Astronomy & Celestial Mechanics
the sky, computed
Stars

The clock that starts by forgetting

An ordinary star tells nothing about its age. It sits on the main sequence for billions of years at almost fixed brightness and colour, and the one property that changes monotonically is how fast it turns — but only because the braking law destroys the initial condition first, and only until it stops.

Assumes Magnetic braking, Periodograms and Asteroseismology.

15 min read 8 figures The sky as a clockMeasuring the unreachable

Ages are the hardest thing to measure about a star, and the reason is that stars are designed not to change. A solar-type star spends nine-tenths of its life on the main sequence, and across that span its luminosity rises by perhaps forty per cent and its colour barely moves. Nothing about the light says whether it has been there for one billion years or eight.

There are exactly three ways round this. A star in a cluster can be dated by the point where its neighbours leave the main sequence, which dates the cluster and not the star. A star with a white dwarf companion can be dated by how far that remnant has cooled. And a star on its own, in the field, with no companion and no cluster, can be dated by how fast it turns — which works only because the braking law forgets where it started.

A period, a colour, and an age. Rotation period against colour for stars of 125, 625, 1000, 2500, 4570 million years, under the empirical relation P = t^0.5189 × 0.7725(B−V − 0.4)^0.601. The isochrones do not cross and are separated at every colour by exactly the age ratio raised to 0.5189, which is what allows a single measured period to be inverted for an age once the colour is known. The Sun, at B−V = 0.653 and 4570 million years, is placed by the relation at 26.8 days against the 25.4 days it is observed to have. Three clusters are marked at a common colour to show the spacing directly. The dashed boundary at the left is where the Rossby number — the period divided by the convective turnover time — passes 2 on the oldest isochrone, at B−V = 1.35: past that point the braking weakens and the relation is known to over-predict the age, which is the one place a rotation period stops being a clock. What the figure cannot show is the scatter, which is a few days at fixed colour and age and is the real error bar on any single star.
Fig. 1 Rotation period against colour for stars of five different ages. The lines do not cross, and at every colour they are separated by exactly the ratio of the ages raised to a fixed power — which is what makes a single measured period invertible for an age once the colour is known. The Sun is placed by the relation at 26.6 days against the 25.4 it has. The dashed boundary on the left is where the braking is known to weaken, and past it the relation over-predicts the age.

Why a clock has to forget

The peculiar thing about using rotation as a clock is that it works because of an erasure rather than in spite of one.

A quantity makes a good clock if it depends on elapsed time and nothing else. Rotation looks like the opposite: a star’s spin ought to depend enormously on the conditions of its birth, on how much angular momentum its parent core happened to have and how much of it survived the disc phase. Young clusters confirm that. In a cluster a hundred million years old, stars of one mass are found turning anywhere from half a day to ten days — a spread of a factor of twenty, at one age and one mass.

By six hundred million years that spread has collapsed to a factor of about two, and by a few billion it is smaller still. The braking law does it: the torque goes as the cube of the rotation rate, so a fast rotator brakes hundreds of times harder than a slow one and catches it. The solution

Ω(t)=Ω0 (1+2KΩ02t)−1/2\Omega(t) = \Omega_0\,(1 + 2K\Omega_0^2 t)^{-1/2}

tends to (2Kt)−1/2(2Kt)^{-1/2}, in which the birth rate does not appear.

Four different beginnings and one ending. Rotation period against age for four stars born turning at 0.3, 1, 3, 8 days, integrated under a braking law that goes as the cube of the rotation rate below a saturation period of 3.4 days and linearly above it. Both axes are logarithmic. The tracks span a factor of 25.7 at ten million years, 1.93 at six hundred million, and 1.09 at the age of the Sun — the initial condition is not merely diluted, it is erased, because the solution Ω = Ω₀(1 + 2KΩ₀²t)^(−1/2) tends to (2Kt)^(−1/2) with no Ω₀ left in it. The late slope measured off the drawn curve is 0.500 against the one half the law demands. The single constant K is fixed by one requirement, that the attractor pass through the Sun at 25.4 days and 4.57 billion years, and the track drawn for the slowest starter arrives at 26.6 days. What the figure cannot show is the saturated branch's physical cause: above a few days' rotation the dynamo stops responding to faster rotation, and that plateau is measured rather than derived.
Fig. 2 The convergence, drawn. Four stars born turning at 0.3, 1, 3 and 8 days, integrated under a braking torque that goes as the cube of the rotation rate below a saturation and linearly above it. The tracks span a factor of eleven at ten million years and a few tens of a per cent at the age of the Sun. Everything that makes the method work is in that collapse — and everything that limits its precision is in the fact that the collapse is fast but not instantaneous, so a period measured at three hundred million years still carries a memory of the birth condition.

