A method and system for shape inversion of rapidly rotating asteroids

Abstract

This application discloses a method and system for shape inversion of a rapidly rotating asteroid. The method includes: determining several equally spaced sampling times within any exposure time of the rapidly rotating asteroid; calculating the position coordinates corresponding to each sampling time; calculating the brightness of each sampling time within each exposure time using a light variation model and taking the average value as the model brightness of the rapidly rotating asteroid; obtaining the residual based on the calculated model brightness and the observed brightness obtained from actual observation; determining whether the iteration stopping condition is met; if so, determining the optimal solution of the Gaussian surface density; otherwise, calculating the iteration step size under the optimization objective function, updating the Gaussian surface density, and continuing the iteration until the optimal solution of the Gaussian surface density is determined; finally, determining the shape of the rapidly rotating asteroid using the optimal solution of the Gaussian surface density according to Minkowski's theorem. This application can effectively improve the accuracy of asteroid shape inversion calculation.

Description

Technical Field

[0001] This application relates to the field of asteroid defense and detection technology, and in particular to a method and system for shape inversion of rapidly rotating asteroids. Background Technology

[0002] In the field of asteroid defense and detection technology, the shape and rotation parameters of an asteroid are two crucial characteristics. Light curves are the primary data foundation for obtaining asteroid shape and rotation parameters, and using light curves to invert these parameters is the most common method. Therefore, designing a method for inverting asteroid shape is a significant technical challenge.

[0003] Current methods for asteroid shape inversion are primarily based on the light variation model and inversion method proposed by Kaasalainen. Specifically, this model proposes that the brightness of an asteroid at any given time t can be represented by the integral of the brightness of sunlight reflected from the visible area of ​​the asteroid's surface at that time: The inversion method discretizes the asteroid surface into several triangles, each with a different normal vector (θ, ψ) and area A. The above equation can then be rewritten as: The following function is used as the objective function for optimization: Finally, the optimal solution for A(θ,ψ) can be obtained by fitting the observation data using the least squares method. According to Minkowski's theorem, under the assumption that the asteroid is a convex body, A(θ,ψ) corresponds to a unique convex body shape, which is the shape of the asteroid.

[0004] However, current asteroid shape inversion methods do not consider exposure time, thus ignoring the integral effect within the exposure time. They rely on the premise that exposure time is ignored in actual observations. But when asteroids rotate rapidly, longer exposure times are required for observing faint asteroids. If current methods are used for asteroid shape inversion, ignoring exposure time, the inversion results will differ significantly from the actual shape, leading to poor accuracy and failing to meet the needs of asteroid detection and research. Summary of the Invention

[0005] This application provides a method and system for shape inversion of rapidly rotating asteroids to solve the problem that existing asteroid inversion methods have poor accuracy in calculation results, which cannot meet the needs of detection and research of rapidly rotating asteroids.

[0006] To address the aforementioned technical problems, the embodiments of this application disclose the following technical solutions:

[0007] A method for shape inversion of a rapidly rotating asteroid, the method comprising:

[0008] S1: Determine several equally spaced sampling times within any exposure time of a rapidly rotating asteroid;

[0009] S2: Calculate the position coordinates corresponding to each sampling moment, where the position coordinates are the position coordinates of the observer and the sun relative to the rapidly rotating asteroid;

[0010] S3: Use the ellipsoid as the initial value for the shape of the asteroid, and use the Gaussian surface density of the ellipsoid as the initial value for the Gaussian surface density;

[0011] S4: Based on the position coordinates, Gaussian surface density, and the brightness of the rapidly rotating asteroid at each sampling time, the model brightness of the rapidly rotating asteroid is calculated using the average value method with the light variation model. The Gaussian surface density is the area A(θ,ψ) of the small triangle corresponding to each normal vector (θ,ψ) in the inversion method.

[0012] S5: Based on the calculated model brightness and the observed brightness obtained from actual observations, use the optimization objective function formula... The residual brightness corresponding to the current Gaussian surface density is calculated, where L model For model brightness, L obs For observing brightness;

[0013] S6: Determine whether the set iteration stop condition is met;

[0014] S7: If so, determine the optimal solution for the Gaussian surface density;

[0015] S8: If not, based on the calculated model brightness and the actual observed brightness, use the formula dA = -(J T J+∈I) -1 J T F is the iteration step size for optimizing the objective function. The Gaussian surface density is updated based on this iteration step size, and the process returns to step S4, where ∈ is the damping coefficient, and F is the difference vector between the model brightness and the observed brightness. k is the number of data observations, and J is the Jacobian matrix. nl represents the number of observation data points on the light curve where a certain observation data point is located. is the average value of the light curve, n is the number of small triangles on the surface, and m is the number of sampling points within the exposure time of a single observation data;

[0016] S9: After determining the optimal solution of the Gaussian surface density, the shape of the rapidly rotating asteroid is determined using the optimal solution of the Gaussian surface density according to the Minkowski theorem.

