Astronomy & Celestial Mechanics
the sky, computed
Starlight

A glow that weighs the grains reddening cannot see

The interstellar extinction curve reads the sizes of dust grains from how much more they dim blue light than red, and it goes blind at the top end: grains larger than about a micron dim every visible colour alike. Those are the grains that hold most of the mass, and the ones from which planets begin. Dense cloud cores glowing faintly at 3.6 microns in scattered starlight are the first direct sign of them — because only grains grown to a micron or more scatter that light at all.

Assumes Extinction and Molecular clouds.

15 min read 5 figures Light is the only messengerMeasuring the unreachableScales that defeat intuition

Interstellar dust dims starlight more in the blue than in the red, and the shape of that dimming against wavelength is a measurement of the grains’ sizes. A grain interacts with light according to the ratio of its circumference to the wavelength. Much smaller than the wavelength, it barely interacts; comparable, it interacts most strongly; much larger, it blocks a fixed fraction of the light that falls on it regardless of colour. The extinction curve of the diffuse interstellar medium — rising steadily from the infrared to the ultraviolet — is well fitted by a power law of grain sizes, many small and few large, that stops at about a quarter of a micron. Larger grains would flatten the curve in the visible, and the curve is not flat.

That upper limit is the least secure number in the whole description, and for a reason built into the method. The correction astronomers apply for dust — which makes everything look further away — works by measuring how much a star has been reddened and scaling that to how much it has been dimmed, and the scaling assumes the grains that redden are the grains that dim. A grain much larger than the wavelength is grey: it dims every visible colour by the same amount. Adding a population of micron-sized grains to the diffuse medium would dim every star a little more without changing its colour, and the extinction curve, which measures colour, would not see them. The size distribution’s upper end is where the extinction curve is blind. It is also where the mass is. In a power law with the interstellar slope, the number of grains falls steeply with size but the mass per grain rises as the cube, and the total mass is dominated by the largest grains present. The part of the distribution that holds most of the solid material in a cloud is exactly the part its colour cannot report.

Inside dense clouds the question matters more. The grains in the cores that will become stars are expected to grow, by colliding and sticking and by accreting mantles of ice, and grains that reach a micron or more before the core collapses carry that head start into the disc of the young star, where planets begin by growing grains further. How large grains get before a star forms is a question about the first step of planet formation, and the extinction curve cannot answer it.

Why the diffuse medium stops at a quarter of a micron

The quarter-micron cut-off of the diffuse medium is not a law of nature but a balance. Grains in the diffuse gas between clouds are struck by the shock waves of supernovae every few hundred million years, and in a shock, grains of different sizes are accelerated to different speeds and collide at kilometres per second — fast enough to shatter them. Shattering grinds the large grains into small ones, and the power law with its slope of three and a half is close to what repeated fragmentation produces. In the dense clouds, where shocks are gentle and the gas is cold, grains collide slowly, stick, and acquire ice; the size distribution grows at its upper end. A grain cycles between the two environments many times over its life, which is several hundred million years, and the diffuse medium’s cut-off is the size at which shattering in one environment balances growth in the other.

That cycle is why grain sizes are hard to measure where they matter most. The diffuse medium is where the extinction curve is measured best — thousands of stars behind modest columns of dust, their light dimmed but not extinguished — and it is the environment where grains are smallest. The dense cores where grains grow are the environment where the extinction curve is hardest to measure, because the stars behind them are dimmed out of sight at visible wavelengths, and the few that can be seen in the infrared are seen through the grey part of the curve. The grains are best characterised where they are least interesting.

Why small grains are invisible at long wavelengths

A grain much smaller than the wavelength scatters light as a small dipole: its scattering efficiency, the cross-section over its geometric area, rises as the fourth power of its circumference over the wavelength. This is the same law that makes the daytime sky blue — and that makes the twilight sky light scattered twice — and it is extraordinarily steep. A grain a tenth of a micron across scatters light of 0.5 microns reasonably well; the same grain scatters light of 3.6 microns, seven times longer, about two and a half thousand times less efficiently.

