计算物理第十一次作业

作者：夏海峰 学号：2013301020094

用数值计算的方法模拟计算行星运行

背景

• 理论基础：
  万有引力定律：万有引力定律是艾萨克·牛顿在1687年于《自然哲学的数学原理》上发表的。牛顿的普适的万有引力定律表示如下： 任意两个质点有通过连心线方向上的力相互吸引。该引力大小与它们质量的乘积成正比与它们距离的平方成反比，与两物体的化学组成和其间介质种类无关。
  

  $$F=\frac {G m_1 m_2} {^2}$$
  *当中心天体的质量不可看做足够大时：
  应采用约化质量：$\mu= \frac {m_1 m_2} {m_1+m-2}$
  并可将轨道运动表示为如下形式：
  $$r=\frac {l}{1-e\cos\theta}$$

• 作业4.7
  Consider a hypothetical solar system consisting of a sun and one planet in which the mass Of the sun is nott much greater than the m ass of the planet, Now you must allow for the motion of both the planet and the sun. Extend your planetary motion progratn to include this effect. You will have to deal with a set of equations such as those in (4.7) for both objects. Investigate the possible types of orbital motion found in such a system. Begin with a double star system in which the two objects are of equal mass. Then explore the behavior when the unequal. Hint: in order to obtain the simplest orbits, it is best to pick initial conditions are such that, the total linear is zero. While this problem can be handled with a stationary sun together with thc concept of a reduced mass， this calculation is all ecessary prelude to the study of orbits of planets in binary star systems， which we Will sonsider in a later exercise.

• 双星系统轨迹如下：

• 轨迹图如下图所示：
  程序
  
• 动态图片显示如下：
  （图片较大可能加载较慢）
  程序

• 若将万有引力改为$\frac{1}{r^{2+\delta x}}$
  

  $$F_G=\frac {GM_1M_2}{r^{2+\delta x}}$$
  可观察到轨迹不闭合的现象：
  (1): $$F_G=\frac {GM_1M_2}{r^{2.1}}$$
  
  (2): $$F_G=\frac {GM_1M_2}{r^{2.2}}$$
  

• 由以上两图可看出，当改变了万有引力的平方反比关系后，行星运动轨迹会出现不闭合的现象。

结论

• 当中心天体质量没有远大于其他天体时，可以明显地观察到其运动轨迹。
• 由以上两图可看出，当改变了万有引力的平方反比关系后，行星运动轨迹会出现不闭合的现象。