CN104477411A

CN104477411A - Double-asteroid system periodic orbit searching method based on speed Poincare section

Info

Publication number: CN104477411A
Application number: CN201410854092.XA
Authority: CN
Inventors: 乔栋; 李翔宇; 崔平远; 尚海滨; 王亚敏
Original assignee: Beijing Institute of Technology BIT
Current assignee: Beijing Institute of Technology BIT
Priority date: 2014-12-31
Filing date: 2014-12-31
Publication date: 2015-04-01
Anticipated expiration: 2034-12-31
Also published as: CN104477411B

Abstract

The invention relates to a method for searching the periodic orbit of a double asteroid system based on the velocity Poincaré cross-section, which belongs to the field of aerospace; it includes the following steps: 1. Obtaining the mass and size of the asteroid and performing normalization; 2. Establishing system rotation coordinates 3. Select the search range, select the initial position of the detector along the X-axis direction, gradually increase the initial velocity, and perform orbit integration, and the termination condition is to pass through a predetermined section; then draw the Poincaré map according to the velocity of the termination point; 4 .Select the two initial points where the curve crosses the axis in the map, and re-integrate by taking the average value of the initial state to obtain a new initial point crossing the axis, and iterate repeatedly to obtain an accurate value; 5. Change the initial position of the detector and repeat Steps 3 and 4 until the value covers the selected search area. Compared with the existing methods, the method of the invention has the characteristics of high efficiency, comprehensive search, simple calculation, etc., and is suitable for orbit design of a detector for detecting a double asteroid system.

Description

Periodic orbit search method for double asteroid system based on velocity Poincaré section

技术领域technical field

本发明涉及一种双小行星系统周期轨道的搜索方法，特别涉及一种基于速度庞加莱截面针对双小行星系统的全局周期轨道的搜索方法，属于航空航天技术领域，适用于探测器对双小行星系统进行探测的轨道设计。The invention relates to a search method for the periodic orbit of a double asteroid system, in particular to a search method for the global periodic orbit of a double asteroid system based on the velocity Poincaré section, which belongs to the field of aerospace technology and is suitable for detectors to search for a double asteroid system. Orbit design for detection of asteroid systems.

背景技术Background technique

对小天体的环绕探测和采样返回是小天体探测的重要组成部分。在探测任务的设计中，需要对小行星附近的轨道进行设计。小行星附近的周期轨道具有良好的动力学特性，可以作为理想的探测轨道和着陆轨道的停泊轨道。The orbital detection and sampling return of small celestial bodies are important components of small celestial body detection. In the design of the detection mission, it is necessary to design the orbit near the asteroid. Periodic orbits near asteroids have good dynamic characteristics and can be used as ideal parking orbits for detection orbits and landing orbits.

双小行星系统是一类特殊的小行星系统，有两颗小行星绕着共同的质心旋转，类似于地球-月球系统的形式，可以采用三体系统类似的模型来研究探测器的运动。但与行星—卫星的形式不同，两颗小行星的质量较接近，质量比较大，同时小行星的尺寸与两小行星间的距离比值也较大，无法将其当作质点来考虑。The double asteroid system is a special type of asteroid system. There are two asteroids revolving around a common center of mass, which is similar to the Earth-Moon system. A model similar to the three-body system can be used to study the motion of the probe. But different from the planet-satellite form, the masses of the two asteroids are relatively close, and the mass is relatively large. At the same time, the ratio of the size of the asteroid to the distance between the two asteroids is also large, so it cannot be considered as a particle.

