Astronomy & Celestial Mechanics
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Concept

Cooling age — where it appears

The time a white dwarf has spent radiating away the thermal energy of its ions, inferred from its luminosity through a cooling model. It is not the star's total age, which also includes however long the progenitor spent on the main sequence.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

7 solar masses of progenitor become 0.56 of white dwarf. Above: the initial–final mass relation. The steep line is what a star would leave if it kept everything; the shallow one is what it actually leaves, M_f = 0.08M_i + 0.489, fitted to white dwarfs in open clusters. Everything between 1 and 8 solar masses ends up between 0.57 and 1.13 — a range of 7 compressed by a factor of 12 — and the fraction thrown away rises from 43% at the bottom to 86% at the top. A star spends most of its mass in the last per cent of its life, at rates no theory predicts from first principles, and this relation is measured by running stellar evolution backwards: a cluster's turn-off gives the age, a white dwarf's temperature gives its cooling time, the difference gives how long its progenitor lived, and a model turns that into the mass it was born with. Below: the white dwarf mass distribution that follows, from a Salpeter initial mass function pushed through the relation above. It peaks at 0.61 solar masses, which is what every survey of white dwarfs measures. Two things make that peak and the flatness of the relation is only one of them: the unsmoothed distribution (the stepped curve) falls monotonically and is largest at its low-mass edge, because the initial mass function is steep. The edge is where it is because no star lighter than about 1.0 solar masses has had time to die yet, and the scatter of the measurement rounds that cliff into a maximum a little above it. A flat relation produces a narrow range; it takes the age of the Galaxy to produce a peak.

The mass a star does not keep

The initial–final mass relation is nearly flat, so everything from one to eight solar masses ends as a white dwarf between 0.57 and 1.13 — and the observed distribution piles up at 0.6 almost regardless of what went in. The relation is measured by running stellar evolution backwards, because nothing predicts the loss.

stars · Stellar evolution 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.

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.

stars · White dwarf cooling A white dwarf cooling track in J−K and absolute V. The track a 0.6 solar-mass hydrogen-atmosphere white dwarf follows as it cools, absolute V magnitude against J−K colour, fainter downward. Dashed, a blackbody of the same radius, which slides steadily redward and fainter. Solid, with collision-induced absorption: the track reddens to J−K = 0.40 at 4250 K and absolute magnitude 16.1, then hooks back to the blue, reaching −0.86 at 3000 K and magnitude 18.1. Temperatures are marked along it. The oldest white dwarfs therefore sit in the same colour range as much younger, hotter ones and are told apart only by being fainter — which is why they were missed by surveys that selected by colour.

A colour that runs backwards at the end of cooling

A white dwarf reddens as it cools, like every other hot body — until its surface falls below about four thousand kelvin. Then pairs of hydrogen molecules, colliding in an atmosphere dense enough to crush them together, begin to absorb its infrared light, and the star turns blue again while it goes on getting fainter. The oldest white dwarfs in the Galaxy have the colours of young ones, and for years they were looked for in the wrong place.

stars · White dwarf cooling The share of white dwarfs whose centres have begun to freeze. The fraction of a population of carbon–oxygen white dwarfs, with masses spread as a Gaussian of mean 0.62 and width 0.12 M☉, whose centres have begun to crystallise, against cooling age, overall and in three mass ranges, in the toy model of the other figures. If a field appears when freezing begins, this is the shape its incidence must take: it switches on first among the heaviest white dwarfs, half of which have begun freezing by 0.5 Gyr, and last among the typical ones near 0.6 M☉, by 2.5 Gyr. A fossil field, present from the white dwarf's birth, would show no such delay and no such ordering by mass.

A field that waits for the core to freeze

A field inherited from the parent star would be present from a white dwarf's birth. Yet magnetic white dwarfs of ordinary mass are rare among the young and common among the old, as if their fields were switched on later. The proposed switch is the freezing of the carbon–oxygen core, which stirs the liquid above it. If that is right, the delay is set by when each mass begins to freeze, and the heaviest should go first.

stars · Flux freezing Three energies a freezing white dwarf radiates late. The age at which a white dwarf has finished crystallising, against its mass, built up from bare Mestel cooling and three energies released at the crystallisation luminosity: the latent heat of freezing (0.85 kT per ion), the gravitational energy of neon-22 settling (0.6), and the gravitational energy of carbon and oxygen separating (0.8). At 0.6 solar masses the three together add 3.20 billion years to 3.56 of cooling; at 1.1 they add 4.36 to 4.85. The separation term is, on these numbers, the second largest of the three. None of them is fuel; all are energy the star already had, released at a luminosity so low that a little of it takes a long time to radiate.

A star that distils itself as it freezes

A carbon–oxygen white dwarf does not freeze at one composition. The crystals that form at its centre take more oxygen than the liquid they form from, the carbon-rich liquid left behind is lighter and rises, and the star slowly sorts its own interior, oxygen down and carbon up. The gravitational energy that releases is radiated at a luminosity so faint that it holds the star up for about a billion years — a delay comparable to the latent heat of freezing, and one of the larger uncertainties in every white-dwarf age.

stars · White dwarf cooling An association that has been flying apart for ten million years. The velocity of each of 220 stars along one direction against its position along the same direction, for a model association whose stars were born 10 million years ago inside a region 1 pc across (one standard deviation), with random velocities of 1 km/s, and that began coasting apart at once. The fast stars have gone furthest, so position and velocity are correlated, and the slope of the line through them is the inverse of the time they have been moving: the fit gives 10.1 million years. A star at 10 pc moving at 1 km/s has been travelling for ten million years — which is what an expansion age is. Measuring it needs the stars' distances and their motions across the sky, both of which space astrometry now supplies for associations within a few hundred parsecs.

An expansion clock read from the stars' motions

The masses of a young cluster's brown dwarfs are read from their brightness, and a brown dwarf's brightness is a mass multiplied by an age. The age can now be measured without any model of a star: the cluster is flying apart, and the time since it began is its size divided by its speed. But the expansion clock counts only the coasting, not the stars' lives, and it runs fast when the stars began spread out — two biases of opposite sign, and nothing in the motions themselves to say which one wins.

stars · Initial mass function

Named alongside it

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

CrystallisationWhite dwarfLatent heatLuminosity functionPhase separationAstrometryAsymptotic giant branchBlackbodyBrown dwarfChandrasekhar limitCollision induced absorptionColour index

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