Generator

The polarisation-curve generator

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.

14 essays call polarisation-curve. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of 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.

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.

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

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.

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 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: 220 µG at a 9-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.

A direction measured by something with no strength in it

An interstellar grain spins with its long axis across the magnetic field, so starlight through a cloud is polarised along the field and the cloud's own emission across it. The observable contains no field strength whatever — and the strength is recovered anyway, from how much the directions disagree.

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.

Two instruments blind in opposite directions

The Zeeman effect measures the component of a field along the line of sight and adds it up, so a field tangled into a hundred cells reports a tenth of one cell's strength. The Hanle effect measures how far an atom precesses between absorbing and re-emitting, does not care about sign, and saturates just where the Zeeman effect becomes useful.

An instrument 1.4 times as polarised as the sky it is measuring. The Stokes plane, with the two linear polarisation parameters as axes. The tight cluster near the centre is a set of stars known to be unpolarised, observed through the same instrument: they should sit at the origin and do not, and their mean is the instrumental polarisation — 0.77 per cent here, which is 1.4 times the real polarisation of the field. The other cluster is the field stars, whose measured values are the sum of their own polarisation and the same instrumental offset. Subtracting one mean from the other recovers 0.59 per cent at the right angle. Two things make this worth doing carefully. The offset is a vector, so leaving it in rotates the measured position angle as well as changing its magnitude — by 31 degrees here — and a calibration that only fixes the scale does not touch that. And the offset depends on where the telescope was pointing, because the reflection angles do, so the standards have to be observed at the same place in the sky and at the same instrument rotation.

An instrument more polarised than the sky

Every oblique reflection polarises. A telescope is a stack of oblique reflections, so it adds a fraction of a per cent of polarisation to everything it looks at — which for most astronomical sources is more than they have themselves, and which is a vector rather than a scale error, so it rotates the answer as well as changing its size.

$A source that erases its own polarisation. The polarised fraction of a uniform slab that both emits and Faraday-rotates, relative to its intrinsic value, against wavelength, for total Faraday depths of 10, 40, 150 rad m⁻². Light from the back of the slab is rotated by the whole depth and light from the front by none, so the polarisation vectors from different depths fan out and cancel. The fraction falls as |sin(Φλ²)/(Φλ²)| and vanishes altogether at λ² = π/Φ — 56.0 cm for 10, 28.0 cm for 40, 14.5 cm for 150 — then returns weakly and vanishes again. A source measured only at wavelengths beyond its first null looks unpolarised, and a source measured across the null shows a polarised fraction that depends on the wavelength in a way no foreground screen can produce. The shape of the fall is a measurement of how the rotating plasma is arranged along the line of sight.$ Starlight

A source that rotates its own light

The rotation-measure method assumes the plasma that turns the polarisation sits in front of the source. When it is mixed into the source instead, light from the far side is turned more than light from the near side, the contributions cancel, and the straight line in wavelength squared bends, halves its slope and then vanishes. The cure is to stop measuring in wavelength at all.

Two lobes, one halo, and two different paths out. A double radio source drawn in the plane containing the line of sight, the observer far to the right. The host galaxy sits at the centre of a hot halo — the rings mark 20, 40, 60, 100 kpc, in a β-model of core radius 40 kpc — and its two lobes lie 60 kpc out along an axis 45° from the line of sight. Light from the near lobe leaves the halo through its outskirts; light from the far lobe has to cross the halo's middle. With a tangled field of 3 µG at the centre in cells of 5 kpc, the rotation measure in front of the near lobe varies across the beam with a dispersion of 9 rad m⁻² and in front of the far lobe 28 rad m⁻². On the sky the two lobes are the same distance from the core, 42 kpc, and nothing in a total-intensity image says which is which.

The fainter polarisation belongs to the lobe further away

The two lobes of a radio galaxy lie at the same distance from the core on the sky, and nothing in a picture of them says which is nearer. Their polarisation does. Light from the far lobe crosses the magnetised hot halo of the host on its way out and light from the near lobe mostly does not, so at long wavelengths the far lobe loses its polarisation first — and the one-sided jet, it turned out, points at the lobe that keeps it.

Polarised dust, synchrotron and the signal between them. The amplitude of B-mode polarisation at angular multipole 80, in microkelvin of CMB temperature on a logarithmic scale, against frequency from 20 to 450 GHz, for a patch of sky as clean as the cleanest large field observed. Dust is a modified blackbody with emissivity index 1.53 at 19.6 K, normalised to 4.6 µK² at 353 GHz; synchrotron falls as frequency to the power −3.1 in antenna temperature, normalised to an illustrative 1 µK² at 23 GHz. The horizontal lines are a primordial signal with tensor-to-scalar ratio 0.01, about 0.048 µK, and the lensing B-modes, about 0.071 µK. The foregrounds together are faintest at 81 GHz, at 0.038 µK, 0.8 times the signal. At 150 GHz the dust alone is 0.105 µK, the amplitude a tensor ratio of 0.046 would produce; at 353 GHz it is 2.23 µK, 47 times the signal. No frequency has the signal above the foreground.