[0017] Optionally, a method for determining several equally spaced sampling times within any exposure time of a rapidly rotating asteroid includes:

[0018] During any given exposure time, samples are taken at equal intervals to determine m data sampling points, where... t e For exposure time, T syn The period of light variation of the rapidly rotating asteroid;

[0019] Using formula The sampling time corresponding to any data observation point is calculated, where 0 <i<m,t jd At the midpoint of the exposure time, t e For exposure duration.

[0020] Optionally, the method for calculating the position coordinates corresponding to each sampling time includes:

[0021] Using linear interpolation, the formula is employed. The position coordinates corresponding to each sampling time are calculated, where, Let r be the position coordinates corresponding to any sampling time. b r represents the coordinates corresponding to the previous observation data. a Let t be the coordinates corresponding to the next observation. a t represents the time corresponding to the previous observation data. b This refers to the time corresponding to the next observation.

[0022] Optionally, based on the position coordinates, Gaussian surface density, and the brightness of the rapidly rotating asteroid at each sampling time, the model brightness of the rapidly rotating asteroid is calculated using the average value method with a light variation model. The Gaussian surface density is calculated using the method of determining the area A(θ,ψ) of the small triangle corresponding to each normal vector (θ,ψ) in the inversion method, specifically as follows:

[0023] Based on the given location coordinates and Gaussian surface density, using the formula... Calculate the sum of the brightness of the rapidly rotating asteroid within the exposure time at each sampling moment and take the average value, where t i For any sampling time, (θ) j ,ψ j Let θ be the normal vector corresponding to each small triangle after triangulation, and let A(θ) be the area of ​​the small triangle corresponding to the normal vector. j ,ψ j Let μ be the Gaussian surface density, P be the parameter of the scattering function in the inversion method, and μ = nor·r. s μ0=nor·r e ,α=r s ·r e, where nor=(sinθcosψ, sinθsinψ, cosθ), r s ,r e These are the unit vectors pointing from the rapidly rotating asteroid towards the Sun and Earth, respectively.

[0024] Optionally, the set iteration stopping condition includes:

[0025] The residual is less than the set value; or,

[0026] The number of iterations has reached the set maximum number of iterations.

[0027] A shape inversion system for rapidly rotating asteroids, the system comprising:

[0028] The sampling time determination module is used to determine several equally spaced sampling times within any exposure time of a rapidly rotating asteroid.

[0029] The position coordinate calculation module is used to calculate the position coordinates corresponding to each sampling moment, wherein the position coordinates are the position coordinates of the observer and the sun relative to the rapidly rotating asteroid;

[0030] The Gaussian surface density initial value determination module is used to take the ellipsoid as the initial value of the asteroid shape and the Gaussian surface density of the ellipsoid as the initial value of the Gaussian surface density.

[0031] The model brightness calculation module is used to calculate the model brightness of the rapidly rotating asteroid using the light variation model and the average value method based on the position coordinates, Gaussian surface density and the brightness of the rapidly rotating asteroid at each sampling time. The Gaussian surface density is the area A(θ,ψ) of the small triangle corresponding to each normal vector (θ,ψ) in the inversion method.

[0032] The residual calculation module is used to calculate the model brightness and the observed brightness obtained from actual observations, using formulas... The residual brightness corresponding to the current Gaussian surface density is calculated, and the residual is used as the objective function, where L model For model brightness, L obs For observing brightness;

[0033] The judgment module is used to determine whether the set iteration stopping condition is met. If it is, the optimal solution of Gaussian surface density is determined. If not, the iteration step size calculation module is started, the Gaussian surface density is updated according to the iteration step size calculated by the iteration step size calculation module, and the model brightness calculation module is restarted.

[0034] The iteration step size calculation module is used to calculate the model brightness and the observed brightness obtained from actual observation, using the formula dA = -(J T J+∈I)-1 J T F is the iteration step size used to optimize the objective function, where ∈ is the damping coefficient and F is the difference vector between the model brightness and the observed brightness. k is the number of data observations, and J is the Jacobian matrix. nl represents the number of observation data points on the light curve where a certain observation data point is located. is the average value of the light curve, n is the number of small triangles on the surface, and m is the number of sampling points within the exposure time of a single observation data;

[0035] The shape determination module is used to determine the shape of the rapidly rotating asteroid by using the optimal solution of the Gaussian surface density according to the Minkowski theorem.

