A small celestial body deflection performance evaluation method based on state transition tensor
By establishing a multibody system dynamics model and calculating the state transition tensor, the problem of insufficient accuracy in the orbital deflection of small celestial bodies in existing technologies has been solved, realizing efficient and accurate orbital deviation assessment and defense strategies.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- DEEP SPACE EXPLORATION LAB
- Filing Date
- 2023-03-02
- Publication Date
- 2026-08-04
AI Technical Summary
Existing technologies fail to effectively consider the effects of Earth's gravity when assessing the orbital deflection of small celestial bodies, resulting in low accuracy of dynamic models and difficulty in determining the optimal kinetic energy impact strategy.
A multibody system dynamics model is established, the orbital offset of the small celestial body is calculated through the state transition tensor, the orbital offset is obtained using the second-order state transition tensor, and the orbital offset effectiveness is evaluated by combining impact geometry and time.
It achieves high-precision and high-speed orbital offset performance evaluation, and can determine the optimal offset direction of small celestial bodies' orbits, making it suitable for small celestial body defense missions.
Smart Images
Figure CN116305557B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a method for evaluating the propagation of orbital errors of small celestial bodies, and more particularly to a method for characterizing the orbital deflection of a small celestial body after being impacted. This method is applicable to evaluating the orbital offset of a small celestial body after being impacted and belongs to the field of spacecraft orbital mechanics. Background Technology
[0002] Currently, numerous studies have been conducted on the risk of potentially hazardous asteroids colliding with Earth. Asteroids whose orbits approach or cross Earth's orbit are classified as near-Earth asteroids. As of February 5, 2015, the number of near-Earth asteroids exceeded 17,000. Asteroid impacts on Earth occasionally occur, causing loss of life and property.
[0003] Researchers have proposed numerous methods for dealing with Earth-threatening asteroids, one of the simplest and most effective deflection methods being the launch of a kinetic impactor (KI), a spacecraft designed to strike an asteroid. Impact velocity is a critical issue in this strategy, as the impactor's speed is limited to 10-15 km / s when the propulsion system is jettisoning fuel. However, studies have shown that some propellant-free propulsion systems that utilize the space environment to generate thrust can overcome this velocity limitation.
[0004] Currently, most simulation studies on KI are based on the theory of heliocentric two-body orbit encounters. Prior art [1] [Izzo D. Optimization of interplanetary trajectories for impulsive and continuous asteroid deflection[J]. Journal of guidance, control, and dynamics, 2007, 30(2): 401-408.] proposed an analytical expression to evaluate the advantages of a given deflection strategy. This expression relates the orbital offset to the impact velocity vector and time, and specifically studies the case of dynamic impactors and long-term thrust deflection.
[0005] Prior technology [2] [Yamaguchi K, Yamakaawa H. Visualization of kinetic-impacteffectiveness for asteroid deflection using impact-geometry maps[J]. Journal of Spacecraft and Rockets, 2018, 55(5): 1181-1197.] In order to study the effectiveness of kinetic impactors in accelerating spacecraft impacts and deflecting asteroids threatening Earth, based on Theories concerning close encounters with small celestial bodies have proposed a graphical representation called impact geometry. The ideal impact of an impactor on an asteroid depends on the shape of the injection trajectory, a characteristic well illustrated by contour plots.
[0006] However, the above technology only starts from the heliocentric two-body orbit and does not take into account the gravitational effect of the Earth after the small celestial body approaches the Earth. The dynamic model is not very accurate, so the results obtained have a large error with the actual situation.
[0007] Some near-Earth asteroids pose a potential risk of collision with Earth, necessitating active manipulation to deflect them from their original orbits away from Earth. Kinetic impacts that instantaneously alter the velocity of asteroids are a crucial method for achieving this orbital shift. However, in the deep-space multi-body environment, the long-term evolution of minute changes in the state of asteroids is unknown, and the mapping relationship between the velocity change and the closest distance between the asteroid and Earth is difficult to determine, posing challenges to the formulation of optimal kinetic impact strategies. Summary of the Invention
[0008] This invention proposes a method for evaluating the deflection effectiveness of small celestial bodies based on state transition tensors. The problem of evaluating the deflection effectiveness of small celestial bodies is transformed into a long-term evolution problem of the perturbed orbit of a small celestial body. This method establishes a multi-body system dynamics model, calculates the impact geometry of the small celestial body relative to Earth, and derives the state transition tensor of the small celestial body's orbit. This allows for rapid calculation of the impact geometry's influence on the initial state changes of the small celestial body, obtaining the orbital deflection caused by the impactor collision when the small celestial body is dominated by Earth's gravity after approaching Earth. The second-order state transition tensor is used to calculate the orbital deflection of the small celestial body after a period of time, achieving an efficient and quantitative evaluation of the deflection effectiveness after impact. This invention, by fixing the velocity deviation and changing the position and direction of the applied deviation, can determine the optimal change direction corresponding to the best deflection effectiveness of the small celestial body's orbit. This invention has the advantages of high evaluation accuracy and efficiency, and is suitable for the selection of deflection strategies and effectiveness evaluation in small celestial body defense missions.
