Simplified Four-Body Problem

Homework
程熹 2013301020038 物基一班

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Abstract

This article discusses the four-body problem by describling the motion of the Sun,Earth,Jupiter and Saturn through computer simulation. By changing the mass of the jupiter and Saturn, we can find some interesting properties about the Four-Body problem.

Key Word

Four-Body Problem,solar system

Introduction

We know that a n-body problem with n larger than 2 can't be solve analytically. However,there are some certain situations that we can solve. Like we have learned, in three-body problem, when the mass of one body is much larger than the other two,the problem can be simplified as two-body problem. In the text book, we solve a three-body system consists of the Sun,the earth and the Jupiter. By taking the center of the mass of the three body system as the origin, we can also simplify the problem, and we find out that the motion of the earth become chaotic.But as the mass of the sun is much larger than the others, its motion will have little influence on the motion of other planets(alhough this "little" inflence make the motion of the other planets become chaotic). So I will add an extral planet,the Saturn,to the orgin system and dicuss their motion with different initial conditions and different masses.

Main Features

In this article,as the mass of the sun is much larger than the other three planets,we assume that the position of the sun wouldn't change with time in all conditions to simplify the problem.That's why the problem is called "simplified Four-Body problem"(just like what we did to the simplified Three-Body problem.And we only discuss the gravitation between the stars,without the relativity effect.
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 $$F_{e,x} =\frac{ -G M_{sun }M_{e}( x_{e} )}{r _ {e} ^{3} }-\frac{ G M_{j }M_{e}( x_{e}- x_{j} )}{r _{ej} ^{3}} - \frac{G M_{s }M_{e}( x_{e}- x_{s} )}{ r _{es} ^{3}}$$ $$F_{e,y} =\frac{ -G M_{sun }M_{e}( y_{e} )}{r _ {e} ^{3} }-\frac{ G M_{j }M_{e}( y_{e}- y_{j} )}{r _{ej} ^{3}} - \frac{G M_{s }M_{e}( y_{e}- y_{s} )}{ r _{es} ^{3}}$$
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.

Results

1.The initail position of the four planets are on X axis, the angular velcocity of the planets are equal.

It's the situation in our solar system where all the three stars moves around the sun with almost round orbits and the sun set in the center of the round.

2.The initial position of the Jupiter and the Saturn are on X axis while the earth is on Y axis, the angular velocity of the planets are same.

The trajectory of the earth become strange while the Jupiter and the Saturn remain the same. And if we set the time t longer.


The trajetory of the earth become even stranger.The earth even hits the sun at last.I think the reason to this consequence is that unlike the first situation, the force between the planets always remain in a line, and the direction of the force is always vertical to its velocity.This time there is an angle between the initial velocity of the earth and the direction of the force on the earth.The periodic motion of the Jupiter and the Saturn make motion of the earth varies largely.

we can see from the trajectory of the earth's motion that the earth firstly move closer and closer to the sun,and after a long time when the earth finally "hit" and "across" the sun, it will begin to leave the sun. So the direction of force on the earth must have changed when the earth "hit" the sun.One more question is: will the earth actually fall into the sun if the initial conditions is changed.

3.The initial position of the four planets are on X axis,but the angular velocity of the three planets are different

The velocity of Saturn is given by

$$v_{s} = 2 \pi \sqrt{ r_{e}/ r_{s} }$$
we can see from the diagram that with the eccentricity become larger with the radius of the planets'orbits become larger.It's easy to understand because the different angular velocity cause the direction of the force on each planets is not vertical to its velocity in most of the time, and the circular motion require the direction of the force is always vertical to its velocity

4.The mass of the Jupiter is 1000 times its true mass and the initial condition is the same as result 1

The motion of the Jupiter as we can predict doesn't change much,as the mass of the Jupiter is on the same scale of the Sun, the force between it and the other two planets is far smaller than it between the sun.On the other hand,as the position of the sun won't change(which is impossible in this case),the motion of the Jupiter is circular,we can combine this two "large" stars to be one star, with the positon on their center of mass,and the trajectory of its motion circular.The other two planets is like doing circular motion around the new "sun". As we can see through the diagram,the orbit of the earth near the orgin can be treated as the combining of two circular motion with each center is on the center of the mass of the Sun and the Jupiter,so does the motion of the Saturn.

5.The mass of the Jupiter is 1000 times its ture mass, and the initial condition is the same as result 2.

In order to find out whether the motion of the earth will change much as it did in result 2, I set a longer time t. However, such a change doesn't happen. Compared to the result 4, the moton of the Earth and the Saturn only change a little angle. I think the reason to this is that the graviton between the new "sun" and the earth is much larger that the gravitaion between the earth and other planets. In the meanwhile, the earth also "lose" the graviton from the Jupiter. So the force between the earth and the new "sun" has become the major factor to its motion. So the different initial between it and the Saturn wouldn't bother much.

6.The mass of the Jupiter is 1000 times its ture mass,and the initial condition is the same as result 3.

Like in the second condition,the trajectory of the Earth and the Saturn have been narrowed,the only difference between this diagram and the diagram in result 3 is that the peroid pattern of the motion of the Earth and the Saturn have changed, they are no longer a circule or ellipse,instead,they are like the combaination of the result 3 and result4.

7.The mass of the Sun is millesimal of its real mass

Same angular velocity

Different angular velocity

Different angular velocity and initial position on different axises

In this condition, the mass of the Sun is millestimal of its real mass,which means that the Sun is on the same level of mass with the other three planets. The force between the Sun and the other planets are no longer much larger that the force between the three planets.Thus,the system have become a three-body system. The only difference is that there is an extral force pointing to the origin on each planet. We know that the free three-body system is chaotic. Each planet's motion is irregular,so I want to find out whether the "three-body system" will be chaotic. However, as you can see from the diagram, no matter how I change the initial condition,the motions of the three planets are still regular.So I believe that the force between each planet and the Sun must have played a important role in violating chaos, although the force is not that strong. It's possible that adding a centripetal force in a chaotic system can make the origin system non-chaos.

Acknowledgement

This article is finished independently.But the idea to simplify a Four-Body system to a Three-Body system is inspired by the classical mechanic textbook.

Reference

Computational Physics(second edition) Nicholas J.Giordano and Hisao Nakanishi