Two instruments blind in opposite directions

Most of the Sun’s magnetic flux is not in sunspots. It is in a tangle of small-scale field filling the quiet photosphere and the chromosphere above it, and by every straightforward measure that field is invisible.

The reason is that the straightforward measure is the Zeeman effect, which reports the component of the field along the line of sight, averaged over whatever the instrument cannot resolve. A field arranged in a hundred independent cells of alternating sign inside one resolution element averages down by ten, and one arranged in ten thousand by a hundred. That is the same cancellation that makes a cloud’s Zeeman detection a two-sigma affair, and for the same structural reason: a signed quantity averaged over a tangle returns its own noise. The measurement returns almost nothing, and returning almost nothing is indistinguishable from there being almost nothing there.

What a scattered photon remembers #

The Hanle effect begins somewhere unexpected, which is that scattering polarises light even with no field present at all.

An atom that absorbs a photon and re-emits it does not re-emit isotropically. Looked at from ninety degrees to the incoming beam, the re-emitted light is linearly polarised, for the same geometric reason that the daytime sky is polarised at ninety degrees from the Sun — the same scattering geometry that makes a polarimeter useful anywhere. In a stellar atmosphere the incoming radiation field is anisotropic — brighter from below than from the sides — so scattering near the limb produces a polarisation parallel to the limb.

That polarisation is small, a fraction of a per cent, and it is present in a great many spectral lines. Observed at high precision it constitutes a spectrum in its own right, structurally unlike the ordinary intensity spectrum: some strong lines show nothing, some weak lines show a great deal, and the pattern follows the atomic physics of the scattering rather than the abundance of the element. It was named the second solar spectrum when it was first surveyed properly in the nineteen-nineties, and much of it is still being identified.

The key point for what follows is that the polarisation has a direction fixed by geometry: it is set by the plane containing the incoming and outgoing rays. That direction is a reference, and a reference is what a measurement needs.

This is worth comparing with the trouble the radio technique has. A rotation measure needs a slope because the emitted angle is unknown; here the emitted angle is known, because scattering geometry fixes it, so a single measurement of the plane carries information. The price is that the geometry has to be known, which restricts the method to places where the illumination is understood — a stellar limb, essentially.

The polarisation’s magnitude is also computable in principle. It depends on the anisotropy of the radiation field at the scattering point and on the atomic transition’s ability to carry alignment, both of which follow from a model atmosphere and from atomic data. That computability is what makes a reduction of the polarisation into a measurement, and it is also, as will become clear, the method’s weakest joint.

What a field does to it #

Now add a magnetic field. Between absorbing the photon and re-emitting it, the atom spends a radiative lifetime — for a typical allowed transition, a hundred nanoseconds or so — in an excited state with an angular momentum.

That angular momentum precesses about the field at the Larmor frequency. If the precession angle in one lifetime is negligible the atom re-emits as though no field were present. If it is large the emission direction of the dipole has been scrambled and the polarisation is destroyed. In between — precession of order one radian per lifetime — the polarisation is partly reduced and its plane is rotated, by an angle that depends on the field.

The characteristic field is set by equating the Larmor precession rate to the inverse lifetime, and for a hundred-nanosecond level with an ordinary Landé factor that is about one gauss.

One gauss is precisely the regime in which the Zeeman effect on a tangled field returns nothing. Two diagnostics, sensitive to disjoint parts of the same range, using the same photons.

The effect is named for Wilhelm Hanle, who found it in 1924 in mercury vapour illuminated by a mercury lamp, and it has a place in the history of quantum mechanics out of proportion to its size: it was among the first demonstrations of coherence between atomic sublevels, and the explanation of it in terms of interference between degenerate states preceded the formalism that would later make it routine. That it later became a stellar magnetometer is the kind of transfer this subject depends on.

The scaling is worth carrying because it is the design rule for choosing a line. The characteristic field is inversely proportional to the level’s lifetime and to its Landé factor, so a long-lived level is sensitive to weak fields and a short-lived one to strong. Lines are chosen for the field range they are wanted to report on, which makes the second solar spectrum a set of magnetometers with different ranges rather than one instrument.

Why the tangle does not cancel #

The cancellation argument is the whole reason the second method exists, and it is worth setting out carefully.

The Zeeman signal in circular polarisation is proportional to the field component along the line of sight, with a sign. Reverse the field and the signal reverses. Sum over many cells of random orientation and the result is a random walk: the expected magnitude falls as one over the square root of the number of cells, and the expected value is zero.

The Hanle depolarisation is not linear in the field and has no sign. A field of either polarity depolarises, and the depolarisation from many cells adds rather than cancelling. So a tangled field of a few gauss, invisible to the Zeeman effect, produces a measurable reduction in the scattering polarisation.

