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A mean dominated by the gaps

The opacity in the equation of radiative transport is not an average of the opacity. It is a harmonic average weighted by the temperature derivative of the Planck function, which makes it a measurement of the transparent windows between the lines rather than of the lines — and that single fact decides what adding a million spectral lines to a table does.

Assumes Opacity, Line formation and Energy transport.

17 min read 8 figures Light is the only messengerSmall perturbations, large consequences

The interior of a star transports energy by diffusion. A photon is absorbed and re-emitted so many times on the way out that its progress is a random walk, and the net flux is proportional to the temperature gradient divided by a single number called the opacity.

There is no such thing as the opacity. What exists is κν\kappa_\nu, a function of frequency that varies by many orders of magnitude across the spectrum — smooth in some places and, where there are atomic transitions, a forest of narrow spikes. Reducing that function to one number is a choice, and the choice that appears in the transport equation is not the obvious one.

Two means of one opacity, 75 times apart, and only the smaller one is in the equation. Above: a synthetic opacity across the frequencies that carry a star's flux, drawn against x = hν/kT, with a continuum falling as ν⁻³ and a forest of 6 lines per unit x on top of it. The shaded curve is the Rosseland weighting function, ∂B_ν/∂T, which peaks at x = 3.83 and is what decides which frequencies matter. The two horizontal lines are the two ways of averaging. The Planck mean is an ordinary average and lands high, among the lines, because that is where most of the opacity is. The Rosseland mean is a harmonic average — it averages 1/κ rather than κ, because what carries the flux out of a star is transparency and transparencies add — and it lands 75 times lower, close to the continuum, because a harmonic mean is dominated by the smallest values in it. In other words the opacity that appears in the equation of radiative transport is a measurement of the gaps between the lines. Below: what that means for a table. The Planck mean rises as the first power of the line density, slope 0.76 as drawn — every line added is another contribution to an ordinary average. The Rosseland mean does almost nothing at first, slope 0.13, and then turns up sharply, slope 0.99, once the lines are close enough to blanket the windows. That is why adding several million atomic transitions to an opacity table in the early 1990s changed nothing for decades and then changed stellar structure: the new lines were not the first lines, they were the ones that finally closed the gaps.
Fig. 1 The two candidates, and the gap between them. Above: a synthetic opacity across the frequencies that carry a star’s flux, with a continuum falling as ν−3\nu^{-3} and a forest of lines on top of it; the shaded curve is the weighting function ∂Bν/∂T\partial B_\nu/\partial T. The Planck mean is an ordinary average and lands high, among the lines. The Rosseland mean is a harmonic average, lands far lower, close to the continuum — and it is the one in the equation. Below: what that means for a table. The Planck mean rises as the first power of the line density; the Rosseland mean barely moves until the lines are dense enough to blanket the windows, and then turns up sharply.

Why a harmonic mean

The rung below this one established what optical depth is and where a photosphere sits. The question here is different: given that the interior transports energy by diffusion, what single opacity does the diffusion equation want?

The diffusion flux at frequency ν\nu is proportional to κν−1 ∂Bν/∂T\kappa_\nu^{-1}\,\partial B_\nu/\partial T times the temperature gradient. The total flux is the integral of that over frequency. Setting the total equal to κ−1\kappa^{-1} times the integral of ∂Bν/∂T\partial B_\nu/\partial T defines

1κR  =  ∫1κν ∂Bν∂T dν∫∂Bν∂T dν,\frac{1}{\kappa_{\rm R}} \;=\; \frac{\displaystyle\int \frac{1}{\kappa_\nu}\,\frac{\partial B_\nu}{\partial T}\,d\nu}{\displaystyle\int \frac{\partial B_\nu}{\partial T}\,d\nu},

which is a harmonic mean of κν\kappa_\nu with a specific weight. Every part of that expression has a physical reading.

It is harmonic because transparencies add. What carries the flux is 1/κ1/\kappa, the mean free path; two frequency channels in parallel carry the sum of their individual fluxes, exactly as two resistors in parallel carry the sum of their currents. An arithmetic average of resistances is the wrong quantity for a parallel network and an arithmetic average of opacities is the wrong quantity here.

The weight is ∂B/∂T\partial B/\partial T rather than BB because the flux is driven by the gradient of the radiation field, not by the field. A frequency at which the Planck function is large but insensitive to temperature carries no net flux however transparent the gas is there.

