A kick that writes over the record of the pulses

A star losing mass slowly leaves its companions’ orbits larger but the same shape; one losing mass suddenly scrambles them. A star on the asymptotic giant branch does neither cleanly. It loses most of its envelope in a few hundred thousand years, in episodes a thousand years or so long after each helium flash, and an orbit feels each episode as slow or sudden depending on whether its period is shorter or longer than the episode. For orbits with periods of thousands of years, the episodes arrive as a train of jolts at random phases, and the eccentricity random-walks while the size settles almost exactly where slow loss would put it.

That prediction suggested a test. The companions of white dwarfs in wide binaries — stars hundreds to thousands of astronomical units away, whose orbits take thousands of years or more — should have their eccentricities stirred by an amount set by the number and sharpness of the pulses, relative to binaries of main-sequence stars at the same separations. Carried through, the test runs into something the pulses do not control.

Three regimes in one curve #

The binary in the figure loses half its total mass: a two-solar-mass star becomes a white dwarf of 0.6 beside a companion of 0.8. Most of the loss is modelled as twelve episodes, each a few hundred years of enhanced wind after a thermal pulse, and the last forty per cent as a final superwind lasting about ten thousand years. Each episode gives the orbit an eccentricity kick of random sign whose size is the episode’s mass fraction, reduced by a factor that falls exponentially once the episode lasts longer than about a sixth of the orbit.

Below a hundred and fifty astronomical units, where the period is under a thousand years, every episode is slow and the eccentricity is untouched to better than a part in a thousand. Between two hundred and a thousand, the thermal-pulse episodes become sudden one after another as the orbit’s period passes their width, and the random walk of twelve kicks of a few per cent each leaves a stirring of about a tenth. Beyond about fifteen hundred, the superwind too becomes sudden, adds a single large kick, and the stirring rises to a quarter or more.

That is the record the earlier argument expected the wide binaries to carry: a stirring that switches on at a separation set by the episodes’ width and climbs in steps set by their sizes. A sample of white-dwarf binaries at different separations would read it off as a function of distance, and the two steps would date the two kinds of episode.

A kick of a few metres a second #

The white dwarf does not leave its giant stationary. If the wind that carried off its envelope was even slightly asymmetric — a little more mass lost in one direction than the opposite one — the remnant recoils like a rocket. The arithmetic is short. A star losing 1.4 solar masses at a wind speed of about fifteen kilometres a second, with one per cent more momentum going one way than the other, leaves a 0.6 solar-mass remnant moving at about a third of a kilometre a second. Asymmetries of that size are expected from rotation, from the companion’s own pull on the wind, and from the magnetic fields of giants, and the outflows round evolved stars are often visibly lopsided.

A kick of a third of a kilometre a second is nothing to a binary at a few astronomical units, whose orbital speeds are tens of kilometres a second. It is a great deal to one at a thousand, where the speed after the loss has doubled the orbit is about eight-tenths of a kilometre a second. A kick changes the eccentricity by about 1.3 times the ratio of the kick to the orbital speed, for a random direction, and the orbital speed falls as the inverse square root of the separation, so the stirring grows as the square root of the separation — steadily, with no threshold. The dashed lines in the opening figure are that stirring for kicks of 0.05 and 0.2 kilometres a second.

The comparison is uncomfortable. A kick of fifty metres a second — a remnant moving at walking pace relative to the binary’s centre of mass — stirs the eccentricity more than all twelve thermal pulses do inside 250 astronomical units, and comparably beyond. A kick of two hundred metres a second stirs it more than everything the mass loss does, at every separation. The pulses’ record is written in the eccentricities; it is written over.

Why a giant’s wind would be lopsided #

Nothing requires a giant’s wind to be symmetric, and the evidence from the outflows round dying stars is that it usually is not. The nebulae that white dwarfs leave behind as their ejected envelopes glow under the remnant’s ultraviolet are round in a minority of cases; most are elliptical, many bipolar, some sharply pinched at the waist, with jets and point-symmetric knots. Those shapes are taken as the imprint of what shaped the outflow: the star’s rotation, which favours mass loss from the equator; a magnetic field, which can channel it; and above all a companion, whose gravity focuses the slow wind of a giant into a disc or a spiral in the orbital plane.

The spiral is the clearest case. A giant losing mass while a companion orbits it at tens of astronomical units sheds its wind into a rotating frame, and the outflow winds into an Archimedean spiral that has been imaged directly round several giants. A wind with that structure carries momentum in the orbital plane, and its net push on the giant is a sum over many windings, mostly cancelling — but not exactly. The residual is the kick.

