## 2. Previous works and description of the new approach## 2.1. Review of the problem and previous works on roughnessThe problem of the scattering of light by particles of small size compared to the incident wavelength (or of the same order) is usually solved using the classical equations of electromagnetism. To solve Maxwell's equations, the crucial problem is that of the boundary conditions (Mugnai & Wiscombe 1980) which characterize the grain surface. The only cases for which we can derive the analytical solutions are those where the grains have exactly the same geometry and symmetry as the coordinate system chosen to develop the equations, among which Mie's spheres are a particular case. On the other hand, size, morphology and composition of dust grains cannot be directly inferred from a comparison between observations and theoretical models. This inversion problem cannot be easily solved for several reasons. One reason is that the equations do not depend linearly on these parameters, at least through the refraction index, and then, the solution is probably not unique. A specific grain candidate with a particulate size, shape, and composition can as well reproduce the results as another grain candidate with different parameters. Another reason is that these parameters characterizing the grain cannot be treated as independent variables, as is generally assumed. For instance : the optical constants of the material depend on the morphology and structure of the grain. One approach devoted to the effect of internal structure is based on the applicability of EMT-type solutions as described in Wolff et al. (1994, 1998); Stognienko et al. (1995); Ossenkopf (1991); Rouleau (1996). In addition to the internal structure that affects the optical properties, grains also have a rough surface which is rarely studied in models. One approach (Mugnai & Wiscombe 1980; Mishchenko & Hovenier 1995) uses expansion with Chebyshev's polynomials to deform a sphere. Despite the advantage that this method does not use any approximations, the shape of the grain is still too regular to represent a real grain. Another approach used by Perrin & Sivan (1991) consists of removing random elements of matter from the surface of a sphere. In their definition of roughness, deep holes on the grain can reach up to of the initial sphere radius. In that case, we consider that the internal structure of the grain is modified so that the effect of roughness cannot be separated from those of volume. While in Perrin & Sivan (1991) the radius of the sphere is comparable to the incident wavelength, McGuire & Hapke (1995) have studied the optical effects of rough spheres with a size much larger than the incident wavelength, so that the irregularities of the sphere are also very large compared to the incident wavelength. In our case, we are mainly interested in the effects of roughness of particles with size comparable to the incident wavelength, for Mie's theory to be valid. The above works on porous and/or rough grains then intend to simulate a collection of grains. There are generally two ways to model a collection of roughly identical grains with a uniform distribution in space: one is based on simulation of numerous grains with the same fractal dimension or same degree of porosity and the other is based on orientation averaging of one grain. Whereas it is commonly admitted that these two approaches are equivalent (Lumme & Rahola 1994), only the second one describes a collection of randomly oriented identical grains (Bohren & Huffman 1983). Nevertheless, both models will result in depolarization of light. Measurements of polarization of starlight in the interstellar medium require that a significant fraction of interstellar grains are aligned (Draine & Weingartner 1997). In this case, averaging over one angle is required to take into account the degree of freedom still available along the line of sight. In our case, we will restrict the discussion to one grain that would represent the case of a perfect alignment of the particles. As our goal is to study in detail the effects of roughness of dust grains on polarization with the statistical method developed here after, it is necessary to first restrict this discussion to a single grain in order to point out some specific feature before considering a collection of grains (which will be done in a forthcoming paper). Previous works have applied different statistical models for the scattering of light by irregular particles. Drossart (1990) proposed a concept of partial coherence of the individual spherical harmonics, assuming a Gaussian distribution for the coherence functions to describe effects of roughness. While the main goal of this work is to provide a new model to approximate scattering by irregular particles, we do not present a model but rather a method to study in detail the dependence of the polarization on the shape of the grains, for which roughness is a first step. Muinonen (1996), Peltoniemi et al. (1989), Muinonen (1998) and Muinonen & Lagerros (1998) used multivariate lognormal statistics, and Schiffer (1985) Gaussian statistics, to deform stochastically large spheres. Then, scattering efficiencies (albedo, polarization) are derived through Monte-Carlo simulations or after randomly orienting grains. Here, we adopt a different statistical point of view: not only is the surface roughness described by a random variable but also the observable (the linear polarization), which is then analyzed using a specific statistical method. ## 2.2. Description of the statistical approachTo quantify the effects of morphology, a comparison with spheres described using Mie's theory came naturally as a starting point. We use a uniform law of exclusion to carve the surface of an homogeneous sphere for which we compute the linear polarization for each simulation of roughness. In order to define the roughness applied on the sphere we need to introduce a term that we call the `thickness' of the surface of the sphere. The surface of the sphere is defined by the external shell of the sphere which has a thickness estimated to about (where is the radius of the initial sphere). Thus, the elements of matter removed from the surface have a fixed height equal to . Then, we define the degree of roughness by the ratio of the elements of matter removed to the initial number of elements of matter which compose the surface of the grain. The chosen degree of roughness applied here is set to , and lies between a few percent that wouldn't be enough to characterize a significant effect and that begins to represent a high degree of roughness. We will see in Sect. 4 that this chosen degree effectively permits us to quantify an effect of roughness. Only the spatial distribution of holes changes from one simulation to another, and is characterized by the uniform law. For each rough sphere, we calculate the degree of linear polarization for various scattering angles. We repeat the simulation for times with the same degree of roughness. This gives us values of linear polarization which are then set as a random variable. A large number of simulations was needed to obtain a sufficient number of significantly different values in order to apply a Gaussian kernel method to this set of random variables. From this method, we can derive, for each scattering angle, the probability density function of the linear polarization which contains all the information about the roughness applied (see Sect. 4). In this first paper, we present the preliminary results of this method for an initially spherical homogeneous grain. We consider a typical interstellar grain composed of water ice with size m (MRN, Pendleton et al. 1990) and size parameter which corresponds to an incident wavelength m. © European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |