Faraday rotation

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The magneto-ionic medium in the interstellar space is birefringent. That means the same medium has different refractive indices depending on the polarization of the propagating wave. So when linearly polarized waves propagate through this plasma their polarization angle is rotated. This is called Faraday rotation (FR).[১]

To calculate RM we need observations at 2 frequencies or if there is ambiguity even a third observation is needed. Preferred observation technique is narrow-band polarimetry around a central frequency. But narrow-band polarimetry has a low SNR. Faraday Rotation Measure Synthesis comes to rescue. By analyzing contributions from multiple sources along the LOS it improves SNR.

Slab model: uniform mixture of thermal emission and synchrotron emission. that describes the distribution of emission as function of frequency. But it is not the case in practice everywhere. There are Faraday screens. Faraday screens are clouds, shells or bubbles embedded in the magneto-ionic medium (MIM) along the LOS with no or very little total intensity emission.

Faraday depth

Brentjens & Bruyn 2005 made a distinction between Faraday depth (FD) and rotation measure. They defined Faraday depth as,

<math>\frac{RM}{[rad\ m^{-2}]} = 0.81 \int_L \frac{n_e}{[cm^{-3}]} \frac{B_\parallel}{[\mu G]} \frac{ds}{[pc]}</math>

Depth is positive when magnetic field direction is towards the observer and vice versa. Different sources may exist at different Faraday depths along the same LOS.

These sources are either Faraday thick (<math>\lambda^2\Delta\phi \gg 1</math>) or Faraday thin (<math>\lambda^2\Delta\phi \ll 1</math>) where <math>\Delta\phi</math> is the extent of the source in Faraday depth (<math>\phi</math>).

Rotation measure

By writing the equation of Faraday rotation we can define RM as the slope of the polarization angle (<math>\chi</math>) versus wavelength squared,

<math>RM = \frac{d\chi(\lambda^2)}{d\lambda^2}</math>

where, <math>\chi = \frac{1}{2} \tan^{-1} \frac{U}{Q}</math> and <math>\Delta\Phi</math> is the amount of Faraday rotation.

If there is only one source along the LOS which has no internal Faraday rotation and does not suffer from beam depolarization than Faraday depth of that source is exactly equal to its RM at all wavelengths,

<math>\chi(\lambda^2) = \chi_0 + \phi \lambda^2 \Rightarrow \Delta\chi = \phi \lambda^2 \equiv RM\ \lambda^2</math>

Faraday dispersion

This is a complex of function of Faraday depth, <math>F(\phi)</math> that is defined as,

<math>P(\lambda^2) = \int_{-\infty}^\infty F(\phi) e^{2i\phi\lambda^2} d\phi</math>

where <math>F(\phi)</math> is the complex polarized surface brightness per unit Faraday depth and <math>P(\lambda^2)=p(\lambda^2) I(\lambda^2)</math> is the complex polarized surface brightness.

See that in this formalism <math>F(\phi)</math> is wavelength independent. Here P can be written as,

<math>P = \lVert p \rVert I e^{2i\chi} \equiv pI = Q+iU</math>

The equation of P is similar to a Fourier transform, by inverting it we can find out F. But for that we also need values of P for <math>\lambda^2 < 0</math> which is not physical. We can assume some negative values based on the positive ones, but than one should take into account the incomplete sampling for the negative values.

Difference between Faraday depth and RM

We said that for a single source along the LOS with no internal FR the FD and RM are the same for all wavelengths. But in general this is not the case. For example,

Imagine a double radio galaxy, the lobe closest to us has a FD of <math>\phi_1\ \text{rad m}^{-2}</math>. The lobe is Faraday thin.
This lobe has an intrinsic polarized flux density of <math>0.25 \text{ Jy beam}^{-1}</math>.
There is galactic foreground (GF) between us and the lobe.

References