Astronomy & Celestial Mechanics
the sky, computed
Concept

Saha equation — where it appears

The balance between ionisation and recombination in a gas, which fixes how much of an element sits in each ionisation stage at a given temperature and pressure. It explains why a spectral sequence is a temperature sequence rather than a composition sequence, and its steepness is what makes a partial ionisation zone a valve.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

Opacity against temperature, and the three things that supply it. The Rosseland mean opacity of a gas of composition X = 0.7, Y = 0.28, Z = 0.02, on logarithmic axes, at 10⁻⁷ g/cm³ and 10⁻⁶ g/cm³. The three faint curves are the separate processes at the first density — electron scattering, the Kramers bound-free and free-free term, and the negative hydrogen ion — and the solid curve is their sum. Every one of them is multiplied by the fraction of hydrogen the Saha equation says is ionised at that temperature and density, or by one minus it for H⁻, which is the only reason the low-temperature end is a picture of a star rather than of a formula outside its range: ungated, Kramers alone gives 10,343 cm²/g at 5,800 K, against the 0.40 drawn here. The peak sits at 15,400 K, where hydrogen is 78% ionised — that bump is not a detail, it is the engine of a Cepheid — and the flat floor at high temperature is electron scattering, which is the one term with no temperature in it at all.

The surface that is a depth

A star has no surface. What looks like one is the level at which the optical depth reaches about two-thirds — and the edge is sharp only because the opacity climbs so steeply that the transition takes a ten-thousandth of the radius.

starlight · Opacity The Balmer maximum is at 9,870 K, and it is a maximum in temperature. What two absorption lines actually count, each normalised to its own maximum. The first curve is the fraction of all hydrogen sitting in n = 2 — the only hydrogen a Balmer line can absorb — which is a Boltzmann factor climbing with temperature multiplied by the neutral fraction falling with it. The product peaks at 9,870 K, where 35.2% of the hydrogen is still neutral and only 8.73·10⁻⁶ of all of it is in n = 2 at all. Below the peak there is plenty of hydrogen and almost none of it excited; above it there is plenty excited and almost none of it neutral. The second curve is the fraction of calcium that is singly ionised, which is what the Ca II K line counts, and it peaks at 6,335 K — cooler, because calcium gives up its first electron at 6.113 eV. At 6,000 K the Balmer curve is at 0.1% of its own maximum while Ca II is near its peak, and calcium is 4.5·10⁵ times rarer than hydrogen in the same gas. A spectrum in which Ca II K is the strongest line is not a spectrum of a calcium star. Reading it as one is precisely the error that had the Sun made of iron until 1925.

Why hydrogen's lines are strongest where hydrogen is not

A Balmer line counts the hydrogen atoms sitting in one particular excited state, and that population peaks near 10,000 K — where a third of the hydrogen has already been ionised away. The strength of a line is a thermometer, and reading it as an abundance put hydrogen at one per cent of the Sun until 1925.

starlight · Ionisation One ionising photon per atom, and where that happens. The number of photons energetic enough to ionise hydrogen, per baryon, against temperature — with temperature falling to the right, so the universe ages to the right. The quantity is the fraction of a Planck distribution above 13.6 eV, integrated numerically, divided by the 6.13·10⁻¹⁰ baryons there are per photon. It crosses one at 5,844 K, which is z = 2143, and it falls by 19 orders of magnitude across the plot because it is the tail of an exponential. That crossing is the answer to why recombination waits until three thousand kelvin. At hydrogen's own ionisation temperature of 157,803 K there are two billion ionising photons per atom and the gas has no chance; the temperature has to fall by a factor of forty before the supply runs out, and it runs out suddenly because an exponential tail does. The number that sets the factor of forty is not an energy at all — it is the photon-to-baryon ratio, which is to say the entropy per baryon, which is to say a number fixed long before any of this and measured today from the second acoustic peak and from deuterium alike.

The surface the background actually is

Hydrogen ionises at 157,800 kelvin and the universe became transparent at 3,000. The factor of fifty between them is not an error, and it is not about energy — it is about there being two billion photons for every atom, so the far tail of the distribution can keep the gas ionised long after the typical photon has become useless.

cosmology · Microwave background Two diagnostics, two bands, and a crossing to 49 kelvin. The plane of effective temperature against surface gravity, with the constraints from two spectroscopic diagnostics drawn as bands. The wings of a hydrogen line are broadened by collisions, so they respond steeply to the gravity and weakly to the temperature: a narrow, steep band. An ionisation balance — requiring that the same element give the same abundance from its neutral and its singly ionised lines — responds to both, and its band is much shallower. Neither diagnostic determines either quantity on its own. Where the two cross is the answer, and the size of the crossing region is set by the band widths divided by the difference of the slopes — so two diagnostics that respond similarly give a long, thin, nearly useless error region however precise each one is. Choosing diagnostics that disagree in their sensitivities is the whole of the art.

A temperature and a gravity that trade against each other

A stellar spectrum contains the star's temperature, its surface gravity and its composition, and no single feature in it contains only one of the three. Every diagnostic is a band in the parameter plane rather than a point, and the answer is where the bands cross — which makes choosing diagnostics that disagree in their sensitivities the whole of the art.

starlight · Spectra 742 kelvin between a dwarf and a supergiant, at the same ionisation. The fraction of calcium still neutral against temperature, for three surface gravities: a dwarf, a giant and a supergiant. The three curves are the same curve slid sideways. The Saha equation carries the electron pressure in its denominator, and the photospheric pressure follows the surface gravity as its square root — hydrostatic equilibrium gives a gas pressure of order g over the opacity, and the opacity in a cool star is set by the electrons themselves. So the pressure runs from 19 newtons a square metre in the dwarf down to 0.48 in the supergiant, a factor of 40, and the half-ionisation point moves from 4427 kelvin to 3685 — 742 kelvin apart. At fixed temperature the ionisation ratio goes as g^-0.50, exactly the square root the algebra requires. A star's luminosity class is a measurement of the density of its photosphere, read off which stage of an element the lines belong to.

