- Email: [email protected]
Lebedev acceleration and comparison of different photometric models in the inversion of lightcurves for asteroids
Planetary and Space Science xxx (2017) 1–10
Contents lists available at ScienceDirect
Planetary and Space Science journal homepage: www.elsevier.com/locate/pss
Lebedev acceleration and comparison of different photometric models in the inversion of lightcurves for asteroids Xiao-Ping Lu a, c, *, Xiang-Jie Huang a, Wing-Huen Ip a, b, Chi-Hao Hsia a a b c
Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau, China Institute of Astronomy, National Central University, Taiwan Key Laboratory of Planetary Sciences, Chinese Academy of Sciences, Nanjing 210008, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cellinoid Lightcurves Asteroids Lebedev Photometric
In the lightcurve inversion process where asteroid's physical parameters such as rotational period, pole orientation and overall shape are searched, the numerical calculations of the synthetic photometric brightness based on different shape models are frequently implemented. Lebedev quadrature is an efficient method to numerically calculate the surface integral on the unit sphere. By transforming the surface integral on the Cellinoid shape model to that on the unit sphere, the lightcurve inversion process based on the Cellinoid shape model can be remarkably accelerated. Furthermore, Matlab codes of the lightcurve inversion process based on the Cellinoid shape model are available on Github for free downloading. The photometric models, i.e., the scattering laws, also play an important role in the lightcurve inversion process, although the shape variations of asteroids dominate the morphologies of the lightcurves. Derived from the radiative transfer theory, the Hapke model can describe the light reflectance behaviors from the viewpoint of physics, while there are also many empirical models in numerical applications. Numerical simulations are implemented for the comparison of the Hapke model with the other three numerical models, including the Lommel-Seeliger, Minnaert, and Kaasalainen models. The results show that the numerical models with simple function expressions can fit well with the synthetic lightcurves generated based on the Hapke model; this good fit implies that they can be adopted in the lightcurve inversion process for asteroids to improve the numerical efficiency and derive similar results to those of the Hapke model.
1. Introduction As the primitive materials at the origin of our solar system, asteroids preserve information regarding planetary formation and their dynamical processes. Demeo and Carry (2014) show the path of solar system evolution in the perspectives of asteroidal composition and dynamical distribution. To more clearly investigate asteroids, including their surface compositions and inner structures, a few space missions were launched over the past decade. For example, Hayabusa 2, following its successful predecessor, was launched at the end of 2014 to visit the C-type asteroid (162173) Ryugu and will return a sample from the asteroid (Tsuda et al., 2013). Recently, in September of 2016, OSIRIX-REx was launched by NASA to visit the B-type asteroid (101955) Bennu (Lauretta et al., 2017). The asteroids will draw increasing attention, especially in the middle of 2018, when both of Hayabusa 2 and OSIRIX-REx will arrive at their respective target asteroids. Compared to the in-situ explorations by space missions, there
are also more surveying observations from both ground-based and space-based telescopes. For example, NEO (Near Earth Objects) surveys, such as LINEAR, Pan-STARRS and so on, have collected vast numbers of photometric lightcurves (Jedicke et al., 2015). Moreover, the Sloan Digital Sky Survey (SDSS) and Wide-field Infrared Survey Explorer (WISE) (Wright et al., 2010) have been used to study asteroids via comprehensive measurements, including the colors and albedos (Michel et al., 2015). In addition, the Gaia satellite, launched at the end of 2013 by ESA, is implementing its 5-year regular observation mission for collecting the accurate positions and photometric sparse data of sources in the solar system, and its data release 1 is now formally published (Gaia Collaboration et al., 2016). Tanga et al. (2016) presented an overview of the asteroid observations by Gaia, covering the data processing and orbital inversion. Cellino and Dell’Oro (2012) indicated that the Gaia observations can be used to derive asteroid physical properties, including masses, sizes, average densities, spin properties, albedos, and reflectance spectra.
* Corresponding author. Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau, China. E-mail address: [email protected] (X.-P. Lu). https://doi.org/10.1016/j.pss.2017.12.001 Received 25 July 2017; Received in revised form 29 November 2017; Accepted 1 December 2017 Available online xxxx 0032-0633/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Lu, X.-P., et al., Lebedev acceleration and comparison of different photometric models in the inversion of lightcurves for asteroids, Planetary and Space Science (2017), https://doi.org/10.1016/j.pss.2017.12.001
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Lommel-Seeliger, Minnaert, and Kaasalainen models, are compared in simulating the synthetic lightcurves generated from the Hapke photometric model and based on various shape models. An appropriate photometric model is expected to be found for use in the lightcurve inversion process that balances efficiency and accuracy. In Section 2, Lebedev quadrature will be introduced in details, as well as the corresponding brightness integral on the unit sphere, based on mapping from the Cellinoid shape model. Next, the four photometric models of Hapke, Lommel-Seeliger, Minnaert and Kaasalainen will be presented in Section 3. Subsequently, the numerical simulations comparing different photometric models will be presented. Section 4 will discuss the implications of the numerical results. The conclusions and plans for future works will conclude the article in Section 5.
Based on the different shape models, the rotational periods and pole orientations of asteroids can be derived from the photometric observations, as well as their overall shapes. Generally, there are three commonly used shape models. The traditional triaxial ellipsoid shape model with three semi-axes is frequently applied in simulating the asteroids for searching their physical properties (Surdej and Surdej, 1978; Drummond et al., 2010; Lu et al., 2013). Furthermore, Muinonen et al. (2015) presented the method of asteroid lightcurve inversion based on application of the Lommel-Seeliger scattering law to the ellipsoid shape, and Cellino et al. (2015) applied this inversion method to the sparse photometric data. For more lightcurves observed in various viewing geometries, Kaasalainen et al. presented an inversion method based on the convex shape models (Kaasalainen et al., 1992; Kaasalainen and Lamberg, 1992). They represent the mapping function from the surface of the convex shape onto a unit sphere by the spherical harmonics, following the Minkowski process to determine the unique shape result (Kaasalainen and Torppa, 2001; Kaasalainen et al., 2001; Lamberg and Kaasalainen, 2001). Moreover, shape models for hundreds of asteroids obtained by this inversion method are available in the DAMIT database (Durech et al., 2010). To consider an intermediate shape between the ellipsoid and the convex shape, the Cellinoid shape model, which accounts for asymmetric shape features, was first presented by Cellino et al. (1989) to simulate asteroids. Lu and Ip (2015) completed the whole lightcurve inversion process based on this intermediate shape and first called it ‘Cellinoid’. Furthermore, Lu et al. (2016) applied the Cellinoid shape model to the Hipparcos data set and confirmed that it can perform particularly well in the case of sparse photometric data, such as the Hipparcos data and the future analogous Gaia data set. In the lightcurve inversion process the numerical routine of simulating the photometric brightness over the specified shape model consumes the most CPU time. Lebedev quadrature is an efficient method to numerically calculate the surface integral on the unit sphere (Lebedev and Laikov, 1999). Kaasalainen et al. (2012) introduced the optimal computation of brightness integrals by adopting the Lebedev quadrature in their convex inversion. Lu et al. (2013) also attempted to apply the Lebedev quadrature to the lightcurve inversion process based on ellipsoid shape model and largely accelerated the algorithm. Before successfully inducing the analytical formula of the brightness integral for the Cellinoid shape model, it should be very useful to apply the Lebedev quadrature to the lightcurve inversion process for reducing computational cost. In this article the mapping from the surface of Cellinoid shape to the surface of unit sphere is presented and the brightness simulation can be accelerated substantially by applying the Lebedev quadrature. The scattering laws, which describe the light reflectance behaviors, can be employed in the lightcurve inversion process of asteroids based on photometric observations. Hapke (2012) described the theory of reflectance in details and introduced the Hapke model to illustrate the bidirectional reflectance of planetary photometry, incorporating the opposition effect, regolith porosity and surface roughness based on the single-particle scattering theory. Based on radiative transfer theory, the Hapke model can describe the physical properties of a planetary surface; however, its complex formula expression is not convenient for use in numerical simulation, especially in lightcurves inversion. There are many other photometric models, such as Lommel-Seeliger (Hapke, 2012) and Minnaert (1941), as well as the scattering function adopted in Kaasalainen's inversion method (Kaasalainen et al., 2001). These models are numerically easy to implement in the lightcurve inversion process. Takir et al. (2015) compared the different photometric models, including the Minnaert and Lommel-Seeliger models, in simulating the ground-based photometric phase curve data of the OSIRIS-REx target asteroid (101955) Bennu. Karttunen and Bowell (1989) concluded that the variations of lightcurves depend very strongly on the body shape by analyzing synthetic lightcurves and phase curves, generated from various asteroid models using the Lumme-Bowell scattering law. Therefore, in this article, three different photometric models, namely, the
2. Lebedev acceleration 2.1. Lebedev quadrature Lebedev and Laikov (1999) presented an efficient tool of the surface integral on the unit sphere that is often applied in the numerical calculation of the surface integral in the spherical coordinate system. Lebedev quadrature can approximately transform the surface integral of the function f over the unit sphere S, π
2π
I ¼ ∬ f ðΩÞ dΩ ¼ ∫ 0 sinðθÞ dθ∫ 0 dφf ðθ; φÞ;
(1)
to a linear combination of the weights wi and the function values f ðθi ; φi Þ at the Lebedev grids with the grid size, i.e., Lebedev degree N, I
N X
wi f ðθi ; φi Þ;
(2)
i¼1
where the sum of the weight wi is equal to the surface area W of the unit sphere, W¼
N X
wi ¼ 4π :
i¼1
Compared with the two-dimensional discretization of the surface integral (1) on the unit sphere, the linearly sum (2) makes the calculation more efficient, and fewer Lebedev grid points are required to obtain similar accuracy to the commonly used triangularization scheme. Fig. 1 shows the discretized unit sphere with Lebedev grids in different degrees N and their corresponding volumes. As the benchmark, the volume of the unit sphere can be derived analytically, V ¼ 4π =3 4:1888. As the Lebedev degree increases, the discretized sphere approaches the unit sphere. In particular, with the degree of only N ¼ 302, the volume of discretized unit sphere is 4.1109, with the deviation of 1:86% to the benchmark value of 4.1888. Fig. 2 shows a comparison between the Lebedev discretization and the traditional triangularization with nearly equal areas.
2.2. Brightness integral As described by Lu and Ip (2015), the brightness integral based on the Cellinoid shape model can be expressed as, BðE0 ; EÞ ¼ ∬ Cþ Sðμ; μ0 ; αÞ ds;
(3)
where Sðμ; μ0 ; αÞ is the scattering function with the definitions of μ and μ0 , representing the projections of viewing and illuminating unit vectors on the unit normal vector of each facet ds, α is the solar phase angle, and Cþ is the part of the Cellinoid surface that is both illuminated by Sun and observable from Earth, i.e., μ0 > 0; μ > 0. Following the definition of the brightness integral (3), the numerical quadrature can be applied to calculate it, as the analytical formula has 2
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Fig. 1. Discretized sphere with Lebedev grids in different degrees and their corresponding volumes (Benchmark value is the volume of unit sphere, 4.1888).
not been successfully deduced to date; however, we have already obtained the formulas for some cases in specific viewing and illuminating geometries. Fortunately, as the Cellinoid shape model consists of eight octants from eight ellipsoids, as shown in Fig. 3, the calculation can be accelerated by parallel computing of the eight octants as follows:
BðE0 ; EÞ
8 X i¼1
! N X Sðμ; μ0 ; αÞΔSi;j ;
(4)
j¼1
where ΔSi;j represents the area of the j-th facet on the i-th octant following the discretization scheme.
Fig. 2. Lebedev discretization (Left) and triangularization (Right).
