Astronomy & Celestial Mechanics
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Stars

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.

Assumes Flux freezing, Pulsars and Degeneracy.

15 min read 6 figures Scales that defeat intuitionMeasuring the unreachable

A magnetic field threading a good conductor moves with it. Where the conductor’s resistance is negligible over the times and lengths that matter — where the magnetic Reynolds number is large — the field lines are carried by the flow as though painted on it, and the magnetic flux through any surface moving with the material stays constant. A collapsing cloud carries its field inward that way, and the arithmetic says it should arrive at the protostar ten thousand times too strong unless most of the flux is shed on the way.

Once a star exists, the arithmetic becomes simpler and more demanding. A star’s interior is an ionised gas with a magnetic Reynolds number of order 101010^{10} or more; a field anchored in it would take longer than the age of the universe to diffuse out by ohmic resistance. So a field present in a young star’s radiative interior stays there for the star’s life, and when the star contracts — to a white dwarf at the end of its life, or through the core collapse that makes a neutron star — the flux through its surface should be conserved. The field then grows as the inverse square of the radius.

The prediction can be checked, because each stage of that contraction has a class of magnetic stars.

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.
Fig. 1 Surface field against radius for five kinds of star, with diagonal lines of constant magnetic flux along which a star shrinking with its field frozen in would slide. The magnetic hot stars, the magnetic white dwarfs and the magnetars all lie near the upper lines, spread across a hundred thousand in radius and a hundred billion in field. The Sun’s global field and the ordinary pulsars carry far less flux.

Three classes and one quantity

The first class is the chemically peculiar Ap and Bp stars, a few per cent to a tenth of all main-sequence stars of two to four solar masses. Their spectral lines split and polarise in fields of three hundred to thirty thousand gauss, organised on a large scale, usually close to a dipole inclined to the rotation axis. The fields do not vary except as the star rotates; they show no cycle and no dependence on the star’s rotation rate or age of the kind a dynamo would produce. And these stars have no convective envelope to run a dynamo in: they are radiative from just below their surfaces to their convective cores. A field in such a star is a relic — a fossil — or it was not made in the way the Sun’s is.

The second class is the magnetic white dwarfs, about a tenth to a fifth of white dwarfs in samples complete to a given distance, with fields from a thousand to a billion gauss. A white dwarf is the exposed core of a star of up to eight solar masses, about a hundredth of the Sun’s radius, and it has no convection, no rotation-driven dynamo and no nuclear burning. Its field, too, is a relic of something.

The third is the magnetars, neutron stars with fields of 101410^{14} to 101510^{15} gauss inferred from how fast they spin down — the same inference that gives every pulsar its field — and confirmed by the energy of their X-ray outbursts. They are perhaps a tenth of the young neutron stars in the Galaxy, possibly more.

Multiplying each field by its star’s cross-sectional area gives the flux.

Magnetic flux, star by star. The magnetic flux πR²B carried by each class, as a range on a logarithmic scale in gauss square centimetres: a star-forming cloud core, 5·10²⁹ to 5·10³⁰; the Sun's dipole, 2·10²² to 3·10²²; Ap and Bp stars, 2·10²⁵ to 7·10²⁷; magnetic white dwarfs, 10²¹ to 3·10²⁷; radio pulsars, 3·10²³ to 5·10²⁵; magnetars, 3·10²⁶ to 5·10²⁷. Three classes that differ in radius by a factor of a hundred thousand — Ap and Bp stars, magnetic white dwarfs and magnetars — share a band of flux from about 10²⁵ to a few times 10²⁷, which is what a field frozen into a contracting star would produce. The cloud core that makes a star carries a thousand to ten thousand times more, so most of the flux is lost before the star exists and little of it afterwards.
Fig. 2 The magnetic flux each class carries, as a range on a logarithmic scale. Ap and Bp stars, magnetic white dwarfs and magnetars overlap in a band from about 102510^{25} to a few times 102710^{27} gauss square centimetres, though their radii differ by a factor of a hundred thousand. Ordinary pulsars and the Sun’s global dipole sit lower. The cloud core that makes a star carries a thousand to ten thousand times more than any of them.

