Spectral index of synchrotron emission: insights from the diffuse and
  magnetised interstellar medium
  The interpretation of Galactic synchrotron observations is complicated by the
degeneracy between the strength of the magnetic field perpendicular to the line
of sight (LOS), $B_\perp$, and the cosmic-ray electron (CRe) spectrum.
Depending on the observing frequency, an energy-independent spectral energy
slope $s$ for the CRe spectrum is usually assumed: $s=-2$ at frequencies below
$\simeq$400 MHz and $s=-3$ at higher frequencies. Motivated by the high angular
and spectral resolution of current facilities such as the LOw Frequency ARray
(LOFAR) and future telescopes such as the Square Kilometre Array (SKA), we aim
to understand the consequences of taking into account the energy-dependent CRe
spectral energy slope on the analysis of the spatial variations of the
brightness temperature spectral index, $\beta$, and on the estimate of the
average value of $B_\perp$ along the LOS. We illustrate analytically and
numerically the impact that different realisations of the CRe spectrum have on
the interpretation of the spatial variation of $\beta$. We find that the common
assumption of an energy-independent $s$ is valid only in special cases. We show
that for typical magnetic field strengths of the diffuse interstellar medium
($\simeq$2$-$20 $\mu$G), at frequencies of 0.1$-$10 GHz, the electrons that are
mainly responsible for the synchrotron emission have energies in the range
$\simeq$100 MeV$-$50 GeV. This is the energy range where the spectral slope,
$s$, of CRe has its greatest variation. We also show that the polarisation
fraction can be much smaller than the maximum value of $\simeq 70\%$ because
the orientation of ${\bf B}_\perp$ varies across the telescope's beam and along
the LOS. Finally, we present a look-up plot that can be used to estimate the
average value of $B_\perp$ along the LOS from a set of values of $\beta$
measured at centimetre to metre wavelengths, for a given CRe spectrum.