[0009] The objective of this invention is achieved through the following technical solution:
[0010] The method for evaluating the performance of small body migration based on state transition tensors disclosed in this invention includes the following steps:
[0011] Step 1: Calculate the geocentric hyperbolic orbit parameters of the small celestial body near Earth based on the six elements of the small celestial body's heliocentric elliptical orbit and Earth's heliocentric elliptical orbit. The hyperbolic orbit parameters include the pericentric radius r. p And the hyperbola enters the remaining velocity U.
[0012] Step 1.1: Based on the known six elements of the two heliocentric elliptical orbits, perform numerical integration of two-body dynamics to obtain the minimum orbital interception distance MOID between the two elliptical orbits. Use this minimum orbital interception distance MOID as the radius r of the pericenter of the hyperbolic orbit. p .
[0013] Step 1.2: Utilize The theory of close encounters models the motion of a small celestial body approaching Earth as a two-body scattering under the planetary center. The relative velocity of the small celestial body with respect to Earth determines the hyperbolic entry velocity U of the geocentric hyperbolic orbit. This velocity is obtained through the elements of the heliocentric elliptical orbit of the small celestial body.
[0014]
[0015] Where: a ast e is the semi-major axis of the heliocentric orbit of the small celestial body. ast Let i be the eccentricity of the heliocentric orbit of a small celestial body. ast The inclination of the heliocentric orbit of the small celestial body.
[0016] Step 2: Establish the B-plane coordinate system. The origin P is chosen at the Earth's center of mass. The x-axis follows the hyperbola of the small celestial body's geocentric orbit into the direction of the remaining velocity. The B-plane passes through the Earth's center of mass and is perpendicular to the direction of the remaining velocity. The y-axis is the opposite direction of the projection of the Earth's heliocentric velocity onto the B-plane. The z-axis is determined according to the right-hand rule.
[0017] Since the perigee lift is only related to the impulse experienced by the small celestial body within the orbital plane, only the orbital state offset of the small celestial body caused by the impulse from the impactor within the orbital plane is considered. The three-dimensional B-plane coordinate system is then transformed to the orbital plane coordinate system, i.e., the x-axis enters the remaining velocity direction along the hyperbola, and the y-axis lies within the hyperbolic orbital plane of the small celestial body, pointing away from the remaining velocity direction. Transforming the three-dimensional B-plane coordinate system to the orbital plane coordinate system improves the efficiency of small celestial body offset performance evaluation.
[0018] Step 3: In the orbital plane coordinate system obtained in Step 2, according to the hyperbolic orbital parameters (U, r) of the small celestial body obtained in Step 1... p Find the velocity-rotation angle δ of the hyperbolic orbit, and further find the orbital state variables at the pericenter of the hyperbolic orbit.
[0019] In the orbital plane coordinate system obtained in step two, the hyperbolic orbital parameters (U, r) of the small celestial body obtained in step one are... p Substituting into equation (2), we obtain the velocity-rotation angle δ, and further, according to equation (3), we obtain the orbital state variables at the pericenter of the hyperbolic orbit: [x per ,y per [] indicates the position at the pericenter of the hyperbolic orbit. This indicates the velocity state at the pericenter of the hyperbolic orbit.
[0020]
[0021]
[0022] Where, μ p is the Earth's gravitational constant.
[0023] Step 4: Calculate the change in velocity of the small celestial body based on the conservation of momentum before and after the collision and the velocity of the impactor relative to the small celestial body; obtain the initial orbital offset based on the change in velocity of the small celestial body and the set position change ΔR.
[0024] Using the conservation of momentum before and after the collision, we obtain equation (4):
[0025] K(m sc v sc +m ast v ast )=(m sc +m ast (v) ast +ΔV) (4)
[0026] Where m represents mass, v represents the velocity vector in the inertial frame, subscript sc represents the impactor, subscript ast represents the small celestial body, and K represents the collision coefficient of the inelastic collision. Because of the influence of ejected material during the impact, the value of K is much greater than 1. Therefore, we take K = 1, thus obtaining the expression for the change in velocity of the small celestial body:
[0027]
[0028] Where V r This represents the velocity vector of the impactor relative to the small celestial body at the time of impact. Changing V... r The angle between the x-axis and the x-axis is [0, 2π), resulting in a series of ΔV values.
