繁忙的平淡

今天读到一篇意大利人写的英文文献, 顿时觉得我国的英语阅读教育太好啦! 以下是三段原文:

In order to avoid any further interference with the geo-stationary protected region, Eq. (1) – specifically the term dealing with solar radiation press...

Title of paper: Dynamics of a particle under the gravitational potential of a massive
annulus: properties and equilibrium


Author of paper:
Eva Tresaco, Antonio Elipe, Andres Riaguas


Magazine:
Celest Mech Dyn Astr (2011) 111:431&n...

男人帮，今年很火的剧，各种搞笑背后，夸张的描述着男人和女人的心思。我很庆幸赵导没有太早的推出来，否则我是不太敢看的，因为男人帮里有太多关于我，关于她的影子。

小闵。在看男人帮之前就看到评论说小闵这个角色有多么的敏感，真正看到了，总是能让我看到另一...

法政先锋3播完了，我是从第一集开始追着看的，昨天终于看完大结局，里面几个人物的设定让我颇有些感慨。

一、布sir

        布sir 是整部剧里唯一的从头到尾保持冷静、理性、客观的男人，这一点让我非常钦佩。他有自己的底线，...

Title of paper:    On quasi-periodic motions around the collinear libration points in the real Earth–Moon system

Author of paper:    X. Y. Hou · L. Liu

Magazine:   Celest Mech Dyn Astr (2011) 110:71–98

1. Background

Since Clark proposed the use of collinear libration points, an upsurge of utilizing these points in space missions arises. The dynamics of the collinear libration points, which are crucial to these missions, have been carefully studied.

Circular restricted three bodies problem(CRTBP) is the most studied model, but it is not a good approximation for some problem in the Solar system, take the earth-moon system for example. Perturbations from the Moons orbit eccentricity and from the Sun are so large that they should be taken into consideration when designing missions in this system.

2. Dynamical substitutes

For real earth-moon system, here 'real' means consider perterbations from main planet and the Sun in the solar system, the three collinear libration points are in fact not equilibrium points anymore, only retaining their geometrical meanings. Special quasi-periodic orbits around them exist. These orbits substitute the geometrical libration points as time-varying equilibrium points. They are often called \({\it dynamical~substitutes}\).

It is easy to write motion equation of small body in Earth-center synodic coordinate. The convert the origin point to collinear libration points, such as \(L_2\). The equation is as follows: $$ {\dot\rho} = \mathbf F_1 + \mathbf F_2 $$ The term \( \mathbf F_1\) can be expanded as a literal series of \(\mathbf{\rho}\) and \(\dot\rho\) being zero when\(\rho,\dot\rho = 0\) holds. The term \(\mathbf F_2\) depends on the motion of major bodies in the solar system except earth, irrelavant to \(\rho,\dot\rho\). Due to the existence of the term \(\mathbf F_2\), the collinear libration point expressed as \(\mathbf\rho,\mathbf{\dot\rho} = 0\) is not an equilibrium point. The term F2 and the coefficients in the literal series of \(\mathbf F_1\) are relevant to the Moon’s motion. They are quasi-periodic under the assumption that the Moon’s motion is quasi-periodic.

Solving the \({\ddot\rho}\) equation, the dynamical substitutes of the three collinear libration points are expressed as: $$ \left\{\begin{aligned} \bar\xi &= \sum_{ijkl}\bar C_{ijkl}\cos\bigg[(i\omega_1 + j\omega_2 + k\omega_3 + l\omega_4)t\bigg] + \bar S_{ijkl}\sin\bigg[((i\omega_1 + j\omega_2 + k\omega_3 + l\omega_4)t\bigg] \\ \bar\eta &= \sum_{ijkl}\bar C'_{ijkl}\cos\bigg[(i\omega_1 + j\omega_2 + k\omega_3 + l\omega_4)t\bigg] + \bar S'_{ijkl}\sin\bigg[((i\omega_1 + j\omega_2 + k\omega_3 + l\omega_4)t\bigg] \\ \bar\zeta &= \sum_{ijkl}\bar C''_{ijkl}\cos\bigg[(i\omega_1 + j\omega_2 + k\omega_3 + l\omega_4)t\bigg] + \bar S''_{ijkl}\sin\bigg[((i\omega_1 + j\omega_2 + k\omega_3 + l\omega_4)t\bigg] \\ \end{aligned}\right. $$ Figure 5 shows the deviation between the analytic solution and the numerically integrated orbit. The initial condition of the integrated orbit is the same as the one used for the analytic solution.

Taking these analytical solutions as initial seeds, the parallel shooting method can be used to generate quasi-periodic orbits which are very close to them.

3. Linearized motions around the dynamical equivalence

Denote the dynamical substitute as \(\bar X = (\bar\xi,\bar\eta,\bar\zeta, \dot{\bar\xi},\dot{\bar\eta},\dot{\bar\zeta})^T\), and the deviation from it as \(\Delta X = X - \bar X = \). It follows: $$ \Delta\dot X = A\Delta X + O(\Delta X) $$ Solving the linear part of the equation, the author have three kinds of solution: the linearized \({\it central, unstable}\) and \(stable\) manifolds. Deviation between the analytic solution of central manifolds and numerical integrated orbit, with the same initial value, are shown in figure 8.

In the circular restricted three-body problem of the Earth–Moon system, linearized central manifolds around the collinear libration points are expressed more simple, its orbit is shown in figure 9. Obviously, the divergence speed is larger than that with author's method.

4. Higher order solutions

4.1 Lissajous and halo orbit

Higher order solutions could be constructed by traditional Lindstedt-Poincare method. Using the analytic solution contain some higher order terms as initial seed, parallel shooting method gives three kinds orbit: \({\it Lissajous~orbit}\) and \({\it halo~orbits}\).

(1) Lissajous orbit.

Deviations between numerical integrated orbit and analytic orbit are shown in figure 10 and Lissajous orbits around \(L_1\) and \(L_2\) are shown in figure 11 and figure 12.

(2) Halo orbit.

Deviations between numerical integrated orbit and analytic orbit are shown in figure 13 and Lissajous orbits around \(L_1\) and \(L_2\) are shown in figure 14 and figure 15.

4.2 Convergence of series

An interesting phenomenon is that the literal expansions of the Lissajous orbit and the halo orbit constructed in this paper are divergent for large amplitude motions. However, for the same amplitude, literal expansions in the CRTBP model are not divergent.

The reason for this phenomenon lies in the resonances between the basic frequencies in the expansions of analytic solutions. Briefly speaking, a set of integers exist such that $$ \omega = i\omega_1 + j\omega_2 + k\omega_3 + l\omega_4 + m\nu_1 \approx 0 $$ In the case of resonances, the well-known small denominator problem will cause the divergence of the literal series.

But for the halo orbit in the CRTBP, there is only one basic frequency, so there are no such problems.

5. Conclusion

The dynamical substitutes of the three collinear libration points in the real Earth-Moon system were computed. They are unstable but with central manifolds around them. For the points \(L_1\) and \(L_2\) , linearized motions around the dynamical substitute were firstly obtained and then higher order analytical solutions of the central manifolds were constructed. Taking the higher order analytical solutions as initial seeds of the parallel shooting method, quasi-periodic orbits including Lissajous orbits and halo orbits were constructed.