Gravity Stars

Gravity Stars

Right, this is where things get complicated and slow. Every star (particle) is affected by every other, so you will have to have 2 nested loops, each looping through every star. This means that the time taken to process the movement of the stars is proportional to the square of the number of stars. So, double the number of stars, and it can take four times longer to calculate.

Every star is attracted to every other by a force which lessens with distance. The force is proportional to 1/(distance squared) between two stars. The force is also affected by the mass of each star.

The full formula is: FORCE = (Mass1 * Mass2) / (distance*distance)

Right so first thing you're going to have to do when comparing two stars to see how much they attract is to calculate the vector between them:

Xvector = star2_x - star1_x
Yvector = star2_y - star1_y

Then you need to find the distance between them by pythagoras:

Distance = square_root_of (Xvextor*Xvector + Yvextor*Yvector)

Now you need to turn the vector into a vector of length 1:

Xvector = Xvector / Distance
Yvector = Yvector / Distance

Now find the force:

Force = (Mass1 * Mass2) / (Distance * Distance)

and multiply the vectors by the force and divide by the mass:

Xvector = Xvector * Force / Mass
Yvector = Yvector * Force / Mass

You will need to do this for every star, summing up the Xvector and Yvector then dividing their values by the mass of the star because Force = Mass * accleleration. Right lets see all that in one mighty big loop: 
Loop a from 1 to n
    x_acceleration = 0
    y_acceleration = 0

    Loop b from 1 to n
	Xvector = star_x(b) - star_x(a)
	Yvector = star_y(b) - star_y(a)
	Distance = square_root_of (Xvector*Xvector + Yvector*Yvector)
	Xvector = Xvector / Distance
	Yvector = Yvector / Distance
	Force = (Mass(a) * Mass(b)) / (Distance * Distance)
	Xvector = Xvector * Force / Mass(a)
	Yvector = Yvector * Force / Mass(a)
	X_acceleration = X_acceleration + Xvector
	Y_acceleration = Y_acceleration + Yvector
    End of b loop

    Star_Xvel(a) = Star_Xvel(a) + X_acceleration
    Star_Yvel(a) = Star_Yvel(a) + Y_acceleration
End of a loop

You may notice that the above loop is slightly innefficient. Twice the necessary number of comparisons are done. Say you had two stars. That code would compare star 1 to star 2, then compare star 2 to star 1. With a slight ammendment, it is possible to double the speed of this algorithm. Note, however, that this is still an n-squared algorithm. Double the number of stars, four times the calculations.


Loop a from 1 to n

   Loop b from a+1 to n
	Xvector = star_x(b) - star_x(a)
	Yvector = star_y(b) - star_y(a)
	Distance = square_root_of (Xvector*Xvector + Yvector*Yvector)
	Xvector = Xvector / Distance
	Yvector = Yvector / Distance
	Force = (Mass(a) * Mass(b)) / (Distance * Distance)
	Xvector = Xvector * Force / Mass(a)
	Yvector = Yvector * Force / Mass(a)
	X_acceleration = Xvector
	Y_acceleration = Yvector

    	Star_Xvel(a) = Star_Xvel(a) + X_acceleration
    	Star_Yvel(a) = Star_Yvel(a) + Y_acceleration
    	Star_Xvel(b) = Star_Xvel(b) - X_acceleration
    	Star_Yvel(b) = Star_Yvel(b) - Y_acceleration
    End of b loop
End of a loop