The least known number in binary evolution

A giant star filling its Roche lobe around a companion of half its mass is on the wrong side of the comparison that decides whether transfer settles or runs away. Its convective envelope swells as it is stripped, its lobe shrinks faster, and within a few orbits the transfer rate has climbed to the point where the companion can no longer swallow what arrives. The envelope flows round it instead. The companion is now orbiting inside the giant, in a gas that neither star owns, and it is losing orbital energy to the wake it raises in that gas.

What happens next is the most consequential event in binary evolution and the least computed. It lasts about a year, nothing has been observed during it with certainty, and three-dimensional simulations of it have not yet followed one from start to finish with the envelope fully removed. Its outcome is nonetheless needed for nearly every kind of close binary containing a compact object, and it is supplied, in every population model, by a two-line energy budget with one unknown number in it.

A budget with one line of credit #

Webbink wrote the budget down in 1984, and in its standard form it equates two energies. On one side is the binding energy of the giant’s envelope — the gravitational energy holding a mass $M_{\rm env}$ to a star of mass $M_d$ and radius $R$, written $G M_d M_{\rm env}/(\lambda R)$, with $\lambda$ a number of order one that encodes how centrally concentrated the envelope is. On the other is the orbital energy released as the companion, of mass $M_a$, falls from the initial separation $a_i$ to the final one $a_f$, with the giant’s core $M_c$ now its partner instead of the whole giant. A fraction $\alpha$ of the released energy goes into unbinding the envelope:

$$\alpha\left(\frac{G M_c M_a}{2a_f} - \frac{G M_d M_a}{2a_i}\right) = \frac{G M_d M_{\rm env}}{\lambda R}.$$

The donor filled its lobe when the spiral began, so its radius is a known fraction $r_L$ of the initial separation, and the budget solves for the final orbit in closed form:

$$\frac{a_f}{a_i} = \frac{M_c}{M_d}\left(1 + \frac{2M_{\rm env}}{\alpha\lambda\, r_L M_a}\right)^{-1}.$$

The two unknowns enter only as their product. That is convenient — one parameter instead of two — and it is also the reason the product cannot be measured by the outcome. Any combination of an inefficient ejection from a loosely bound envelope and an efficient one from a tightly bound envelope gives the same answer.

Nothing is paid until the orbit has shrunk by a factor of three #

The figure has a feature the formula hides. The released energy is not positive from the start. It is zero when $a_f = a_i M_c/M_d$: the companion has to spiral in by the ratio of the giant’s mass to its core’s — here from 118 solar radii to 34 — before the orbital energy of the pair it ends up in, core and companion, exceeds the orbital energy of the pair it started in, giant and companion. The envelope’s mass leaves with the envelope, and an orbit around a lighter partner at the same separation is less bound. The first factor of three in the spiral merely pays for the change of partner.

After that the release grows as one over the separation, which is a slow function, and the binding energy to be paid is fixed. So the final orbit is very nearly proportional to $\alpha\lambda$: the three crossings in the figure are at 1.29, 2.48 and 4.63 solar radii for products of a quarter, a half and one. A factor of four in an unknown is a factor of 3.6 in the separation, and five in the period.

The three curves are the same giant at three moments, and they show the second variable that sets the outcome. A giant on the red-giant branch grows as its helium core grows, by a relation tight enough that a core mass is a radius: the radius at contact rises from about six solar radii at a 0.2 M☉ core to 85 at 0.4. A larger giant at contact means a wider initial orbit and a less tightly bound envelope, so the same αλ leaves a wider final orbit. The core is also what the binary is later seen to contain — the giant’s core is the white dwarf — so the white dwarf’s mass in a post-envelope binary is a record of how large the giant was when the companion entered it.

Why it takes a year #

The budget says where the spiral ends. How fast it gets there is a separate question, and the answer explains why no common envelope has been watched from start to finish.

The companion inside the envelope is a mass moving through a gas at roughly its orbital speed, which is supersonic in the giant’s outer layers. It raises a wake behind it, and the wake’s gravity pulls back on it — the same drag that makes a massive body sink through a cluster of stars, operating here on gas instead. The drag force goes as the square of the companion’s mass and the local density, divided by the square of its speed, and the density of a giant’s envelope rises steeply inwards. So the decay accelerates as it proceeds. The outer orbits take years, the plunge through the bulk of the envelope takes days to weeks, and the end of the plunge — where the density is high and the drag would be ferocious — is reached only if the envelope has not already been heated and expanded out of the way.

That is what produces the stall in the simulations. The energy deposited in the first plunge puffs the envelope up, the density around the companion falls, the drag weakens, and the orbit stops shrinking at a separation set by how much gas is left near it rather than by the energy budget. The budget assumes the spiral continues until the account balances; the drag stops it when the gas has gone, whether or not the account balances. In systems where those two coincide the formalism is right. In systems where they do not, the final orbit is wider than the budget predicts and the remaining envelope is removed later, more slowly, by other means.

