Modeling Binary Asteroids: Integrating Orbital and Rotational Motion for Physical Property Inversion

The field of space science places significant emphasis on deep space exploration, with a particular focus on asteroids as a potential hazard to humanity. Inverting their physical characteristics from photometric observations is essential for uncovering their origins and evolution. This article attempts to present a solution to the challenging task of estimating the physical properties of binary asteroids, which are common in near-Earth asteroids larger than 200 meters. A novel model for binary asteroids is proposed, which integrates orbital and rotational motions to simulate brightness variations based on two Cellinoid shapes. The model combines the projection and occultation effects of the shapes to generate the simulated brightness. The inversion of determining physical properties is optimized based on the Levenberg–Marquardt algorithm through a simulation process involving several parameter corrections. Finally, the performance of the proposed model is demonstrated through numerical experiments and applications to two real binary asteroids, namely, asteroid (317) Roxane and asteroid (624) Hektor. The derived results are nearly identical to those from other publications, which confirms that the proposed model provides reliable and accurate estimations of the physical properties of binary asteroids. Additionally, this method has a potential application in supporting the development of effective strategies for the Double Asteroid Redirection Test (DART) project, the first planetary defense experiment in space undertaken by humans.


Introduction
The Universe is immense and contains numerous mysteries that have yet to be discovered.As humanity's technological advancements continue to accelerate, we have been able to push the boundaries of deep space exploration.In fact, the quest for understanding the farthest reaches of space has become a cornerstone in the progressive journey of space science.Significant strides have been made in the direction of studying our own planet's system, as well.Tesch et al. (2023) played a pivotal role in this by developing causal deep learning models, thereby expanding our understanding of Earth's mechanisms.In parallel, the complexity and importance of oceanic interactions have also been brought into focus.Jin et al. (2022) proposed an innovative explicit tidal scheme in their revised LICOM2.0 model.Their focus on the critical role tides play in the energy transfer and mixing processes within the oceans was unprecedented, and the promising results their scheme produced in recreating observed tidal characteristics are a testament to their revolutionary work.While these advancements are centered on our home planet, exploration beyond Earth is just as crucial.One of the many intriguing aspects of our solar system is asteroids.DeMeo & Carry (2014) hold the view that asteroids consist of primitive materials from the early stages of the solar system and can therefore furnish insights into the beginning and development of the solar system.However, asteroids pose a risk to humanity.Their impacts on Earth have led to geological changes in the climate, which have ultimately caused the extinction of the dinosaurs one by one (Brusatte et al. 2015).A significant number of asteroids have been discovered in the solar system in recent times.Acquiring knowledge of the physical characteristics of asteroids, such as their rotational period, orbital pole, and eccentricity, is critical in gaining an understanding of their origins and development.Additionally, important experiments in the investigation of binary asteroids are currently being conducted.As an illustration, (65803) Didymos is a binary asteroid.It served as an ideal subject for the Double Asteroid Redirection Test (DART) project, which was the first planetary defense experiment undertaken by humankind.A spacecraft was launched on 2021 November 24 with the objective of colliding with the satellite of the asteroid (65803) Didymos.To maximize the quality of telescopic observations, it was deemed necessary to minimize the distance between Earth and Didymos.This resulted in the scheduling of the DART impact for 2022 September 26.
The characterization of the physical attributes of asteroids can be achieved through the examination of both radar imagery and lightcurves.The efficacy of radar techniques in determining the physical properties of asteroids is limited to instances of close proximity, as the radar echo power is inversely proportional to the fourth power of the distance between the target and the observer (Scheirich & Pravec 2009).Therefore, radar data are comparatively scarce, while photometric observations in the form of lightcurves serve as the principal means of gaining insight into asteroids.The extensive utilization of lightcurves in asteroid research over the past several decades has resulted in numerous breakthroughs and advancements.The simulation of a synthetic lightcurve based on a triaxial ellipsoid was performed utilizing both Lambert's law and the Lommel-Seeliger law (Surdej & Surdej 1978).Karttunen (1989) and Karttunen & Bowell (1989) proposed a method for generating lightcurves based on a triaxial ellipsoid through the application of the Lumme-Bowell scattering law (Karttunen & Bowell 1989;Lumme & Bowell 1981).To generate lightcurves more similar to real observed data, Cellino et al. (1985Cellino et al. ( , 1987Cellino et al. ( , 1989) used a special shape based on a composite of eight ellipsoidal octants.Additionally, Lu et al. (2014) originally referred to this model as the Cellinoid shape model and presented a thorough proof of its derivation, as well as a method for inversion.Kaasalainen et al. (1992bKaasalainen et al. ( , 1992a)), Kaasalainen (2001), Kaasalainen & Torppa (2001), Kaasalainen et al. (2001Kaasalainen et al. ( , 2002aKaasalainen et al. ( , 2003Kaasalainen et al. ( , 2005)), Kaasalainen & Lamberg (2006), and Kaasalainen et al. (2012) devised a methodology for calculating the shape and related parameters of an asteroid and subsequently validated the efficiency of this method through its application to a substantial number of real observed lightcurves.Ďurech et al. (2010) and Ďurech et al. (2011) utilized Kaasalainen's technique to determine the physical parameters of hundreds of asteroids.Lu & Ip (2015), Lu et al. (2016Lu et al. ( , 2017Lu et al. ( , 2018)), and Lu & Jewitt (2019) utilized the Cellinoid model for inverting asteroids and proposed the adoption of the Lebedev quadrature method to enhance the inversion process.The disk-integrated brightness method combined with the Lommel-Seeliger ellipsoid model has been employed by Muinonen & Lumme (2015) and Muinonen et al. (2015) to effectively invert physical parameters such as rotational period and shape using both sparse and dense photometric data.Inversion tests conducted by Cellino et al. (2015) using the Muinonen method have further advanced the field.Additionally, Muinonen et al. (2020) introduced the first Markov chain Monte Carlo method, accounting for uncertainties in spin, shape, and scattering parameter solutions.Comparative studies involving hundreds of asteroids have been facilitated by Martikainen et al. (2021), who developed a method to derive reference absolute magnitudes and phase curves from Gaia data.Furthermore, Muinonen et al. (2022) provided error models for different lightcurve classes and explored phase angle behavior.These studies collectively enhance our understanding of asteroid properties and contribute valuable insights into photometric data analysis.