3
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2ffi sin θ cos φ sin θ sin φ cos θ Gðθ; φÞ ¼ abc þ þ : a b c
(8)
Ultimately, the discretized formula (4) of calculating the brightness can be accelerated by the Lebedev grids as BðE0 ; EÞ
8 X i¼1
! N X Sðμ; μ0 ; αÞ G θj ; φj wj ;
(9)
j¼1
where, the (θj ; φj ) and wj are the corresponding Lebedev grids and weights in different octants. 2.3. Computational efficiency An arbitrary Cellinoid shape model with six semi-axes a1 ¼ 6; a2 ¼ 3; b1 ¼ 4; b2 ¼ 2; c1 ¼ 3:5; c2 ¼ 2:5 is adopted to generate the synthetic lightcurves based on the two discretized integrations (4) and (9). As shown in Fig. 4, the synthetic lightcurves generated by the Lebedev scheme with degree N ¼ 590 are similar to the lightcurves generated by the triangularization scheme with 8 10000 facets in total. In particular, the synthetic lightcurves with the normalized brightness are almost identical for the two discretization schemes, as shown in the right figure. The computational efficiency can be increased by a factor of 10 via the Lebedev scheme compared to the triangularization scheme in this case. For choosing an appropriate degree for the Lebedev scheme that balances accuracy and efficiency, we apply the two integration schemes to calculate the surface area of an arbitrary Cellinoid shape model with the 6 semi-axes a1 ¼ 5; a2 ¼ 4; b1 ¼ 4; b2 ¼ 3; c1 ¼ 3:5; and c2 ¼ 2:5. The number of triangular facets for triangularization and Lebedev will vary from more than 100 to less than 6000. As the benchmark, the standard surface area for the Cellinoid shape is obtained from the algorithm of the surface area for the triaxial ellipsoid model. The deviations from the benchmark of the derived surface areas using triangularization and the Lebedev scheme with various facet numbers N are shown in the left of Fig. 5. The result shows that the deviations of both integration schemes are less than 2% for facet number N greater than 590, as indicated by the red dot in the figure. Note that the first several points for the Lebedev scheme are not stable as their degrees are less than 200, which is too small to eliminate a random disturbance. In addition, as the Lebedev degree increases, its deviation from the standard area doest not decrease significantly, suggesting that Lebedev degree of approximately N ¼ 590
Fig. 3. Cellinoid shape model.
Before employing the Lebedev quadrature in the discretized formula (4) for the brightness integral, the curvature function Gðθ; φÞ mapping the surface of ellipsoid E to the surface of unit sphere S should be built, as illustrated by Lu et al. (2013): ! ! ∂ r ∂θ ∂∂rφ ;! r ðθ; φÞ 2 E: Gðθ; φÞ ¼ sin θ
(5)
Thus, the surface brightness integral (3) can be transformed to the surface integral on the unit sphere S: BðE0 ; EÞ ¼ ∬ Sþ Sðμ; μ0 ; αÞGðθ; φÞ ds;
(6)
where Sþ has an analogous definition to that of Cþ . Following the standard parameterization of the ellipsoid surface E, x1 ¼ a sin θ cos φ; x2 ¼ b sin θ sin φ; x3 ¼ c cos θ; θ 2 ½0; π ; φ 2 ½0; 2π ; (7) the curvature function Gðθ; φÞ will have the form,
Fig. 4. Synthetic lightcurves with the brightness (Left) and relative brightness (Right), generated by different integration schemes.
4
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Fig. 5. Comparison of calculating the surface area of the Cellinoid shape by triangularization and Lebedev (Left) and comparison of computational efficiency for triangularization and Lebedev (Right).
Furthermore, the Hapke photometric parameters for different taxonomic types of asteroids are derived (cf., Li et al. (2015) and references therein). Although it is difficult and complicated to describe the light reflectance behaviors on the asteroid surface using parameterized models, the Hapke model can perform well in the photometric simulations (Jin et al., 2015; Fink, 2015). In this article, the Hapke scattering function Sðμ; μ0 ; αÞ derived from the corresponding Hapke bidirectional reflectance formula with 5 parameters, namely, average particle single scattering albedo (ω0 ), asymmetric factor (g), opposition surge amplitude (B0 ), opposition surge width (h), and roughness parameter (θ), has the following complex form:
may be a good choice in real application for an acceptable accuracy. In the right of Fig. 5, a comparison of the computational efficiency for the two schemes is shown. The computational cost for Lebedev degree N ¼ 74 is set to be the benchmark. This benchmark is only used to provide a reference for numerically comparing the computational cost of two schemes with different number of facets N. The first three points are not meaningful for comparison because the computer will designate the memory and balance the CPU task at the startup. The computational efficiency for each of the two schemes with different facet number N is estimated by the quotients of their computational cost to the benchmark. It is apparent that the Lebedev scheme has much higher computational efficiency compared to triangularization.
Sðμ; μ0 ; αÞ ¼
3. Photometric models
ω0 μ0 μ ½pðαÞð1 þ BðαÞÞ þ HðμÞHðμ0 Þ 1S ðμ; μ0 ; ψ Þ; 4π μ þ μ0 (10)
3.1. Description of models
where the shadow hiding opposition surge function BðαÞ, the average particle single-scattering phase function pðαÞ, and the AmbartsumianChandrasekhar function HðxÞ for multiple scattering are listed as follows,
Asteroid photometric models are generally adopted to simulate the light reflectance of the surface of asteroids, which is important in the photometric inversion process. The scattering function Sðμ; μ0 ; αÞ in the brightness integrals (4) and (9) can be derived simply from the bidirectional reflectance rðμ; μ0 ; αÞ with the multiplication of the term μ. Li et al. (2015) tabulated the forms of bidirectional reflectances for the commonly used photometric models, such as the Hapke, Lommel-Seeliger, Minnaert and Kaasalainen models.
BðαÞ ¼
B0 ; 1 α 1 þ tan h 2
pðαÞ ¼ HðxÞ ¼
3.1.1. Hapke model Hapke (2012) thoroughly described the theory of light reflectance, including the applications to planetary photometry, and the reflectance spectroscopy. Li et al. (2015) reviewed the advances in asteroid photometry from the perspectives of observations, laboratory experiments, and theory. Derived from radiative transfer theory, the Hapke model incorporates many physical parameters, such as surface roughness and porosity, as well as accounting for the opposition effect.
1 g2 1 þ 2g cos α þ g2
3=2 ;
1 þ 2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 þ 2 1 ω0 x
In particular, S ðμ; μ0 ; ψ Þ in (10) simulates the photometric effects of large-scale roughness with the roughness parameter (θ) given by equation (12.28) in Hapke (2012). Considering the complex form of the Hapke model in the photometric inversion process of asteroids, the comparison between the Hapke model and other commonly used empirical photometric models is implemented
Table 1 Parameters list for four different photometric models. Hapke
ω B0 h g θ
Lommel-Seeliger 0.031 3.9 0.11 0:32 20∘
Als β γ δ
Minnaert 0.037 0:0612 1:6166 104 9:7942 107
AM β γ δ k0 b
0.0130 0.0603 0:0013 1:7219 105 0.2495 0.0029
5
Kaasalainen A0 D k b c
0.0132 15.7858 8:3446 104 0.0646 0:3225
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
to find an appropriate model with a balance of computational efficiency and accuracy.
In total, there are 6 parameters in Minnaert photometric model, including AM ; k0 ; b, and β; γ; δ in the phase function.
3.1.2. Lommel-Seeliger model The Lommel-Seeliger photometric model only considers the single scattering with the form
3.1.4. Kaasalainen model By combining single (Lommel-Seeliger term SLS ) and multiple scattering (Lambert term SL ) with a weight factor c for the latter, Kaasalainen et al. (2001) presented a scattering function with the simple form,
Sðμ; μ0 ; αÞ ¼ ALS
μ0 μ f ðαÞ; μ þ μ0
(11)
Sðμ; μ0 ; αÞ ¼ f ðαÞ½SLS ðμ; μ0 Þ þ c SL ðμ; μ0 Þ μμ0 ¼ f ðαÞ þ cμμ0 : μ þ μ0
where ALS is a constant, and f ðαÞ is the phase function, for which Takir et al. (2015) adopted an exponential empirical function with the following form f ðαÞ ¼ e
βαþγ α2 þδα3
:
In addition, they reported a four-parameter form
α f ðαÞ ¼ A0 exp þ kα þ b; D
(12)
As Lommel-Seeliger model is applicable to dark, low-albedo C-type asteroids, Schr€ oder et al. (2017) applied this model to analyze the surface properties of Ceres from the Dawn mission images. Furthermore, Muinonen and Lumme (2015) derived the analytical formula for the brightness integral based on the ellipsoid shape model using Lommel-Seeliger photometric model. In total, there are 4 parameters in the Lommel-Seeliger photometric model, including ALS and β; γ; δ in the phase function.
tional to ðμμ0 Þ1 . In addition to the solar phase function presented by Takir et al. (2015): f ðαÞ ¼ 10
3.2. Numerical comparison
;
First, as listed in Table 1, the five parameters of the Hapke model for the B-type asteroid (101955) Bennu predicted by Takir et al. (2015) are adopted to generate six synthetic lightcurves. Without loss of generality, we have chosen the ellipsoid shape model with the semi-axes of ½5; 4; 3 and the solar phase angles from 5∘ to 30∘ with an increment of 5∘ to generate the six lightcurves, as shown in Fig. 6. Subsequently, the corresponding parameters for three different empirical photometric models, namely, the Lommel-Seeliger, Minnaert, and Kaasalainen models, are
the Minnaert model has the following form with the consideration of limb-darking behavior of the surface: Sðμ; μ0 ; αÞ ¼ AM f ðαÞðμ0 μÞk0 þbα ;
(15)
where A0 and D are the amplitude and scale length of the opposition effect, and k is the slope of phase curve. It should be noticed that the parameter b in (15) is normalized to unity at opposition for the relative brightness in Kaasalainen et al. (2001). Here, we want to keep the parameter a free variable for better fitting of the synthetic lightcurves in the following numerical experiments. Therefore, there are 5 parameters in the Kaasalainen photometric model: c; A0 ; D; k and b in the phase function. This photometric model is adopted in their lightcurve inversion method, which allows determination of asteroid's convex shape model and its rotation state properties. These shape models are usually available in the DAMIT database (Durech et al., 2010).
3.1.3. Minnaert model The Minnaert model, firstly presented by Minnaert (1941), is based on the Lambert law S ¼ AL μμ0 , in which the reflected power is propor-
2 3 βαþγ α2:5þδα
(14)
(13)
where AM is the so-called Minnaert albedo.
