CN113962119A - High-precision and high-efficiency modeling method for interaction of irregularly-shaped double-asteroid system - Google Patents

High-precision and high-efficiency modeling method for interaction of irregularly-shaped double-asteroid system Download PDF

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CN113962119A
CN113962119A CN202111019996.7A CN202111019996A CN113962119A CN 113962119 A CN113962119 A CN 113962119A CN 202111019996 A CN202111019996 A CN 202111019996A CN 113962119 A CN113962119 A CN 113962119A
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asteroid
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尚海滨
卢榉承
韦炳威
喻志桐
徐瑞
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Beijing Institute of Technology BIT
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Abstract

The invention discloses a high-precision and high-efficiency modeling method for interaction of irregular-shaped double-small planetary systems, and belongs to the technical field of aerospace. The implementation method of the invention comprises the following steps: establishing a main asteroid composite gravitational field model, wherein the composite gravitational field model improves the interaction modeling efficiency through a high-efficiency numerical interpolation method of a residual gravitational field; establishing a minor asteroid finite element model, and calculating an interpolation coefficient of a domain element node to a corresponding domain element; according to the composite gravitational field model, determining gravitational field parameters of each domain element node in the main asteroid gravitational field; determining that each minor asteroid domain element is acted by a main asteroid based on an interpolation coefficient and the gravitational field parameters, and realizing the quick modeling of the interaction of the double asteroid systems through domain element superposition; and carrying out interaction rapid modeling according to the state of the current double-asteroid system, substituting the modeling result into the kinematic equation to obtain the system state at the next moment, and predicting the evolution process of the double-asteroid system.

Description

High-precision and high-efficiency modeling method for interaction of irregularly-shaped double-asteroid system
Technical Field
The invention relates to a modeling method for interaction of double-small-planet systems, in particular to a high-precision and high-efficiency modeling method for interaction of double-small-planet systems, which relates to evolution prediction of the double-small-planet systems, and belongs to the technical field of aerospace.
Background
Asteroid refers to a celestial body in the solar system that moves around the sun like a planet, but is much smaller in volume and mass than a planet and cannot clear the area near the orbit. It is estimated that there are over 127 ten thousand asteroids in the solar system, of which about 15% are double asteroid systems. How to calculate the interaction force and moment in the double-asteroid system is a key technology for researching the evolution of the double-asteroid system. The interaction inside the double asteroid system is one of the hot problems concerned by researchers at present.
In a developed method of studying the internal interaction forces and moments of a bipartite planetary system, the interaction forces are considered as the mass integral of one asteroid in the other asteroid gravitational field (i.e. the mass integral of the secondary asteroid in the main asteroid gravitational field) in prior art [1] (Shi, Yu, Yue Wang, and Shijie Xu. "Mutual kinetic potential, force, and torque of a homogenetic group polyhedron and an extended body: an application to binary fields." Celestial Mechanics and dynamic advancement 129.3(2017): 307-. And performing Taylor expansion on the expression of the mutual potential at the centroid position of the minor asteroid, and expressing the mutual potential of the double-minor-planet system by using the finite-order partial derivative of the gravitational potential energy function of the major asteroid, the inertia integral of the minor asteroid and the relative position and posture between the two asteroids. The disadvantage of this method is that when the distance between two asteroids is close, the taylor series truncation error will increase significantly and the effect of the uneven mass distribution of the asteroids cannot be fully described.
Prior art [2] (Hou, Xiyun, Daniel J.Scheers, and Xiiaosheng Xin. "Mutual potential between two perpendicular bodies with arbitrary profiles and mass distributions" (cellular mechanisms and dynamic asymmetry 127.3(2017): 369. 395.)), the Mutual potentials of the bipartite planetary systems are calculated according to the basic definitions of the Mutual potentials. And (3) expanding the distance term between the two asteroid mass infinitesimals according to a Legendre series, and describing the mass uneven distribution caused by the irregular geometric configuration of the asteroid by using generalized inertia integral. The method has the disadvantages that the method is limited by item numbers of Legendre grades, has truncation errors and cannot completely describe the influence caused by uneven distribution of the internal mass of the asteroid.
Prior art [3] (Yu, Yang, et al. "A finite element method for computational full two-body protocol: I. the mechanical potential and derivative over bilinear tetrahedron elements." cellular mechanisms and dynamic asteromy 131.11(2019):1-21.), the interaction force and moment between two asteroids are calculated according to the definitions of mutual potential, force and moment. Two asteroids are established into a finite element model based on a finite element division method, and numerical calculation forms of mutual potential, force and moment are given according to a bilinear shape function interpolation method. The disadvantage of this method is that the finite element method requires a large number of field elements to ensure the accuracy, so that the method has a large computational burden in practical operation.
Disclosure of Invention
The invention discloses a high-precision high-efficiency modeling method for interaction of irregular-shaped double-small planetary systems, which aims to solve the technical problems that: the method is based on a composite gravitational field model and is combined with a finite element method to realize rapid modeling of interaction, describe uneven mass distribution of two asteroids, control the truncation error of a result and ensure prediction accuracy. The interaction of the double-asteroid system refers to the internal interaction force and moment of the double-asteroid system.
The purpose of the invention is realized by the following technical scheme:
the invention discloses a high-precision and high-efficiency modeling method for interaction of irregular-shaped double-small planetary systems, which is used for building a composite gravitational field model of a main small planetary and improving the efficiency of prediction of the interaction of the double-small planetary systems by a numerical interpolation method; establishing a finite element model of the minor planet, calculating an interpolation coefficient of the domain element node to the corresponding domain element, wherein the division precision of the domain elements can effectively control truncation errors; determining and storing gravitational field parameters of the domain element nodes in the main asteroid gravitational field through the composite gravitational field model; based on the interpolation coefficient and the gravitational field parameters, the domain elements are determined to be acted by the main asteroid, the finite element model can fully utilize the gravitational field of the main asteroid in the process of predicting the interaction, the calculation efficiency is improved, and the rapid modeling of the interaction force and the moment between the two asteroids is realized; and carrying out interaction rapid modeling according to the state of the current double asteroid system, substituting the modeling result into the kinematic equation to obtain the state of the system at the next moment, and predicting the evolution process of the double asteroid system.
The invention discloses a high-precision high-efficiency modeling method for interaction of irregular-shaped double-small planetary systems, which comprises the following steps of:
step one, establishing a composite gravitational field model of the main planet and the small planet, wherein the composite gravitational field model describes a real gravitational field through superposition of a nominal gravitational field and a residual gravitational field. Because the residual gravitational field uses a numerical interpolation method, the composite gravitational field model has higher numerical calculation efficiency, can reflect the influence of uneven mass distribution of the main planets and the asteroids on the gravitational field, and improves the prediction precision. By establishing the composite gravitational field to describe the gravitational field of the main small planet, the calculation load can be reduced, and the interaction modeling efficiency of the double small planet system is improved.
The composite gravitational field model can efficiently describe the gravitational field of a single small planet, and when the mass distribution of the small planet is not uniform, the composite gravitational field is still effective. The composite gravitational field model passes through a nominal gravitational field VNDescription of true gravitational field V by superposition with residual gravitational field T
V＝VN+T (1)
Nominal gravitational field VNIs a triaxial ellipsoidal gravitational field and is used for describing the main characteristics of a single asteroid gravitational field.
The composite gravitational field effectively covers the spatial region near the asteroid by a layered stack of wedge meshes. Each wedge-shaped mesh comprises six vertexes, and the position of the ith vertex of the k-th layer meshPosition vector
Figure BDA0003241520440000031
And a vector A of the vertex position of the surface of a polyhedron triangle for describing the geometric shape of a small celestial bodyiIn a relationship of
Figure BDA0003241520440000032
Where Θ is the average solid angle across the surface of the polyhedron.
After the wedge-shaped grid structure of the composite gravitational field is established, the real gravitational field V and the nominal gravitational field V on each grid node need to be calculatedNThe residual value omega, i.e. the difference u epsilon R of the gravitational potential energy function and the difference g [ g ] of the gravitational potential energy gradientx,gy,gz]∈R1 ×3. Defining a polyhedral model describing the shape of the main asteroid with nFacetA surface, when the wedge-shaped grid has nLayersWhen a layer, it contains n in totalLaypers×nFacetA grid, if each wedge grid is nNodePoint wedge units, then all contain nLaypers×nFacet×nNodeAnd (4) each node. Save all residual values as nLaypers×nFacet×nnodeA matrix of rows and 4 columns, each row of the matrix being a row of data of a residual value ω on a node, i.e., ω ═ u, gx,gy,gz]。
The residual gravitational field T is represented by a numerical interpolation method, and the residual value in the local area of the wedge-shaped grid unit can be effectively determined
Figure BDA0003241520440000033
Wherein the coefficient ωiAnd piThe real residual values and the basis functions of the ith interpolation node in the wedge-shaped grid unit are respectively, i is the natural coordinate of the designated field point in the current wedge-shaped grid, and n is the current wedge-shaped unit. The composite gravitational field model comprises a residual gravitational field,therefore, the composite gravitational field model can quickly calculate the gravitational field change of the real gravitational field caused by uneven distribution of the masses of the asteroids.
In order to further improve the prediction efficiency, preferably, a second-order spherical harmonic function is adopted to describe the gravitational field of the triaxial ellipsoid
Figure BDA0003241520440000041
Wherein mu is the gravitational constant of an ellipsoid,
Figure BDA0003241520440000042
is the distance of the field point from the center of the ellipsoid, C20And C22Is the spherical harmonic coefficient, r ═ x, y, z]TIs the position vector of the field point.
