Modelling the cosmic ray electron propagation in M 51
  Cosmic ray electrons (CREs) are a crucial part of the ISM and are observed
via synchrotron emission. While much modelling has been carried out on the CRE
distribution and propagation of the Milky Way, little has been done on normal
external star-forming galaxies. Recent spectral data from a new generation of
radio telescopes enable us to find more robust estimations of the CRE
propagation. We model the synchrotron spectral index of M 51 using the
time-dependent diffusion energy-loss equation and to compare the model results
with the observed spectral index determined from recent low-frequency
observations with LOFAR. This is the first time that this model for CRE
propagation has been solved for a realistic distribution of CRE sources, which
we derive from the observed star formation rate, in an external galaxy. The
radial variation of the synchrotron spectral index and scale-length produced by
the model are compared to recent LOFAR and older VLA observational data and
also to new observations of M 51 at 325MHz obtained with the GMRT. We find that
propagation of CREs by diffusion alone is sufficient to reproduce the observed
spectral index distribution in M 51. An isotropic diffusion coefficient with a
value of $6.6\pm0.2\,\times 10^{28}\mathrm{cm}^2\,\mathrm{s}^{-1}$ is found to
fit best and is similar to what is seen in the Milky Way. We estimate an escape
time of $11\,\mathrm{Myr}$ from the central galaxy to $88\,\mathrm{Myr}$ in the
extended disk. It is found that an energy dependence of the diffusion
coefficient is not important for CRE energies in the range
$0.01\,\mathrm{GeV}$--$3\,\mathrm{GeV}$. We are able to reproduce the
dependence of the observed synchrotron scale-lengths on frequency, with $l
\propto \nu^{-1/4}$ in the outer disk and $l \propto \nu^{-1/8}$ in the inner
disk.
