A Model for the Polarimetric and Photometric Characteristics of the Moon
at Moderate Phase Angles Based on Realistic Assumptions on Regolith Microstructure

Yu. I. Velikodsky, V. V. Korokhin, and L.A. Akimov

 

Astronomical Institute of Kharkov University, 35, Sumskaya Ul., Kharkov 61022, Ukraine,
 e-mail: velikodsky@astron.kharkov.ua

web: http://www.univer.kharkov.ua/astron/dslpp/moon/polar/

 

Abstract

 

There are known two potential mechanisms for explaining positive polarization that is observed for atmosphereless celestial bodies at large phase angles. The first one, Fresnel's reflection, may take place due to existing of large (in comparison with wavelength) smooth surfaces in regolith (glasses). The second one, Rayleigh's scattering, supposes the presence of particles with sizes smaller than wavelength, i.e. with submicron sizes. It is known that typical size of regolith particles is about tens of micrometers. The particles are mainly aggregates of smaller grains of micron- submicron sizes. Such particles cannot produce a pure Rayleigh polarization. On the other hand, Fresnel's reflection yields too large phase angles of the positive polarization maximum and cannot explain this effect by oneself.

Taking into account that the principal contribution to scattering (and, hence, to positive polarization) may produce small particles with subwavelength sizes we propose a combinative heuristic model using the Rayleigh-Gans approximation, Fresnel's reflection on large surfaces, shadow-hiding effect, and multiple scattering on micro- and macro-scales. The model shows a good agreement with observational data for the Moon for both brightness and polarization degree phase dependences (Figs. 5 and 6) with the same set of parameters. Also the model explains Umov's law and decreasing of the phase angle of polarization maximum with albedo increasing, which is observed for the Moon and asteroids.

Phase dependence of brightness was approximated at moderate phase angles (Fig. 6), where a good approximation with exponential function exists (Akimov, 1988). At large phase angles the model yields higher values of brightness than the exponent. It is known that phase dependence at large phase angles differs from Akimov's exponent, and our model can explain this difference. To check this we plan to study phase dependence at large phase angles by absolute observations and investigation of mare-highlands contrast.

 

Introduction

 

There are a lot of observations of positive polarization and its maximum. A lot of woks were published about this property of light reflecting by surfaces of atmosphereless bodies (e.g. Shkuratov, Opanasenko, 1992). But there is not satisfactory model to describe this effect including its dependence on albedo and wavelength. There is a hypothesis that the positive polarization is due to a mirror reflection on surfaces of regolith particles (Shkuratov, Opanasenko, 1992). This reflection is described by Fresnel's formulas, which, however, yield too large phase angle of maximum polarization, which is not observed. Some additional mechanisms are needed to reduce this angle (Shkuratov, Opanasenko, 1992).

We propose another approach. We suppose that surfaces of atmosphereless bodies (Moon, asteroids, etc.) are mature and contain too small quantity of large (in comparison with a wavelength) facets giving Fresnel's reflection. And a principal part of surface brightness is light scattered on particles, which are comparable with wavelength. However, we permit presence of small deposit of Fresnel’s reflection and consider this case parallel with only scattering case.

 

Regolith microstructure

 

Regolith during a long time was exposed to powerful meteoric and micrometeoric bombardment. It leads to destruction of large smooth surfaces, to creation of microparticles and conglomerates with non-homogeneities in microscale. To estimate the deposit of scattering on microparticles into regolith brightness we should get to know size distribution for these particles. Electron microscope images of regolith samples (Rode et al., 1979) show a large quantity of micron and submicron particles covering surfaces of larger particles. We propose two typical size distributions to attempt to describe the real distribution. The first one is based on self-similarity principle for surface and yields distribution function f(R)=R-3 (number of particles with radii between R and R+dR is proportional to f(R)dR). The second one is equilibrium distribution (when frequencies of processes of decay and fusion of particles with volume invariance are equal). In this case distribution function is f(R)=R-4. One can suppose that the real size distribution may be close to f(R)=R-s with exponent s about 3-4.

