Orbit

This is the collection of equations needed to perform a simple two body orbit propagation. 

  
Kepler's Third Law tells us that orbits are elliptical (shaped like an ellipse).  The forces working on an orbit include force of gravity, the force of drag, and potentially magnetism.  

The simplest orbit predictions are called the two body orbits where the masses are considered point masses and the only force at work is gravity.  Furthermore, the ratio of the masses are such that one mass is negligible compared to the second.  

To compute the future position of a satellite (P2) using the two body method, one needs to know the  initial position (P1)  and velocity (V1) vectors of the satellite and the mass of the planet.  In our example we will use the mass of the Earth.

The steps are as follows:

1. The initial position is P1
2. Choose the time increment for estimating the next position (delta t)
3. Compute the acceleration due to gravity at the current position.  The acceleration vector is in the direction of the planet and of a magnitude equal to acceleration due to gravity.
4. Use the formula that velocity = acceleration / time  (v = a/t) to compute the velocity vector in the direction of the planet.  The time is delta t, and the acceleration is that due to gravity.  
5. Compute the unit vector of P1, and scalar multiply it by -1 * velocity created by the acceleration due to gravity.  We will call this vector Vg. 
6. Use vector addition to add the velocity vector (V) with the velocity from gravity (Vg) to have the true velocity (Vt) over time delta t. 
7. Compute the change in position (delta P) of the satellite over delta t, but multiplying the true velocity vector (Vt) by the time delta (delta t).
8. Now compute the final position (P2) by vector addition of the initial position (P1) and the change in position (delta P).

Small time periods (delta t) are recommended, and one can repeat the process to sum up to a desired time in the future with a prediction of the position and velocity vectors at that time.