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

Abstract

The invention relates to a double-asteroid system periodic orbit searching method based on the speed Poincare section and belongs to the field of aerospace. The method comprises the following steps that 1, the mass and size of each asteroid are obtained and are normalized; 2, a system rotating coordinate system is built; 3, the searching range is selected, the initial position of a detector is selected in the X-axis direction, the initial speed is gradually increased, orbit integration is carried out, the stopping condition is that the detector penetrates through the predetermined section, and then a Poincare map is drawn according to the speed at the termination point; 4, two adjacent initial points, penetrating through the axis, in the curve in the map are selected, an initial state average value is obtained to be subjected to integration again, a new initial point penetrating the axis is obtained, repeated iteration is carried out, and an accurate value is obtained; 5, the initial position of the detector is changed, and the third step and the four step are carried out repeatedly until the values cover the searching area. Compared with the prior art, the method has the advantages of being high in efficiency, comprehensive in searching, simple in calculation and the like, and is suitable for orbit design occasions where the detector detects a double-asteroid system.

Description

Based on speed Poincare section binary asteroid system cycle track seek method

Technical field

The present invention relates to a kind of searching method of binary asteroid system cycle track, in particular to a kind of based on the searching method of speed Poincare section for the Global Periodic track of binary asteroid system, belong to field of aerospace technology, be applicable to the Track desigh that detector detects binary asteroid system.

Background technology

The important component part of small celestial body exploration to returning around detection and sampling of small feature loss.In the design of detection mission, the track near to minor planet is needed to design.Periodic orbit near minor planet has good dynamics, can as the parking orbit of desirable detection track and landing track.

Binary asteroid system is the special minor planet system of a class, has two minor planets to rotate around common barycenter, is similar to the form of the earth-moon system, three-system can be adopted to unite similar model to study the motion of detector.But different from the form of planet-satellite, two asteroidal quality are more close, quality comparation is large, and the distance ratio simultaneously between asteroidal size and two minor planets is also comparatively large, cannot be used as particle to consider.

In searching method at present for periodic orbit near minor planet, in first technology [1] (see Richardson D L.Analytic construction of periodic orbits about thecollinear points [J] .Celestial mechanics, 1980, 22 (3): 241-253.) the periodic orbit searching method near equilibrium point in three-system system is given, microvariations are applied at system balancing point place, disturbance quantity relation is over time provided according to linearizing equation, provide the initial value of periodic orbit near equilibrium point, the three rank analytic solution that can obtain equation are replaced according to variable, utilize analytic solution as initial value, accurate periodic orbit can be obtained by differential correction.Utilize the Halo track that the method can obtain in three-system system near Collinear Poi nt, Li Sa is as track, plane and vertical Liapunov track, the method can obtain periodic orbit comparatively accurately, and is applicable to different mass ratioes, but can only be used for the periodic orbit search near equilibrium point, when disturbance far from equilibrium point, the method cannot obtain periodic orbit, and cannot find the periodic orbit of the system overall situation, analytical expression calculation of complex.Intrasystem multi-turn periodic orbit also cannot be obtained by the method.

In first technology [2] (see Yu Y, Baoyin H.Generating families of 3Dperiodic orbits about asteroids [J] .Monthly Notices of the RoyalAstronomical Society, 2012, 427 (1): 872-881.) give monomer minor planet and rotate the periodic orbit searching method under being that is connected, being connected at body is that lower employing Poincare interface is analyzed the track passed, consider to carry out differential correction through the point that planimetric position is close, obtain accurate periodic orbit, use the method can search many races periodic orbit of minor planet 216, but this kind of track is only applicable to produce because of minor planet out-of-shape and is connected the track seek under being at body, inapplicable for binary asteroid system.

Comprehensively, the basis of to be the research detector key of moving near minor planet be also following ASTEREX task that fast speed obtains the intrasystem periodic orbit of binary asteroid.In the system that current method provides, periodic orbit is not comprehensive, and only can obtain individual pen periodic orbit, does not restrain multi-turn periodic orbit searching method.

