Astronomy & Celestial Mechanics
the sky, computed
Starlight

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.

Assumes Extinction, Interstellar medium and Opacity.

16 min read 9 figures Light is the only messengerMeasuring the unreachableScales that defeat intuition

The word extinction is a small lie. Nothing is extinguished. A photon absorbed by an interstellar grain deposits its energy in the grain, the grain’s temperature rises until it radiates as fast as it absorbs, and the energy leaves again — at a wavelength two or three orders of magnitude longer, and in a direction that has nothing to do with where the photon was going.

So the extinction curve and the far-infrared spectrum of a region of the sky are not two independent measurements. They are the two sides of one equation.

Treating them as one equation rather than as two observations changes what can be asked. It means an infrared measurement constrains what an ultraviolet one is missing; it means a galaxy’s total output can be assembled without seeing all of it; and it means an internal check exists — if the absorbed and emitted energies fail to balance, something has been left out of the census.

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.
Fig. 1 One energy budget drawn twice, over four and a half decades of wavelength. The shaded region is the starlight removed by a magnitude of visual extinction, weighted towards the ultraviolet exactly as the extinction law requires; the curve on the right is what the grains do with it, a modified blackbody at twenty kelvin normalised so the two areas agree. Integrating the drawn curves returns a ratio of one. The starlight peaks near half a micron and the re-emission near a hundred and twenty-four — nothing about the two is recognisable as the same photons and everything about them is the same joules.

Why twenty kelvin

A grain in interstellar space absorbs starlight and radiates. Its equilibrium temperature is set by balancing the two, and the balance has an asymmetry in it that decides the answer.

Absorption happens at optical and ultraviolet wavelengths, where the grain — a tenth of a micron across — is comparable to or larger than the wavelength, and absorbs efficiently. Emission happens in the far infrared, where the grain is a hundred times smaller than the wavelength it is trying to radiate at, and is therefore a poor emitter: a body much smaller than the wavelength radiates far less than a blackbody of the same area.

That inefficiency is what makes the grain cold. To radiate the absorbed power at a wavelength where it is a hundred times worse than a blackbody, it has to be hotter than a blackbody would be — but it also has the enormous advantage of the fourth power of temperature working the other way, and the net result is an equilibrium at fifteen to twenty-five kelvin in the general interstellar radiation field.

The temperature is remarkably insensitive to almost everything. Because emission goes as roughly the sixth power of temperature for a modified blackbody with β\beta near two, a factor of ten in the intensity of the starlight falling on a grain changes its temperature by only about forty per cent. That is why dust temperatures across an entire galaxy span fifteen to forty kelvin while the radiation field spans four orders of magnitude, and it is also why the temperature is a poor diagnostic and the luminosity is a good one.

The inefficiency is parameterised as an emissivity index: the emitted spectrum is νβ\nu^{\beta} times a Planck function, with β\beta near two for silicates. So the emission is a modified blackbody, and its shape is not that of any body at any temperature.

What the balance is used for

Stated as a conservation law it sounds like bookkeeping. It is an instrument, and it is the only instrument that reaches the star formation that is hidden. The stars whose light is being intercepted are the same ones that make the ionising photons a birth rate is usually counted from, so obscuration hits exactly the population the measurement depends on.

A galaxy forming stars produces most of its light in the ultraviolet, from the small number of massive stars. Those stars form in dense clouds and are heavily obscured, so a large fraction of that ultraviolet never escapes — and the fraction is not small: for a typical star-forming galaxy roughly half of the total stellar output has been absorbed and re-emitted, and for the most vigorous ones it is over ninety per cent.

A survey in the ultraviolet therefore measures the star formation that happened to be visible, and a survey in the far infrared measures the rest. Neither is the total, and the total is the sum — which is why the cosmic history of star formation could not be established until both had been surveyed.

The history of the subject makes the point better than the argument does. Ultraviolet surveys in the 1990s produced a star-formation history of the universe that peaked modestly and was thought to be roughly right. Submillimetre surveys later that decade found a population of galaxies invisible in the optical, radiating hundreds of times the Milky Way’s luminosity entirely in the far infrared, and the accepted history moved by a factor of several at the epochs where obscuration matters most. Nothing about the earlier measurements was wrong; they were measurements of the escaping half.

