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A slope that needs no source

Polarised light passing through magnetised plasma has its plane rotated, by an amount proportional to the square of the wavelength. A single measurement of the angle is worthless, because nobody knows the angle the source emitted at. A measurement of the slope against wavelength squared does not need to know.

Assumes Polarimetry and Interstellar medium.

18 min read 10 figures Light is the only messengerMeasuring the unreachable

Every magnetic measurement in astronomy has to solve the same problem, which is that the quantity has no emission of its own. The devices available are few, and this essay is about the one that works on the path rather than on the source: a field between here and there changes what happens to light on the way, and the change has a signature no other effect shares.

The signature is a wavelength dependence. That sounds like a technicality and it is the entire method.

A slope, not an angle. Above: the polarisation angle of a background source against the square of the observing wavelength, at the five wavelengths a radio survey actually uses. The plane rotates as it passes through magnetised plasma, by an amount proportional to the integral of the electron density times the field along the path — and to λ². The intercept is the angle the source emitted at, which nobody knows, so a measurement at one wavelength contains no information whatever; the slope is the rotation measure, here 42 radians per square metre, and it needs no knowledge of the source at all. Below: the fractional polarisation that survives. A telescope beam covers many lines of sight with slightly different rotation measures, their angles disagree by more at longer wavelengths, and the vector sum collapses — which is why the useful band has a long-wavelength edge that has nothing to do with sensitivity. Divide the rotation measure by the dispersion measure of the same path, 26.8 in the usual units, and the electron density cancels: the mean line-of-sight field is 1.93 µG, obtained without knowing the distance, the density, or where along the path the field was.
Fig. 1 Above: the polarisation angle of a background source against the square of the observing wavelength, at five wavelengths a radio survey actually uses. The intercept is the angle the source emitted at, which nobody knows; the slope is the rotation measure, and it needs no knowledge of the source at all. Below: the polarised fraction that survives, which collapses at long wavelengths because a telescope beam averages over lines of sight whose rotations disagree.

Why a magnetised plasma is birefringent

A free electron in a magnetic field does not respond to a passing electromagnetic wave the same way for both circular polarisations. One sense of rotation drives it around the field in the direction it already wants to gyrate, and the other opposes it, so the two circular polarisations see slightly different refractive indices and travel at slightly different speeds.

Linear polarisation is a sum of two circular polarisations in phase. Give them different speeds and the phase between them drifts, which rotates the plane of the linear polarisation. That is the whole mechanism, and it is called Faraday rotation because Michael Faraday found it in a piece of lead borate glass in 1845 — the first demonstration that light and magnetism were connected at all, made twenty years before Maxwell wrote down why.

The size of the rotation follows from the size of the index difference, and the index difference goes as the electron density times the field component along the line of sight, divided by the square of the frequency. Turning that into a rotation of a plane over a path introduces one more factor of wavelength, and the result is that the angle rotates as the square of the wavelength, with a coefficient that is the path integral of the electron density times the parallel field.

Two features of that coefficient decide how it can be used. It is linear in the field rather than quadratic, so a reversal along the path subtracts rather than adds. And it involves only the component along the line of sight, so a field lying across the sky contributes nothing whatever.

Both features are inconvenient and both are what make the technique honest. A quantity quadratic in the field would always be positive and would therefore always report something, which sounds better and is worse: a tangled field would give a large answer with no way of knowing it was tangled. Linearity means a tangled field averages toward zero and says so. The measurement’s willingness to return nothing is what makes a non-zero return meaningful, and the same logic runs through the Zeeman measurements of clouds, which are linear in the field for the same reason and cancel for the same reason.