That is the observation that has been made, and its consequence was substantial: the quiet Sun’s photosphere turns out to carry a turbulent field of order a hundred gauss in strength, hidden in cells far below any telescope’s resolution, containing more magnetic energy than all the active regions put together at solar minimum.

The comparison with the active regions is the part worth dwelling on. A sunspot carries three kilogauss over a thousandth of the surface; a hundred gauss over the whole surface, if it fills a reasonable fraction of the volume, is a larger total energy — and it is present at solar minimum, when there are no spots at all. So the Sun’s magnetism does not switch off between cycles. It becomes invisible, which is a different thing, and only a diagnostic insensitive to sign could have told the difference. The origin of that hidden field is a separate question from the cycle that produces the spots. The leading account is a small-scale dynamo driven by granular convection, which would operate independently of the cycle and produce a field with no preferred polarity — consistent with the observation that the turbulent flux shows little cycle variation.

The number came from a comparison rather than from a direct reading. Compute the scattering polarisation a line should have with no field, measure what it has, and attribute the difference to Hanle depolarisation — which requires the field-free calculation to be right, and that is where the difficulty lives.

What the method costs #

The Hanle effect gives up a great deal in exchange for seeing what the Zeeman effect cannot.

It gives up the sign. A depolarisation is a scalar, so nothing about the field’s direction along the line of sight survives, and a map made this way has no polarity information whatever.

It gives up strength above saturation. Once the precession exceeds a few radians per lifetime the polarisation is destroyed as completely as it can be, and further increases in the field change nothing. So the method has a ceiling at a few tens of gauss and reports “strong” for everything above it.

And it depends on a calculation. The field-free polarisation of a line is not measured anywhere; it is computed, from the atomic physics of the transition and a model of the atmosphere’s anisotropy, and the computation involves quantities — collisional depolarisation rates, in particular — that are known to a factor of two at best.

That last dependence is the honest weakness. A Hanle field strength is the difference between a measurement and a calculation, and the calculation’s error propagates directly into the answer. Where the two methods overlap, they can be compared; where only Hanle applies, the result is as good as the atomic data.

There is an additional confusion that is specific to this technique and has caused real disagreement in the literature. A reduction in the observed scattering polarisation can be caused by a magnetic field, by collisional depolarisation, by the line forming over a range of heights with different anisotropies, or simply by the resolution element containing a mixture of quiet and active atmosphere. The published turbulent field strengths from different lines and different groups have ranged over a factor of several, and the spread is dominated by which of those effects each analysis included.

Two variants that do better #

Two refinements exist that reduce the dependence on the calculation, and both are in routine use.

The first is differential Hanle. Two lines of the same atom with different Landé factors and different lifetimes have different Hanle sensitivities but nearly the same radiative transfer, so their ratio of polarisations is sensitive to the field and much less sensitive to the atmosphere. A field measured this way needs the two calculations to be wrong in the same way rather than to be right.

The second is the forward-scattering geometry. At disc centre the scattering geometry is symmetric, so there should be no linear polarisation at all — and a magnetic field inclined to the line of sight breaks the symmetry and creates one. A signal where zero is expected is a far cleaner detection than a reduction of a signal whose expected value is computed, and forward-scattering Hanle signatures are among the more convincing measurements of chromospheric fields.

The forward-scattering case also comes with an internal check that the depolarisation case lacks: the induced polarisation’s direction depends on the field’s azimuth, so a measurement returns an angle as well as an amplitude, and the two together over-determine the geometry.

A third refinement is to use both effects at once on the same line. In an intermediate regime — a field strong enough to split the level appreciably and weak enough that the Hanle rotation is not saturated — the two effects are present together and produce a joint signature in all four Stokes parameters. Fitting that jointly is harder than either alone and returns more: a strength, an inclination and an azimuth from one profile. The technique is young and it is where the field-measuring effort in solar physics is currently concentrated.

What it has been used for #

Three results are worth recording because they are things nothing else could have supplied.

The turbulent photospheric field. The estimate of a hundred gauss or so of hidden flux, from Hanle depolarisation of molecular and atomic lines, changed the accounting of the Sun’s magnetism: most of its flux is not in the visible structures. Later infrared Zeeman observations at high resolution found consistent flux, which is the cross-check that made the result stick — infrared because the splitting grows as the square of the wavelength while the line width grows only linearly, so the same field is easier to see further to the red.

The chromospheric field. Above the photosphere the Zeeman effect weakens — the lines are broad, the fields are weaker — and the Hanle effect in strong chromospheric lines is one of the few routes to a field there. Measurements in the calcium and helium lines give tens of gauss in quiet regions, which is the input to every calculation of how energy is stored and released above.

Prominences. A prominence is cool material suspended in the hot corona, visibly held up by something, and the Hanle effect in the helium triplet gives fields of a few to a few tens of gauss with a measurable orientation. The result — that prominence fields are mostly horizontal and cross the filament channel at a shallow angle — constrains the magnetic topology that supports them — the only direct measurement of a structure whose stability is otherwise argued about entirely from models.

