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A field strength read off a line that will not split

A few hundred gauss splits a spectral line by a ten-thousandth of its own Doppler width, which no spectrograph will ever resolve. The two components are circularly polarised with opposite handedness, so the split survives in the difference of two polarisations — where it is linear in the field rather than quadratic.

Assumes Polarimetry, Line formation and Spectra.

17 min read 9 figures Light is the only messengerSmall perturbations, large consequences

The rung below took a direction out of light that a photometer discards: a photon count records how many arrived and throws away a two-component quantity that survives every attenuation on the way, carrying a grain size, a magnetic field direction and the shape of an unresolved explosion.

This rung takes a number out of the same discarded quantity, and the number is a magnetic field strength — measured on stars where the effect it is measured from is four orders of magnitude too small to see.

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. 1 The whole argument in two panels. Above: an iron line at 6173 ångströms in a longitudinal field of three gauss. The unmagnetised profile and the magnetised one are drawn on top of each other because they are not distinguishable — the splitting is a third of a per cent of the line’s own width. Underneath them, magnified a hundredfold, is the circular polarisation of that same field, which is not small at all. Below: why. The polarised signal is linear in the field and every signature in the intensity is quadratic, so the two run apart by orders of magnitude as the field gets weaker.

The splitting, and why it is hopeless

An atomic energy level in a magnetic field splits into sublevels separated in frequency by

Δν=1.4 MHz per gauss\Delta\nu = 1.4\ \text{MHz per gauss}

times an effective Landé factor of order one, which is a property of the particular transition. In wavelength,

ΔλB=4.67×10−13 λ2gB,\Delta\lambda_B = 4.67\times10^{-13}\,\lambda^2 g B,

with λ\lambda in ångströms and BB in gauss.

Put numbers in. The line a solar magnetograph uses is Fe I 6173 Å, chosen partly because its effective Landé factor is an unusually large 2.5. At 3 gauss — roughly the mean longitudinal field of a quiet solar-type star — the splitting is 1.3×10−41.3\times10^{-4} Å.

The line itself is about 0.041 Å wide, from thermal motion and microturbulence together. So the splitting is 0.33 per cent of the line width, and no spectrograph that has ever been built or is likely to be would see a broadening of a third of a per cent as anything but a change in the temperature.

What survives the merger

Now the part that rescues it. The three groups of Zeeman components are not equivalent. Viewed along the field, the unshifted π\pi component vanishes entirely, and the two shifted σ\sigma components are circularly polarised with opposite handedness.

So the observed intensity — the sum of the two polarisations — is the mean of two profiles displaced by ±ΔλB\pm\Delta\lambda_B. Take two profiles a hair apart and average them and the result is the original profile broadened at second order: the first-order displacements cancel, and what is left goes as ΔλB2\Delta\lambda_B^2 times the curvature of the line.

Take their difference and the second-order terms cancel instead. What is left is 2ΔλB2\Delta\lambda_B times the derivative of the line, which is first order.

That is the entire mechanism. A symmetric splitting can only broaden an intensity profile, and broadening is quadratic in the splitting; but the two halves of the splitting carry opposite polarisation, so subtracting the polarisations recovers the displacement itself.

The measured quantity is the Stokes parameter VV, and in the weak-field limit

V(λ)=−4.67×10−13λ2gB∥dIdλ,V(\lambda) = -4.67\times10^{-13}\lambda^2 g B_\parallel \frac{dI}{d\lambda},

which has three properties worth stating separately. It is proportional to BB rather than B2B^2. It is proportional to the longitudinal component only — a field across the line of sight produces no circular polarisation at all. And it is antisymmetric about line centre, because a derivative of a symmetric function is.

Why the antisymmetry matters as much as the linearity

The antisymmetry is what separates a magnetic signal from an instrumental one, and instruments producing spurious polarisation at the 10−410^{-4} level are the normal case rather than the exception.

A polarimeter’s systematic errors are almost all symmetric in wavelength across a line, or slowly varying across a spectrum: a mirror’s reflectivity difference between two polarisations, a detector’s response, a filter’s leak. A signal with a sign reversal exactly at line centre, appearing in every magnetically sensitive line and in no insensitive one, in the ratio their Landé factors predict, is not any of those things.