目前对于小行星附近周期轨道的搜索方法中，在先技术【1】(参见Richardson D L.Analytic construction of periodic orbits about thecollinear points[J].Celestial mechanics,1980,22(3):241-253.)给出了三体系统中平衡点附近的周期轨道搜索方法，在系统平衡点处施加小扰动，根据线性化的方程给出扰动量随时间的变化关系，给出平衡点附近周期轨道的初值，根据变量替换可以得到方程的三阶解析解，利用解析解作为初值，通过微分修正可以得到精确的周期轨道。利用该方法可以得到三体系统内共线平动点附近的Halo轨道，李萨如轨道，平面和垂直李雅普诺夫轨道，该方法可以较为精确的获得周期轨道，且适用于不同的质量比，但只能用于平衡点附近的周期轨道搜索，当扰动远离平衡点时，该方法无法得到周期轨道，且无法找到系统全局的周期轨道，解析表达式计算复杂。系统内的多圈周期轨道也无法通过该方法得到。Among the current search methods for periodic orbits near asteroids, the prior art [1] (see Richardson D L. Analytic construction of periodic orbits about the collinear points [J]. Celestial mechanics, 1980,22(3):241-253. ) gives the periodic orbit search method near the equilibrium point in the three-body system. A small disturbance is applied at the equilibrium point of the system, and the relationship of the disturbance with time is given according to the linearized equation, and the initial periodic orbit near the equilibrium point is given. value, according to the variable substitution, the third-order analytical solution of the equation can be obtained, using the analytical solution as the initial value, and the accurate periodic orbit can be obtained through differential correction. Halo orbits, Lissajous orbits, plane and vertical Lyapunov orbits near the collinear translation point in the three-body system can be obtained by using this method. This method can obtain periodic orbits more accurately and is suitable for different mass ratios. However, it can only be used to search for periodic orbits near the equilibrium point. When the disturbance is far away from the equilibrium point, this method cannot obtain periodic orbits, and cannot find the global periodic orbit of the system, and the calculation of the analytical expression is complicated. The multi-turn periodic orbit in the system cannot be obtained by this method.

在先技术【2】(参见Yu Y,Baoyin H.Generating families of 3Dperiodic orbits about asteroids[J].Monthly Notices of the RoyalAstronomical Society,2012,427(1):872-881.)给出了单体小行星旋转固连系下的周期轨道搜索方法，在本体固连系下采用庞加莱界面对穿过的轨道进行分析，考虑穿过平面位置接近的点进行微分修正，获得精确的周期轨道，运用该方法可以搜索到小行星216的多族周期轨道，但这类轨道仅适用于因小行星形状不规则而产生在本体固连系下的轨道搜索，对于双小行星系统并不适用。The prior art [2] (see Yu Y, Baoyin H. Generating families of 3D periodic orbits about asteroids [J]. Monthly Notices of the Royal Astronomical Society, 2012, 427(1): 872-881.) gives the monomer small The periodic orbit search method under the solid connection system of the planetary rotation uses the Poincaré interface to analyze the passing orbit under the fixed connection system of the body, and considers the differential correction of the points close to the crossing plane to obtain an accurate periodic orbit. This method can search for the multi-families periodic orbits of the asteroid 216, but this type of orbit is only suitable for the orbit search under the fixed system of the body due to the irregular shape of the asteroid, and it is not suitable for the double asteroid system.

全面，快速的得到双小行星系统内的周期轨道是研究探测器在小行星附近运动的关键也是未来小行星探测任务的基础。当前的方法给出的系统内周期轨道不全面，且仅能得到单圈周期轨道，对多圈周期轨道搜索方法不收敛。Comprehensive and rapid acquisition of the periodic orbit in the binary asteroid system is the key to the study of the motion of the probe near the asteroid and the basis for future asteroid detection missions. The current method gives incomplete periodic orbits in the system, and can only obtain single-turn periodic orbits, and the search method for multi-turn periodic orbits does not converge.

发明内容Contents of the invention

本发明的目的是为了解决上述双小行星系统中周期轨道搜索受限制的问题，提出一种基于速度庞加莱截面针对双小行星系统的全局周期轨道的搜索方法，该方法利用模型的对称性和轨道演化的连续性对沿轴线方向的对称轨道进行搜索，可以实现系统内具有对称性的全局周期轨道搜索，使周期轨道的范围从平衡点附近扩大至系统内全区域。The purpose of the present invention is to solve the problem of limited periodic orbit search in the above-mentioned double asteroid system, and propose a search method for the global periodic orbit of the double asteroid system based on the velocity Poincaré section, which uses the symmetry of the model Searching for symmetrical orbits along the axial direction can realize the search for global periodic orbits with symmetry in the system, and expand the range of periodic orbits from near the equilibrium point to the entire area of the system.

本发明的思想是基于双小行星系统旋转坐标系，采用速度庞加莱截面和模型的对称性对双星系统内的全局周期轨道进行搜索，利用二分法提高搜索效率，同时可以得到环绕系统多圈的周期轨道，该方法具有效率高，搜索全面，计算简单等特点，可适用于不同尺寸和质量比的双小行星系统周期轨道搜索。The idea of the present invention is based on the rotating coordinate system of the double asteroid system, using the velocity Poincaré section and the symmetry of the model to search the global periodic orbit in the double star system, using the dichotomy method to improve the search efficiency, and at the same time can get multiple circles around the system The method has the characteristics of high efficiency, comprehensive search, simple calculation, etc., and can be applied to the periodic orbit search of double asteroid systems with different sizes and mass ratios.