The light dust removes at 0.15 µm comes back at 69 µm. One energy budget drawn twice, on a wavelength axis spanning four and a half decades. The upper curve is starlight from a 25000 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 35 K with an emissivity index of 2, 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 0.15 microns and the re-emission at 69, a factor of 468, 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.
Fig. 2 The same construction for a hotter star, more extinction and warmer grains — the conditions inside a star-forming region rather than in the diffuse medium. The absorbed fraction is far larger, the re-emission peaks at seventy microns rather than a hundred and twenty, and the energy balance is unchanged. The dust temperature is a measurement of the radiation field’s intensity, not of anything about the dust, which is why a starburst’s dust is warm and a quiescent disc’s is cold.

The two curves are not the same curve

There is a distinction here that is easy to slide past and that matters a great deal in practice.

An extinction law describes what happens to a single star behind a screen of dust: every photon either passes or does not, and the attenuation is a clean exponential in the column. An attenuation law describes what happens to a whole galaxy, where the stars and the dust are mixed, and where some stars are in front of the dust, some behind it, and some inside it.

The two laws are different, and not only in normalisation. Scattering returns some photons to the line of sight; the least obscured stars dominate the escaping ultraviolet, so the effective law is greyer than the true one; and the geometry of the mixture matters more than the properties of the grains.

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.
Fig. 3 The extinction law itself, in the clean case. It rises steeply into the ultraviolet — which is why reddening and extinction are one measurement — and carries a bump at 2,175 ångströms that appears on almost every sight line in this galaxy and whose carrier is still not agreed. That the bump exists at all is one of the strongest constraints on what the grains are made of.

There is one more asymmetry that makes the infrared side the more reliable of the two. Absorbed energy has only one place to go, so the far-infrared luminosity is a nearly assumption-free measurement of how much starlight was intercepted. The ultraviolet side is the opposite: what escapes depends on geometry, on clumping and on the viewing angle, so a correction applied to an ultraviolet flux is a correction for a quantity that is not a property of the galaxy at all. A brightness is a distance only if something is known, and here the something is the arrangement of the dust.

The reddening vector, at E(B−V) = 0.45. A main sequence in a colour–magnitude diagram, and the same sequence behind 0.45 magnitudes of colour excess. Dust moves a star down and to the right along a fixed direction whose slope is R_V = 3.1 by definition — 1.40 magnitudes of dimming for 0.45 of reddening. That direction is not parallel to the sequence, which is what makes it possible to say how much of a star's faintness is distance and how much is dust.
Fig. 4 And how the law is measured. A reddened star moves along a vector in a colour–magnitude diagram whose slope is set by the ratio of extinction to reddening, and that slope is not parallel to the main sequence — so a star’s intrinsic colour and its reddening can be separated. Doing it star by star across the sky is how an extinction map is built where nothing was measured.

The consequence for the energy budget is that the attenuation law cannot be assumed. A galaxy’s infrared luminosity is a direct measurement of the absorbed energy and its ultraviolet slope is an indirect one, and comparing the two is how the attenuation law of a distant galaxy is measured rather than adopted. Where only the ultraviolet is available — which is the situation for the faintest and most distant sources — the correction rests on an assumed law and is the largest uncertainty in the result.

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.
Fig. 5 The extinction law for three kinds of dust, spanning a wider range of RVR_V than the standard set: 2.0, the diffuse interstellar medium at its steepest, up to 6.0, the interiors of dense clouds. The curves are normalised at V and fan out in the ultraviolet, where the difference between them is a factor of two. RVR_V is a measurement of grain size, and the reason it is quoted with every extinction correction is that using the wrong one is a systematic that grows towards the blue.

Using the emission as a mass

If the emission is a modified blackbody, its normalisation is a mass — because the amount of radiating material and the amount of light are proportional once the temperature and the emissivity per unit mass are known.

That is how nearly every dust mass in the literature is obtained, and through it, most gas masses. The dust-to-gas ratio in this galaxy is about one to a hundred by mass, and if that ratio holds elsewhere then a far-infrared measurement of a distant galaxy is a measurement of how much interstellar gas it contains — which is otherwise obtainable only from a molecular line, at a hundred times the observing cost, with its own conversion factor to argue about.

The chain has three assumed numbers in it: the emissivity per unit mass of the grains, the dust-to-gas ratio, and the temperature. All three are calibrated locally and applied globally, and the second is known to vary with metallicity by an order of magnitude. It is a serviceable measurement and it is not a clean one, and it is worth knowing which of the numbers in a paper came through it.

There is a version of the argument that avoids the worst of the assumptions, and it works because dust emission is optically thin at long wavelengths: the far-infrared and submillimetre emission escapes from the whole column rather than from a surface, so the measurement is genuinely of the total, with no hidden interior. That is the opposite of the situation in the optical, where what is measured is a surface and everything behind it is invisible.

The reddening vector, at E(B−V) = 0.45. A main sequence in a colour–magnitude diagram, and the same sequence behind 0.45 magnitudes of colour excess. Dust moves a star down and to the right along a fixed direction whose slope is R_V = 5.5 by definition — 2.48 magnitudes of dimming for 0.45 of reddening. That direction is not parallel to the sequence, which is what makes it possible to say how much of a star's faintness is distance and how much is dust.
Fig. 6 The reddening vector drawn for RV=5.5R_V = 5.5 rather than 3.1. The vector is steeper: the same colour excess corresponds to a larger extinction in V, because AV=RVE(B−V)A_V = R_V E(B-V) and that is what RVR_V means. A colour excess is not an extinction until a value of RVR_V has been assumed, and the difference between 3.1 and 5.5 is nearly a factor of two in the correction applied to every magnitude.

Where the grains come from and what they are

The grains are made in the outflows of evolved stars and in supernova ejecta — the mass a star does not keep is where most of them condense — and they are processed in the interstellar medium by shocks and by ultraviolet photons, and destroyed by both. The grains carry a substantial fraction of the heavy elements. Comparing the abundance of an element measured in the gas phase against its total cosmic abundance shows a deficit — a depletion — and the missing atoms are in the solid phase. Iron is depleted by a factor of a hundred, carbon and oxygen by factors of a few, and the pattern of depletions is a composition measurement of a solid nobody has a sample of. It is the same style of inference as reading a composition from what is missing in a spectrum, applied one level up: the absences in the gas phase are the presences in the solid one.

The budget at the scale of the universe

Integrate the argument over everything and it produces a number that can be checked against an entirely independent measurement.

All the starlight ever emitted either escaped or was absorbed by dust and re-emitted. Both components accumulate as diffuse backgrounds — the optical and near-infrared background from the escaped light, the far-infrared background from the reprocessed part. Measuring both and adding them gives the total energy released by stars over cosmic time.

The check is not trivial to make. Both backgrounds are diffuse, faint and sitting under foregrounds far brighter than themselves — zodiacal light for the optical, the galaxy’s own dust emission for the far infrared — so measuring either is a subtraction of two large numbers. The far-infrared background was established first, from a satellite designed for the microwave background, and the optical one is still the harder of the two.

The two components turn out to be comparable. Roughly half of all the starlight ever produced has been through a grain. That is a striking statement about a component that makes up under one per cent of the mass of the interstellar medium and about a ten-thousandth of the mass of a galaxy.

The reddening vector, at E(B−V) = 0.45. A main sequence in a colour–magnitude diagram, and the same sequence behind 0.45 magnitudes of colour excess. Dust moves a star down and to the right along a fixed direction whose slope is R_V = 3.1 by definition — 1.40 magnitudes of dimming for 0.45 of reddening. That direction is not parallel to the sequence, which is what makes it possible to say how much of a star's faintness is distance and how much is dust.
Fig. 7 The same construction in a denser line of sight — three magnitudes of visual extinction per kiloparsec rather than 1.8. Every star is displaced further along the same vector and the vector’s direction is unchanged. That constancy of direction is what makes the correction possible: the amount of dust varies enormously between sight lines and the shape of what it does varies far less, so one law fitted once corrects for a quantity nobody measured.

Where the picture stops

One temperature is not enough. A real sight line contains grains of many sizes in radiation fields of many intensities, so the emission is a superposition of modified blackbodies. Fitting a single one returns a luminosity-weighted temperature that is biased warm, and the mass inferred from it is biased low — by factors of two in the worst cases, because the coldest grains hold most of the mass and contribute least to the emission.

