EDP Sciences
Free Access
Volume 552, April 2013
Article Number A40
Number of page(s) 12
Section Interstellar and circumstellar matter
DOI https://doi.org/10.1051/0004-6361/201219567
Published online 20 March 2013

Online material

Appendix A: Derivation of used formulas

Appendix A.1: Rotation and kinetic temperature

As described in Sect. 3, the hyperfine fit routine returns the best fit parameters of the brightness temperature, Tmb, optical depth, τ, velocity of rest, Vlsr, and line width FWHM, Δv, (with errors).

The brightness temperature Tmb describes the brightness temperature of a source and depends on the beam this source is observed with by returning the mean value over this beam. It is needed to calculate the excitation temperature.

The excitation temperature Tex is no physical temperature, but describes the ratio between two population levels u(p) and l(ow) via (A.1)where ni is the numbers of particles in the state i and gi its statistical weight. ΔE represents the energy difference between the states and k the Boltzmann constant. For the IRDCs temperatures above 10 K are expected. Thus, (by applying the Rayleigh-Jeans law) the brightness and excitation temperatures are simplified connected by (A.2)with τ being the source’s optical depth and TBG the background temperature (being about 2.7 K). ηf represents the beam-filling factor. It describes the fraction of the antenna pattern being received from the source. In the case of extended and not clumpy sources, the beam-filling factor is equal to 1. We assume this case for the sample. The excitation temperatures are calculated for both transition lines.

The rotation temperature Trot is defined similar to the excitation temperature by (A.3)But in contrast to the excitation temperature, the rotation temperature does not describes the ratio between different levels being split by inversion, but between the levels of quantum numbers J and K (total angular momentum and its absolute projection along the z-axis). We are interested in the rotation temperatures of the metastable inversion levels with J = K (non-metastable inversion levels: J > K). These levels cannot be populated or depopulated by radiation. Therefore, their population numbers are high enough to emit measurable line intensities. Furthermore, metastable levels interact only via collisions (Ho & Townes 1983; Schilke 1989) and are useful for the calculation of gas temperatures.

The remaining problem is that we do not know the exact population numbers and have to approximate them by the column densities. For lines being observed in the same region the ratio of the population numbers should be equal to the ratio between the corresponding column densities depending on the ratio of the lines’ optical depths: (A.4)where τJK is the optical depth of the (J,K)-inversion line, ΔvJ its line width FWHM, Nl(J,K) its column density and its excitation temperature. Because only the (1,1)- and (2,2)-inversion lines have been studied, it is J = K = 2 and J′ = K′ = 1 in the following calculations.

Inserting this into Eq. (A.3), one gets: (A.5)where K (Ho & Townes 1983) and (A.6)Following the instruction of Schilke (1989) and Ho & Townes (1983), there are four cases one has to consider for calculating the rotation temperature. These cases differ in the optical depth of each inversion line:

  • 1.

    both inversion lines are optical thick:(A.7)where τii is the optical depth of the (i, i)-inversion line and fi is the relative intensity of the main hyperfine component. For the given transitions, there is f1 = 0.5 and f2 = 0.796 (Ho & Townes 1983).

  • 2.

    only the (1,1)-inversion line is optical thick and the (2,2)-inversion line is optical thin: (A.8)

  • 3.

    both inversion lines are optical thin: (A.9)where represents the brightness temperature of the (i, i)-inversion line.

  • 4.

    only the (1,1)-inversion line is detected, the (2,2)-inversion line is not: (A.10)In this case, the (2,2)-inversion line lies within the noise. To be able to continue with the calculations, has been given the triple value of the average root-mean-square noise (rms = 0.0925 K). It is important to emphasise that in this case it is not possible to derive exact rotation temperatures, but only upper limit estimations!

Having the rotation temperature of ammonia, we were able to calculate the kinetic temperature of a source’s gas by using the approximation of Tafalla et al. (2004): (A.11)This approximation has been derived with Monte Carlo models and gives an accuracy of 5% in the range between 5 and 20 K. Because the majority of the sample IRDCs are within this range, their errors are set to this.

Appendix A.2: Column density

As mentioned before in Sect. A.1, the column density is related to a source’s optical depth (Schilke 1989). If one neglects the background radiation and expresses the excitation temperature in terms of the optical depth, following Schilke (1989), the column density of the (1,1)-inversion level N11 can by calculated by (A.12)where μ = 1.476 Debye is the electric dipole moment and ν1 = 23   694.496 MHz the laboratory frequency of the NH3 (1,1)-inversion transition. To derive the total column density,

NNH3, we assumed that the sources are in thermal equilibrium. In this case, the column density follows a Boltzmann distribution. Additionally, we took into account that Eq. (A.12) calculates the column density of para-NH3 being the ammonia inversion levels with K ≠ 3n (n being an integer) where the hydrogen spins are not parallel. In contrast, the states with parallel hydrogen spins are called ortho-NH3 (with K = 3n). The statistical weight gJK of ortho-NH3 is twice the one of para-NH3. Therefore, for calculating the total column density of ammonia, one has to take the triple of column density of para-NH3 (cf. Schilke 1989): (A.13)

thumbnail Fig. B.1

ATLASGAL dust emission map with indicated sample IRDCs in the range of l = 27° to 51° (Schuller et al. 2009). The red triangles indicate the IRDCs with detected ammonia lines, the green triangles the ones without detected ammonia lines.

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thumbnail Fig. B.2

ATLASGAL dust emission map with indicated sample IRDCs in the range of l = 12° to 27° (Schuller et al. 2009). The red triangles indicate the IRDCs with detected ammonia lines, the green triangles the ones without detected ammonia lines.

Open with DEXTER

© ESO, 2013

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