A direction measured by something with no strength in it

In 1949 two groups looking for something else found that starlight is polarised, by a per cent or two, and that the polarisation is larger for more reddened stars. The obvious inference — that the polarising agent is the dust doing the reddening — was immediate. The next inference took longer and is the subject of this essay: dust polarises light only if the grains are lined up, and the only thing available to line them up over hundreds of parsecs is a magnetic field. That is the same reasoning by which a cloud’s field is inferred from a splitting nobody can see: the field is never the observable, only the one agent capable of producing what is.

That accident gave the subject its first map of the galactic field, and the technique it started is still the only one that reports the field’s direction in the plane of the sky.

Why an aligned grain polarises #

An elongated grain absorbs more efficiently when the electric field of the light lies along its long axis. Light passing through a cloud of aligned grains therefore loses more of the component along the long axes, and what emerges is polarised along the short axis — that is, along whatever direction the long axes are not.

The same grain, radiating its own heat in the far infrared, emits more efficiently along its long axis. So the thermal emission is polarised parallel to the long axes, which is perpendicular to the polarisation of the transmitted starlight.

That perpendicularity is the technique’s internal check. A cloud observed in optical polarisation of background stars and in far-infrared polarisation of its own emission should give field directions that agree, with the two polarisation angles ninety degrees apart. They do, which is a stronger statement than it sounds: it confirms that the same grains are doing both jobs and that the alignment is with respect to a common axis.

The two halves also complement each other operationally. Absorption polarimetry needs background stars, so it works in diffuse regions and fails where the extinction is so high that nothing shines through; emission polarimetry needs the dust to be warm enough to radiate, so it works in dense regions and is swamped by the background in diffuse ones. Between them they cover the whole column density range, and the overlap is where the ninety-degree check is made.

There is one more geometric fact that governs everything below. The polarisation depends on the projected field, and it is proportional to the square of the sine of the angle between the field and the line of sight. A field pointing at the observer produces no polarisation at all, and no amount of alignment changes that — so a low polarisation fraction is ambiguous between a disordered field, a poorly aligned population and a field pointing the wrong way.

What actually does the aligning #

The alignment mechanism took fifty years to settle and the wrong answer held the field for most of them.

The original proposal, from Leverett Davis and Jesse Greenstein in 1951, was paramagnetic relaxation: a spinning grain with unpaired electron spins dissipates energy as its magnetisation lags the field, and the dissipation drives the rotation axis toward the field direction. The physics is right and the timescale is wrong — for ordinary grains in ordinary clouds it is too slow by orders of magnitude against the randomising effect of gas collisions.

What works is radiative torques. An irregular grain is chiral, so it scatters left- and right-handed light differently, and an anisotropic radiation field therefore exerts a systematic torque on it. That torque spins the grain up to rotation rates far above thermal, and a suprathermally rotating grain is stiff against collisional disorientation. Its angular momentum then precesses about the magnetic field — because the grain is slightly magnetised — and settles with the angular momentum along the field, which puts the long axis across it.

The mechanism has a testable consequence that is unusual for this subject: alignment requires a radiation field, so it should fail deep inside dense cores where starlight cannot penetrate. It does. Polarisation fractions fall in the densest regions, and the effect appears at roughly the depth the radiative torque theory predicts — one of the few clean confirmations in the whole area.

It has a second consequence that is stranger and also confirmed. The torque depends on the grain being larger than the wavelength doing the torquing, so alignment should be efficient for large grains and absent for small ones — and the observed polarisation should therefore be produced by a different, larger population than the extinction as a whole. The wavelength of peak polarisation is indeed longer than a naive reading of the extinction curve would give, and the offset is in the right direction and of roughly the right size.

What is worth carrying from all this is that the alignment is with the field rather than with anything mechanical. A grain in a wind lines up with the flow; a grain in a magnetised medium lines up with the field, because the precession about the field is far faster than any other torque can reorient it. The field wins by being fast rather than by being strong, which is the same argument that makes the adiabatic invariants of a trapped particle hold.

The direction is honest and the strength is not there #

What comes out of a polarisation map is a set of directions on the sky: at each point, the projected orientation of the field, with a hundred-and-eighty-degree ambiguity because a direction and its reverse polarise identically.

Nothing in that map is a field strength. A cloud with a one-microgauss field perfectly aligned and a cloud with a one-milligauss field perfectly aligned give the same picture. The polarisation fraction is not a strength either: it depends on the grain shape, the alignment efficiency, the grain composition and how much of the field lies along the line of sight, and those unknowns are not separable.

