A macroscopic assessment method for long-term collision risk of spacecraft

Abstract

The present invention discloses a method for assessing the long-term collision risk of spacecraft, which belongs to the field of aerospace. The method is implemented by dividing space objects into several groups based on their distance from the Earth's center of gravity, right ascension, declination, orbital inclination, and surface mass fraction. The density of each group is selected as a macroscopic representation of the space debris environment. Taking into account multi-source environmental factors such as atmospheric drag, a continuity equation is used to establish a macroscopic representation and rapid evolution model of the space debris environment, and a numerical solution to the evolution model is provided. The density of the space object, the density of the spacecraft, the collision cross-sectional area, and the collision velocity within a certain volume element are calculated, and the long-term collision probability between the spacecraft and the space object is calculated. Based on the long-term collision risk macroscopic 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. This supports the optimization of satellite and constellation launches and their post-mission disposal strategies, space traffic management, and space debris management.

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 deteriorating space debris environment will greatly increase the collision risk of spacecraft in orbit. Therefore, conducting long-term collision risk assessments for target spacecraft is a prerequisite for the safe implementation of space missions.

[0003] Spacecraft collision risk analysis is based on the evolution of the debris environment, primarily by calculating the probability of spacecraft collision. Methods for calculating 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 a space object and is often used as a criterion for collision warning. However, the short-term method is significantly affected by orbital prediction errors, meaning it is ineffective for assessing the long-term collision risk of spacecraft. The long-term method, originally proposed by Opik and later improved and generalized by Wetherill and Kessler, is used to assess the collision risk between asteroids, comets, and planetary debris. This method assumes that a, e, and i are fixed for each object, while Ω and ω are uniformly distributed. By randomly or uniformly sampling all geometric shapes of close encounters 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 address this issue, NASA proposed the CUBE model. At a microscopic scale (a cube), this model employs gas kinetic theory and uses the spatial density of objects within the cube to estimate collision probability. Compared to traditional long-term collision probability algorithms, algorithms like CUBE can dynamically assess collision probability during spacecraft orbit evolution by uniformly sampling time and updating orbital elements at each time step. However, these algorithms rely on predictions of the motion state of individual debris, which can lead to high computational resource requirements. Summary of the Invention

[0004] The present invention aims to address the high computational complexity and low efficiency of current long-term collision probability algorithms, which prevent efficient analysis of spacecraft collision risks. This paper proposes a macroscopic assessment method for long-term collision risk of spacecraft. This method divides space objects into several groups based on their distance from the Earth's center of gravity, right ascension, declination, orbital inclination, and surface mass fraction. The density of each group is used as a macroscopic representation of the space debris environment. Taking into account multiple environmental factors such as atmospheric drag, a continuity equation is used to establish a macroscopic representation and rapid evolution model of the space debris environment, and a numerical solution is provided for the evolution model. The density of space objects, the density of spacecraft, the collision cross-sectional area, and the collision velocity within a given volume element are calculated, respectively. The long-term collision probability between spacecraft and space objects is then calculated. Based on the proposed long-term collision probability algorithm, the long-term collision risk of the Chinese Space Station (CSSS) is assessed and the safest orbit for the CSSS is discussed. This method supports the optimization of satellite and constellation launch and post-mission disposal strategies, space traffic management, and space debris management, addressing related engineering and technical issues.

