Astronomy & Celestial Mechanics
the sky, computed
Starlight

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.

Assumes Extinction, Photometric systems and Collisional cascade.

15 min read 6 figures Light is the only messengerMeasuring the unreachable

Starlight that crosses a cloud of interstellar dust arrives dimmer and redder. The reddening is what makes the dimming correctable: a star’s intrinsic colour is known from its spectrum, the excess redness measures how much dust lies in front of it, and the total dimming follows from a fixed ratio between the two. That ratio, and the whole curve of dimming against wavelength that it summarises, is called the extinction law, and for most sight lines in the Galaxy it is the same curve, scaled.

The curve has a shape worth looking at. Drawn against inverse wavelength — frequency, in effect — the extinction rises from near zero in the infrared, climbs almost in a straight line through the visible, bends, shows a bump in the ultraviolet at 2175 ångströms, and rises again steeply into the far ultraviolet. The straight part is the puzzle, because it is not what any single kind of particle does.

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.
Fig. 1 Extinction relative to the V band against inverse wavelength, through the infrared and visible to the near ultraviolet. The thick grey curve is the observed Galactic average. A population of grains all of one size makes a curve with a knee — rising, then flattening or even falling — at a position set by the size. Only a mixture, with many more small grains than large ones, produces the nearly straight observed rise.

A grain sees the wavelength in units of itself

The interaction of light with a small particle depends on one number, the ratio of the particle’s circumference to the wavelength, x=2πa/λx = 2\pi a/\lambda. A grain much smaller than the wavelength is nearly invisible: it scatters and absorbs weakly, with an efficiency that rises as the fourth power of its size relative to the wavelength. That is Rayleigh scattering, the process that makes the daytime sky blue and that steepens the transmission spectrum of a hazy exoplanet. A grain much larger than the wavelength is simply opaque; it removes the light that falls on it and, by diffraction around its edge, an equal amount again, so its efficiency tends to two whatever the wavelength.

How efficiently one grain removes light, against its size. The extinction efficiency Q of a single spherical grain of refractive index 1.5 — its extinction cross-section divided by its geometric area — against the size parameter 2πa/λ, the grain's circumference in wavelengths, on a logarithmic axis, in the anomalous diffraction approximation. A grain much smaller than the wavelength hardly interacts: Q rises as the fourth power of size, the Rayleigh regime that makes the sky blue. Near a size parameter of 4.1 it peaks at 3.17, and for grains much larger than the wavelength it settles toward 2 — twice the geometric area, because a large grain removes the light it blocks and diffracts as much again around its edge. Only grains comparable to the wavelength remove light selectively by colour.
Fig. 2 The extinction efficiency of one spherical grain — its cross-section for removing light over its geometric area — against the grain’s circumference in wavelengths, for a refractive index of 1.5. Small grains barely interact, rising as the fourth power of size; the efficiency peaks at about 3.2 near a circumference of four wavelengths and settles toward 2 for large grains. Only in the middle, over a factor of ten in size, does a grain remove more blue light than red.

The approximation drawn — anomalous diffraction, due to van de Hulst — ignores absorption and is exact only for materials whose refractive index is close to one. For silicates it misplaces the details by tens of per cent. What it gets right, and what matters here, is that selectivity lives in a band of sizes. A grain of 0.02 microns reddens light strongly but removes very little of it; a grain of two microns removes a great deal of light, equally at all colours. Reddening, which is extinction that depends on colour, is done by grains a few tenths of the wavelength across.

It follows that each wavelength sees a different part of the grain population. Light at 0.3 microns is selectively dimmed by grains around 0.1 microns; light at one micron by grains of a few tenths; light at two microns by grains of a micron, if there are any. A grain population of a single size would redden only across the band of wavelengths near its own size and be grey or transparent elsewhere — the knee in the first figure.

A power law of sizes makes a straight line

The observed curve is nearly straight over a factor of three in wavelength, which means the grains that dominate each wavelength must be present in the right proportion to each other at every size. In 1977 John Mathis, William Rumpl and Kenneth Nordsieck showed that a simple power law does it: a number of grains per unit radius falling as a−3.5a^{-3.5}, from about five thousandths of a micron to about a quarter of a micron, mixing silicate and graphite grains. The distribution has come to be known by their initials.

The power law works because it makes the contribution per logarithmic interval of size nearly scale-free. The number of grains per logarithmic interval of radius goes as a−2.5a^{-2.5}, and each grain’s geometric cross-section as a2a^2, so the geometric area per logarithmic interval falls only as a−0.5a^{-0.5} — slowly. Every decade of size contributes an area within a factor of three of its neighbours, and since each decade of size dominates its own decade of wavelength, the extinction per decade of wavelength changes slowly too. A nearly power-law distribution of sizes projects into a nearly power-law extinction curve, and in the visible that is a nearly straight line against inverse wavelength.

Which grains do the dimming at each wavelength. How the extinction of an MRN grain population, 0.005 to 0.25 microns, is shared among grain sizes at 0.15 µm, 0.3 µm, 0.55 µm, 2.2 µm, per logarithmic interval of radius. Each wavelength is dominated by grains whose radius is about 0.6 of the wavelength, where the efficiency peaks — or by the largest grains present, if those are smaller: the extinction per size interval peaks at radii of 0.085 µm, 0.169 µm, 0.245 µm, 0.245 µm respectively, and half of it comes from grains smaller than 0.071, 0.117, 0.144, 0.155 microns. So the extinction curve read across wavelength is the size distribution read across radius: the ultraviolet counts small grains, the visible the largest few tenths of a micron, and the infrared sees only the tail of the largest, in the Rayleigh regime where the curve becomes a power law.
Fig. 3 How the extinction at four wavelengths is shared among grain sizes, per logarithmic interval of radius, for the power-law mixture. At 0.15 microns it peaks near 0.085-micron grains; at 0.3 microns near 0.17. At 0.55 and 2.2 microns the dominant grains are the largest present, at the quarter-micron cut-off: those wavelengths are longer than any grain can match, and the extinction they see is the Rayleigh tail of the biggest grains.

This is the precise sense in which the extinction curve is the size distribution read in a mirror. The ultraviolet counts small grains, the visible counts the largest few tenths of a micron, and the infrared counts nothing selectively — it sees only the tail of the largest grains, deep in the Rayleigh regime, where the extinction falls as a power of wavelength regardless of the details. That is why the infrared part of the extinction law is nearly universal from one sight line to another while the ultraviolet part varies: the infrared measures a limit, the ultraviolet measures a population.

One number that weighs the largest grains

The optical extinction law is usually summarised by a single number, RVR_V, the ratio of the total extinction in the V band to the reddening between B and V. It is about 3.1 averaged over the diffuse interstellar medium and rises to five or six in dense molecular clouds. The change is the most common variation of the law, and a model of grain sizes says at once what it means.

The ratio of total to selective extinction, set by the largest grains. Rᵥ, the ratio of total extinction in V to the reddening B − V, computed for an MRN size distribution from 0.005 microns up to a largest grain size aₘₐₓ, against aₘₐₓ on a logarithmic axis. Small grains redden strongly and dim little in the visible, so a distribution cut off at small sizes has a low Rᵥ; adding large grains adds grey extinction that dims without reddening, and Rᵥ rises. The diffuse Galactic average, Rᵥ = 3.1, comes out at aₘₐₓ ≈ 0.35 microns; the value in dense molecular clouds, about 5.5, at aₘₐₓ ≈ 0.55 microns. A single number measured from the reddening of stars is therefore an estimate of the size of the largest grains along the line of sight — grains a thousand times smaller than a human hair's width, weighed by the colour of the starlight behind them.
Fig. 4 RVR_V computed for a power-law grain mixture from 0.005 microns up to a largest radius amax⁡a_{\max}, against that largest radius. Small grains redden without dimming much, so a distribution cut off low has a small RVR_V. Adding large grains adds grey extinction, which dims without reddening, and RVR_V rises: in this approximation to 3.1 at a largest radius near 0.35 microns and to 5.5 near 0.55.