How much light a fixed mass of dust scatters, as its largest grains grow. The scattering cross-section per unit mass of dust, on a logarithmic scale relative to the largest value drawn, against the size of the largest grains in a power-law distribution n(a) ∝ a^−3.5 from 0.005 µm, at wavelengths of 1.25, 2.2, 3.6 µm. With the largest grains at a quarter of a micron — the size the interstellar extinction curve implies — the dust scatters 3.6 µm light 33 times less efficiently than it does once they reach a micron: grains much smaller than the wavelength scatter as the fourth power of their size over the wavelength and are all but invisible. At 3.6 µm the scattering per mass peaks when the largest grains are 3.5 µm, and falls beyond, as the mass moves into grains too large to scatter efficiently per gram. A dense core that glows at 3.6 µm in scattered starlight is therefore a core whose grains have grown to a micron or more.
Fig. 1 Scattering per unit mass of dust against the radius of the largest grains in a power-law distribution from 0.005 µm, at 1.25, 2.2 and 3.6 µm, on logarithmic scales. With the largest grains at a quarter of a micron, as the diffuse extinction curve implies, the dust scatters 3.6 µm light 33 times less efficiently than once they reach a micron. At 3.6 µm scattering per mass peaks when the largest grains are about 3.5 µm.

The figure asks how much light a fixed mass of dust scatters at three near- and mid-infrared wavelengths, as the size distribution’s upper end is extended. At a quarter of a micron — the diffuse-medium value — the dust scatters 3.6-micron light feebly, thirty-three times less than it would if its largest grains reached a micron. The curve then rises steeply as the largest grains approach the wavelength, peaks when they are a few microns, and falls slowly beyond, because a fixed mass shared among ever larger grains has less total surface, and grains much larger than the wavelength are no more efficient per unit area. At every wavelength drawn, the scattering per unit mass is highest when the grains are comparable to the wavelength; at 3.6 microns, that means grains of a micron or more.

The scattering efficiency used is a bridge between two limits — the dipole law for small grains and the anomalous diffraction formula for large ones — rather than a full calculation for real grain materials. The anomalous diffraction formula alone rises only as the square of the size at the small end, which is the behaviour of absorption, not of scattering by a transparent grain; used on its own, it would make small grains scatter the mid-infrared hundreds of times too efficiently and hide the very effect drawn here.

The colour of scattered light

The colour of light scattered by small grains and by grown ones. Scattering per unit dust mass against wavelength, both logarithmic, for size distributions whose largest grains are 0.25, 1, 5 µm, relative to the largest value drawn. Grains all smaller than the wavelength scatter as λ⁻⁴ at long wavelengths, and between 2.2 and 3.6 µm the distribution ending at 0.25 µm falls with slope −3.8: its scattered light at 3.6 µm is 15 per cent of its light at 2.2. Grains grown to a micron are comparable to those wavelengths, and their scattering falls with slope −2.7; grown to 5 µm, with slope −0.8, close to grey. The colour of scattered light between the near and the mid-infrared is therefore a measurement of the largest grains: steeply blue scattered light means small grains, flatter means grown ones.
Fig. 2 Scattering per unit dust mass against wavelength, for distributions whose largest grains are 0.25, 1 and 5 µm, relative to the largest value drawn. Between 2.2 and 3.6 µm the quarter-micron distribution falls with slope −3.8, so its 3.6 µm scattered light is 15 per cent of its 2.2 µm light; the one-micron distribution falls with slope −2.7, the five-micron one with −0.8, close to grey.

Scattered light has a colour, and the colour reports the grains; the dust is not lost light but moved light, and where it moves each wavelength is set by the grains’ size. Small grains scatter infrared light with nearly the steepest possible wavelength dependence, falling almost as the inverse fourth power between 2.2 and 3.6 microns: their scattered light is strongly blue. Grains grown to a micron scatter with a slope of about −2.7 over the same range; grains of five microns are nearly grey. Measuring scattered light at two infrared wavelengths therefore gives a colour, and the colour is a measure of where the size distribution ends — the complement, at the large end, of what the extinction curve does at the small end.

A colour ratio that reads off the size of the largest grains. The ratio of scattered light per unit dust mass at 3.6 to 2.2 µm and at 4.5 to 3.6 µm, logarithmic, against the size of the largest grains. For interstellar grains ending at a quarter of a micron, 3.6 µm light is scattered 0.152 as efficiently as 2.2 µm light; for grains grown to a micron, 0.27; for three microns, 0.63. The ratio rises steadily with the largest grains' size and levels off only when they are several times larger than both wavelengths. A measured ratio of scattered light between two infrared bands — corrected for the different depths the two wavelengths penetrate and for the spectrum of the light being scattered — is a measure of where the size distribution ends.
Fig. 3 The ratio of scattering per unit dust mass at 3.6 to 2.2 µm and at 4.5 to 3.6 µm, against the radius of the largest grains. For the diffuse-medium quarter-micron cut-off, 3.6 µm light is scattered 0.15 as efficiently as 2.2 µm light; for grains to a micron, 0.27; to three microns, 0.63. The ratio rises steadily and levels off only when the largest grains exceed both wavelengths.

