The surface a star stops at

A single star has no outer boundary that means anything. Its material thins outward until it is indistinguishable from the interstellar medium, and where the surface is said to be is a convention about optical depth rather than a place where anything stops.

Put a second star nearby and that changes. In the frame rotating with the pair there is a genuine surface around each — the last equipotential that closes around one star without enclosing the other — and it is not a convention. Material inside it belongs to that star. Material that reaches it does not.

Where the surface comes from #

The setting is the circular restricted three-body problem: two masses on circular orbits about their common centre, and a test particle. In the frame that rotates with them, the particle moves in an effective potential

$$\Omega = \frac{Gm_1}{r_1} + \frac{Gm_2}{r_2} + \tfrac{1}{2}\omega^2 s^2,$$

with $s$ the distance from the rotation axis. The last term is the centrifugal one, and it is what makes the problem interesting: without it the equipotentials would be simple ovals around each mass and there would be nothing to say. The lobes are traditionally called the Roche lobes, after Édouard Roche, who worked out this geometry in the 1850s for a different question — how close a moon can come to a planet before tides pull it apart. That question and this one are both about the same potential and are not the same question, and the two answers are routinely confused because they share his name. The Roche limit is a distance at which a body’s self-gravity loses to the tidal field. The Roche lobe is a surface, and a star can fill it without being pulled apart at all — it simply spills over the edge.

How big it is #

The number that matters for whether a star fills its lobe is the lobe’s volume-equivalent radius: the radius of a sphere with the same volume, since a star’s own radius is a sphere’s radius and the comparison has to be made somehow. There is no closed form, and the standard is Peter Eggleton’s 1983 fit,

$$\frac{R_L}{a} = \frac{0.49\,q^{2/3}}{0.6\,q^{2/3} + \ln\left(1+q^{1/3}\right)},$$

with $q$ the star’s mass divided by its companion’s. It is quoted as accurate to one per cent for every mass ratio, which is a strong claim about a fit with three constants in it.

A word about how that check was made, because it is the kind of thing that goes wrong quietly. Counting every cell where the potential exceeds its value at $L_1$ is not the lobe: that condition is also satisfied inside the companion’s lobe and, because the centrifugal term grows without limit, throughout the outer region beyond $L_2$. A first attempt at this figure did exactly that and returned a lobe 46 per cent too large, produced entirely by the corners of the bounding box. A lobe is a connected component, not a level condition.

Filling it #

A star fills its lobe by growing, and stars grow. A main-sequence star of one solar mass has a radius of one solar radius; the same star on the giant branch has a radius of a hundred, because its envelope swells as its core contracts. At a separation of ten solar radii — an orbital period of about a day for a pair of solar masses — the lobe radius is about four solar radii, and the star crosses it long before it reaches the tip of the giant branch. So close binaries transfer mass, and they transfer it in a particular way: not as a wind blown off in all directions, but as a stream leaving through the neighbourhood of $L_1$, with the angular momentum of the rotating frame, which is why it does not fall directly onto the companion but goes into orbit around it as a disc.

The transfer changes the lobe #

Here is the part that makes this a mechanism rather than a description. When mass moves from one star to the other, three things change at once: the donor’s mass, the companion’s mass, and — because angular momentum is conserved — the separation.

For conservative transfer, with total mass and total angular momentum both held fixed, the algebra is short. The orbital angular momentum is $J = m_1 m_2\sqrt{Ga/(m_1+m_2)}$, so at fixed $J$ and fixed total mass,

$$a \propto \frac{1}{(m_1 m_2)^2}.$$

The product $m_1m_2$ is largest when the masses are equal. So transfer towards equality widens the orbit and transfer away from equality shrinks it — and which of those is happening depends on which star is the donor.

The runaway case does not run away forever, because it is running towards equal masses and past equality the sign reverses. What it does is pass through the unstable phase quickly — on a thermal or even a dynamical timescale rather than a nuclear one — and emerge on the other side with the mass ratio inverted.

The paradox this explains #

Algol is a bright eclipsing binary that has been watched since 1670, and by the 1950s its two components were well characterised: a hot main-sequence star of about 3.7 solar masses, and a cooler subgiant of about 0.8 that has clearly left the main sequence.

That is impossible. Stellar lifetimes fall steeply with mass — as roughly $M^{-2.5}$ — so in any pair formed at the same time the heavier star evolves first. Algol has the lighter star evolved and the heavier one still on the main sequence, and both are the same age.

The resolution is the previous figure. The evolved star was originally the heavier of the two. It reached the giant branch first, filled its lobe, and began transferring mass — in the unstable direction, so the transfer accelerated, and it lost most of its envelope to its companion. What is now the 0.8-solar-mass subgiant is the stripped remnant of what was once the more massive star, and what is now the 3.7-solar-mass main-sequence star has been fattened on its companion’s outer layers.

The check is that the evolved star should be overluminous for its present mass, because its luminosity is set by the core it built when it was heavy, and it is. The check is also that such systems should be found overwhelmingly with the lighter star filling its lobe, because that is the stable configuration and the unstable one is passed through fast — and they are.

Two surfaces with one name #

It is worth separating the lobe from the limit properly, because Roche’s name is on both and the two are used within a paragraph of each other in most accounts. A star can fill its Roche lobe while sitting comfortably outside its Roche limit, and that is the normal case: the lobe is a few tenths of the separation and the limit for a body of stellar density is around two and a half stellar radii, which for a close binary is well inside the companion. The material leaving through $L_1$ is not being torn off. It is walking out through a door.

What these systems look like #

The observational payoff is that lobe-filling systems announce themselves, and they do it in several unrelated wavebands.

If the accreting star is a main-sequence star, the result is an Algol — an eclipsing binary with an evolved secondary and a peculiar mass ratio, visible in ordinary photometry.

If it is a white dwarf, the disc reaches temperatures at which it is unstable, and the system brightens by several magnitudes every few weeks or months: a dwarf nova. The instability is in the disc rather than in the transfer, and the transfer rate is what sets the recurrence time.

If it is a neutron star or a black hole, the inner disc reaches $10^7$ K and the system is an X-ray binary. The luminosity is then bounded by the brightness at which radiation pressure blows the accreting material back out, which is the Eddington limit — so a quantity computed from the electron-scattering opacity of hydrogen turns out to cap what a geometrical argument about a saddle point can deliver.

And if enough material accumulates on a white dwarf for its core to reach carbon ignition, the result is a supernova whose peak brightness is a mass of nickel. Every one of those outcomes begins with a star growing into a surface that was there all along.

When the transfer cannot be absorbed #

The account so far has one star handing material to another in an orderly way. There is a regime where it cannot be orderly, and what happens instead is the least understood phase in binary evolution and the one that produces most of the interesting objects.

If the donor’s envelope is convective and it is the more massive of the pair, losing mass makes matters worse rather than better: the envelope expands as it loses mass, and the lobe shrinks as mass moves to the companion, so the overflow deepens. The transfer rate runs away, rising by orders of magnitude in a few orbits.

The accretor cannot take it. The material arrives faster than it can be radiated away or incorporated, and it piles up until it fills the accretor’s own lobe and overflows it. At that point the two stars are orbiting inside a shared envelope of gas — a common envelope.

What follows is dominated by drag. The two cores move through the envelope, which does not corotate with them, so they lose orbital energy and angular momentum to it and spiral inwards. The energy released heats and eventually unbinds the envelope, which is expelled; what is left is the two cores in a much tighter orbit than before, or, if the envelope was not ejected in time, a single merged object.

The whole episode takes of order a year, against the millions of years the stars spent evolving to it, and it can reduce an orbital separation by a factor of a hundred.

That is where the close pairs come from. A pair of white dwarfs orbiting in hours, a neutron star with a low-mass companion in a two-hour binary, a hot subdwarf with a compact companion — none of them could have formed at their present separation, because the progenitor stars were larger than the current orbit. Every such system is evidence that a common envelope happened.

And almost nothing about it is calculable. The standard treatment is a single efficiency parameter — the fraction of the released orbital energy that goes into unbinding the envelope — fitted to the observed population rather than derived, and hydrodynamic simulations have not converged on it. A phase that lasts a year and sets the properties of every compact binary is parameterised by one number nobody can compute, which is the honest state of the subject.

The lobe that changes size #

The geometry throughout assumes a circular orbit and a donor rotating in step with it, and both assumptions are needed for the surface to be static in the rotating frame. Where they fail, the lobe is not a fixed shape.

An eccentric orbit brings the two stars closer at periastron, and the lobe scales with the separation — so a donor that just fits its lobe at apastron overflows it near periastron and not otherwise. The result is episodic transfer, in bursts once per orbit, and there are systems whose outbursts recur on exactly the orbital period for this reason.

Asynchronous rotation changes the effective potential itself, since the centrifugal term depends on the rotation rate rather than on the orbital rate. A donor spun up by accretion, or one not yet tidally locked, has a lobe of a different size and shape from the classical one.

Both are ignored in the standard treatment because tides are efficient: they circularise the orbit and synchronise the rotation on timescales short compared with the evolution, so a system that has been interacting for a while satisfies both assumptions. The exceptions are the systems that have just begun to interact, which are exactly the ones a survey catches in the act.

There is a further reason those exceptions matter out of proportion to their number. A system caught at the onset of transfer still carries the orbital elements it had before the interaction began, and everything about the pair’s earlier history has to be reconstructed from systems that have already been reshaped by it. The eccentric, unsynchronised cases are the only ones whose initial conditions are still legible, which is why they receive attention that their rarity would not otherwise justify.

The observational handle on them is the same in both cases: an interaction that switches on and off with a period gives a light curve modulated at the orbital period, so the phase at which the brightening occurs says where in the eccentric orbit the overflow begins. That is a measurement of the lobe’s size at a particular separation, which is the only direct check the geometry of this essay has ever been given.

The shape a lobe-filling star has #

One consequence of the geometry is directly observable and needs no transfer at all: a star that fills its lobe is not a sphere, and the departure is large.

The lobe is a teardrop, elongated towards the companion and flattened at the poles, and a star filling it takes that shape. The long axis exceeds the short by tens of per cent for a pair of comparable masses, so the projected area presented to an observer changes continuously as the system rotates — largest at quadrature, when the elongation is across the line of sight, and smallest at conjunction.

The light curve therefore varies smoothly through the whole orbit rather than only during eclipses, with two maxima and two minima per revolution. That signature is called an ellipsoidal variation, and its amplitude measures how close the star is to filling its lobe.

Two further effects ride on the same geometry. The elongated end is at lower effective gravity, and a lower gravity means a lower surface temperature — gravity darkening — so the elongated tip is not merely larger but dimmer per unit area than the poles. And the face turned towards a hot companion is heated by it, so that hemisphere is brighter, which adds a single maximum per orbit rather than two and breaks the symmetry between the two peaks.

A light curve with unequal maxima is therefore a measurement of two things at once: how distorted the star is, from the ellipsoidal term, and how much reflected and reprocessed light the companion delivers, from the asymmetry. Fitting both is how a system’s geometry is recovered when it does not eclipse — which is most of them, since a lobe-filling star’s elongation is visible at any inclination and an eclipse requires alignment.

The lobe’s size and its response to transfer are the two things the surface is used for, and both are worth reading over a wider range of mass ratio than the essay’s own figures cover.

What the lobe does not decide #

Two limits are worth stating, because the surface is a clean piece of geometry and the physics around it is not.

The lobe is defined in a frame that rotates rigidly, which assumes a circular orbit and a synchronously rotating star. Neither is guaranteed. An eccentric binary has no such frame at all and its “lobe” changes through the orbit; a star spinning faster than the orbit has a different effective potential entirely. Tides drive both towards the assumed state on a timescale that is short for close pairs and not short for wide ones, so the geometry is exact where it is used and approximate at the edges of where it is used.

And the lobe says nothing about the rate. It says when transfer begins, not how fast — that depends on how the donor’s radius responds to losing mass, which is a question about its internal structure and has opposite answers for radiative and convective envelopes. A star whose envelope expands when mass is removed will overflow further and transfer more, which is a second runaway with the same sign as the orbital one. And the response of the lobe to the transfer itself, started from a more extreme ratio than the one above.

Where the ladder goes next #

The first rung of this anchor was the five points themselves and their stability. This one is the surface through the first of them. The rung after is the stream: where the material leaving $L_1$ actually goes, which is set by the Coriolis force in the rotating frame and produces an accretion disc with a radius that can be computed from the mass ratio alone — and which is the reason a cataclysmic variable flickers.