Two geometries and one bit to choose between them

A galaxy’s magnetic field has a direction as well as a strength, and the direction is the harder of the two to measure and the more informative once measured.

The reason it is informative is that dynamo theory makes a prediction about it. Different modes of a galactic dynamo produce fields with different symmetries, and the symmetries are distinguishable — in principle — by an observation that no photograph can make.

What the two geometries are #

Imagine looking down on a spiral galaxy from above. The field follows the arms — that much is established by polarised radio emission, which shows the field’s plane-of-sky orientation everywhere in a face-on galaxy and shows it tracing the spiral.

The question is which way it points along the arms. In an axisymmetric field the answer is the same everywhere at a given radius: follow the arm around and the field always points the same way relative to the direction of rotation. Rotate the galaxy by any angle about its axis and the field is unchanged.

In a bisymmetric field the answer alternates. On one side of the galaxy the field points outward along the arm and on the other it points inward, so a rotation by a hundred and eighty degrees reverses it. It has the symmetry of a sine wave in azimuth rather than of a constant.

Polarised radio emission cannot distinguish them, and the reason is fundamental rather than instrumental. Synchrotron polarisation reports the field’s orientation modulo a hundred and eighty degrees — an axis, not a vector — because a field and its reverse produce identical radiation. Every plane-of-sky measurement in this subject has that ambiguity, and so does the alignment of dust grains.

The ambiguity is not a defect of the instruments and cannot be engineered away. Synchrotron emissivity depends on the square of the field’s perpendicular component, and grain alignment on the field’s axis rather than its direction, so both observables are even under reversing the field. A quantity that is even in the field cannot report an odd property of it, however precisely it is measured.

There is a third geometry worth naming, because it is the one dynamo theory expects to lose. A field with no large-scale order at all — purely turbulent, correlated over a hundred parsecs and random beyond that — produces polarised emission too, with the polarisation fraction reduced by the averaging. Distinguishing “ordered and axisymmetric” from “disordered” is a question of polarisation fraction, which is measurable; distinguishing the two ordered geometries from each other is not.

Where a sign comes from #

Only one common measurement in astronomy is linear in the magnetic field rather than quadratic, and it is Faraday rotation.

The rotation of a polarisation plane through magnetised plasma is proportional to the field component along the line of sight, with a sign: a field pointing toward the observer rotates one way and a field pointing away rotates the other. Reverse the field and the rotation measure changes sign. So a catalogue of rotation measures across the sky is a map of signs. Each entry is one number, mostly of one sign in one part of the sky and the other elsewhere, and the pattern of the boundaries is the geometry.

The counting is straightforward for a face-on external galaxy. In an axisymmetric field the line-of-sight component is a cosine of the azimuth, which changes sign twice around a circuit. In a bisymmetric field it is a cosine times a second cosine, which changes sign four times. Two against four, from a sequence of pluses and minuses.

The face-on case needs one refinement, because a strictly face-on axisymmetric field would have no line-of-sight component anywhere. What supplies one is the pitch angle: the field follows the arms rather than running in circles, so it has a radial component, and the radial component projects onto the line of sight. The amplitude of the rotation measure pattern is therefore proportional to the sine of the pitch angle — which means the same data that determine the geometry also measure the pitch, and that the measurement is hardest for the galaxies with the tightest arms.

An inclined galaxy is easier for this purpose than a face-on one, because inclination projects the azimuthal field onto the line of sight directly and the signal is far larger. The trade is that an inclined galaxy’s near and far sides overlap in projection, so the disc’s own emission is rotated by its own field before it leaves — the internal-rotation problem again, and the reason moderately inclined galaxies are the preferred targets.

What has been found in other galaxies #

The measurement is easier in external galaxies than in the Milky Way, because a face-on galaxy presents its whole disc at once and the geometry is not confused by the observer sitting inside it.

The results are not clean. Most of the galaxies mapped well enough show fields that are predominantly axisymmetric, a few show bisymmetric components, and several show mixtures — a dominant axisymmetric mode with a bisymmetric perturbation, or an axisymmetric disc with a reversal at some radius.

That is roughly what dynamo theory expects. The axisymmetric mode is the fastest growing in a disc with ordinary turbulence and rotation, so it should dominate after enough e-foldings; bisymmetric modes are excited by non-axisymmetric forcing, such as a bar, a companion or a strong spiral pattern.

The number of galaxies mapped to the necessary depth is small — a few dozen — because each requires polarisation observations at several frequencies over an extended source, and the polarised flux is a few per cent of a signal that is itself faint. That is the real limitation, and it is why a question posed clearly in the nineteen-seventies is still answered statistically.

There is one class of external galaxy where the answer is unambiguous, and it is the interacting ones. A galaxy with a close companion has a strongly non-axisymmetric potential and a field to match, and the polarised structures in interacting pairs are visibly distorted — field lines drawn out along tidal bridges, compressed at the leading edges, and in some cases tracing structures with no counterpart in the light at all. That last point is worth its own note: polarised radio emission sometimes shows a bridge between two galaxies where no gas or starlight is detected, which makes the field a tracer of interaction history rather than merely a passenger — much as a tidal stream records an encounter that the light no longer shows.

The Milky Way, from inside it #

The galaxy the measurement is best made in is the one where interpreting it is hardest.

An observer inside a disc sees every rotation measure as an integral along a path that may traverse several arms, several field reversals and a range of electron densities. The mapping from a sky map of signs to a three-dimensional geometry is an inversion with more unknowns than data.

What has been established, from several thousand extragalactic rotation measures and several hundred pulsars, is that the field is broadly azimuthal, follows the arms with a pitch angle around ten degrees, and reverses at least once inside the solar circle. That is consistent with what the arms themselves look like from inside, and it was arrived at with the same difficulty — an observer with no vantage point. The pulsars are what make the Milky Way tractable at all. Because a pulsar’s rotation measure and dispersion measure give the mean field along the path to a source at a known distance, differencing two pulsars in nearly the same direction gives the field in the segment between them. The line-of-sight integral is differentiated, and reversals can be localised in radius rather than merely detected. That is exactly the trick that makes a pulsar the most valuable single object in this subject: it is the only source at a known place along the path.

Whether there is one reversal or several is disputed, and the dispute is about the electron density model as much as about the field. A reversal and a change in electron density along the path produce similar signatures, and the density model is calibrated on the same pulsars.

The disagreement is substantial rather than technical. Published models of the galactic field over the last two decades have proposed one reversal, two, three and a continuous winding, all fitted to overlapping datasets, and the reduced chi-squared values do not separate them. That is the signature of a problem with more free functions than the data constrain, and the standard response — adding more data of the same kind — has not resolved it.

What would resolve it is a different kind of constraint. Rotation measures of pulsars in the far half of the galaxy, where almost none are known, would double the path length sampled; and rotation measures of extragalactic sources seen through the disc at low latitude, which are hard because the polarised background is depolarised there, would supply the total column independently.

What a dynamo mode actually is #

The word “mode” has been used freely and is worth unpacking, because the two geometries are not arbitrary alternatives but solutions of the same equation.

A mean-field dynamo is described by a linear equation for the large-scale field, with coefficients set by the turbulence and the differential rotation. Like any linear equation on a disc, it has a family of solutions labelled by their azimuthal dependence: no dependence, one wave around the circuit, two waves, and so on. Each grows at its own rate, and after enough time the fastest-growing one dominates whatever the initial condition was. For a smooth, axisymmetric disc the mode with no azimuthal dependence grows fastest, so the axisymmetric field wins. Introduce a non-axisymmetric forcing — a bar, a companion, a strong two-armed spiral — and the one-wave mode is driven directly rather than having to grow, so it can be present at appreciable amplitude even though it would not have won a growth-rate contest.

That is why the observational picture of “mostly axisymmetric, with exceptions in barred and interacting galaxies” is the expected one. It also means the amplitude of the bisymmetric component is a measurement of the forcing rather than of the dynamo — a quantity that reports on the galaxy’s dynamical environment.

The signal the whole inference rests on depends on the arm’s pitch angle, and it is worth seeing at the two ends of the observed range.

Why the answer matters #

It would be reasonable to ask why the geometry is worth this effort when the field strength is not in doubt, and there are two answers.

The first is that geometry is a test of the dynamo and strength is not. The strength is set by saturation, which is a statement about the turbulence; the geometry is set by which mode grew, which is a statement about the dynamo’s structure. So a galaxy’s field pattern is the only observable that reports on how the field is generated. The second is parity. A dynamo can produce a field that is symmetric about the disc’s midplane or antisymmetric — quadrupolar or dipolar, in the usual terms. A quadrupolar field points the same way above and below the plane; a dipolar one reverses. The distinction is again a sign, and it is again invisible to any measurement of a magnitude.

Quadrupolar is what the standard disc dynamo produces, and it appears to be what most galaxies have. Halo fields are a different matter — several edge-on galaxies show X-shaped polarised structures extending out of the plane, and the parity of those structures is not consistently determined. The parity question connects to something more tangible than symmetry, which is why it is pursued. A dipolar halo field has open lines running away from the disc on both sides and is a candidate route for a galactic wind; a quadrupolar one closes back and is not. So the sign that distinguishes them decides whether the field helps a galaxy lose its gas, which is one of the outstanding problems in galaxy evolution and is bound up with what holds a gas disc open in the first place.

The field beyond the optical disc #

One result from the last decade deserves separating out, because it changes what the geometry is a geometry of.

Radio observations sensitive enough to detect polarised emission well outside a galaxy’s visible extent find ordered fields there — at ten to twenty kiloparsecs in galaxies whose stars run out at ten, and in some cases in the gas-rich outer discs where star formation has essentially stopped.

That is awkward for the standard picture. A dynamo runs on differential rotation and turbulence, and the outer disc has plenty of the first and very little of the second, because the turbulence is driven by supernovae and there are no young stars out there. Yet the field is ordered. Two explanations are on offer and neither is settled. The field may have been generated further in and transported outward, by a wind or by the same turbulent diffusion that spreads anything else. Or the outer disc may sustain its own turbulence by a different mechanism — the magnetorotational instability, which needs only rotation and a weak field, and which in a galactic disc grows on the orbital timescale.

The second option is attractive because it closes a loop: the instability requires a field and generates the turbulence that maintains one. Whether it operates at the required amplitude in a galactic disc’s low ionisation is precisely the question, and it is the same question that decides whether a protoplanetary disc’s midplane is active.

What could go wrong with the inference #

Three problems undermine the counting argument, and it is worth being explicit because the clean two-against-four statement above is an idealisation.

The first is that the disc is not the only magnetised medium along the path. A halo field, a local bubble of the observer’s own, and intervening material all contribute rotation measure, and the local contribution in particular is substantial and structured on the sky.

The second is that the emitting and rotating material are mixed. If a galaxy’s own relativistic electrons are inside the same volume as its thermal ones, the observed polarisation is a sum over depth with different rotations, and the resulting angle is not linear in wavelength squared at all — so the fitted rotation measure is not the path integral. The third is that a turbulent field of comparable strength to the ordered one adds a random rotation measure to every line of sight. The ordered pattern has to be extracted from a scatter of the same size, which requires many sources and makes the result statistical.

And two more readings of the kinematic problem the magnetic one sits inside.

What would settle it #

More rotation measures, by a large factor, and better electron densities.

The next generation of low-frequency radio arrays is expected to deliver something like ten million rotation measures over the sky, against the tens of thousands available now. At that density the sky is sampled finely enough that the ordered pattern can be separated from the turbulent scatter by averaging, and the reversals located rather than argued about. More pulsars would help more per object, and the same surveys will find them: a few thousand are known and tens of thousands are expected, distributed through the disc, each supplying a rotation measure at a distance. Their distances come from the same electron density model the field measurement depends on, so a survey that improves the pulsar census improves both halves of the problem at once.

The likely outcome is that the clean dichotomy dissolves. A real galaxy probably has an axisymmetric mode, a bisymmetric perturbation, a radial dependence, a halo component of different parity and a turbulent field comparable to all of them — and the useful question becomes the amplitude of each rather than which one is present.

That is a familiar trajectory and it is not a disappointment. The dichotomy was always a way of asking whether the field is generated by a mechanism with the disc’s own symmetry, and the answer — mostly yes, with exceptions where something breaks the symmetry — is the answer that a dynamo predicts. What remains is to measure the exceptions well enough to say what breaks it.

There is a last thing worth saying about why a one-bit observable is worth this much apparatus, because it looks disproportionate. A rotation measure’s sign costs the same to obtain as its magnitude — the same observation, the same fit, the same source — and the magnitude is the part everybody quotes. The sign is thrown away in most analyses, averaged over, or folded into an absolute value.

But the magnitude is a density-weighted path integral through an unknown medium, carrying a systematic nobody can bound, and the sign is not. A sign cannot be biased by a density model, cannot be inflated by a filling factor, and cannot be wrong by a factor of two. It is either plus or minus, and the only way to get it wrong is to get the polarisation angles wrong by ninety degrees — which is a calibration error, and calibration errors are the kind of thing an instrument can be tested for.

So a catalogue of rotation measures contains two datasets of very different quality sharing one column. The magnitudes are worth a factor of a few; the signs are worth what they say. Building the large-scale field’s geometry out of the second rather than the first is not a limitation being made the best of — it is the correct use of the data.

The same reasoning applies to the parity question above, and to the sign of a Zeeman signal, and to every other place in this collection where a magnetic diagnostic happens to be linear rather than quadratic in the field. Linearity is what buys a sign, a sign is what carries geometry, and geometry is what a dynamo predicts. Everything quadratic — a synchrotron brightness, a polarisation fraction, a magnetic pressure — reports a magnitude and forgets which way the field was pointing. That forgetting is not a defect of the instruments; it is a property of the coupling, and the whole art of magnetic astronomy is choosing which coupling to look through. A subject whose object emits nothing has to be built out of couplings, and the choice of coupling is the choice of what will be knowable — which is why the same field ends up described in four incompatible ways by four instruments, none of them wrong.