Abstract

This article will discuss some problems in solar system again, including three-body motion, resonance of asteroids in the gap and chaotic tumbling of Hyperion.

Three-body motion

Problem 4.16

Three-body motion equations:


Use Euler-Cromer method to solve this equation,

code1

1.when the value of

is real mass of Jupiter,both of the planets follow stable circular orbits.

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if we add a time axis, it forms a 3D figure

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2.When the value of

is 10 times of real mass of Jupiter

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3D figures:

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Jupiter has a little effect on Earth and Sun will move and its orbit is a small circle.

3.When the value of MJ is 100 times of real mass of Jupiter,

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3D graph:

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The orbit of Earth will precess obviously and forms beautiful pattern.Sun also distinctly moves along a circular orbit.

3.

When the value of MJ is 1000 times of real mass of Jupiter,

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3D

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They can't form a bound state Jupiter will be far away from Earth and Sun,leaving Earth rotating around Sun. If we decrease the initial speed of Jupiter, which can make Jupiter and Sun form a binary system. Let the initial speed of Jupiter become two-thirds of that value before.

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The trajectory of Earth are chaotic.

2.Resonance

Similarly, we research the effect of Jupiter on asteroid whose period is a half of period of Jupiter

code2

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3D figure

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Upper part and lower part of the trajectory of asteroid are somewhat thicker than other part of it, which demonstrates the resonance effect of Jupiter on asteroid.

3.Hyperion

Problem4.20

Equation


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Its motion is also regular and periodic.

If we plot the difference between two calculated results for

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We can find that the absolute value of

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Its motion is weird and has no regulation that can be found.

Similarly,we plot the difference between two calculated results for

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The absolute value of


is a larger and larger in general,thus the motion of this system can't be predicted.

Conclusions

•The tracjectory of one of three-body can be chaotic.
•The resonance effect of Jupiter on asteroid may be the reason that make that there is no asteriod in the 2/1 gap.
•In the problem of Hyperion, if the orbit of Hyperion is circular, its rotation is regular and periodic; if the orbit of Hyperion is elliptical, its rotation is chaotic and unpredicted

Acknowledgements

Thanks for 郭潇学长's code and pictures.