Asteroid rotation parameter measuring method based on speckle tracking technology

Abstract

The invention discloses an asteroid rotation parameter measurement method based on a radar speckle tracking technology, and the method comprises the steps: firstly calculating an observation distance range of a target asteroid, and determining a plurality of observation windows through combining with ephemeris; receiving speckle echoes by using more than three base stations, and performing mutual correlation pairwise to obtain the time required by the speckles to sweep the base lines so as to invert speckle velocity vectors in the observation windows; and finally, using the geometrical relationship between the speckle velocity vector and the rotation vector to invert the rotation parameter. The method has the advantages that long-time-span observation is not needed, and the rotation parameters of the asteroid can be accurately estimated only through observation in a short time window (such as about one week); the autorotation parameter can be directly solved on the premise of not having the asteroid shape, the influence of the north-south blur of the ground-based radar delay Doppler image inversion method is avoided, and the calculation complexity is low. The method can be used as an effective means for measuring the rotation state of the near-earth asteroid.

Description

Technical Field

[0001] The present invention relates to an asteroid rotation parameter measurement method based on radar speckle tracking technology, and belongs to the field of radar technology. Background Art

[0002] The physical properties of asteroids primarily include their rotation state, shape, and surface scattering properties. By measuring these properties, we can accurately model their orbital evolution, providing essential information for asteroid defense missions. Furthermore, based on the asteroid's true shape, we can analyze its gravitational properties, which can then be combined with orbital information to plan survey missions. If the physical properties of a large number of asteroids were available, statistical studies could be conducted. This would have two scientific implications: first, analyzing the evolution of asteroid rotation, such as the influence of the YORP effect on the orientation of the spin axis; and second, understanding the history and current status of asteroid populations, revealing the patterns of formation and evolution of the asteroid family, the asteroid belt, and ultimately the solar system.

[0003] Generally speaking, it is impractical to conduct close-up studies of asteroids. In most cases, we can only use data from long-distance observations to analyze the physical properties of asteroids. The process of using observational data to obtain one or several physical properties of an asteroid is called inversion, such as rotation inversion, shape inversion, etc. The inversion technology first needs to parameterize the physical properties to be inverted, and then search for the optimal parameters. During the search process, the rotation state parameters, especially the rotation period, usually have many local minima, so special attention needs to be paid. The current mainstream rotation inversion technologies are divided into inversion technology based on optical data (referred to as optical inversion) and inversion technology based on radar data (referred to as radar inversion).

[0004] Optical inversion primarily uses light curve data, using convex inversion techniques to obtain a convex hull model of an asteroid's rotational state and shape. Light curves, which represent the temporal variation of an asteroid's brightness, are the richest source of data for studying asteroid physical properties. Light curve inversion has been used to obtain shape models and rotation parameters for over 10,000 asteroids, and these data are now included in the DAMIT database.

[0005] Radar inversion mainly uses delayed Doppler image data to jointly solve the asteroid's shape model and rotation parameters through the iterative algorithm of SHAPE software. Compared with optical data, radar data has the following advantages: First, the radar provides its own illumination, so when using radar for observation, it is only necessary to consider the observation geometry of the Earth and the asteroid, without considering the position of the sun; second, delayed Doppler images have fine spatial resolution, which can provide detailed information on the shape of the object, thereby improving the accuracy of the inversion results. However, because radar is expensive and difficult to observe targets at long distances, as of April 7, 2025, radar has only observed a total of 1,281 asteroids, which is far less than the number of asteroids observed using optical observations.

[0006] Current inversion techniques all have limitations. For optical inversion, accurate measurements require light curves covering a wide range of observation geometries and a sufficient number of data points. This requires long-term, repeated observations, which not all asteroid orbits meet. For Delayed Doppler image inversion, first, obtaining Delayed Doppler images with a high signal-to-noise ratio requires the asteroid to be very close to Earth. Second, inverting the rotation state using Delayed Doppler images requires iterative calculations, which are computationally intensive and sensitive to initial parameter values. If the initial rotation state values differ significantly from the actual state, the inversion process can easily become trapped in a local optimum. Third, Delayed Doppler image inversion is prone to north-south ambiguity. Due to the Delayed Doppler imaging mechanism, every point in the northern and southern hemispheres of an object has the same Delayed Doppler position, making it impossible to distinguish north from south in the image. Consequently, in many cases, the Delayed Doppler image yields two mirror-image solutions with opposite rotation axes. Furthermore, current mainstream rotation inversion techniques all require a joint inversion of the asteroid's shape, rotation, and scattering properties. Inversion techniques that skip the shape modeling process and directly determine the rotation state have not been fully explored. This provides an opportunity for speckle tracking technology to be used. Summary of the Invention

[0007] In view of this, this patent proposes a method for measuring asteroid rotation parameters based on speckle tracking technology, which can conveniently and efficiently obtain the rotation axis direction of an asteroid with a known rotation period.