The smallest grains are not in equilibrium at all. A grain of a few hundred atoms has so little heat capacity that a single ultraviolet photon raises its temperature by hundreds of kelvin, after which it cools before the next photon arrives. Those grains radiate in spikes at near- and mid-infrared wavelengths rather than at an equilibrium temperature, and the mid-infrared emission features that dominate many galaxy spectra come from them.

And β\beta is degenerate with temperature. The emissivity index and the dust temperature are both fitted from the shape of the same curve, and they trade off against each other almost exactly. Published dust masses depend on which was held fixed, and comparisons between studies that made different choices are not comparisons of the same quantity.

Two further points belong to the budget rather than to the list of caveats, and the second is not about light at all.

The floor a distant grain cannot get below

There is a limit on how cold interstellar dust can be, and at high redshift it becomes the dominant complication in every measurement of the kind this essay describes.

A grain is in equilibrium with everything that irradiates it, and one of the things irradiating it is the microwave background. Locally that is a 2.7-kelvin bath and it is irrelevant, since grains sit at fifteen to twenty-five. At redshift six the background is at nineteen kelvin, and at redshift ten it is thirty.

So a grain at high redshift cannot be colder than the background it sits in, and the observed dust temperatures at those epochs are bounded below by a number that rises steeply with redshift.

There is a second and more awkward effect. What a telescope measures is the contrast between the source and the background behind it, and if the dust is only slightly warmer than the background then most of its emission is indistinguishable from the background and is subtracted away along with it.

The correction for that is large and it goes the wrong way: the observed flux underestimates the true emission, by a factor that depends on the dust temperature, which is the quantity being measured. A cold, high-redshift galaxy can have most of its dust emission hidden this way, and the corrections applied in the literature range from a few per cent to a factor of two depending on the assumed temperature.

The energy budget’s cleanest half acquires a systematic at exactly the epoch it is most wanted for, and it is a systematic that pushes every inferred obscured star-formation rate downward.

Neither correction is optional at the redshifts where obscured star formation matters most.

The reddening vector, at E(B−V) = 0.45. A main sequence in a colour–magnitude diagram, and the same sequence behind 0.45 magnitudes of colour excess. Dust moves a star down and to the right along a fixed direction whose slope is R_V = 3.1 by definition — 1.40 magnitudes of dimming for 0.45 of reddening. That direction is not parallel to the sequence, which is what makes it possible to say how much of a star's faintness is distance and how much is dust.
Fig. 8 And the same reddening line carried out to twelve kiloparsecs. Distant stars pile up at the far end with several magnitudes of extinction, which is why the Galactic plane is opaque in the visible beyond a few kiloparsecs and why the map of the Galaxy was made at twenty-one centimetres and in the infrared. The correction is linear in distance and the error in it is too, so the furthest points on this line are the least trustworthy, and that is the practical limit of optical astronomy in the disc.

The surface the molecules are made on

The grains do one more job that has nothing to do with light, and the whole sequence this collection describes depends on it.

Molecular hydrogen cannot form efficiently in the gas phase. Two hydrogen atoms colliding have nowhere to put the binding energy — a two-body collision cannot conserve both energy and momentum and end in a bound state — so the reaction requires a third body, and at interstellar densities three-body collisions essentially never happen.

What happens instead is that a hydrogen atom sticks to a grain surface, migrates across it, meets another, and the two combine — with the grain absorbing the binding energy and the resulting molecule desorbing.

So the dust is the catalyst for the molecular medium, and the rate at which a cloud becomes molecular is proportional to the amount of dust in it. That is why molecular gas and dust track each other closely, and why the two are used interchangeably as tracers of each other.

It also explains an observed threshold. A cloud becomes molecular only where the dust column is enough to shield the interior from the ultraviolet radiation that would dissociate the molecules faster than the grains can make them, and the transition happens at a visual extinction of about one magnitude. Below that a cloud is atomic; above it, molecular.

The consequence runs through the whole subject. Low-metallicity systems have less dust per unit gas, so they form molecules more slowly and shield them less effectively, so their star formation is inefficient in a way that has nothing to do with the availability of gas.

The component that carries one per cent of the mass decides whether the other ninety-nine can cool, and the energy budget this essay is about is one of two jobs it does.

