Derivation of the rotation measure

When a wave passes through an ionized gas containing a magnetic field, there is a slight difference in refractive index for the left and right circulairly polarized waves. As a result of this the direction of polarization changes with frequency. To be more precise: with the difference in polarization direction, the so called rotation measure and the wave length. By looking at the polarization at different frequencies (at least 3 to do a good job) it is possible to solve this equation at least in a consistant way, although, you can miss radians (with an integer) due to the ambiguity in the rotation measure distribution.

The rotation measure is defined as

Here is a constant, is the thermal electron density and is the magnetic field projected along the line of sight . is usually measured in . If foreground rotation (due to Faraday dispersion in our own Galaxy and the intergalactic medium) can be neglected, then is equal to the depth of the source along the line of sight. For Faraday rotation in a slab with uniform field on can write

(with . The changing of polarization direction becomes smaller at high frequencies (small wavelengths). A good estimate of the true direction of can be made by looking at the 2 cm polarization maps of the sources. The vectors should be approximately perpendicular to the magnetic field directions.