On fig.1 we present scattering cross section of spheres, multiplied by several size distribution function, i.e. this is an effective deposit of different sizes into scattering. Instead radius R we use here and below the size parameter x=2pR/l (where l - wavelength). One can see that if exponent s is 3-4 or larger this function has a strong maximum at x=2 or less. So for such size distributions a principal deposit in scattering is made by submicron particles.

Fig.2 shows a phase dependences of linear polarization degree for different size distributions. One can see that when s³4, this dependence have a strong maximum of positive polarization near phase angle 90° looking like lunar observational data.

Of course, this is a very approximate estimation (for specific size distributions and for specific particle shape), but it shows that microparticles are able to make a significant deposit into surface brightness and polarization.

Let us consider a medium of such microparticles and describe single and multiple scattering in a such medium. We will assume that these particles are spherical.

 

Rayleigh-Gans theory

 

For scattering on single particle we used Rayleigh-Gans approximation (Bohren, Huffman, 1983). This approximation was obtained for case of small particles or particles with small optical density:

x·|m-1|<<1,                                                         (1)

where x=2pR/l - size parameter, m - refractive index, R - size (radius) of particle, l - wavelength. Actually, for m=1.55 (typical for quartz), Rayleigh-Gans approximation is close to exact Mie theory at x<1.5. If the particles have a wide size distribution, this limit shifts to about x≈2 (for moderate scattering angles), although significant depolarization arises for such x. As one can see below, we obtained values of x about 0.5-2. And we suppose that this approximation is applicable to our microparticles, at least as first approximation.

The phase function (indicatrix) of single particle can be written like this:


Fig. 1. Size dependence of scattering cross section of spheres s multiplied by several size distributions f(x)

 

Fig. 2. Phase dependence of linear polarization degree of spheres with several size distributions f(x).

,                                                 (2)

where I|| and I^ - intensity components, parallel and perpendicular to scattering plane correspondingly; q - scattering angle. For spherical particles function F can be written like this:

,                                                   (3)

,                                                                  (4)

normalizing factor for indicatrix:

                          (5)

Note that function F increases toward forward scattering (q®0), and the slope of the curve increases with size parameter x.

 

Multiple scattering on microparticles

 

Formula (2) describes single scattering on microparticles. To take into account multiple scattering one need to use radiative transfer theory. But we considered an approximate solution, where total intensity of scattering light consists of two components, describing single and multiple scattering:

,                               (6)

where I(a) - intensity at mirror point (i=e=a/2) in relative units, a=p-q – phase angle, i – incidence angle, e – emergence angle, a and b - some coefficients. In (6) the term with coefficient a approximately describes polarized part of multiply scattering light, and the term with coefficient b describes nonpolarized part of it.

 

Shadow effect and multiple scattering on macrorelief

 

To take into account the shadow effect we should multiply the intensity (6) by shadow function, which determines relative area of simultaneously illuminated and visible part of surface. We use Akimov’s shadow function (Akimov, 1988):

,                                                                (7)

where I(a) - intensity in mirror point, I0 - intensity without taking into account shadow effect (formula (6)), h - roughness factor.

Light, multiply scattered by macrorelief, can be approximately described without taking into account shadow effect. Let us for this add to intensity a constant nonpolarized function with some coefficient c:

.               (8)

Polarization degree in this case is like this:

.                                      (9)

 

Taking into account Fresnel's reflection

 

Let us suppose that some part of regolith surface reflects light like mirror. Let us write it like this:

     (10)

where k - weighting coefficient, R|| and R^ - Fresnel’s reflectances (parallel and perpendicular to scattering plane correspondingly) as functions of refractive index and scattering angle. Polarization degree in this case is like this:

.                 (11)

This function contains six parameters: x, h, a, b, c, k. They are not independent: parameters a, b, c should correlate with albedo. But we must find them independently to obtain these dependences.