The overlap is the observation behind the fossil-field hypothesis. Three kinds of object that have nothing in common but a history of contraction carry the same magnetic flux within the scatter of each class. A field frozen into a contracting star predicts exactly that; a field made afresh in each object, by a mechanism that depends on its own size and rotation, has no reason to. The simplest reading is that one population of magnetic stars, a tenth or so of each generation, carries the same flux through its main-sequence life, its white-dwarf end and, for the most massive, its core collapse.

A population with a gap in it

The fossil fields have a second property that a dynamo would not give them. Among the hot, radiative stars, the magnetic ones do not grade smoothly into the non-magnetic ones. Surveys of Ap stars find almost no dipole fields weaker than about three hundred gauss, even though fields of a few gauss would be easily detected; below that there is a desert, and then, in a handful of bright stars measured with great care, fields of under a gauss — Vega’s is about half a gauss. The distribution is bimodal: a tenth of stars with strong, ordered fields, and the rest with nothing or almost nothing.

That gap has a natural explanation in the stability argument. A frozen field in a rotating radiative star is stable only if its energy is large enough to resist being wound up by the star’s own differential rotation; a weaker one is sheared into a tangle of small, alternating patches whose net flux is nearly zero. So a star emerges from formation with either a field strong enough to organise itself or one that is shredded, and nothing in between survives. The gap is at roughly the field strength where the magnetic energy and the energy of the rotational shear are comparable.

The same bimodality appears one step down the sequence. Ordinary radio pulsars, with fields of 101210^{12} gauss, carry a hundred times less flux than the magnetars and sit below the fossil band in the second figure. Whether they are the collapsed cores of the nine stars in ten that carried no strong fossil, with their modest fields made by the collapse itself, or fossils whose fields have partly decayed in the crust over millions of years, is not settled, and the two readings predict different trends of field with age that the pulsar population only partly shows.

The ratio a collapse cannot move

There is a second quantity that flux freezing conserves, and it is the one that sets what a fossil field can do. A star’s magnetic energy is roughly the field’s energy density times its volume, about B2R3/6B^2R^3/6. Its gravitational binding energy is about GM2/RGM^2/R. With the flux πR2B\pi R^2 B fixed, the field goes as R−2R^{-2} and the magnetic energy as R−1R^{-1} — the same power as the binding energy. Their ratio is independent of radius.

The one ratio a frozen collapse cannot change. The ratio of a star's magnetic energy, about B²R³/6, to its gravitational binding energy, about GM²/R, on a logarithmic scale, against radius. Both energies grow as one over the radius when a star contracts with its flux frozen in, so their ratio is constant: the horizontal line follows a 2.5 solar-mass star with a 3 kG field at three solar radii from ten solar radii down to neutron-star size, and the ratio stays at 1.7·10⁻⁹ throughout. The bars are the observed classes: the Sun's dipole, 10⁻¹⁷ to 6·10⁻¹⁷; Ap and Bp stars, 10⁻¹¹ to 10⁻⁷; magnetic white dwarfs, 3·10⁻¹⁹ to 3·10⁻⁷; radio pulsars, 5·10⁻¹⁵ to 5·10⁻¹¹; magnetars, 5·10⁻⁹ to 5·10⁻⁷. Even the magnetars, with the strongest fields known, hold magnetic energy of about a millionth of their binding energy. A collapse makes a field enormous; it cannot make it important, unless it was important already or something other than the collapse amplifies it.
Fig. 3 The ratio of magnetic energy to gravitational binding energy against radius. A star contracting with its flux frozen in moves along a horizontal line: the dashed line follows a 2.5 solar-mass star with a 3 kilogauss field from ten solar radii to neutron-star size, and its ratio never changes. The bars are the observed classes. Even magnetars hold magnetic energy of about a millionth of their binding energy, far below the dashed ceiling where the field would rival gravity.

That is a strong constraint, and it cuts against intuition. A hundred trillion gauss is the strongest field known in the universe, strong enough to distort atoms into needles and to split photons in two. It is still, relative to the gravity holding the star together, a millionth part. A collapse that shrinks a star by a factor of a hundred thousand multiplies its field by ten billion and its field’s energy by a hundred thousand, and multiplies its binding energy by exactly the same factor. The field is no more important to the star’s structure after the collapse than before.

