A falling polarisation and the angle that says why

Interstellar dust grains are elongated, and in the Galaxy’s magnetic field they spin with their long axes across the field lines. Their thermal glow is therefore polarised perpendicular to the field, and a map of that polarisation is a map of the field’s direction projected on the sky — the only such map that exists for the cold, dense gas where stars form. The polarised glow of the diffuse Galaxy reaches twenty per cent and more, which says that the grains that glow are the grains that align and that they align very well.

Into the clouds, the polarisation falls. Plotted against the column density of gas along the sightline, the polarisation fraction declines almost everywhere it has been measured, from twenty per cent in the thinnest gas to a few per cent at the column densities of molecular clouds and below one per cent in the densest cores. On a logarithmic plot the decline is close to a straight line, and its slope is the number every study of it reports: about minus a half in some clouds, closer to minus one in others, and in between in most. The same slope appears in the polarisation of starlight shining through clouds, where it was measured first.

The slope looks like a measurement of something, and it is natural to read it as one. It is in fact the sum of two quite different processes, which produce the same kind of decline for different reasons — and the figures below show that the slope alone cannot separate them.

A sightline is a sum #

The polarisation measured along a sightline is the sum of the polarised emission from every piece of dust along it, and polarisations add as vectors. The quantities that add are the Stokes parameters $Q$ and $U$, which encode the polarised intensity and twice the polarisation angle, and the measured fraction is the length of the summed $(Q, U)$ vector divided by the total intensity. Two cells polarised along the same direction add their polarisation; two polarised at right angles cancel it.

The opening figure builds four clouds from that rule. Each cell of dust, a column of $10^{21}$ atoms per square centimetre, polarises its own emission by twenty per cent along its own local field direction. The clouds differ only in how the cells’ contributions are oriented and weighted, and each mechanism has a distinctive fingerprint in how the summed fraction depends on how many cells the sightline crosses.

The first cloud has a single uniform field and every grain aligned. Every cell adds its full contribution in the same direction, the vector grows in proportion to the column, and the fraction stays at twenty per cent at every column density. This is the baseline against which every decline is measured, and no real cloud shows it.

The walk makes the mechanisms visible. The fraction measured at the end is the distance from the origin divided by the number of steps. A sightline through a uniform field walks straight. A sightline through a tangled field takes steps of the same length in directions that vary, and arrives somewhere shorter. A sightline through a cloud whose interior grains have stopped aligning takes its steps only in the outer layers and stands still while it crosses the dark middle, arriving at the same place it would have reached through a thin cloud — and the fraction falls because the denominator, the total emission, kept growing while the numerator did not.

A decline first seen in starlight #

The fall was found before anyone could measure the polarised glow at all. In 1949 William Hiltner and John Hall, looking for polarisation from eclipsing binaries, found instead that the light of distant stars was polarised by a per cent or two by the dust it had crossed, in directions that lined up across large stretches of sky. Within a few years that was understood as the dust’s alignment with a Galactic magnetic field, and for forty years starlight through dust clouds was the only way to map such fields.

When the technique was pushed into the dark clouds in the 1980s and 1990s, the background stars dimmed by many magnitudes of extinction showed less polarisation per magnitude than the stars at the clouds’ edges. The polarisation of starlight is a measurement of the extinction the dust imposes and of how much of it is aligned; the ratio of the two, polarisation per magnitude, fell as the extinction rose, roughly as its inverse square root. Those sightlines were called polarisation holes, and they raised the question this essay is about, in its oldest form: whether the dust inside dark clouds stops aligning, or whether the field there is too tangled for the alignment to show.

Starlight could not settle it. A star behind ten magnitudes of extinction is ten thousand times fainter than it would otherwise be, and the stars bright enough to measure through the densest cores are few. The polarised thermal glow of the dust itself, measured at submillimetre wavelengths, has no such limit — the densest part of a core is the brightest part of its emission — and it is in emission that the slopes near minus one have been found.

A tangled field levels off, unless there is no field to level at #

The second cloud’s field varies from cell to cell by forty degrees about a mean direction, the kind of turbulent tangling a molecular cloud’s internal motions would produce. Its fraction falls at first, as cells pointing in different directions cancel one another, and then stops falling. The mean direction is still there, and every cell contributes, on average, a component along it; the components perpendicular to it cancel, but the component along it accumulates in proportion to the column just as in a uniform field. The fraction levels off at the share of the polarisation the mean direction carries — for Gaussian tangling with an angular spread $\delta$, a factor $e^{-2\delta^2}$ below the cells’ own, which is 7.5 per cent here.

