B-plane-based near-earth asteroid collision probability analysis method, storage medium and device
By constructing a right-handed coordinate system and an error covariance matrix, mapping and transforming the coordinate system, and establishing an elliptical model, the accuracy problem of near-Earth asteroid collision probability assessment was solved, enabling effective assessment and monitoring of potential threats.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- ZIJINSHAN ASTRONOMICAL OBSERVATORY CHINESE ACAD OF SCI
- Filing Date
- 2023-03-28
- Publication Date
- 2026-05-15
AI Technical Summary
Current technologies are insufficient to effectively assess the probability of near-Earth asteroids colliding with Earth, affecting the accuracy of subsequent monitoring and early warning systems.
A near-Earth asteroid collision probability analysis method based on the B-plane is adopted. By constructing a right-handed coordinate system, calculating the error covariance matrix, mapping it to the rendezvous time, transforming the coordinate system and projecting it into 2D data, an elliptical model is established to evaluate the collision probability.
It provides an accurate assessment of the probability of near-Earth asteroids colliding with Earth, providing a reliable basis for subsequent monitoring and early warning work.
Smart Images
Figure CN116341248B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of astronomy technology, and in particular to a method, storage medium, and device for analyzing the probability of near-Earth asteroid collisions based on the B-plane. Background Technology
[0002] In recent years, several asteroids have been observed passing close to the Earth, near the Moon or even closer, making asteroid impacts a real and present threat to human survival. Although the probability of such an event is extremely low, the destructive range it would cause is wide and the threat is significant. Hazard assessments of potentially hazardous asteroids are necessary to determine subsequent monitoring methods and technological countermeasures. The probability of an asteroid impact is a key factor in assessing the threat level of near-Earth objects and enabling effective monitoring and early warning systems.
[0003] One of the most common and convenient frameworks for asteroid hazard assessment is the target plane, also known as the B plane. Whenever there is a close encounter between an asteroid and Earth, the target plane is defined as the plane with Earth as the origin and perpendicular to the asymptotic relative incident velocity of the nominal asteroid's unperturbed relative velocity.
[0004] This invention is based on the target plane. According to the error covariance of the initial epoch state vector, the error covariance at the rendezvous time is calculated, and then the probability density distribution in the target plane is calculated to obtain the collision probability between the near-Earth asteroid and Earth. Summary of the Invention
[0005] This invention addresses the shortcomings of existing technologies by providing a method, storage medium, and device for analyzing the probability of near-Earth asteroid collisions based on the B-plane. This method analyzes the probability of collisions between near-Earth asteroids and Earth, helping to assess the risks of potentially threatening asteroids and determine subsequent monitoring methods and technical countermeasures.
[0006] To achieve the above objectives, the present invention adopts the following technical solution:
[0007] A method for analyzing the probability of near-Earth asteroid collisions based on the B-plane includes the following steps:
[0008] S1: Define a right-handed coordinate system (ξ, η, ζ) under the target plane, and construct the error covariance matrix P model of the near-Earth asteroid angle measurement observations under this coordinate system based on the uncertainty of the near-Earth asteroid orbit;
[0009] S2: Based on the content of step S1, a pseudo-sequential processing method is used to determine the time t of the near-Earth asteroid. k The error covariance matrix under the given conditions is called the epoch error covariance matrix P. k The epoch error covariance matrix P k Perform a state transition so that the epoch covariance matrix P kThe covariance matrix C(t) is obtained by mapping it to the rendezvous time t between the near-Earth asteroid and Earth.
[0010] S3: Establish a new coordinate system (x, y, z) in the target plane, and transform the covariance matrix C(t) obtained in step S2 into this coordinate system to obtain C. xyz (t);
[0011] S4: C xyz (t) Projected onto the target plane, the 3D data is transformed into 2D data, which is a 2×2 matrix. By rotating the coordinates of this 2D data by a certain angle, a new matrix C is obtained. new (t), where one of the diagonals of the matrix has all values of 0, and the two values of the other diagonal are used as the semi-major axis and semi-minor axis, thus constructing an ellipse;
[0012] S5: Analyze the probability of a collision between a near-Earth asteroid and Earth based on the constructed ellipse.