为实现上述目的，本发明通过以下技术方案实现：To achieve the above object, the present invention is achieved through the following technical solutions:

一种基于速度庞加莱截面双小行星系统全局周期轨道的搜索方法，包括以下步骤：A search method for the global periodic orbit of a double asteroid system based on the velocity Poincaré section, comprising the following steps:

步骤一、获取双小行星的质量和尺寸，并进行归一化；Step 1. Obtain the mass and size of the double asteroid, and perform normalization;

步骤二、建立双小行星系统旋转坐标系；Step 2, establishing the rotating coordinate system of the double asteroid system;

步骤三、选定周期轨道搜索范围，沿X轴方向选择探测器的初始位置，逐步增大初始速度，根据下式进行轨道积分，积分终止条件为穿过预定截面；然后根据终止点的速度参数绘制庞加莱映射图；Step 3. Select the periodic orbit search range, select the initial position of the detector along the X-axis direction, gradually increase the initial velocity, and perform orbit integration according to the following formula. The integration termination condition is to pass through a predetermined section; then according to the velocity parameter of the termination point Draw the Poincaré map;

$\{\begin{matrix} \overset{\cdot &Center Dot; \cdot \cdot}{x x} - - 22 \overset{\cdot &Center Dot;}{y the y} = = x x + + \frac{((11 - - μ μ)) ((x x + + μ μ))}{{r r}_{11}^{33}} + + \frac{μ μ ((x x - - 11 + + μ μ))}{{r r}_{22}^{33}} \\ \overset{\cdot \cdot \cdot &Center Dot;}{y the y} + + 22 \overset{\cdot &Center Dot;}{x x} = = y the y + + \frac{((11 - - μ μ)) y the y}{{r r}_{11}^{33}} + + \frac{μy μy}{{r r}_{22}^{33}} \\ \overset{\cdot \cdot \cdot \cdot}{z z} = = \frac{((11 - - μ μ)) z z}{{r r}_{11}^{33}} + + \frac{μz μz}{{r r}_{22}^{33}} \end{matrix}$

其中μ＝M₂/(M₁+M₂)，M₁为系统中质量较大的小行星质量，M₂为系统中质量较小的小行星质量；r₁,r₂分别表示探测器到两者的距离，计算公式如下：Among them, μ=M ₂ /(M ₁ +M ₂ ), M ₁ is the mass of the asteroid with larger mass in the system, and M ₂ is the mass of the asteroid with smaller mass in the system; r ₁ and r ₂ respectively represent the The distance between the two is calculated as follows:

${r r}_{11} = = \sqrt{{((x x + + μ μ))}^{22} + + {y the y}^{22} + + {z z}^{22}};;$

${r r}_{22} = = \sqrt{{((x x - - 11 + + μ μ))}^{22} + + {y the y}^{22} + + {z z}^{22}};;$

探测器的速度满足下述公式：The speed of the detector satisfies the following formula:

$V V = = \sqrt{22 Ω Ω - - C C};;$

其中，C为雅各比常数，表示系统的能量特性；Among them, C is the Jacobian constant, which represents the energy characteristics of the system;

$Ω Ω = = \frac{11}{22} [[(({x x}^{22} + + {y the y}^{22})) + + μ μ ((11 - - μ μ))]] + + \frac{11 - - μ μ}{{r r}_{11}} + + \frac{μ μ}{{r r}_{22}};;$

步骤四、选择映射图中曲线穿过轴线临近的两个初始点，采用二分法，通过取初始状态的平均值重新积分，得到新的穿越轴线的初始点，反复迭代，获得穿过轴线的精确值；Step 4. Select two initial points where the curve crosses the axis in the map, and use the dichotomy method to obtain a new initial point crossing the axis by taking the average value of the initial state and re-integrate. Repeat iterations to obtain the precise value of crossing the axis. value;

步骤五、改变探测器的初始位置重复步骤三和步骤四,直至取值涵盖所选搜索区域。Step 5. Change the initial position of the detector and repeat steps 3 and 4 until the value covers the selected search area.

作为优选，所述小行星的质量和尺寸，以及二者间的距离，可根据天文观测数据确定；归一化可通过将双星的质量和作为单位质量，双星间的距离作为单位长度，对小行星质量和尺寸进行归一化。As preferably, the quality and size of described asteroid, and the distance between the two, can be determined according to astronomical observation data; Planet mass and size are normalized.

