Fascinating. Basic physics question: the linked Wikipedia page on barycentre suggests that the further away a planet (in a 1 sun 1 planet system), the further the barycentre is from the centre of the sun. How does that work in the context of the declining force of gravity as a function of distance? I would have thought that the further a planet , the less pull it can exert on the sun. Does gravity not work that way?
Think of it like this: if the planet were inside the sun, at the very center, that's where the barycenter would be. If the planet were then to move steadily away from the center of the sun, the barycenter would also move steadily (though more slowly) away from that point. As that planet steadily moves away from the sun, there is no point at which the barycenter will stop and start moving back toward the sun.
The barycenter is a balancing point. If you were trying to balance two objects on the ends of a see-saw, if you move one object farther away, the balance point has to move toward it (you're actually balancing moments of inertia, which are functions of radial distance.)
If your gravitating objects are in an orbit, they also have angular momentum and moments of inertia.
You're right that a nearer small body pulls the large body with a greater force, but after one full orbit the large body has been pulled rightward just as much as it has been pulled leftward. The missing piece is: for how long is the force exerted before the situation repeats?
A distant small body exerts a smaller force for a longer time, and a nearby one exerts a greater force for a shorter time.
It comes down to how much time and how much force. The law of graviton plus Kepler's third law plus some algebra ought to show that you get a larger circle with a weaker force over a longer time (not an obvious result).
Suppose that you have a 2 planet system: A sun type star (mass 210^30 kg), an Earth type planet 1 AU from the sun (weight 610^24 kg), and a Jupiter type planet (mass 2*10^27 kg) that is 1000 AU away from the sun.
The distance between the Earth type planet the center of the star could be nearly constant.
The distance between the Earth type planet and the barycenter would not be nearly constant -- it would not "orbit" the barycenter in a circular orbit.
I may be wrong and I would be very happy if someone corrected me.
(Note: I think the barycenter would be about 5 AU from the star!)
I believe this is false in the sense that Mercury's orbit is more centered about the center of the Sun rather than the barycenter of the solar system (center of mass).
That might be, but the real statement would be more like that the solar system rotates around the barycenter of the solar system, for sure the moon revolves "more" around the barycenter of the earth-moon pair than that of the whole solar system.
The planet's pull on the sun and the sun's pull on the planet decline at the same rate (they're both proportional to 1/r^2). So the barycenter will always be the same proportion along the line between their centers of mass.
https://en.m.wikipedia.org/wiki/Barycenter