In addition, concurrent research on binary asteroids is also in progress.André (1901) was the first to propose the idea that (433) Eros, an asteroid, was a binary asteroid based on its lightcurve analysis.Belton & Carlson (1994) reported that the first detection of a satellite of the asteroid (243) Ida was made by the Galileo spacecraft.Merline et al. (1999) recounted the details of the discovery of a satellite orbiting around (45) Eugenia, which was made using adaptive optics on a groundbased telescope.The near-Earth asteroid (NEA) 1994 AW1 was the first binary system revealed through photometry (Pravec & Hahn 1997).The initial radar detection of a binary system was that of NEA 2000 DP107 (Ostro et al. 2000;Margot et al. 2002), which was subsequently verified by photometry (Pravec et al. 2006).By the time Pravec et al. finished studying this case (Pravec et al. 2006), a total of fifteen binary systems had been observed using both methods, showing the reliability of photometry as a dependable detection process.Subsequently, a considerable amount of research findings have been acquired by scholars concerning binary asteroids.Ferrari et al. (2016) presented a dynamical model for binary asteroid systems using a combination of patched threebody problems.They delved into the basic principles of astrodynamics and the overall dynamics of the restricted full three-body problem.Additionally, they analyzed the stability of relative equilibria in the simpler two-body problem, offering insights into spacecraft motion around binary asteroids.Jean et al. (2019) investigated the influence of modeling solar radiation pressure on the orbital behavior around binary asteroids.Additionally, they delved into how this modeling can offer insight into the formation and composition of the solar system.Wang & Hou (2021) introduced a model of synchronous binary asteroid systems that incorporates a coplanar averaged ellipsoid-ellipsoid design, as well as thermal and tidal effects.Utilizing this model, they investigated the mechanism of breakup in synchronous binary asteroid systems.However, within the binary asteroid research field, various theoretical models have been introduced by scholars, yet they infrequently utilize real lightcurves to assess their model validity.By contrast, utilizing real lightcurve data, Scheirich & Pravec (2009) created an efficient model for a binary asteroid, which has been implemented in various subsequent studies.For example, Scheirich et al. (2015) analyzed the properties of the binary near-Earth asteroid (175706) 1996 FG3 using photometric observations from April 1996 to January 2013, providing an observational constraint on its orbital evolution and a detailed analysis of related published data.Scheirich & Pravec (2022) delved into the preimpact mutual orbit of the binary asteroid (65803) Didymos, which was obtained through observations of mutual events between 2003 and 2021.Their results indicate that the preimpact mutual orbit is sturdy and can be utilized to anticipate future orbital changes.
The model proposed by Scheirich & Pravec (2009) utilizes a Fourier series technique to calculate the brightness and perform least-square inversion for physical properties.However, the model's calculation process is complicated and not intuitive, making it challenging for other scholars to comprehend.To overcome these limitations, this article introduces a novel model for binary asteroids that integrates orbital and rotational motions to simulate brightness variations based on two Cellinoid shapes, providing a simpler and more intuitive alternative.Karttunen (1989) and Karttunen & Bowell (1989) found that an asteroid's shape, not its scattering law, is the primary factor affecting its brightness change.Kaasalainen & Torppa (2001) and Kaasalainen et al. (2001) derived physical asteroid parameters from lightcurves using inversion methods and a "convex hull" approximation with a universal albedo assumption.The triaxial ellipsoidal shape, convex polyhedral shape, and Cellinoid shape can be used as "convex hulls," but the ellipsoid's symmetry makes it challenging to model asymmetrical asteroids accurately.Lu et al. (2017) revealed that the convex polyhedral shape model is too complex compared to the Cellinoid shape model, with over 50 parameters requiring much observational data to fit.Although the convex inversion can reduce the total parameters for shape representation by utilizing low-degree spherical harmonics models, it is worth noting that the Cellinoid shape provides a simple representation for asymmetric shapes, using only six parameters.Thus, the Cellinoid shape is used in this article to estimate the binary asteroid's physical properties instead.
The physical property parameters of a binary asteroid obtained by inversion are illustrated in Figure 1.The lightcurve data were gathered by a ground-based observer, which captured the reflected light from the binary asteroid.Then, the simulated lightcurve data were generated using the proposed model in this article.Finally, to obtain the optimal physical property parameters, the Levenberg-Marquardt (LM) algorithm was used to fit the observed data by regenerating the lightcurve data through a simulation process that involved making several parameter corrections.
This work makes the following significant contributions: (1) A novel binary asteroid model, which integrates orbital and rotational motions, is proposed to generate lightcurve data that mimic the real motion trajectory of binary asteroids.This model is intuitive and easy to understand, and the proposed algorithm reproduces experimental results similar to Scheirich & Pravec (2009) The rest of this article unfolds as follows: Section 2 outlines the Cellinoid shape model along with the method used for brightness computation.Section 3 delves into the binary asteroid system.The numerical experiments are showcased and evaluated in Section 4. Finally, the article ends with Section 5 summarizing the conclusions.

