A disc that stores its inflow and spends it in a century

In 1937 a faint star in Orion brightened by a factor of several hundred over a few months and has stayed bright ever since. FU Orionis was at first taken for a slow nova, but novae fade and it did not, and when a second star did the same in 1969 and a third in 1974 it became clear that these were a class: young stars, still embedded in the remains of the clouds they formed from, whose discs had abruptly begun to deliver mass at ten thousand times their previous rate. Their spectra are not a star’s. They are the spectrum of a disc, hotter towards the centre, with the absorption lines broadened by orbital motion and changing with wavelength exactly as rings at different temperatures should. Something had switched a disc on.

The explanation now favoured begins with the fact that most of a young disc cannot accrete. The instability that turns any field into turbulence needs the gas to be ionised enough to hold the field, and the dense, cold midplane of a disc between a tenth of an astronomical unit and several is not: the field slips through the neutral gas faster than the instability grows. Only a thin surface layer, ionised by the star’s X-rays and by cosmic rays, stays active. Gas arriving faster than that layer can pass it on has nowhere to go.

A skin with a fixed capacity #

Gammie set out the layered disc in 1996, and its central quantity is simple. If the active layer has a fixed column $\Sigma_a$ — about a hundred grams per square centimetre, the depth to which cosmic rays and X-rays keep the gas ionised — then the accretion rate it can carry is the ordinary viscous rate applied to that column alone:

$$\dot M_a = 3\pi\,\alpha\, c_s H\, \Sigma_a.$$

The sound speed and the scale height both depend on radius. In a disc warmed by its star, with temperature falling as the inverse square root of radius, the product $c_s H$ rises in proportion to the radius, and so does the layer’s capacity.

The hundred grams is not arbitrary. A cosmic-ray proton of the energies that dominate ionisation loses its energy to the gas it passes through over a column of about ninety-six grams per square centimetre, so cosmic rays arriving at the disc’s two faces ionise roughly that much on each side and very little below it. The star’s X-rays, which are more intense near the star but softer, penetrate only about ten grams per square centimetre before they are absorbed; they dominate the ionisation of the skin in the inner disc and the cosmic rays dominate it further out. The radioactive decay of short-lived isotopes in the disc’s dust gives a weak floor of ionisation everywhere, too weak to couple the field to the gas in the dense midplane. So the active column is set by an attenuation length, which depends on the gas’s composition and not on the disc’s mass — a heavier disc has a deeper dead zone, not a thicker skin.

The capacity is smallest exactly where the gas has to end up. A protostar in its embedded phase is being fed from an infalling envelope at about a millionth of a solar mass a year — it has to be, to assemble half a solar mass in the half-million years that phase lasts — and the layer can carry that only at radii of tens of astronomical units or more. Inside that, each annulus receives more from outside than its skin can pass on, and the excess settles into the dead midplane below. A layered disc with a supply larger than its inner capacity is a disc that accumulates.

The accumulation is not small. At a millionth of a solar mass a year, with a hundredth of it getting through, a disc stores a hundredth of a solar mass every ten thousand years. That cannot go on for long: within a few tens of thousands of years the dead zone holds more than the active parts of the disc, and something has to give.

What gives way first #

Two things can switch the dead zone on. One is heat: if the midplane reaches about a thousand kelvin, potassium and sodium begin to ionise thermally, the conductivity jumps, and the instability can run through the whole column. The other is gravity: a column heavy enough becomes gravitationally unstable, grows spiral arms, and the arms both carry mass inward and dissipate energy as heat. Which comes first decides how the outburst is triggered.

The heat available in a quiescent dead zone is only what the skin’s own small accretion dissipates, blanketed by the dust in the column above and below. At an astronomical unit, the column needed for that to raise the midplane to a thousand kelvin is four times the column at which the disc goes gravitationally unstable. So the dead zone cannot ignite itself. It piles up until gravity stirs it.

The spiral arms of a gravitationally unstable disc are not quiet. They carry mass inward and shock the gas as they move through it, and the heating they supply is set by the inflow they drive, not by the trickle through the skin. Simulations by Armitage, Livio and Pringle in 2001, and by Zhu, Hartmann and Gammie later in the decade, found the sequence that the figure implies: the dead zone builds until its arms switch on, the arms heat the inner disc past a thousand kelvin, the magnetorotational instability ignites through the whole thickness, and the stored mass drains. Once it is draining, the drain heats the disc, and the dashed line shows how far that heating carries: the column stays hot until it has fallen by a factor of two thousand. The outburst sustains itself until it has spent its store.

