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Astron. Astrophys. 342, 233-256 (1999)

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4. Chemical and thermal modelling

In order to understand its physical, thermal, chemical and dynamical structure, a simplified chemical and thermal model of one of the fingers was developed. This is used to understand factors which will help predict its future evolution and potential for forming star(s). [FORMULA] was chosen as a basis for this modelling, since it seems to have a simpler structure than [FORMULA], whose appearance seems to be affected by overlapping finger material. The geometrical form of the cloud model is shown in Fig. 15. The cloud model consists of a cylindrical column of gas with a spherical core embedded towards its tip. The spherical core has an internal density profile with a maximum number density of n (H 2) [FORMULA] 2 105 cm-3, decreasing to n (H 2) [FORMULA] 2 104 cm-3, which is the average value of material in the finger. These estimates allow a simple first order model of the dense clumps observed near the fingertips to be computed.

[FIGURE] Fig. 15. Geometric configuration adopted for the model finger, showing the size and shape used, along with the co-ordinates of the sample points used for the calculation grid.

It was assumed that this column of gas is subjected to anisotropic heating by the FUV field, with an incident ionising flux corresponding to [FORMULA] 2 103 G 0 on the top of the finger, and [FORMULA] 10 G 0 on the sides. The finger receives no illumination from below, since it appears to be well shielded from that direction. Starting from the density and UV field, the chemical and thermal structure of the cloud can be estimated, using the thermal and chemical models described below.

4.1. Thermal modelling

The thermal model used in the calculations presented here is similar to that used in Nelson & Langer (1997), but with some extensions described in 4.1.1 and the Appendix. We describe each contributor to the thermodynamic balance of the gas in turn, starting with those processes responsible for heating the gas, followed by the cooling mechanisms. We note that these models are designed to examine the cloud properties interior to the ionisation front, so we do not attempt to model the outer region of the cloud.

4.1.1. Grain photoelectric heating

Ignoring gas heating by the EUV field at the ionisation front, the dominant contribution to the heating of the gas in the outer layers of the M16 nebula is then through the photo-ejection of electrons from dust grains. This is caused by the incident FUV flux from the group of O stars located about 2 pc above the three gaseous columns. We adopt the approach used in Nelson & Langer (1997) for modelling this process, with the modification that the effect of grain charging on the grain's work function is now included, since significantly higher UV fields are being considered in this work.

The photoelectric heating rate is given by (Hollenbach et al. 1991):

[EQUATION]

where n 0 = n (HI) + 2n (H 2), the total number density of hydrogen atoms, G 0 is the standard interstellar UV field (Habing 1968), A v is the visual extinction, [FORMULA] is a factor which accounts for the increased extinction of the UV field due to dust grain scattering, and [FORMULA] accounts for the reduced efficiency of heating when the grains become positively charged, leading to an increase in the grain work function. The above formula is usually applied to 1-D slab models of clouds, but in this work we are interested in developing a cloud model in which the 3-D nature of the object is approximated. We therefore adopt the approach used in Nelson & Langer (1997), and calculate the heating at an arbitrary point in the cloud as if it were coming from six directions, namely along the positive and negative Cartesian co-ordinate directions. When computing the photoelectric heating rate at a point in the cloud, we integrate along each of these Cartesian co-ordinate directions to the cloud surface in order to calculate six values of Av. An approximation to the previous equation is then used to calculate the heating:

[EQUATION]

The heating at any point within the cloud is then estimated by considering the six heating rates coming from the positive and negative x , y , and z directions. Note that the value of G 0 clearly will change with direction, so that anisotropic UV fields may be modelled using this formula. When running the models, we assume that the cloud is primarily illuminated from above (the positive z direction), to a lesser degree from the sides of the cloud (x and y directions), and receives no illumination from below (negative z direction). The values of G0 have typical values of G 0 (z+) = 2000, G 0(x+/-) = G 0(y+/-) = 10, G 0(z-) = 0, where the +/- superscripts refer to the positive and negative co-ordinate directions.

