Three clocks and a runaway

The first rung of this anchor established the valve: two stars close enough share a critical surface, the surface pinches at a point between them, and what comes through it changes the orbit — with a sign that reverses at equal masses.

That reversal is a statement about a derivative, and a derivative is not an outcome. Knowing that a valve opens further when the flow increases does not say whether the flow settles at a value or tears the pipe apart; that depends on how fast the opening responds compared with how fast the pressure behind it falls, and both of those are properties of the star rather than of the geometry.

So this rung asks two questions the first left open. Does the transfer settle or run away, and — separately, and with a quite different answer — how fast does it run? Whether the transfer settles down or runs away depends on a comparison, and the comparison is between two logarithmic responses to the same loss of mass: how the donor’s radius answers, and how its Roche lobe’s does.

Why it is exponential and not a threshold #

The feedback is worth writing down because it is two lines and it explains the shape of everything that follows.

Let Δ be the logarithmic overfill, $\ln(R/R_L)$. The rate at which material squeezes through the inner Lagrange point rises steeply with Δ — approximately as a power of it, since the flow is through a nozzle whose throat area grows with the overfill — so to a sufficient approximation

$$\frac{\dot M}{M} \;=\; -\frac{\Delta}{H\tau}$$

with τ the timescale of whatever is driving the transfer and H a small number setting how sharply the rate responds. Meanwhile the overfill itself changes as mass is lost, at

$$\frac{\mathrm{d}\Delta}{\mathrm{d}\ln M} \;=\; \zeta_{\rm ad} - \zeta_L$$

where the two ζ are the logarithmic derivatives of the donor’s radius and of its lobe. Combining them,

$$\dot\Delta \;=\; (\zeta_L - \zeta_{\rm ad})\,\frac{\Delta}{H\tau}$$

which is exponential growth or exponential decay according to a sign. The stability criterion is not a threshold the system crosses; it is the sign of an exponent, and once the sign is unfavourable the outcome is fixed and only the rate remains to be settled.

Where the donor’s response comes from #

The number that decides everything is a property of the donor’s interior, and it takes two values for two structurally different kinds of star.

A star with a deep convective envelope is very nearly an $n = 3/2$ polytrope, because a fully convective region has an adiabatic structure and a monatomic adiabat is exactly that polytrope. Polytropes of that index obey $R \propto M^{-1/3}$ — so removing mass makes the star larger. The response is −1/3, it is a mathematical property of the polytrope rather than a fit, and it applies to red giants and to low-mass main-sequence stars alike.

A star with a radiative envelope has no such simple structure, and removing mass from it exposes deeper, denser layers whose scale is smaller. The star contracts, and it contracts sharply: values between +1 and +4 appear in the literature depending on the mass and the evolutionary stage.

The distinction is therefore between two kinds of star and not a continuum, which is why the critical mass ratios in this essay come in two families rather than one range. It is also why the evolutionary state of the donor matters as much as its mass: a two solar-mass star is radiative on the main sequence and convective as a giant, and it is a different problem in each.

The three clocks #

Stability says which way the feedback points. It says nothing about how fast, and the available rates are not close together.

Those three are not points on a continuum. They correspond to three physically distinct situations, and the stability comparison decides which.

Nuclear. If the transfer is stable and the donor is driven only by its own slow expansion as it burns — the same expansion that makes a star swell because its centre shrank — the rate is set by the nuclear clock: 10⁻¹⁰ solar masses a year, a hundredth of an Earth mass per century. The companion accretes this comfortably, radiates it, and the system is a stable interacting binary that lasts for a nuclear time.

Thermal. If the donor is knocked out of thermal equilibrium — which happens when the transfer is stable adiabatically but not thermally, so it runs until the star has readjusted — the rate is set by the thermal clock: 10⁻⁸ to 10⁻⁶ solar masses a year. This is far more than the companion can radiate, so the arriving material piles up into an envelope around it.

Dynamical. If the adiabatic response is unfavourable, the overfill runs away and the rate climbs until it is limited by the dynamical time — not because anything intervenes, but because the star cannot deliver mass faster than it can rearrange itself. There is no accretion at this rate. The donor’s envelope engulfs the companion, and what follows is a common envelope.