Fig. 6. Synthetic lightcurves based on Hapke model, and the fitted lightcurves from other photometric models. (Hapke: Black dots; Lommel-Seeliger: Blue squares; Minnaert: Red circles; Kaasalainen: Magenta diamonds). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
6
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Fig. 7. Synthetic lightcurves, generated by different photometric models in two different cases. (Left, Cellinoid: [7, 3, 5, 3, 3, 3], Pole: (40 ; 50 ), Solar phase: 30 , Earth: (0:28; 0:96; 0), Sun:(0:24; 0:97; 0); Right, Convex, Pole: (20 ; 30 ), Solar phase: 25 , Earth: (0:66; 0:75), Sun: (0:24; 0:97; 0)).
derived by fitting the six synthetic lightcurves based on the identical ellipsoid shape model and the observation geometries, as listed in Table 1. Finally, the derived parameters of the three numerical models are employed to generate the respective synthetic lightcurves, as shown in Fig. 6. The results confirm that the empirical photometric models of Minnaert and Kaasalainen can fit well to the Hapke model, especially for the synthetic lightcurves close to opposition, although there is little deviation among the fitted lightcurves from the Hapke model for the large phase angles. Nevertheless, the Lommel-Seeliger model cannot well fit the exact maximum and minimum of the synthetic lightcurves of the Hapke model, although the overall morphologies of the fitted lightcurves are consistent with the Hapke model. The primary reason for this result is that the scattering function of Lommel-Seeliger has only 4 parameters, whereas the Hapke and Kaasalainen models have 5 parameters, and the Minnaert model has 6 parameters. Furthermore, the so-called disk function in the Lommel-Seeliger model (11) has a fixed form, whereas the other empirical models having an adjustable parameter in the disk functions. To examine whether the derived parameters of three numerical photometric models can perform as well as the Hapke model in simulating the lightcurves, more synthetic lightcurves are generated in different cases, such as various shape models, pole orientations, and the larger solar phase angles. In Fig. 7 the synthetic lightcurves of four photometric models are generated respectively based on an arbitrary Cellinoid shape with six semi-axes of a1 ¼ 7; a2 ¼ 3; b1 ¼ 5; b2 ¼ 3; c1 ¼ 3; c2 ¼ 3 and the convex shape for asteroid (4179) Toutatis from Lu et al. (2017). In addition, the pole orientations in the ecliptic frame are (40∘ ; 50∘ ) and (20∘ ; 30∘ ) for the synthetic lightcurves of the two groups. Furthermore, Earth and Sun positions are randomly given at the asteroid-centric ecliptic frames, with
the constraint of the solar phase angle. As the pole orientations vary, at some point, the viewing geometries of the numerical simulations will be equivalent to the real situations. The results show that the performances of both the Minnaert model and the Kaasalainen model are consistent with the performance of the Hapke model, whereas the Lommel-Seeliger can roughly fit the other three models. Intuitively, the Minnaert model can obtain the best fitting to the synthetic lightcurves generated by Hapke model because its disk function has 1 more parameter than the Kaasalainen model. Following the comparison of the two different cases in Fig. 7, we extend the phase angle to 70∘ . As most of the main belt asteroids are observable from ground-based telescopes with the solar phase angle of less than 50∘ , the phase angle with larger than 70∘ is not essential although some Near-Earth-Asteroids (NEAs) can have a larger phase angle. In Fig. 8, synthetic lightcurves are shown for another two cases with the same shape models and larger solar phase angles, as well as at different viewing geometries. The results confirm that the empirical photometric models can well fit the case of large solar phase angle, although the deviations of the fitted lightcurves increase with a very small magnitude. Finally, the relative Mean Residual Sum of Squares (MRSS) value, MRSS ¼
2 1 X LCHapke LCmodel ; N LCHapke
(16)
is defined for quantitative comparison of the three empirical models to fit the synthetic lightcurves generated by the Hapke model. For a better estimation of the three models in the cases of various shape models, viewing geometries, and pole orientations, we implement the numerical simulations for the various cases as listed in Table 2, and the MRSS values are calculated for quantitative comparison of three numerical models.
Fig. 8. Synthetic lightcurves, generated by different photometric models in two different cases. (Left, Cellinoid: [7, 3, 5, 3, 3, 3], Pole: (40 ; 50 ), Solar phase: 50 , Earth: (0:59; 0:81; 0), Sun:(0:24; 0:97; 0); Right, Convex, Pole: (20 ; 30 ), Solar phase: 70 , Earth: (0:83; 0:56; 0), Sun:(0:24; 0:97; 0)). 7
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Table 2 Quantitative comparison for three different numerical photometric models.
C (7,3,5,3,3,3), P (40 ; 80 ), α ¼ 30 Convex1, P (20 ; 30 ), α ¼ 25 C (7,3,5,3,3,3), P (40∘ ; 80∘ ), α ¼ 50∘ Convex1, P (20∘ ; 30∘ ), α ¼ 70∘ C (4,5,6,5,6,7), P (30∘ ; 60∘ ), α ¼ 20∘ C (4,5,6,5,6,7), P (120∘ ; 85∘ ), α ¼ 60∘ C (4,5,6,5,6,7), P (60∘ ; 60∘ ), α ¼ 40∘ C (4,5,6,5,6,7), P (250∘ ; 25∘ ), α ¼ 5∘ C (4,5,6,5,6,7), P (150∘ ; 70∘ ), α ¼ 15∘ C (6,8,5,7,6,4), P (80∘ ; 80∘ ), α ¼ 35∘ C (6,8,5,7,6,4), P (60∘ ; 70∘ ), α ¼ 65∘ C (6,8,5,7,6,4), P (120∘ ; 75∘ ), α ¼ 20∘ C (6,8,5,7,6,4), P (80∘ ; 70∘ ), α ¼ 10∘ C (6,8,5,7,6,4), P (160∘ ; 80∘ ), α ¼ 55∘ C (7,5,6,6,4,3), P (55∘ ; 70∘ ), α ¼ 25∘ C (7,5,6,6,4,3), P (130∘ ; 40∘ ), α ¼ 40∘ C (7,5,6,6,4,3), P (200∘ ; 35∘ ), α ¼ 40∘ C (7,5,6,6,4,3), P (30∘ ; 80∘ ), α ¼ 30∘ C (7,5,6,6,4,3), P (40∘ ; 75∘ ), α ¼ 10∘ Convex2, P (60∘ ; 70∘ ), α ¼ 60∘ Convex2, P (50∘ ; 60∘ ), α ¼ 40∘ Convex3, P (60∘ ; 60∘ ), α ¼ 60∘ Convex3, P (40∘ ; 70∘ ), α ¼ 80∘
Convex4, P (10 ; 60 ), α ¼ 30 Convex4, P (100 ; 60 ), α ¼ 40
Lommel-Seeliger
Minnaert
Kaasalainen
269:1 105 363:7 105 405:9 105 926:9 105 34:5 105 145:5 105 47:6 105 26:8 105 47:4 105 98:8 105 131:2 105 71:2 105 56:3 105 135:3 105 37:0 105 325:2 105 356:4 105 19:2 105 19:5 105 30:5 105 27:5 105 199:4 105 526:0 105 117:9 105 150:2 105
34:0 105 99:1 105 113:0 105 806:0 105 5:1 105 88:7 105 18:2 105 1:1 105 5:7 105 16:2 105 61:8 105 7:4 105 4:0 105 60:3 105 9:0 105 160:6 105 177:3 105 1:2 105 1:6 105 7:5 105 8:5 105 75:1 105 348:7 105 25:7 105 36:8 105
76:6 105 146:9 105 170:4 105 908:7 105 13:5 105 95:5 105 23:2 105 7:8 105 18:6 105 39:9 105 108:4 105 35:7 105 20:8 105 86:5 105 17:2 105 200:9 105 191:0 105 2:2 105 6:4 105 11:3 105 16:9 105 127:7 105 485:3 105 42:0 105 57:1 105
Notations: C(7,3,5,3,3,3) for Cellinoid shape model with six semi-axes, Convex1 for the shape model of (4179) Toutatis, Convex2 for the shape model of (2) Pallas, Convex3 for the shape model of (6) Hebe, Convex4 for the shape model of (7) Iris, P for pole orientation in ecliptic frame, and α for solar phase angle.
inversion method based on the Cellinoid shape model, such as the Minnaert and Kaasalainen models. In particular, based on the Lommel-Seeliger photometric model, the analytical formula of the brightness integral based on the Cellinoid shape model may be deduced in the future similar to the finished work by Muinonen and Lumme (2015) based on the ellipsoid shape model.
Generally, the Minnaert model has the best performance, whereas the Lommel-Seeliger model with the simple scattering function can provide an approximate fitting to the Hapke model, compared to other models. 4. Discussions 4.1. Different taxonomic types
5. Conclusions As described by Li et al. (2015), asteroids with different taxonomic types have the corresponding Hapke parameters. The B-type asteroid (101955) Bennu is extremely dark, with the albedo ω ¼ 0:031. To better investigate the differences of photometric models in synthetic lightcurves, we perform analogous comparisons for S-type asteroids with moderately albedo. As shown in Fig. 9, synthetic lightcurves generated by the Hapke model with the mean parameters for S-type asteroids are fitted well by the other numerical models. The parameters for the four photometric models are listed in Table 3. Furthermore, we also simulated the cases of various shape models, pole orientations and viewing geometries. In Table 4, the various cases are tabulated, and the corresponding MRSS values, as defined in (16), are presented for quantitative comparison of three numerical photometric models. The results confirm that, numerically, the three models can fit well to the Hapke model.
In this article, Lebedev quadrature technique is applied to the lightcurve inversion process for substantially accelerating the bright simulation. Numerical comparisons of Lebedev quadrature applied in the volumes of unit sphere and the surface area of Cellinoid shape for the synthetic lightcurves were implemented to confirm its efficiency. The inversion process based on Lebedev acceleration was described in details. In addition, we released the Matlab codes of the Cellinoid shape model on GitHub; everyone can download the codes for free from the website.1 To better compare the different photometric models used in simulating the scattering laws of lightcurves inversion, the synthetic lightcurves were generated from the Hapke model based on the parameters of the B-type asteroid (101955) Bennu, and the corresponding parameters of the other three numerical models of Lommel-Seeliger, Minnaert and Kaasalainen were derived by fitting the synthetic lightcurves. Furthermore, the synthetic lightcurves of the four photometric models were compared for various cases, covering different viewing geometries, different shape models, and different pole orientations. The results confirm that the scattering function will not affect the morphologies of the synthetic lightcurves as much as the shape variations, which is a finding that has already been extensively accepted. In addition, the numerical photometric models with simple expressions can be adopted in the inversion method to substitute the complex photometric models, such as in the Hapke model.
4.2. Shape variations and photometric models We numerically compared the Hapke photometric model and other numerical models in different cases. The results suggest that the morphologies of synthetic lightcurves generated from the derived parameters for different photometric models are very similar. In other words, the different photometric models will not change the morphologies of lightcurves significantly, whereas the variations of the lightcurves morphologies are primarily caused by shape variations. These results are consistent with the conclusions made by Kaasalainen et al. (2001) and Karttunen and Bowell (1989). This conclusion implies that we can employ the numerical photometric model with the simple form in the
1
8
https://github.com/XPLU/CellinoidModel.git.
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Fig. 9. Simulations for S-type asteroids in two different cases (Left, Cellinoid: [7, 3, 5, 3, 3, 3], Pole: (120 ; 20 ), Solar phase: 30 Earth: (0:28; 0:96; 0), Sun:(0:24; 0:97; 0); Right, Convex, Pole: (40 ; 60 ), Solar phase: 55 , Earth: (0:66; 0:75; 0), Sun:(0:24; 0:97; 0)).
Table 3 Parameters of S-type asteroid for four different photometric models. Hapke
ω B0 h g θ
Lommel-Seeliger 0.23 1.6 0.08 0:27 20∘
Als β γ δ
Minnaert 0.0469 0:0612 1:6171 104 9:8702 107
AM β γ δ k0 b
0.0163 0:0091 3:1507 105 1:8824 106 0.2495 0.0031
Kaasalainen A0 D k b c
0.0123 14.4880 8:1379 104 0.0649 0:3192
Table 4 Quantitative comparison of S-type asteroids for three different numerical photometric models.