And step two, establishing a finite element model of the minor planet. And dividing the minor planets into a limited number of domain elements, and calculating interpolation coefficients of the domain element nodes to the corresponding domain elements. According to the precision of the domain element division, the truncation error of the interaction of the double asteroid systems, which is increased along with the reduction of the distance between the small celestial bodies, can be effectively controlled.
Presetting finite element division precision of minor planets, utilizing finite element division algorithm to disperse target minor planets into finite field element set, defining finite element model containing nFEMIndividual field element, nFEMVertAnd (4) a domain metanode. Within each field element, calculating the interpolation coefficient of the field element node for the current field element
κ＝fκ(x,y,z) (5)
Wherein f isκThe method is an interpolation coefficient calculation method corresponding to the type of the domain element, and x, y and z are the vertex position parameters of the domain element in the current domain element.
Preferably, the finite element partition precision of the minor planet is set, the target polyhedron is dispersed into a set of a finite number of tetrahedral field elements by using a finite element partition algorithm, and the interpolation coefficient of the vertex of the field element for the current field element is calculated in each tetrahedral field element
Figure BDA0003241520440000043
Wherein x isi,yi,ziAnd i is the position coordinate of the ith node of the tetrahedral field element 1,2,3 and 4.
And step three, determining the gravitational field parameters of each domain element node in the main and asteroid gravitational fields according to the composite gravitational field model in the step one, wherein the gravitational field parameters comprise gravitational potential energy and gravitational potential energy gradient values. And the gravitational potential energy gradient value are stored to prepare for modeling of the interaction of the double-small planetary system, so that the repeated calculation of the domain element node gravitational field parameters is avoided, and the interaction modeling efficiency of the double-small planetary system is improved.
Defining the position vector of any domain element node in the main asteroid fixed connection coordinate system as r, and then the gravitational potential energy of the domain element node in the composite gravitational field model is
Figure BDA0003241520440000051
The gradient of gravitational potential energy is
Figure BDA0003241520440000052
Wherein, UN(r) gravitational potential energy of a nominal gravitational field, uiIs the residual value of the gravitational potential energy on the ith interpolation node of the wedge-shaped unit where the domain element node is positioned, pi(r) is the basis function of the wedge-shaped element,
Figure BDA0003241520440000053
is the gravitational potential energy gradient of the nominal gravitational field, gi＝[gxi,gyi,gzi]TIs the residual error value of the gravitational potential energy gradient on the ith interpolation node of the wedge-shaped unit where the finite element node is located, and n is the wedge-shaped unit where the domain element node is located.
Each domain sectionGravitational potential energy value U (R) E R at point, gravitational potential energy gradient value
Figure BDA0003241520440000054
Is stored as nFEMVertA row 4 column matrix, each row of which is a row data of gravitational field parameters on a field element node, i.e. f ═ fp,fx,fy,fz]. And storing the gravitational potential energy and the gravitational potential energy gradient value to prepare for modeling of the interaction of the double small planetary systems, thereby avoiding the repeated calculation of the gravitational field parameters of the domain element nodes.
And step four, determining that each minor asteroid domain element is acted by the main asteroid based on the interpolation coefficient obtained in the step two and the gravitational field parameters stored in the step three, wherein the action acted on each domain element can reflect the uneven density distribution of the minor asteroids at the domain element, so that the interaction prediction precision of the double-minor planet system is improved. By establishing a finite element model of the minor asteroid, the gravitational field of the main asteroid can be fully utilized in the process of calculating the interaction force and the moment, and the calculation efficiency is improved. And the fast modeling of the interaction force and the moment between the two asteroids is realized by comprehensively considering the action of the main asteroid on all the minor asteroid field elements.
The interaction of the double asteroid system comprises the mutual potential P in the double asteroid system, the force F of the main asteroid to the secondary asteroid, and the moment mu of the main asteroid to the secondary asteroid1Moment mu of secondary asteroid to primary asteroid2. According to the finite element model established in the second step, the interaction between the two asteroids is the sum of the action of the main asteroid on each minor asteroid area element
Figure BDA0003241520440000055
Figure BDA0003241520440000056
Figure BDA0003241520440000061
μ2＝-R×F-μ1 (12)
Wherein the content of the first and second substances,
Figure BDA0003241520440000062
is a finite element field element of which the number of elements,
Figure BDA0003241520440000063
the finite element field element set is a finite element field element set of a minor asteroid finite element model, D is a projection of a position vector of a minor asteroid field element node in a minor asteroid fixed coordinate system, k is an interpolation coefficient of the field element node in the current field element, sigma is a density value of a minor asteroid at the finite element field element node, a first summation symbol refers to summation of all finite element field elements in the finite element field element set, a second summation symbol refers to summation of all field element nodes in a single field element, and a third summation symbol refers to summation of all interpolation nodes in a single wedge-shaped grid element.
