The companion survives, and it is moving

Most massive stars are in binaries, and most massive stars explode — and which kind of explosion is told from which line is missing rather than from the star’s own history. Those two facts together mean that the commonest fate of a close pair is for one member to be removed abruptly, and the question of what happens to the other has a clean answer with almost no astrophysics in it.

The criterion, in one line #

Treat the explosion as instantaneous. That is a good approximation: the ejecta leave at ten thousand kilometres a second and cross the orbit in hours, while the orbital period is days to years.

Before the explosion the pair is bound: the kinetic energy of the relative motion is half the magnitude of the potential energy, which is the virial statement for a circular orbit. Afterwards the positions and velocities are unchanged and the total mass is smaller, so the potential energy has shrunk while the kinetic energy has not.

The pair remains bound if the new total energy is still negative, and working that through for an initially circular orbit gives a strikingly simple condition: the system is unbound if more than half the total mass is lost.

That is worth pausing on. It does not depend on the separation, on the period, or on which star exploded. It is a statement about a ratio, and it comes from the fact that a circular orbit sits at exactly the energy where halving the mass takes it to zero.

What the survivor takes with it #

Suppose the pair is unbound. The companion leaves with the velocity it had at that instant, which is its orbital velocity about the common centre.

That velocity is set by the pre-explosion orbit: it is the mass of the exploding star divided by the total, times the relative orbital speed. For a close pair of massive stars the relative speed can be several hundred kilometres a second, so the survivor can leave at a hundred or more — far above the velocity dispersion of the disc it was born in — and far above the escape speed from the cluster that made it.

Such stars are called runaways, and there are enough of them to constitute a population: a few per cent of O and B stars have space velocities above forty kilometres a second, which for stars of those masses and lifetimes means they cannot have been born where they are.

The kick, which is the part that is not bookkeeping #

The mass-loss argument is exact and it is not the whole story. A neutron star born in a core collapse receives a kick, from the asymmetry of the explosion itself, of a few hundred kilometres a second.

The evidence for that is direct: young pulsars have space velocities averaging four hundred kilometres a second, far more than the mass-loss argument alone can produce, and their proper motions point away from the remnants they were born in. The kick changes the criterion in both directions. A kick aligned with the star’s orbital motion can unbind a system that mass loss alone would have left bound; one directed against it can bind a system that mass loss would have unbound, and can even leave the pair on a tighter orbit than before.

That second case is what produces the double neutron stars — pairs in which both components exploded and the system survived twice. Their existence is evidence that kicks are sometimes small, and the distribution of their orbital eccentricities is the sharpest constraint on the kick distribution available. Several of them contain a recycled pulsar, spun up by the transfer that preceded the second explosion.

The eccentric case, and why it is a probability #

The half-the-mass criterion is exact for a circular orbit and only for a circular orbit, and real pre-explosion binaries are not all circular.

For an eccentric orbit the survivor’s speed at the moment of the explosion depends on where in the orbit it is. Near pericentre it is moving fast and is harder to unbind; near apocentre it is moving slowly and is easier. So a given amount of mass loss unbinds the system with a probability set by the fraction of the orbital period spent in the vulnerable part — which by the equal-area law is weighted toward apocentre.

The practical consequence is that the criterion becomes a distribution. A system losing slightly less than half its mass survives if the explosion catches it near pericentre and does not otherwise, and the branching ratio is computable from the eccentricity alone.

Most close massive binaries are close to circular, because tides circularise them on timescales short compared with their lives, so the sharp criterion is a good approximation for the systems that matter. The eccentric cases are the wide ones, and those are the ones that unbind most easily anyway.

Eccentricity is only half of what sets the branching ratio, though, and the other half is the mass ratio the pair had before anything exploded. The criterion is about the fraction of the system’s mass that leaves, so the same explosion is survivable in one pairing and fatal in another.

That is the sense in which the criterion is a statement about the pair rather than about the star: nothing in it refers to the explosion’s energy, and everything in it refers to how the mass was divided beforehand. A twenty-solar-mass star exploding beside a two-solar-mass companion almost always unbinds the pair; the same star exploding beside an eighteen-solar-mass companion frequently does not. The observed runaway fraction is therefore a statement about the distribution of mass ratios among massive binaries at birth, read out through an explosion, and it is one of the few handles on that distribution that does not require resolving the pair.

Reading the history backwards #

The survivor carries information about a system that no longer exists, and extracting it is the point of the exercise.

