Mass that leaves takes the orbit's spin with it

Whether mass transfer between two stars settles or runs away is the sign of an exponent, and the exponent is a difference of two logarithmic derivatives: how the donor’s radius answers a loss of mass, and how its Roche lobe does. The second of those was computed there on an assumption that is almost never true. Every gram that left the donor was assumed to arrive at the companion and stay, so that the binary’s total mass and total angular momentum were both fixed, and the lobe’s response followed from those two conservation laws alone.

Real accretors refuse most of what they are offered. A neutron star can take in only as much as its own radiation lets fall onto it, about a hundred-millionth of a solar mass a year, and a donor on its thermal clock supplies a hundred times that. A main-sequence star receiving material faster than it can adjust swells out of equilibrium. A white dwarf ignites what lands on it and blows it off again. In each case most of the transferred gas leaves the binary, and the question the conservative calculation never had to ask becomes the deciding one: from where?

A gram carries the spin of the place it left #

The orbit is a reservoir of angular momentum. For two masses $M_d$ and $M_a$ on a circular orbit of separation $a$ and angular velocity $\Omega$, it holds $J = \mu a^2 \Omega$, with $\mu$ the reduced mass. Anything that leaves the binary takes some of that reservoir with it, and the amount per gram is fixed by where it was when it left. Gas launched from the accretor’s surface was moving with the accretor round the centre of mass, at a distance $a\,M_d/(M_d+M_a)$ from it. Gas blown off the donor was moving with the donor, at $a\,M_a/(M_d+M_a)$. Gas parked in a ring outside both stars, at radius $\gamma a$, orbits the whole binary and carries the angular momentum of a circular orbit there.

Writing $q$ for the donor-to-accretor mass ratio and $\beta$ for the fraction of the transferred mass that is lost, the logarithmic response of the separation to the donor’s mass comes out of differentiating $J$:

$$\frac{d\ln a}{d\ln M_d} = 2\left[\, h\beta(1+q) - 1 + (1-\beta)q + \frac{\beta q}{2(1+q)} \right],$$

where $h$ is the lost matter’s specific angular momentum in units of $a^2\Omega$: $q^2/(1+q)^2$ from the accretor, $1/(1+q)^2$ from the donor, $\sqrt{\gamma}$ from a ring. Setting $\beta = 0$ recovers the conservative $2(q-1)$, as it must, and the figures here check that identity rather than assume it.

The three values of $h$ are the whole of the physics, and their sizes are easy to state. A light donor sits far from the centre of mass and the accretor close to it, so gas leaving the accretor carries very little angular momentum per gram — less than the binary’s average — and gas leaving the donor carries nearly as much as the donor’s own orbit. A ring carries more than either star, because it is further out than both.

The conservative orbit turns round at equal masses for a reason that has nothing to do with stars: at fixed total mass and angular momentum, the separation is smallest when the product of the masses is largest. Remove mass from the binary and the reason dissolves. Expelled from the accretor, the gas takes away mass but little angular momentum, so the remaining binary has more spin per unit mass than before and the orbit opens more readily; the turning point moves to a donor 1.28 times the accretor’s mass. Blown off the donor, the gas takes the donor’s own orbital angular momentum and the orbit widens throughout — it is Jeans’s case for a star losing mass in a wind, where the separation simply grows as the total mass falls. Fed to a ring, it takes more angular momentum than it brings, and the orbit closes whatever the masses are.

Two losses that both stabilise #

The surprising part is what the two stellar channels do to stability. The intuition that losing material should be dangerous — that a binary leaking gas is a binary in trouble — has the sign wrong. Both channels move the critical mass ratio up.

For a donor with a deep convective envelope, whose radius grows as the third root of one over its mass, conservative transfer is stable only if the donor is less than 0.63 of the accretor’s mass. Expel everything from the accretor and the limit rises to 0.96; blow it off the donor and the limit rises to 1.24. The donor has not changed. Its envelope answers the loss of a gram exactly as before. What has changed is the lobe’s answer, because the departing gas takes away mass while carrying less angular momentum per gram than the binary holds, and a binary that keeps more spin per unit mass has a wider orbit and a larger lobe than one that does not.

The two limits are often said to differ from each other by a factor of two. They do not: computed, loss from the accretor and loss from the donor give 0.96 and 1.24, a third apart. The factor of two is between either of them and the conservative case, which is the more useful statement, since conservative transfer is what a reader of a stability criterion usually has in mind and what the simplest population models assume. For a radiative donor the numbers are 1.72 conservatively and 2.25 with loss from the accretor, and loss from the donor pushes the limit past twenty, which is to say that a radiative star blowing off what it cannot pass on is effectively never dynamically unstable.

