Exercise_11:The Three-body motio
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}{dt2}=-\frac{GM_S}{r3_{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}{dt2}=-\frac{GM_S}{r3_{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}{dt2}=-\frac{GM_E}{r3_{SE}}\vec r_{SE}-\frac{GM_J}{r^3_{SJ}}\vec r_{SJ})
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.
Paste_Image.pngif we add a time axis, it forms a 3D figure
Paste_Image.png2.When the value of
is 10 times of real mass of Jupiter
Paste_Image.png3D figures:
Paste_Image.pngJupiter 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.png3D graph:
Paste_Image.pngThe 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.png3D
Paste_Image.pngThey 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 Paste_Image.pngThe 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
Paste_Image.png3D figure
Paste_Image.pngUpper 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})
Its motion is also regular and periodic.
If we plot the difference between two calculated results for
We can find that the absolute value of
Paste_Image.pngIts motion is weird and has no regulation that can be found.
Similarly,we plot the difference between two calculated results for
Paste_Image.pngThe 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