HOMEWORK11——Two-body System

homework

author ： 2013301020086 Yuqi Wang

1.Abstract

In this paper I studied the two-body system under the Newton's framework. I explored the behaviers of the system in the case where the two celestial bodies have the same and different mass, respectively. In both condition the orbitss can be circles and ellipses.

2.Introduction

The law of gravity written by Isaac Newton is:

$$\vec F_G=\frac{Gm_1m_2\vec r}{r^3}$$
where $G$ is the gravitational constant, and $\vec r$ is the displacement from $m_1$ to $m_2$.
The potential energy between $m_1$ and $m_2$ is:
$$V(r)=-\frac{Gm_1m_2}{r}$$

3.Two-body system

This is the complete code
We can choose a coordinate system where the centroid of the system is always localts at the origin, thus the momentums of the two bodies are in the opposite direction with the same magnitude, which means the orbits of the two bodies are in the same flat.

Provided the masses of the two bodies are the same, they are in the same orbit when $\frac{Gm_1m_2}{r_0^2}=\frac{p_0^2}{m_1r_1}$, where $p_0$ is the magnitude of the momentums of the two bodies, and $r_1$ is the length between $m_1$ and the centroid.

If the masses of the two bodies are different, they will in different circle orbits with the length between $m_1$, $m_2$ and the centroid are $r_1,r_2$ respectively, and $\frac{r_1}{r_2}=\frac{m2}{m1}$.

The total energy of the system is:

$$E=\frac{p_0^2}{2m_1}+\frac{p_0^2}{2m_2}-\frac{Gm_1m_2}{r_0}$$
When $E$ is lower than, equal to, and higher than $0$, the orbits will be ellipses, parabola, and hyperbola, respectively. Here I only plot ellipse.

The masses of the two bodies are the same:

The masses of the two bodis are different:

If we don't choose the centroid system, the physics is nothing different but the system has a overall momentum.