C (7,3,5,3,3,3), P (40∘ ; 80∘ ), α ¼ 30∘ Convex1, P (20∘ ; 30∘ ), α ¼ 25∘ C (7,3,5,3,3,3), P (40∘ ; 80∘ ), α ¼ 50∘ Convex1, P (20∘ ; 30∘ ), α ¼ 70∘ C (4,5,6,5,6,7), P (30∘ ; 60∘ ), α ¼ 20∘ C (4,5,6,5,6,7), P (120∘ ; 85∘ ), α ¼ 60∘ C (4,5,6,5,6,7), P (60∘ ; 60∘ ), α ¼ 40∘ C (4,5,6,5,6,7), P (250∘ ; 25∘ ), α ¼ 5∘ C (4,5,6,5,6,7), P (150∘ ; 70∘ ), α ¼ 15∘ C (6,8,5,7,6,4), P (80∘ ; 80∘ ), α ¼ 35∘ C (6,8,5,7,6,4), P (60∘ ; 70∘ ), α ¼ 65∘ C (6,8,5,7,6,4), P (120∘ ; 75∘ ), α ¼ 20∘ C (6,8,5,7,6,4), P (80∘ ; 70∘ ), α ¼ 10∘ C (6,8,5,7,6,4), P (160∘ ; 80∘ ), α ¼ 55∘ C (7,5,6,6,4,3), P (55∘ ; 70∘ ), α ¼ 25∘ C (7,5,6,6,4,3), P (130∘ ; 40∘ ), α ¼ 40∘ C (7,5,6,6,4,3), P (200∘ ; 35∘ ), α ¼ 40∘ C (7,5,6,6,4,3), P (30∘ ; 80∘ ), α ¼ 30∘ C (7,5,6,6,4,3), P (40∘ ; 75∘ ), α ¼ 10∘ Convex2, P (60∘ ; 70∘ ), α ¼ 60∘ Convex2, P (50∘ ; 60∘ ), α ¼ 40∘ Convex3, P (60∘ ; 60∘ ), α ¼ 60∘ Convex3, P (40∘ ; 70∘ ), α ¼ 80∘ Convex4, P (10∘ ; 60∘ ), α ¼ 30∘ Convex4, P (100∘ ; 60∘ ), α ¼ 40∘
Lommel-Seeliger
Minnaert
Kaasalainen
257:7 105 351:8 105 391:0 105 919:3 105 33:2 105 142:1 105 46:3 105 25:8 105 45:4 105 94:3 105 126:7 105 67:7 105 53:6 105 131:9 105 35:8 105 317:9 105 348:6 105 18:6 105 18:9 105 28:7 105 26:7 105 194:1 105 520:8 105 113:9 105 145:6 105
33:7 105 98:5 105 114:1 105 800:8 105 5:1 105 88:6 105 18:1 105 1:3 105 5:8 105 16:1 105 62:3 105 7:5 105 4:2 105 60:6 105 8:9 105 160:0 105 176:5 105 1:2 105 1:6 105 7:7 105 8:5 105 76:1 105 352:0 105 25:5 105 36:7 105
77:1 105 146:0 105 171:1 105 906:9 105 13:4 105 94:8 105 23:1 105 7:6 105 18:3 105 40:3 105 110:2 105 35:7 105 20:5 105 86:5 105 17:2 105 199:8 105 189:4 105 2:2 105 6:3 105 11:3 105 16:8 105 127:7 105 485:6 105 41:8 105 56:9 105
Notations: C(7,3,5,3,3,3) for Cellinoid shape model with six semi-axes, Convex1 for the shape model of (4179) Toutatis, Convex2 for the shape model of (2) Pallas, Convex3 for the shape model of (6) Hebe, Convex4 for the shape model of (7) Iris, P for pole orientation in ecliptic frame, and α for solar phase angle.
Acknowledgements
Although the Lebedev technique can largely accelerate the computational efficiency in the lightcurves inversion process based on the Cellinoid shape model, it is not comparable to the analytical formula of the brightness integral. In the future, we plan to focus on this topic, following an approach similar to that used for the ellipsoid shape model by Muinonen and Lumme (2015).
We thank the anonymous reviewers for their professional comments and significant contribution on the scattering function. This work is funded under the grant NO. 095/2013/A3 and 039/2013/A2 from the Science and Technology Development Fund, MSAR. The project is supported partly by the Key Laboratory of Planetary Sciences, Purple
9
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Mountain Observatory, Chinese Academy of Sciences. W.-H. Ip is supported by MSAR Science and Technology Fund (Project No. 017/2014/ A1) and NSC 101-2111-M-008-016.
Lauretta, D.S., Balram-Knutson, S.S., Beshore, E., Boynton, W.V., Drouet dAubigny, C., et al., 2017. OSIRIS-REx: sample return from asteroid (101955) Bennu. Space Sci. Rev. 1, 1–60. Lebedev, V., Laikov, D., 1999. A quadrature formula for the sphere of the 131st algebraic order of accuracy. Dokl. Math. 59, 477–481. Li, J.Y., Helfenstein, P., Buratti, B.J., Takir, D., Clark, B.E., 2015. Asteroid photometry. In: Michel, P., et al. (Eds.), Asteroids IV. Univ. of Arizona, Tucson, pp. 129–150. Lu, X.P., Ip, W.H., 2015. Cellinoid shape model for multiple light curves. Planet. Sp. Sci. 108, 31–40. Lu, X.P., Zhao, H.B., You, Z., 2013. A fast ellipsoid model for asteroids inverted from lightcurves. Res. Astron. Astronphysics 13, 465–472. Lu, X.P., Cellino, A., Hestroffer, D., Ip, W.H., 2016. Cellinoid shape model for Hipparcos data. Icarus 267, 24–33. Lu, X.P., Ip, W.H., Huang, X.J., Zhao, H.B., 2017. Analysis for Cellinoid shape model in inverse process from lightcurves. Planet. Sp. Sci. 135, 74–85. Michel, P., DeMeo, F.E., Bottke, W.F., 2015. Asteroids:Recent advances and new perspectives. In: Michel, P., et al. (Eds.), Asteroids IV. Univ. of Arizona, Tucson, pp. 3–10. Minnaert, M., 1941. The reciprocity principle in lunar photometry. Astronphys. J. 93, 403–410. Muinonen, K., Lumme, K., 2015. Disk-integrated brightness of a Lommel-Seeliger scattering ellipsoidal asteroid. A&A 584, A23. Muinonen, K., Wilkman, O., Cellino, A., Wang, X., Wang, Y., 2015. Asteroid lightcurve inversion with Lommel–Seeliger ellipsoids. Planet. Sp. Sci. 118, 227–241. Schr€ oder, S.E., Mottola, S., Carsenty, U., Ciarniello, M., Jaumann, R., Li, J.Y., Longobardo, A., Palmer, E., Pieters, C., Preusker, F., Raymond, C.A., Russell, C.T., 2017. Resolved spectrophotometric properties of the Ceres surface from Dawn framing camera images. Icarus 288, 201–225. Surdej, A., Surdej, J., 1978. Asteroid lightcurves simulated by the rotation of a three-axes ellipsoid model. A&A 66, 31–36. Takir, D., Clark, B.E., d'Aubigny, C.D., Hergenrother, C.W., 2015. Photometric models of disk-integrated observations of the OSIRIS-REx target Asteroid (101955) Bennu. Icarus 252, 393–399. Tanga, P., Mignard, F., Oro, A.D., Muinonen, K., Pauwels, T., Thuillot, W., Berthier, J., Cellino, A., Hestroffer, D., Petit, J.M., Carry, B., David, P., Delbo, M., Fedorets, G., Galluccio, L., Granvik, M., Ordenovic, C., Pentik€ainen, H., 2016. The daily processing of asteroid observations by Gaia. Planet. Sp. Sci. 123, 87–94. Tsuda, Y., Yoshikawa, M., Abe, M., Minamino, H., Nakazawa, S., 2013. System design of the Hayabusa 2—asteroid sample return mission to 1999 JU3. Acta Astronaut. 91, 356–362. Wright, E.L., Eisenhardt, P.R.M., Mainzer, A.K., Ressler, M.E., Cutri, R.M., Jarrett, T., Kirkpatrick, J.D., Padgett, D., McMillan, R.S., Skrutskie, M., Stanford, S.A., Cohen, M., Walker, R.G., Mather, J.C., Leisawitz D III, T.N.G., McLean, I., Benford, D., Lonsdale, C.J., Blain, A., Mendez, B., Irace, W.R., Duval, V., Liu, F., Royer, D., Heinrichsen, I., Howard, J., Shannon, M., Kendall, M., Walsh, A.L., Larsen, M., Cardon, J.G., Schick, S., Schwalm, M., Abid, M., Fabinsky, B., Naes, L., Tsai, C.W., 2010. The wide-field infrared survey explorer (wise): mission description and initial on-orbit performance. Astron. J. 140, 1868–1881.
References Cellino, A., Dell'Oro, A., 2012. The derivation of asteroid physical properties from Gaia observations. Planet. Sp. Sci. 73, 52–55. Cellino, A., Zappal a, V., Farinella, P., 1989. Asteroid shapes and lightcurve morphology. Icarus 78, 298–310. Cellino, A., Muinonen, K., Hestroffer, D., Carbognani, A., 2015. Inversion of sparse photometric data of asteroids using triaxial ellipsoid shape models and a Lommel–Seeliger scattering law. Planet. Sp. Sci. 118, 221–226. Demeo, F.E., Carry, B., 2014. Solar System evolution from compositional mapping of the asteroid belt. Nature 505, 629–634. Drummond, J.D., Conrad, A., Merline, W.J., Carry, B., Chapman, C.R., Weaver, H.a., Tamblyn, P.M., Christou, J.C., Dumas, C., 2010. Physical Properties of the ESA Rosetta target asteroid (21) Lutetia I. The triaxial ellipsoid dimensions, rotational pole, and bulk density. A&A 523, A93. Durech, J., Sidorin, V., Kaasalainen, M., 2010. DAMIT: a database of asteroid models. A&A 513, A46. Fink, U., 2015. Considerations regarding the colors and low surface albedo of comets using the Hapke methodology. Icarus 252, 32–38. Gaia Collaboration, Brown, A.G.A., Vallenari, A., Prusti, T., de Bruijne, J.H.J., Mignard, F., et al., 2016. Gaia data release 1-summary of the astrometric, photometric, and survey properties. A&A 595, A2. Hapke, B., 2012. Theory of Reflectance and Emittance Spectroscopy, second ed. Wiley Online Library. Jedicke, R., Granvik, M., Micheli, M., Ryan, E., Spahr, T., Yeomans, D.K., 2015. Surveys, astrometric follow-up, and population statistics. In: Michel, P., et al. (Eds.), Asteroids IV. Univ. of Arizona, Tucson, pp. 795–813. Jin, W., Zhang, H., Yuan, Y., Yang, Y., 2015. In situ optical measurements of Chang’E-3 landing site in Mare Imbrium: 2. Photometric properties of the regolith. Geophys. Res. Lett. 42, 8312–8319. Kaasalainen, M., Lamberg, L., 1992. Interpretation of lightcurves of atmosphereless bodies. II-Practical aspects of inversion. A&A 259, 333–340. Kaasalainen, M., Torppa, J., 2001. Optimization methods for asteroid lightcurve inversion I. Shape determination. Icarus 153, 24–36. Kaasalainen, M., Lamberg, L., Lumme, K., Bowell, E., 1992. Interpretation of lightcurves of atmosphereless bodies. I-General theory and new inversion schemes. A&A 259, 318–332. Kaasalainen, M., Torppa, J., Muinonen, K., 2001. Optimization methods for asteroid lightcurve inversion II. The complete inverse problem. Icarus 153, 37–51. Kaasalainen, M., Lu, X.P., V€ anttinen, A., 2012. Optimal computation of brightness integrals parametrized on the unit sphere. A&A 539, A96. Karttunen, H., Bowell, E., 1989. Modelling asteroid brightness variations. II-The interpretability of light curves and phase curves. A&A 208, 320–326. Lamberg, L., Kaasalainen, M., 2001. Numerical solution of the minkowski problem. J. CoAM 137, 213–227.
