Black holes, white dwarfs and neutron star - Shapiro, Teukolsky

[Vrbic]

Homework Statement

Exercise 2.6 (page 28)
Consider completely ionized matter consisting of hydrogen, helium, and heavier atomic species i>2. Let X and Y denote the fractions by mass of hydrogen and helium, respectively. Show that
$\mu_e=\frac{2}{1+X}.$
Approximate $m_i=A_i m_u$ for all i, and take $Z_i/A_i=1/2$ for i>1.

Homework Equations

$\mu_e=\frac{m_B}{m_uY_e} \\ m_B=\frac{\sum{n_i m_i}}{\sum{n_i A_i}}$ baryon rest mass, where $m_u$ is mass of nucleon, $Y_e=Z/A$ is number of electrons per baryon.

The Attempt at a Solution

I didn't find definition of X and Y but I suppose $X=m_H/m_{tot} =$. I am quit confused so I can't realize how proceed.

 

[mfb]

I didn't find definition of X and Y

It is defined in the problem statement, and you found the right formula.

$$\mu_e=\frac{m_B}{m_uY_e}$$

This is for a single element. For the total mixture, you'll need a weighted average.

Here is an easier version of the problem: if you have 200 hydrogen nuclei and 100 helium nuclei, how many electrons per baryon do you have, and what is the mass fraction of hydrogen?
What about 100 and 100 nuclei? Or arbitrary numbers?

 

[Vrbic]

Or arbitrary numbers?

For Hydrogen: $$M_H=n_H m_u$$, for Helium: $$M_{He}=n_{He}A_{He}m_{He}$$ Than
$$X=\frac{n_H m_u}{n_H m_u+n_{He}A_{He}m_{u}}=\frac{n_H}{n_H+n_{He}A_{He}}$$ and $$Y=\frac{n_{He} m_u}{n_H m_u+n_{He}A_{He}m_{u}}=\frac{n_{He}}{n_H+n_{He}A_{He}}$$.
So than I mean amount eletroncs per baryon is $$Y_e=\frac{n_{H}+Z_{He}n_{He}}{n_H +n_{He}A_{He}}=\frac{n_{H}+n_{He}A_{He}/2}{n_H+n_{He}A_{He}}$$. Ok?