Summary of the invention

The object of the invention is to search for restricted problem to solve periodic orbit in above-mentioned binary asteroid system, propose a kind of based on the searching method of speed Poincare section for the Global Periodic track of binary asteroid system, the method utilizes the symmetry of model and the continuity of Orbital Evolution to search for symmetry orbital in the axial direction, can realize that there is in system symmetric Global Periodic track seek, the scope of periodic orbit is extended in system near equilibrium point region-wide.

Thought of the present invention is based on binary asteroid system rotating coordinate system, the symmetry of employing speed Poincare section and model is searched for the Global Periodic track in binary-star system, dichotomy is utilized to improve search efficiency, the periodic orbit of surrounding system multi-turn can be obtained simultaneously, it is high that the method has efficiency, search comprehensively, calculates the features such as simple, is applicable to the binary asteroid system cycle track seek of different size and mass ratio.

For achieving the above object, the present invention is achieved through the following technical solutions:

Based on a searching method for speed Poincare section binary asteroid system Global Periodic track, comprise the following steps:

Step one, the quality obtaining binary asteroid and size, and be normalized;

Step 2, set up binary asteroid system rotating coordinate system;

Step 3, selected periodic orbit search coverage, select the initial position of detector along X-direction, progressively increase rate of onset, carry out orbit integration according to following formula, integration end condition is through predetermined cross-sectional; Then Poincaré map figure is drawn according to the speed parameter of terminating point;

$\left\{\begin{array}{c}\stackrel{\·\·}{x}-2\stackrel{\·}{y}=x+\frac{(1-\mathrm{\μ})(x+\mathrm{\μ})}{r_1^3}+\frac{\mathrm{\μ}(x-1+\mathrm{\μ})}{r_2^3}\\ \stackrel{\·\·}{y}+2\stackrel{\·}{x}=y+\frac{(1-\mathrm{\μ})y}{r_1^3}+\frac{\mathrm{\μy}}{r_2^3}\\ \stackrel{\·\·}{z}=\frac{(1-\mathrm{\μ})z}{r_1^3}+\frac{\mathrm{\μz}}{r_2^3}\end{array}\right.$

Wherein μ=M ₂/ (M ₁+ M ₂), M ₁for the minor planet quality that quality in system is larger, M ₂for the minor planet quality that quality in system is less; r ₁, r ₂represent the distance of detector to both respectively, computing formula is as follows:

$r_1=\sqrt{{(x+\mathrm{\μ})}^2+y^2+z^2};$

$r_2=\sqrt{{(x-1+\mathrm{\μ})}^2+y^2+z^2};$

The speed of detector meets following formula:

$V=\sqrt{2\mathrm{\Ω}-C};$

Wherein, C is Jacobi constant, represents the energy response of system;

$\mathrm{\Ω}=\frac{1}{2}[(x^2+y^2)+\mathrm{\μ}(1-\mathrm{\μ})]+\frac{1-\mathrm{\μ}}{r_1}+\frac{\mathrm{\μ}}{r_2};$

Two initial points that in step 4, Choose for user figure, curve closes on through axis, adopt dichotomy, by getting the aviation value integration again of initial condition, obtaining the new initial point passing through axis, iterating, obtaining the exact value through axis;

Step 5, the initial position repetition step 3 changing detector and step 4, until value contains selected region of search.

As preferably, described asteroidal quality and size, and the distance between the two, can determine according to celestial observation data; Normalization method is by the quality using double star with as unit mass, and the distance between double star, as unit length, is normalized minor planet quality and size.

As preferably, described binary asteroid system rotating coordinate system of setting up is by setting up with under type:

Using the barycenter of binary-star system as the origin of coordinates, X-axis and amphiaster barycenter line coincident, the larger planet of quality is as primary, the less planet of quality is as satellite, it is just that primary points to satellite direction, and Z-direction is system spin direction, and Y direction meets right-hand screw rule.

As preferably, search coverage is selected to be that described predetermined cross-sectional is the plane simultaneously comprising X-axis perpendicular to Y-axis along X-direction [-2,2] interval.