[0029] Integrating the trajectory backward from the pericenter to the moment of impact yields the trajectory state at the moment of impact, which serves as the initial condition for integration. The initial trajectory state bias ΔX instantaneously after impact is:
[0030] ΔX=[ΔR,ΔV], ΔR=[0,0,0] (6)
[0031] Step 5: Construct a multibody dynamics model of the Sun-Earth-small celestial body that reflects the gravitational influence of Earth, making the model more consistent with the actual gravitational environment after the small celestial body approaches Earth. Based on the initial orbital values of the small celestial body obtained in Step 3, construct a state transition tensor for the subsequent calculation of the orbital offset in Step 6, using the multibody dynamics model of the Sun-Earth-small celestial body.
[0032] Since the gravitational influence of Earth becomes dominant after a small celestial body approaches Earth, the accuracy of two-body dynamics considering only the Sun's gravity is relatively low. The multibody dynamics model of the Sun-Earth-small celestial body is constructed as shown in equation (7):
[0033]
[0034] Among them, the right-hand side of the second equation Represents the gravitational pull of Earth on small celestial bodies. This represents the gravitational pull of the sun on small celestial bodies.
[0035] Since the multibody dynamics model of the Sun-Earth-small celestial body as shown in formula (7) includes a reaction to the gravitational effect of the Earth, This makes the multibody dynamics model of the Sun-Earth-small celestial body more consistent with the actual gravitational environment after the small celestial body approaches the Earth.
[0036] Because the multibody dynamics model of the Sun-Earth-small body is highly nonlinear, linearization methods, such as the orbital state transition matrix, have low accuracy in calculating orbital offsets and cannot meet the mission requirements. Therefore, for the multibody dynamics model of the Sun-Earth-small body shown in Equation (7), a state transition tensor suitable for the strongly nonlinear multibody dynamics model of the Sun-Earth-small body is constructed. This facilitates the subsequent step six in calculating the orbital offset of the small body using the state transition tensor, thereby improving the accuracy of orbital offset prediction under strongly nonlinear dynamics.
[0037] For the multibody dynamics model of the Sun-Earth-small body as shown in Equation (7), a state transition tensor suitable for the strongly nonlinear multibody dynamics model of the Sun-Earth-small body is constructed. The specific implementation method is as follows:
[0038] Initial conditions of the orbital state of small celestial bodies The solution to the above multibody dynamics model of the Sun-Earth-small celestial body is expressed as:
[0039] x(t)=φ(t;x 0 ,t 0 (8)
[0040] Where x(t) is the orbital state at time t, taking the derivative of both sides of the equation with respect to time, we get:
[0041]
[0042] Where f represents the right-hand side of equation (7). Since the initial conditions are independent of time, we have dx 0 / dt=0, then due to the initial time t of the orbit 0 State deviation δx at time 0 The resulting orbital state deviation δx(t) and its derivative at time t for:
[0043] δx(t)=φ(t;x 0 +δx 0 ,t 0 )-φ(t;x 0 ,t 0 (10)
[0044]
[0045] For equations (10) and (11) above, respectively, in the initial orbital state x 0 Performing a Taylor expansion to order m at point m, we obtain:
[0046]
[0047]
[0048] Where k j ∈{1,...,6}, representing the k-th orbital state variable. j There are elements. And there are:
[0049]
[0050]
[0051] Where: the superscript "*" indicates the value of the state quantity along the nominal orbit, and This is the state transition tensor STTs.
[0052] Because only Since it is related to time, taking the derivative of equation (12) with respect to time, we obtain the derivative of the orbital state deviation:
[0053]
[0054] Meanwhile, substituting equation (12) into equation (13), we obtain the derivative of the orbital state deviation:
[0055]
[0056] Combining equations (17) and (16), since δx0 The corresponding coefficients are equal, thus we obtain the system of ordinary differential equations (19) for the state transition tensor:
[0057]
[0058] The state transition tensor in the ordinary differential equation system (19) can be solved by only knowing the initial value of the state transition tensor and the nominal orbit.
[0059] The initial value of the state transition tensor is given by equation (14):
[0060]
[0061] The nominal orbit x(t) * The orbital integral is obtained directly from the initial orbital state with zero deviation under the multibody dynamics model (7) of the Sun-Earth-small celestial body.
[0062] Step 6: Using the initial state offset of the small celestial body's orbit obtained in Step 4 and the state transition tensor obtained in Step 5, the orbit offset of the small celestial body moving under the Sun-Earth-small celestial body multibody dynamics model at any time is analytically obtained, that is, the small celestial body orbit offset is predicted based on the state transition tensor.
