Astronomy & Celestial Mechanics
the sky, computed
Concept

White dwarf — where it appears

The exposed degenerate core left when a star of low or intermediate mass sheds its envelope, supported by electron pressure alone. It cools rather than evolves, so the faintest ones in a population are the oldest, and the turnover in their luminosity function is a clock.

Named by 21 essays across 5 fields — each of them below, with the objects they name alongside it.

The Hertzsprung–Russell diagram. Luminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius.

The diagram that sorted the stars, by plotting two things against each other

Plot brightness against colour for a few thousand stars and they do not scatter. They fall on a narrow band with two islands off it, and explaining that structure is most of stellar astronomy.

stars · The HR diagram 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.

stars · The mass–luminosity relation How long a star lasts, against its mass. Main-sequence lifetime against mass, on logarithmic axes. Fuel grows in proportion to mass and consumption grows as its three-and-a-half power, so the lifetime falls steeply — a star of thirty solar masses lives for a few million years.

Why the biggest stars die first, and take the galaxy with them

A star thirty times the Sun's mass has thirty times the fuel and burns it forty thousand times faster. It lasts a few million years, and everything heavier than iron exists because of it.

stars · Stellar lifetimes 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.

stars · Degeneracy From 1.667 to 1.343, and the limit is in the second number. The local slope Γ = d ln P / d ln ρ of the exact degenerate electron equation of state, against density, differentiated numerically from the drawn pressure rather than quoted. It is 1.6666 at 10⁴ kg/m³ — the 5/3 of a non-relativistic Fermi gas — and 1.3430 at 10¹² — the 4/3 of a relativistic one — falling monotonically between, and passing 1.5 at 3.63·10⁹ kg/m³. Nothing gives way at either end; the electrons simply run out of room to go faster, because they are already near c. The mass limit is in the exponent and not in the strength. Hydrostatic balance for a polytrope of index Γ gives M ∝ ρ_c^(3Γ−4)/2, so at Γ = 4/3 the exponent is zero: the mass no longer depends on the central density at all, and there is exactly one mass a fully relativistic degenerate star can have — 1.459 M☉ at μ_e = 2. Above it, squeezing harder buys pressure that rises more slowly than gravity does, and there is nothing left to stop at.

The mass a cold star cannot exceed

A degenerate star's radius falls as its mass rises, and nothing in that relation suggests a limit. Making the electrons relativistic softens the pressure law until pressure and gravity scale the same way with radius — and the radius drops out of the balance, leaving one mass and no room to argue.

stars · Degeneracy The mirror: an envelope 62 times larger around a core 1.48 times smaller. Envelope radius and core radius against helium-core mass for a 1 solar-mass star, both in solar radii on one logarithmic axis. Two curves with opposite slopes at every point: the envelope grows from 2.0 to 122 solar radii — 0.57 astronomical units — while the degenerate core contracts from 13,848 to 9,368 kilometres, and the two are separated by a factor of 364 in the middle of the range. The figure is drawn from two stated relations rather than from a stellar model: a core-mass–luminosity law of the sixth power, calibrated to 2,500 solar luminosities at 0.48 solar masses of core, and a Hayashi temperature falling from 5000 to 3700 kelvin across the range. Everything else follows exactly — the radius by the Stefan–Boltzmann law, the core radius by the cold degenerate relation R ∝ M⁻¹ᐟ³. Note what the tip radius is and is not: it is the red-giant branch, and the later asymptotic-giant tip is a different and larger number.

The star that swells because its centre shrank

A helium core contracts, and the envelope around it expands by a factor of sixty. The central density rises at the same time as the radius, which is the opposite of what a self-gravitating body is expected to do, and a burning shell between the two is the whole reason it happens.

stars · Stellar evolution A light curve that is a decay chain. Bolometric luminosity against days since explosion, for three Type Ia models synthesising 0.3 M☉, 0.6 M☉, 0.9 M☉ of ⁵⁶Ni, from Arnett's one-zone diffusion model. Nothing in the shape is fitted: the two timescales are the laboratory half-lives of ⁵⁶Ni and ⁵⁶Co, 8.8 and 111.3 days, and the rise is those decays seen through an envelope that takes 11.3, 15.5, 18.6 days to leak. Each peaks at 14, 18, 21 days, and at maximum the luminosity equals the instantaneous deposited power to within 0.6% — Arnett's rule, and the only reason a peak brightness can be read as a mass of nickel. After maximum the curves settle to nearly straight lines on this logarithmic axis, declining at 0.0360 magnitudes a day against ⁵⁶Co's own 0.0098. The difference is gamma-ray escape: the ejecta become transparent to the very photons that are supposed to be heating them, and the tail is therefore steeper than the isotope. The one thing the tail is not is a property of the star — it is a half-life, plus a column density that is falling as t⁻².

