Astronomy & Celestial Mechanics
the sky, computed
Generator

The energy-transport generator

Which gradient is steeper, and where
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 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.

9 essays call energy-transport. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

The limb is 40% as bright as the centre, and that is a temperature gradient. Left: a stellar disc shaded by the grey-atmosphere law I(μ)/I(1) = (2 + 3μ)/5, in 26 steps, with μ = cos θ read from the centre outwards. Right: that law against μ, with linear laws at the measured solar coefficients from 400 to 1600 nm. The grey law's coefficient is exactly 3/5 and its very limb is exactly 2/5 of the central brightness — both read off the drawn curve rather than quoted — because the Eddington–Barbier relation makes the emergent intensity at angle μ the source function at optical depth τ = μ, and in radiative equilibrium that source function is linear in τ. The limb is not a cooler part of the star. A sight line entering at the edge reaches unit optical depth higher up, where the gas is cooler, so what the darkening measures is the run of temperature with depth; a star with an isothermal atmosphere would show a uniform disc, and one with a steeper gradient a darker limb. The measured coefficients fall from 0.9 at 400 nm to 0.35 at 1600 — the same gradient seen through a less steep Planck function — which is why a radius measured from a transit or a fringe null has to say which colour it was measured in. Starlight

The light that is missing from the edge

The Sun's limb is forty per cent as bright as its centre, and the reason is not that the edge is cooler. A sight line entering at the edge stops higher up, so what the darkening measures is the temperature gradient — and it is worth seventeen per cent on Betelgeuse's 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. Stars

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.

 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. Stars

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.

 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. Stars

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.

 A sight line stops where its own optical depth is one. The source function of a grey atmosphere in radiative equilibrium — a straight line, S = (3/4)(τ + 2/3) in units of the flux — with the depth each sight line reads off it. The emergent intensity at an angle is the source function integrated along the ray with everything in front of it attenuating, and for a source function linear in optical depth that integral is exactly the source function evaluated at τ = μ. Not approximately: the Eddington–Barbier relation is an identity for a linear source function, and the numerical integral agrees with it here to a part in ten thousand, which is the quadrature's error and not the relation's. A vertical ray reads the source function at τ = 1 and a ray at 78 degrees reads it at τ = 0.2, which is higher in the atmosphere and therefore cooler — so the limb is dimmer than the centre by the amount the source function has risen between those two depths. Everything about limb darkening is the slope of this one line, and the famous 2/3 is the intercept: the average ray, over the whole disc, reads the source function at two thirds of a unit of optical depth, which is the depth a stellar "surface" actually means. Starlight

A sight line stops at two thirds

The whole of limb darkening is one sentence: what leaves a star at an angle is the source function evaluated where the slant optical depth is one. For a source function linear in depth that statement is exact rather than approximate, and it produces the grey atmosphere's (2+3μ)/5 with no fitting anywhere in it.

 The sign of one gradient decides which way the disc is shaded. The emergent intensity across a stellar disc, against the fractional radius, for four gradients of the source function with depth. Every curve is the Eddington–Barbier relation evaluated at the appropriate μ and checked against a numerical integration of the transfer equation. The contrast between centre and limb is the gradient, and its sign is the gradient's sign. A source function rising inward — the ordinary case, since temperature rises inward and the source function follows it — darkens the limb: a sight line at the edge leaves from higher up, where the gas is cooler. A flat source function produces a uniformly bright disc whatever the geometry. And a source function that falls inward brightens the limb, because now the shallower ray is reading a hotter layer. That last case is not hypothetical: above a star's photosphere the temperature stops falling and begins to rise, so at wavelengths that see those layers — millimetre continuum, the cores of strong lines, the ultraviolet — the Sun's limb is brighter than its centre. Limb darkening is therefore not a property of stars but of a gradient, and the same atmosphere shows both signs at different wavelengths. Starlight

A limb brighter than the middle

Limb darkening is not a property of stars. It is a property of a gradient, and the same relation that produces it produces the opposite whenever the source function falls inward — which it does above every stellar photosphere. The Sun's limb is darker in the visible and brighter at a millimetre, and both are the same equation.

 Larger and cooler, along a line of constant light. The fractional change in effective temperature against the fractional change in radius for a star whose luminosity is unchanged, and for luminosities −10% and +10%, from L = 4πR²σT⁴. Holding the luminosity, a radius 10 per cent larger needs a temperature 4.7 per cent lower: about half the radius change, with the sign reversed. The shaded box is the range reported for red dwarfs in eclipsing binaries against standard models — radii 5 to 15 per cent too large, temperatures 3 to 5 per cent too low — and it lies along the line of constant luminosity. Stars

Too large and too cool by just enough to keep the light

Red dwarfs in eclipsing binaries, whose radii are measured directly, come out several per cent larger than stellar models predict for their masses and a few per cent cooler. They are not brighter or fainter. The light is decided deep inside, by mass and opacity, and whatever disturbs the convecting outer layers can change only the radius and temperature through which that light leaves.

 A feature read back from the limb. A bump in the source function at optical depth 0.1, a quarter of a decade wide (black dashed), and what an inversion of the limb profile recovers on average when the profile is measured at 24 angles with errors of 1, 0.1, 0.01, 0.001 per cent, the inversion assuming only that the departure from a grey atmosphere is smooth. At one per cent the recovered bump has 56 per cent of its height and is spread over about twice its width; at a thousandth of a per cent, 82 per cent, and still wider than the truth. A thousandfold gain in precision buys a modest gain in resolution — each factor of ten a step, not a factor. Starlight

A profile that averages a decade of depth

Each point on a star's disc reports the temperature at a different depth, so a limb-darkening profile looks like a depth sounding of the atmosphere waiting to be read. It is one, but a blurred one. Every angle averages a whole decade of optical depth, the averages overlap, and recovering the temperature structure from them is the inversion of a Laplace transform — a problem whose resolution grows only as the logarithm of the data's precision.

 A reservoir so large that it does not notice. The fractional increase in the heat of the Sun's convective envelope if a fraction 0.001, 0.003, 0.01 of the surface is held under spots, against time. The envelope holds about 7×10⁴⁵ erg, 58 thousand years of the Sun's light. Spots covering 0.3% of the surface for an eleven-year cycle add 4.2e-7 of that, a temperature change of 0.4 parts per million at most, spread over the whole envelope. So the blocked heat simply remains in the star, and the surface flux shows the deficit. It levels off only after the envelope's thermal time, when the star has adjusted. Stars

A spot whose missing heat goes nowhere

A sunspot blocks most of the energy trying to come up beneath it, and a large group visibly dims the whole Sun by a thousandth. Energy is conserved, so the blocked flux ought to come out somewhere else, and for a century observers looked for the bright ring around every spot. There is none to speak of. The heat has gone into the rest of the convection zone, spread so fast and so widely that the surface cannot notice, and it will stay there for longer than human history. For the time being the Sun simply is fainter, without any contradiction.

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