There is a fourth possibility hiding in the list, and it is the one that produces the most spectacular objects. If the transfer is adiabatically stable but thermally unstable, the donor is out of thermal equilibrium and transfers at the thermal rate for a thermal time, and then stops — leaving a system in which the originally less massive star is now the heavier and the more evolved star is the lighter. That configuration was called the Algol paradox before anybody understood mass transfer, and it is the best evidence there is that any of this happens.

What the orbit does meanwhile #

The orbit is not a bystander. Transferring mass moves it, and which way depends on the same mass ratio the stability does.

The separation is not what an observer has. Kepler’s third law turns it into a period, which in an eclipsing system is measured to many decimal places, so the reversal above is in principle a watchable event — and the boundary the donor actually feels is not the separation either but its lobe, which is a fraction of it.

There is one more term in the orbital response that the conservative picture leaves out and that changes the sign of the answer for some systems. If the accretor is a compact object it can only take material through a disc, and a disc can only process what its own viscosity allows — anything above the Eddington rate is driven back out. Material leaving the system from near the accretor carries a large specific angular momentum, so it shrinks the orbit far more effectively than conservative transfer does, and a system that would have been stable becomes unstable. That is why the stability of transfer onto a black hole is a different calculation from the stability of transfer onto a main-sequence star of the same mass.

What a common envelope is #

The unstable outcome deserves a paragraph because it is the most consequential process in binary evolution and the least calculable.

Once the donor’s envelope has engulfed the companion, the companion is orbiting inside a gas cloud, and the drag it feels is the wake it makes rather than any friction with a surface. Drag removes orbital energy and angular momentum, the orbit shrinks rapidly, and the energy released heats and expands the envelope. Two outcomes are possible: the envelope is ejected before the two cores merge, leaving a very close binary; or it is not, and they merge.

Which happens is decided by an energy budget — the orbital energy released against the envelope’s binding energy — with an efficiency parameter that nobody can compute and everybody fits. The phase lasts of order a year, which is a thousandth of the thermal time and a ten-billionth of the nuclear one — so a process that determines the fate of a substantial fraction of all binaries occupies a vanishing fraction of their existence, and nothing has ever been observed in it. Essentially every close binary containing a compact object went through one, and the population of such systems is the main constraint on the efficiency, which makes the reasoning circular in a way the field is explicit about.

What decides between the outcomes, in practice #

Assembling the pieces gives a decision procedure with three inputs and no free parameters, which is worth stating because it is used in every population-synthesis calculation ever run.

Take the donor’s mass and evolutionary stage; that gives ζ_ad. Take the mass ratio; that gives ζ_L. Compare them. If ζ_ad exceeds ζ_L the transfer is adiabatically stable, and the question moves to the thermal comparison — the same test with the thermal-equilibrium radius response instead of the adiabatic one. If that is also favourable, the transfer runs on the nuclear clock; if not, on the thermal one. And if the adiabatic comparison fails, the transfer runs away and the outcome is a common envelope.

Three comparisons, two of them of the same form, and a decision tree with three leaves. What makes it uncomfortable is that the leaves are so different: the same binary with the mass ratio changed by ten per cent can end as a stable X-ray source lasting a hundred million years or as a merger lasting a year.

What is actually measured #

None of the rates above is observed directly for any system, and the honest accounting is short.

What is observed is a period, a set of radial velocities, a light curve where the system eclipses, and — for the systems with a compact accretor — an X-ray flux. Everything else is inference.

Transfer rates are inferred from accretion luminosity: material falling onto a compact object releases a known fraction of its rest energy, so a measured X-ray luminosity divided by that fraction is a rate. That works for the fast cases and gives values in the expected range. For the slow cases the accretion luminosity is negligible against the stars themselves and nothing is measurable.

Stability is not observed at all. What is observed is a population of survivors, and the inference runs backwards: systems that exist are ones that were stable, systems with very short periods are ones that had a common envelope, and the relative numbers constrain the criteria. The only stars whose masses are known are the eclipsing binaries, and the interacting ones among them are precisely the systems whose masses have been changed by the process being studied.