The candle that has to be standardised

A Type Ia's light is the decay of half a solar mass of nickel-56 seen through an expanding envelope. Their peak brightnesses span six-tenths of a magnitude — and it is the width of the light curve, not anything else, that says which.

stars · Supernovae Two responses to the same transfer, turning round at q = 0.79 and q = 1. What conservative mass transfer does to the orbit and to the lobe, plotted against the mass ratio of donor to accretor on a logarithmic axis. Both curves are logarithmic derivatives with respect to the donor's mass, so a positive value means the quantity shrinks as the donor loses mass and a negative one means it grows. The orbit's response is exactly twice the mass ratio less one, which follows from holding the total mass and the total angular momentum fixed and nothing else, and it crosses zero at equal masses: transfer from the heavier star draws the orbit in, transfer from the lighter one pushes it out. The lobe's response adds to that the change in the lobe's shape, and it crosses zero earlier, at a mass ratio of 0.788. Between those two crossings the orbit is still widening while the lobe is already closing. To the right of both, a donor that loses mass finds its lobe shrinking around it, which is the runaway the essay is about: the transfer narrows the valve it is flowing through.

The flow that narrows its own channel

Two stars close enough share a surface, and the point where that surface pinches is a valve. What comes through it changes the orbit, and the orbit changes the valve — with a sign that reverses at equal masses, which is why some binaries transfer quietly for a hundred million years and others tear themselves apart in a thousand.

stars · Mass transfer 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.

stars · White dwarf cooling The horizon and the orbit that cannot come back, against the hole's spin. Two radii round a rotating black hole, in gravitational radii GM/c², against its spin a from −1 (an orbit against the rotation) to +1 (with it), for orbits in the equatorial plane. The lower curve is the horizon, 1 + √(1 − a²), which is the same whichever way the orbit goes. The upper one is the marginally bound orbit, 2 − a + 2√(1 − a): the closest a body falling in from far away can pass and still escape back out. For a hole with no spin the horizon is at 2 and the marginally bound orbit at 4 — twice as far out. With maximal spin the marginally bound orbit comes in to 1.09 for a prograde orbit (a = 0.998) and moves out to 5.83 for a retrograde one. A star whose tidal radius lies inside this curve is swallowed whole, so the curve, not the horizon, is the line a disruption flare is measured against — and it moves by a factor of 5.3 with the spin.

The line a star is swallowed at is not the horizon

A star torn apart by a black hole makes a flare, and a star swallowed whole makes nothing, so the heaviest hole that can produce a flare is a measurement. That ceiling is set not at the horizon but at the closest orbit from which infalling matter can still come back out — twice as far out for a hole that does not spin, and moved by a factor of five by the hole's rotation.

galaxies · Tidal disruption A power law has no timescale: half the Type Ia supernovae by 740 Myr and a tail to 13.7 Gyr. The fraction of all the Type Ia supernovae a single burst of star formation will ever produce that have exploded by a given delay, on a logarithmic time axis, for t⁻¹ from 40 Myr; t⁻¹·⁴ from 40 Myr; single delay of 1 Gyr; Gaussian, 3 ± 1 Gyr. The power law is what rates measured against host-galaxy ages and against the cosmic star-formation history both favour, and its cumulative fraction rises as the logarithm of the delay — equal numbers per decade of time. Half have exploded by the geometric mean of its limits, 740 Myr, 55 per cent by 1 Gyr, and the last are still exploding after a Hubble time. A single delay turns the whole population on at once; a Gaussian concentrates it at a characteristic age. The power law's shape has a physical reading: if white dwarfs explode when a pair of them merges by emitting gravitational waves, the merger time goes as the fourth power of their separation, and a broad distribution of separations becomes a distribution of delays with no preferred scale. What the drawing cannot say is which progenitors are involved — the measured rates constrain the shape and the normalisation, about one Ia per thousand solar masses of stars formed, and not the mechanism.