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 two regimes the same line has. At three thousand gauss the Zeeman components are separated by more than the line is wide and the splitting is visible directly; at three gauss they overlap completely and the line looks single. What survives in the second case is not the splitting but the polarisation: the two components are circularly polarised in opposite senses, so the difference between the two circular polarisations is proportional to the field even when the sum shows nothing at all. That difference is the whole measurement.
A position angle to 0.7° and a polarisation that is biased upwards. The Stokes plane, with 60 simulated measurements of one star at a true polarisation of 0.05 per cent and a position angle of 40°, each Stokes parameter carrying an independent error of 0.01 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 39.3° against the true 40°. But averaging the lengths gives 0.051 per cent against a true 0.05: a length cannot be negative, so noise can only push it up, and the excess here is 0.001 against the large-signal expectation σ²/2p = 0.001. At zero true polarisation the same effect returns 0.01 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 What a weak polarisation measurement looks like before it is a number. The Stokes parameters are measured with noise and the degree of polarisation is a quadrature sum of them, so it is positively biased: noise alone produces a non-zero degree, and a five-per-cent signal measured at one per cent per component is not a five-per-cent detection. Every field strength here comes through that debiasing, and at low signal it dominates the error budget.

What the two methods actually reach

Put a detection threshold on each. A good stellar spectropolarimeter reaches a fractional circular polarisation of about 10−410^{-4} per line, and far better after combining hundreds of lines. A line width measured photometrically is good to about a part in a thousand, at best.

Feeding those into the two scalings gives the lower panel of the first figure: the polarised route reaches a few gauss and the intensity route reaches a few hundred. That factor is not an instrumental accident. It is the difference between a linear and a quadratic dependence, evaluated at a place where both signals are very small.

A 10-gauss field moves the line by 0.94% of its width and is measured anyway. Above: the Fe I 5250 Å line, Landé factor 3, 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 10 gauss, which splits it by 3.9·10⁻⁴ Å — 0.94 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 43, 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. 4 The same measurement on a more sensitive line. The splitting is proportional to the Landé factor and to the square of the wavelength, so a line with a larger factor at a longer wavelength gives a bigger signal for the same field — which is why iron 5250 Å and 6173 Å are the workhorses, and why infrared lines are preferred for the weakest fields. Choosing the line is choosing the sensitivity; the atomic physics is fixed.

Choosing the line, and then choosing a thousand of them

The signal is proportional to the effective Landé factor, and the Landé factor varies from zero to about three across the lines in a stellar spectrum. So the choice of line is worth a factor of several before anything else is done — and a line with geff=0g_{\text{eff}} = 0 is worth having too, as a control that should show nothing.

The Fe I line at 6173 Å is a workhorse for three reasons at once: its Landé factor is 2.5, it is strong enough to have a steep derivative and weak enough not to be saturated, and it forms high enough in the photosphere to be free of the worst of the granulation that is itself a floor on a velocity measurement. Once the shape of the expected signal is known, the obvious next step is to use all of them at once. Least-squares deconvolution treats the observed Stokes V spectrum as one common profile convolved with a line list weighted by depth and Landé factor, and solves for the common profile. Combining a few thousand lines improves the effective signal-to-noise by a factor approaching the square root of the number of lines, which is how a measurement that needs 10810^8 photons per line becomes an hour rather than a week.

The assumption it rests on is that every line has the same shape, differing only in scale. That is true in the weak-field limit and it is not true in a spot, which is another reason the technique is quoted for mean fields and not for structure.

What was actually measured

The technique was invented on the Sun. Hale’s 1908 detection of Zeeman splitting in sunspots is the first magnetic field measured anywhere outside a laboratory, and it did not need any of the argument above, because a sunspot’s field is a few thousand gauss and does resolve: at 3,000 gauss the splitting is a third of the line width and the three components are visible as three components.

The extension to weak fields is what needed the polarisation. The modern solar magnetograph measures Stokes V in a single line across the whole disc and produces a map of the longitudinal field — which is how the reversal of the Sun’s polar field at each activity maximum was found, and how the polarity rule that doubles the length of the sunspot cycle was established.

