Right the joke is about the barycenter about which two objects orbit each other. It gets even more complicated when you remember that these aren't point masses orbiting each other. So the elliptical approximation of the orbit is for one instant in time, even if we are just accounting for gravity and no other forces. What you end up with when you view the state in terms is an osculating element set over time. If you graph the terms you'll see a larger trend in the orbit state and a much faster oscillating looking component along the the curve as well. The barycenter is constantly changing as well. For systems with very large mass discrepancies, say a man-made satellite around the earth, you can essentially say it is the center of the larger mass.
The "common center" isn't the barycenter about which Earth and the Sun orbit "each other".
The Sun's position is more or less unaffected by puny Earth.
It's Jupiter and Saturn that causes the Solar System barycenter to sometimes be outside of the Sun. And Uranus and Neptune have an outsized effect because they're further away (the effect is proportional to distance and mass).
@isaackuo Well I was referring to barycenter of two body orbits generically. It's the center of mass of the two objects. And yes, when we calculate that for ridiculously lopsided bodies it essentially collapses to the center of the biggest body. But yes when we calculate solar system barycenter the earth is not really driving it much. While we do often use, if not generally use, solar system barycenter for sun orbiting objects I do believe for cis-lunar we still just stick with the earth center and treat the moon as a secondary gravitational field.
@isaackuo I'm referring to the coordinate system origin for state and force calculations when saying "use solar system barycenter" or "stick with earth center".
@hankg Yeah, when it comes right down to it, the only way to put it in a small number of words is, "It's complicated."
For example, for cis-lunar it depends on the context, what we use. For most BOE calcs for most orbits, you want to ignore the Moon. Beyond that, for some BOE purposes, the Sun can be ignored but for others it can't.
When it comes to actual mission planning and navigation, BOE type simplifications aren't appropriate. So ... yeah, there's no short way to summarize it.
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