In this simplified model the largest grain radius needed for the diffuse average comes out a little larger than the quarter micron of the original fit, because the approximation underweights the absorption of real grains. The trend is what survives any refinement: RVR_V is a measurement of the largest grains along the line of sight. In dense clouds, where grains collide gently at low speeds, coagulate, and grow icy mantles, the largest grains are larger, and RVR_V rises accordingly. The extinction law’s variation across the Galaxy is a map of where grains grow.

That turns a number measured from the colours of stars behind a cloud into a statement about particles a fraction of a micron across, a thousand times smaller than the width of a hair, inside a cloud hundreds of light years away. The polarisation of the same starlight gives an independent reading: its peak wavelength, measured through a sequence of filters, scales with the size of the aligned grains, and the ratio of RVR_V to that wavelength is close to the one the grain models predict.

The dense-cloud case can be drawn directly, as a test of the reading.

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 = 5.5. The coloured curves are computed: power law to 0.55 µm, Rᵥ = 5.5; power law to 0.25 µm, Rᵥ = 2.4. Extending the power law to larger grains flattens the curve and raises Rᵥ; the first model follows the observed curve to an r.m.s. of 0.07 in these units. The ultraviolet bump beyond this range needs a separate population of small carbonaceous grains and is not modelled.
Fig. 5 The observed extinction law for RV=5.5R_V = 5.5, typical of dense molecular clouds, against two power-law mixtures. Extending the distribution to 0.55 microns flattens the computed curve onto the dense-cloud law; the diffuse-medium distribution, stopping at a quarter micron, rises far too steeply toward the blue. The whole difference between the two laws is carried by grains larger than a quarter micron.

The flatter curve is what grain growth looks like from outside. Nothing about the small grains has to change: the same power law, extended to larger sizes, reproduces the dense-cloud law, because the extra grains contribute nearly grey extinction that dilutes the reddening without changing its shape at short wavelengths. The ultraviolet part of the dense-cloud law is also weaker relative to the visible, which requires that some of the smallest grains have been removed — stuck onto larger ones in the slow collisions of a cold cloud — and that is the other half of growth.

Where the mass is

The dust’s mass sits in a different place from its extinction. A grain’s mass goes as the cube of its radius, so the mass per logarithmic interval of radius, for the power-law distribution, goes as a0.5a^{0.5} — it rises toward large sizes. Half the mass of the distribution is in its largest grains.

Where the dust's mass is, and where its extinction is. The share of the dust's mass per logarithmic interval of radius for an MRN distribution up to 1 microns, which rises as the square root of radius so that half the mass is in grains larger than 0.29 microns, against the shares of the extinction at 0.15 µm, 0.3 µm, 0.55 µm, 2.2 µm. The visible extinction is carried mostly by grains of 0.03 to 0.2 microns; the mass is carried by the largest grains present, which contribute a nearly grey and colourless extinction. A dust mass inferred from reddening therefore depends on a cut-off the reddening barely constrains.
Fig. 6 The same shares of extinction, now for a distribution extending to one micron, drawn against the share of the dust’s mass per logarithmic size interval, shaded. Half the mass is in grains larger than about 0.3 microns, where the extinction at visible wavelengths is already nearly grey. The mass rises toward the cut-off whatever the cut-off is.

That is an uncomfortable property for a measurement. The mass of dust in a cloud, or in a galaxy, is often inferred from its extinction, and the extinction is dominated by grains whose total mass is a minority. The grains that hold the mass add almost uniformly to the dimming at every colour and so leave almost no trace in the reddening. A dust mass computed from reddening therefore depends directly on where the size distribution is assumed to stop, a number the reddening barely constrains. The far-infrared emission of the same grains weighs them more fairly, since every grain radiates the energy it absorbs, but the emissivity of a large grain at long wavelengths has its own uncertainty of a factor of two.