And a harmonic mean is dominated by its smallest values. If a tenth of the spectrum is a hundred times more transparent than the rest, that tenth carries most of the flux and the harmonic mean sits close to its opacity. The Rosseland mean is therefore a measurement of the windows between the lines.

Two means of one opacity, 58 times apart, and only the smaller one is in the equation. Above: a synthetic opacity across the frequencies that carry a star's flux, drawn against x = hν/kT, with a continuum falling as ν⁻³ and a forest of 2 lines per unit x on top of it. The shaded curve is the Rosseland weighting function, ∂B_ν/∂T, which peaks at x = 3.83 and is what decides which frequencies matter. The two horizontal lines are the two ways of averaging. The Planck mean is an ordinary average and lands high, among the lines, because that is where most of the opacity is. The Rosseland mean is a harmonic average — it averages 1/κ rather than κ, because what carries the flux out of a star is transparency and transparencies add — and it lands 58 times lower, close to the continuum, because a harmonic mean is dominated by the smallest values in it. In other words the opacity that appears in the equation of radiative transport is a measurement of the gaps between the lines. Below: what that means for a table. The Planck mean rises as the first power of the line density, slope 0.76 as drawn — every line added is another contribution to an ordinary average. The Rosseland mean does almost nothing at first, slope 0.13, and then turns up sharply, slope 0.99, once the lines are close enough to blanket the windows. That is why adding several million atomic transitions to an opacity table in the early 1990s changed nothing for decades and then changed stellar structure: the new lines were not the first lines, they were the ones that finally closed the gaps.
Fig. 2 The same construction with a third as many lines, which is the clean case for reading what the harmonic mean does. With two lines per unit xx the windows are wide, the flux goes round every obstacle, and the Rosseland mean sits essentially on the continuum — while the Planck mean, which counts each line in proportion to its strength, is still nearly sixty times higher. Sixty for a spectrum that is mostly window. That ratio is not a measure of how much absorption there is; it is a measure of how structured the absorption is, and a grey opacity would give two means that agree exactly.

The other mean, and where it belongs

The Planck mean,

κP  =  ∫κνBν dν∫Bν dν,\kappa_{\rm P} \;=\; \frac{\int \kappa_\nu B_\nu\, d\nu}{\int B_\nu \,d\nu},

is an ordinary average and does what an ordinary average does: every line contributes in proportion to its strength. It is the right quantity for a thin medium, where each photon either escapes or is absorbed once and the question is what fraction is absorbed. It is the wrong quantity for a diffusive interior, and the two differ by a large factor whenever the spectrum has structure — for the synthetic case drawn at the top of this essay, by about seventy.

That factor is not a curiosity. It is the reason a single number quoted as “the opacity” is meaningless without saying which, and the reason a stellar atmosphere and a stellar interior use different ones for the same gas.

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.
Fig. 3 What supplies the opacity in the first place. Electron scattering is grey — the same at every frequency — and sets a floor that nothing can go below. The bound-free and free-free processes give the Kramers behaviour, steeply temperature-dependent. And the negative hydrogen ion dominates in the outer layers of a cool star, an ion that exists at all only because a neutral hydrogen atom is polarisable enough to bind a second electron by three quarters of an electronvolt. The metals contribute the lines, which are absent from this figure and are the whole subject of this one.

What that does to a table

The practical consequence is a statement about when adding data changes an answer.

Consider building an opacity table by including atomic transitions one species at a time. Each new species adds lines. In the Planck mean, every line added raises the answer in proportion to its strength, so the mean rises roughly linearly with the number of lines and the improvement is steady and unsurprising.

In the Rosseland mean, almost nothing happens. A new line covers a narrow interval of frequency and blocks it completely — but the flux simply goes round, through the windows either side, and the harmonic mean is unchanged. Adding lines to an opacity table is like adding obstacles to a wide river: until they overlap, the flow is the same.

And then they overlap. Once the lines are dense enough that the windows between them are themselves narrow, there is nowhere for the flux to go, and the Rosseland mean starts rising steeply. The transition is sharp, and it happens at a line density that depends on temperature and density, because both control how wide each line is.

