The "Two-Body Problem" is a very common approach for Keplerian mechanics. In this problem, we will make a few assumptions such as:
Here, the word mechanics is not being referred to as the person who fixes your car, but rather the branch of mathematics that deals with motion and the forces that cause motion. Mechanics can be broken up onto two smaller parts:
It can be thought that mechanics is the combination of both kinematics and kinetics. This is what we will be utilizing throughout the two-body problem.
Let's now talk about reference frames. The curious thing about reference frames is everyone is currently in a different reference frame from everyone else. I sit here, writing this tutorial in my own reference frame as you are reading this in a different location, with a different reference frame of your own. A car could zoom by my location and I could observe its motion to be different from what you observe.
You see, we all live in different reference frames, but for problems we can share the same reference frame by defining a specific location, known as the origin, and treat that as the center point about where to take our measurements from. We can define a reference frame with a caligraphy I, or $ \mathcal{I} $. $ \mathcal{I} $ will have 4 components: the origin, O, and the 3 position components, $ \hat{i}$, $ \hat{j} $, $ \hat{k} $. We can in fact write a velocity vector in this frame as
$$ v_{\mathcal{I}} = v_{\mathcal{I}_x} \hat{i} + v_{\mathcal{I}_y} \hat{j} + v_{\mathcal{I}_z} \hat{k} $$
Now I should also mention the two types of reference frames. There are what's called inertial and non-inertial reference frames. Inertial means an object or frame is resistive to any change in its motion, hence, non-accelerating. As for non-inertial, this means the object or frame is non resistive to change in its motion, so it is accelerating. If we go back to the car passing example from before. If there are two different inertial (non-accelerating) reference frames observing a car passing by, as long as the two frames have a velocity difference of zero relative between them, then they will observe the car to be moving at the same speed! However, if one of the reference frames were to be moving with the car at a constant velocity, it would still be inertial because it is not accelerating, but the velocity observed in the stationary reference frame would no longer match the other. This is why keeping all of these reference frames straight is important!
Next we will build off of the reference frames discussed in the tutorial Coordinate Systems. One of the mentioned frames was ECI (Earth centric inertial), and from the name, you can tell it is an inertial reference frame. ECI tends to have its coordinate system's origin at the center of the Earth and its x and y axes span in the equatorial plane while the z axis spans through the North and South Poles. The x axis in ECI is aligned with distant stars so its coordinates are non-rotating. This reference frame is often used for spacecraft.
The other reference frame discussed was EFEC (Earth-fixed Earth-centric). This reference frame is non-inertial because it rotates with the Earth. The x axis is aligned with the Greenwich Meridian and the z is aligned with the North and South Poles.
A new type of reference frame that we will talk about is TRF (Topographic Reference Frame). TRF is non-inertial and has an origin either fixed at or near the Earth's surface. One axis points upwards in the zenith direction (directly overhead), and the two two axes point along the local plane, or local horizon.
The next type is the orbit reference frame. The orbit RF has its origin on an orbit, its x axis pointing towards the velocity vector, and z axis pointing towards the center of mass of the Earth. The last type is the body reference frame. The body RF is with respect to the orbiting body itself.