We have reached a point where the theory accounts for many of the phenomena
expected to be observed and, in one case, we have been able to verify the approach at
least in the single scattering approximation. There are however several issues. The
resulting theory has eight free parameters (which can be reduced if some assumptions can be made). Four of these parameters (the mean slope angle, the single
scattering albedo, and the H-G parameters) have clear physical meanings with
respect to the material under investigation. The primary question is whether the
problem can be inverted to derive the properties of the surface from photometric
measurements. It is important to appreciate here that fitting photometric data from
multiple observations of a target surface is not the problem. The Finnish astronomer
and photometry expert, Kari Lumme, is reputed to have said in this context, “give me
7 parameters and I can fit an elephant”. (This was almost certainly a deliberate
mis-quotation of John von Neumann who said, “With four parameters I can fit an
elephant, and with five I can make him wiggle his trunk.” as attributed by Enrico
Fermi—students might be amused to learn that some mathematicians have actually
tried to do this—successfully—with complex numbers). An example of the issues is
given in Gunderson et al. (2006). Here, a highly accurate goniometer was used under
optimum conditions to acquire photometric data of a surface from many angles.
When fitting the data, correlation coefficients were determined and it was shown that
there are correlations between several of the parameters (e.g. between ω, ξ, and c)
basically confirming the trends first reported by Helfenstein and Veverka (1987).
Subsequently, Shepard and Helfenstein (2007) (who followed this further in 2011—
Shepard and Helfenstein (2011), Helfenstein and Shepard (2011)) have looked
closely at the relationship between the parameters and physical quantities and have
assessed the mutual coupling between the parameters to provide better error
determination.
At present, the inversion of Hapke parameters derived from remote-sensing data
to produce meaningful material properties still seems challenging. One can also ask
why. The fundamental assumption is that scatterers are sufficiently dispersed such
that each particle interacts with the incoming beam in isolation. This might be valid
if the particles were even only a few wavelengths apart. However, the particles are in
contact and this affects the interaction of the wave.
One of the results illustrated later in Fig. 4.7 shows the single scattering albedo
computed from Mie theory. Two points are relevant here. Firstly, the single scattering albedo is particle size dependent. The sizes of particles on cometary surfaces are
highly unlikely to be a single size and the size distribution is poorly constrained. The
size dependency is not treated in the Hapke formulation. (The asymmetry parameter
is also size dependent.) Secondly, if we assume that the particles are typically larger
than a few microns, then the single scattering albedo in Fig. 4.7 is an order of
magnitude larger than the results for cometary surfaces derived so far and shown in
Table 2.1. The large particle value of the single scattering albedo is influenced by the
material properties (through the complex refractive index) but it is generally larger
than 0.5. The reason is that diffraction plays a very strong role in determining the
single scattering albedo and this is not influenced by the material. Hence, what is
being derived is not the single particle single scattering albedo—despite the fact that
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2 The Nucleus
expected to be observed and, in one case, we have been able to verify the approach at
least in the single scattering approximation. There are however several issues. The
resulting theory has eight free parameters (which can be reduced if some assumptions can be made). Four of these parameters (the mean slope angle, the single
scattering albedo, and the H-G parameters) have clear physical meanings with
respect to the material under investigation. The primary question is whether the
problem can be inverted to derive the properties of the surface from photometric
measurements. It is important to appreciate here that fitting photometric data from
multiple observations of a target surface is not the problem. The Finnish astronomer
and photometry expert, Kari Lumme, is reputed to have said in this context, “give me
7 parameters and I can fit an elephant”. (This was almost certainly a deliberate
mis-quotation of John von Neumann who said, “With four parameters I can fit an
elephant, and with five I can make him wiggle his trunk.” as attributed by Enrico
Fermi—students might be amused to learn that some mathematicians have actually
tried to do this—successfully—with complex numbers). An example of the issues is
given in Gunderson et al. (2006). Here, a highly accurate goniometer was used under
optimum conditions to acquire photometric data of a surface from many angles.
When fitting the data, correlation coefficients were determined and it was shown that
there are correlations between several of the parameters (e.g. between ω, ξ, and c)
basically confirming the trends first reported by Helfenstein and Veverka (1987).
Subsequently, Shepard and Helfenstein (2007) (who followed this further in 2011—
Shepard and Helfenstein (2011), Helfenstein and Shepard (2011)) have looked
closely at the relationship between the parameters and physical quantities and have
assessed the mutual coupling between the parameters to provide better error
determination.
At present, the inversion of Hapke parameters derived from remote-sensing data
to produce meaningful material properties still seems challenging. One can also ask
why. The fundamental assumption is that scatterers are sufficiently dispersed such
that each particle interacts with the incoming beam in isolation. This might be valid
if the particles were even only a few wavelengths apart. However, the particles are in
contact and this affects the interaction of the wave.
One of the results illustrated later in Fig. 4.7 shows the single scattering albedo
computed from Mie theory. Two points are relevant here. Firstly, the single scattering albedo is particle size dependent. The sizes of particles on cometary surfaces are
highly unlikely to be a single size and the size distribution is poorly constrained. The
size dependency is not treated in the Hapke formulation. (The asymmetry parameter
is also size dependent.) Secondly, if we assume that the particles are typically larger
than a few microns, then the single scattering albedo in Fig. 4.7 is an order of
magnitude larger than the results for cometary surfaces derived so far and shown in
Table 2.1. The large particle value of the single scattering albedo is influenced by the
material properties (through the complex refractive index) but it is generally larger
than 0.5. The reason is that diffraction plays a very strong role in determining the
single scattering albedo and this is not influenced by the material. Hence, what is
being derived is not the single particle single scattering albedo—despite the fact that
62
2 The Nucleus