That is a limitation and it is also the technique’s strength, because a measurement that reports only geometry cannot be biased by a wrong field strength. The morphological results — hourglass shapes around collapsing cores, fields perpendicular to dense filaments and parallel to diffuse ones, ordered fields across whole spiral arms — are robust in a way that no number derived from them is.

The scatter is the measurement #

The strength is recovered from a quantity the map contains almost by accident: how much the directions disagree from point to point.

The argument is due to Davis and, independently, to Chandrasekhar and Enrico Fermi, both in 1953, and it is a two-line estimate. Turbulent motions in a magnetised medium bend the field lines. A stiff field is bent less. The angular displacement of a field line is roughly the ratio of the turbulent velocity to the Alfvén speed, and the Alfvén speed is the field strength divided by the square root of the density.

Rearranged, the field strength is the square root of the density times the turbulent velocity divided by the angular dispersion. Every quantity on the right is measurable: the density from molecular line ratios or dust emission, the turbulent velocity from the non-thermal width of a spectral line, and the angular dispersion from the polarisation map itself.

For a core at ten thousand molecules per cubic centimetre with a kilometre-a-second line width and a ten-degree dispersion, the answer is a few tens of microgauss — which is the right order, and which agrees with the Zeeman detections where both exist.

It is worth noticing the shape of the inference, because it recurs. The observable that carries the information is a dispersion rather than a mean, and dispersions are usually the part of a dataset that gets called noise. Here the mean of the angles is the geometry, which is interesting, and the scatter about it is the strength, which is more interesting still. A survey that reported only the mean orientation per cloud would have thrown the field strength away without noticing.

The same logic supports a refinement that is now standard. Rather than a single dispersion, the angle differences are computed as a function of separation on the sky — a structure function — which separates the large-scale ordered field from the turbulent component and gives the correlation length of the turbulence as a by-product. That is a considerable amount of physics extracted from a map of line segments.

The factor of a half #

The estimate as stated overestimates the field, and by a lot. Written with no correction it gives values two or three times too high, and the standard practice is to multiply by a factor of about a half.

That factor is not derived. It comes from running the analysis on simulations where the field is known, and asking what number would have to be inserted to recover it.

The reasons it is needed are understood in outline. The polarisation angle at a point on the sky is an average along the line of sight through many turbulent cells, and averaging reduces the observed dispersion below the true one, which biases the field high. The telescope beam averages further, in the plane of the sky. And the derivation assumes small angular displacements, which fails exactly where the dispersion is large enough to measure well.

It is worth being explicit about what that means for the status of the result. A field strength from this method is a calibrated quantity, not a derived one, and the calibration is against numerical experiments whose turbulence may or may not be the turbulence in a cloud. Quoting three significant figures from it is not defensible; quoting an order of magnitude, and a comparison between two clouds analysed identically, is.

The comparison is the part worth defending. Systematic errors that are common to two clouds cancel in their ratio, so a statement that one core is twice as strongly magnetised as another is far better supported than either absolute value. Most of what the method has actually established is of that form: dense filaments are more strongly magnetised than their surroundings, cores at the ends of filaments more than cores in the middle, and the field strength rises with density in the same way the Zeeman relation says it does.

That last agreement is worth its own sentence. Two techniques with nothing in common — a circular polarisation in a radio line, and the scatter of angles in a far-infrared map — return the same power-law relation between field and density over four decades. Neither is trustworthy alone at the factor-of-two level, and the agreement between them is much better than that.

What the maps have actually shown #

Set the numbers aside and the morphology has been decisive in several arguments.

Diffuse interstellar gas shows filaments of neutral hydrogen aligned with the field. Dense molecular filaments are oriented across it. The transition happens at a column density around ten to the twenty-one and a half per square centimetre, and the reading is that below it gas flows along field lines and accumulates into field-parallel structures, while above it gravity has won and the material has been able to drag the field with it. The hourglass is the other decisive shape. A core that has begun to collapse drags the field inward with it in the plane perpendicular to the field and not along it, so the field lines pinch — and the pinch has been mapped in a dozen cores. Fitting the hourglass gives the mass-to-flux ratio without the dispersion argument at all, from geometry alone, and the answers agree with the dispersion method to within the factor of two that everything in this subject agrees to.

The galactic-scale result is that the field follows the spiral arms with a pitch angle of about ten degrees, ordered over kiloparsecs, with a turbulent component comparable to the ordered one. The polarisation of starlight established that in the nineteen-fifties from a few hundred stars, and eighty thousand stars later the picture has not changed, in agreement with what a galaxy measured from inside it says about the arms themselves.