[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. The method establishes a macroscopic representation and rapid evolution model of the space debris environment, divides the LEO orbit into several spatial volume elements according to the distance from the Earth's center of gravity, right ascension, declination, and area-to-mass ratio, and then divides the space objects within each volume element into several space object groups according to the orbital inclination and area-to-mass ratio. The density of the object group is selected as the macroscopic representation of the space debris environment. Multi-source environmental factors such as atmospheric drag and J2 perturbation are considered to improve the accuracy of the space debris evolution prediction model. The space debris is layered and discretized, and the continuity equation is used to model the evolution of the space debris environment, saving computing resources and improving the efficiency of evolution prediction. The method numerically solves the corresponding first-order two-dimensional hyperbolic partial differential equations and provides numerical solutions to the macroscopic representation and rapid evolution model of the space debris environment. The method calculates the density of space objects, the density of spacecraft, the collision cross-sectional area, and the collision velocity within a certain volume element, and then calculates the long-term collision probability between the spacecraft and the space object. Based on the proposed long-term collision probability algorithm for spacecraft, the long-term collision risk of the space station is assessed and the safest orbit for 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. 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 used as a state variable to improve the computational 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 Divided 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 A group of space objects; a 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 evolution of the debris environment, establish the total perturbation motion equation of the spacecraft under the combination of the Earth's oblateness perturbation and atmospheric perturbation, use the average orbital decay velocity of space objects to characterize the influence of atmospheric drag on the density evolution of the space object group; use the average transfer velocity of the space object group in the same latitude but different longitude intervals to characterize the influence of J2 perturbation on the density evolution of the space object group; and construct an expression for the influence of continuity factors on the density evolution of the space object group, 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 and rapid evolution model of the space debris environment in step 4.

[0013] Step 2.1: The total perturbation equation of motion of the spacecraft under the Earth's oblateness 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 perigee depression angle; 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 mean angular velocity; ρ is the atmospheric density; S is the frontal area of ​​the space object; n represents the mean angular velocity; m o is the mass of the space object; J2 is the Earth's oblateness J2; C d is the atmospheric drag coefficient; R e is the equatorial radius of the Earth; t is time;

[0016] Step 2.2: For Average decay rate of space objects within the group Expressed as:

[0017]

[0018] For an object running on a circular orbit, the orbital eccentricity e = 0, which can be substituted into formula (2) to obtain:

[0019]

[0020] Where ρ is the atmospheric density; (A / m o ) q represents the median value of the q surface mass fraction layer; μ is the earth's gravitational constant; thus, 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 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 expression for atmospheric density 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 number of 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 of the ascending node right ascension 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 Earth's equatorial radius.

[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 follows a uniform distribution within this range. The number of future launches per month / year is obtained by random sampling N L ;

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

[0035] 3) Finally, the Markov Chain Monte Carlo method (MCMC) is used to extract N from the Gaussian mixture result obtained in step 2). L launch samples, and ultimately 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, 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 debris / debris (N). Then, divide the three types of objects into nodes according to their orbital altitude and face value. Represents three types of objects in The density of the group, the dynamic equations of each node are (10) to (12);

[0037]

[0038] Where: for The density of new failed satellites generated within 100 km is τ, which is the success rate of PMD. PMD is the PMD's descending speed; is the natural orbit descent speed; is the object transfer speed under the action of J2 perturbation; therefore, the model of space activities, that is, the discontinuity factor, is:

[0039]

[0040] Step 4: Combining 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 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; Therefore, 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 the formula 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;

[0044]

[0045] Among them, V i,j,k is the volume element U i,j,k The volume, 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 object group 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;

[0046]

[0047]

[0048] Substitute equations (16) to (18) into (15); then divide both sides of the equation by Δr·Δλ·r 2 ; Then 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: Numerical solution is performed on the macroscopic representation 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 macroscopic representation of the space debris environment and the numerical solution of the rapid evolution model are given.

[0051] The evolution state of the debris environment is obtained by numerically solving equation (19); at this time, the solution of the evolution state of various objects can be regarded as a numerical solution. A first-order two-dimensional hyperbolic partial differential equation can be written as a standard system of equations as Equation (20):

[0052]

[0053] Among them, n is included indivual , f is a function of the right-hand side of the equation, and both 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 rate 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 rate of space objects in the orbital inclination interval is is the average transfer velocity 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 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 transfer velocity of space objects in the orbital inclination interval.

