计算物理第11次作业

Name: 贺一珺
Student Number: 2014302290002

Question

-4.19 Study the behavior of our model for Hyperion for different initial conditions. Estimate the Lyapunov exponent from calculations of Δθ, such as those shown in Figure 4.19. Examine how this exponent varies as a function of the eccentricity of the orbit.

-4.20 Our results for the divergence of the two trajectories θ1(t) and θ2(t) in the chaotic regime, shown on the right in Figure 4.19, are complicated by the way we dealt with the angle θ. In Figure 4.19 we followed the practice employed in Chapter 3 and restricted θ to the range -π to +π, since angles ouside this range are equivalent to angles within it. However, when during the course of a calculation the angle passes out of this range and is then 'reset' (by adding or subtracting 2π), this shows up in the results for Δθ as a discontinuous (and distrcting) jump. Repeat the calculation of Δθ as in Figure 4.19, but do not restrict the value of θ. This should remove the large (Δθ ～ 2π) jumps in Δθ in Figure 4.19, but the smaller and more frequent dips will remain. What is the origin of these dips? Hint: Consider the behavior of a pendulum near one of its turning points.

Abstract

In last homework I investigate the trajectory of planets and the precession of the Perihelion of Mercury. Today I will try to study the model of Hyperion for different initial conditions and redo the calculation in Figure 4.19. The problem of Hyperion is stimulating and it can be seen as another way to chaos. Since the motion of asteroids located near the Kirkwood gaps is believed to be chaotic, you may wanna stimulate that motion. However, it's not that easy to do that. Fortunately, the motion of Hyperion is chaotic too and that is accessible for us. It's extremely small and we can make use of this character to begin our study. Now let me introduce you into the world of Hyperion!

Background

Constellation_Fornax,_EXtreme_Deep_Field

Three-body problem

In physics and classical mechanics, the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then determining the motions of the three bodies, in accordance with the laws of classical mechanics (Newton's laws of motion and of universal gravitation). The three-body problem is a special case of the n-body problem.
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:

Equations

Hyperion

Hyperion, also known as Saturn VII (7), is a moon of Saturn discovered by William Cranch Bond, George Phillips Bond and William Lassell in 1848. It is distinguished by its irregular shape, its chaotic rotation, and its unexplained sponge-like appearance. It was the first non-round moon to be discovered.

Hyperion_false_color

The Voyager 2 images and subsequent ground-based photometry indicated that Hyperion's rotation is chaotic, that is, its axis of rotation wobbles so much that its orientation in space is unpredictable. Its Lyapunov time is around 30 days. Hyperion, together with Pluto's moons Nix and Hydra, is among only a few moons in the Solar System known to rotate chaotically, although it is expected to be common in binary asteroids. It is also the only regular planetary natural satellite in the Solar System known not to be tidally locked.

Hyperion is unique among the large moons in that it is very irregularly shaped, has a fairly eccentric orbit, and is near a much larger moon, Titan. These factors combine to restrict the set of conditions under which a stable rotation is possible. The 3:4 orbital resonance between Titan and Hyperion may also make a chaotic rotation more likely. The fact that its rotation is not locked probably accounts for the relative uniformity of Hyperion's surface, in contrast to many of Saturn's other moons, which have contrasting trailing and leading hemispheres.

Plotting

4.19

Initial angle: 0
Initial speed: 2pi
Here is the code
Here are plots:

theta versus time

speed versus time

speed versus theta

You can see the repid jump in the picture which is due to the progrem's resetting the angle from -pi to pi, and later I will show you the picture without adjustion.

Now I will change the initial speed to 5:
Here is the code

angle versus time

speed versus time

phase plot

Here I will show you the difference of theta:
Here is the code

The difference of angle when initial speed is 2pi:

difference of angle

The difference of angle when initial speed is 5:

difference of angle

It can be erived that the picture is the situation of chaos. We can see from picture that the value increase repidly, which is approximately exponential.

4.20

Here is the code
Now I will remove the adjustion of angle in my code and here I will show you the plots again:

Initial speed = 2pi
Initial angle = 0

Difference of angle

You can see that the extrem value of the curve becomes extremely stable and the picture shows the condition without any phenomenon of chaos.

Initial speed = 5
Initial angle = 0