[0063] Based on the initial orbital offset ΔX obtained from equation (6), the state transition tensor is used... Get t s Momentary deviation of small celestial body orbital state:
[0064]
[0065] in t s The orbital state deviation δx of small celestial bodies at any given moment i (t s ) represents the offset of the perigee of a small celestial body relative to the Earth, that is, the prediction of the orbital offset of a small celestial body is realized based on the state transition tensor.
[0066] It also includes step seven: selecting the impactor's velocity V relative to the small celestial body. r The angle with the x-axis is the impact geometry, and the magnitude of the impactor's velocity relative to the small celestial body is fixed at ||V. r By changing the impact geometry and impact time, the orbital offset of the small celestial body at the pericenter is predicted according to steps one through six, based on the corresponding impact geometry and impact time. The orbital offset δx(t) of the small celestial body is then used to... sThe direction and magnitude of the impactor's impact on the small celestial body are used to evaluate the orbital deflection effectiveness of the impactor. Based on the orbital deflection effectiveness, the impact geometry and impact time corresponding to the maximum offset can be found. The impactor impacts the small celestial body according to the impact geometry and impact time, so that the small celestial body's orbit changes direction with the best offset effectiveness, thereby achieving defense against potentially dangerous small celestial bodies.
[0067] To facilitate the visual evaluation of the orbital deflection performance, it is preferable to use contour lines to represent the impact geometry for visualization.
[0068] Beneficial effects:
[0069] 1. Some near-Earth asteroids pose a potential risk of collision with Earth, requiring active manipulation to deflect them from their original orbits away from Earth. Using kinetic impact to instantaneously change the velocity of the asteroid is a crucial method for achieving orbital deflection. However, in deep-space multi-body environments, the long-term evolution of minute changes in the state of asteroids is unknown, and the mapping relationship between the velocity change and the closest distance between the asteroid and Earth is difficult to determine, posing a challenge to the formulation of the optimal kinetic impact strategy. This invention discloses a method for evaluating the deflection effectiveness of asteroids based on state transition tensors. This method transforms the problem of evaluating the deflection effectiveness of asteroids into a problem of the long-term evolution of the perturbed orbit of asteroids. A multi-body system dynamics model is established to calculate the impact geometry of the asteroid relative to Earth, deriving the state transition tensor of the asteroid's orbit. This allows for rapid calculation of the impact geometry effect of initial state changes on the asteroid, obtaining the orbital deflection caused by the impactor collision when the asteroid is dominated by Earth's gravity after approaching Earth. The second-order state transition tensor is used to calculate the orbital deflection after a period of time, achieving efficient and quantitative evaluation of the deflection effectiveness of asteroids after impact.
[0070] 2. The small body migration efficiency evaluation method based on state transition tensor disclosed in this invention can determine the optimal change direction corresponding to the best migration efficiency of a small body orbit by fixing the velocity deviation and changing the position and direction of the applied deviation.
[0071] 3. The small body migration performance evaluation method disclosed in this invention is based on the state transition tensor. It uses the second-order state transition tensor to obtain the small body orbital migration after a period of time, realizes semi-analytical orbital migration prediction for strongly nonlinear multibody dynamics systems, and achieves a balance between prediction accuracy and computational efficiency.
[0072] 4. The small body offset performance evaluation method based on state transition tensor disclosed in this invention transforms the Earth's orbit in the heliocentric system and the orbit of a potentially hazardous small celestial body to the geocentric coordinate system. Using the B-plane coordinate system, the geocentric hyperbolic orbit of the small celestial body is obtained. Using the pericenter of the hyperbolic orbit as a reference point, impacts of different directions are applied to the small celestial body at different times before it reaches this point. The orbital offset of the small celestial body at the pericenter is calculated according to the impact geometry and impact time. s The direction and magnitude of the impactor's impact on the small celestial body are used to evaluate the orbital deflection effectiveness of the impactor. Based on the orbital deflection effectiveness, the impact geometry and impact time corresponding to the maximum offset can be found. The impactor impacts the small celestial body according to the impact geometry and impact time, so that the small celestial body's orbit changes direction with the best offset effectiveness, thereby achieving defense against potentially dangerous small celestial bodies. Attached Figure Description
[0073] Figure 1 Use the B-plane coordinate system;
[0074] Figure 2 A contour plot of the x-axis positional deviation with respect to the impactor velocity direction and impact time;
[0075] Figure 3 A contour plot of the positional deviation along the y-axis with respect to the impactor's velocity direction and impact time;
[0076] Figure 4 A contour plot showing the positional deviation along the x-axis with respect to the velocity direction of the impactor relative to the small celestial body and the impact time.
