The flow that narrows its own channel

Half the stars in the sky have a companion, and the pairs are the only stars whose masses anybody actually knows. For most of them the companion is a bookkeeping convenience: two spheres in orbit, each evolving as though the other were not there. This essay is about the minority for which that is false — the pairs close enough that one star can hand material to the other, and the surprising fact that once it starts, the handover changes the geometry that permits it.

The two curves are the whole essay, and they are the answer to a question that sounds as though it should have no answer at all: what happens next.

The surface two stars share #

Put two stars in a circular orbit and go into the frame that turns with them. In that frame nothing moves, so gravity and the centrifugal term can be added into a single potential, and the surfaces of constant potential are surfaces a stationary fluid would settle onto. Near each star those surfaces are almost spheres. Far from both they are almost a single sphere around the pair. In between there is one surface that does something neither of those does.

The pinch point is the inner Lagrange point, and it is one of the five places where a small body can keep station — the one between the two masses rather than ahead of or behind them. What matters here is not that a satellite could sit there but that the potential has a saddle there: it is the lowest level at which the two lobes communicate.

A star whose surface lies well inside its lobe is a star in isolation. A star whose surface reaches the lobe is a valve.

One number, and a fit good to a per cent #

The lobe is not a sphere, so quoting a radius for it means choosing what a radius is. The convention is the volume-equivalent radius: the radius of the sphere with the same volume. That number depends on the mass ratio and the separation and on nothing else, and it has a famous closed-form approximation.

The shape matters more than the values. Over a factor of five hundred in mass ratio the lobe radius changes by less than four, so it is a weak function of the mass ratio and a strong function of the separation — which is one over the separation exactly, since the whole curve is drawn in units of it. That asymmetry decides which of two effects dominates when both change at once, and both do.

There is also a warning in the figure that is worth stating plainly. The fit has been quoted in a thousand papers, and quoting a fit is not the same as checking one. The dots exist because a generator that plots a published expression and then asserts that it equals itself has asserted nothing at all; the volume integral is a second, independent route to the same number, and it is what licenses everything that follows.

Why a star reaches its lobe #

Nothing in the geometry so far says a star will ever touch that surface. Two things drive it there, and they come from opposite directions.

The first is that stars swell. A star that has finished burning hydrogen in its core expands enormously as the core contracts, by a factor of tens to hundreds in radius, and a binary that was comfortably detached on the main sequence need not stay so. The second is that orbits shrink. Tidal friction bleeds spin into orbit or orbit into spin, and in the closest pairs gravitational radiation removes angular momentum outright — a mechanism that needs no gas, no friction and no contact.

Either way the lobe and the star are brought into contact, and the moment they touch, matter begins to flow through the saddle. The rate is set by how far the star overfills: a photosphere a scale height above the critical surface pushes material through at a rate that rises steeply with the overfill, so the flow is a thermostat that will hold the star very close to exact contact.

The orbit answers, and its answer changes sign #

Now the interesting part. Material moving from one star to the other carries angular momentum with it, and if none of it leaves the system, the total is fixed. Write the orbital angular momentum of a circular pair,

$$J = M_1 M_2 \sqrt{\frac{G a}{M_1 + M_2}},$$

and hold both $J$ and $M_1 + M_2$ constant while mass moves from the second star to the first. Differentiating gives

$$\frac{d\ln a}{d\ln M_2} = 2\left(\frac{M_2}{M_1} - 1\right),$$

which is the dashed curve in the hero. It is exactly zero at equal masses, and the sign either side of that is the first thing worth taking away. Transfer from the heavier star draws the orbit in. Transfer from the lighter star pushes it out. The reason is a competition between two effects that both act on the same integral: moving mass toward the centre of mass of the pair reduces the moment arm, and equalising the masses increases the product $M_1 M_2$ that multiplies it. When the donor is heavier the second effect wins and the separation must fall to keep the product fixed.