A 3-gauss field moves the line by 0.33% of its width and is measured anyway. Above: the Fe I 6173 Å line, Landé factor 2.5, at a Doppler width of 0.041 Å. The solid curve is the unmagnetised profile; the dashed curve is the same line in a longitudinal field of 3 gauss, which splits it by 1.3·10⁻⁴ Å — 0.33 per cent of its own width — and is drawn on top of it because the two are not distinguishable. The double-lobed curve underneath them is the Stokes V profile of that same field, magnified 100 times: the two σ components are circularly polarised with opposite handedness, so what is lost in the sum survives in the difference, and the difference of two profiles a hair apart is the derivative of one of them. Below: what that buys. The V amplitude is linear in the field, because it is a first derivative; every signature of the same field in the intensity is quadratic, because a symmetric splitting can only broaden. At a polarimetric precision of 10⁻⁴ the first reaches 0.1 gauss; at a line-width accuracy of 0.001 the second reaches 38, a factor of 420 worse. A quantity far too small to resolve is measured because it is the only thing in the signal that carries a sign — and the same argument run the other way says what polarimetry cannot do: a field of mixed polarity inside the resolution element cancels in V and does not cancel in I, so the 3000-gauss field of a sunspot, which does resolve, is measured the other way round.
Fig. 2 The local diagnostic that shares the linearity and nothing else. A three-gauss field displaces the two circular polarisations of a line by a third of a per cent of its own width — undetectable in the intensity, and recoverable in the difference of the two circular components because that difference is proportional to the derivative of the profile with one free amplitude. Both techniques are linear in the field and both therefore cancel on a reversal; what separates them is where they look. This one reports the field in the volume the line forms in, and the rotation measure reports an integral along the whole path with no way to say where.

The intercept nobody knows, and why it does not matter

A radio source emits polarised light at some angle. What that angle is depends on the orientation of the field in the source, which is exactly as unmeasurable as everything else about a magnetic field.

So a polarisation angle measured at one frequency is a sum of two numbers, one of which is unknown. It contains no information at all about the intervening medium. This is a real trap and not a hypothetical one: early attempts to detect interstellar rotation compared angles between sources and got nowhere, because the scatter between sources was the intrinsic scatter of the sources.

The escape is that the unknown enters as a constant and the quantity of interest enters as a slope. Measure the angle at several wavelengths, plot against wavelength squared, and fit a line: the intercept absorbs whatever the source did, and the gradient is the rotation measure. Nothing about the source needs to be known, or even assumed, beyond the requirement that its intrinsic angle does not itself depend on wavelength.

A position angle to 1.0° and a polarisation that is biased upwards. The Stokes plane, with 200 simulated measurements of one star at a true polarisation of 1.2 per cent and a position angle of 40°, each Stokes parameter carrying an independent error of 0.35 per cent. The polarisation is the length of the vector and the position angle is half its azimuth — which is why a rotation of 180° in this plane is a rotation of 90° on the sky, and why polarisation has no sign. Averaging the two components recovers 39.0° against the true 40°. But averaging the lengths gives 1.241 per cent against a true 1.2: a length cannot be negative, so noise can only push it up, and the excess here is 0.044 against the large-signal expectation σ²/2p = 0.051. At zero true polarisation the same effect returns 0.44 per cent from a star that has none, which is the reason a polarimetric detection is quoted in σ and almost never in per cent alone.
Fig. 3 How the angle at each wavelength is obtained in the first place. Polarisation is measured as a pair of Stokes parameters rather than as an angle, because the pair are linear in the incoming field and the angle is not; the angle is recovered afterwards, and its uncertainty grows without bound as the polarised fraction falls toward the noise.

That last requirement is not vacuous, and it is the standard failure mode. If a source is itself a mixture of regions at different rotation measures, or if some of the rotation happens inside the emitting volume, the angle stops being linear in wavelength squared and the fitted slope means something else. The diagnostic is straightforward and is always applied: check that the points lie on a line. Sources that do not are set aside, and there are many of them.

It is worth noticing what kind of check that is. The physics predicts not merely a relation but a functional form, and the form has one parameter more than the measurement needs. Any two angles determine a slope; the third and fourth wavelengths are spent testing whether the slope is real. A technique that consumes most of its data on verification rather than on estimation is unusual and is the reason this one is trusted: a source that passes has demonstrated the wavelength dependence that the mechanism requires and that no competing effect produces.

Two path integrals, and their ratio

The rotation measure is an integral of the electron density times the field. That is a product of two unknowns, and on its own it is not a field strength.

What rescues it is that the same path is available in a second integral. A pulsar’s radio pulse arrives later at lower frequencies, because the plasma’s refractive index departs from one by an amount proportional to the electron density over the frequency squared, and the delay accumulated over the path is the dispersion measure — an integral of the electron density alone. Divide one by the other and the electron density cancels — not exactly, since the ratio is a density-weighted mean of the field rather than a plain mean, but well enough that the quotient is quoted as an average line-of-sight field strength, with a numerical coefficient of about 1.23 in the usual units. For a pulsar with a rotation measure of forty-two radians per square metre and a dispersion measure of twenty-seven, that is about two microgauss.