Two means of one opacity, 112 times apart, and only the smaller one is in the equation. Above: a synthetic opacity across the frequencies that carry a star's flux, drawn against x = hν/kT, with a continuum falling as ν⁻³ and a forest of 6 lines per unit x on top of it. The shaded curve is the Rosseland weighting function, ∂B_ν/∂T, which peaks at x = 3.83 and is what decides which frequencies matter. The two horizontal lines are the two ways of averaging. The Planck mean is an ordinary average and lands high, among the lines, because that is where most of the opacity is. The Rosseland mean is a harmonic average — it averages 1/κ rather than κ, because what carries the flux out of a star is transparency and transparencies add — and it lands 112 times lower, close to the continuum, because a harmonic mean is dominated by the smallest values in it. In other words the opacity that appears in the equation of radiative transport is a measurement of the gaps between the lines. Below: what that means for a table. The Planck mean rises as the first power of the line density, slope 0.77 as drawn — every line added is another contribution to an ordinary average. The Rosseland mean does almost nothing at first, slope 0.07, and then turns up sharply, slope 1.09, once the lines are close enough to blanket the windows. That is why adding several million atomic transitions to an opacity table in the early 1990s changed nothing for decades and then changed stellar structure: the new lines were not the first lines, they were the ones that finally closed the gaps.
Fig. 4 The other knob on the same transition, and the reason it is a physical statement rather than a bookkeeping one. Here the line count is unchanged and each line is half as wide, so the same transitions blanket half as much of the spectrum — and the Rosseland mean falls back towards the continuum while the Planck mean, which does not care where the absorption sits, barely moves. Line width is set by temperature, by density through pressure broadening, and by the microturbulence; so a table’s approach to the blanketed regime depends on thermodynamic conditions and not only on how many transitions the atomic physicists have managed to compute.

What was actually measured, and what was computed

Almost nothing about a stellar interior opacity is measured. It is computed — from atomic structure calculations, level by level, for every ionisation stage of every element at every relevant temperature and density — and the reason this essay can talk about it as an observation is that the computation was checked against stars and failed.

Through the 1980s the standard tables under-predicted the opacity in a temperature range near 200,000 kelvin. Two long-standing discrepancies followed from it, and both were about pulsation:

The Cepheid mass discrepancy. The mass a Cepheid’s pulsation implied and the mass its position on the Hertzsprung–Russell diagram implied differed by twenty to forty per cent, consistently and in the same direction.

The β Cephei stars. A class of hot, massive, pulsating stars existed and no calculation could make them pulsate. Every model was stable.

Both were resolved between 1991 and 1994 by two independent recomputations of the opacities — the Livermore OPAL tables and the international Opacity Project — which included vastly more transitions of iron-group elements than their predecessors. The Rosseland mean near 200,000 K went up by a factor of two to three, and both discrepancies closed at once.

Where the light gets out, and how thin that is. Left: the contribution to the emergent intensity, e^−τ dτ/dz, through an isothermal atmosphere of scale height 140 km. It peaks at τ = 1 and is negligible above τ ≈ 0.1 and below τ ≈ 10, so essentially all the light a telescope receives leaves from a layer 645 km thick — H ln 100, fixed by the scale height and nothing else. The τ = 2/3 level, which is what "the photosphere" means and what a stated radius refers to, sits 57 km above the peak. Right: that thickness against a solar radius of 695,700 km, drawn to scale — 92.7 parts in a hundred thousand, or 0.0927%. A star has no surface and looks as though it has one, and the reason is that opacity climbs so steeply with depth that the transition takes a ten-thousandth of the radius. What this cannot show is the wavelength dependence: the depth reached is different in every colour, which is what makes a limb dark rather than merely edged.
Fig. 5 The layer where it happened. Iron is a trace element by mass and an enormous contributor by line count, because a partially ionised iron atom has thousands of accessible transitions where hydrogen has a handful. Near 200,000 kelvin iron is in exactly the ionisation stages with the richest spectra, so that is where a table with too few lines is most wrong — and it is deep enough to matter for the star’s structure rather than only for its surface.

The absorber that decides where convection starts

The iron bump is the case where a change in the tables changed the answer. There is a second feature of the opacity curve that has always been in the tables and that decides something more basic, and it is worth setting out because it shows the harmonic mean behaving in the opposite way.

In the outer layers of a cool star the dominant absorber is not an atom but an ion that barely exists: a hydrogen atom with a second electron attached, bound by three quarters of an electronvolt. It is fragile, its abundance depends on the supply of free electrons — which at those temperatures come from the metals rather than from hydrogen — and its absorption is a smooth continuum rather than a forest of lines.

