# Calculating the velocity planets need to have to be in circular orbit around a star

I'm toying with the idea of making a little 2D space orbiter game so I've implemented Newton's universal gravity law into this little app. It works really well, even.

The problem I'm having is when I want to create an asteroid-belt. I spawn little asteroids randomly around an area around the sun and give them a starting velocity vector v that is based on their position relative to the sun. Then I rotate v 90 degrees by multiplying with a transformation matrix.

After that I'm just guessing basically. Though I have tried many different ideas based on some calculations. At the moment I multiply v by r^-2 (where r is the distance to the center of the sun) but that sure isn't it. Help?

## Answers and Replies

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phyzguy
Science Advisor

An object is in circular orbit when its centripetal acceleration is equal to the gravitational acceleration:
$$\frac{v^2}{r} = \frac{G M}{r^2}$$
solving for v:
$$v = \sqrt{\frac{G M}{r}}$$

Hah! It works!
Now the problem turned out to be that my vectors were rotated incorrectly, but your answer helped me realize what wasn't wrong so thank you very much!

Here's a li'l pic of it at the moment:
http://img814.imageshack.us/img814/3983/gasim.png" [Broken]

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