Astron. Astrophys. 348, 594-599 (1999)

## 2. Discrete Dipole Approximation (DDA) and porous grain models

Validity Criteria for Discrete Dipole Approximation:

The discrete dipole approximation (DDA) method is described by Purcell & Pennypacker (1973) and Draine (1988). The DDA replaces the solid particle by an array of N point dipoles. When a grain is exposed to an electromagnetic wave, each dipole is under the radiation field of the incident wave as well as the fields due to all other dipoles. There are two validity criteria for DDA (Draine & Flatau, 1994 and Wolff et al., 1994); viz. (i) where m is the complex refractive index, is the wave number, and d is the lattice dispersion spacing and (ii) d should be small enough (N should be sufficiently large) to describe the shape of the particle satisfactorily. We have used the DDA program `DDSCAT.4c' (Draine & Flatau, 1995) to generate the porous grains. In this program it is assumed that the dipoles are located on a cubic lattice. Initially we assume a large number of dipoles Nx, Ny, Nz along the axis x, y, z for the spheroidal target grain. This would result in a certain number of N dipoles in the solid grain (e.g. N = 4088 in the present case). Then we reduce Nx, Ny, Nz to generate the porous grains. These dipoles are reduced such that the shape of the grain does not change. The assumed shape of the grain is a prolate spheroid with axial ratio of 1.3. If the semi-major axis and semi-minor axis of the prolate spheroids are denoted by and respectively then ; where a is the radius of a sphere whose volume is the same as that of a spheroid.

The porosity is defined as ; where is the volume of the solid material inside the grain and is the total volume of the grain (Greenberg, 1990; Hage & Greenberg, 1990). The porosity P of the grain varies between . Accordingly, using DDSCAT.4c the porous grain models with the number of dipoles N = 4088, 1184 and 152 are generated (Vaidya & Desai, 1996 and Paper I). These levels of porosities are selected to maintain the spheroidal shape of the grain i.e. axial ratio of 1.3).

As an illustrative example we show in Fig. 1 the porous grain model with N = 1184. This figure is produced by using the `calltarget' and `dtarget' programs (Draine & Flatau, 1995).

 Fig. 1. A model of the porous dust grain with N = 1184 dipoles

Tables 1 & 2 show the maximum radius of the grain that satisfies the validity criteria for DDA (viz. ) at several wavelengths for N = 4088, 1184 and 152 for silicates and graphite respectively. The complex refractive indices m for silicates and graphite are obtained from Draine (1985, 1987). In the case of graphite, results for both the dielectric functions i.e. parallel (Ell) and perpendicular (Elr) are shown.

Table 1. Validity Criteria for silicate Grains

Table 2. Validity criteria for graphite grains

It is seen from the Table 1 & 2 that in the UV region the DDA is valid for grain sizes of about for N = 1184 and about for N = 4088 whereas in the IR it is valid for larger grain sizes, between and for N = 1184 and between and for N = 4088. Porous grains with N = 152 need to be very small (less than ) in order to satisfy the DDA validity criteria in the UV. In Paper I we had given a table as well as two figures showing the range of applicability of DDA in the spectral range, -, for the porous silicate and graphite (Elr) grains.

Using DDA, Wolff et al., (1994) have investigated the effects of porosity on electromagnetic scattering and have compared their results with those obtained using the effective medium theory (EMT). Vaidya & Desai (1996) have used DDA to study the scattering properties of porous grains. They have used the porous grain model to explain the low density and low albedo observed in the dust coma of the comet Halley. DDA has also been utilized to study composite particles and to determine the limits of EMT (Bazell & Dwek, 1990; Perrin & Lamy, 1990; Ossenkopf, 1991; Stognienko et al., 1995 and Wolff et al., 1993).

© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999