Astronomy & Celestial Mechanics
the sky, computed
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.

Assumes The mass–luminosity relation and Tides.

17 min read 9 figures Scales that defeat intuitionMeasuring the unreachable

The habitable zone is defined by a flux and not by a distance. A planet is in it if the starlight falling on each square metre is within a certain range — enough to keep water liquid somewhere on the surface, not so much that the oceans evaporate.

That definition has one enormous consequence, and it makes the whole diagram: since flux falls as the inverse square of distance, the zone’s radius scales as L\sqrt{L}. And luminosity scales as roughly the fourth power of stellar mass. So the zone moves inward faster than the mass falls: a star of a fifth the Sun’s mass is about a two-hundredth as luminous, and its habitable zone sits at 0.08 to 0.16 AU — a twelfth of the Earth’s distance, inside where Mercury would be.

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.
Fig. 1 The conservative habitable zone as a band in stellar mass against orbital distance, from the runaway-greenhouse and maximum-greenhouse limits. For the Sun it runs from 0.99 to 1.70 AU, which is the published result and this figure’s check on its own arithmetic. The dashed line is the distance inside which a planet is tidally locked within 4.5 billion years, and it crosses the inner edge of the zone at 0.67 solar masses.

Where the edges come from

The inner edge is the runaway greenhouse. Raise the flux on an ocean planet and more water evaporates; water vapour is a greenhouse gas, so the surface warms and more evaporates still. Past a threshold the feedback runs away, the entire ocean enters the atmosphere, and the surface reaches temperatures at which nothing recognisable survives. That threshold is a computed quantity — the point at which the outgoing infrared radiation of a moist atmosphere stops rising with surface temperature — and for a Sun-like star it lands at about 1.01 times the Earth’s insolation, or 0.99 AU.

The Earth is one per cent inside the inner edge of its own habitable zone. The margin is not comfortable, and it is one of the least widely known numbers in the subject.

The outer edge is the maximum greenhouse. Move a planet outward and it needs more carbon dioxide to stay warm; the carbonate–silicate cycle is expected to supply it, weathering slowing as the planet cools and letting volcanic CO₂ accumulate. But CO₂ also scatters sunlight, and past a certain column the scattering beats the greenhouse effect. That optimum is the outer edge: about 0.34 times the Earth’s insolation, or 1.70 AU.

Both edges are outputs of one-dimensional climate models, and the numbers here are Kopparapu’s parameterisation of them, with the effective flux written as a polynomial in the star’s effective temperature. The temperature enters because a cooler star emits more of its light in the near infrared, where water and carbon dioxide absorb more strongly and snow reflects less — so the same total flux is more effective at warming a planet around a red star.

Why the band tilts so steeply

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 10 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 0.73 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 0.61 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 9.
Fig. 2 The same zone ten billion years on, which is what makes it a moving target rather than a place. A star brightens through its main-sequence life as helium accumulates and the core contracts, so the zone migrates outward — and a planet that begins comfortably inside it can be left behind on the hot side. The continuously habitable zone, the band that stays inside the limits for the whole interval, is narrower than either snapshot, and for a solar-type star it is narrow enough that the Earth’s own position in it is not obviously secure.

The steepness of the mass–luminosity relation is why the habitable zone is a diagonal band rather than a horizontal one, and it drives everything observational about the subject.

A planet in the habitable zone of a 0.2 M☉ star orbits between 0.08 and 0.16 AU, with a period between about three and seven weeks. That is a gift to observers, and every advantage compounds:

  • The transit probability is R⋆/aR_\star/a, and although the star is small the orbit is far smaller, so it rises to about 1.6 per cent — more than three times the Earth’s 0.47.
  • A period of weeks means dozens of transits inside a survey, rather than three or four.
  • The transit depth for an Earth-sized planet is thirteen times deeper, because the star is 0.28 solar radii and the depth goes as the inverse square of that.
  • The radial-velocity amplitude is nearly eight times larger, because the star is lighter and the orbit tighter, and KK carries M⋆−2/3P−1/3M_\star^{-2/3}P^{-1/3}.

