Chapter 4 problem 4.16

python.assignment

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name: Li FangYing
number: 2013301020027

Abstract

We consider one of the simplest three-body problems, the Sun, Jupiter and Earth. We know without Jupiter, Earth's orbit will keep stable. Our objective is to observe how much effect the gravitational force from Jupiter has on Earth's motion.

Introduction

Problem with two body interacting through the inverse-square law can be solved exactly, however, if we introduce one more planet, which is known as $three-body$ $problem$, the analytic theory will become much more difficult. In fact, there are much less results even when this kind of problems have been studyed for centuries. What's more, we need to pay attention that three-body problem or more general, the n-body problem, is the true problem of celestial mechanics.

Nummerical Approach

The force between Jupiter and Earth can be described as:

$$F_E,J=\frac{GM_JM_E}{r_{EJ}^2}$$
where $M_J$ is the mass of Juipiter and $M_E$ is the mass of Earth. We assume they are coplanar, thus the gravity Jupiter has on Earth is: $$F_{EJ,x}=-\frac{GM_JM_E(x_e-x_j)}{r_{E,J}^3}$$ We should pay attention to the direction of the force, which points to Juipiter.
Considering the effect of the Sun, the total force on Earth in the x direction is:
$$\frac{dv_{x,e}}{dt}=-\frac{GM_Sx_e}{r^3}-\frac{GM_J(x_e-x_j)}{r_{EJ}^3}$$ And force in the y direction can be written in the same way.

Data analysis

Using parameters appropriate for Earth and Jupiter, we find that both of the planets follow stable circular orbits, which means Jupiter has negligible effect on Earth:

Figure 1: Simulation of a solar system with Jupiter and Earth, Jupiter has its true mass.

Then we mutiply the mass of Jupiter to see what would happen:

Figure 2: The mass of has been set to be 100 times of its true mass

Unexpectedly, the orbit of Earth is generally stable. It seems that Jupiter just give Earth a perturbation. So we set the mass of Jupiter much more larger:

Figure 3:The mass of Jupiter has been set to be 1000 times of its true mass.

This time the orbit of Earth becomes completely unsteady, as it is eventually ejected from the solar system.

Conclusion

Jupiter with its true mass is too small to have a major influence on Earth, only when its mass reach 1000 times or more, it can make Earth's orbit unstable.

program codes

threebody.py

Acknowledgement

Thank 2013301020066 for his guidance of program codes