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

[0057]

[0058] 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 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 objects in the i-th altitude interval and the j-th longitude interval at 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 objects in the i+1th altitude interval and the jth longitude interval at 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, 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 aspect 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), and the average cross-sectional area A of the object group is used. 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 velocity of the space object group, v p is the spacecraft velocity; since the volume element scale is small enough, v P It can be represented by the velocity at the midpoint of the volume element, satisfying the following conditions:

[0067]

[0068] 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:

[0069]

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

[0071]

[0072] Among them, dU i,j,k Represents the 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. This supports the optimization of satellite and constellation launch and post-mission disposal strategies, space traffic management and space debris management, and solves related engineering and technical problems.

[0077] Beneficial effects:

[0078] 1. This invention discloses a macroscopic assessment method for long-term spacecraft collision risk. This method discretizes space debris in a layered manner by establishing a macroscopic representation and rapid evolution model of the space debris environment. The LEO orbit is divided into several spatial volume elements based on distance from the Earth's center of gravity, right ascension, declination, and area-to-mass ratio. The space objects within each volume element are then divided into several groups based on orbital inclination and area-to-mass ratio. The density of the object group is selected as a macroscopic state variable, and the continuity equation from fluid mechanics is used to model the space debris environment. This method conserves computing resources and makes the evolution model highly efficient.

[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 evaluates 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 Schematic diagram of 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 4Schematic 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 element numbers in the embodiment;

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

[0087] In order to better illustrate the purpose, technical solutions and advantages of the present invention, the invention is further described below with reference to 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, with the remainder being space debris. The deteriorating space debris environment will significantly increase the collision risk for spacecraft in orbit. Therefore, analyzing the evolution of debris within the target spacecraft's operating space and conducting long-term collision risk assessments are prerequisites for the safe conduct of space missions.

[0089] Currently, models of space debris environment evolution fall into two main categories. The first category involves microscopic models that track the motion of individual debris fragments, such as NASA's LEGEND, ESA's DELTA, the Italian National Research Institute's SDM, the UK Space Agency's DAMAGE, the Chinese Academy of Sciences' SOLEM, and Harbin Institute of Technology's SDEEM. These models rely on Monte Carlo techniques to simulate debris events, requiring significant computational resources and time. The other category involves macroscopic models that use debris population and density as variables. These models are constructed using differential equations (systems of differential equations), which can be solved to obtain information on the evolution of the debris population. These models offer significantly higher computational efficiency than microscopic models.

[0090] Current macroscopic models have limitations. Almost all suffer from simplistic spatial partitioning (grouping only by orbital altitude, or even failing to distinguish between altitude layers), making it impossible to predict the state of space objects at arbitrary spatial locations (latitude and longitude). Furthermore, most models employ relatively simple perturbation models, for example, failing to account for Earth's oblateness perturbations, and atmospheric drag models failing to consider the influence of solar and geomagnetic activity. However, the impact of these environmental factors on debris evolution cannot be ignored. The shift in the ascending node's right ascension caused by Earth's oblateness perturbations can cause space objects to migrate between different longitude regions at the same latitude. Extreme weather events, such as geomagnetic storms, can cause a sudden increase in atmospheric drag, accelerating the decay of objects' orbits. For example, the 40 Starlink satellites launched on February 3, 2022, were damaged by a geomagnetic storm upon reentering the Earth's atmosphere. This suggests that while macroscopic models offer significant advantages in efficiency, they still require improvement in terms of accuracy in spatial partitioning and environmental perturbations.

[0091] Spacecraft collision risk analysis is based on the evolution of the debris environment, primarily by calculating the probability of spacecraft collision. Methods for calculating 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 a space object and is often used as a criterion for collision warning. However, the short-term method is significantly affected by orbital prediction errors, meaning it is ineffective for assessing the long-term collision risk of spacecraft. The long-term method, originally proposed by Opik and later improved and extended by Wetherill, Kessler, and Greenberg, is used to assess the collision risk between asteroids, comets, and planetary debris. This method assumes that a, e, and i are fixed for each object, while Ω and ω are uniformly distributed. By randomly or uniformly sampling all geometric shapes of close encounters 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 address this issue, NASA proposed the CUBE model. At a microscopic scale (a cube), this model employs gas kinetic theory and uses the spatial density of objects within the cube to estimate collision probability. Compared to traditional long-term collision probability algorithms, algorithms like CUBE can dynamically assess collision probability during spacecraft orbital evolution by uniformly sampling time and updating orbital elements at each time step. However, these algorithms typically rely on predictions from microscopic debris evolution models, thus also requiring high computational resources.