A luminosity class is a density measurement

The Saha equation has an electron pressure in its denominator, and a supergiant's photosphere is forty times less dense than a dwarf's at the same temperature. So the same element sits in different ionisation stages in the two, the spectrum says which, and the second axis of stellar classification is a barometer.

starlight · Ionisation Where titanium oxide comes apart, and what the pressure does to it. The fraction of titanium oxide dissociated into Ti + O, against temperature, for a gas of equal parts at 10, 300, 6000 N/m² of total pressure. The equilibrium is Saha's with a free atom in place of the free electron and the pair's reduced mass in place of the electron's — the two atoms' partial pressures multiplied together, divided by the molecule's, equal an equilibrium constant Kₚ(T) — which for equal abundances gives a dissociated fraction of √(Kₚ/(Kₚ+P)) exactly. A molecule is one particle becoming two, so pressure suppresses the reaction exactly as it suppresses ionisation — the half-dissociation point moves from 3145 K at 10 N/m² to 4164 K at 6000. The bond energy is 6.87 eV against hydrogen's ionisation potential of 13.6, which is why this happens at a third of the temperature. Rotational and vibrational partition functions are included, classically and harmonically; the atoms keep their ground-term weights.

A spectrum with no continuum left in it

Below about four thousand kelvin a photosphere stops being a gas of atoms. Titanium oxide, water and carbon monoxide take over, the same equilibrium settles how much of each survives with a bond energy where an ionisation potential used to be — and the bands are so crowded that the level every line depth is measured against is nowhere on the plate.

starlight · Ionisation Where a photosphere's free electrons come from. The share of the free electrons donated by metals rather than by hydrogen, against temperature, at a gas pressure of 12000 N/m² and at solar abundance, 1 dex below solar, 2 dex below solar. The electron pressure is not assumed here — it is solved for, as the value at which the ionisation of the gas supplies exactly the electrons the gas contains. Below about 5897 K at solar abundance every free electron in the gas comes from an element present at one part in ten thousand, because hydrogen's 13.6 eV keeps it neutral while magnesium's 7.6 does not. The handover is fast: solar crosses a half at 5897 K, 1 dex down crosses a half at 5088 K, 2 dex down crosses a half at 4467 K. At 5,772 K and this pressure the solved electron pressure is 1.78 N/m².

The continuum is made by one atom in ten thousand

The Sun's light leaves through an ion that exists only because a hydrogen atom will hold a second electron by three-quarters of an electronvolt. The electrons it holds come almost entirely from magnesium, silicon and iron — so the level against which every solar line depth is measured is rationed by elements present at one part in ten thousand.

starlight · Ionisation Elements sorted by how easily they lose an electron. The abundance of each element in the solar wind relative to the photosphere, normalised to oxygen, against its first ionisation potential, logarithmic: in the slow wind (filled) and the fast wind from coronal holes (open). Representative values. Elements easy to ionise — K, Na, Ca, Mg, Fe, Si, all below about 9 eV — are about three times over-abundant in the slow wind and about one and a half in the fast; those harder to ionise sit near one. The step comes at about 10 eV, which is the energy of a photon from the Lyman-α line of hydrogen: the sorting happens where hydrogen's own light sets the ionisation.

A wind that carries the ledger of its loops

The slow solar wind is richer in iron, magnesium and silicon than the Sun's surface is — about three times richer, relative to oxygen — and the elements that are enriched are exactly the ones that lose an electron easily. That sorting happens low in the chromosphere, inside closed magnetic loops, and it takes days. The fast wind from coronal holes barely shows it. So the composition of the wind arriving at the Earth is a record of where it was held, and for how long, before the field let it go.

stars · Plasma beta Almost everything is ionised, and then almost nothing is. The ionised fraction of hydrogen against distance from an O7 star in gas of 10 per cubic centimetre, from integrating photoionisation balance outward, in units of the counted radius of 14.62 parsecs. The fraction is near one until about the edge and then falls from 90% to 10% within 0.192 parsecs, 1.3% of the radius. That width is about 18 mean free paths of an ionising photon in neutral gas (0.011 pc). Nothing about the star sets it. A region with no gradual boundary is a surprising result of a smooth input, and it comes from the fact that a photon that gets as far as the neutral gas is absorbed at once.

An ionised sphere that is a count of photons

A hot star sits in a cloud of hydrogen and ionises it, out to a radius, and there the ionisation stops within a tiny fraction of the distance. That edge was predicted in 1939 from a single count: every recombination uses up one photon, so the star can hold ionised only as many atoms as its light can pay for. By the equation that describes a stellar atmosphere at the same temperature, the same gas would hold ten orders of magnitude fewer neutral atoms than it does. In a nebula the temperature is not what decides the ionisation. A budget is.

starlight · Ionisation

Named alongside it

The objects these essays reach for when they reach for this one.

Electron pressurePartition functionSpectral classificationH minus opacityIonisation equilibriumLuminosity classOpacityOptical depthPhotosphereRecombinationSurface gravityAbundance

All concepts