Its velocity gives the pre-explosion orbital speed, hence the separation for an assumed mass. Its rotation is often extremely fast, because it was spun up by tidal locking to a close orbit or by accretion during a prior phase of mass transfer. Its surface composition is often enriched in helium and nitrogen, because the material it accreted came from the interior of a star that had already burned hydrogen.

Three lines of evidence, all pointing at the same vanished companion.

The other channel #

Runaway stars have a second possible origin, and distinguishing them is a live problem.

A three- or four-body encounter in a young cluster can eject a star at high speed with no explosion involved at all — the same energy exchange that hardens a binary and throws out the intruder, operating on stars massive enough to matter — and the biggest stars die first, so the window in which a cluster can eject one dynamically is only a few million years wide.

The two channels make different predictions. The dynamical channel ejects stars in pairs and can eject binaries intact; it produces no chemical anomalies; and it works only in dense clusters, so the runaways should trace back to them. The binary channel produces single stars with anomalous compositions and fast rotation, and it can operate anywhere. Observationally both channels are populated. The clearest cases for the binary channel are runaways with a compact companion still attached — high-mass X-ray binaries that also have large space velocities, which is exactly what a system that survived the explosion but received a kick should look like.

What was actually measured #

Four quantities, and one of them was not measurable until recently.

Space velocities require proper motions, and for a star at two kiloparsecs a hundred-kilometre-a-second transverse motion is ten milliarcseconds a year. Before precise astrometry that was measurable only for the nearest examples; it is now measurable for tens of thousands of stars, which turned runaway identification from a case-by-case exercise into a population study.

Radial velocities come from spectra, and they are the easy half.

The pre-explosion orbit is never measured for the system in question, because it is gone. It is inferred from the survivor’s velocity under an assumption about the masses.

The kick distribution is inferred from the population of surviving systems — the double neutron stars, the X-ray binaries with measurable eccentricities — and it is the quantity with the largest uncertainty, because the systems that received the largest kicks are exactly the ones that did not survive to be counted.

The one system where all of it is visible #

Every strand of this essay comes together in a small class of objects: a massive star with a compact companion, a large space velocity, an anomalous surface composition and a fast rotation.

Such a system has necessarily been through the whole sequence. It formed as a close pair; the more massive star transferred its envelope to the other, which is where the composition anomaly and the fast rotation came from; the stripped star exploded, leaving a neutron star; the pair survived the mass loss and the kick, which is why the compact object is still there; and the recoil of the surviving system carries it away from its birthplace, which is why it is moving.

There are a few dozen of them. Each one is a complete reconstruction of a binary’s history from a single snapshot, and the constraint they place is not on any one step but on the joint probability of all of them — which is the quantity population synthesis calculations predict and the quantity that is hardest to check any other way. One caution goes with all of it. Every reconstruction above runs backwards from a present state to a history, and backwards reconstructions in dynamics are rarely unique. A star’s velocity is consistent with a range of pre-explosion orbits, its composition with a range of transfer histories, and its rotation with either. What makes the class useful is not that any individual object is solved but that the constraints are different for each of the three observables, so a population of a few dozen pins down parameters that no single system could.

That is the same logic the whole anchor runs on. A supernova is an event nobody sees the beginning of and a binary is a system nobody sees the end of, so every statement about how the two interact is assembled from what survived — and the survivors are a biased sample of a population that has to be modelled rather than counted.

There is a final asymmetry in the bookkeeping worth recording, because it decides which systems are ever seen at all. A runaway star from a disrupted binary is bright and long-lived: an O or B star travelling at a hundred kilometres a second remains visible for millions of years and crosses hundreds of parsecs, so the population accumulates. Its companion, the neutron star, is faint unless it happens to be a radio pulsar beamed toward the observer, and the two are no longer in the same place. So the two products of a single event are catalogued separately, by different instruments, with no way of pairing them except by tracing both back to a common origin — which requires proper motions for both and a birthplace for neither. A handful of such pairings have been made, by extrapolating a runaway star’s motion and a pulsar’s motion backwards to a common point inside a known association. Each is a complete reconstruction of one explosion, and there are fewer than a dozen.

Two further pieces of the accounting are worth setting down, and neither is about the star that leaves.

The kick’s direction, read off a spin #

The kick has been described as a velocity, and it has an orientation, and the orientation carries information about the mechanism that no speed can.

A neutron star’s spin axis is measurable. The radio emission is beamed and polarised, and the position angle of the linear polarisation sweeps through the pulse in a pattern determined by the geometry — so fitting that sweep gives the projected direction of the rotation axis on the sky. Independently, the remnant a pulsar was born in often has a visible symmetry axis, and some remnants show jets or elongations aligned with something.

Comparing the spin axis with the proper motion is then a measurement of whether the kick was along the spin or across it, and the answer for the objects where both are measured is that the two are aligned more often than chance allows.

That alignment is a constraint on timing rather than on strength. A kick delivered over an interval long compared with the star’s rotation period would be smeared over all azimuths and would average to a residual along the spin axis only weakly; a kick delivered in a fraction of a rotation would point wherever the asymmetry happened to be. Alignment therefore suggests either that the star was already rotating fast when the kick was delivered, so the transverse components averaged away, or that the same asymmetry produced both the spin and the kick.

The two explanations differ in what they say about the newborn star’s rotation, which is a quantity the explosion models predict and the observations constrain badly.

There is a complication that keeps the inference honest. The polarisation sweep gives the spin axis projected on the sky, and the proper motion gives the velocity projected on the sky, so what is compared is two projections — and two vectors can appear aligned in projection while being far apart in three dimensions. The statistical statement survives that; the individual cases do not.

A direction measured from a polarisation and a direction measured from a position over decades are compared to constrain what happened in a second, four hundred years ago, which is a fair summary of how this subject works.

That timing constraint is the one thing about the explosion mechanism that the surviving population supplies, and it is supplied by an angle rather than by a speed.

The pairs that stay together #

The essay has followed the systems that come apart. The ones that do not are a population in their own right, and they are how the surviving fraction is measured.

A binary that stays bound after one explosion consists of a normal star and a compact object on an orbit that the mass loss and the kick have left eccentric. Nothing much happens for a while. Then the surviving star evolves, expands, and begins to lose mass — either through a wind that the compact object captures a fraction of, or by filling its Roche lobe and transferring directly.

Material falling onto a neutron star or a black hole releases a substantial fraction of its rest energy as it arrives, so the system becomes an X-ray source. That is a high-mass X-ray binary, and there are of order a hundred known in the Galaxy.

Their properties are the record of the explosion that made them. The orbital eccentricity is what the mass loss and the kick left behind, and it is measurable from the timing of the pulsar if there is one. The space velocity is the recoil of the surviving pair, which is smaller than a runaway’s because the system carries more mass. And the orbital period says how close the pair was when the explosion happened.

Comparing the number of such systems with the number of runaways gives the branching ratio — what fraction of massive binaries survive the first explosion — and the answer is of order a fifth to a third, which is consistent with the mass-loss criterion plus a kick distribution of a few hundred kilometres a second.

The extreme survivors are worth drawing on their own, because they are the ones that carry the most information and the ones a criterion stated for circular orbits describes worst.

Systems in that corner of the parameter space are how the kick distribution is constrained from above. A pair that survived with an eccentricity near unity did so because the kick happened to point in nearly the right direction, and the fraction of surviving systems found at high eccentricity is therefore a statement about how often a kick is large. The pulsar–companion binaries with measured eccentricities above 0.8 are a handful, and every one of them is a data point in that argument.

The comparison is harder than it sounds, because the two populations are found by different means and have different lifetimes. A runaway is visible for as long as its star lives; an X-ray binary is visible only while the transfer is happening, which is a short phase. Correcting for that is where most of the uncertainty in the branching ratio comes from.

One thing the branching ratio cannot settle on its own is how much of the surviving fraction was never at risk. A binary wide enough that its components never interact loses mass from a star that has evolved on its own, and a binary close enough to transfer mass before the explosion arrives at the explosion with a mass ratio it was not born with — sometimes reversed, the original secondary having become the more massive of the two. The two channels produce survivors with different orbital periods and different companion masses, and the observed population is a mixture of them in unknown proportion. Untangling it needs the pre-explosion mass ratio, which is precisely the quantity the explosion destroyed, so the argument has to run through population synthesis rather than through any single system. What population synthesis needs in return is a kick distribution, a mass-transfer efficiency and a common-envelope parameter, none of which is measured independently, and the branching ratio is one of the numbers used to fit them. The reasoning is therefore circular at the level of the population and sound at the level of the individual system, which is why the essays in this direction keep returning to the handful of objects where every quantity is separately observable.

Where the ladder goes #

The next rung is the kick mechanism itself: what makes a core-collapse explosion asymmetric, and why the asymmetry is large enough to give a neutron star a velocity comparable to its own escape speed. That is a question about hydrodynamics inside a collapsing core, and the observational constraint comes entirely from the population described here.

The other direction is the double-explosion systems, where the bookkeeping runs twice and the survivors are the gravitational-wave sources. The rate at which those merge is one of the few predictions of binary evolution that can now be checked against a direct measurement, and the check does not currently agree.