[0013] To optimize the above technical solution, the specific measures also include:
[0014] Furthermore, the specific content of step S1 is as follows:
[0015] S1.1: Define the target plane: a plane perpendicular to the asymptote of the geocentric hyperbola of the near-Earth asteroid's kissing orbit, that is, a plane perpendicular to the direction of the near-Earth asteroid's unperturbed relative velocity;
[0016] S1.2: Establish a right-handed coordinate system (ξ, η, ζ) on the target plane with the Earth's center as the origin, where the η axis is the direction of the unperturbed relative velocity, which is perpendicular to the target plane; the ξ axis is the opposite direction of the projection of the Earth's heliocentric velocity onto the target plane; and the ζ axis is determined by the right-handed system.
[0017] S1.3: The formula for calculating the error covariance matrix P in the right-handed coordinate system (ξ, η, ζ) is expressed as follows:
[0018] P = (A T WA) -1 (1)
[0019] In the formula, W is the weight matrix, representing the error of near-Earth asteroid angular measurement observations in the right-hand coordinate system (ξ, η, ζ); A is the derivative matrix of angular measurement observations in the right-hand coordinate system (ξ, η, ζ), which is an m×6 partial derivative matrix. Each value in this matrix represents the partial derivative of the asteroid's angular measurement observation with respect to the epoch state value.
[0020] Furthermore, the specific content of step S2 is as follows:
[0021] S2.1: Calculate the near-Earth asteroid at t k Angular observation value z over time k Partial derivative with respect to the near-Earth asteroid epoch state x0
[0022]
[0023] In the formula, e1,...,e6 are the components of the near-Earth asteroid's epoch state x0; x1,...,x6 are the components of the near-Earth asteroid's epoch state t. k Time-based angular observation value z k The components; where matrix A has a total of k rows is called matrix A. k And let in t k Values calculated over time As the derivative matrix of angle measurement observation A k Similarly, the angle measurement observation derivative matrix A is obtained by using formula (2) to calculate the last k-th row value. k The values of each of the first k-1 rows are used to determine the complete angle measurement observation derivative matrix A. k ;
[0024] S2.2: Construct a model of the error covariance matrix P and the observation derivative matrix A based on formula (1). k The epoch error covariance matrix P is obtained. k =(A k T WA k ) -1 ;
[0025] S2.3: The epoch error covariance matrix P is determined by the following relationship. k Mapping this to the rendezvous time t between the near-Earth asteroid and Earth, we obtain C(t):
[0026] C(t)=Φ(t)P k Φ(t) T (3)
[0027] In the formula, Φ(t) represents the transition matrix at rendezvous time t, which maps the angle measurement error at the near-Earth asteroid epoch to the angle measurement error at rendezvous time t.
[0028] Furthermore, the specific content of step S3 is as follows:
[0029] S3.1: In the target plane, establish a new coordinate system (x, y, z) with the center of the near-Earth object as the origin; where the x-axis lies in the target plane and its direction points from the center of the near-Earth object towards the Earth's center; the z-axis is the normal to the target plane; the y-axis is determined by the right-hand rule.
[0030] S3.2: Transform the covariance matrix C(t) to the coordinate system (x,y,z) to obtain C xyz (t):
[0031] C xyz (t)=MC(t)M T (4)
[0032] In the formula, M represents the transformation matrix from the right-hand coordinate system (ξ, η, ζ) to the coordinate system (x, y, z).
[0033] Further, in step S4, the new matrix C new The specific content of (t) is as follows:
[0034]
[0035] In the formula, c1 represents the semi-major axis of the constructed ellipse; c2 represents the semi-minor axis of the constructed ellipse.
[0036] Furthermore, step S5, which describes analyzing the probability of a collision between a near-Earth asteroid and Earth based on the constructed ellipse, specifically states that theoretically, the ellipse intersects with the Earth on an arc, but in reality, the intersecting arc is extremely large, essentially covering the entire Earth's sphere. The Earth's sphere is taken as the object of study.
[0037] (a) Evaluate the normal density function at the center of the circle and multiply it by the area of the circle to obtain the value P1. The specific calculation formula is as follows:
[0038]
[0039] In the formula, The meaning of c1 is the diameter of the Earth; d represents the distance between the projection of the near-Earth object and the projection of the Earth on the target plane; σ1 and σ2 represent the major and minor axes of the ellipse, respectively, which are determined by the values of c1 and c2.
