Magnetic fields in the Milky Way from pulsar observations: effect of the
  correlation between thermal electrons and magnetic fields
  Pulsars can act as an excellent probe of the Milky Way magnetic field. The
average strength of the Galactic magnetic field component parallel to the line
of sight can be estimated as $\langle B_\parallel \rangle = 1.232 \,
\text{RM}/\text{DM}$, where $\text{RM}$ and $\text{DM}$ are the rotation and
dispersion measure of the pulsar. However, this assumes that the thermal
electron density and magnetic field of the interstellar medium are
uncorrelated. Using numerical simulations and observations, we test the
validity of this assumption. Based on magnetohydrodynamical simulations of
driven turbulence, we show that the correlation between the thermal electron
density and the small-scale magnetic field increases with increasing Mach
number of the turbulence. We find that the assumption of uncorrelated thermal
electron density and magnetic fields is valid only for subsonic and transsonic
flows, but for supersonic turbulence, the field strength can be severely
overestimated by using $1.232 \, \text{RM}/\text{DM}$. We then correlate
existing pulsar observations from the Australia Telescope National Facility
with regions of enhanced thermal electron density and magnetic fields probed by
${^{12} \mathrm {CO}}$ data of molecular clouds, magnetic fields from the
Zeeman splitting of the 21 cm line, neutral hydrogen column density, and
H$\alpha$ observations. Using these observational data, we show that the
thermal electron density and magnetic fields are largely uncorrelated over kpc
scales. Thus, we conclude that the relation $\langle B_\parallel \rangle =
1.232 \, \text{RM}/\text{DM}$ provides a good estimate of the magnetic field on
Galactic scales but might break down on sub - kpc scales.
