Spacecraft long-term collision risk macroscopic assessment method

Abstract

The invention discloses a spacecraft long-term collision risk assessment method, and belongs to the field of aerospace. The implementation method comprises the following steps: dividing space objects into a plurality of space object groups according to a geocentric distance, a right ascension, a declination, an orbit inclination angle and an area-mass ratio, selecting the density of the object groups as the macroscopic representation of the space debris environment, considering multi-source environmental factors such as atmospheric resistance, and establishing a space debris environment macroscopic representation and rapid evolution model by adopting a continuity equation; giving a numerical solution of the evolution model; the density of a space object in a certain volume element, the density of a spacecraft, the collision sectional area and the collision speed are calculated respectively, then the long-term collision probability of the spacecraft and the space object is calculated, the long-term collision risk of the space station is evaluated based on a long-term collision risk macroscopic evaluation method, and the safest orbit of the space station is determined according to the evaluation result. Satellite and constellation launching and task post-processing strategy optimization, space traffic management and space debris treatment are supported.

Description

Technical Field

[0001] The invention relates to a macroscopic assessment method for long-term collision risk of a spacecraft, and belongs to the field of aerospace. Background Art

[0002] The increasingly deteriorating space debris environment will greatly increase the collision risk of spacecraft in orbit. Therefore, long-term collision risk assessment for target spacecraft is a prerequisite for the safe implementation of space missions.

[0003] The analysis of spacecraft collision risk is based on the evolution of the debris environment, which is mainly reflected in the solution of the spacecraft collision probability. The calculation methods of spacecraft collision probability can be divided into short-term (instantaneous) and long-term methods. The short-term collision probability is calculated based on the position ellipsoid error of the space object and is usually used as a criterion for collision warning. However, the short-term method is greatly affected by the orbit prediction error, which means that it cannot play a role in the long-term collision risk of spacecraft. The long-term method was originally proposed by Opik, and was later improved and extended by Wetherill and Kessler to evaluate the collision risk between asteroids, comets and planetary fragments. This method assumes that a, e, i of each object are fixed, and Ω and ω are uniformly distributed. By randomly or uniformly sampling all geometric shapes of the close encounter between the two objects, the long-term average collision probability between the two objects is obtained. However, this assumption is invalid for objects with rapidly changing a, e, and i. To solve this problem, NASA proposed the CUBE model, which uses gas kinetic theory at a microscopic scale (a cube) and uses the spatial density of objects in the cube to evaluate the collision probability. Compared with traditional long-term collision probability algorithms, algorithms such as CUBE can dynamically evaluate the collision probability during the evolution of the spacecraft orbit by uniformly sampling in time and updating the orbital elements at each time step. However, it needs to rely on the prediction results of the motion state of a single fragment, which has the problem of large computing resource requirements. Summary of the invention

[0004] The purpose of the present invention is to solve the problem that the current long-term collision probability algorithm has high computational complexity and low efficiency, which leads to the inability to efficiently analyze the spacecraft collision risk, and to provide a macroscopic assessment method for the long-term collision risk of spacecraft. The method divides space objects into several space object groups according to the distance from the center of the earth, right ascension, declination, orbital inclination and surface mass ratio, selects the density of the object group as the macroscopic representation of the space debris environment, considers multi-source environmental factors such as atmospheric resistance, and uses the continuity equation to establish a macroscopic representation and rapid evolution model of the space debris environment, and gives the numerical solution of the evolution model; respectively calculates the density of space objects, the density of spacecraft, the collision cross-sectional area and the collision speed in a certain volume element, and then calculates the long-term collision probability of spacecraft and space objects. Based on the proposed long-term collision probability algorithm for spacecraft, the long-term collision risk of the Chinese space station (space station) is evaluated and the safest orbit of the space station is discussed, supporting the optimization of satellite and constellation launch and post-mission disposal strategies, space traffic management and space debris governance, and solving related engineering and technical problems.

[0005] The objectives of the present invention are achieved through the following technical solutions.

[0006] The present invention discloses a macroscopic assessment method for the long-term collision risk of spacecraft, establishes a macroscopic representation and rapid evolution model of the space debris environment, divides the LEO orbit into a number of space volume elements according to the distance from the center of the earth, right ascension, declination and surface mass ratio, and then divides the space objects in each volume element into a number of space object groups according to the orbital inclination and surface mass ratio, selects the density of the object group as the macroscopic representation of the space debris environment, considers atmospheric resistance, J 2 Perturbations and other multi-source environmental factors are taken into account to improve the accuracy of the space debris evolution prediction model, the space debris is discretized in layers, and the continuity equation is used to model the evolution of the space debris environment, which saves computing resources and improves the efficiency of evolution prediction; the corresponding first-order two-dimensional hyperbolic partial differential equations are numerically solved, and the macroscopic representation of the space debris environment and the numerical solution of the rapid evolution model are given; the density of space objects, the density of spacecraft, the collision cross-sectional area and the collision velocity in a certain volume element are calculated respectively, and then the long-term collision probability between the spacecraft and the space object is calculated; based on the proposed long-term collision probability algorithm for spacecraft, the long-term collision risk of the space station is evaluated and the safest orbit of the space station is discussed.

[0007] The present invention discloses a macroscopic assessment method for long-term collision risk of spacecraft, comprising the following steps:

[0008] Step 1: Divide the LEO orbit into several space volume elements according to the distance from the center of the Earth, right ascension and declination, and then divide the space objects in each volume element into several space object groups according to the orbital inclination and surface mass ratio. The density of the space object group is a state variable to improve the calculation efficiency of steps 2 to 4.

[0009] First, the LEO orbit is calculated according to the distance from the center of the earth r, right ascension λ and declination Divide into The space volume elements are divided into Δr, Δλ, For the space volume element U i,j,k , the spatial range it contains is determined by formula (1);

[0010]

[0011] Among them, r i is the median value of the ith geocentric distance layer, is the median value of the jth declination layer, λ k is the median value of the kth right ascension layer; the space objects in each volume element are divided into N according to the orbital inclination and surface mass fraction i ×N a interval; therefore The total number of spatial volume elements is Space object group; volume element U i,j,k The space object group within the pth orbital inclination interval and the qth surface mass ratio interval The spatial density as a state variable.

