While reading Stanislaw Lem's essays on advanced civilizations, I had a question: When did the earliest generation of population 1 star systems form? How much older could they reasonably be than our star system?
[Physics] Oldest population 1 star system
astrophysicscosmologystars
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I work with stellar models, so I thought I'd chip in here. My instant reaction is that you shouldn't worry too much: determining the age of a star is difficult and different models will disagree (sometimes significantly!) on that age.
How reliable is this research?
I can't see an obvious reason to doubt the conclusion.
What method do they use to measure the age of such a star as Methuselah?
Basically, one tries to measure as many properties about the star as accurately as possible, and then find the best fitting stellar model. These models are solutions to a set of differential equations (in time and one spatial dimension) that tries to capture all the relevant physics that determines how stars evolve. The bulk physics is a fairly well-defined problem but there are several potentially important components that are lacking in these models. (I'll expand on this if desired...)
The usual difficulty here is breaking down the degeneracy between brightness and distance. That is, a distant object is fainter, so it's hard to know whether a certain object is intrinsically faint or just further away. The principal result in this paper is the Hubble-based parallax measurement, which makes a big improvement on that distance measurement and, therefore, the brightness of the star. The other things they use are proxies for the surface composition and the effective temperature of the star, as far as I can see.
Incidentally, this is where I would suspect the tension can be resolved. If you look at Fig. 1 of the paper, they show the evolution of different stars for different compositions. What you're looking for, roughly speaking, is lines that go through the observed points. That figure shows that if the oxygen content is underestimated, then the best fit is actually about 13.3 Gyr, which is no longer at odds with the age of the Universe.
Take note of Table 1, where the sources of error (at 1$\sigma$) are listed. It's interesting that, not only is the star's oxygen content the largest source of error, but even the uncertainty of the oxygen content of the Sun is a contributor!
Which is more likely to be wrong, the age of Methuselah or the current estimate of the age of the universe?
The age of Methuselah, definitely. I would describe our estimates of the age of the Universe as in some way "converegent": different methods point to consistent numbers. Sure, Planck shifted the goalpost by 80 Myr or so, but it'd be a real shock to see that number change by, say, half a billion years.
Could relativistic effects account for some of the age?
I have no idea and haven't really thought about it. Since I'm pretty sure this isn't a big problem, I don't think relativistic effects are necessary to explain the discrepancy.
The ejecta of a supernova does indeed move at a fraction of the speed of light (somewhere around the 10% mark). However, it does not remain at this speed forever. As the supernova ejecta expands outwards, it creates a shell of material that is actually gathering up particles in the ambient medium (typical interstellar densities are around 1 particle per cubic-centimeter, much higher in molecular clouds).
After a few hundred years, the supernova remnant enters the Sedov phase in which the velocity of the ejecta moves at approximately $$ v(t)=\beta\left(\frac{E_0}{n_0}\right)^{1/5}t^{-3/5}\,{\rm pc/s} $$ After a few thousand years, the remnant's velocity slows down to approximately the speed of sound of the interstellar medium (a few km/s)--at this point we cannot distinguish the supernova remnant from the interstellar medium. The material that was part of the star is mixed in with the surrounding interstellar medium, thus seeding it with heavier elements.
As for first-generation stars, typically this means the metal-poor stars (where metal-poor typically means $[Fe/H]=\log_{10}(N_{Fe}/N_H)<-1$) that we call the population II stars, as opposed to the more metal-rich population I stars. Rarely does it mean the cosmologically-old population III stars (note that we have not actually observed these, so they're still hypothetical; James Webb Space Telescope might be able to catch the remnants of these) which have a metallicity of approximately zero (purely H & He).
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The oldest Population I stars are about 10 billion years old. Those stars have 0.1 times the metal abundance of the Sun (source).