Ionization and the Spectral Sequence

In the early 1900s Annie Jump Cannon sorted hundreds of thousands of stellar spectra into the classes O, B, A, F, G, K and M, purely by the look of their lines. Were these different kinds of stars, made of different stuff? In 1925 Cecilia Payne showed that their compositions are much alike, dominated by hydrogen and helium; the differences come from temperature. (We now know stars are about three-quarters hydrogen and one-quarter helium by mass.) The key was ionization, which follows on from the Boltzmann equation.

Why it works

Freeing an electron costs the ionization energy $\chi$ (13.6 eV for hydrogen), so ionization also carries a factor $e^{-\chi/kT}$. But a free electron can be anywhere, with any momentum, so it has a vast number of states open to it, and that number grows as $T^{3/2}$ and as the electron density $n_e$ falls. This huge count of states means atoms ionize at temperatures where $kT$ is only a small fraction of $\chi$, typically a tenth to a twentieth. The result is the Saha equation; for hydrogen it reads $$\frac{N_{\text{II}}}{N_{\text{I}}} = \frac{C\,T^{3/2}}{n_e}\,e^{-\chi/kT}, \qquad C = 2.415 \times 10^{21}\text{ m}^{-3}\text{K}^{-3/2}.$$

Key idea

Each element and ion has a narrow range of temperature where its lines are strong: hot enough to excite it, not so hot that it is ionized away. So the spectral classes O B A F G K M are a temperature sequence, not a sequence of composition.

Approximate surface temperatures of main-sequence stars in each class.
Class	Temperature	Strongest features
O	above 30,000 K	ionized helium (He II)
B	10,000–30,000 K	neutral helium (He I), hydrogen strengthening
A	7,500–10,000 K	the strongest hydrogen Balmer lines
F	6,000–7,500 K	hydrogen weakening, ionized calcium appears
G	5,200–6,000 K	strong ionized calcium (Ca II), many metal lines
K	3,700–5,200 K	neutral metals, some molecules
M	below 3,700 K	molecular bands such as titanium oxide (TiO)

Line strengths across the spectral classes. Each kind of line rises and falls with temperature.

Worked examples

Worked example: How ionized is hydrogen at 10,000 K?

In a stellar atmosphere at $T = 10{,}000$ K with $n_e = 1.0 \times 10^{20}$ m⁻³, find $N_{\text{II}}/N_{\text{I}}$ for hydrogen.

$kT = 0.862$ eV, so $e^{-13.6/0.862} = e^{-15.78} = 1.40 \times 10^{-7}$.
$C\,T^{3/2} = 2.415 \times 10^{21} \times 10^{6} = 2.42 \times 10^{27}$ m⁻³.
$$\frac{N_{\text{II}}}{N_{\text{I}}} = \frac{(2.42 \times 10^{27})(1.40 \times 10^{-7})}{1.0 \times 10^{20}} \approx 3.4.$$

Answer: About 3.4 ionized atoms per neutral one: roughly 77% of the hydrogen is ionized, even though $kT$ is only $1/16$ of 13.6 eV.

Worked example: A little cooler

Repeat at $T = 8{,}000$ K with the same $n_e$.

$kT = 0.689$ eV, so $e^{-13.6/0.689} = e^{-19.73} = 2.70 \times 10^{-9}$.
$T^{3/2} = 7.16 \times 10^{5}$, so $C\,T^{3/2} = 1.73 \times 10^{27}$ m⁻³.
$$\frac{N_{\text{II}}}{N_{\text{I}}} = \frac{(1.73 \times 10^{27})(2.70 \times 10^{-9})}{1.0 \times 10^{20}} \approx 0.047.$$

Answer: About 0.047: only 4.5% ionized. Between 8,000 K and 10,000 K hydrogen goes from mostly neutral to mostly ionized.

Worked example: Reading a spectrum

A star shows strong lines of ionized helium. What can you say?

Ionizing helium once takes 24.6 eV, and exciting He II enough to absorb visible light takes even more.
Only the hottest atmospheres have enough energetic collisions and photons.

Answer: It is an O star, hotter than about 30,000 K. It need not hold any more helium than the Sun.

Common mistake

Thinking stars of different spectral classes are made of different elements. An M star shows titanium oxide bands not because it is rich in titanium, but because it is cool enough for molecules to survive. The Sun shows weak hydrogen lines not because it lacks hydrogen (it is about 74% hydrogen by mass), but because at 5,800 K almost no hydrogen is in level $n = 2$. Check any claim about composition against temperature first.

Ionization and the Spectral Sequence

Worked examples

Practice

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