So the clock has a dead zone at the start. Before a few hundred million years a rotation period says more about a star’s origin than about its age, and gyrochronology is not applied there. That is the opposite of every other stellar clock, all of which are most precise when young.

The relation, and what kind of statement it is

The working form is a fit, and it is worth being blunt about that.

P=t n a (B−V−c) b,P = t^{\,n}\,a\,(B-V-c)^{\,b},

with nn near one half, and aa, bb and cc chosen so that the relation reproduces the rotation periods measured in clusters of known age. The separation of the age dependence from the colour dependence — the assumption that the equation factorises — is the substantive claim, and it is an assumption rather than a derivation. What theory supplies is the exponent nn: the braking law predicts one half, and the fits return values close enough to it that the agreement is evidence rather than coincidence.

The colour term is a stand-in for the depth of the convective envelope. A redder star has a deeper envelope, a longer convective turnover time, a more effective dynamo, and therefore a stronger brake at the same rotation rate. Colour is used because it is measurable; the physical variable is the turnover time, and where the two disagree the fit inherits the disagreement.

What is actually measured

The observable is not a rotation period. It is a brightness that varies, and the interpretation of that variation as rotation is a step with its own failure modes.

A cool star has spots. Spots are darker than the surrounding photosphere and they are not distributed uniformly in longitude, so as the star turns, the disc-integrated brightness rises and falls by anything from a hundredth of a per cent to several per cent. Find that period and the star’s rotation follows — which is the same inversion that recovers a pulsation period from a sparsely sampled record.

A true peak at 0.3103 and a false one at 0.6897, from the same data. The Lomb–Scargle periodogram of 162 simulated observations taken over 419 nights from one site, of a star carrying a 4.2-unit sinusoid at 0.3103 cycles per day under 3 units of Gaussian noise per point. The injected signal is recovered at 0.3103 cycles per day, within the 0.0024 resolution element the baseline allows. The second peak, at 0.6899, is 87 per cent as tall and corresponds to nothing: it is 1 − f, the signal reflected in the one-cycle-per-day spike of the sampling. Neither peak is more real than the other in this picture — deciding between them needs a second site at a different longitude, or a run long enough for the seasonal window to separate them. The dashed line is the power a pure-noise series would exceed once in 1000 trials, computed from 586 independent frequencies rather than from the 4691 grid points searched; using the grid count instead would put the line 2.08 higher and reject a real detection.
Fig. 3 How a period is extracted from an unevenly sampled brightness record. Each frequency is scored by how much of the variance a sinusoid at that frequency explains, and the highest peak is the candidate. The width of the peak is set by the length of the record and the height by the amplitude against the noise — so a four-year photometric mission resolves a twenty-five day period very sharply, and the difficulty is never the resolution.

The difficulty is which peak. A spot group on each hemisphere produces a light curve with two minima per rotation and a periodogram whose largest peak is at half the true period. Spots at different latitudes on a differentially rotating star produce peaks a few per cent apart, and which one dominates depends on which spots happened to be present. And the sampling itself imprints structure. There is also a selection effect built into the method: a star is measurable only if it has spots, and it has spots only if it has a dynamo, which is the same machinery that makes the solar cycle. Stars that have already spun down far enough to become inactive are the ones the technique cannot see, and they are exactly the old stars whose ages would be most valuable.

Calibration, and what it is calibrated against

Every number in the relation traces back to clusters, because clusters are the only place where a large number of stars have a known age and a measurable period at once. The Pleiades at 125 million years, the Hyades and Praesepe at around 625 million, NGC 6811 at a billion, NGC 6819 at two and a half, M67 at four — each contributes one column of the relation.