[0077] Figure 5 A contour plot showing the positional deviation of the impactor along the y-axis with respect to the velocity direction of the impactor relative to the small celestial body and the impact time.
[0078] Figure 6 This is a flowchart of the small body migration performance evaluation method based on state transition tensor disclosed in this invention. Detailed Implementation
[0079] To better illustrate the purpose and advantages of the present invention, the invention will be further described below in conjunction with the accompanying drawings and examples.
[0080] Example 1: This example uses a small body migration effectiveness evaluation method based on state transition tensors to calculate the orbital migration of the potentially hazardous small celestial body Apophis (parameters are shown in the table below) after impact. The specific implementation steps are as follows:
[0081] Table 1. Orbit and physical parameters of the Apophis asteroid.
[0082]
[0083] Step 1: Calculate the geocentric hyperbolic orbit parameters of the small celestial body near Earth based on the six elements of the small celestial body's heliocentric elliptical orbit and Earth's heliocentric elliptical orbit. The hyperbolic orbit parameters include the pericentric radius r. p And the hyperbola enters the remaining velocity U.
[0084] Step 1.1: Based on the six roots of the heliocentric orbit of Apophis and the heliocentric orbit of Earth, perform numerical integration of two-body dynamics to obtain the minimum orbital interception distance MOID between the two elliptical orbits. Use this minimum orbital interception distance MOID as the radius r of the pericenter of the hyperbolic orbit. p .
[0085] Step 1.2: Utilize The theory of close encounters models the motion of a small celestial body approaching Earth as a two-body scattering under the planetary center. The relative velocity of the small celestial body with respect to Earth determines the hyperbolic entry velocity U of the geocentric hyperbolic orbit. This velocity is obtained through the elements of the heliocentric elliptical orbit of the small celestial body.
[0086]
[0087] Where: a ast e is the semi-major axis of the heliocentric orbit of the small celestial body. ast Let i be the eccentricity of the heliocentric orbit of a small celestial body. ast Inclination of the heliocentric orbit of a small celestial body
[0088] Step 2: Establish the B-plane coordinate system, such as... Figure 1 As shown. The origin P is chosen at the Earth's center of mass. The x-axis enters the direction of the remaining velocity along the hyperbola of the small celestial body's geocentric orbit. The B plane passes through the Earth's center of mass and is perpendicular to the direction of the remaining velocity. The y-axis is the opposite direction of the projection of the Earth's heliocentric velocity onto the B plane. The z-axis is determined according to the right-hand rule.
[0089] Since the perigee lift is only related to the impulse experienced by the small celestial body within the orbital plane, only the orbital state offset of the small celestial body caused by the impulse from the impactor within the orbital plane is considered. The three-dimensional B-plane coordinate system is then transformed to the orbital plane coordinate system, i.e., the x-axis enters the remaining velocity direction along the hyperbola, and the y-axis lies within the hyperbolic orbital plane of the small celestial body, pointing away from the remaining velocity direction. Transforming the three-dimensional B-plane coordinate system to the orbital plane coordinate system improves the efficiency of small celestial body offset performance evaluation.
[0090] Step 3: In the orbital plane coordinate system obtained in Step 2, according to the hyperbolic orbital parameters (U, r) of the small celestial body obtained in Step 1... p Find the velocity-rotation angle δ of the hyperbolic orbit, and further find the orbital state variables at the pericenter of the hyperbolic orbit.
[0091] In the orbital plane coordinate system obtained in step two, the hyperbolic orbital parameters (U, r) of the small celestial body obtained in step one are... p Substituting into equation (2), we obtain the velocity-rotation angle δ, and further, according to equation (3), we obtain the orbital state variables at the pericenter of the hyperbolic orbit: [x per ,y per [] indicates the position at the pericenter of the hyperbolic orbit. This indicates the velocity state at the pericenter of the hyperbolic orbit.
[0092]
[0093]
[0094] Where, μ p is the Earth's gravitational constant.
[0095] Step 4: Calculate the change in velocity of the small celestial body based on the conservation of momentum before and after the collision and the velocity of the impactor relative to the small celestial body; obtain the initial orbital offset based on the change in velocity of the small celestial body and the set position change ΔR.
[0096] Using the conservation of momentum before and after the collision, we obtain equation (4):
[0097] K(m sc v sc +m ast v ast )=(m sc +m ast (v) ast +ΔV) (4)
[0098] Where m represents mass, v represents the velocity vector in the inertial frame, subscript sc represents the impactor, subscript ast represents the small celestial body, and K represents the collision coefficient of the inelastic collision. Because of the influence of ejected material during the impact, the value of K is much greater than 1. Therefore, we take K = 1, thus obtaining the expression for the change in velocity of the small celestial body:
[0099]
[0100] Where V r Let m be the velocity vector of the impactor relative to the small celestial body at the time of impact. Given the mass m of the impactor. sc =500kg, K=30, ||V r ||=5km / s, changing V r The angle between the x-axis and the x-axis is [0, 2π), and a series of ΔV values are obtained.
