I'm basically saying what Hornblower said. I wrote
all below except the final sentence before he posted.
"... it doesn't really orbit the barycenter ..."
When I wrote "really, truly" I had in mind little kids
who are struggling to differentiate between fiction
and fact. One in particular who seemed not to know
what information to trust.
I know that one focus of Earth's elliptical orbit is
approximately at the center of the Sun. I also know
that all the other matter in the Universe affects that
orbit, and thus the location of the focus. Obviously
all the matter inside Earth's orbit acts to increase
Earth's centrepital acceleration. The sum is the
barycenter of the "inner" part of the Solar System.
But all the matter outside Earth's orbit also affects
Earth's centrepital acceleration. The directions of
their forces cancel each other almost perfectly, so
for many purposes they can be and are ignored.
I don't how to describe it both simply and correctly.
If everything outside Earth's orbit was a perfectly
uniform shell, the barycenter of the rest would be
the the nearest thing to an exact focal point of
Earth's elliptical orbit.
-- Jeff, in Minneapolis