Name: 贺一珺
Student Number: 2014302290002

Question

• 4.9 Orbits that starts out as nearly circular will remain almost stable for a longer period than those that are highly elliptical. Investigate this by studying orbits with the same value of beta(say, beta=2.05) and comparing the behavior with different values of the ellipticity of the orbit.
• 4.10 Calculate the precession of the perihelion of Mercury, following the approach described in this section.
• 4.11 Investigate how the precession of the perihelion of a planet's orbit due to general relativity varies as a function of the eccentricity of the orbit. Study the precession of different elliptical orbits with different eccentricities, but with the same value of the perihelion. Let the perihelion have the same value as for Mercury, so that you can compare it with the results shown in this section.

Abstract

In last chapter we investigated the phenomenion of chaos and in this chapter we turn to a totally different topic. I will put my eye in the Solar system. Yhah! In this article I use Euler-Cromer method to investigate the motion of planets in Solar system. Also, I must stress that the problem I solve in this homework is far from the reality since it's too hard to take all factors into consideration. I will show you the beautiful trajectory of planets moving around the sun. Tricks are used also to geretee the effectivness of the progrem. Let's get started now !!

Background

Solar system

The Solar System is the gravitationally bound system comprising the Sun and the objects that orbit it, either directly or indirectly. Of those objects that orbit the Sun directly, the largest eight are the planets, with the remainder being significantly smaller objects, such as dwarf planets and small Solar System bodies. Of the objects that orbit the Sun indirectly, the moons, two are larger than the smallest planet, Mercury.
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.


Perihelion precession of Mercury

Under Newtonian physics, a two-body system consisting of a lone object orbiting a spherical mass would trace out an ellipse with the spherical mass at a focus. The point of closest approach, called the periapsis (or, because the central body in the Solar System is the Sun, perihelion), is fixed. A number of effects in the Solar System cause the perihelia of planets to precess (rotate) around the Sun. The principal cause is the presence of other planets which perturb one another's orbit. Another (much less significant) effect is solar oblateness.

Perihelion precession of Mercury

Plotting

Here is my code.

• Problem 4.9
  I will show you the situation when beta=2.05, 2.10 and compare them with the condition when beat=2.00. I will show the trejectory under various initial speed to see if they are stable.
  Now I will show you the trejectory when beta=2 & 2.1
  Initial velocity=2pi

  
  trejectory
  

  Initial velocity=2.25pi

trejectory

Initial velocity=2.5pi

trejectory

Initial velocity=2.75pi

trejectory

Initial velocity=3pi

trejectory

Now I will show you the trejectory when beta=2 & 2.05
Initial velocity=2pi

trejectory

Initial velocity=2.25pi

trejectory

Initial velocity=2.5pi

trejectory

Initial velocity=2.75pi

trejectory

Initial velocity=3pi

trejectory

Now we can conclude that when initial velocity is 2pi, the trejectory is a circle. When initial velocity is 2.5pi, the planet begins its perihelion. When initial velocity is 2.75pi, the velocity of the rotating of the trejectory begins to increase.

Let's get further now! Now about the situation when we set beta to be 3? Let's do it!

trejectory trejectory trejectory trejectory

In this situation it's interested to see that something unusual happens. The orbits are distorted! It can be easily seen by comparing with the situation when beta=2.

• Problem 4.10
  Here is the trejectory of Mercury.

  
  trejectory

• Problem 4.11
  To investigate the perihelion of a planet under different, I will show you the precession velocity versus eccentricity.

Here is the result. It must have some problems since the curve is not smooth. But I can't find it out. Sorry...

Conclution

I investigated several situation in Solar system in this homework, but as I said before, it's far from perfection.

Acknowledgement

[1]wikipedia
[2]Yuqiao Wu(吴雨桥)
[3]Prof. Cai

How to contact me

• Wechat ID: bestsola
• Phone number: 18827628190
• E-mail: 2014302290002@whu.edu.cn