[0101] The orbit is integrated backward from the pericenter to the moment of impact, with a time series of 0–500 days, yielding multiple sets of orbital states at the moment of impact, which serve as the initial conditions for integration. The initial orbital state bias ΔX instantaneously after impact is:
[0102] ΔX=[ΔR,ΔV],ΔR=[0,0,0] (6)
[0103] Given the mass m of the impactor sc =500kg, K=30, ||V r ||=5km / s, changing V r The angle between the x-axis and the x-axis is [0, 2π), and a series of ΔV values are obtained.
[0104] Integrating the orbit backward from the pericenter to the impact time, using a time series of 0–500 days, yields multiple sets of orbital states at the impact time, which serve as the initial conditions for integration. The initial orbital state bias is then:
[0105] ΔX=[ΔR,ΔV],ΔR=[0,0,0] (7)
[0106] Step 5: To balance computational accuracy and efficiency, as a preferred approach, a second-order state transition tensor is selected, and the system of ordinary differential equations of the state transition tensor with second-order accuracy is solved:
[0107]
[0108] in:
[0109]
[0110]
[0111] The superscript "*" here indicates the value of the state variable along the nominal orbit. To solve equation (8), only the initial value of the state transition tensor, i.e., the nominal orbit, is known. The initial value of the state transition tensor is given by the following equation:
[0112]
[0113] Step Six: Analyze the orbital offset using the state transition tensor. This involves calculating the initial orbital deviation obtained in Step Four. t is obtained using the second-order state transition tensor s Approximate deviation of orbital state at any given time:
[0114]
[0115] Take t s Given the time for the inverse integration of the orbital path in step four, we obtain δx. i (t sThe deviation represents the orbital state of the small celestial body at the pericenter. The positional deviations along the x and y axes can be used to evaluate the orbital deflection effectiveness caused by the impactor striking the small celestial body.
[0116] Step 7: Use contour maps (such as...) Figures 2-5 Find the impact time and relative velocity direction that bring about the maximum orbital offset under the above conditions.
[0117] When the impact time is 500 days before the small celestial body reaches its pericenter, and the angle between the impactor's velocity relative to the small celestial body and the x-axis (along the small celestial body's entry velocity U) is 34°, the maximum orbital offset distance is 6.1857 km; when the impact time is 500 days before the small celestial body reaches its pericenter, and the angle between the impactor's velocity relative to the small celestial body and the x-axis is 0°, the maximum orbital offset distance in the x-axis direction is 6.1813 km; when the impact time is 37 days before the small celestial body reaches its pericenter, and the angle between the impactor's velocity relative to the small celestial body and the x-axis is 270°, the maximum orbital offset distance in the -y-axis direction is 6.1779 km.
[0118] When the absolute velocity of the impactor is set to 10 km / s, changing its direction yields the impact time and absolute velocity direction that result in the maximum orbital deviation. The maximum orbital deviation occurs when the impact time is 500 days before the small celestial body reaches its pericenter, and the angle between the impactor velocity and the x-axis (along the small celestial body's entry velocity U) is 180°. The maximum orbital deviation occurs when the impact time is 500 days before the small celestial body reaches its pericenter, and the angle between the impactor velocity and the x-axis is 0°. The maximum orbital deviation occurs when the impact time is 52 days before the small celestial body reaches its pericenter, and the angle between the impactor velocity and the x-axis is 270°.