A last consequence of the energy balance is worth drawing out, because it is what makes the far-infrared sky worth surveying at all. The luminosity a galaxy emits in the infrared is not an addition to its starlight; it is a part of that starlight, moved. So the ratio of the two — infrared to ultraviolet — is a measurement of how much of the galaxy’s star formation is hidden, and it can be made without resolving anything. A galaxy’s obscured fraction is a photometric quantity, which is why the census of star formation across cosmic time is an infrared census and why it disagreed with the optical one until the infrared existed.

One more reading covers the cool end, where the re-emitted light is nearly all of what is seen.

The light dust removes at 0.92 µm comes back at 175 µm. One energy budget drawn twice, on a wavelength axis spanning four and a half decades. The upper curve is starlight from a 4000 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 15 K with an emissivity index of 1.5, 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 0.92 microns and the re-emission at 175, a factor of 191, 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.
Fig. 9 The absorbed and re-emitted energy for cool dust around a cool star. The emission peak moves to longer wavelengths as the fourth root of the heating and the integrated energies still balance exactly, which is the accounting the essay is about: nothing is lost, and everything is moved.

The accounting closes exactly, which is what makes the correction trustworthy in principle and difficult in practice: the light that was removed is measurable, and it is measurable at a wavelength a thousand times longer and with a completely different instrument.

Where this ladder goes next

Later rungs on this anchor: the grain size distribution and how it is inferred from the shape of the extinction curve; stochastic heating and the mid-infrared features; the attenuation law as a function of galaxy type and inclination, and what an assumed law costs at high redshift; depletion patterns as a composition measurement; and the far-infrared background, where the budget above is closed against an observation of the whole sky.

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 Bouguer line, and the intercept nobody observed. Instrumental magnitude against airmass for one star of magnitude 10 outside the atmosphere, observed at 5 airmasses in 5 bands. Each slope is that band's extinction coefficient, computed from Rayleigh scattering, an aerosol term and ozone rather than assumed: U 0.493, B 0.246, V 0.126, R 0.061, I 0.027 magnitudes per airmass. Every line is fitted through its points and extended to X = 0, and the intercept there is the published magnitude — a measurement made at an airmass no observation is ever taken at, because the smallest airmass available is 1 and that is already a whole atmosphere. Two consequences follow and neither is a detail. The slope has to be re-measured every night, because the aerosol term changes with the weather and is not a property of the site. And the U-band line is 3.9 times steeper than the V-band one, so the extrapolation is 3.9 times longer in exactly the band where photons are scarcest — which is why ultraviolet photometry from the ground was always the least trustworthy part of a magnitude system. A magnitude measured where nothing was measured Every published brightness is an extrapolation off the end of a graph. A star is observed through one atmosphere at least, a line is fitted against airmass, and the number quoted is its intercept at zero — a place no observation is ever taken from. Two densities in thermal balance at one pressure, 121 times apart. Thermal equilibrium for interstellar gas, drawn as pressure against density with both axes logarithmic. Every point on the curve is a temperature between 40 and 9000 K at which cooling exactly balances the 2·10⁻²⁶ erg per second per hydrogen nucleus that grain photoelectrons deliver. The curve is not monotonic: it rises to 5007 K cm⁻³, falls to 1597, and rises again, so a horizontal line anywhere between those two crosses it three times. At the 3000 K cm⁻³ of the local medium the three crossings are a warm phase at 0.47 cm⁻³ and 6354 K, a cold phase at 57 cm⁻³ and 52 K, and one in between drawn dashed because it cannot survive — a parcel there that is squeezed cools and keeps contracting, and one that expands heats and keeps expanding. The two survivors differ by a factor of 121 in density and by exactly the same factor in temperature — necessarily the same, because their product is the pressure both are held at — and yet they press on one another equally, which is why they can share the same volume of the disc indefinitely rather than mixing. Two temperatures, and nothing in between The gas between the stars is not one gas at one temperature. Thermal balance has three solutions at any ordinary interstellar pressure, the middle one cannot survive being nudged, and the two that can differ by a factor of over a hundred in both density and temperature — which is why a map of the interstellar medium looks like clouds rather than like fog.

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.

Attenuation lawDust temperatureDust to gas ratioEmissivity indexEnergy balanceFar infrared backgroundGrain alignmentInfrared excessInterstellar dustModified blackbodyObscured star formationRadiative equilibrium