However, polarimetric data are not enough to obtain all 6 parameters, and we should use, in addition, photometric data. For brightness phase dependence (at moderate phase angles: 30-60°) a good approximation with exponential function exists (Akimov, 1988):

,                                                                 (12)

where m - effective roughness factor. This is a generalization of function (7) by taking into account multiple scattering. We have normalized our intensity images using absolute catalogue of Akimov (1986) at phase angles 35-45° and have found average m=0.66. Analyzing dependence "m - albedo" we have found roughness factor h=0.70. Parameter m, which is not appear in (11), can be recalculated to parameter c using approximation at small phase angles:

.                               (13)

So, we should find only 4 parameters: x, a, b, k. Results of approximation for 4 lunar areas are presented on fig.3 (without Fresnel’s reflection: k=0) and fig.4 (with Fresnel’s reflection). At the second case the parameter a has turned out to be equal zero. So at both cases there were only 3 fitted parameters (by polarization data).

The figures show that the second case (with Fresnel’s reflection) has better agreement with observational data: the dispersion is smaller by coefficient 1.2-1.5.

Fig.5 shows the same as fig.4, but we have added here two points at very large phase angles 145 and 153° (it is fresh data). Fitted parameters have not changed essentially, that confirms our model.

Approximations of brightness phase curves with formulas (10) and (12) are presented at fig.6. Both formulas have a good agreement with observational data at a<60°. For larger phase angles brightness phase function needs more detailed study, including absolute observations.

 

Conclusions

 

We considered an approximate model of positive polarization. It needs more accurate taking into account some effect, in particular, multiple scattering. It is necessary also obtain dependences between model parameters to reduce its number.

However, even such approximate model has a good agreement with observational polarimetric and photometric data and can be used for approximation of them.

 

These results and other information are presented at our website:

http://www.univer.kharkov.ua/astron/dslpp/moon/polar/.

References

 

Akimov L. A., et al. (1986). Reference catalogue of optical characteristics of selected areas of lunar surface // -Kiev. -32 p. [in Russian]

Akimov L. A. (1988). Light reflection by the Moon. II, Kinematika i Fizika Nebesnykh Tel 4, No 2, 10-16 [in Russian].

Bohren C. F., Huffman D. R. (1983). Absorption and Scattering of Light by Small Particles (Wiley, New York)

Korokhin V.V., Akimov L.A. (1997). Mapping of the Phase Parameters of the Lunar Surface Brightness // Astronomicheskiy Vestnik 31, No 2, pp.143-152 [Solar System Research (Engl. transl.), 1997, 31, No 2, pp.128-136]

Rode O.D., Ivanov A.V., Nazarov M.A., Cimbálníková A., Jurek K., Hejl V. (1979) Atlas of photomicrographs of the surface structures of lunar regolith particles // Academia, Prague, ed. K.P.Florenskij.

Shkuratov Yu. G., Opanasenko N. V. (1992). Polarimetric and Photometric Properties of the Moon: Telescope Observation and Laboratory Simulation. 2. The Positive Polarization, Icarus 99, 468-484


Fig. 3. Approximation of lunar observational data with formula (9) (without taking into account Fresnel's reflection). Negative polarization effect at small phase angles is not considered. A – albedo.


Fig. 4. Approximation of lunar observational data with formula (11) (with taking into account Fresnel's reflection). Negative polarization effect at small phase angles is not considered.


Fig. 5. Model approximation of lunar observational data (including large phase angles).


Fig. 6. Two approximation curves of phase dependence of lunar brightness (at area with mirror reflectance geometry): a model curve (10) and exponential function (12) (proposed by Akimov, 1988). Approximation was performed at phase angles 35-45°. Opposition effect at small phase angles is not considered.