10
Contents lists available at ScienceDirect
Planetary and Space Science journal homepage: www.elsevier.com/locate/pss
Lebedev acceleration and comparison of different photometric models in the inversion of lightcurves for asteroids Xiao-Ping Lu a, c, *, Xiang-Jie Huang a, Wing-Huen Ip a, b, Chi-Hao Hsia a a b c
Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau, China Institute of Astronomy, National Central University, Taiwan Key Laboratory of Planetary Sciences, Chinese Academy of Sciences, Nanjing 210008, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cellinoid Lightcurves Asteroids Lebedev Photometric
In the lightcurve inversion process where asteroid's physical parameters such as rotational period, pole orientation and overall shape are searched, the numerical calculations of the synthetic photometric brightness based on different shape models are frequently implemented. Lebedev quadrature is an efficient method to numerically calculate the surface integral on the unit sphere. By transforming the surface integral on the Cellinoid shape model to that on the unit sphere, the lightcurve inversion process based on the Cellinoid shape model can be remarkably accelerated. Furthermore, Matlab codes of the lightcurve inversion process based on the Cellinoid shape model are available on Github for free downloading. The photometric models, i.e., the scattering laws, also play an important role in the lightcurve inversion process, although the shape variations of asteroids dominate the morphologies of the lightcurves. Derived from the radiative transfer theory, the Hapke model can describe the light reflectance behaviors from the viewpoint of physics, while there are also many empirical models in numerical applications. Numerical simulations are implemented for the comparison of the Hapke model with the other three numerical models, including the Lommel-Seeliger, Minnaert, and Kaasalainen models. The results show that the numerical models with simple function expressions can fit well with the synthetic lightcurves generated based on the Hapke model; this good fit implies that they can be adopted in the lightcurve inversion process for asteroids to improve the numerical efficiency and derive similar results to those of the Hapke model.
1. Introduction As the primitive materials at the origin of our solar system, asteroids preserve information regarding planetary formation and their dynamical processes. Demeo and Carry (2014) show the path of solar system evolution in the perspectives of asteroidal composition and dynamical distribution. To more clearly investigate asteroids, including their surface compositions and inner structures, a few space missions were launched over the past decade. For example, Hayabusa 2, following its successful predecessor, was launched at the end of 2014 to visit the C-type asteroid (162173) Ryugu and will return a sample from the asteroid (Tsuda et al., 2013). Recently, in September of 2016, OSIRIX-REx was launched by NASA to visit the B-type asteroid (101955) Bennu (Lauretta et al., 2017). The asteroids will draw increasing attention, especially in the middle of 2018, when both of Hayabusa 2 and OSIRIX-REx will arrive at their respective target asteroids. Compared to the in-situ explorations by space missions, there
are also more surveying observations from both ground-based and space-based telescopes. For example, NEO (Near Earth Objects) surveys, such as LINEAR, Pan-STARRS and so on, have collected vast numbers of photometric lightcurves (Jedicke et al., 2015). Moreover, the Sloan Digital Sky Survey (SDSS) and Wide-field Infrared Survey Explorer (WISE) (Wright et al., 2010) have been used to study asteroids via comprehensive measurements, including the colors and albedos (Michel et al., 2015). In addition, the Gaia satellite, launched at the end of 2013 by ESA, is implementing its 5-year regular observation mission for collecting the accurate positions and photometric sparse data of sources in the solar system, and its data release 1 is now formally published (Gaia Collaboration et al., 2016). Tanga et al. (2016) presented an overview of the asteroid observations by Gaia, covering the data processing and orbital inversion. Cellino and Dell’Oro (2012) indicated that the Gaia observations can be used to derive asteroid physical properties, including masses, sizes, average densities, spin properties, albedos, and reflectance spectra.
* Corresponding author. Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau, China. E-mail address: [email protected] (X.-P. Lu). https://doi.org/10.1016/j.pss.2017.12.001 Received 25 July 2017; Received in revised form 29 November 2017; Accepted 1 December 2017 Available online xxxx 0032-0633/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Lu, X.-P., et al., Lebedev acceleration and comparison of different photometric models in the inversion of lightcurves for asteroids, Planetary and Space Science (2017), https://doi.org/10.1016/j.pss.2017.12.001
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Lommel-Seeliger, Minnaert, and Kaasalainen models, are compared in simulating the synthetic lightcurves generated from the Hapke photometric model and based on various shape models. An appropriate photometric model is expected to be found for use in the lightcurve inversion process that balances efficiency and accuracy. In Section 2, Lebedev quadrature will be introduced in details, as well as the corresponding brightness integral on the unit sphere, based on mapping from the Cellinoid shape model. Next, the four photometric models of Hapke, Lommel-Seeliger, Minnaert and Kaasalainen will be presented in Section 3. Subsequently, the numerical simulations comparing different photometric models will be presented. Section 4 will discuss the implications of the numerical results. The conclusions and plans for future works will conclude the article in Section 5.
Based on the different shape models, the rotational periods and pole orientations of asteroids can be derived from the photometric observations, as well as their overall shapes. Generally, there are three commonly used shape models. The traditional triaxial ellipsoid shape model with three semi-axes is frequently applied in simulating the asteroids for searching their physical properties (Surdej and Surdej, 1978; Drummond et al., 2010; Lu et al., 2013). Furthermore, Muinonen et al. (2015) presented the method of asteroid lightcurve inversion based on application of the Lommel-Seeliger scattering law to the ellipsoid shape, and Cellino et al. (2015) applied this inversion method to the sparse photometric data. For more lightcurves observed in various viewing geometries, Kaasalainen et al. presented an inversion method based on the convex shape models (Kaasalainen et al., 1992; Kaasalainen and Lamberg, 1992). They represent the mapping function from the surface of the convex shape onto a unit sphere by the spherical harmonics, following the Minkowski process to determine the unique shape result (Kaasalainen and Torppa, 2001; Kaasalainen et al., 2001; Lamberg and Kaasalainen, 2001). Moreover, shape models for hundreds of asteroids obtained by this inversion method are available in the DAMIT database (Durech et al., 2010). To consider an intermediate shape between the ellipsoid and the convex shape, the Cellinoid shape model, which accounts for asymmetric shape features, was first presented by Cellino et al. (1989) to simulate asteroids. Lu and Ip (2015) completed the whole lightcurve inversion process based on this intermediate shape and first called it ‘Cellinoid’. Furthermore, Lu et al. (2016) applied the Cellinoid shape model to the Hipparcos data set and confirmed that it can perform particularly well in the case of sparse photometric data, such as the Hipparcos data and the future analogous Gaia data set. In the lightcurve inversion process the numerical routine of simulating the photometric brightness over the specified shape model consumes the most CPU time. Lebedev quadrature is an efficient method to numerically calculate the surface integral on the unit sphere (Lebedev and Laikov, 1999). Kaasalainen et al. (2012) introduced the optimal computation of brightness integrals by adopting the Lebedev quadrature in their convex inversion. Lu et al. (2013) also attempted to apply the Lebedev quadrature to the lightcurve inversion process based on ellipsoid shape model and largely accelerated the algorithm. Before successfully inducing the analytical formula of the brightness integral for the Cellinoid shape model, it should be very useful to apply the Lebedev quadrature to the lightcurve inversion process for reducing computational cost. In this article the mapping from the surface of Cellinoid shape to the surface of unit sphere is presented and the brightness simulation can be accelerated substantially by applying the Lebedev quadrature. The scattering laws, which describe the light reflectance behaviors, can be employed in the lightcurve inversion process of asteroids based on photometric observations. Hapke (2012) described the theory of reflectance in details and introduced the Hapke model to illustrate the bidirectional reflectance of planetary photometry, incorporating the opposition effect, regolith porosity and surface roughness based on the single-particle scattering theory. Based on radiative transfer theory, the Hapke model can describe the physical properties of a planetary surface; however, its complex formula expression is not convenient for use in numerical simulation, especially in lightcurves inversion. There are many other photometric models, such as Lommel-Seeliger (Hapke, 2012) and Minnaert (1941), as well as the scattering function adopted in Kaasalainen's inversion method (Kaasalainen et al., 2001). These models are numerically easy to implement in the lightcurve inversion process. Takir et al. (2015) compared the different photometric models, including the Minnaert and Lommel-Seeliger models, in simulating the ground-based photometric phase curve data of the OSIRIS-REx target asteroid (101955) Bennu. Karttunen and Bowell (1989) concluded that the variations of lightcurves depend very strongly on the body shape by analyzing synthetic lightcurves and phase curves, generated from various asteroid models using the Lumme-Bowell scattering law. Therefore, in this article, three different photometric models, namely, the
2. Lebedev acceleration 2.1. Lebedev quadrature Lebedev and Laikov (1999) presented an efficient tool of the surface integral on the unit sphere that is often applied in the numerical calculation of the surface integral in the spherical coordinate system. Lebedev quadrature can approximately transform the surface integral of the function f over the unit sphere S, π
2π
I ¼ ∬ f ðΩÞ dΩ ¼ ∫ 0 sinðθÞ dθ∫ 0 dφf ðθ; φÞ;
(1)
to a linear combination of the weights wi and the function values f ðθi ; φi Þ at the Lebedev grids with the grid size, i.e., Lebedev degree N, I
N X
wi f ðθi ; φi Þ;
(2)
i¼1
where the sum of the weight wi is equal to the surface area W of the unit sphere, W¼
N X
wi ¼ 4π :
i¼1
Compared with the two-dimensional discretization of the surface integral (1) on the unit sphere, the linearly sum (2) makes the calculation more efficient, and fewer Lebedev grid points are required to obtain similar accuracy to the commonly used triangularization scheme. Fig. 1 shows the discretized unit sphere with Lebedev grids in different degrees N and their corresponding volumes. As the benchmark, the volume of the unit sphere can be derived analytically, V ¼ 4π =3 4:1888. As the Lebedev degree increases, the discretized sphere approaches the unit sphere. In particular, with the degree of only N ¼ 302, the volume of discretized unit sphere is 4.1109, with the deviation of 1:86% to the benchmark value of 4.1888. Fig. 2 shows a comparison between the Lebedev discretization and the traditional triangularization with nearly equal areas.
2.2. Brightness integral As described by Lu and Ip (2015), the brightness integral based on the Cellinoid shape model can be expressed as, BðE0 ; EÞ ¼ ∬ Cþ Sðμ; μ0 ; αÞ ds;
(3)
where Sðμ; μ0 ; αÞ is the scattering function with the definitions of μ and μ0 , representing the projections of viewing and illuminating unit vectors on the unit normal vector of each facet ds, α is the solar phase angle, and Cþ is the part of the Cellinoid surface that is both illuminated by Sun and observable from Earth, i.e., μ0 > 0; μ > 0. Following the definition of the brightness integral (3), the numerical quadrature can be applied to calculate it, as the analytical formula has 2
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Fig. 1. Discretized sphere with Lebedev grids in different degrees and their corresponding volumes (Benchmark value is the volume of unit sphere, 4.1888).
not been successfully deduced to date; however, we have already obtained the formulas for some cases in specific viewing and illuminating geometries. Fortunately, as the Cellinoid shape model consists of eight octants from eight ellipsoids, as shown in Fig. 3, the calculation can be accelerated by parallel computing of the eight octants as follows:
BðE0 ; EÞ
8 X i¼1
! N X Sðμ; μ0 ; αÞΔSi;j ;
(4)
j¼1
where ΔSi;j represents the area of the j-th facet on the i-th octant following the discretization scheme.
Fig. 2. Lebedev discretization (Left) and triangularization (Right).