Why dust is not the sky

The daytime sky is blue for the reason that makes the smallest grains redden: particles much smaller than the wavelength scatter as its inverse fourth power. Air molecules are a ten-thousandth of a micron across, far into that regime, and the Earth’s atmosphere extinguishes starlight with a steep λ−4\lambda^{-4} law that makes a star near the horizon orange. Interstellar dust extinguishes with a law close to λ−1\lambda^{-1} in the visible, four times shallower in its exponent, and that difference is the whole of the information about grain sizes. A law as steep as the sky’s would mean grains much smaller than the wavelength; a flat law would mean grains much larger; a law that is exactly in between says the grains straddle the wavelength.

That is also why interstellar reddening is so much weaker than a sunset’s for the same dimming. A star behind a magnitude of interstellar dust in V is dimmed by about 1.3 magnitudes in B — its colour changes by a third of a magnitude. A star seen through the same dimming by air near the horizon changes colour by far more, because Rayleigh scattering discriminates so strongly. The ratio RVR_V of about three, for dust, would be below one for pure Rayleigh scattering — about 0.7; the grains are larger than molecules by a factor that the ratio measures.

And the same grains, aligned in the Galaxy’s magnetic field, polarise the starlight most strongly at wavelengths near their own size, and glow in polarised millimetre light that is now the chief obstacle to measuring the early universe. The size distribution read from the extinction curve is the input to both calculations, and the grain models that fit extinction, polarisation and emission together are the tightest constraints on what interstellar dust is made of.

The exponent that grinding makes

The exponent −3.5-3.5 appears elsewhere in astronomy, and the coincidence is not a coincidence.

In 1969 J. S. Dohnanyi calculated the size distribution of a population of bodies that collide and fragment, each collision breaking bodies into smaller pieces with a distribution of fragment sizes, the population as a whole in a steady state in which mass flows from large sizes to small. If the strength of a body does not depend on its size, the steady-state distribution is a power law with exponent −3.5-3.5. The asteroid belt and debris discs around other stars follow it over the sizes where that assumption holds, and it is the reason those discs are bright in small grains and massive in large bodies.

Interstellar grains are not in a steady collisional cascade of the same kind; they are made in the winds of cool giant stars and the ejecta of supernovae, shattered in the shock waves of supernova remnants, grown in dense clouds and destroyed again when the clouds disperse. But shattering in shocks is a fragmentation process, and a population repeatedly shattered and regrown drifts toward the distribution a cascade produces. That the dust between the stars carries the same exponent as the rubble of the asteroid belt is evidence that interstellar grains, too, are ground down by collisions for most of their lives.

The part of the curve that needs something else

Beyond the range the figures draw, the extinction curve does two things a population of quarter-micron silicates cannot. At 2175 ångströms it rises into a broad bump, centred at the same wavelength to within a per cent on almost every sight line in the Galaxy while its width and strength vary. And beyond about 1700 ångströms it climbs again, steeply, into the far ultraviolet. Both require grains much smaller than the ones the visible curve counts — a few thousandths of a micron, a few hundred atoms — and the bump requires a specific material with a resonance at that energy.

The constancy of the bump’s position is the strongest clue. A resonance whose wavelength does not move while the particles’ sizes and environments vary must be intrinsic to the material, not to the particle size, and the favoured candidates are small graphitic grains or large aromatic molecules, whose delocalised electrons resonate near that energy. The same molecules, heated by single ultraviolet photons to hundreds of kelvin for an instant, are thought to produce the emission bands that dominate the mid-infrared spectra of star-forming galaxies.

So the complete extinction curve is at least two populations read together: silicate and carbonaceous grains of a few hundredths to a few tenths of a micron for the visible and near-infrared, and a separate population of very small carbonaceous particles for the ultraviolet. The power law with its single exponent is a description of the first, and the observation that fixes its exponent is the straightness of the visible curve — which is the part of the extinction law every correction for dust relies on.

