Intuitive connection between the periods of oscillation of Betelgeuse and the elemental concentrations at its core? (Betelgeuse; Saio et al. (2023))

Question

Preamble (yes it's long, but it's part of this question's premise, so need to spell it out)

Dr. Becky's recent video New study claims Betelgeuse supernova IMMINENT (decades not centuries!) | Night Sky News June 2023 links to the preprint Saio et al. (2023) The evolutionary stage of Betelgeuse inferred from its pulsation periods and between about 12:20 and 13:30 discusses Figure 6, how to read it, and how to get from estimated mass fractions of carbon at the center ($X_C(C)$ in the last column of Table 2) to an estimate time to supernova, the news^a being that it could be of order 10² to 10¹ years away.

Looking at Figure 6 I noticed that between about 10^4.2 and 10³ years before collapse (a period of about 15,000 years) there are no discernible changes in the mass fractions (at the core) of C, O or Ne (nor anything else). There are two more progressively shorter plateaus in concentrations once core C and O reach zero.

I assume that what's happening here is that nucleosynthesis and heat production is happening outside the core in some shell, and when that stops there's a partial collapse, pressure and temperature of the core increases, and the next element in the A=4n chain begins to burn.

Begin actual question

Saio et al. (2023) infers the evolutionary stage of Betelgeuse from its pulsation periods, and shown in Table 1 $P_1$ through $P_4$ (this work) are 2190 ± 270, 417 ± 24, 230 ± 29 and 185 ± 4 days.

Question: Intuitively, how can one understand the connection between the periods of oscillation of Betelgeuse and the elemental concentrations at its core, in the case of Betelgeuse as discussed in Saio et al. (2023)?

I understand they use a stellar numerical model, but what is it about the periods that gives us information about the star's internal structure that then correlates to the core concentrations?

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^aDr. Becky reminds us that this is a new preprint and has not yet been peer reviewed.

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Figure 6. Central abundances of various elements versus time (in logarithm of base 10) to collapse for the model of $M_i = 19 M ⊙$ with the initial rotation velocity $\nu_i = 0.2 \nu_{crit}$.

Answer 1

Sensationalist stuff and the $10^1$ to $10^2$ years till supernova claim is not made by the authors.

The radial pulsation frequencies of a ball of gas depend on mass and radius. For a given mass, the natural pulsation frequencies (the resonant frequencies and overtones of the effective cavity formed by the star's envelope) get smaller, and the pulsation periods get longer, for larger radii.

As Betelgeuse approaches the end of its life, theoretical models predict that its envelope will get larger$^1$ and its various pulsation periods will get longer. The connection with the composition of the core is indirect - it is merely that Betelgeuse would get larger during those later core burning stages.

The authors have predicted the pulsation periods for Betelgeuse in models that are consistent with its position in the HR diagram. The latter is quite uncertain because of its variability and uncertain distance. What they find is a good match with models at the high end of the luminosity possibilities. Since temperature isn't changing greatly during the evolution, we can directly translate high luminosity to large radius.

From the models, this high luminosity, large radius state is reached during core carbon burning. i.e. There is carbon in the core, so we are looking at the red line in Fig. 6. The authors say that nothing much happens to the surface of the star during and after core carbon burning, so there is no evidence of how close to the end it is there. The models they say are a good match to the observed pulsation periods covered a range of core carbon mass fractions from about zero to 0.17.

If we look at the red line in Fig.6 and take 0.0 to 0.17 as the maximum possible range of core carbon mass fractions then that I think is the origin of to 10-100 years before supernova claim (I would say more like 200 years). This is either naive or sensationalist. The authors do not say in their paper that 0.17 is the largest possible core carbon fraction that might fit the periods, which themselves have error bars. There must also be considerable uncertainties and assumptions that go into the models (and they only show one model at one mass and rotation rate). For example a core carbon mass fraction of just 0.21 would appear to allow Betelgeuse to be 10,000 years till supernova. A Bayesian fitting approach would surely pick that as more likely than that we are fortunate enough to see Betelgeuse with 10 years left to go.

If I were refereeing this paper, I would like the authors to be more definite about which models and carbon core mass fractions could be ruled out. So yes, it could be 10 years to go, but my reading would be that it is more likely to be thousands of years from a Bayesian point of view and possibly even tens of thousands of years unless higher carbon core fractions can be absolutely ruled out.

$^1$ Why does Betelgeuse get larger? The core pressure is proportional to density and temperature and inversely proportional to mean particle mass. The luminosity is strongly dependent on core temperature via the nuclear fusion reaction rates. As the core turns to heavier elements, the mean particle mass increases and the core temperature rises to maintain pressure and hydrostatic equilibrium. This increases the luminosity. The envelope has work done upon it and expands. The radius increases sufficiently to radiate the increased luminosity.