Age determination of stars In the report 'New observations of the most distant galaxies close in on cosmic dawn', an astronomer states: 'Using a well-understood age indicator based on the amount of hydrogen absorption seen in the atmospheres of different stars, we were able to infer that, even at these early observation times, these galaxies were already between 200 and 300 million years old'.
Question: how does this age indicator work? how can the age of a star (or a galaxy) be estimated from the amount of hydrogen absorption in the star's atmosphere?
report: ras.ac.uk/news-ans-press 2021.06.24
 A: The age discriminator in question is the "Balmer jump" or "Balmer break". This is a discontinuity in flux from stars that is seen across the limit of the Balmer series, at a (rest) wavelength of 365 nm. The discontinuity is caused by the ionisation of hydrogen atoms out of the $n=2$ level. This causes a photoionisation absorption continuum with an abrupt edge at 365 nm.
This absorption feature is seen in stars in a temperature-dependent way. For hot O-stars, the $n=2$ level is depopulated and the edge is not seen. For cooler B-stars the absorption edge gets stronger and it is at maximum strength in A-stars (which also have the strongest Balmer absorption lines) because the $n=2$ population is maximised. For cooler stars the atoms occupy the $n=1$ level and the edge is also not seen.
Now, given the inverse mass-dependence of the lifetimes of high-mass main-sequence stars, we can see that if the light from a galaxy is dominated by late-B and A-type stars then the integrated light of that galaxy would show a strong Balmer-break, but obviously that would be redshifted by a factor of $(1+z)$, where $z$ is the redshift of the galaxy. But late-B and A-type stars are at most a few hundred million years old. In fact, to become the dominant population after a burst of star formation that likely produces stars over a range of masses, then because the more massive stars would be more luminous, it would need these more massive O-stars to have died off, otherwise they would dominate the light$^\dagger$ and the Balmer-break would not be seen. Note that it is the integrated spectrum of the stellar population that is age sensitive, not that of any individual star.
Therefore, the presence of a (redshifted) Balmer-break is telling you that the light is coming from a stellar population that is old enough for the late-B and A-stars to have become dominant, but not so old that the A-stars have died. In practice, what you do is compare the spectrum (or spectral energy distributions in the case of these high redshift galaxies) with the results of stellar population synthesis models (see picture below), match the magnitude of the Balmer-break and hence infer when star formation must have begun (in the past) in order to have an A-star dominated population when the light from the galaxy was emitted.

This plot taken from Laporte et al. (2021) shows the basic idea. The blue points are the data, the grey spectrum is the best fit model spectrum produced by a stellar population synthesis shifted to the redshift of the galaxy. The red points show this spectrum folded through the photometric band responses. The insets show the probability distribution vs the redshift (right) and a contour plot of likelihood, varying the age and total mass of the stellar population. In this case, the best-fit redshift is about 9.6 with a stellar population of a bit more than a billion solar masses that formed 0.3 Gyr before the light was emitted our way. The Balmer break is the obvious discontinuity at about 4 microns ($(1+9.6)\times 365$ nm). The big drop-off to the left is the Lyman-break, the position of which is what constrains the redshift.
$^\dagger$ A typical birth mass distribution might be that $N(M) \propto M^{-2.3}$ (a Salpeter mass function). But the luminosity of main sequence stars has $L \propto M^4$ at these masses, so we see that even though O-type stars might be fewer, their luminosity would still dominate over the more numerous lower mass stars unless the majority have died.
