English – i.e. what people actually mean when they say one thing orbits another.

Before any physicists agreeing with you jump on me, strictly speaking both the Sun and Earth are tracing elliptical paths around a barycenter. The barycenter of the Sun-Earth system is about 5000 km from the exact center of the sun’s core- roughly the distance from New York to California, or less than about 0.3% of the Sun’s radius.

But the Sun-Earth barycenter is irrelevant given the presence of Jupiter. Orbital calculations are more strongly affected by the Sun-Jupiter barycenter. That one is just below the surface of the sun, and the sun pivots around it once every 12 years.

NASA worries more about that one.

But the barycenter offset changes nothing. Even if the barycenter is in space, like it is between a pair of neutron stars, we still say the stars “orbit each other”. And in our case, we’re orbiting a barycenter that’s inside the sun. Therefore “the Earth orbits the Sun” qualifies as a fact, I think, unless you’re being ridiculously strict with your definition of “orbits”.