Since the gravitational field parameter values at the nodes have already been calculated in step three, the node parameters need only be calculated in step four according to the node density values of the different minor asteroid territories. The gravitational field parameters stored in the composite gravitational field can describe the mass uneven distribution of the main asteroid, and the domain element node parameter values in the third step can reflect the mass uneven distribution of the secondary asteroid, so that the mass uneven distribution of the main asteroid and the secondary asteroid in the double asteroid system can be described on the basis, and the modeling precision of the interaction force and the moment of the double asteroid system is further improved.
All field elements of the secondary asteroid are traversed and summed to obtain the mutual potential P, the force F of the primary asteroid to the secondary asteroid and the moment mu of the primary asteroid to the secondary asteroid in the double asteroid system1Moment mu of secondary asteroid to primary asteroid2Therefore, the interaction of the double asteroid system can be rapidly predicted.
And step five, substituting the state of the double-asteroid system at the current moment into the step four to carry out rapid modeling and prediction on the interaction force and the moment of the double-asteroid system, and substituting the modeling result of the interaction force and the moment of the double-asteroid system obtained in the step four into a differential equation of relative motion of the double-asteroid system to obtain the state of the double-asteroid system at the next moment.
Differential equation of relative motion of double minor planet system is
Figure BDA0003241520440000064
Wherein P is the bus momentum of the system, R is the position vector pointing from the centroid of the minor celestial body 2 to the centroid of the minor celestial body 1, and ΓiIs the angular momentum of the small celestial body, omegaiIs the angular velocity of the small celestial body, A is the attitude matrix transferred from the fixed coordinate system of the small celestial body 2 to the fixed coordinate system of the small celestial body 1, A2Is an attitude matrix which is formed by fixedly connecting a coordinate system with the small celestial body 2 and transferring the coordinate system to an inertial coordinate system. F is the force of the celestial body 2 on celestial body 1, μ1(μ2) Is the moment of the small celestial body 2 (small celestial body 1) to the small celestial body 1 (small celestial body 2).
And step six, repeating the step five, predicting the evolution process of the double-asteroid system, and solving the technical problem of the related engineering of the double-asteroid system according to the evolution result of the double-asteroid system.
Has the advantages that:
1. the invention discloses a high-precision and high-efficiency modeling method for interaction of irregular-shaped double-small planetary systems.
2. The invention discloses a high-precision and high-efficiency modeling method for interaction of irregular-shaped double small planetary systems.
3. The invention discloses a high-precision high-efficiency modeling method for interaction of irregular-shaped double-asteroid systems, which is based on a composite gravitational field model and a finite element model, can describe the uneven distribution of the mass of a main asteroid and a secondary asteroid in the double-asteroid system at the same time, and improves the prediction precision of the interaction of the double-asteroid systems.
4. The invention discloses a high-precision and high-efficiency modeling method for interaction of double-small planetary systems with irregular shapes, which is combined with a finite element method, can effectively control the truncation error of the interaction of the double-small planetary systems, which is increased along with the reduction of the distance between small celestial bodies, and further improve the interaction modeling and prediction precision of the double-small planetary systems.
Drawings
FIG. 1 is a flow chart of a high-precision and high-efficiency modeling method for interaction of an irregularly-shaped double-asteroid system;
FIG. 2 is a schematic diagram of a dual-small planetary system interaction;
FIG. 3 is a graph of the change in the semimajor axis of orbital motion of the secondary asteroid relative to the primary asteroid over time in a BIA-1999KW4 double asteroid system in a particular embodiment;
FIG. 4 is a graph of eccentricity over time of the secondary asteroid relative to the orbital motion of the primary asteroid in a BIA-1999KW4 double asteroid system in a particular embodiment;
FIG. 5 is a graph of energy conversion over time in a BIA-1999KW4 double small planetary system in a particular embodiment.
Detailed Description
For better illustration of the objects and advantages of the present invention, the following description will be made in detail with reference to the drawings by taking the BIA-1999KW4 double asteroid system as an example.
As shown in fig. 1, the interaction high-precision high-efficiency modeling method for the irregular-shaped dual-small planetary system disclosed in this embodiment includes the following specific steps:
step one, establishing a composite gravitational field model of the main planet and the small planet, wherein the composite gravitational field model describes a real gravitational field through superposition of a nominal gravitational field and a residual gravitational field. Because the residual gravitational field uses a numerical interpolation method, the composite gravitational field model has higher numerical calculation efficiency, can reflect the influence of uneven mass distribution of the main planets and the asteroids on the gravitational field, and improves the prediction precision. By establishing the composite gravitational field to describe the gravitational field of the main small planet, the calculation load can be reduced, and the interaction modeling efficiency of the double small planet system is improved.
