4. Results and discussion
To quantify statistically the effects of roughness on polarization we compute the degree of linear polarization for each stochastic simulation. We made stochastic simulations which give a set of the random variable : . For instance, Fig. 4 shows versus the number of stochastic simulations n, reduced to stochastic removal processes for better clarity. In Fig. 4 we have plotted the average value of the polarization in the simulations , and the standard deviation . We stress on the fact that we do not average the intensity involved in the definition of polarization but average directly the polarization over the simulations. From a physical point of view, this average () will never represent the net polarization of a collection of grains for which only the individual intensity can be added, but the information we get from this particular study will represent the specific effect of roughness on polarization.
The average value can have no meaning although usually used without verifying its validity. In order to determine whether the average value is physically relevant for use in computations, we have to calculate the probability density function of . In addition to the mean value, the probability density function contains the number of modes, i.e., the number of maxima of the function. Even in the case of only one maximum, if the mean value doesn't co"ncide with the maximum of the function, i.e. the most probable value, the average value has no physical meaning. If the density function exhibits more than one maximum, the interpretation becomes difficult.
In practice, we do not know the probability function of the random variable. The only value that is available is the empirical repartition function which, we remind, corresponds to the integrated probability density function over a restricted interval. Then, the probability density function deduced from the empirical repartition function will be a sum of Dirac functions. Nevertheless, if the number of values is large enough, one can approximate the density function by a continuous function using a kernel method. The details concerning the construction of this method is given in Appendix. Here, we choose a Gaussian kernel method.
We have calculated the probability density function of the linear polarization for several scattering angles in the interval and for 3 values of : (only results for and are shown). Figs. 5-19 show the probability density functions of obtained for several scattering angles. We have used exactly the same scale along the x-axis for all these figures.
The general aspect of the density function shows the specific effect of roughness at least in two ways: the first one is the shape of the density function which characterizes the type of roughness applied and, among other things, contains the number of maxima. The other way is the highest maximum which reflects the mean effect of roughness on . What first appears is that the density functions are rarely symmetrical around the most probable value, i.e. the principal mode of , in view of the randomness of the roughness. Moreover, in some cases, a secondary maximum clearly appears, as for instance at the scattering angle in Fig. 14.
Consequently, a Gaussian distribution may not provide a realistic approximation to describe the effects of roughness, even in the case of a uniform roughness as modeled here.
All these density functions exhibit very close values for the most probable value of the density function and the average value , and also the median value. That means that the closeness of the values of the most probable value, the average value and the median value do not imply the symmetry of the density function, although the near equality of the most probable value and the average value reflects certainly that the law of abrasion applied is a uniform law.
The fact that the most probable value nearly equals the average value of the density function demonstrates the possibility of taking, to a first order, the average value as the mean indicator of the effect of roughness on polarization.
Whereas the polarization of a sphere is symmetrical with respect to the angle and is invariant through a rotation, we find that the density functions are not symmetrical with respect to (see Figs. 17-19), and are very different for the three values of (given ) (see Figs. 14-15). However, the value of the first maximum (which is considered as the mean effect of roughness on polarization) is kept the same for a fixed . It is also symmetrical with respect to . Thus, over all the simulations (n=1000), whatever the scattering angles, the mean value of the density functions follows exactly the symmetry of a sphere.
Any value of the polarization different to the first maximum will not be preserved for different , given . Indeed, the polarization obtained with a rough sphere for a specific scattering angle (,) has no reason to be equal to the polarization for another couple (,) because of the anisotropy of the rough sphere to the incident light. This explains why the graphs of the density function are not identical for different values of (for a fixed ), and also means that the density functions accurately contain the information on the dependence of the polarization on the anisotropy of the rough sphere, due to its roughness.
As a first conclusion, we can say that the mean value of the density function represents the average effect of roughness on polarization.
The secondary maximum which appears in some cases suggests the coexistence of two effects of roughness on polarization.
We have calculated, using Mie's theory, the linear polarization of the equivalent sphere, i.e., the sphere that would have the same number of dipoles: , and, the linear polarization for the interior sphere of radius . The interior sphere corresponds to the initial sphere from which all the dipoles of the surface have been removed (see Fig. 3 and Sect. 3.2). All the graphs of the density function also show the value of the linear polarization of the initial sphere of radius m, and the average value and the median value, both pointed out by vertical lines.
