# Harvard Astronomy 201b

## CHAPTER: Spitzer Notation

In Book Chapter on March 5, 2013 at 3:19 am

(updated for 2013)

We will use the notation from Spitzer (1978). See also Draine, Ch. 3. We represent the density of a state j as $n_j(X^{(r)})$, where

• n: particle density
• j: quantum state
• X: element
• (r): ionization state
• For example, $HI = H^{(0)}$

In his book, Spitzer defines something called “Equivalent Thermodynamic Equilibrium” or “ETE”. In ETE, $n_j^*$ gives the “equivalent” density in state j. The true (observed) value is $n_j$. He then defines the ratio of the true density to the ETE density to be $b_j = n_j / n_j^*$.

This quantity approaches 1 when collisions dominate over ionization and recombination. For LTE, $b_j = 1$ for all levels. The level population is then given by the Boltzmann equation: $\frac{n_j^\star(X^{(r)})}{n_k^\star(X^{(r)})} = (\frac{g_{rj}}{g_{rk}})~e^{ -(E_{rj} - E_{rk}) / kT }$,

where $E_{rj}$ and $g_{rj}$ are the energy and statistical weight (degeneracy) of level j, ionization state r. The exponential term is called the “Boltzmann factor”‘ and determines the relative probability for a state.

The term “Maxwellian” describes the velocity distribution of a 3-D gas. “Maxwell-Boltzmann” is a special case of the Boltzmann distribution for velocities.

Using our definition of b and dropping the “r” designation, $\frac{n_k}{n_j} = \frac{b_k}{b_j} (\frac{g_k}{g_j})~e^{-h \nu_{jk} / kT }$

Where $\nu_{jk}$ is the frequency of the radiative transition from k to j. We will use the convention that $E_k > E_j$, such that $E_{jk}=h\nu_{jk} > 0$.

To find the fraction of atoms of species $X^{(r)}$ excited to level j, define: $\sum_k n_k^\star (X^{(r)}) = n^\star(X^{(r)})$

as the particle density of $X^{(r)}$ in all states. Then $\frac{ n_j^* (X^{(r)}) } { n^* (X^{(r)})} = \frac{ g_{rj} e^{-E_{rj} / kT} } {\sum_k g_{rk} e^{ -E_{rk} / kT} }$

Define $f_r$, the “partition function” for species $X^{(r)}$, to be the denominator of the RHS of the above equation. Then we can write, more simply: $\frac{n_j^\star}{n^\star} = \frac{g_{rj}}{f_r} e^{-E_{rj}/kT}$

to be the fraction of particles that are in state j. By computing this for all j we now know the distribution of level populations for ETE.