Concept

Rotation measure — where it appears

The coefficient of the wavelength-squared law by which magnetised plasma rotates the plane of polarised light. It is the path integral of the electron density times the line-of-sight magnetic field. It is one of the few magnetic observables carrying a sign, which is what makes it a measurement of direction.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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

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.

starlight · Faraday rotation

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.

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

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.

starlight · Faraday rotation

Named alongside it

The objects these essays reach for when they reach for this one.

Depolarisation Faraday rotation Interstellar medium Magnetic field Polarimetry Pulsars Synchrotron radiation Axisymmetric field Birefringence Bisymmetric field Dispersion measure Dynamo

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