A binary that goes silent for an hour of period

A cataclysmic variable is a white dwarf with a small star so close to it that the star overflows its Roche lobe. Matter falls on the white dwarf through a disc, and the disc flickers, brightens in outbursts by a few magnitudes, and occasionally detonates on the white dwarf’s surface as a nova. Thousands of them are known. Their orbital periods, which are measured directly from photometry and spectroscopy, are extremely short, from about eighty minutes up to about ten hours.

When enough periods had been collected to make a histogram, by the late 1970s, they showed something odd. There were systems above about three hours and many below about two hours. Between the two was a clear dip in which hardly any systems lay. The hole has held up for half a century of larger samples, between about 2.15 and 3.18 hours, and it is well established that it is not a selection effect. Selection could not produce it. Faint systems are hard to find, but nothing makes a binary of 2.5 hours harder to find than one of 2 hours or 3.

Since a binary’s period falls continuously with time as it loses angular momentum, any binary above the gap on its way to below must cross it. So why are there none to be seen there? The accepted answer, which was worked out independently around 1983 by Spruit and Ritter, and by Rappaport, Verbunt and Joss, depends on what happens to a star’s magnetic field at a particular mass.

Two ways to lose an orbit #

A binary whose orbit shrinks has to get rid of angular momentum, and these systems have two known ways to do so. Gravitational waves are one. Their rate, from general relativity and the masses and separation alone, was written down by Peters in 1964. It is exact and it applies at all times. It gets stronger as the orbit shrinks, and it is the only effect that can be calculated without assumptions.

The other is the one that slows every single star like the Sun. A low-mass star loses gas in a magnetised wind, and the field forces the wind to co-rotate far out, so the wind carries away much more angular momentum than its mass suggests. In a close binary the donor is tidally locked to the orbit. Any spin it loses is replaced immediately from the orbit by the tide, and so the wind brakes the orbit.

The difference in size matters. The brake is much stronger than gravitational radiation above the gap, by a factor of tens. So it is the brake that drives the evolution down from long periods, and that means that mass transfer is rapid. Its strength depends on the same empirical law that fits the slowing of isolated solar-type stars, $\dot J\propto M R^4\omega^3$, a relation due to Skumanich’s observations, extrapolated by a large factor to much faster rotators.

A star driven out of its own size #

A star in a close binary fills its Roche lobe, whose size is decided by the orbit:

$$R_L\approx0.462\,a\left(\frac{M_2}{M_1+M_2}\right)^{1/3}.$$

Combined with Kepler’s third law, that gives a nearly unique link between period and the donor’s density, $P\approx8.75\,(M_2/R_2^3)^{-1/2}$ hours in solar units. So a donor of given size is at a given period. A star shrinking gets to shorter periods.

When a star is losing mass, its radius changes in two ways. A small convective star responds adiabatically by getting a bit bigger, by $d\ln R/d\ln M=-1/3$, on a time scale that is essentially instantaneous. And it tries to settle to the thermal size for its mass, $R_{\rm eq}(M)$, on its Kelvin–Helmholtz time, which at $0.2$ solar masses is a few times $10^8$ years. If mass goes faster than the star can readjust, then it stays bigger than it should be.

That’s just what the brake forces. The mass-transfer time is comparable to the thermal time, so the star never catches up.

This bloating has independent evidence: observed donor stars above the gap are indeed larger, for their masses, than ordinary main-sequence stars, by the amount expected, as measured in eclipsing systems. Knigge, Baraffe and Patterson used this in 2011 to construct a semi-empirical donor sequence. The radius excess of about 20 to 30 per cent just above the gap is what allows the observed edge to be located, and it is close to the value computed in this simplified treatment.

The brake stops #

Now comes the critical part of the argument. Below a mass of about 0.2 to 0.3 solar masses, a star has no radiative core; it is convective from centre to surface. If the Sun’s kind of magnetic field depended on the shear layer at the base of its convective zone, then a fully convective star might have no such layer, and might not make the same field. The idea was that the brake would shut down, or at any rate become much weaker, at this point.

So at about three hours, with the donor reaching that mass, the brake stops. The thermal relaxation of the star, which had been fighting the brake’s forced transfer, now wins. The star shrinks on its thermal time and it shrinks faster than the orbit can follow, since only gravitational waves are left to close the orbit. It falls inside its lobe and the transfer stops.

It remains a close binary, but with no accretion there is nothing to see except an ordinary faint red dwarf and a cooling white dwarf, an object that does not get catalogued as a cataclysmic variable. As soon as the orbit has closed enough for the star to touch its lobe again, now at its thermal size, transfer resumes. Gravitational waves drive it at a much lower rate, and the system reappears as a dwarf nova at about two hours.

The model used in the figures is about as simple as it could be, with just the star’s mass and radius and the orbit. It has no stellar interior at all. Only one number has been fitted, the thermal time at 0.2 solar masses, and yet the lower edge comes out within about a tenth of an hour of the observed value, because it depends only on the thermal size of the star at the mass where the brake stops, and not on the brake. The period at the upper edge comes from how much the star has been bloated, and the period at the lower edge from what it has returned to. So this gap really is the ratio of two sizes of the same star, to the 3/2 power. The width is the size of the bloat.

A population with a hole in it #

Then a steady birth rate of such binaries gives the observed distribution of periods as the time spent at each period.

This single track already shows the three main features of the actual distribution: the gap, a sparsely populated region above it, and an accumulation below. Real systems enter at a range of donor masses and they will include other classes as well. Systems that form within the gap, with their donors at about the same mass, are born as contact binaries directly, so the gap isn’t strictly empty. And the observed peak has been reshaped by the need to detect them, since long-period systems are brighter and short-period systems are faint. It took deep surveys such as SDSS, and more recently those using Gaia parallaxes, to find the faintest of the population. These have confirmed that most cataclysmic variables lie near the minimum, as evolution predicts.

There is a shortfall here that the simplified model shares with its better-equipped relatives. The orbit reaches a minimum when the donor’s thermal time gets long compared with the mass-transfer time and the star, now substellar and degenerate, stops shrinking in response to the loss of mass. In this figure, the minimum period is about 68 minutes, as it is in early full models. The observed one is about 80 minutes. Gravitational radiation alone predicts too short a minimum period, and agreement requires an additional sink of angular momentum, about 2.5 times stronger than gravitational radiation at those periods. Something other than gravitational waves seems to be at work in those systems as well, and the proposals include the mass ejected in nova eruptions, or a residual weak brake that has not actually fallen to zero. So the assumption that the brake completely stops cannot be quite right, in detail.

A gap that measures the strength of the brake #

There is one more quantitative test, the gap’s dependence on the brake’s strength.

The lower edge is pinned by the thermal size, so it should be the same everywhere. The upper edge goes up with the strength of the brake. The observed width thus constrains the rate of angular momentum loss, and in that sense the gap is a measurement of magnetic braking in a part of the mass range where nothing else measures it. The strength that it wants, close to the standard law of Rappaport and collaborators, turns out to be several times stronger than the rate implied by the spin-down of single stars of similar masses. This is the subject of a long-running argument. Single stars at those masses and rapid rotation are thought to saturate, so their spin-down has a much shallower dependence on spin than the old law assumes. The tidally locked donors, at very rapid rotation, should be in that saturated regime too, which would make the brake much weaker than the old law.

Magnetic white dwarfs provide a natural experiment. In a polar, the white dwarf’s strong magnetic field is thought to trap part of the donor’s wind, which reduces the brake. Observed polars do have a less pronounced gap, in accordance with this idea. So one has a binary whose brake is reduced by a known mechanism, and whose gap is correspondingly reduced. That’s much the same effect as the strength axis of the figure shows.

Why the brake stops is the part in doubt #

The most awkward fact has emerged recently. Fully convective stars do not stop being magnetic, and it is not obvious that they stop having winds. Single stars below the mass where the core disappears are strongly active and do have dynamos, with X-ray activity as strong as stars above it, and no visible break in the X-ray relation at that mass. Wright and Drake showed in 2016 that the activity of fully convective stars follows the same rotation relation as that of partially convective ones. And it is known that fully convective field stars can spin down, slowly at first and then rapidly.

So the mechanism cannot be that fully convective stars have no field. The current proposals seek a more subtle change: the geometry of the field may change, with less of it in the large-scale dipole that matters for the lever arm of a wind. A complex field holds a wind less far out, as one might expect, and hence brakes much less. Or the change may be connected with some feature in the structure of a rapidly rotating star at that mass.

There is a further proposal. Baraffe and colleagues, and Ivanova and Taam, have argued that the star’s structure itself responds to the disappearance of its core, with a change in radius that contributes to detachment even if the brake only weakens. In this version the gap would be partly a structural matter, and it would not demand a magnetic explanation alone. But the fact that the gap’s location coincides with that mass makes any explanation connect at this point.

Thus the theory has an empirical strength that is unusual for its subject and a physical basis that is not quite secure. The gap is an observation, and the explanation gives its position, its width and its dependence on magnetic field with a few parameters, while the cause of the central event, the sudden reduction in braking, remains something of a postulate.

What the one-zone model leaves out #

A star is reduced to a mass and a radius. All the physics of stellar interiors is condensed into a single scaling of radius with mass and a thermal time. A full evolution code gets the radii at any given mass better, and it gives a period minimum and edges that depend on details of the opacities in cool atmospheres.

All donors begin at the same mass. Actual binaries begin at a variety of donor masses, and the ones that come from above the gap with an evolved core don’t follow this track. Systems with evolved donors can be at periods below the minimum drawn here, and these do exist, as the AM CVn stars and others show.

The white dwarf has fixed total mass with the donor. Mass is conserved here, when in reality each nova ejects what accumulated and a bit more. Ejecting matter on a very long time scale changes the orbit, and might also drive consequential angular momentum loss, which is a candidate for the extra sink.

The switch is abrupt. The brake is taken to turn off at exactly 0.2 solar masses and to zero, and a real transition has some width. That choice makes the corners sharp in these figures, in a way that the actual population does not show.

A test in the gap #

If binaries in the gap are really detached, then they should be there, as a dim population. They should appear as white dwarfs with low-mass companions, not accreting, with periods between about 2 and 3 hours, at the mass that the theory says. Such pairs cannot be distinguished from others of the same kind by any single property, but a statistical excess should appear in the right part of the period distribution. In 2016 Zorotovic and colleagues reported an excess of just this type in the Sloan survey’s detached white dwarf–M dwarf binaries, at the required companion masses and periods, and it fits the expected population of “detached cataclysmic variables”.

It is an unusually satisfying test of the idea, because it finds the missing objects themselves, at the place predicted. The sample is small, and a statistically stronger test is coming from surveys that combine Gaia and large spectroscopic programmes. If the excess is verified the argument above about silent binaries will have been checked directly, and with it the fact that they are passing through the gap on their way to resuming transfer.

How the donor’s orbit is measured at all #

All of this rests on periods and masses, and these come from a surprisingly direct set of observations. The period is the easiest number in astronomy to get for such an object. An eclipse, or a hump in the light curve from the bright spot where the stream hits the disc, recurs with a precision of parts in a million, and it can be measured with a small telescope. That is why the period histogram was the first thing known about the population, well before anyone knew what the binaries were doing.

The masses are harder, and the essential work has been done on eclipsing systems. A velocity curve of a star whose spectrum is drowned in the disc’s light is seldom available, but the timing of the eclipses of the white dwarf and the bright spot gives the geometry with high accuracy, because the stream’s trajectory is known ballistically. That has yielded masses and radii of donors for many dozen systems, and they show the donor’s oversize above the gap and its near-absence below it, as the figure of radius against mass has it.

The white dwarf’s mass has a surprise of its own. Those in cataclysmic variables average about 0.8 solar masses, considerably above the 0.6 of single white dwarfs. Selection accounts for only part of this. It may be telling something about which binaries survive their first bout of mass transfer: systems with a low-mass white dwarf may merge rather than settle down, through the runaway a donor can undergo as it loses mass, and the mass that leaves carries the orbit’s spin with it. The same consequential loss of angular momentum is a candidate for what closes the orbit faster than gravitational waves at the period minimum. So the distribution of masses and the location of the minimum may turn out to be one problem.

Still open: how a field changes when a core disappears #

Why the brake should change at that mass is the question that the gap has asked of stellar physics since 1983, and the essay’s account is not an answer to it. Single stars at the transition are now studied much more closely, because of an observed narrow gap in the Gaia colour–magnitude diagram at about the mass where stars become fully convective, a feature found in 2018. This is known as the Jao gap, and it is connected with instabilities as the core disappears. Models show the convective core and envelope merging repeatedly. A similar kind of event may affect the field at that moment.

Direct measurements of the fields of rapidly rotating stars at those masses have been made by spectropolarimetry, which shows a tendency to strong dipolar fields in many of the coolest stars and for more complex fields in others. A brake that depends on the dipole has a clear prediction, but its value at a given mass and rotation rate remains unpredicted by any theory of the dynamo. A binary system in which a star, spun up beyond anything a single star might reach, is locked for a billion years has the best prospect for showing how that dynamo works. Thousands of those binaries are waiting to be measured, and the most informative of them are the ones that have gone silent.