• 作业 4.16 4.18 4.20

Homework 12 - Three-Body System(Jupiter, Earth, and Sun)

ComputationalPhysics_HW 岳绍圣2013301020033

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CONTENT:

• Homework 12 - Three-Body System(Jupiter, Earth, and Sun)
  • Abstract
  • Background3
  • Three-Body System(Jupiter, Earth, and Sun)
  • Codes
  • Reference

Abstract

This article studied a three-body system made up of the Sun, the Earth and the Jupiter is built to investigate the effect on the Earth's motion introduced by the gravitational force between the Earth and the Jupiter. The mass of the Jupiter is changed to illustrate this influence.

Background3

The problem is to determine the possible motions of three point masses m1 , m2 , and m3 , which attract each other according to Newton's law of inverse squares. It started with the perturbative studies of Newton himself on the inequalities of the lunar motion. In the 1740s it was constituted as the search for solutions (or at least approximate solutions) of a system of ordinary differential equations by the works of Euler, Clairaut and d'Alembert (with in particular the explanation by Clairaut of the motion of the lunar apogee). Much developed by Lagrange, Laplace and their followers, the mathematical theory entered a new era at the end of the 19th century with the works of Poincaré and since the 1950s with the development of computers. While the two-body problem is integrable and its solutions completely understood3, solutions of the three-body problem may be of an arbitrary complexity and are very far from being completely understood.

The following form of the equations of motion, using a force function $U$ (opposite of potential energy), goes back to Lagrange, who initiated the general study of the problem: if $\vec r_i$ is the position of body $i$ in the Euclidean space $E\equiv R^p$ (scalar product $\langle,\rangle\ ,$ norm $||.||$),

$$m_i{d^2\vec r_i\over dt^2}=\sum_{j\not=i}{m_im_j}\frac{\vec r_j-\vec r_i}{||\vec r_j-\vec r_i||^3}=\frac{\partial U}{\partial\vec r_i},\; i=1,2,3,\;\hbox{where}\; U=\sum_{i<j}\frac{m_im_j}{||\vec r_j-\vec r_i||}\cdot$$
Endowing the configuration space $\hat{\mathcal X}=\{x=(\vec r_1,\vec r_2,\vec r_3)\in E^3,\; \vec r_i\not=\vec r_j\;\hbox{if}\; i\not=j\}$ (or rather its closure ${\mathcal X}$) with the mass scalar product $$x'\cdot x''=\sum_{i=1}^3{m_i\langle\vec r'_i,\vec r''_i\rangle}$$ we can write them $${d^2 x\over dt^2}=\nabla U(x),$$ where the gradient is taken with respect to this scalar product. In the phase space $T^*\hat{\mathcal X}\equiv\hat{\mathcal X}\times{\mathcal X}\ ,$ that is the set of pairs $(x,y)$ representing the positions and velocities (or momenta) of the three bodies, the equations take the Hamiltonian form (where $|y|^2=y\cdot y$): $${dx\over dt}={\partial H\over\partial y},\quad {dy\over dt}=-{\partial H\over \partial x}, \quad\hbox{where}\quad H(x,y)={1\over 2}|y|^2-U(x).$$

For simplicity, we set the initial location of each planets as in the rest frame.
As for initial velocity, set it with the magnitude with which the planet circles around the Sun in a stable orbit with no extra influence.
Because
we have

Three-Body System(Jupiter, Earth, and Sun)

The Jupiter, Earth and Sun composed a three-body system, and as a real situation, the sun is not stable, thus we change the mass of the Jupiter to see its effect on the motion of the Earth. The initial condition is and
When the mass of the Jupiter equals $m_J$.

　　　　　　　　　　　　fig.1 Three-Body Simulation with mass of Jupiter=1

Analysis of Fig.1:
The motion of the system is almost stable with three planets circling around the Sun.

When the mass of the Jupiter equals $50m_J$:

　　　　　　　　　　　　fig.2 Three-Body Simulation with mass of Jupiter=50
　　　　　　　　　　　　

Analysis of Fig.3:
we can clearly see that the orbit of Earth and Jupiter are not stable.

Animations are made in the Rest Frame as follows. The variance of the mass of the Jupiter is marked in the figures.
In the Rest Frame: Figure 1.

fig.3 Three-Body Simulation with mass of Jupiter changed dynamically
　　　　　　　　　　　　

Codes

All these codes used can be found in my github.

Reference

1: Cai Hao. https://github.com/caihao/computational_physics_whu.
2: Nicholas J.Giordano, Hisao Nakanishi.Computational physics.清华大学出版社
3: Alain Chenciner (2007) Three body problem. Scholarpedia, 2(10):2111., revision #152224