What is the elemental composition of the Sun overall, rather than at the photosphere?

Question

Many sources claim that the Sun is around $70-74\%$ hydrogen and $25-27\%$ helium by mass, or $91\%$ and $9\%$ by atom count, without any further caveats - e.g. 1, 2, 3. I believe these sources are incorrect. (Especially bad ones will report photospheric atom ratios as mass ratios, like space.com here.)

More careful sources like this (from which I think many of the less-good sources above derive) or Wikipedia (citing this page) give the same numbers, but clarify that they only measure the composition in the photosphere, where we can analyze the spectral lines. I'm curious about the overall distribution across all radii (which ought to have more helium, since it's produced in the core and my impression is that there isn't a ton of mixing between layers - sources like this paper seem to suggest that would be true, but don't mention concrete numbers).

This helpful answer linked to some informative plots of the distribution by radius from this paper, which mostly answers my question, but the paper doesn't actually provide the relevant integrals of their plots weighted by radial density - just going off of the fact that the core contains around $34\%$ of the mass within $0.2$ solar radii and eyeballing the plot, it looks to me like the overall composition is something like $65\%$ hydrogen and $35\%$ helium, but if someone has worked out the actual numbers from our current best models I'd love to see those instead.

Answer 1

Both your estimate and @ProfRob's answer are roughly in the right area. I've done the integration on an older standard solar model, Model S (Christensen-Dalsgaard J., et al., 1996, Sci, 272, 1286) and get average values for the hydrogen, helium and metal mass fractions of $$ \bar{X}=0.671 \\ \bar{Y}=0.309 \\ \bar{Z}=0.020 \\ $$

Model S admittedly has the older, higher, Grevesse & Noels (1993) abundance of $Z\approx0.02$, compared to the more recent (e.g. Asplund et al. 2009) values of $Z\approx0.014$, so there's about $0.006$ wiggle room there, but the metallicity remains contentious.

If anyone is especially interested and knows Python, I created a Python package called tomso for working with stellar models (in their obscure formats) that makes this calculation quite straightforward.

import numpy as np
from tomso import fgong

# fetch Model S
S = fgong.load_fgong('https://users-phys.au.dk/jcd/solar_models/fgong.l5bi.d.15c', G=6.67232e-8)

# integrate the X and Z abundances over the fractional mass co-ordinate q
# negative because stellar models are usually ordered from surface to centre
X = -np.trapz(S.X, x=S.q)                                                                                                                                                          
Z = -np.trapz(S.Z, x=S.q)

print('X =', X)
print('Y =', 1-X-Z)
print('Z =', Z)

Answer 2

I think you are right.

Metals are not produced in the Sun (though lithium is destroyed), so will be distributed throughout. To first order you can ignore radiative diffusion and chemical stratification - these take a long time in a star like the Sun.

The core is radiative and will not be well-mixed. That is why the Sun's lifetime will be about 10 billion years, even though it has enough hydrogen to burn for 5 times as long at its main sequence luminosity. Thus at this roughly halfway stage in its life, about 10% of the hydrogen has been turned into helium. So the overall composition will be about 63% H, 36% He and the rest heavier elements.

A more exact number would have to come from detailed evolutionary models. Such models exist, but don't usually report what you want to know.

Edit:

Warwick has supplied the "right answer" from the standard solar model. It is interesting to think about why it differs a bit from the numbers I argued above - 67% H, 31% He and 2% metals.

Ignore the difference between 1% and 2% metals - that just depends on whether you use the older (Warwick) or newer (me) estimates of the solar metallicity. It is basically whatever the initial metallicity of the Sun was and it's also the photospheric abundance.

The interesting difference is that less hydrogen has turned into helium. I think this is simply because the Sun is actually a little less than halfway through its main sequence life and, because the luminosity increases by a factor of 2 during the full main sequence, it is in fact only about a third of the way through its use of hydrogen.

Answer 3

The composition is… pretty close to the photospheric composition.

Red dwarves are well-mixed, being too small and light to support ‘differentiation’ in the stellar sense. Plasma dynamics is able to overcome stratification. Supergiants are able to stratify, which is partly why they burn quicker and die sooner.

So, where is our Sun along this scale? Decades of study, both of the Sun and other, nearby “boring” stars, has put us closer to the red-dwarf regime. This study includes general plasma physics, boundary conditions from spectrometries and isotopes, helioseismology, and more recently asteroseismology. Stars are ‘poked’ and we watch them jiggle. Only certain interiors will give certain jiggles.

All told, heterogenity doesn’t appear to be high enough to be an issue.