第十二次作业

姓名：周辉
班级：天眷班
学号：2013301020171

摘要

The problem of two objects interacting through the inverse-square law can be exactly, leading to Kepler's laws. However, if we add just one more planet to give what is known as the three-body problem, an analytic theory becomes much more difficult.We will consider one of the simplest three-body problems, the Sun, the Earth and the Jupiter.

背景

4.16 Carry out a true three-body simulation in which the motions of Earth, Jupiter, and the Sun are all caculated. Since all three bodies are now in motion, it is useful to take the center of mass of the three-body system as the origin, rather than the position of Sun. We also suggest that you give Sun an initial velocity which makes the total momentum of the system exactly zero (so that the center of the mass will remain fixed). Study the motion of Earth with different initial conditions. Also, try increasing the mass of Jupiter to 10, 100 and 1000 times its true mass.
该三体问题中，我们主要来分析木星对地球的影响。我们知道，如果没有木星，地球的轨迹是不会随时间变化的。地球的轨迹得变化因为木星的存在而改变。我们将研究木星的质量变为现在的10,100,1000倍时，地球的轨迹变化。
木星对地球的作用力：
$F_{E,J}=\frac{GM_JM_E}{r_{E,J}^2}$
建立以太阳为原点的x-y正交系：
$F_{EJ,x}=-\frac{GM_JM_E}{r_{EJ}^2}cos\theta_{EJ}=-\frac{GM_JM_E(x_e-x_j)}{r_{EJ}^3}$
$\frac{dv_x,e}{dt}=-\frac{GM_Sx_e}{r^3}-\frac{GM_J(x_e-x_j)}{r_{EJ}^3}$
$F_{EJ,y}=-\frac{GM_JM_E}{r_{EJ}^2}sin\theta_{EJ}=-\frac{GM_JM_E(y_e-y_j)}{r_{EJ}^3}$
$\frac{dv_y,e}{dt}=-\frac{GM_Sy_e}{r^3}-\frac{GM_J(y_e-x_j)}{y_{EJ}^3}$
Euler-Cromer方法处理如下：
$r_e(i)=\sqrt{x_e(i)^2+y_e(i)^2}$
$r_j(i)=\sqrt{x_j(i)^2+y_j(i)^2}$
$r_{EJ}(i)=\sqrt{[x_e(i)-x_j(i)]^2+[y_e(i)-y_j(i)]^2}$
$v_{e,x}(i+1)=v_{e,x}(i)-\frac{4\pi^2x_e(i)}{r_e(i)^3}\Delta{t}-\frac{4\pi^2(M_J/M_S)[x_e(i)-x_j(i)]}{r_{EJ}(i)^3}\Delta{t}$
$v_{j,x}(i+1)=v_{j,x}(i)-\frac{4\pi^2x_j(i)}{r_j(i)^3}\Delta{t}-\frac{4\pi^2(M_E/M_S)[x_j(i)-x_e(i)]}{r_{EJ}(i)^3}\Delta{t}$
$x_e(i+1)=x_e(i)+v_{e,x}(i+1)\Delta{t}$
$y_e(i+1)=y_e(i)+v_{e,y}(i+1)\Delta{t}$
$x_j(i+1)=x_j(i)+v_{j,x}(i+1)\Delta{t}$
$y_j(i+1)=y_j(i)+v_{j,y}(i+1)\Delta{t}$
$t_{i+1}=t_i+\Delta{t}$

正文

代码
1.现实情况，未木星的质量的变化，用作对比

太阳基本静止不动，地球的运动轨迹基本稳定。
2.木星的质量变为原来的十倍

太阳有些许运动，地球的运动轨迹发生变化，轨道的半径有了一定的取值范围。
木星质量变为原来的100倍：

太阳的运动已经很明显，地球的运动轨迹向内部挤压。
木星质量变为原来的1000倍，即与太阳质量相同：

太阳已经不能束缚住木星了，木星飞离地球，地球仍然在绕太阳运动。

总结

在太阳，地球，木星这一三体系统，木星质量的增大最终会导致这一系统瓦解。

致谢

参考书籍：
Nichoulas J.Giordano, hisao Nakanishi.计算物理.清华大学出版社
参照了Markdown数学公式的输入
感谢吴雨桥同学的代码模板