What is the correlation between stellar abundance by mass and by number?

Let’s take the Sun and Helium for example. This paper mentions an abundance by mass for helium as Y=0.275 and by number as A=10.99, which I believe is derived by assuming N(He)/N(H)=8.5%.

  1. How do you get this 8.5%? Is there a standard value for N(H)?
  2. Why are abundances given both by mass and by number? Is there a historic reason?
  3. Is there a relation between the two, that is can you estimate one “type” by knowing the value of the other?

According to Lodders (2003, https://arxiv.org/pdf/1010.2746 ) the relative abundance of helium to hydrogen is $A({\rm He})=10.925$, on a logarithmic scale where the hydrogen number abundance is 12. So this would mean a helium to hydrogen ratio, by number, of $10^{10.925-12}=0.08414$. i.e. 8.4% (your source uses 10.93, not 10.99, hence a very slightly different percentage).

Why have the two systems? The appearance of the spectrum - absorption lines, emission lines etc. depends on their number density. So it is relative numbers of the elements that count. On the other hand when you are doing structural calculations on stars, it is usually the mass fractions that matter.

The relationship between the two?

The relative numbers of helium and hydrogen are easily translated into relative mass fractions of helium ($Y$) and hydrogen ($X$): $$\frac{Y}{X} = 0.08414 \times 4.0026/[(1-0.08414)\times 1.0078] =0.3649$$, where $4.0026/1.0078$ is the ratio of atomic masses of helium to hydrogen.

Now there is also a small fraction by mass of heavier elements $Z\simeq 0.014$, where $1 = X + Y +Z$.

If we substitute $X= Y/0.3649$ and $Z=0.014$, then $$ Y \left(1 + \frac{1}{0.3649}\right) = 1-Z$$ $$ Y = 0.264$$

To sum up: $$Y = \frac{1-Z}{\left(1 + \frac{1.0078(1 - 10^{A({\rm He})-12})}{4.0026\times 10^{A({\rm He})-12}}\right)} = \frac{1-Z}{\left(0.7482 + 0.2518\times10^{12-A({\rm He})}\right)}$$

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