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 $l \propto ν^{-1/4}$ in the outer disk and $l \propto ν^{-1/8}$ in the inner disk.