The surface that is a depth

The Sun has an edge. Look at it through anything that dims it enough and the disc ends abruptly: there is light, and then within an angular distance too small to resolve there is none. Every photograph of the Sun has a hard rim.

There is nothing there. The Sun is gas from the core to the corona, and the density at the apparent edge is about $10^{-7}$ grams per cubic centimetre — ten thousand times thinner than the air in this room. Nothing changes state, nothing is bounded, and no property of the material has a discontinuity anywhere near it.

What produces the edge is a rate: the opacity of stellar material rises so steeply with temperature and density that the transition from transparent to opaque happens over a few hundred kilometres out of seven hundred thousand. The Sun looks as though it has a surface for the same reason a fog bank has an edge, and for no better reason.

Optical depth, and why two-thirds #

The quantity that decides whether a photon gets out is the optical depth, measured inward from the top:

$$\tau(z) = \int_z^\infty \kappa\,\rho\,dz',$$

with $\kappa$ the opacity in area per unit mass. It is dimensionless, it counts mean free paths, and a photon emitted at depth $\tau$ has a probability $e^{-\tau}$ of escaping without being absorbed.

Radiation emerging from a semi-infinite atmosphere therefore comes from a range of depths, weighted by $e^{-\tau}\,d\tau$, which peaks at $\tau = 1$ and has its mean at $\tau = 1$ as well. The often-quoted $\tau = 2/3$ is a slightly different statement: it is the depth whose source function equals the emergent intensity, when the source function varies linearly with $\tau$. That is the Eddington–Barbier relation, and it is what makes a darkened limb a read-out of the temperature gradient rather than a property of a surface.

For present purposes the difference between 1 and 2/3 does not matter. What matters is that the number is of order one, and that it defines a level rather than a place.

Why the layer is thin #

The scale height is what does it, and it is short for a reason worth spelling out. In hydrostatic equilibrium the pressure falls with height as $e^{-z/H}$ with $H = kT/\mu m_{\rm H} g$, and at the solar photosphere $T$ is 5,800 K, $\mu$ about 1.3, and $g$ is 274 metres per second squared — twenty-eight times the Earth’s. That gives about 140 km, which is $2\times10^{-4}$ of the radius.

The density therefore falls by a factor of $e$ every 140 km, and the opacity falls with it, so the optical depth falls faster than exponentially. Two decades in $\tau$ — from opaque to transparent — take $\ln 100 = 4.6$ scale heights.

What actually absorbs #

The three processes in the opening figure divide the temperature range between them, and each is worth naming because they behave differently.

Electron scattering is the floor. A free electron scatters a photon with the Thomson cross-section, and per gram that gives $\kappa = 0.2(1+X)$ square centimetres — about 0.34 for cosmic composition — with no dependence on temperature, density or wavelength at all. It dominates above about a million kelvin, and it is the opacity that sets the brightness at which radiation pressure blows a star apart — the one place in astrophysics where an opacity appears in an answer with no other quantity beside it.

Kramers opacity covers bound-free and free-free absorption by ions, and goes as $\rho T^{-3.5}$. Steeply falling with temperature, it dominates through the interiors of most stars, and it is what makes the temperature gradient steep enough to boil once the material cools on the way out.

And in the outer layers of a cool star, neither of them. At 5,800 K hydrogen is one part in six thousand ionised, so there are almost no free electrons for the first two processes and almost no ions for the second. What absorbs instead is an ion that has no business existing.

An ion that ought not to exist #

A hydrogen atom can hold a second electron. The binding energy is 0.754 electronvolts — a twentieth of the energy that binds the first — and the ion exists at all only because the neutral atom is polarisable: the extra electron induces a dipole in the atom it is approaching and is bound by the field of the dipole it made.

$\mathrm{H}^-$ has no bound excited states, so it has no spectral lines. What it has is a bound-free continuum: any photon with more than 0.754 eV can detach the extra electron, and that is every photon with a wavelength shorter than 1.64 microns. The cross-section peaks near 850 nm at about $4\times10^{-17}$ square centimetres, which is enormous per ion.