For a giant’s wind, slow and pushed by radiation on dust at ten to twenty kilometres a second — far slower than the line-driven winds of hot stars — the rocket arithmetic turns a fractional asymmetry of one per cent into a third of a kilometre a second, and a tenth of a per cent into thirty metres a second. The difference between those two is the difference between a kick that writes over the pulses everywhere and one that leaves them a window. A tenth of a per cent is a very symmetric wind.

Where the pulses could still be read #

The question can be turned into a single curve: for each kick speed, the separations at which the pulses still dominate.

For kicks of a few metres a second, the pulses’ signature is readable from a couple of hundred astronomical units outward, as the original argument assumed. Above about ninety metres a second there is no separation at which it is. The test of the pulses is therefore conditional on the white dwarfs’ kicks being smaller than a tenth of a kilometre a second, and that is exactly what the wide binaries have been used to measure, with a result that points the other way.

The binaries a kick breaks #

A kick comparable to the orbital speed does not merely stir an orbit; it unbinds it.

The edge where half the binaries break moves inward as the square of the kick speed, from ten thousand astronomical units at a quarter of a kilometre a second to six hundred at one. Surveys of wide binaries from the astrometric catalogues have compared the separations of pairs containing a white dwarf with those of pairs of main-sequence stars, and found that the white-dwarf pairs thin out relative to the others beyond a few thousand astronomical units, by an amount that fits kicks of somewhat under a kilometre a second. If that reading is right, the kicks are several times larger than the ninety metres a second below which the pulses could be read, and the wide white-dwarf binaries’ eccentricities are a record of kicks.

That is a different test, and a more useful one than the original. The kick’s size is the measure of how asymmetric the star’s mass loss was, which is itself an open question about how giants shed their envelopes, and the eccentricities of the surviving binaries just inside the edge carry it in full, independent of the separations of the broken ones.

The widening every orbit undergoes #

One effect of the mass loss is not stirred at all, and it matters for the separations on which every comparison rests.

Every binary within a few hundred astronomical units doubles its separation, exactly, whatever the history of the loss. Beyond that, each sudden episode leaves the orbit slightly larger than a slow loss would, because a sudden loss always raises the orbit’s energy relative to the new well, and the excess accumulates over episodes rather than averaging away. The widest binaries end a quarter larger again.

So a white-dwarf binary at two thousand astronomical units today was a main-sequence binary at about a thousand. Comparing white-dwarf pairs with main-sequence pairs at the same present separation compares orbits that began at different separations, and since the eccentricities of main-sequence binaries themselves depend on separation, the comparison has to shift one sample by the expansion factor before any stirring can be read. The factor is known from the masses, and for most binaries it is exactly the slow one.

Eccentricity from an angle #

A binary wide enough to have a period of thousands of years cannot be followed round its orbit. What an astrometric catalogue gives for each pair is its separation on the sky and the difference in the two stars’ motions across it, both projected. The angle between those two vectors carries the eccentricity statistically: in a circular orbit the relative velocity is perpendicular to the separation, so the projected angle is usually large; in an eccentric one, much of the motion is along the separation, and the angle is often small.

The method has been applied to hundreds of thousands of main-sequence pairs, and it found that their eccentricities depend on separation: close to uniform at a hundred astronomical units, close to thermal — the distribution a population reaches when its orbits have been thoroughly randomised — at a few thousand, and more eccentric than thermal beyond. The same statistic applied to white-dwarf pairs, shifted by the expansion, measures how much more their orbits have been stirred than their main-sequence twins’.

How many binaries a stirring needs #

The signal is a few per cent of the probability, spread across the whole range of angles, and it needs many binaries.

Stirring of two or three tenths, the size a kick of a few tenths of a kilometre a second gives at several hundred astronomical units, needs a few hundred to a couple of thousand white-dwarf binaries in a band of separation. The astrometric catalogues already contain thousands of wide white-dwarf pairs, so that test is within reach. The pulses’ own signature, a few per cent at the separations where a tiny kick would let it dominate, would need tens of thousands, and only if the kicks were tiny.

Kicks everywhere a star leaves a remnant #

Remnants receiving kicks is not a strange idea; it is the rule at the other end of the mass range. Neutron stars are born moving at hundreds of kilometres a second, from asymmetries in the supernova that made them, and their kicks are measured from the companions left moving on wild orbits or set free and from the proper motions of pulsars. A white dwarf’s kick, if it is a third of a kilometre a second, is a thousand times smaller, and the mechanism is gentler — a wind’s recoil over hundreds of thousands of years, not an explosion’s over a second — but the logic is the same: the remnant carries away the imbalance of what it lost.

