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Astron. Astrophys. 362, 697-710 (2000)

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4. Astrophysically relevant examples

A first example of the application of our code has already been given in Sects. 2 and 3.4. This model is a spherically symmetric (one-dimensional) model; many authors have illustrated the capability of Monte Carlo and other methods in solving one-dimensional problems (Bernes, 1979; Zhou, 1995; Choi et al., 1995, e.g.). This section presents a number of astrophysically relevant examples, again drawn from star formation studies, to illustrate the distinguishing properties of our code - in addition to accelerated convergence: the capability to calculate axisymmetric source models with a wide range of spatial scales, and the effects of dust continuum on radiative transfer and excitation.

The models presented in the following sections all include continuum radiation fields from dust. For these star-forming regions, we have chosen the model of Ossenkopf & Henning (1994) for the dust emissivity, which includes grain growth for a period of [FORMULA] yr at an ambient density of [FORMULA] cm-3 and thin ice mantles. Except for the calculations in Sect. 4.3 where we specifically examine the effect of dust on the excitation including infrared transitions, only (sub) millimetre transitions were included in the excitation calculations which are not significantly influenced by the relatively weak continuum field.

4.1. A young stellar object with rotation

Observations of nearby young stellar objects often show flattened structures rather than the spherical symmetry of models like that of Shu (1977), presumably caused by ordered magnetic fields and/or rotation (Hogerheijde et al., 1998, e.g.). These mechanisms probably influence the accretion behaviour, and may give rise to a rotationally supported circumstellar disk. To test these ideas against observations, cylindrically symmetric source models need to be considered. This section examines a model of protostellar collapse that includes flattening due to rotation following the description of Terebey, Shu, & Cassen (1984), and its appearance in aperture synthesis maps.

The model of Terebey et al. (1984) treats rotation as a small perturbation to the solution of Shu (1977) for a collapsing envelope, joined smoothly to the description of a circumstellar disk by Cassen & Moosman (1981). In addition to the sound speed and age, which we take identical to the values of Sect. 2 of [FORMULA] km s-1 and [FORMULA] yr, this model is parameterized by a rotation rate [FORMULA]. This gives rise to a centrifugal radius [FORMULA], within which the infalling material accretes onto a thin disk. Here, [FORMULA] is a numerical constant. We choose [FORMULA] s-1, so that [FORMULA] AU. We assume that inside [FORMULA] the material accretes onto a thin disk, and that most molecules rapidly freeze out onto dust grains (cf. Sect. 4.2). Effectively, we assume the region within [FORMULA] to be empty for this calculation. Fig. 4 (top left) shows the adopted density structure. All other parameters are similar to the model of Sect. 2.

[FIGURE] Fig. 4. Left: Density distribution (top) and temperature distribution (bottom) of the collapse model including rotation of Sect. 4.1. Middle: Resulting excitation temperature of HCO+ 1-0 and 3-2. The adopted grid is visible in these panels, with small cells at the center were the density is large, and larger cells in the outer regions of the object. Right: Images of integrated intensity of HCO+ 1-0 and 3-2 for an edge-on source orientation, after sampling on spatial frequencies corresponding to interferometric baselines between 15 and 300 m and subsequent image reconstruction. This results in synthesized beam sizes of [FORMULA] and [FORMULA] for the 1-0 and 3-2 lines, respectively, as indicated in the lower left corner of the panels.

To follow the power-law behaviour of the density in the model, a total of [FORMULA] cells are spaced exponentially in the R and z directions. To reach a final signal-to-noise ratio of 10, with 300 rays initially making up the radiation field in each of the cells, the Monte Carlo code requires 5 hours CPU time on a UltraSparc 10 to converge on the HCO+ solution. For comparison, the optically thin and more readily thermalized 13CO problem takes only 10 minutes. The resulting excitation temperature distribution (Fig. 4 ; middle panels) does not deviate much from that of the spherically symmetric model of Sect. 2, apart from the flattened distribution of the material at the center: rotation is only a small perturbation on the overall source structure. As a result, the appearance is mostly unaffected in single-dish observations which do not resolve scales comparable to [FORMULA] where flattening is important.

