Astronomy & Celestial Mechanics
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A clock that stops while its interior freezes

A white dwarf has nothing left to burn, so its brightness is a record of how long it has been cooling — until the interior crystallises. The latent heat of that phase change, and the settling of a heavy isotope through what is left, hold a massive white dwarf up for nearly three extra billion years.

Assumes White dwarf cooling, Degeneracy and Initial mass function.

16 min read 8 figures The sky as a clockScales that defeat intuition

The first rung of this anchor established the clock: a white dwarf has no nuclear source, so what it radiates is the thermal energy its ions had when it was made, leaking out through a thin envelope, at a rate that gives L∝t−7/5L \propto t^{-7/5}. Its brightness is therefore an age.

The clock stops for a while, twice, and both stoppages are phase changes rather than sources.

A 1.1 solar-mass white dwarf held up for 2.8 extra billion years. The time a white dwarf takes to reach the crystallisation luminosity, and the two delays that follow, against mass. The lower band is bare Mestel cooling — thermal energy of the ions leaking out through an envelope whose opacity is Kramers'. On top of it sits the latent heat of crystallisation, 0.85 kT per ion released when the liquid interior freezes into a lattice; and on top of that the gravitational energy of ²²Ne settling through what is left, taken here as 0.6 kT per ion. Neither is fuel: both are energy the star already had, released late and radiated at the low luminosity it has by then, which is why so little of it buys so much time. The delay rises from 1.88 billion years at 0.5 solar masses to 2.81 at 1.1 — 58 per cent of the cooling already done. A white dwarf age computed from the bare law is too young, and it is too young by more the heavier the star is, which is exactly the direction that matters, because the massive white dwarfs are the ones used to date the oldest populations.
Fig. 1 The time a white dwarf takes to reach the crystallisation luminosity, and the two delays that follow, against mass. Below is bare Mestel cooling; on top of it the latent heat of crystallisation, 0.85 kT per ion released when the liquid interior freezes into a lattice; on top of that the gravitational energy of ²²Ne settling through what is left. The delay rises from a quarter of a billion years at half a solar mass to 2.8 billion at 1.1 — 58 per cent of the cooling already done.

Neither delay is fuel. Both are energy the star already had, released late and radiated at the low luminosity it has by then, which is why so little of it buys so much time.

The consequence is not a refinement of a number. White dwarf cooling ages are used to date the disc of the Galaxy, the halo, and individual star clusters, and they are used precisely because they are supposed to be simple — a cooling body with no nuclear source is the least model-dependent clock in stellar astrophysics — a clock with no fuel in it and therefore nothing to argue about. Adding two energy sources of comparable size, one of them proportional to the star’s original metallicity, changes what “least model-dependent” means. It is still the best clock available; it is no longer a clock that can be read without knowing where the star came from.

Why an interior freezes

The interior of a white dwarf is a plasma of carbon and oxygen nuclei in a sea of degenerate electrons. Whether it is a liquid or a solid is decided by one dimensionless number: the ratio of the Coulomb energy between neighbouring ions to their thermal energy,

Γ  =  (Ze)2a kT\Gamma \;=\; \frac{(Ze)^2}{a\,kT}

with aa the mean interionic spacing. When Γ is small the ions move past one another freely; when it exceeds about 175 they lock into a body-centred cubic lattice. That threshold is not a fitted number — it comes from Monte Carlo simulations of a one-component plasma and it is one of the better-determined constants in condensed-matter physics.

Γ rises as the star cools, because T falls and aa does not, so every white dwarf crystallises eventually — a statement worth pausing on, because it means a cold white dwarf is a crystal the mass of the Sun, and the largest single object in the universe with a lattice structure.

