A star that outgrows the light pushing it away

The lifetime of a star stops shortening once it passes a certain mass, because its luminosity has hit a ceiling and a heavier star has more fuel at the same rate of burning. That answered how long the most massive stars live. It left open how massive they can get in the first place, and that is a much harder question than it appears, as the most obvious calculation says they cannot exist.

The difficulty is one of timing, peculiar to the massive end of the distribution. A star like the Sun takes millions of years to contract to the main sequence, and it finishes gathering its mass well before that. So it spends its growing phase as a cool, modestly luminous object, and nothing it emits can seriously impede the gas falling on it. A massive star does not get that grace period, and it cannot ignore its own light. Its contraction is so rapid that it reaches full luminosity while gas is still on its way in, and light from a star of tens of solar masses is powerful enough to stop a cloud falling. An attempt to build a heavier star turns out to be an attempt to drop gas into a searchlight.

The calculation was first done in the 1970s, by Franz Kahn, and by Richard Larson and Sumner Starrfield, and worked out with care by Mark Wolfire and Joseph Cassinelli in 1987. It gives a firm upper limit of about twenty to forty solar masses. Observation says that stars with masses well over a hundred solar masses exist. This disagreement lasted several decades, and its resolution is a useful lesson in how a correct calculation can fail because of an assumption nobody thought was important, which here was a matter of shape.

A star that is shining while it is being fed #

Compare two clocks. One is the time a protostar takes to contract, the Kelvin–Helmholtz time, the gravitational energy divided by the luminosity. The other is the time needed to gather the mass.

Below about ten solar masses these clocks work in the familiar order. A protostar accumulates its mass first and then slowly contracts along a pre-main-sequence track, with a low luminosity until it arrives. Above that, the steep rise in luminosity makes the contraction very quick indeed, and the two clocks cross. The star reaches the main sequence as it is still growing, already burning hydrogen. Every further solar mass has to be accreted against the full output of a massive star. At the same time, its lifetime has started to get short, so it does not have long to do the job.

There is a deeper way of phrasing this. Most of a low-mass star’s formation takes place in the dark, and its growth is a problem of angular momentum and gas supply. A massive star’s formation takes place in full light, and its growth is a problem of radiation.

A limit written in dust #

Every reader of this subject has met the Eddington limit, in which light pushing on free electrons balances gravity pulling on protons. For hydrogen it comes to about 32,000 solar luminosities per solar mass, which is high enough that a star needs to be well above a hundred solar masses to reach it. Infalling gas is another matter entirely. It is cold, and it carries dust grains, and dust has an opacity to the light of a hot star that is enormously larger than the opacity of electrons. Even when that opacity is averaged per gram of all gas, it is a few to several square centimetres per gram, against 0.34 for electrons.

The limit therefore scales down in proportion, for the same balance between force and gravity:

$$\Gamma = \frac{\kappa L}{4\pi G M c}.$$

So a star that would be comfortably below its own Eddington limit reaches the same limit in its dusty surroundings at about twenty solar masses. Dusty gas falling towards a heavier star feels less force pulling it in than pushing it out. This is not sensitive to the details of a calculation, for even a generous lower opacity would only double the limit. Note also how this differs from the limit on a star’s own atmosphere, whose interior has long since lost its dust, and so is safe. The limit in dust applies only to what falls in from outside, and that’s the limit relevant to how a star grows.

An immediate objection arises from the fact that dust does not survive close to a star. Grains sublimate at about 1,500 kelvin, and any massive star has a dust-free zone around it, a few hundred astronomical units in radius. Gas that reached this zone would have no dust left to feel a force. So perhaps the infall only needs to carry itself through that zone.

How hard the infall has to press #

That was the question Wolfire and Cassinelli addressed. An infall reaching the sublimation radius has momentum, set by the mass accretion rate and the free-fall speed there. The star’s output has momentum too, a rate of $L/c$ if every photon is absorbed once. If the gas’s momentum flux exceeds the light’s, it can punch through the shell of dust and arrive at the star.