This is worth pausing on. A field strength has been obtained without knowing the distance to the source, the density along the path, or where along the path the field was. Two integrals over the same unknown density, divided. It is one of the cleanest measurements in the subject, and the price is that it requires a pulsar — a source that both is polarised and emits a pulse sharp enough to time.

A position angle to 0.3° and a polarisation that is biased upwards. The Stokes plane, with 200 simulated measurements of one star at a true polarisation of 3 per cent and a position angle of 110°, each Stokes parameter carrying an independent error of 0.35 per cent. The polarisation is the length of the vector and the position angle is half its azimuth — which is why a rotation of 180° in this plane is a rotation of 90° on the sky, and why polarisation has no sign. Averaging the two components recovers 110.3° against the true 110°. But averaging the lengths gives 2.995 per cent against a true 3: a length cannot be negative, so noise can only push it up, and the excess here is 0.019 against the large-signal expectation σ²/2p = 0.020. At zero true polarisation the same effect returns 0.44 per cent from a star that has none, which is the reason a polarimetric detection is quoted in σ and almost never in per cent alone.
Fig. 4 What the angle at one wavelength costs, in the favourable case. A three-per-cent polarisation measured against the same noise gives a position angle good to three tenths of a degree, and the ensemble of Stokes pairs is a tight cloud well away from the origin. Five such angles determine a rotation measure to a fraction of a radian per square metre — which is what makes the pulsar quotient above worth quoting to two figures. The bias visible even here is worth noting: the polarised fraction is the length of a two-component vector and cannot be negative, so noise pushes it up whatever the truth is, and it is the angle rather than the fraction that this technique should be read from.

A few thousand pulsars are known and most of them lie in the galactic plane, so the sample is a set of skewers through the disc at known and unequal depths. That is more useful than a set of complete paths would be. Two pulsars in nearly the same direction at different distances give a difference of rotation measures and a difference of dispersion measures, and the quotient of the differences is the mean field in the segment between them. The line-of-sight integral has been differentiated, and the field’s variation with distance along one direction becomes accessible in a way no extragalactic source could provide.

The long-wavelength edge

The rotation grows as the square of the wavelength, so a naive reading says to observe at the longest wavelength available and get the largest signal. The opposite is true, and the reason is instructive.

A telescope beam covers a patch of sky, not a line. Within that patch the rotation measure varies, because the electron density and the field vary. Each line of sight rotates by a different amount, and what the instrument records is the vector sum of polarisation vectors pointing in different directions. When the spread of angles across the beam approaches a radian the sum collapses toward zero.

The spread of angles is the spread of rotation measures times the wavelength squared, so the depolarisation sets in as the fourth power of wavelength. That is brutally steep: doubling the wavelength can take a source from comfortably polarised to undetectable. The consequence is a working band with edges at both ends for different reasons. Too short a wavelength and the rotation is too small to measure against the errors on the angles; too long and there is no polarisation left to measure an angle of. The useful decade or so in between is why radio polarimetry lives where it does.

A position angle to 3.1° and a polarisation that is biased upwards. The Stokes plane, with 200 simulated measurements of one star at a true polarisation of 0.4 per cent and a position angle of 40°, each Stokes parameter carrying an independent error of 0.35 per cent. The polarisation is the length of the vector and the position angle is half its azimuth — which is why a rotation of 180° in this plane is a rotation of 90° on the sky, and why polarisation has no sign. Averaging the two components recovers 36.9° against the true 40°. But averaging the lengths gives 0.563 per cent against a true 0.4: a length cannot be negative, so noise can only push it up, and the excess here is 0.165 against the large-signal expectation σ²/2p = 0.153. At zero true polarisation the same effect returns 0.44 per cent from a star that has none, which is the reason a polarimetric detection is quoted in σ and almost never in per cent alone.
Fig. 5 The long-wavelength end of that band, where depolarisation has taken the signal down to four tenths of a per cent. The Stokes cloud now straddles the origin, the recovered position angle is uncertain by three degrees, and the upward bias in the fraction has become the dominant feature rather than a footnote — a source with no polarisation at all would report a positive one here. This is why the depolarisation limit is a hard edge rather than a gradual loss: past it the measurement does not merely become imprecise, it becomes biased in a direction that looks like a detection.
Two effects that are blind in opposite directions. The sensitivity of two magnetic diagnostics against field strength, on a logarithmic axis spanning five decades. The Zeeman effect measures the line-of-sight component and adds it up along the path, so a field tangled into 100 independent cells inside one resolution element averages down by a factor of 10 and reports almost nothing — which is exactly the situation in a chromosphere or a turbulent cloud. The Hanle effect is a different device altogether: a field precesses the atom between absorption and re-emission, so a scattering line's polarisation is rotated and reduced, and the amount depends on how far the precession gets in one radiative lifetime. That makes it sensitive around 1 gauss for a 100-nanosecond level, and — the useful part — it does not care about sign, so a tangled field does not cancel. Above the crossing at 27.7 gauss the Hanle signal has saturated and carries no strength information, and the Zeeman effect is the instrument. Neither is a measurement of the field; each is a measurement of what the field did to something else.
Fig. 6 The same problem in a different diagnostic. Every magnetic technique has a range of field strengths it can report and blind spots on either side of it, and choosing an instrument means choosing which blind spot to accept — a comparison worked out in detail for the two optical methods.

There is a modern way around part of this, and it changes the character of the measurement. Observing across a wide band and Fourier transforming the polarisation against wavelength squared gives the distribution of rotation measures along the line of sight rather than a single fitted slope — so a path with two magnetised screens at different rotation measures shows two peaks rather than one meaningless average. That technique turns a number into a spectrum, and it recovers information that the fitted-line method throws away by construction.

The one case where the path is known

Almost every application of the technique suffers from not knowing where along the path the rotation happened. There is one case where it is known exactly, and it is worth setting out because it is the closest thing to a laboratory demonstration the subject has.

When a spacecraft passes behind the Sun as seen from Earth, its radio signal crosses the corona at a known distance from the solar surface, and the crossing point sweeps inward and out again over days. The signal is transmitted at a known and stable polarisation, so the intercept in the fit is not merely unknown-but-constant — it is known. Every ambiguity that troubles an astronomical source has been removed by controlling the transmitter. What comes back is the coronal field as a function of height, from a few solar radii out to a few tens, in a regime where no other measurement reaches: the fields are too weak and the plasma too hot for the Zeeman effect, and the corona is too faint for anything else. The rotation measures run to hundreds of radians per square metre near the Sun and fall steeply outward, tracing a field that declines roughly as the inverse square of distance in the outer corona — which is the signature of a field being dragged out by the wind rather than falling off as a dipole would. That is the same field that, further out, is wound into a spiral by the star’s rotation.

A slope, not an angle. Above: the polarisation angle of a background source against the square of the observing wavelength, at the five wavelengths a radio survey actually uses. The plane rotates as it passes through magnetised plasma, by an amount proportional to the integral of the electron density times the field along the path — and to λ². The intercept is the angle the source emitted at, which nobody knows, so a measurement at one wavelength contains no information whatever; the slope is the rotation measure, here 300 radians per square metre, and it needs no knowledge of the source at all. Below: the fractional polarisation that survives. A telescope beam covers many lines of sight with slightly different rotation measures, their angles disagree by more at longer wavelengths, and the vector sum collapses — which is why the useful band has a long-wavelength edge that has nothing to do with sensitivity. Divide the rotation measure by the dispersion measure of the same path, 120 in the usual units, and the electron density cancels: the mean line-of-sight field is 3.08 µG, obtained without knowing the distance, the density, or where along the path the field was.
Fig. 7 A rotation measure of three hundred radians per square metre, which is the coronal regime rather than the interstellar one. The slope is seven times the hero’s and the consequence is visible in the top panel: between the two longest wavelengths the angle has turned through more than a full rotation, so the points can only be joined into a line if the intermediate wavelengths are close enough together to rule out a wrap. That is the nπn\pi ambiguity, and at coronal rotation measures it is the design requirement on the experiment rather than a caveat on the analysis — the wavelengths are chosen so that consecutive angles cannot have wrapped, which is a decision made before the observation.

What a sky of rotation measures is a map of

Thousands of extragalactic sources have measured rotation measures, and each of them is one number describing the whole path from there to here. Almost all of that path is intergalactic and contributes very little; the Milky Way’s disc contributes most of it.

So the catalogue is a map of the galaxy’s own field, made with background sources as illumination. The sources are not the object of study — they could be replaced by any others and the map would be the same.

A peak at 0.55 µm, and therefore a grain size. Interstellar polarisation against wavelength — the Serkowski law, p(λ) = p_max exp[−K ln²(λ_max/λ)] with K = 1.66 λ_max. The heavy curve peaks at 0.550 µm, measured off the drawing rather than read back from the parameter, and falls to half its peak at 1.314 µm on the red side, against the closed form λ_max exp√(ln2/K) = 1.315. That peak wavelength is the measurement. It is set by the size of the grains doing the aligning — bigger grains, longer λ_max — and it is tied to the shape of the extinction curve along the same sight line by R_V ≈ 5.5 λ_max, which gives 3.03 here against the diffuse-medium value of 3.1. The two faint curves are populations peaking at 0.35 µm and 0.75 µm: the same amount of polarisation, distributed differently, and a different dust. Nothing in a photometric measurement of the same star distinguishes them.
Fig. 8 The wavelength dependence that says the aligning is grains rather than anything else. Polarisation from aligned dust peaks near half a micron and falls away on both sides, following a curve with one free parameter — the wavelength of the maximum, which tracks the typical grain size. Nothing about a synchrotron slope produces that shape, and nothing about scattering does either, so the shape is the identification. It is also the reason the effect is measured in the optical while the emission it is a foreground to is measured in the submillimetre.

The sign is the useful part. Because the rotation measure is linear in the field, its sign says whether the field points toward the observer or away, and a map of signs is a map of directions with no calibration in it at all. Reversals of sign across the sky locate places where the large-scale field turns round, and the pattern of those reversals distinguishes field geometries that no photograph could tell apart.

The same catalogue does something else that is worth stating: it puts limits on fields where none has ever been detected. A magnetic field in the intergalactic medium would add a small rotation to every distant source, and its absence in the statistics of the catalogue bounds that field at the level of a nanogauss or so — a constraint on the primordial field obtained entirely from the fact that a scatter plot is no wider than the galaxy alone would make it.

Where the method breaks

Three failure modes are worth naming, because each one has produced a published result that later turned out to be an artefact.

The first is the ambiguity of an angle. Polarisation angles are defined modulo a hundred and eighty degrees, so a fit through widely spaced wavelengths can wrap: the same set of angles is consistent with rotation measures differing by a fixed step, and picking the wrong one gives a field of the right magnitude and the wrong value. The defence is to sample the band densely enough that consecutive points cannot have wrapped, which is a requirement on the observing setup rather than on the analysis.

The second is internal rotation. If the plasma doing the rotating is mixed with the plasma doing the emitting, light from the far side of the source is rotated more than light from the near side, and the angle is not linear in wavelength squared at all — it turns over and the polarisation oscillates. Fitting a line to the short-wavelength end of such a source gives a number that is neither the source’s field nor the path’s.

Zero forwards, zero backwards, and exactly one at a right angle. The degree of linear polarisation produced by a single scattering, against the angle through which the light was turned. The curve is (1 − cos²θ)/(1 + cos²θ): it is exactly zero at 0° and 180° and exactly one at 90°, and neither of those is a fitted number — they are the geometry of a dipole seen end-on and side-on. The lower curves are the same shape divided down by an unpolarised component, for slabs of optical depth 0.2, 1, 3 in which some of the light has scattered more than once; the peak falls to 16 per cent of its single-scattering value at the thickest. The angular position of the maximum does not move, which is why a polarisation map of a reflection nebula locates the illuminating star even when the star is hidden: every position angle is perpendicular to the line back to the source, and the pattern converges on it.
Fig. 9 The other way a photon acquires a direction, and how it is told apart from the first. Scattering polarises light perpendicular to the scattering plane, so a source seen through an optically thin envelope shows a polarisation that grows with optical depth and then collapses again as multiple scatterings randomise it. Three depths are drawn and the turnover is at about unity. Alignment saturates instead of turning over, which is what separates a dusty foreground from a scattering envelope when only one number is available.

The third is that a mean weighted by density is not a mean. In a medium where the dense regions and the strong-field regions coincide, the ratio of the two integrals overestimates the volume-averaged field; where they anti-correlate, it underestimates. In the local interstellar medium the two are believed to be weakly anti-correlated, which biases pulsar-derived fields low by some tens of per cent. That is a systematic, it is not small, and it is the reason field strengths from this technique are usually quoted to one significant figure.