[0036] Optionally, the set iteration stopping condition includes:

[0037] The residual is less than the set value; or,

[0038] The number of iterations has reached the set maximum number of iterations.

[0039] Optionally, the sampling time determination module includes:

[0040] The data observation point division unit is used to sample at equal intervals within any given exposure time to determine m data sampling points, wherein... t e For exposure time, T syn The period of light variation of the rapidly rotating asteroid;

[0041] The sampling time determination unit is used to determine the sampling time using the formula. Calculate the sampling time corresponding to any data observation point, where 0 <i<m,t jd At the midpoint of the exposure time, t e For exposure duration.

[0042] Optionally, the model brightness calculation module is used to calculate the brightness based on the position coordinates and Gaussian surface density using the formula... Calculate the sum of the brightness of the rapidly rotating asteroid within the exposure time at each sampling moment and take the average value, where t i For any sampling time, (θ) j ,ψ j Let θ be the normal vector corresponding to each small triangle after triangulation, and let A(θ) be the area of ​​the small triangle corresponding to the normal vector. j ,ψ jLet μ be the Gaussian surface density, P be the parameter of the scattering function in the inversion method, and μ = nor·r. s μ0=nor·r e ,α=r s ·r e , where nor=(sinθcosψ, sinθsinψ, cosθ), r s ,r e These are the unit vectors pointing from the rapidly rotating asteroid towards the Sun and Earth, respectively.

[0043] The technical solutions provided by the embodiments of this application may include the following beneficial effects:

[0044] This application provides a method for shape inversion of a rapidly rotating asteroid. After obtaining the exposure times corresponding to multiple observation data of the rapidly rotating asteroid, the method first determines several equally spaced sampling times within any given exposure time and calculates the position coordinates corresponding to each sampling time. Next, based on these position coordinates, the Gaussian surface density, and the brightness of the rapidly rotating asteroid at each sampling time, the model brightness of the asteroid is calculated. Then, the residual is calculated using the formulas for the calculated model brightness and the actual observed brightness. Next, according to a set iteration stopping condition, it is determined whether the iteration stopping condition is met. If the condition is met, the optimal solution for the Gaussian surface density is determined; otherwise, based on the calculated model brightness and the actual observed brightness, the iteration step size for optimizing the objective function is calculated using a formula. The Gaussian surface density is updated according to the iteration step size, and the iteration continues until the optimal solution for the Gaussian surface density is determined. Finally, based on Minkowski's theorem, the shape of the rapidly rotating asteroid is determined using the optimal solution for the Gaussian surface density. This embodiment takes exposure time into account when performing shape inversion calculations for rapidly rotating asteroids, fully considering the integral effect within the exposure time, thereby significantly improving the accuracy of asteroid inversion calculations. Especially for the observation of faint asteroids, where the exposure time is longer than that of typical rapidly rotating asteroids, the method in this embodiment can further improve the accuracy of asteroid shape inversion calculations, thus meeting the needs of asteroid exploration and research.

[0045] In addition, this embodiment also performs equally spaced sampling within any exposure time. When calculating the model brightness of the rapidly rotating asteroid using its position coordinates, Gaussian surface density, and the brightness of the asteroid at each sampling time, the formula is used. The method calculates the sum of the brightness of rapidly rotating asteroids within the exposure time at each sampling moment and takes the average value. This method can further improve the accuracy of asteroid shape inversion calculation results because it collects multiple observation moments within an exposure time and takes the average value of multiple sampling moments.

[0046] This embodiment determines the optimal solution for the area A(θ,ψ) of the small triangle in the inversion method by setting a certain iterative method and determining the iteration step size under the optimization objective function. Then, the shape of the fast-rotating asteroid is determined by matching the optimal solution of the area A(θ,ψ). The shape of the fast-rotating asteroid obtained by this method of repeated calculation is closer to the actual shape of the asteroid, which is conducive to further improving the accuracy of the shape inversion method of fast-rotating asteroid.

