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In reading this article about updated estimates for the number of exoplanets in the Milky Way, I am curious how to get an estimate of the mean distance between them. The Milky Way is ~50,000 light years in radius and an average of ~1000 ly in thickness, and the article estimates there to be ~500 million exoplanets in the habitable zone around their stars.

To simplify this greatly (grotesquely!), I think the galaxy can be thought of as a plane, since I want to know the method of doing this for a disc anyway, and there are so many estimates here, it shouldn't affect the results greatly. So, the area of this disc would be:

$\pi (50000ly)^2 \approx 7853981634 ly^2$

However, beyond this, to find an even distribution of 500,000,000 points, I'm not really sure how to go about that. Given this, I can later apply it to an oblate disk, but I'd like to know the general method of evenly distributing points in a circle, and possibly finding the distance between them if it's not straightforward.

Thank you in advance.

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    No problem. It's a nice question in itself. ;)2011-02-20

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Insofar as the number of points is practically infinite it is enough to calculate the average distance $a(D)$ of two independently chosen and uniformly distributed points in the unit disc $D$. The result is $a(D)= {128\over 45\pi}.$ I found this on page 1 of the following document:

http://www.math.uni-muenster.de/reine/u/burgstal/d18.pdf

where also references are given.

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It seems a rough approximation is useful here, as the distance from one planet to the nearest will vary considerably. I would just calculate the volume of interest (taking account of the bulge and arms as best you can), then divide by the number of planets to get the volume each one "occupies". The expected distance will then be something like 1-2 times the radius of a sphere of this volume. Very rough, but doable.

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    Haha, I missed that thread. I wish I'd used that; it's sure a lot easier than the old transcendental value! ;) It kind of goes with all the other fudge factors I've used though, so I guess I shouldn't laugh..2019-05-07