## 2. ModellingThe model configurations for an extrasolar system were assumed in imitation of our solar system - with a solar-like central star ( K, m) and two planets at distances and with emissions like Earth and Jupiter (planetary photon fluxes were taken from Burke (1992, p. 78). Star and planets are embedded in a circumstellar dust disk which represents the remnant material from planet formation. ## 2.1. Circumstellar dustAssumptions about the properties and the distribution of the circumstellar dust and the temperature distribution in the debris disk are mainly made on the basis of observational hints from Vega phenomenon dust as it was detected around Lyr, PsA, and, as a particularly clear case, for Pic. ## 2.1.1. Debris dust diskThe dust was modelled as orbiting about the central star in a Keplerian disk. The density distribution of such a disk is given by (Shakura & Sunyaev 1973): with For the disk parameters we used radius , thickness , and the midplane density in the range of . Ground based and IUE spectroscopic studies (Lagrange-Henri et al. 1989) of the Pic debris disk have provided hints at the precense of a central hole and gaps. We assumed the inner edge of the disk to be at a radius of 0.25 AU. Gaps in the disk along the orbits of the planets are introduced in our third model. The width of these gaps was taken from the relation which predicts the onset of chaotic behaviour of a test particle in
the planar and circular restricted three-body problem inside the
semimajor axis region (see Duncan et al. 1989,
Holman & Wisdom 1993). The quantity The total disk masses assumed in our models are in the range of
(for a gas/dust ratio of 100, dust masses are
in the range of about ), corresponding to masses
possible for debris disks around G-type main-sequence stars
(André 1994). The midplane density of the solar system
interplanetary dust of kg/m Because each run of the Monte Carlo simulation of the radiative transfer is done for only one wavelength, we need an approximation of the temperature distribution in the disk. As a first attempt we modelled the emission from the mid-size grains by (Artymowicz et al. 1989). For a distance of 1 AU we took 150 K.
## 2.1.2. Dust propertiesThe Pic spectrum indicates the
existence of micron-sized silicate particles (Knacke et al. 1993). For
the dust population in our debris disk model we used grains with radii
m and a distribution .
For the dust composition we applied the main components of the
interplanetary dust particle (IDP) population proposed by Sandford
(1988), which give quit well a fit to the
Pic spectrum - 60% olivines (MgFeSiO In a second model variant we introduced an ice component to the
dust in form of a mantle. For the core/mantle grains we assumed a
mantle-to-core volume ratio of 1 which lies in
the range of values found on the line of sight to the
Becklin-Neubebauer object (Lee & Draine 1985). On the basis of
and outer (core+mantle) grain radii of
m we deduced core radii of
m. The functional dependence of the grain size
distribution is the same as for the population of bare grains. For the
grain cores we used the bare grain dust. For the ice mantles mixed
molecular ices were applied. Hudgins et al. (1993) calculated the
optical constants for a variety of pure and mixed molecular ices. From
their data we selected the so-called weak interstellar mix
(H ## 2.2. Monte Carlo simulation of radiative transferFor our investigation we performed monochromatic radiative transfer
calculations at 11 wavelength points in the range
m for the O The principal task of the Monte Carlo simulation of radiative transfer is to construct a stochastic model in which the expected values of the intensities have to be determined. This model consists of the random walk of a weighted photon. The random path is determined by the random quantities: starting-point, free path lengths and propagation directions. We started weighted photons from four sources: the central star, the two planets, and the thermally emitting dust (photon fluxes of the sources are shown in Fig. 3, luminosity ratios star/planet and disk/planet are given for and m in Table 1). During the course of the random walk of a weighted photon, the initial intensity (weight) will be changed as the result of scattering events and absorption. After the last scattering (or without any scattering event), the weighted photon becomes "observable". In our 3D-model (Fig. 2), we can "observe" the configuration from 480 positions around it. Therefore, the full solid angle of all possible observer directions is divided into 480 equal sized solid angle intervalls of . The centre of each intervall is determined by the angles und . The photons are accumulated in circles of different size ( AU in diameter) centred on the lines of sight to the planets (comparable with beam sizes).
The accuracy of the resulting intensities growths with increasing
number The large demand of computing time is an essential disadvantage of the Monte Carlo method. The computations were performed on -DEC-Workstations (200 series, 3000 series, SMP A2100/4). Dependent on the type, a CPU time of several hours was necessary for the transport of weighted stellar photons per wavelength. More details about the Monte Carlo simulation of radiative transfer can be found in Fischer et al. (1994) and Fischer (1995). © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |