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.

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