[0119] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A method for evaluating the performance of small body migrations based on state transition tensors, characterized in that: Includes the following steps, Step 1: Calculate the geocentric hyperbolic orbit parameters of the small celestial body near Earth based on the six elements of the small celestial body's heliocentric elliptical orbit and Earth's heliocentric elliptical orbit. The hyperbolic orbit parameters include the pericentric radius r. p And the hyperbola enters the remaining velocity U; Step 2: Establish the B-plane coordinate system; the origin P is chosen at the Earth's center of mass, the x-axis follows the hyperbola of the small celestial body's geocentric orbit into the direction of the remaining velocity, the B-plane passes through the Earth's center of mass and is perpendicular to the direction of the remaining velocity; the y-axis is the opposite direction of the projection of the Earth's heliocentric velocity onto the B-plane; the z-axis is determined according to the right-hand rule. Since the perigee lift is only related to the impulse experienced by the small celestial body within the orbital plane, only the orbital state offset of the small celestial body caused by the impulse of the impactor within the orbital plane is considered. The three-dimensional B-plane coordinate system is transformed to the orbital plane coordinate system, that is: the x-axis enters the remaining velocity direction along the hyperbola, and the y-axis is located in the hyperbolic orbital plane of the small celestial body, pointing away from the remaining velocity direction. The efficiency of small celestial body offset performance evaluation is improved by transforming the three-dimensional B-plane coordinate system to the orbital plane coordinate system. Step 3: In the orbital plane coordinate system obtained in Step 2, according to the hyperbolic orbital parameters (U, r) of the small celestial body obtained in Step 1... p Find the velocity-rotation angle δ of the hyperbolic orbit, and further find the orbital state variables at the pericenter of the hyperbolic orbit; Step 4: Calculate the change in velocity of the small celestial body based on the conservation of momentum before and after the collision and the velocity of the impactor relative to the small celestial body; obtain the initial orbital offset based on the change in velocity of the small celestial body and the set position change ΔR. Step 5: Construct a multibody dynamics model of the Sun-Earth-small celestial body that reflects the gravitational influence of Earth, making the multibody dynamics model of the Sun-Earth-small celestial body more consistent with the actual gravitational environment after the small celestial body approaches Earth; For the multibody dynamics model of the Sun-Earth-small celestial body, based on the initial value of the small celestial body's orbit obtained in Step 3, construct a state transition tensor to facilitate the calculation of the small celestial body's orbital offset in Step 6. Step 6: Using the initial state offset of the small celestial body's orbit obtained in Step 4 and the state transition tensor obtained in Step 5, the orbit offset of the small celestial body moving under the Sun-Earth-small celestial body multibody dynamics model at any time is analytically obtained, that is, the small celestial body orbit offset is predicted based on the state transition tensor.
2. The method for evaluating the performance of small body migration based on state transition tensors as described in claim 1, characterized in that: It also includes step seven: selecting the impactor's velocity V relative to the small celestial body. r The angle with the x-axis is the impact geometry, and the magnitude of the impactor's velocity relative to the small celestial body is fixed at ||V. r By changing the impact geometry and impact time, the orbital offset of the small celestial body at the pericenter is predicted according to steps one through six, based on the corresponding impact geometry and impact time. The orbital offset δx(t) of the small celestial body is then used to... s The direction and magnitude of the impactor's impact on the small celestial body are used to evaluate the orbital deflection effectiveness of the impactor. Based on the orbital deflection effectiveness, the impact geometry and impact time corresponding to the maximum offset can be found. The impactor impacts the small celestial body according to the impact geometry and impact time, so that the small celestial body's orbit changes direction with the best offset effectiveness, thereby achieving defense against potentially dangerous small celestial bodies.
3. The method for evaluating the performance of small body migration based on state transition tensors as described in claim 1 or 2, characterized in that: The implementation method for step one is as follows: Step 1.1: Based on the known six elements of the two heliocentric elliptical orbits, perform numerical integration of two-body dynamics to obtain the minimum orbital interception distance MOID between the two elliptical orbits. Use this minimum orbital interception distance MOID as the radius r of the pericenter of the hyperbolic orbit. p ; Step 1.2: Utilize The theory of close encounters models the motion of a small celestial body approaching Earth as a two-body scattering under the planetary center. The relative velocity of the small celestial body with respect to Earth determines the hyperbolic entry velocity U of the geocentric hyperbolic orbit. This velocity is obtained through the elements of the heliocentric elliptical orbit of the small celestial body. Where: a ast e is the semi-major axis of the heliocentric orbit of the small celestial body. ast Let i be the eccentricity of the heliocentric orbit of a small celestial body. ast The inclination of the heliocentric orbit of the small celestial body.
4. The method for evaluating the performance of small body migration based on state transition tensors as described in claim 3, characterized in that: The method for implementing step three is as follows: In the orbital plane coordinate system obtained in step two, the hyperbolic orbital parameters (U, r) of the small celestial body obtained in step one are... p Substituting into equation (2), we obtain the velocity-rotation angle δ, and further, according to equation (3), we obtain the orbital state variables at the pericenter of the hyperbolic orbit: [x per ,y per [] indicates the position of the center point of the hyperbola. This indicates the velocity state at the midpoint of the hyperbolic orbit. Where, μ p is the Earth's gravitational constant.
