Errata: Corrigendum Volume 50, Issue 12, 2142, Article first published online: 17 November 2015
We have simulated asteroid lightcurves for simple shape models using a realistic surface scattering law. The scattering law includes a shadowing function computed with numerical ray-tracing. We computed lightcurves in a variety of illumination geometries for both the traditional Lommel–Seeliger law and our seminumerical law. We observe a shift in the rotational phase of the lightcurves, which depends on the parameters of the scattering law as well as the illumination geometry and the direction of the spin axis of the asteroid. This phase shift is always zero at opposition, and can be as large as 10° for illumination geometries typical for Main Belt asteroids. The phase shift has implications on the accuracy of other results which are based on asteroid lightcurve analysis, such as spin-state or shape determination.
Diffuse reflection of light from a surface element on an atmosphereless solar system object is described by the reflection coefficient. The coefficient relates the incident flux density to the outgoing specific intensity. Alternatively, the relation can be given by using the bi-directional reflectance distribution function or the scattering law. The three alternative ways differ from each other only by certain simple trigonometric expressions that depend on the cosines of the angles of incidence and emergence as measured from the outward normal direction of the surface element.
Reflection coefficients of varying rigor have been put forward for photometric analyses of asteroid surfaces. In lightcurve inversion for asteroid rotation periods, pole orientations, and shapes, it is customary to utilize a linear combination of the Lommel–Seeliger (LS) and Lambert reflection coefficients (L; Kaasalainen et al. 2001). The former is a first-order multiple-scattering approximation of the radiative-transfer equation and thus applicable to dark particulate surfaces. The latter is typically taken as a model for bright particulate surfaces, although it does not follow from radiative transfer. For a review on asteroid lightcurve inversion, the reader is referred to Kaasalainen et al. (2002).
However, the LS–L reflection coefficient, as usually applied in lightcurve inversion, does not account for, e.g., angle-dependent shadowing effects caused by the particulate character of the surface element and the rough interface between the surface element and free space. These shadowing effects are not fully accounted for in the popular models by Lumme and Bowell (1981), Hapke (1986), and Shkuratov (1999); whereas the effects are accounted for, to varying degree of completeness, in the works by Lumme et al. (1990), Peltoniemi and Lumme (1992), Parviainen and Muinonen (2009), Muinonen et al. (2011), and Wilkman (in preparation).
The importance of the direct problem for asteroid lightcurve computation has been recognized already by, e.g., Muinonen (1998), Kaasalainen and Torppa (2001) and Kaasalainen et al. (2001). An in-depth study of the effects due to a physical reflection coefficient has, however, never been carried out. The present work sets the stage for such a study through an analysis of scattering effects in lightcurves of dark particulate media.
In the Theoretical Methods section, we describe the theoretical framework of the present study. The Numerical Methods section includes a review of the numerical methods required for the computation of lightcurve phase shifts. The Results section describes the outcome for a selection of realistic illumination and observation geometries, followed by a discussion section. Conclusions are presented in the last section.
The illumination geometry is defined by the unit vectors from the asteroid toward the Sun and the observer. For lightcurve studies, also the pole direction of the asteroid is relevant. These vectors are usually given in ecliptic latitude and longitude. We denote the pole position by (β, λ).
In a reference frame centered on the asteroid, the illumination geometry can also be characterized with three angles (after Muinonen 1998): the angles between the spin axis of the asteroid and the incident and emergent directions, and the longitudinal difference between the directions along the asteroid's equatorial plane. This last angle, which we denote Δφ, is of special interest in the present work. This angle can be written as a function of the direction of the spin axis and the solar phase angle. This angle should not be confused with the local azimuth angle φ, defined below.
The local illumination geometry at any given point on the surface is characterized by three angles. The angle of incidence θi and the angle of emergence θe are the angles between the surface normal and the vectors toward the light source and the observer. The azimuth angle φ is the angle between the two planes defined by the surface normal and the incidence/emergence vectors. In most situations, only the cosines of the incidence and emergence angles are needed. These are denoted as μ0 = cos θi and μ = cos θe.
The intensity of light scattered by the surface can be given in terms of the incident flux density πF0 and the reflection coefficient
Simple and commonly used reflection coefficients are the Lambertian and Lommel–Seeliger reflection coefficients. For Lambertian scattering RL = const., and for Lommel–Seeliger scattering RLS ∝ 1/(μ + μ0). For asteroid lightcurves, a linear combination of Lambertian and Lommel–Seeliger scattering is also often used.
We have studied a semianalytical scattering law, which has the Lommel-Seeliger form, and additionally takes into account the effects of shadowing as well as a phase function:
where P(α) is the phase function and S(μ, μ0, φ) is the shadowing function. The shadowing function describes the effects of mutual shadowing of particles in a medium composed of round particles. It is computed numerically and is described in more detail in the next section.