It follows that any class of stars in which magnetic energy is dynamically important must have been born with an important field, or amplified it after the collapse by something other than contraction. The magnetars’ ratio, a few times 10−710^{-7}, is at the top of what the Ap stars carry and a hundred times the Ap stars’ typical value. If magnetars are fossils, they are the fossils of the most strongly magnetised massive stars. If they are not, something added energy to their fields, and the only reservoir large enough is rotation.

Where the cloud’s flux went

The flux ledger has one very large entry that the stars do not account for.

The flux ledger, from a cloud to a remnant. Magnetic flux at each stage of a massive star's life, on a logarithmic scale: cloud core, 2.2·10³⁰; young star, 2 kG at 2 R☉, 1.2·10²⁶; Ap star, 3 kG at 3 R☉, 4.1·10²⁶; white dwarf, 10⁸ G, 1.5·10²⁶; magnetar, 3 × 10¹⁴ G, 1.4·10²⁷ gauss square centimetres. The radius shrinks by about 11 orders of magnitude from the first bar to the last. The flux drops by a factor of 2·10⁴ between the cloud and the young star, the magnetic flux problem of star formation, and then stays within a factor of a few through the star's life and its collapse. Whatever happens to the flux happens before the star is made.
Fig. 4 Magnetic flux at five stages of a massive star’s life. The cloud core carries about 2×10302 \times 10^{30} gauss square centimetres; the young star, the main-sequence star, the white dwarf and the magnetar each carry 102610^{26} to 102710^{27}. The radius shrinks by eleven orders of magnitude from the first bar to the last. The flux falls by a factor of twenty thousand in the first step and stays within a factor of ten through all the others.

The first step is the magnetic flux problem of star formation, and it is solved, in outline, by processes that only work while the gas is cold and mostly neutral. In a molecular cloud the field is tied only to the few ions — one in ten million particles — and the neutral gas can drift through the field lines under gravity, slipping out of the field’s support. Later, in the disc around the forming star, turbulence and reconnection shed flux as well. Once the gas is hot enough to be thoroughly ionised, as it is in any star, those escape routes close. From then on the flux is locked in, and the star keeps what the formation process left it.

That makes a fossil field a record of star formation, not of stellar evolution. Why one star in ten emerges with a strong, ordered field while nine emerge with almost none is a question about the cloud, the disc and the first few million years, and it is not answered by anything that happens afterwards. The fossil hypothesis moves the origin problem back to the stage where flux is lost, which is also the stage where the observations are hardest.

A core that needs a white dwarf’s field

A neutron star is not made from a whole star. It is made from the iron core at the centre of a massive star, which is a few thousand kilometres across when it collapses. The flux that matters is the core’s, and the core’s field is hidden beneath a star’s whole envelope. What it has to be can be calculated backwards from the neutron star.

The field a collapsing core needs to make a given neutron star. The mean field an iron core must carry before it collapses, on a logarithmic scale in gauss, against the core's radius before collapse, for neutron stars of 10¹², 10¹⁴, 10¹⁵ gauss at twelve kilometres. An iron core is a few thousand kilometres across when it collapses, and its interior is the remnant of a star's whole burning history, so the relevant comparison is not the star's surface but its core. For a core of 3000 km, a 10¹⁴ G magnetar needs 2·10⁹ G in the core, and an ordinary 10¹² G pulsar 2·10⁷ G. Those are the fields of magnetic white dwarfs, or just above them — the other object made of a stellar core — and that match is the fossil-field hypothesis's strongest numerical argument.
Fig. 5 The mean field a collapsing core must carry to make a neutron star of a given field, against the core’s radius before collapse. For a core of three thousand kilometres, a magnetar of 101410^{14} gauss needs about 2×1092 \times 10^9 gauss in the core, and an ordinary pulsar about 2×1072 \times 10^7. The shaded band is the range of fields seen on magnetic white dwarfs.

The match is the most striking number in the subject. An iron core and a white dwarf are the same kind of object — the degenerate, burnt-out centre of a star, held up by the pressure of electrons that cannot share a state — at similar sizes and densities. The fields a core needs to make the observed range of neutron stars are the fields that the observed range of white dwarfs actually carry. If a tenth of white dwarfs are strongly magnetic because a tenth of stellar cores are, then a tenth of core collapses should produce strongly magnetic neutron stars, and that is roughly the magnetars’ share.