Cellinoid Shape Model
The Cellinoid model, which is composed of eight octant surfaces derived from ellipsoids defined by six semiaxes, is shown in Figure 2. A detailed description of this model can be found in Lu et al. (2014).
By taking each octant from an ellipsoid, the surface of every octant in the Cellinoid shape model can be defined using the same coordinate system employed for an ellipsoidal shape.Hence, defining the surface of each octant is a straightforward process.
According to the findings presented in Karttunen (1989) and Karttunen & Bowell (1989), the primary factor influencing the fluctuation in an asteroid's luminosity is its shape rather than the scattering principles.In a similar vein, Lu et al. (2018) conducted numerical simulations comparing the Hapke model with three other numerical models, namely the Lommel-Seeliger, Minnaert, and Kaasalainen models.Their results demonstrated that the numerical models with simpler function expressions can effectively fit synthetic lightcurves based on the Hapke model.This suggests that these alternative models can be utilized in the lightcurve inversion process for asteroids, improving numerical efficiency while still yielding comparable results to those obtained from the Hapke model.Hence, choosing a suitable scattering law can make it easier to simulate brightness.Therefore, this article employs the scattering function, S(μ, μ 0 , α), defined below, as documented in Kaasalainen & Torppa (2001) and Kaasalainen et al. (2001Kaasalainen et al. ( , 2002a)).
The phase angle, denoted as α, represents the Sun-objectobserver relationship.This article utilizes the phase function f (α), characterized by the four parameters presented in the equation below: Furthermore, μ and μ 0 are defined as follows: Here, η represents the outward unit normal vector of the surface, while the unit vectors E and E 0 indicate the directions to the Earth and the Sun, respectively, as viewed from the asteroid.
The brightness of the asteroid can be calculated based on the Cellinoid model through surface integration.This calculation is demonstrated as follows: Here, C + refers to the portion of the Cellinoid shape model's surface that is both illuminated and observed.
The discretization of a Cellinoid shape enables an approximate calculation of the surface integral in Equation (4).This approximation is presented as follows: Here, i denotes the index of the octants, j denotes the index of the triangular facets in each octant, and !s i,j represents the area of the jth facet in the ith octant.

Binary Asteroid System
The Cellinoid model has been employed in previous studies (Zhang et al. 2022) to invert the period and pole parameters for over 50 real asteroids, resulting in accurate periods and nearaccurate poles.Lu et al. (2017) concluded that more observations obtained from various viewing geometries could enhance the accuracy of the inverted pole orientation.However, using the Cellinoid model for binary asteroids presents challenges due to mutual occultation, which affects the brightness data and impedes the acquisition of accurate physical properties, such as the pole.Furthermore, the Cellinoid model lacks the capability to directly determine the orbital radius, orbital period, and eccentricity of satellites.To address these limitations, this research proposes a novel binary asteroid model based on the Cellinoid shape.In this section, two crucial algorithms are introduced, the projection algorithm and the occultation algorithm, and a detailed method for calculating the brightness of binary asteroids is expounded.Finally, an inversion method and parallel acceleration schemes that are specifically designed to overcome the challenges encountered in binary asteroids are introduced.

Projection Algorithm
Before determining the mutual occultation of binary asteroids, it is necessary to project the center points of all triangular facets of the two Cellinoid shapes that have been triangulated onto a hyperplane with the Sun or Earth as the normal direction and passing through the origin of the coordinate axis.If the hyperplane N(A, B, C) passes through the origin of the coordinate system and (x 0 , y 0 , z 0 ) is an arbitrary point in space, the projected point (x, y, z) can be obtained as follows: The projected points (x, y, z) need to be converted to the XOY plane of the three-dimensional coordinate system.As shown in Figure 3, the first step is to rotate the point around the Z-axis by angle α and then rotate the resulting point around the Y-axis by angle β.The angles α and β are obtained from the normal vector N(A, B, C).To perform these rotations, the conversion matrix is represented as follows: Algorithm 1. Projection algorithm ¬ project the 3D_points to the plane of normal vector n

