# Two-Body Gravitating System

By

Dr Lim Zhi Han

●

Published 2022-07-28

Let us now apply what we have learnt in an application of differential equation that is perhaps the most famous work of Newton: How does a planet orbit he Sun. We will use a familiar 2-body system, the Sun and the Earth to illustrate.

First fill in the following:

(We can get p_{e} by knowing that the Earth circles around the Sun in about 365 days. How?)

Let {\bf r}_{s} and {\bf r}_{e} be the position vectors of the Sun and Earth respectively. Express {\bf r}_{se} in terms of the two postition vectors.

{\bf r}_{se}=\phantom{{xxxxxxxxx}}

Newton's Law of Gravitation:

{\bf F}=-\frac{GMm}{r^{2}}\hat{{ \bf r }}

Force that the Sun exerts on the Earth:

{\bf F}_{se}=

Force that the Earth exerts on the Sun:

{\bf F}_{es}=

Newton's second law (express in terms of rate of change of momentum)

{\bf F}=\frac{d{\bf p}}{dt}

Applying Euler's scheme for solving differential equation numerically

\begin{align*}
d{\bf p} & ={\bf F}dt\\
{\bf p}_{i+1} & ={\bf p}_{i}+d{\bf p}\\
 & ={\bf p}_{i}+{\bf F}_{i}dt
\end{align*}

Relationship between momentum and position vector is given by {\bf p}=m{\displaystyle \frac{d{\bf r}}{dt}}

Applying Euler's scheme for solving differential equation numerically

\begin{align*}
d{\bf r} & =\\
{\bf r}_{i+1} & ={\bf r}_{i}+d{\bf r}\\
 & =
\end{align*}

Go to Activity 4.