[0047] This application also provides a shape inversion system for rapidly rotating asteroids. The system mainly includes: a sampling time determination module, a position coordinate calculation module, a Gaussian surface density initial value determination module, a model brightness calculation module, a residual calculation module, a judgment module, an iteration step size calculation module, and a shape determination module. The sampling time determination module fully considers the asteroid's exposure time and the integral effect within that exposure time, effectively improving the accuracy of asteroid inversion calculations. Especially for the observation of faint asteroids, whose exposure times are longer than those of typical rapidly rotating asteroids, this system further enhances the accuracy of asteroid shape inversion calculations, thus meeting the needs of asteroid exploration and research.

[0048] The configuration of the model brightness calculation module, residual calculation module, and iteration step size calculation module in this embodiment enables the acquisition of multiple observation times within an exposure time and the calculation of the average value of multiple sampling times, which is beneficial to further improve the accuracy of asteroid shape inversion calculation results.

[0049] The judgment module is configured with certain iteration stopping conditions. Based on the judgment result, it determines whether to continue iterative calculation until the optimal solution of Gaussian surface density is determined. In this embodiment, the iteration process is as follows: determine the model brightness based on the initial value of Gaussian surface density; determine the iteration step size based on the model brightness and the observed brightness; determine the new Gaussian surface density based on the iteration step size; and then determine the new model brightness based on the new Gaussian surface density. This process is repeated multiple times until the set iteration stopping condition is reached, thereby determining the optimal solution of Gaussian surface density. The shape of the rapidly rotating asteroid obtained through multiple iterations is closer to the actual shape of the asteroid, which is beneficial to further improving the accuracy of the shape inversion method for rapidly rotating asteroids.

[0050] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and do not limit this application. Attached Figure Description

[0051] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this application and, together with the description, serve to explain the principles of this application.

[0052] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0053] Figure 1 A schematic flowchart illustrating a method for shape inversion of a rapidly rotating asteroid provided in an embodiment of this application;

[0054] Figure 2 A schematic diagram showing the experimental results of comparing the shape inversion method in this invention with the inversion method proposed by Kaasalainen using the shape distribution method;

[0055] Figure 3 This is a schematic diagram of the structure of a shape inversion system for a rapidly rotating asteroid provided in an embodiment of this application. Detailed Implementation

[0056] To enable those skilled in the art to better understand the technical solutions in this application, the technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments in this application, all other embodiments obtained by those of ordinary skill in the art without creative effort should fall within the scope of protection of this application.

[0057] To better understand this application, the embodiments of this application will be explained in detail below with reference to the accompanying drawings.

[0058] Example 1

[0059] See Figure 1 , Figure 1 This is a schematic flowchart illustrating a method for shape inversion of a rapidly rotating asteroid provided in an embodiment of this application. Figure 1 As can be seen, the shape inversion method for rapidly rotating asteroids in this embodiment mainly includes the following processes:

[0060] S1: Determine several equally spaced sampling times within any exposure time of a rapidly rotating asteroid.

[0061] The observational data of rapidly rotating asteroids includes exposure times; each observation corresponds to one data point, and each data point corresponds to a specific exposure time. In this embodiment, when performing shape inversion for rapidly rotating asteroids, the impact of exposure time on the shape inversion calculation is considered. This is especially true for faint asteroids with relatively long exposure times, effectively improving the accuracy of the final determined shape of the rapidly rotating asteroid, making the inverted asteroid shape closer to the actual shape of the asteroid.

[0062] After obtaining multiple exposure times for the rapidly rotating asteroid, several equally spaced sampling times are determined within any given exposure time. Specifically, step S1 includes the following process:

[0063] S11: Sample at equal intervals within any exposure time to determine m data sampling points.

[0064] in, t e For exposure time, T syn The light variation period of the rapidly rotating asteroid is given.

[0065] S12: Using the formula The sampling time corresponding to any data observation point is calculated, where 0 <i<m,t jd At the midpoint of the exposure time, t e For exposure duration.

[0066] In this embodiment, one observation data point corresponds to an exposure time period, which is equivalent to integrating the brightness change curve over the exposure time and taking the average value. m observation times are equivalent to sampling the brightness at equal intervals over the exposure time and taking the average value, which is equivalent to discretizing the integral.

[0067] See also Figure 1 It can be seen that after determining several equally spaced sampling times within any exposure time, step S2 is executed: calculate the position coordinates corresponding to each sampling time, where the position coordinates are the position coordinates of the observer and the sun relative to the rapidly rotating asteroid.