[0008] The technical solution of the present invention comprises the following steps:

[0009] Step 1: Select the observation target and calculate the change in the distance between the target asteroid and the Earth in the next few years using the formula Calculate the upper and lower limits of the observation distance and combine them with the observation range of the fixed length baseline Take the intersection of the two, select at least two appropriate observation windows, design an observation experiment, and receive speckle echoes;

[0010] Step 2: Cross-correlate the echoes received by different base stations and delay the time corresponding to the maximum value of the cross-correlation function by t lag As the time required for the speckle to sweep across each baseline;

[0011] Step 3: Use t lag The apparent velocity vector of the speckle along each baseline is calculated by the baseline length, and then the objective function is searched. The minimum value of is used to obtain the estimated value of the actual velocity vector of the speckle;

[0012] Step 4: Fill the estimated values of the speckle velocity vectors measured in all observation windows into a three-column matrix as row vectors, and perform singular value decomposition on the matrix. The line containing the right singular vector corresponding to the singular value closest to 0 is selected as the estimated value of the rotation axis direction.

[0013] Beneficial effects:

[0014] 1. Compared to current mainstream inversion techniques, this method can skip the complex asteroid shape modeling process and directly estimate the asteroid's rotation parameters without the need for iterative calculations. This has two advantages: first, the computational complexity is low, which allows the method to measure the rotation state of ten or more targets per year; second, there is no need to set initial values, and the inversion results can also provide initial values for other inversion techniques, accelerating the asteroid shape modeling process;

[0015] 2. Compared with optical inversion technology, the present invention does not require long-term tracking observations. Instead, it only requires observations within a shorter time window (e.g., about a week) to more accurately estimate the asteroid's rotation parameters.

[0016] 3. Compared with the delayed Doppler image inversion technology, the present invention has less stringent requirements on target distance than the delayed Doppler image, can observe more radar targets, and the inversion result will not have the north-south ambiguity phenomenon. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 ,Schematic diagram of speckle formation process;

[0018] Figure 2 ,Schematic diagram of speckle movement mechanism;

[0019] Figure 3 ,Schematic diagram of related measurement mechanism;

[0020] Figure 4 , Flowchart of the asteroid rotation parameter measurement method based on speckle tracking technology;

[0021] Figure 5 , schematic diagram of speckle velocity vector inversion. DETAILED DESCRIPTION

[0022] The present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0023] The speckle formation process is as follows Figure 1 As shown in the figure, the radar transmits electromagnetic waves toward the asteroid, which then reflects off the asteroid and returns to Earth. Due to the uneven surface of the asteroid, echoes from different scattering points interfere constructively or destructively on Earth's surface, forming a large-scale speckle pattern. The speckle pattern acts like a distant, invisible shell of the asteroid, moving in sync with the asteroid's rotation. Therefore, the speckle pattern moves perpendicular to the projection of the asteroid's rotation axis onto the celestial plane, sweeping across the radar base station. If different radar base stations receive the same speckle pattern, then theoretically, there should be a strong correlation between the echo signals. This is the speckle formation process and the principle of speckle tracking technology.

[0024] The speckle movement mechanism is as follows Figure 2 As shown in the figure. The line of sight vector r points from the observation station to the center of the asteroid, and its modulus is recorded as r, which represents the distance between the observation station and the asteroid; the asteroid's rotation period is recorded as P, and the latitude of the radar point on the asteroid is recorded as θ. The asteroid's rotation causes the speckle to move. From the geometric relationship in the figure, we can know that the radial distance from the observation station to the asteroid is rcosθ. According to the physical relationship between angular velocity and linear velocity, the speckle velocity vector v can be derived. speckle The calculation formula of modulus value is:

[0025]

