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Appendix A: Linewidth of recombination lines

RLs are broadened by several mechanisms: a natural broadening from the quantum uncertainty ΔE of the energy level E, a thermal/microturbulence Gaussian broadening from the motions of the emitting particles and of parcels of gas that are much smaller than the beam, the Stark (or “pressure”) broadening from the perturbation of the atomic energy levels by the electric field of neighboring charged particles, and a dynamical broadening from bulk flows (e.g., infall, rotation, outflow) in the gas. A detailed discussion of these processes and the physics of RLs can be found in Gordon & Sorochenko (2002).

For n > 20 (where Hnα is a n + 1 → n transition), the natural linewidth in velocity units is (A.1)where c is the speed of light. For H30α and H53α, Δv_N is 1.3 × 10^-3 km s^-1 and 0.5 × 10^-3 km s^-1, respectively. The natural broadening is negligible for radio and (sub)mm RLs.

The thermal distributions of velocities of both the individual particles and small pockets of gas (“microturbulence”) produce a Gaussian contribution to the broadening. Neglecting microturbulence, the thermal FWHM is (A.2)where k_B is the Boltzmann constant, m_H is the hydrogen-atom mass, and it is assumed that all the gas is thermalized to the electron temperature T_e. For a pure-hydrogen gas with T_e = 9000 K, Δv_th = 20.3 km s^-1.

The pressure broadening has a Lorentzian shape, and it increases with density and quantum number. Collisions with ions and electrons make different contributions. The contribution from ions has the form (Gordon & Sorochenko 2002) (A.3)where γ_i = 6−2.7 × 10^-5   T_e−0.13(n + 1)/100. For T_e = 9000 K and N_i = 10⁷ cm^-3, the H30α line is virtually free from ion broadening (0.04 km s^-1, and decreases close to linearly with decreasing T_e), whereas the H53α line is broadened by 5.1 km s^-1.

The collisions with electrons dominate the collisions with ions under most conditions. They produce a broadening of width (Smirnov et al. 1984): (A.4)For N_e = 10⁷ cm^-3, the electron broadenings for the H53α and H30α are ≈ 37.3 km s^-1 and 0.6 km s^-1, respectively.

Finally, the last source of broadening is bulk motions (Δv_dy) of the ionized gas, which may be in the form of outflows/winds, infall/accretion, and/or rotation. For unresolved observations, the method outlined here permits estimating the magnitude of these motions from the nonthermal linewidth of the line that is free of pressure broadening. Of course, ideally one wishes to spatially resolve these motions. ALMA in its Cycle 1 is now able to resolve the ionized-gas motions in sources as faint as ~10 mJy at the line peak at subarcsecond resolution.

Including all the contributions to the broadening, the total linewidth will have a Voigt profile with FWHM given by Eq. (1) in the text.

Appendix B: Cm and (sub)mm recombination lines

Here we discuss the advantages and limitations that cm and (sub)mm RLs have compared to each other. The cm lines are intrinsically fainter and optically thicker, and can be partially absorbed by the relatively high continuum opacity at their wavelengths (see e.g., Wilson et al. 2009). A possible drawback of (sub)mm RLs is that the free-free continuum may be contaminated by dust emission. Another problem is that they are out of LTE more easily than cm lines, so their interpretation may require careful modeling (e.g., Jiménez-Serra et al. 2011; Peters et al. 2012). As shown below, our simple LTE interpretation seems to be reasonable, but it is possible that a significant fraction of the 1.3-mm continuum comes from dust in the line of sight.

Let b_n be the ratio of the population of level n to its LTE population. Then the actual line absorption coefficient κ_L is related to the LTE coefficient κ_L,LTE by (Gordon & Sorochenko 2002) (B.1)with (B.2)While 0 < b_n < 1 expresses the non-LTE level depopulation, β_n can be negative when there are conditions for maser amplification.

In the optically-thin case (τ_C,1.3mm ~ 0.08 from the data), the non-LTE corrected line intensity I_L is (Gordon & Sorochenko 2002) (B.3)Using the tabulated values in the calculations of Walmsley (1990), b₃₀ ≈ 0.98 and β₃₀ ~−1.5 for a density N_e = 10⁷ cm^-3. Therefore, the intensity of the H30α line is only amplified by ~3%. The H53α is even closer to LTE, with b₅₃ ≈ 0.999 and β₅₃ ~ 0.5.

Sub(mm) RLs are more opticaly thin and intrinsically brighter than cm RLs. For a given T_e and N_e, and in the Rayleigh-Jeans regime (hν ≪ k_BT_e), the line absorption coefficient decreases linearly with frequency κ_L ∝ ν^-1, so the LTE line emission coefficient j_L = κ_LB_ν(T_e) ∝ ν. Similarly, the continuum absorption coefficient κ_C ∝ ν^-2.1, so the LTE continuum emissivity j_C ∝ ν^-0.1. Therefore, in the optically thin regime, the line-to-continuum ratio increases almost linearly with frequency S_LΔv/S_C ∝ ν^1.1. Under these assumptions, the expected ratio of the H30α to H53α line-to-continuum ratios is 6.4. However, the observed value is 1.6. If we assume that only ~32 mJy of the 1.3-mm continuum are due to free-free then the line-to-continuum ratio would increase to the expected value. This suggests that dust contributes a significant fraction to the (sub)mm continuum of BN. However, it is known (and we have confirmed this with the ALMA data set) that BN is not embedded in dense molecular gas. Therefore, it is unlikely that this line-of-sight dust arises in a dense core around BN.

© ESO, 2012