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Fig. 1. A low-mass X-ray binary, ray-marched through the Roche potential: the companion fills its lobe, gas pours through L₁ and is swung by the Coriolis force onto the disc. X-rays from the compact object are scattered by the star (its lit face is a mirror) and by the stream and the disc, and come out polarized perpendicular to the plane of scattering: the red strokes circle the compact object. Turn the orbit with φ, tilt it with i, weigh the companion with q.

Fig. 2. The inner disc around a Kerr black hole, by analytical ray-tracing as in the Schwarzschild paper and artpol: light bending (Beloborodov's approximation) lifts the far side over the shadow; the spin moves the disc's inner edge (the ISCO) inwards and drags the orbits, so the approaching side brightens. Red strokes: the plane of polarization of an electron-scattering atmosphere, their length the degree, parallel-transported to us. Through the filter the brightness is modulated by 1 + p cos 2(χ − θ), exaggerated ×4.

Fig. 3. The pulsar as seen from Earth. Grey ticks: the magnetic field on the sphere of the polarization-limiting radius r_pl (dashed; really tens of stellar radii). On the way out a photon's polarization follows the field (the magnetized vacuum is birefringent) and freezes at r_pl, where every ray sees the same dipole: so the red strokes (the X-mode, ⟂ field) all agree. ψ: the angle between the magnetic axis and the line of sight.

Fig. 4. Polarization angle against phase: the rotating-vector model, tan χ = sin θ sin φ / (sin i cos θ − cos i sin θ cos φ), plus 90° for the X-mode; fitting it to IXPE data is how the 4U 1538-52 paper weighs the geometry. Drag across the graph to turn the star by hand.

To include the effects of non-adiabatic regions and accretion-flow birefringence, see Mushtukov et al. (2026).