[0026] Base station related measurement mechanisms such as Figure 3 As shown. β is the normal plane of the sight vector r, which is the perspective from the asteroid to the Earth, that is, the reverse direction of the radar sight line; points A and B are the projections of base stations 1 and 2 on β; B is the projection of the base station spacing vector on β, that is, the baseline vector, pointing from A to B; the angle between the projection of the asteroid's rotation vector p on the β plane and B is α, v speckle The projection perpendicular to p points to the upper right in the figure according to the right-hand spiral rule. The shaded area in the figure represents a part of the speckle. As shown in the figure, the actual distance that this part of the speckle passes through the two base stations is the baseline at v. speckle The projection length in the direction |B|sinα. Assume that the time it takes for this spot to move from the first base station to the second base station is t lag , then:

[0027]

[0028] The equation is not strictly equal. This is because, in reality, the measured speckle velocity is not solely due to the asteroid's rotation, due to the Earth's rotation and the relative motion between the Earth and the asteroid. Generally speaking, the linear velocity caused by the Earth's rotation is <0.5 km / s, and the relative motion speed between the Earth and the near-Earth asteroid is always <90 km / s, usually <10 km / s. Therefore, in most cases, a speckle velocity greater than 1000 km / s is used as the applicable condition for equation (2). At this time, the contribution of other factors to the speckle movement speed on the ground can be ignored.

[0029] The flow chart of the asteroid rotation parameter measurement method based on speckle tracking technology of the present invention is as follows: Figure 4 As shown, the specific steps include:

[0030] Step 1: Baseline configuration and observation window selection.

[0031] Select an asteroid with a known rotation period and ephemeris as the observation target, and calculate the time window suitable for speckle tracking observation of the target. According to the approximate conditions of formula (2), the speckle speed should be as high as 1000km / s. min = 1000km / s. Since the speckle velocity is related to the latitude of the radar point, and the latitude of the radar point cannot be measured when the direction of the rotation axis is unknown, the speckle velocity when the latitude of the radar point is 0 must at least reach v min ,Right now

[0032]

[0033] Where r is the distance between the asteroid and the radar base station.

[0034] t in formula (2) lag Measured using the correlation function, in order to make the data of the correlation function valid, it is necessary to ensure that the correlation signal-to-noise ratio conSNR>>1. The calculation formula of the correlation signal-to-noise ratio is

[0035]

[0036] Where C is the cross-correlation amplitude, P Eff is the effective echo power, T Eff is the effective temperature of the receiver (assuming the effective temperature of the two stations is the same), b is the echo bandwidth, t Int is the integration time, and k is the Boltzmann constant. The relationship between the effective echo power and the radar transmission power is:

[0037]

[0038] Among them, P T is the transmitter power, G Tis the transmitter gain, σ is the radar cross-section of the target, and A1 and A2 are the effective areas of the antenna.

[0039] From conSNR>>1, we can get

[0040]

[0041] The cross-correlation amplitude C≤1, from which we can get

[0042]

[0043] The right side of formula (3) is denoted as r lowerlimit , the right side of formula (7) is denoted as r upperlimit , from this we can get the upper and lower limits of the observation distance of the speckle tracking technology, which is recorded as R1.

[0044] R1={r|r lowerlimit ≤r<<r upperlimit} (8)

[0045] In addition to the inherent limitations of speckle tracking technology on the range of observation distance, the range of distance that can be observed with a fixed-length baseline is also limited. In order to perform effective speckle tracking observations, the baseline length needs to meet

[0046]

[0047] Where Δt lag t lag The 1-σ uncertainty of

[0048]

[0049] L speckle is the speckle size, and the calculation formula is

[0050]

[0051] From this we can deduce the distance range that can be observed with a baseline of length B

[0052]

[0053] The left side of formula (12) is denoted as r closest , the right side is denoted as r farthest , the distance range that can be observed by the fixed length baseline is recorded as R2,

[0054] R2={r|r closest <<r<<r farthest} (13)

[0055] Take the intersection R of R1 and R2, where R is the distance range where effective speckle tracking observation can be performed. Use the target ephemeris data and the Earth ephemeris data to calculate the time period that meets the range conditions, and select the time when the asteroid is not blocked by the Earth as the observation window. A baseline of appropriate length needs to be selected.

[0056] Step 2: Cross-correlation to obtain time delay.

[0057] The power signals received by all base stations are cross-correlated with each other, and the time delay corresponding to the maximum correlation value is the time t that the speckle scans the baseline in formula (2). lag Each baseline can obtain a corresponding t lag However, the reliability of the data measured by each baseline varies. lag The 1-σ uncertainty of Δt lag , only when t lag >>Δt lag In addition, if the cross-correlation function of a baseline has multiple peaks with similar amplitudes, the t lag It is usually invalid and needs to be discarded. In actual observation, it is necessary to select a valid baseline based on the specific measurement results and proceed to the next step.