Fig. 20 shows the variation of the polarization for the equivalent sphere and the interior sphere related to those for the initial sphere versus the scattering angle , so that the values for the initial sphere are represented by the horizontal line . Fig. 20 also shows the most probable values obtained from the density functions for each angle where they have been calculated. Fig. 20 represents the effect of small variations of the volume of spheres on polarization. The values plotted for the three sphere () have been calculated using Mie's theory.
Fig. 20 clearly shows that the polarization of the rough sphere given by the most probable value of the density function behaves like a sphere with some "effective " radius. Together with the spherical symmetry of the most probable value, this implies that this value mirrors the main information about the polarization of the rough sphere due to its volume. Thus, we can say that the polarization of a sphere due to roughness can be characterized by an effective polarization equal to the mean value of the density functions for each scattering angle.
Although neither of the equivalent sphere nor the interior sphere correctly approximate the rough sphere, the interior sphere seems to provide a better approximation of the rough sphere. Moreover, whereas the equivalent sphere represents a sphere of equal mass, this study shows that, since the effective polarization of a rough sphere as a function of angle follows a variation of the volume, the polarization of the interior sphere corresponds better to the polarization of a rough sphere. Thus, the effective polarization of the rough sphere could be considered as the sum of the polarization of the interior sphere plus a correction which would correspond to the rough surface, so that the sum is equal to the mean values of the density functions for each angle.
Thus, we attribute the most probable value of the density function mainly to the effect of the global volume of the rough sphere, characterizing only the average effect of roughness. Similarly, we attribute the secondary maximum to the details of roughness and the dependence of the polarization on the anisotropy of the grain with the scattering angle.
Among the density functions Figs. 5-16, we note that near the scattering angle , the density function is very symmetrical and narrow about the most probable value and is also clearly unimodal. The shape of the density function near does not depend on . This implies that for the values of the scattering angle near , the polarization is not sensitive to the type of roughness.
Moreover, in Fig. 20, we can see that even the average effect of roughness is small compared to the polarization of the initial sphere . At this angle, we cannot distinguish between any of the three spheres: , , .
In conclusion, near , the polarization is probably mostly due to the overall shape of the grain, and more precisely to the geometrical symmetry of the global volume related to the incident light. Thus, at , the polarization is not affected by roughness of grains and only depends on the general characteristics of the volume of the grain. Thus, is not the best angle to observe effects of roughness on polarization.
Kozasa et al. (1993) have also observed that for non-absorbing small particles the maximum of linear polarization (which is obtained near ) is more closely related to the chemical composition of the particle than to its structure. Nevertheless, Perrin & Sivan (1991) have observed a small shift of the maximum of polarization near . This could be explained by the fact that as they modeled roughness deep into the grain, this kind of roughness may affect the geometrical volume of the grain, as in porous grains, and then modify significantly the observed polarization. Indeed, their polarization curves for porous grains are very similar to those for rough grains.
In the same paper, they also observed that the effect of roughness increases the amount of scattered light for , and decreases for . This will be reflected into the degree of polarization by an opposite effect (see definition of in Sect. 3) which is visible in Fig. 20.
From this paper, it also follows that approximations by spheres with equivalent mass correspond to neither rough grains nor porous grains. This is in agreement with Fig. 20, where the variation of polarization due to roughness follows changes of volume and not to changes of mass (which would correspond to the sphere of equivalent mass ).
With our statistical approach (with simulations), the accuracy of the results for the most probable value, and then the average value , is found to about which is very accurate. However, the actual accuracy of measurements does not give access to measurements of the effects of this degree of roughness on polarization. The greatest effect we have obtained for gives while very accurate observations (Aitken et al. 1997) are obtained within . Recent experimental results (Worms et al. 1999) lead to an error of about , which depends on the intensity received and on since the light is not scattered isotropically. Nevertheless, such effects of roughness, with such size parameter of grains, could be detectable for some angles in the future.
© European Southern Observatory (ESO) 2000
Online publication: December 5, 2000