That is a statement about the orbit. What the star cares about is its lobe, and the lobe follows the separation but also follows the mass ratio through the curve two figures above:

$$\zeta_L \equiv \frac{d\ln R_L}{d\ln M_2} = 2\left(\frac{M_2}{M_1} - 1\right) + \left(1 + \frac{M_2}{M_1}\right)\frac{d\ln f}{d\ln q},$$

with $f$ the fitted function. The second term is negative and the sum crosses zero before the first does — at a mass ratio near 0.79 rather than at 1. Between those two crossings the orbit is still widening while the lobe has already begun to close.

Runaway, and how to tell whether it will happen #

Now put the star back in. When the donor loses a little mass, two radii change: the lobe’s, by the expression above, and the star’s own, by whatever its interior does. If the star shrinks faster than the lobe does, contact is broken, the flow stops, and the star has to expand again on its own timescale before anything more happens — a stable, self-limiting drip. If the lobe shrinks faster, the star overfills further, the flow increases, the mass ratio moves further in the wrong direction, and the process accelerates.

The threshold below equal masses is the result worth carrying, because it contradicts the natural guess. A giant transferring to a lighter companion is already unstable before the mass ratio has reversed, and by a wide margin. The instability does not wait for the orbit to start shrinking; it begins as soon as the lobe starts shrinking, which happens sooner.

What the donor does is not a detail of stellar structure that can be looked up once. A star with a convective envelope has almost all of its mass in a nearly adiabatic region, and removing the top of it lets the rest expand; a star with a radiative envelope is stratified and the same removal lets it contract sharply. The same star can be either at different points in its life, so the same binary can be stable and then not, without anything about the orbit changing.

What was actually measured #

None of this is inferred from a picture of two stars touching. Nothing of the kind has ever been resolved. The chain runs through three measurements, and they are worth separating.

The first is the orbit. In an eclipsing pair the light curve gives the inclination and the fractional radii, and the two radial-velocity curves give the mass ratio and the projected semi-major axes; between them the masses and radii of both stars come out in absolute units, which is the whole reason these are the only stars whose masses are known. The second is the anomaly those measurements produce. In a family of eclipsing binaries the star that is clearly more evolved — larger, cooler, filling its lobe — turns out to be the less massive of the pair. That ordering is impossible for two stars that formed together and evolved independently, because the more massive star burns out first by a wide margin. The resolution is that the evolved star was once the heavier one and has given most of itself away.

The third is what the material does when it arrives. It has far too much angular momentum to fall straight onto the accretor, so it settles into a ring and then into a disc that has to export angular momentum outward in order to let mass move inward. That disc is the reason these systems are found at all. A binary quietly exchanging material would otherwise look like a slightly odd single star.

Where it goes when the accretor is degenerate #

The case that has been studied hardest is the one where the accretor is not an ordinary star. A white dwarf is held up by a rule about counting rather than by heat, and it has a maximum mass it cannot exceed — so a white dwarf being fed by a companion is a system with a deadline in it. Whether that deadline is ever reached depends on details this essay has deliberately kept out: how much of the transferred material is retained rather than blown back off, and whether the hydrogen accumulating on the surface burns steadily or in flashes. But the framework is the one above. The lobe sets whether flow happens, the orbital response sets whether it accelerates, and the donor’s own structure decides which.

The assumption that has been doing quiet work #

Every number in the hero figure assumes that all the mass leaving one star arrives at the other and that the system’s angular momentum is unchanged. That is the conservative case, and it is a limit rather than a fact.

Real transfer loses material — through the outer Lagrange points, in a wind driven off the disc, in the radiation-driven outflow that appears when the accretion rate approaches the limit at which radiation pressure balances gravity. Material leaving the system carries angular momentum with it, and how much depends on where it leaves from. Mass lost from near the accretor takes far more specific angular momentum than mass lost from the centre of mass, so the orbit’s response can be pushed either way by assumptions about a wind nobody has measured.

So the honest summary is that the sign of the effect is robust and the magnitude is not. Transfer from the more massive star shrinks the orbit under any accounting; how fast, and whether the acceleration ends in a merger or in a long-lived semi-detached system, depends on a mass-loss rate that is inferred rather than observed.

The period that gives the donor away #

There is a relation hiding in the geometry which turns an orbital period into a statement about the donor’s interior, and it needs no masses at all.

A star that fills its Roche lobe has a radius equal to the lobe’s radius, and the lobe’s radius is a function of the mass ratio times the separation. The separation is related to the total mass and the period by Kepler’s third law. Combine the two and the masses very nearly cancel, leaving the donor’s mean density as a function of the orbital period alone:

$$\bar\rho \approx \frac{110}{P_{\rm hr}^2}\ \mathrm{g\,cm^{-3}},$$

with the coefficient varying by only a few per cent across the whole range of mass ratios because the lobe function is so flat.

That is a strong statement. An orbital period measured from a light curve is a measurement of the density of a star nobody has resolved, and it identifies what kind of star the donor is without any spectrum.

A binary with a period of a few hours has a donor of density near ten grams per cubic centimetre — a main-sequence M dwarf or a degenerate object. One with a period of a few days has a donor near a hundredth of that, which is a main-sequence star of about a solar mass. One with a period of a year has a giant.

The relation is what makes the period distribution of interacting binaries interpretable. A pile-up of systems at a particular period is a pile-up at a particular donor density, and therefore at a particular stage of the donor’s evolution — which is how the observed gap in the periods of cataclysmic variables between about two and three hours was recognised as a change in the donors rather than a selection effect.

It also explains why the shortest-period interacting binaries have degenerate donors. A main-sequence star cannot be dense enough to fill a lobe at a period below about eighty minutes, so any system found below that has a donor which is not a main-sequence star — and the systems with periods of ten minutes have white dwarfs on both sides.

The critical surface and the response are the two objects the argument turns on, and each is worth drawing at a second configuration.

The transfer that needs no contact #

The whole essay has assumed the donor fills its lobe. There is a second transfer mechanism that does not require it, and it is how a large class of systems works.

A massive star loses mass through a wind — a fast, roughly spherical outflow driven by radiation pressure on its own spectral lines, carrying away a substantial fraction of the star’s mass over its lifetime. A companion sitting in that wind captures part of it.

How much depends on a competition between the companion’s gravity and the wind’s speed. Material passing at a distance small enough that the companion’s escape speed exceeds the relative velocity is captured; material further out is not. That defines a capture radius, and the accretion rate is the wind’s mass flux through the disc of that radius.

The rate is far smaller than lobe overflow can deliver — typically a thousandth of the wind rather than all of it — because the capture radius is small compared with the orbital separation and the wind is spreading over a whole sphere.

Two things follow. Wind accretion is inefficient, so a system fed this way is far fainter than one fed by overflow at the same evolutionary stage. And it is insensitive to the geometry: the companion need not be anywhere near its lobe, so the mechanism operates over a wide range of separations rather than at a threshold.

The observational class this produces is the high-mass X-ray binary in which a compact object orbits a luminous companion at a comfortable distance and shines by eating a fraction of its wind. Such systems are variable on the wind’s own clumping timescale rather than on any orbital one, which distinguishes them from the overflow-fed cases.

A valve and a leak are different mechanisms, and the observational signature that separates them is the steadiness of the flow rather than its magnitude.

And the response at a mass ratio close to unity, which is the case where the two curves that decide stability nearly meet.

Where the ladder goes #

The mechanism in this essay is the origin of a long list of objects that look nothing like each other: the millisecond pulsars, spun up by a companion’s gift of angular momentum; the X-ray binaries, where the accretor is a neutron star or a black hole and the disc’s inner edge is hot enough to radiate in X-rays; the cataclysmic variables, where a white dwarf’s surface layer ignites. All of them are the same valve with a different thing on the other side of it.

What the next rung has to add is what happens when the flow is not slow. A dynamically unstable transfer does not settle into a disc; it engulfs the accretor in the donor’s envelope, and the pair spirals inside a common atmosphere that neither of them is in orbit around any more. That process shortens periods by factors of a hundred in a few hundred years, and it is the single largest uncertainty in predicting how many close binaries of any kind should exist.