Supplemental Calculations

Appendices

C.1 Early Estimates of Stark Broadening

It is interesting to revisit the calculations of the effects of line broadening with regard to the detectability of RRLs as considered by van de Hulst. We try to reproduce his calculations below following the suggestions of Sullivan (1982).

The full width of a Gaussian line at half-intensity, Δν _T , due to thermal broadening is given by

$$\Delta \nu_T = \nu_0 \left(4 \ln 2 \frac{2kT}{Mc^2}\right)^{1/2},$$

((C.1))

where ν₀ is the rest frequency of the line and M is the mass of the radiating atom or molecule. Working from the van de Hulst (1945) and the Inglis and Teller (1939) papers, we derived the simplified expression for the Stark width, Δν _vdh , calculated by van de Hulst to be

$$\Delta\nu_{vdh} \approx \nu_0\left(\frac{3\times 10^6}{\nu_0}\right)^{3/5},$$

((C.2))

where we have used the inverted exponent suggested by Sullivan. In this case, the correct, simplified expression for the Stark width would be

$$\Delta\nu_S \approx \nu_0\left(\frac{3\times 10^6}{\nu_0}\right)^{5/3}.$$

((C.3))

The generalized expression for the line-to-continuum ratio used by van de Hulst to estimate the detectability of RRLs was

$$\frac{I_L}{I_C} = \frac{\nu_0}{\Delta\nu}\frac{h\nu_0}{kT}g,$$

((C.4))

where g is the ratio of the appropriate Gaunt factors and was taken to be ≈ 0.1. Substituting the above expressions for Δν and plotting the results gives Fig. C.1.

Fig. C.1 The line/continuum ratio of Hnα RRLs plotted against frequency. Calculations are based upon approximations used by van de Hulst (1945). Solid line assumes that line broadening is only thermal for a 10⁴ K environment. Dashed line assumes Stark broadening estimated by the alleged “incorrect” formula with the inverted exponent of 3/5. Dotted line: same Stark broadening formula but with the “correct” exponent, i.e., 5/3. Also shown are observations of the H220α and H109α RRLs from the Orion nebula with appropriate error bars

Inspection shows that the I _L /I _C ratio is indeed low for the calculation of Stark broadening using the 3/5 exponent, in fact, lower than the thermal case by at least two orders of magnitude in the radio wavelength regime from, say, 10⁷ to 10¹¹ Hz. Furthermore, the VdH values of I _L /I _C are so low to constitute unrealistic detection prospects for equipment available in 1945 when the paper appeared. From the presumed calculations shown here, we can easily understand why van de Hulst rejected the possibility of detecting RRLs.

On the other hand, his approximate calculations make sense if we “correct” them by inverting the exponent to 5/3. In these calculations, Fig. C.1 shows that thermal broadening dominates the line widths at frequencies above 900 MHz. Actual observations of the H220α and H109α lines from the Orion nebula fall in the appropriate positions. The former falls below the thermal line and the latter falls on that line. Stark broadening diminishes the I _L /I _C ratio for the H220α line but has virtually no effect on the H109α line.

We conclude that Sullivan's claim may be correct. An accidental inversion of an exponent would have changed van de Hulst's conclusion of whether or not RRLs would be detectable in radio astronomy.

C.2 Refinements to the Bohr Model

While the equations derived in Sect. 1.3 describe the salient features of recombination spectra, they need additional refinements for generality. For example, the electron orbits need not be circular in a classical sense; elliptical orbits are also possible. von Sommerfeld (1916a; 1916b) extended Bohr's work by generalizing the quantization of angular momentum to radial and to azimuthal coordinates:

$$L_r = n_r\frac{h}{2\pi}; \quad n_r=0,1,\ldots$$

((C.5))

and

$$L_\phi = k\frac{h}{2\pi}; \quad k=0,1,\ldots,$$

((C.6))

where n _r and k are called the radial and azimuthal quantum numbers, respectively. Accordingly, the diameters of the major and minor axes of an elliptical electron orbit can be written, respectively,

$$2a = \frac{2h^2}{4\pi^2 m_R e^2} \frac{{n}^2}{Z}$$

((C.7))

and

$$2b = \frac{2h^2}{4\pi^2 m_R e^2} \frac{nk}{Z},$$

((C.8))

where n = n _r + k. Dividing (C.7) by (C.8) shows the axial ratio 2a/2b = n/k.

While, classically, the energy of the orbiting electron does not depend upon the ellipticity and is still given by (1.14) after the substitution of the reduced mass m _R for m, the application of relativity (von Sommerfeld, 1916a) leads to a slight modification:

$$\begin{array}{rcl} E_{n,k} & = & -\frac{2\pi^2 m_R e^4}{h^2} \frac{Z^2}{n^2} \left[1 + \frac{\alpha^2Z^2}{n} \left(\frac{1}{k}-\frac{3}{4n}\right)+ \ldots\right] \\ & \approx & -2.17987\times 10^{-11} \left(1-\frac{5.48580\times10^{-4}}{M_A}\right) \frac{Z^2}{n^2} \end{array}$$

((C.9))

$$\times \left[1 + \frac{5.32514\times 10^{-5}Z^2}{n} \left(\frac{1}{k}-\frac{3}{4n}\right)\right] \quad {\rm ergs},$$

((C.10))

where M _A is the mass of the atom in amu and the calculated dimensionless “fine-structure constant” is

$$\alpha = \frac{2\pi e^2}{hc} = 7.29735337 \times 10^{-3},$$

((C.11))

compared with the currently accepted value of 7.297352533 × 10⁻³ (Audi and Wapstra, 1995).

The spectral line frequencies would then result in the usual way from

$$\nu = \frac{E_{n_2}{k_2}-E_{n_1}{k_1}}{h}.$$

((C.12))

The higher-order terms of α² in (C.9) are usually very small and can be neglected.