Orbital Mechanics

Question

Does anyone have an example of implementing Orbital Mechanics (preferably in XNA)? The code I am currently using is below, but it doesn't "feel right" when it executes. The object just bends ever so slightly to the planet, and no matter how much I tweak the variables I cant get it to enter an orbit, or even a partial orbit.

shot.Position += shot.Velocity;  

foreach (Sprite planet in planets)  
{  
  Vector2 directionToPlanet = (planet.Position - shot.Position);  
  directionToPlanet.Normalize();  

  float distance = Vector2.DistanceSquared(shot.Position, planet.Position);  

  float gPull = (float)(planet.gravityStrength * (planet.Mass * shot.Mass) / distance) + planet.gravityField;  
  shot.Position += new Vector2(directionToPlanet.X * gPull, directionToPlanet.Y * gPull);  
}

Edit Marking Mendelt's answer correct for pointing out that I need to update the velocity, not the position. I also needed to change the calculation of gPull to

float gPull = shot.Mass * planet.Mass / distanceSqr * planet.gStr;

Answer 1

Newton-Raphson iteration is not a stable way to solve this problem (that is you can't get it right using so simple an integrator for the differential equation). Consider using a second (or higher) order solution: Runge-Kutta is good and is fairly easy to implement in this case.

From the point of view of numeric analysis, the problem of orbital mechanics reduces to that of solving the set of coupled differential equations:

x_i'' + G m_i \sum_{i != j} m_j r_ji/(|r_ji|)^3 = 0

where the x's are three-vectors representing the positions of the bodies, the m's are the masses of the same bodies, and r_ji = x_j - x_i is the vector displacement between bodies j and i.

Answer 2

In the last line you're updating the position of the shot. You should be updating the velocity.

You might want to take a look at the code in this blogpost http://blog.mendeltsiebenga.com/post/Fun-with-planets.aspx No xna, but working orbital mechanics. (although i never got rid of the screen-flicker)

Answer 3

A) We have no idea what your input values are.

B) You might want to use a better approximation than Newton-Raphson.

C) Passing objects do not generally fall into orbit IRL, gravity is extremely weak, it takes equally weak velocities or truly exceptional masses to get much more than curvature.

Answer 4

A passing object will not enter orbit. One characteristic of an orbit is that you will return to the same point (relative to the body being orbited) with the same velocity. If you started from effective infinity, you'll go back to effective infinity.

In order to enter orbit, you will need to change the velocity at some point in a way unrelated to gravity, or perhaps have additional large bodies. Similarly, you can't launch an object into orbit from the surface: you have to have something (like a last rocket burn) once the satellite reaches the desired altitude. Otherwise it will try to return to the launch point, which is on the surface.

Some of my worst debugging experiences were when the program was fine and my test data or calculations were off. Make sure you know what to look for.