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I ran across this paper's title η Carinae: high angular resolution continuum, H30α and He30α ALMA images (arXiv) and see that the body of the paper also mentions "H40α, H30α and H29α".

What notation is this, in which contexts is it used, and what exactly to the numbers and greek letters denote? (I'm assuming that H and He are elements).

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These are the names of recombination lines. In terms of the Bohr model of the atom:

The electron in a Bohr atom can fall from the level ($n+\Delta n$) to $n$, where $\Delta n$ and $n$ are any natural numbers (1, 2, 3, …) by emitting a photon whose energy equals the energy difference $\Delta E$ between the initial and final levels. Such spectral lines are called recombination lines because formerly free electrons recombining with ions quickly cascade to the ground state by emitting photons. Astronomers label each recombination line by the name of the element, the final level number $n$, and successive letters in the Greek alphabet to denote the level change $\Delta n$: α for $\Delta n=1$, β for $\Delta n=2$, γ for $\Delta n=3$, etc. For example, the recombination line produced by the transition between the $n=92$ and $n=91$ levels of a hydrogen atom is called the H91α line.

So H30α indicates the transition from levels 31 to 30 of the hydrogen atom.

As noted by Peter Erwin in the comments, the series with low $n$ have traditional names. In particular, the Balmer series (transitions to $n=2$) are traditionally known as Hα, Hβ, etc. The first four of these are in the visible part of the spectrum.

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    $\begingroup$ And of course the low-$n$ transitions have traditional names: H1$\alpha$ is Lyman-$\alpha$, H2$\alpha$ is H$\alpha$, H3$\alpha$ is Paschen-$\alpha$, etc. $\endgroup$ – Peter Erwin Oct 29 at 16:02
  • $\begingroup$ @PeterErwin I wonder which of those low-n transitions would also be referred to as resonant lines by astronomers as well? $\endgroup$ – uhoh Oct 29 at 22:31
  • $\begingroup$ Thank you for this concise yet thorough answer! $\endgroup$ – uhoh Oct 29 at 22:36
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    $\begingroup$ @uhoh Only transitions directly to/from the ground state $n = 1$) of the atom are traditionally considered "resonant" lines. So, of the transitions I listed, only Lyman-$\alpha$ is a resonant line. $\endgroup$ – Peter Erwin Oct 30 at 8:09

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