[0068] Specifically, step S2 can be performed using the following method:

[0069] Using linear interpolation, the formula is employed. The position coordinates corresponding to each sampling time are calculated. Where r ti Let r be the position coordinates corresponding to any sampling time. b r represents the coordinates corresponding to the previous observation data. a Let t be the coordinates corresponding to the next observation. a t represents the time corresponding to the previous observation data.b This refers to the time corresponding to the next observation.

[0070] S3: Use the ellipsoid as the initial value for the shape of the asteroid, and use the Gaussian surface density of the ellipsoid as the initial value for the Gaussian surface density.

[0071] The Gaussian surface density is the area A(θ,ψ) of the small triangle corresponding to each normal vector (θ,ψ) in the inversion method.

[0072] In the inversion method, the initial value of the Gaussian surface density is determined based on the initial value of the asteroid's shape. Specifically, the Gaussian surface density corresponding to the ellipsoid is set as the initial value of the Gaussian surface density. This ellipsoid represents the initial shape of the rapidly rotating asteroid, and the Gaussian surface density corresponding to the ellipsoid is the initial value of the Gaussian surface density. In other words, the surface of the ellipsoid is triangulated to determine the initial value of the Gaussian surface density in the rapidly rotating asteroid inversion method.

[0073] The subsequent Gaussian surface density is calculated based on the iteration step size. That is, the Gaussian surface density in the next iteration is the sum of the Gaussian surface density in the previous iteration and the iteration step size.

[0074] S4: Based on the position coordinates, Gaussian surface density, and the brightness of the rapidly rotating asteroid at each sampling time, the model brightness of the rapidly rotating asteroid is calculated using the average value method based on the light variation model. The Gaussian surface density is the area A(θ,ψ) of the small triangle corresponding to each normal vector (θ,ψ) in the inversion method.

[0075] The light variation model here is the light variation model proposed by Kaasalainen.

[0076] Specifically, step S4 is implemented as follows:

[0077] Based on the given location coordinates and Gaussian surface density, using the formula... Calculate the sum of the brightness of the rapidly rotating asteroids during the exposure time at each sampling moment and take the average value.

[0078] Among them, t i For any sampling time, (θ) j ,ψ j Let θ be the normal vector corresponding to each small triangle after triangulation, and let A(θ) be the area of ​​the small triangle corresponding to the normal vector. j ,ψ j Let μ be the Gaussian surface density, P be the parameter of the scattering function in the inversion method, and μ = nor·r. s μ0=nor·r e ,α=r s ·r e , where nor=(sinθcosψ, sinθsinψ, cosθ), r s ,re These are the unit vectors pointing from the rapidly rotating asteroid towards the Sun and Earth, respectively.

[0079] S5: Based on the calculated model brightness and the observed brightness obtained from actual observations, use the optimization objective function formula... The residual brightness corresponding to the current Gaussian surface density is calculated, where L model For model brightness, L obs For observing brightness.

[0080] S6: Determine whether the set iteration stop condition is met.

[0081] In this embodiment, the iteration stopping conditions include: the residual is less than a set value, that is, the value of the optimized objective function is less than a set value; or, the number of iterations reaches the set maximum number of iterations.

[0082] The residual is less than the set value, which is the objective function of optimization.

[0083] If the set iteration stopping condition is met, proceed to step S7: determine the optimal solution for the Gaussian surface density.

[0084] The optimal solution for Gaussian surface density is also the optimal solution for the area A(θ,ψ) of the small triangle in the inversion method.

[0085] If the set iteration stopping condition is not met, proceed to step S8: Based on the calculated model brightness and the observed brightness obtained from actual observation, use the formula dA = -(J T J+∈I) -1 J T F is used to calculate the iteration step size when optimizing the objective function, and the Gaussian surface density is updated according to the iteration step size. Then, the process returns to step S4 to recalculate the model brightness of the rapidly rotating asteroid.

[0086] In the formula, ∈ represents the damping coefficient, and F is the vector difference between the model brightness and the observed brightness. k is the number of data observations, and J is the Jacobian matrix. nl represents the number of observation data points on the light curve where a certain observation data point is located. is the average value of the light curve, n is the number of small triangles on the surface, and m is the number of sampling points within the exposure time of a single observation.

[0087] In this embodiment, k represents the number of data observations, which is the number of observation data. Each observation corresponds to one observation data. After k observations, there are k observation data. Each observation data point has one exposure. One exposure is equivalent to one integration. Then, m data sampling points are determined within each exposure time period, and samples are taken m times to approximately calculate the value of the integration.

[0088] Return to step S4, and execute steps S5 and S6 sequentially until the set iteration stopping condition is met, and then calculate and execute step S7: determine the optimal solution of Gaussian surface density.