The consequence is an opacity that rises extremely steeply with temperature, roughly as the ninth or tenth power over the relevant range, because raising the temperature both ionises more metals and therefore supplies more electrons, and increases the population of the states that can capture them. That steepness is what makes the outer envelope of a cool star convective.

The argument is short. Convection begins where the temperature gradient required to carry the flux radiatively exceeds the adiabatic one, and the radiative gradient is proportional to the opacity. An opacity rising as a high power of temperature therefore makes the radiative gradient rise steeply with depth, and it crosses the adiabatic value at a definite place. Above that place the star is radiative and below it, it boils.

So the depth of a star’s convection zone is set by the behaviour of a weakly bound ion, and the fact that cool stars have deep convective envelopes and hot ones do not is a statement about where that ion can exist. It is also why the boundary between the two behaviours falls where it does on the main sequence: above about 7,500 kelvin at the surface, hydrogen is ionised, the negative ion has nothing to form from, and the envelope stays radiative.

Opacity against temperature, and the three things that supply it. The Rosseland mean opacity of a gas of composition X = 0.75, Y = 0.2498, Z = 0.0002, 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 535.5 cm²/g at 5,800 K, against the 0.40 drawn here. The peak sits at 13,000 K, where hydrogen is 42% 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.
Fig. 6 The same three sources in gas a hundred times poorer in metals, which is the test of the argument above. The negative hydrogen ion needs free electrons, and at these temperatures the free electrons come from the metals rather than from hydrogen — so removing the metals removes the electrons and the H⁻ hump collapses. A metal-poor star therefore has a shallower convection zone than a solar-composition star of the same mass, and the effect is on the donor of the electron rather than on the absorber, which is as indirect as a dependence gets. It is also measurable: the halo subdwarfs sit below the solar-metallicity main sequence partly for this reason.

A mean opacity is usually a summary of many contributions and occasionally the property of one species, and where it is the second, everything downstream inherits that species’ peculiarities.

The test that is not circular

There is a fair objection to all of this: the opacities were adjusted until the stars worked, so of course the stars work.

Three things answer it.

The recomputations were not fits. OPAL and the Opacity Project were atomic-physics calculations undertaken for their own reasons, with no pulsation constraint anywhere in them, and they agreed with each other to a few per cent while disagreeing with the older tables by a factor of two.

They made a prediction that had not been used. The same tables predicted a further, weaker opacity feature from nickel and iron near 100,000 K, which drives the pulsations of subdwarf B stars — a class whose variability was discovered in 1997, after the tables, and which the older opacities cannot produce.

And helioseismology tested them somewhere else entirely. The Sun’s oscillation frequencies fix its internal sound speed to better than a part in a thousand over most of its radius, and the sound speed depends on the opacity through the temperature gradient. The new tables improved the agreement between the standard solar model and the seismic profile; the old ones could not be made to fit.

A better measurement that made the model worse: 0.9 per cent in the sound speed. The fractional difference between the Sun's sound speed as its own oscillations measure it and as a structural model predicts it, against fractional radius. Zero would be agreement. The lower curve is the model built on the solar abundances used until the mid-2000s, and it hugs the axis: a part in a thousand across most of the interior, which was for a long time the best-tested piece of stellar physics anybody had. The upper curve is the same model with the abundances re-measured using three-dimensional atmospheres and without assuming local thermodynamic equilibrium — better measurements by every methodological standard, which lowered carbon, nitrogen and oxygen by around thirty per cent. The disagreement grows to 0.9 per cent, and it is not spread through the star: it peaks at 0.683 of the radius, just beneath the base of the convection zone at 0.713. The same substitution moves the model's own convection-zone base from 0.715 to 0.729, against a seismic value known to about a thousandth. What is being tested here is not really the abundances but what converts a composition into a structure, which is the opacity: the metals whose abundances fell are exactly the ones whose bound–free absorption dominates at those temperatures, and an opacity larger by some fifteen per cent near that boundary would restore the agreement. Laboratory measurements of iron at those conditions have since come in high by about that much, which is a satisfying result to have arrived at by way of a discrepancy in the sound speed of the Sun. The curves are published inversions and model differences rather than anything computed here; what the figure adds is where they peak and by how much.
Fig. 7 The seismic test, and the reason it cuts both ways. The lower curve is the sound-speed residual for a model built on the older solar abundances, and it hugs zero across most of the radius — a part in a thousand, which is the agreement the recomputed opacities bought. The upper curve is the same model with the abundances revised downwards in the 2000s on the strength of better atmospheric physics, and it is worse by nearly a per cent near the base of the convection zone. A better measurement of what the Sun is made of made the model of its interior worse, and the opacity is the term with enough freedom to absorb the difference. That is the live discrepancy this essay ends on, drawn as the thing that is actually observed.

