Brownian motion of the Sun

[chornedsnorkack]

How fast is the Sun undergoing brownian motion due to weak collisions with stars?

The relaxation time of Milky Way is quoted as in the order of 10^14 years. The orbital period of Sun is 10^8 years, so about 10^6 times smaller. Since the velocity grows with square root of time, the uncertainty of Sun´s velocity after 1 orbital period is 10^-3 times of... what? The translational orbital speed (250 km/s) or the peculiar velocity (mere 20 km/s)?

Also, how does the position uncertainty of Sun scale with time? To the power of square root of time like speed, to the power of square root of the cube of time, or to some other power?

[antoniseb]

I have no answer, but I like the question. This is something I've wondered about, but never expressed so clearly.

[StupendousMan]

Consider this thought experiment: put a bunch of stars in a giant box in space. Give each star some random initial velocity, with an RMS of, say, 10 km/s. Then let the stars interact gravitationally, and watch their velocities slowly change. Can you model this as Brownian motion? If so, what velocity goes into the calculations?

Now put a second bunch of stars in a giant box in space, give each star some random initial velocity as before, but, on top of that, give the entire box and all the stars in it a velocity of 250 km/s. Let the stars interact gravitationally. Can you model the change in their initial velocities as Brownian motion? If so, what velocity goes into the calculations?