The rival: a field made in the first minute

The alternative explanation does not deny flux freezing; it adds an amplifier. A newly formed neutron star is hot, convecting violently for the first few tens of seconds, and — if its core was rotating fast before collapse — spinning at a millisecond period. Convection and rapid rotation together make a dynamo of enormous power, and calculations show that such a dynamo could raise the field to 101510^{15} gauss or more in seconds, drawing on the star’s rotational energy, before the convection dies away. The field is then frozen in as the star cools.

The energy ratio decides what this requires. A field at 101510^{15} gauss has a magnetic energy of a few times 104710^{47} ergs; a neutron star spinning at a millisecond has a rotational energy of about 105210^{52}. There is plenty of energy. What there may not be is enough rotation: most massive stars’ cores are thought to be slowed by magnetic coupling to their envelopes before they collapse, and neutron stars are generally born turning more slowly than a millisecond dynamo needs. Magnetars today spin in seconds, having lost their rotation to the very field in question, so their birth spin cannot be read off.

The two explanations predict different things about magnetars’ surroundings. A dynamo-born magnetar would have poured its rotational energy, 105210^{52} ergs, into its supernova remnant in the first hours; the remnants around known magnetars show ordinary explosion energies, which argues against it for those objects. A fossil magnetar needs a strongly magnetic massive progenitor, and those exist: a few per cent of O and B stars have measured fields of the Ap type.

The third possibility: two stars made one

There is a mechanism that makes a strong fossil field without needing to inherit one from the cloud, and it has recently moved from speculation to observation. When two stars merge — which a significant fraction of massive stars in close binaries eventually do — the merger is a violent, differentially rotating event that can generate a strong field by dynamo action and then leave it frozen into a radiative star. Simulations of such mergers produce stars with ordered fields of the Ap type. A massive star in a nearby cluster has been found that looks younger than its neighbours, is magnetic, and is surrounded by a nebula of the kind a merger ejects.

For white dwarfs the case is stronger. Magnetic white dwarfs are almost never found in wide binaries with ordinary companions, while non-magnetic white dwarfs often are — as though whatever made a white dwarf magnetic also removed its companion. Mergers inside a shared envelope, or of two white dwarfs, would do exactly that. On this reading, the flux band the three classes share is a property of mergers, and the fossil is younger than the star.

Why the Sun is not a fossil

The Sun’s global field is a different kind of object, and the figures show it in a different place. Its dipole is one or two gauss, a flux of about 102210^{22} gauss square centimetres, a thousand times below the fossil band. It reverses every eleven years, on the schedule the sunspot butterfly records, and it is made continuously by a dynamo in the convective envelope, where the flow is turbulent and the field is regenerated faster than it decays. Convection destroys an ordered fossil: whatever field the Sun was born with was mixed and replaced within its first few million years.

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 = 10²², 10²⁴, 10²⁶ 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.
Fig. 6 The same diagram with flux lines two orders of magnitude lower. The Sun’s global dipole sits on the lowest line, and its field is regenerated rather than inherited. A white dwarf formed from a star like the Sun, if the flux of its dipole were frozen in, would have a field of about ten thousand gauss — weak for a magnetic white dwarf, and near the limit of detection.

So the fossil and the dynamo are not competing explanations for the same stars. They apply to different interiors: a convective envelope makes and unmakes fields on the timescale of a cycle, a radiative interior keeps whatever it was given. Stars below about one and a half solar masses have convective envelopes and dynamo fields that scale with their rotation; stars above it are radiative outside their cores and are either fossils or nothing. A dynamo can erase any seed it starts from, which is why the Sun’s field carries no memory of its birth and an Ap star’s carries little else.

What the figures cannot show

Every field in these figures is a surface field, a single number standing for a structure. The Ap stars’ fields are measured from the splitting and polarisation of spectral lines, averaged over a visible hemisphere; white dwarfs’ from line splitting in fields so strong the atomic structure is rebuilt; neutron stars’ from spin-down, which measures only the dipole at the equator and is blind to anything smaller or buried. Interior fields — which carry the energy and set the flux — are unobserved in every class.

Stable fossil fields also have to exist, which is less obvious than it sounds. A purely poloidal field, or a purely toroidal one, is unstable inside a star and decays in a few Alfvén crossing times, which for an Ap star is years. Only mixed configurations — a twisted torus of field lines wrapped around a poloidal loop — have been shown in simulations to settle and persist. That a fossil field can survive for a star’s life is therefore a result of those simulations, not an assumption, and it is one of the reasons the hypothesis became respectable.

And the flux band is a band, not a line. Its width of two orders of magnitude within each class is as large as the gaps between them, so the overlap is suggestive rather than decisive. A population that shares a flux distribution is what fossil fields predict; it is not, by itself, proof that any individual magnetar was a magnetic massive star.

Still open: which magnetic white dwarfs are fossils

The three candidate origins — inherited from the cloud, amplified at a core’s birth, or made in a merger — make different predictions for the populations, and white dwarfs are where they can be separated, because white dwarfs can be dated. A white dwarf’s cooling age says how long ago it formed, and its mass says roughly what its parent star was. Recent surveys have found that strongly magnetic white dwarfs are rare among the youngest and common among older ones of the same mass, as though fields emerged gradually — which neither simple fossil nor merger theory predicts, and which has been attributed to a dynamo driven by the crystallisation of the white dwarf’s own core, the phase change that also delays its cooling. If that is right, some magnetic white dwarfs are neither fossils nor mergers but made late, and the flux band shared with Ap stars and magnetars is partly a coincidence of three mechanisms that happen to reach the same flux.

Where this goes next

The field a collapse would arrive with, and the one it has. Field strength against hydrogen density, both logarithmic, over the eight decades between diffuse gas and a protostellar core. The steeper line is what perfect flux freezing demands: a sphere collapsing conserves both mass and flux, so B goes as R⁻² while ρ goes as R⁻³, and therefore B goes as the two-thirds power of the density exactly — an exponent with no free parameter in it. The shallower locus is what Zeeman measurements find: a flat branch at about 10 µG up to 300 per cubic centimetre, where the density is rising and the field is not, and a rise as the 0.65 power of the density above it. By the density of a core the two differ by a factor of 1, and a star built at the frozen-flux value would carry a field four orders of magnitude beyond anything measured on one. The flat branch is the important half: it says the gas is moving along field lines without dragging them, which is what gravity does to a cloud that is still magnetically supported, and it locates where the freezing has to break. A field that would have arrived ten thousand times too strong A collapsing cloud carries its magnetic field with it, and the arithmetic of that is not negotiable — the field grows as the two-thirds power of the density. Run it from a diffuse cloud to a protostellar core and the answer is four orders of magnitude above anything measured on a young star. Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway. A clock that runs down and says what it is A pulsar hands over two measured numbers, a period and its derivative. Everything else usually quoted about it — an age, a magnetic field, a luminosity — is derived from those two through a model, and the one measurement that tests the model refutes it. Thirty e-foldings erase the memory of a seed. Field strength against time for three seed fields 8 orders of magnitude apart, amplified at one e-folding every 3·10⁸ years — a galactic dynamo's measured turnover rate — and stopped at the 3·10⁻⁶ gauss the disc actually has. In 10 billion years the budget is 33 e-foldings, which is a factor of 3·10¹⁴. That is the finding: the three tracks reach the same ceiling within 5.5 billion years of one another, so the field a galaxy has today carries essentially no information about the field it started with. Any seed above about 10⁻²⁰ gauss will do, and mechanisms that produce far less than that are the only ones ruled out. The measurement that does constrain a seed has to be made where no dynamo ever ran, which means the voids between clusters — and the limit there comes from gamma rays that never arrived. The seed that cannot be remembered A galactic dynamo affords something like thirty e-foldings over the age of a galaxy, which multiplies any seed field above a ten-thousandth of a billionth of a microgauss up to the microgauss actually observed. The field a galaxy has today therefore says nothing about the field it started with, and the only place a seed survives unamplified is the emptiness between clusters.

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Ap starsFlux freezingFossil fieldMagnetarMagnetic fluxMagnetic reynolds numberStellar mergerWhite dwarf