The ratio is not a clean measurement in practice. The light a core scatters depends on the light falling on it — the interstellar radiation field, which has its own spectrum and its own direction, dominated by the bright Galactic centre and the plane — and on how deep into the core each wavelength penetrates. A ratio of two observed brightnesses is corrected for both before it can be read against the curve, and the corrections are model-dependent. But the curve says what the ratio would mean, and its steepness between a quarter of a micron and a few microns is why even an imperfect measurement is informative.

A window in the infrared

Scattered light has to get out of the cloud to be seen, and a dense core is opaque at most wavelengths.

The wavelengths at which a dense core is transparent enough to shine. The optical depth through a dense core against wavelength, both logarithmic, for hydrogen columns of 1 × 10²², 5 × 10²², 2 × 10²³ cm⁻², from the average interstellar extinction law. Scattered light can escape a core only where its optical depth is less than about one: the dashed line. At visible wavelengths a core of 5 × 10²² cm⁻² has an optical depth of 24, and the light scattered inside it never gets out; the depth reaches one at 1.5 µm, 4.2 µm, 9.8 µm for the three columns. The mid-infrared bands at 3.6 and 4.5 µm are the window in which dense cores are both illuminated through and seen through — and it is only there that coreshine appears. The extinction law used is the diffuse interstellar one; inside a core with grown grains the infrared extinction is flatter, which raises the optical depth at these wavelengths and moves the window towards longer ones.
Fig. 4 The optical depth through a dense core against wavelength, for hydrogen columns of 10²², 5 × 10²² and 2 × 10²³ cm⁻², from the average interstellar extinction law. Scattered light escapes only where the depth is below about one (dashed): for 5 × 10²² cm⁻², the visible depth is 24 and the depth reaches one at 4.2 µm. The 3.6–4.5 µm bands (shaded) are the window in which cores are both lit through and seen through.

A core with a hydrogen column of five times ten to the twenty-two per square centimetre — typical of the densest parts of a starless core — has an optical depth in visible light of about twenty-four. Light scattered inside it at those wavelengths is scattered again and again and absorbed before it can escape; the core is a dark silhouette, and at its edges it glows faintly in scattered light only from a thin skin. In the near-infrared at two microns the depth is still several, and the core is dark at its centre. Only in the mid-infrared, near three and a half to five microns, does the depth fall to about one, so that starlight from outside penetrates to the centre of the core, is scattered there, and escapes to the observer.

That window explains when the effect appears. At shorter wavelengths the core is too opaque for its centre to glow; at longer ones, beyond about five microns, the scattering by any plausible grains has fallen too far, and in addition emission from very small grains heated by starlight begins to dominate the images, along with a strong absorption feature of silicates near ten microns. The glow of a core’s centre in scattered light is therefore a mid-infrared phenomenon — and only possible at all if the grains there scatter mid-infrared light, which quarter-micron grains barely do.

What was seen

The glow was found in 2010, in images taken at 3.6 and 4.5 microns by the Spitzer Space Telescope. Dense cores that appear as dark silhouettes against the background at visible and near-infrared wavelengths appeared at 3.6 microns as faint, diffuse brightenings following the cores’ shapes, peaking towards their centres. The effect was named coreshine. A survey of about a hundred cores found it in roughly half, with its presence depending on where the cores are in the Galaxy — commonly seen in cores in the direction of the Galactic centre, whose bright radiation field supplies the light to be scattered, and absent from many in other directions.

Models that fit the observed brightness and its distribution across a core require grains of about a micron or larger in the cores’ dense interiors. Grains with the diffuse-medium upper cut-off produce no detectable glow at 3.6 microns; with the cut-off extended to a micron or more, the predicted brightness matches. The number the figures give for the change in scattering per mass, a factor of thirty between the two, is the heart of that argument: there is no reasonable way to make the diffuse-medium grain population glow at those wavelengths, whatever the illumination.

The same growth in the polarisation

Grain growth leaves a second signature, in the polarised light of the same cores. Non-spherical grains spin and align with their long axes across the magnetic field, so that starlight passing through them is polarised along the field and their own thermal emission across it. The mechanism now thought to align them — radiative torques, in which the anisotropic starlight falling on an irregular grain spins it up — works only for grains comparable in size to the wavelength of the light falling on them. Deep inside a dense core, the light has been reddened by the dust outside; only long wavelengths reach the centre, and only large grains can be aligned by them.