作为优选，所述建立双小行星系统旋转坐标系可通过以下方式建立：As a preference, the establishment of the rotating coordinate system of the double asteroid system can be established in the following manner:

将双星系统的质心作为坐标原点，X轴与两星体质心连线重合，质量较大的行星作为主星，质量较小的行星作为卫星，主星指向卫星方向为正，Z轴方向为系统自旋方向，Y轴方向满足右手螺旋定则。Take the center of mass of the binary star system as the origin of the coordinates, the X-axis coincides with the line connecting the centers of mass of the two stars, the planet with a larger mass is used as the main star, and the planet with a smaller mass is used as a satellite, the direction of the main star pointing to the satellite is positive, and the direction of the Z-axis is the system spin direction, the Y-axis direction satisfies the right-hand spiral rule.

作为优选，选择搜索范围为沿X轴方向[-2，2]区间，所述预定截面为垂直于Y轴同时包含X轴的平面。Preferably, the selected search range is the interval [-2, 2] along the X-axis direction, and the predetermined section is a plane perpendicular to the Y-axis and including the X-axis.

本发明方法的原理解释如下：The principle of the inventive method is explained as follows:

双小行星系统的动力学模型在双星旋转坐标系下可表示为：The dynamic model of the double asteroid system can be expressed as:

$\{\begin{matrix} \overset{\cdot &Center Dot; \cdot &Center Dot;}{x x} - - 22 \overset{\cdot &Center Dot;}{y the y} = = x x + + \frac{((11 - - μ μ)) ((x x + + μ μ))}{{r r}_{11}^{33}} + + \frac{μ μ ((x x - - 11 + + μ μ))}{{r r}_{22}^{33}} \\ \overset{\cdot &Center Dot; \cdot \cdot}{y the y} + + 22 \overset{\cdot \cdot}{x x} = = y the y + + \frac{((11 - - μ μ)) y the y}{{r r}_{11}^{33}} + + \frac{μy μy}{{r r}_{22}^{33}} \\ \overset{\cdot \cdot \cdot &Center Dot;}{z z} = = \frac{((11 - - μ μ)) z z}{{r r}_{11}^{33}} + + \frac{μz μz}{{r r}_{22}^{33}} \end{matrix} - - - - - - ((11))$

其中μ＝M₂/(M₁+M₂)，M₁为系统中质量较大的小行星质量，M₂为系统中质量较小的小行星质量；r₁,r₂分别表示探测器到两者的距离。Among them, μ=M ₂ /(M ₁ +M ₂ ), M ₁ is the mass of the asteroid with larger mass in the system, and M ₂ is the mass of the asteroid with smaller mass in the system; r ₁ and r ₂ respectively represent the the distance between the two.

${r r}_{11} = = \sqrt{{((x x + + μ μ))}^{22} + + {y the y}^{22} + + {z z}^{22}},, {r r}_{22} = = \sqrt{{((x x - - 11 + + μ μ))}^{22} + + {y the y}^{22} + + {z z}^{22}}$

根据方程形式可知系统具有对称特性According to the form of the equation, it can be seen that the system has symmetric properties

即若轨道状态积分时间t经过状态则对称的状态经过相同的时间逆向积分，将经过状态 That is, if the orbital state Integral time t elapsed state then the state of symmetry Reverse integration over the same time will pass through the state

同时系统存在一个积分常数，称为雅各比常数At the same time, there is an integral constant in the system, called the Jacobian constant

$C C = = 22 Ω Ω - - (({\overset{\cdot \cdot}{x x}}^{22} + + {\overset{\cdot \cdot}{y the y}}^{22} + + {\overset{\cdot \cdot}{z z}}^{22})) - - - - - - ((22))$

其中 $Ω = \frac{1}{2} [(x^{2} + y^{2}) + μ (1 - μ)] + \frac{1 - μ}{r_{1}} + \frac{μ}{r_{2}}$ in $Ω = \frac{1}{2} [(x^{2} + {the y}^{2}) + μ (1 - μ)] + \frac{1 - μ}{r_{1}} + \frac{μ}{r_{2}}$

探测器的速度大小为 The velocity of the detector is

C可以表示系统的能量特性。固定C，当探测器在系统的位置确定后，探测器的速度也可确定 C can represent the energy properties of the system. Fixed C, when the position of the detector in the system is determined, the speed of the detector can also be determined