Only when the mean direction is gone altogether does the fall continue indefinitely. A field with no preferred direction adds its cells as a pure random walk, and the length of a random walk of $n$ equal steps grows as $\sqrt{n}$. Divided by $n$, the fraction falls as $n^{-1/2}$: the minus-one-half slope. That is the slope usually quoted for depolarisation by tangling, and the figure says it is the limiting case. A turbulent field with any mean component flattens out once the column is large enough for the mean to dominate, and the column at which that happens depends on how strong the tangling is.

This matters for reading a measured slope of minus a half. It requires either no mean field across the cloud — which the large-scale order seen in the polarisation maps of the Galaxy argues against for most clouds — or a mean field whose direction itself changes over the depth of the cloud, which is plausible for a cloud deep enough to include several turbulent structures stacked along the line of sight.

Grains that stop aligning fall faster #

The fourth cloud has an orderly field and does something else. Its grains align only where starlight reaches.

The process that aligns interstellar grains is now thought to be radiative torque: an irregular grain scatters an anisotropic beam of starlight unevenly, the recoil spins it up, and the spinning grain’s angular momentum then settles along the magnetic field. The mechanism needs light, and specifically light with wavelengths comparable to the grain’s size. Deep inside a cloud, where the visual extinction from the surface is a few magnitudes, that light is gone — absorbed or scattered into other directions by the same grains — and grains there spin slowly and randomly and stop contributing polarisation.

A sightline through such a cloud receives polarised emission only from the starlit skin on its near and far sides, and unpolarised emission from the whole column. As the cloud thickens, the polarised part stays fixed and the total grows, so the fraction falls as $1/N$: a slope of minus one. Every grain in the interior is still glowing, so the dust still weighs the column correctly; it has simply stopped saying anything about the field.

Every slope, by either route #

Real clouds are not one of these idealisations. They have fields tangled to some degree about a mean, and grains whose alignment falls off with depth more gradually than a step. What matters is whether the measured slope picks out which of the two dominates.

It does not. Tangling about a mean takes the slope over a fixed range of column from zero, for an orderly field, toward minus a half as the tangling grows — reaching minus 0.42 at an angular spread of 69 degrees, by which point the mean direction is almost gone. A skin of aligned grains takes it from minus one, when only the outermost layer aligns, toward zero as the aligned depth approaches the middle of the cloud. Any slope between zero and minus a half can be made either way. A cloud measured at minus 0.4 might be a tangled cloud with all its grains aligned, or an orderly cloud whose interior grains have stopped aligning a few magnitudes in, or any mixture of the two.

Only the steepest slopes are diagnostic. A slope steeper than about minus 0.6 cannot be produced by tangling of any degree, and needs the grains to stop aligning. Starless cores, the dense centres where stars are about to form and where the cloud’s own support is giving way, have shown slopes near minus one in polarised starlight and in emission, and those are among the strongest evidence that alignment fails inside them. Most of the slopes measured in clouds are shallower than that, and for them the slope is a statement about two unknowns at once.

The angle that separates them #

The two mechanisms lower the polarisation fraction for different reasons, and the difference shows up in a quantity the fraction discards: the polarisation angle.

A sightline through a tangled field ends its walk in a direction that depends on the particular cells it crossed, and neighbouring sightlines, crossing different cells, end in different directions. The angle wanders across a map. For a field tangled about a mean, the wandering shrinks as the column grows, because the mean dominates the sum; for a field with no mean, the angle is random at every column. A sightline through a cloud whose interior has stopped aligning ends its walk exactly along the field, however thick the cloud, because the only steps it took were in the starlit layers where the field is orderly. Its fraction falls and its angle does not move.

That is the test, and it is the one the maps make possible. A polarisation map gives the angle at every point, and the local spread of the angle — measured over a small circle around each point, a statistic the Planck analyses called the dispersion function — can be compared with the polarisation fraction point by point. Across the whole sky, those analyses found that the two are anticorrelated: where the polarisation fraction is low, the angle wanders more. That is the signature of tangling, and it was read as evidence that, over most of the diffuse Galaxy, the fraction is lowered mainly by the field’s structure along the sightline rather than by any failure of the grains.

