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Radio relics are diffuse, extended synchrotron sources that originate fromshock fronts generated during cluster mergers. The massive merging galaxycluster 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) andband 4 (550-850 MHz) observations. These new observations, combined withpublished 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 offrequencies. The integrated spectrum of the relic closely follows a power-lawbetween 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 otherisolated filaments also follow power-law behaviors, and show similar spectralslopes. Assuming Diffusive Shock Acceleration, we estimate a dominant Machnumber of $\sim 3.7$ for the shocks that make up the relic. Comparison withrecent numerical simulations suggests that in the case of radio relics, theslopes of the integrated radio spectra are determined by the Mach number of theaccelerating shock, with $\alpha$ nearly constant, namely between $-1.13$ and$-1.17$, for Mach numbers $3.5 - 4.0$. The spectral shapes inferred fromspatially resolved regions show curvature, we speculate that the relic isinclined along the line-of-sight. The locus of points in the simulatedcolor-color plots changes significantly with the relic viewing angle. Weconclude that projection effects and inhomogeneities in the shock Mach numberdominate the observed spectral properties of the relic in this complex system.Based on the new observations we raise the possibility that the relic and anarrow-angle-tailed radio galaxy are two different structures projected alongthe same line-of-sight.

Understanding the radio relic emission in the galaxy cluster MACS J0717.5+3745: spectral analysis