[0058] Step 3: Speckle velocity vector estimation.

[0059] Get the t corresponding to each baseline lag After excluding invalid data, although the baseline length is known, the speckle velocity vector cannot be directly calculated because the angle α in formula (2) is unknown. In order to invert the speckle velocity vector, the present invention defines the physical quantity "speckle apparent velocity" and hereinafter refers to the speckle velocity as "speckle actual velocity" for distinction. The speckle apparent velocity is the velocity of the speckle along the baseline direction, denoted as v B .v B It is collinear with B, and its direction is from the projection of the first base station scanned by the speckle to the projection of the second base station scanned by the speckle. The formula for calculating its modulus is:

[0060]

[0061] It can be proved from simple geometric relationships that the actual velocity vector of the speckle is the projection vector of the apparent velocity vector of the speckle to the direction of the speckle movement, as shown in Figure 5 Based on this geometric relationship, the unit vector estimate of the actual speckle velocity is recorded as The apparent velocity vector of the speckle along the i-th baseline is denoted as v Bi , the number of baselines is recorded as M, and the optimization function is constructed:

[0062]

[0063] The physical meaning of this function is to transform all the speckle apparent velocity vectors to Project in the direction of the projected image and make the difference between the two projection lengths. When the sum of the squares of the differences is the smallest, it is considered that is the optimal estimate of the actual velocity vector direction of the speckle. The nonlinear constraint is is a unit vector in the β plane, which can be described in mathematical language as:

[0064]

[0065] By solving the above optimization function, we can get the estimated value of the actual velocity direction of the speckle in the current observation window.

[0066] Step 4: Inversion of the rotation axis.

[0067] Let the actual speckle velocity estimation vector obtained in the kth observation window be recorded as All After being transposed into a row vector, it is placed into a 3-column matrix A. For N observations, we have

[0068]

[0069] Perform singular value decomposition on matrix A and get

[0070] A=UΣV T (18)

[0071] The columns of the right singular vector matrix V correspond to the standard orthogonal basis of the row space of A. When N = 2, the rank of A is 2, which is less than the number of columns 3, so there is a singular value of 0. The singular vector corresponding to this singular value points to the direction of the null space of A, that is, the direction orthogonal to all rows in A; when N>2, since the speckle velocity is always perpendicular to the rotation axis, are almost coplanar, which means that there is a singular value very close to 0, and its corresponding singular vector is the same as all As orthogonal as possible. Therefore, the line where the right singular vector corresponding to the singular value of matrix A that is closest to 0 or exactly 0 is located is the best estimate of the asteroid's rotation axis. The rotation vector p has two possible directions on the rotation axis, and it is also necessary to combine the rotation vector p, the line of sight vector r and the speckle velocity vector v speckle The right-handed spiral relationship formed determines whether the direction of p is the same as or opposite to the direction of the right singular vector.

[0072] Example

[0073] In order to verify the feasibility of this method, the parameters in Table 1 are set to carry out a simulation experiment for estimating the asteroid rotation parameters.

[0074] Table 1 Simulation experiment parameters

[0075]

[0076] The appropriate observation dates for 2001WN5 were calculated, and the time periods during which the target asteroid was not blocked by the Earth were calculated for each observation date. The results are shown in Table 2.

[0077] Table 2 Observable time period of 2001WN5

[0078] Observation date Unobstructed time period 2028-06-21 10:38～19:53 2028-06-22 10:38～19:47 2028-06-23 10:42～19:39 2028-06-29 00:00~09:35 and 19:34~23:59 2028-06-30 00:00~09:37 and 19:29~23:59 2028-07-01 00:00~09:36 and 19:24~23:59 2028-07-02 00:00~09:34 and 19:19~23:59

[0079] For each observation date, the midpoint of the period when the asteroid is not obscured by the Earth is calculated. Speckle tracking observations are performed every 10 minutes, one hour before and after the midpoint. This yields 13 estimated speckle velocity vectors for each observation date. All estimated speckle velocity vectors for each observation date are averaged to form the measured speckle velocity vector result for that observation date. The seven speckle velocity vector results are placed in matrix A and subjected to singular value decomposition. The resulting rotation axis direction results are shown in Table 3.