The number of ions is tiny. Its abundance follows its own Saha balance,

$$\frac{n(\mathrm{H}^-)}{n(\mathrm{H}^0)} = \frac{n_e}{4\,(2.4147\times10^{15})\,T^{3/2}e^{-8750/T}},$$

and at the solar photosphere that ratio is about $2\times10^{-8}$: one hydrogen atom in fifty million carries a second electron. Multiply the tiny abundance by the enormous cross-section and the result is 0.4 square centimetres per gram — which is the opacity the Sun’s continuum actually has.

The visible appearance of the Sun is set by the rarest thing in it. And the electrons that make those ions do not come from hydrogen, which is neutral; they come from the metals, whose ionisation potentials are 5 to 8 eV rather than 13.6, so a trace constituent supplies the electrons that a trace ion needs. A solar spectrum is a picture of what iron, magnesium and sodium have donated.

The depth depends on the colour #

That last point is the general one. Optical depth is computed with an opacity, and the opacity is wavelength-dependent, so the level at which the star becomes opaque is a different level in every colour.

Where the opacity is high — in the core of a strong absorption line, for instance — $\tau = 1$ is reached higher up, in cooler material, and the emergent intensity is lower. That is what an absorption line is: not material removed from the beam, but a wavelength at which the observer is seeing a shallower and cooler layer. The wavelength dependence is also why a single number can be used at all. The Rosseland mean is the harmonic mean of the opacity weighted by the temperature derivative of the Planck function — chosen so that the flux through a thick layer comes out right, which means it is dominated by the wavelengths at which the material is most transparent. It is the correct average for radiative transport in an interior and the wrong one for anything happening near the surface, which is why atmosphere models do not use it.

What was actually measured #

Nothing above is measurable directly. Opacity is a laboratory quantity that cannot be produced in a laboratory at stellar conditions, so every number in the opening figure is calculated, and the check has to come from somewhere else.

There are three places it comes from, and they are worth ranking by how directly they bear.

The solar limb. The run of intensity from disc centre to edge is a measurement of the temperature at a sequence of optical depths, because looking at angle $\mu$ from the centre means $\tau = 1$ is reached at a height $\mu$ times shallower. Measuring limb darkening in several colours gives the temperature structure of the photosphere; comparing that with what a model atmosphere produces from an assumed opacity is a test of the opacity. Helioseismology. The speed of sound inside the Sun depends on the temperature and composition, which depend on the opacity through the equation of hydrostatic and radiative equilibrium. Inverting the observed p-mode frequencies gives the sound speed as a function of radius to a part in a thousand, and comparing it with a model is the most stringent test of interior opacities there is. It is the test that flagged the solar abundance problem: when the measured solar oxygen abundance was revised downward in the 2000s, the resulting models stopped fitting the seismology, and one of the ways out proposed is that the opacities are five to fifteen per cent low near the base of the convection zone.

Pulsating stars. As above, an instability strip in the wrong place is an opacity bump in the wrong place, and that is what corrected the iron-group opacities.

None of these measures $\kappa$. Each measures a consequence of it, in a system with several other things going on, and the honest statement is that stellar opacities are known to a few per cent where they have been tested and to rather worse where they have not.

The bump that makes stars pulsate #

The peak in the opening figure, at around 15,000 K where hydrogen is passing through half ionisation, is not decoration. It is the mechanism behind an entire class of variable star.

In an ordinary layer of a star, compression raises the temperature and lowers the opacity, so the layer leaks heat faster and damps any oscillation. In a partial-ionisation zone the opposite happens: compression goes into ionising more hydrogen rather than into raising the temperature, the opacity rises, the layer dams the flux, and it is pushed outward on the next cycle. A layer that gains heat when compressed is a heat engine, and it drives. Both halves of the argument move if the atmosphere is different, and it is worth seeing how far.

Above the surface, the temperature turns around #

The photosphere is where the star stops being opaque, and it is natural to read that as where the star ends. The temperature says otherwise.

Going outward from the interior, the temperature falls steadily — it has to, since energy is flowing outward — and reaches a minimum a few hundred kilometres above the photosphere, at about 4,100 K for the Sun. Above that it rises. Through the chromosphere it climbs to twenty or thirty thousand; then, across a transition region a few hundred kilometres thick, it jumps to a million; and the corona beyond sits at one to three million kelvin, extending outward for millions of kilometres and eventually becoming the solar wind.