3
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2ffi sin θ cos φ sin θ sin φ cos θ Gðθ; φÞ ¼ abc þ þ : a b c
(8)
Ultimately, the discretized formula (4) of calculating the brightness can be accelerated by the Lebedev grids as BðE0 ; EÞ
8 X i¼1
! N X Sðμ; μ0 ; αÞ G θj ; φj wj ;
(9)
j¼1
where, the (θj ; φj ) and wj are the corresponding Lebedev grids and weights in different octants. 2.3. Computational efficiency An arbitrary Cellinoid shape model with six semi-axes a1 ¼ 6; a2 ¼ 3; b1 ¼ 4; b2 ¼ 2; c1 ¼ 3:5; c2 ¼ 2:5 is adopted to generate the synthetic lightcurves based on the two discretized integrations (4) and (9). As shown in Fig. 4, the synthetic lightcurves generated by the Lebedev scheme with degree N ¼ 590 are similar to the lightcurves generated by the triangularization scheme with 8 10000 facets in total. In particular, the synthetic lightcurves with the normalized brightness are almost identical for the two discretization schemes, as shown in the right figure. The computational efficiency can be increased by a factor of 10 via the Lebedev scheme compared to the triangularization scheme in this case. For choosing an appropriate degree for the Lebedev scheme that balances accuracy and efficiency, we apply the two integration schemes to calculate the surface area of an arbitrary Cellinoid shape model with the 6 semi-axes a1 ¼ 5; a2 ¼ 4; b1 ¼ 4; b2 ¼ 3; c1 ¼ 3:5; and c2 ¼ 2:5. The number of triangular facets for triangularization and Lebedev will vary from more than 100 to less than 6000. As the benchmark, the standard surface area for the Cellinoid shape is obtained from the algorithm of the surface area for the triaxial ellipsoid model. The deviations from the benchmark of the derived surface areas using triangularization and the Lebedev scheme with various facet numbers N are shown in the left of Fig. 5. The result shows that the deviations of both integration schemes are less than 2% for facet number N greater than 590, as indicated by the red dot in the figure. Note that the first several points for the Lebedev scheme are not stable as their degrees are less than 200, which is too small to eliminate a random disturbance. In addition, as the Lebedev degree increases, its deviation from the standard area doest not decrease significantly, suggesting that Lebedev degree of approximately N ¼ 590
Fig. 3. Cellinoid shape model.
Before employing the Lebedev quadrature in the discretized formula (4) for the brightness integral, the curvature function Gðθ; φÞ mapping the surface of ellipsoid E to the surface of unit sphere S should be built, as illustrated by Lu et al. (2013): ! ! ∂ r ∂θ ∂∂rφ ;! r ðθ; φÞ 2 E: Gðθ; φÞ ¼ sin θ
(5)
Thus, the surface brightness integral (3) can be transformed to the surface integral on the unit sphere S: BðE0 ; EÞ ¼ ∬ Sþ Sðμ; μ0 ; αÞGðθ; φÞ ds;
(6)
where Sþ has an analogous definition to that of Cþ . Following the standard parameterization of the ellipsoid surface E, x1 ¼ a sin θ cos φ; x2 ¼ b sin θ sin φ; x3 ¼ c cos θ; θ 2 ½0; π ; φ 2 ½0; 2π ; (7) the curvature function Gðθ; φÞ will have the form,
Fig. 4. Synthetic lightcurves with the brightness (Left) and relative brightness (Right), generated by different integration schemes.
4
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Fig. 5. Comparison of calculating the surface area of the Cellinoid shape by triangularization and Lebedev (Left) and comparison of computational efficiency for triangularization and Lebedev (Right).
Furthermore, the Hapke photometric parameters for different taxonomic types of asteroids are derived (cf., Li et al. (2015) and references therein). Although it is difficult and complicated to describe the light reflectance behaviors on the asteroid surface using parameterized models, the Hapke model can perform well in the photometric simulations (Jin et al., 2015; Fink, 2015). In this article, the Hapke scattering function Sðμ; μ0 ; αÞ derived from the corresponding Hapke bidirectional reflectance formula with 5 parameters, namely, average particle single scattering albedo (ω0 ), asymmetric factor (g), opposition surge amplitude (B0 ), opposition surge width (h), and roughness parameter (θ), has the following complex form:
may be a good choice in real application for an acceptable accuracy. In the right of Fig. 5, a comparison of the computational efficiency for the two schemes is shown. The computational cost for Lebedev degree N ¼ 74 is set to be the benchmark. This benchmark is only used to provide a reference for numerically comparing the computational cost of two schemes with different number of facets N. The first three points are not meaningful for comparison because the computer will designate the memory and balance the CPU task at the startup. The computational efficiency for each of the two schemes with different facet number N is estimated by the quotients of their computational cost to the benchmark. It is apparent that the Lebedev scheme has much higher computational efficiency compared to triangularization.
Sðμ; μ0 ; αÞ ¼
3. Photometric models
ω0 μ0 μ ½pðαÞð1 þ BðαÞÞ þ HðμÞHðμ0 Þ 1S ðμ; μ0 ; ψ Þ; 4π μ þ μ0 (10)
3.1. Description of models
where the shadow hiding opposition surge function BðαÞ, the average particle single-scattering phase function pðαÞ, and the AmbartsumianChandrasekhar function HðxÞ for multiple scattering are listed as follows,
Asteroid photometric models are generally adopted to simulate the light reflectance of the surface of asteroids, which is important in the photometric inversion process. The scattering function Sðμ; μ0 ; αÞ in the brightness integrals (4) and (9) can be derived simply from the bidirectional reflectance rðμ; μ0 ; αÞ with the multiplication of the term μ. Li et al. (2015) tabulated the forms of bidirectional reflectances for the commonly used photometric models, such as the Hapke, Lommel-Seeliger, Minnaert and Kaasalainen models.
BðαÞ ¼
B0 ; 1 α 1 þ tan h 2
pðαÞ ¼ HðxÞ ¼
3.1.1. Hapke model Hapke (2012) thoroughly described the theory of light reflectance, including the applications to planetary photometry, and the reflectance spectroscopy. Li et al. (2015) reviewed the advances in asteroid photometry from the perspectives of observations, laboratory experiments, and theory. Derived from radiative transfer theory, the Hapke model incorporates many physical parameters, such as surface roughness and porosity, as well as accounting for the opposition effect.
1 g2 1 þ 2g cos α þ g2
3=2 ;
1 þ 2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 þ 2 1 ω0 x
In particular, S ðμ; μ0 ; ψ Þ in (10) simulates the photometric effects of large-scale roughness with the roughness parameter (θ) given by equation (12.28) in Hapke (2012). Considering the complex form of the Hapke model in the photometric inversion process of asteroids, the comparison between the Hapke model and other commonly used empirical photometric models is implemented
Table 1 Parameters list for four different photometric models. Hapke
ω B0 h g θ
Lommel-Seeliger 0.031 3.9 0.11 0:32 20∘
Als β γ δ
Minnaert 0.037 0:0612 1:6166 104 9:7942 107
AM β γ δ k0 b
0.0130 0.0603 0:0013 1:7219 105 0.2495 0.0029
5
Kaasalainen A0 D k b c
0.0132 15.7858 8:3446 104 0.0646 0:3225
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
to find an appropriate model with a balance of computational efficiency and accuracy.
In total, there are 6 parameters in Minnaert photometric model, including AM ; k0 ; b, and β; γ; δ in the phase function.
3.1.2. Lommel-Seeliger model The Lommel-Seeliger photometric model only considers the single scattering with the form
3.1.4. Kaasalainen model By combining single (Lommel-Seeliger term SLS ) and multiple scattering (Lambert term SL ) with a weight factor c for the latter, Kaasalainen et al. (2001) presented a scattering function with the simple form,
Sðμ; μ0 ; αÞ ¼ ALS
μ0 μ f ðαÞ; μ þ μ0
(11)
Sðμ; μ0 ; αÞ ¼ f ðαÞ½SLS ðμ; μ0 Þ þ c SL ðμ; μ0 Þ μμ0 ¼ f ðαÞ þ cμμ0 : μ þ μ0
where ALS is a constant, and f ðαÞ is the phase function, for which Takir et al. (2015) adopted an exponential empirical function with the following form f ðαÞ ¼ e
βαþγ α2 þδα3
:
In addition, they reported a four-parameter form
α f ðαÞ ¼ A0 exp þ kα þ b; D
(12)
As Lommel-Seeliger model is applicable to dark, low-albedo C-type asteroids, Schr€ oder et al. (2017) applied this model to analyze the surface properties of Ceres from the Dawn mission images. Furthermore, Muinonen and Lumme (2015) derived the analytical formula for the brightness integral based on the ellipsoid shape model using Lommel-Seeliger photometric model. In total, there are 4 parameters in the Lommel-Seeliger photometric model, including ALS and β; γ; δ in the phase function.
tional to ðμμ0 Þ1 . In addition to the solar phase function presented by Takir et al. (2015): f ðαÞ ¼ 10
3.2. Numerical comparison
;
First, as listed in Table 1, the five parameters of the Hapke model for the B-type asteroid (101955) Bennu predicted by Takir et al. (2015) are adopted to generate six synthetic lightcurves. Without loss of generality, we have chosen the ellipsoid shape model with the semi-axes of ½5; 4; 3 and the solar phase angles from 5∘ to 30∘ with an increment of 5∘ to generate the six lightcurves, as shown in Fig. 6. Subsequently, the corresponding parameters for three different empirical photometric models, namely, the Lommel-Seeliger, Minnaert, and Kaasalainen models, are
the Minnaert model has the following form with the consideration of limb-darking behavior of the surface: Sðμ; μ0 ; αÞ ¼ AM f ðαÞðμ0 μÞk0 þbα ;
(15)
where A0 and D are the amplitude and scale length of the opposition effect, and k is the slope of phase curve. It should be noticed that the parameter b in (15) is normalized to unity at opposition for the relative brightness in Kaasalainen et al. (2001). Here, we want to keep the parameter a free variable for better fitting of the synthetic lightcurves in the following numerical experiments. Therefore, there are 5 parameters in the Kaasalainen photometric model: c; A0 ; D; k and b in the phase function. This photometric model is adopted in their lightcurve inversion method, which allows determination of asteroid's convex shape model and its rotation state properties. These shape models are usually available in the DAMIT database (Durech et al., 2010).
3.1.3. Minnaert model The Minnaert model, firstly presented by Minnaert (1941), is based on the Lambert law S ¼ AL μμ0 , in which the reflected power is propor-
2 3 βαþγ α2:5þδα
(14)
(13)
where AM is the so-called Minnaert albedo.
