          Astron. Astrophys. 363, 1155-1165 (2000)

## 5. Conclusions and perspective

This statistical study leads to the probability density functions of the linear polarization for various scattering angles. The probability density function characterizes the state of roughness of grains. Indeed, since this density function depends on the law of abrasion applied to the grain, it contains information on the type of erosion which the grains have undergone.

We found that the first maximum of the density function, which corresponds to the most probable value, nearly equals the average value of the function. We think that this is in close relation with the random law of abrasion. However, the density function is symmetrical with respect to neither the most probable value nor the average value, and furthermore it sometimes shows a secondary maximum. Consequently, the details of the density functions show the dependence of the polarization on the overall geometrical shape of grains (in the present case, on their roughness). Moreover, this study shows that the law of dependence of the polarization on the shape of grains cannot be described by a simple Gaussian distribution, even in the case of a uniform roughness.

The mean indicator of the effect of roughness is the most probable value. Nevertheless, the closeness of the most probable value and the average value leads us to interpret the most probable value as the average effect, or global effect, of roughness on polarization.

According to the variations of the most probable value with the scattering angle (Fig. 20), we conclude that the rough sphere still behaves like a sphere. We conclude that roughness does not destroy the general properties of polarization, which are essentially due to the global volume of the grain, i.e. its geometrical shape, relative to the incident light.

The effects of roughness are important around and negligible near (Fig. 20, see also the density functions). Indeed, the density functions at and are very narrow and symmetrical about the most probable value. At , the effects of roughness could be approximated by a Dirac function. We conclude that the effect of volume is major near , and that this angle is adequate to study in particular these kind of effects, for example, those of porous grains on polarization.

Finally, we explain the shape of the density function, in particular the secondary maximum, by the coexistence of two effects on polarization: the first and most important one associated with the most probable value of the density function comes from the general shape of the grain, (its geometrical volume), and the second one, associated with the secondary maximum, is attributed to the state of roughness of the grain. We emphasize that these two effects could have been separated only through the probability density function of polarization.

Neither the mass equivalent sphere nor the interior sphere fit the polarization of the rough sphere. Nevertheless, since the interior sphere seems to be a better approximation of the rough sphere (Fig. 20), together with the fact that the behavior of the polarization for a rough sphere is that of a sphere, we suggest that effects of roughness on polarization could be approximated by an effective polarization. This effective polarization could be composed of a mean part deduced from the interior sphere, and a correction which corresponds to the roughness, so that the sum of both is given by the most probable value of the density function of polarization for each scattering angle.

All these preliminary results were obtained with a purely scattering grain of water ice. In a following paper, we will apply this method to study the effect of roughness when water ice is absorbing, in particular at the wavelength m. In addition, as we observe not only one grain but a collection of grains in circumstellar medium, a forthcoming paper will discuss a collection of randomly oriented grains, again with this kind of approach using a kernel method.

We also plan to analyze the evolution of the secondary maximum with the degree of roughness. To achieve this, we intend to increase the degree of roughness modeled, still with a uniform law, until all the dipoles of the shell constituting the surface are removed, i.e. until the final grain is in fact the interior sphere.

A more general goal is, of course, to get more information about the effect of morphology and internal structure of grains on polarization. However, we consider that a complete and careful study of roughness on polarization is a necessary step before dealing with complex shapes of grains, such as fractals, where effects of external morphology and internal structure are mixed (as already mentioned by Ossenkopf (1991)).    © European Southern Observatory (ESO) 2000

Online publication: December 5, 2000 