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We discuss a new technique to constrain models for the origin of radio relicsin galaxy clusters using the correlation between the shock Mach number and theradio power of relics. This analysis is carried out using a sample of relicswith information on both the Mach numbers derived from X-ray observation,$\mathcal{M}_X$, and using spectral information from radio observations of thepeak and the average values of the spectral index along the relic,$\mathcal{M}_R$. We find that there is a lack of correlation between$\mathcal{M}_X$ and $\mathcal{M}_R$; this result is an indication that thespectral index of the relic is likely not due to the acceleration of particlesoperated by the shock but it is related to the properties of a fossil electronspopulation. We also find that the available data on the correlation between theradio power $P_{1.4}$ and Mach numbers ($\mathcal{M}_R$ and $\mathcal{M}_X$) inrelics indicate that neither the DSA nor the adiabatic compression can simplyreproduce the observed $P_{1.4}-\mathcal{M}$ correlations. Furthermore, we findthat the radio power is not correlated with $\mathcal{M}_X$, whereas it is notpossible to exclude a correlation with $\mathcal{M}_R$. This also indicatesthat the relic power is mainly determined by the properties of a fossilelectron population rather than by the properties of the shock. Our resultsrequire either to consider models of shock (re)acceleration that go beyond theproposed scenarios of DSA and adiabatic compression at shocks, or to reconsiderthe origin of radio relics in terms of other physical scenarios.

The correlation between radio power and Mach number for radio relics in galaxy clusters