[0089] See also Figure 1 It can be seen that after determining the optimal solution of Gaussian surface density in the inversion method, step S9 is executed: based on Minkowski's theorem, the shape of the rapidly rotating asteroid is determined using the optimal solution of Gaussian surface density.

[0090] According to Minkowski's theorem, assuming the asteroid is a convex body, the area A(θ,ψ) corresponds to a unique convex body shape. Therefore, the shape of the rapidly rotating asteroid corresponding to the optimal solution of the area A(θ,ψ) of the small triangle is the final determined shape of the asteroid.

[0091] The shape distribution method is used to compare the shape obtained by the shape inversion method in this embodiment with the asteroid shape obtained by the inversion method in the prior art. The data comparison results can be found in [reference needed]. Figure 2 As shown. By Figure 2 It is evident that the longer the exposure time, the more significant the accuracy advantage of the method in this embodiment becomes. When the exposure time is 0.4 times the period, the dissimilarity of the result obtained by the method in this embodiment compared to the result obtained by the Kaasalainen method is approximately 1 / 5 of the latter. Therefore, for rapidly rotating asteroids, using the same observational data, the inversion method in this embodiment yields more accurate results, better meeting the needs of detecting and studying rapidly rotating asteroids.

[0092] Example 2

[0093] exist Figure 1 and Figure 2 Based on the illustrated embodiment, see also Figure 3 , Figure 3 This is a schematic diagram of the structure of a shape inversion system for a rapidly rotating asteroid provided in an embodiment of this application. Figure 3 As can be seen, the shape inversion system for a rapidly rotating asteroid in this embodiment mainly includes: a sampling time determination module, a position coordinate calculation module, a Gaussian surface density initial value determination module, a model brightness calculation module, a residual calculation module, a judgment module, an iteration step size calculation module, and a shape determination module.

[0094] The system comprises several modules: a sampling time determination module to determine several equally spaced sampling times within any exposure time of a rapidly rotating asteroid; a position coordinate calculation module to calculate the position coordinates corresponding to each sampling time, where the position coordinates are those of the observer and the Sun relative to the rapidly rotating asteroid; a Gaussian surface density initial value determination module to use the ellipsoid as the initial value for the asteroid's shape and the Gaussian surface density of the ellipsoid as the initial value for the Gaussian surface density; a model brightness calculation module to calculate the model brightness of the rapidly rotating asteroid using the light variation model and the average value method based on the position coordinates, Gaussian surface density, and the brightness of the rapidly rotating asteroid at each sampling time, where the Gaussian surface density is the area A(θ,ψ) of the small triangle corresponding to each normal vector (θ,ψ) in the inversion method; and a residual calculation module to calculate the residual based on the calculated model brightness and the observed brightness obtained from actual observations, using the optimization objective function formula. The residual brightness corresponding to the current Gaussian surface density is calculated, and the residual is used as the objective function, where L model For model brightness, L obs For observing brightness;

[0095] The judgment module determines whether the set iteration stopping condition is met. If so, it determines the optimal solution for the Gaussian surface density; otherwise, it starts the iteration step size calculation module, updates the Gaussian surface density based on the iteration step size calculated by the iteration step size calculation module, and restarts the model brightness calculation module. The iteration step size calculation module is used to calculate the model brightness and the observed brightness obtained from actual observation, using the formula dA = -(J T J+∈I) -1 J T F is the iteration step size used to optimize the objective function, where ∈ is the damping coefficient and F is the difference vector between the model brightness and the observed brightness. k is the number of data observations, and J is the Jacobian matrix with k data observations. nl represents the number of observation data points on the light curve where a certain observation data point is located. is the average value of the light curve, n is the number of small triangles on the surface, and m is the number of sampling points within the exposure time of a single observation data; the shape determination module is used to determine the shape of the rapidly rotating asteroid by using the optimal solution of the Gaussian surface density after determining the optimal solution of the Gaussian surface density according to the Minkowski theorem.

[0096] Furthermore, the iteration stopping conditions set in the judgment module include: the residual is less than a set value; or, the number of iterations reaches the set maximum number of iterations.

[0097] The sampling time determination module includes a data observation point division unit and a sampling time determination unit. The data observation point division unit is used to sample at equal intervals within any exposure time, determining m data sampling points. t e For exposure time, T syn The light variation period of the rapidly rotating asteroid; the sampling time determination unit, used to utilize the formula Calculate the sampling time corresponding to any data observation point, where 0 <i<m,t jd At the midpoint of the exposure time, t e For exposure duration.

