Astronomy & Celestial Mechanics
the sky, computed
Concept

Convection — where it appears

Energy carried by the bulk motion of gas, which takes over wherever radiation would need a steeper temperature gradient than a rising parcel follows. The criterion for its onset compares two gradients rather than two temperatures, and it takes over in the outer envelopes of cool stars and in the cores of hot ones.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

stars · Energy transport 14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here.

The interior read from a comb of frequencies

A star's surface moves by about twenty centimetres a second, in thousands of overlapping sound modes at once. Two numbers off that spectrum give a mass and a radius with almost no stellar model in the chain, and a third gives an age.

stars · Asteroseismology A furnace with the thermostat taken out. Left, the two pressures in a red giant's helium core at 10⁶ g/cm³. The ideal-gas pressure rises with temperature, as the whole regulation of a star depends on it doing; the degenerate electron pressure is a horizontal line, because the exclusion principle does not know the temperature. At the ignition point near 10⁸ K the degenerate term is 5.1 times the gas term, so a temperature rise there raises the total pressure by almost nothing and the core does not expand. Right, what that costs. Both curves start at the same temperature with the same triple-alpha rate, whose logarithmic slope is 41 — the famous T⁴⁰, differentiated here rather than quoted. The regulated core settles; the degenerate one, with nowhere to put the extra energy, goes vertical. The flash reaches 10¹⁰ solar luminosities for a few seconds and not one photon of it is seen: every erg goes into lifting the degeneracy, and the star's visible response is to become fainter and settle on the horizontal branch.

The explosion that never reaches the surface

When helium ignites in a degenerate core the thermostat every other burning stage runs on is disconnected. The runaway reaches ten billion solar luminosities for a few seconds, and the star's visible response is to get fainter.

stars · Stellar evolution 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.

stars · Variable stars The Sun's internal rotation: differential above 0.693 R, rigid below, and the 15 nHz between two modes. Angular velocity against fractional radius, in nanohertz, from the inversion of hundreds of thousands of measured rotational splittings. Two features had to be discovered rather than deduced. The convection zone rotates differentially in latitude — 473 nanohertz at the equator, 330 at sixty degrees — and that latitude dependence persists all the way down through it, in surfaces that are very nearly radial rather than the cylinders a rotating convective fluid was expected to produce. And below about 0.693 of the radius the latitude dependence stops: the radiative interior turns as a single rigid body at 430 nanohertz, which is a remarkable thing for a fluid with no strength to do, and requires something — most likely a weak internal magnetic field — to be enforcing it. The two regimes are joined by a shear layer a few per cent of the radius thick, the tachocline, and it is where the solar magnetic cycle is generally thought to be generated, because it is the only place in the Sun with the shear a dynamo of that strength needs. The horizontal marks are what individual modes would report: the rotation averaged over the cavity each one occupies, weighted by the time the wave spends at each radius. A mode of degree 100 is trapped near the surface and reports 473 nanohertz; one of degree 1 passes through the deep interior and reports 458. The 15-nanohertz difference between them is the measurement, and splittings are measured to about a nanohertz — which is why the profile can be resolved at all. The kernels used here have the right support and the right sense; a real inversion uses computed eigenfunctions, and its resolution below 0.2 R is poor for the reason the previous figure gives.

A fluid that turns as one piece

The Sun's surface turns faster at its equator than at its poles, and everyone expected the inside to do something similar. Below seven-tenths of the way down it does not — the radiative interior rotates as rigidly as a bell, and nothing in hydrodynamics makes a fluid do that.

stars · Internal rotation Where each mode turns back: ℓ = 0 through the centre, ℓ = 300 in the outer 2 per cent. Why a set of frequencies is a depth profile and a single frequency is not. An acoustic wave travelling into a star meets a rising sound speed and is refracted back; it turns where its horizontal phase speed matches the local sound speed, which happens at c(r)/r = 2πν/√(ℓ(ℓ+1)). The horizontal axis is the angular degree on a logarithmic scale and the vertical axis is the fractional radius of that turning point, drawn at 2000, 3090, 4000 microhertz. The ordering is the content. A radial mode, ℓ = 0, has no horizontal phase speed at all and passes straight through the centre. Degrees one and two turn deep in the core. By ℓ = 300 the mode is trapped in the outer 2 per cent and knows nothing about anything below. So a frequency measured to a part in ten thousand constrains an average of the interior weighted in a way the mode itself decides, and measuring thousands of modes of different degree gives thousands of differently weighted averages — which is a solvable inverse problem, and is how the base of the convection zone was located at 0.713 of the radius rather than assumed. The sound speed here is a polytrope's rather than a tabulated solar model's, so the curve is the right shape and the wrong star in its outer tenth, where the real Sun is convective and this one is not. What the picture cannot show is the frequency dependence at fixed degree, which is weaker but not negligible: a higher-frequency mode of the same degree turns slightly deeper, and the three curves separating toward the right is that effect.