[0080] Table 3 Estimation results of the rotation axis direction

[0081]

[0082] It can be seen that this method can accurately measure the direction of the asteroid's rotation axis when the asteroid's rotation period is known and the orbital distance meets the requirements of speckle tracking observation.

[0083] In summary, the above are only preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for measuring the direction of an asteroid's rotation axis based on speckle tracking technology, characterized in that: The method comprises the following steps: Step 1: Select the observation target, select at least two observation windows according to the proposed observation window selection principle, design the observation experiment, and receive speckle echoes; Step 2: Cross-correlate the echoes received by different base stations to obtain the time required for the speckle to scan each baseline; Step 3: Calculate the speckle apparent velocity vector along each baseline, and then estimate the actual speckle velocity vector by solving the optimization function; Step 4: Estimate the direction of the rotation axis through singular value decomposition.

2. The method for measuring the asteroid's rotation axis direction based on speckle tracking technology according to claim 1, characterized in that: The observation targets selected in step 1 are asteroids with known rotation periods and ephemeris.

3. The method for measuring the asteroid's rotation axis direction based on speckle tracking technology according to claim 1, characterized in that: The distance between the asteroid and the Earth within the observation window selected in step 1 must meet the following conditions: Where r represents the distance between the asteroid and the receiving base station; R1 is the upper and lower limits of the observation distance of the speckle tracking technology, P on the left is the rotation period of the asteroid, and v min =1000km / s; right side P T is the transmitter power, G T is the transmitter gain, σ is the radar cross section of the target, k is the Boltzmann constant, T Eff is the effective temperature of the receiver (assuming the effective temperature of the two stations at both ends of the baseline is the same), A1 and A2 are the effective areas of the antennas of the base stations at both ends of the baseline, b is the echo bandwidth, and t Int is the integration time; R2 is the observable distance range of the fixed-length baseline, d on the left is the diameter of the asteroid as it appears to the radar, λ is the carrier wavelength, and B is the baseline length; the distance range for effective speckle tracking observations can be obtained by taking the intersection of R1 and R2.

4. The method for measuring the asteroid's rotation axis direction based on speckle tracking technology according to claim 1, characterized in that: In step 1, multiple observation dates should be selected, and multiple observation windows should be selected within each date.

5. The method for measuring the direction of an asteroid's rotation axis based on speckle tracking technology according to claim 1, characterized in that: In step 1, there are no less than three receiving base stations and the distance between the base stations is smaller than the speckle size of the asteroid.

6. The method for measuring the direction of an asteroid's rotation axis based on speckle tracking technology according to claim 1, characterized in that: The definition of the optimization function in step 3 is: In the formula is the estimated value of the unit direction vector of the actual speckle velocity, v Bi (1≤i≤M) is the apparent velocity vector of the speckle along the i-th baseline, and M is the number of baselines. Its physical meaning is to move all the speckle apparent velocity vectors to Project in the direction of the projected image and make the difference between the two projection lengths. When the sum of the squares of the differences is the smallest, it is considered that is the best estimate of the actual velocity vector direction of the speckle.

7. The method for measuring the direction of an asteroid's rotation axis based on speckle tracking technology according to claim 1, characterized in that: The constraints of the optimization function in step 3 are is a unit vector in the normal plane of the sight vector, which can be described in mathematical language as 8. The method for measuring the asteroid's rotation axis direction based on speckle tracking technology according to claim 1, characterized in that: The matrix A that needs to be decomposed into singular values in step 4 is: In the formula is the unit vector of the actual speckle velocity obtained at the kth observation moment, and is a column vector; N is the number of observations.

9. The method for measuring the direction of an asteroid's rotation axis based on speckle tracking technology according to claim 1, characterized in that: In step 4, the singular value decomposition of the matrix A composed of the unit vectors of the actual speckle velocity is performed. If the number of observations is 2, then there must be a singular value of 0; if the number of observations is greater than 2, then there is only one singular value close to 0. The column vector in the right singular vector matrix corresponding to this singular value is the direction vector of the asteroid's rotation axis.

10. The method for measuring the direction of an asteroid's rotation axis based on speckle tracking technology according to claim 1, wherein: The rotation vector estimated in step 4 has two possible directions on the line where the rotation axis is located. It is also necessary to combine the right-hand spiral relationship formed by the rotation vector, the line of sight vector and the speckle velocity vector to determine whether to perform the inversion operation on the estimated value of the rotation vector.

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