Two Body Equation

Consider the two body problem where the Moon orbits the Earth. To be general, let’s take the coordinate system \(S\) where \(\vec r_1\) is the distance boldsymboltor to the Earth and \(\vec r_2\) is the distance boldsymboltor to the Moon. The boldsymboltor between the Earth and Moon is \(\Delta \vec r\)

Let’s consider the geocentric coordinate system \(S'\) (centered at the Earth) where the gravitational force acting on the Moon by the Earth is given by

\[ F_g = -\frac{G m_1 m_2}{r^2}~ \hat r \]

• \(\vec r\) :The boldsymboltor between the Earth and Moon. Naturally, it is equivalen to \(\vec r = \Delta \vec r\)

• \(r\) : The distance betwen the Earth and Moon, \(r = \lvert{\Delta r}\rvert\)

The force of gravity felt by each body is then,

\[ F_{g1} = -\frac{Gm_1m_2}{r^2} \hat r = m_1 \ddot{\vec{r}}_1 \]

\[ F_{g2} = \frac{Gm_1m_2}{r^2} \hat r = m_2 \ddot{\vec r}_2 \]

The relative acceleration seen in \(S'\) is \(\ddot{\vec r}\) given by,

\[ \ddot{\vec{r}} = - \frac{G(m_1 + m_2)}{r^2} \hat r \]

Defining the gravitational parameter \(\mu = G(m_1 + m_2)\), we end up with the Two Body Equation.

\[ \ddot{\vec{r}} = -\frac{\mu}{r^2} \hat r \]