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

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3. Numerical method

3.1. Numerical modeling and accuracy of the model DDA

We performed the numerical simulations using the DDA model.

In the model, the grain is replaced by a set of electric dipoles which are small compared to the wavelength and the grain size. The dipoles are larger than atomic dipoles in order for the classical equations of electromagnetism to be valid. The main approximation of this method is that the electric dipole interactions are taken into account in the limit of long-wavelengths and large distance between dipoles. Then, the dipoles are located at the nodes of a cubic array (Draine 1988).

The DDA code internally computes the scattering properties of the dipole array in terms of a complex scattering matrix, so that the scattered electric field [FORMULA] is related to the incident electric field [FORMULA] by a [FORMULA] complex amplitude scattering matrix [FORMULA]. The elements of [FORMULA] are a function of [FORMULA] the scattering angle, and [FORMULA] the azimuth angle, and depend on the characteristic parameters of the grain (size, composition, shape). [FORMULA] and [FORMULA] (respectively incident and scattered field) are split in components parallel and perpendicular to the scattering plane (see Fig. 1). In Fig. 1 the incident wave vector [FORMULA] is propagating along the z-axis.

[FIGURE] Fig. 1. Scattering by a grain of any shape.

It is more convenient to describe the scattering properties in terms of the [FORMULA] Mueller matrix [FORMULA] connecting the Stokes parameters [FORMULA] and [FORMULA] of the incident and scattered radiation at the distance [FORMULA]:


where the [FORMULA] are related to the amplitude scattering matrix elements [FORMULA] through analytical relations (Bohren & Huffman 1983) which are straightforwardly implemented in the code.

The quantity we are interested in is the degree of linear polarization [FORMULA] of the scattered light, which is defined by:


which in the case of incident unpolarized light is reduced to:


The linear polarization [FORMULA] is then angle ([FORMULA],[FORMULA]) dependent.

The main advantage of the DDA is that it is a flexible method regarding the geometry of the target. This approximation of a continuous target is limited only by the condition that the distance between two neighboring dipoles d be small compared to, first, any structural lengths in the target, and second, the wavelength [FORMULA] of the incident wave. Furthermore, d has to be small compared to the wavelength within the grain and be small also compared to the attenuation of the electromagnetic wave. This can be written by:


where m is the complex refractive index of the material and k the wave number [FORMULA].

A validity criterion is then given by the dimensionless parameter [FORMULA] (Draine & Goodman 1993; Draine & Flatau 1994):


If the target is represented by an array of N dipoles large enough, located on a cubic lattice with lattice spacing d, then the target volume can be taken as [FORMULA]. If the size of the target is characterized by an effective radius [FORMULA] of an equal volume sphere, then the size parameter x can be related to N and [FORMULA] by:


which with 5 gives:


Numerical studies (Draine & Goodman 1993; Draine & Flatau 1994) show that a stringent criterion with [FORMULA] in relation (5) should be satisfied for accurate calculations.

Criterion (7) is easily satisfied in calculations, with [FORMULA] and [FORMULA], if [FORMULA].

Nevertheless, the DDA method introduces some artificial granularity in modeling a sphero"dal shape on a cubic array (Draine & Flatau 1994): a numerical artifact is caused by the dipoles in regions near the target boundaries. To study the particular effect of the roughness of grains on scattering, and in order to reduce this numerical artifact, we use the largest number of dipoles allowed by the computer memory. However, the choice of the number of dipoles depends on the acceptable CPU time, and also on the quantum limit reached by the dipoles for a given V. We fix the quantum limit to 100Å, i.e. [FORMULA]m. We take [FORMULA]m in our calculations as the lower limit acceptable for d. This allows us to use a number of dipoles for the sphere equal to [FORMULA] (for a [FORMULA] memory) to describe an initial sphere with radius [FORMULA]m. We take the size parameter [FORMULA] and consider a typical astrophysical grain of ice at the wavelength [FORMULA]m. Under such conditions, the complex refractive index is [FORMULA] (Pollack et al. 1991).

All these values lead to [FORMULA] which is a more satisfactory criterion of (4) than [FORMULA].

We verify the accuracy of this method by comparison with the results of Mie's theory with spherical grains. We have compared the degree of linear polarization computed with the DDA method to the theoretical value obtained with Mie's theory in the case of spheres: [FORMULA]. Fig. 2 represents the relative error of the linear polarization [FORMULA] versus the scattering angle [FORMULA]. The results of Mie's theory have been obtained with the numerical code given in Appendix A of Bohren & Huffman (1983). The parameters we used in the DDA model for this comparison are: [FORMULA]m, [FORMULA]m (which give [FORMULA]), [FORMULA] for water ice at this wavelength, and [FORMULA] dipoles in the sphere. We find that the fractional error due to the approximation is less than about [FORMULA] for any scattering angle greater than about [FORMULA], and is very small near [FORMULA] ([FORMULA]). We note that the error is approximately a constant for [FORMULA]. We also found that this method tends to overestimate the polarization of the sphere in the range [10o;80o] and underestimate the polarization in the range [100o;170o]. We consider that the error of the model is systematic over both intervals.

[FIGURE] Fig. 2. Relative error of [FORMULA] calculated by the DDA method in comparison with Mie's theory versus the scattering angle [FORMULA]

3.2. Generation of the grain roughness

We start from a spherical grain and remove dipoles from its surface only. The removed dipoles will be counted as holes that will model a certain roughness. The depth of the surface of the spherical grain is defined as about [FORMULA] of [FORMULA] which corresponds to 3 layers of dipoles. The dipoles are never removed from layers deeper than these three layers. All holes are then three dipoles deep. The width of each element of matter removed is fixed to one dipole, which then corresponds to the distance d between two dipoles. As the dipoles are removed at random, the average width of holes on the surface is larger than one dipole. Then, the degree of roughness is defined as the ratio of the total number of dipoles removed to the initial number of dipoles on the surface. As the depth of holes is fixed for all holes, and corresponds to the thickness of the surface we have defined, we take into account only the outer layer for the determination of the degree of roughness. We fix the degree of roughness to [FORMULA].

With a total number of dipoles within the sphere equal to [FORMULA], the number of dipoles of the outer layer is only [FORMULA]. We randomly remove about [FORMULA] of these [FORMULA] dipoles, say, [FORMULA] that are now counted as holes on the surface with a depth of [FORMULA].

As we intend to repeat the simulations [FORMULA] times, a large number of dipoles is necessary to ensure that the spatial distribution will be different from one simulation to another.

[FIGURE] Fig. 3. Schema of the three spheres: [FORMULA] is the radius of the initial sphere, [FORMULA] that of the sphere with mass equivalent to that of the remaining dipoles, and [FORMULA] that of the interior sphere. Note that the dipoles represented by the small circles are out of scale for better clarity.

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000