Millimetre interferometers, on the other hand, can resolve these scales at the distances of the nearest star-forming regions, [FORMULA] pc. Fig. 4 (right panels) shows the integrated emission in HCO+ J=1-0 and 3-2 after sampling at the same spatial scales as real interferometer observations and subsequent image reconstruction. Delays between 100-1000 ns were used, corresponding to angular scales of [FORMULA]-[FORMULA] for the 1-0 line and [FORMULA]-[FORMULA] for 3-2. Hence, emission on scales [FORMULA] AU ([FORMULA] AU at 3-2) is filtered out. This results in a reconstructed (`cleaned') image that is dominated by the central, flattened regions of the envelope when the object is seen edge-on. Aperture synthesis observations of embedded protostars in Taurus show similar structures (Ohashi et al., 1997; Hogerheijde et al., 1998, e.g.).

4.2. A circumstellar disk

Planetary systems form from the disks that surround many young stars (Beckwith, 1999; Mannings et al., 2000). Observational characterization of these disks is of prime importance to increase our understanding of the processes that shape planetary systems. Here, we present simulations of molecular line observations of a circumstellar disk around a T Tauri star as obtained with current and planned millimetre-interferometric facilities.

The physical structure of the model disk is that of a passive accretion disk in vertical hydrostatic equilibrium as described by Chiang & Goldreich (1997). This description includes the effect of `backwarming' of the disk by thermal radiation of a thin, flared surface layer that intercepts the stellar light. The total amount of material in the superheated surface layer is too small to be detectable, but the overall effect of increased mid-plane temperature is significant. The effective temperature of the central star is 4000 K and its luminosity is 1.5 [FORMULA]. The outer radius of the disk is 700 AU, with a total mass of 0.04 [FORMULA].

The chemical structure of the disk follows Aikawa & Herbst (1999), who describe the radial and vertical composition of a flared disk. Freezing out of molecules onto dust grains is one of the dominant processes influencing the gas-phase composition in disks, and strongly depends on temperature and density. In the dense and cold midplane, many molecules will be depleted onto grains. However, close to the star where temperatures are raised, and away from the midplane where densities are lower and depletion time scales longer, gas-phase abundances will be significant. In addition, ultraviolet radiation and X-rays may penetrate the upper layers of the disk, photodissociating molecules and increasing the abundance of dissociation products like CN. Fig. 5 shows the distribution of the density, temperature, and abundances of CO, HCO+, HCN and CN in the adopted model. We have used the results presented in Aikawa & Herbst (1999, their Figs. 6 and 7; high ionization case), and parameterized the abundances as function of scale height.

[FIGURE] Fig. 5. The density, temperature, and molecular abundance distribution of the circumstellar disk model described in Sect. 4.2. Density and abundances are plotted on logarithmic scales; the temperature is plotted on a linear scale. The `superheated' layer of the disk model of Chiang & Goldreich is not included in our model because only a very small amount of dust and gas is present in this layer. Its `backwarming' effect on the disk interior is included, however.

For the Monte Carlo calculations, a gridding is adopted that follows the radial power-law density profile in 14 exponentially distributed cells and the vertical Gaussian profile in 13 cells linearly distributed over 3 scale heights. Convergence to a signal-to-noise ratio of 10 requires approximately 6 hours CPU time per model on an UltraSparc 10 workstation, starting with 100 rays per cell and limiting the spatial dynamic range to 36 (i.e., the smallest cell measures 10 AU on the side). The resulting excitation and emission depends on the competing effects of increased abundance and decreased density with distance from the midplane. Fig. 6 shows the excitation temperature of selected transitions and molecules. Fig. 7 shows a number of representative simulated observations, at resolutions of [FORMULA], [FORMULA], and [FORMULA] attainable with current and planned (sub) millimetre interferometers. Van Zadelhoff et al. (in prep.) present a simpler analysis of similar models.