[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. 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 used as a state variable to improve the computational 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 Divided 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 kis 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 A group of space objects; a 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 evolution of the debris environment, establish the total perturbation motion equation of the spacecraft under the combination of the Earth's oblateness perturbation and atmospheric perturbation, use the average orbital decay velocity of space objects to characterize the influence of atmospheric drag on the density evolution of the space object group; use the average transfer velocity of the space object group in the same latitude but different longitude intervals to characterize the influence of J2 perturbation on the density evolution of the space object group; and construct an expression for the influence of continuity factors on the density evolution of the space object group, 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 and rapid evolution model of the space debris environment in step 4.

[0098] Step 2.1: The total perturbation equation of motion of the spacecraft under the Earth's oblateness 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 perigee depression angle; 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 mean angular velocity; ρ is the atmospheric density; S is the frontal area of ​​the space object; n represents the mean angular velocity; m o is the mass of the space object; J2 is the Earth's oblateness J2; C d is the atmospheric drag coefficient; R e is the equatorial radius of the Earth; t is time;

[0101] Step 2.2: For Average decay rate of space objects within the group Expressed as:

[0102]

[0103] For an object running on a circular orbit, the orbital eccentricity e = 0, which can be substituted into formula (2) to obtain:

[0104]

[0105] Where ρ is the atmospheric density; (A / m o) q represents the median value of the q surface mass fraction layer; μ is the earth's gravitational constant; thus, 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 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 expression for atmospheric density 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 number of 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 of the ascending node right ascension 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 Earth's equatorial radius.

[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 follows a uniform distribution within this range. The number of future launches per month / year is obtained by random sampling N L ;

[0119] 5) Using a Gaussian mixture model (GMM), we fit the actual launch information of payloads and rocket bodies from 2020 to 2023, including orbital parameters and dimensions, and obtain Gaussian mixture results of the actual launch information of payloads and rocket bodies;

[0120] 6) Finally, the Markov Chain Monte Carlo method (MCMC) is used to extract N from the Gaussian mixture result obtained in step 2). L launch samples, and ultimately 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, 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 debris / debris (N). Then, divide the three types of objects into nodes according to their orbital altitude and face value. Represents three types of objects in The density of the group, the dynamic equations of each node are (10) to (12);

[0122]

[0123]

[0124] Where: for The density of new failed satellites generated within 100 km is τ, which is the success rate of PMD. PMD is the PMD's descending speed; is the natural orbit descent speed; is the object transfer speed under the action of J2 perturbation; therefore, the model of space activities, that is, the discontinuity factor, is:

[0125]

[0126] Step 4: Combining 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 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; Therefore, 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 the formula 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;

[0130]

[0131] Among them, V i,j,k is the volume element U i,j,k The volume, 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 object group 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;

[0132]

[0133] Substitute equations (16) to (18) into (15); then divide both sides of the equation by Δr·Δλ·r 2 ; Then 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: Numerical solution is performed on the macroscopic representation 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 macroscopic representation of the space debris environment and the numerical solution of the rapid evolution model are given.

[0136] The evolution state of the debris environment is obtained by numerically solving equation (19); at this time, the solution of the evolution state of various objects can be regarded as a numerical solution. A first-order two-dimensional hyperbolic partial differential equation can be written as a standard system of equations as Equation (20):

[0137]

[0138] Among them, n is included indivual , f is a function of the right-hand side of the equation, and both 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 rate 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 rate of space objects in the orbital inclination interval is is the average transfer velocity 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 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 transfer velocity of space objects in the orbital inclination interval.

