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Astron. Astrophys. 320, 957-971 (1997)

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4. Excitation models and physical properties derived from [FORMULA]

[FORMULA] is a symmetrical top molecule which has been successfully used to determine the kinetic temperature of molecular clouds (Cummins et al. 1983; Loren & Mundy 1984; Churchwell et al. 1991; Olmi et al. 1993). As in the case of other symmetrical top molecules, transitions, between J levels in different K ladders are radiatively forbidden. Within a J ladder, collisional excitation is thought to dominate radiative excitation. Thus the relative population of the K components of a given J depends mainly on the kinetic temperature of the molecular gas. Since different K components of a given rotational line can be observed simultaneously, the calibration should do not affect the kinetic temperature determinations. Furthermore, [FORMULA] is also a molecule with a relatively high dipole moment, which can also be used to estimate hydrogen densities from measurements of different rotational transitions. Therefore [FORMULA] is one of the few molecules which one can use to determine both the kinetic temperature and the density structure in molecular clouds.

Different methods have been used to derive the kinetic temperatures in molecular clouds from [FORMULA] (Churchwell & Hollis 1983, Cummins et al. 1983, Loren & Mundy 1984). In the first approach, the kinetic temperature was derived from rotational diagrams in which a single excitation temperature was used to describe the population distribution for all levels. A second approach (Churchwell & Hollis 1983) considered the previous method too approximate and two temperatures were obtained, a rotational temperature connecting different rotational levels in the same K ladder and the kinetic temperature that describes the relative population between states in different K ladders. Cummins et al. (1983), compared the results obtained from this method with those obtained for a statistical equilibrium model for which two sets of collisional coefficients were derived from an analogous molecular species such as OCS. They concluded that when observing high rotational transitions ([FORMULA]) the two temperature approach gives reliable estimates for the kinetic temperatures but less accurate estimates for hydrogen densities.

To derive the kinetic temperature and density in Sgr B2, we have made a multitransition analysis of the [FORMULA] data by solving the statistical equilibrium equation under the assumption that the Large Velocity Gradient approximation (LVG) is valid. Due to the complexity of the line profiles in Sgr B2 (see 3.2), with several broad velocity components, our model also incorporates a more complete treatment than previous LVG analyses. Our model considers both species, A and E (ortho and para) separately with blending between the K=0 and K=1 lines. The model also incorporates the most recent collisional cross sections derived by Green (1986). The blending between the K=0 and K=1 components was taken into account by considering total overlapping to calculate both the opacity and the source function for the [FORMULA] and [FORMULA] lines. If [FORMULA] is the opacity for the [FORMULA] line and [FORMULA] for the [FORMULA] line, [FORMULA] is the velocity difference between both lines and [FORMULA] the linewidth, the final opacities [FORMULA] and [FORMULA] for lines [FORMULA] and [FORMULA] were estimated by,


and the final source function [FORMULA] and [FORMULA] for each line are,


where [FORMULA] and [FORMULA] are the source functions for the [FORMULA] and [FORMULA] lines respectively.

The complex velocity field of the molecular gas in Sgr B2 has been taken into account by the following: we have derived densities and kinetic temperatures by simultaneously fitting intensities and line profiles of all the K components for all rotational lines, as well as the corresponding lines of [FORMULA]. Since the observed line profiles are not single gaussians, we have also considered that the observed profiles could also be caused by two velocity components whose radial velocities and line widths were derived from our [FORMULA] data. The free parameters used in the fits were [FORMULA] column density, hydrogen density and kinetic temperature. A/E ratios used in the fits were always within 20% of the expected LTE value of 3 to 1. For the [FORMULA] isotope line we have used the typical [FORMULA] ratio of 20 for the Galactic Center (see for example, Wilson & Rood 1994)

Fig. 7 shows a sample of profiles of the J=5-4 and J=8-7 line with the fits superimposed. In order to improve the signal to noise ratio and to obtain the same angular resolution for all transitions, we have averaged several spectra with similar profiles, radial velocity and linewidth. This averaging provided an effective resolution of [FORMULA]. Since the emission in the envelope is more extended than the beam we have used the [FORMULA] scale to determine the physical conditions for this component. For the hot cores, where the intensities are larger, we were able to obtain an effective angular resolution of [FORMULA]. In this case, the sources have smaller sizes and we used the main beam temperature scale.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998