[FIGURE] Fig. 6. The distribution of excitation temperatures in K for the J=1-0 and selected submillimetre lines of CO, HCO+, HCN, and CN. In all panels the grey scale ranges between 0 K and 30 K on a linear scale. The adopted grid, exponentially spaced in radius and linearly in height, is reflected in the excitation distribution.

[FIGURE] Fig. 7. Appearance of integrated intensity of selected lines in the circumstellar disk model at resolutions attainable with current and planned interferometric facilities. The first three columns from the left show CO 1-0 at [FORMULA] resoltuion, CO 1-0 at [FORMULA] resolution, and CO 6-5 at [FORMULA] resolution. A distance of 140 pc is adopted for the source, and inclination increases from [FORMULA] to [FORMULA] from top to bottom. The last three columns from the left show the emission in HCO+ 4-3, HCN 4-3, and CN [FORMULA] at [FORMULA] resolution. All panels are shown with a linear grey scale ranging from 0 to the image maximum in K km s-1. Contours are drawn at 1%, 5%, 10%, 15%, 20%, 30%, 40%, 50%, 70%, and 90% of maximum.

4.3. A high mass young stellar object

The above examples referred to the formation of stars with masses of up to a few [FORMULA] and luminosities [FORMULA] [FORMULA]. Stars of higher mass spend their first [FORMULA] yr in envelopes of [FORMULA] [FORMULA]. With their luminosities of [FORMULA]-[FORMULA] [FORMULA], these stars heat significant parts of their envelopes to several hundred K, shifting the peak of the Planck function to the wavelengths of the vibrational transitions of many molecules. Stars of lower mass and luminosity only heat small regions to a few hundred K, and the impact on the excitation is correspondingly smaller. As an example, Fig. 8 shows two models of the HCN submillimetre line emission, with and without pumping through the bending ("[FORMULA]") mode at 14.02 µm. For computational convenience, only energy levels up to J=10 within the first vibrationally excited and ground states have been included, including l-type doubling in the excited state. This doubling occurs due to the two possible orientations of the rotational and vibrational motions with respect to each other. Collisional rate coefficients between rotational levels are from Green (1994, priv. comm., see http://www.giss.nasa.gov/data/mcrates <); between vibrational levels, they were set to [FORMULA] cm3 s-1. The source model is that of the young high-mass star GL 2136 by van der Tak et al. (2000). Based on its luminosity of [FORMULA] [FORMULA] and dust mass of [FORMULA] [FORMULA], the star has heated a region of radius [FORMULA] AU to [FORMULA] K, making pumping through the 14  µm HCN bending mode important. We have assumed that [FORMULA], as is a good approximation for high density regions. The dust emissivity, from Ossenkopf & Henning (1994), is the same as in the previous sections. As seen in Fig. 8, the effect of dust is especially strong for the rotational lines within the [FORMULA]=1 band, which occur at frequencies slightly offset from the ground state transitions. These lines have indeed been detected towards similar objects (Ziurys & Turner1986; Helmich & van Dishoeck1997, e.g.).

[FIGURE] Fig. 8. Submillimetre emission lines in a [FORMULA] beam of HCN in the vibrational ground state (top panels) and first excited state (bottom panels). Thin line: only collisional excitation; thick line: model with pumping through the [FORMULA] band at 14  µm.

The shells around evolved stars is another area where inclusion of infrared pumping by dust is essential to understand the rotational line emission (Ryde et al., 1999, e.g.). Many molecules that are commonly observed through rotational lines at millimetre wavelengths have ro-vibrational bands in the mid-infrared, and can be pumped by warm dust. In a few cases, pumping through rotational lines at far-infrared wavelengths is important as well, for example CS (Hauschildt et al., 1993) and all hydrides, most notably water (Hartstein & Liseau, 1998).

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© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000