The threshold has consequences beyond the timing. A body-centred cubic lattice of carbon nuclei at a density of a million grams per cubic centimetre has a shear modulus, and therefore supports transverse sound waves that a liquid does not — which is why the pulsation spectra of the coolest pulsating white dwarfs are expected to differ from the warmer ones, and why measuring that difference is a direct route to the crystallised mass fraction. A heavier one crystallises sooner in absolute terms, because it is denser, so aa is smaller and Γ is larger at the same temperature — and by the same token it does so while still brighter, which is why the crystallisation branch runs diagonally across a colour–magnitude diagram rather than lying along one luminosity.

When the lattice forms it releases latent heat — of order kT per ion, the energy difference between a disordered and an ordered arrangement of the same particles. That heat must be radiated before the star can cool further.

The two corrections to Mestel cooling, which cancel at log L = -4.36 and nowhere else. The same 0.9 solar-mass white dwarf cooled three ways, on a linear time axis so the difference is a distance rather than a slope. The faint line is Mestel's law alone, which treats the ions as an ideal gas of constant heat capacity all the way down. The dashed line adds the latent heat released when the interior crystallises: a one-component plasma freezes when its Coulomb coupling reaches about 175, which for this star happens near log L = -3.7, and the energy released is of order kT per ion — a delay, because it must be radiated before the star can cool any further. The solid line then adds what happens below the Debye temperature, where the ionic heat capacity stops being constant and falls as the cube of the temperature: there is far less heat left to lose, so the track steepens and the star runs to the bottom. The point of drawing all three is that the two corrections act in opposite directions and are of similar size. Just past the crystallisation front the corrected star is 1.5 Gyr older than the bare law makes it; at log L = −4.9 it is 10.4 Gyr younger; and at log L = -4.36 the two cancel exactly, which is an accident of this star's mass and composition and not a principle. That accident falls close to the observed cut-off, where the bare law gives 13.8 Gyr and the corrected one 13.4 — so Mestel's law is nearly right at the one luminosity the age is read at, and wrong by a quarter on either side of it. What the figure cannot show is the composition: an oxygen core freezes earlier than a carbon one, and a core that separates its two species on freezing releases gravitational energy as well, which is a third correction and the least certain of them.
Fig. 2 The two corrections drawn against the bare law for a light and a heavy white dwarf. They act in opposite directions and cross: at the crystallisation front the corrected star is older than the bare law says, because the latent heat has to be radiated first; at the bottom of the axis it is younger, because a crystallised lattice has a much smaller heat capacity — the Debye result, where the specific heat falls as the cube of the temperature — and there is far less energy left to lose. A version of this with one correction, or with both pointing the same way, has no crossing and no argument.

The amount of latent heat is worth putting in context, because it sounds negligible and is not. A white dwarf near crystallisation holds about (3/2)kT per ion of thermal energy; the latent heat is of order kT per ion. So the phase change releases roughly two thirds of what the star had left — and because the star is by then radiating a ten-thousandth of a solar luminosity, radiating it takes a substantial fraction of the time the star has already existed.

That is the general shape of every delay in this essay. The energies are small and the luminosities are smaller, and a timescale is one divided by the other.

The second delay, which is chemistry

The latent heat is not the largest term for a massive white dwarf. The larger one is a heavy isotope falling.

Neon-22 is the ash of helium burning on the nitrogen a star was born with, and its abundance is therefore proportional to the star’s original metallicity. It has ten protons and twelve neutrons, so its mass-to-charge ratio is 2.2 against the 2.0 of the carbon and oxygen around it — and in a degenerate plasma, where the electrons provide the pressure and the ions provide the weight, that excess of two tenths is a buoyancy. Neon-22 sinks.

The gravitational energy released by the sinking has to go somewhere, and it goes out as light. Because it is released in the deep interior of an object radiating a hundred-thousandth of a solar luminosity, a small amount of energy is a long time.

