Abstract
This chapter derives the physics ab initio that underlie the formation of radio recombination lines in astronomical objects. It includes natural, thermal, and pressure broadening of the spectral lines; the radiation transfer of the spectra and their underlying freeāfree emission (Bremsstrahlung) through the ionized media; the excitation of the atomic levels; the frequency range over which lines may be detected from astronomical objects; and the sizes of excited atoms that can exist within interstellar environments.
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Notes
- 1.
N.B. One axis only. See Sect. 4.11 of Chapman and Cowling (1960) for a full discussion of MaxwellāBoltzmann statistics.
- 2.
This formula works well when v x āŖ c. Gordon et al. (1992) discuss the relationship between spectral red shift and velocity for Euclidean (Special Relativity) and cosmological (General Relativity) models.
- 3.
A useful expression is that the area I of a Gaussian line is
$$I = \left( \frac{\pi}{4\,\ln 2}\right)^{1/2} I(\nu_0)\, \Delta \nu_G$$((2.23))$$\approx 1.064\, I(\nu_0)\, \Delta \nu_G,$$((2.24))in terms of the intensity measured at line center I(Ī½0) and the full width of the line at half-intensity, the expression resulting from the integration of (2.19).
- 4.
A tertiary Stark broadening can also occur.
- 5.
Griem (1967) shows that they can be as large as 20% in some circumstances.
- 6.
For calculation of perturbations induced by an external field, parabolic coordinates can be more useful than spherical coordinates because of the asymmetrical nature of the charge distribution within the atom. See Sect. 6 of Bethe and Salpeter (1957) or similar texts.
- 7.
See, e.g., Sect. 4.11 of Chapman and Cowling (1960) or any other text on statistical mechanics for the general technique for deriving weighted speeds of a MaxwellāBoltzmann gas.
- 8.
Griem (1967) used this method by comparing the threshold QS frequency shift from the center of an RRL with the Doppler width. If this ratio ā« 1, then the validity region of the QS approximation falls far from the Doppler core, and the QS approximation cannot be used. Substitution of T = 104 K ā typical for an astronomical H ii region ā into his expression gave a ratio of several thousand for both ions and electrons. Griem thereby concluded that the QS approximation was an inappropriate model for the Stark broadening of RRLs by either ions or electrons. This is the same conclusion we reach above with (2.48) and (2.50).
- 9.
Some authors define a differently.
- 10.
The parameter Ī½ is sometimes listed as b.
- 11.
See also very recent observations described at the end of this section.
- 12.
The integration uses the integrating factor e āĻ and integration by parts.
- 13.
Oster (1961) discusses the history and problems of calculating the freeāfree emission coefficient in detail.
- 14.
In thermodynamic equilibrium, all temperatures are the same. In reality, various temperatures can differ from each other.
- 15.
At millimeter and submillimeter wavelengths where calibration procedures involve measurements of atmospheric emission, the units of antenna temperature can be quite different and unintuitive ā often symbolized by \(T_A^\ast\), \(T_R^\ast\), or \(T_{mb}^\ast\). Appendix F describes their relationship to astrophysical units.
- 16.
Some authors (cf. Spitzer (1978)) define the Einstein coefficient B in terms of energy density rather than specific intensity, the latter being the form commonly used by astronomers and the form we use here (cf. Sect. 90.1 of Chandrasekhar (1950), Sect. 4.1 of Mihalas (1978), or Sect. 6.5.3 of Allen's Astrophysical Quantities (Hjellming, 1999)). These two definitions are not interchangeable. In brief, for a transition of n = 2 ā 1, B 21 (specific intensity) = (4Ļ/c) Ć B 21(energy density).
- 17.
This term evolved from the āLadenburg f,ā a vestige of classical physics when the intensity of a spectral line was characterized in terms of the number of dispersion electrons per atom or, āoscillators.ā See Appendix D for detailed information. Also, see the discussion relating the quantum mechanical to the classical form of the oscillator strength (Kardashev, 1959).
- 18.
Here, B is again defined in terms of specific intensity rather than energy density. See the earlier footnote 16 for details.
- 19.
Equation (2.124) does not include the second term of the expansion for the line frequency given by (1.21). This omission will lead to an overestimate of I L by a few percent, depending on the frequency. The reader can improve the accuracy by adding a multiplicative factor like (1 ā 3Īn/2n 1) to the right-hand side of (2.124).