Occultation Algorithm
To convert all the centroids of the two Cellinoid-shaped triangulated facets of the binary asteroid into two-dimensional points in the XOY plane of the three-dimensional coordinate system, Algorithm 1 is utilized.Once these two-dimensional points are obtained, they are used to implement the occultation effect.Figure 4 illustrates how the occultation points are identified and how the object points mark the occluded object points based on the occultation area formed by the occultation points.end for 42: [

= -
´----- ´- ´-´----- ´-43: Algorithm 2 is responsible for implementing the occultation effect and returns an array of Tag_array.This array contains either 0 or 1, with 0 indicating no occultation and 1 indicating occultation.If the sum of this array is greater than 0, it indicates the presence of an occultation effect.To visualize the occultation effect, Algorithm 2 is used along with Open Graphics Library (OpenGL), as depicted in Figure 5. Algorithm 2 takes two-dimensional object points and twodimensional occultation points as input.The method for identifying an occluded object and occultation object is described, following which Algorithm 1 can be used to obtain the two-dimensional object points and two-dimensional occultation points.
The distance between point P 0 (x 0 , y 0 , z 0 ) and hyperplane Ax + By + Cz + D = 0 can be calculated using the following formula: In a binary asteroid system, the terms "Primary" and "Secondary" are used to refer to the larger and smaller components, respectively.Specifically, the satellite is referred to as the Secondary.
To determine the occluded object and occultation object in a binary asteroid system, the following method is used: First, for either the Sun or the Earth, the normal vector pointing to its position is used to construct a hyperplane at that position.The distance from the center of the Primary and the center of the Secondary to this hyperplane is then calculated.
If the distance from the Primary is greater than that of the Secondary, the Primary is the occluded object and the Secondary is the occultation object.If the distance from the Primary is smaller than that of the Secondary, the Primary is the occultation object and the Secondary is the occluded object.
Therefore, to summarize, the occultation and occluded objects are determined by comparing their distances to a hyperplane constructed at the position of either the Sun or Earth, depending on which is used as the reference point.
Based on the aforementioned details, the integration of Algorithm 1 and Algorithm 2 effectively enables the simulation of occultation and eclipse events in astronomical studies.10

Brightness Calculation
To construct a binary asteroid system, certain approximations are necessary.As indicated by Zhang et al. (2022), the occurrence of an opposite surge, when the phase angle α is near 0°, necessitates the inclusion of parameters A and D in Equation (2).However, given the predominance of nonzero angles in a wide range of sample data sets, along with the priority of computational efficiency, it is often feasible to exclude the parameters A and D. Additionally, Scheirich & Pravec (2009) described that the standard errors for the Primary's rotational pole and mutual orbit pole are less than 3°, leading to the assumption that the Primary's rotational pole and mutual orbit pole share the same direction.To simplify the calculation process and reduce the number of required parameters, it is assumed that the Primary has a Cellinoid shape and the Secondary is an oblate spheroid (D a = D b D c ).In the realm of geometric shapes, it can be observed that an oblate spheroid constitutes a particular case of a Cellinoid sphere.In total, the binary asteroid system relies on 19 parameters as a result of these assumptions.These parameters are as follows: According to Lu et al. (2014) and Zhang et al. (2022), the Cellinoid shape model is defined in the Cellinoid-frame coordinate system, while the positions of the asteroid and Earth are specified in the ecliptic coordinate system.To determine the lightcurve brightness of the asteroid in the Cellinoid-frame coordinate system, an asteroidcentric coordinate system is used as an intermediary system.This helps determine the solar direction E0 and Earth direction E using the real positions of the asteroid and Earth in the ecliptic coordinate system.Using Equation (5) in the Cellinoid-frame coordinate system, the brightness of an individual asteroid with a Cellinoid shape model can be calculated.The Jet Propulsion Laboratory (JPL) Horizons On-Line Ephemeris System maintained by  NASA provides the three-dimensional positions of Earth and an asteroid of interest in the ecliptic coordinate system. 11he spherical center of the Secondary is defined under the asteroidcentric coordinate system since it orbits the Primary axially.To calculate the spherical center, this article utilizes the following formula: cos , 1 ecc sin , 0 .9 Here, R represents the longest distance between the centers of the Primary and Secondary, f represents the orbital phase angle, and ecc represents the orbital eccentricity.
To convert the spherical center of the Secondary from the asteroidcentric coordinate system to the Cellinoid-frame coordinate system, the following formula can be used: Secondary .10 center sc

= +
The orthogonal matrix Q and the center of mass , , T can be obtained by referring to Lu et al. (2014) or Zhang et al. (2022).
The process of determining the brightness of a binary asteroid is more complex than that of an individual asteroid.The calculation entails several steps that require careful consideration of the positions of both the real asteroid and Earth.Specifically, the first step involves calculating E0 and E in the Cellinoid-frame coordinate system based on the positions of the two bodies.Next, the position of the spherical center of the Secondary is determined in the asteroidcentric coordinate system using Equation (9).This position is then converted to the Cellinoid-frame coordinate system using Equation (10).In the third step, the brightness of all triangular facets of both the Primary and Secondary bodies is computed separately using Equation (5).The fourth step involves using Algorithm 1 and Algorithm 2 to identify occluded triangular facets by setting their brightness to 0. Finally, the total brightness of the binary asteroid is obtained by summing the brightness of all triangular facets of both the Primary and Secondary bodies.This methodology, which accounts for the complex geometry and mutual occultation of binary asteroids, provides a reliable means of determining their brightness and can be used to better understand their physical properties.

Inversion Process
Deriving the physical property parameters of binary asteroids from lightcurves is a challenging nonlinear optimization problem.In this study, the LM algorithm, which is based on the damping coefficient method, is used to solve this problem.
The physical parameters of a binary asteroid can be obtained by minimizing the χ 2 function defined as follows: where L i represents the observed brightness of the ith lightcurve, while Li represents the calculated brightness corresponding to the observing geometry.The summation is taken over all M data points.Minimizing χ 2 yields the physical parameters of the binary asteroid.
If the observed lightcurve is uncalibrated, a modified version of the χ 2 function is used, given by the following formula: Here, 〈L i 〉 and Li á ñ represent the mean observed and synthetic brightness, respectively.The summation is again taken over all M data points.
To obtain the physical parameters of binary asteroids, the χ 2 function is minimized by adjusting the parameters.Its modified version can be used when the observed lightcurve is uncalibrated.The LM algorithm is used, which involves iteratively updating the parameters until the minimum of the χ 2 function is reached.The damping coefficient method is used to control the step size during the iterative updates, which improves the convergence of the algorithm.
The LM algorithm typically acquires a solution that is locally optimal, which may result in inaccurate inverted parameters at times.To overcome this limitation, the current study implements a step-by-step inversion scheme to attain the globally optimal solution.The rotational period is the first parameter to be inverted, followed by the pole after the inverted period has been fixed.Finally, other parameters can be inverted once the inverted period and pole have been fixed.

Parallelization Schemes
The process of inverting the physical property parameters of binary asteroids using the LM algorithm is known to be timeconsuming and may lack efficiency without parallel computing.Previous research studies (Zhang & Lu 2021;Zhang et al. 2022) have demonstrated the effectiveness of using CPU-or GPU-based parallelization techniques to enhance the inversion process.Hence, parallelization technology, such as OpenMP, CUDA C, and OpenACC, is essential for achieving optimal acceleration.
OpenMP is a parallelization technology that utilizes multicore CPUs, while CUDA C is a GPU-based parallelization technology.By contrast, OpenACC can be implemented using either multicore CPUs or GPUs.Although CUDA C generally provides a better performance than OpenACC, it can be challenging to implement.For beginners with limited experience in either OpenACC or CUDA C, utilizing OpenACC instead of CUDA C may yield a better performance.

Numerical Experiments and Discussion
According to the above analysis, the methodology for calculating the brightness of binary asteroids and the inversion method has been described.In this section, a proposed model is used to reproduce the findings of published experiments.Next, the self-consistency of this model inversion is verified.The section culminates in demonstrating the successful inversion of two real binary asteroids, namely asteroid (317) Roxane and asteroid (624) Hektor.Utilizing the OpenMP technique in combination with a pair of Hygon C86 7185 CPUs, the inversion process of physical property parameters for binary asteroids is effectively implemented.Pravec et al. (2006) reported that the Primary has a rotational period of 2.2593 hr.Observational lightcurves of asteroid (65803) Didymos were collected on 2003 December 20, as reported in Pravec et al. (2006) and Scheirich & Pravec (2009).The JPL Horizons On-Line Ephemeris System provided the position data of Earth and the asteroid for that date.The Primary and Secondary are assumed as oblate spheroids; hence, a proposed model with 19 parameters is used to reproduce the synthetic data.These parameters are a 1 = 1.00, a 2 = 1.00, b 1 = 1.00, b 2 = 1.00, c 1 = 0.8, c 2 = 0.8, λ = 330°, β = −70°, prd = 2.2593 hr, θ 0 = 0°, R = 3.0, P = 11.92 hr, f 0 = 245°, r = 0.21, ecc = 0.02, sdratio = 0.8, K = −0.0023,B = 0.42, and γ = 0.13.The intensity of the synthetic lightcurve can be calculated using this binary model.Miles (2007), Grayson & Graney (2011), Graney (2009), and Agrawal (2016a, 2016b, 2018) provided equations to calculate the apparent magnitude from the intensity, given by the following formula: As illustrated in Figure 6, the lightcurve generated by the proposed model in this study exhibits a morphology that is highly similar to the model described in Pravec et al. (2006) and Scheirich & Pravec (2009).This result implies that utilizing both the parameters obtained from Pravec et al. (2006) and Scheirich & Pravec (2009) and the proposed model can reproduce the experimental findings, which confirms the feasibility and validity of the proposed model.

Synthetic Data and Inversion Process
Primary and Secondary bodies with Cellinoid and oblate spheroid shapes are used to corroborate the proposed model.To accurately simulate the asteroid parameter inversion process, positional data from the JPL Horizons On-Line Ephemeris System were obtained for Earth and asteroid (65803) Didymos between 2019 November 1 and 2022 October 31.
The proposed model inversion method yields the following physical parameters: a 1 = 1.02, a 2 = 0.90, b 1 = 0.37, b 2 = 0.76, c 1 = 0.53, c 2 = 0.47, λ = 31.83°,β = −15.28°,prd = 4.28 hr, θ 0 = 217.61°,R = 6.64,P = 30.91hr, f 0 = 166°.46,r = 0.38, ecc = 0.14, sdratio = 1.08,K = −0.0021,B = 0.34, and γ = 0.14.The results presented in Figure 7 demonstrate a high degree of consistency between the fitted data and the real data.Notably, six occultation effects were observed at specific times throughout the observation period.Leveraging sufficient viewing geometries, we were able to accurately infer several key parameters, including the rotational period of the Primary, pole, initial rotational phase angle of the Primary, orbital radius, orbital period, orbital eccentricity, and initial orbital phase angle of the Secondary, as outlined in Table 1.However, it is worth noting that the shape parameters of the binary asteroid showed a slight deviation due to the nonlinear relationship between the shape of the Primary asteroid and the parameters K and B. Additionally, the shape of the Secondary asteroid is influenced by that of the Primary asteroid.The presence of noise in the experimental data had a discernible impact on γ in Equation (1).Despite these challenges, the inversion method successfully inferred several key parameters, including the rotational period of the Primary, and the pole, orbital radius, orbital period, and orbital eccentricity of the Secondary.This outcome underscores the feasibility of the inversion method and the self-consistency of the model, even in the presence of experimental noise, provided that sufficient viewing geometries are available.

Real Asteroids
Although the application of the inversion process to synthetic data is crucial for initial validation, it can lead to a phenomenon known as the 'inverse crime'.This happens when the model used to generate the synthetic data is also used for inversion, potentially leading to overly optimistic validation results.To further validate the model and address this concern, the inversion process is now applied to real-world scenarios.This subsection discusses the application of the inversion process to two real binary asteroids, (317) Roxane and (624) Hektor.By leveraging real data, the aim is to test the model under less controlled and more complex conditions, thereby providing a more robust validation of its performance.The Gaussian fitting method is employed to evaluate the uncertainty of a specific parameter.Initially, optimal results are obtained through model inversion.Subsequently, Gaussian noise is introduced to the optimal inversion result of this specific parameter, and this operation is replicated 1000 times, resulting in 1000 unique sets of parameters.These sets of parameters are then input into the model, and optimization operations are conducted on them.The ultimate outputs from these 1000 iterations are subsequently analyzed to calculate the mean and standard deviation of the specific parameter in question.These statistical metrics provide the necessary values for the uncertainty analysis of this specific parameter.

Asteroid (317) Roxane
Information regarding the pole ((220°, −62°) or (40°, −70°)) and the rotational period (8.16961 hr) of asteroid (317) Roxane .This research analyzes 16 instances of synthetic lightcurves and presents four types of data: noise, superimposed, real, and fitted.The noise data are derived from experimental observations and include random variations and errors.The superimposed data represent the combined brightness data of all triangular facets of the Primary and Secondary objects without accounting for the mutual occultation effect between them.The real data are generated based on specific parameters and represent the asteroid lightcurves without any noise.Finally, the fitted data refer to the combined brightness data of all triangular facets of the Primary and Secondary objects, taking into account the mutual occultation effect.
Two models were utilized in this study to invert the real lightcurves of asteroid (317) Roxane: the individual Cellinoid model in Lu et al. (2014)  including prd, pole(λ, β), R, P, r, and ecc, were estimated using a Gaussian fitting method, as elaborated in Table 3.Both the individual and binary models provided accurate estimates of the rotational period of the Primary.However, the pole parameters derived by the binary model were superior to those obtained using the individual model, and additional parameters were obtained that could not be obtained with the individual model.These parameters were found to be approximately consistent with the reference values, as described in Table 2. Notably, the binary model yielded a better fit to the observed lightcurve morphology than the individual model, as shown in Figure 8.It is worth noting that while the binary model provides a more comprehensive solution, it comes at the cost of greater computational complexity.Thus, the individual model, despite its lower accuracy, serves as a quick and efficient alternative for obtaining preliminary pole parameters, which can subsequently be refined using the binary model.
In this study, two models were employed to invert the real lightcurves of asteroid (624) Hektor.The first model used was the individual model with Cellinoid described in Lu et al. (2014) and Zhang et al. (2022).The novel model proposed in this article was also employed.Physical parameters of the asteroid were extracted from the inversion using the individual model.These parameters were found to be a 1 = 1.20, a 2 = 0.19, b 1 = 0.21, b 2 = 0.40, c 1 = 0.17, c 2 = 0.43, λ = 327°.66,β = −27°.88, prd =6.920511 hr, θ 0 = 345°.47,K = 0.103665, B = 0.20, and γ = 0.01.The proposed model was then utilized to extract physical parameters of the binary asteroid, resulting in a 1 = 0.89, a 2 = 0.11, b 1 = 0.19, b 2 = 0.23, c 1 = 0.25, c 2 = 0.06, λ = 332°.80,β = −31°.05,prd = 6.920509 hr, θ 0 = 241°.20,R = 2.45, P = 71.17hr, f 0 = 115°.20, r = 0.01, ecc = 0.30, sdratio = 0.98, K = −0.0018,B = 0.38, and γ = 0.07.Uncertainties in certain parameters, such as prd, pole(λ, β), R, P, r, and ecc, were estimated using a Gaussian fitting method, as detailed in Table 5.Both the individual and binary models provided accurate estimates of the rotational period of the primary asteroid.However, the pole parameters derived using the binary model were superior to those obtained by the individual model, and additional parameters were obtained that could not be obtained with the individual model.These parameters were found to be approximately consistent with the reference values, as described in Table 4.It is worth noting that the binary model provided a better fit to the observed lightcurve morphology than the individual model, as illustrated in Figure 9.However, the binary model is computationally more complex, while it offers higher accuracy and can refine the initial pole parameters obtained from the individual model.