That predicts a fall in the polarisation fraction towards the centres of cores, where small grains are no longer aligned, and such a fall — a “polarisation hole” — is widely observed. The observed polarisation that remains at the centres of the densest cores therefore requires aligned large grains, and in some cores it requires grains of about a micron, the same size coreshine does. Two independent observables, one in scattered light and one in polarised emission, reaching the same grain size from different physics is a stronger argument than either alone — and the polarised emission, as the cosmologists who have to remove it know, is the foreground that traces the field through exactly the grains whose sizes are in question.

Where the mass is

The grains that hold the mass, and the light that reports them. For size distributions ending at 0.25, 1, 3, 10 µm, the share contributed by grains larger than 0.5 µm to three things: the dust mass, the visual extinction, and the scattering at 3.6 µm. With grains up to 1 µm, the large grains hold 31 per cent of the mass but make only 17 per cent of the visual extinction, while they make 86 per cent of the 3.6 µm scattering. With grains up to 10 µm the large grains hold 79 per cent of the mass. The extinction curve is written by the small grains, which hold little of the mass; the mid-infrared scattered light is written by the large ones, which hold most of it. Only the second can say how much solid material a cloud has gathered into grains big enough to begin building planets.
Fig. 5 For size distributions ending at 0.25, 1, 3 and 10 µm, the share from grains larger than half a micron of the dust mass, the visual extinction and the 3.6 µm scattering. With grains up to a micron, the large grains hold 31 per cent of the mass, make 17 per cent of the visual extinction and 86 per cent of the 3.6 µm scattering; with grains up to ten microns they hold 79 per cent of the mass.

The last figure puts the three quantities side by side. With grains up to a micron, the grains above half a micron hold nearly a third of the dust mass, make only a sixth of the visual extinction, and make nearly nine-tenths of the scattering at 3.6 microns. The visual extinction is dominated by the smaller grains, which is why the extinction curve reports the small end; the mid-infrared scattering is dominated by the large ones, which is why coreshine reports the large end. And as the distribution extends, the large grains’ share of the mass grows towards four-fifths, while their share of the visible extinction stays small. The observable that sees the large grains is the one that sees the mass.

This is the same division that appears in a debris disc, where the mass is in the large bodies and the light in the small ones, turned round. In a debris disc the small grains are bright because they present the most area for their mass; here, at mid-infrared wavelengths, the small grains are invisible because their area is spent on a wavelength far larger than they are. What an observation weighs depends on the wavelength it is made at compared with the objects that make it, and a size distribution has to be read through several wavelengths, each weighting a different end.

What the grains’ growth means

A grain that grows from a tenth of a micron to a micron in a dense core does so by colliding with others and sticking, helped by the ice mantles that coat grains in cold, dense gas and make them stickier. The time it takes depends on the density, the grains’ relative velocities — set by the turbulence in the core — and how efficiently they stick, and simple estimates give hundreds of thousands to a million years at the densities of starless cores. That is comparable to the lifetimes of the cores themselves, measured from how many are seen compared with how many young stars, and it means growth to a micron is plausible before collapse but not guaranteed. Coreshine’s presence in some cores and absence from others may be partly a matter of illumination and partly a matter of age.

If the grains do reach a micron before collapse, the young star’s disc begins with grains ten times larger than the interstellar ones. The next steps in forming planets — growth to millimetres, then to pebbles, then to bodies large enough to hold together by gravity — face a series of barriers where colliding grains bounce or shatter rather than stick, and where growing bodies drift inwards and are lost to the star. A head start of a factor of ten in size at the beginning shortens the first steps and changes where in the disc the barriers bite. How big the grains are when the core collapses is, in that sense, an initial condition for planet formation, and coreshine is the only direct measurement of it made before a disc exists.

What the figures leave out

The grains drawn are spheres of a single refractive index, non-absorbing, with a power-law size distribution. Real grains in cores are aggregates — fluffy clusters of smaller grains stuck together, coated with ice — and aggregates scatter differently from compact spheres of the same mass: they present more area and scatter more efficiently at long wavelengths for the same mass, which would lower the grain size needed to explain a given brightness. The ice mantles change the refractive index and add absorption features, one of them at three microns, close to the 3.6-micron band. Carbonaceous grains absorb as well as scatter. And the size distribution in a core need not be a single power law at all; growth by coagulation tends to deplete the smallest grains and pile material up at the large end, making the distribution curved rather than straight.

The optical depths use the diffuse interstellar extinction law, which the grain growth being argued for would itself change; the infrared extinction of cores is indeed measured to be flatter than the diffuse law, as larger grains would make it. And the illumination is treated as a given, when in practice it is the least certain part of the modelling: the radiation field at 3.6 microns in the direction of each core, including the contribution from the Galaxy behind it, has to be estimated before the scattered brightness can be read as a grain property.

