= Discharge spectra and the Bohr model
In https://doi.org/10.1080/14786441308634955[On the Constitution of Atoms and Molecules, Part I] (1913), pp. 8–9, <Niels Bohr> interprets existing <hydrogen> discharge and astronomical spectra. Electrical discharge in a vacuum tube removes <Electrons>; recombination supplies <spectral lines>. This is a theoretical interpretation, without a reported <spectrograph> of his own.<On the Constitution of Atoms and Molecules, Part I by Niels Bohr>[Part I, section 2, pp. 8–9: existing spectra]{ref}
An actual optical obstacle appears in <Theodore Lyman>'s measurements: <hydrogen> discharge light entered an evacuated <spectrograph> directly through a slit, avoiding a transmitting window. A concave reflection <diffraction grating> dispersed the light onto photographic film. Pumping preserved the vacuum; a second slit supplied a known spectral displacement for <wavelength> calibration. Gas contamination and vacuum leaks limited the measurement. This apparatus resolved ultraviolet <spectral lines>; it did not image <orbits>.<Theodore Lyman, Pioneer of Ultraviolet Vacuum Spectroscopy by Paul Horowitz>[Paul Horowitz, Theodore Lyman, Pioneer of Ultraviolet Vacuum Spectroscopy, pp. 3–4, figures 2–5: windowless discharge connection, grating, film, pumping and dual-slit calibration]{ref}
For a separate original calibration exercise, suppose an idealized <spectrograph> returns synthetic vacuum line positions $x_n=\lambda_{n\to2}+b$ for $n=3,4,5,6$, with chosen additive offset $b=0.200\,\mathrm{nm}$. Assume its dispersion scale is already known and only this offset is uncertain. Compare the finite-<mass> prediction with the fixed-nucleus prediction $\lambda_n^{(\infty)}=(\mu/m_e)\lambda_{n\to2}$. Calibrate each model using only the $3\to2$ line: $\widehat b=x_3-\lambda_3^{\rm model}$. Then hold that offset fixed and test the other lines with residuals
$$
d_n=x_n-\widehat b-\lambda_n^{\rm model}.
$$
For the finite-<mass> model every $d_n=0$. The fixed-nucleus model has $\widehat b=0.55733\,\mathrm{nm}$ and residuals $d_4=-0.09264\,\mathrm{nm}$, $d_5=-0.12100\,\mathrm{nm}$, $d_6=-0.13400\,\mathrm{nm}$. Algebraically $d_n=(1-\mu/m_e)(\lambda_n-\lambda_3)$: one additive calibration cannot absorb a multiplicative change of the <wavelengths>. Allowing an unknown multiplicative dispersion factor as well would make the two models degenerate in this exercise, so independent scale calibration matters.
\Image[/bohr-balmer-line-calibration-comparison.png]
{title=<Balmer series> predictions tested after calibrating one line}
{description=Original synthetic calibration exercise with chosen $b=0.200\,\mathrm{nm}$. Left: finite-<mass> vacuum predictions and deliberately offset synthetic positions, displayed on a separate local <wavelength> axis for each line. Right: model residuals after anchoring the $3\to2$ line; the remaining three lines test the prediction. Symbols are exact synthetic inputs and calculated results, not historical observations, error bars or an intensity trace.}
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