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:

![](http://latex.codecogs.com/png.latex?\frac{d^2 \vec r}{dt^{2}=-\frac{GM_S}{r}3_{ES}}\vec r_{ES}-\frac{GM_J}{r^3_{EJ}}\vec r_{EJ})
![](http://latex.codecogs.com/png.latex?\frac{d^2\vec r_J}{dt^{2}=-\frac{GM_S}{r}3_{JS}}\vec r_{JS}-\frac{GM_E}{r^3_{JE}}\vec r_{JE})
![](http://latex.codecogs.com/png.latex?\frac{d^2\vec r_S}{dt^{2}=-\frac{GM_E}{r}3_{SE}}\vec r_{SE}-\frac{GM_J}{r^3_{SJ}}\vec r_{SJ})

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.

Paste_Image.png

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

Paste_Image.png

2.When the value of

is 10 times of real mass of Jupiter

Paste_Image.png

3D figures:

Paste_Image.png

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,

Paste_Image.png

3D graph:

Paste_Image.png

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,

Paste_Image.png

3D

Paste_Image.png

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.

Paste_Image.png

Exercise_11:The Three-body motion_第10张图片

Paste_Image.png

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

Exercise_11:The Three-body motion_第11张图片

Paste_Image.png

3D figure

Exercise_11:The Three-body motion_第12张图片

Paste_Image.png

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

![](http://latex.codecogs.com/png.latex?\dot{v_x}=-\frac{4 \pi ^2 x}{r^3})

![]( http://latex.codecogs.com/png.latex?\dot{v_y}=-\frac{4\pi^2 y}{r^3})

![]( http://latex.codecogs.com/png.latex?\dot{\omega} \approx \frac{12\pi ^2}{r5}(xsin\theta -ycos\theta)(xcos\theta+ysin\theta))

Use Euler-Cromer method to solve this equation,

code3

When

its orbits is a circle.

Exercise_11:The Three-body motion_第13张图片

Paste_Image.png

Its motion is also regular and periodic.

If we plot the difference between two calculated results for

with different initial conditions

![]( http://latex.codecogs.com/png.latex?\Delta \theta -t)

Exercise_11:The Three-body motion_第14张图片

Paste_Image.png

We can find that the absolute value of

is always smaller thean 0.01. It's not chaonic.Nevertheless,when

![](http://latex.codecogs.com/png.latex?v_0=2\pi +1)
,#### it's orbit is an ellipse,

Exercise_11:The Three-body motion_第15张图片

Paste_Image.png

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

Similarly,we plot the difference between two calculated results for

with different initial conditions

![]( http://latex.codecogs.com/png.latex?\Delta \theta -t)

Exercise_11:The Three-body motion_第16张图片

Paste_Image.png

The absolute value of

![](http://latex.codecogs.com/png.latex?\Delta \theta )

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

Exercise_11:The Three-body motion

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,

1.when the value of

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

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

2.When the value of

is 10 times of real mass of Jupiter

3D figures:

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,

3D graph:

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,

3D

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.

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

3D figure

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

Use Euler-Cromer method to solve this equation,

When

Its motion is also regular and periodic.

We can find that the absolute value of

is always smaller thean 0.01. It's not chaonic.Nevertheless,when

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

Similarly,we plot the difference between two calculated results for

with different initial conditions

The absolute value of

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

Conclusions

Acknowledgements

Thanks for 郭潇学长's code and pictures.

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