Everything that makes a temperate rocky planet nearly undetectable around a Sun-like star is relieved by a factor of ten or more around an M dwarf. That is why every temperate rocky planet currently known orbits a small star, and it is a selection effect of unusual severity rather than a fact about where such planets exist.

The line that crosses the band

And here is the cost, drawn as the dashed line in the first figure.

A planet close to its star is despun by the tide the star raises on it. The timescale for that is

τ∼ω0 αmQa63GM⋆2k2R3,\tau \sim \frac{\omega_0\,\alpha m Q a^6}{3 G M_\star^2 k_2 R^3},

with the sixth power of the orbital distance doing what sixth powers do. Computed for an Earth-like planet with a ten-hour initial spin, Q=100Q = 100 and k2=0.3k_2 = 0.3, the locking radius is 0.53 AU for the Sun — so the Earth, at 1 AU, is comfortably outside it and rotates freely. For a 0.089 M☉ star it is 0.24 AU, and TRAPPIST-1e at 0.029 AU is inside it by a factor of eight — so the innermost planets of that system have been locked since long before the star finished contracting.

The line crosses the inner edge of the habitable zone at about 0.67 solar masses. Below that, which is the great majority of stars, a planet in the habitable zone is tidally locked, with one hemisphere in permanent daylight and the other in permanent night.

Whether that is fatal is an open question and a lively one. Early work assumed the atmosphere would freeze out on the night side, ending the discussion. Three-dimensional climate models since have found that a modest atmosphere transports enough heat to prevent collapse, and that a locked planet can have a habitable band or a substellar “eyeball” ocean. What was once a disqualifying objection is now a parameter.

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 1 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 0.57 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 0.41 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 6.
Fig. 3 The same band with the tidal-locking radius computed for a system one billion years old rather than four and a half. The habitable zone has not moved — it is set by the star’s present luminosity — and the locking radius has, inward, because locking takes time and less of it has passed. Around the lowest-mass stars the two curves cross at a different place for every age, so whether a habitable-zone planet is tidally locked is a question about the age of the system as much as about the mass of its star.

Where the planets in it actually are

The observational payoff of the whole diagram is that it says where to point, and the answer has moved sharply toward small stars.

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 1 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 0.57 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 0.41 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 6.
Fig. 4 The same band with the locking line drawn for a one-billion-year-old system rather than a four-and-a-half. The zone does not move; the locking radius does, inward, because there has been less time to despin. Even so it still crosses the zone above half a solar mass — the sixth power of distance means the boundary is insensitive to almost everything, including a factor of four in age.

Two things follow from the arithmetic in the previous section. Around an M dwarf, a habitable-zone planet is detectable by every method at once; around a Sun-like star it is at the edge of the best of them. So the occurrence rate of habitable-zone planets is well measured for small stars and barely measured for Sun-like ones, and the two rates are not obviously the same.

The current numbers put the rate around M dwarfs at a few tens of per cent — of order one habitable-zone rocky planet for every two to four such stars. Since M dwarfs are about three-quarters of all stars, the great majority of temperate rocky planets in the galaxy orbit stars whose habitable zone lies inside the locking radius. That is a statement about the typical case rather than about the solar system’s, and it is why the question of whether a tidally locked planet can hold an atmosphere is not a curiosity.

What the zone does not include

The list of things this definition ignores is longer than the definition.

It assumes an atmosphere. The limits are computed for an Earth-like atmosphere with an active carbonate–silicate cycle. A planet with no atmosphere has no greenhouse, and its surface temperature is the equilibrium temperature — 255 K for the Earth’s insolation, which is below freezing everywhere.

It assumes water and a surface. A gas giant in the zone has neither. A moon of that gas giant might have both, and would not appear on this diagram at all.

