Astronomy & Celestial Mechanics
the sky, computed
Generator

The giant-mirror generator

The mirror: an envelope 62 times larger around a core 1.48 times smaller
The mirror: an envelope 62 times larger around a core 1.48 times smaller. Envelope radius and core radius against helium-core mass for a 1 solar-mass star, both in solar radii on one logarithmic axis. Two curves with opposite slopes at every point: the envelope grows from 2.0 to 122 solar radii — 0.57 astronomical units — while the degenerate core contracts from 13,848 to 9,368 kilometres, and the two are separated by a factor of 364 in the middle of the range. The figure is drawn from two stated relations rather than from a stellar model: a core-mass–luminosity law of the sixth power, calibrated to 2,500 solar luminosities at 0.48 solar masses of core, and a Hayashi temperature falling from 5000 to 3700 kelvin across the range. Everything else follows exactly — the radius by the Stefan–Boltzmann law, the core radius by the cold degenerate relation R ∝ M⁻¹ᐟ³. Note what the tip radius is and is not: it is the red-giant branch, and the later asymptotic-giant tip is a different and larger number.

Envelope radius and core radius against helium-core mass for a 1 solar-mass star, both in solar radii on one logarithmic axis. Two curves with opposite slopes at every point: the envelope grows from 2.0 to 122 solar radii — 0.57 astronomical units — while the degenerate core contracts from 13,848 to 9,368 kilometres, and the two are separated by a factor of 364 in the middle of the range. The figure is drawn from two stated relations rather than from a stellar model: a core-mass–luminosity law of the sixth power, calibrated to 2,500 solar luminosities at 0.48 solar masses of core, and a Hayashi temperature falling from 5000 to 3700 kelvin across the range. Everything else follows exactly — the radius by the Stefan–Boltzmann law, the core radius by the cold degenerate relation R ∝ M⁻¹ᐟ³. Note what the tip radius is and is not: it is the red-giant branch, and the later asymptotic-giant tip is a different and larger number.

5 essays call giant-mirror. 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.

A body that heats up because it is losing energy. A uniform self-gravitating sphere of 1.0 solar masses radiating at 1.0 solar luminosities, with no nuclear source at all, from 3.0 solar radii. Everything is in units of the starting energy, and the two curves that matter run in opposite directions: the total energy falls, and the temperature rises. That is not a paradox and it is not a special case. The virial theorem makes 2K = −U for any self-gravitating gas in equilibrium, so E = U + K = −K, and −dE/dt = +dK/dt: energy leaving as light is energy arriving as heat. The bookkeeping is exact and is measured here rather than quoted — over the run 5.465·10⁴⁰ J of gravitational energy is released and 2.733·10⁴⁰ J is radiated, a ratio of 2.0000. Half the release is spent on the star's own heat and only half escapes. The consequence is a body with a negative heat capacity, which is why a contracting protostar gets hotter until it ignites, why a globular cluster's core runs away instead of settling, and why nothing self-gravitating ever comes to thermal equilibrium. Gravitation

The system that gets hotter as it loses energy

The virial theorem makes a self-gravitating body's total energy equal to minus its kinetic energy. Radiating heat away therefore raises the temperature, there is no equilibrium to settle into, and every star and every cluster is running away from one.

 The mirror: an envelope 62 times larger around a core 1.48 times smaller. Envelope radius and core radius against helium-core mass for a 1 solar-mass star, both in solar radii on one logarithmic axis. Two curves with opposite slopes at every point: the envelope grows from 2.0 to 122 solar radii — 0.57 astronomical units — while the degenerate core contracts from 13,848 to 9,368 kilometres, and the two are separated by a factor of 364 in the middle of the range. The figure is drawn from two stated relations rather than from a stellar model: a core-mass–luminosity law of the sixth power, calibrated to 2,500 solar luminosities at 0.48 solar masses of core, and a Hayashi temperature falling from 5000 to 3700 kelvin across the range. Everything else follows exactly — the radius by the Stefan–Boltzmann law, the core radius by the cold degenerate relation R ∝ M⁻¹ᐟ³. Note what the tip radius is and is not: it is the red-giant branch, and the later asymptotic-giant tip is a different and larger number. Stars

The star that swells because its centre shrank

A helium core contracts, and the envelope around it expands by a factor of sixty. The central density rises at the same time as the radius, which is the opposite of what a self-gravitating body is expected to do, and a burning shell between the two is the whole reason it happens.

 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. Stars

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.

 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 free parameter worth 42 per cent of an age. The main-sequence lifetime of a star against its mass, for four values of the convective overshoot parameter. A convective core has a boundary where buoyancy vanishes, and a rising blob arriving there still has momentum, so it penetrates into the stable region above and mixes fresh hydrogen into the core. How far it penetrates is not computed — it is a parameter, quoted in pressure scale heights, and the values in current use range from zero to about a third. A larger core is a larger fuel supply, so the lifetime rises and the turn-off at a given age is more massive. Across the plausible range the lifetime of a two-solar-mass star changes by 42 per cent, and an age read off a cluster's turn-off changes by the same amount. That is larger than the quoted uncertainty on almost every cluster age in the literature. Stars

A convective boundary with no theory to fix it

A convective core has an edge where buoyancy vanishes. A blob arriving there still has momentum, so it carries on, mixing fresh hydrogen into the core and extending the star's life. How far it carries on is a fitted parameter, and across its plausible range every stellar age changes by nearly half.

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