The iron clock has no single delay

The α-element knee is drawn as though Type Ia supernovae switched on a billion years after the stars that made them. Measured rates say otherwise — the delays are spread evenly over every decade from forty million years to a Hubble time, as a power law with no timescale in it — and a clock with no timescale bends where a clock with one would break.

galaxies · Chemical evolution Field strength against radius, from the Sun to a magnetar. Surface magnetic field against radius, both on logarithmic scales, radius in solar radii and field in gauss, for five kinds of star: the Sun's dipole, Ap and Bp stars, magnetic white dwarfs, radio pulsars, magnetars. The diagonal lines are constant magnetic flux, πR²B = 10²³, 10²⁵, 10²⁷ gauss square centimetres, along which a star shrinking with its field frozen in would slide. The Ap and Bp stars, the magnetic white dwarfs and the magnetars all fall close to the upper lines: a field of a few kilogauss on a star of three solar radii, squeezed to the size of a white dwarf, becomes a hundred million gauss, and squeezed to twelve kilometres becomes a hundred trillion. The Sun's dipole and the ordinary pulsars carry a hundred to a thousand times less flux.

The same flux in three sizes

A few per cent of hot main-sequence stars carry kilogauss fields that do not change over decades. So do a similar share of white dwarfs, at a million times the strength, and of young neutron stars, at a hundred billion times it. Multiply each field by its star's area and the three classes carry nearly the same magnetic flux — what a field frozen into a star would keep as the star shrank by a factor of a hundred thousand. A collapse makes a field enormous, but it cannot make it important.

stars · Flux freezing 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 One history of mass loss, three orbits, three outcomes. The eccentricity of three planets' orbits against the fraction of the star's mass lost so far, as the star sheds 45 per cent of its mass in 6 equal pulses at irregular intervals. Each planet starts with e = 0.05. The pulses are the same events in the star's own time, and the planets differ only in their periods, so each pulse lasts 3, 0.3, 0.03 orbital periods respectively. For the innermost planet, to which every pulse is slow, the eccentricity ends at 0.050: the orbit swells and keeps its shape. For the outermost, to which every pulse is sudden, each pulse kicks the eccentricity by an amount set by where the planet happened to be, and it ends at 0.298. The middle planet ends at 0.090. The same total loss leaves three different shapes, and only one of them is predicted by either limit alone.

The size forgets how the mass was lost, and the shape does not

A dying star loses half its mass, and a planet round it keeps its orbit's shape if the loss is slow and has it scrambled if the loss is sudden. Real stars lose mass in pulses — each one sudden or slow depending on the planet — and what survives a train of pulses is neither answer: the orbit's size settles to the slow value quickly, while its eccentricity random-walks and keeps a record of the history.

orbits · Effective potential A period set by the donor's density, running in two directions. The orbital period at which a donor exactly fills its Roche lobe, against the donor's mass, onto a 0.7 M☉ white dwarf. A lobe-filling star's period is fixed by its mean density alone, so the period follows the donor's mass–radius relation. A main-sequence donor shrinks as it is stripped and grows denser, and the period falls: from 6.0 hours at 0.6 M☉ to 1.5 hours at the hydrogen-burning limit. A white-dwarf donor swells and grows less dense, and the period rises — 3.5 minutes at 0.2 M☉, 13 at 0.05, 47 at 0.01. The binaries with helium-rich spectra and periods between five minutes and an hour lie along the second curve and nowhere near the first, and their period is a mass gauge for a star that cannot be seen.

A donor that grows as it is eaten

A white dwarf that loses mass gets bigger, so a binary in which one white dwarf feeds another runs backwards — its period lengthens as the donor shrinks. Whether the pair survives first contact at all depends less on the donor than on whether the stream finds room to form a disc, and for most pairs it does not.

stars · Mass transfer Orbital energy spent against the envelope it has to lift. The orbital energy a 0.6 M☉ companion releases by spiralling in from the onset of a common envelope down to a final separation, against that separation (curve), and the binding energy of the 0.85 M☉ envelope of a 1.2 M☉ giant with a 0.35 M☉ core, divided by the efficiency, for αλ of 0.25, 0.5, 1 (horizontal lines). Energies are in units of GM☉²/R☉. The giant fills its lobe at a separation of 118 R☉, and the companion releases nothing net until the orbit has shrunk below 34 R☉, because the envelope's mass leaves with the envelope; from there it falls until the energy released pays for the envelope, so each crossing is a final orbit — 1.29, 2.48, 4.63 R☉. The released energy grows only as one over the separation, so halving the efficiency halves the final orbit: a factor of four in αλ is a factor of four in the outcome, and there is no measurement of αλ that does not run through outcomes like these.