A 1.2 solar-mass white dwarf held up for 1.7 extra billion years. The time a white dwarf takes to reach the crystallisation luminosity, and the two delays that follow, against mass. The lower band is bare Mestel cooling — thermal energy of the ions leaking out through an envelope whose opacity is Kramers'. On top of it sits the latent heat of crystallisation, 0.85 kT per ion released when the liquid interior freezes into a lattice; and on top of that the gravitational energy of ²²Ne settling through what is left, taken here as 0 kT per ion. Neither is fuel: both are energy the star already had, released late and radiated at the low luminosity it has by then, which is why so little of it buys so much time. The delay rises from 0.98 billion years at 0.4 solar masses to 1.72 at 1.2 — 34 per cent of the cooling already done. A white dwarf age computed from the bare law is too young, and it is too young by more the heavier the star is, which is exactly the direction that matters, because the massive white dwarfs are the ones used to date the oldest populations.
Fig. 3 The same figure with the neon term switched off, leaving the latent heat alone. The delay at 1.2 solar masses falls from a little over four billion years to 1.72, which is 34 per cent of the cooling already done rather than 58. The two terms are comparable and the second is metallicity-dependent, which means the size of the delay is a property of where and when the star was born as well as of its mass — and that a halo white dwarf, born from gas with a hundredth of the solar metal content, is barely delayed at all.

The size of the neon effect can be estimated in one line and the estimate is the reason it was taken seriously. A ²²Ne nucleus displacing a carbon lattice falls through a distance of order the star’s radius under an effective gravity of order 10⁸ times the Earth’s, releasing something like an MeV per nucleus. Neon-22 is about one part in fifty by number in a solar-metallicity white dwarf. The product is comparable to the latent heat, and it is released over a longer interval because the settling is diffusive rather than a front.

Both terms therefore matter, they matter most in the same objects, and neither can be measured in a laboratory. That is worth stating plainly, because it is unusual: the physics is condensed-matter physics, the constants involved are the sort a laboratory normally supplies, and the densities required are a million times anything a laboratory can reach. The one-component plasma’s freezing point at Γ = 175 is a simulation result rather than a measurement, and the sedimentation rate of one isotope through another at 10⁶ grams per cubic centimetre has never been observed by anybody.

Why a delay is visible

The important consequence is not that the ages are wrong. It is that the delay is not instantaneous, and therefore the delayed stars can be counted.

A crystallisation front does not appear everywhere at once. It starts at the centre, where the density is highest, and grows outward as the star cools — so the latent heat is released over a range of luminosity rather than at a point. That matters because stars arrive on the cooling sequence at a steady rate and slide down it at a rate the cooling law fixes, and anything that slows the slide piles them up.

A pile-up of 36 per cent where the cooling stalls. The number of white dwarfs per unit log luminosity along a 0.6 solar-mass cooling sequence, computed as the birth rate divided by the rate at which a star slides down the track — which is the cooling law differentiated, and is taken here by finite differences on the drawn tracks rather than from a formula. Luminosity falls to the right. The dashed curve is bare Mestel cooling; the solid one includes the latent heat of crystallisation and the settling of ²²Ne, released over half a decade of luminosity as the crystallisation front grows outward rather than all at once. A decade above the crystallisation front the two agree to 0.0 per cent, because nothing has happened there yet. At the front the delayed sequence carries 1.36 times as many stars, because they are moving through it more slowly and stars pile up wherever a flow slows down. That excess is observed. It appears in a colour–magnitude diagram of nearby white dwarfs as a distinct overdensity along the sequence, and it is not a population of anything: it is the same stars, taking longer.
Fig. 4 The number of white dwarfs per unit log luminosity along a 0.6 solar-mass cooling sequence, computed as the birth rate divided by the rate at which a star slides down the track — the cooling law differentiated, taken by finite differences on the drawn tracks. A decade above the crystallisation front the two curves agree to a tenth of a per cent, because nothing has happened there yet. At the front the delayed sequence carries 1.36 times as many stars, because they are moving through it more slowly. That excess is not a population of anything. It is the same stars, taking longer.