The same relation also measures how ordered the field is, which is what the Davis–Chandrasekhar–Fermi method converts into a field strength. A cloud whose depolarisation turns out to be loss of alignment rather than tangling would give a misleadingly small angle spread at high column — not because its field is strong and orderly, but because its interior is silent. The method assumes that the angles it measures come from every depth; the skin model says they may come only from the outside.

Two clouds with one slope #

The decisive case is a pair of clouds the slope cannot tell apart.

On the left, the two polarisation curves lie almost on top of each other; a measurement of the fraction against column, with any realistic scatter, would not separate them. On the right, they are fifty degrees apart. The first cloud’s angles are random; the second’s are steady to a degree. A map of either would show the fraction falling into the cloud. Only the angle map would say whether the field was tangled or the grains had gone dark.

In real clouds the separation will rarely be that clean, because both mechanisms act together, and because a cloud’s projected field can genuinely curve across it, which adds angle structure unrelated to either. But the direction of the test is clear. The angle spread at high column density, compared with the angle spread at low column density in the same cloud, measures how much of the depolarisation is tangling. Whatever is left over in the fraction is alignment loss.

Where the knee sits, and what it would measure #

If the angle test shows a cloud’s depolarisation to be mostly alignment loss, the fraction-against-column curve stops being a nuisance and becomes a measurement.

A skin model has a knee: the column below which the whole sightline is starlit and the fraction keeps its full value, and above which it falls. The knee sits at twice the depth to which aligning light penetrates — the skin on both sides — and so it measures that depth directly. Radiative-torque theory predicts the depth from the grains’ sizes and the spectrum of the radiation that reaches them: larger grains are aligned by redder light, which penetrates further, so a cloud whose grains have grown by coagulation should align deeper. A knee measured at a few magnitudes of extinction in one cloud and at ten in another would then say something about grain growth in the second.

That reading assumes the field is orderly enough not to contribute its own depolarisation, which is exactly what the angle spread checks first. The two measurements together — the fraction’s knee and the angle’s spread — are the minimum that separates the field’s structure from the grains’ physics in a single cloud.

What the model leaves out #

The cells in these figures are independent and identical, and three simplifications follow from that.

Real turbulence is correlated over a range of scales, not organised in independent cells. A field whose correlation length is comparable to the cloud’s depth does not average down the way the walk does; a sightline crosses only a few independent structures, and the fraction can fall and then level off at a column set by the correlation length rather than by the tangling amplitude. The analyses that fit polarisation maps with correlated turbulent fields find the same qualitative behaviour with different numbers.

The field’s inclination to the sky changes the fraction too. A field pointing towards the observer produces almost no polarisation in projection, and a cloud whose field bends from the plane of the sky to the line of sight over its depth would show a falling fraction with a steady angle — which looks like alignment loss in the angle test. The test separates tangling from silence, not silence from geometry.

And the dust properties themselves may change into the cloud, and the instrument’s own polarisation, which can exceed the sky’s, has to be removed to a fraction of a per cent before a fraction of one per cent at a core’s centre means anything. Grains that grow, acquire ice mantles or change shape polarise differently per unit column, and the cell polarisation of twenty per cent is a property of diffuse dust that is not guaranteed in a dense core.

Still open: how deep the grains keep aligning #

The slope of polarisation against column density is the most commonly reported number about dust polarisation in clouds, and the figures say it is two numbers fused. Any slope between zero and minus a half can be a tangled field or a dark interior; only slopes near minus one require the interior to be dark.

The angle spread separates them, and the instruments that would do it cloud by cloud now exist: the balloon-borne and airborne polarimeters and the large submillimetre interferometers map polarisation angles at resolutions fine enough to measure the local spread inside individual cores. Where both the fraction and the spread have been measured, cores show the angle steady while the fraction falls, which favours alignment loss at their centres. Whether that is general — whether every dense core goes dark to its field beyond a few magnitudes, or whether some keep aligning to their centres because their grains have grown large enough to use the redder light that gets there — is what decides how far into a forming star the field can be traced at all.