根据系统的以上特性可以设计周期轨道的搜索方法，由于探测器在Z方向的运动相对独立，暂不考虑Z方向的位置和速度，仅考虑探测器在平面的运动，将探测器的初始状态选择为X轴，即y＝0，同时确定探测器的初始速度方向沿Y轴正方向，即对于确定的初始位置(x_c,0)，Ω为常值，减小C，相应的速度大小V也会变化,初始速度为(0,V)。According to the above characteristics of the system, the search method of the periodic orbit can be designed. Since the movement of the detector in the Z direction is relatively independent, the position and velocity in the Z direction are not considered for the time being, and only the movement of the detector in the plane is considered, and the initial state of the detector is selected as is the X axis, that is, y=0, and at the same time determine that the initial velocity direction of the detector is along the positive direction of the Y axis, that is For the definite initial position (x _c ,0), Ω is a constant value, if C is reduced, the corresponding velocity V will also change, and the initial velocity is (0,V).

选择截面为垂直于Y轴同时包含X轴的平面，由于仅考虑平面情况，探测器穿过截面等同于穿过X轴。记录探测器轨道穿过X轴时的沿X轴方向的速度如图1所示。绘制轨道初始状态速度与穿过截面时X轴向速度的庞加莱映射图如图2所示，由图可知固定点初始速度发生变化时，穿过截面时X轴向速度的相应变化。若则根据对称特性，以同样的初始状态逆向积分也将经过截面的同一点，即可以找到周期轨道。The section is selected as a plane perpendicular to the Y axis and includes the X axis. Since only the plane is considered, the detector passing through the section is equivalent to passing through the X axis. Record the speed along the X-axis direction when the detector track passes through the X-axis As shown in Figure 1. Plot the initial state velocity of the orbit and the X-axis velocity when passing through the section The Poincaré map of As shown in Figure 2, it can be seen from the figure that when the initial velocity of the fixed point changes, the corresponding change of the X-axis velocity when passing through the section. like Then, according to the symmetry property, the reverse integration with the same initial state will also pass through the same point of the section, that is, the periodic orbit can be found.

由于系统为连续系统，因此探测器的速度也应连续变化，均匀增大探测器的初始速度，曲线穿过X轴，说明在两组初始速度间存在初始值使采用二分法可以获得穿过X轴的精确值，即为周期轨道初始值。Since the system is a continuous system, the speed of the detector should also change continuously, increasing the initial speed of the detector uniformly, The curve crosses the x-axis, illustrating the initial velocities in the two sets There is an initial value between The exact value across the X axis can be obtained by using the dichotomy method, which is the initial value of the periodic orbit.

考虑到双星系统的小行星尺寸，若轨道积分过程中距离两小行星的距离小于相应的小行星半径，则认为轨道与小行星相交，忽略该组初值。Considering the size of the asteroid in the binary star system, if the distance from the two asteroids during orbit integration is less than the corresponding asteroid radius, the orbit is considered to intersect the asteroid, and this set of initial values is ignored.

若曲线多次穿越X轴,说明在初始位置存在多组周期轨道的初始值，如图3所示。对于超过1圈的周期轨道，可以选择轨道多次穿越X轴截面时的与初始状态绘制庞加莱映射图，穿越点除去单次相交的情况即为多圈周期轨道的初值，如图4所示。like The curve crosses the X axis multiple times, indicating that there are multiple sets of initial values of periodic orbits at the initial position, as shown in Figure 3. For a periodic orbit with more than 1 circle, you can choose the time when the orbit crosses the X-axis section multiple times with initial state The Poincaré map is drawn, the initial value of the multi-circle periodic orbit is the initial value of the multi-turn periodic orbit except for the single intersection of the crossing point, as shown in Figure 4.

有益效果Beneficial effect

根据本发明给出的全局周期轨道搜索方法，得到的周期轨道位于系统全区域内，搜索效率高，可以对非平衡点附近的周期轨道和多圈周期轨道进行搜索。对比已有方法，本发明方法具有效率高，搜索全面，计算简单等特点，可以对非平衡点附近的周期轨道和多圈周期轨道进行搜索，可适用于不同尺寸和质量比的双小行星系统周期轨道搜索，适用于探测器对双小行星系统进行探测的轨道设计。According to the global periodic orbit search method provided by the present invention, the obtained periodic orbit is located in the whole area of the system, the search efficiency is high, and the periodic orbit and multi-turn periodic orbit near the non-equilibrium point can be searched. Compared with the existing methods, the method of the present invention has the characteristics of high efficiency, comprehensive search, simple calculation, etc., can search for periodic orbits and multi-circle periodic orbits near the non-equilibrium point, and is applicable to double asteroid systems with different sizes and mass ratios Periodic orbit search, suitable for the orbit design of the detector to detect the double asteroid system.