It ignores the star’s history. An M dwarf is far more luminous during its first few hundred million years than on the main sequence, so a planet in its eventual habitable zone spends its youth inside the runaway-greenhouse limit and may lose its water entirely before the star settles. Then there is the flaring: M dwarfs produce X-ray and ultraviolet flares thousands of times stronger relative to their output than the Sun’s, and the cumulative effect on an atmosphere over billions of years is not established.

It ignores everything about the planet except its orbit. Mass, composition, rotation, obliquity, magnetic field and the presence of a large moon are all absent from the calculation, and each of them plausibly matters more than a ten per cent difference in flux.

The zone moves while the star ages. A main-sequence star brightens slowly — the Sun is about 30 per cent brighter now than when it formed — so the zone sweeps outward over billions of years, and the continuously habitable zone, the band that stays inside the limits for a planet’s whole life, is narrower than the instantaneous one. The Earth was near the outer edge when life began and is near the inner edge now, which is a consequence of the same nuclear physics that sets a star’s whole life. And “habitable” is a very strong word for it. What the zone marks is where a particular kind of surface water could exist under a particular kind of atmosphere. It is a targeting criterion — where to point a telescope — and it has been repeatedly criticised for being read as anything more.

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 10 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 0.73 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 0.61 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 9.
Fig. 5 And at ten billion years, older than the Sun will be when it leaves the main sequence. The locking radius has moved outward and now crosses the habitable zone at a higher stellar mass than before. Given enough time every habitable-zone planet around a star below about half a solar mass ends up locked, and since those stars live for hundreds of billions of years, “enough time” is not a restriction on any of them.

The temperature a planet would have without an atmosphere

Underneath the climate models there is one quantity that requires no modelling at all, and it is worth separating out because it is the thing actually determined by the orbit.

A body absorbing a fraction (1−A)(1-A) of the flux falling on its cross-section and radiating from its whole surface reaches

Teq=T⋆R⋆2a (1−A)1/4,T_{\text{eq}} = T_\star\sqrt{\frac{R_\star}{2a}}\,(1-A)^{1/4},

which for the Earth’s albedo of 0.3 gives 255 K. The Earth’s actual mean surface temperature is 288 K, so 33 kelvin of the Earth’s habitability is atmosphere and none of it is orbit.

Blackbody curves at 5772, 288, 255 K. Thermal emission against wavelength, each curve scaled to its own peak so the shift can be seen on one plot. The peak moves to shorter wavelengths as the temperature rises, which is why colour is a thermometer.
Fig. 6 Why that 33 K exists. The star’s output peaks in the visible, where the atmosphere is largely transparent; the planet radiates in the mid-infrared, where water and carbon dioxide absorb strongly. Energy enters through a window and leaves through a wall, and the surface warms until the leak balances the input. Every number in this essay’s habitable zone is a statement about how far that imbalance can be pushed in either direction.

That separation is the honest way to read the whole diagram. The orbit fixes TeqT_{\text{eq}} exactly, given the star; everything between TeqT_{\text{eq}} and a surface temperature is climate, and climate is where the models and the arguments are.

What was actually measured

The solar system’s own three data points. Venus receives 1.91 times the Earth’s flux and is at 737 K under 92 bars of carbon dioxide — the runaway greenhouse, observed. Mars receives 0.43 and is frozen, with clear evidence of ancient liquid water and no atmosphere left to keep it. The Earth is between them. Any model of the limits has to reproduce those three, and the conservative limits do: Venus is inside the inner edge and Mars is inside the outer, which is precisely the awkward part — Mars is in the conservative zone and is not habitable, because it lost its atmosphere.

Proxima Centauri b, 2016. msin⁡i=1.27m\sin i = 1.27 Earth masses at 0.0485 AU around a 0.122 M☉ star: an insolation of about 0.65 times the Earth’s, squarely inside the zone, around the nearest star to the Sun. It is also inside the locking radius, and Proxima flares violently.

TRAPPIST-1 e, f and g. Three planets of the seven receive between 0.66 and 0.25 times the Earth’s flux, with radii and masses giving densities consistent with rock. JWST observations of the inner planets have so far found no atmospheres, which does not settle the outer ones but does raise the question of whether any of the system retained one.