The principle of the inventive method is explained as follows:

The kinetic model of binary asteroid system can be expressed as under double star rotating coordinate system:

$\left\{\begin{array}{c}\stackrel{\·\·}{x}-2\stackrel{\·}{y}=x+\frac{(1-\mathrm{\μ})(x+\mathrm{\μ})}{r_1^3}+\frac{\mathrm{\μ}(x-1+\mathrm{\μ})}{r_2^3}\\ \stackrel{\·\·}{y}+2\stackrel{\·}{x}=y+\frac{(1-\mathrm{\μ})y}{r_1^3}+\frac{\mathrm{\μy}}{r_2^3}\\ \stackrel{\·\·}{z}=\frac{(1-\mathrm{\μ})z}{r_1^3}+\frac{\mathrm{\μz}}{r_2^3}\end{array}\right.---\left(1\right)$

Wherein μ=M ₂/ (M ₁+ M ₂), M ₁for the minor planet quality that quality in system is larger, M ₂for the minor planet quality that quality in system is less; r ₁, r ₂represent the distance of detector to both respectively.

$r_1=\sqrt{{(x+\mathrm{\μ})}^2+y^2+z^2},r_2=\sqrt{{(x-1+\mathrm{\μ})}^2+y^2+z^2}$

According to the known system of equation form, there is symmetry characteristic

Even track condition integration time, t was through state then symmetrical state through reverse integration of identical time, will through state

There is an integral constant in simultaneity factor, is called Jacobi constant

$C=2\mathrm{\Ω}-({\stackrel{\·}{x}}^2+{\stackrel{\·}{y}}^2+{\stackrel{\·}{z}}^2)---\left(2\right)$

Wherein $\mathrm{\Ω}=\frac{1}{2}[(x^2+y^2)+\mathrm{\μ}(1-\mathrm{\μ})]+\frac{1-\mathrm{\μ}}{r_1}+\frac{\mathrm{\μ}}{r_2}$

The velocity magnitude of detector is

C can represent the energy response of system.Fixation of C, when detector is after the position of system is determined, the speed of detector also can be determined

Can the searching method of design cycle track according to the above characteristic of system, because detector is relatively independent in the motion of Z-direction, put aside position and the speed of Z-direction, only consider the motion of detector in plane, the initial condition of detector is chosen as X-axis, i.e. y=0, determines that the rate of onset direction of detector is along Y-axis positive dirction, namely simultaneously for the initial position (x determined _c, 0), Ω is constant value, and reduce C, corresponding velocity magnitude V also can change, and rate of onset is (0, V).

Select cross section to be the plane simultaneously comprising X-axis perpendicular to Y-axis, owing to only considering planar environment, detector is equal to through X-axis through cross section.Record detector track is through the speed along X-direction during X-axis as shown in Figure 1.Draw track initial condition speed with through X axis speed during cross section poincaré map figure as shown in Figure 2, when attachment point rate of onset changes as seen from the figure, through the respective change of X axis speed during cross section.If then according to symmetry characteristic, with the reverse integration of same initial condition also by the same point through cross section, namely periodic orbit can be found.

Because system is continuous system, therefore the speed of detector also answers consecutive variations, evenly increases the rate of onset of detector, curve, through X-axis, illustrates at two groups of rate of onset between there is initial value and make employing dichotomy can obtain the exact value through X-axis, is periodic orbit initial value.

Consider the minor planet size of binary-star system, if the asteroidal distance of orbit integration process middle distance two is less than corresponding minor planet radius, then think that track is crossing with minor planet, ignore this group initial value.

If curve repeatedly passes through X-axis, the initial value that there is many group periodic orbits is described, as shown in Figure 3 at initial position.For the periodic orbit more than 1 circle, when track can be selected repeatedly to pass through X-axis cross section with initial condition draw Poincaré map figure, pass through the initial value that a situation that removing single intersects is multi-turn periodic orbit, as shown in Figure 4.

Beneficial effect

According to the Global Periodic track seek method that the present invention provides, it is region-wide that the periodic orbit obtained is positioned at system, and search efficiency is high, can search for the periodic orbit near non-equilibrium point and multi-turn periodic orbit.The existing method of contrast, it is high that the inventive method has efficiency, search comprehensively, calculate the features such as simple, can search for the periodic orbit near non-equilibrium point and multi-turn periodic orbit, be applicable to the binary asteroid system cycle track seek of different size and mass ratio, be applicable to the Track desigh that detector detects binary asteroid system.

