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I'm playing with orbits in a simple 2-d game where a ship flies around in space and is attracted to massive things. The ship's velocity is stored in a vector and acceleration is applied to it every frame as appropriate given Newton's law of universal gravitation. The point masses don't move (there's only 1 right now) so I would expect an elliptical orbit.

Instead, I see this:

This is what I see

I've tried with nearly circular orbits, and I've tried making the masses vastly different (a factor of a million) but I always get this rotated orbit.

Here's some (D) code, for context:

void accelerate(Vector delta)
    velocity = velocity + delta; // Velocity is a member of the ship class.

// This function is called every frame with the fixed mass. It's a
// method of the ship's.
void fall(Well well)
    // f=(m1 * m2)/(r**2)
    // a=f/m
    // Ship mass is 1, so a = f.
    float mass = 1;
    Vector delta = well.position - loc;
    float rSquared = delta.magSquared;
    float force = well.mass/rSquared;
    accelerate(delta * force * mass);
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woo. yeaaaah. D. Unit test that math code against known results; and all will be well. – dcousens Jun 23 '12 at 6:43
up vote 7 down vote accepted

The bug is in the fall function. We have

  1. delta: a vector from the well to the ship
  2. force: the magnitude of the gravity between these two bodies.

|force| is G * m1 * m2 / r^2

but |delta| is already r! so you are actually accelerating too fast. You need to divide by r again (basically normalizing the delta vector) before calling accelerate.

accelerate(delta * well.mass * mass / rSquared / Math.sqrt(rSquared))
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Note that even with the math bug(s) fixed, you're using Euler integration (i.e. velocity += delta and presumably position += velocity), so you're probably going to get some odd effects like rotation of the orbital ellipse over time, and perhaps the ellipse getting larger/smaller since Euler integration isn't guaranteed to conserve energy.

You might want to switch to leapfrog integration, which is energy-conserving and should work better for orbital mechanics. Also, you should include the frame time in your equations so that the speed of your simulation is framerate-independent.

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