5. The method for evaluating the performance of small body migration based on state transition tensors as described in claim 4, characterized in that: Step four is implemented as follows: Using the conservation of momentum before and after the collision, we obtain equation (4): K(m sc v sc +m ast v ast )=(m sc +m ast )(v ast +ΔV) (4) Where m represents mass, v represents the velocity vector in the inertial frame, subscript sc represents the impactor, subscript ast represents the small celestial body, and K represents the collision coefficient of the inelastic collision; because of the influence of ejected material during the impact, the value of K is much greater than 1; therefore, we take K = 1, thus obtaining the expression for the change in velocity of the small celestial body: Where V r The velocity vector of the impactor relative to the small celestial body at the time of impact; changing V r With the angle between the x-axis and the x-axis in the range [0, 2π), a series of ΔV values are obtained; Integrating the trajectory backward from the pericenter to the moment of impact yields the trajectory state at the moment of impact, which serves as the initial condition for integration; the initial trajectory state bias ΔX instantaneously after the impact is: ΔX=[ΔR,ΔV],ΔR=[0,0,0] (6).
6. The method for evaluating the performance of small body migration based on state transition tensors as described in claim 5, characterized in that: Step five is implemented as follows: Since the gravitational influence of Earth becomes dominant after a small celestial body approaches Earth, the accuracy of two-body dynamics that only considers the Sun's gravity is relatively low; a multi-body dynamics model of the Sun-Earth-small celestial body is constructed as shown in formula (7): Among them, the right-hand side of the second equation Represents the gravitational pull of Earth on small celestial bodies. This represents the Sun's gravitational pull on small celestial bodies; Since the multibody dynamics model of the Sun-Earth-small celestial body as shown in formula (7) includes a reaction to the gravitational effect of the Earth, This makes the multibody dynamics model of the Sun-Earth-small celestial body more consistent with the actual gravitational environment after the small celestial body approaches the Earth; For the multibody dynamics model of Sun-Earth-small body as shown in Equation (7), a state transition tensor suitable for the multibody dynamics model of Sun-Earth-small body with strong nonlinearity is constructed. This facilitates the use of the state transition tensor to calculate the orbital offset of the small body in the subsequent step six, thereby improving the accuracy of orbital offset prediction under strong nonlinear dynamics. For the multibody dynamics model of the Sun-Earth-small body as shown in Equation (7), a state transition tensor suitable for the strongly nonlinear multibody dynamics model of the Sun-Earth-small body is constructed. The specific implementation method is as follows: Initial conditions of the orbital state of small celestial bodies The solution to the above multibody dynamics model of the Sun-Earth-small celestial body is expressed as: x(t)=φ(t;x 0 ,t 0 ) (8) Where x(t) is the orbital state at time t, taking the derivative of both sides of the equation with respect to time, we get: Where: f represents the right-hand side of equation (7); since the initial conditions are independent of time, we have dx 0 / dt=0, then due to the initial time t of the orbit 0 State deviation δx at time 0 The resulting orbital state deviation δx(t) and its derivative at time t for: δx(t)=φ(t;x 0 +δx 0 ,t 0 )-φ(t;x 0 ,t 0 ) (10) For equations (10) and (11) above, respectively, in the initial orbital state x 0 Performing a Taylor expansion to order m at point m, we obtain: Where k j ∈{1,...,6}, representing the k-th orbital state variable. j There are elements; and we have: Where: the superscript "*" indicates the value of the state quantity along the nominal orbit, and This refers to the state transition tensor STTs; Because only Since it is related to time, taking the derivative of equation (12) with respect to time, we obtain the derivative of the orbital state deviation: Meanwhile, substituting equation (12) into equation (13), we obtain the derivative of the orbital state deviation: Combining equations (17) and (16), since δx 0 The corresponding coefficients are equal, thus we obtain the system of ordinary differential equations (19) for the state transition tensor: The state transition tensor in the ordinary differential equation system (19) can be solved by only knowing the initial value of the state transition tensor and the nominal orbit. The initial value of the state transition tensor is given by equation (14): The nominal orbit x(t) * The orbital integral is obtained directly from the initial orbital state with zero deviation under the multibody dynamics model (7) of the Sun-Earth-small celestial body.
7. The method for evaluating the performance of small body migrations based on state transition tensors as described in claim 6, characterized in that: In step six, Based on the initial orbital offset ΔX obtained from equation (6), the state transition tensor is used... Get t s Momentary deviation of small celestial body orbital state: in t s The orbital state deviation δx of small celestial bodies at any given moment i (t s ) represents the offset of the perigee of a small celestial body relative to the Earth, that is, the prediction of the orbital offset of a small celestial body is realized based on the state transition tensor.
8. The method for evaluating the performance of small body migration based on state transition tensors as described in claim 7, characterized in that: Contour lines are used to represent the impact geometry, enabling visualization of orbital deflection performance assessment.