That means gyrochronology is anchored to the main-sequence turn-off ages of those clusters, and inherits their systematic errors. If cluster ages were revised by ten per cent, every gyrochronological age would move with them. This is the same dependence that runs through most of astronomy: a distance is measured with the last distance, and an age is measured with the last age.

A period, a colour, and an age. Rotation period against colour for stars of 125, 625, 2500, 10000 million years, under the empirical relation P = t^0.5189 × 0.7725(B−V − 0.4)^0.601. The isochrones do not cross and are separated at every colour by exactly the age ratio raised to 0.5189, which is what allows a single measured period to be inverted for an age once the colour is known. The Sun, at B−V = 0.653 and 4570 million years, is placed by the relation at 26.8 days against the 25.4 days it is observed to have. Three clusters are marked at a common colour to show the spacing directly. The dashed boundary at the left is where the Rossby number — the period divided by the convective turnover time — passes 2 on the oldest isochrone, at B−V = 1.30: past that point the braking weakens and the relation is known to over-predict the age, which is the one place a rotation period stops being a clock. What the figure cannot show is the scatter, which is a few days at fixed colour and age and is the real error bar on any single star.
Fig. 4 The same relation drawn out to ten billion years, past every cluster that calibrates it. The spacing between isochrones shrinks as the square root of the age, so the same measurement error in a period buys a larger error in the age as the star gets older: at half a billion years a five per cent period error is a ten per cent age error, and the fractional relation stays fixed while the absolute uncertainty grows. Beyond the oldest calibrating cluster the curves are extrapolation, and the figure draws them because that is where the method was found to break.

Where it stops being true

The failure was found by a method that does not use rotation at all.

Asteroseismology reads an interior off a comb of frequencies: it measures a star’s mean density from the spacing of those frequencies and its surface gravity from where the oscillation power peaks, and together with a model those give a mass and an age. It is not a precise age — ten to fifteen per cent at best — but it is independent of rotation, and a four-year photometric mission delivered both the oscillations and the spot modulation for the same stars. The comparison found that stars older than about the Sun rotate faster than the relation says they should. Not by a little: a star seismically dated at seven or eight billion years turns in something like thirty days when the extrapolated relation demands forty-five or more. The discrepancy is one-sided, it grows with age, and it appears at a well-defined place — at a Rossby number, the ratio of the rotation period to the convective turnover time, of about two. Each of those three properties matters for believing it. A two-sided scatter would be measurement error; a discrepancy that did not grow would be a calibration offset; and one appearing at no particular place would be hard to attribute to anything physical.

The interpretation is that the braking switches off. Something about the field’s large-scale organisation changes when a star turns slowly enough, the Alfvén radius collapses towards the surface, and the lever that had been removing angular momentum stops working. The Sun sits close to that transition, which is uncomfortable: the star whose age is best known is the one nearest the place where the method fails.

A lever 30 radii long, and the spin it removes. A magnetised stellar wind, drawn with the Alfvén surface at 30 stellar radii — the schematic distance at which the wind's inertia finally beats the field. Inside it the gas is forced to turn with the star, so every gram that leaves carries the specific angular momentum of the radius at which it broke free rather than of the surface it came from, and the lever squares: J̇ = (2/3) Ṁ Ω r_A². Beyond the surface the streamlines curve backwards, because angular momentum conservation makes the azimuthal speed fall as 1/r while the radial speed does not. With a moment of inertia coefficient of 0.073 and a mass loss of 2.3·10⁻¹⁴ solar masses a year, the star loses a fraction 2.3·10⁻¹⁴ of its mass and a fraction 1.9·10⁻¹⁰ of its angular momentum in the same year — a ratio of 8,219, which is (2/3)(r_A/R)²/k² and nothing else. The e-folding time for the spin is 5.3·10⁹ years against 4.3·10¹³ years for the mass. Nothing here is to scale in one respect that matters: the wind's density falls by more than ten orders of magnitude across the drawn region, so the streamlines are drawn as though the flow were visible when almost none of it is.
Fig. 5 The torque that does the forgetting. A magnetised wind removes angular momentum at a lever arm set by where the field can still enforce corotation, and the field strength itself grows with rotation — so a fast rotator brakes harder, which is exactly the feedback that erases the initial condition. Thirty stellar radii against twelve is a factor of six in torque. Without that dependence the braking would preserve the ordering of initial periods and there would be no clock at all.