[0040] (ii) Evaluate the normal density function at four points along the two axes of the circle, and then multiply it by the area of the circle / 4 to obtain the value P2. The specific calculation formula is as follows:
[0041]
[0042] (III) Evaluate the normal density function at the four endpoints of the diagonals along the two axes of the circle, and then multiply it by the area of the circle / 4 to obtain the value P3. The specific calculation formula is as follows:
[0043]
[0044] Based on the values obtained from (I), (II), and (III) above, the probability density is obtained, that is, the probability p of a near-Earth asteroid colliding with Earth:
[0045]
[0046] A computer-readable storage medium storing a computer program, characterized in that the computer program causes a computer to execute the near-Earth asteroid collision probability analysis method as described in any of the preceding claims.
[0047] An electronic device includes: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it implements the near-Earth asteroid collision probability analysis method as described in any of the preceding claims.
[0048] The beneficial effects of this invention are: This application constructs an analytical method for predicting the probability of a near-Earth asteroid colliding with Earth, which can assess the probability of a future asteroid encountering Earth and causing a collision, providing a basis for subsequent monitoring and early warning work. Attached Figure Description
[0049] Figure 1 This is a schematic diagram illustrating the relationship between the target plane, the Earth, near-Earth objects, geocentric hyperbolas, and asymptotes as defined in this invention.
[0050] Figure 2 This is a schematic diagram showing the positional relationship between the right-handed coordinate system (ξ, η, ζ) and the coordinate system (x, y, z) of this invention.
[0051] Figure 3 This is a schematic diagram of how the present invention constructs an ellipse. Detailed Implementation
[0052] The present invention will now be described in detail with reference to the accompanying drawings.
[0053] The overall technical solution of this application is as follows:
[0054] Approximate definition of the target plane: a plane perpendicular to the asymptote direction of the geocentric hyperbola of the near-Earth object's orbit (also known as the direction of the relative velocity without perturbation), such as... Figure 1 As shown, a right-handed coordinate system (ξ, η, ζ) is established on the B plane with the Earth's center as the origin. Here, η represents the direction of the unperturbed relative velocity v. ∞ ξ is perpendicular to plane B; ξ is the opposite direction of the projection of the Earth's heliocentric velocity onto plane B. Let the coordinates of the intersection point of the asymptote and plane B be (ξ′, ζ′), where ζ′ is the rendezvous distance between the near-Earth object and Earth. By adjusting the rendezvous time, we can obtain its minimum value on plane B, which is the minimum distance between the asteroid and Earth's orbital contact.
[0055] In addition, the assessment of collision probability mainly considers the uncertainty of asteroids, that is, it is calculated through the error covariance matrix:
[0056] P = (AT WA) -1 (1)
[0057] Where W is the weight matrix, representing the angular measurement error of the asteroid, composed of the reciprocals of the measurement variance along the main diagonal, and A is an m×6 partial derivative matrix; the specific steps are as follows:
[0058] S1: Perform a state transition on the error covariance matrix, mapping the epoch error covariance matrix to the rendezvous time t. First, calculate the near-Earth asteroid at t. k Angular observation value z over time k Partial derivative with respect to the near-Earth asteroid epoch state x0
[0059]
[0060] z k For time t k The obtained angular measurement observation data, e1,...,e6 are the components of the initial epoch state, and x1,...,x6 are the components of the near-Earth asteroid at t k Time-based angular observation value z k The amount. This indicates that the state error at the initial epoch is mapped to t. k Time error. Here It is the k-th row of matrix A. Therefore, a recursive relation can be used, employing a pseudo-sequential processing method, utilizing the k-th row of matrix A. and (A) k-1 T WA k-1 )Calculate (A k T WA k Thus, P is obtained. k =(A k T WA k ) -1 Here, the prior information matrix A0 is zero. Then, the epoch error covariance matrix P... k This is mapped to the rendezvous time t through the following relationship:
[0061] C(t)=Φ(t)P k Φ(t) T (3)
[0062] In the formula, Φ(t) represents the transition matrix at the intersection time t, which is obtained by mapping the initial epoch state vector to the intersection time state vector.
[0063] S2: Perform coordinate transformation on the covariance matrix C(t) obtained from the mapping. Establish a new coordinate system (x, y, z) on the target plane. The reference vector x will lie in the target plane, and its direction will be defined by the vector from the near-Earth object's position to the Earth. The reference vector z will be the normal to the target plane. The reference vector y will lie in the target plane, and its definition is to complete the right-handed system, such as... Figure 2 As shown. Calculate the coordinate transformation matrix M from the right-handed coordinate system (ξ, η, ζ) to the coordinate system (x, y, z). Then the matrix of C(t) in the new coordinate system is:
[0064] C xyz (t)=MC(t)M T (4)
[0065] S3: C xyz (t) Projected onto the target plane, transforming the 3D problem into a 2D problem. The projection process involves removing the z-related terms from the original matrix to obtain a 2D matrix. Let C xyz (t) becomes a diagonal matrix after coordinate rotation. The rotation angle θ can be calculated, and an appropriate angle is chosen to diagonalize the 2D matrix.
