Chapter 4 Problem 4.18: The Kirkwood Gap

计算物理

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Name:陈锋
Student Number:2013301020145
May 14,2016

Abstract

The article is focus on the gap width of Kirkwood gap especially for the case of ratio equals to 2

Introduction

A Kirkwood gap is a gap or dip in the distribution of the semi-major axes (or equivalently of the orbital periods) of the orbits of main-belt asteroids. They correspond to the locations of orbital resonances with Jupiter.
For example, there are very few asteroids with semimajor axis near 2.50 AU, period 3.95 years, which would make three orbits for each orbit of Jupiter (hence, called the 3:1 orbital resonance). Other orbital resonances correspond to orbital periods whose lengths are simple fractions of Jupiter's. The weaker resonances lead only to a depletion of asteroids, while spikes in the histogram are often due to the presence of a prominent asteroid family.

Figure 1:Histogram showing the four most prominent Kirkwood gaps.
The gaps were first noticed in 1866 by Daniel Kirkwood, who also correctly explained their origin in the orbital resonances with Jupiter while a professor at Jefferson College in Canonsburg, Pennsylvania.

The motion of Asteroid number 2

Firstly, I consider the equations of motion of the asteroid. Actually, it is just the same as we discussed in the pervious problem -- the sun, earth, jupiter system. For convenience, I set the mass of asteroid is one.

Figure 2: The left figure is the orbit of the asteriod number 2 whose radius is 3.276AU. The right one is the fine structure of the orbit. The total simulation time of this programme is twenty years.

The resonance of Asteriod number 2

In this part, I consider the resonance of asteroid number 2. Firstly, I will show you the long-term simulation result.(750 years).

Figure 3: The curve is the distance distribution of the asteroid. (The distance is defined as the distance between the sun and the asteroid.) Thus, in this situation, the orbit varies from a circular orbit to an elliptical one. If we define the distance between the furthest aphelion and nearest perhelion as the resonance amplitude, the magnitude of which is about 0.5 AU. This is a dramatically strong resonance.

By contrast, if the initial radius is 3.100 AU, the resonance will vanish.

Figure 4: The figure shows the situation of this case. The resonance amplitude is about 0.07AU, which is much smaller than the previous case. Thus, the asteroid is in the quasi-circular orbit.

The mean orbital Energy

In order to analyze this phenomenon, we introduce orbital energy, which is defined:

$$\begin{equation} E_o = \frac{p^2}{2m}-G\frac{M_sM_a}{r_a}-G\frac{M_jM_a}{r_{aj}} \end{equation}$$
The minus orbital energy means the object is in a circular orbit or elliptical one.

Figure 5: The left one is the orbital energy of initial radius equal to 3.1AU and the right one is that of initial radius equal to 3.276AU (the resonance case). Therefore, if the asteroid is not in the resonance orbit, its mean orbital energy is almost a constant. In the resonance case, the mean orbital energy will fluctuate. The is the consipicuous feature of resonance.
Thus, I use the fast Fourier transformation to figure out this frequency.

Figure 6: Obviously, the low frequency part is which we want to make a discussion.

The width of the resonance

In this section, I will use the that feature of resonance to find out the width of Kirkwood gap. Figure 6 tells us that when the initial radius is smaller 3.22AU or larger than 3.42AU, the fluctuation of mean orbital energy can be ignored.

Figure 7: The left figure shows the distance distribution of case of 3.22AU and 3.42AU. The right one shows the fluctuation of mean orbital energy of both cases.

Conclusion

This is one way to find out the gap width. Apparently, there are many methods to work out this problem. For example, we can set a large amount asteroids in that area and simulation their motions. After a long simulation time, we count the number of asteroid in each orbit.

Reference

1.Wikipedia contributors. "Kirkwood gap." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 28 Feb. 2016. Web. 14 May. 2016.
2.Giodano, N.J., Nakanishi, H. Computational Physics. Tsinghua University Press, December 2007.

Programme Code

-Chapter4 Problem 4.18