I'm working on a 2D physics engine for a game. I have gravity and masses working, using a simple iterative approach (that I know I'll have to upgrade eventually); I can push the masses around manually and watch them move and it all works as I'd expect.
Right now I'm trying to set up the game world in advance with a satellite in a simple circular orbit around a planet. To do this I need to calculate the initial velocity vector of the satellite given the mass of the planet and the desired distance out; this should be trivial, but I cannot for the life of me get it working right.
Standard physics textbooks tell me that the orbital velocity of an object in circular orbit around a mass M is:
v = sqrt( G * M / r )
However, after applying the appropriate vector the satellite isn't going anything like fast enough and falls in in a sharply elliptical orbit. Random tinkering shows that it's off by about a factor of 3 in one case.
My gravity simulation code is using the traditional:
F = G M m / r^2
G is set to 1 in my universe.
Can someone confirm to me that these equations do still hold in 2D space? I can't see any reason why not, but at this point I really want to know whether the problem is in my code or my assumptions...
Update: My physics engine works as follows:
for each time step of length t:
reset cumulative forces on each object to 0.
for each unique pair of objects:
calculate force between them due to gravity.
accumulate force to the two objects.
for each object:
calculate velocity change dV for this timestep using Ft / m.
v = v + dV.
calculate position change dS using v * t.
s = s + dS.
(Using vectors where appropriate, of course.)
Right now I'm doing one physics tick every frame, which is happening about 500-700 times per second. I'm aware that this will accumulate errors very quickly, but it should at least get me started.
(BTW, I was unable to find an off-the-shelf physics engine that handles orbital mechanics --- most 2D physics engines like Chipmunk and Box2D are more focused on rigid structures instead. Can anyone suggest one I could look at?)