Understanding the radio relic emission in the galaxy cluster MACS
  J0717.5+3745: spectral analysis
  Radio relics are diffuse, extended synchrotron sources that originate from
shock fronts generated during cluster mergers. The massive merging galaxy
cluster MACS J0717.5+3745 hosts one of the more complex relics known to date.
We present upgraded Giant Metrewave Radio Telescope band 3 (300-500 MHz) and
band 4 (550-850 MHz) observations. These new observations, combined with
published VLA and the new LOFAR HBA data, allow us to carry out a detailed,
high spatial resolution spectral analysis of the relic over a broad range of
frequencies. The integrated spectrum of the relic closely follows a power-law
between 144 MHz and 5.5 GHz with a mean spectral slope $\alpha=-1.16\pm0.03$.
Despite its complex morphology, the subregions of the relic and the other
isolated filaments also follow power-law behaviors, and show similar spectral
slopes. Assuming Diffusive Shock Acceleration, we estimate a dominant Mach
number of $\sim 3.7$ for the shocks that make up the relic. Comparison with
recent numerical simulations suggests that in the case of radio relics, the
slopes of the integrated radio spectra are determined by the Mach number of the
accelerating shock, with $\alpha$ nearly constant, namely between $-1.13$ and
$-1.17$, for Mach numbers $3.5 - 4.0$. The spectral shapes inferred from
spatially resolved regions show curvature, we speculate that the relic is
inclined along the line-of-sight. The locus of points in the simulated
color-color plots changes significantly with the relic viewing angle. We
conclude that projection effects and inhomogeneities in the shock Mach number
dominate the observed spectral properties of the relic in this complex system.
Based on the new observations we raise the possibility that the relic and a
narrow-angle-tailed radio galaxy are two different structures projected along
the same line-of-sight.
