A donor that grows as it is eaten

Every star that has appeared as a donor in this collection so far shrinks when it is stripped, or at least swells only slowly. A star on the main sequence, robbed of its outer layers, settles into a smaller, denser main-sequence star of the lower mass. A giant with a deep convective envelope grows as the third root of one over its mass, and that modest growth is enough to make it dangerous to a companion of similar mass. The white dwarf is a different object. Its radius is set not by a balance of heating and cooling but by a rule about how many electrons can share a state, and that rule makes a heavier white dwarf smaller. Take mass off it and it gets larger.

A binary containing two white dwarfs is therefore a binary in which mass transfer, if it starts, runs the opposite way to the ordinary case. The donor swells as it is eaten, the orbit widens to make room, the period lengthens rather than shortens, and the whole system drifts from minutes towards an hour over a billion years. Those systems exist — a few dozen are known, with helium-rich spectra and periods between five and sixty-five minutes — and their existence is not to be taken for granted, because the moment of first contact is, for most pairs of white dwarfs, a moment at which the transfer should run away.

A period that is a density #

The figure rests on one fact about Roche lobes that makes it almost free to draw. The lobe’s volume-equivalent radius is a fixed fraction of the separation, and the fraction depends only weakly on the mass ratio. Kepler’s third law ties the separation to the period and the total mass. Put the two together for a star that exactly fills its lobe, and almost everything cancels: the period comes out as a function of the donor’s mean density alone, $P \propto \bar\rho^{-1/2}$, with a weak dependence on the mass ratio. A lobe-filling star’s period is a measurement of how dense it is.

That turns the direction of the evolution into a question about the mass–radius relation. For a main-sequence star below a solar mass the radius goes roughly as the mass to the 0.8, so the density rises as mass is lost and the period falls; cataclysmic variables, with main-sequence donors, evolve towards shorter periods for most of their lives, down to about eighty minutes, where the donor stops being a hydrogen-burning star. For a white dwarf the radius goes roughly as the mass to the minus one-third, so the density falls as the square of the mass and the period, from the density relation, rises as one over the mass. Halve the donor and the period doubles. The 0.7 M☉ accretor in the figure was chosen as typical, and changing it barely moves either curve, because the dependence on the mass ratio is weak.

The degenerate star’s answer #

The logarithmic response of a donor’s radius to its mass is the number the stability of transfer turns on, and for a white dwarf it can be read straight off the mass–radius relation.

Over the range that matters for donors, a few hundredths to half a solar mass, a white dwarf is a non-relativistic degenerate electron gas, and such a gas is an $n = 3/2$ polytrope, whose radius goes as the mass to the minus one-third. The coincidence with the convective giant is exact rather than approximate: a fully convective envelope is an ideal gas on an adiabat with exponent five-thirds, a non-relativistic degenerate gas has a pressure that goes as the density to the five-thirds, and both are therefore the same polytrope. The two kinds of star answer a loss of mass in the same way, for the same mathematical reason and for quite different physical ones.

The resemblance ends at the two extremes. Heavy white dwarfs have electrons moving at nearly the speed of light, the effective index rises towards three, and the mass at which the star ceases to have a radius sends the response to minus infinity: near the Chandrasekhar mass a white dwarf losing a little mass would swell enormously. That limit is irrelevant as a donor, because the lighter star is the donor and heavy white dwarfs are always the accretors. At the other end, below a few hundredths of a solar mass, the ions’ electrostatic attraction begins to matter beside the electrons’ degeneracy, and the response climbs towards zero, on its way to the nearly flat mass–radius relation of a giant planet near a thousandth of a solar mass. The donors at the end of these binaries’ lives are heading into exactly that regime, and some of the longest-period systems may have donors closer to planets than to stars.

Two ways for the stream to land #

With the donor’s response at −1/3, the conservative stability criterion is the convective giant’s: a white-dwarf donor transfers stably onto a companion more than about one and a half times its mass. That would make most pairs of white dwarfs that come into contact stable, since most such pairs have quite unequal masses. It leaves out where the transferred gas goes.

In an ordinary interacting binary the stream from the inner Lagrange point swings round the accretor, collides with itself and forms a ring at the circularisation radius, which spreads into a disc that passes its angular momentum outward. The angular momentum carried by the stream ends up at the disc’s outer edge, and tides from the donor take it back into the orbit. Nothing is lost from the orbit, and the conservative calculation is right.

Two white dwarfs at contact are so close together that the accretor, though tiny, can be larger than the closest approach of the stream. Then there is no ring and no disc. The stream strikes the accretor’s surface directly, and the angular momentum it carried — the specific angular momentum of a circular orbit at the circularisation radius — is deposited in the accretor’s spin. The orbit loses it. Marsh, Nelemans and Steeghs worked out the consequence in 2004: the orbit’s response to the donor’s mass loss acquires an extra term, $2\sqrt{(1+q)\,r_h}$ with $r_h$ the circularisation radius in units of the separation, and it is destabilising.

The cost is large. A disc lets a white-dwarf donor transfer stably onto a companion about one and a half times its mass. Direct impact demands a companion four or five times its mass. The difference between the two is not a property of either star; it is whether the gas has room to circle before it lands.

A plane of pairs, sorted by fate #

Putting the two criteria and the geometry of the stream together sorts every pair of white dwarfs by what happens when the lighter one first fills its lobe.

The band in the middle is where the argument lives. Its pairs are stable if the gas forms a disc and unstable if it does not, and nearly all of them begin in direct impact, because the separation at contact is so small. So the question for them is not whether a disc can form — at contact it cannot — but whether the angular momentum dumped into the accretor’s spin comes back to the orbit before the instability grows. It can come back by tides: a spinning accretor raises a tide that the donor’s gravity torques, and if that coupling is strong enough the orbit recovers what it lost and the transfer behaves as if a disc were there. If the coupling is weak, the spin is gone, the overfill grows, and the pair merges within a few orbits.

The synchronisation timescale of a white dwarf by tides is not known to within orders of magnitude. The dissipation that converts a tidal bulge’s motion into heat is set by processes in a degenerate interior — the excitation and damping of internal waves in the white dwarf’s non-degenerate outer layers — that are hard to compute and only recently constrained by the heating of white dwarfs in detached binaries. Whether a few per cent of the pairs that reach contact survive, or most of them, depends on that coupling, and the predicted number of helium binaries in the Galaxy swings by a factor of ten or more between the two assumptions.

A rate set by the waves the pair radiates #

Once a pair has survived contact, the donor is held at its lobe by a process that removes angular momentum steadily from outside: gravitational radiation. Two white dwarfs orbiting in minutes are among the strongest sources of gravitational waves in the Galaxy, and the waves carry off orbital angular momentum at a rate that rises steeply as the orbit shrinks. Each loss shrinks the lobe a little, the donor overfills it a little, and mass flows until the lobe and the donor are back in step.

The rate is the radiation’s angular-momentum loss divided by the gap between the donor’s radius response and its lobe’s. That denominator is the stability margin, so the rate runs fastest exactly where the margin is smallest — at contact, when the mass ratio is largest — and it is a reminder that a binary near its stability limit is a binary transferring fast. The early phase is brief and bright: six million years from a fifth of a solar mass to a twentieth, at rates high enough that the accreted helium on the surface can ignite. The late phase is long and faint: more than a billion years spent at periods of forty minutes to an hour, transferring a millionth of a solar mass every million years.

That is why the known systems are distributed as they are. Most sit at long periods with faint, quiescent discs. Between about twenty and forty-five minutes the discs flare in outbursts like those of the dwarf novae, and between ten and twenty they are held permanently bright by the higher supply. A handful at the shortest periods are bright in X-rays and the ultraviolet, and the two shortest — at 5.4 and 9.5 minutes — show no disc at all and an X-ray spot that brightens once an orbit, which is what direct impact looks like from outside.

Two populations at the same period, moving in opposite directions #

The shortest of those systems is the place where the argument turns into something measurable. At 5.4 minutes, the orbit of the binary known as HM Cancri is shrinking: its frequency rises by a measurable amount from year to year, at about the rate gravitational radiation alone predicts. That is the opposite of what a mass-transferring helium binary should do, since stable transfer from a degenerate donor lengthens the period.

Two explanations are possible, and they correspond to the two sides of the stability boundary. The pair may not yet be transferring mass at all — a detached double white dwarf spiralling inwards under radiation, its X-rays from something other than accretion — and the period will reverse when contact is reached. Or it may be at the very start of transfer, in direct impact, with the orbit still dominated by the radiation’s pull and the donor only beginning to respond. In the second case HM Cancri is a system in the shaded band of the plane above, being decided now. A detached pair at seven minutes, ZTF J1539+5027, is known to be shrinking towards contact in the same way and should reach it in about 130,000 years.

The sign of the period derivative is the one observable that tells a pair approaching contact from a pair that has survived it, and it is measured from the arrival times of eclipses or X-ray pulses over years. A space gravitational-wave detector will measure it for thousands of these systems directly, from the drift of each one’s frequency — the chirp that gives a mass combination with no distance in it — and the population will divide into the ones climbing towards higher frequency and the ones sliding down.

The eclipsing few that weigh the donor #

The first figure turns a period into a donor mass through a mass–radius relation, and the relation is an assumption. A handful of helium binaries eclipse, and an eclipse gives radii from timing alone: the duration of the accretor’s disappearance and the shape of the ingress fix the donor’s size relative to the orbit, and the eclipse of the bright spot where the stream meets the disc fixes the mass ratio through the stream’s trajectory. Those measurements test the gauge directly.

The answer so far is that the donors are larger than cold white dwarfs of their masses. In the eclipsing system Gaia14aae, with a period of fifty minutes, the donor comes out near three-hundredths of a solar mass with a radius that would belong to a cold white dwarf of about half that mass. A donor that is too large for its mass is a donor that is still warm, or one that was never fully degenerate — a helium core stripped from a star before it had finished contracting. Either way its radius response is not the cold −1/3, its period at a given mass is longer than the curve in the first figure, and the masses inferred from periods alone are too small. The bias is largest at long periods, where most systems are.

That matters beyond the bookkeeping, because the mass gauge is how the donor population is counted. A period distribution read through a cold mass–radius relation gives one history of how these binaries formed; read through a warm one, it gives another, with the donors starting from less evolved stars. The eclipsing systems are few, but each is a calibration point for every non-eclipsing one, which is the same structure as a distance chain: a small number of direct measurements anchoring a large number of inferred ones.

The same donor around a neutron star #

Replace the accretor with a neutron star and the physics is the same with one change. The neutron star cannot accept gas faster than its Eddington rate, and at the shortest periods a degenerate donor supplies more than that, so the transfer begins non-conservative — through the accretor channel that raises the stability limit — and becomes conservative as the rate falls. These are the ultracompact X-ray binaries, with periods from eleven to about sixty minutes. The brightest, 4U 1820−30 in the globular cluster NGC 6624, has an eleven-minute period and a helium donor of a few hundredths of a solar mass.

Its period should be lengthening. It is measured to be shortening, slowly, and the most economical reading is not that the theory is wrong but that the binary is being accelerated by the cluster’s own gravity: a system near the centre of a dense cluster falls towards the core, and the changing Doppler shift of its orbit appears as a period derivative of either sign. That is a reminder about every period derivative in this essay. A measured change of period is the sum of the binary’s evolution and its acceleration along the line of sight, and the second is not always small.

What the figures assume #

The mass–radius relation drawn is for a zero-temperature white dwarf. Real donors are warm when contact begins, especially if contact comes soon after the common envelope that made the pair, and a warm low-mass white dwarf is larger than a cold one by tens of per cent. A larger donor fills its lobe at a longer period, transfers at a lower rate and responds to stripping differently while it cools. Some of the longest-period helium binaries have donors too luminous for cold white dwarfs, which is evidence that the donor’s thermal history matters.

The direct-impact term uses fitted forms for the circularisation radius and for the stream’s closest approach, from ballistic integrations of single particles in the Roche potential. A real stream has width and pressure, and at the highest rates it may puff up enough to wrap round the accretor rather than strike it. Everything here is also conservative: the helium that lands on the accretor is assumed to stay there. Helium accumulating on a white dwarf can ignite in a flash that blows much of it off, which makes the transfer non-conservative in bursts, with the accretor-loss channel that raises the stability limit rather than lowering it. Some models of the first flash after contact predict that it is powerful enough to be seen as a faint, fast thermonuclear transient — and transients of roughly that kind have been found.

A merger that is not an ending #

The pairs above the solid line merge, and they are not lost to observation. Two white dwarfs merging over a few orbits produce a single object whose nature depends on the combined mass. Light pairs make a helium-burning star, a hot subdwarf; pairs of helium and carbon-oxygen white dwarfs are the leading explanation for the R Coronae Borealis stars, hydrogen-poor giants that periodically vanish behind clouds of their own soot; heavy pairs may make a single massive white dwarf spinning fast, a neutron star by collapse, or — if the combined mass is near the Chandrasekhar limit — a thermonuclear supernova. Each of those populations has a rate, and each rate is a count of the pairs on the wrong side of a line that depends on whether a stream found room to form a disc.

Still open: how strongly a white dwarf’s spin is tied to its orbit #

The fate of every pair in the shaded band rests on a tidal coupling in degenerate stars whose strength is uncertain by orders of magnitude. The detached white-dwarf binaries found in the last decade, several with periods under twenty minutes, now give a handle on it: tidal heating warms the white dwarfs measurably, and the spin of a white dwarf can be measured from the broadening of its spectral lines. If the pairs that are spiralling in turn out to be spinning in step with their orbits already, the coupling is strong, the band survives, and the helium binaries are numerous; if not, most pairs merge at contact, and the known systems are the lucky few whose masses were unequal enough to need no help.