[tel-00726959, v1] Caractériser le milieu interstellaire ... - HAL - INRIA

A&A 526, A47 (2011)tel-00726959, version 1 - 31 Aug 2012Fig. 1. Spectra between 524.2 and 525.5 GHz observed towards Orion-KL with Herschel-HIFI (filled histogram) and LTE model produced withWeeds (continuous black line). The rest frequencies of several detected methanol lines are indicated.where c is the light speed and D is the diameter of the telescope.T bg is the brightness temperature of the background emission,i.e. the physical temperature that would have a black body producingthe same background continuum emission (e.g. 2.73 Kfor the cosmic microwave background). T ex is the excitation temperature,and the opacity τ (ν) is:τ (ν) =c2 N tot∑(8πν 2 A i g i u e −Ei u /kT ex ehν i 0 /kT ex− 1 ) φ i (4)Q(T ex )iwhere the summation is over each line of the considered species.Here N tot is the total column density of the species considered,Q(T ex ) is the partition function, A i is the Einstein coefficient ofthe i line, g i u and E u are the upper level degeneracy and energyof the i line, and φ i is the i line profile function. The latter isgiven by:φ i 1=σ √ 0) 2 /2σ 2 (5)2π e−(ν−νiwhere ν i 0the is i line rest frequency and σ the line width in frequencyunits at 1/e. σ can be expressed as a function of the lineFWHM in velocity units ΔV as follows:ν i 0σ =c √ ΔV. (6)8ln2Note that some of the model parameters may be degenerate incertain cases. In the optically thick or thin limits, the source sizeand temperature or the size and column density are degenerate,respectively (see Eqs. (1) and(3)). This degeneracy can be usuallylifted if both thick and thin lines are present in the survey,or if lines from an rare isotopologue are detected together withthe main one (e.g. 13 CH 3 OH and CH 3 OH). The source size mayalso be constrained from interferometric observations.Several components with e.g. different kinetic temperature orcolumn density can be included in the computation. For this, weassume that the various components are not coupled radiatively– that is a photon from one component can not be absorbed bya another, foreground component – in which case the emergingspectrum is simply the sum of the brightness temperature ofeach components given by Eq. (1). Each of these component canbe Doppler-shifted with respect to each other, which is usefulwhen modeling sources with several components at different velocities.It is also possible to compute the spectra from severalspecies; this is done by a summation of Eq. (1) over each specie.The column densities, kinetic temperatures, Doppler widthand source sizes for each species and components are read froma text file. Einstein coefficients, upper level degeneracy and energiesas well as the partition functions are taken from spectralline catalogs. Because these catalogs usually give the partitionfunctions at a few temperatures only, the partition function atthe user temperature is computed from a linear interpolation (orextrapolation if the user given temperature is outside the rangeof temperature provided in the catalog). When computing thesynthetic spectrum, a frequency sampling corresponding to theminimum ΔV divided by 10 is taken (or a frequency samplingequal to that of the observed spectra, if it smaller than the minimumΔV divided by 10). This ensures that the sampling at allfrequencies and for all species and components is sufficient. Atthe end of the computation, the synthetic spectrum is re-sampledto the same channel spacing than the observed spectrum in orderto take the channel dilution factor into account. This allows fora direct comparison between the synthetic and observed spectra.In Fig. 1, we show an example of such a modeling. The figureshows a spectrum between 524.2 and 525.2 GHz observedtowards Orion-KL with Herschel-HIFI as part of the HEXOSguaranteed time key program (Bergin et al. 2010). These datahave been already presented by Wang et al. (2010). SeveralA47, page 4 of 5