Still open: how large, and how many

Coreshine establishes that micron-sized grains exist in some dense cores; it does not establish how the size distribution continues beyond them, or what fraction of the dust mass has grown. Grains of tens or hundreds of microns would be grey even at 3.6 microns and would be detected instead by their thermal emission at millimetre wavelengths, whose spectral slope flattens when grains exceed about a tenth of a millimetre — a change that has been reported in some protostellar envelopes and disputed in others, since the slope also depends on the grains’ temperature and composition. Whether the grains in the densest starless cores have already grown that far, or whether the micron is roughly where growth stands at the moment of collapse, is what the combination of scattered light at several infrared wavelengths and emission at several millimetre wavelengths is now being used to measure, core by core.

Where this goes next

The extinction curve of a mixture, and of single sizes. Extinction relative to the V band against inverse wavelength, from the infrared at 0.3 inverse microns to the near ultraviolet at 3.3. The thick grey curve is the observed Galactic average, the Cardelli–Clayton–Mathis law at Rᵥ = 3.1. The coloured curves are computed: MRN, 0.005–0.25 µm, Rᵥ = 2.4; all grains 0.05 µm, Rᵥ = 1.8; all grains 0.3 µm, Rᵥ = 18.4. Grains of any single size make a curve with a knee: steep where the wavelength is longer than the grain, flat where it is shorter. Only a spread of sizes, with many more small grains than large ones, makes the nearly straight rise with inverse wavelength that is observed, and the first model follows the observed curve to an r.m.s. of 0.19 in these units. The ultraviolet bump beyond this range needs a separate population of small carbonaceous grains and is not modelled. A size read from the slope of a colour Interstellar dust dims blue light more than red, and the way the dimming rises toward the blue is nearly a straight line in inverse wavelength. No grain of any single size makes that line. A grain only removes light selectively when it is comparable in size to the wavelength, so the extinction curve read across wavelength is the population of grains read across size — and it says there are vastly more small grains than large ones, in the same proportion that grinding rocks together produces. The light dust removes at 0.56 µm comes back at 124 µm. One energy budget drawn twice, on a wavelength axis spanning four and a half decades. The upper curve is starlight from a 6500 K photosphere; the shaded region under it is the part removed by a magnitude of visual extinction, computed from the same CCM law the other modes here draw, and weighted towards the ultraviolet exactly as that law says. The curve on the right is what the grains do with it: a modified blackbody at 20 K with an emissivity index of 1.8, normalised so that the energy under it equals the energy under the shaded region. Integrating the two curves actually drawn returns a ratio of 1.000. The starlight peaks at 0.56 microns and the re-emission at 124, a factor of 220, so nothing about the two is recognisable as the same photons and everything about them is the same joules. The practical consequence is a rule about arithmetic: a galaxy's ultraviolet luminosity and its far-infrared luminosity are not two independent measurements of how many young stars it has. One is the light that escaped and the other is the light that did not, and adding them without noticing counts part of the population twice. The dust is not lost light, it is moved light An extinction curve says how much starlight dust removes. It does not say the light is gone. Every photon a grain absorbs heats the grain, which radiates it back out at twenty kelvin — so an extinction measurement and a far-infrared spectrum are two halves of one energy budget, and the area under them is the same area. The critical mass falls as the cloud contracts, with a slope of −0.50. The Jeans mass — the least mass of molecular gas that its own pressure cannot support — against number density, for gas at 10, 20, 50 K, both axes logarithmic. The curves are straight because the mass goes as T^(3/2) ρ^(−1/2), and the slope measured off the drawn 10 K curve is −0.500 against the −0.5 that exponent requires. The consequence is the one that matters and it is a matter of sign: a cloud collapsing at constant temperature moves to the right along one of these lines, so the mass it takes to be unstable keeps falling, and sub-regions that were individually stable when the collapse began become individually unstable during it. A giant molecular cloud at 100 particles per cubic centimetre has a Jeans mass of 99 solar masses; a dense core at 10⁵ has one of 1.70. That is why a cloud of ten thousand solar masses makes a cluster rather than a star, and why the question about star formation is not what makes gas collapse but what stops the fragmenting. The cloud that cannot hold itself up A cold cloud collapses when gravity beats pressure, and the mass at which that happens falls as the square root of the density — so the collapse makes the condition for collapse easier, over and over, until the gas can no longer get rid of the heat. That is why a cloud of ten thousand solar masses makes a cluster and not a star.

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CoreshineDense coreExtinction curveGrain growthGrain size distributionInterstellar dustOptical depthRayleigh scattering