light elements abundance in a static toy universe

TrickyDicky

As an exercise in cosmology-fiction (I actually got curious about this from an actual cosmology textbook problem), taking into account the stellar nuclear reactions that involve the fusion of hydrogen into helium, what would (roughly) be the proportion (in mass) between Hydrogen and He-4 in a static universe?
Would it resemble the proportions of primordial nucleosynthesis given that these are produced close to thermodynamic equilibrium?

Chronos

Chemical abundances in 'pristine' [primordial] gas clouds would be very difficult to explain if the universe was significantly more ancient than we suspect. It is not uncommon that what you don't see tells you as much or more than what you do see.

TrickyDicky

A static universe can't be more or less "ancient", it is symply time-invariant as a whole.
It is totally discarded by science so I'm not sure what your comment means. My question is purely theoretical.

BillSaltLake

Are you assuming that one begins with nothing but pure hydrogen, and that any transmutation was stellar in origin ?

TrickyDicky

Yes, just the stellar nuclear reactions, only in a static universe makes little sense to say what one begins with, since time is invariant.

BillSaltLake

Then ⁴He, ³He, and ²H would be much rarer than observed. Most of these isotopes would be inside active stars and would be consumed during the stars' lives. The initial collapse into stars would be different (I'm not sure of the details) because the initial gas would not be composed of a quarter ⁴He. I assume there would still be supernovae and metal enrichment, though. There would be probably 1/100 or less ⁴He than ¹H, and most of that locked up in stars. Lower-mass stars (< ~1.5 solar masses) will usually become composed for a limited period of mostly ⁴He in later life, but this He will eventually fuse to heavier elements.

TrickyDicky

I guess you are ignoring that there is no global evolution in the kind of scenario I'm talking about. There can only be a permanent equilibrium distribution. I'm asking what would that distribution be according to the stellar nuclear reactions and core temperature conditions. Maybe it's easier to think in terms of proportion of neutrons and protons in thermodynamical equilibrium at the stars core temperatures.

BillSaltLake

The meaning of steady state might be a problem. After a very long time, everything might be photons and neutrinos (after black hole evaporation, and after any possible proton decay with the resulting positrons annihilating the existing electrons). I still doubt that the He could ever be over 1% of the H.
I'm assuming that H was distributed (with some density fluctuations) uniformly at the same density as baryonic matter now, and then it interacted, while the expansion factor remained constant. However, in a true steady state model, nothing would change, and the He:H ratio wouldn't change. There is no a priori equlibrium ratio unless there is a high-temp period or perhaps proton decay.

TrickyDicky

Steady state and static spacetime are different notions, the most known steady state universe , that of Gold, Bondi and Hoyle was an expanding spacetime, it wasn't static.
In static spacetimes there is no global change wrt time (although there may be locally).
There can be no periods nor "after a very long time"s. No expansion factor either and if you bother to look up some GR text no possibility for the existence of black holes, so no bh evaporation either. As anyone can see it is a completely unrealistic cosmology.

I'm only trying to learn something about stellar nuclear reactions in equilibrium and in a time invariant situation to contrast it with the factual primordial nucleosynthesis context and have a better understanding of it.

TrickyDicky

Being more specific how much would the Nn/Np ratio in the time invariant situation in star's cores differ from the neutron freeze-out ratio in primordial nucleosynthesis (of around 1/6-1/7)?

twofish-quant

In any sort of time invariant universe, everything would end up as iron. The difficulty is that with light elements, there isn't a chemical equilibrium. Everything wants to be iron, and there are no cosmologically significant reverse reactions to break down iron into lighter elements.

However, this has been known since the 1940's, and the idea behind the steady state model was that there was a "magic source" of hydrogen to replace anything that got burned to He4. However, if we lived in a universe in which the magic source of hydrogen kept H/He ratios constant, we'd see no deuterium or He3, and a lot more Carbon-12 and heavier elements.

Very different. He4 to C12 has 1:1 and heavier elements are have more neutrons than proton culminating in neutron stars which are all neutrons.

TrickyDicky

For some reason you guys keep mixing static with eternal steady state models, even though I tried to clarify the difference in a previous post.
In steady state models there is time dependency, it is a expanding model. In static spacetimes nothing "ends up", there must be just an equilibrium distribution related to temperature, density and mass difference of protons and neutrons but independent of time.

phyzguy

There can be no equlibrium as you envision it unless there is a two-way pathway. So in your hypothetical static universe you need to answer twofish's question about what moves things "uphill" from iron back to hydrogen. Otherwise the equilibrium distribution in the universe you are asking about is a universe filled with black holes and iron. The steady-state universe answers this by postulating continuous creation of hydrogen and expansion.

TrickyDicky

What moves things "uphill" is the fact that in a static universe both ways of the two-way path have the same probability by definition, or at least that is what time-reversible reactions seem to imply.

mathal

Clearly in the model you present there is no entropy and yes theoretically all reactions are time invariant so without entropy there would be no way to discern a 'direction' to time. To answer your question, in such a universe you can have whatever ratio of elements you like since all reactions are matched with their counterpart (or your universe won't remain static).
mathal

TrickyDicky

Thanks for your input mathal, finally someone sees what I mean.
You are right, that was in fact my initial thought, that in such a universe any ratio would be possible, but I'm trying to introduce some constraints in the form of typical stellar core temperature , pressure and density and supposing the usual stellar nucleosynthesis reactions would also work so that some equilibrium distribution can be given that would make more sense than some other.

Chronos

An interstellar medium enriched by stellar ejecta would be anything but static. A universe without entropy would be not even wrong, but, I naively suspect it would remain in its original state. A universe without BB nuclosynthesis still needs a source of hydrogen for primordial stars to form. Once the stellar formation process began, the ISM would be continuously enriched by stellar ejecta. I fail to see how that could ever achieve equilibrium. It would merely continue to be enriched until so heavily metallized it could no longer support stellar fusion. The answer seems to be stars would not form in a universe without entropy.