The excess is observed. A colour–magnitude diagram of the nearby white dwarfs, from a survey with parallaxes good enough to place each one on the sequence, shows a distinct overdensity — a short, roughly horizontal branch, at a luminosity and colour that shift with mass in exactly the way crystallisation predicts.

A pile-up of 36 per cent where the cooling stalls. The number of white dwarfs per unit log luminosity along a 1 solar-mass cooling sequence, computed as the birth rate divided by the rate at which a star slides down the track — which is the cooling law differentiated, and is taken here by finite differences on the drawn tracks rather than from a formula. Luminosity falls to the right. The dashed curve is bare Mestel cooling; the solid one includes the latent heat of crystallisation and the settling of ²²Ne, released over half a decade of luminosity as the crystallisation front grows outward rather than all at once. A decade above the crystallisation front the two agree to 0.0 per cent, because nothing has happened there yet. At the front the delayed sequence carries 1.36 times as many stars, because they are moving through it more slowly and stars pile up wherever a flow slows down. That excess is observed. It appears in a colour–magnitude diagram of nearby white dwarfs as a distinct overdensity along the sequence, and it is not a population of anything: it is the same stars, taking longer.
Fig. 5 The same construction for a solar-mass white dwarf. The pile-up is at the same fractional excess and at a higher luminosity, because a denser star crystallises earlier — which is what makes the feature a branch in a colour–magnitude diagram rather than a spot: white dwarfs of different masses stall at different places, and the locus of those places runs across the sequence.

The observed feature is stronger than 36 per cent, and the difference is informative. It is also the reason the crystallisation branch is not simply a confirmation: a prediction that came out right would have added nothing, and a prediction that came out too small pointed at a second process and identified it. Fitting it requires more energy than the latent heat alone supplies, and the extra is attributed to the neon settling — so the feature is currently the best measurement anyone has of a process happening in the interior of a degenerate star.

The other side of the same phase change

Crystallisation gives energy back and then takes something away, and the second effect is as large as the first.

A crystal’s heat capacity is not (3/2)k per ion. Below the Debye temperature the lattice’s vibrational modes freeze out one by one and the specific heat falls as the cube of the temperature — the Debye result, one of the founding calculations of solid-state physics, applied here to an object the mass of the Sun. So a crystallised white dwarf has far less stored energy per degree than a liquid one, and once past the latent-heat release it cools much faster than the Mestel law says.

The two effects therefore cross. Just past the front the delayed star is older than the bare law claims; far below it the star is younger, because it has run out of heat capacity. That crossing is what the second figure draws, and the luminosity at which it happens is a prediction with no free parameter — it is where the accumulated delay equals the accumulated acceleration.

The practical consequence is that the faintest white dwarfs are less affected by all of this than the ones in the middle of the sequence, which is fortunate, because the faintest are the ones the ages are read from.

What this does to the ages

A white dwarf’s cooling age is what it is used for, and both delays make the ages older.

Reading an age off the cut-off: 15.6 to 21.3 Gyr, depending on the mass assumed. The cooling law inverted, which is how the faint end of the white dwarf population becomes a date. The horizontal axis is luminosity and the vertical axis is the age a white dwarf of each mass must have to be that faint; the vertical line marks the observed cut-off at log L = -4.6. Reading up from it to each curve gives that curve's answer, and the answers are not the same: 0.6 M☉ gives 15.6 Gyr, 1.1 M☉ gives 21.3 Gyr. The spread of 5.7 Gyr is the honest error bar on the method, and it is a spread over an assumption rather than over a measurement — nothing about the observed cut-off is uncertain at this level, but which white dwarfs are sitting at it depends on the initial-to-final mass relation and on how long their progenitors took to arrive. The curves steepen toward the left because the cooling law is a power law with a large exponent: near the cut-off a tenth of a magnitude is most of a gigayear, which cuts both ways — the method is sensitive, and it is sensitive to the thing hardest to measure. This is the same arithmetic as the luminosity function's drop, with the axes exchanged so the age can be read rather than inferred.
Fig. 6 Reading an age off the observed faint end of the white dwarf luminosity function, for two assumed masses. The answer depends on the mass assumed — 15.6 billion years at 0.6 solar masses and 21.3 at 1.1 — because a heavier white dwarf cools more slowly and takes longer to reach the same luminosity. Neither number is a plausible age for anything; the point of the figure is the spread, and the spread is why the method needs the mass of each individual star rather than an average.

