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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?

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  • $\begingroup$ 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. $\endgroup$ – user21 Mar 7 '20 at 16:57
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The average str is about half the size of the Sun. So I get 44 km.

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.

Check.

I have no explanation for you as to why it doesn't agree wth your answer or with Joshua Winn's. But I suggest you read his problem carefully, because I rather fancy he might have done something like say imagine 1 year is equal to a second or something... perhaps.

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