What the method is worth

It survives the failure, with a boundary drawn round it. Between about half a billion and five billion years, for stars cooler than the convective boundary, a rotation period gives an age good to perhaps ten or fifteen per cent — which is better than anything else available for a single field star, and good enough to ask whether a planet-hosting star is young or old, whether a moving group is real, or how the disc’s stars are distributed in age.

Outside that window it does not work, and the two edges fail for opposite reasons. Too young and the initial condition has not been forgotten. Too old and the mechanism that does the forgetting has stopped.

The right way to hold it is as a clock with a stated running range rather than a formula with an error bar. An error bar invites extrapolation, because the arithmetic keeps returning a number and the number keeps getting a plausible uncertainty attached to it; a running range says that outside these two dates the instrument is not measuring the thing it is calibrated for. The seismic comparison did not find that the relation was imprecise beyond five billion years. It found that it was biased, in one direction, by an amount that grows — which is the failure mode an uncertainty cannot express.

There is a practical consequence for how the ages get used. A gyrochronological age quoted for one star is a statement about where that star sits in a converged family, and it is at its best when the question is comparative: is this group of stars older than that one, is this planet host young. It is at its worst when a single number is wanted for a single object near either boundary — which, because the interesting objects are usually old, is where it is most often asked for. The convergence the whole method depends on is a property of the braking law, so it is worth drawing at a second saturation threshold and a second lever arm.

Four different beginnings and one ending. Rotation period against age for four stars born turning at 0.3, 1, 3, 8 days, integrated under a braking law that goes as the cube of the rotation rate below a saturation period of 5 days and linearly above it. Both axes are logarithmic. The tracks span a factor of 26.3 at ten million years, 7.51 at six hundred million, and 1.16 at the age of the Sun — the initial condition is not merely diluted, it is erased, because the solution Ω = Ω₀(1 + 2KΩ₀²t)^(−1/2) tends to (2Kt)^(−1/2) with no Ω₀ left in it. The late slope measured off the drawn curve is 0.500 against the one half the law demands. The single constant K is fixed by one requirement, that the attractor pass through the Sun at 25.4 days and 4.57 billion years, and the track drawn for the slowest starter arrives at 26.6 days. What the figure cannot show is the saturated branch's physical cause: above a few days' rotation the dynamo stops responding to faster rotation, and that plateau is measured rather than derived.
Fig. 6 Four starting periods converging under a saturation threshold at five days rather than 3.4. The convergence still happens and it takes longer, so the age below which gyrochronology is unusable moves with a parameter that is itself fitted to the clusters the method is calibrated on.
A lever 60 radii long, and the spin it removes. A magnetised stellar wind, drawn with the Alfvén surface at 60 stellar radii — the schematic distance at which the wind's inertia finally beats the field. Inside it the gas is forced to turn with the star, so every gram that leaves carries the specific angular momentum of the radius at which it broke free rather than of the surface it came from, and the lever squares: J̇ = (2/3) Ṁ Ω r_A². Beyond the surface the streamlines curve backwards, because angular momentum conservation makes the azimuthal speed fall as 1/r while the radial speed does not. With a moment of inertia coefficient of 0.073 and a mass loss of 2.3·10⁻¹⁴ solar masses a year, the star loses a fraction 2.3·10⁻¹⁴ of its mass and a fraction 7.6·10⁻¹⁰ of its angular momentum in the same year — a ratio of 32,877, which is (2/3)(r_A/R)²/k² and nothing else. The e-folding time for the spin is 1.3·10⁹ years against 4.3·10¹³ years for the mass. Nothing here is to scale in one respect that matters: the wind's density falls by more than ten orders of magnitude across the drawn region, so the streamlines are drawn as though the flow were visible when almost none of it is.
Fig. 7 The angular-momentum accounting with the Alfvén radius at sixty stellar radii. The torque scales as the square of that radius, so the spin-down rate rises by a factor of twenty-five — from a quantity that has never been measured for any star but the Sun.

