For an orbit, as others have said, the easiest thing to use is polar coordinates. First construct a reference frame fixed to the large body that you'll use to approximate an inertial frame, and then construct a rotating reference frame pointing toward the orbiting satellite. Suppose the inertial frame is defined by the three orthonormal vectors \hat{\boldsymbol \imath}_1, \hat{\boldsymbol \imath}_2, and \hat{\boldsymbol \imath}_3, and that the rotating frame is defined by \hat{\boldsymbol e}_1, \hat{\boldsymbol e}_2, and \hat{\boldsymbol e}_3.
If we say that \hat{\boldsymbol e}_1 points to the mass center of the satellite, we can define the inertial position vector as {\boldsymbol p} = r \hat{\boldsymbol e}_1. Now, to make calculating the inertial velocity and acceleration in the {\boldsymbol e}^+ frame, we need a result called the kinematic transport theorem. Consider two reference frames {\boldsymbol a}^+ and {\boldsymbol b}^+, where the {\boldsymbol b}^+ frame is rotating with respect to the {\boldsymbol a}^+ frame with angular velocity {\boldsymbol \omega}_\text{a/b}. The derivative of a vector {\boldsymbol p} in the {\boldsymbol a}^+ frame, coordinatized in the {\boldsymbol b}^+ frame, comes from two components:
- The rate of change of {\boldsymbol p} in the {\boldsymbol b}^+ frame
- The angular velocity of {\boldsymbol b}^+ relative to {\boldsymbol a}^+
This gives us the
kinematic transport theorem, which states that \frac{{}^a\text{d}}{\text{d}t}\left({\boldsymbol p}\right) = \frac{{}^b\text{d}}{\text{d}t}\left({\boldsymbol p}\right) + {\boldsymbol \omega}_\text{a/b} \times {\boldsymbol p}. This is an incredibly powerful result, as it let's us easily use as many reference frames as we want while still being able to find how observers in different frames see vectors changing.
Why do we want to do this? The ultimate goal is to find the inertial acceleration. Recall that it is an axiom of Newtonian mechanics that inertial reference frames exist and that Newton's laws hold in inertial reference frames. The KTT makes it almost trivial to use non-inertial frames and still find the inertial velocity and acceleration.
Let's apply the KTT to find the inertial velocity of our {\boldsymbol p} vector. Let's assume we are in the plane spanned by \hat{\boldsymbol e}_1, \hat{\boldsymbol e}_2 and that \hat{\boldsymbol \imath}_3 = \hat{\boldsymbol e}_3. (That is, we are considering planar motion, which we can do since orbits are planar.) Suppose the angular velocity vector is {\boldsymbol \omega}_\text{e/i} = \dot{\theta}\hat{\boldsymbol e}_3
<br />
\begin{align*}<br />
{\boldsymbol v} = \dot{\boldsymbol p} &= \dot{r} \hat{\boldsymbol e}_1 + \dot{\theta}\hat{\boldsymbol e}_3 \times r \hat{\boldsymbol e}_1 \\<br />
{\boldsymbol v} &= \dot{r} \hat{\boldsymbol e}_1 + r\dot{\theta}\hat{\boldsymbol e}_2 <br />
\end{align*}
If we repeat this to find the inertial acceleration, we find that
{\boldsymbol a} = \dot{\boldsymbol v} =\ddot{\boldsymbol p} = \left(\ddot{r} - r\dot{\theta}^2\right) \hat{\boldsymbol e}_1 + \left(r\ddot{\theta} + 2\dot{r}\dot{\theta}\right)\hat{\boldsymbol e}_2(Verifying this expression is left as an exercise for the reader.)
Now, we can apply Newton's second law, {\boldsymbol F} = m{\boldsymbol a}. Using Newton's law of gravitation, we find that
{\boldsymbol F} = -\frac{GMm}{r^2}\hat{\boldsymbol e}_1 = -\frac{\mu m}{r^2}\hat{\boldsymbol e}_1where \mu=GM is the gravitational parameter. This results in the equations of motion for an orbiting satellite[1],<br />
\begin{align*}<br />
\ddot{r} - r\dot{\theta}^2 &= -\frac{\mu}{r^2} \\<br />
r\ddot{\theta} + 2\dot{r}\dot{\theta} &= 0 <br />
\end{align*}<br />
[1] More specifically, this is the position of the satellite's mass center. For a rigid body, you can attach a third reference frame to the body and use various attitude coordinatizations to fully define the pos and attitude of the satellite.
