Velocity Verlet Algorithm: Cannot seem to determine correct velocity for stable Orbit? - objective-c

Question

Like many that post about this topic, I too am busy trying to write myself an accurate simulator for the movement of objects in a 2D gravitation field.
I decided early on that I would settle on Velocity Verlet Integration, as I want my objects to maintain stable orbits and conserve energy even if the timestep is rather large. So, what might the problem be?
Well, so far, everything seems to behave correctly, except for one component. When I try to calculate the correct velocity for a stable orbit at a certain distance, the resulting velocity sends them into odd elliptical orbits that quickly increase in magnitude each time.
So, to begin, here are the following methods that determine an objects next position, velocity, and acceleration in scene: (Objective C)
Acceleration:
-(CGVector)determineAccelerationFor:(SKObject *)object
{ // Ok, let's find Acceleration!
CGVector forceVector = (CGVector){0,0}; // Blank vector that we will add forces to
for (SKObject *i in self.sceneObjects)
{
if (![i isEqual:object]) // Just make sure we're not counting ourselves here
{
CGPoint distance = [self getDistanceBetween:i.position And:object.position];
float hypotenuse = sqrtf(powf(distance.x, 2)+ powf(distance.y, 2));
float force = ((self.gravity * object.mass * i.mass)/powf(hypotenuse, 3));
float xMagnitude = (force * distance.x);
float yMagnitude = (force * distance.y);
forceVector.dx += xMagnitude;
forceVector.dy += yMagnitude;
}
}
CGVector acceleration = (CGVector){forceVector.dx/object.mass, forceVector.dy/object.mass};
return acceleration;
}
Cool, so basically, I just take an object, add all the other forces that each other object imposes on it together then divide the X & Y factor by the mass of the current object to get the acceleration!
Next up is Velocity. Here I use the following equation:
The method for it is pretty straightforward too:
-(CGVector)determineVelocityWithCurrentVelocity:(CGVector)v OldAcceleration:(CGVector)ao NewAcceleration:(CGVector)a
{
float xVelocity = (v.dx + ((ao.dx + a.dx)/2) * self.timeStep);
float yVelocity = (v.dy + ((ao.dy + a.dy)/2) * self.timeStep);
CGVector velocity = (CGVector){xVelocity,yVelocity};
return velocity;
}
And finally, position! The equation for this is:
And it is determined with the following method!
-(CGPoint)determinePositionWithCurrentPosition:(CGPoint)x CurrentVelocity:(CGVector)v OldAcceleration:(CGVector)ao
{
float xPosition = (x.x + v.dx * self.timeStep + ((ao.dx * powf(self.timeStep, 2))/2));
float yPosition = (x.y + v.dy * self.timeStep + ((ao.dy * powf(self.timeStep, 2))/2));
CGPoint position = (CGPoint){xPosition,yPosition};
return position;
}
This is all called from the below method!!
-(void)refreshPhysics:(SKObject *)object
{
CGPoint position = [self determinePositionWithCurrentPosition:object.position CurrentVelocity:object.velocity OldAcceleration:object.acceleration]; // Determine new Position
SKAction *moveTo = [SKAction moveTo:position duration:0.0];
[object runAction:moveTo]; // Move to new position
CGVector acceleration = [self determineAccelerationFor:object]; // Determine acceleration off new position
CGVector velocity = [self determineVelocityWithCurrentVelocity:object.velocity OldAcceleration:object.acceleration NewAcceleration:acceleration];
NSLog(#"%# Old Velocity: %f, %f",object.name,object.velocity.dx,object.velocity.dy);
NSLog(#"%# New Velocity: %f, %f\n\n",object.name,velocity.dx,velocity.dy);
[object setAcceleration:acceleration];
[object setVelocity:velocity];
}
Okay, so those methods above dictate how objects are moved in scene. Now onto the initial issue, the ever present problem of achieving a stable orbit!
In order to determine what velocity an object should have to maintain an orbit, I use the following equation:
And I implement that as follows:
-(void)setObject:(SKObject *)object ToOrbit:(SKObject *)parent
{
float defaultSeparation = 200;
// Move Object to Position at right of parent
CGPoint defaultOrbitPosition = (CGPoint){parent.position.x + (parent.size.width/2)+ defaultSeparation,parent.position.y};
[object setPosition:defaultOrbitPosition];
// Determine Orbital Velocity
float velocity = sqrtf((self.gravity * parent.mass)/(parent.size.width/2+defaultSeparation));
CGVector vector = (CGVector){0,velocity};
[object setVelocity:vector];
}
And for some reason, despite this, I get abysmal results. Here is some of the output:
Information:
Gravity(constant) = 1000 (For test purposes)
Mass(Parent) = 5000 units
Mass(Satellite) = 1 units
Separation = 224 pixels
It determines that in order for the Satellite to Orbit the Parent, a velocity of:
149.403580 pixels/timeStep
is required. And that checks out on my calculator.
So this has left me a little confused as to what could be going wrong. I log all the output concerning new velocities and positions, and it does use the velocity I set it to, but that just doesn't seem to make a difference. If anyone could possible help spot what's going wrong here I would be immensely grateful.
If anyone believes I have left something out, tell me and I will edit this right away. Thanks!