Accompanying drawing explanation

Fig. 1 is the inventive method initial condition and section selection schematic diagram;

Fig. 2 is the inventive method speed Poincaré map schematic diagram;

Fig. 3 is that the inventive method is by speed Poincaré map the periodic orbit initial value schematic diagram determined;

Fig. 4 is that the inventive method is by speed Poincaré map the multi-turn periodic orbit initial value schematic diagram determined;

Fig. 5 is the track schematic diagram that embodiment of the present invention periodic orbit 1 initial value is corresponding;

Fig. 6 be embodiment of the present invention periodic orbit 2 initial value corresponding track schematic diagram;

Fig. 7 be embodiment of the present invention periodic orbit 3 initial value corresponding track schematic diagram;

Fig. 8 be embodiment of the present invention periodic orbit 4 initial value corresponding track schematic diagram.

Detailed description of the invention

Below in conjunction with drawings and Examples, the present invention is described in detail, also describe technical matters and the beneficial effect of technical solution of the present invention solution simultaneously, it is pointed out that described embodiment is only intended to be convenient to the understanding of the present invention, and any restriction effect is not play to it.

According to summary of the invention, this periodic orbit searching method is divided into following components: the quality constant of certainty annuity and minor planet size, select detector initial position and rate of onset direction integral, adopt dichotomy to obtain exact solution according to speed Poincaré map.

Embodiment

1) asteroidal quality and size can be determined according to celestial observation data, are normalized system, the quality of double star and be unit quality, and the distance between double star is unit length, can obtain the minor planet quality after normalization method and size;

Select binary asteroid system 1996FG3 to be that example carries out periodic orbit search as target, can obtain two star distances in system according to observation is 3km, primary diameter 1.69km, and satellite diameter 0.49km, the quality of primary is 3.5 × 10 ¹²kg, satellite quality 1.0 × 10 ¹¹kg, after normalization method, two interstellar distances are 1, primary diameter 0.5633, satellite diameter 0.1633, primary quality 0.972, satellite quality 0.0278.

2) selected periodic orbit search coverage, selects the initial position of detector, progressively increases rate of onset, carry out orbit integration according to formula (1) along X-direction, set integration end condition as through predetermined cross-sectional; Speed parameter according to terminating point draws Poincaré map figure.

Do not consider the change of Z-direction, the initial position selecting detector is (0.395,0), and the variation range of characteristic energy C is 3.53 ~ 2.03.According to formula (2), corresponding speed V variation range is 1.083 ~ 1.632, and direction is Y-axis positive dirction.Draw and once pass through cross section Poincaré map figure as Fig. 3, secondary passes through cross section Poincaré map figure as Fig. 4.There are 3 intersection points in Fig. 3 curve and axis, Fig. 4 curve and axis exist 4 intersection points.

3) two initial points that in Choose for user figure, curve closes on through axis, by getting the aviation value integration again of initial condition, obtaining the new initial point passing through axis, iterating, obtaining the exact value through axis.

In Fig. 3, the speed range of first intersection point is between 1.180 ~ 1.184, and obtaining accurate periodic orbit initial velocity through iteration is 1.182, corresponding periodic orbit 1; The speed range of Fig. 3 second intersection point is between 1.397 ~ 1.401, and obtaining accurate periodic orbit initial velocity through iteration is 1.400, corresponding periodic orbit 2; In Fig. 3, the speed range of the 3rd intersection point is between 1.563 ~ 1.566, and obtaining accurate periodic orbit initial velocity through iteration is 1.565, corresponding periodic orbit 3.In Fig. 41,2,4 intersection point speed are respectively in corresponding diagram 31,2,3 intersection points.In Fig. 4, second larger speed range is between 1.335 ~ 1.339, and obtaining accurate periodic orbit initial velocity through iteration is 1.337, corresponding periodic orbit 4.

4) initial position changing detector repeats 2,3 steps, until value contains selected region of search.

From above-mentioned steps, the inventive method can be searched for the periodic orbit in region-wide scope.The track schematic diagram that in Fig. 4, periodic orbit 1 initial value is corresponding as shown in Figure 5, its track presents spindle, track schematic diagram corresponding to periodic orbit 2 initial value as shown in Figure 6, its track presents circle, track schematic diagram corresponding to periodic orbit 3 initial value as shown in Figure 7, its track presents crescent, and track schematic diagram corresponding to periodic orbit 4 initial value as shown in Figure 8, and its track presents the dicyclo alternate shaped of rule.Wherein periodic orbit 1,2,3 is individual pen periodic orbit, and periodic orbit 4 is multi-turn periodic orbit.