附图说明Description of drawings

图1是本发明方法初始状态及截面选择示意图；Fig. 1 is a schematic diagram of initial state and section selection of the inventive method;

图2是本发明方法速度庞加莱映射示意图；Fig. 2 is the speed Poincaré mapping of the method of the present invention schematic diagram;

图3是本发明方法由速度庞加莱映射确定的周期轨道初值示意图；Fig. 3 is that the inventive method is mapped by velocity Poincaré Schematic diagram of the initial value of the determined periodic orbit;

图4是本发明方法由速度庞加莱映射确定的多圈周期轨道初值示意图；Fig. 4 is that the inventive method is mapped by velocity Poincaré Schematic diagram of the initial value of the determined multi-turn period orbit;

图5是本发明实施例周期轨道1初值对应的轨道示意图；Fig. 5 is a schematic diagram of the orbit corresponding to the initial value of periodic orbit 1 according to the embodiment of the present invention;

图6是本发明实施例周期轨道2初值对应的的轨道示意图；Fig. 6 is a schematic diagram of the orbit corresponding to the initial value of the periodic orbit 2 in the embodiment of the present invention;

图7是本发明实施例周期轨道3初值对应的的轨道示意图；Fig. 7 is a schematic diagram of the orbit corresponding to the initial value of the periodic orbit 3 in the embodiment of the present invention;

图8是本发明实施例周期轨道4初值对应的的轨道示意图。Fig. 8 is a schematic diagram of the orbit corresponding to the initial value of the periodic orbit 4 according to the embodiment of the present invention.

具体实施方式Detailed ways

下面将结合附图和实施例对本发明加以详细说明，同时也叙述了本发明技术方案解决的技术问题及有益效果，需要指出的是，所描述的实施例仅旨在便于对本发明的理解，而对其不起任何限定作用。The present invention will be described in detail below in conjunction with accompanying drawing and embodiment, also described the technical problem and beneficial effect that the technical solution of the present invention solves simultaneously, it should be pointed out that described embodiment is only intended to facilitate the understanding of the present invention, and It has no limiting effect on it.

根据发明内容所述，该周期轨道搜索方法分为以下几个部分：确定系统的质量常数和小行星尺寸，选择探测器初始位置和初始速度方向积分，根据速度庞加莱映射采用二分法获得精确解。According to the content of the invention, the periodic orbit search method is divided into the following parts: determine the mass constant of the system and the size of the asteroid, select the initial position of the detector and the integral of the initial velocity direction, and use the dichotomy method to obtain accurate untie.

实施例Example

1)小行星的质量和尺寸可根据天文观测数据确定，对系统进行归一化处理，双星的质量和为单位质量，双星间的距离为单位长度，可以得到归一化后的小行星质量和尺寸；1) The mass and size of the asteroid can be determined according to the astronomical observation data, and the system is normalized. The mass sum of the double stars is unit mass, and the distance between the double stars is unit length. The normalized asteroid mass sum can be obtained size;

选择双小行星系统1996FG3为例作为目标进行周期轨道搜索，根据观测可以得到系统中两星距离为3km，主星直径1.69km，卫星直径0.49km，主星的质量为3.5×10¹²kg，卫星质量1.0×10¹¹kg，通过归一化后，两星间距离为1，主星直径0.5633，卫星直径0.1633，主星质量0.972，卫星质量0.0278。Select the double asteroid system 1996FG3 as an example to carry out periodic orbit search. According to the observation, the distance between the two stars in the system is 3 km, the diameter of the main star is 1.69 km, and the diameter of the satellite is 0.49 km. The mass of the main star is 3.5×10 ¹² kg, and the mass of the satellite is 1.0 ×10 ¹¹ kg, after normalization, the distance between the two stars is 1, the diameter of the main star is 0.5633, the diameter of the satellite is 0.1633, the mass of the main star is 0.972, and the mass of the satellite is 0.0278.

2)选定周期轨道搜索范围，沿X轴方向选择探测器的初始位置，逐步增大初始速度，根据式(1)进行轨道积分，设定积分终止条件为穿过预定截面；根据终止点的速度参数绘制庞加莱映射图。2) Select the periodic orbit search range, select the initial position of the detector along the X-axis direction, gradually increase the initial velocity, perform orbit integration according to formula (1), and set the integration termination condition to pass through a predetermined section; The velocity parameter plots the Poincaré map.