Fig. 6. Synthetic lightcurves based on Hapke model, and the fitted lightcurves from other photometric models. (Hapke: Black dots; Lommel-Seeliger: Blue squares; Minnaert: Red circles; Kaasalainen: Magenta diamonds). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
6
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Fig. 7. Synthetic lightcurves, generated by different photometric models in two different cases. (Left, Cellinoid: [7, 3, 5, 3, 3, 3], Pole: (40 ; 50 ), Solar phase: 30 , Earth: (0:28; 0:96; 0), Sun:(0:24; 0:97; 0); Right, Convex, Pole: (20 ; 30 ), Solar phase: 25 , Earth: (0:66; 0:75), Sun: (0:24; 0:97; 0)).
derived by fitting the six synthetic lightcurves based on the identical ellipsoid shape model and the observation geometries, as listed in Table 1. Finally, the derived parameters of the three numerical models are employed to generate the respective synthetic lightcurves, as shown in Fig. 6. The results confirm that the empirical photometric models of Minnaert and Kaasalainen can fit well to the Hapke model, especially for the synthetic lightcurves close to opposition, although there is little deviation among the fitted lightcurves from the Hapke model for the large phase angles. Nevertheless, the Lommel-Seeliger model cannot well fit the exact maximum and minimum of the synthetic lightcurves of the Hapke model, although the overall morphologies of the fitted lightcurves are consistent with the Hapke model. The primary reason for this result is that the scattering function of Lommel-Seeliger has only 4 parameters, whereas the Hapke and Kaasalainen models have 5 parameters, and the Minnaert model has 6 parameters. Furthermore, the so-called disk function in the Lommel-Seeliger model (11) has a fixed form, whereas the other empirical models having an adjustable parameter in the disk functions. To examine whether the derived parameters of three numerical photometric models can perform as well as the Hapke model in simulating the lightcurves, more synthetic lightcurves are generated in different cases, such as various shape models, pole orientations, and the larger solar phase angles. In Fig. 7 the synthetic lightcurves of four photometric models are generated respectively based on an arbitrary Cellinoid shape with six semi-axes of a1 ¼ 7; a2 ¼ 3; b1 ¼ 5; b2 ¼ 3; c1 ¼ 3; c2 ¼ 3 and the convex shape for asteroid (4179) Toutatis from Lu et al. (2017). In addition, the pole orientations in the ecliptic frame are (40∘ ; 50∘ ) and (20∘ ; 30∘ ) for the synthetic lightcurves of the two groups. Furthermore, Earth and Sun positions are randomly given at the asteroid-centric ecliptic frames, with
the constraint of the solar phase angle. As the pole orientations vary, at some point, the viewing geometries of the numerical simulations will be equivalent to the real situations. The results show that the performances of both the Minnaert model and the Kaasalainen model are consistent with the performance of the Hapke model, whereas the Lommel-Seeliger can roughly fit the other three models. Intuitively, the Minnaert model can obtain the best fitting to the synthetic lightcurves generated by Hapke model because its disk function has 1 more parameter than the Kaasalainen model. Following the comparison of the two different cases in Fig. 7, we extend the phase angle to 70∘ . As most of the main belt asteroids are observable from ground-based telescopes with the solar phase angle of less than 50∘ , the phase angle with larger than 70∘ is not essential although some Near-Earth-Asteroids (NEAs) can have a larger phase angle. In Fig. 8, synthetic lightcurves are shown for another two cases with the same shape models and larger solar phase angles, as well as at different viewing geometries. The results confirm that the empirical photometric models can well fit the case of large solar phase angle, although the deviations of the fitted lightcurves increase with a very small magnitude. Finally, the relative Mean Residual Sum of Squares (MRSS) value, MRSS ¼
2 1 X LCHapke LCmodel ; N LCHapke
(16)
is defined for quantitative comparison of the three empirical models to fit the synthetic lightcurves generated by the Hapke model. For a better estimation of the three models in the cases of various shape models, viewing geometries, and pole orientations, we implement the numerical simulations for the various cases as listed in Table 2, and the MRSS values are calculated for quantitative comparison of three numerical models.
Fig. 8. Synthetic lightcurves, generated by different photometric models in two different cases. (Left, Cellinoid: [7, 3, 5, 3, 3, 3], Pole: (40 ; 50 ), Solar phase: 50 , Earth: (0:59; 0:81; 0), Sun:(0:24; 0:97; 0); Right, Convex, Pole: (20 ; 30 ), Solar phase: 70 , Earth: (0:83; 0:56; 0), Sun:(0:24; 0:97; 0)). 7
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Table 2 Quantitative comparison for three different numerical photometric models.
C (7,3,5,3,3,3), P (40 ; 80 ), α ¼ 30 Convex1, P (20 ; 30 ), α ¼ 25 C (7,3,5,3,3,3), P (40∘ ; 80∘ ), α ¼ 50∘ Convex1, P (20∘ ; 30∘ ), α ¼ 70∘ C (4,5,6,5,6,7), P (30∘ ; 60∘ ), α ¼ 20∘ C (4,5,6,5,6,7), P (120∘ ; 85∘ ), α ¼ 60∘ C (4,5,6,5,6,7), P (60∘ ; 60∘ ), α ¼ 40∘ C (4,5,6,5,6,7), P (250∘ ; 25∘ ), α ¼ 5∘ C (4,5,6,5,6,7), P (150∘ ; 70∘ ), α ¼ 15∘ C (6,8,5,7,6,4), P (80∘ ; 80∘ ), α ¼ 35∘ C (6,8,5,7,6,4), P (60∘ ; 70∘ ), α ¼ 65∘ C (6,8,5,7,6,4), P (120∘ ; 75∘ ), α ¼ 20∘ C (6,8,5,7,6,4), P (80∘ ; 70∘ ), α ¼ 10∘ C (6,8,5,7,6,4), P (160∘ ; 80∘ ), α ¼ 55∘ C (7,5,6,6,4,3), P (55∘ ; 70∘ ), α ¼ 25∘ C (7,5,6,6,4,3), P (130∘ ; 40∘ ), α ¼ 40∘ C (7,5,6,6,4,3), P (200∘ ; 35∘ ), α ¼ 40∘ C (7,5,6,6,4,3), P (30∘ ; 80∘ ), α ¼ 30∘ C (7,5,6,6,4,3), P (40∘ ; 75∘ ), α ¼ 10∘ Convex2, P (60∘ ; 70∘ ), α ¼ 60∘ Convex2, P (50∘ ; 60∘ ), α ¼ 40∘ Convex3, P (60∘ ; 60∘ ), α ¼ 60∘ Convex3, P (40∘ ; 70∘ ), α ¼ 80∘
Convex4, P (10 ; 60 ), α ¼ 30 Convex4, P (100 ; 60 ), α ¼ 40
Lommel-Seeliger
Minnaert
Kaasalainen
269:1 105 363:7 105 405:9 105 926:9 105 34:5 105 145:5 105 47:6 105 26:8 105 47:4 105 98:8 105 131:2 105 71:2 105 56:3 105 135:3 105 37:0 105 325:2 105 356:4 105 19:2 105 19:5 105 30:5 105 27:5 105 199:4 105 526:0 105 117:9 105 150:2 105
34:0 105 99:1 105 113:0 105 806:0 105 5:1 105 88:7 105 18:2 105 1:1 105 5:7 105 16:2 105 61:8 105 7:4 105 4:0 105 60:3 105 9:0 105 160:6 105 177:3 105 1:2 105 1:6 105 7:5 105 8:5 105 75:1 105 348:7 105 25:7 105 36:8 105
76:6 105 146:9 105 170:4 105 908:7 105 13:5 105 95:5 105 23:2 105 7:8 105 18:6 105 39:9 105 108:4 105 35:7 105 20:8 105 86:5 105 17:2 105 200:9 105 191:0 105 2:2 105 6:4 105 11:3 105 16:9 105 127:7 105 485:3 105 42:0 105 57:1 105
Notations: C(7,3,5,3,3,3) for Cellinoid shape model with six semi-axes, Convex1 for the shape model of (4179) Toutatis, Convex2 for the shape model of (2) Pallas, Convex3 for the shape model of (6) Hebe, Convex4 for the shape model of (7) Iris, P for pole orientation in ecliptic frame, and α for solar phase angle.
inversion method based on the Cellinoid shape model, such as the Minnaert and Kaasalainen models. In particular, based on the Lommel-Seeliger photometric model, the analytical formula of the brightness integral based on the Cellinoid shape model may be deduced in the future similar to the finished work by Muinonen and Lumme (2015) based on the ellipsoid shape model.
Generally, the Minnaert model has the best performance, whereas the Lommel-Seeliger model with the simple scattering function can provide an approximate fitting to the Hapke model, compared to other models. 4. Discussions 4.1. Different taxonomic types
5. Conclusions As described by Li et al. (2015), asteroids with different taxonomic types have the corresponding Hapke parameters. The B-type asteroid (101955) Bennu is extremely dark, with the albedo ω ¼ 0:031. To better investigate the differences of photometric models in synthetic lightcurves, we perform analogous comparisons for S-type asteroids with moderately albedo. As shown in Fig. 9, synthetic lightcurves generated by the Hapke model with the mean parameters for S-type asteroids are fitted well by the other numerical models. The parameters for the four photometric models are listed in Table 3. Furthermore, we also simulated the cases of various shape models, pole orientations and viewing geometries. In Table 4, the various cases are tabulated, and the corresponding MRSS values, as defined in (16), are presented for quantitative comparison of three numerical photometric models. The results confirm that, numerically, the three models can fit well to the Hapke model.
In this article, Lebedev quadrature technique is applied to the lightcurve inversion process for substantially accelerating the bright simulation. Numerical comparisons of Lebedev quadrature applied in the volumes of unit sphere and the surface area of Cellinoid shape for the synthetic lightcurves were implemented to confirm its efficiency. The inversion process based on Lebedev acceleration was described in details. In addition, we released the Matlab codes of the Cellinoid shape model on GitHub; everyone can download the codes for free from the website.1 To better compare the different photometric models used in simulating the scattering laws of lightcurves inversion, the synthetic lightcurves were generated from the Hapke model based on the parameters of the B-type asteroid (101955) Bennu, and the corresponding parameters of the other three numerical models of Lommel-Seeliger, Minnaert and Kaasalainen were derived by fitting the synthetic lightcurves. Furthermore, the synthetic lightcurves of the four photometric models were compared for various cases, covering different viewing geometries, different shape models, and different pole orientations. The results confirm that the scattering function will not affect the morphologies of the synthetic lightcurves as much as the shape variations, which is a finding that has already been extensively accepted. In addition, the numerical photometric models with simple expressions can be adopted in the inversion method to substitute the complex photometric models, such as in the Hapke model.
4.2. Shape variations and photometric models We numerically compared the Hapke photometric model and other numerical models in different cases. The results suggest that the morphologies of synthetic lightcurves generated from the derived parameters for different photometric models are very similar. In other words, the different photometric models will not change the morphologies of lightcurves significantly, whereas the variations of the lightcurves morphologies are primarily caused by shape variations. These results are consistent with the conclusions made by Kaasalainen et al. (2001) and Karttunen and Bowell (1989). This conclusion implies that we can employ the numerical photometric model with the simple form in the
1
8
https://github.com/XPLU/CellinoidModel.git.
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Fig. 9. Simulations for S-type asteroids in two different cases (Left, Cellinoid: [7, 3, 5, 3, 3, 3], Pole: (120 ; 20 ), Solar phase: 30 Earth: (0:28; 0:96; 0), Sun:(0:24; 0:97; 0); Right, Convex, Pole: (40 ; 60 ), Solar phase: 55 , Earth: (0:66; 0:75; 0), Sun:(0:24; 0:97; 0)).
Table 3 Parameters of S-type asteroid for four different photometric models. Hapke
ω B0 h g θ
Lommel-Seeliger 0.23 1.6 0.08 0:27 20∘
Als β γ δ
Minnaert 0.0469 0:0612 1:6171 104 9:8702 107
AM β γ δ k0 b
0.0163 0:0091 3:1507 105 1:8824 106 0.2495 0.0031
Kaasalainen A0 D k b c
0.0123 14.4880 8:1379 104 0.0649 0:3192
Table 4 Quantitative comparison of S-type asteroids for three different numerical photometric models.