[0066]
[0067] At this point, c1 represents the semi-major axis of the ellipse to be constructed; c2 represents the semi-minor axis of the ellipse to be constructed.
[0068] S4: After obtaining the ellipse intersecting the target plane by the confidence ellipsoid constructed from the error covariance matrix, the probability of a collision can be calculated. The integral can be approximated using the following method: the integral covers the entire circle (in the theoretical case, the ellipse intersects the Earth on an arc surface, such as...). Figure 2 As shown, but in reality, the intersecting arc is extremely large, basically covering the entire Earth's sphere, such as... Figure 3 As shown, taking the Earth's sphere as the research object, it can be approximated in three different ways. First, by evaluating the normal density function at the center of the circle and multiplying it by the area of the circle. Second, by evaluating the normal density function at four points along both axes and then multiplying it by the area / 4. Third, by evaluating the normal density function at four points along the diagonals of the axes and multiplying it by the area / 4. Finally, these approximations are averaged to obtain the probability density. The specific calculation formula is as follows:
[0069] (I) Evaluate the normal density function at the center of the ellipse and multiply it by the area of the circle to obtain the value P1. The specific calculation formula is as follows:
[0070]
[0071] In the formula, The meaning of c1 is the diameter of the Earth; d represents the distance between the projection of the near-Earth object and the projection of the Earth on the target plane; σ1 and σ2 represent the major and minor axes of the ellipse, respectively, which are determined by the values of c1 and c2.
[0072] (ii) Evaluate the normal density function at four points along the two axes of the circle, and then multiply it by the area of the circle / 4 to obtain the value P2. The specific calculation formula is as follows:
[0073]
[0074] (III) Evaluate the normal density function at the four endpoints of the diagonals along the two axes of the circle, and then multiply it by the area of the circle / 4 to obtain the value P3. The specific calculation formula is as follows:
[0075]
[0076] Based on the values obtained from (I), (II), and (III) above, the probability density is obtained, that is, the probability p of a near-Earth asteroid colliding with Earth:
[0077]
[0078] This method is applied to an example of asteroid collision probability calculation to predict the impact of asteroid 1997XR2. Table 1 shows the future encounter time, nominal encounter distance, minimum possible distance, and collision probability p of 1997XR2 with Earth.
[0079] Table 1
[0080]
[0081] A computer-readable storage medium storing a computer program, characterized in that the computer program causes a computer to execute the near-Earth asteroid collision probability analysis method as described in any of the preceding claims.
[0082] An electronic device includes: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it implements the near-Earth asteroid collision probability analysis method as described in any of the preceding claims.
[0083] The above are merely preferred embodiments of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principles of the present invention should be considered within the scope of protection of the present invention.
Claims
1. A method for analyzing the probability of near-Earth asteroid collisions based on the B-plane, characterized in that, Includes the following steps: S1: Define a right-handed coordinate system in the target plane. In this coordinate system, the error covariance matrix of near-Earth asteroid angular measurement observations is constructed based on the uncertainty of near-Earth asteroid orbits. Model; S2: Based on the content of step S1, a pseudo-sequential processing method is used to determine the time from the epoch of the near-Earth asteroid. The error covariance matrix under the given conditions is called the epoch error covariance matrix. The epoch error covariance matrix Perform a state transition so that the epoch covariance matrix Mapped to the time of near-Earth asteroid encounter with Earth t The covariance matrix is obtained from the above. ; S3: Establish a new coordinate system (x, y, z) in the target plane, and use the covariance matrix obtained in step S2. Transform to this coordinate system to obtain ; S4: Will Projecting the data onto the target plane transforms the 3D data into 2D data, which is a 2×2 matrix. A new matrix is obtained by rotating this 2D data by a certain angle. In this matrix, one of the diagonals has all values of 0, and the two values of the other diagonal serve as the semi-major axis and semi-minor axis, thus constructing an ellipse; S5: Based on the constructed ellipse, the probability of a collision between a near-Earth asteroid and Earth is analyzed: In theory, the ellipse intersects with Earth on an arc, but in reality, the intersecting arc is extremely large, basically covering the entire Earth's sphere. Taking the Earth's sphere as the research object: (a) Evaluate the normal density function at the center of the circle and multiply it by the area of the circle to obtain the value. The specific calculation formula is as follows: （6） In the formula, The term indicates the diameter