Zero forwards, zero backwards, and exactly one at a right angle. The degree of linear polarisation produced by a single scattering, against the angle through which the light was turned. The curve is (1 − cos²θ)/(1 + cos²θ): it is exactly zero at 0° and 180° and exactly one at 90°, and neither of those is a fitted number — they are the geometry of a dipole seen end-on and side-on. The lower curves are the same shape divided down by an unpolarised component, for slabs of optical depth 0.5, 2, 6 in which some of the light has scattered more than once; the peak falls to 1 per cent of its single-scattering value at the thickest. The angular position of the maximum does not move, which is why a polarisation map of a reflection nebula locates the illuminating star even when the star is hidden: every position angle is perpendicular to the line back to the source, and the pattern converges on it.
Fig. 10 The competing polarising mechanism at optical depths where it has stopped being a competitor. Three envelopes are drawn, at half, two and six scatterings deep, and the polarisation falls as the depth rises past unity because each additional scattering randomises the direction the previous one imposed. That turnover is the identification. A quantity that grows and then collapses is scattering; a quantity that grows and saturates is alignment; a quantity proportional to the square of the wavelength is Faraday rotation. Three mechanisms, three functional forms, and the whole discipline of separating them is reading which curve a source lies on rather than how large its polarisation is.

What the technique is for

It is worth being clear about what this method is and is not good at, because it is the workhorse of galactic magnetism and it is routinely asked for things it cannot supply.

It measures the line-of-sight field, integrated, weighted by density. It cannot see a field in the plane of the sky at all — for that a wholly different instrument is needed, and aligned dust reports a direction with no strength in it. It cannot localise anything along the path without additional information. It is exquisitely sensitive to sign, and therefore to geometry, which is why the large-scale structure of the galactic field is known almost entirely from it.

It is also, uniquely among the magnetic diagnostics, insensitive to what the field is doing to the matter. The Zeeman effect needs atoms in the right state; grain alignment needs grains and a radiation field to spin them; an instability’s growth rate needs a disc to grow in. Faraday rotation needs only free electrons and a path, which is why it works equally well in a supernova remnant, a cluster of galaxies, the solar corona and the space between here and a pulsar. A method with no requirements on its target is rare, and it is the reason this one is applied to objects that share nothing else.

And it is cheap in a way that matters at scale. A polarisation survey of the radio sky delivers a rotation measure for every polarised source in it, tens of thousands at once, each one an independent probe of a different line of sight. Nothing else in magnetic astronomy comes with that kind of multiplex advantage, which is why the next generation of radio arrays is expected to raise the count of measured rotation measures by two orders of magnitude — and why the map of the galaxy’s field is expected to improve faster than the map of anything else about it.

There is one more property that is easy to overlook and is the reason the technique keeps finding new uses. Everything about it scales with wavelength squared, so an instrument observing at a longer wavelength is a more sensitive magnetometer by a fixed and calculable factor — up to the depolarisation limit. Arrays working at metre wavelengths therefore reach rotation measures of a fraction of a radian per square metre, which is small enough to detect the magnetic field of the diffuse gas well outside a galaxy’s disc. The measurement stays what it always was: an angle that means nothing, measured several times, and a line drawn through the results.

One further comparison sets the technique in its place among the others in this collection. A rotation measure is a path observable: it reports an integral over everything between the source and here, with no ability to say where along the path anything happened. A Zeeman splitting is a local observable: it reports the field in the volume where a particular line forms, and nowhere else. Neither is better; they answer different questions, and a great deal of confusion comes from comparing a number produced by one with a number produced by the other and expecting agreement.

The place where the two are forced to agree is the Sun, and there they do. The coronal field measured by rotation of a spacecraft’s signal, the photospheric field measured by splitting, and the field extrapolated from one to the other by a potential-field model form a chain that closes — the extrapolation predicts the rotation measures within the errors. That is the only object in the sky where all three exist, and it is the reason the extrapolation technique is trusted when it is applied to stars whose coronae nobody can probe.