C (7,3,5,3,3,3), P (40∘ ; 80∘ ), α ¼ 30∘ Convex1, P (20∘ ; 30∘ ), α ¼ 25∘ C (7,3,5,3,3,3), P (40∘ ; 80∘ ), α ¼ 50∘ Convex1, P (20∘ ; 30∘ ), α ¼ 70∘ C (4,5,6,5,6,7), P (30∘ ; 60∘ ), α ¼ 20∘ C (4,5,6,5,6,7), P (120∘ ; 85∘ ), α ¼ 60∘ C (4,5,6,5,6,7), P (60∘ ; 60∘ ), α ¼ 40∘ C (4,5,6,5,6,7), P (250∘ ; 25∘ ), α ¼ 5∘ C (4,5,6,5,6,7), P (150∘ ; 70∘ ), α ¼ 15∘ C (6,8,5,7,6,4), P (80∘ ; 80∘ ), α ¼ 35∘ C (6,8,5,7,6,4), P (60∘ ; 70∘ ), α ¼ 65∘ C (6,8,5,7,6,4), P (120∘ ; 75∘ ), α ¼ 20∘ C (6,8,5,7,6,4), P (80∘ ; 70∘ ), α ¼ 10∘ C (6,8,5,7,6,4), P (160∘ ; 80∘ ), α ¼ 55∘ C (7,5,6,6,4,3), P (55∘ ; 70∘ ), α ¼ 25∘ C (7,5,6,6,4,3), P (130∘ ; 40∘ ), α ¼ 40∘ C (7,5,6,6,4,3), P (200∘ ; 35∘ ), α ¼ 40∘ C (7,5,6,6,4,3), P (30∘ ; 80∘ ), α ¼ 30∘ C (7,5,6,6,4,3), P (40∘ ; 75∘ ), α ¼ 10∘ Convex2, P (60∘ ; 70∘ ), α ¼ 60∘ Convex2, P (50∘ ; 60∘ ), α ¼ 40∘ Convex3, P (60∘ ; 60∘ ), α ¼ 60∘ Convex3, P (40∘ ; 70∘ ), α ¼ 80∘ Convex4, P (10∘ ; 60∘ ), α ¼ 30∘ Convex4, P (100∘ ; 60∘ ), α ¼ 40∘
Lommel-Seeliger
Minnaert
Kaasalainen
257:7 105 351:8 105 391:0 105 919:3 105 33:2 105 142:1 105 46:3 105 25:8 105 45:4 105 94:3 105 126:7 105 67:7 105 53:6 105 131:9 105 35:8 105 317:9 105 348:6 105 18:6 105 18:9 105 28:7 105 26:7 105 194:1 105 520:8 105 113:9 105 145:6 105
33:7 105 98:5 105 114:1 105 800:8 105 5:1 105 88:6 105 18:1 105 1:3 105 5:8 105 16:1 105 62:3 105 7:5 105 4:2 105 60:6 105 8:9 105 160:0 105 176:5 105 1:2 105 1:6 105 7:7 105 8:5 105 76:1 105 352:0 105 25:5 105 36:7 105
77:1 105 146:0 105 171:1 105 906:9 105 13:4 105 94:8 105 23:1 105 7:6 105 18:3 105 40:3 105 110:2 105 35:7 105 20:5 105 86:5 105 17:2 105 199:8 105 189:4 105 2:2 105 6:3 105 11:3 105 16:8 105 127:7 105 485:6 105 41:8 105 56:9 105
Notations: C(7,3,5,3,3,3) for Cellinoid shape model with six semi-axes, Convex1 for the shape model of (4179) Toutatis, Convex2 for the shape model of (2) Pallas, Convex3 for the shape model of (6) Hebe, Convex4 for the shape model of (7) Iris, P for pole orientation in ecliptic frame, and α for solar phase angle.
Acknowledgements
Although the Lebedev technique can largely accelerate the computational efficiency in the lightcurves inversion process based on the Cellinoid shape model, it is not comparable to the analytical formula of the brightness integral. In the future, we plan to focus on this topic, following an approach similar to that used for the ellipsoid shape model by Muinonen and Lumme (2015).
We thank the anonymous reviewers for their professional comments and significant contribution on the scattering function. This work is funded under the grant NO. 095/2013/A3 and 039/2013/A2 from the Science and Technology Development Fund, MSAR. The project is supported partly by the Key Laboratory of Planetary Sciences, Purple
9
X.-P. Lu et al.
Planetary and Space Science xxx (2017) 1–10
Mountain Observatory, Chinese Academy of Sciences. W.-H. Ip is supported by MSAR Science and Technology Fund (Project No. 017/2014/ A1) and NSC 101-2111-M-008-016.
Lauretta, D.S., Balram-Knutson, S.S., Beshore, E., Boynton, W.V., Drouet dAubigny, C., et al., 2017. OSIRIS-REx: sample return from asteroid (101955) Bennu. Space Sci. Rev. 1, 1–60. Lebedev, V., Laikov, D., 1999. A quadrature formula for the sphere of the 131st algebraic order of accuracy. Dokl. Math. 59, 477–481. Li, J.Y., Helfenstein, P., Buratti, B.J., Takir, D., Clark, B.E., 2015. Asteroid photometry. In: Michel, P., et al. (Eds.), Asteroids IV. Univ. of Arizona, Tucson, pp. 129–150. Lu, X.P., Ip, W.H., 2015. Cellinoid shape model for multiple light curves. Planet. Sp. Sci. 108, 31–40. Lu, X.P., Zhao, H.B., You, Z., 2013. A fast ellipsoid model for asteroids inverted from lightcurves. Res. Astron. Astronphysics 13, 465–472. Lu, X.P., Cellino, A., Hestroffer, D., Ip, W.H., 2016. Cellinoid shape model for Hipparcos data. Icarus 267, 24–33. Lu, X.P., Ip, W.H., Huang, X.J., Zhao, H.B., 2017. Analysis for Cellinoid shape model in inverse process from lightcurves. Planet. Sp. Sci. 135, 74–85. Michel, P., DeMeo, F.E., Bottke, W.F., 2015. Asteroids:Recent advances and new perspectives. In: Michel, P., et al. (Eds.), Asteroids IV. Univ. of Arizona, Tucson, pp. 3–10. Minnaert, M., 1941. The reciprocity principle in lunar photometry. Astronphys. J. 93, 403–410. Muinonen, K., Lumme, K., 2015. Disk-integrated brightness of a Lommel-Seeliger scattering ellipsoidal asteroid. A&A 584, A23. Muinonen, K., Wilkman, O., Cellino, A., Wang, X., Wang, Y., 2015. Asteroid lightcurve inversion with Lommel–Seeliger ellipsoids. Planet. Sp. Sci. 118, 227–241. Schr€ oder, S.E., Mottola, S., Carsenty, U., Ciarniello, M., Jaumann, R., Li, J.Y., Longobardo, A., Palmer, E., Pieters, C., Preusker, F., Raymond, C.A., Russell, C.T., 2017. Resolved spectrophotometric properties of the Ceres surface from Dawn framing camera images. Icarus 288, 201–225. Surdej, A., Surdej, J., 1978. Asteroid lightcurves simulated by the rotation of a three-axes ellipsoid model. A&A 66, 31–36. Takir, D., Clark, B.E., d'Aubigny, C.D., Hergenrother, C.W., 2015. Photometric models of disk-integrated observations of the OSIRIS-REx target Asteroid (101955) Bennu. Icarus 252, 393–399. Tanga, P., Mignard, F., Oro, A.D., Muinonen, K., Pauwels, T., Thuillot, W., Berthier, J., Cellino, A., Hestroffer, D., Petit, J.M., Carry, B., David, P., Delbo, M., Fedorets, G., Galluccio, L., Granvik, M., Ordenovic, C., Pentik€ainen, H., 2016. The daily processing of asteroid observations by Gaia. Planet. Sp. Sci. 123, 87–94. Tsuda, Y., Yoshikawa, M., Abe, M., Minamino, H., Nakazawa, S., 2013. System design of the Hayabusa 2—asteroid sample return mission to 1999 JU3. Acta Astronaut. 91, 356–362. Wright, E.L., Eisenhardt, P.R.M., Mainzer, A.K., Ressler, M.E., Cutri, R.M., Jarrett, T., Kirkpatrick, J.D., Padgett, D., McMillan, R.S., Skrutskie, M., Stanford, S.A., Cohen, M., Walker, R.G., Mather, J.C., Leisawitz D III, T.N.G., McLean, I., Benford, D., Lonsdale, C.J., Blain, A., Mendez, B., Irace, W.R., Duval, V., Liu, F., Royer, D., Heinrichsen, I., Howard, J., Shannon, M., Kendall, M., Walsh, A.L., Larsen, M., Cardon, J.G., Schick, S., Schwalm, M., Abid, M., Fabinsky, B., Naes, L., Tsai, C.W., 2010. The wide-field infrared survey explorer (wise): mission description and initial on-orbit performance. Astron. J. 140, 1868–1881.
References Cellino, A., Dell'Oro, A., 2012. The derivation of asteroid physical properties from Gaia observations. Planet. Sp. Sci. 73, 52–55. Cellino, A., Zappal a, V., Farinella, P., 1989. Asteroid shapes and lightcurve morphology. Icarus 78, 298–310. Cellino, A., Muinonen, K., Hestroffer, D., Carbognani, A., 2015. Inversion of sparse photometric data of asteroids using triaxial ellipsoid shape models and a Lommel–Seeliger scattering law. Planet. Sp. Sci. 118, 221–226. Demeo, F.E., Carry, B., 2014. Solar System evolution from compositional mapping of the asteroid belt. Nature 505, 629–634. Drummond, J.D., Conrad, A., Merline, W.J., Carry, B., Chapman, C.R., Weaver, H.a., Tamblyn, P.M., Christou, J.C., Dumas, C., 2010. Physical Properties of the ESA Rosetta target asteroid (21) Lutetia I. The triaxial ellipsoid dimensions, rotational pole, and bulk density. A&A 523, A93. Durech, J., Sidorin, V., Kaasalainen, M., 2010. DAMIT: a database of asteroid models. A&A 513, A46. Fink, U., 2015. Considerations regarding the colors and low surface albedo of comets using the Hapke methodology. Icarus 252, 32–38. Gaia Collaboration, Brown, A.G.A., Vallenari, A., Prusti, T., de Bruijne, J.H.J., Mignard, F., et al., 2016. Gaia data release 1-summary of the astrometric, photometric, and survey properties. A&A 595, A2. Hapke, B., 2012. Theory of Reflectance and Emittance Spectroscopy, second ed. Wiley Online Library. Jedicke, R., Granvik, M., Micheli, M., Ryan, E., Spahr, T., Yeomans, D.K., 2015. Surveys, astrometric follow-up, and population statistics. In: Michel, P., et al. (Eds.), Asteroids IV. Univ. of Arizona, Tucson, pp. 795–813. Jin, W., Zhang, H., Yuan, Y., Yang, Y., 2015. In situ optical measurements of Chang’E-3 landing site in Mare Imbrium: 2. Photometric properties of the regolith. Geophys. Res. Lett. 42, 8312–8319. Kaasalainen, M., Lamberg, L., 1992. Interpretation of lightcurves of atmosphereless bodies. II-Practical aspects of inversion. A&A 259, 333–340. Kaasalainen, M., Torppa, J., 2001. Optimization methods for asteroid lightcurve inversion I. Shape determination. Icarus 153, 24–36. Kaasalainen, M., Lamberg, L., Lumme, K., Bowell, E., 1992. Interpretation of lightcurves of atmosphereless bodies. I-General theory and new inversion schemes. A&A 259, 318–332. Kaasalainen, M., Torppa, J., Muinonen, K., 2001. Optimization methods for asteroid lightcurve inversion II. The complete inverse problem. Icarus 153, 37–51. Kaasalainen, M., Lu, X.P., V€ anttinen, A., 2012. Optimal computation of brightness integrals parametrized on the unit sphere. A&A 539, A96. Karttunen, H., Bowell, E., 1989. Modelling asteroid brightness variations. II-The interpretability of light curves and phase curves. A&A 208, 320–326. Lamberg, L., Kaasalainen, M., 2001. Numerical solution of the minkowski problem. J. CoAM 137, 213–227.
10