不考虑Z轴方向的变化，选择探测器的初始位置为(0.395，0)，特征能量C的变化范围为3.53～2.03。根据式(2)，对应的速度V变化范围为1.083～1.632，方向为Y轴正方向。绘制一次穿越截面庞加莱映射图如图3，二次穿越截面庞加莱映射图如图4。图3曲线与轴线存在3个交点，图4曲线与轴线存在4个交点。Regardless of the change in the Z-axis direction, the initial position of the detector is selected as (0.395, 0), and the variation range of the characteristic energy C is 3.53-2.03. According to formula (2), the corresponding speed V ranges from 1.083 to 1.632, and the direction is the positive direction of the Y axis. Draw the Poincaré map of the first crossing section as shown in Figure 3, and the Poincaré map of the second crossing section as shown in Figure 4. There are 3 intersections between the curve in Figure 3 and the axis, and there are 4 intersections between the curve in Figure 4 and the axis.

3)选择映射图中曲线穿过轴线临近的两个初始点，通过取初始状态的平均值重新积分，得到新的穿越轴线的初始点，反复迭代，获得穿过轴线的精确值。3) Select two initial points in the map where the curve crosses the axis and re-integrate the average value of the initial state to obtain a new initial point crossing the axis, and iterate repeatedly to obtain the precise value of crossing the axis.

图3中第一个交点的速度范围在1.180～1.184之间，经过迭代得到精确的周期轨道初速为1.182，对应周期轨道1；图3第二个交点的速度范围在1.397～1.401之间，经过迭代得到精确的周期轨道初速为1.400，对应周期轨道2；图3中第三个交点的速度范围在1.563～1.566之间，经过迭代得到精确的周期轨道初速为1.565，对应周期轨道3。图4中的1，2，4交点速度分别对应图3中1，2，3交点。图4中第二个较大的速度范围在1.335～1.339之间，经过迭代得到精确的周期轨道初速为1.337，对应周期轨道4。The velocity range of the first intersection point in Figure 3 is between 1.180 and 1.184, and the accurate initial velocity of the periodic orbit is 1.182 after iteration, which corresponds to periodic orbit 1; the velocity range of the second intersection point in Figure 3 is between 1.397 and 1.401, after The initial velocity of the periodic orbit obtained through iteration is 1.400, corresponding to periodic orbit 2; the velocity range of the third intersection point in Figure 3 is between 1.563 and 1.566, and the accurate initial velocity of periodic orbit is 1.565 after iteration, corresponding to periodic orbit 3. The velocities of intersections 1, 2, and 4 in Figure 4 correspond to intersections 1, 2, and 3 in Figure 3, respectively. The second larger velocity range in Figure 4 is between 1.335 and 1.339. After iteration, the accurate initial velocity of the periodic orbit is 1.337, which corresponds to the periodic orbit 4.

4)改变探测器的初始位置重复2，3步,直至取值涵盖所选搜索区域。4) Change the initial position of the detector and repeat steps 2 and 3 until the value covers the selected search area.

由上述步骤可知，本发明方法可以对全区域范围内的周期轨道进行搜索。图4中周期轨道1初值对应的轨道示意图如图5所示，其轨道呈现纺锤形，周期轨道2初值对应的轨道示意图如图6所示，其轨道呈现圆形，周期轨道3初值对应的轨道示意图如图7所示，其轨道呈现月牙形，而周期轨道4初值对应的轨道示意图如图8所示，其轨道呈现规则的双环交替形状。其中周期轨道1，2，3为单圈周期轨道，周期轨道4为多圈周期轨道。It can be seen from the above steps that the method of the present invention can search for periodic orbits in the whole area. The schematic diagram of the orbit corresponding to the initial value of periodic orbit 1 in Fig. 4 is shown in Fig. 5, and its orbit is spindle-shaped. The schematic diagram of the orbit corresponding to the initial value of periodic orbit 2 is shown in Fig. 6. The corresponding orbit diagram is shown in Figure 7, and its orbit is crescent-shaped, while the orbit diagram corresponding to the initial value of periodic orbit 4 is shown in Figure 8, and its orbit presents a regular double-ring alternating shape. Among them, periodic orbits 1, 2, and 3 are single-turn periodic orbits, and periodic orbit 4 is a multi-turn periodic orbit.

综上所述，本发明提供的一种基于速度庞加莱截面双小行星系统全局周期轨道的搜索方法计算简单、搜索全面，可以对非平衡点附近的周期轨道和多圈周期轨道进行搜索。In summary, the present invention provides a search method for the global periodic orbit of a double asteroid system based on the velocity Poincaré section, which is simple in calculation and comprehensive in search, and can search for periodic orbits and multi-turn periodic orbits near non-equilibrium points.

