Astronomy & Celestial Mechanics
the sky, computed
Concept

Hayashi track — where it appears

The nearly vertical line on the Hertzsprung–Russell diagram along which fully convective stars must lie. Its steepness comes from the negative hydrogen ion's opacity rising as about the ninth power of temperature, which pins the surface temperature.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A planet in the middle of a 0.08 M☉ star's zone spends 94 Myr too hot to keep an ocean. How long a planet spends receiving more flux than the runaway-greenhouse limit while its star contracts onto the main sequence, against stellar mass, for planets at the inner edge, in the middle, at the outer edge of the zone the star will have once it settles. The luminosity is the contraction law of a fully convective star, falling as t^(−2/3) from an age of 1 Myr until arrival; the limit is the runaway flux for the star's temperature. A planet at the inner edge is too hot for 313 Myr round a 0.08 M☉ star and 17.8 Myr round a 0.6 M☉ one; a planet in the middle is too hot for 94 Myr round a 0.08 M☉ star and 5.5 Myr round a 0.6 M☉ one; a planet at the outer edge is too hot for 39 Myr round a 0.08 M☉ star and 2.0 Myr round a 0.6 M☉ one. A runaway greenhouse is not a hot climate; it is a state in which the ocean is entirely in the atmosphere as steam, where ultraviolet light splits it and hydrogen escapes. A planet in the eventual zone of the commonest stars in the galaxy begins its life in that state for tens to hundreds of millions of years — a span comparable with the whole assembly of the Earth. The durations are measured from 1 Myr; a rocky planet may take tens of millions of years to finish forming, and one that formed later misses the start of its exposure, while the smallest stars' arrival times are somewhat short in this model, which lengthens the end of it.

Steam before the zone existed

The smallest stars take hundreds of millions of years to contract onto the main sequence, shining at many times the luminosity they will settle at. A planet in the habitable zone such a star will eventually have spends that time with its ocean in the air as steam, while starlight splits the water and the hydrogen leaves — so the zone of the commonest star in the galaxy is a place that had to survive being too hot first.

exoplanets · Habitable zone Hayashi lines: a luminosity that moves and a temperature that does not. The effective temperature a fully convective star must have, against its luminosity, for 0.5, 1, 2 solar masses, drawn as a Hertzsprung–Russell diagram with temperature falling to the right. The temperature follows from matching the star's convective adiabat to a photosphere whose opacity, from the negative hydrogen ion, rises as the ninth power of temperature. Over a factor of thirty thousand in luminosity each line moves by only 363 K, 398 K, 436 K respectively: the lines are nearly vertical, at 3650 K, 4000 K, 4383 K for one solar luminosity. A contracting young star, or a swelling giant, moves up or down its line almost without changing colour. To the right of the lines no star in hydrostatic equilibrium can sit.

A wall built by a steep opacity

A star that is convective all the way to its surface cannot choose its surface temperature. The interior fixes a relation between pressure and temperature, the photosphere fixes another through its opacity, and the negative hydrogen ion's opacity rises so steeply with temperature that the two meet at nearly the same temperature whatever the star's size. Young stars contracting and old stars swelling slide up and down a line in the diagram almost without changing colour — and nothing in equilibrium can sit to its right.

starlight · Opacity The gradient radiation would need, as a young star shrinks. The radiative temperature gradient at the centre of a contracting, fully convective pre-main-sequence star — the gradient radiation alone would need to carry the luminosity — against the star's radius, which shrinks to the right, for 0.3, 0.5, 1, 2 solar masses, logarithmic. The centre stays convective while the gradient exceeds the adiabatic 0.4 (dashed). As the star shrinks its centre gets hotter and denser, Kramers opacity, ∝ ρT⁻³·⁵, falls, and the gradient falls as the radius to the power 2.5; where it crosses 0.4 a radiative core appears. 0.3 M☉: not before the main sequence; 0.5 M☉: not before the main sequence; 1 M☉: at 1.94 R☉; 2 M☉: at 7.62 R☉. Dots: each mass's main-sequence radius, where contraction stops. A Sun-like star leaves its convective line at 1.9 solar radii; the lightest never do.

Where a shrinking star stops boiling at its centre

A young star descends the Hertzsprung–Russell diagram at almost fixed temperature, convective from centre to surface, until something at its centre changes and it turns sharply left toward the main sequence. The something is the opacity. As the star shrinks its centre grows hotter and denser, the Kramers opacity there falls, and radiation becomes able to carry the luminosity without the steep gradient that drives convection. A Sun-like star crosses that line at about twice its present radius, two million years old; a star of a third of the Sun's mass never crosses it at all.

starlight · Opacity

Named alongside it

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

Pre-main-sequenceConvectionOpacityAbiotic oxygenEnergy-limited escapeFully convective starHabitable zoneHenyey trackKelvin helmholtz contractionKelvin helmholtz timescaleKramers opacityM dwarf

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