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

[0142]

[0143] 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 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 objects in the i-th altitude interval and the j-th longitude interval at 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 objects in the i+1th altitude interval and the jth longitude interval at 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, 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 aspect 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), and the average cross-sectional area A of the object group is used. 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 velocity 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 represented by the velocity at the midpoint of the volume element, satisfying the following conditions:

[0152]

[0153] 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:

[0154]

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

[0156]

[0157] Among them, dU i,j,k Represents the 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 launch and post-mission disposal strategies, space traffic management and space debris management, and solve related engineering and technical problems.

[0162] In this example, long-term collision risk analysis and assessment are conducted for important large spacecraft, such as the space station. This is of great significance for spacecraft operational safety, collision protection strategies, and optimal orbit selection. Two simulation scenarios are set up to evaluate the long-term collision probability during the operation of the Chinese space station. Based on the collision probability results, the safest orbit for the space station is discussed.

[0163] A. Long-term collision risk of the Chinese space station

[0164] The Chinese space station operates in a near-circular low Earth orbit of 340-450 km with an orbital inclination of 41-42°. Therefore, the orbital altitude range of [340,450] km is selected as the operating space of the space station, and the intervals of the space volume elements in the radial, right ascension and declination directions are Δh=10 km, Δλ=10°, The system is divided into 7128 volume elements. A three-dimensional network model is used to predict the evolution of objects within the Chinese space station's operating space. Based on this prediction, a seven-day collision risk assessment of the Chinese space station is conducted. Table 1 shows a set of orbital elements of the Chinese space station, which serve as the initial state of the simulation.

[0165] Table 1 Orbital elements of the Chinese 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 of space objects passing through the volume element during the operation of the Chinese space station is in the range of [0,3.5×10 -8 ] / km 3 . Figure 3 The collision probabilities of the Chinese space station with three different objects, namely, payload, rocket body, and debris, are shown. It can be seen that the collision probability of the Chinese space station with payload and space debris is one order of magnitude higher than that with rocket body, and the collision risk with payload is the highest on the fourth day, reaching 2.79×10 -8 / year / m 2 The risk of collision with space debris is highest on day 2.5, reaching 1.45×10 -8 / year / m 2 .

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

[0169] Figure 4The distribution of objects in the [340,450] km region is shown (using the 340km, 390km, and 440km slices as examples). It can be seen that the higher the orbital altitude, the greater the density of space objects, with a significant clustering occurring at latitudes of [-70,-90]°. Based on this spatial environment, the long-term collision probability of Chinese space stations operating on different orbits within this region is analyzed, and the safest orbit for 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, while the other orbital elements, the spatial volume element division method, and the simulation time remain unchanged.

[0170] Figure 5 The figure shows the average collision probability of space stations with different orbital elements (a, i) within 7 days. In general, the collision risk of the Chinese space station is the lowest 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 distribution of space objects displayed 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 / m 2 , 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 scope of protection of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection 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. 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 and establish the total perturbation motion equation for spacecraft under the combination of Earth oblateness perturbation and atmospheric perturbation. Use the average orbital decay velocity of space objects to characterize the influence of atmospheric drag on the density evolution of the space object group. Use the average transfer velocity of the space object group in intervals of the same latitude but different longitudes to characterize the influence of J2 perturbation on the density evolution of the space object group. Finally, 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 velocity and the average transfer velocity, respectively, to facilitate improving the accuracy of the three-dimensional network evolution model of the space debris environment in Step 4. 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, and obtain a model of the discontinuity factors. This will help improve the accuracy of the three-dimensional network evolution model of the space debris environment in step 4. Step 4: Combining the expression for the effect of the continuity factor on the density evolution of the space object group in Step 2 with the model for the discontinuity factor in Step 3, and using the continuity equation in fluid mechanics, a three-dimensional network evolution model of the space debris environment is established. This model improves the efficiency of evolution prediction while ensuring the accuracy of evolution prediction. Step 5: Numerical solution is performed on the three-dimensional network evolution model of the space debris environment in step 4 to obtain A first-order two-dimensional hyperbolic partial differential equation is solved, and a numerical solution of a three-dimensional network model of the space debris environment is given; Step 6: Use the numerical solution obtained in step 5 to calculate the density of space objects, the density of spacecraft, the collision cross-sectional area and the collision velocity within a certain volume element, and then macroscopically calculate the long-term collision risk between spacecraft and space objects.