Given a polyhedral geometric model of the main asteroid of the BIA-1999KW4 double asteroid system, which contains 4586 nodes and 9168 facets. The polyhedral gravitational field model is a real gravitational field, and the real gravitational field can be directly calculated according to the geometric model. The nominal gravitational field in the composite gravitational field is a triaxial ellipsoid gravitational field, and the triaxial lengths of the ellipsoids are [0.7007,0.6681,0.5870 ]]km, attraction constant 1.5711 × 10-7km3/s2Coefficient of spherical harmonics C20＝-0.0248km2,C22＝0.0022km2。
Given that the number of grid layers is 50, the composite gravity field of the main small planet of the BIA-1999KW4 double small planet system comprises 50 × 9168 ═ 458400 unit grids, and considering the wedge with each unit being 15 nodes, the interpolation unit of the composite gravity field has 6876000 nodes.
And step two, establishing a finite element model of the minor planet. And dividing the minor planets into a limited number of domain elements, and calculating interpolation coefficients of the domain element nodes to the corresponding domain elements. According to the precision of the domain element division, the truncation error of the interaction of the double asteroid systems, which is increased along with the reduction of the distance between the small celestial bodies, can be effectively controlled. By establishing a finite element model of the minor asteroid, the gravitational field of the main asteroid can be fully utilized in the process of calculating the interaction force and the moment, and the calculation efficiency is improved.
Given a polyhedral geometric model of the minor asteroid of the BIA-1999KW4 double asteroid system, 4296 nodes and 8588 facets. The minor planet is divided by using a finite element domain element division method to obtain 4494 domain elements and 921 nodes. Taking the first finite element field as an example, the interpolation coefficient k in the finite element is 7.97930155756224 × 10-8km3。
And step three, determining the gravitational field parameters of each domain element node in the main and asteroid gravitational fields according to the composite gravitational field model in the step one, wherein the gravitational field parameters comprise gravitational potential energy and gravitational potential energy gradient values. And the gravitational potential energy and the gradient value of the gravitational potential energy are stored and are used as the mutual interaction of the double small planetary systemsAnd the used prediction is prepared, so that the repeated calculation of the domain element node gravitational field parameters is avoided, and the interaction modeling efficiency of the double small planetary system is improved. And D, determining a grid unit where the finite element nodes are located according to the composite gravitational field model established in the step one, and calculating gravitational field parameter values on the nodes by utilizing superposition of the nominal gravitational field and the residual gravitational field. Taking a node located at a position of 1km on an x axis of a fixed coordinate system of the main small celestial body as an example, the gravitational potential energy of a nominal gravitational field on the node is-1.5772 multiplied by 10-7km2/s2Acceleration of gravitational force is [ -1.5920 × 10-7,-7.7767×10-10,8.0254×10-10]km/s2. The node is positioned in the 217205 th grid cell, and the gravitational potential energy of the residual gravitational field is-3.8524 multiplied by 10-9km2/s2Acceleration of gravity of [ -1.4043 × 10-8,1.6442×10-9,1.4062×10-9]km/s2. The gravitational potential energy calculated by the composite gravitational field is-1.6157 multiplied by 10-7km2/s2Acceleration of gravity of [ -1.7325 × 10-7,8.66598767621932×10-10,2.2087×10-9]km/s2。
And step four, determining that each minor asteroid domain element is acted by the main asteroid based on the interpolation coefficient obtained in the step two and the gravitational field parameters stored in the step three, wherein the action acted on each domain element can reflect the uneven density distribution of the minor asteroids at the domain element, so that the interactive modeling precision of the double-minor-planet system is improved. By establishing a finite element model of the minor asteroid, the gravitational field of the main asteroid can be fully utilized in the process of calculating the interaction force and the moment, and the calculation efficiency is improved. And the fast modeling of the interaction force and the moment between the two asteroids is realized by comprehensively considering the action of the main asteroid on all the minor asteroid field elements.
If the attitude angle of the main asteroid with respect to the inertial space is (27.04 °, 10 °, -83.93 °), the attitude angle of the slave star with respect to the inertial space is (0 °,0 °,180 °), and the distance of the slave star with respect to the main star is 2.5659km, the mutual potential is-8176 kgkm2s-2The mutual force is 3208kN, the moment of the master satellite to the slave satellite is 1.8121kNkm, and the slave satellite to the master satelliteThe moment is 22.3937 kNkm.
And step five, substituting the state of the double-asteroid system at the current moment into the step four to quickly predict the interaction force and the moment of the double-asteroid system, and substituting the prediction result of the interaction force and the moment of the double-asteroid system obtained in the step four into a differential equation of relative motion of the double-asteroid system to obtain the state of the double-asteroid system at the next moment.
Taking the state of the four-step double-asteroid system as an example, the bus momentum P of the system is [ -2.6933, -1.7503, -0.0590]T×107kgkm/s, position vector R [ -1.3840,2.1147,0.4431 ] from the centroid of the small celestial body 2 to the centroid of the small celestial body 1]Tkm, angular momentum of small celestial body Γ1＝[0.2969,-0.5818,3.7043]×105kgkm2/s，Γ2＝[0,0，2.9123]×108kgkm2S, angular velocity Ω of the small celestial body1＝[0,0,0.0001]rad/s，Ω2＝[0,0,0.0006]rad/s, from the coordinate system fixed by the small celestial body 2 to the attitude matrix of the coordinate system fixed by the small celestial body 1
Figure BDA0003241520440000101
Attitude matrix for transferring coordinate system fixedly connected with small celestial body 2 to inertial coordinate system
Figure BDA0003241520440000102
And sixthly, repeating the step five, and predicting the evolution process of the double asteroid system.
After 200h, the bus momentum P of the system is [3.1103,0.7523,0.2859 ═]T×107kgkm/s, position vector R from the centroid of the small celestial body 2 to the centroid of the small celestial body 1 is [0.5457, -2.4787,0.3788]Tkm, angular momentum of small celestial body Γ1＝[-0.4878,0.4835,3.7297]×105kgkm2/s，Γ2＝[-0.0031,-0.0061，2.9122]×108kgkm2S, angular velocity Ω of the small celestial body1＝[0.0005,-0.0018,0.1010]rad/s，Ω2＝[-0.0008,-0.0015,0.6313]rad/s, from the coordinate system fixed by the small celestial body 2 to the attitude matrix of the coordinate system fixed by the small celestial body 1
Figure BDA0003241520440000103
Attitude matrix for transferring coordinate system fixedly connected with small celestial body 2 to inertial coordinate system
Figure BDA0003241520440000104
The variation with time of the semimajor axis of orbital motion of the minor asteroid relative to the major asteroid in the BIA-1999KW4 double asteroid system is shown in fig. 3.
The eccentricity of the secondary asteroid relative to the orbital motion of the primary asteroid in the BIA-1999KW4 double asteroid system as a function of time is shown in fig. 4.
The energy conversion over time in the BIA-1999KW4 double small planetary system is shown in FIG. 5.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (9)

1. The interactive high-precision high-efficiency modeling method of the irregularly-shaped double-asteroid system is characterized by comprising the following steps of: comprises the following steps of (a) carrying out,
step one, establishing a composite gravitational field model of a main planet and a small planet, wherein the composite gravitational field model describes a real gravitational field through superposition of a nominal gravitational field and a residual gravitational field; because the residual gravitational field uses a numerical interpolation method, the composite gravitational field model has higher numerical calculation efficiency, can reflect the influence of uneven mass distribution of the main planets and the asteroids on the gravitational field, and improves the prediction precision; the gravitational field of the main small planet is described by establishing the composite gravitational field, so that the calculation load is reduced, and the interaction modeling efficiency of the double small planet system is improved;
step two, establishing a finite element model of the minor planet; dividing the minor planets into a limited number of domain elements, and calculating interpolation coefficients of the domain element nodes for the corresponding domain elements; according to the precision of the domain element division, the truncation error of the interaction of the double asteroid systems, which is increased along with the reduction of the distance between the small celestial bodies, can be effectively controlled;
determining a gravitational field parameter of each domain element node in a main and asteroid gravitational field according to the composite gravitational field model in the step one, wherein the gravitational field parameters comprise gravitational potential energy and a gravitational potential energy gradient value; the gravitational potential energy and the gravitational potential energy gradient value are stored to prepare for modeling of the interaction of the double small planetary systems, so that the repeated calculation of the domain element node gravitational field parameters is avoided, and the interaction modeling efficiency of the double small planetary systems is improved;
step four, determining that each minor asteroid domain element is acted by the main asteroid based on the interpolation coefficient obtained in the step two and the gravitational field parameters stored in the step three, wherein the action acted on each domain element can reflect the uneven density distribution of the minor asteroid at the domain element, so that the interactive prediction precision of the double-minor-planet system is improved; by establishing a finite element model of the minor asteroid, the gravitational field of the main asteroid can be fully utilized in the process of calculating the interaction force and the moment, and the calculation efficiency is improved; the method comprehensively considers the action of the main asteroid on all secondary asteroid field elements, and realizes the rapid modeling of interaction force and moment between two asteroids;
and step five, substituting the state of the double-asteroid system at the current moment into the step four to carry out rapid modeling and prediction on the interaction force and the moment of the double-asteroid system, and substituting the modeling result of the interaction force and the moment of the double-asteroid system obtained in the step four into a differential equation of relative motion of the double-asteroid system to obtain the state of the double-asteroid system at the next moment.
2. The method for modeling the interaction of the irregularly-shaped double-asteroid system with high precision and high efficiency as claimed in claim 1, characterized in that: and sixthly, repeating the step five, predicting the evolution process of the double-asteroid system, and solving the technical problem of the related engineering of the double-asteroid system according to the evolution result of the double-asteroid system.
3. The irregular double small planetary system interaction high-precision high-efficiency modeling method as claimed in claim 1 or 2, wherein: the first implementation method comprises the following steps of,
the composite gravitational field model can efficiently describe the gravitational field of a single small planet, and when the mass distribution of the small planet is not uniform, the composite gravitational field is still effective; the composite gravitational field model passes through a nominal gravitational field VNDescription of true gravitational field V by superposition with residual gravitational field T
V＝VN+T (1)
Nominal gravitational field VNIs a triaxial ellipsoid gravitational field used for describing the main characteristics of a single asteroid gravitational field;
the composite gravitational field effectively covers a spatial region near the asteroid through the layered stacking of the wedge-shaped grids; each wedge-shaped mesh comprises six vertexes, and the position vector of ith vertex of k-th layer mesh
Figure FDA0003241520430000021
And a vector A of the vertex position of the surface of a polyhedron triangle for describing the geometric shape of a small celestial bodyiIn a relationship of
Figure FDA0003241520430000022
Wherein Θ is the average solid angle of the surface of the entire polyhedron;
after the wedge-shaped grid structure of the composite gravitational field is established, the real gravitational field V and the nominal gravitational field V on each grid node need to be calculatedNThe residual value omega, i.e. the difference u epsilon R of the gravitational potential energy function and the difference g [ g ] of the gravitational potential energy gradientx,gy,gz]∈R1×3(ii) a Defining a polyhedral model describing the shape of the main asteroid with nFacetA surface, when the wedge-shaped grid has nLayersWhen a layer, it contains n in totalLaypers×nFacetA grid, if each wedge grid is nNodePoint wedge units, then all contain nLaypers×nFacet×nNodeA node; save all residual values as nLaypers×nFacet×nnodeA matrix of rows and 4 columns, each row of the matrix being a row of data of a residual value ω on a node, i.e., ω ═ u, gx,gy,gz]；
The residual gravitational field T is represented by a numerical interpolation method, and the residual value in the local area of the wedge-shaped grid unit can be effectively determined
Figure FDA0003241520430000023
Wherein the coefficient ωiAnd piRespectively real residual values and basis functions on the ith interpolation node in the wedge-shaped grid unit, wherein l is a natural coordinate of a designated field point in the current wedge-shaped grid, and n is the current wedge-shaped unit; because the composite gravitational field model comprises the description of the residual gravitational field, the composite gravitational field model can quickly describe the gravitational field change of the real gravitational field caused by uneven distribution of the masses of the asteroids.
4. The method for modeling the interaction of the irregularly-shaped double-asteroid system with high precision and high efficiency as claimed in claim 3, characterized in that: the second step is realized by the method that,
presetting finite element division precision of minor planets, utilizing finite element division algorithm to disperse target minor planets into finite field element set, defining finite element model containing nFEMIndividual field element, nFEMVertA domain metanode; within each field element, calculating the interpolation coefficient of the field element node for the current field element
κ＝fκ(x,y,z) (4)
Wherein f isκThe method is an interpolation coefficient calculation method corresponding to the type of the domain element, and x, y and z are the vertex position parameters of the domain element in the current domain element.
5. The irregular-shape double-asteroid system interaction high-precision high-efficiency modeling method as claimed in claim 4, wherein: the third step is to realize the method as follows,
defining the position vector of any domain element node in the main asteroid fixed connection coordinate system as r, and then the gravitational potential energy of the domain element node in the composite gravitational field model is
Figure FDA0003241520430000031
The gradient of gravitational potential energy is
Figure FDA0003241520430000032
Wherein, UN(r) gravitational potential energy of a nominal gravitational field, uiIs the residual value of the gravitational potential energy on the ith interpolation node of the wedge-shaped unit where the domain element node is positioned, pi(r) is the basis function of the wedge-shaped element,
Figure FDA0003241520430000033
is the gravitational potential energy gradient of the nominal gravitational field, gi＝[gxi,gyi,gzi]TIs the residual error value of the gravitational potential energy gradient on the ith interpolation node of the wedge-shaped unit where the finite element node is located, and n is the wedge-shaped unit where the domain element node is located;
the gravitational potential energy value U (R) epsilon R and the gravitational potential energy gradient value on each domain element node are calculated
Figure FDA0003241520430000034
Is stored as nFEMVertA row 4 column matrix, each row of which is a row data of gravitational field parameters on a field element node, i.e. f ═ fp,fx,fy,fz](ii) a And storing the gravitational potential energy and the gravitational potential energy gradient value to prepare for modeling of the interaction of the double small planetary systems, thereby avoiding the repeated calculation of the gravitational field parameters of the domain element nodes.