A foreground drawn by the field it traces

The grains that polarise starlight along the Galaxy's magnetic field glow, at millimetre wavelengths, with light polarised across it — and that glow is brighter than the primordial signal cosmologists are looking for at every frequency above eighty gigahertz and in every patch of sky. Its pattern on the sky is set by how loosely the interstellar filaments follow the field, and removing it rests on a bet that the clouds along each line of sight share one temperature.

A lobe, the shell it pushed out, and two sight lines through it. A radio lobe of radius 20 kpc drawn in the plane containing the line of sight, the observer to the right, 60 kpc from the host in the same halo as before. Growing at Mach 2, it swept the gas that filled its volume into a shell compressed 2.29 times and 2.9 kpc thick, and the field in that shell was amplified along the boundary. A sight line through the lobe's centre crosses the shell head-on, where the amplified field lies across the line of sight and contributes nothing: the shell there turns the light by about 6 rad m⁻². A sight line at 85 per cent of the radius crosses it obliquely, along 4.8 kpc of shell with the amplified field partly along the path, and the shell turns it by about 18 rad m⁻². The rotation the halo model attributes to the gas in front of the lobe is partly written by the lobe itself, and it is written most strongly around its edge.

A rotation measure the lobe writes on its own boundary

The magnetic fields of hot group and cluster gas are mostly measured through the Faraday rotation of the radio galaxies that live in it, on the assumption that a lobe is a passive lamp behind a screen. But a lobe made its own room. The gas that filled it is now a shell around it, compressed if the lobe grew faster than sound, and the shell rotates the light too — so the inferred field comes out too low for a lobe that grew gently and up to three times too high for one that drove a shock. Only sources behind the halo, which carry no shell, can say which.

$A polarisation fraction that should change with frequency. The polarisation fraction of dust emission relative to its value at 353 GHz, against frequency on a logarithmic axis. A single population at a single temperature gives a fraction that does not change at all. If the carbon grains are unaligned and warmer than the silicates, they take a growing share of the emission toward high frequency and dilute the polarisation: at 1200 GHz the fraction falls to 0.90 of its 353 GHz value for a 3 K difference and 0.85 for 5 K. Half-aligned carbon halves the effect. The shaded band is flat to within ten per cent between 100 and 1200 GHz, the precision with which the satellite and balloon measurements across this range agree with no change.$ Galaxies

The grains that glow and the grains that align

Interstellar dust polarises starlight and polarises its own glow, and for decades the standard account gave the two jobs to different grains — aligned silicates, unaligned carbon. Two populations at two temperatures predict a polarisation fraction that falls toward high frequency; what satellites and balloons measured was a fraction that does not change. That flatness, which no grain shape can fake, says the grains that glow are the grains that align.

How fast to switch. The rms false polarisation left by a 1 per cent rms brightness fluctuation, against the modulation frequency, both on logarithmic scales, for fluctuations whose spectrum turns down at 5, 20, 100 Hz. Below the turnover the false signal is the whole fluctuation, about 0.7 per cent; above it, it falls steeply as the pairs are taken faster than the air changes. To reach a part in ten thousand (dashed) — the precision at which stellar and planetary polarisation becomes interesting — the switching must run at 3,311 Hz, 13,183 Hz, 66,069 Hz respectively: switching at hertz is useless, and a single beam needs kilohertz or more.

A polarisation measured faster than the air changes

Polarisation is a difference between two brightnesses, and the brightness of a star seen through the air changes by a per cent from one millisecond to the next. Measure the two one after the other and the air's flicker becomes a false polarisation a hundred times larger than the signals worth having. The cure is to switch between them faster than the air can change, or to measure both at once and then swap them — turning a calibration problem into a differential one.

$How a cloud's glow loses its polarisation as it thickens. The polarisation fraction of dust emission against the column density through a cloud, both on logarithmic scales, for four clouds built from cells of 10²¹ hydrogen atoms per square centimetre, each polarising its glow by 20%. A single uniform field with every grain aligned keeps 20% at every column. A field tangled from cell to cell by 40 degrees about a mean direction loses polarisation at first and then levels off at 7.5%, the share its mean direction carries: between 10²² and 10²³ its slope is −0.05. A field with no mean direction at all adds its cells as a pure random walk and falls with a slope of −0.55. Grains that align only in the outer 3 × 10²¹ cm⁻² on each side, where starlight still reaches, contribute nothing from the interior, and the fraction falls with a slope of −1.00.$ Galaxies

A falling polarisation and the angle that says why

The glow of dust in a molecular cloud is less polarised the thicker the cloud, almost everywhere it has been measured. Two explanations fit that fall equally well — a magnetic field tangled along the line of sight, or grains in the dark interior that have stopped lining up with it — and between them they can produce any slope a cloud has shown. What separates them is not how much polarisation is lost but whether the direction of what remains wanders.

A polarisation that a round star cancels

In 1946 Chandrasekhar worked out that light leaving a hot star's atmosphere, where scattering by free electrons dominates, is polarised parallel to the star's edge — not at all at the centre of the disc, 11.7 per cent at the very limb. Over a round, unresolved star the pattern adds up to exactly nothing. It took seventy years to see it in the light of a whole star, because something has to break the circle first: a companion crossing the disc, a planet, or a star spun so fast that its outline is no longer round.

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