以上所述的具体描述，对发明的目的、技术方案和有益效果进行了进一步详细说明，所应理解的是，以上所述仅为本发明的具体实施例而已，并不用于限定本发明的保护范围，凡在本发明的精神和原则之内，所做的任何修改、等同替换、改进等，均应包含在本发明的保护范围之内。The specific description above further elaborates the purpose, technical solution and beneficial effect of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not used to limit the protection of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A search method based on the velocity Poincaré cross-section double asteroid system periodic orbit, is characterized in that: comprise the following steps:

Step 1. Obtain the mass and size of the double asteroid, and perform normalization;

Step 2, establishing the rotating coordinate system of the double asteroid system;

Step 3. Select the periodic orbit search range, select the initial position of the detector along the X-axis direction, gradually increase the initial velocity, and perform orbit integration according to the following formula. The integration termination condition is to pass through a predetermined section; then according to the velocity parameter of the termination point Draw the Poincaré map;

\{\begin{matrix} \overset{\cdot \cdot \cdot \cdot}{x x} - - 22 \overset{\cdot \cdot}{y the y} = = x x + + \frac{((11 - - μ μ)) ((x x + + μ μ))}{{r r}_{11}^{33}} + + \frac{μ μ ((x x - - 11 + + μ μ))}{{r r}_{22}^{33}} \\ \overset{\cdot &Center Dot; \cdot \cdot}{y the y} + + 22 \overset{\cdot &Center Dot; \cdot &Center Dot;}{x x} = = y the y + + \frac{((11 - - μ μ)) y the y}{{r r}_{11}^{33}} + + \frac{μy μy}{{r r}_{22}^{33}} \\ \overset{\cdot &Center Dot; \cdot &Center Dot;}{z z} = = \frac{((11 - - μ μ)) z z}{{r r}_{11}^{33}} + + \frac{μz μz}{{r r}_{22}^{33}} \end{matrix}

Among them, μ=M ₂ /(M ₁ +M ₂ ), M ₁ is the mass of the asteroid with larger mass in the system, and M ₂ is the mass of the asteroid with smaller mass in the system; r ₁ and r ₂ respectively represent the The distance between the two is calculated as follows:

{r r}_{11} = = \sqrt{{((x x + + μ μ))}^{22} + + {y the y}^{22} + + {z z}^{22}};;

{r r}_{22} = = \sqrt{{((x x - - 11 + + μ μ))}^{22} + + {y the y}^{22} + + {z z}^{22}};;

The speed of the detector satisfies the following formula:

V V = = \sqrt{22 Ω Ω - - C C};;

Among them, C is the Jacobian constant, which represents the energy characteristics of the system;

Ω Ω = = \frac{11}{22} [[(({x x}^{22} + + {y the y}^{22})) + + μ μ ((11 - - μ μ))]] + + \frac{11 - - μ μ}{{r r}_{11}} + + \frac{μ μ}{{r r}_{22}};;

Step 4. Select the two initial points where the curve crosses the axis in the map, and re-integrate the average value of the initial state to obtain a new initial point crossing the axis, and iterate repeatedly to obtain the precise value of crossing the axis;

Step 5. Change the initial position of the detector. Repeat steps 3 and 4 until the values cover the selected search area.

2. A kind of search method based on velocity Poincaré cross-section double asteroid system periodic orbit according to claim 1, it is characterized in that: the mass and size of described asteroid, and the distance between the two can be determined according to astronomical Observational data is determined; normalization can normalize the mass and size of the asteroid by taking the mass sum of the binary star as the unit mass and the distance between the binary stars as the unit length.

3. a kind of search method based on velocity Poincaré cross-section double asteroid system periodic orbit according to claim 1, is characterized in that: described establishment double asteroid system rotating coordinate system can be set up in the following way:

The center of mass of the binary star system is taken as the origin of the coordinates, the X-axis coincides with the line connecting the centers of mass of the two stars, the planet with a larger mass is used as the main star, and the planet with a smaller mass is used as a satellite, the direction of the main star pointing to the satellite is positive, and the direction of the Z-axis is the system spin direction, the Y-axis direction satisfies the right-hand spiral rule.

4. According to any one of claims 1-3, a search method based on the velocity Poincaré cross-section double asteroid system periodic orbit, characterized in that: the predetermined cross-section is a plane perpendicular to the Y-axis and simultaneously containing the X-axis .