External galaxies are where the arm-following is really established, because they can be seen from outside. Radio polarimetry of nearby spirals shows the field tracing the arms, and — the part that surprised people — often strongest between the arms rather than in them, where the gas is smoother and the turbulence has had less opportunity to tangle the field. A quantity that measures order rather than strength will do that, and disentangling the two remains one of the standing difficulties.

Where it goes wrong #

Three failure modes recur and each has produced a wrong published field.

Depolarisation by superposition. If two clouds at different field orientations lie along the same line of sight, their polarisations partly cancel, and the residual points in a direction that is neither — the same superposition problem that makes two temperatures along one sight line hard to separate. Nothing in the data flags this; it is diagnosed by finding an implausibly low polarisation fraction and looking for a second velocity component in a spectral line.

Alignment failure. In the densest cores the polarisation drops, and the drop can be read either as a disordered field or as grains that have stopped being aligned. The two are distinguishable in principle by whether the polarisation fraction falls with column density in the way radiative torque theory predicts, and in practice by observing at a wavelength where the relevant grains still emit.

And geometry. Everything measured is projected, so a field pointing mostly along the line of sight gives a small, noisy polarisation whose direction is nearly meaningless. There is no way to detect that from the polarisation alone; it takes a Faraday rotation measurement, which is sensitive to precisely the component this technique cannot see.

What a map costs to make #

The technique’s other virtue is practical, and it decides what is measured rather than what could be.

An optical polarimeter is a filter and a rotating element in front of an ordinary camera. Measuring one star’s polarisation to a tenth of a per cent takes minutes on a small telescope, and catalogues of tens of thousands of stars have been assembled by exactly that route over seventy years, with instruments that would not now be considered research-grade.

Submillimetre polarimetry of a cloud’s own emission is a different proposition: it needs a cold telescope, a large detector array, and careful control of instrumental polarisation, which is why full maps of nearby clouds only became routine in the last fifteen years. What those maps added was the dense regions, which is where the physics is, and the resulting picture of fields threading filaments is essentially a product of that decade.

The next increment is not resolution but depth. Polarised emission from a cloud is a few per cent of an already faint signal, and mapping the field in a low-mass core at a few hundred astronomical units takes an interferometer working at the limit of its sensitivity. That is the measurement the collapse question needs, and it exists for a handful of objects.

Two instruments that need each other #

The natural summary is that neither magnetic diagnostic is any good alone and the pair is powerful.

Faraday rotation gives a signed, density-weighted line-of-sight component along a path. Grain alignment gives an unsigned, emission-weighted plane-of-sky direction, and — through the dispersion argument, with a calibration factor — a magnitude. The weights are different, the components are orthogonal, and the systematic errors have nothing in common.

That last caveat is real and is the frontier. Rotation measures weight by electron density, which lives in the warm ionised medium; dust polarisation weights by dust emission, which lives in the cold neutral one. Combining them assumes the field is the same in both phases, and there is no strong reason it should be. The honest position is that the galaxy’s field is known in projection in two complementary ways, that the two agree on large scales, and that the assembled three-dimensional field is a model rather than a measurement.

What is not in doubt is the original accident: grains line up, they line up with respect to a field, and a photograph taken through a polarising filter is therefore a magnetic instrument. That was not obvious in 1949 and it remains, seventy-five years later, the cheapest magnetic measurement in astronomy.

There is a structural point in that cheapness which is easy to miss. Every other magnetic diagnostic in this collection needs the field to do something energetic — split a level, precess an atom, rotate a plane, drive an instability — and the signal is correspondingly small and hard-won. Grain alignment needs the field only to supply a direction for something else to line up with, and the something else is a solid particle whose optical cross-section is enormous compared with any atomic transition. The signal is a per cent, not a part in ten thousand, and it comes off a photograph.

That is why the galactic field’s geometry was mapped decades before its strength was measured anywhere, and why the maps have barely changed since. A technique whose sensitivity comes from geometry rather than from a coupling constant does not improve much with better instruments, and does not need to.

The corresponding weakness is the one this essay has laboured: geometry is all it gives. The strength has to be extracted from the scatter, with a calibration factor from simulations, and that step is where every quoted number’s uncertainty lives. A direction is measured; a strength is estimated; and the two should not be quoted with the same number of significant figures. The literature usually does, which is how a calibration factor derived from four simulations acquired three decimal places. The factor is a correction for projection, beam averaging and a small-angle approximation, none of which is known to better than tens of per cent. The discovery itself is a reminder of how the observational side of this subject usually goes. Hiltner and Hall were looking for intrinsic polarisation in early-type stars, expecting a stellar effect; what they found was an interstellar one, correlated with reddening rather than with spectral type, and the correlation was what identified it. The dust that reddens turned out to be a magnetometer, and nobody had gone looking for one.