In sum, a kind of searching method based on speed Poincare section binary asteroid system Global Periodic track provided by the invention calculates simply, search is comprehensive, can search for the periodic orbit near non-equilibrium point and multi-turn periodic orbit.

Above-described specific descriptions; the object of inventing, technical scheme and beneficial effect are further described; be understood that; the foregoing is only specific embodiments of the invention; the protection domain be not intended to limit the present invention; within the spirit and principles in the present invention all, any amendment made, equivalent replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (4)

1., based on a speed Poincare section binary asteroid system cycle track seek method, it is characterized in that: comprise the following steps:

Step one, the quality obtaining binary asteroid and size, and be normalized;

Step 2, set up binary asteroid system rotating coordinate system;

Step 3, selected periodic orbit search coverage, select the initial position of detector along X-direction, progressively increase rate of onset, carry out orbit integration according to following formula, integration end condition is through predetermined cross-sectional; Then Poincaré map figure is drawn according to the speed parameter of terminating point;

$\left\{\begin{array}{c}\stackrel{\·\·}{x}-2\stackrel{\·}{y}=x+\frac{(1-\mathrm{\μ})(x+\mathrm{\μ})}{r_1^3}+\frac{\mathrm{\μ}(x-1+\mathrm{\μ})}{r_2^3}\\ \stackrel{\·\·}{y}+2\stackrel{\·\·}{x}=y+\frac{(1-\mathrm{\μ})y}{r_1^3}+\frac{\mathrm{\μy}}{r_2^3}\\ \stackrel{\·\·}{z}=\frac{(1-\mathrm{\μ})z}{r_1^3}+\frac{\mathrm{\μz}}{r_2^3}\end{array}\right.$

Wherein μ=M ₂/ (M ₁+ M ₂), M ₁for the minor planet quality that quality in system is larger, M ₂for the minor planet quality that quality in system is less; r ₁, r ₂represent the distance of detector to both respectively, computing formula is as follows:

$r_1=\sqrt{{(x+\mathrm{\μ})}^2+y^2+z^2};$

$r_2=\sqrt{{(x-1+\mathrm{\μ})}^2+y^2+z^2};$

The speed of detector meets following formula:

$V=\sqrt{2\mathrm{\Ω}-C};$

Wherein, C is Jacobi constant, represents the energy response of system;

$\mathrm{\Ω}=\frac{1}{2}[(x^2+y^2)+\mathrm{\μ}(1-\mathrm{\μ})]+\frac{1-\mathrm{\μ}}{r_1}+\frac{\mathrm{\μ}}{r_2};$

Two initial points that in step 4, Choose for user figure, curve closes on through axis, by getting the aviation value integration again of initial condition, obtaining the new initial point passing through axis, iterating, obtaining the exact value through axis;

Step 5, the initial position repetition step 3 changing detector and step 4, until value contains selected region of search.

2. a kind of searching method based on speed Poincare section binary asteroid system cycle track according to claim 1, is characterized in that: described asteroidal quality and size, and the distance between the two, can determine according to celestial observation data; Normalization method is by the quality using double star with as unit mass, and the distance between double star, as unit length, is normalized minor planet quality and size.

3. a kind of searching method based on speed Poincare section binary asteroid system cycle track according to claim 1, is characterized in that: described binary asteroid system rotating coordinate system of setting up is by setting up with under type:

Using the barycenter of binary-star system as the origin of coordinates, X-axis and amphiaster barycenter line coincident, the larger planet of quality is as primary, the less planet of quality is as satellite, it is just that primary points to satellite direction, and Z-direction is system spin direction, and Y direction meets right-hand screw rule.

4., according to the arbitrary described a kind of searching method based on speed Poincare section binary asteroid system cycle track of claim 1-3, it is characterized in that: described predetermined cross-sectional is the plane simultaneously comprising X-axis perpendicular to Y-axis.