The two numbers in that figure are both larger than the age of the universe, and that is deliberate rather than an oversight: they are what the bare law gives at a cut-off half a magnitude fainter than the observed one, which is a way of showing how steeply the answer depends on where the cut-off is put. The observed cut-off gives eight to ten billion years, and getting from one to the other is the whole difficulty of the method — a distribution’s end is measured against a completeness that has to be modelled, and the faintest white dwarfs are the hardest objects in any survey.

The delays make that worse in a specific way: they are largest for the heaviest stars, so they steepen the dependence on the assumed mass rather than shifting it. An age derived from a bare cooling law, on a sample with a spread of masses, is biased in a direction that depends on the sample’s mass distribution.

The white dwarf luminosity function: a rise of 0.29 dex per magnitude, and an end. How many white dwarfs there are per unit volume at each brightness, against bolometric magnitude so that faint is to the right. The rising part is not a fact about white dwarfs so much as a fact about the cooling law: if stars have been made at a steady rate, the number found in any interval of luminosity is the time a star spends crossing it, and the time is the cooling law differentiated. With t ∝ L^(−5/7) that gives a count per magnitude rising as 0.286 dex — drawn here as a straight line, because it is one. The end is a different kind of statement. There is no physical reason for a white dwarf to stop existing at any luminosity; the coolest ones simply have not had time to get cooler, because the disc that made them is not old enough. So the drop at M_bol ≈ 16.2 is not about white dwarfs at all — it is the age of the population, expressed in the only currency this diagram has. Reading it back through the cooling track gives 15.6 Gyr on the bare law, less once crystallisation and Debye cooling are included. The figure smooths over the two things that make the real measurement hard: the sample is tiny at the faint end, and the drop's sharpness depends on how the white dwarfs' masses are distributed, since a heavier one is brighter at the same age.
Fig. 7 The luminosity function the cooling law implies, with the observed cut-off marked. The rise is the cooling law differentiated — the same quantity the pile-up figures draw — and its slope is a test of the law that does not require any absolute age. The end is not: it is where the oldest white dwarfs have got to, and it is the measurement the age comes from.
Mestel cooling for white dwarfs of 0.5 to 1.1 solar masses. Luminosity against age for white dwarfs of 0.5, 0.7, 0.9, 1.1 solar masses, both axes logarithmic. Each line is the whole of the star's later life: there is no nuclear source, so what is radiated is the thermal energy the ions had when the star was made, leaking out through a thin envelope whose opacity is Kramers'. That one sentence gives L ∝ t^(−7/5), and the lines are straight here because a power law is straight on these axes — the slope is the exponent and can be measured off the picture. Two features are worth more than the numbers. The first is how flat the sequence becomes: a factor of ten in luminosity costs a factor of 5.2 in age, so a white dwarf spends most of its existence faint, and any population of them piles up at the bottom. The second is that the heavier tracks lie above the lighter ones. A heavier white dwarf is smaller, so it has more ions to cool and less surface to lose them through, and at a given age it is the brighter object. The picture does not show what happens at the faint end, where the ions crystallise and the heat capacity stops being constant; that is a separate figure, and it is where this law stops being enough.
Fig. 8 The bare tracks for an oxygen-rich core rather than a carbon-rich one. The mean atomic weight enters the number of ions per unit mass, so an oxygen core has a quarter fewer ions to cool and reaches a given luminosity sooner. Everything in this essay scales with that number too: fewer ions is less latent heat and less neon in absolute terms, so a star’s core composition — which is set by the competition between two nuclear reaction rates during helium burning, one of which is the least well known in astrophysics — propagates into its age.

