## 3. Numerical method## 3.1. Numerical modeling and accuracy of the model DDAWe 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 is related
to the incident electric field by a
complex
It is more convenient to describe the scattering properties in terms of the Mueller matrix connecting the Stokes parameters and of the incident and scattered radiation at the distance : where the are related to the amplitude scattering matrix elements 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 of the scattered light, which is defined by: which in the case of incident unpolarized light is reduced to: The linear polarization is then angle (,) 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 where A validity criterion is then given by the dimensionless parameter (Draine & Goodman 1993; Draine & Flatau 1994): If the target is represented by an array of Numerical studies (Draine & Goodman 1993; Draine & Flatau 1994) show that a stringent criterion with in relation (5) should be satisfied for accurate calculations. Criterion (7) is easily satisfied in calculations, with and , if . 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 All these values lead to which is a more satisfactory criterion of (4) than . 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:
. Fig. 2 represents the relative
error of the linear polarization
versus the scattering angle . 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:
m, m
(which give ),
for water ice at this wavelength,
and dipoles in the sphere. We find
that the fractional error due to the approximation is less than about
for any scattering angle greater
than about , and is very small near
().
We note that the error is approximately a constant for
. We also found that this method
tends to overestimate the polarization of the sphere in the range
[10
## 3.2. Generation of the grain roughnessWe 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 of
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 With a total number of dipoles within the sphere equal to , the number of dipoles of the outer layer is only . We randomly remove about of these dipoles, say, that are now counted as holes on the surface with a depth of . As we intend to repeat the simulations times, a large number of dipoles is necessary to ensure that the spatial distribution will be different from one simulation to another.
© European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |