Astronomy & Celestial Mechanics
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Generator

The habitable-zone generator

The habitable zone, and the radius inside which a day never ends
The habitable zone, and the radius inside which a day never ends. The conservative habitable zone — the runaway-greenhouse and maximum-greenhouse limits of Kopparapu's parameterisation — as a band in stellar mass against orbital distance, both logarithmic. For the Sun it runs from 0.99 to 1.71 AU, which is the published result and the check on the arithmetic here. The band moves inward far faster than the mass falls, because luminosity goes as roughly the fourth power of mass: a 0.2 M☉ star's zone is at 0.082–0.158 AU. The dashed line is the distance inside which a planet is tidally locked within 4.5 billion years, computed from τ = ω₀αmQa⁶/(3GM⋆²k₂R³) with an initial ten-hour spin, Q = 100 and k₂ = 0.3. It crosses the inner edge of the zone at 0.67 M☉ — so around every star below that, which is the great majority of stars, a planet in the habitable zone has one hemisphere in permanent daylight. The Earth is outside its own locking radius of 0.53 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 8.

The conservative habitable zone — the runaway-greenhouse and maximum-greenhouse limits of Kopparapu's parameterisation — as a band in stellar mass against orbital distance, both logarithmic. For the Sun it runs from 0.99 to 1.71 AU, which is the published result and the check on the arithmetic here. The band moves inward far faster than the mass falls, because luminosity goes as roughly the fourth power of mass: a 0.2 M☉ star's zone is at 0.082–0.158 AU. The dashed line is the distance inside which a planet is tidally locked within 4.5 billion years, computed from τ = ω₀αmQa⁶/(3GM⋆²k₂R³) with an initial ten-hour spin, Q = 100 and k₂ = 0.3. It crosses the inner edge of the zone at 0.67 M☉ — so around every star below that, which is the great majority of stars, a planet in the habitable zone has one hemisphere in permanent daylight. The Earth is outside its own locking radius of 0.53 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 8.

9 essays call habitable-zone. 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 habitable zone, and the radius inside which a day never ends. The conservative habitable zone — the runaway-greenhouse and maximum-greenhouse limits of Kopparapu's parameterisation — as a band in stellar mass against orbital distance, both logarithmic. For the Sun it runs from 0.99 to 1.71 AU, which is the published result and the check on the arithmetic here. The band moves inward far faster than the mass falls, because luminosity goes as roughly the fourth power of mass: a 0.2 M☉ star's zone is at 0.082–0.158 AU. The dashed line is the distance inside which a planet is tidally locked within 4.5 billion years, computed from τ = ω₀αmQa⁶/(3GM⋆²k₂R³) with an initial ten-hour spin, Q = 100 and k₂ = 0.3. It crosses the inner edge of the zone at 0.67 M☉ — so around every star below that, which is the great majority of stars, a planet in the habitable zone has one hemisphere in permanent daylight. The Earth is outside its own locking radius of 0.53 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 8. Exoplanets

One number sets the zone

The habitable zone is a band of stellar flux, so it scales as the square root of luminosity and moves inward far faster than mass falls. For most stars it lies inside the radius at which a planet is tidally locked.

 The ice line at 2.7 AU, and the 3.4-fold jump in solid material across it. Two temperature thresholds turned into radii, against stellar mass, both axes logarithmic. The shaded band is the habitable zone, where water can be liquid on a planet's surface. The heavy line is the snow line of the disc the planets formed in — the distance at which a passively heated disc, whose temperature falls as the inverse square root of radius, reaches 170 K and water freezes. For a solar-luminosity star it sits at 2.71 AU, just outside the asteroid belt, and it is 1.6 times further out than the outer edge of the habitable zone; around a 0.15 solar-mass star both have moved inwards and the ratio is 5.5. What makes the line matter is what happens as it is crossed. Water is by far the most abundant condensable material after hydrogen and helium, so freezing it raises the surface density of solids by roughly 3.4 times in one step. Everything about the architecture of a planetary system follows from that step: a core massive enough to capture gas can be assembled outside the line and not inside it, which is why the solar system has small rocky planets in and giant ones out, and why a giant planet found at 0.05 AU is a statement about migration rather than about formation. The line is drawn where a mature disc puts it; a young, accreting disc is hotter and its line is several times further out, sweeping inwards as the disc drains. Exoplanets

The line beyond which ice counts as rock

A disc of gas around a young star gets colder outwards, and at about a hundred and seventy kelvin water stops being vapour and becomes a building material. Crossing that one line multiplies the solid mass available by roughly three and a half, in a single step, and the architecture of every planetary system is downstream of it.

 The habitable zone of a 1 M☉ star, sweeping outwards. The inner and outer edges of the liquid-water zone against time, for a 1 solar-mass star whose main sequence lasts 10.0 Gyr. The star brightens as it burns hydrogen — a heavier core needs a hotter centre to hold the star up — so both edges move outward by a factor of 1.63 across the whole main sequence, and the band drawn here sweeps past any fixed orbit rather than containing it. Two quite different zones can be read off. The instantaneous zone at the age of the present-day Sun is 0.99 to 1.71 AU, which is the band a survey means by "in the habitable zone". The continuously habitable zone over the 10.0 Gyr drawn is the overlap of every instant in it — outside the inner edge at the end and inside the outer edge at the beginning — which is 1.37 to 1.44 AU, 10 per cent of the instantaneous width. The horizontal line is an orbit at 1 AU. It leaves the zone at 4.72 Gyr, when the inner edge overtakes it. None of these edges is a measurement: both come from one-dimensional climate models, and the inner one in particular is where a runaway greenhouse begins in a model whose clouds are prescribed. Exoplanets

The band moves and the orbit does not

A star brightens as it burns, so the distance at which water can be liquid sweeps outwards by a factor of one and a half across a main sequence. The band a survey quotes is an instant; the band a planet needs is the overlap of every instant, and for the Sun it is a tenth as wide and does not contain the Earth.

 The solid mass available, as a staircase of 5 fronts. The share of the condensable material that is solid, against distance from a 1 solar-mass star, on a logarithmic radius axis. Each riser is one species freezing out, at the radius where the disc's temperature — falling as the inverse square root of the distance — reaches that species' condensation point. silicates and iron at 1400 K and 0.04 AU; water ice at 170 K and 2.71 AU; carbon dioxide at 70 K and 16.00 AU; methane and ammonia at 30 K and 87.11 AU; carbon monoxide at 20 K and 196.00 AU. Inside every front the solid surface density is 22 per cent of what is available, which is the refractories alone; water alone contributes 53 per cent, more than twice everything else combined, which is why one of these steps is called the snow line and the others are not. A body's composition is decided by which pair of risers it formed between, and the steps are narrow because a vapour pressure is exponential in the inverse temperature. Exoplanets

There is not one line, there is a staircase

Water freezing is the biggest step in a protoplanetary disc and it is one of five. Each condensable species has its own temperature and therefore its own radius, and what a body is made of is decided not by which side of a line it formed on but by which pair of risers it formed between.

 The ice line sweeps from 6.4 to 2.7 AU while the disc drains. The radius at which a disc around a 1 solar-mass star reaches 170 K, against the disc's age, both axes logarithmic. The temperature is the fourth root of the sum of two fluxes: starlight, which does not change, and the disc's own accretion, which releases gravitational energy at a rate set by how fast material is flowing inward. The accretion rate decays as the disc drains — taken here as 5e-5 solar masses a year falling off as t^(−3/2) beyond 0.1 million years — so the viscous term fades and the line sweeps in. It starts at 6.38 AU and ends at 2.73, the passive value the closed form gives. A body at three astronomical units formed dry if it formed early and icy if it formed late, so a composition dates a formation rather than locating one, and the dating is only as good as the accretion history assumed. Exoplanets

A composition that dates a formation rather than placing it

The ice line in a young disc starts six astronomical units out and sweeps inward to under three as the disc drains. A body at four astronomical units therefore formed dry or wet depending only on when — so what it is made of is a clock, and reading it as a map is the mistake the moving line makes easy.