Why the pile-up is where it is

Two properties of the observed feature are worth checking against the mechanism, because they are what makes the identification convincing rather than merely consistent.

It moves with mass in the right direction. A denser star has a larger Γ at a given temperature, so it crystallises while still brighter. The observed branch runs from fainter at low mass to brighter at high mass, and the slope matches.

It is a branch and not a gap. A process that removed stars would leave a gap; a process that slows them leaves an excess. The feature is an excess, counted against the smooth sequence either side, and the number of stars in it is what the energy released has to account for.

There is a third property that does not follow from crystallisation alone, and it is where the neon term entered. The excess contains more stars than the latent heat can hold up — by a factor that depends on the mass, and by more for the more metal-rich. That is a signature of an energy source proportional to metallicity, and there is only one candidate.

What is actually measured

Nothing above is observed directly, and the chain is short enough to set out.

What is measured is a colour and a magnitude for each of a few hundred thousand white dwarfs with good parallaxes, and a spectrum for a much smaller subset. That sample is a recent thing: before it, the number of white dwarfs with distances good enough to place them individually on a cooling track was a few thousand, and the crystallisation branch is a feature of a few per cent that cannot be seen in a few thousand objects spread over a two-dimensional diagram. The colour and magnitude give a position on the cooling sequence; a model atmosphere converts them into an effective temperature and a surface gravity; the gravity plus a mass–radius relation gives a mass; and the temperature plus the mass gives an age through a cooling model.

Three of those four steps are models. The mass–radius relation for a degenerate star is as close to a first-principles result as stellar astrophysics has and is not in doubt; the atmosphere models are good and are the reason spectroscopic and photometric masses agree; the cooling model is where every effect in this essay lives, and it is the one being tested by the pile-up.

The parallaxes deserve a sentence of their own, because they are what made this subject observational rather than theoretical. Before a survey with microarcsecond astrometry there was no way to put a white dwarf on a cooling sequence except by assuming its mass, so the sequence was a band rather than a set of tracks and a 36 per cent overdensity along it was invisible. The crystallisation branch was predicted in the 1960s and detected in 2019, and what changed in between was a parallax.

The crystallisation feature is therefore doing double duty. It is a confirmation that the physics is there, and it is a calibration of how much energy the two processes actually release — a number that could not be got any other way and that changes every white dwarf age computed afterwards.

The generalisation

The structure worth extracting is that a delay and a pile-up are the same fact seen in time and in space, and that only the second is observable.

Nothing about a single white dwarf reveals that its cooling has been delayed: it is at a luminosity, and the luminosity is consistent with a range of ages. What reveals the delay is a population, because a flow that slows down leaves an excess where it slowed, and an excess is a count rather than an inference.

The same shape recurs in every part of this collection where a population is used to measure a rate. A mass function corrected by an age is a count converted into a birth rate through a lifetime. A count with a knee in it is the same conversion for galaxies. And the corner of the HR diagram that has to be earned is the same statement about the main sequence: a dense region of a diagram is a slow stage of an evolution, not a preferred kind of star. In each case the number density is the rate divided by the speed, and a bump in the first means a dip in the second.

There is a second reading about what makes a delay measurable at all. A delay that were instantaneous — all the energy released at one luminosity — would shift every star’s age by the same amount and change no observable count, because a constant offset in time does not change a number density. It is precisely because the release is spread that it is visible. A process is observable in a population when it changes a rate, not when it changes a total.

The corollary is a rule for reading any sequence along which objects evolve. An overdensity is a slow patch. It is almost never a place where more objects were made, because the making happened long ago and elsewhere; it is a place where the ones already made are lingering.

