# Q: star scaling calculation

If in our neighborhood of the Milky Way, the typical spacing between stars is 1 parsec, and the relative velocities between neighboring stars are of order 20 km/sec, what would the spacing between stars be in a scaled-down model in which stars are replaced by grains of sand (1 mm in diameter)? How fast would these sand grain sized stars be moving? [This is a quiz question from The Teaching Company's Introduction to Astrophysics by Joshua Winn.] The given answers are 44 km and 0.9 m/s.

I used the size of the Sun as a typical star (perhaps not accurate) and calculated the ratio of a sand grain's diameter to the Sun's diameter. I then multiplied this by 1 parsec (= 3.09 x 10^16 m) to get 21 km. This is about half the given answer so I thought maybe our Sun is not average sized.

I used the same calculated ratio and multiplied it by the give star velocity and got 1.43 x 10^-5 m/s. This is way off the given answer. What am I doing wrong? Is my calculation for the first part also wrong?

• I vaguely remember HA Rey saying in his book that the Sun is the 3rd or 4th largest star of the 25 nearest stars. There are a lot more very small stars nearby. – user21 Mar 7 '20 at 16:57

20 km/s is equal to $$6.47 \times 10^{-13}$$ pc/s, but we've just established that a pc is about 44 km in your sand grain model. So the speed is $$2.85\times 10^{-10}$$ m/s.
Just a double check - to travel 1 pc at 20 km/s would take $$1.5\times 10^{10}$$s.
To travel 44 km at $$2.85\times 10^{-10}$$ m/s takes $$1.5 \times 10^{10}$$ s.