4.1.2. Gas-dust thermal exchange

When the kinetic temperature of the gas molecules differs from that of the dust grains, there is a thermal energy exchange through collisions. We assume that when a molecule hits a dust grain, it sticks to it, and is later re-emitted carrying away energy [FORMULA] 3/2 T d, where T d is the dust temperature. The rate of energy exchange between the gas and dust can then be simply estimated following Goldsmith & Langer (1978), where the energy exchange rate is given by:

[EQUATION]

It should be noted that gas-dust interactions only play a major role in the thermal evolution of the gas where the number density n 0 [FORMULA] 105 cm-3, as a consequence of the quadratic dependence on n0.

We employ the expression suggested by Hollenbach et al. (1991) to calculate the dust temperature:

[EQUATION]

where [FORMULA] = 3 1015 Hz, T 0 = 12.2 G[FORMULA] K, and [FORMULA] = 0.001.

The model of dust heating used in deriving this expression assumes that the radiation incident on a grain is made up of contributions from the attenuated UV field, the 2.7 K background, and an infrared radiation field produced by dust emission at the surface of the cloud. This infrared emission is produced in a layer of thickness [FORMULA] [FORMULA] 1, which corresponds to an optical depth of [FORMULA] [FORMULA] 0.001 at 100 µm, and an equilibrium temperature T 0 =12.2 G[FORMULA] K. However, since there is thermal exchange between gas molecules and dust grains, dust temperatures derived using the previous equation should be interpreted as upper limits. High-density gas in shielded regions is usually cooler than the dust in clouds suffused with large UV radiation fields, and so will tend to cool the dust. We find that the dust temperature in the shielded regions is T d [FORMULA] 20 K, which agrees well with the values inferred from the observations.

4.1.3. Heating by cosmic rays and H2formation

The cosmic ray heating term is:

[EQUATION]

where [FORMULA](H 2) is the primary cosmic ray ionisation rate of H 2, and [FORMULA] is the energy deposited as heat after ionisation. Following Goldsmith & Langer (1978), values of [FORMULA](H 2) = 2.0 10-17 s-1 and [FORMULA] = 20 eV were adopted, leading to a total cosmic ray heating rate [FORMULA] = 6.4 10-28 n(H 2) ergs cm-3 s-1.

An additional source of heating in molecular clouds is H 2 formation on grains. Some fraction of the 4.48 eV binding energy of the released H 2 molecule is released as kinetic energy, and transferred to the gas as heat. In a steady state, under conditions where only cosmic rays destroy H 2 molecules, and at high densities where most of the hydrogen is in molecular form, H 2 formation heating can be considered as another term in the cosmic ray heating term (see discussion in Goldsmith & Langer 1978). Accordingly, the heating rate can be expressed as:

[EQUATION]

where Q [FORMULA] is the energy released as heat. We adopt a value of Q [FORMULA] = 2.0 eV in the work presented here.

4.1.4. CO, CI, CII, and OI line emission

The cooling due to CO, CI, and OI line emission is the same as that employed in Nelson & Langer (1997). The CO cooling function used is an analytical fit to the cooling curves of Goldsmith & Langer (1978), and has the correct limiting behaviour in both the optically thin and optically thick limits. The cooling formulae for CII and O I were originally taken from the work of Chieze & Pineau des Forêts (1987). The cooling due to CII has been modified, however, to account for the effects of collisional de-excitation. We note that the cooling due to CII and OI dominates in the warm, photoionised outer layers of molecular clouds, but contributes little to the thermal evolution in the cooler, shielded regions in the cloud interiors. The CI emission was calculated according to the formula given in Nelson & Langer (1997), using collision rates and Einstein A coefficients taken from Genzel (1991).

One heating mechanism that has been omitted is through collisional de-excitation of IR pumped OI (63 µm) fine-structure transitions. The work of Hollenbach et al. (1991) indicates that the heating due to this process is similar to the dust heating rate, however in the case of the Eagle Nebula, simple calculations indicate that its inclusion has a negligible effect on the calculated gas temperatures.