There is one class of system where a rate is measured rather than inferred, and it is worth naming because it is the exception. In a few cataclysmic variables the accretion disc’s brightness responds to the transfer rate on a timescale of days, and the long-term light curve therefore tracks it directly; and in a few eclipsing systems the period’s slow change gives the mass-transfer rate through the orbital response drawn above. Those give values around 10⁻⁹ to 10⁻⁸ solar masses a year, between the nuclear and thermal clocks, which is what the theory predicts for a low-mass donor driven by angular-momentum loss rather than by its own expansion.

The donor’s adiabatic response — the ζ that everything turns on — is a calculation, not a measurement. It comes from taking a stellar model, removing mass from it faster than it can readjust thermally, and asking how the radius changes. For a convective envelope the answer is close to the polytropic −1/3 and is robust; for a radiative one it depends on the envelope’s structure and published values span a wide range.

The generalisation #

The structure worth extracting is that a stability criterion and a rate are independent questions, and that confusing them is the commonest error in reading any feedback system.

The sign of the exponent decides the outcome; its magnitude decides when. A marginally unstable system and a violently unstable one reach the same end state, and the only difference is that the first takes longer — which matters enormously for whether anything else intervenes and not at all for the answer.

The same shape appears everywhere in this collection. A collision rate that needs no collision is a quadratic overtaking a linear: the crossing decides the outcome and the coefficients decide the date. A threshold with no free parameter in it is a criterion whose sign is the whole answer, and whose magnitude says only how quickly. And the mass a star does not keep is the single-star version of the same accounting, with a wind in place of a companion.

There is a second reading, about what makes a system’s fate computable. Everything decisive in this essay is a logarithmic derivative — how a radius responds to a mass, how a lobe responds to a mass ratio — and logarithmic derivatives are exactly the quantities that survive being ignorant of the scale. It does not matter what the star’s radius is; it matters how the radius answers. That is why the criterion can be stated once and applied to systems spanning ten orders of magnitude in size, and it is why the one genuinely uncomputable step — the common envelope’s efficiency — is the one that is a ratio of energies rather than a derivative.

The corollary is a habit. When a system has a threshold, integrate through it rather than quoting it. A threshold quoted as a number invites the reading that the system sits near it; integrating shows that a system near a threshold does not stay near it, because the feedback that defines the threshold is the same feedback that moves it away.

What the outcomes look like from outside #

It is worth naming what each leaf of the decision tree produces, because the observable populations are what the theory is tested against and they are very different objects.

The nuclear branch gives the long-lived interacting binaries: the cataclysmic variables, the low-mass X-ray binaries, the Algols after their fast phase. They last a nuclear time, so most of the interacting binaries in existence at any moment are in this state, and they are the ones with measured rates.

The thermal branch gives the systems caught in the act, which are rare in proportion to how short the phase is: a thermal time is a thousandth of a nuclear time, so a thousandth of the population. The Algol paradox is the fossil of a completed thermal phase rather than an observation of one in progress.

The dynamical branch gives nothing observable at all, and then gives everything: the phase lasts about a year, so no system has been caught in it, and its products are the close binaries whose separations are far smaller than either star’s original radius — the double white dwarfs, the cataclysmic variables’ progenitors, and the compact binaries whose mergers are detected as gravitational waves — where the chirp gives a distance with no ladder under it and, incidentally, a mass ratio.

The last of those is the reason this arithmetic has become urgent rather than academic. Every predicted merger rate runs through the common-envelope efficiency, and the measured rates are now precise enough to constrain it.

Where the ladder goes next #

The next rung takes the non-conservative case. Transfer is assumed conservative throughout the figures here — every gram that leaves the donor arrives at the companion, and the total angular momentum is unchanged — and it very often is not. Material leaving the system carries angular momentum away, at a rate that depends on where it leaves from, and that changes the orbital response and therefore the stability criterion. The two limiting cases, loss from near the donor and loss from near the companion, give critical mass ratios differing by a factor of two.

Further rungs on this anchor: the thermal-timescale phase, which is where the Algol systems are and which explains why the less massive star in those systems is the more evolved; the response of a degenerate donor, which expands as it loses mass and is therefore always unstable in the naive comparison and is stabilised by the orbit instead; the case where the accretor cannot accept the material and drives it out, which is its own feedback loop; and the common envelope’s energy budget, which is the least constrained number in binary evolution and the one that most of the interesting outcomes depend on.