This model has been previously applied to photometric studies of the lunar surface (Muinonen et al. 2011; Wilkman, in preparation) and is able to fit photometric observations of lunar mare regions well. In this article, we call it the lunar mare scattering law.
Our asteroid models are represented as 3-D surfaces discretized into triangular facets. We limit ourselves to convex shape models for computational simplicity. This way, we do not need to worry about parts of the model surface obstructing other parts, and are able to quickly calculate the projected area and the local illumination geometry for all visible facets.
As our main shape model we used a prolate spheroid rotating about its short axis. The most elongated model spheroid had an axis ratio of 2:1 while the least elongated has a ratio of 11:10. The spheroid surface has been discretized into 3200 triangular facets. Verification with a finer discretization made no significant difference in the results.
The shadowing functions for particulate media have been computed through a ray-tracing simulation. Our regolith model consists of a simulation volume packed with a large number of spherical particles. The packing density of the particles, as well as the topography of the surface, can be controlled within a realistic range.
The volume is first filled with particles at the desired packing density by a sphere-dropping algorithm. Then, the surface topography is generated by clipping the volume with a random surface, and all the spheres above the clipping surface are removed. The statistics of the random field can be chosen in several different ways. We choose to use a fractional Brownian motion (fBm) surface, whose statistical topography is controlled by two parameters, the Hurst exponent H, which controls the horizontal correlation, and the height variance or amplitude σ. See Parviainen and Muinonen (2007, 2009) for comparisons between Gaussian roughness and fBm roughness for both solid and particulate surfaces.
The values of the shadowing function are computed as a function of the local illumination geometry (μ0, μ, φ). The function is sampled in an efficient hemispherical grid over μ and φ for several values of μ0 (Parviainen and Muinonen 2009). For each sampled geometry, a large number of random points in the model regolith are selected. From each point, rays are then traced to the directions of the light source and the observer. If both rays extend to infinity without intersecting other particles of the medium, the point is both lit and visible. The value of the shadowing function is then estimated as the ratio of lit and visible points to the total number of points tested.
This computation of the shadowing function samples for given surface roughness parameters is expensive, taking up to several hours of computation time, but it only needs to be carried out once. In the scattering law computations, the value of the shadowing function is linearly interpolated between the nearest sampled values.
Three different shadowing functions for different sets of surface roughness parameters were computed.1 The roughest surface model has a packing density of ν = 0.15 and topography parameters H = 0.5, σ = 0.1. The smoothest has a packing density of ν = 0.55 and perfectly planar topography. The third one is roughly half-way between the other two in all parameters. Figure 1 illustrates the shadowing function for the smoothest and roughest surface.
The lightcurves were computed by rotating the asteroid shape model under known illumination conditions. The illumination geometry for each individual facet has been computed and the contribution of the chosen scattering law was summed over all visible facets:
where Ai is the area of the ith facet and the summation is performed over all facets i for which both μ > 0 and μ0 > 0.
The lightcurves were computed in steps of 2° of rotation, so that each lightcurve consists of 181 points. In the zero rotation position, the longest axis of the body is in the direction of the observer and the object's cross section is at its minimum.
In this study we only consider the shapes and rotational phase of the lightcurves. Therefore, the brightnesses are converted to magnitudes and the mean of each lightcurve is subtracted from it. This transformation from linear brightness units to logarithmic magnitudes allows us to disregard all multiplicative factors to the brightness which are constant over one lightcurve, such as the phase function, the albedo and absolute size of the asteroid, and the solar flux density.
We computed lightcurves for our model shapes in a variety of observational geometries. In the simplest case, the spin axis of the model asteroid is perpendicular to the Sun-asteroid-observer plane, and therefore the longitudinal angle Δφ is equal to the solar phase angle α. The following results are computed for this geometry.
Figure 2 shows lightcurves of the most elongated spheroidal body with the Lommel–Seeliger and the lunar mare scattering laws at increasing solar phase angle. As the phase angle increases, the lightcurve amplitude increases. This effect is known as the amplitude–phase relationship. Figure 3 shows the lightcurve amplitude in the same simulation. The overall shape and magnitude of the amplitude-phase relationship agree with previous studies (Zappala et al. 1990; Gutiérrez et al. 2006). The amplitude with the lunar mare scattering law is slightly higher than with that with the plain Lommel–Seeliger law.
The most interesting feature, however, is a shift in rotational phase of the lightcurve with the lunar mare scattering law relative to the regular Lommel-Seeliger law. The brightness extrema of the Lommel-Seeliger lightcurve correspond almost exactly to the minimum and maximum of visible and lit area. The lunar mare law lightcurve, however, is shifted in rotational phase as a function of the longitudinal angle Δφ.
