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Astron. Astrophys. 325, 551-558 (1997)

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2. Modelling

The model configurations for an extrasolar system were assumed in imitation of our solar system - with a solar-like central star ([FORMULA]  K, [FORMULA]  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 dust

Assumptions 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 [FORMULA]  Lyr, [FORMULA]  PsA, and, as a particularly clear case, for [FORMULA]  Pic.

2.1.1. Debris dust disk

The 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):




For the disk parameters we used radius [FORMULA], thickness [FORMULA], and the midplane density in the range of [FORMULA].

Ground based and IUE spectroscopic studies (Lagrange-Henri et al. 1989) of the [FORMULA]  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 [FORMULA] (see Duncan et al. 1989, Holman & Wisdom 1993). The quantity µ is the ratio of the masses of the planet and the Sun. From (1) we got gaps in the disk at [FORMULA]  AU for the Earth and [FORMULA]  AU for Jupiter.

The total disk masses assumed in our models are in the range of [FORMULA] (for a gas/dust ratio of 100, dust masses are in the range of about [FORMULA]), 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 [FORMULA] kg/m3 at 1 AU (Leinert & Grün 1990) is about [FORMULA] times lower than in our less evolved debris disk.

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 [FORMULA] (Artymowicz et al. 1989). For a distance of 1 AU we took [FORMULA] 150 K.

[FIGURE] Fig. 1a and b. Cross sections (absorption, scattering) and albedos for the two different populations of circumstellar dust. The bare grains consist of 60 % olivines and 40 % pyroxenes with radii [FORMULA] m ([FORMULA]). The core/mantle grains are made of bare grain material in the core and the weak interstellar mix (H2 O:CH3 OH:CO:NH3 = 100:10:1:1, Hudgins et al. 1993) in the mantle. The core radii are [FORMULA] m, the core+mantle radii [FORMULA] m, ([FORMULA]).

2.1.2. Dust properties

The [FORMULA]  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 [FORMULA] m and a distribution [FORMULA]. 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 [FORMULA]  Pic spectrum - 60% olivines (MgFeSiO4) and 40% pyroxenes (Mg0.8 Fe0.2 SiO3). The optical constants for the bare silicate grains are taken from Dorschner et al. (1995) who prepared silicate glasses as laboratory analogues of circumstellar silicate dust.

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 [FORMULA] 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 [FORMULA] and outer (core+mantle) grain radii of [FORMULA] m we deduced core radii of [FORMULA] 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 (H2 O:CH3 OH:CO:NH3 = 100:10:1:1) at a medium temperature of 80 K (Hudgins et al. 1993, Table 2C).

2.2. Monte Carlo simulation of radiative transfer

For our investigation we performed monochromatic radiative transfer calculations at 11 wavelength points in the range [FORMULA] m for the O3 band and at 16 wavelength points in the range [FORMULA] m for the CO2 band.

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 [FORMULA] and [FORMULA] 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 [FORMULA]. The centre of each intervall is determined by the angles [FORMULA] und [FORMULA]. The photons are accumulated in circles of different size ([FORMULA]  AU in diameter) centred on the lines of sight to the planets (comparable with beam sizes).

[FIGURE] Fig. 2. Geometry of the 3D radiative tranfer model.
[FIGURE] Fig. 3. Sources of emission in the model. The thermal dust emission is shown for both the circumstellar dust populations: bare silicate grains - upper curve, core+mantle grains - lower curve.


Table 1. Ratios of luminosities (star/planet, dust disk/planet). In case of the disk, three types, each with two densities were assumed in the modelling (see Sect. 2.1.1).

The accuracy of the resulting intensities growths with increasing number N of weighted photons accumulated in the circles (error [FORMULA]).

The large demand of computing time is an essential disadvantage of the Monte Carlo method. The computations were performed on [FORMULA] -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 [FORMULA] 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).

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Online publication: April 28, 1998