[0012] Step 2: Consider the continuity factor in the process of debris environment evolution, establish the total perturbation motion equation of the spacecraft under the combination of the Earth's flattening perturbation and atmospheric perturbation, use the average orbital decay rate of space objects to characterize the influence of atmospheric drag on the density evolution of the space object group; use the average transfer speed of the space object group in the same latitude and different longitude intervals to characterize J 2 The influence of perturbations on the density evolution of space object groups; and construct an expression for the influence of continuity factors on the density evolution of space object groups, that is, construct expressions for the average orbital decay velocity and the average transfer velocity respectively, so as to improve the accuracy of the macroscopic representation of the space debris environment and the rapid evolution model in step four.

[0013] Step 2.1: The total perturbation equation of motion of the spacecraft under the Earth's flattening perturbation and atmospheric perturbation is:

[0014]

[0015] Where a is the semi-major axis of the orbit; i is the orbit inclination; Ω is the right ascension of the ascending node; e is the orbit eccentricity; ω is the depression angle of the perigee; f is the true anomaly; M is the mean anomaly of the space object; r is the distance from the center of the earth; n is the average angular velocity of motion; ρ is the atmospheric density; S is the frontal area of ​​the space object; n represents the average angular velocity of motion; m o is the mass of the space object; J 2 is the Earth's oblateness J 2 Item; Cd is the atmospheric drag coefficient; R e is the equatorial radius of the earth; t is the time;

[0016] Step 2.2: For The average decay rate of space objects in the group It is expressed as:

[0017]

[0018] For an object running on a circular orbit, the orbital eccentricity e = 0. Substituting it into equation (2), we get:

[0019]

[0020] Where ρ is the atmospheric density; (A / m o ) q represents the median value of the surface mass fraction layer of q; μ is the earth's gravitational constant; thus, we can know that It is a function related to atmospheric density, orbital altitude, and object surface-to-mass ratio. For a certain space object with a known orbital altitude, the atmospheric density ρ can be determined to obtain Taking into account solar and geomagnetic activities, the atmospheric density model is expressed as:

[0021]

[0022] Where T is the temperature; F 10.7 is the solar radio flux at 10.7 cm; A p is the geomagnetic index; m is the molecular mass, which is defined as a function of the altitude h; thus, the atmospheric density expression is expressed as:

[0023] ρ=6×10 -10 exp(0.012h 2 -31.5h+5145 / T) (6)

[0024] Substituting (6) into (4), we obtain the average orbital decay rate:

[0025]

[0026] Step 2.3: The relationship between the longitude and latitude of a point on the orbit of a space object and the orbital elements is as shown in equation (8);

[0027]

[0028] It can be seen that the longitude drift speed of any point in space is Drift velocity with right ascension of ascending node equal; for The average transfer speed of space objects in the group From equation (9), we get:

[0029]

[0030] Among them, i p represents the median value of the p-th orbital inclination layer; a e is the equatorial radius of the Earth.

[0031] Step 3: Consider the discontinuity factors in the evolution of the debris environment and model the space launch activities; then use the topological network model to further characterize and model the space activities to obtain a model of discontinuity factors; this will help improve the accuracy of the macroscopic characterization and rapid evolution model of the space debris environment in step 4.

[0032] Step 3.1: Consider the discontinuity factor in the evolution of the debris environment and model the space launch activities to obtain the model

[0033] 1) Based on the actual launch data from 2020 to 2023, the upper and lower bounds of the number of payloads and rocket bodies launched each month / year are obtained. The number of future launches is uniformly distributed within this range. The number of future launches per month / year is obtained by random sampling. L ;

[0034] 2) Using a Gaussian mixture model (GMM), we fit the actual launch information of the payload and rocket body from 2020 to 2023, including orbital parameters and dimensions, and obtain Gaussian mixture results of the actual launch information of the payload and rocket body;

[0035] 3) Finally, the Markov Chain Monte Carlo method (MCMC) is used to extract N from the Gaussian mixture results obtained in step 2). L launch samples, and finally obtain a prediction model for future launch activities Indicates that the emission Satellite density;

[0036] Step 3.2: Construct a physical layer topology network model that considers three activities: space launch, satellite post-mission disposal (PMD), and natural orbit reduction; consider three types of space objects related to space activities, namely, working satellites (S), failed satellites (F), and satellite remains / debris (N), and then divide the three objects into nodes according to orbital altitude and face value. Represents three types of objects in The density of the group, the dynamic equations of each node are (10)~(12);

[0037]

[0038] Where: for The density of new failed satellites generated in the system; τ is the success rate of PMD; υ PMD is the PMD descending speed; is the natural orbit descent speed; It's J 2 The speed of object transfer under perturbation; therefore, the model of space activity, that is, the discontinuity factor, is:

[0039]

[0040] Step 4: Combined with the expression of the influence of continuity factors on the density evolution of space object groups in step 2 and the model of discontinuity factors in step 3, the continuity equation in fluid mechanics is used to establish a macroscopic representation and rapid evolution model of the space debris environment, so that the established space debris evolution prediction model can improve the efficiency of evolution prediction while ensuring the accuracy of evolution prediction.

[0041] The continuity equation (14) in fluid mechanics is used to model the density evolution of space debris;

[0042]

[0043] in, represents the continuity factor acting on the system, represents the discontinuous factor; the continuous factors considered are the atmospheric drag perturbation and J 2 perturbation, and the discontinuous factors are space launch activities and post-satellite mission disposal; therefore, considering the atmospheric drag, J 2 Perturbation, space launch and satellite mission post-processing The group density evolution equation changes from (14) to (15); the left side of the formula is the change in the number of space objects, and the first and second terms on the right side represent the atmospheric drag and J, respectively. 2 Group of objects under the influence of perturbation The third term represents the change in the number of objects caused by space activities;

[0044]

[0045] Among them, V i,j,k is the volume element U i,j,k The volume of is the cross-sectional area in the direction of atmospheric resistance, For J 2 The cross-sectional area in the direction of perturbation is expressed as follows; Space objects caused by atmospheric drag The average decay rate of the orbital altitude is For J 2 The average transfer speed of space objects caused by perturbations in the same latitude but different longitude intervals, It is caused by space activities The amount of change in density of objects within the group;

[0046]

[0047]

[0048] Substitute equations (16) to (18) into (15); then divide both sides of the equation by Δr·Δλ·r 2 ; Let Δr→0, Δλ→0, Formula (15) is simplified to Formula (19), which gives the macroscopic representation and rapid evolution model of the space debris environment:

[0049]

[0050] Step 5: Numerically solve the macroscopic characterization and rapid evolution model of the space debris environment in step 4 to obtain A first-order two-dimensional hyperbolic partial differential equation is solved, and the numerical solution of the macroscopic characterization and rapid evolution model of the space debris environment is given.

[0051] The evolution state of the debris environment is obtained by numerically solving equation (19); at this point, the solution to the evolution state of each object can be regarded as a numerical solution A first-order two-dimensional hyperbolic partial differential equation, written in the form of a standard system of equations, is given by equation (20):

[0052]

[0053] Among them, n is included indivual A and B are diagonal matrices:

[0054]

[0055] in, is the average decay rate of space objects in the first altitude interval, the first right ascension interval, the first declination interval, the first aspect ratio interval, and the first orbital inclination interval, is the average decay velocity of the space object in the i-th altitude interval, the k-th right ascension interval, the j-th declination interval, the q-th aspect ratio interval, and the p-th orbital inclination interval, It is the Nth h height interval, Nth λ Right ascension interval, Declination interval, Nth a The face value ratio interval and the Nth i The average decay speed of space objects in the orbital inclination interval is is the average transfer speed of the space object in the first altitude interval, the first right ascension interval, the first declination interval, the first aspect ratio interval and the first orbital inclination interval, is the average transfer speed of the space object in the i-th altitude interval, the k-th right ascension interval, the j-th declination interval, the q-th aspect ratio interval, and the p-th orbital inclination interval, It is the Nth h height interval, Nth λ Right ascension interval, Declination interval, Nth a The face value ratio interval and the Nth i The average transfer speed of space objects in the orbital inclination interval.

[0056] For a space object group with k latitude interval, p orbital inclination interval, and q surface value ratio interval, the difference format of the evolution state is written as:

[0057]

[0058] Among them, τ, h 1 and h 2 are the discrete time interval, altitude interval and longitude interval in differential approximation respectively. The superscript δ represents the δth discrete time step. When selecting the discrete interval, it should be ensured that in each step of calculation, the fragments in the same volume element are transferred to the adjacent volume element at most. represents the density of the object in the i-th altitude interval and the j-th longitude interval at the time δ+1, represents the density of objects in the i-th altitude interval and the j-th longitude interval at time δ, represents the density of the object in the i+1th altitude interval and the jth longitude interval at the time δ+1, is the right side term of the equation (19), It can be expressed as:

[0059]

[0060] in, represents the density of objects in the i-th altitude interval and the j-1-th longitude interval at time δ, It represents the density of objects in the i-th altitude interval and the j+1-th longitude interval at time δ.

[0061] Step 6: Use the numerical solution of step 5 to calculate the density of space objects, density of spacecraft, collision cross-sectional area and collision velocity in a certain volume element, and then macroscopically calculate the long-term collision risk between spacecraft and space objects.

[0062] The volume element U is obtained by numerical calculation in step 5. i,j,k Space object group in the Pth orbital inclination interval and the Qth face value ratio interval Density And calculate the spacecraft density in this volume element, the pth orbital inclination interval, and the qth surface value ratio interval Object Group The collision cross-sectional area with the spacecraft is expressed as σ(Q,q), using the average cross-sectional area A of the object group Q and the spacecraft cross-sectional area A q To define:

[0063]

[0064] Space object group The relative collision velocity with the spacecraft reflects the relative motion state of the spacecraft and the space object group within the volume element and is determined by formula (26):

[0065]

[0066] Among them, v P is the speed of the space object group, v p is the spacecraft velocity; since the volume element scale is small enough, v P It can be expressed by the velocity at the midpoint of the volume element, satisfying the following conditions:

[0067]

[0068] n 1 is the vector from the center of the earth to the center of the volume element, n 2 is the normal vector to the orbital plane of the object group, n 3 are the normal vectors to the equatorial plane, and they mutually satisfy the following conditions:

[0069]

[0070] Then, the long-term collision risk between spacecraft and space objects is calculated macroscopically. i,j,k The average collision probability of spacecraft per unit time is:

[0071]

[0072] Among them, dU i,j,k Represents volume element U i,j,k Volume;

[0073] Therefore, the number of collisions during the spacecraft mission can be expressed as:

[0074]

[0075] Among them, T mission is the task time, U t is the volume element where the spacecraft is located at time t, is the density of the object group in the volume element where the spacecraft is located at time t, The spatial density of spacecraft, is the collision velocity between the spacecraft and the space object group at time t.

[0076] Step 7: Based on the long-term collision risk macro-assessment method, the long-term collision risk of the space station is assessed, and the safest orbit of the space station is determined based on the assessment results, to support the optimization of satellite and constellation launches and their post-mission disposal strategies, space traffic management and space debris management, and to solve related engineering and technical problems.

[0077] Beneficial effects:

[0078] 1. The present invention discloses a macroscopic assessment method for the long-term collision risk of spacecraft. By establishing a macroscopic representation and rapid evolution model of the space debris environment, the space debris is discretized in layers. The LEO orbit is divided into a number of space volume elements according to the distance from the center of the earth, right ascension, declination and surface mass ratio, and the space objects in each volume element are divided into a number of space object groups according to the orbital inclination and the surface mass ratio. The density of the object group is selected as the state variable, and the continuity equation in fluid mechanics is used to model the space debris environment, which can save computing resources and make the evolution model have high computing efficiency.

[0079] 2. The present invention discloses a method for assessing the long-term collision risk of spacecraft, which is implemented based on the macroscopic representation and rapid evolution model of the space debris environment. Therefore, it has the advantage of beneficial effect 1. Based on the proposed macroscopic representation and rapid evolution model of the space debris environment, the density of space objects, the density of spacecraft, the collision cross-sectional area and the collision speed in a certain volume element are calculated respectively, and then the long-term collision probability between the spacecraft and the space object is calculated, which can greatly reduce the computational complexity of the traditional model and has high computational efficiency.

[0080] 3. The present invention discloses a method for assessing the long-term collision risk of spacecraft. Based on the proposed algorithm for the long-term collision probability of spacecraft, the method assesses the long-term collision risk of the Chinese space station and discusses its safest orbit, supports the optimization of satellite and constellation launches and their post-mission disposal strategies, space traffic management and space debris management, and solves related engineering and technical problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 A schematic diagram of the spacecraft speed in the embodiment;

[0082] Figure 2 The orbit of the Chinese space station and the density distribution of space objects in its operating space in the embodiment;

[0083] Figure 3 is the collision probability between the Chinese space station and objects of different categories in the embodiment;

[0084] Figure 4 Schematic diagram of the distribution of objects at different orbital altitudes in the embodiment;

[0085] Figure 5 Schematic diagram of the average collision probability of the Chinese space station at different orbital elements in the embodiment;

[0086] Figure 6 The present invention is a flow chart of a method for assessing the long-term collision risk of a spacecraft. DETAILED DESCRIPTION

[0087] In order to better illustrate the purpose, technical solutions and advantages of the present invention, the invention is further described below in conjunction with the accompanying drawings and embodiments.

[0088] As of February 2024, the number of cataloguable objects in the space environment has reached nearly 28,000, of which spacecraft account for only one-third, and the rest are space debris. The deteriorating space debris environment will greatly increase the collision risk of spacecraft in orbit. Therefore, for the target spacecraft, analyzing the evolution of debris in its operating space and conducting long-term collision risk assessment are prerequisites for the safe implementation of space missions.

[0089] At present, there are two main types of space debris environment evolution models. One type is the microscopic model that tracks the motion state of a single debris, such as NASA's LEGEND, ESA's DELTA, Italian National Research Institute's SDM, UK Space Agency's DAMAGE, Chinese Academy of Sciences' SOLEM, Harbin Institute of Technology's SDEEM and other engineering models. The debris events of this type of model are implemented with Monte Carlo technology, which relies on a lot of computing resources and time. The other type is the macroscopic model with the number and density of debris as variables. The macroscopic model is established using differential equations (groups), and the evolution of the number of debris can be obtained by solving the differential equations, which has considerable computational efficiency compared to the microscopic model.

[0090] The current macro models have certain limitations. Almost all macro models have the problem of simple spatial division (grouping only according to orbital altitude, and even not distinguishing between altitude layers), and cannot predict the state of space objects at any spatial position (latitude and longitude); moreover, the perturbation models of most models are relatively simple, such as not considering the perturbation of the earth's flattening, and the atmospheric drag model does not consider the influence of solar activity and geomagnetic activity. The impact of these environmental factors on the evolution of debris cannot be ignored. The right ascension drift of the ascending node caused by the earth's flattening perturbation will cause the transfer of space objects between different longitude areas at the same latitude; and extreme weather such as geomagnetic storms will cause a sudden increase in atmospheric drag and accelerate the decay of the object's orbit. For example, the 40 Starlink satellites launched on February 3, 2022 were affected by geomagnetic storms and re-entered the earth's atmosphere and were damaged. It can be seen that although the macro model has great advantages in efficiency, the accuracy of spatial division and environmental perturbations still needs to be improved.

[0091] The analysis of spacecraft collision risk is based on the evolution of the debris environment, which is mainly reflected in the solution of the spacecraft collision probability. The calculation methods of spacecraft collision probability can be divided into short-term (instantaneous) and long-term methods. The short-term collision probability is calculated based on the position ellipsoid error of the space object and is usually used as a criterion for collision warning. However, the short-term method is greatly affected by the orbit prediction error, which means that it cannot play a role in the long-term collision risk of spacecraft. The long-term method was originally proposed by Opik, and was later improved and extended by Wetherill, Kessler and Greenberg to evaluate the collision risk between asteroids, comets and planetary fragments. This method assumes that a, e, i of each object are fixed, and Ω and ω are uniformly distributed. By randomly or uniformly sampling all geometric shapes of close encounters between two objects, the long-term average collision probability between the two objects is obtained. However, this assumption is invalid for objects with rapidly changing a, e, and i. To solve this problem, NASA proposed the CUBE model, which uses gas kinetic theory at a microscopic scale (a cube) and uses the spatial density of objects in the cube to evaluate the collision probability. Compared with traditional long-term collision probability algorithms, algorithms such as CUBE can dynamically evaluate the collision probability during the evolution of the spacecraft orbit by uniformly sampling in time and updating the orbital elements at each time step. However, they usually rely on the prediction results of micro-debris evolution models, and therefore also have the problem of large computing resource requirements.

[0092] like Figure 6 As shown, the present embodiment discloses a method for assessing the long-term collision risk of a spacecraft, and the specific implementation steps are as follows:

[0093] Step 1: Divide the LEO orbit into several space volume elements according to the distance from the center of the Earth, right ascension and declination, and then divide the space objects in each volume element into several space object groups according to the orbital inclination and surface mass ratio. The density of the space object group is a state variable to improve the calculation efficiency of steps 2 to 4.

[0094] First, the LEO orbit is calculated according to the distance from the center of the earth r, right ascension λ and declination Divide into The space volume elements are divided into Δr, Δλ, For the space volume element U i,j,k , the spatial range it contains is determined by formula (1);

[0095]

[0096] Among them, r i is the median value of the ith geocentric distance layer, is the median value of the jth declination layer, λ k is the median value of the kth right ascension layer; the space objects in each volume element are divided into N according to the orbital inclination and surface mass fraction i ×N a interval; therefore The total number of spatial volume elements is Space object group; volume element U i,j,k The space object group within the pth orbital inclination interval and the qth surface mass ratio interval The spatial density as a state variable.

[0097] Step 2: Consider the continuity factor in the process of debris environment evolution, establish the total perturbation motion equation of the spacecraft under the combination of the Earth's flattening perturbation and atmospheric perturbation, use the average orbital decay rate of space objects to characterize the influence of atmospheric drag on the density evolution of the space object group; use the average transfer speed of the space object group in the same latitude and different longitude intervals to characterize J 2 The influence of perturbations on the density evolution of space object groups; and construct an expression for the influence of continuity factors on the density evolution of space object groups, that is, construct expressions for the average orbital decay velocity and the average transfer velocity respectively, so as to improve the accuracy of the macroscopic representation of the space debris environment and the rapid evolution model in step four.

[0098] Step 2.1: The total perturbation equation of motion of the spacecraft under the Earth's flattening perturbation and atmospheric perturbation is:

[0099]

[0100] Where a is the semi-major axis of the orbit; i is the orbit inclination; Ω is the right ascension of the ascending node; e is the orbit eccentricity; ω is the depression angle of the perigee; f is the true anomaly; M is the mean anomaly of the space object; r is the distance from the center of the earth; n is the average angular velocity of motion; ρ is the atmospheric density; S is the frontal area of ​​the space object; n represents the average angular velocity of motion; m o is the mass of the space object; J 2 is the Earth's oblateness J 2 Item; C d is the atmospheric drag coefficient; R e is the equatorial radius of the earth; t is the time;

[0101] Step 2.2: For The average decay rate of space objects in the group It is expressed as:

[0102]

[0103] For an object running on a circular orbit, the orbital eccentricity e = 0. Substituting it into equation (2), we get:

[0104]

[0105] Where ρ is the atmospheric density; (A / m o ) q represents the median value of the surface mass fraction layer of q; μ is the earth's gravitational constant; thus, we can know that It is a function related to atmospheric density, orbital altitude, and object surface-to-mass ratio. For a certain space object with a known orbital altitude, the atmospheric density ρ can be determined to obtain Taking into account solar and geomagnetic activities, the atmospheric density model is expressed as:

[0106]

[0107] Where T is the temperature; F 10.7 is the solar radio flux at 10.7 cm; A p is the geomagnetic index; m is the molecular mass, which is defined as a function of the altitude h; thus, the atmospheric density expression is expressed as:

[0108] ρ=6×10 -10 exp(0.012h 2 -31.5h+5145 / T) (36)

[0109] Substituting (6) into (4), we obtain the average orbital decay rate:

[0110]

[0111] Step 2.3: The relationship between the longitude and latitude of a point on the orbit of a space object and the orbital elements is as shown in equation (8);

[0112]

[0113] It can be seen that the longitude drift speed of any point in space is Drift velocity with right ascension of ascending node equal; for The average transfer speed of space objects in the group From equation (9), we get:

[0114]

[0115] Among them, i p represents the median value of the p-th orbital inclination layer; a e is the equatorial radius of the Earth.

[0116] Step 3: Consider the discontinuity factors in the evolution of the debris environment and model the space launch activities; then use the topological network model to further characterize and model the space activities to obtain a model of discontinuity factors; this will help improve the accuracy of the macroscopic characterization and rapid evolution model of the space debris environment in step 4.

[0117] Step 3.1: Consider the discontinuity factor in the evolution of the debris environment and model the space launch activities to obtain the model

[0118] 4) Based on the actual launch data from 2020 to 2023, the upper and lower bounds of the number of payloads and rocket bodies launched each month / year are obtained. The number of future launches is uniformly distributed within this range. The number of future launches per month / year is obtained by random sampling. L ;

[0119] 5) Using a Gaussian mixture model (GMM), the real launch information of the payload and rocket body from 2020 to 2023 is fitted, including orbital parameters and dimensions, and the Gaussian mixture results of the real launch information of the payload and rocket body are obtained;

[0120] 6) Finally, the Markov Chain Monte Carlo method (MCMC) is used to extract N from the Gaussian mixture results obtained in step 2). L launch samples, and finally obtain a prediction model for future launch activities Indicates that the emission Satellite density;

[0121] Step 3.2: Construct a physical layer topology network model that considers three activities: space launch, satellite post-mission disposal (PMD), and natural orbit reduction; consider three types of space objects related to space activities, namely, working satellites (S), failed satellites (F), and satellite remains / debris (N), and then divide the three objects into nodes according to orbital altitude and face value. Represents three types of objects in The density of the group, the dynamic equations of each node are (10)~(12);

[0122]

[0123]

[0124] Where: for The density of new failed satellites generated in the system; τ is the success rate of PMD; υ PMD is the PMD descending speed; is the natural orbit descent speed; It's J 2 The speed of object transfer under perturbation; therefore, the model of space activity, that is, the discontinuity factor, is:

[0125]

[0126] Step 4: Combined with the expression of the influence of continuity factors on the density evolution of space object groups in step 2 and the model of discontinuity factors in step 3, the continuity equation in fluid mechanics is used to establish a macroscopic representation and rapid evolution model of the space debris environment, so that the established space debris evolution prediction model can improve the efficiency of evolution prediction while ensuring the accuracy of evolution prediction.

[0127] The continuity equation (14) in fluid mechanics is used to model the density evolution of space debris;

[0128]

[0129] in, represents the continuity factor acting on the system, represents the discontinuous factor; the continuous factors considered are the atmospheric drag perturbation and J 2 perturbation, and the discontinuous factors are space launch activities and post-satellite mission disposal; therefore, considering the atmospheric drag, J 2 Perturbation, space launch and satellite mission post-processing The group density evolution equation changes from (14) to (15); the left side of the formula is the change in the number of space objects, and the first and second terms on the right side represent the atmospheric drag and J, respectively. 2Group of objects under the influence of perturbation The third term represents the change in the number of objects caused by space activities;

[0130]

[0131] Among them, V i,j,k is the volume element U i,j,k The volume of is the cross-sectional area in the direction of atmospheric resistance, For J 2 The cross-sectional area in the direction of perturbation is expressed as follows; Space objects caused by atmospheric drag The average decay rate of the orbital altitude is For J 2 The average transfer speed of space objects caused by perturbations in the same latitude but different longitude intervals, It is caused by space activities The amount of change in density of objects within the group;

[0132]

[0133] Substitute equations (16) to (18) into (15); then divide both sides of the equation by Δr·Δλ·r 2 ; Let Δr→0, Δλ→0, Formula (15) is simplified to Formula (19), which gives the macroscopic representation and rapid evolution model of the space debris environment:

[0134]

[0135] Step 5: Numerically solve the macroscopic characterization and rapid evolution model of the space debris environment in step 4 to obtain A first-order two-dimensional hyperbolic partial differential equation is solved, and the numerical solution of the macroscopic characterization and rapid evolution model of the space debris environment is given.

[0136] The evolution state of the debris environment is obtained by numerically solving equation (19); at this point, the solution to the evolution state of each object can be regarded as a numerical solution A first-order two-dimensional hyperbolic partial differential equation, written in the form of a standard system of equations, is given by equation (20):

[0137]

[0138] Among them, n is included indivual A and B are diagonal matrices:

[0139]

[0140] in, is the average decay rate of space objects in the first altitude interval, the first right ascension interval, the first declination interval, the first aspect ratio interval, and the first orbital inclination interval, is the average decay velocity of the space object in the i-th altitude interval, the k-th right ascension interval, the j-th declination interval, the q-th aspect ratio interval, and the p-th orbital inclination interval, It is the Nth h height interval, Nth λ Right ascension interval, Declination interval, Nth a The face value ratio interval and the Nth i The average decay speed of space objects in the orbital inclination interval is is the average transfer speed of the space object in the first altitude interval, the first right ascension interval, the first declination interval, the first aspect ratio interval and the first orbital inclination interval, is the average transfer speed of the space object in the i-th altitude interval, the k-th right ascension interval, the j-th declination interval, the q-th aspect ratio interval, and the p-th orbital inclination interval, It is the Nth h height interval, Nth λ Right ascension interval, Declination interval, Nth a The face value ratio interval and the Nth i The average transfer speed of space objects in the orbital inclination interval.

[0141] For a space object group with k latitude interval, p orbital inclination interval, and q surface value ratio interval, the difference format of the evolution state is written as:

[0142]

[0143] Among them, τ, h 1 and h 2 are the discrete time interval, altitude interval and longitude interval in differential approximation respectively. The superscript δ represents the δth discrete time step. When selecting the discrete interval, it should be ensured that in each step of calculation, the fragments in the same volume element are transferred to the adjacent volume element at most. represents the density of the object in the i-th altitude interval and the j-th longitude interval at the time δ+1, represents the density of objects in the i-th altitude interval and the j-th longitude interval at time δ, represents the density of the object in the i+1th altitude interval and the jth longitude interval at the time δ+1, is the right side term of the equation (19), It can be expressed as:

[0144]

[0145] in, represents the density of objects in the i-th altitude interval and the j-1-th longitude interval at time δ, It represents the density of objects in the i-th altitude interval and the j+1-th longitude interval at time δ.

[0146] Step 6: Use the numerical solution of step 5 to calculate the density of space objects, density of spacecraft, collision cross-sectional area and collision velocity in a certain volume element, and then macroscopically calculate the long-term collision risk between spacecraft and space objects.

[0147] The volume element U is obtained by numerical calculation in step 5. i,j,k Space object group in the Pth orbital inclination interval and the Qth face value ratio interval Density And calculate the spacecraft density in this volume element, the pth orbital inclination interval, and the qth surface value ratio interval Object Group The collision cross-sectional area with the spacecraft is expressed as σ(Q,q), using the average cross-sectional area A of the object group Q and the spacecraft cross-sectional area A q To define:

[0148]

[0149] Space object group The relative collision velocity with the spacecraft reflects the relative motion state of the spacecraft and the space object group within the volume element and is determined by formula (26):

[0150]

[0151] Among them, v P is the speed of the space object group, v p is the spacecraft speed, such as Figure 1 As shown; since the volume element scale is small enough, v P It can be expressed by the velocity at the midpoint of the volume element, satisfying the following conditions:

[0152]

[0153] n 1 is the vector from the center of the earth to the center of the volume element, n 2 is the normal vector to the orbital plane of the object group, n 3 are the normal vectors to the equatorial plane, and they mutually satisfy the following conditions:

[0154]

[0155] Then, the long-term collision risk between spacecraft and space objects is calculated macroscopically. i,j,k The average collision probability of spacecraft per unit time is:

[0156]

[0157] Among them, dU i,j,k Represents volume element U i,j,k Volume;

[0158] Therefore, the number of collisions during the spacecraft mission can be expressed as:

[0159]

[0160] Among them, T mission is the task time, U t is the volume element where the spacecraft is located at time t, is the density of the object group in the volume element where the spacecraft is located at time t, The spatial density of spacecraft, is the collision velocity between the spacecraft and the space object group at time t.

[0161] Step 7: Based on the long-term collision risk macro-assessment method, the long-term collision risk of the space station is assessed and the safest orbit of the space station is discussed to support the optimization of satellite and constellation launches and their post-mission disposal strategies, space traffic management and space debris management, and solve related engineering and technical problems.

[0162] In this embodiment, long-term collision risk analysis and assessment are carried out for important large spacecraft such as space stations, which is of great significance for the operation safety, collision protection strategy, and optimal orbit selection of spacecraft. Two simulation scenarios are set up to evaluate the long-term collision probability during the operation of the Chinese space station, and the safest orbit of the space station is discussed based on the collision probability results.

[0163] A. Long-term collision risk of China's space station

[0164] The Chinese space station operates in a near-circular low-Earth orbit of 340 to 450 km with an orbital inclination of 41 to 42°. Therefore, the orbital altitude range of [340,450] km is selected as the operating space of the space station, and the intervals of the division of the space volume element along the radial, right ascension and declination directions are Δh = 10 km, Δλ = 10°, It is divided into 7128 volume elements, and a three-dimensional network model is used to predict the evolution of objects in the operating space of the Chinese space station. On this basis, a collision risk assessment of the Chinese space station is conducted for 7 days. Table 1 gives a set of orbital elements of the Chinese space station as the initial state of the simulation.

[0165] Table 1 Orbital elements of China's space station

[0166]

[0167] Figure 2 The orbit of the Chinese space station and the density distribution of space objects in its operating space are shown. The density range of space objects in the volume element that the Chinese space station passes through during its operation is [0,3.5×10 -8 ] / km 3 . Figure 3 The collision probabilities of the Chinese space station with the payload, rocket body and debris are shown in Figure 2. It can be seen that the collision probability of the Chinese space station with the payload and space debris is one order of magnitude higher than that with the rocket body, and the collision risk with the payload is the highest on the fourth day, reaching 2.79×10 -8 / year / m 2 The highest risk of collision with space debris is at day 2.5, reaching 1.45×10 -8 / year / m 2 .

[0168] B. Safest orbit selection based on long-term collision probability

[0169] Figure 4 The distribution of objects in [340,450] km is shown (taking 340km, 390km, and 440km slices as examples). It can be seen that the higher the orbital altitude, the greater the density of space objects, and there is a clear clustering at the latitude of [-70,-90]°. Based on this space environment, the long-term collision probability of Chinese space stations in different orbits in this area is analyzed, and the safest orbit of the Chinese space station is discussed. In this case, the orbital altitude (semi-major axis) and orbital inclination of the space station in Case A are changed, and other orbital elements, space volume element division methods, and simulation time remain unchanged.

[0170] Figure 5 The average collision probability of space stations with different orbital roots (a, i) within 7 days is shown. In general, the collision risk of the Chinese space station is the smallest when the orbital altitude is 340km and 370km. As the orbital altitude increases, the collision risk gradually increases and reaches the maximum at the orbital altitude of 450km. At the same orbital altitude, the collision risk of the Chinese space station is the highest when the orbital inclination is 90°, which is consistent with the Figure 4 The displayed distribution of space objects can be verified with each other. In addition, the average collision probability of the Chinese space station is the lowest at an orbital altitude of 340km and an inclination of 50°, which is 1.81×10 -8 / year / m 2 ; The average collision probability is the highest at 450 km and an inclination of 90°, reaching 2.667×10 -7 / year / m2 , which is 14.7 times the former.