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. 5 And the effect that reaches the fields the Zeeman method cannot. Below about a gauss the circular polarisation falls below anything measurable, but a weak field still depolarises resonantly scattered light by rotating the atomic alignment during the excited state’s lifetime — the Hanle effect, sensitive exactly where Zeeman is blind, over a range set by that lifetime and the Landé factor. Two effects on one line, each useless in the other’s regime, and between them eight orders of magnitude in field strength.

On other stars the measurement is harder by the ratio of the light collected and easier by nothing at all, and it was not routine until the 1980s. A mean longitudinal field of a few gauss is now measured on hundreds of solar-type stars, and the results have a shape: activity, rotation and field strength are correlated, slow rotators have weak fields, and the youngest stars have fields tens of times the Sun’s.

The thing the method cannot see

There is a failure mode built into the linearity, and it is severe.

Stokes V measures the net longitudinal field averaged over the resolution element — which for a star is the entire visible hemisphere. A field of mixed polarity within that element cancels. Two kilogauss patches of opposite sign contribute nothing at all.

The intensity signature does not cancel, because it is quadratic: both signs broaden the line. So the two methods measure different things, and the difference between them is informative rather than contradictory.

The Sun is the demonstration. Its mean longitudinal field, measured the way a distant star’s would be, is a few gauss. Its actual photospheric field is kilogauss in flux tubes covering perhaps one per cent of the surface. The ratio of the two — the filling factor — is what the comparison of a polarised and an unpolarised measurement recovers, and it is the reason both are made.

Rotation, which helps and hurts

The stars with the strongest fields are the fast rotators, and fast rotation is also what makes the measurement hardest — which is the tension the whole subject of stellar magnetism runs on.

It helps because rotation is what drives the dynamo. A star’s magnetic activity correlates tightly with its rotation rate divided by the convective turnover time of the outer layer that boils, across three orders of magnitude in activity, and the correlation saturates above a threshold. A young solar-type star rotating in three days has a mean field tens of times the Sun’s; the Sun rotating in twenty-five days sits far down the relation; and the whole of stellar magnetism’s usefulness as a clock — spin-down as an age indicator — is that relation read backwards.

It hurts because rotation broadens every line. The Stokes V amplitude goes as the derivative of the line profile, and a rotationally broadened line has a shallower derivative in exactly the proportion that it is broader — so a star rotating at 50 km s⁻¹ gives a signal an order of magnitude weaker than the same field on a star rotating at 5.

What to do about the field that cancels

The cancellation described above is not a nuisance to be corrected; it is a hole in the measurement, and two quite different techniques exist to see through it. Both are worth knowing because they measure the quantity Stokes V is blind to.

The first works in the intensity after all, by comparing lines rather than by resolving one. Take two lines of the same element and ionisation stage, formed at the same depth, with nearly the same strength and very different Landé factors. A magnetic field broadens the sensitive one and leaves the insensitive one alone, and since both are affected identically by temperature, turbulence and rotation, the ratio of their widths isolates the magnetic broadening. That signal is quadratic in the field, so it is hopeless at a few gauss — and it does not cancel, so on a star whose surface is covered in kilogauss patches of both polarities it works where the polarimetry returns nothing.

What it returns is not a field but a product: the field strength inside the patches multiplied by the fraction of the surface they cover. Combining that with a Stokes V measurement, which returns the field times the net signed filling factor, separates the two. The technique established that the surfaces of active M dwarfs carry kilogauss fields over a large fraction of their area while their net longitudinal fields are a hundred times smaller, which is a statement no single measurement could have made.

The second technique goes after even weaker fields and works through scattering rather than through splitting. Light scattered in a stellar atmosphere is linearly polarised, with a direction set by the scattering geometry, and a magnetic field causes the scattering atoms to precess between absorption and re-emission. The polarisation is then rotated and partly destroyed, by an amount that depends on the ratio of the precession rate to the radiative decay rate — which for the relevant transitions is a match at a few gauss.

