The aggregate model and the numerical method to compute the
optical properties of coagulated particles are adopted from HS.
The aggregates are assumed to be in the form of PCA (50%) and CCA
(50%) particles consisting of m spherical subgrains.
The spectral representation of inhomogeneous media
(Bergman [1978]) and the discrete multipole method (DMM) are
elaborated to calculate effective dielectric functions of
aggregates (Stognienko et al. [1995]). At first, we use the
DMM to calculate the spectral function of the aggregated particles
of a special topology, when their subgrains touch each other only
at one point. Then, we account for the interaction strength
between the subgrains (percolation), which varies with the size
and type of the aggregates, by an analytical expression (see
Equation
in HS). Finally, the effective dielectric functions
of the aggregates are obtained by the spectral representation (HS,
Eq.
). In the case of the composite aggregates, we compute the
optical constants of the composite material by the Bruggeman
effective medium theory (Bruggeman [1935]), generalized to
many components by Ossenkopf ([1991]). The optical properties
of the dust aggregates are calculated with the usual Mie theory.
It should be noted that this numerical approach is valid in the
static limit only, which means that the scale of inhomogeneities
within the particles must be small compared to the wavelength.
Given the
m size of the subgrains and the shortest
wavelength of
m we considered, this condition is
fulfilled.
We use the numerical approach of VM to model the (porous)
composite and multishell dust particles. In this method, the
composite grains are represented as spheres with many
concentric shells, where each shell includes several layers of
randomly distributed dust materials. The multishell grains
are modelled exactly as the composite ones but each shell includes
only one layer of a dust constituent. Then a generalized multilayered
Mie theory can be applied to calculate their optical properties. As it
has been shown by VM, a convergence in the optical behavior of the
multishell particles is achieved if the number of shells exceeds
and dust materials within shells are randomly ordered. In our
calculations, we found that this number must be at least
,
because highly absorbing materials, like troilite and iron, are
used, which induces interference within the shells and
prevents fast convergence. Thus, in our case, a typical composite
grain is modelled as a spherical particle with about hundred
shells. On the contrary, a multishell grain is represented by a
spherical particle with only a few shells.
We modify somewhat the dust model for the case of multishell and composite spherical particles. We mixed the silicates and iron into one material with the Bruggeman mixing rule. A similar mixture of silicates, sulphides, and metals (GEMS, Glass with Embedded Metals and Sulphur) is found to be a common component of interplanetary dust particles (Rietmeijer & Nuth [2000]). However, the main reason for this change is the convergence failure of the applied numerical method for the case of multishell grains with iron layers. This is due to numerical uncertainty which arises in the calculation procedure for the Mie scattering coefficients in the case of highly absorbing materials, like iron (for further explanation, see Gurvich, Shiloah, & Kleiman [2001]).
We assume that each dust component has a total volume fraction in
a particle according to its mass fraction and density, as
specified in Table 1. For example, for the first
temperature region and in the case of IPS silicate mineralogy, the
mixture of iron and silicates occupies %, troilite -
%, refractory organics -
%, volatile organics -
%, and water ice -
% of the entire particle volume,
respectively. These values are similar in the case of NRM and IRS
models. Thus, if the temperature is low, organics and ice are the
dominant components of the dust grains.
Unlike to the composite particles, it is assumed in the case of
multishell spherical grains that the distribution of dust
materials is not random but follows their evaporation sequence.
Thus, for the first temperature region in the protoplanetary disc,
multishell spherical grains consist of a refractory core made of
a mixture of silicates and iron and subsequent shells of
troilite, refractory organics, volatile organics, and water ice.
For higher temperature ranges, the number of shells is smaller
since some materials are evaporated. In total, the number of shells in
the case of multishell spherical particles varies from to
.
For the fifth temperature range (
K), where all troilite is
converted to solid iron, we let this iron form an additional
layer on the grain surface. It is an extreme and probably physically
unjustified case, but this allows us to study the influence of the
distribution of highly absorbing materials, like iron, within the
grains on resulting opacities.
The porosity of particles is treated in a simple manner by the
addition of of vacuum (by volume) inside. For the case of
the porous composite spheres, we consider vacuum to be one of the
grain constituents, which forms additional "empty" layers. In the
case of the multishell grain model, we mix each shell with vacuum in
the same way which is applied to create the composite spherical
particles. That is, we subdivide each individual shell in many
layers and fill some of them with vacuum according to the
requested porosity degree.
With the two computational approaches described above, we calculated
the ensemble averaged absorption and scattering cross section as
well as albedo and the mean cosine of the scattering angle for all
kinds of the grains. Applying Eqs. -
from Pollack et
al. ([1985]), the dust monochromatic opacity and,
consequently, the Rosseland and Planck mean opacities were
obtained for temperatures below roughly
K and a density
range between
gcm
and
gcm
. A
convenient analytical representation of the Rosseland and Planck
mean opacities for every temperature region is provided
as a
-order polynomial fit. This representation allows to calculate
the Rosseland and Planck mean dust opacities accurately (
%) and quickly for any given temperature and density values
within the model applicability range, which is important for
computationally extensive hydrodynamical simulations. The corresponding
fit coefficients can be found on the web page.