Three-Body system

程熹 2013301020038 物基一班
Homework

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Abstract

This article investigate the three-body system where the sun the earth and the jupiter are including. We set different initial conditions and different mass jupiter to see if there is some interesting results. To achieve in numberical way we use the Euler-Cromer method.

Key Word

three-body system,Newton=law,euler-cromer method.

Introduction

A prominent example of the classical three-body problem is the movement of a planet with a satellite around a star. In most cases such a system can be factorized, considering the movement of the complex system (planet and satellite) around a star as a single particle; then, considering the movement of the satellite around the planet, neglecting the movement around the star. In this case, the problem is simplified to two instances of the two-body problem. However, the effect of the star on the movement of the satellite around the planet can be considered as a perturbation.
A three-body problem also arises from the situation of a spacecraft and two relevant celestial bodies, e.g. the Earth and the Moon, such as when considering a free return trajectory around the Moon, or other trans-lunar injection. While a spaceflight involving a gravity assist tends to be at least a four-body problem (spacecraft, Earth, Sun, Moon), once far away from the Earth when Earth's gravity becomes negligible, it is approximately a three-body problem.
The general statement for the three body problem is as follows. At an instant in time, for vector positions x and masses m, three coupled second-order differential equations exist.
n the circular restricted three-body problem, two massive bodies move in circular orbits around their common center of mass, and the third mass is negligible with respect to the other two.With respect to a rotating reference frame, the two co-orbiting bodies are stationary, and the third can be stationary as well at the Lagrangian points, or orbit around them, for instance on a horseshoe orbit. It can be useful to consider the effective potential.

Main Features

According to Newton's law of gravitation, the magnitude of the gravitational force between two objects is given by

$$F_{12} = \frac{G M_{1 }M_{2}} {r _{12} ^{2}}$$
We assume that the orbits of the two planets are coplanar,and we take this to be the xy-plane.Writing F in terms of components we have $$F_{12,x} =\frac{-G M_{1 }M_{2}( x_{1}- x_{2} )}{ r _{12} ^{3} }$$ $$F_{12,y} =\frac{-G M_{1 }M_{2}( y_{1}- y_{2} )}{r _ {12} ^{3}}$$
we take account for the total force on the earth $$F_{e,x} =\frac{ -G M_{sun }M_{e}( x_{e} )}{r _ {e} ^{3} }-\frac{ G M_{j }M_{e}( x_{e}- x_{j} )}{r _{ej} ^{3}}$$ $$F_{e,y} =\frac{ -G M_{sun }M_{e}( y_{e} )}{r _ {e} ^{3} }-\frac{ G M_{j }M_{e}( y_{e}- y_{j} )}{r _{ej} ^{3}}$$
with $$r_{e} = \sqrt{ x_{e} ^{2}+y_{e} ^{2} }$$ $$r_{ej} = \sqrt{ (x_{e}-x_{j}) ^{2}+(y_{e}-y_{j}) ^{2} }$$ $$r_{es} = \sqrt{ (x_{e}-x_{s}) ^{2}+(y_{e}-y_{s}) ^{2} }$$
combining with Newton's second law we get the acceleration of the earth in x and y direction.Then we use the Euler-Cromer method to translate the equations into equations suitable to design program $$v_{e,x}(i+1)= v_{e,x}(i)- \frac{4 \pi ^{2} x_{e}(i) }{ r_{e } ^{3} } \Delta t - \frac{4 \pi ^{2} ( M_{j}/ M_{sun}) [x_{e}(i) - x_{j}(i)] }{ r_{e j} ^{3} } \Delta t-\frac{4 \pi ^{2} ( M_{s}/ M_{sun}) [x_{e}(i) - x_{s}(i)] }{ r_{e s} ^{3} } \Delta t$$ $$v_{e,y}(i+1)= v_{e,y}(i)- \frac{4 \pi ^{2} y_{e}(i) }{ r_{e } ^{3} } \Delta t - \frac{4 \pi ^{2} ( M_{j}/ M_{sun}) [y_{e}(i) - y_{j}(i)] }{ r_{e j} ^{3} } \Delta t-\frac{4 \pi ^{2} ( M_{s}/ M_{sun}) [y_{e}(i) - y_{s}(i)] }{ r_{e s} ^{3} } \Delta t$$
where we use the astronomical unit here.
Using the same way, we can describle the change of the position and velocity of the Jupiter and the Saturn.
Then we can choose different initial conditions(different initial positions and velocity of these stars) to get different results.

Result

1.The mass of the jupiter is nomal with the sun's initial velocity is 0.

2.The mass of the jupiter is 10 times its mass with the sun's initial velocity is 0.

3.The mass of the jupiter is 100 times its mass with the sun's initial velocity is 0.

4.The mass of the jupiter is 1000 times its mass with the sun's initial velocity is 0.

5.The mass of the jupiter is normal with the sun's initial velocity is not 0.

6. The mass of the jupiter is 10 times its mass with the sun's initial velocity is not 0.

7. The mass of the jupiter is 100 times its mass with the sun's initial velocity is not 0.

8. The mass of the jupiter is 1000 times its mass with the sun's initial velocity is not 0.

Conclusion

1. From these diagram we will find that when the Jupiter is at the normal mass, the motion of the sun won't be affectted much by the other two planets, and the earth and the Jupiter will continue to circle around the sun.But when we increase the mass of the jupiter, the motion of the sun began to grow intense. At the beginning, the sun is moving in a straight line, but as the mass of the jupiter increase, the sun began to do circle motion,with the radius become larger and larger.
2. From the diagram 5, diagram 6, diagram 7 and diagram 8, we will find that when the jupiter's is growing bigger, the influence its initial velocity to the future motion of the sun will become greater.As you can see, the larger the jupiter's mass is, the more vertical the diagram will be.

Acknowledgement

This program is finished by myself. But I have refer some contents from last report of my own in the Main Feature part.

Reference

Computational Physics(second edition) Nicholas J.Giordano and Hisao Nakanishi
Wikipedia Three body problem
https://en.wikipedia.org/wiki/Three-body_problem