How a table is actually used

A stellar model does not evaluate the integral in the definition. It looks the answer up, and the mechanics of that lookup contribute an error term that is rarely quoted and is not always negligible.

An opacity table is a grid: the Rosseland mean tabulated against temperature and against a density-like variable, for a set of compositions specified by the hydrogen fraction and the metal fraction. A model integrating a star’s structure needs the opacity at whatever temperature, density and composition it has arrived at, which is almost never a grid point, so the value is interpolated — bilinearly in the two thermodynamic variables and then between composition tables.

Two things make that harder than it sounds. The surface being interpolated is not smooth: the ionisation bumps described above are genuine sharp features, and an interpolation across one of them either smooths it, which changes the structure, or overshoots, which introduces oscillations. And the derivatives of the opacity matter as much as its value, because whether a layer is convective depends on comparing two gradients — so an interpolation scheme that reproduces the opacity to a per cent and its logarithmic derivative to twenty is inadequate for the one thing the model most needs.

The standard practice is therefore to interpolate in the logarithms of everything, on a grid fine enough that the bumps are resolved, with a scheme that preserves monotonicity where the underlying function is monotone. Different stellar-evolution codes make different choices, and comparisons between codes run on identical physics have found differences in derived quantities — convective boundary positions, main-sequence lifetimes — that trace back to interpolation rather than to any disagreement about the physics.

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 10⁻⁸ g/cm³ and 10⁻⁴ 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 1,034 cm²/g at 5,800 K, against the 0.09 drawn here. The peak sits at 13,100 K, where hydrogen is 81% 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.
Fig. 8 What the grid has to span, drawn as the two densities four decades apart that a single stellar model visits between its photosphere and its core. The curves are not parallel: electron scattering is density-independent and the Kramers and H⁻ terms are not, so the ordering of the three sources changes with depth and the bumps move. That is why interpolation in this table is harder than interpolation in a smooth function of two variables — the feature being interpolated across is at a different temperature on every row, and a scheme accurate in the value can be poor in the derivative exactly where two curves cross. The derivative is what decides convection, so the crossings are where the interpolation error matters most.

There is a further seam where two tables meet. Interior tables and atmospheric tables are computed by different groups with different assumptions, and a model has to join them somewhere in the outer envelope. The join is a discontinuity in a quantity whose derivative is structurally important, and smoothing it is a matter of convention rather than of physics.

The composition dimension has a seam of its own, and it is the one that has moved most. A table is computed for a fixed pattern of heavy elements — so much oxygen relative to iron, so much neon relative to oxygen — and scaled by an overall metallicity. A star whose pattern differs from the assumed one, which is every star that is not a young disc star, is being described by a table computed for something else.

That matters because the elements contributing most to the opacity at a given temperature are not the ones contributing most to the mass. Near the base of a solar-type convection zone, oxygen and neon carry much of the bound-free absorption while iron carries much of the line blanketing deeper in, so a composition revision that lowers oxygen and leaves iron alone changes the opacity by a different factor at every depth. Rescaling a solar-pattern table by a single metallicity cannot represent that, and the tables now used for the revised solar composition were recomputed rather than rescaled — which is the correct response and is not always what is done for other stars.

The practical shortcut, where a recomputation is not available, is to carry two metallicities — one for the α elements and one for iron — and interpolate between two table families. That reproduces the leading effect and not the ordering of the ionisation bumps, which depends on the pattern in detail.

A quantity computed to a few per cent and used through a grid, an interpolation and a join is not known to a few per cent, and the difference between the two figures is one of the quieter systematic errors in stellar astrophysics.

Where the model stops

Line lists are still incomplete. The number of transitions in a modern table runs to hundreds of millions, and completeness matters in exactly the regime this essay is about — the windows. A missing line in a window costs more than a missing line in a crowded region, and missing lines are by construction the ones nobody has computed.

