L06 Mean molecular weight

L06 Mean molecular weight#

Week 3, Thursday

Material covered and references#

Now we want to consider more realistic models where the pressure is described by e.g. the ideal gas law (we will talk about other sources of pressure later on in the course). The ideal gas pressure is a function of the concentration of free particles (which includes free electrons!). However, the equation for hydrostatic equilibrium is a function of density.

We therefore relate the concentration of free particle with the density using a quantity called the mean molecular weight (\(\mu\)). The mean molecular weight conceptually represents the mean mass (in units of the mass of one hydrogen atom \(m_H\)) per free particle. The mean molecular weight thus depends on:

  • The distribution of element abundances (usually measured in mass fraction \(X_i\))

  • The ionization level of each element (\(y_i\)) – we will see how this is highly dependent on temperature in the next class meeting

Micro-objectives:

  • I can state the ideal gas law in terms of the concentration of free particles.

  • I can explain the concept of “free” particles in a plasma (gas with ionization)

  • I can use logic to relate the concentration of free particles with density, for simple cases

  • I can state the definition of the mean molecular weight

In the textbooks:

  • Leblanc 5.6.2

  • McD 2.5 (fully ionized only)

  • Kip 13.1

We defined two mean molecular weights

  • \(\mu_\mathrm{ion}\), which is the mean mass (in units) per ion

  • \(\mu_\mathrm{e}\), which is the mean mass (in \(m_H\) units) per free electron (note, all the mass is coming from the ions – if you have 10 hydrogen atoms, and only two of them are ionized, then you have 10 \(m_H\) / 2 free electron, and \(\mu_e=5\)).

The mean molecular weight per free particle is then:

\[\frac{1}{\mu} = \frac{1}{\mu_\mathrm{ion}} +\frac{1}{\mu_\mathrm{e}}\]

In your notebook, we found out how to calculate the mean molecular weight given a known distribution of abundances for each element, in the completely neutral and completely ionized cases.

  • I can state the mass fraction of the sun.

  • I can sketch the mass fraction as a function of element for the sun.

  • I can explain the conceptual meaning of \(\mu_\mathrm{ion}\) and \(\mu_\mathrm{e}\) using explanatory examples.

  • I can derive the expressions for \(1/\mu_\mathrm{ion}\) and \(1/\mu_\mathrm{e}\) as a function of the mass fraction.

  • I can sketch the variation of the mean molecular weight as a function of radial coordinate for the current-day sun, and explain the different zones.

In the textbooks:

  • Leblanc 5.6.2

  • Hansen 1.4.1

In astronomy, we often refer to the abundance of elements in terms of their mass fraction. We also usually refer to the mass fraction of hydrogen by \(X\), the mass fraction of helium by \(Y\), and the mass fraction of everything else by \(Z\) (also called the “metallicity”). In class I pointed out that we can write an approximation to calculate the mean molecular weight based on \(X\), \(Y\) and \(Z\), instead of the complete calculation. The derivation of these formulas (and the approximations made) are in the textbooks

In the textbooks:

  • Leblanc, equations 5.126 and 5.127

  • Kip 8.1 for abudances

Python skills:

  • Practice adding scatter points to graph

  • Learn about reading data text files

  • Practice using a MESA stellar structre file