The numbers are demanding. A collapsing cloud in a dark cloud at ten kelvin supplies gas at a thousandth of what a hundred-solar-mass star needs. Massive stars do form in clumps that are much hotter, denser and more turbulent than the low-mass case, with rates of up to about a thousandth of a solar mass per year inferred from observations of massive protostellar outflows. So this argument does not by itself forbid a very large star. But it does require conditions very far from those in a quiet molecular cloud, and it implies that the most massive stars are made only in specific environments. Yet more force is available in a real cloud, since the absorbed light is re-emitted in the infrared and can be absorbed many times, each time passing on its momentum. A cloud that is opaque even to its own infrared re-emission gets pushed a good deal harder than the single-scattering calculation allows.

Where light goes when it has a choice #

The way out took a while to appear, and it is to do with geometry. Almost nothing in the universe falls spherically, and gas with any angular momentum settles into a disc. Treating this properly in three dimensions, with radiation, took computing power that was not available in 1987.

A thick disc around a massive protostar is opaque, and the radiation that strikes its inner edge is absorbed and re-emitted. What happens next is that the re-emitted infrared light has a choice of paths. It can diffuse through the great optical depth of the disc in its midplane, or it can leave vertically through the disc’s thin atmosphere. It goes vertically, as light takes the easy way. So the disc beams its own radiation out along the rotation axis, leaving the disc’s interior comparatively shadowed. This has been called the flashlight effect, and the numbers are striking.

In a calculation by Rolf Kuiper and collaborators in 2010, a two-dimensional disc around a protostar could carry on accreting and form a star of well over a hundred solar masses, as the radiation escapes through the polar cavity and the disc is largely protected. Their conclusion was unambiguous: the radiation pressure barrier can be avoided through this anisotropy alone. A lower limit in a sphere simply does not apply to a disc.

There is a second mechanism of a very different kind. Mark Krumholz and colleagues, in three-dimensional calculations, found that radiation trapped in a bubble beneath infalling gas is unstable in the same manner as a heavy fluid above a light one. Fingers of dense gas sink through the radiation-filled bubbles and feed the star, while the radiation escapes through low-density channels between them. The result is that accretion continues, and that a core of a hundred solar masses forms a pair of stars of about 40 and 30 solar masses, with no limit in sight. Kuiper’s group has disputed whether this particular instability is needed or really operates in that way, finding the flashlight enough on its own and the fingers an artefact of how radiation is treated. Later calculations with better treatments of radiation have been divided. On the core conclusion they agree, though the process differs: radiation pressure does not set the upper limit in the way the sphere suggests.

Massive protostellar outflows add a further release valve. Jets launched from the disc clear a cavity through the surrounding envelope, as is widely observed, and radiation pours out through this as well. So three distinct routes exist by which light may escape without pushing all the gas away. Each of them depends on breaking spherical symmetry.

What is known of the top #

The obvious check is how massive the most massive stars are. It is not an easy measurement.

Nobody has weighed the most massive stars directly, as a binary orbit does for lower masses, except for a few systems that reach about a hundred solar masses. The largest values rest on models of the star’s spectrum and luminosity, and then on evolutionary models that convert these into masses. A difficulty with the luminosity is whether a bright object is really one star. R136a1, the current title holder, lies in a very dense cluster in the Tarantula Nebula, where close companions are hard to separate. A value of about 265 solar masses, reported by Paul Crowther and collaborators in 2010, drew much attention. Later analyses, including speckle imaging at higher resolution in 2022, gave lower values, about 150 to 230. There is a clear direction to this: resolving a source tends to reduce the mass assigned to its brightest member, as the luminosity is divided between stars.

The numbers have a further meaning. A count of massive stars in a cluster should show an upper limit if there is a physical cut-off. Otherwise, the most massive star found should simply depend on how many stars the cluster has, by sampling the initial mass function. Studies of that kind have suggested a limit near 150 solar masses, for a metal-rich population, with the caveat that the most massive stars die quickly and lose mass rapidly. Whether a limit near 150 is set by formation, by instability, or merely by rarity has not been decided. The mechanism that limits formation is no longer radiation pressure on dust, at any rate. It may be the star’s own instability, or the reservoir, or a final phase of photoionisation that disperses the gas around it.

Discs that can now be seen #

The flashlight argument makes a prediction that can be tested directly: massive stars ought to form with discs, and those discs ought to be feeding them. For a long time this was hard to check, since massive protostars are rare, distant and buried in dust. Interferometers working in the near-infrared and millimetre wave bands have begun to change that. In 2010, Stefan Kraus and collaborators resolved a disc round a young star of about twenty solar masses, IRAS 13481−6124, on scales of a few tens of astronomical units, in the plane perpendicular to its outflow. Millimetre arrays have since found rotating structures round several protostars of higher mass, some of them with spiral arms.

