Exercise 10: The Solar System

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We will discuss a simple modle —— Earth-Sun system. Change the initial velocity and time step. Study the differeces in the senstivity to increasing time step.

Background and Introduction

The Solar System formed 4.6 billion years ago from the gravitational collapse of a giant interstellar molecular cloud. The vast majority of the system's mass is in the Sun, with most of the remaining mass contained in Jupiter. The four smaller inner planets, Mercury, Venus, Earth and Mars, are terrestrial planets, being primarily composed of rock and metal. The four outer planets are giant planets, being substantially more massive than the terrestrials. The two largest, Jupiter and Saturn, are gas giants, being composed mainly of hydrogen and helium; the two outermost planets, Uranus and Neptune, are ice giants, being composed mostly of substances with relatively high melting points compared with hydrogen and helium, called volatiles, such as water, ammonia and methane. All planets have almost circular orbits that lie within a nearly flat disc called the ecliptic.

Here I found a 3D simulation of solar system

Content

• Problem 4.1
  Investigate the results obtained from the planetary motion program with different values of the time step. Show that for simulations of Earth, time steps greater than about △t=0.01 yr do not lead to satisfactory results. For such large time steps the orbits are not stable(i.e., not repeating). This is in accord with our general rule of thumb that the time step should be no larger than 1 percent or so of the characteristic time scale of the problem. In this case the characterisctic time scale is period of one orbit. Now use other initial speeds, bith larger and smaller than the value $2\pi$ which was needed for a circular orbit and again study the stability of the elliptical orbit to different values of the time step. Do you see differences in the sensitivity to increasing time steps.
• Earth orbiting the Sun
  According to Newton's law of gravitation:
  $$F_G=\frac{GM_SM_E}{r^2}$$
  We have:
  $$\frac{d^2x}{dt^2}=\frac{F_{G,x}}{M_E}$$
  $$\frac{d^2y}{dt^2}=\frac{F_{G,y}}{M_E}$$
  Use Eular-Cromer method and change the units to astronamical units
  $$v_{x,i+1}=v_{x,i}-\frac{4\pi^2x_i}{{r_i}^3}△t$$
  $$x_{i+1}=x_i+v_{x,i+1}△t$$
  $$v_{y,i+1}=v_{y,i}-\frac{4\pi^2y_i}{{r_i}^3}△t$$
  $$y_{i+1}=y_i+v_{y,i+1}△t$$
  Figure
  
  Click 'Figure' to get code.
  Then I make time step larger.
  △t=0.04:
  
  △t=0.08:
  
  Then I take $v_y=4$ △t=0.01:
  
  We can see that the elliptical orbit is quiet unstable.
  Then take different values of △t：
  
  Finally, I take $v_y=8$ △t=0.01. Because the initial velocity changes, I change the total time to 6 years.
  
  For different value of △t
  

Conclusion

From the result we can see that the time step should be no longer 1 percent of the total time. The smaller time step I take, the more precise the orbit is.