of the Earth. It represents the distance between the near-Earth object projection and the Earth projection on the target plane; and They respectively represent the major axis and minor axis of the ellipse, which are... and The value is known; (ii) Evaluate the normal density function at four points along the two axes of the circle, and then multiply it by the area of the circle / 4 to obtain the value. The specific calculation formula is as follows: （7） (iii) Evaluate the normal density function at the four endpoints of the diagonal along the axis of the circle, and then multiply it by the area of the circle / 4 to obtain the value. The specific calculation formula is as follows: （8） Based on the values obtained from (I), (II), and (III) above, the probability density is obtained, which is the probability of a near-Earth asteroid colliding with Earth. : 。 2. The method for analyzing the probability of near-Earth asteroid collisions based on the B-plane according to claim 1, characterized in that, The specific content of step S1 is as follows: S1.1: Define the target plane: a plane perpendicular to the asymptote of the geocentric hyperbola of the near-Earth asteroid's kissing orbit, that is, a plane perpendicular to the direction of the near-Earth asteroid's unperturbed relative velocity; S1.2: Establish a right-handed coordinate system on the target plane with the Earth's center as the origin. ,in The axis represents the direction of the relative velocity without perturbation, and it is perpendicular to the target plane; The ζ-axis is the opposite direction to the projection of the Earth's heliocentric velocity onto the target plane; the ζ-axis is determined using the right-hand rule. S1.3: In a right-handed coordinate system Constructing the error covariance matrix The model calculation formula is expressed as: （1） In the formula, Let be the weight matrix, representing the weights in the right-handed coordinate system. Errors in near-Earth asteroid angular measurement observations; In a right-handed coordinate system The derivative matrix of the angle measurement observations is... The partial derivative matrix, where each value represents the partial derivative of the asteroid's angular observation with respect to the epoch state value.
3. The method for analyzing the probability of near-Earth asteroid collisions based on the B-plane according to claim 2, characterized in that, The specific details of step S2 are as follows: S2.1: Calculate the near-Earth asteroid's position in... Angle measurement observations over time Relative to the epochal state of near-Earth asteroids partial derivatives : （2） In the formula, Near-Earth asteroid epoch status The amount; It is a near-Earth asteroid in Time-based angle measurement observations The components; where the matrix Total number of rows is This is called a matrix. and order in Values calculated over time As the derivative matrix of angle measurement observation The last one in the middle Similarly, the derivative matrix of the angle measurement observation can be obtained by formula (2) for row numerical values. The former The values in each row are used to determine the complete angle measurement observation derivative matrix. ; S2.2: Constructing the error covariance matrix based on formula (1) The model, and the observation derivative matrix The epoch error covariance matrix is obtained. ; S2.3: The epoch error covariance matrix is determined by the following relationship. Mapped to the time of encounter between near-Earth asteroids and Earth t Up, get : （3） In the formula, The meaning is the meeting time. t The transition matrix is used to map the angular observation error of the near-Earth asteroid epoch to the rendezvous time. t Error in angle measurement observations.
4. The method for analyzing the probability of near-Earth asteroid collisions based on the B-plane according to claim 3, characterized in that, The specific content of step S3 is as follows: S3.1: In the target plane, establish a new coordinate system (x, y, z) with the center of the near-Earth object as the origin; where the x-axis lies in the target plane and its direction points from the center of the near-Earth object towards the Earth's center; the z-axis is the normal to the target plane; the y-axis is determined by the right-hand rule. S3.2: Convert the covariance matrix Transform to the coordinate system (x, y, z) to obtain : （4） In the formula, The meaning of is from the right-hand coordinate system The transformation matrix to the coordinate system (x, y, z).
5. The method for analyzing the probability of near-Earth asteroid collisions based on the B-plane according to claim 1, characterized in that, In step S4, the new matrix The specific content is as follows: （5） In the formula, This represents the semi-major axis of the constructed ellipse; This represents the semi-minor axis of the constructed ellipse.
6. A computer-readable storage medium storing a computer program, characterized in that, The computer program causes the computer to execute the near-Earth asteroid collision probability analysis method as described in any one of claims 1-5.
7. An electronic device, characterized in that, include: The memory, the processor, and the computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it implements the near-Earth asteroid collision probability analysis method as described in any one of claims 1-5.