Kepler-186 f, 2014. The first Earth-sized planet found in a habitable zone: 1.11 Earth radii, 130-day period, 0.48 M☉ star, at about 0.29 times the Earth’s insolation — near the outer edge. It is 180 parsecs away and its mass has never been measured.

The ice line at 3.5 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 150 K and water freezes. For a solar-luminosity star it sits at 3.48 AU, just outside the asteroid belt, and it is 2.0 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 7.1. 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.
Fig. 7 The ice line drawn at a condensation temperature of 150 kelvin rather than 170. It moves outward to 3.5 AU, because the disc’s temperature falls with radius and a lower threshold is reached further out. The habitable zone underneath it does not move at all. The gap between the two is what a terrestrial planet’s water has to cross, and how wide that gap is depends on a condensation temperature that is a function of the disc’s pressure — which no observation of a mature system can recover.

The one thing that would settle it

Every statement in this essay is an inference from an orbit and a stellar model. None of it observes a planet.

What would observe one is a spectrum: carbon dioxide bands to establish there is an atmosphere at all, water to establish it is wet, and — much harder — a combination of gases that chemistry alone would not maintain. That measurement is transmission spectroscopy for a transiting planet and reflected-light spectroscopy for an imaged one, and both are at the limit of what is possible.

The arithmetic is unforgiving. A temperate rocky planet around an M dwarf has an atmospheric signal of tens of parts per million even with the small-star advantage, and it sits on a star whose spots contaminate the spectrum at a comparable level. Around a Sun-like star, reflected-light imaging needs a contrast of 10−1010^{-10}. Both are the design drivers of instruments that do not yet exist.

So the habitable zone is currently doing the job it was designed for and no more: it tells a committee which stars to observe. Whether the concept survives its first real atmospheric measurement is one of the things the next twenty years will decide.

A moon has two heat sources

The diagram treats starlight as the only thing warming a planet, and there is a class of body for which that is badly wrong.

A large moon of a giant planet in the habitable zone receives starlight like any other body at that distance, and it also receives tidal heat — because it is close to a massive primary and its orbit is unlikely to be exactly circular, for the same resonant reasons that keep Io’s eccentricity from damping.

That gives such a moon an energy budget with two terms, and the second one does not depend on the star at all. So a moon can be warm outside the habitable zone, and it can be far too warm inside it.

The second case is the interesting one. Tidal heating falls very steeply with distance from the primary — as the inverse sixth power, through the same dependence that sets a locking radius — so there is a distance inside which a moon receives more tidal heat than it can radiate at a habitable temperature. Io is the worked example: it receives about a fortieth of the Earth’s starlight and radiates twenty times the Earth’s internal heat flux, and the result is a body resurfaced continuously by volcanism.

So a moon has an inner edge of its own, set by the primary rather than by the star, and it is a hard edge: the flux rises by a factor of sixty-four for every halving of the orbital radius, so the transition from temperate to Io-like is fast.

The outer edge for a moon is different too. Beyond a distance of roughly a third of the primary’s Hill radius the orbit is not stable over the age of the system, so there is a bounded annulus in which a large moon can persist at all.

Between the two there is a band, and whether it overlaps the star’s habitable zone depends on the primary’s mass. For a Jupiter-mass planet at one astronomical unit the two bands do overlap, and a moon there would be warmed by both.

None of this appears on the diagram, because the diagram’s horizontal axis is distance from the star and a moon’s temperature depends on a distance from something else. It is a reminder that a habitable zone computed for planets is a statement about planets, and that the largest reservoir of liquid water in this solar system is under the ice of moons far outside it.

The ice line at 2.7 AU, and the 8.5-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 8.5 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.
Fig. 8 The same two curves with the inner disc’s solid fraction cut from 0.5 per cent to 0.2. The jump in available solid material across the ice line goes from 3.4-fold to 8.5-fold — the same step, read against a smaller starting value. That ratio is what decides whether a giant planet’s core can be assembled before the gas disperses, and it is why the ice line matters to the habitable zone at all: the planets that end up in the zone were assembled from material whose supply changes discontinuously somewhere outside it.