 A carbon-to-oxygen ratio that steps at every front. The carbon-to-oxygen ratio of the gas and of the solids against distance from a 1 solar-mass star, each in units of the star's own ratio. Crossing a condensation front moves one element or both out of the gas and into the solids, so the two curves step in opposite directions at water at 2.7 AU, carbon dioxide at 16.0 AU, carbon monoxide at 196.0 AU. Water takes oxygen and no carbon, so beyond it the gas is carbon-rich — 3.20 times the stellar ratio — and the solids are oxygen-rich. Carbon monoxide takes both, in a ratio of one to one, so beyond that front the gas ratio rises again. A giant planet's atmosphere is made mostly of gas it accreted, so measuring its ratio and inverting this staircase gives a formation radius — and the inversion is not unique, because more than one interval returns the same value once the solids a planet also swallowed are allowed for. Exoplanets

Two elements in a ratio, and a birthplace read off it

Carbon and oxygen freeze out at different places, so the gas between the fronts is carbon-rich and the solids are oxygen-rich. A giant planet is made mostly of gas it accreted, so measuring the ratio in its atmosphere and inverting the staircase should give the radius it formed at — and the inversion turns out not to be unique.

 A thermostat that passes 40 per cent of the change it is meant to cancel. Surface temperature against the flux a planet absorbs, in units of the Earth's, for a planet round a 1 M☉ star, with and without the carbonate–silicate cycle. With carbon dioxide held at 280 µbar the temperature follows the flux directly. With weathering allowed to adjust — rock dissolves faster when it is warm and when there is more CO₂, and in the steady state it must remove exactly what volcanoes supply at 1 times today's rate — a colder planet accumulates CO₂ until the balance is restored. The feedback is real and it is not a set point. Near S = 1 it passes 40 per cent of a flux change through to the surface: the loop gain is k s / β = 1.49, with weathering rising one e-fold for every 9.7 K, a greenhouse of 4.33 K per e-folding of CO₂ and a CO₂ exponent of 0.3. The required CO₂ would reach 8.8 bar — the point at which more of it scatters sunlight faster than it traps heat, and the controller has nothing left to add — at S = 0.249. The steady state reaches 273 K at S = 0.531, before the CO₂ has run out — the outer edge of this planet's habitable zone is where the thermostat saturates or freezes, whichever comes first. The climate law is logarithmic in CO₂, which is right near today's values and only a calibration at several bar; ice-albedo feedback, which makes a cooling planet able to jump to a frozen state, is left out. Exoplanets

A thermostat that only halves the error

The carbonate–silicate cycle is credited with keeping a planet's water liquid across the whole width of its habitable zone. Written down with its own measured exponents, it is a proportional controller that cancels about three-fifths of a change in sunlight, takes half a million years to do it — and reaches the published outer edge only on a planet with an order of magnitude more volcanism than the Earth.

 An eccentricity of 0.4 swings the surface by 126 K or by 0.1, depending on the length of the year. The peak-to-trough swing in surface temperature over one orbit, against orbital period, for a planet with eccentricity 0.4 receiving on average the flux the Earth does, for surface layers of 1, 10, 50 metres of water. The dashed line is the 126 K the surface would swing through with no heat capacity. Every curve rises from near zero at short periods, where the orbit is over before the layer can respond and the planet feels only the average flux, towards the full swing at long periods, where every part of the orbit lasts long enough to be felt in full. The crossover is where the orbital period is comparable with the layer's thermal time. The vertical marks are the orbital periods of the Earth-flux orbit round stars of 0.1 M☉ (7 days), 0.5 M☉ (0.2 years), 1 M☉ (0.9 years). At a fixed eccentricity a planet in the habitable zone of a small star is thermally averaging almost regardless of how much water it has, and one round a Sun-like star is not unless it has an ocean — the ordering is set by the star through the period, which is the one quantity the flux-averaged habitable zone discards. Exoplanets

A year too short to feel its own eccentricity

A planet on an eccentric orbit can have a comfortable average and murderous extremes, and the habitable zone is drawn from the average. Whether the surface lives on the average or on the extremes is not decided by the flux at all — it is the ratio of how long the surface takes to change temperature to how long the year lasts, and the star sets the year.

 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. Exoplanets

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.

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