Where the ladder goes next

The next rung follows the crystallisation further. As the interior freezes it does not freeze uniformly: carbon and oxygen have different charges, so the solid that forms is oxygen-enriched and the liquid left behind is carbon-enriched — a distillation, driving a slow convective redistribution of the whole interior that releases a further amount of gravitational energy and may be the largest term of all in the most massive white dwarfs.

Further rungs on this anchor: the Debye branch below the crystallisation front, where the heat capacity collapses and cooling accelerates enough to compensate for the delay; the white dwarfs in globular clusters, whose ages can be compared with the main-sequence turn-off age of the same cluster, which is the only external check the cooling clock has; the ultra-massive white dwarfs, which crystallise while still bright and whose delays are the largest anywhere; and the coolest and oldest white dwarfs of all, whose atmospheres are so cold that molecular hydrogen absorbs their own infrared and makes them bluer as they age — a colour that runs backwards, on a sequence where colour is otherwise a thermometer.

Where this goes next

Mestel cooling for white dwarfs of 0.4 to 1 solar masses. Luminosity against age for white dwarfs of 0.4, 0.6, 0.8, 1 solar masses, both axes logarithmic. Each line is the whole of the star's later life: there is no nuclear source, so what is radiated is the thermal energy the ions had when the star was made, leaking out through a thin envelope whose opacity is Kramers'. That one sentence gives L ∝ t^(−7/5), and the lines are straight here because a power law is straight on these axes — the slope is the exponent and can be measured off the picture. Two features are worth more than the numbers. The first is how flat the sequence becomes: a factor of ten in luminosity costs a factor of 5.2 in age, so a white dwarf spends most of its existence faint, and any population of them piles up at the bottom. The second is that the heavier tracks lie above the lighter ones. A heavier white dwarf is smaller, so it has more ions to cool and less surface to lose them through, and at a given age it is the brighter object. The picture does not show what happens at the faint end, where the ions crystallise and the heat capacity stops being constant; that is a separate figure, and it is where this law stops being enough. A clock with no fuel in it A white dwarf has nothing left to burn, so its brightness is a record of how long it has been cooling. The faintest ones in the Galaxy are not faint because they are small — they are as faint as anything has had time to become, and the place where they stop is a date. Radius against mass for a degenerate star. The mass–radius relation for electron-degenerate matter. More mass gives a smaller star, and the radius reaches zero at 1.46 solar masses — the Chandrasekhar limit, solved from the same expression that draws the curve rather than quoted alongside it. A star held up by a rule about counting A white dwarf makes no energy and does not collapse. What holds it up is not heat or pressure in any ordinary sense — it is a quantum rule forbidding two electrons from occupying the same state. A high end that is 2.5 in slope steeper than the one stars were born with. The mass function a star count measures against the one stars were born with, for a population that has been forming stars steadily for 10 billion years. Below about a solar mass nothing has had time to die, so the two coincide exactly. Above it the fraction still alive is the main-sequence lifetime divided by the age, and since the lifetime falls as the two-and-a-half power of the mass, the present-day function is steeper than the initial one by exactly that exponent. A count of massive stars in an old population therefore under-represents them by orders of magnitude, and reading it as an initial mass function gives a slope far too steep. The correction is large, it is calculable, and it depends on the star-formation history — which is usually the thing the mass function was going to be used to constrain. A mass function corrected by an age Counting stars by mass gives the stars that are alive. What every argument needs is the stars that were born, and the two differ by the fraction of each mass still on the main sequence — which is a lifetime divided by an age, and which for massive stars in an old population is a very small number.

The objects this essay names

Each one links to every other essay that touches it.

Cooling ageCoulomb couplingCrystallisationDebye coolingLatent heatLuminosity functionNeon-22 sedimentationNumber densityPhase transitionWhite dwarf cooling