Accretion onto them has also been caught in a more dramatic form. In 2019, maser monitoring followed a heat wave passing through the disc round the massive protostar G358.93−0.03, the sign of a sudden burst of accretion. This is the behaviour familiar from low-mass stars, and episodes of the kind suggest that growth at high mass, like that at low mass, comes unevenly through discs that fragment. What these observations do not yet reach is the top of the range. No disc has been resolved around a protostar well over thirty solar masses, where the spherical limit would be most clearly passed, and the sample remains small. Still, the qualitative prediction has fared well. Discs are present, they rotate as they should, and they lie across the outflow, as the geometry requires.

Dust, metals and the first stars #

The limit depends on the amount of dust, and that opens a route to testing it.

If radiation on dust mattered much, the top of the mass function should depend on the metal content of the gas, with more massive stars in metal-poor regions. The observed evidence is inconclusive. The Tarantula, with about half the Sun’s metallicity, does hold the most massive stars known, but it is also the largest star-forming region in the neighbourhood. The Milky Way’s Arches cluster, with metals at or above solar levels, has a similar top end to clusters of lower metallicity. There is no clear indication of metallicity dependence so far in the data, which is consistent with the idea that geometry rather than dust opacity decides the limit.

The opposite limit is where the argument has most consequence. In primordial gas with no metals and no dust, there is nothing to absorb the star’s light except atoms, and the conditions in the collapsing gas are also different, so that the accretion rates are much higher. The first stars might therefore have grown to hundreds of solar masses, stopped by photoionisation and not by a dust limit, and in that case, they would differ in outcome, including the supernova behaviour of a star tipped over at a thousand times its horizon. That also bears on the origin of early massive black holes. So a question which began as a puzzle about one kind of star turns out to bear on the universe’s earliest stars as well.

A wrong limit that was not wrong #

It is easy to look back on a limit like this and call the old calculation a mistake, but this one was correct as stated. For a spherical cloud with interstellar dust, the limit stands, and it can be demonstrated. What misled was the idea that a sphere is the simplest approximation to any object. It is the simplest to compute. It is also the one geometry that gives radiation no way of escaping except through the gas. Once a disc is present, the star’s light has a preferred direction, and a large fraction of the problem disappears.

A comparison with black holes is instructive here. A quasar can exceed its Eddington limit in a disc for a similar reason: radiation and accretion occupy different directions. The same Eddington factor that seems absolute in a sphere turns into a statement about angle-averaged flux. Accretion physics on two scales, differing by many orders of magnitude in mass and in energy source, uses one trick to beat one limit.

What the picture cannot show #

The figures give a smooth luminosity law that is a simple fit, ignoring the star’s actual state during accretion, when it may be bloated and cooler than a main-sequence object of the same mass. Dust opacity is a single number per gram of gas, whereas it really depends on wavelength, on how large the grains are and on temperature, and grains might be destroyed well before a formal sublimation radius or grow in dense regions. The momentum calculation counts each photon’s momentum once, and a dense envelope that traps infrared gives a force greater by a factor of the infrared optical depth, a difference that has been the central point of dispute among simulators.

The disc factor is a free parameter standing in for a calculation of radiative transfer that needs three dimensions. Nothing in the figure says what its value should be. The mass ranges are rounded, are often not independent, and come from very different approaches, so they should be taken only as a sense of the trend. Nor can a picture of this kind address whether the gas supply itself runs out first, or what the star’s magnetic field and outflows do, and those may be as important as the radiation.

Still open: what does set the upper limit #

Radiation on dust no longer has a strong claim to set the top of the mass function, but no other mechanism has convincingly taken over. A limit near 150 solar masses could result from a star’s own pulsational instability, or from the feedback of ionising radiation that photoevaporates the disc and the reservoir, or merely from how much gas can be gathered in the time available. Telescopes that resolve the discs round massive protostars have recently begun to show signs of accretion onto them, including some bursts, and they might measure the gas’s inward rate directly near a star above thirty solar masses, where any limit would first make itself felt. Whether massive stars in the present universe are formed as a smooth extension of low-mass star formation, or need collisions and mergers in the dense centres of clusters to reach the very top, is the next question, and the answer could be read, in part, from whether very massive stars exist anywhere outside such dense clusters.