That is the Hanle effect, and its virtue is exactly the one the Zeeman effect lacks: it depends on the field’s strength but not on its sign, so a tangled field does not cancel. Its weakness is the mirror image. It saturates once the precession is fast compared with the decay, so it is blind above a few tens of gauss, and it requires a scattering geometry, which means observing near the limb of a resolved star. The Sun is the only object where it can be used at all, and on the Sun it says that the quiet photosphere carries a turbulent field of order a hundred gauss, carrying more magnetic energy than everything the magnetograms show.

The two methods the essay compares are both worth reading at settings well away from the solar case, because each of them fails in a different direction and the failures are what set the choice of line.

A 30-gauss field moves the line by 3.26% 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 30 gauss, which splits it by 0.0013 Å — 3.26 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 10 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. 6 The same line in a thirty-gauss field rather than three. The splitting is still far below the line width and the antisymmetric signature is ten times larger, so the measurement is linear over the whole range in which it is used — a property that holds only because the splitting stays small.
A 3-gauss field moves the line by 0.28% of its width and is measured anyway. Above: the Fe I 5250 Å line, Landé factor 3, 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 1.2·10⁻⁴ Å — 0.28 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 43, 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 1000-gauss field of a sunspot, which does resolve, is measured the other way round.
Fig. 7 A different line with a larger Landé factor at the same weak field. The signal scales with the product of the factor and the square of the wavelength, which is the whole of how a line is chosen — and it is why the same instrument is redesigned around a different line when it moves to the infrared.

Three complications that are the current work

The weak-field approximation fails where the field is strongest. The linear relation between VV and BB holds while the splitting is much smaller than the line width. In a sunspot it is not, and the profile has to be modelled rather than differentiated — which means solving the transfer of polarised radiation through a stratified atmosphere, a considerably larger undertaking.

The transverse field needs the other Stokes parameters. Linear polarisation carries the field component across the line of sight, and it is second order in the field, so it is where circular polarimetry’s advantage disappears. Maps of the full vector field are correspondingly worse than maps of the longitudinal one, and the ambiguity in the transverse direction — a field and its reverse produce identical linear polarisation — has to be resolved by an assumption.

And a stellar map needs rotation. A star is a point, so a single spectrum gives one number for the whole disc. Watching the Stokes V profile change shape as the star rotates, and inverting the sequence, gives a map — the same trick as reading a velocity off a distorted line profile, applied to polarisation instead, and it depends on rotation carrying a surface feature across the profile. It works, it requires a full rotation’s worth of good spectra, and it cannot see anything on the hemisphere that never comes into view.

The number depends on the pixel

There is a consequence of the cancellation that deserves stating separately, because it makes “the magnetic field of a star” an ill-posed quantity rather than merely a hard one.

A solar magnetogram is a map, and every pixel of it reports the net longitudinal field averaged over the area that pixel covers. Halve the pixel size and the reported field strengths rise, because less cancellation happens inside a smaller box. That is not a calibration error; it is the measurement doing what it is defined to do, and it means a magnetogram’s numbers cannot be compared between instruments of different resolution without a conversion that depends on the unresolved structure.

The Sun’s photospheric field is the standing example. At the resolution of a routine full-disc magnetogram the quiet Sun reads a few gauss. At the resolution of the best ground-based telescopes, working on a small patch in good seeing, the same region resolves into kilogauss flux tubes with almost nothing in between. Neither number is wrong and they differ by three orders of magnitude.

For a star there is no pixel at all — the whole hemisphere is one resolution element — so a stellar longitudinal field is the extreme end of that sequence, and the quantity it reports is the large-scale, spatially coherent component and nothing else. That is genuinely worth measuring: the large-scale field is what shapes a stellar wind, what governs angular momentum loss, and what reverses over an activity cycle. It is simply not the same quantity as the field that fills the atmosphere, and the two are related by a factor that has to be measured separately for every star.

And the other effect, at a much shorter atomic lifetime, which is the regime where it stops working.

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 3 gauss for a 30-nanosecond level, and — the useful part — it does not care about sign, so a tangled field does not cancel. Above the crossing at 61.8 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. 8 The Hanle effect for an atomic lifetime of thirty nanoseconds. The field strength at which the effect is sensitive scales inversely with the lifetime, so a short-lived level is diagnostic of a strong field and a long-lived one of a weak field — the two effects between them cover the range neither reaches alone.

