Scattering Law Analysis Based on Hapke and LommelSeeliger Models for Asteroidal Taxonomy
1. Introduction
Being composed of primitive matter from the early solar system, an asteroid can reveal the evolution mechanism of planets and provide information on the formation of our solar system (DeMeo & Carry 2014 ). By using groundbased observations, the physical properties of asteroids can be derived from their photometric data such as rotational periods, pole orientations and overall shapes (Lu et al. 2013 , 2014 ; Lu & Ip 2015 ; Lu et al. 2016 ).
In deriving the physical properties of asteroids from their photometry data, the scattering law plays an important role, although the shape variations of asteroids result in the main variations in their lightcurves (Karttunen
1989
; Karttunen & Bowell
1989
). As shown in Figure
However, application of the Hapke model is very complicated when applying a numerical inversion process to investigate the physical properties of asteroids. There are many equivalent methods that have attempted to numerically replace it such as the joint linear exponential model published by Muinonen et al. ( 2009 ), the Minnaert model proposed by Minnaert ( 1941 ), and the LommelSeeliger (LS) model constructed by Seeliger ( 1884 ). In particular, the LS model is very efficient in numerical calculations of synthetic brightnesses of asteroids. Recently,many applications have been developed to derive the physical parameters for asteroids with the LS model (Cellino et al. 2015 ; Muinonen & Lumme 2015 ).
Parameters in the Hapke model have corresponding physical meanings such as albedo, which is a measure of the amount of reflected light from the surface, and others as described in Table


The paper will be arranged as follows. First of all, in Section
2. Scattering Models
The Hapke model is based on a semiphysical model that uses the analytical solutions of radiative transfer on an asteroidal surface with simple assumptions, coupled with empirical models that describe the scattering behavior of the particulate surface (Hapke 2012 ).
There are at least five parameters in the Hapke model, including the single scattering albedo (ω), photometric roughness (
The LS model is a widelyused scattering model (Besse et al.
2013
), developed by Seeliger (
1884
) and improved by Hapke (
2012
). The RADF (Takir et al.
2015
) of the LS model with four free parameters is shown in Equation (
Here in both Equation (
3. Scattering Law Analysis for Hapke and LS models
Both the Hapke and LS models are explained in the previous section. Intuitively, the LS model has a much simpler form than the Hapke model. In this section, first, the equivalent parameters of the LS model are numerically fitted to the given parameters of the Hapke model. By applying the two equivalent parameters to generate synthetic lightcurves based on a Cellinoid shape model, their similar morphologies of synthetic lightcurves confirm that the LS model can provide the same result as the Hapke model. Then the PCA technique is applied to analyze equivalent parameters of the LS model, derived from the Hapke parameters of C and S type asteroids. Then the result shows that different types of asteroids can be identified from parameters of the LS model.
To better simulate the real circumstances, the derived parameters of different scattering models for (101955) Bennu by Takir et al. ( 2015 ) are exploited for comparison of the equivalent parameters.
The phase curves of (101955) Bennu for the Hapke model with ω = 0.031, B
_{so} = 3.9, h
_{s} = 0.11, g = −0.32 and
Additionally, the fitted equivalent parameters of the LS model are listed in Table


Furthermore, the parameters of the Hapke model and the fitted equivalent parameters of the LS model listed in Table
As the parameters in the Hapke model can reveal information about the surface of asteroids, Helfenstein & Veverka (
1989
) introduced the taxonomy of C and S type asteroids according to the five parameters. The mean parameters are listed in Table


In order to find taxonomic relations in the LS model for the two types of asteroids, we build test sets containing 1200 sample points for each type, which are selected respectively from the C and S type asteroids with the five corresponding Hapke parameters. Subsequently, equivalent parameters of the LS model for two types of asteroids are derived.
Then the PCA technique (Abdi & Williams
2010
) is applied to determine the correlation of the four variables in the LS model and the results are presented in Table
Cluster distribution map of C (blue ‘∗’) and S (red ‘∗’) type asteroids. The regression line of C type is represented as a green line and the one of S type is shown as a pink line.
PCA Results
Principal Component  Z _{1}  Z _{2}  Z _{3}  Z _{4} 
Percent Variance (%)  70.74  25.24  3.59  0.43 
Cumulative Percent (%)  70.74  95.98  99.57  100 
The PCA results show that the parameters of the LS model for C and S type asteroids appear clustered, which can be applied in asteroid taxonomy. Following this, we try to find an efficient way to classify the C and S type asteroids based on the LS model parameters in the subsequent subsection.
As previously described, the principal components of four parameters in the LS model show clear clustering. Here we present an efficient technique to identify the taxonomy of an asteroid from its LS parameters. The LASSO method (Tibshirani 1996 ; Mei & Ling 2009 ; Musoro et al. 2014 ) in machine learning is commonly applied to search the bestfit solution with sparse nonzero variables.
Asteroid taxonomy from the LS parameters can be expressed in the LASSO format as a linear system of equations,
As the LASSO method minimizes the following optimization problem with the l _{1} regularization term,
First, the training set including a total of 2400 sample points is built by merging the LS model parameters for known C and S type asteroids with the respective 1200 sample points, where the LS model parameters are calculated equivalently from Hapke model parameters for known C and S type asteroids. Then we also generate a test set containing 200 points for known C and S type asteroids. Finally, by applying the gradient projection method (Figueiredo et al.
2007
) to solve the problem in Equation (
As shown in Figures
Successfully derived indicators for C type asteroids by the LASSO method.
Successfully derived indicators for S type asteroids by the LASSO method.
Indeterminate indicators for C and S type asteroids by the LASSO method.
4. Conclusions
In this article, we analyze the two scattering models, Hapke and LS. Considering that the Hapke model is deduced with physical meanings and the LS model is efficient in numerical calculation, we numerically compare the two scattering models and confirm that the LS model can fit the Hapke model well in terms of phase curves. Additionally, for the shape determination of asteroids, the LS model can generate similar lightcurves with the same morphology as the Hapke model. Moreover, we primarily investigate two types asteroids, C and S type, classified based on the parameters of the Hapke model. By calculating the equivalent parameters of the LS model, we apply the PCA technique to show clear clustered features of two principal components of the LS model for C and S type asteroids. Moreover, we also introduce the method based on LASSO to classify an asteroid from LS scattering parameters to C or S type. This is very useful in real application. The shape, pole, rotational period, as well as the four LS scattering parameters can be derived from photometric data by an inversion process such as the Kaasalainen inversion method or the Cellinoid inversion process. Then by applying the technique presented in this article, an asteroid can be classified as C or S type in a fast way.
As now we only test C and S type asteroids, we want to explore more general taxonomy of asteroids and try to apply our technique to classify them in the future. We also expect that a new taxonomy method based on the LS model can be developed in the future.
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