The stars that never converge

The relation is drawn against colour, and it stops working at both ends of that axis for different reasons. The hot end is straightforward: a star hotter than about the middle of the F class has almost no convective envelope, no dynamo worth the name, and no brake — so it keeps whatever rotation it was born with and carries no clock at all.

The cool end is more interesting, because the stars there do brake and do not converge.

Below about a third of a solar mass a star is convective throughout: there is no radiative core and therefore no interface between a rotating core and a convective envelope. Whatever the dynamo is doing in such a star, it is not doing what it does in the Sun.

What is observed is that the rotation periods of such stars are not distributed smoothly. There are fast rotators, turning in less than about ten days, and slow ones turning in more than about thirty, and comparatively few in between — a gap in the distribution rather than a spread around a mean.

A gap is not what a convergent braking law produces. A convergent law makes every star approach one track, so a population of one age and one mass should be tightly clustered; a bimodal distribution means either that the braking is not convergent for these stars or that it accelerates sharply somewhere in the middle of the range, so that stars cross the gap quickly and are rarely caught in it.

The second explanation is the favoured one, and it puts the mechanism back in the field’s geometry: a star above the gap has one kind of magnetic topology and one below it has another, and the transition between them is fast.

Whichever it is, gyrochronology does not apply. The most numerous stars in the Galaxy, and the ones whose planets are easiest to find, are the ones for which the only single-star clock in the subject does not work.

The rotation measured from a line instead

Photometric rotation periods are recent. Before them, and still for stars that show no spots, rotation is measured spectroscopically, and the difference between the two measurements is worth setting out because they are not the same quantity.

A rotating star has one limb approaching and the other receding, so every absorption line in its spectrum is broadened — the light from the approaching side is blueshifted and from the receding side redshifted, and the sum is a line whose width carries the equatorial velocity.

What the width gives is the projection: the equatorial speed times the sine of the angle between the rotation axis and the line of sight. That is a lower bound on the speed, and converting it to a period needs the radius as well as the inclination.

So the spectroscopic measurement gives a velocity with an unknown projection factor, and the photometric one gives a period directly. For gyrochronology the second is what is wanted, and the first is what was available for most of a century.

The spectroscopic method has one advantage that keeps it in use: it works on stars with no spots. A hot star, or a very quiet cool one, produces no photometric modulation at all and can still be measured from its line widths — and the population of quiet old stars that the photometric method selects against is precisely the population the spectroscopic method can reach.

Its limitation at the slow end is severe. Rotational broadening of a few kilometres a second sits below the other broadening mechanisms in a cool star’s spectrum, so a solar-type star turning in twenty-five days is at or below the detection threshold of all but the best instruments. The two methods are therefore good in opposite regimes, and the overlap where both work is narrow.

The period that has to be known before a planet can be claimed

There is a use for a stellar rotation period that has nothing to do with ages and is now the more common reason for measuring one.

A star’s spots produce a radial-velocity signal as well as a photometric one: a dark spot removes light from the approaching or receding limb as it crosses, distorting the line profile and shifting its measured centroid. The shift is periodic at the rotation period, its amplitude can be metres per second, and it looks exactly like a planet.

It has been mistaken for one repeatedly. Several announced planets around active stars have been withdrawn when the claimed orbital period turned out to equal the star’s rotation period, or half of it, or a third.

So a modern radial-velocity analysis measures the rotation period first, from photometry or from an activity indicator, and treats any velocity signal at that period or its harmonics as suspect until shown otherwise. The rotation period is a piece of prior information the planet search needs rather than an incidental result.

The same applies with the opposite sign to transit searches, where spots crossed by a planet distort the light curve and spots not crossed by it change the baseline — both of which bias the derived planetary radius, and both of which vary at the rotation period.

A quantity introduced here as a clock is used far more often as a veto, and the second use has driven more of the observing than the first.

One more calibration set covers a younger cluster than the essay’s own sequence begins with.

