In this document we are going to derive the two body equation for a satellite in orbit about earth.

The two body problem is a simplified version of the n-body problem where the gravitational forces of all other bodies are neglected. The two body problem is a special case of the n-body problem where n=2. The two body problem is a classical problem in celestial mechanics and is used to describe the motion of two massive bodies under the influence of their mutual gravitational attraction.

Newton’s law of gravitation gives -

\[ F = \frac{Gm_1m_2}{r^2} \]

where \(F\) is the gravitational force between two bodies, \(m_1\) and \(m_2\) are the masses of the two bodies, \(r\) is the distance between the two bodies, and \(G\) is the gravitational constant.

The gravitational force acting on a body of mass \(m\) due to another body of mass \(M\) is given by -

\[ F = \frac{GMm}{r^2} \]

where \(G\) is the gravitational constant, \(M\) is the mass of the body exerting the force, \(m\) is the mass of the body

The acceleration of the body of mass \(m\) due to the gravitational force is given by -

\[ a = \frac{F}{m} = \frac{GM}{r^2} \]

The acceleration of the body of mass \(m\) is given by -

\[ a = \frac{d^2\bold{r}}{dt^2} \]

where \(\bold{r}\) is the distance between the two bodies and \(t\) is the time.