Non-thermal $X$-rays from pulsation-driven shocks in Cepheids

Rapid X-ray phase-dependent flux enhancement in the archetype classical Cepheid star $\delta$~Cep was observed by XMM-Newton and Chandra. We jointly analyse thermal and non-thermal components of the time-resolved X-ray spectra prior to, during and after the enhancement. A comparison of the time scales of shock particle acceleration and energy losses is consistent with the scenario of a pulsation-driven shock wave traveling into the stellar corona and accelerating electrons to $\sim$ GeV energies and with Inverse Compton (IC) emission from the UV stellar background leading to the observed X-ray enhancement. The index of the non-thermal IC photon spectrum, assumed to be a simple power-law in the $[1-8]$ keV energy range, radially integrated within the shell $[3 - 10]$ stellar radii, is consistent with an enhanced X-ray spectrum powered by shock-accelerated electrons. An unlikely $\sim$100-fold amplification { via turbulent dynamo} of the magnetic field at the shock propagating through density inhomogeneities in the stellar corona is required for the synchrotron emission to dominate over the IC; the lack of time-correlation between radio synchrotron and stellar pulsation contributes to make synchrotron as an unlikely emission mechanism for the flux enhancement. Although current observations cannot rule out a high-flux two-temperature thermal spectrum with a negligible non-thermal component, this event might confirm for the first time the association of Cepheids pulsation with shock-accelerated GeV electrons.