[0171] In summary, the above are only preferred embodiments of the present invention and are not intended to limit the protection scope of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A macroscopic assessment method for long-term collision risk of spacecraft, characterized by: The following steps are included: Step 1: Divide the LEO orbit into several space volume elements according to the distance from the center of the Earth, right ascension and declination, and then divide the space objects in each volume element into several space object groups according to the orbital inclination and surface mass fraction. The density of the space object group is the state variable; Step 2: Consider the continuity factor in the evolution of the debris environment, establish the total perturbation motion equation of the spacecraft under the combination of the Earth's oblateness perturbation and the atmospheric perturbation, use the average orbital decay rate of space objects to characterize the influence of atmospheric drag on the density evolution of the space object group; use the average transfer rate of the space object group in the same latitude and different longitude intervals to characterize the influence of the J2 perturbation on the density evolution of the space object group; and construct an expression for the influence of the continuity factor on the density evolution of the space object group, that is, construct expressions for the average orbital decay rate and the average transfer rate respectively, so as to improve the accuracy of the three-dimensional network evolution model of the space debris environment in step 4; Step 3: Considering the discontinuity factors in the evolution of the debris environment, model the space launch activities; then use the topological network model to further characterize and model the space activities to obtain a model of discontinuity factors; this is convenient for improving the accuracy of the three-dimensional network evolution model of the space debris environment in step 4; Step 4: Combined with the expression of the influence of the continuity factor on the density evolution of the space object group in step 2 and the model of the discontinuity factor in step 3, the continuity equation in fluid mechanics is used to establish a three-dimensional network evolution model of the space debris environment, so that the established space debris evolution prediction model can improve the evolution prediction efficiency while ensuring the evolution prediction accuracy; Step 5: Numerically solve the three-dimensional network evolution model of the space debris environment in step 4 and obtain A first-order two-dimensional hyperbolic partial differential equation is solved, and a numerical solution of a three-dimensional network model of space debris environment is given; Step 6: The density of space objects, the density of spacecraft, the collision cross-sectional area and the collision velocity within a certain volume element are calculated through the numerical solution obtained in step 5, and then the long-term collision risk between spacecraft and space objects is calculated macroscopically.

2. A macroscopic assessment method for long-term collision risk of spacecraft according to claim 1, characterized in that: It also includes step seven: Based on the long-term collision risk macro-assessment method, the long-term collision risk of the space station is assessed, and the safest orbit of the space station is determined based on the assessment results, to support the optimization of satellite and constellation launches and their post-mission disposal strategies, space traffic management and space debris management, and to solve related engineering and technical problems.

3. A macroscopic assessment method for long-term collision risk of spacecraft according to claim 1 or 2, characterized in that: The implementation method of step one is: Discretize the space debris in layers; divide the LEO orbit into several space volume elements according to the distance from the center of the Earth, right ascension, declination and surface-to-mass ratio, and then divide the space objects in each volume element into several space object groups according to the orbital inclination and surface-to-mass ratio, and select the density of the object group as a macroscopic quantity as the state variable to improve the calculation efficiency of steps 2 to 4; The LEO orbit is calculated according to the distance from the center of the Earth r, right ascension λ and declination Divide into The space volume elements are divided into Δr, Δλ, For the space volume element U i,j,k , the spatial range it contains is determined by formula (1); Among them, r i is the median value of the ith geocentric distance layer, is the median value of the jth declination layer, λ k is the median value of the kth right ascension layer; the space objects in each volume element are divided into N according to the orbital inclination and surface mass fraction i ×N a interval; therefore The total number of spatial volume elements is Space object group; volume element U i,j,k The space object group within the pth orbital inclination interval and the qth surface mass ratio interval The spatial density as a state variable.

4. A three-dimensional network evolution method for space debris environment as claimed in claim 3, characterized in that: The implementation method of step 2 is: Step 2.1: The total perturbation equation of motion of the spacecraft under the Earth's flattening perturbation and atmospheric perturbation is: Where a is the semi-major axis of the orbit; i is the orbit inclination; Ω is the right ascension of the ascending node; e is the orbit eccentricity; ω is the depression angle of the perigee; f is the true anomaly; M is the mean anomaly of the space object's orbit; r is the distance from the center of the earth; n is the average angular velocity of motion; ρ is the atmospheric density; S is the windward area of ​​the space object; n represents the average angular velocity; m o is the mass of the space object; J2 is the earth's flattening J2 term; C d is the atmospheric drag coefficient; R e is the equatorial radius of the earth; t is the time; Step 2.2: For The average decay rate of space objects in the group It is expressed as: For an object running on a circular orbit, the orbital eccentricity e = 0. Substituting it into equation (2), we get: Where ρ is the atmospheric density; (A / m o ) q represents the median value of the surface mass ratio stratification q; μ is the Earth's gravitational constant; It is a function related to atmospheric density, orbital altitude, and object surface-to-mass ratio. For a space object with a known orbital altitude, the atmospheric density is determined by ρ You can get Taking into account solar and geomagnetic activities, the atmospheric density model is expressed as: Where T is the temperature; F 10.7 is the solar radio flux at 10.7 cm; A p is the geomagnetic index; m is the molecular mass, which is defined as a function of the altitude h; the atmospheric density expression is expressed as: ρ=6×10 -10 exp(0.012h 2 -31.5h+5145 / T) (6) Substituting (6) into (4), we obtain the average orbital decay rate: Step 2.3: The relationship between the longitude and latitude of a point on the orbit of a space object and the orbital elements is as shown in equation (8); Longitude drift velocity of any point in space Drift velocity with right ascension of ascending node equal; for The average transfer speed of space objects in the group From equation (9), we get: Among them, i p represents the median value of the p-th orbital inclination layer; a e is the equatorial radius of the Earth.