[0098] The model brightness calculation module is used to calculate the brightness based on the position coordinates and Gaussian surface density using the formula... Calculate the sum of the brightness of the rapidly rotating asteroid within the exposure time at each sampling moment and take the average value, where t i For any sampling time, (θ) j ,ψ j Let θ be the normal vector corresponding to each small triangle after triangulation, and let A(θ) be the area of ​​the small triangle corresponding to the normal vector. j ,ψ j Let μ be the Gaussian surface density, P be the parameter of the scattering function in the inversion method, and μ = nor·r. s μ0=nor·r e ,α=r s ·r e ,nor=(sinθcosψ,sinθsinψ,cosθ),r s ,r e These are the unit vectors pointing from the rapidly rotating asteroid towards the Sun and Earth, respectively.

[0099] The working principle and method of the shape inversion system for rapidly rotating asteroids in this embodiment are explained below. Figure 1 and Figure 2 The embodiments shown have been described in detail and will not be repeated here.

[0100] The above description is merely a specific embodiment of this application, enabling those skilled in the art to understand or implement this application. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of this application. Therefore, this application is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method of shape inversion of fast-rotating asteroids, characterized by, The method comprises: S1: determining a plurality of equally spaced sampling time points within any exposure time of the fast-rotating asteroid; S2: calculating position coordinates corresponding to each of the sampling time points, the position coordinates being position coordinates of an observer and the sun relative to the fast-rotating asteroid; S3: taking an ellipsoid as an initial value of the shape of the asteroid, and taking a Gaussian surface density of the ellipsoid as an initial value of the Gaussian surface density; S4: according to the position coordinates, the Gaussian surface density, and the brightness of the fast-rotating asteroid at each sampling time point, calculating a model brightness of the fast-rotating asteroid by using a light variation model and an average value method, the Gaussian surface density being a small triangle area A(θ, ψ) corresponding to each normal vector (θ, ψ) in an inversion method; S5: Using the optimal objective function formula The residual error of the brightness corresponding to the current Gaussian surface density is calculated, wherein, L model is the model brightness, L obs is the observed brightness; S6: judging whether a set iteration stop condition is met; S7: if yes, determining an optimal solution of the Gaussian surface density; S8: If no, according to the model brightness calculated and the observed brightness observed, using the formula dA = -(J T J+∈I) -1 J T F, the iteration step length when the optimization objective function is calculated, updating the Gaussian surface density according to the iteration step length, and returning to step S4, wherein ∈ is a damping coefficient, F is the difference vector of the model brightness and the observed brightness, k is the number of data observations, J is the Jacobian matrix, nl is the number of observation data points of a certain observation data point on the light curve, is the average value of the light curve, n is the number of surface small triangles, and m is the number of sampling points in the exposure time of the first observation data. S9: after the optimal solution of the Gaussian surface density is determined, determining the shape of the fast-rotating asteroid according to the Minkowski theorem and by using the optimal solution of the Gaussian surface density.

2. The shape inversion method of a fast-rotating asteroid according to claim 1, wherein The method for determining a plurality of equally spaced sampling time points within any exposure time of the fast-rotating asteroid comprises: m data sampling points are determined at equal intervals in any of the exposure times, wherein, t e is the exposure time, T syn is the light variation period of the fast-rotating asteroid; The sampling time corresponding to any data observation point is calculated by the formula where 0 < i < m, t jd is the midpoint of the exposure time, and t e is the exposure duration.

3. The shape inversion method of a fast-rotating asteroid according to claim 1, wherein The method for calculating position coordinates corresponding to each of the sampling time points comprises: The linear interpolation method is used to calculate the position coordinates corresponding to each sampling time by using the formula The position coordinates corresponding to any sampling time are denoted as r The position coordinates corresponding to any sampling time are denoted as r b The coordinates corresponding to the previous observation data are denoted as r a The coordinates corresponding to the next observation data are denoted as r a The time corresponding to the previous observation data is denoted as t b The time corresponding to the next observation data is denoted as t.

