mathematics - Elliptical orbit modeling

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:

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);
}

Answer

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))