Two body problem

In astronomy, two body problems normally refer to the study of the gravitational interaction of two celestial objects.

Classic form: planetary/satellite orbit (point mass approximation)

Approximate a star/planet as point mass $M$, and a planet/satellite orbiting around it as point mass $m$.

The orbit of the planet in the reference frame of the star satisfies the equation:

\ddot{\mathbf{r}} = \mu\frac{\mathbf{r}}{r^3}

Where $\mu=G(M+m)$.

The orbit described by the equation is usually elliptical:

Derivation

Let

$$ \mathbf{r}=\mathbf{r}_m-\mathbf{r}_M $$

be the vector from $M$ to $m$.

Let $r=|\mathbf{r}|$.

Gravity pulls $m$ toward $M$:

$$ m\ddot{\mathbf{r}}_m=-GMm\frac{\mathbf{r}}{r^3} \text{ (1)} $$

Gravity pulls $M$ toward $m$:

$$ M\ddot{\mathbf{r}}_M=+GMm\frac{\mathbf{r}}{r^3} \text{ (2)} $$

Therefore:

$$ \begin{cases} \ddot{\mathbf{r}}_m=-GM\frac{\mathbf{r}}{r^3} \text{ (1)/m} \\ \ddot{\mathbf{r}}_M=+Gm\frac{\mathbf{r}}{r^3} \text{ (2)/M} \end{cases} $$

The relative acceleration of $m$ according to $M$, $\ddot{\mathbf{r}}$, is

$$ \ddot{\mathbf{r}} = \ddot{\mathbf{r}}_m-\ddot{\mathbf{r}}_M $$

Substitute:

$$ \ddot{\mathbf{r}} = -GM\frac{\mathbf{r}}{r^3} - Gm\frac{\mathbf{r}}{r^3} $$

$$ \boxed{ \ddot{\mathbf{r}} = -G(M+m)\frac{\mathbf{r}}{r^3} } $$

Usually, we note

$$ \boxed{ \mu=G(M+m) } $$