4. The shape inversion method of a fast-rotating asteroid according to claim 1, wherein The method for calculating a model brightness of the fast-rotating asteroid by using a light variation model and an average value method according to the position coordinates, the Gaussian surface density, and the brightness of the fast-rotating asteroid at each sampling time point, the Gaussian surface density being a small triangle area A(θ, ψ) corresponding to each normal vector (θ, ψ) in an inversion method, specifically comprises: According to the position coordinates, the Gaussian surface density, and by using the formula the sum of the brightness of the fast-rotating asteroid in the exposure time at each sampling time is calculated and averaged, wherein t i is any sampling time, (θ j , ψ j ) is the normal vector of each small triangle after triangulation, the area of the small triangle corresponding to the normal vector is A(θ j , ψ j ), the Gaussian surface density is P, and μ = nor·r s , μ0 = nor·r e , α = rs·re, wherein nor = (sinθcosψ, sinθsinψ, cosθ), r s , r e are unit vectors of the fast-rotating asteroid pointing to the sun and the earth, respectively.

5. The shape inversion method of a fast-rotating asteroid according to claim 1, wherein The set iteration stop condition comprises: a residual is less than a set value; or an iteration number reaches a set maximum iteration number.

6. A shape inversion system for fast-rotating asteroids, characterized by, The system comprises: a sampling time point determination module configured to determine a plurality of equally spaced sampling time points within any exposure time of the fast-rotating asteroid; a position coordinate calculation module configured to calculate position coordinates corresponding to each of the sampling time points, the position coordinates being position coordinates of an observer and the sun relative to the fast-rotating asteroid; a Gaussian surface density initial value determination module configured to take an ellipsoid as an initial value of the shape of the asteroid, and take a Gaussian surface density of the ellipsoid as an initial value of the Gaussian surface density; a model brightness calculation module configured to calculate a model brightness of the fast-rotating asteroid by using a light variation model and an average value method according to the position coordinates, the Gaussian surface density, and the brightness of the fast-rotating asteroid at each sampling time point, the Gaussian surface density being a small triangle area A(θ, ψ) corresponding to each normal vector (θ, ψ) in an inversion method; a residual calculation module, configured to calculate a residual of the brightness corresponding to the current Gaussian surface density according to the calculated model brightness and the observed brightness obtained from the actual observation, by using the formula calculate the residual of the brightness corresponding to the current Gaussian surface density, and take the residual as a target function, wherein, L model is the model brightness, L obs is the observed brightness; a judgment module configured to judge whether a set iteration stop condition is met, and if yes, determine an optimal solution of the Gaussian surface density, and if not, start an iteration step calculation module, update the Gaussian surface density according to an iteration step calculated by the iteration step calculation module, and restart the model brightness calculation module; The iteration step calculation module is configured to calculate an iteration step according to the calculated model luminance and the observed luminance obtained by observation, by using a formula dA = -(J T J+∈I) -1 J T F, wherein ∈ is a damping coefficient, F is a difference vector of the model luminance and the observed luminance, and dA is the iteration step when the optimization objective function is calculated. k is the number of data observation values, J is a Jacobian matrix, nl is the number of observation data points of a certain observation data point on a light variation curve, is the average value of the light variation curve, n is the number of surface small triangles, and m is the number of sampling points in one observation data exposure time. a shape determination module configured to, after the optimal solution of the Gaussian surface density is determined, determine the shape of the fast-rotating asteroid according to the Minkowski theorem and by using the optimal solution of the Gaussian surface density.

7. The shape inversion system of a fast-rotating asteroid according to claim 6, wherein The set iteration stop condition comprises: a residual error is less than a set value; or a number of iterations reaches a set maximum number of iterations.

8. The shape inversion system of a fast-rotating asteroid according to claim 6, wherein The sampling time determination module comprises: The sampling time determination module comprises: The data observation point division unit is configured to sample at equal intervals in any of the exposure times to determine m data sampling points, wherein, t e is the exposure time, T syn is the light variation period of the fast-rotating asteroid; A sampling time determination unit is configured to calculate the sampling time corresponding to any data observation point by using the formula where 0 < i < m, t jd is the midpoint of the exposure time, and t e is the exposure duration.

9. The shape inversion system of a fast-rotating asteroid according to claim 6, wherein The model brightness calculation module is configured to calculate the sum of the brightness of the fast-rotating asteroid in the exposure time at each sampling time according to the position coordinates and the Gaussian surface density, and take an average value, wherein t is the exposure time, and t i is any sampling time. j , ψ j ) is a normal vector corresponding to each small triangle after triangulation, the area of the small triangle corresponding to the normal vector is A(θ j , ψ j ), the Gaussian surface density is P, μ = nor·r s , μ0 = nor·r e , α = r s · r e , wherein nor = (sin θ cos ψ, sin θ sin ψ, cos θ), r s , r e are unit vectors of the fast-rotating asteroid pointing to the sun and the earth, respectively.

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