The other side of that agreement is worth stating: the potential-field extrapolation is an assumption, not a measurement, and it is known to be wrong where the corona holds currents. Where it is checked against rotation measures it fails in exactly the places currents are expected — above active regions, and along the boundaries of coronal holes. A model tested by a path integral and found wanting in a predictable place is a model doing its job. The currents it is missing are the ones that store the energy a flare releases, so the discrepancy is not a nuisance but the observable of interest. A coronal model that matched every rotation measure exactly would be describing a corona with nothing stored in it, and therefore a corona that cannot flare. The residual is the energy budget, read through a polarisation angle. What makes it powerful is not the physics of the rotation, which is elementary, but the structure of the inference — an unknown pushed into an intercept, a wanted quantity left in a slope, and a functional form with enough spare degrees of freedom to check itself.

Where this goes next

A peak at 0.55 µm, and therefore a grain size. Interstellar polarisation against wavelength — the Serkowski law, p(λ) = p_max exp[−K ln²(λ_max/λ)] with K = 1.66 λ_max. The heavy curve peaks at 0.550 µm, measured off the drawing rather than read back from the parameter, and falls to half its peak at 1.314 µm on the red side, against the closed form λ_max exp√(ln2/K) = 1.315. That peak wavelength is the measurement. It is set by the size of the grains doing the aligning — bigger grains, longer λ_max — and it is tied to the shape of the extinction curve along the same sight line by R_V ≈ 5.5 λ_max, which gives 3.03 here against the diffuse-medium value of 3.1. The two faint curves are populations peaking at 0.35 µm and 0.75 µm: the same amount of polarisation, distributed differently, and a different dust. Nothing in a photometric measurement of the same star distinguishes them. The direction a photon count throws away A photometer records how many photons arrived. It discards a two-component quantity that survives every attenuation on the way, and that quantity carries a magnetic field direction, a grain size, and the shape of an exploding star nobody can resolve. Three ways for a timing model to be wrong, and three shapes that say which. Timing residuals over 6 years for the Crab pulsar, one curve per kind of error in the model, in microseconds. A position error of 1.2 mas leaves a sinusoid of period exactly one year — measured off the drawn curve as 1.000 — with amplitude (a/c)·δθ·cos β = 2.7 μs, because the error is being projected onto a baseline that is the Earth's own orbit and nothing about the pulsar. An unmodelled proper motion of 0.9 mas/yr leaves the same sinusoid with an envelope growing linearly: twice as large at 6 years as at 3. An error of one part in 10⁹ in Ṗ leaves a parabola, the second integral of a frequency drift, whose second derivative is constant to 4e-12 across the span — and that fractional error is deliberately minute, because anything larger produces a residual thousands of times the other two and draws them as flat lines. The shapes do not resemble each other, which is the whole reason a pulsar is an instrument rather than a clock: fitting them simultaneously delivers a position, a proper motion and — from the annual curvature term, not drawn here — a parallax, all from the arrival times of pulses and no image of anything. What is left when every known shape has been removed is the science: glitches, red noise, and the correlated residual between pairs of pulsars that a timing array exists to find. A clock read against a model of everything in between A pulsar delivers arrival times and nothing else. Everything else is a model, and what is measured is the difference between the model and the arrivals — a residual whose shape says which term is wrong, and whose annual sinusoid is a position measured from pulses rather than from an image. Two geometries, one bit of data, and the bit decides. Rotation measure against galactic longitude for two field geometries that look identical in every image ever taken. An axisymmetric field runs the same way round the disc at every azimuth, so the component along the line of sight changes sign exactly 2 times as the longitude goes round — once where the field turns toward the observer and once where it turns away. A bisymmetric field reverses its own direction from one side of the galaxy to the other, and that doubles the count to 4. Nothing about the brightness, the arms or the colour distinguishes the two; the sign of a rotation measure does, and a sign is one bit. The measurement is made against several hundred background sources, each contributing one rotation measure through the whole disc — the sources are not the object of study, they are the illumination. The Milky Way's own answer is untidy: broadly axisymmetric with at least one reversal inside the solar circle, which is neither clean case, and is why the field's origin is still argued about. Two geometries and one bit to choose between them A galaxy's large-scale magnetic field either runs the same way round the disc everywhere or reverses from one side to the other. No image distinguishes the two. The sign of a rotation measure does — two reversals around the sky for one geometry and four for the other — and a sign is one bit of information.

What this makes readable

Essays that name this one as a prerequisite.

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.

BirefringenceDepolarisationDispersion measureFaraday rotationInterstellar mediumMagnetic fieldPlasma frequencyPolarimetryPulsarsRotation measureSynchrotron radiation