A limit cycle #

Put the pieces together and the disc does not settle to a steady state at all. It cycles: a long quiescent phase in which the star accretes only what the skin passes on, while the dead zone fills; a trigger when the store reaches the gravitational threshold; a short outburst in which the inner disc, fully active and hot, drains on its own viscous time; and then quiescence again. The first figure is that cycle, run for a supply of a millionth of a solar mass a year with the store released at an astronomical unit.

The numbers in it are worth reading against the FU Orionis stars. The burst rate, $8 \times 10^{-5}$ solar masses a year, is the rate their disc spectra imply. The burst duration, a little under two centuries, is the viscous time of a hot inner disc with a fully active α of a tenth; FU Orionis itself has now been bright for nearly ninety years and is fading slowly. The interval between bursts, sixteen thousand years, is close to the recurrence time that counts of outbursting stars imply: so few stars have been seen to erupt among the thousands of young stars monitored that each must do so only every five to fifty thousand years.

None of those three numbers was tuned to match. They come from the layer’s capacity, the gravitational threshold and the viscous time, each evaluated with standard values, and the agreement is the best evidence that the layered picture is roughly right.

A slower supply makes rarer bursts, not smaller ones #

The cycle has a property that a steady disc does not. Changing the supply does not change the outbursts.

The store is released when it reaches a column set by the disc’s temperature and rotation, not by the supply, and it drains at a rate set by the hot disc’s viscous time. So the bursts are the same size whatever feeds them, and the supply sets only how long the dead zone takes to refill. A protostar late in its embedded phase, its envelope thinning, erupts less often but no less violently. That is a testable statement: the outbursting stars should have similar peak rates across a range of envelope masses, with the envelope mass correlated with how often outbursts happen rather than how bright they get. The small sample of known outbursts does not yet decide it, but the peak rates of the classical FU Orionis stars do lie within a factor of a few of one another.

The protostars that are too faint #

The cycle explains a discrepancy that was noticed before anyone proposed it.

Gas falling onto a protostar releases its gravitational energy, and the luminosity this produces is fixed by the accretion rate and the star’s mass and radius. A star that must grow by half a solar mass in a few hundred thousand years must, on average, shine at several to tens of solar luminosities from accretion alone. Kenyon and collaborators pointed out in 1990 that embedded protostars in Taurus are mostly far fainter than that, and the infrared surveys of the following decades confirmed it across many clouds: the median embedded protostar shines at one to a few solar luminosities. Either the stars are not growing at the rate their numbers require, or the growth is not steady.

The cycle answers that a protostar observed at a random moment is almost always in quiescence, accreting a hundredth of the supply and shining at a hundredth of the luminosity steady growth would give. It spends one per cent of its time in outburst and gains nearly all its mass there. A census of luminosities counts time; the mass is delivered in a fraction of the time it counts. The luminosity problem is what episodic accretion looks like when sampled.

A second, smaller cycle closer in #

FU Orionis has a smaller cousin. EX Lupi, and the stars named after it, brighten by a factor of a few to a hundred for months to a year, and do it again within years or decades. Their discs reach rates of $10^{-7}$ to $10^{-6}$ solar masses a year at peak rather than $10^{-4}$, and their spectra during outburst still show the star, with hot emission from gas falling onto it, rather than a disc swamping everything.

They are best understood as a second limit cycle with the same structure on a smaller scale. In the innermost tenth of an astronomical unit the disc can be hot enough that hydrogen itself is partly ionised, and there the opacity rises so steeply with temperature that the disc has two stable states at the same column — a cool one and a hot one — with no stable state between. Gas accumulates in the cool state until the column exceeds what it can hold, flips to the hot state, drains, and flips back. That is the instability that drives the outbursts of the dwarf novae, in which a white dwarf’s disc brightens every few weeks, transplanted to a young star. It operates in a small region, so its store is small and its cycle is fast.

The two cycles can be nested. A gravitationally triggered outburst at an astronomical unit floods the inner disc with gas, which then cycles through the hydrogen instability on its way to the star, and some models produce FU Orionis events whose light curves have structure on the timescale of the inner cycle superposed on the slow decline of the outer. The observed outbursts span the whole range of amplitudes and durations between the two classes, which suggests a continuum of stores released at different radii rather than two separate mechanisms.

The ages read off a cluster #

Episodic accretion changes the star as well as the disc. A protostar’s radius and luminosity at the end of its embedded phase depend on how it was built: material arriving slowly is accreted cold, radiating away its heat at the shock where it lands, while material arriving in a burst carries heat into the star. Stars built from different accretion histories arrive at the visible stage with different radii, and a star’s radius and luminosity are what its position on the Hertzsprung–Russell diagram is read from.

That matters because a young cluster’s age is read from exactly those positions, against tracks computed for stars that started from a standard initial radius. The stars of a single young cluster are observed to spread across a range of luminosities at fixed temperature, and if the spread is read as ages it implies star formation lasting ten million years in clusters whose dynamics suggest one. Baraffe and collaborators showed in 2009 that a history of episodic accretion produces a spread of that size in coeval stars: two stars of the same mass and age, one built mostly in bursts and one mostly between them, sit at different luminosities for millions of years afterwards. Some of what has been read as an age spread may be the scatter of accretion histories — the outbursts’ fingerprint on the diagram that sorted the stars.

What the bursts leave behind #

An outburst heats the whole inner disc, and some of its effects outlast it.

The line beyond which water freezes sits at a few astronomical units around a quiescent young star. During an outburst it moves outward by an order of magnitude. Around V883 Orionis, an outbursting star in Orion, millimetre images have caught the water snow line at about forty astronomical units, where the dust’s emission changes character, and the sublimated ices have released complex organic molecules into the gas where they can be observed. Every burst repeats this, so the ice in the inner tens of astronomical units of a young disc is sublimated and refrozen many times during the embedded phase — which complicates any attempt to read a planet’s composition as a record of where it formed, since the line it is read against has been moving back and forth.

The jets record the bursts too. Herbig–Haro jets are not smooth streams but chains of knots, each a shock where faster material has caught up with slower, and the spacing of the knots along a jet, divided by the jet’s speed, gives intervals of decades to centuries for the small knots and thousands of years for the large bow shocks. If the outflow rate follows the accretion rate, as the magnetocentrifugal lever says it should, the large bow shocks are the jets’ memory of outbursts, written out along a parsec of sky.

Where the picture is uncertain #

The layered disc depends on the active column, and the active column is uncertain by at least an order of magnitude. Cosmic rays may be excluded from the inner disc by the young star’s own wind, as the Sun’s wind shields the inner Solar System today, which would shrink the layer; X-ray flares from the star would expand it. Small dust grains soak up free electrons and shrink the active layer further; grain growth releases them. A disc that has grown its dust into pebbles is more ionised than one that has not, so the dead zone shrinks as planet formation proceeds.

The figures also treat the disc as accreting only by turbulence. A disc threaded by enough vertical flux accretes through a wind from its surface, and a wind-driven layer can carry far more than the turbulent skin assumed here. Whether a disc is wind-driven depends on whether it kept its flux, which depends on its history. The cycle drawn here is what a disc does if it has too little flux for a wind and too little ionisation for turbulence — a disc squeezed between the two transport mechanisms, with only gravity left.

The thresholds are idealised too. The gravitational column is computed for a passive disc’s temperature, the heating for a single opacity, and the store for a single radius; real models follow the column through radius and time, and find bursts that begin at the inner edge and propagate outward or begin further out and propagate inward, with durations and shapes that depend on which. The observed outbursts show both kinds of rise, fast and slow.

Still open: how many of a star’s grams arrive in a burst #

The cycle says nearly all. The observations can say only that outbursts happen and how often they are seen, and the conversion to a fraction of the mass needs the peak rate, the duration and the interval for the same stars, none of which is measured over a full cycle for any of them. The infrared and optical surveys of the whole sky now running find new outbursts every year, and each adds a measured rise time and peak; the millimetre images of the outbursting discs give their masses and the extent of their snow lines. If the bursts turn out to carry most of a star’s mass, then the protostellar collapse is a story told mostly in centuries separated by quiet millennia, and the stars seen at any moment are almost all between chapters.