What the second solar spectrum turned out to be #

The survey that made all this possible deserves its own section, because it was an unusual piece of observational work: a systematic measurement of something nobody had a use for yet.

Between 1994 and 2000 the linear polarisation of the solar spectrum near the limb was atlassed at a precision of a few parts in a hundred thousand, wavelength by wavelength, across the visible. What came back looked nothing like the intensity spectrum. Strong iron lines that dominate the intensity often showed no polarisation at all; obscure lines of rare-earth elements showed a per cent; molecular bands of the diatomic carbon and magnesium hydride showed structure across their whole extent. The pattern is explained by the quantum mechanics of the scattering. A transition can carry alignment only if the levels involved have enough angular momentum to be aligned, and the polarisability follows from angular-momentum algebra rather than from anything about the star. So the second solar spectrum is largely a readout of atomic structure, with the atmosphere supplying only the anisotropy and the field supplying the modification.

That is what makes it usable. A quantity determined by atomic physics can be computed once and applied everywhere, so the same line has the same intrinsic polarisability in any star — and the departure from it is the measurement.

The general lesson about instruments #

The pair is a good example of something that recurs throughout magnetic astronomy, which is that no diagnostic reports the field.

Each reports what the field did to something else, and each therefore inherits that something’s blind spots. The Zeeman effect reports an energy shift, so it is linear, signed and cancels. The Hanle effect reports a precession during a lifetime, so it is unsigned, saturates, and needs the lifetime. Faraday rotation reports a phase accumulated along a path, so it is weighted by electron density and blind to the plane of the sky. Grain alignment reports an orientation, so it carries no strength at all.

Nothing in that list is a defect. Each is what a particular physical coupling can carry, and the combinations are what make the subject work — a field measured by two methods with different blind spots is far better known than one measured twice by the same method.

The particular reason the Hanle effect is worth the trouble is that it fills the specific hole the others leave: weak, tangled, unresolved field, which is where most of the magnetic flux in a quiet stellar atmosphere actually is. A method that measures the majority of something is worth some awkwardness, and this one measures a majority that was thought for decades not to be there.

Both of the complementary effects are worth reading at second settings, because what each is blind to is set by a parameter rather than by the physics.

Where it goes next #

The obvious extension is to other stars, and it is hard for a simple reason: scattering polarisation is a limb effect, and a star is not resolved.

Integrated over a disc the limb polarisation largely cancels by symmetry, leaving a signal at the level of parts in a million for a spherically symmetric star. What breaks the symmetry is rotation, a companion eclipsing the disc, or the star’s own oblateness — and in each of those cases a polarisation signal has been detected and used to constrain geometry.

The magnetic application is harder still and has been achieved only in a few cases, mostly for stars whose fields are strong enough that the Zeeman effect would work anyway. The quiet-star analogue of the Sun’s hidden flux — the tangled field of an ordinary main-sequence star — remains out of reach, which means the one thing known about the Sun that could not be learned any other way is the one thing not yet known about any other star.

That asymmetry is worth dwelling on, because it recurs across this whole subject and it decides what solar physics is for. The Sun is not an interesting star. It is an ordinary main-sequence dwarf of unremarkable mass, age and composition, and nothing about it would attract attention if it were ten parsecs away. What makes it the most studied object in astronomy is that it is resolved — by a factor of a hundred thousand over the next best star — and that resolution is what every technique in this essay depends on.

Scattering polarisation is a limb effect. Its whole geometry is the difference between looking straight down into an atmosphere and looking across it, and a star integrated over its disc averages those two together and returns almost nothing. The Hanle effect on the Sun is measured at a specific position on a specific limb, in a specific line, at a specific height; none of those qualifiers is available anywhere else.

So the hidden turbulent field, which is most of the Sun’s magnetic flux, is a fact about one star. Whether every solar-type star carries the same thing is not known, is important — it bears on chromospheric heating, on the coronae that drive winds, and on the winds that brake rotation — and is not currently answerable.

What can be done instead is to check the mechanism rather than the measurement. A small-scale dynamo driven by granular convection is a generic process: it needs turbulence, a conducting fluid and a seed, all of which any convective star has. If it operates in the Sun, it operates everywhere, and the argument transfers even though the observation does not. That is a weaker kind of knowledge than a measurement and it is the kind this subject usually has to settle for.

The nearest thing to progress is indirect. A star’s integrated Hanle signal is out of reach, but the same physics operating in a circumstellar disc or an exoplanet’s atmosphere produces polarisation at levels a few parts in a hundred thousand, and instruments now reach that. What is measured there is geometry rather than field, and the magnetic version waits on another factor of ten.