Relativistic ions with power-law spectra explain radio phoenixes

Radio phoenixes are filamentary sources in the intracluster medium (ICM) of galaxy clusters, often extending over $>100$ kpc, arising from fossil radio lobes. Their soft, curved spectrum is widely attributed to aged relativistic electrons recently accelerated or compressed, but at high frequencies is shown to approach a power-law. Moreover, the full, curved spectrum is naturally reproduced by secondary $e^{\pm}$ from a pure power-law spectrum of relativistic ions, radiating in highly-magnetized filaments; this model provides a better fit to all phoenixes, with only three free parameters. Weaker magnetization shifts the curvature to low frequencies, explaining pure power-law phoenixes. Hadronic high-curvature phoenixes require $e^{\pm}$ heating, by a factor $\gtrsim 15$ if at ICM pressure. The $\sim$keV Compton- and $\sim$GeV $\pi^0$-decay-peaked counterparts of hadronic phoenixes may be detectable as non-thermal X-rays and $\gamma$-rays.