A massive star’s envelope is a harder debt #

Everything above is a low-mass giant engulfing a lighter companion. The same budget, applied to a massive star engulfing a neutron star or a black hole, runs into a problem of a different size.

The binding energy of an envelope is dominated by its innermost layers, where the gas is closest to the core. For a red giant of a solar mass the envelope is extended and loosely held, and $\lambda$ is of order one. For a supergiant of twenty solar masses the envelope is more centrally concentrated in its binding, and calculations from stellar models give $\lambda$ of a few hundredths for the deepest layers — a binding energy tens of times the naive estimate. A neutron star of 1.4 solar masses releasing orbital energy as it spirals towards a ten-solar-mass helium core cannot pay that debt before it reaches the core, unless the envelope is ejected from where the binding is weak and the deep layers are left behind on the core.

That is why the formation of the black-hole binaries now detected by their gravitational waves is argued over. If they formed through common envelopes, the envelopes had to be ejectable from massive stars whose binding energy the budget says is formidable, and the outcome is extremely sensitive to where exactly the core’s boundary is drawn. If they formed through stable transfer instead — possible for more systems once the transfer is allowed to be non-conservative — the orbits come out wider, and the merger times longer. The two routes predict different distributions of mass ratio and spin, and the detected population is beginning to be large enough to tell them apart.

The periods that come out, and the ones that are seen #

That record turns the budget into a prediction that can be compared with a population.

Hundreds of white dwarfs with low-mass main-sequence companions in orbits of a few hours to a few days have been found in spectroscopic surveys, and their white-dwarf masses, companion masses and periods are measured. They do not lie on one curve. At a given white-dwarf mass the periods scatter across a factor of ten or more, and fitting each system separately returns values of αλ that differ from one system to the next.

That is not a failure of the formalism, because nothing in it says αλ should be the same for every system. The structure parameter $\lambda$ certainly is not: a giant near the tip of its branch has a far more extended envelope than one just starting to climb, and calculations of $\lambda$ from stellar models give values from a few tenths to well over one depending on the star’s mass and radius at contact. Whether to count the envelope’s thermal energy as available to help unbind it changes $\lambda$ again, by a factor of two for extended giants. And the ionised hydrogen and helium in a giant’s envelope hold recombination energy — released when the expanding envelope cools enough for electrons to rejoin their nuclei — that is comparable to the binding energy of the outer layers. Whether that energy goes into lifting the envelope or is radiated away before it can matter is one of the questions the simulations are still answering.

The measured systems give a narrower statement than they seem to. Fits to the white-dwarf-plus-main-sequence binaries consistently prefer small values, αλ of a few tenths, with no clear dependence on the companion’s mass. But the systems are the survivors, and the curves above show what that selects: at each efficiency, the systems with the lightest cores do not survive at all.

Light companions and the merger line #

Survival depends on the companion as well as the giant.

A light companion releases little orbital energy per unit of separation — the release is proportional to its mass — so to pay for the same envelope it must fall further. A long fall ends close to the core, and a companion close to a core has a small Roche lobe. The lightest companions therefore need the most efficient ejection to avoid ending inside their own lobes, which is the opposite of the naive expectation that a small object slips through. A brown dwarf of a twentieth of a solar mass sits further still up this line, and several white dwarfs with brown-dwarf companions in orbits of an hour or two are known. Each of those requires a value of αλ that a heavier companion would not, and they are among the most constraining systems there are.

The merged systems are not lost to sight either; they become single stars with odd properties. A companion that merges with a giant’s core inside its envelope spins the envelope up, and rapidly rotating red giants — rotating far faster than any single giant should after the spin-down of its own expansion — are one population whose numbers are a count of the systems below the line.

The fourth power #

The consequence that has made this number urgent is downstream of a second common envelope.

Double white dwarfs are made by two episodes of mass transfer: the first star becomes a white dwarf, and later the second star, now a giant, engulfs it. The pair that comes out of the second envelope is two white dwarfs in an orbit of hours or less, and from then on the only thing that changes its orbit is gravitational radiation, which shrinks it at a rate that goes as the inverse cube of the separation. The time to merge goes as the separation to the fourth power.