The generalisation

Defining a region by a threshold in a computed quantity, and then discovering that the threshold is where the interesting physics is, is a recurring move.

The snow line is defined by a temperature at which water condenses, and it organises the entire theory of planet formation. The Roche limit is where a tidal difference beats self-gravity, and it separates rings from moons. The Chandrasekhar mass is where degeneracy pressure stops winning. In every case the boundary is computed from a model, is approximate, and is nevertheless the most useful line on the diagram.

What is distinctive about this one is how much is being asked of it. The Roche limit predicts whether an object is torn apart, which is a mechanical question with a mechanical answer. The habitable zone is being asked to predict whether a planet has a biosphere, from its distance from a star, and the gap between what is computed and what is claimed is the widest in the subject.

One more reading separates the zone’s boundaries from the tidal-locking line that is usually drawn across 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 6 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 0.69 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 0.56 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 9.
Fig. 9 The zone at six billion years with the locking boundary removed. What is left is a band whose position depends only on the stellar flux, and the worlds inside it sit there for reasons that have nothing to do with their rotation — which is the honest version of the diagram, and the less interesting one.

One number sets the zone and a great many others decide whether anything in it is habitable, which is why the boundary is drawn from stellar flux alone and read as though it meant something about planets.

Where this goes next

Everything above is an inference from an orbit. What would actually settle whether a planet has liquid water is a measurement of its atmosphere — and the technique for that begins with subtracting two brightnesses to isolate the planet’s own light.

Later rungs on this anchor: the runaway greenhouse derived. The carbonate–silicate cycle. Three-dimensional climate models of locked planets. M-dwarf flares and atmospheric loss. The pre-main-sequence luminosity problem. Habitable zones of binaries. Moons in habitable zones. Biosignatures and their false positives. The continuously habitable zone over a star’s lifetime. And whether the concept survives contact with the first real atmospheric spectrum of a temperate rocky planet.

Where this goes next

Luminosity against mass, against a slope of 3.5. Main-sequence luminosity against mass, both in solar units, on logarithmic axes, over the range 0.079 to 63 solar masses. The measured curve comes from the eclipsing binaries and is the same in every drawing of it; what changes here is what it is compared against. The dashed line is a pure power law of exponent 3.5, and the curve crosses it rather than following it — the local slope runs from about 2.3 at the bottom of the range, where the interiors are convective, through nearly 4 near a solar mass where bound-free opacity dominates, to about 3 among the massive stars where electron scattering does. Quoting one exponent across the whole sequence is a convenience and the places it fails are the places the interior physics changes. Because the slope is between three and four across most of the range, a small spread in mass becomes an enormous spread in output: the 63-solar-mass end is 3.0e+9 times brighter than the 0.079-solar-mass end. Mass decides everything, by a power of three and a half Two stars of the same mass are almost the same star. Double the mass and the output multiplies by eleven — which is why a modest range of masses produces a colossal range of stars. The tidal field is a difference. The pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one. The tide is a difference, which is why there are two of them The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting. A gap where planets should be. The number of planets per star per interval of log radius, for orbital periods under a hundred days, corrected for detection efficiency. There are two peaks — super-Earths near 1.3 R⊕ and sub-Neptunes near 2.4 R⊕ — and a deficit between them at 1.89 R⊕, where the occurrence falls to 33% of the peak. The gap is not a gap in what can be detected: detection efficiency rises smoothly through it, so a smooth underlying distribution could not produce a dip there. The planets that were not seen An occurrence rate is a count divided by a probability, and the probability can be a five-hundredth. Everything difficult about saying how common planets are lives in that denominator.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 12 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

AlbedoClimate modelEffective fluxEquilibrium temperatureHabitable zoneInsolationM dwarfRunaway greenhouseStellar luminosityTidal locking