The mean assumes local thermodynamic equilibrium. The derivation above put a Planck function in the weight, which requires the radiation field to be very nearly thermal. Deep in a star it is. In an atmosphere, and in the outer layers of a hot star where the density is low, it is not, and the Rosseland mean loses its meaning there rather than merely losing accuracy.

And a mean is a scalar where the truth is a spectrum. Two gases with the same Rosseland mean and entirely different κν\kappa_\nu behave identically in a diffusive interior and completely differently anywhere the flux is not diffusive. A model that carries only the mean cannot know which it has.

There is also a live discrepancy. The solar composition was revised downwards in the 2000s on the strength of better atmospheric models, and the revised composition does not reproduce the seismic sound speed. Raising the opacity near the base of the convection zone by fifteen per cent would fix it, and a 2015 laboratory experiment on iron plasma at the right conditions measured an opacity that high — against a calculation that says otherwise. The disagreement is unresolved, and it is the only case in this subject where a stellar interior opacity has been measured in a laboratory at all.

The generalisation

The lesson is about which average a physical law asks for, and it is not confined to radiation.

Thermal conductivity through a layered composite is a harmonic mean across the layers and an arithmetic mean along them, and the two differ by orders of magnitude for the same material. Electrical resistance in a network is the same statement. The effective viscosity of a suspension, the effective permeability of a rock, the effective yield of a galaxy that loses gas — each is a case where a heterogeneous quantity is summarised by one number, and each has a different right answer depending on whether the paths are in series or in parallel.

The rule that covers all of them: average the quantity that adds. For a flux through parallel channels that is the conductance, so the mean is harmonic. For an absorption along one path it is the opacity itself, so the mean is arithmetic. Getting it backwards does not give a slightly wrong answer; for a spectrum with structure in it, it gives an answer wrong by a factor of seventy.

Where this ladder goes next

Later rungs on this anchor: opacity in the non-equilibrium regime, where a Rosseland mean does not exist and the transfer equation has to be solved frequency by frequency; conductive opacity, which takes over in degenerate matter and makes a white dwarf’s interior nearly isothermal; molecular and dust opacities in the coolest atmospheres, where the absorbers are no longer atoms; the opacity of the intergalactic medium, which is a different physical problem with the same formalism; and the laboratory measurements, which are the only external check on a quantity that otherwise reaches the observations only through a stellar model.

Where this goes next

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. Which gradient is steeper, and where. The temperature gradient radiation would need in order to carry the whole luminosity, against the gradient a rising blob of gas follows, through an n = 3 polytrope with a Kramers opacity and an energy generation going as ρT^4. Wherever the first exceeds the second the layer is unstable and convects. This model puts the outer boundary at 89 per cent of the radius; helioseismology measures the base of the Sun's convection zone at 71.3 per cent, and the gap between the two numbers is the price of a polytrope with one opacity law rather than a solar model. The curve also rises above the adiabatic value inside 4 per cent of the radius, which is a convective core — it appears when the energy generation is concentrated enough, and it is the structure a star burning by the CNO cycle has. The part of a star that boils Energy leaves a star's interior either by radiation or by bulk motion, and which one happens is settled by comparing two temperature gradients. The comparison decides that the Sun's outer third churns and its core does not, and that a massive star does exactly the opposite. The period spans 492× and the pulsation constant 2.1×. The pulsation constant Q = P√(ρ̄/ρ̄_⊙) for 7 radial pulsators, against their periods. A radial pulsation is a standing sound wave across a star and the sound speed in a self-gravitating body is set by its own gravity, so the period should go as (Gρ̄)^−1/2 and Q should be the same for every star pulsating in the same mode. Across a factor of 492 in period — from 1.3 hours to 27 days — Q varies by 2.07, with a mean of 0.0393 days. That is the whole reason a period–luminosity relation can exist: a period is a density, a density is a mass and a radius, and a radius with a temperature is a luminosity. The masses here are the weak link and the caption should say so — for the Cepheids they come from evolutionary models rather than from a dynamical measurement, and the drift of Q with period is where that assumption is showing. The valve that has to sit at the right depth A period–luminosity relation only exists because a period is a density in disguise. Which stars have a period at all is decided by whether a partial ionisation zone sits deep enough to have mass behind it and shallow enough that convection has not taken it over.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 9 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Bound bound absorptionFlux weightingHarmonic meanIron bumpLine blanketingMean free pathOpacityOpacity projectOptical depthPlanck meanRadiative diffusionRosseland mean