Because the separation is proportional to αλ, the merger time goes as its fourth power, and a factor of two in an unknown becomes a factor of sixteen in a time. The age of the universe crosses the curves at values of αλ within the range the fits allow, so whether a given kind of pair merges within a Hubble time is decided by the uncertainty itself. Every predicted rate of thermonuclear supernovae from merging white dwarfs, every predicted count of the helium binaries whose donors grow as they are eaten, and every forecast of how many double white dwarfs a space gravitational-wave detector will hear in the Galaxy passes through this exponent. The same arithmetic applies, with neutron stars and black holes in place of white dwarfs and a massive star in place of the giant, to the compact binaries whose mergers are now detected — and the measured merger rates are one of the constraints on the efficiency that the electromagnetic samples cannot give.

A second bookkeeping #

There is a rival prescription, and the reason it exists is a measurement.

Several double white dwarfs have both masses measured, and working the history backwards through two envelopes runs into trouble with the first. The second envelope shrank the orbit as the energy budget says it should. The first apparently did not: the orbit after it must have been wide enough for the second star to grow into a giant of the right core mass before filling its lobe, and an energy budget with any efficiency below one shrinks it too far. Nelemans and collaborators proposed in 2000 to describe that first episode not by energy but by angular momentum — the departing gas carries away γ times the binary’s average angular momentum per unit mass — and found γ near 1.75 fitted the known pairs.

The figure shows what that number means in the other currency. For this system, γ = 1.75 leaves an orbit of 22 solar radii, which the energy budget would reach only if αλ were about twelve — more energy than the orbit could have supplied, unless the envelope was extraordinarily loosely bound. And the prescription is steep where the energy one is shallow: γ = 2, fourteen per cent larger, leaves an orbit ten times smaller. The two forms are not different measurements of one parameter. They are different hypotheses about what controls the outcome, both fitted to the same few dozen systems, and they diverge for systems unlike those. The first envelope in the double white dwarfs may not have been a common envelope at all, but a stable phase of transfer from a giant onto a companion that could accept it — the kind of transfer the non-conservative criteria make possible for more systems than the conservative one allows.

What the simulations have and have not shown #

Three-dimensional hydrodynamic simulations of common envelopes have been run since the 1990s and have improved greatly, and they agree on the first act. The companion plunges from the giant’s surface to a separation of a few to a few tens of solar radii within a few orbits, the drag transfers its energy and angular momentum to the envelope, and the envelope is spun up and puffed out. They disagree on, or fail to reach, the end. Most simulations unbind only part of the envelope, and the orbit stalls at a separation larger than the post-envelope binaries show, with much of the gas still bound and expanding slowly. Including recombination energy unbinds more; including the jets the companion might launch as it accretes unbinds more still; and the long phase after the plunge, in which the remaining envelope may be removed over decades or centuries, is too long for the codes to follow.

So the formalism is a parameterisation of an outcome the physics has not yet produced from first principles. Its value is that it gives the right shape — the proportionality between final orbit and αλ, the dependence on core mass, the merger line — and a single place to put the ignorance. Its danger is the same: a single number, fitted to the survivors, applied to systems that did not survive and so could not have contributed to the fit.

A year that is never seen #

There are now a handful of candidates for common envelopes caught in the act. Luminous red novae — transients brighter than ordinary novae, fainter than supernovae, red and cooling over months — are thought to be the ejection of an envelope at the end of a plunge, and one of them, V1309 Scorpii, was observed beforehand as an eclipsing binary whose period shortened over several years before it brightened ten-thousandfold in 2008. That is a merger rather than a survival, and it is the only case in which the pre-event binary is known. The light curves of these events carry the mass and speed of the ejected gas, and so in principle the energy budget of a specific system, but the inference runs through models of the transient’s radiation that are themselves uncertain.

The ejected envelopes of survivors are seen too, later. About a fifth of the central stars of planetary nebulae are close binaries with periods of hours to a few days — the nuclei of systems that have just come through a common envelope, their ejected gas still glowing around them. Those nebulae are preferentially bipolar or ring-shaped, with their axes perpendicular to the binary’s orbit where it can be measured, which is the geometry an envelope spun up by a companion and ejected mostly in the orbital plane should leave. Their masses and expansion ages give, for a few systems, the mass and the energy of the ejected envelope directly — the one place the right-hand side of the budget is observed rather than calculated.

Still open: whether αλ is one number or a function #

The fits to the survivors find small values with large scatter, and the scatter is the interesting part. If it is the structure parameter varying with the giant’s radius at contact, as stellar models say it must, then αλ is a known function of the donor’s state and the efficiency underneath it may be roughly constant. If recombination energy contributes to some envelopes and not others, the scatter is a measure of how much of that energy is kept. The test is a large sample of post-envelope binaries with measured masses spanning a wide range of white-dwarf mass, read against the radius each giant had at contact. Surveys combining astrometric distances with spectroscopic orbits are assembling such samples now, and the gravitational-wave detectors add the compact mergers at the other end of the mass range. Between them they should say whether the least known number in binary evolution is one number at all.