A period, a colour, and an age. Rotation period against colour for stars of 50, 300, 1500, 4570 million years, under the empirical relation P = t^0.5189 × 0.7725(B−V − 0.4)^0.601. The isochrones do not cross and are separated at every colour by exactly the age ratio raised to 0.5189, which is what allows a single measured period to be inverted for an age once the colour is known. The Sun, at B−V = 0.65 and 4570 million years, is placed by the relation at 26.6 days against the 25.4 days it is observed to have. Three clusters are marked at a common colour to show the spacing directly. The dashed boundary at the left is where the Rossby number — the period divided by the convective turnover time — passes 2 on the oldest isochrone, at B−V = 1.35: past that point the braking weakens and the relation is known to over-predict the age, which is the one place a rotation period stops being a clock. What the figure cannot show is the scatter, which is a few days at fixed colour and age and is the real error bar on any single star.
Fig. 8 The period–colour relation calibrated against clusters from fifty million to four and a half billion years. The youngest sequence is broad and the oldest is tight, which is the convergence the method depends on drawn as a set of snapshots rather than as a set of tracks.

Where the ladder goes

The open question is not whether the braking weakens but why. The candidate explanations are all about the field’s geometry rather than its strength — that a slowly rotating dynamo produces a field with more small-scale structure, which closes over near the surface and stops holding the wind out — and none of them predicts the transition Rossby number from first principles.

There is also a question the method raises about itself. Its precision comes from the convergence, and the convergence comes from a torque that is very steep in rotation rate. A steeper braking law converges faster and dates better; but a steeper law also means that the surface must be able to hand its angular momentum to the rest of the star quickly, or the surface would brake and then be spun back up from below. The tightness of the observed period–age relation is therefore also a measurement of how strongly a star’s interior is coupled to its surface, and it says the coupling is nearly rigid — which is the thing no proposed transport mechanism produces at the right strength.

Where this goes next

A lever 12 radii long, and the spin it removes. A magnetised stellar wind, drawn with the Alfvén surface at 12 stellar radii — the schematic distance at which the wind's inertia finally beats the field. Inside it the gas is forced to turn with the star, so every gram that leaves carries the specific angular momentum of the radius at which it broke free rather than of the surface it came from, and the lever squares: J̇ = (2/3) Ṁ Ω r_A². Beyond the surface the streamlines curve backwards, because angular momentum conservation makes the azimuthal speed fall as 1/r while the radial speed does not. With a moment of inertia coefficient of 0.073 and a mass loss of 2.3·10⁻¹⁴ solar masses a year, the star loses a fraction 2.3·10⁻¹⁴ of its mass and a fraction 3·10⁻¹¹ of its angular momentum in the same year — a ratio of 1,315, which is (2/3)(r_A/R)²/k² and nothing else. The e-folding time for the spin is 3.3·10¹⁰ years against 4.3·10¹³ years for the mass. Nothing here is to scale in one respect that matters: the wind's density falls by more than ten orders of magnitude across the drawn region, so the streamlines are drawn as though the flow were visible when almost none of it is. A wind that takes no mass and all the spin The Sun loses about a ten-thousandth of itself to its own wind over its whole life. It loses most of its rotation to the same wind, and the whole of that asymmetry is one geometric factor — the gas is forced to keep turning with the star until it is a dozen or more radii out from the surface it left. Three clusters, three ages, one diagram. Isochrones for populations of 100 Myr, 1 Gyr, 12 Gyr, each drawn as the main sequence up to its own turnoff and then the post-main-sequence track of the turnoff mass. The turnoffs are at 5.52 M☉, 2.20 M☉, 0.94 M☉, from t = 10¹⁰ M/L with this file's own mass–luminosity relation. Nothing here is a track along which a star moves. Every point is a different star of a different mass, all the same age, and the bend is simply where the population runs out of stars that have had time to leave. That is why a cluster has an age and a field star does not: the bend needs a population, and one star is not one. The oldest globular clusters sit near the 12 Gyr line, and in the 1990s the same construction gave them ages of 16 to 18 Gyr against a universe measured at 10 — a two-standard-deviation contradiction that was resolved from the distance side, by Hipparcos, and not from this one. An age read off a bend A cluster is a population of one age and many masses, so the place where its stars leave the main sequence is a clock. The bend needs a population — which is why a cluster has an age and a single star does not. 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.

About the same objects

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

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

AsteroseismologyCalibrationConvective envelopeConvective turnover timeGyrochronologyIsochroneMagnetic brakingMain sequence turn-offRossby numberRotation periodSkumanich lawStarspotsStellar ageWeakened magnetic braking