Discussion
To illustrate the efficacy of the proposed model, simulated lightcurves of asteroid (65803) Didymos were generated using the proposed model.The lightcurves generated by the proposed model exhibited a morphology that was highly similar to that of the lightcurves described in Pravec et al. (2006) and Scheirich & Pravec (2009).This result suggests that by utilizing the parameters obtained from Pravec et al. (2006) and Scheirich & Pravec (2009) and the proposed model, the experimental findings can be accurately reproduced.
The self-consistency of the model inversion has been verified.Additionally, the reliability and effectiveness of the proposed model have been demonstrated by the successful inversion of two real binary asteroids, namely, asteroid (317) Roxane and asteroid (624) Hektor.These results suggest that the proposed model can be used as an effective tool for the inversion of physical property parameters of binary asteroids.
However, there are some limitations to this approach.The proposed model assumes that the binary system is rigid and oblate, which may not always be true.In addition, the model assumes that the physical properties of the binary system are homogeneous, which may not always be the case.These assumptions may lead to inaccuracies in the physical property parameter estimation, particularly for complex binary systems with nonuniform physical properties.
Nevertheless, the implementation of the physical property parameter inversion process for binary asteroids utilizing the OpenMP technique along with a pair of Hygon C86 7185 CPUs environment has been demonstrated.This implementation allows for a faster and more efficient inversion process, which is necessary for the future success of the DART project.
The proposed method has the potential to improve our understanding of binary asteroids and aid in the development of effective strategies for the DART project.By accurately determining the physical property parameters of binary asteroids, we can better predict their behavior and take appropriate measures to prevent potential collisions with Earth.

Conclusions
The exploration of asteroids has significant scientific and practical implications.Advancements in technology have facilitated the understanding of their physical characteristics and origins, which have provided valuable insights into the early stages of the formation of the solar system.Furthermore, the recent development of the DART project, the first planetary defense experiment, demonstrates the need for continued research into asteroids.To achieve a more comprehensive understanding of these celestial bodies, the study of lightcurves has become increasingly important.This article proposes a novel model to generate lightcurve data for binary asteroids, which offers a simpler and more intuitive alternative to the Fourier series technique proposed by Pravec et al. (2006) and Scheirich & Pravec (2009).Utilizing the proposed inversion method can facilitate the estimation of several critical physical properties of binary asteroids.Specifically, this approach enables the determination of the rotational period of the primary asteroid, its pole orientation, and the orbital parameters of the secondary asteroid, including its radius, period, and eccentricity.
The research presented in this article has made significant contributions to the study of binary asteroids.The proposed model is shown to be effective in generating synthetic lightcurve data, and the inversion method presented provides a detailed approach to determining physical property parameters.The successful inversion of the physical properties of asteroids (317) Roxane and (624) Hektor showcases the potential of the proposed model for future research and applications.However, challenges remain in accurately determining the physical properties of binary asteroids due to mutual occultation and the Cellinoid model's limitations.Future    research could investigate more observations from various viewing geometries to enhance the accuracy of the inverted parameters.
Overall, this article offers a novel approach to studying binary asteroids and presents an effective method for generating synthetic lightcurve data and determining physical property parameters.Simultaneously, the model adeptly simulates the occultation and eclipse events associated with binary asteroid systems within the domain of astronomical research.The results of this research provide valuable insights into the study of binary asteroids and may contribute to future developments in planetary defense strategies.The proposed model could also be extended to the study of other celestial bodies, enabling a more comprehensive understanding of the formation and evolution of the solar system.

Figure 1 .
Figure 1.The inversion process is used to determine the physical property parameters of a binary asteroid.
the pseudocode for the projection process.