The asymmetry between the channels has a geometric reading. The accretor is the heavier star whenever the question is interesting, so it sits near the centre of mass and its surface gas is nearly at rest in the binary’s frame. The donor is lighter and sits far out, so its gas is carried round on a wide circle. A gram leaving the donor has more angular momentum than a gram leaving the accretor by a factor of $1/q^2$ — four, for a donor half the accretor’s mass — and the stability limit moves with it.

What the bookkeeping does to a whole evolution #

A stability criterion is a statement about an instant. Following the transfer through, the choice of channel accumulates.

The conservative track is the textbook one, and the integration reproduces its closed form — the separation varies as the inverse square of the product of the masses — to a part in ten thousand, which is the check that the step-by-step version is computing what it claims. The other three are what the textbook form cannot give. They begin from the same donor in the same orbit and remove the same 1.2 solar masses of envelope, and the final separations span a factor of nearly three. Periods go as the separation to the three-halves power, so the final periods span a factor of four. Nothing in the stars’ structure distinguishes them; only the line in the ledger where the departing angular momentum is written.

The loss through a ring deserves its own look, because it works with very little mass.

A ring is the most destabilising place a small amount of mass can go. Each gram in it carries $\sqrt{\gamma}$ times $a^2\Omega$ — at four separations, twice the specific angular momentum of the orbit itself and four times that of a light donor — and the binary must supply it. Losing ten per cent of the transferred gas this way removes a third of the convective donor’s stable range. The gas need not be launched into a ring deliberately. Material spilling through the outer Lagrange point behind the lighter star leaves at about 1.2 separations with the corotating angular momentum of that point, and some of it settles into a disc around the binary rather than escaping. Such discs are seen: many post-asymptotic-giant-branch binaries are surrounded by dusty circumbinary discs whose rotation has been measured by interferometry and in molecular lines, and their central binaries have eccentricities and periods that the conservative picture does not explain.

An accretor that cannot eat #

The cleanest example of non-conservative transfer is the one in which the accretor is a neutron star, because there the fraction lost is fixed by a known limit rather than by a free parameter.

A neutron star’s surface sits deep in its own gravity, so gas falling onto it releases about a tenth of its rest-mass energy, and the radiation from a hundred-millionth of a solar mass a year is enough to hold back further infall. A donor supplying more than that cannot have the excess accreted. It is driven off from near the neutron star — through a wind from the inner accretion disc, or along jets — and leaves with roughly the neutron star’s orbital angular momentum. That is the accretor channel, and it is switched on by the supply rate itself: below the limit the transfer is nearly conservative, above it nearly everything is lost.

The consequence is a class of binary that the conservative criterion forbids. An intermediate-mass star of three or so solar masses, with a radiative envelope, filling its lobe around a 1.4 M☉ neutron star, has a mass ratio above two. Conservatively it is past the limit of 1.72 and should engulf the neutron star. With the excess expelled the limit is 2.25, and the transfer is stable — fast, at the thermal rate, with almost nothing accreted, but stable. Such systems exist. Cygnus X-2 contains a neutron star and a donor of about half a solar mass that is far too luminous and too large for its mass, which is what a star of three or four solar masses looks like after it has been stripped to its core and part of its envelope; reconstructing its history gives exactly this path. Its present donor is lighter than the neutron star, and its past one was more than twice as heavy.

SS 433 may be the same process caught in the act. A massive donor feeds a compact object far above any Eddington rate, and the excess leaves in two jets moving at a quarter of the speed of light, precessing every 162 days, with a mass-loss rate near $10^{-4}$ solar masses a year. The binary has survived that for at least thousands of years, which on the conservative criterion it should not have.

The neutron stars that come out of these systems are the pulsars spun up by a companion’s gift, and the gift is small. A neutron star that accreted conservatively from a three-solar-mass donor would collapse to a black hole long before the donor was exhausted. That millisecond pulsars are neutron stars of 1.4 to 2 solar masses rather than black holes is itself evidence that most of what their donors gave them left again.

An accretor that swells instead of refusing #

A neutron star refuses gas because of its own luminosity. An ordinary star refuses it for a different reason, and the refusal sets the loss fraction just as firmly.

A main-sequence star that gains mass must readjust its whole structure to the new mass, and it can do so only on its own thermal timescale — the time it takes to radiate away the energy of rearranging itself, some tens of millions of years for a star like the Sun and less for a heavier one. Gas arriving faster than that piles onto the surface without being absorbed into an equilibrium star. The outer layers are heated by the arriving material, the star swells, and if the supply is fast enough it swells until it fills its own Roche lobe. The binary is then in contact: both stars fill their lobes, they share an envelope at the level of the inner Lagrange point, and anything more the donor sends overflows the pair through the outer Lagrange point, behind the lighter star.

So the loss fraction for a stellar accretor is set by a ratio of clocks: the accretor’s thermal time against the timescale of the transfer. When the donor transfers on its own thermal clock and the accretor is the more massive star — which it is, at the start, by the stability argument — the accretor’s thermal time is shorter and it can mostly keep up. When the donor is the heavier star and its thermal clock is faster, the accretor cannot, and the transfer becomes heavily non-conservative through the one channel that takes the most angular momentum per gram, the outer Lagrange point.

That is the route by which a criterion computed for a stable binary can be undone by the binary’s own response. The donor’s adiabatic answer said stable; the accretor’s thermal answer turned the transfer non-conservative; and the channel the excess found was the ring-like one that lowers the stability limit. The Algol systems, in which a less massive, more evolved star is transferring to a more massive main-sequence companion, are the survivors of this phase, and the mass ratios and periods of observed Algols are generally better fitted by models that lose part of the transferred mass than by conservative ones.

Where the gas actually goes #

The calculation turns on a quantity nobody measures directly. Every term in it is a statement about the specific angular momentum of gas that has already left, and gas that has left is faint, cool, diffuse and moving away.

The three channels are idealisations. Real outflows leave from somewhere between the stars, through the outer Lagrange points, from the accretion disc’s surface at a range of radii, or from a disc wind whose lever arm can give each gram several times its own angular momentum. A disc wind is not the accretor channel; it carries more angular momentum than the accretor’s orbital value by the square of its magnetic lever arm, and it moves the critical ratio in the ring’s direction rather than the accretor’s. Which of these dominates in a given system depends on the accretor’s magnetic field, on the disc’s ionisation and on how far above the Eddington rate the supply is, and none of that is in the figures.

The figures also treat the donor’s response as fixed — −1/3 for a convective envelope, +2 for a radiative one — and treat the loss fraction as known. The first is an approximation that real stellar models refine: a giant with a small core is close to the polytropic value, a giant whose core is a large fraction of its mass is not, and the response of a radiative envelope depends on how deep the transfer has eaten. The second is the point of this essay. In nearly every population-synthesis calculation the loss fraction is a single number, chosen once and applied to every binary in the synthetic population, along with a single choice of channel. The criteria above show what that choice buys: a factor of two in the largest stable donor, a factor of four in the final periods, and the difference between a binary that becomes an X-ray source and one that becomes a merger.

What has been measured that bears on it #

None of the angular momentum is observed leaving. What is observed is consequences.

Orbital periods change, and in some systems the rate of change has been measured precisely enough to separate the channels. A conservative transfer from the lighter star widens the orbit at a rate set by the transfer rate; the same transfer with loss from the accretor widens it more slowly, and with loss to a ring it shrinks it. For the eclipsing X-ray binaries with decades of eclipse timing, the measured period derivatives are generally larger or of the opposite sign to the conservative prediction, which in several cases requires more angular-momentum loss than the accretor channel alone supplies.

Masses of the survivors are measured from the radial velocities of both stars where both are visible, and the stripped donors in systems like Cygnus X-2, overluminous for their masses, carry their history in their structure. The pulsars’ masses come from relativistic timing, and the narrow range they occupy is a direct upper bound on how much they accreted.

And the outflows themselves are seen in the systems where they are densest — the jets of SS 433, the circumbinary dust around post-giant binaries, the winds from accretion discs absorbed in X-ray spectra at hundreds to thousands of kilometres a second. Each is an existence proof for a channel. None is a measurement of how the channels divide the lost mass in any one binary.

A ledger with a column nobody fills in #

The pattern here recurs throughout the subject and is worth naming. A conservation law gives a clean answer as long as nothing escapes. When something does escape, the answer depends on what the escaping material took with it, and the conservation law cannot say, because the escaped material is by construction no longer part of the system being described. The stability of mass transfer, the spin-down of a star by its own wind, the accretion of a disc that has to lose angular momentum to let anything fall — all three are stated as conservation and decided by a loss term whose size depends on the lever arm the escaping matter had.

In the magnetic-braking case the lever arm is set by a field and can, at least in the Sun, be measured. In mass transfer it is set by where gas happened to be when something gave it escape speed, and in most systems nothing records that. The result is a criterion that is exact in form and uncertain by a factor of two in its one decisive number. Degenerate donors show the same lever arm acting the other way, with angular momentum that never leaves the binary at all but is diverted from the orbit into the accretor’s spin, and the common envelope is the limit in which the departing gas carries off everything at once.

Still open: how the lost mass divides between the channels #

For any individual binary the split between accretor loss, donor loss, outer-Lagrange-point spillage and disc winds is unknown, and the critical mass ratio inherits that ignorance. The observations that would narrow it are of two kinds. Long baselines of eclipse timing on X-ray binaries and Algol-type systems measure period derivatives that the channels predict differently, and the number of systems with decades of timing is growing. And resolved spectroscopy of outflows — the velocity at which gas leaves, measured from absorption lines against the accretion disc or the donor — gives the specific angular momentum of the departing gas directly in the systems bright enough for it. Until enough binaries have both, the most consequential number in the stability of mass transfer will remain a parameter, held fixed across systems that almost certainly differ.