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I've got a few questions about the first stars to form in the universe. First off how might metalicity have impacted the formation of the first stars and also what effect would the absence of metals impacted the evolution of these stars?

Why were these stars able to be so massive? I've heard that metallicity impacts opacity so would this maybe inhibit mass loss from radiation pressure?

Would the Eddington limit come into play with stars these large?

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Metallicity impacts star formation and stellar evolution in many different ways.

Star formation

A dense core of a molecular cloud becomes unstable and starts collapsing under its own gravity. During the collapse, the cloud fragments into small clumps. Each of these clumps will become a star, so the size of the clumps will determine the typical mass of the formed stars. The size of the clumps depends strongly on the cooling efficiency of the cloud during the collapse and the cooling efficiency depends on metallicity.

Primordial gas is very difficult to cool. Its main cooling mechanism is through emission of radiation from roto-vibrational line of molecular hydrogen ($H_2$), but it is not very efficient and moreover, $H_2$ is extremely fragile. Easy to break (by Lyman-Werner radiation) and difficult to reassemble (in absence of dust). On the other hand, metals (and dust!) are great at cooling the gas. Lots of emission lines, they also work at high gas density, where $H_2$ would not have a chance.

Omukai et al. (2005) have shown that higher cooling efficiency from dust and metals produces clumps with about the mass of the Sun, and this would explain why this is about the typical mass of a star today. On the contrary, in absence of metals and dust, the clumps at the end of fragmentation would be larger, hundreds or even a thousand of solar masses.

This has been later confirmed in numerous other studies, see for example Hirano et al. (2014) or Chon et al. (2021).

Of course, metallicity not only impacts the mass of the stars, but other stellar parameters as well, such as the binary fraction (and system multiplicity), the initial semi-major axis distribution, the rotation, etc. but I believe that our constraints on these are still vague.

Stellar evolution

Here also metallicity has great impact. You are right to point out that low metallicity inhibits stellar mass loss. In fact, when $Z<0.001 Z_\odot$ I believe we can safely ignore mass loss due to line driven winds for the whole stellar evolution. Mass loss in metal poor stars is driven by other mechanisms, such as binary interactions (if any), Eddington limit, and pulsational pair instability (in some mass ranges).

Other effects of metallicity are present in the core of the star. No metals means that the star is unable to start the CNO cycle. It will need to produce all its energy with the PP chain, which is extremely less efficient. As a consequence, the star will need to burn at a higher temperature, to supply the required energy. They will be bluer and will also have a smaller radius (this is another reason why mass losses are negligible, zero-metallicity stars do not become red giants with a loosely bound atmosphere). Additionally, the high temperature reached in the core can in some cases reach the threshold to activate the 3-alpha reaction and start burning helium in the same place where it is still currently burning hydrogen! This means that carbon, oxygen and nitrogen will be produced and the CNO cycle will begin.

But at this point the star is already nearing the end of its life. We are talking about a massive (~$100 M_\odot$) star that lost little mass due to winds. In just a couple million years (the blink of an eye for stellar astrophysicists) it has exhausted its hydrogen (and helium) reserves. In a moment it will be over. Depending on the mass, this star can explode as a core collapse supernova, collapse directly into a black hole (a very massive black hole, impossible to produce from solar metallicity stars that loose most of their mass through winds), or trigger pair instability and explode as a spectacular pair instability supernova (never observed yet!)

Unfortunately, our current observational constraints on the phenomena I am talking about are almost nonexistent. We have never directly observed one of the first stars, although we are getting closer to it (see A highly magnified star at redshift 6.2). Maybe the next generation of space and ground telescopes will be able to actually see one of these amazing objects.

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