The least known number in binary evolution

When a giant swallows its companion, the companion spirals inward until the orbital energy it has released is enough to throw the envelope off. The final orbit is set by one product of two unknowns, it moves in proportion to that product, and the merger time of the white dwarfs it leaves moves as its fourth power.

stars · Mass transfer 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 Dark companions' masses with and without a gap, as an astrometric survey would see them. The distribution of the masses of dark companions in a sample found by their stars' astrometric orbits, per unit of the logarithm of mass, for two hypotheses: neutron stars near 1.35 solar masses and black holes near 8 in equal numbers with nothing between 2 and 5 (solid), or the same with the gap filled at the density the black holes have at 5 (dashed; 16 per cent of the sample in it). Each is drawn as measured with mass errors of 10 per cent (blue) and 30 per cent (orange). With 10 per cent errors the empty gap stays empty: 0.2 per cent of the sample lands between 2.5 and 4.5 solar masses, against 10 per cent if it is filled. With 30 per cent errors the edges spill in — 4.3 per cent against 12 per cent — and the gap becomes a shallow dip that a modest sample cannot distinguish from a filled one.

A gap that counting can test and impostors can fill

The black holes in X-ray binaries leave a gap between the heaviest neutron stars and about five solar masses, and nobody knows whether it is real or an artefact of how those masses are measured. Astrometric orbits give masses that need no light curve, and with errors of ten per cent a few dozen of them would settle it. The difficulty is not the counting. It is that pairs of white dwarfs and neutron stars, dark and close, can weigh exactly what a gap object would.

gravitation · The two-body problem How much a dying star stirs its companion's orbit, against their separation. The rms change in orbital eccentricity of a 0.8 M☉ companion when a 2 M☉ star becomes a 0.6 M☉ white dwarf, against the binary's separation before the loss, both logarithmic. Blue: the mass loss alone — twelve thermal-pulse episodes 500 years wide and a superwind 10,000 years wide, each adding a kick of random sign whose size falls off exponentially once it lasts longer than about a sixth of an orbit. Below 100 au the orbit is slow to every episode and the eccentricity is untouched. The pulses switch on near 229 au, and by 3,000 au the change is 0.26 and the superwind too has become sudden. Dashed: the change a velocity kick to the white dwarf of 0.05 or 0.2 km/s would make on its own, which grows as the square root of the separation. A 0.05 km/s kick outweighs the pulses out to 251 au; a 0.2 km/s kick outweighs the pulses at every separation drawn.

A kick that writes over the record of the pulses

A star that becomes a white dwarf loses half its mass in pulses, and a companion far enough away feels each pulse as a sudden jolt to its orbit's shape. The wide binaries of white dwarfs should therefore carry the pulses' record in their eccentricities. They would, except that the white dwarf itself is born moving at a few tenths of a kilometre a second, and a kick that small writes more eccentricity than all the pulses together.

orbits · Effective potential 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 Rise, steady state and decline. The mass of magnesium (solid) and iron (dashed) in a white dwarf's mixing layer, each as a fraction of its steady-state value for magnesium, during an accretion episode that delivers both at equal rates for 5 magnesium sinking times and then stops; time in units of the magnesium sinking time. Iron sinks 1.6 times faster. While they rise, both grow together and their ratio is the accreted ratio; once accretion has run for a few sinking times each element saturates at its rate times its own sinking time, so the layer holds 0.63 times the accreted iron-to-magnesium ratio; after accretion stops iron vanishes first, and 7 sinking times later the layer holds 0.009 of the accreted ratio. The same body seen at three moments reads as three compositions.

A planet read in the ashes of its star

A white dwarf's gravity drags every heavy element out of its visible surface, so a white dwarf showing magnesium, iron and silicon is swallowing rock now. Its spectrum is the only place where the bulk composition of a rocky body round another star can be read element by element. But each element sinks at its own rate, so the ratio seen depends on when in the meal the spectrum was taken — and only the hottest hydrogen white dwarfs say.

exoplanets · Planet composition

Named alongside it

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

Chandrasekhar limitDegeneracy pressureBlack holeCooling ageCrystallisationGravitational wavesHydrostatic equilibriumMain-sequenceMass lossMass radius relationMass transferRoche lobe overflow

All concepts