- 20.
The original mathematical expansion of Ī² given by Goldberg (1968) as his equation (22) has
$$\beta \approx \frac{b_{n_2}}{b_{n_1}}\left(1-\frac{kT_e}{h\nu} \frac{d\ln b_{n_2}}{dn}\, \Delta n\right),$$((2.128))which would create a cofactor of \(b_{n_2}\) in (2.132). Note the difference in subscript. Because that would be unphysical for an absorption coefficient, because \(b_{n_1} \approx b_{n_2}\), and because it is an approximation in any case, we write (2.132) as it stands.
Also, the definition of the correction factor Ī² has evidently evolved slightly. Brocklehurst and Seaton (1972) (BS) use
$$\beta \equiv \beta_{n_1,n_2} = \left(1-\frac{kT_e}{h\nu}\frac{d\,\ln b_{n_1}}{dE_{n_1}}\right),$$((2.129))which differs in the argument of the logarithm. Because n 2 ā n 1 ā” Īn ā 1 in many RRL observations of interest, it is a common practice to ignore the difference between \(b_{n_1}\) and \(b_{n_2}\) in applications of the corrective factors. Using the generic symbol n as a subscript usually involves no significant loss of accuracy and, in fact, BS adopt this simplification. Here, we include all subscripts for completeness, however.
- 21.
Goldberg (1968) introduced the symbol \(\gamma\equiv d\,\ln b_{n_2}\,\Delta n/dn\), which has now passed from common usage.
- 22.
Equation (2.132) shows that Ļ L will be negative in non-LTE situations where Ī² < 0. Hence, we use the absolute value of Ļ L in the criteria for this approximation.
- 23.
A more accurate correction factor (the term following the Ć symbol in (2.144)) is
$$\frac{b_{n_2}\left(1-\frac{\tau_C}{2}\beta\right)}{1-\tau_C}, \qquad \tau_c, |\tau_L|,\,(h\nu/kT_e) \ll 1.$$((2.145)) - 24.
Actually, Baker and Menzel (1938) subdivided Case A into a Case A1 and a Case A2, distinguished by the Gaunt factors that are used.
- 25.
The Storey and Hummer calculations of departure coefficients for a wide range of T e and N e are available from the Centre de DonnĆ©es Astronomique de Strasbourg via anonymous FTP to the directory /pub/cats/VI/64 of cdsarc.u-strasbg.fr. These files do not contain Ī² n values, however.
- 26.
In the equations above, the freeāfree optical depth Ļ C is also a measure of the freeāfree emission because of Kirchhoff's law of thermodynamics.
- 27.
In this particular model, because Ī· and ĻĪ½ change sign at n ā 95, our simple calculations of I L /I C exhibit a computational discontinuity due to insufficient significant figures, and we therefore show this region as a gap in the plot.
- 28.
Microwave amplification by stimulated emission radiation.
- 29.
Moran (2002) notes that this argument can also be reversed. The equation illustrates that āthe phase requirements are so stringent that no cosmic maser can operate as a spatially coherent amplifier. As a result, cosmic masers have virtually no intrinsic beaming properties ā¦ [and] are essentially temporally incoherent and spatially incoherent, unlike laboratory lasers with parallel mirrors.ā
- 30.
Watson et al. (1980) suggested the term dielectronic ācaptureā for this low-temperature process.
- 31.
This relationship is the specific intensity form. The relationship between B n ,m and A n ,m coefficients is a numerical rather than a directional one. The relationship results from the detailed balance requirement in TE that
$$n_1 JB_{1,2} = n_2A_{2,1} + n_2JB_{2,1}$$((2.171))such that the dimensions of JB 1,2 are the same as A 2,1, i.e., sā1.
- 32.
Using the approximate expression of (2.176) for the f mn reduces the numerical accuracy to two significant figures, at most.
- 33.
Figure 2.40 does not contain the more recent observations of Stepkin et al. (2007), which were obtained after the original edition of this book had been published.
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Gordon, M., Sorochenko, R. (2009). RRLs and Atomic Physics. In: Radio Recombination Lines. Astrophysics and Space Science Library, vol 282. Springer, New York, NY. https://doi.org/10.1007/978-0-387-09691-9_2
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