Every note turns back at its own depth

A sound wave heading into a star is refracted by the rising sound speed and turns around before it reaches the middle. Where it turns depends on which mode it is — so a list of frequencies is not one average of the interior but thousands of differently weighted ones, and that is a solvable problem.

stars · Asteroseismology 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.

A better measurement that made the model worse

The Sun's composition was re-measured with better atmospheres and better physics, and the carbon, nitrogen and oxygen abundances fell by about thirty per cent. The improved model then disagreed with the Sun's own oscillations by ten times as much as the model it replaced, and it still does.

starlight · Opacity A spot is dark because it is squeezed. Pressure against depth below the quiet photosphere's optical surface, drawn as a ratio to the pressure there. The rising curve is the surrounding gas, which grows exponentially with a 140-kilometre scale height because that is what hydrostatic equilibrium in an ideal gas produces. The flat pair of bands is the spot's own budget: a magnetic pressure of 3.58·10⁵ dyn/cm² from a 3000-gauss field, which is 25 per cent of the total, plus the gas pressure left over. Horizontal balance requires the two columns to reach the same total at the same geometric level, and the level at which they do is 348 kilometres below the quiet surface — so the spot's own optical surface sits in a hollow. Measured Wilson depressions, obtained from the foreshortening of a spot near the limb, are four to six hundred kilometres. Nothing about the darkness was assumed: a field strength read off a Zeeman splitting fixes the magnetic share, the share fixes the depression, and the depression fixes the temperature the deeper layer must have to carry the reduced flux.

The darkness a field pays for

A sunspot is not cool because something is missing. It is cool because a three-thousand-gauss field supplies part of the pressure that holds the column up, the gas therefore supplies less, and the level at which that gas becomes opaque sits some hundreds of kilometres deeper than the surface around it.

stars · Plasma beta A free parameter worth 88 kelvin across its plausible range. The effective temperature a stellar model predicts, against mass, for three values of the mixing-length parameter. The parameter has no derivation: it is the distance a convective blob is supposed to travel before dissolving, in units of the local pressure scale height, and it is fixed by requiring that a model of the Sun reproduce the Sun. The three curves span 88 kelvin, which at fixed luminosity is a radius difference of 1.5 per cent — comparable to the precision with which radii are now measured by interferometry and by eclipsing binaries. Every stellar age, every isochrone and every mass inferred from a position in the temperature–luminosity plane depends on the value chosen, and there is no reason beyond convenience to expect the solar value to apply to a red giant or to a metal-poor dwarf.

A length nobody derived, fitted to one star

Convection in a star is turbulent, three-dimensional and impossible to compute inside an evolution code. What is used instead is one number — how far a blob of gas travels before dissolving — fixed by requiring that a model of the Sun come out with the Sun's radius, and then applied to every star ever modelled.

stars · Energy transport An exponent between 2.3 and 4, not a number. The local logarithmic slope of the mass–luminosity relation against mass — the exponent that "L ∝ M^3.5" stands in for — read off the same fit to eclipsing binaries that the relation itself is drawn from. It is a step function with two corners, and the steps are 2.3 below 0.43 M☉, 4 0.43 to 2 M☉, 3.5 2 to 55 M☉. The horizontal lines are what homology predicts: a star in radiative equilibrium has L ∝ μ⁴M³/κ̄, which gives an exponent of 3 when the opacity is electron scattering and a constant, 5.5 when it is Kramers and depends on density and temperature, and 1 where radiation pressure holds the star up and the luminosity is pinned to the Eddington value. The measured steps sit between those predictions rather than on them, which is what a real star does — no star is homologous, the low-mass ones are convective throughout and the heavy ones have convective cores, and the fitted exponents are what is left after all of that. The useful statement is not that the exponent is 3.5 but that it is between 2 and 5 everywhere and 4 in the middle, and that the reason it moves is a change of which opacity is carrying the energy out.

An exponent that is a slope, not a law

The phrase “L goes as M to the three and a half” stands in for a curve with three straight pieces and two corners. The exponent is 2.3 below half a solar mass, 4 in the middle and falls towards 1 at the top — and each change of exponent is a change in which opacity is carrying the energy out.

stars · The mass–luminosity relation

Named alongside it

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

HelioseismologyOpacityAsteroseismologyGranulationHydrostatic equilibriumRotational splittingSolar cycleStellar radiusSystematic errorAbundanceAcoustic cavityAdiabatic gradient

All concepts