Figure 3 .
Figure 3. Visualization of rotating a hyperplane defined by the normal vector N(A, B, C).(a) The initial hyperplane and normal vector.(b) Rotation of the hyperplane around the z-axis by an angle α.(c) Further rotation of the hyperplane around the y-axis by an angle β.The hyperplane is represented by a translucent surface plot and the normal vector is shown as a red arrow.
1.The axes of the Primary are represented by (a 1 ,a 2 , b 1 , b 2 , c 1 , c 2 ), where a 1 + a 2 b 1 + b 2 c 1 + c 2 .2. The pole of the Primary and the mutual orbit pole are represented by (λ, β). 3. The rotational period of the Primary is represented by prd, and its initial rotational phase angle is represented by θ 0 .4. The longest distance between the centers of the Primary and Secondary is represented by R, the orbital period of the Secondary by P, and its initial orbital phase angle by f 0 .5. The long axis of the Secondary is represented by r, the ratio of the short axis to the long axis for the Secondary is represented by sdratio, and the orbital eccentricity is represented by ecc. 6.The scattering coefficient is represented by K and B, and the weight factor in Equation (1), is represented by γ.

Figure 5 .
Figure 5. (a) The smaller celestial body in the binary asteroid appears partially obscured.(b) The larger celestial body in the binary asteroid appears partially obscured.

Figure 4 .
Figure 4. (a) The points marked with an asterisk represent object points, while the points marked with a triangle represent occultation points.The occultation area is the smallest area that contains all occultation points and is enclosed by a line segment formed by the outermost occultation points.(b) The object points in the occluded area transition from blue to green, indicating that these object points are now occluded.

,
Pravec et al. (2006) andScheirich & Pravec (2009) introduced a dependable model for the inversion of physical property parameters of binary asteroids using lightcurves.Their model has proven to be robust and reliable, as demonstrated by its successful application to the inversion study of physical property parameters of multiple binary asteroids.The effectiveness of their approach serves to reinforce the dependability of their model in the domain of asteroid physical property parameter inversion via lightcurve analysis.As described inScheirich & Pravec (2009), the estimated parameters of two solutions for asteroid (65803) Didymos are as followsD 2 /D 1 = 0.22 ± 0.01, a/D 1 = 1.4 ± 0.1, e 0 denotes the intensities (brightnesses) measured from Earth.

Figure 6 .
Figure 6.The synthetic lightcurve can be derived using the proposed model.

Figure 7
Figure7.This research analyzes 16 instances of synthetic lightcurves and presents four types of data: noise, superimposed, real, and fitted.The noise data are derived from experimental observations and include random variations and errors.The superimposed data represent the combined brightness data of all triangular facets of the Primary and Secondary objects without accounting for the mutual occultation effect between them.The real data are generated based on specific parameters and represent the asteroid lightcurves without any noise.Finally, the fitted data refer to the combined brightness data of all triangular facets of the Primary and Secondary objects, taking into account the mutual occultation effect.

Figure 8 .
Figure8.The study analyzes 16 instances of real lightcurves of asteroid (317) Roxane, fitting the data using both an individual model and a binary model.The individual model employs the Cellinoid model(Lu et al. 2014;Zhang et al. 2022), while the binary model utilizes the proposed model proposed in this article.
Note.D a = 416, D b = 131, and D c = 120.The rotational period of the Primary is represented by prd.The pole of the Primary and the mutual orbit pole are represented by pole(λ, β).The rotational phase angle of the Primary is represented by θ 0 .The longest distance between the centers of the Primary and Secondary is represented by R, and the orbital period of the Secondary by P. The long axis of the Secondary is represented by r, and the orbital eccentricity by ecc.
Note.D a = a 1 + a 2 , D b = b 1 + b 2 , and D c = c 1 + c 2 .The rotational period of the Primary is represented by prd.The pole of the Primary and the mutual orbit pole are represented by pole(λ, β).The rotational phase angle of the Primary is represented by θ 0 .The longest distance between the centers of the Primary and Secondary is represented by R, and the orbital period of the Secondary by P. The long axis of the Secondary is represented by r, and the orbital eccentricity by ecc.

Table 1
Comparison between Real Parameters and Inverted Parameters The axes of the Primary are represented by (a 1 , a 2 , b 1 , b 2 , c 1 , c 2 ).The rotational period of the Primary is represented by prd.The pole of the Primary and the mutual orbit pole are represented by pole(λ, β).The rotational phase angle of the Primary is represented by θ 0 .The longest distance between the centers of the Primary and Secondary is represented by R, the orbital period of the Secondary by P, and its initial orbital phase angle by f 0 .The long axis of the Secondary is represented by r, the ratio of the short axis to the long axis for the Secondary is represented by sdratio, and the orbital eccentricity by ecc.The scattering coefficient is represented by K and B, and the weight factor is represented by γ.

Table 2
Reference Parameters of Asteroid (317) Roxane from Other Publications Note.D a = 29, D b = 17, and D c = 14.4.The rotational period of the Primary is represented by prd.The pole of the Primary and the mutual orbit pole are represented by pole(λ, β).The rotational phase angle of the Primary is represented by θ 0 .The longest distance between the centers of the Primary and Secondary is represented by R, and the orbital period of the Secondary by P. The long axis of the Secondary is represented by r, and the orbital eccentricity by ecc.

Table 3
Inverted Parameters of Asteroid (317) Roxane using the Individual and Binary Models

Table 4
Reference Parameters of Asteroid (624) Hektor from Other Publications