One more method estimates the same quantity without any polarimetry of the line at all.

A field strength read off the scatter of directions. The Davis–Chandrasekhar–Fermi estimate: field strength against the dispersion of polarisation angles across a cloud, at a density of 3·10⁴ molecules per cubic centimetre and a turbulent velocity of 0.9 kilometres a second. The polarisation directions themselves contain no field strength — an aligned grain reports which way the field points and nothing about how hard it is pulling. The strength is in the disorder. Turbulence of a given energy bends a stiff field less than a weak one, so the angular scatter is the ratio of the turbulent velocity to the Alfvén speed, and inverting it gives the field: 343 µG at a 10-degree dispersion. The relation is exactly inverse, so the estimate is most trustworthy where the field is most ordered and worst where it matters most. The factor of 0.5 in front is not theory — it is what simulations of a known field, analysed this way, turn out to need, and the honest statement is that this method is calibrated rather than derived.
Fig. 9 The Davis–Chandrasekhar–Fermi estimate at three times the density. The field strength comes out as the square root of the density times a velocity dispersion divided by an angular dispersion, so it is a statistical argument about a turbulent medium rather than a measurement of any one line.

Where this ladder goes next

This rung has taken a splitting far below any spectrograph’s resolution and measured it anyway, by using the one property of the signal that the instrument’s errors do not share: a sign.

The rung above is the full inversion. Given Stokes I, Q, U and V across several lines, the vector field and its stratification with height can be recovered by fitting a model atmosphere — which is where solar physics spends most of its instrumental effort, and which is limited by the same degeneracy every atmospheric inversion has.

Beside it lies the other way of measuring a field entirely: the Zeeman effect in the radio, where the splitting is compared against a line width that is a thousand times narrower, so that the same field is a far larger fractional signal — which is how the fields in molecular clouds are measured, and why they are known better than the fields in stars.

And below it, the habit: when a signal cannot be resolved, look for a quantity in which it is first order. The splitting is unmeasurable in the sum and measurable in the difference, and the difference between those two statements is a factor of ten thousand — bought not with a better instrument but with a different quantity.

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 profiles of equal equivalent width, 28.6 mÅ. Three absorption profiles with the same equivalent width — 28.6 milliångström, 1.72 km/s at 500 nm, matched to better than 0.1% by root-finding over the quadrature — differing in nothing but shape. Left: the cores, on a common velocity axis. Right: the same three normalised to their own half widths, on a logarithmic depth scale. The thermal profile is a Gaussian set by Fe's mass at 6000 K, 1.34 km/s; the collisional one a Lorentzian of γ = 2.28e-3 nm; the rotational one the classical kernel of a disc turning at v sin i = 1.73 km/s, which is exactly zero beyond 1.28 half widths and is the only one of the three with an edge. Matching the areas does not match the widths: the half widths are 1.11 km/s (thermal), 1.35 km/s (rotational), 0.68 km/s (collisional), a factor of 1.98 between the widest and the narrowest. At three half widths the collisional wing is 49 times the thermal one and at five it is 1.27·10⁶ times — six decades, which is why a line's shape stays diagnostic long after its width has stopped being so. 11% of the Lorentzian's own equivalent width lies beyond the right-hand panel's edge and is not drawn anywhere. The same width for three different reasons Thermal motion, rotation and collisions each widen an absorption line, and they can be tuned to areas that agree to a part in a thousand. What separates them is the shape, and the shape carries a rotation speed from one profile and a surface gravity from another. An absorption spectrum at 5772 K. A blackbody continuum at 5772 K with absorption lines cut out of it, each line's depth computed from how much of the gas is in a state that can absorb it. 8 of the 8 lines are strong enough to see at this temperature, which is the whole reason the spectral sequence is a temperature sequence. Composition, read from what is missing The dark lines in a stellar spectrum are wavelengths that never arrived. Which ones are absent names the elements present — and the strength of a line says more about temperature than about abundance.

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.

Circular polarisationDoppler widthFilling factorFlux cancellationLande factorLongitudinal fieldMagnetogramPolarimetric precisionSpectropolarimetryStokes parametersWeak-field approximationZeeman effect