That is thermodynamically startling. Heat does not flow from a cold body to a hot one, so the corona cannot be heated by the photosphere’s radiation — something must be carrying energy outward in a form that is not heat and depositing it there. The candidates are magnetic: waves propagating along field lines and dissipating where the density is low, or the reconnection of tangled small-scale fields in a continuous storm of tiny flares. Which of them dominates has been argued since the corona’s temperature was established in the 1940s.

The reason the corona is not obvious is the same reason the photosphere is the surface: it is very hot and very thin, so it radiates almost nothing in visible light. Its density is a trillionth of the photosphere’s, and its total visible brightness is about a millionth — which is why it is seen only when the disc is blocked, and why its discovery had to wait for eclipses.

So the star’s outer boundary depends on which photons are asked. In visible light the Sun ends sharply at the photosphere; in extreme ultraviolet and X-rays the photosphere is dark and the corona is the source; in radio the emission comes from higher still. The single surface of this essay is the answer to a question asked at one wavelength.

What “the radius of the Sun” means #

If the surface is a depth rather than a boundary, then the star’s radius requires a convention, and it has one.

The photosphere is defined as the layer at optical depth two-thirds at a specified wavelength, and the radius follows from that. Since the depth of that layer varies with wavelength by hundreds of kilometres — the previous section’s point — the number depends on the choice, at the level of a part in ten thousand.

That is not negligible for a body measured as carefully as the Sun, and it was resolved by decree rather than by measurement: the nominal solar radius is now a defined constant, 695,700 kilometres, adopted so that published stellar radii expressed in solar units mean the same thing from one paper to the next.

The same problem arrives with less tidy solutions elsewhere. A gas giant has no surface at any wavelength, so its radius is quoted at the level where the pressure is one bar — a convention chosen because it is roughly where the visible clouds are, and one that makes a planet’s radius a function of its atmosphere rather than of anything solid. A radius is a measurement for a rocky body and a definition for everything else, and the fluid bodies are the majority.

The convention has one further consequence worth stating, because it affects every stellar radius ever quoted. A radius measured by interferometry is an angular diameter converted with a distance, and what the interferometer measures is a limb-darkened disc whose edge is not sharp; a radius derived from an eclipsing binary is the radius of the layer that blocks the light. Neither is exactly the optical-depth surface this essay defines, and the differences are at the level of a per cent — which is larger than the formal errors on the best measurements and is therefore, quietly, the dominant uncertainty in comparing one to another.

The lesson is the one this essay began with, carried to its conclusion: a star has no edge, so every quantity that presupposes one is a choice about which layer to name, and the choices only agree to the accuracy at which the layer is thin.

That is also why interferometric and eclipse radii are compared only after both have been corrected to a common definition, and why a paper reporting a stellar radius to better than a per cent has to say which surface it means.

The Sun is the one star where the difference has been measured rather than estimated, because its disc can be resolved and the shape of its limb profile observed directly at many wavelengths — and the spread across those wavelengths is what fixed the size of the correction everywhere else.

What the figures cannot show #

Two things, and both matter for how much weight the numbers here can carry.

The opacity curves are composites of three analytic approximations, each valid over part of its range, gated by an ionisation fraction. Real opacity tables are computed from millions of atomic transitions and are not smooth: the OPAL and OP projects produced them in the 1990s and their revision of the iron-group bound-bound contributions changed the opacity around 200,000 K by a factor of three, which resolved a long-standing failure to reproduce the pulsation periods of $\beta$ Cephei stars. A curve like the one above is right about the shape and can be a factor of a few wrong in places, and the places where it is wrong have historically been where the interesting problems were.

And the atmosphere in the edge figure is isothermal, which no atmosphere is. A real photosphere has a temperature gradient — that is precisely what makes the limb dark — so the contribution function is not symmetric and the emergent spectrum is not a single Planck curve. The 645 km is a scale, not a measurement.

The opacity itself is worth reading twice more, at densities either side of the ones used above.

Where the ladder goes next #

The rung above is convection: below about $\tau = 1$ in a cool star the opacity is so high that radiation cannot carry the flux, the gradient steepens past the adiabatic value, and the outer third of the Sun boils. That transition is set by the same curve drawn here, and granulation is what it looks like from outside. Past that is the reverse problem: what an opacity that depends on wavelength does to a spectrum as opposed to a continuum, which is where the shape of a line stops being about the line and starts being about the atmosphere it forms in.