AI-generated illustration for Galileo Whispers. A symbolic radio sightline through magnetized plasma; the path stays straight, while the glimmers only suggest changing polarization orientation.
Faraday rotation changes the orientation of linearly polarized radio light as it crosses magnetized plasma. In a simple foreground screen, the rotation grows with wavelength squared. Measuring that slope probes the magnetic field along the line of sight, weighted by free-electron density and accumulated over the path.
It does not yield a unique three-dimensional magnetic map. A strong field through sparse plasma and a weaker field through denser plasma can produce similar rotation. Field reversals can also cancel contributions.
Two circular components travel differently
Linear polarization describes the orientation of a radio wave’s electric-field oscillation. The same wave can be represented as a combination of left- and right-circularly polarized components. That decomposition is a way to describe the electromagnetic field, not two separately emitted radio beams.
Electrons in a magnetic field respond differently to the two circular components. Their phase propagation differs slightly; as the components recombine, their changing relative phase rotates the linear polarization direction. The NRAO treatment of propagation in a magnetized plasma derives this effect from the plasma’s two refractive responses.
The radio ray need not trace a visible spiral across an image. What rotates is the polarization position angle. Ordinary plasma dispersion instead produces a frequency-dependent pulse-arrival delay; a delay measurement and a polarization-angle measurement carry different information.
Reading the wavelength-squared slope
For one simple external screen, the model is
χ(λ) = χ₀ + RM × λ².
Here χ is the polarization angle in radians, λ is wavelength in meters, χ₀ is the angle extrapolated to zero wavelength under this model, and RM is rotation measure in radians per square meter. The NRAO Faraday-rotation primer connects the effect with magnetic-field measurements.
As a worked example, assume a single screen with RM = 20 rad/m². Between wavelengths of 0.10 and 0.20 meter, the expected angle difference is
20 × (0.20² − 0.10²) = 0.60 rad, or about 34.4°.
This is an invented demonstration, not a measurement of a celestial source. Doubling wavelength quadruples the accumulated RM contribution relative to χ₀; it does not necessarily quadruple the observed total angle, which includes the source’s intrinsic orientation.
Polarization orientation repeats after 180°, so fitting only two widely spaced wavelengths can leave an ambiguity about how many half-turns occurred. Measurements across multiple frequency channels help test the slope and resolve such ambiguities.
What the integral includes
In commonly used astronomical units, the simple-screen relation is approximately
RM = 0.81 ∫ nₑ B∥ dl,
with electron density nₑ in particles per cubic centimeter, the line-of-sight field B∥ in microgauss and path length dl in parsecs. The output is in rad/m². Under the NRAO convention, a field directed toward the observer contributes positive RM.
For an illustrative uniform region with nₑ = 0.03 cm⁻³, B∥ = 2 microgauss and length 100 parsecs, the result is about 4.9 rad/m². This example assumes no reversals and no additional foregrounds. A region twice as long would give twice the rotation measure only if its density and signed field stayed the same.
A positive result establishes a signed weighted contribution under the adopted convention, not that the field points toward Earth everywhere. Independent density or path information is needed to turn the integral into a field estimate.
A complicated source can break the simple picture
Radio emission and Faraday rotation may occur in the same volume. Different emitting depths then accumulate different rotations, and neighboring paths within the telescope beam may disagree. Their polarized signals can partly cancel, causing depolarization and a more complicated wavelength dependence.
The Milky Way, Earth’s ionosphere and material near the source can all contribute. Instrumental polarization and the limited wavelength range must also be addressed before an inferred slope is assigned to a particular galaxy or cloud.
Faraday rotation is powerful precisely because it couples radio polarization to matter along the path. Its interpretation becomes reliable when that coupling—and the assumptions behind the simple straight-line fit in λ²—remains part of the answer.
