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xkcd #2898: Orbital Argument
(imgs.xkcd.com)
Alt text:
"Some people say light is waves, and some say it's particles, so I bet light is some in-between thing that's both wave and particle depending on how you look at it. Am I right?" "YES, BUT YOU SHOULDN'T BE!"
And that point is inside the sun.
No actually. Due to Jupiter, the centre of mass of the solar system is actually very slightly outside of the sun
Wouldn’t the center of mass constantly be shifting by the planets’ varying positions in orbit?
Yes, but it's mostly shifting because of Jupiter. It's just so dang heavy. Like, a couple times heavier than every other planet put together. I don't have the brain wattage to do the cool math right now, but a quick google search says that while the barycenter of the solar system does depend on all the planets, more often than not, it is outside the sun
Easy reminder:
sun ~ 10^30 kg
jupiter ~ 10^27 kg
earth ~ 10^24 kg
so the ratio is always 1000:1