2. The method for macroscopically assessing the long-term collision risk of a spacecraft according to claim 1, wherein: 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 space debris in layers; divide the LEO orbit into several spatial volume elements based on the distance from the Earth's center, right ascension, declination, and area-to-mass ratio. Then, divide the space objects within each volume element into several space object groups based on orbital inclination and area-to-mass ratio. Select the macroscopic quantity of the object group density as the state variable to improve the computational 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 Divided 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 A group of space objects; a 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. The method for macroscopically assessing the long-term collision risk of a spacecraft according to claim 3, wherein: The implementation method of step 2 is: Step 2.1: The total perturbation equation of motion of the spacecraft under the Earth's oblateness 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 perigee depression angle; 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; m o is the mass of the space object; J2 is the Earth's oblateness J2; C d is the atmospheric drag coefficient; R e is the equatorial radius of the Earth; t is time; Step 2.2: For Average decay rate of space objects within the group Expressed as: For an object running on a circular orbit, the orbital eccentricity e = 0, which can be substituted into formula (2) to obtain: Where ρ is the atmospheric density; (A / m o ) q represents the median value of the surface mass fraction layer 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 ρ You can get Taking into account solar and geomagnetic activities, the atmospheric density model is expressed as: Where T is 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 number of orbital elements is as shown in equation (8); Longitude drift velocity of any point in space Drift velocity of the ascending node right ascension 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 Earth's equatorial radius.

5. The macroscopic assessment method for long-term collision risk of spacecraft according to 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 launches N per month / year in the future is obtained by 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 2017, The launch information includes orbital parameters and dimensions, and a Gaussian mixture result of actual launch information of the payload and the rocket body is obtained; 3) Using the Markov Chain Monte Carlo method MCMC, extract N from the Gaussian mixture result obtained in step 2) L launch samples, and ultimately 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, post-mission disposal (PMD), and natural orbit reduction. Consider three types of space objects related to space activities: working satellites (S), failed satellites (F), and satellite debris / debris (N). Then, divide the three objects into nodes according to their orbital altitude and face value. Represents three types of objects in The density of the group, the dynamic equations of each node are (10) to (12); Where: for The density of new failed satellites generated within 100 km is τ, which is the success rate of PMD. PMD is the PMD's 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. The macroscopic assessment method for long-term collision risk of spacecraft according to 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, 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 object group 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 ; Then 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. The macroscopic assessment method for long-term collision risk of spacecraft according to claim 6, characterized in that: In step five, The evolution state of the debris environment is obtained by numerically solving equation (19); at this time, the solution of the evolution state of various objects can be regarded as a numerical solution. A first-order two-dimensional hyperbolic partial differential equation can be written as a standard system of equations as Equation (20): Among them, n is included , f is a function of the right-hand side of the equation, and both 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 rate 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 rate of space objects in the orbital inclination interval is is the average transfer velocity 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 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 transfer speed of space objects in the orbital inclination interval; For a space object group with a latitude interval of k, an orbital inclination interval of p, and a surface value ratio interval of q, 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 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 objects in the i-th altitude interval and the j-th longitude interval at 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 objects in the i+1th altitude interval and the jth longitude interval at time δ+1, is the right side term of the equation (19), Expressed as: in, represents the density of objects in the i-th altitude interval and the j-1-th longitude interval at time δ, represents the density of objects in the i-th altitude interval and the j+1-th longitude interval at time δ.

8. The macroscopic assessment method for long-term collision risk of spacecraft according to 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 aspect 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), and the average cross-sectional area A of the object group is used. 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 velocity of the space object group, v p is the spacecraft velocity; since the volume element scale is small enough, v P It can be represented 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, and the volume element U i,j,k The average collision probability of spacecraft per unit time is: Among them, dU i,j,k Represents the 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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