5. A three-dimensional network evolution method for space debris environment as claimed in claim 4, characterized in that: The implementation method of step three is: Step 3.1: Consider the discontinuity factor in the evolution of the debris environment and model the space launch activities to obtain the model 1) Based on the past N y The actual launch data of the year is used to obtain the upper and lower bounds of the number of payloads and rocket bodies launched each month / year. The number of future launches follows a uniform distribution within this range. The number of future launches N per month / year is obtained through random sampling. L ; 2) Use Gaussian mixture model GMM to fit the past N y The actual launch information of payloads and rocket bodies in 2018, The launch information includes orbital parameters and dimensions, and obtains a Gaussian mixture result of the actual launch information of the payload and the rocket body; 3) Using the Markov chain Monte Carlo method MCMC, extract N from the Gaussian mixture result obtained in step 2) L launch samples, and finally obtain a prediction model for future launch activities Indicates that the emission Satellite density; Step 3.2: Construct a physical layer topology network model that considers three activities: space launch activities, post-mission disposal of satellites (PMD), and natural orbit reduction; consider three types of space objects related to space activities, namely, working satellites (S), failed satellites (F), and satellite remains / debris (N), and then divide the three objects into nodes according to orbital altitude and face value. Represents three types of objects in The density of the group, the dynamic equations of each node are (10)~(12); Where: for The density of new failed satellites generated in the system; τ is the success rate of PMD; υ PMD is the PMD descending speed; is the natural orbit descent speed; is the object transfer speed under the action of J2 perturbation; the space activity is expressed by the differential equation shown in formula (13):

6. A three-dimensional network evolution method for space debris environment as claimed in claim 5, characterized in that: In step four, The continuity equation (14) in fluid mechanics is used to model the density evolution of space debris; in, represents the continuity factor acting on the system, represents discontinuous factors; the continuous factors considered are atmospheric drag perturbation and J2 perturbation, and the discontinuous factors are space launch activities and post-satellite mission disposal; considering atmospheric drag, J2 perturbation, space launch and post-satellite mission disposal The group density evolution equation changes from (14) to (15); the left side of formula (15) is the change in the number of space objects, and the first and second terms on the right side represent the group density of objects under the influence of atmospheric drag and J2 perturbation, respectively. The third term represents the change in the number of objects caused by space activities; Among them, V i,j,k is the volume element U i,j,k The volume of is the cross-sectional area in the direction of atmospheric resistance, is the cross-sectional area in the direction of J2 perturbation, and its expression is as follows; Space objects caused by atmospheric drag The average decay rate of the orbital altitude is is the average transfer speed of space objects in the same latitude but different longitude intervals caused by J2 perturbation, It is caused by space activities The amount of change in density of objects within the group; Substitute equations (16) to (18) into (15); divide both sides of the equation by Δr·Δλ·r 2 ; Let Δr→0, Δλ→0, Formula (15) is simplified to Formula (19), which gives the three-dimensional network evolution model of space debris environment:

7. A macroscopic assessment method for long-term collision risk of spacecraft as claimed in claim 6, characterized in that: In step five, The evolution state of the debris environment is obtained by numerically solving equation (19); at this point, the solution to the evolution state of each object can be regarded as a numerical solution A first-order two-dimensional hyperbolic partial differential equation, written in the form of a standard system of equations, is given by equation (20): Among them, n is included A and B are diagonal matrices: in, is the average decay rate of space objects in the first altitude interval, the first right ascension interval, the first declination interval, the first aspect ratio interval, and the first orbital inclination interval, is the average decay velocity of the space object in the i-th altitude interval, the k-th right ascension interval, the j-th declination interval, the q-th aspect ratio interval, and the p-th orbital inclination interval, It is the Nth h height interval, Nth λ Right ascension interval, Declination interval, Nth a The face value ratio interval and the Nth i The average decay speed of space objects in the orbital inclination interval is is the average transfer speed of the space object in the first altitude interval, the first right ascension interval, the first declination interval, the first aspect ratio interval and the first orbital inclination interval, is the average transfer speed of the space object in the i-th altitude interval, the k-th right ascension interval, the j-th declination interval, the q-th aspect ratio interval, and the p-th orbital inclination interval, It is the Nth h height interval, Nth λ Right ascension interval, Declination interval, Nth a The face value ratio interval and the Nth i The average transfer speed of space objects in the orbital inclination interval; For a space object group with k latitude interval, p orbital inclination interval, and q surface value ratio interval, the difference format of the evolution state is written as: Among them, τ, h1 and h2 are the discrete time interval, altitude interval and longitude interval of the differential approximation, respectively. The superscript δ represents the δth discrete time step. When selecting the discrete interval, it should be ensured that at each step of calculation, the fragments in the same volume element are transferred to the adjacent volume element at most. represents the density of the object in the i-th altitude interval and the j-th longitude interval at the time δ+1, represents the density of objects in the i-th altitude interval and the j-th longitude interval at time δ, represents the density of the object in the i+1th altitude interval and the jth longitude interval at the time δ+1, is the right side term of the equation (19), It is expressed as: in, represents the density of objects in the i-th altitude interval and the j-1-th longitude interval at time δ, It represents the density of objects in the i-th altitude interval and the j+1-th longitude interval at time δ.

8. A macroscopic assessment method for long-term collision risk of spacecraft as claimed in claim 7, characterized in that: In step six, The volume element U is obtained by numerical calculation in step 5. i,j,k Space object group in the Pth orbital inclination interval and the Qth face value ratio interval Density And calculate the spacecraft density in this volume element, the pth orbital inclination interval, and the qth surface value ratio interval Object Group The collision cross-sectional area with the spacecraft is expressed as σ(Q,q), using the average cross-sectional area A of the object group Q and the spacecraft cross-sectional area A q To define: Space object group The relative collision velocity with the spacecraft reflects the relative motion state of the spacecraft and the space object group within the volume element and is determined by formula (26): Among them, v P is the speed of the space object group, v p is the spacecraft velocity; since the volume element scale is small enough, v P It can be expressed by the velocity at the midpoint of the volume element, satisfying the following conditions: n1 is the vector from the center of the earth to the center of the volume element, n2 is the normal vector of the orbital plane of the object group, and n3 is the normal vector of the equatorial plane, which satisfy the following conditions: Then, the long-term collision risk between spacecraft and space objects is calculated macroscopically. i,j,k The average collision probability of spacecraft per unit time is: Among them, dU i,j,k Represents volume element U i,j,k Volume; Therefore, the number of collisions during the spacecraft mission can be expressed as: Among them, T mission is the task time, U t is the volume element where the spacecraft is located at time t, is the density of the object group in the volume element where the spacecraft is located at time t, The spatial density of spacecraft, is the collision velocity between the spacecraft and the space object group at time t.

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