4.2. Chemical model

The chemical model used here is the same as that used by Nelson & Langer (1997) in dynamical collapse calculations for isolated molecular clouds. This model is designed to capture the essentials of the carbon and oxygen chemistry, since these species and their molecules are responsible for the majority of the cooling that occurs in the cloud. In the model, the following species are evolved: CO, CI, CII, HCO+, O I, He+, [FORMULA], OHx, CHx, M+ and e . Here OHx represents the species H2O, OH and O2, and CHx represents CH2 and CH. These species are grouped together for the sake of simplicity, and because their reactions occur at similar rates and along similar pathways. The M+ term represents ionised Fe, Na, Mg and Ca, which provide a background source of electrons in highly shielded regions.

The chemical reactions which are evolved with time, and their associated reaction rates [FORMULA] are listed in the Appendix.

4.3. Approach

The model that we have described is by its nature a fully time-dependent one. In developing a model for the M16 gas column, we take as a starting premise that the columns have only been exposed to the ionising radiation of the nearby O stars for the last 105 years. This is approximately the crossing time for the ionisation-shock front to propagate into the cloud.

Our approach to the chemical and thermal modeling has been, therefore, to evolve the chemical state of the clouds for 2 Myr, using the initial conditions described in the Appendix, and in the absence of an UV field. After this time a steady state is attained. We then use this steady state as the initial conditions for a calculation in which the UV field is switched on, and the chemical state is evolved for 105 years. We note that the chemical time scale is substantially shorter than this, so that the chemical profile of the cloud evolves to a steady state within this time.

Fig. 16 shows the density and temperature, and the chemical abundance profiles along a cut across the finger located 0.1 pc behind the photo-ionised outer edge of the fingertip. This computation corresponds to 10 5 years after the switch-on of the UV field. Fig. 16a) shows the density and temperature profile through the dense clump located close to the tip of the finger. We note that the temperature tends towards a value of [FORMULA] 20 K inside the dense clump, close to that estimated from the SCUBA data. This follows from the fact that the dust and gas are strongly coupled at these relatively high densities. The chemical profiles (Fig. 16b-d) are such that CO is dominant throughout the Finger, with CI and C II becoming prominent towards the edges of the cloud. The OHx species (H2O, O2, and OH) are very abundant in the cloud interior.

[FIGURE] Fig. 16a-d. A cut across the modelled finger at a distance of 0.1 pc in from the photoionised edge (i.e. across the core at the tip of the finger). Panel a  shows the adopted density and calculated temperature profiles, b -d  show the computed molecular abundances 105 years after switching on the UV field. The column density matches that which is observed when convolved to the resolution of the J= 3-2 C18O data.

The tip of the finger is located at z = 0.325 pc from the base of the model (the calculation does not attempt to model the whole finger - but just concentrates on its upper portion). The temperature is [FORMULA] 200 K at the cloud tip, and decreases rapidly to [FORMULA] 20 K in the cloud interior. The gas outside of the dense core, but deep in the cloud interior, has a slightly lower temperature because of gas-dust thermal coupling in the denser regions. Fig. 17b shows the extinction, and Figs. 17c-e show the computed chemical abundances.

[FIGURE] Fig. 17a-e. This shows in a  the density, temperature, b  the extinction and in c -e the chemical abundance profiles along the symmetry z -axis of the modelled finger with the base of the finger on the left, and the tip at the right. The tip of the finger is located at z = 0.325 pc from the base of the model, so this is what we take as `average material' lying shielded from the main radiation field by the core at the tip of the Finger.

The change in visual extinction moving into the cloud from its tip along the symmetry axis is plotted in Fig. 17b, and the chemical profiles are shown in Figs. 17c-e. Fig. 17c indicates that CO becomes abundant within a few Av of the edge of the cloud, and that the between this region and the edge, carbon is primarily locked in CII and CI. The OHx species (H2O, O2, and OH) are also observed to be in abundant within a few A v of the edge the cloud. The CO profile rises more steeply than that of some other species due to CO self-shielding.

Integrating the emission from the models, we predict that the total CO emission from the finger = 6.7 1031 ergs s-1 = 1/60 [FORMULA], and the total CI emission from the finger = 2.3 1030 ergs s-1 = 1 / 1700 [FORMULA]. Thus the CI cooling is therefore relatively insignificant in comparison to the situation seen elsewhere (Israel et al. 1995a, b).

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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