What the model leaves out

The grains here are spheres of one refractive index, with no absorption. Real interstellar grains are at least two populations — silicates and carbonaceous material — of irregular shape, partly aligned, with complex refractive indices that absorb strongly in the ultraviolet. The 2175-ångström bump, which the model does not reproduce, is attributed to small carbonaceous grains or large aromatic molecules with a resonance at that wavelength, and it varies independently of RVR_V in ways no single-parameter description captures.

The power law itself is a fit, not a law. Modern models of interstellar dust, constrained by extinction, emission and the abundances of the elements missing from the gas, use distributions that are curved rather than straight, with a separate population of very small grains and molecules responsible for mid-infrared emission bands. They agree with the power law over the sizes the optical curve constrains, a few hundredths to a few tenths of a micron, and differ outside it.

And the whole construction assumes that the extinction is measured along a line to a single star. For a galaxy seen from outside, the light comes from stars mixed with the dust at every depth, some heavily obscured and some not, and the effective curve — the attenuation law — is flatter than the extinction law of any single sight line, for geometric reasons that have nothing to do with the grains.

The observation behind the curve

An extinction curve is measured by the pair method: a reddened star is compared with an unreddened star of the same spectral type, and the ratio of their fluxes at each wavelength, after allowing for distance, is the extinction at that wavelength. The ultraviolet part came from satellites; the infrared from ground-based and space photometry. Every point on the observed curve in the first figure is an average over hundreds of such pairs. The size distribution is then an inference from that curve through a model of how particles remove light — a model that is exact in principle, since it is Maxwell’s equations applied to a sphere, and uncertain in practice, since the grains are not spheres and their materials are not certain.

What the pair method measures directly, with no grain model at all, is that the curve’s shape varies, that the variation is correlated with the cloud’s density, and that it is summarised well by one number. That the number is a size is the inference, and it is an inference from the physics of how a particle sees a wave, which is not in doubt.

Still open: how large the largest grains get

The size distribution’s upper end is the part the extinction curve reads worst, because large grains are grey, and it is the part that holds the mass and seeds the growth of planetesimals in discs around young stars. Observations of dense cores in scattered mid-infrared light — a glow called coreshine, which requires grains of about a micron to scatter efficiently at those wavelengths — suggest that grains grow to a micron or more before the cloud has collapsed into a star. If they do, the first step of planet formation begins inside the interstellar cloud rather than inside the disc, and the size distribution’s upper cut-off is the first measurement of it. The extinction curve cannot settle it, since grains of a micron are grey across the whole visible range; what can is the scattered light, the far-infrared and millimetre emission of the same cores, whose spectral slope flattens when grains grow past a hundred microns, and the polarisation of that emission, which changes its sign of alignment when grains become large enough to scatter their own thermal light.

Where this goes next

The extinction law, for three kinds of dust. How much of a star's light dust removes, against inverse wavelength, normalised to one at the V band. Blue light is extinguished more than red, which is why reddening and extinction are the same measurement — and the hump at 4.6 inverse microns is a feature of the grains themselves, present on almost every sight line and still without an agreed carrier. Larger grains give a flatter law and a larger R_V. Dust makes everything look further away A star behind dust is fainter, so a distance taken from its brightness comes out too large. The same dust also makes it redder, and the reddening is measurable where the dimming is not — which is the only reason the correction can be applied at all. 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. Mass at the top, area at the bottom, 10 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 10 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.5. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.5, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.5, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 10⁵ times the area of the large end and 10⁻⁵ times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line. Where the mass is and where the light is A population that grinds itself up settles into a size distribution with a fixed slope, and that slope puts almost all the mass in the largest bodies and almost all the cross-section in the smallest. So a debris disc's brightness measures a population whose mass it says nothing about, and the two are connected only by the exponent.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Collisional cascadeExtinctionExtinction efficiencyGrain size distributionInterstellar dustRayleigh scatteringReddening