Kicks of a few kilometres a second have also been proposed for white dwarfs from where they sit in old star clusters, whose escape speeds are tens of kilometres a second and whose young white dwarfs appear less concentrated towards the centre than their progenitors should be. That evidence is indirect and contested, since the clusters’ own dynamics redistribute their stars. The wide binaries, which a kick of a fraction of a kilometre a second breaks, are a far more sensitive scale, and the two measurements can be compared directly.

The pulses’ effect and the kick’s add as two independent random walks in the eccentricity vector, so they are not exclusive; the kick is simply larger. The random walk from pulses is the same arithmetic as rounding error accumulating in a long calculation, and a single larger step from the kick dominates any number of smaller ones once it exceeds their root-sum-square.

The pairs that are pairs #

Every statistic here assumes that the two stars of a wide pair are bound to each other, and at thousands of astronomical units that is not automatic. Two unrelated stars at the same distance, moving with similar velocities through the Galaxy, can sit near each other on the sky by chance, and at large separations the chance pairs outnumber the real ones. They are weeded out by requiring the two stars’ parallaxes to agree and their proper motions to differ by no more than the orbital speed their separation allows, and the remaining contamination is estimated by counting pairs whose parallaxes or motions have been deliberately scrambled.

That weeding interacts with the measurement in an awkward way. A kick that leaves a white-dwarf binary barely bound, on a wide, eccentric orbit, gives it a relative velocity near the limit the selection allows; a kick that unbinds it gives it one slightly above. The binaries just inside the edge, where the kick’s eccentricity is largest, are the ones closest to being excluded, and an analysis that cuts too tightly on velocity removes exactly the signal. A velocity difference projected on the sky can be measured to tens of metres a second only for pairs within a few hundred parsecs, where the parallaxes and motions of white dwarfs are precise enough; beyond that, the test works on binaries already selected by the quantity it is trying to measure.

What the model leaves out #

The mass-loss history is a caricature: twelve equal Gaussian episodes and a superwind, when real thermal pulses differ in size, rise steeply and decay slowly, and last for different times in stars of different mass. The eccentricity kick per episode uses the circular-orbit result, which is right in size for moderate eccentricities but not in detail. The kick is applied once, instantaneously, after the loss, when in reality the asymmetric wind pushes the star throughout and the impulse is spread over the same episodes as the mass loss; for the widest orbits that makes no difference, and for the closest it softens the kick into something partly adiabatic. The companion is assumed to have lost no mass and to have accreted none from the wind, and the binary is assumed to be alone, with no encounters with passing stars or the Galaxy’s tide, which disrupt the widest pairs over billions of years regardless of what their white dwarfs did.

None of these restores the pulses’ record against kicks of a few tenths of a kilometre a second. The kick’s stirring grows as the square root of the separation and the pulses’ saturates; any plausible refinement of either leaves the kick ahead wherever the kick is large enough to break the widest pairs.

A record of the wind instead #

The original prediction is not wrong; it is overtaken. The random walk of an orbit’s shape under pulsed mass loss happens as described, and in a binary whose white dwarf received no kick it would be measurable. What the wide binaries offer instead is a cleaner measurement of something else: the asymmetry of a giant’s wind, read from the remnant’s recoil, through the separation at which the binaries break and the eccentricity of those that survive. That is a quantity the pulses never encoded, and one that bears directly on how giant stars lose their envelopes — whether the wind is shaped by rotation, by a companion or by magnetic fields — which the mass a star does not keep depends on and nothing else in the remnant records.

The pulses’ record survives where kicks cannot reach it: in orbits fast enough that no plausible kick matters but slow enough that the episodes are sudden. That window is between about a hundred and fifty and two hundred and fifty astronomical units for kicks of a few tens of metres a second, and it closes altogether for faster ones. It is the regime of the planets and debris belts of white dwarfs at tens to hundreds of astronomical units — too far for tides, near enough to feel the episodes — and it is where the pulses’ signature, if anywhere, will be found.

Still open: how fast white dwarfs are born moving #

The separation at which white-dwarf binaries thin out gives a kick speed of a few tenths of a kilometre a second only through a chain of assumptions: about the separation distribution of the progenitor binaries, the expansion, the disruption of wide pairs by the Galaxy, and the completeness of the catalogues at large separations. The eccentricities of the binaries that survive just inside the edge measure the same kicks independently, and with a few thousand pairs the stirring they predict is measurable. Whether the two agree — and whether the kicks depend on the white dwarf’s mass, as a recoil from an asymmetric wind would — is the measurement that turns the wide binaries from a test of the pulses into a measurement of how lopsidedly stars die.