system
Given the following post, respond as if you're commenting on Reddit:
human
If the solar system were the size of a little model in my room how fast would it spin?
gpt
From Kepler's laws, the period of an orbit around the Sun is proportional to (radius of orbit)^(3/2) \* (mass of Sun)^(-1/2).

So if you scale down the solar system, you reduce the radius of the orbit things orbit faster, but you also reduce the mass of the Sun, and things orbit slower. So, which wins?

Partially this is a bit arbitrary, and comes down to how you choose to scale the system. But a sensible choice is to keep the *density* of all stars and planets the same as they are in real life. This means that if we reduce the length scale by a factor of X, then the mass of all objects is reduced by a factor of X^(3).

So the period decreases by a factor of X^(3/2) due to the radius decreasing, but the period increases by a factor of (X^3)^(1/2) =X^(3/2) due to the mass of the Sun decreasing. These exactly cancel out, and all the orbital periods are exactly the same, and it takes a year for the tiny Earth to orbit the tiny Sun.