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How do astronomers know a star's surface temperature when nobody can put a thermometer there? A large part of the answer came from Meghnad Saha in 1920. He showed how the share of atoms that have lost electrons depends on temperature and pressure, which let the lines in a star's spectrum be read as a temperature scale. Below is what he did, the equation in plain terms, and a calculation you can check with a calculator.
Who was Meghnad Saha?
Saha lived from 1893 to 1956. D. S. Kothari's biographical memoir for the Royal Society says his name "would always remain associated with the theory of thermal ionization and its application to the interpretation of stellar spectra." The same memoir describes a much wider life. He built research groups at Allahabad and Calcutta and played a central role in establishing the Institute of Nuclear Physics at Calcutta. He also started and edited the journal Science and Culture, chaired the Indian Calendar Reform Committee, and was elected an independent member of Parliament.
What problem was Saha solving?
By 1920, astronomers had sorted the spectra of a huge number of stars into a sequence. Saha's 1921 paper in the Proceedings of the Royal Society opens by noting that the spectra of more than 100,000 stars had been photographed and classified at Harvard. It then quotes the astronomer H. N. Russell: more than 99 per cent fell into six groups, B, A, F, G, K, M, forming a continuous series. What was missing was a physical reason for the series.
His answer, set out in "Ionization in the solar chromosphere" (Philosophical Magazine, 1920), was that the differences reflect physical conditions in each star's atmosphere, above all how far heat has ionised its atoms. Kothari's memoir puts the key step simply: it followed from recognising that "the laws of thermodynamics and the kinetic theory of gases can be extended to a gas of free electrons."
What does the Saha equation say?
In a hot gas, collisions knock electrons off atoms while free electrons recombine with ions. At equilibrium, the Saha equation (as written in Max Pettini's Cambridge lecture notes) gives the ratio of ions to neutral atoms:
ne × N(i+1) / N(i) = 2 × [Z(i+1) / Z(i)] × (2π me kT)^(3/2) / h^3 × e^(−χ/kT)
You do not need every symbol to see how it behaves:
χ is the ionisation energy. It sits inside an exponential, so small changes in χ or T make huge changes in the ratio.
ne is the electron density. More free electrons push the balance back towards neutral atoms, so pressure matters too.
T appears twice, in the power term and in the exponential, and the exponential dominates.
Worked example: why sodium is ionised in the Sun but hydrogen is not
Pettini's notes give the ionisation energies: 5.16 eV for sodium and 13.6 eV for hydrogen. They also take the Sun's surface temperature as 5770 K. Compare the exponential factors only.
kT at 5770 K: using k = 8.617 × 10^-5 eV/K from NIST, kT = 8.617 × 10^-5 × 5770 ≈ 0.497 eV.
The ratio of the two factors is e^(−5.16/kT) / e^(−13.6/kT) = e^((13.6 − 5.16)/kT) = e^(8.44/0.497) = e^(17.0).
e^17.0 ≈ 2.4 × 10^7.
So, other things equal, the ionised share of sodium is tens of millions of times higher than that of hydrogen at the Sun's surface. Pettini's notes give the same order of magnitude, about 10^7. They also explain the consequence: although the Sun has far less sodium than hydrogen, most of the free electrons in its atmosphere come from easily ionised metals such as sodium and calcium.
How does this turn spectra into temperatures?
The same notes walk through hydrogen's Hα line. It needs hydrogen atoms already excited to the second energy level. In cool stars too few atoms are excited, so the line is weak. In stars hotter than about 10,000 K the line weakens again, because more and more hydrogen is ionised. The line therefore peaks near 10,000 K.
Other elements and ions behave in a similar way, each strongest in its own temperature range. By comparing lines from different elements and different ionisation stages, you can read off the temperature of a star's atmosphere. Hotter stars show lines from highly ionised atoms; the coolest can even hold molecules.
How to practise this kind of reasoning
Rank before you calculate. Lower χ means more ionised at the same T. Say which is bigger before using a calculator.
Keep units consistent. If χ is in eV, use k in eV/K. Our guide to units and significant figures covers the common slips.
Connect to the next idea. What happens to stars after they run out of fuel is the subject of Chandrasekhar's limit.
If Boltzmann factors or ionisation energy are still shaky, the Fermi Concept Library has a lesson for every concept in physics and chemistry, with its place in the map, so you can fill the gap before tackling problems. Then redo the worked example at 10,000 K and see how the ratio changes.