Figure 4 shows the amount of phase shift as a function of Δφ, for the three different shadowing functions. At Δφ = 0, zero solar phase angle, there is no shifting. The shift increases steeply at low angles, turning until reaching a maximum between 45 and 60°. The region of steepest change coincides with the phase angles most relevant for Main Belt asteroids.
The amount of phase shift is also, to a lesser degree, a function of the surface roughness parameters, but the roughness does not seem to affect the amplitude-phase relationship. On the other hand, the elongation of the shape affects the lightcurve amplitude strongly, but does not significantly change the phase shift.
The above case of Δφ = α is true only when the spin axis is perpendicular to the Sun-asteroid-observer plane. Computations in a number of more general observational geometries show that the phase shift is indeed mainly a function of Δφ. In the general case, the angle Δφ, and therefore the phase shift, are functions of the solar phase angle α and the direction of the asteroid's spin axis (βλ). For Main Belt asteroids, which are observed at fairly low solar phase angles, the difference between the solar phase angle and the longitudinal angle is generally small when the spin axis is more or less vertical (i.e., perpendicular to the observational plane). The pole distribution of observed asteroids also shows that fewer asteroids have poles close to the ecliptic plane (Pravec et al. 2002).
Figure 5 shows the phase shift as a function of solar phase angle in five situations where the spin axis is tilted 45° from the vertical, and the phase angle is no longer equal to the longitudinal angle. The phase angle behavior of the phase shift is still quite similar to the previous case, but its magnitude is strongly dependent on the spin axis direction. As the axis points farther away from the solar direction, the phase shift is smaller.
If the sign of the longitudinal angle or, equivalently, the direction of rotation is changed, the phase shift changes sign accordingly. Therefore, as an asteroid passes opposition during a single apparition, the phase shift will first approach zero and then grow again in the other direction.
For a given lightcurve, the only difference between the Lommel-Seeliger and the lunar mare scattering laws is the shadowing function. This shows that the phase shift is entirely due to the asymmetric effects of shadowing. Figure 6 shows an asteroid model rendered with both scattering laws.
The main effect of the shadowing function is that facets with high incidence angle become darker away from the opposition. When the solar longitudinal angle increases, this causes the parts of the asteroid away from the solar direction to darken. The surface roughness affects the steepness of this darkening.
The observed phase shift has implications for studies of asteroid rotation. If the lightcurve rotational phase depends on both the solar phase angle and the direction of the spin axis, a search for the rotational state of the asteroid will invariably have an additional systematic error when the set of observations cover several different geometries.
Asteroid shape inversion requires lightcurves observed at different solar phase angles to produce well-constrained solutions (Kaasalainen et al. 1992). Phase-angle-dependent uncertainty in the rotational phase of the asteroid will translate into uncertainty of the shape modeling. Exploring the actual effect of such a phase shift on asteroid spin-shape modeling is beyond our current scope.
The phase shift also has some implications for the detection of effects such as YORP (Bottke et al. 2002) that change the spin rate. Errors in spin rates due to changes in the rotational phase could lead to spurious differences in spin rates.
With highly accurate astrometric observations, such as those from the upcoming Gaia mission, the photocenter effect, the offset between the center of mass of the asteroid and the observed center of its disk, becomes relevant for the study of asteroid orbits. The shadowing function also affects the location of the photocenter, but our simulations indicate that the difference between the Lommel-Seeliger scattering law and our lunar mare scattering law is very small for Main Belt asteroids.
Multiple scattering between surface volume elements may change the shadowing effects, but is beyond the scope of our present work. Exploring the effects of more detailed light scattering models is planned for the future. Laboratory and field measurements of particulate mineral surfaces show effects dependent on the illumination geometry similar to our numerical shadowing functions (Peltoniemi et al. 2007). As the phase shift is entirely due to the shape of the shadowing function, such shapes seen in real surfaces suggest the existence of the phenomenon.
A meticulous study of existing asteroid lightcurves at different solar phase angles is required to demonstrate the existence and magnitude of the phase shift. A set of lightcurves as accurate as possible, in as wide a range of solar phase angles as possible, for an individual asteroid would be ideal for investigating the lightcurve phase shift.
We have simulated asteroid lightcurves for simple shape models in a variety of illumination geometries, emphasizing the choice of scattering law. With a realistic scattering law, we observed a shift in the rotational phase of the lightcurves which depends on the surface properties and the observational geometry. The implications of this phase shift on various applications of lightcurve analysis warrant more detailed studies. Future work includes comparisons to those established scattering models which take into account roughness. The existing inventory of lightcurves also needs to be examined for signs of such an effect.
We thank professor emeritus Kari Lumme for his comments. The work is supported by the Academy of Finland (contract 127461).
Dr. Dina Prialnik
The shadowing function data are available from the corresponding author on request.