6. The irregular-shaped double-small planetary system interaction high-precision high-efficiency modeling method as claimed in claim 5, wherein: the implementation method of the fourth step is that,
the interaction of the double asteroid system comprises the mutual potential P in the double asteroid system, the force F of the main asteroid to the secondary asteroid, and the moment mu of the main asteroid to the secondary asteroid1Moment mu of secondary asteroid to primary asteroid2(ii) a According to the finite element model established in the second step, the interaction between the two asteroids is the sum of the action of the main asteroid on each minor asteroid area element
Figure FDA0003241520430000035
Figure FDA0003241520430000041
Figure FDA0003241520430000042
μ2＝-R×F-μ1 (10)
Wherein the content of the first and second substances,
Figure FDA0003241520430000043
is a finite element field element of which the number of elements,
Figure FDA0003241520430000045
is a set of finite element field elements of a finite element model of the minor asteroid, D is a projection of a node position vector of the finite element field elements of the minor asteroid in a fixed coordinate system of the minor asteroid, kappa is an interpolation coefficient of the nodes of the finite element field elements in the current field element, sigma is a density value of the minor asteroid at the nodes of the finite element field elements, a first summation symbol means that all the finite element field elements are summed in the set of finite element field elements, a second summation symbol means that all the nodes of the finite element field elements are summed in a single finite element field element, and a third summation symbol means that the nodes of the finite element field elements are summed in the set of the finite element field elementsSumming all interpolation nodes in a single wedge-shaped grid unit;
because the gravitational field parameter values on the nodes are already calculated in the third step, the node parameters are only required to be calculated according to the node density values of different minor planet domain elements in the fourth step; the gravitational field parameters stored in the composite gravitational field can describe the uneven mass distribution of the main asteroid, and the domain element node parameter values in the third step can reflect the uneven mass distribution of the secondary asteroid, so that the uneven mass distribution of the main asteroid and the secondary asteroid in the double asteroid system can be described on the basis, and the modeling precision of the interaction force and the moment of the double asteroid system is further improved;
all field elements of the secondary asteroid are traversed and summed to obtain the mutual potential P, the force F of the primary asteroid to the secondary asteroid and the moment mu of the primary asteroid to the secondary asteroid in the double asteroid system1Moment mu of secondary asteroid to primary asteroid2Therefore, the quick modeling of the interaction of the double asteroid system is realized.
7. The irregular-shape double-asteroid system interaction high-precision high-efficiency modeling method as claimed in claim 6, wherein: the fifth step is to realize that the method is that,
differential equation of relative motion of double minor planet system is
Figure FDA0003241520430000044
Wherein P is the bus momentum of the system, R is the position vector pointing from the centroid of the minor celestial body 2 to the centroid of the minor celestial body 1, and ΓiIs the angular momentum of the small celestial body, omegaiIs the angular velocity of the small celestial body, A is the attitude matrix transferred from the fixed coordinate system of the small celestial body 2 to the fixed coordinate system of the small celestial body 1, A2Is an attitude matrix which is formed by fixedly connecting a coordinate system with a small celestial body 2 and transferring the coordinate system to an inertial coordinate system; f is the force of the celestial body 2 on celestial body 1, μ1(μ2) Is the moment of the small celestial body 2 (small celestial body 1) to the small celestial body 1 (small celestial body 2).
8. The irregular double small planetary system interaction high-precision high-efficiency modeling method as claimed in claim 1 or 2, wherein: in order to further improve the prediction efficiency, a second-order spherical harmonic function is adopted to describe the gravitational field of the triaxial ellipsoid
Figure FDA0003241520430000051
Wherein mu is the gravitational constant of an ellipsoid,
Figure FDA0003241520430000052
is the distance of the field point from the center of the ellipsoid, C20And C22Is the spherical harmonic coefficient, r ═ x, y, z]TIs the position vector of the field point.
9. The irregular-shaped double-small planetary system interaction high-precision high-efficiency modeling method as claimed in claim 8, wherein: in the second step, the finite element partition precision of the minor planet is set, the target polyhedron is dispersed into a set of finite tetrahedral domain elements by using a finite element partition algorithm, and the interpolation coefficient of the vertex of the domain element for the current domain element is calculated in each tetrahedral domain element
Figure FDA0003241520430000053
Wherein x isi,yi,ziAnd i is the position coordinate of the ith node of the tetrahedral field element 1,2,3 and 4.
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CN114485620A (en) * 2022-01-29 2022-05-13 中国科学院国家空间科学中心 Orbital dynamics fused asteroid detector autonomous visual positioning system and method
CN115640720A (en) * 2022-10-14 2023-01-24 中国科学院国家空间科学中心 Self-gravitation simulation method based on distance control grid encryption

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CN114485620A (en) * 2022-01-29 2022-05-13 中国科学院国家空间科学中心 Orbital dynamics fused asteroid detector autonomous visual positioning system and method
CN114485620B (en) * 2022-01-29 2023-07-28 中国科学院国家空间科学中心 Autonomous visual positioning system and method for asteroid detector fused with orbit dynamics
CN115640720A (en) * 2022-10-14 2023-01-24 中国科学院国家空间科学中心 Self-gravitation simulation method based on distance control grid encryption
CN115640720B (en) * 2022-10-14 2023-10-10 长安大学 Self-attraction simulation method based on distance control grid encryption

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